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0-hspaper develops a systematic approach to select 8 time-dependent state

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an equivalent filter having diagonal state-feedback matrices. Due to the struct-

simplicity of the diagonal systems. this time-dependent state transformation

is & convenient toal for analyzing recursive LTV filters expressible in the

state-variable form. tn this paper, we discuss both the theoretical basis and

the application of this diagonalization procedure. The properties of two

types of recursive LTV filters are examined by using this state transformation

technique. Based upon the separable properties of the iplersosswt

major class of recursive LTV filters. This technique, through suboptimal can

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INT. J. CONTROL, 1986, VOL. 44, No. 4, 1103-1124

Approximate and local linearizability of non-linear discrete-time

systems

HONG-GI LEEt and STEVEN 1. MARCUSt

We consider a single-input non-linear discrete-time system of the form

E: x(Q + 1)-f(x(t), 1t))

where x E P', u e R and f(x,u): P + I -+ PN is a C ® PN-valued function. Necessaryand sufficient conditions for approximate linearizability are given for IT. We also givenecessary and sufficient conditions for local linearizability. Finally, we presentanalogous results for multi-input non-linear discrete-time systems.

*1. IntroductionWe consider a single-input non-linear discrete-time system of the form

Z: x(t + I) =f(x(t), u(t)) (I)

where x e P1, u e K and f(x, u): pI _ pN is a C' N-valued function.Many authors have studied (local or global) linearization (Cheng et al. 1985, Hunt

and Su 1981, Jakubczyk and Respondek 1980, Krener 1973, Su 1982) and approxi-mate linearization (Krener 1984) by state feedback and coordinate change for non-linear continuous-time systems. In this paper we discuss necessary conditions andsufficient conditions for local linearization and approximate linearization by statefeedback and coordinate change for non-linear discrete-time systems. A necessary andsufficient condition for local linearization has recently been found by Grizzle (1985 c);! Ace- I ": Fora result equivalent to this is proved in our Theorem 5. These conditions are very I;- ; :" ---similar to those available for continuous-time systems, but they are more difficult to T-.:v, 7

calculate than our sufficient condition in Theorem 4. Other related work on non- I'linear discrete-time systems can be found in Grizzle (1985 a, b), Grizzle and Nijmeijer .

(1985), Monaco and Normand-Cyrot (1983 a, b, 1984).

Definition I . .A point (x., u.) such that f(x., u.) = x. is called an equilibrium point. 1'VrNow consider the following linear discrete-time system Eo:

1)It+) =At) +bt)=g t),t4t))

Received 23 January 1986.t Department of Electrical and Computer Engineering, University of Texas at Austin, DTIO

Austin, Texas 78712, U.S.A. COPYINSPECTED

NO

W~*~ *"J~~*~? ~ %

1104 H.-G. Lee and S. 1. Marcus

where

0 1 0... 0

A=0 (N x N matrix)

00

b= [j(N x I matrix)

Similarly to the continuous-time case (Krener 1984, Su 1982), we can define locallinearizability and approximate linearizability for a discrete-time system. Let (x., u,)be an equilibrium point of E.

Definition 2E is said to be locally linearizable at (x., u.) if there exist an open neighbourhood

U ((- FPN+') of (xe, u,) and a diffeomorphism T: U -*T(U) such that

(i) T = (T1 , T2, ... , TN) are functions Of X 1 , X2 , .. ,XN Only,

(ii) T(x., Ue) = 0 (N 1)x 1'

(iii) Tof =go-T

If we let (yAt)'tvt))T = T(x(t), u(t)) then yAt) and t#) satisfy IE0. Definition 2 indicatesthat we want to find a diffeomorphism T such that the following diagram commutes:

pN. f pN+l

P4 jTN+

Once we find such a diffeomotrphism, we can apply linear system theory instead ofnon-linear system theory.

Definition 3I is said to be approximately linearizable with order p if there exist an open

neighbourhood U (c PN + 1) of (x., u.) and a diffeomorphism T: U --+ T(U) such that

(i) T= (TI , T2, ... , TN) are functions Of X 1 , X2 , .. ,XN only,

00i T(X., U.) = 0

(N +4 1) .1, and(iii) T-~-+(-x,-."

Thus in Definition 3 we consider the following nearly linear discrete-time system:

E6 At + 1) A-)+b~)+O- x. u .)

Linearizability of non-linear discrete-time systems 1105

where the N x N matrix A and N x I matrix b are the same as To. Clearly, locallinearizability at (x,, u,) implies approximate linearizability with arbitrary order.

In § 2 some background material is reviewed and notation is defined. In § 3necessary and sufficient conditions for approximate linearizability will be given for thesystem (1). Also, we shall give necessary and suflicient conditions for local linear-izability. We can define local linearizability and approximate linearizability for multi-input discrete-time systems similarly to Definitions 2 and 3. Then the multi-input casewill be discussed in § 4.

2. PreliminariesV- In this section notations and definitions to be used later will be mentioned. The

Kronecker product is very useful in the field of matrix calculus (Graham 1981). First,define the Kronecker product ® by

- ajB a12 B ... ajB

a2,B a 2 2 B ... a2 ,BA (

a, aB ap2B ... a-B (pM) (qnl

where a1j is the (i,j)-component of the p x q matrix A.Define the derivative of a matrix with respect to a matrix by

B. B .

Oa I (a1 2 Oa,q

_ _BaB BDAB= Oa2 , a2 2 ' a2 B

L Oa B p_2_B . _ B (mp(nq

We also define

DO B = B

%I DA1B = DAB

D''B=DA(D'B) fori ;>l

Let h(x) be a scalar real-valued function of xE P . Then (Dlh)(x) and (D'Th)(x) areN' x I and I x N' vectors respectively.

Fact (Vetter 1970, 1971)Using the definition of Kronecker product and derivative operations on matrices,

Taylor's formula can be expressed by

h(x) = h(0) + I=2,

where R,+ ,(x*) is a remainder term.

J%-f,,

~%.7A

1106 H.-G. Lee and S. . Marcus

Now define the N" x N' permutation matrix U1 ,.2. ,k as follows: the((a,, - I)Nh- I + (a,2 - l)NI - 2 + + (a, - I)N + aijth column of U1, 2 . ,k is the((a, - I)Nk- 1 + (a 2 - l)Nk -2 + ... + (a 1 I I)N + ak)th column of the N' x N' iden-tity matrix (IN- .N) for I <a , a 2 ..., ak < N (the {a,} are related to the 'base N'representation of the column). Here {i1, i2, .... li} is a permutation of (1, 2'..., k}. Forexample, when N =2 and k = 3

U 1 2 3 = 8x8

and1I 0 0 0 0 0 0 0-

0 0 0 0 1 0 0 0

0 0 1 0 0 0 0 0

0 0 0 0 0 0 1 0U3 2 1 =U31 0 1 0 0 0 0 0 0

0 0 0 0 0 1 0 0

0 0 0 1 0 0 0 0

L0 0 0 0 0 0 0 1

Let A be a p x N" matrix. Define the operator 0 byk!6) = AE.,IJ....,.)

kt! | • permutations('t1,..k of 11.2.....k)

i~i For example, when A is a p x N' matrix

(DA = A(UI23 + U132 + U213 +.. U231 +4 U312 + U321)

30

Let

au. TXx ] o0olt, OUlo.olJ

)(.) I 00 00f0100V

be linearly independent; that is, they form a basis for N.Define t: opera to by

FM = plN, where v is a I x N row vector and

vT ='f f

. i=l ex)(0,0) Oz lO.O}

ILe

. That is, C(v) is the last coefficient Of VT with respect to the basis (wI, W 2 ... WN}), wherebe lieaxl inenda;u),, ta i, te form. Also deine fo P"N. R by

,(Cm &(2)

where v, is the ith row of and

N af-l.f

I.in ,ari:zihilitv " o ti i,'-linear di110l7c-I tue s vlrtll. I I 07

3. Single-input caseIn this section our main results will be given. If I (v.. i) has an eluilibrium point.

without loss of gencralitN. we can assume that .l(. 0) = 0: for, if not, let i = \ \,. andt= =. it. Then i( + I) = f(¢(). i7(:)) 4 I + v,. ii + it,) \ with / o. 0)- o.

Let

I '(\. ,) f (x. ,)

1i '(v.i)= f(Ah'. \). 0) for I i N I

it.. represents the elect of an input ii at = 0 on (lhe state at = . I'(v. u) isessential for solving many problems arising in discrete-time non-linear systems.

L4'Illlia I

X is locally linearizable at (0.0) if and only if there exists a C' functionh: IV ( c P") - F such that

(i) It' is an open neighbourhood of 0e cR

(ii) D,(h f') 0 on some neighbourhood of 0 e R' for I < i N- I

(iii) det ~ oo 0

.

(iv ) ( D .(Ih " f +") o~ o1 # 0

(v) I1(0) = 0

Proof

Necessity. Suppose that E is locally lineariiable. Then we have a diffeomorphismT Let h(.) = T,.(x). (Since T,(.v. it) depends only on v. we can write T,(v) instead ofT,(x. u).) Note that T, = T, I Since D.(T) = 0 on some neighbourhood of the origin.D(TI 1 = 0 on some neighbourhood of the origin. From now on, for convenience,we shall omit 'on some neighbourhood of the origin'. Note that T% = T, f= Tf 2.(Actually, we can write T, = T, j. , because T, 'f depends only on v. But 2 is used.for consistency of notation.) Since D,(T-) 0, D,(T, ,)-0. Proceeding in thismanner, since T= T.,'f ..... T ,N 1 and D.(T.)-0. D.(T, ')=0. Thuswe have shown that D.(TI .) 0, for I < i < N - I. Since T is a diffeomorphism,

..V~ '4 ... ***~***J. p *~ p ~~-.-'~-~V ".~ ~

1108 H.-G. Lee and S. 1. Marcus

T = T 1 f for 2 i N + 1, and TI, T2 ,..., TN depend only on x,

det ( X( 0 0oo

( x (0o,)

and D.(h of N) #0 . Since T,(0, 0) = 0, h(0) = 0.Sufficiency. Suppose that there exists h: RN -+ R satisfying the given conditions. Let

T(x) = h of -' for I < i < N + 1. Then it can be easily checked that Tof = g o T andT(0, 0) = 0. Since det ((aT/(x, u))(oo)) # 0 there exists an open neighbourhood U of(0,0) such that T: U -. T(U) is a diffeomorphism by the inverse function theorem. El

Let

Lemma 2I. is approximately linearizable with order p if and only if there exists a C function

h: W(c: ) -+R such that

(i) W is an open neighbourhood of 0 c PN

(ii) (DjD,(h ofI))(0.o) = 0 (N+ 1)J. , for I i < N - I and 0 <j p - 1

(Oh)

(iii) det \ x (o o 960

aI ____-___)

ax ) (o.o )

(iv) (D.(ho'))(o0 o) # 0

(v) h(O) = 0

ProofNecessity. Suppose 1: is approximately linearizable with order p. Let h(x) = T (x).

By definition, T, of(x, u) = T(x) + 0(x, u)' '. So (D4(D.(h of))(o.0) = 0 for 0 <j < p - 1.Note that T, of2 = T of + 0(x, u)y'. Since T 2 °f(x, u) = T3(x) + 0(x, u)"' by defi-nition, Tof 2 = T3(x)+0(x,u)P'. Thus (DICD.(hof 2 ))(oo)=0 for 0<j <p- l.Proceeding in this manner, we can show that (DJCD.(h o ')(o.0) =0 for I < i < N - I

04

l-in'arizabilitv of non-linear discrete-lime syslnem. 1109

and 0 -<j p - I. Note that ('T/tx),).o, = (O)(h' )/]x)u for I i -< N. Since

(, ( /

i l)Tdet \kx],o o, #0. dct \(Ax],.o, # 0

- I'I T, i ~

\' /x 10.01

It can easily be shown that T; f (x, u) = 7N' (x, u) + (x, uY' . Thus (D1 ,(h . .(Du7N + , (x. u)),,, # 0. Finally, h(O) = T1 (0) = 0.

Sufi(ciencY (hy c'mstruction). Let

T, (x) = h(x)

,/.x = (D.,(h , r'h (x 0 x 0 ..0 .c

k-I

-I T(X)= - (D',(h ../2))".)( (H®x X . x

7N(x) = (D)ho(x®x® . ®x)k I "- .1

TN. 15+ ,(x, u) = T, j U(x. )

Then it can easily be checked that T as defined above satisfies the conditions ofDefinition 3.

Now note that

. .(D D (h h )(0,.01 (B=) )()D h).- (2)5-',)

where

B..,= D (')

and

,8.. .,E= Y ... Y D7 '+' ',(DiJ® k ,Dfii ® A)

x (Dr ... I k'(Dfh® D' & D f .)) for I< I<m

j'%

1110 H.-G. Lee and S. 1. Marcus

(For a proof of (2) see the Appendix.) Let

All A 12 ... Ak

Ak= A 2 1 A 2 2 ... Azt".Ak=

[Aki Ak 2 ... Ak

where A,, is an (N - 1)(N + 1)' x N +' submatrix defined by

A= (B 2),oo, if i >jL N 1) I

(Bu )ho~o

Let the (N - 1)(N + 1) x I vector ft' be defined by

(B20)06.0(B,.o')(o.o,

\LBNO - J/

(See § 2 for the definitions of D and .) Also, let Ilk = (flit 1.t flkr)r.

With these preliminaries, we can state our main theorems.

Theorem 3

1 is approximately linearizable with order p (> 2) if and only if

! .... O(i) {(-j) ,(of' (f), ... (¢)fIS(l) f } are linearly independentu'. Xoo OX(00)OA', N-,,-.,ff. ii) ,l - Image (A, _ )

ProofNecessity. Suppose that 1; is approximately linearizable with order p. Then there

exists a function h(x) satisfying (i)-(v) of Lemma 2; in particular.

..... _)o~o o=0 for ,<,<,N- II,', .'o ~(O,(h 0'/ 11)oo If \ax,< o(0 .0~oo

and

-OX -X (0.O (-. ( 0I 0) 70'q ,,o

Linearizabilif' of'non-linear discrete-lime s.y.lem.s I I

Assume that

1"41 ( '("Uare not linearly independent. Then there exists k such that I <- k <- N -- I and

(= ! kfor some constants '2,:,

Thus(1i N1 IN ly 3.(

0." I,,. ,), "l 0J ,,.0

and

1 0

This is a contradiction, which implies that

a r e li e a l ) ~ ) ( .0) , . .( I~ ) 2 ( Iare linearly independent.

Recall that (D.(h I ))~m=(D7D.(h .[)).,= 0 for m 0 and I < i - N 1.Since

are linearly independent, (),h), i is uniquely determined up to a constant multiple (i.e.(D,h), ac. where the scalar o 2 : (0) is arbitrary and the N x I column vector c satis-

lies 4.r(iy (.\){o 34 33(4j ,).3 3 = 0 for 0 N i N 2, and c(rlf, ., ''ul04, = I).Now by (ii) and (iv) of Lemma 2. (DT'D,(h I")),4(, 0 for I < m f - I and

I <- i -- N I. From (2) we obtain

B:., t0 0 .. 0 jB2 0 0 ... 0 B2,0

.1 0I 0 ... 0 B .' ). B, W, _. , 0 . . 0 =B o 1 .

: :".. ( ,'h) ,,(3)B"' B ' 0 ... 0 (2.0 (

P4

,.,, 0 Bi, 1.o

) B,1 3.2 .. Bl.lB 111

P .0 (0.01B4," I.

1 . 1 p p

'p." '-.3,, .' '.'." "•" .1'•"". -"" . .' - •'" . "- ," '- ."" *,p•- ." " ' ° -"- ' " " " -" -". .% ."•"-",% ," " , . -

1112 H-G Lee and S. I. Marcus

Since (B,,o)o0.oD'h)y D o = (D . oP) o-((PP )(oo)

.o ( h) =o at# '

J.0(0.0)Thus the right-hand side of (3) is - t[(fl) )T (.. P I )T]T - -__. It follows thatfl, -, is in the image of the matrix on the left-hand side of(3). However, the {Dh} are

1constrained because, for example, O'h/8x dxj = O2h/Oxjaxj. Hence the stronger

condition P, e Image (A, -) holds, as is proved in the Appendix in Lemma A.2.

Sufficiency. Suppose that (i) and (ii) above are true. By (i) there exists an N x I

vector C, such that Cj(Of/ax)'oo0 (af/u)o.O)=O for 0 < i<N-2 andCT(af/Ox)N - I (Of/,Ou)(o0 = 1. By (ii) there exist C 2 , C 3 , .... C, such that

C2

-C,

where C, is an N i x I vector. Let

hx) Cr(x ® x ®...® x)

"times

Then it can be easily checked that (DD,(hfi))(0oo) =0 for 1< i < N - I and

0 <j p - 1. Clearly (D.(h fN))4O.O, = (D Ih)T= o(Of/Ox)No-oI(Of/Iu),oo = 9 #0.

Now assume that

(ah\ =

axL~

det ax 0 =0

x )(0.0)Then there exists k such that I < k < N - I and

(D Ih)T ( o) -i f

( ho ,(0 \oX )),oo for some {o'j}_-d

* Ox

* .AP .. ,r? .

Linearizability of non-linear discrete-time systems 1113

Thus

Miw~hT= 0 ~,DhT I -, (L- . .+,o

This is a contraction, which implies (iii) of Lemma 2. Hence, by Lemma 2, E isapproximately linearizable with order p. E

RemarkI is approximately linearizable with order I if and only if (i) of Theorem 3 holds,

just as in the continuous case (Krener 1984).

Now a sufficient condition for local linearizability is given in the followingtheorem.

Theorem 4Suppose that f(x, u) of .is an analytic RN-valued function. I is locally linearizable

at (0, 0) if

(it 0.),o (Lf t ),o.o,,f)"o( ..... , (Lf)N -I (,f/ .o~ji (OU~ (ax)1 0 (AU( . ax)1 0 .0 ) ( 1U) 00

are linearly independent,

(ii) there exists k (< oo) such that fP e span (CI) for all I > 1, where C' is composedof the first k columns of A.

Proof

By (i) there exists an N x I vector c1 such that c(af/ax)'O.o0 (0f/1Ou) 0 .0 = 0 for0 < i < N - 2 and cI(ef/X)No-o'(Of/ u)o.o = I. By (ii) there exist c 2, c3 , ... , cj suchthat j < oc and

_C22

A, = -fl for I>j-Ci

0

0

where c, is an N i x I vector. Let

h(x) = C c (x ®... ®xl

Oi

: ...

la hN kin, k%...A

1114 H.-G. Lee and S. 1. Mart-us

Then it can easily be checked that (iD.D(h f i)),00 ) = 0 for I <- i -< N - I and s >- 0.Since both h(x) and f' are analytic, h f(x, u) are analytic, for I <, i < N - 1. ThusD.(h 'f') =-0 for I < i <- N - 1. As in the sufficiency proof in Theorem 3. it can beshown that h satisfies the other conditions of Lemma 1.H

2R It is easy to see that (ii) of Theorem 4 implies (ii) of Theorem 3.

RemarkConditions (i) and (ii) are also necessary for local linearizability at (0, 0) whenever

f(x. u) is polynomial and a polynomial T(x, u) is sought.

In the following theorem we give necessary and sufficient conditions for locallinearizability.

Theorem 5E is locally linearizable at (0, 0) if and only if

IG (),..... ( -f),oo(I(., (Jo (0.) (0.0)1

are linearly independent,(ii) there exists an open neighbourhood U ofOc FN such that Ai =f.,(A0 ) + .(0 0)

+ ... + f,(Ao) are well-defined i-dimensional involutive distributions on U for Ii <N - 1,where AO -=span 11/au).

Proof'Necessity. For condition (i) see Theorem 3. By Lemma I there exists h(x) such that

conditions (i)-(v) of Lemma I are satisfied. Let yi = h(x), Y2 = hi -f(x, u), ... , and YN =

h!-f "(x, u). Then by (ii) and (iii) of Lemma I we can choose (y1(x), y2(X), ... , yN(x)) asnew coordinates on an open neighbourhood U of OE W.N By (ii) and (iv) of Lemma 1,f.(Ao) = span ' illey, }. Similarly, f,(A,) + f 2(Ao) = span (010y, -, 010y 1~(Y, and byind uction, A, span I 010Y 2, O/MY3 . .,01y,). Hence A, - is an involutivedistribution.

Sufficiency. By Frobenius' Theorem there exists a C"2 function h: PFN- P such thatAN - I(h) =-0 and (Oh/~x), .0 #60. Therefore f,(A0)(h) -=0 for I < i <, N - 1. By (i), since(?/i/ix).., #0, (0/?u)(hofN(x, u))1001 #0. It is easy to see that h(x) satisfies theconditions of Lemma 1.

Remarks'(a) Condition (ii) of Theorem 5 can be replaced by (ii)' kerf, + (X 'f,)(AO) is

involutive for 0 <, i <, N - 2, where ir(x, u) =-x, (n* 'f*)0 (Ao) =-A0, and(n - 'f,, '(A0) = i 'f*)(0r* 'f*K '(Ao)) for] j I (for this see Grizzle 1985 c,Lemma 2.1)~.

% (b) A more geometric necessary and sufficient condition for approximate lineari-0 F e-kzability of order p can also be obtained (Lee 1986), however, the conditions of

Theorem 3 are much easier to check.

I% &-,

*

L,,h'are :ahiit t of vI non li , it . l

Example IConsider the folio-Aing di%4crete-rznm F1.111, A1-1

= ) )

Since i+ ,. and

Note that

and /1P = O, for -'

,U IB))<. . I a 1 . ..i l

.I 4 = I [ ,. .. tI I i1

(B I )10 1) 1).

It can be easil) checked that all element-, of the 4th ,olumin I 4 arc 1 h1r 2Thus l, Espan (C') for I/ I %here (" is composed of the tiri four 'olUmn, at 4Therefore fit) of Theorem 4 is also satisfied Hence 1 is locall, lineariablc at it. tActually we can construct a diffeomorphism i = i F 1, in the %%a, that i, gien inthe proof of Theorem 4. Since

0

I1 0

.4 I - , for l3!08.,

I Tp

5 _____-- 0,, ,(1 + 1)! " "

c. =(0 0 0 -2). Clearly cT 0 0). Thus

T,(x) = x, - jX2 =x, -

T2(x) = T" +f(x- U) = X 2 -x

T3(x, u) = T, f 2(x. u) x1 + u - (x, + 2x u + 1')2

-A,.

. 4 A

.- ,.-%

*1116 H.-G. Lee and S. 1. Marcus

~ V Example 2Consider

E: [X, (t + 1)1 [X2(0) + 2x I(t)u(t) + X1(t)2U(t) + U(t)21 =fXt, U(t)Lxz( + 1)] L I1(t) + U(t) I

Clearly (i) of Theorem 3 is satisfied, because (Of/ieu)(0.0, and (ef/iOx)(0.O)(ief/iu)(0.0) arethe same as in Example 1. Since

(D Dj TIO0 01 - ((DDf T )0) 0

Since2 0

0 0.. 0 0

0 0

0 0

0 0

0 0

Also we have

(BT)oo )(0 ])01 (3 (Du f)(o.o,

14 j0 [ 1A](0 1)

0 0 0 0 0 0 0 1i

0 0 00 0 0001

0p 0 0010010 0

0 0 00 000 1

0~ 0 001

-0000

Linearizability of non-linear discrete-time systems 1117

therefore A22 - (B2)oo3

00000006

00020220

00000006

00020220

-02 202000

00020220

00000006I0002 02 2 0

00000006

(B~ )bo.o 0 = (D DeiT 0 DJ " ))O.O + (DJ 0 D0D0fT) oo

0DJT 0XDJ 1

=Dfs®D~fT'°'°1 + DJTfs® DX DMST i+ (DJT® D@ JT)o~so)oLD lf~i D.,D~fT J poo,"

00402000 0

0 02 40

* 2000

=0 0 0 00006

-"2000j

0020 2 2 0

2000

0440 S

4000

0660

4O000

A22 =@(B~)(oo= 0 0 0 0

4000

0660

4000

06601

D,. fT •9 D., D.?

llS 1H.-G. Lee and S. I. Marcus

Since flc Image (A,), E is approximately linearizable with p = 2. However, sinceP2$Image(41). it is not approximately linearizable with p = 3. Thus it is also notlocally linearizable. Let

T, = x, -

T3 = xI + u -(x 2 + 2xju + 2U + U2)2

Thenv-(tI + 1)4 T, (x(t + I)) = x 2(t) + 2xl(t)ut) + xt(t)2ujt) + u(t) 2

- (x,(t) + l)) 2

X0= 2t) - x1(t) 2 + xI(t)2u(t) = T2(x(t)) + O(x, u)3

= 2 (t) + 0(X, U)3

Y2(0 + 1) T2(x(t + I)) = T3(x0, u(t)) 4 tt)

4. Multi-input case,. The results in § 3 can be easily generalized to the multi-input case. Thus in this

section we give (without proof) a sufficient condition for local linearizability and anecessary and sufficient condition for approximate linearizability by state feedbackand coordinate change for a multi-input non-linear discrete-time system (for proof seeLee 1986).

Consider a multi-input non-linear discrete-time system of the form.. T5: Xlt + 1) = f (xlt), 14 t)) (4)

.4.." where x(c1t}rn. u(Ot- P'. and _f(x, u): F'*-._ is a C' PN-valued function. Also,consider the following multi-input linear discrete-time system F-o:

1 E0 : )jt + 1)= A.t)t+ Bilt)=g(t),v il))

where ,t)E P', zitE P, A = block diag :A11. 4 22, ..., A.I

.. -. 0 I 0 ... 0 0

0 0 I ... 0 0

A,, (K, x K, matrix)

0 0 0 ... 0 0

-0 0 0 ... 0 0-

ZK,=N.

B =block diag h, h 2 .... h,

bi =(0 ... 0 I)' (Kx I matrix)

Definition 4

p. Eu is said to be locally linearizable at (x4, u,) if there exist indices K, = , an openneighbourhood U (c: P+,) of an equilibrium point (x,. u,) and a diffeomorphismT: U - T(U) such that

(i) T=(TI, T2 ... , TN) are functions of x1 , x x. only,

(ii) T(X,, U ) = ON+m ).I

(iii) Tf= g T

W - ,-. .--.•'"

"

4 ./ , d € ." ,". "., " " ." '..,, " ..... " .' .'. " ." ,1 .. .. ... .. ... -. . ...- .. • * . " . ..-. "

Lineari:ability of non-linear discrete-time systems 1119

If we let

then )it) and it) satisfy the relation Eo

Definition 5E is said to be approximately linearizable with order p if there exist indices 1K,} } .,

an open neighbourhood U (c P' + ) of an equilibrium point (x,, u,) and a dif-feomorphism T: U -. T(U) such that

(i) T = (T, T2,._ T.,) are functions of x,, x 2. _ xN only,

(ii) T(x,, u,) = 0is +,,) I

(iii) T f =gVT+O(x-x,u-uF +

Thus in Definition 5 we consider the following nearly linear multi-input discrete-time system:

Eo: )t+ l)=Aylt)+ Brt)+O(x-x,,,u-u, ' '

where the N x N matrix A and N x m matrix B are as in Zo.Now we state the generalized version of Lemmas I and 2 and Theorems 3 and 4.

Just as in the single-input case, we can assumef(0, 0) = 0 without loss of generality, iffhas an equilibrium point. Also, we define f'(x, u) in the same way as in the previoussection.

Lemma 5E is locally linearizable at (0, 0)if and only if there exist {Kj) } 1. and C' functions

hi(x), h2(x) .... h.(x): W(c P).-+ P such that

(i) W is an open neighbourhood of 0 N P,

(ii) D.(h,-f')=O for I <j<m and I -i<k j - I(iii)

ex 0o.o

det # 0

ex 10o.01

%%

1120 H.-G. Lee and S. i. Marcus

(iv)

u )(.01

det 960

U ")0.0)-

(v) h(O) -0 for I j < m

Lm Let = (x' uT)T. Thus is a (N + m) x I vector.

i Lemma 6

o is approximately linearizable with order p if and only if there exist {K} -'= andC' functions hI(x), h2(x), .... h(x): W(C PN)-+ fP such that

(i) W is an open neighbourhood of 0c PN,

(ii) (DD(hj))o= for I -<fort < I <i<kj- 1, 0<k<p- 1

(iii), (iv) and (v) of Lemma 5 are satisfied.

Let

E ( )o.o (b0.0( )0.0) .\ (J )0 u .o

(~lf)K.~Z~of (f:)00 _,__)

Suppose that the elements of E are linearly independent; that is, they form a basis

for N. Let o, = K1 for 1 <i < m. Define I(v):Rfn ..Pf by C(v)- <,, where v is a

I x N row vector andO + (f(0

(X/, CA \0u 1 , (X( O1 a2(,)

lod~efine N = +fr+, i 2 b...y

)K"<,,o.o t ),o.o), (if~,,oo,X 4 1 2 U (0 0-1u

K'.s-, , (IfLs)I

II

ItxN ow ecoro , oo

I x(0 (7U2 ( ,) ) .. . ( 10)

ef)00' ,"i, .'.' """-..i%'' ' '-' ".--e,.. '. ." .-+'. . ",;$. " "lRC~'' 'G- - -(0.-01 f ... . -.-.- ,.,-.- ... ,

Linearizability of non-linear discrete-time systems 1121

where v, is the jth row of V Let

(WjD,,fT)(o.oI

4 (oD 2T),o.0o

'..1KX - I T(0 1 Ir.

Also, let y = T((fl)t (f 2 )T ... (fl)T)T. Let

D,', 0D, ... D,, "D2' I D22 ... D2,

D 2

D= , D 2 ... D , 1

whereD!j 0,.01 1(N +,' m x M+ I if i<jr 1

(Bh)(oO)

D= (Bj)(°'°} if i >j

(BJ- ')J.o

Theorem 7

: is approximately linearizable with order p (>2) if and only if there exist {Kj} 1 Isuch that

(i) the elements of E are linearly independent,

(ii) span E =span(EnE) for I < im

(iii) " 'c- Ilmage (D ) for I <lm

Remark

1 is approximately linearizable with order I if and only if (i) and (ii) of Theorem 7

hold,just as in the continuous-time case (Krener 1984). If m = I (single-input case) thenK, = N. Thus (i) of Theorem 7 is the same as (i) of Theorem 3. Since E, = Er E,(ii) of Theorem 7 is trivially satisfied. Since the operator C' is the same as the opera-tor C in the previous section, y,, = - . Since D" _ = A, (iii) of Theorem 7is the same as (ii) of Theorem 3. Therefore Theorem 7 is a generalized version ofTheorem 3.

Now a sufficient condition for local linearizability is given in the followingtheorem.

Theorem 8

Suppose that f(x, u) of E is an analytic R-valued function. T is locally linearizable

'.I ,,,. ,-.: :7 ,, :, :'.:.: . ... : . . . . : . : .. . ... : : .::4 . : > .. . : . ... . . ..

1122 H.-G. Lee and S. I. Marcus

at (0, 0) if

(i) the elements of E are linearly independent,

(ii) span E =span(E, nE) for I <i <m

(iii) there exists k (< o) such that Y! E span ((Fi)k) for I I m and i> 1, where(F')' is composed of the first k columns of D!.

Given the system (4), we choose the Kronecker indices {Ki}' , in a similar way tothe continuous-time case (Hunt and Su 1981). First we form the matrix

Lt be the number of lieryidpnetvcosin the first i +! rw for0

4 <N- I. Take 0= and ,a a 1 for I (i< N- 1, and define K, to be thenumber of' 1 with 00 i.

A('KNOWLEIXiMENT

The authors would like to thank A, Arapostathis and J. W. Grizzle for manyhelpful discussions involving the material presented in this paper.

This research was supported in par! by the Air Force Office of Scientific Researchunder grant AFOSR-86-0029. in part by the National Science Foundation undergrant ECS-8412100, and in part by the Joint Services Electronics Program undercontract F49602-82-C-0033.

Appendix

Lemma A.l1Equation (2) holds.

Proof

Clearly.

=x 10 (u ( B l .l~ O (0. 0

IMt

I=0

l',, .,i' G . -U ... OOX ( (.1 G )( .ll=.lD L.",.,

wer2e th are to be determined. Then

B"+~ ,.o= Dz(B,.,o) (AlI)N - 1.B Take y+ 0andfor I < i <<m (A 2)

nube of wit >, i.l|

Thell ats oul k o=D ha nk A. = rps i an J.W rzlefrmn

u ( isssinc i n Bm, t D D p t for m in I. Note that (2) is true when

L-em A. 1. , .- .. --. . -

Linearizability of non-linear discrete-time systems 1123

m 1. Now suppose that (2) is true form < p. Let I < I < p. Then, by (A 2),

B',,.= D(B-p.) + Df I BA,.,

=P K K Df.' fiT ( &Dtl '(DjfI ®~D A2 g , I I1 ...

,'It - V2= Al ,'t l 3

Changing the dummy variables k, k 2, ..., k, I of the second term into k2 , k3 ...., k,respectively. the second term becomes

P, i T I I -I k2 AI .'®e I JI k (D111

T®gD!-' *l(DJ iT® (D . (M i-k,k 2 -I k 3 - I

1D Tp+ I I I-I At A Al(D J.'T ®D 'DDD) ... ))= F E ... D D -P 1-k,

.T (9D!-k,(D fT k

(Dj-O O¢- AIIDOfIT® ®D!.' IAI(DOtD 'Dj T) ...))

Thus (2) is true for m = p + I and I < I p. By (A 3), it is easy to see that (2) is true form = I = p + 1. Hence (2) is true for m p + I. By induction, (2) is true for in > I. Z

Let h(x): R be a C function.

Lemma A.2

if

[ (DIh), -o(D~h}, -,

(S I S 2 ... S O ) .. = p I 4

4 (Dkh )h Ih o

where S, is a p x N' matrix for I k, i k then dE Image (B), where

ProofSi Equation (A 4) is equivalent to

"S, S(D 2h),,- + S2iDlh)x-,+ + S,(DA"'lh), _(,d

Consider

S, (D.,2 0 = d'

Note that (,'2 h .x,j) 3 -o - h,1 = hJ1 1 ( 2 h, i .xx,), -, for I < i < N and I <j < N.S-.,, VLet (sl), be the ith column of S. Then, since h,1 = hj.

NN N

WD S1 Dh)(-o, = I (s',),,, , I... + I (('I, ), ,.",, ~ ~ a/ aa~ I2a~

-+ ( 1 ) IIN+")h" 2 d

.1%

1124 Linearizabilitv of non-linear discrete-fime .sterns

Therefore dec-span (Q). where

Q -U l, :Nla I N .+N.' ' ,:W

A Now consider the matrix S,1 defined by

S S SI([-'12 + l' 21(

It is easy to see that the ((a1, - I (N + ath column of S,

(~21 11 2s )1,, 1Ii.%+ if a, a,

S 11 i) +4, + 's lW2 I, lIN' I if a, 6a,

where I -<a, <_ N and I <_ al N. Clearly Image (S, = span (Q. Similar argumentscan be applied for (D-,h), 0 . D 'h) _o. Therefore dE Image IBI.

.e 1RFIFRUNCtS

CHuNG, D., TARN. ., and ISjIX)Ri. A.. 1985. I.E.E.E. Trans. ajtom. Control. 30. 80KGRAH Am. A.. 1981. Kronecker Product.s and Afatrix Calcuilus vit h pplit at io,% tNev% York. Ell is

-e Horwood).GRIZZLE, J. W.. 1985 a. J.E.E.L.. Trans. aiaom. Control, 30, 868; 1985 K. Local input output

decoupling of discrete time nonlinear systems. Preprint University of Illinois, 1985 c.Feedback linearization of discrete-time systems. Presented at the 7th tnt Conf. onA nalv-sis and Optimnization of Syvstems. Antibes, France.

GRIZZLE,. W., and NuIJiER, H., 1985. Zeros at infinity for nonlinear discrete time systemsPreprint Memorandum No. 500, Department of Applied Mathematics, TWenteUniversity of Technology.

HUT.r, L.. and Su, R-, 1981. Local transformations for multi-input nonlinear ssstems ProcJoint -1utotnatic Control Conf ', Charlottesville, Virginia.

JAKtBCiYK. B., and RtsPNLM'K, W.. 1980. Bull. Atcad. Polon S4i I Sr M~ath -l srmn. Phoisc s. 28,517.

* KRENI&R, A- J.. 1973. SIA1M J. Control. 11, 670; 1984. Stist. Cont rol Let i, . 1 1li11, H.-G. 1986, On discrete time nonlinear control systems. Ph.D) dissertation. Department of

Electrical and ( omputcr Engineering. Unisersits of Texas at Austin%\o--A( 0. S.. and NOiRMAxN-C~t ROIl. 1). 1983 a. Init. J Control. .38, 245:' 1983 h. Formal power

series and input output lineari/ation of nonlinear discrete-time svstems. Pro( 22,idI.E.E.E ConfI on Decision and Control. San Antonio, Texas. pp 65 660, 1984, SsControl IAtt.. 5S, 191.

St , R.. 1982.5 .v. Control Ia'tt.. 2, 48.* VIrTIIR. W. J., 1970. I.E.E,. Trans autom. Control. 15. 24L; 197f. IMid.. 16, II 3.

%I'p..

e,

lilt 110j%.( iO.S ON (i llUtis AND)'. Si Si'.l. Vol (AS-33. No 7, it 149

A Structure-Independent Approach to the2Analysis and Synthesis of Recursive Linear

Time-Variant Digital FiltersJ,

TZONG-YEU LEOU, SrUDENr MEMBER, ILEir AND J. K A(iGARWAL. I'LLOW. tilI:F

4btract -This paper develop% a sy tematic approach to select a time- discrete-time counterpart have also been reported 111. [8].dependent %tate tran.f, rwation which can map a linear time-variant (I.TV) But the previous studies of discrete-time recursive LTVdigital filter to an equivalent filter having diagonal %tate-feedback matrices. filters are limited to those filters realized with time-variantDue to the structural %implicit) of the diagonal systems, this time-depen-

""* dent %tate transformation is a convenient tool for analting recursive VI difference equations. Since there are many other conceiv-filter expre ible in the %tate-variable form. In this paper, we discuss both able structures which can be utilized to implement an LTVthe theoretical basi' and the application of this diagonaliiation procedure. filter, a structure-independent approach to the analysis'The properties of two ripe' of recursive LTV filters are examined by using and synthesis of recursive LTV filters is desirable. Unlike

.* thi% %late transformation technique. Based upon the separable properties ofthe imptlse responses. vwe have explored a new algorithm for %%nthesiiing the implementation of a recursive LTI filter, the structure

deired impule responses vilh a major class of recursive LIF filters. 'i% selected for implementing a recursive LTV filter is an

technique, though suboptimal, can substantialb reduce the computation important factor in determining the characteristics of therequired in the snthesi% procedure. realized filter.

The main objective of this paper is to develop a struc-: . INTRODU(IOr ture-independent approach to analyze and synthesize re-

1N RECENT YEARS. there has been considerable inter- cursive LTV filters. Therefore, we express the basic modeljest in the analysis and svnthesis of linear time-variant of recursive LTV filters in the state-variable form. which is

S(LIV) digital filters for processing signals whose character- capable of representing most recursive LTV filters. Aistics change significantly with time. Generally speaking, a time-dependent state transformation is devised to reducestraightforward extension of a synthesis technique for lin- the complexity of the state-variable model of LTV filters.car time-invariant (LTI) filters is sufficient for implement- We then examine the basic properties of recursive LTVtog I. [V filters which have finite-duration impulse re- filters by utilizing the time-dependent state transforma-spnses. tlowever, the use of a recursive structure has the tion. Further, we explore the solutions to the synthesisadvantage, of saving computation time and storage space, problem of realizing an LTV impulse response with a

, if the duration of the desired impulse response of an LTV finite-order recursive digital filter.Sfilter is relatively long. But the synthesis of a recursive In Section il, we first introduce several descriptions of

LIV filter is difficult because the characteristics of an LTV digital filters, and discuss the properties of thoseI.[V filter are related to the filter coefficients in a com- representations relevant to the synthesis of recursive LTVpliatcd fashion, except for certain filters implemented filters. In Section III, attention is devoted to the analysiswith some special structures [1]. Some researchers have of LTV filters represented in the state-variable form. Asuggested a very simple but somewhat heuristic synthesis time-dependent state transformation that diagonalizes themethod, which is based on the implementation of the state-feedback matrices has been developed so that an

- fro/en-time transfer function of an LTV filter [2], [3]. LTV filter expressed in the state-variable form can belfoshve eer. as illustrated in [4], noticeable differences be- transformed into a filter consisting of K parallel first-ordero.Iecen the desired and the realized filter characteristics of filters. This leads to a general expression for the impulsei. n hI V filter exist, unless the filter coefficients change response realizable via a recursive LTV filter. In Section-cr, slovsly with time. IV, we illustrate how this diagonalization procedure can be ,

I he basic properties of a continuous-time LTV system applied to analyze the properties of LTV filters realizedrcahitahlc as a differential equation have been investigated with different filter structures. In Section V, we formulate :1in the literature [51 [7]. Recently, the properties of the the time-domain synthesis problem of a recursive I-TV

filter by minimizing the squared difference between the.M,inu,,rlpi received November 26. 19X4: revised November 14. 1995 desired impulse response and the impulse response realiz-

-h. srk va, supported hN the t)epartment of Defense Joint Sctmices able as a major class of recursive LTV filters. The numer;-SI lco rfnl, Program under the Air Force Offic of Scientific Research

S4.n 20 . X2.(-(3 I cal difficulty of obtaining the optimal solution is ex-rh(. J'h,,r, arc with the Department of E-lectrical and Computcr amined. An efficient suboptimal algorithm based on the

Sngine,-rin, The Lniversitv of Texis at Austin, Aumin. TX 79712IIt I .g Numhcr X1(iSQ52 minimization of the localized squared difference between

(X98-4094/86/0700-068701.00 1986 IEEE

-...4. . .. .,•f' .,r "' ; %' .. ,',." ". . ' e"d ": ." '" -V,, ,.-",. . ,, . , ,- -".,, .--

688 IEEE TRANSACTIONS ON CIRCUITS AND SYLSTEMS, VOL CAS-33, NO. 7. J:LY 198M

the desired impulse response and the realized impulse (ii)

response is also developed. A numerical example has been C,(n)

selected as an illustration of this synthesis algorithm.

11. CHARACTERIZATIONS OF LTV FILTERS

Generally speaking, most methods for describing LTV ,n in- I win K)

filters are evolved from those used for LTI filters. A X(n) _-- -.(f - .i yin)common time-domain description for LTV filters is the -a

'n)

time-variant impulse response, which is defined as theoutput measured at the instant n in response to a unit-sample input applied at instant m. Then, the input x(n)and the output y(n) are related to the impulse responseh(n, m) by the summation Fig. 1. Block diagram of the direct form It realization of a recursivc

LTV filter.

v(n)= h(n,m)x(mn) (1)tively, the input and the output of the filter, the input-

where the filter is said to be causal if the impulse response output relationship of a recursive LTV filter can be ex-h(n. m) satisfies pressed in terms of the state equations

h(n, m) = 0, for n < n. (2) W(n)=A(n)W(n-1)+B(n)x(n)

If one considers the computation and storage require-ments of implementing an LTV filter having a long-dura- y(n) = C(n)W(n) (5)tion impulse response, it is desirable to synthesize the filterusing some recursive filter structure. A widely used struc- where W(n) is the state vector and A(n), B(n) and C(n)ture for implementing LTV filters is a time-variant dif- are matrices of appropriate dimensions. It is clear that aference equation. which relates the output sequence v(n) recursive LTV filter realized via a difference equation canto the input sequence x(n) by be easily expressed in the state-variable form by choosing

K an appropriate set of variables a- he state vector. Further,Y a,(n)y(n - i)= b,(n)x(n -i) (3) the state-variable form is very suitable for representing a,-o ,=0 recursive LTV filter with multiple inputs and multiple

where ao(n) * 0 for all n and the order of the difference outputs.

equation is equal to K if aA.(n) 0 0 for some n. By The basic properties of an LTV filter realizable as a

adopting the direct form II structure [91 used in the time-variant difference equation has been explored by

synthesis of LTI filters, one can define another LTV filter Huang and Aggarwal [1]. The main result of Huang andstructure in terms of a two-stage difference equation Aggarwal's work relevant to our present study is statedbelow. The time-variant impulse response of a recursive

K LTV difference equation given in (3) is a Kth-order causalw(n)- Ea,(n)w(n -i) +x(n) separable sequence of the form

'I

y(n)= Y c,(n)w(n-i). (4) h(n, m)= u,(n),(m), n>_ m (6)

1-0 0-O, elewer

The block diagram of the direct form II realization of LTV 0 elsewherefilters is shown in Fig. 1. Many other filter structures can where u,(n), i =1,2,.•., K are K independent solutions ofbe developed in the same way. However, unlike the case of Krepresenting recursive LTI filters, it is difficult to establish a, (n).(n - ) = 0 (7)the explicit relationships among the recursive LTV filters ,-0realized with different structures. and v,(m) is given by

For the purpose of analyzing recursive LTV filters I . D , ( m +kksynthesized with a variety of structures, we focus our v,(m) f b(m + k)D(m+k) (8)attention on the state-variable representation in the pres- A- 0 D(m + k)ent study. Assuming that x(n) and y(n) denote, respec- where D(m) denotes the determinant of[wlr)u2m ' hr D(m) deoeste eeminn of

u(m-l) u 2(m-l) ... UK (m-l) uA(m-)

• - U ( ... UK ((M0)(9)

-U,(m-K+ 1) .. u. (m- K+l1) u, (n -AK'4l)

~ - . *' .

..

I hl 'Ni) \GG5R5 %I- kt( t'RSI5% It% DIGI IAL iJ1IItS o8

and D(?m) is the cofactor of the element u(m) of the characterized b" the state equationsdeterminant D(r). An implicit formulation of this result R'*(n )=A*(n)W*(n - I)+ B*(n)x(n)can also he found in [8]. However, after careful examina-tion, we find that the expression in (6' holds true for all n v(n) n ( )W *(n) (12)

ad m onls % hen I.1 K - I and aA (n) * 0 for all Pi. whereI herefore. e suspect that there exist-, a more generalizedeprcin for the impulse response of an LTV difference Al(n P (n)A(n)P(n - l)equation In addition to the time-%ariant difference equa- B*( n) = P '(n)B(n)ion,. ,.e are also interested in analyzing LTV filters real- n ) = C( n)P(n (13)tied \%ith lthcr filter structures (ionsequently, the desel-opmcnt in this paper is based upon the state-,,ariahle ('onsequentl\. the expression for the impulse responseMndcl for i. [, filters Vhe detailed deri\ations ass(oiated gisen in 10) can also be rewritten for the equivalent filter,.,ith the state-%,ariable model are discussed in the following as

*sectionh ( t. rn)

Ill. I) I \60.At Its I R idt~R~ I tON MRik

S I1A f-AR IABI I MODI I ('tt) A*ln - B*lm n "m I

S[he technque of state !ransforination isi a ser, con\.ni- --

Ont tool f ,r anal\ zing linear models represented in the C * n )B'i)

state--ariable form. In most cases, \,e want to reduce the 0. Pt

.ittc-feedha.-k matrix to a diagonal matrix or a diagonal- (14).hape matrix consisting of Jordan hMocks. Methods forNelectinc state transformation matrices for a continuous- If all the state-feedback matrices A*n) are of diagonaltille ss\ teni have been discussed in 1101. But these proce- form. then the original filter has been decomposed into A,ures are not applicable to discrete-time filters hasing parallel first-order filters. In this was, ,e can represent

5iiiii1:ul1r state-transition matrices. Therefore, we devote our each recursie LIV filter in terms of an equivalent di-

cl fi rt to des clop a new, procedure for selecting tranforma- agonahzed filter. Hence the diagonahzing transformation

tion mair:ces for a general discrete-time LTV filter. is an useful tool for the analysis of recursive LTV filters of

l t , ,'.r,,t restrict our considcration to a single-input different structures. Next. we V%) show that such a di-Si-|L-ui t~t L rv filter. For the K th-order filter given by agonalizing transformation exists for evers recursive LTVi-. the niatritccs A(n). Bin) and ('P,) are of dimension filter represented in the state-variable form.

A x K. A . 1. and 1 X K. respectisel.. After the input Assuming that the system matrices of an LTV filter are

X(n) in , is substituted with a shifted unit impulse defined for all n such that M, < n < ., we should be able

S( Pi Ci. the impulse response of the fiher can be derived to choose the transformation matrices P( n for M,, - I < n

as _< N, in order to completely define the equivalent filter.lct us define the forward state-transition matrix 1( n, m

0n[ A(n- j) B(m), n~m+l of the original filter as

C(n)B(n). n pit H A(n- ) n ii. n < . , -- (15 )

(10) \I. n=m

and let S,(n) denote the linear vector space consisting of

Because the impulse response gisen in (10) is a com- all the column %ector q such thatplicated expression involving multiple matrix multipli- *(n + . n)q 0.cations. it is difficult to examine the properties of theimpulse response without further simplification. We also choose A( Mo - I) = A( No + 1) = 0 to facilitate

One way to circumvent this difficulty is to select a the illustration of this algorithm. Then the transformation

time-dependent state transformation that can transform all matrices P(n) may be chosen in sequence by carrying outthe state-feedback matrices A(n) into diagonal matrices, the following steps from n = - to n = No:Assume the new state vector W*(n) is related to theoriginal state vector W(n) by 1) Let P,(n) denote the ith column of the transforma-

tion matrix P(n). Then choose P,(n)= A(n)P,(n-1) if

W(n)=P(n)W*(n) (11) A(n)P,(n-1)*0, where i=1,2...K. Let 1(n) denotethe set of index i such that A(n)P,(n -1) * 0 and let

where P(n) is a nonsingular transformation matrix. After U(n) denote the linear expansion of the column vectorssubstituting (11) into the original state equation (5) and P,(n) for I C, 1(n). The rest of the column vectors in P(n)rearranging the result, we can define an equivalent filter are to be determined in the next step.

P-"%" . ' . •, . . "., ".. . . .<,. ~~~ % -%.,..:;..:

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS. %O1. CAS-33. NO 7. 19

2) For= 1,2- -. N - n + 1, define bitn) 1(t c)n

S,( ) = the linear space generated by the vectors

in S(n)U12(n) (17) b;(n) w;() c;(n)

and 0.0)

'6 ( n ) = s , ( n ) S ," ,( . ) *t. -- • ).

qq ES,(n) and q'p=o

b,, (n) cK, (n)for every p e Sr f(n)) (18) C() (

Let (g,()1_ k < K,(n)) be a basis of the subspace 401,M)

T,(n) and {I7,(n), k =1,2,.. K,(n)) be an arbitraryvector set in S I( n). For j = 1.2,. IN, - n + 1, we choose Fig 2 Block diagram of the equivalent filter having a diagonal statc-

gA,(n)+ i%(n). I <k < K,(n) as column vectors of the feedback matrix.

transformation matrix P(O) if T(n) * (0). The columnvectors selected in this step will exactly fill the empty freedom in choosing the column vectors and assigningcolumns of P(n) which have not been determined in them to respective column positions. Even though selectingstep 1. different sets of transformation matrices may generate

different diagonalized filters, the input-output characteris-The basic propertie!: of the state transformation defined tics of these filters remain the same.

by the procedure stated above can be summarized in the A general expression for the impulse response can be. following two theorems. The proofs are given in the Ap- derived from the result in Theorem 2. Since the state-

pendix. feedback matrix of the equivalent filter has the form ofTheorem I: Assuming that a discrete LTV filter repre- (20), the corresponding forward state-transition matrix of

sented in the state-variable form of (5) is defined in the the equivalent filter is given asinterval M,, < n _ No. then the state transformation matrixP(n) selected with the iterative procedure stated above has 4)*(n. in) = diag[/ 1 ( n. m) f,. n) .. #,(n, P)]

the following properties: (21)

I ) The matrix P(n) is of full rank. where

2) Each column of P(n) belongs to one of the dis- ,

jointed sets H a,(n-j). n > I + I

AS,(n) S,(n)-S, ,(n) 0,(n. it) = /=) (22)

( = {qlq S,(n)andq TS, ,(n)} (19) 0' n<i

where j = 1.2... %, - n + 1. The number of the column for i = 1.2. . K. After substituting (21) into (14). we canvectors of P(n) belonging to AS, is equal to the difference derive the impulse response of the LTV filter asbetween the dimensions of S (n) and S, (n). h(n, m) = C*(n)V*(n. ,n)B*(m)

Theorem 2: Under the same assumptions of Theorem 1. K

there exists an equivalent filter given in (12) such that the = ,*(n),8, ( n, m)b,*(m) (23)new state-feedback matrix is of the form

where C*(n) = C(n)P(n). B*(m) = P '(m)B(m) andA*(n) = diag[(n) a 2 (n) .. *(n)] c'(n) and b*(tn) denote ith elements of C*(n) and

a,( n) B0B*(m). respectively. This result can also be easily verified

I ] )by examining the block diagram of the equivaent LTV

= I,)(20) filter shown in Fig. 2.The expression for the time-variant impulse response

0 aA(n) shown in (23) somewhat resembles the result given in (6).which is obtained specifically for time-variant difference

where a,(n) = 0 or I for t=1.2.. •. K. equations. However, an additional weighting factorThe essence of these two theorems is that once an LTV fl,(n, m), which is a function of m and n. has been

filter of the state variable form is completely specified in included in the summation of (23). Further, for some LTVan interval, there exists a time-dependent state transforma- filters. certain terms in (23) might cancel. This leads to thetion which can define an equivalent filter having diagonal situation that fewer terms in (23) really contribute to thestate-feedback matrices in the same interval. However, the impulse response. This property will be illustrated later bystate transformation thus determined is not unique be- applying the diagonalization procedure to an LTV filtercause in step 2 of the selection procedure, we have some realized as a time-variant difference equation.

LEOU AND AGGARWAL: RECURSIVE LTV DIGITAL FILTERS 691

IV. ANALYSIS OF LTV FILTERS VIA whereDIAGONALIZING TRANSFORMATION 0 0

Now, the usefulness of this diagonalizing transformationis illustrated by applying it to several realization schemes AI,(n)for recursive LTV filters. The analysis of LTV filters using 0the diagonalizing transformation is usually accomplishedby the following steps. First, we represent an LTV filter in - aA(n) . a2 (n) - a1(n)the state-variable form. Then we can obtain a set of 0 ]transformation matrices by using the diagonalizing proce- A 2(n) = bL(n) ... b2(n) b(n) Idure discussed in Section III. The properties of the originalfilter can be extracted from the system matrices of the A21(n) = [01corresponding diagonalized filter. 0 1

The first example is an LTV filter whose state-feedback 1matrix A (n) is nonsingular for all n such that M, < n < No. jFollowing the selection procedure discussed in Section III, A 22(n) =

we can choose the transformation matrix P(Mo - 1) as 0 1any nonsingular K X K matrix P0 . Then, just by carrying 0out step I of the selection procedure. we can select the restof the transformation matrices as By using the diagonalization procedure discussed earlier,

%. the transformation matrix P(n) may be chosen as

00

t P(n)= H A(n -j) Po, for Mo <n < No . (24) P,,(n) P,2(n)SP( n) M (29)

0 Is( nAfter substituting (24) into (13), we can derive the system

matrices of the equivalent filter as where[.Mo

A*(n) =P-(n)A(n)P( n - Ij), n>M o

B*(n) = P-'(n)B(n) 11(n) 0

C*(n)= C(n)P(n) (25) n = -1

where Mo< n <No. The impulse response of such a filter Is() [is(O)]=

can be obtained from (23) as 0 1 0'K 10

h(n.m) c*(n)b(n, n (26) 01- 0 0

,0, nl m . MIN

A typical example of such an LTV filter is the direct form P 2(n)=- PI(n) N Pn(n+ j)bL-((,-)(n + j )

I1 realization of the recursive LIN filter given in (4). ,The second example is the time-variant difference equa- "A[L, ((n + i - M0)) + 1]

tion shown in (3). where ao(n) is assumed to be 1. In this MINcase, the state vector may be chosen as + Q(n) I - o(n + I - Mo)

)[(n-r+) "" y -) (n)] r P(n) Pi'(i)A12(i)Is(i - MO -1),

x(n-L+l) "" x(n-1) x(nT (27) Q(n) -M 0

Note that the dimension of the state vector is K + L. No > n > M0Then, the corresponding system matrices of the filter are [0, n = M0 - Ifound to be

MIN = minimum { L, No - n)B(n)=[0 ... 0 bo(n) 10 0 0 1] r Io(j)='(j)[K, KIs(j)

C(n) = [ I 1 where A[i, j] is an L x L matrix having a single nonzero

and element I at the position (i, j) and ((i)) denotes then[ A12() (28)L-modulus of i.,,," A 2(n (28) After substituting (29) into (13), we obtain the corre-

[A 21(n) A22 (n) sponding system matrices of the equivalent diagonalized

,~~~ %. % %.- ,% '.1:.<-.2.:. ,. '"..2.<-:. -.-. ,.,-,,--.'.-.-.'- .. '.-.'.-"...,- ,-'."..:

692 WL-. IPRANSACIIONS ON CIRCUITS AND SYSTEMS. VOL (AS-33, NO. 7. JULY 1986

4 filter as according to some error criterion. In a practical problem.A*(n) = P '(n)A(n)P( n -1) the desired impulse response may be determined either by

solving a statistical filtering problem or by using some= [I 0 ] empirical rules. Since the computation for obtaining the

-0 1- lI(n- M0 ) true optimal solution grows rapidly as the order of thefilter or the duration of the impulse response increases, it

B*(n) = P '(n)B(n) is necessary to develop efficient solutions for the synthesisC*(n ) = C(n ) P( n). (30) of recursive LTV filters. In this section, we introduce an

Then the impulse response of this filter can be obtained by efficient suboptimal technique which is based upon thesubstituting (30) into (23). After some simplification of the minimization of the squared difference between the desiredresulting expression the impulse response is of the form impulse response and the synthesized impulse response inlocalized regions. A numerical example is selected to il-min , . m ... . lustrate this new synthesis technique.

.Y I _ [0'. 0 l]Pl1(n)P,,'(?+j) It is quite difficult to synthesize a desired impulse re-

h(n.-m) - -0 sponse in terms of impulse responses given in (23) thatb(m + j)[O ... 0 1]. n > m have arbitrary,,(n, n)'s. In order to obtain a manageable

10. n <m formulation of the synthesis problem. we restrict our con-(31) sideration to a subset of recursive LTV filters whose

impulse responses are causal Kth-order separable func-%'"' tions. i.e.,

which is somewhat different from the expression given in(6). But. with further simplification, it can be shown that - ,)(31) is equivalent to (6) for L < K -1. From the definition h( ) n P11)) = in). n "1 (34)of P1 1(n) given in (29), it is clear that the upper K-I",. " ,elsewhererows of P,1 (n) are obtained by shifting each of the corre-sponding rows of P,,(n-l) upward by one row. This where {u,(n): i= 1. .. K and v,(mn): I=I.- .K) areleads to the relation two sets of independent functions. Even though (34) is a

(I 0... IIP,,(n ) P,,'(i + j)0 0~ ... I]'=0 (32) special case of the expression given in (23). (34) is stillcapable of representing a major class of the recursive LTV

for n - n + 1 < j n - rn + K - 1. After substituting (32) filters. In particular. the impulse responses of most recur-for n - m + I << L into (31). we have sive LTV filters discussed in Section IV satisf, (34). As-

suming that the desired causal impulse response is

,z + j)[0 ... ] P( n) P,'(tt + j) hn. the s\nthesis of the recursive LTV filter canh'n )= , __ then be formulated as the minimization of the squared

[0..1]r. error function1' .1] ,,< 1. 3 ,' A V<= h,(n. ) _ u,(, )i,(n (35)

(33) , LThis expression is equivalent to the result given in (6). where 6 = { u,( n ). v,(n )I < i < K. 0 < t _< N } is a set of

From the above discussion, we have shown that the unknown variables and K is the order of the recursive. diagonalizing transformation is a valuable tool for the LTV filter. Once we determine the optimal 9 which mini-- analysis of digital LTV filters. Therefore, a systematic mizes the cost function, the coefficients of the recursive

approach based upon the diagonalizing transformation can LTV filter synthesized with a particular filter structure can- -be developed to reduce the complexity of the LTV filters be derived from the optimal ). An unrestricted nonlinear

synthesized with a variety of structures. Since the original optimization algorithm may be applied to find the solutionfilter is state-to-state equivalent to the corresponding di- of this optimization problem. The coefficients u,(n)'s andagonalized filter, the use of the diagonalizing transforma- v,(m)'s are considered as the unknown variables in the

% tion is promising in such areas as the stability analysis and optimization process. However, since the number of un-the roundoff noise analysis of LTV filters, known parameters is proportional to the order of the filter

times the duration of the impulse response, it is impracti-'-J.."V. SYNTHESIS OF R ( URSlWF. LTV FIITIARSV T ORT Tcal to find the true optimal solution of (35) for an LTV

In this section, we examine the time-domain solutions to filter with a large K or a large N. Therefore. it is useful tothe deterministic synthesis problem of recursive LTV filters. develop some efficient suboptimal techniques for solving

- Our main objective is to develop techniques for synthesiz- this nonlinear minimization problem.ing a desired time-variant impulse response with a recur- In Huang and Aggarwal's work 111. the causal condi-sive LTV filter such that the difference between the desired tion in (35) is removed so that a straightforward algorithmand the synthesized impulse responses is minimized can be applied to solve this filter synthesis problem. The

Itrl7,r2%

.IOU AND AGGARWAL: RECURSIVE LTV DIGItAL FILTERS 693

new error function is defined asm

A' IV [ K 2

n'(8 = IhD(nm)- u, (n) v, (m) (36)

The minimization of (36) can easily be solved by using aprocedure that was originally developed for finding thespectral decomposition of a matrix 112]. The requiredcomputation is approximately equivalent to that of obtain-ing K most dominant eigenvalues and eigenvectors of an(N + 1)X(N+ 1) matrix. Iv

The main drawback of using the noncausal error func-tion is that the performance of this synthesis procedure II N1.K n j

depends largely on how we choose the desired impulse Fig. 3. Regions of the impulse response where the error function isresponse in the noncausal region. In order to make the evaluated in different steps of the localized techniqueresult obtained by the spectral decomposition techniqueclose to the optimal one, the function hi,(n, on) in the for u,(n)'sregion ((n. in); 0 < n < ni < N ) must be selected such that

V N K- K

[h,(n.ni)- ho,(n.i)] 2 (37) u,(n) IM)I"(M)

P. ' Kwhere hiop1 .(n,n) is the noncausal separable impulse re- = Y h,(n.in)c,(in), i=l.....K. (42)sponse defined by the optimal C- that minimizes (35). ,,-0

Since there are no explicit rules for making a good guess ofthe noncausal part of the desired impulse response. the Note that the coefficients u,(n)'s in (42) can be easilyspectral decomposition technique often achieves less than calculated once we determine the values of (m) forsatisfactory result. It usually takes a high-order recursive = K.. and n = 0, 1.-. n - K. Therefore, by usingLTV filter to make a good approximation of the desired (40) and (42) iteratively, we can determine a suboptimalimpulse response. set of u, 1 n )'s and v,,( n)'s.

To circumvent the problems in the nonlinear optimiza- The constraints given in (39) ensure that the synthesizedtion technique and the spectral decomposition technique, a impulse response is the same as the desired one in thenew suboptimal technique for solving the synthesis prob- region 0 < n - in < K - 1. Hence, the localized synthesislem of recursive LTV filters is formulated bN minimIii ng technique will favor the synthesis of a impulse response(35) in a localized sense. The solution of u,(n) for 1 having dominant components along the diagonal line n =1...., K is obtained b mininu,'g the loc.licd error it. NoS let us summarize the complete algorithm of thisfunction suboptimal solution as follows:

I1) Appl% thle nonlinear optimization technique to findD = h(On.S) u)nlt,(t) 138) the optimal solution of (35) in a small interval 10..N 1. For

. example, the Fletcher- Powell technique [ 3] works well forunder the conditions that a small N,

. A 2) Use (42) to find u,(n) for n = N + 1.. , + K and* h,)(n. ) = Y'L u,(n),(.) V I.. NK.

3) For= 1.2.. N- N1. use (40) and (42) to obtainfori=n.n-I..t - K +1. (39) the solutions for I',(N 1 +j) and u,(N,+ K+) i=

With an index change in (39). we can express the linear L

equations for obtaining vt, In. i = 1. K as In fact, this localized minimization technique has dividedA the domain of the impulse response into four distinct

J, + j. i ) = j u,(In + )" (m). regions as shown in Fig. 3. Regions I and II denote,,-i respectively, the areas where steps I and 2 of the algorithm

for 1 = 0. 1.-. K - . (40) are applied. And regions Ill and IV represent. respectively.the areas where (40) and (42) are applied at step 3 of the

After substituting (39) into (38). w'e have that algorithm. The computation of this localized minimization,~A algorithm at each sampling point is equivalent to that of

D,(n) = Y h,(n. m)- Y u,(n)t,,(m (41) solving two K th-order linear equations..m. - 0 - We have tested both the spectral decomposition tech-

Differentiating (41) with respect to u,(n) and ,etting the nique and the localized minimization technique with aresult equal to zero, we can find a set of linear equations variety of time-variant impulse responses. The results show

* S.%

9:64 [III IKANAC WIINS O IN AIfCiISAND SlSUEMS. %01 AS-I;. NO "7. 'lI yh

IABLE I" PtlRORMANt I NII( ES Of TH& SYN IHISI A iuoRuiHms fOR I TV

IFII Itt D

SPECTRAL DECO14POXSITION LlOCALIZ LJED H tATLOW

I 0.74114 0.17202

0.1 2 0.',8230 0.0180- 0.46670 ).00552 J \

-

4 0.5b559 0.00215

10.88489 0.1954920,77557 o.0386

* 4 0.60011 0.00074 .(a)

that the localized minimization technique consistentlv per-forms better than the spectral decomposition technique. Anumerical example is selected to illustrate this situation.The impulse response of the desired filter is chosen as

h n, m) = h( (n/16, m/16) (43) V

where

(t'T) = 0.sinc[2(-t)'r(1-yt)]. lehrt>' ! tfexp( - [0.02(1 - )2+0.itJ

(h4 I) .- in 2( T( -yfl'0, elsewhere

and sinct )= sin(lrx)/(rx). The domain of the desiredimpulse response under consideration is limited to theregion where 0 < n < 128 and 0 < m < 128. Basically, thedesired impulse response is a truncated sine function along (hithe axis n + m = 0. but with time-variant frequency con-tents and an accelerated decay. A normalized squarederror function

b()= (h(n,m)-hz,(n,m)) ]

b t ) = ( l=(

I h (..,,)j (45)"-. ..

is selected as the measure of the performance of thesynthesis algorithm. A lower value of the normalized errorfunction means a better approximation of the desiredimpulse response. Two sets of the normalized errors havebeen obtained: the first set is for the spectral decomposi-tion technique in which the noncausal half of the impulseresponse is assumed to be symmetncal to the causal half:

S.., and the second set is for the localized minimization tech-nique. Table I lists the normalized error indices obtainedfor the desired impulse response functions with "y values0.1 and 0.05. The desired impulse response with y - 0.05,the impulse response of a fourth-order filter synthesizedwith the localized minimization technique, and the dif-ference between the two are shown in Fig. 4. Fig 4(a) ,Dered impulse respond" for Y - II the ih- nuilicriA1 r5aMo

It is clearly seen from Table I that the localized svnthe- pie (hi Impul.t' rcsponsw synihsited Aiih A fou-h order t TV filtcthieh uing the lo.aized mmnimization lechnque 1, 1Iifference h trn iht

sis technique achieves much better results than the spectral dewred impulke r sponw and ihe %nlhesiwedt imipui.e r eponse

%. .• .% ,%

." ." '. _, -' ,' '. '. '. " .. '. -' +' .' - ,% .' .' .% ." " . . .' -" . ,. ." . " .." . .i', . . ". ". ".. ".. ..". "/'€ . ..". -L.. .. . . - , .

11111 %0 .5Id.%KI.A-Al I ( Ki, sis 1 is D1uIiAI~ IIII

*decomposition technique does We alst, attempt ito find the columns are independent. Because the dimension of the set1puimal solution of minimizing ( 5) with a general nonlin- 1it Al, - I) is the difference betw~een the dimensions of

car miuinimi/ation algorithm No Lhat Ace can es aluate the S, Al,, I ) and S, 01A,, - 1), it is clear that the number of*result% obtained bK those suihoptinial teolhniques After column sectors of P( M,, 1) belonging to AS, ( Al", -- 1) is

experimenting with a numbhei of initial conditions, we find equal to the difference between the dimensions of the0that the perlformance index settles at a much larger salue subspace% S,( M, - 1 ) and S, 01'l,, 1than that obtained K the localized technique. I his seem,, Assunie the statements in Theorem I are true for n Vt0 c(Infirmn the conjecture that it is, quite diffikult ito use a A e need to shovw that theN are also true for it + .N#. Ingzeneral nnlinear miiifni/atii n technique thr fi nding the step I of the selection procedure. w~e choosef)\± -

Optimal solution of a s5 nthesis probhlemi Aluoh has mod- At IA + I1) /,, ( %r ) if A( N , I )P( A\ r 0. Assumec that thecrate %alue, ofI filter order A and Impulse respoinse d'jra- dimnension of the suhspace S, N )is, equal to AK, .Sinclion \St. -~t is equis alent to the set .A S,( N ), there are A

colJumn srectors of Pt N' ) belonging to S,( N ). Therefore.VI a t Nl \5there are A' - AK column sectors of P( N -I) selected in

*It has been demonstrated that the dia'rtal1111i-,i Iran',- the step I And these A - KA ectors must he independentformation is, a helpful tool for the anal\ Nis aind ss nthesis of because the matrixs AtO + I)t has rank K - A, and matixrecursive [TVI filters. Because there Is, a state-to-state P( A ) has rank A' Further. each (of these A' - A'sctr

equivalence between the original filter and the correspond- .elected in step 1 satisfs one oif the conditionsing diagonali/ed filter, the diagonali.'i g tranisfllmnatanca n be useful in other research areas related to the recur- + / I + I 0 -f and 4)(,r '\/\ It q tI

s i \e I.1 V s~stemns. In particular. the obsersabilits and the (46)controllabilit\ properties of a recursisc e -iT% filter are fr 12 ,-NTee r.ec i h etr

4casils obtainable from the ssstem matrices oif the equis,- fr 1 ,NTeer.ec fte co

alent diagonalized filter. Since the diagonaliuing transfor-sectdisepIblnsooeofhees (N+I.

miatioln procedure introduced in this paper Ailrks onI for 12...-Nadtenme fscosblnigtI.TV filters defined in a finite inters al. further insestiga- ec e seult h ifrnebt~e h iesonion is m~arranted oin sshether sse: can generali/e the of the intersection of the sets S 1Nt + I1) and S,( NV + It

diagonaliiing procedure such that it is, applicable itI ilters n h iesono h e vI h tp2odefined in ,in infinite intersajl the selection procedure. each column %ector selected also

belongs to one of the sets AS, ( V + I) for I, = I1.2.- NI,N. From the discussion at the beginning of this piaof, thenumber of column vectors belonging to AS,(N + 1) is the

Art \ii\difference betwseen and the dimension of Sl V +~ It) and the

1r'P,1 loi /1/1he 1',t I dimension of the intersection of S, ( N.\ It) and S,( N +- I)After reviewing the properties of the column vectors in

it i, caisi\ Iseen thait k, .n '+uI and I it are linear both steps of the selection process. swe have that the'Uihspaccs lf Rio for , i.I.V it I. and that (1 } number of column vectors belonging to AS,( N + I ) Is

1 - ~ ~ ~ ~ ~ t 1 11 N,(' S 1 . I 1nbecmesa ollctin f ds:equal to the difference between the dimensions, of S,(N

Sr 'i I~~~~~~. ( anV (N±1 o .. I- . Further, the ne%itlinted sets Further, if g,titt. v ,t it). . i,'( ti I ) is / N

* hsi ofit;i an a~ et ~~. ~~. . , ~isvectors selected at each stage of step 2. together with theIN*at contaie it nd n. sthe yp) 17 1, ti column vectors of P( N t- I) pre% iousl\ chosen, still con-

I) s t it) 1 v,, (P1 is I oillection of independent stitute a set of independent sectors. [bus the matrixP( V + I) is a nons ingular matrixs No\A we ba\se showkn that

I lluniri \ectors in the set AN t i And the number ofthe statements in Theorem I holds, true for the matrix

ccttIirs in this collection is, equal toi the dimnensioln (i the . t.linuiohesamntnTermIisre

Ipit~pc FT n . which is equis .ilent to the difterenceIfor all n such that I 1 PI nhet\ c cni lie dimension of S,ftir and the diMW11 Olclililf

the interseClion of N! Pi and Vi it I urthernil ro it is Ir,,,/l for 1/u-unrn 2* ~ ~ lcar that the sector set ii If it r it ( ' n II Pi sn h cut bandi ere ecnfn

(Itt n ,(i) + r1t 1 to agether %kith ins sci -4 I&lesn.h eul )tie nIit~cn1 Ccnfn

penident sectors in thle transforniatilln miatriLcs Pt 'it )for it ,-I.

( it constitute: .ii, k indic pc ucli t and these matrices has e the properties statedsector set.

IIIT I ireili I -\N ssUre that miarrito +4 n it has rank A,, FlirThen. Theorem I can be shosk ii hs i ndili t i i i N, sehseta hr ieA. needn II

'4,1. we base that A( V., I 1 0 Asht, !I Icac t,, hliat ui ~o'r P u 1ta10A,--~~ ) and that SYt V4 Ii V I NsIIIe

- ~ ~~~~ ft -4, 0l'4 for all q. all the t.l uriun \.-Ts ,I Pi'' Vtt .ttIti liit~* I1) are selected in step 2 (if the diagn- ,ro .. i dl (m ti

procedure. From the diSssil f in the prcs Ii, pa.gr~i~iph iilth s Itcluissast ht*se know that each column of Pt VaJ I I hl I- fl-' 1' 'nc) -4the disjointed sets AS,(ni i 1 1.1~ ttc5. .41 i iP t r It Ij)S

% * -

*From these tv-o equations. \4e halie that P 'i , n )P (n I rin I,- 19M) h, 1w rkcd a, A Tca~hin..

-1) Al if Ai(n FP,(ln -1) * 0. and P I (n)A(Pi)P,(n -i at \2a1ona iai~a hi %r\ Tal)- 1~0 if At n F P1 n - 1) 0. \Nhere Alti] denotes a column A'si~lanl at the Decpartmecnt tif I IVitrlLaI and

% ector ha% ing the oni' nonzero element I at its i th posi- ( 'rniputer F rci~at Oil I nii, rit Aition. frherefore. the equi~dalent feedback matrix Lorre- Jt 'Nuir1n 'ill~ n Tl, \. Sin I'Jw i h ti- 1,

spondino ito the time-s anant state transformation obtained a Red Vw taNnt Ii i Ii iilift 1w lm

in Theorem I is a diagonal matrix w&ith the diagonal Au-tin. Au'rin tc~a, Ill tinrain rccii-

elements of the % alue 0) or I - arc %inJ iilcor,, Plk.~iClit

RFFuRfNC i-S

11 ( Nuang arid I K Agadr a. (In lneat 'hilt aianit digital

itcr, III 'rt, ( iruti;% t i- 1 ( AiS -' pp t'2 619, Aug,-2; / I Nikoi, A- reuri f irne,,.a''-nd: hand pam flter." ,

.4. .l . 'I 440. PP l2o sO. Junc PjI

\ariahle pam-hand'. Pr- 111 1 /,,, S, Pitir .o (l tddi adt J, k. A ganii (S'O' (1,s SM 14 F "ttemn ( h1L.,. Ft.. pp 4'4 42- '\P' PJI cidth nnIIinai-adp \4 T N loIu and I K *Wd.mrvl a1 R,,,.r,,c imtpicritn titn of I T\ ciIdteBS: alctt n hI? Itc r, r,-/cfl-tiirc i i'Tjr fun,!in 'Icr~u, gc:n~ralicd transh'r - the Unier,it% M it Hnhj iNS6) ilil It In

ful ;, ,, /I I I -i pp YX 9tst 1, 1. JUI %1 from the ULnic r, it~ \ fI i \ crp .,I IFngland (IwJ It rul 4 n t- rcalia, :%' lf linear differential ~~cn. I/I and the MI d- and Ph 1) it the L ni%,it'A.

I \ 1,dc ii a'\ n4 iL I., /R Ju I ,, Il - PP, He jrifled The L i.ci -if Ic~a, in 1-4l%45. j "! "' %i arnd\-itantP F1-c.' r and ha, %intc hcld p-

Ii u., -. ,., %Itnc rn lflet 'i~ t,- a,A - I I ", tc - i 9 X ar i a. a'f* . ' R a' ci . n H (I' I ll I . -ai I'a ,, 'L- li d \-rl 'utr ihr tc , 1 ga' tir-i n1 il' 'l~ J. M, Kr ' tne s

R4 I- 'F rn BA. Irn.'. ;rrt' ll- 'I 1h I'td I rtilne hett ifti- .'r %.:in a- tn r- ., ,, r ,:~ I I a. n pn~c

''1 ~ ,rn~w~ ,-o !j i. 5~,, r i ,' l, 1 1 - ni- ~ t c, J-l ". !; ii, 'ill/i i g-. Ii-h:11.1-11 Fli t. al " ,l,ari SI th.' -t ir, a~r ,ttI tio 4, 11 he 1110,L' TO.~ t n alh-, ,,I ''4 ''

I F I'~ flir Ind I I A '--Aetl 1 rap-id eL rc ii: l l, Cit r-t. ar ' ' 'd ' ilin I a r- F11

- , F-i. '1. '(11 aIt., III I'r. ;:r, n- fi .'i it , .. r~

' 'ntiI :I i i IF I ll rar at ' - : I t I I ,t~ ll'

J I R HIdit-, in1 ',rI r'', IFercn F,%Ln F, F, r.t Ftd '1 \1' !r ct- T -,t rtr 1-i 11

'4. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1 t/)l-e !o SX a -' n 1t..tt '1Yii.i )I~( FI -rple - cr. Of d M NA'S! ic-ld ('viih 1nd914 .:

lana ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ i it nnsr. i1-w~e.' a'at nI ''adIl~ V e ncs r- ,I Fiti, ne .rn Pratnni1 .l P,tw, m- l itnai t: ild Jap hII

.. ,,~i B -w411 [t:, l . (. (j n r l (1a i a (I

lnfcIn, i4P o riR - 0,i- n na' h arr5.f~ il-I

A1ii a nv.~!''1 A ,'"Id hJ o gNu'en54'I %%,bn i i, n - ~ i 9 9(1 uc , 1,-n AN od t 9.

ree w th 41n IS d gc i ~~t-:c~n c i rr D n rh r . V \ A i T eIi~n in 11, ii'.~a Un k ri% ap i x...i n X .r-1l ) t - i e rna P te n R .,C~ill 11-N ilIi a p L

%I

%~1.4.

K1epF1 n.rd f- JOL RN4L OF T'HE ELECTRIOCHEMICAL SOCIETY

VoI 13. No 2. Ie-bnn 1986Prnnted -, ,t A

* Temperature Response of GaAs in a Rapid Thermal AnnealingSystem

T. R. Block, C. W. Farley,* and B. G. Streetman"Microelectronics Research Center, Department of Electrical and Computer Engineering, The University of Texas, Austin,

Texas 78712

Rapid thermal annealing (RTA) systems are used to sponse. Comparison of doped vs. semi-insulating GaAs inheat semiconductors for short periods of time, usually Fig. 2 shows the tremendous effect of free carrier concen-seconds. This technique has a variety of applications in tration on the temperature response, an effect also foundthe processing of Si and III-V compound semiconductors in silicon by Seidel et al. (5). However this is not enough(1). In an RTA system which uses radiant energy to heat to explain the data completely. At 7500C the free carrierthe semiconductors, one might expect to find variations concentration of Si is greater than that of the dopedin the temperature response of materials due to their dif- GaAs, yet GaAs is at a higher temperature. This effectferent absorption properties. An investigation of GaAs must be due to basic differences in the bulk material

, and Si shows that this is indeed the case, and that the properties, for example direct vs. indirect bandgap, size oftemperature '.ariation is quite pronounced.

Experimental Thermocouple a)The RTA system used in these experiments is similar to

that described by Davies and Kennedy (2), and consists ofheating chamber containing two elliptically shaped cav- GCAs7

ities which focus light produced by two 2 kW tungsten Silicafilament quartz lamps onto a sample suspended at thecenter of the chamber. A microprocessor controls powerto the lamps with a thermocouple for temperature feed-back. Samples are suspended on very thin (-250 uimthick) silica slides inside a silica tube which allows an- Them-rocoupie b)nealing in a selected gas ambient. / T

Semi-insulating (100) GaAs doped with Cr, p-type (100)GaAs doped with 7 x 10" cm-' Zn, and n-type (100) Sidoped with 4 x 101' P were cut into samples -8 mm X 8 S1 GaAsmm. Holes were etched into the GaAs using a 20% bro. r S'mine-methanol etch and a mask of CVD deposited SiO,,, Silicaand into the Si using a solution of pyrocatechol, ethylenediamine, and water (3). 200 nm of CVD SiO containing Fig. 1. Schematic diagram of sample configuration far (a) thermally7% P was then deposited on each sample. K-type thermo- isolated and (b) thermally connected cases.couples (0.002 in. diam wire) were glued into the holeswith a mixture of Aremco no. 516 cement (ZrO) and eitherground GaAs or Si using a procedure similar to that sug- (a) Setpoint

u gested by Cohen et al. (4). Such small thermocouples 1000 - (b) SIP -- -

were used to insure that the temperature response of the (C) GaAsn --

samples would be little affected by the thermocouples. (d) GaAs:Cr - -Throughout the experiments, the GaAs samples were 800 -- . ,'.

either thermally isolated from each other or placed on a 0larger piece of silicon to thermally connect them, as

. shown in Fig. 1. Temperature feedback to the controller LtJwas provided by the Si sample. These two configurations a600-

4. were then submitted to a heat cycle consisting of a pre- - -0

heat at 300'C for 30s to stabilize initial conditions, then an < / --...instantaneous change of setpoint to 750 C, a hold at this : -temperature for 30s (counted after the sample is within L 400 /30*C of 750*C), and then a change of the setpoint back to

* zero. The heat cycling was done in a stagnant N, atmo- L .sphere. t200 "

Results and DiscussionTemperature response curves for the thermally isolated

samples are shown in Fig. 2. Curve (a) is the programmed -

setpoint and curve (b) is the thermocouple output of the 0 20 40 60 80 1 0Si sample. The Zn doped and semi-insulating GaAs 06

% samples are shown as curve (c) and (dJ, respectively. TIME (SECONDS)Clearly, GaAs couples to the radiant flux differently

than Si, exhibiting a drastically different temperature re- Fig. 2. Temperature response curve for thermally isolated samples:* (a) temperature setpoont, (b) P doped So, (c) Zn doped GaAs, and (d)

*Electrochemical Society Active Member Cr doped GaAs.

% %• . .-• , ,r,----------------------------------"-"------------ -- "r , ,s -- . ," AA,. -.- _-., '-A*A - "A, ,"- -. " "-- -",-- -

Vol. 133, No. 2 GaAs IN A RAPID THERMAL ANNEALING SYSTEM 451

(a) Setpoint would contain the thermocouple. In such a setup, we1000 - (b) Si:P - have observed negligible temperature overshoot and the

(C) GaAs'n - - temperature difference between GaAs and Si is usually(d) GaAs:Cr - - less than 15°C at 750*C. It should be noted that this an-

-nealing arrangement requires good thermal contact of the0 800- sample to the large Si susceptor. We also found the tem-0perature response to be sensitive to gas flow in the sys-tem. A high flow rate resulted in a GaAs temperature

" 600 50*C higher than the temperature of the Si susceptor at7500 C. The source of this pheno.nenon is not clear; the an-

- neals were therefore performed in a stagnant atmosphere.cc These results point to an important fact: GaAs behavesJ 400 quite differently from Si in a rapid thermal annealing sys-n , tem. Consequently, the temperature of the GaAs must be

used to control the system. This can be done either di-200 rectly, by using a pyrometer looking at the GaAs or indi-

I 0 rectly, by having the GaAs in isothermal contact to a sus-ceptor to which a thermocouple is attached. The lattermethod is the less complicated of the two and avoids

/1 -r problems pyrometers themselves face in a radiant RTAsystem (7).

0 20 40 60 80 100 AcknowledgmentsTIME (SECONDS) The authors acknowledge support from the Joint Ser-

Fig. 3. Temperature response curve for thermally connected vices Electronics Program under contract No. AFOSRsamples: (a) temperature setpoint, (b) P doped Si, (c) Zn doped GaAs, F49620-82-C-0033 and the National Science Foundation

* and (d) Cr doped GaAs. under grant no. ECS-8420002. We also wish to thank D.Neikirk and S. Lester for useful discussion during the

bandgap, optical absorption parameters, and thermal con- course of the experiments.... ductivity. Theoretical work by Borisenko et al. (6) pre- Manuscript submitted Oct. 4, 1985.

dicts different heating rates for materials with differentemissivities. Another factor may be a shift in the fre- The University of Texas assisted in meeting the publica-quency distribution of the tungsten lamps' output at dif- tion costs of this article.ferent power levels (due to blackbody radiation at differ-ent temperatures). As the lamp power is cut back to hold REFERENCESthe sample at temperature, the light shifts to longer wave- 1. J. F. Gibbons, D. M. Dobkin, M. E. Greiner J. L. Hoyt,lengths, which may affect the relative amounts of band- and W. G. Opyd, Mater. Res. Soc. Symp. Poc., 23, 37

4, to-band vs. free carrier excitation. (1984).Results for the thermally connected samples are shown 2. D. E. Davies and E. F. Kennedy, Electronics Lett., 18,no. 7, 282 (1982).in Fig. 3. Curves (a) and (b) are again the programmed 3. K. E. Bean, IEEE Trans. Electron. Devices, ed-25, 1185setpoint and the Si sample response, (c) and (d) the Zn (1978).doped and semi-insulating GaAs samples. 4. S. A. Cohen, T. 0. Sedgewick, and J. L. Speidell, Ma-

When the materials are put in thermal contact, their ter. Res. Soc. Symp. Proc., 23, 321 (1984).temperature behavior becomes similar. The significant 5. T. E. Seidel, D. E. Lischner, C. S. Pai, and S. S. Lau, J.temperature overshoot of setpoint is due to thermal lags Appi. Phys., 57, (4), 1317 (1985).

6. V. E. Borisenko, V. V. Gribkovskii, V. A. Labunov, andin the control loop from the configuration in Fig. lb. In S. G. Yudin, Phys. Status Solidi A, 86, 573 (1984).actual anneals, the large piece of Si pictured in Fig. lb 7. C. Russo, Nucl. Instrum. Methods, B6, 298 (1985).

, J

5'p~,

R~,,pIl-tid llll t, ... {I + M OF M El Ml I RIK. LElI l A t, I IIi , \ -, , n.n, -1-n6

1Ijh.

Factors Influencing the Photoluminescence Intensity of InP

S. D. Lester,* T. S. Kim, and B. G. Streetman**

Dt'pa rtmen t of Electrtcal and Computer Engineering. Microelectronics Research Center. The L'nt'ersty q/ Texas.Austin. Texas 78712

Room temperature photoluminescence (PL) has recently Another example of surface effects on PL intensity isbeen used as a tool to assess the quality of InP substrates the case of lnP exposed to various gas ambients. Fig. I showsduring and after various processing steps and the quality of the band-edge PL intensity of an undoped ( n=5xlO15 cm-3 )InP/insulator interfaces It has generally been suggested that sample repeatedly exposed to oxygen and nitrogen. As thehigh PL intensities reflect high quality bulk material or high figure shows, the PL intensity is reduced in oxygen, increasedquality interfaces. For example, band edge PL intensity has in nitrogen, and can be cycled repeatedly. The size of these

- been used to assess the effectiveness of annealing treatments for ambient-induced PL changes are influenced by many factors,activating ion-implanted dopants I and has been used in a including pressure (flowrate), substrate doping concentrationnumber of studies to monitor Jrocessing steps used during the and type, the laser intensity, humidity, and the history of the

. fabrication of MIS devices.- In the former studies it was sample. Some of these effects have been reportedsuggested that high PL intensities reflect high quality annealing previously, 7' 8 and a more detailed description will be presented(a bulk property) and in the latter studies it was suggested that elsewhere.9 We also note that GaAs is sensitive to ambient*,he PL intensity of n-type lnP yields a reliable estimate of the effects, but to a lesser degree than JnP. Like the effects ofinterface state density in the upper part of the band gap. In this chemical solutions, these ambient effects are very substantialcommunication we discuss factors which influence band-edge (in certain cases the intensity can be change by >5x) andPL intensity and point out that great care must be taken in illustrate the high surface sensitivity of band edge PL intensity.interpreting such data. The fact that PL intensity is so strongly influenced by these

surface effects (including interface effects at InP/dielectricAlthough many factors are involved in determining PL interfaces) indicates that great care must be taken in extracting

intensity, three important influencing factors are bulk bulk information from PL intensity data. We also note that PLparameters (i.e., mobility, lifetimes, doping level, etc.), the intensities measured at low temperatures (samples immersed in

" surface recombination velocity (S), and the presence of a space liquid He) are sensitive to sample surface properties prior tocharge region (electric field) at the surface. The first of these cooling. To safely compare the PL intensities of a number ofreflects bulk crystal quality and the second two are determined samples it is therefore important to insure that the samplesby surface properties. The relative influence of these factors in have nearly identical surfaces.determining the PL intensity of a given sample is, in general,

, very difficult to determine; thus, processing-induced changes inPL intensity can easily be misinterpreted. We note that PL N ,intensity changes have been used as a quantitative measure of 2 02 2 2GaAs surface properties. 3'4

Although PL has long been recognized as a near-surface " ,probe of bulk material, surface effects can very strongly -influence the intensity of band-edge luminescence. This surface Csensitivity is dramatic in the case of n-type lnP where both

" .- liquid and gas ambients have been shown to have pronouncedeffects on PL intensities. In the case of n-InP immersed inchemical solutions, it has been demonstrated. 5 and confirmed in 10 5 min. oour laboratory, that PL intensities can be changed by threeorders of magnitude. For example, in-situ measurements of

- n-lnP alternately flushed with DI water and dilute HF show that Timethe band-edge intensity can be reversibly varied by -1000x in avery short time. Apparently, different chemical treatments also Fig. 1. Band-edge PL response of n=5x 1015 cm3 lnP toleave the InP surface with varying degrees of "stability" to ambient changes.subsequent changes in PL intensity caused by exposure to air orlow temperature annealing.

5'6

Another notable feature of Fig. I is the gradualElectrochemical Socicty Student Member reduction of PL intensity. This slow trend is at least partiallyElectrochemical Socity Atuivc Member reversible and has previously been attributed to oxidation. 7 We.. *$Electrochemical Society Active Member

"II

2208

"~~~ ~ .+ d""'"•

"*"*"P""""", - z . """""" + ".. . . . . . . "Y/ " + ' 'mm % d+

-% ' ' ' ' ..-.. " -.-. ".-." ' ' "

I%,%

Vol. 133. No. 10 PHOTOLUMINESCENCE INTENSITY OF InP 2209

have found that this slow trend is a direct result of the in or a change in band bending is responsible for (he PLillumination process and can substantially reduce the PL intensity and resistance changes. However, Fig. 2b clearlyintensity under high laser power levels. Therefore, it should be indicates that ambient changes result in changes in the surface

recognized that the illumination process itself can markedly Fermi level (depletion depth). This change in EFs and the

,~,r alter the surface properties and consequently the PL intensity of depletion depth then results in the PL intensity response

InP, so care should be taken to account for it when taking room shown in Fig. 1. Such PL intensity variations have

temperature PL data (i.e., samples should receive comparable previously been attributed solely to changes in S.7 Our d. ta,amounts of laser excitation prior to the start of the of course, does not exclude a change in S, and in fact S shouldmeasurement). be a function of Fermi level position. However, this does

demonstrate that the surface Fermi level is affected by ambientThe previous examples have shown that surface changes and that changes in EFs must be considered when

properties have an important influence on PL intensity and need interpreting PL intensity variations.to be considered if bulk information is to be extracted from PL

" ~ intensity data. On the other hand, these surface effects can be In summary, room temperature PL can be an extremelyused as a tool for studying lnP surfaces and interfaces. If useful technique for investigating bulk and surface properties ofsurface information is to be obtained it is important to InP. However, since the intensity of the band-edge transition isdistinguish between the effects of band bending changes and strongly affected by a number of factors, great care must besurface recombination velocity changes on measured PL taken in interpreting intensity data. If bulk information is tointensities. As suggested by Aspnes, 10 to do this, it is be obtained, samples must be prepared with identical surfacesextremely helpful to have a second measurement technique and if meaningful surface information is sought, a secondwhich can give an independent measure of the surface Fermi measurement should be used to separate the effects of bandlevel (EFs). Possible techniques for this include Raman or bending from the effects of changes in the surfacephotoemission spectroscopy or the use of PL with two recombination velocity.excitation wavelengths; however, the simplest example of such

_ a technique is to measure the resistance of a thin film resistor Acknowledgement:which will have a resistivity that is a function of the depth ofthe space charge region at the surface. The authors acknowledge support from the Joint

Services Electronics Program under contract No. AFOSRFigure 2 shows the resistivity of an n-type resistor F49620-82-C-0033 and the Texas Advanced Technology

structure in oxygen and nitrogen. The structure was made by Program.implanting Si (1012 cm "2 @ 150 keV) into a -3 x 5 mmInP:Fe sample and alloying lnfSn ohmic contacts. Figure 2z" References:

M... shows the resistance under illumination and Fig. 2b shows the

resistance in complete darkness. It is clear from these figures 1. D. Kirillov, .. L. Merz, R. Kalish, and S. Shatas, I- A.ppl.that the resistance of this structure, like the PL intensity, is Phys. 57, 531 (1985).reversibly changed with ambient. Since the resistance 2. S. Krawczyk, B. BaillO, B. Sautreuil, R. Blanchet, and P.under illumination depends on the surface recombination Viktorovitch, Electron. Lett. 20, 255 (1984).velocity, Fig. 2a is not sufficient to indicate %.,-ther a change 3. J. M. Woodall. G. D. Pettit, T. Chappell. and H. J. Hovel.

J. Vac. Sci. Technol. 16. 1389 (1979).4. S. D. Offsey, 1. M. Woodall, A. C. Warren, P. D.

Kirchner, T. 1. Chappell, and G. D. Pettit. Appl. Phys.N 2 02 2 0Lett. 48, 475 (1986).

(b) I I 5. S. K. Krawczyk and G. Hollinger. Appl. Phys. Lett. 45.870(1984).

6. H. Nagai and Y. Noguchi, J. Appl. Phys. 50. 1544 (1979).(a) 7. H. Nagai and Y. Noguchi. Appl. Phys. Lett. 33, 312

) i (1978).

8. H. Nagai. S. Tohno, and Y. Mizushima, J. Appl. Phys.*50, 5446 (1979).

9. S. D. Lester, T. S. Kim, and B. G. Streetman, un-

" mm. '- published.5 min. %d - -,10. D. E. Aspnes, Surf. Sci. 132, 406 (1983).

TimeMaruscril t sulir ittel Ma' 1, 1

reviseI manuscrij t receive 7'l v , 1 .

Fig. 2 Resistivity response of an n-type resistor (a) underillumination and, (b) in complete darkness. The n i vers i tv of Texa oss t of iS

meeting the ublication co';ts of this~article.

'4,

PHYSICAL REVIEW B VOLUME 33, NUMBER 2 15 JANUARY 1986

Experimental observation of adsorbate orbital splitting at single-crystal metal surfaces

Marshall Onellion and J. L. ErskineDepartment of Ph.isLcs, ,niversin of Texas, Austin, Texas "8712

(Received 30 May 19851

Splitting of the 5PI,, component of the photoexcited Xe ion doublet is observed on the (110) planes ofseeral metal surfaces This effect is shown to originate from a true "crystal-field" effect, not as a conse-quence of adatom-adatom interactions The splitting therefore provides a probe of local fields at thescreened ion

One of the more important parameters associated with negligible (i.e., low coverages), leading to the conclusionsurface phenomena is the local potential. Recent efforts to that the splitting is similar in nature to that reported by Opi-more thoroughly under,tand local surface potentials include la and Gomer. We have conducted additional experimentscalculations of the orbital splitting of atoms approaching a on Ni(110) and Ni(100) surfaces which yield the samemetal surface,' studies of dipole moments and polarizabili- result, i.e., that the lower-symmetry surface produces split-ties of adsorbed atoms' (both ground-state effects), and ting of the 5P I12 line. The purpose of the present Brief Re-analysis of charge transfer and screening effects at metal port is to present these results which suggest that splittingsurfaces which accompan various photon and electron exci- of the j = 3 component of photoexcited Xe atoms phy-tation phenomena.' - One particular focus of work on this sisorbed at low coverage on (110) surfaces results from theproblem has involved excitation properties of rare-gas lower coordination symmetry of the adsorption site.atoms' - ! on metal surfaces, i.e., systems which have well- Experiments reported here were conducted using andefined ground states. Auger-photoelectron spectrometer' equipped with low-

Nkaclawski and ilerbst' conducted one of the first photo- energy electron diffraction (LEED) optics and a cold stageemission inestigations of a rare-gas monolayer on a metal manipulator capable of sample temperatures ranging fromsurface. They reported significant broadening of the 5P312 1200 to below 30 K. The --in.-diamx 1 -in.-thickcomponent of the spin-orbit split 5p level of Xe physisorbed NiAI(110) samples were aligned to ± 1 using x-ray Laueon W(100), and attributed the broadening to unresolved techniques, and spark cut and mechanically polished usingplitting resulting from the surface crystal field. This inter- alumina powder to 0.05-.Lm grit. In stu cleaning using Ne

pretation stimulated several model calculations -9 and addi- ion sputtering (500 eV, 10 ,iA/cm 2) and annealing totional experiments"' aimed at testing the hypothesis of sur- 800'C yielded clean, well-ordered surfaces. Auger analysislace crystal fields in more detail. These calculations, and of the clean surfaces, our work14 in which work-functionsubsequent experiments,'10' 2 which were conducted under changes were studied, and chemisorption experiments" in-more carefully controlled conditions, have shown that the volving CO indicate that the well-annealed NiAI(I10) sur-broadening of the 5PV,2 line observed by Waclawski and faces exhibit a stoichiometry (ratio of Ni to Al) equal to thetterbst at full monolayer coverages was not due to crystal- bulk value (i.e.. I to ). Recent LEED studies of this sur-field or image charge effects.'-' In this case, the line face suggest a reconstruction involving atomic rippling.1 "

broadening resulted from Xe-Xe interactions as shown by which consists of small displacement of the surface atomsangle-resolved photoemission determination of the Xe-band ( - 0.08 A) perpendicular to the surface. Extensive angle-dispersion throughout the surface Brillouin zone and com- resolved photoemission studies 8 of the NiAI(l10) surfaceparison with results of simple tight-binding calculations. using synchrotron radiation have yielded bulk band struc-

More recent experiments by Opila and Gomer" have ture in good agreement with calculations. These expert-again raised the issue of surface image dipole or crystal-field ments constitute additional characterization of theeffects in relation to the photoemission spectra of phy- NiA(I 110) surface.sisorbed rare-gas atoms on metal surfaces. In these careful- Our interpretation of the splitting of the 5P,, peak inly conducted experiments a very convincing case is present- terms of local-substrate-related fields rather than adsorbate-ed in support of the existence of a mechanism, unrelated to adsorbate coupling relies on accurate knowledge of the sur-adatom-adatom interactions, which splits the 5P312 line of face conditions, including substrate order and composition,rare-gas atoms on W(l10). This result is in contrast to the which was just discussed, as well as adatom coverage andnull result obtained by Erskine") under similar experimental spatial distributions. Adatom concentration was accuratelyconditions for Xe on W(I100) calibrated at integral monolayer coverages by analysis of

We ha've recently conducted extensive photoemission multipeak spectra resulting from layer-dependent VO0studies of physisorbed rare-gas atoms on single crystal NiAI Auger energy shifts, 1 and checked using 5P valence levelalloy surfa.Cesi to investigate surface stoichiometry and local shifts observed in photoemission spectra. tniform Xewoirk functions as probed by the photoemission of adsorbed layers of' n = . 2. and 3 monolavers could be obtained byxenon IPAX) technique " During this study we ohserved adsorption followed by carefully monitored annealing. Sub-splitting of the SP,, component of photoexcited Xe atoms monolaycr co,,erages were determined from work-functionon the NiAI)I lO) surface, but not on the (100) surface of changes. which are roughly linear in the 0-0.5-ML range,the same ordered alloy The splitting was observed using and friim the intensity of the angle-integrated photoemis-experimental conditions under which Xe-Xe interactions are sion peaks relatie to the NiAI d bands measured in the

33 1440 c 1986 The American Physical Society

%*

.,. ,, .".

33 BRIEF REPORTS 1441

same configuration used in calibration experiments. Sample ponent in each case is composed of two Gaussians havingtemperatures during and after adsorption were maintained the same width as the 5PI12 component. This analysisbelow 40 K to ensure low surface mobility of the adsorbed shows that splitting of the 5P312 line is approximately 0.35rare gases. No evidence of island or cluster formation of Xe eV.

- atoms was observed in photoemission (i.e., k0 dispersion of There are no detectable shifts in peak positions as a func-peaks) or by LEED analysis at these temperatures. Weak tion of Xe coverage at low coverages. This coverage in-halos having sixfold symmetry were observed in LEED dependence suggests that splitting of the 5P31 2 line resultsstudies of low coverage Xe films only after annealing to from coupling between the screened ion and the substrate

100 K, indicating that temperatures in this range are re- rather than with neighboring Xe atoms. The annealing ex-quired to induce island formation. periments which established the temperature at which is-

Figure I displays angle-resolved electron energy distribu- lands did form also eliminated the possibility that the in-, tion curves (EDC's) for various coverages of physisorbed dependence of binding energy with coverage resulted from

Xe on NiAI(110) at T -30 K. Various features 8 of the islands or clusters of constant density at all coverage.d bands of NiAI along the A direction of the three- We have carried out corresponding experiments involving

. dimensional Brillouin zone are apparent in the energy range low coverage Xe layers on Ni(100) and Ni(110). These ex-within 5 eV of the Fermi energy, EF. Submonolayer Xe periments yield a similar splitting of the 5P3/2 peak for Xespectra exhibit two primary peaks corresponding to the on the Ni(110) surface but not on the Ni(100) surface.

' - 5PI12, 5P 3/2 states of the ion. Spectra for coverages greater Close inspection of experimental results of Jacobi andthan one monolayer exhibit two additional peaks which in- Rotermund,' 9 which were obtained under similar experi-crease in strength with coverage. These are the second mental conditions, also reveals a split 5P312 level for lowlayer peaks, which are shifted to higher binding energy due coverages of Xe on Ni(l10), in agreement with our results.to less effective screening of the ion (relaxation shifts), and Our previous search for crystal-field splitting on the W(100)which were used in thickness calibration, surface revealed no splitting of the 5P31 2 level,"0 but photo-

Close inspection of the 5P3/2-derived peaks corresponding emission studies of Xe of Opila and Gomer on the W(110)to low coverage reveals that it is not symmetric as is the surface exhibit splitting of this level.' 9 Based on these ex-5PI12 peak at equal coverage. Figure 2 illustrates, on an ex- perimental data. 7 it appears that the splitting of the 5P31 2panded scale, the two peaks for equal coverages of Xe on state could be related to the reduced symmetry of the ad-NiAI(110) and on NiAI(100). One does not expect to be sorption site on the (110) surfaces.able to resolve the actual crystal-field splitting of the P312 To validate this possibility, one must be convinced thatlevel because the intrinsic broadening of the lines due to re- the (.O) and the (100) faces of W, Ni, and 3-NiAI can andlaxation mechanisms related to the presence of the metal probably do yield qualitatively different local environmentssurface is approximately equal to the splitting. The inset ofFig. 2 compares results of curve fitting the two EDC's using

a three Gaussian functions, assuming that the 5P 312 com-p p12 3/2

Xe/NiAl NAI(O0), T-30 K

...-.. hv: 21.22 eV NAI (11O)

Xe / NiAKI10) 0 ,

T- 30 K C') ENERGY (eV)

hv~ 21,22eV t-

COVERAGE

r (MONENLAYERS) Z5

EIG Xe Na I t i)

Z 0CLEAN - IO

005

070

CLEAN- 53CLA-046 -1 75 -5 21 --

0 30

02 ELECTRON BINDING ENERGY We)0

*-10 -5 07E FIG. 2. Expanded scale of angle-resolved photoemission spectra* ELECTRON BINDING ENERGY (eV) for Xe adsorbed on NiAI(l 10) and NiAI(l00) surfaces. Inset, line

intensities as determined by Gaussian fitting of the data. Values ofFIG. 1. Angle-resolved photoemission spectra for Xe adsorbed the splitting are Xe on NiAI(I 10), A- 1.22 ±0.04 eV, %'-0.37 ±O 4

on NiAI(II0) as a function of Xe coverage, eV; Xe on NiAI(100), A - 1.20 ±0.04 eV, %'- 0.06 ±0.06 eV

"- -.. "..... .. ,. -.. .. '.... .. .. .,......'-,, . - ..- . .. .. ..

1442 BRIEF REPORTS 33

TABLE 1. Polar-angle dependence of SPI1 2 and 5P312 linewidths of photoemission spectra for Xe adsorbedon NiAI( 10). Photon energy hv -21.22 eV, sample temperature T - 30 K Angle is measured from thesample normal along the [ITO) direction, center and width energies are in eV Angular resolution

±40.

SPI12 5P312

Polar Center Width Center Widthangle (below EF) (below s)

0. 7.05 0.54 5.83 0 7510* 7.05 0.53 5.83 070200 7.05 0.52 5.83 0.67300 7.05 0.50 5.83 0.63400 7.05 0.50 5.83 0.63

for adsorbed Xe. For physisorbed atoms, at low coverage, field split level will be different.the preferred site will most likely be the deepest hollow sites Table I illustrates the experimentally determined polar an-in a surface unit cell. For bcc W(100), Xe should physisorb gle dependence of the 5P/ 2, 5P3, 2 linewidths for Xe onat the C4, fourfold hollow site; for W(10) the correspond- NiAl(110). The intrinsic broadening is too large to clearlying site has C2, symmetry. In the case of fcc Ni, again, the resolve the 5P312 state splitting (as shown in Fig. 2). How-preferred surface site on the (100) face will have C4, sym- ever, two features are clear from our polar-angle-variationmetry, and C2, symmetry on the (110) face. The crystal data. First, the binding energies of the 5P,,2 and 5P3/2structure of 8-NiAI is the CaF 2 (cubic) structure. Here states are independent of polar angle, confirming that lateralagain, the (100) surface offers only C4. sites, whereas the interactions (which would produce band dispersion) are not(110) surface offers C2, sites. Qualitatively, the argument present. Second, the 5PI1 2 linewidth is nearly constant,of a symmetry based origin of the splitting appears valid, whereas the 5P3,/2 linewidth changes significantly, as would

Herbst 9 has investigated theoretically the angular depen- be expected if the intensity ratio of the component linesdence of photoelectrons from atoms adsorbed on metal sur- changed. This constitutes additional evidence of a localfaces, taking into account the effects of the substrate atoms, crystal-field origin of the splitting.Although none of the specific results obtained by Herbst ap-ply directly to Xe adsorbed in the C2. site on NiAI, one ofthe general characteristics of the model should apply. This This work was supported by the National Science Founda-characteristic is that the polar-angle variation of photoelec- tion, Grant No. DMR-83-04368, and by the Joint Servicestron emission associated with component lines of a crystal- Electronics Program.

1P. V. S. Rao and J. T. Waber, Surf. Sci. 28, 299 (1971). 822 (1978).2T. C. Chiang, G. Kaindl, and D. E. Eastman, Solid State Commun. 12M. Scheffmer, K. Horn, A. M. Bradshaw, and K. Kambe, Surf. Sci.

41, 661 (1982). $0, 69 (1979).3N. D. Lang, Phys. Rev. Lett. 46, 842 (1981). 13R. Opila and R. Gomer, Surf. Sci. 127, 569 (1983).4N. C. Giles and C. M. Varma, Phys. Rev. B 23, 5600 (1981). 14M. Onellion and J. L. Erskine (unpublished).5N. D. Lang and A. R. Williams, Phys, Rev. B 16, 2408 (1977). 15K. Wandelt, J. Vac. Sci. Technol. A 2, 802 (1984).6B J. Waclawski and J. F. Herbst, Phys. Rev. Lett. 35, 1594 (1975). 16W. K. Ford (private communication).7P. R. Antoniewicz, Phys. Rev. Lett. 38, 374 (1977). 17H. L. Davis and J. R. Noonan, Phys. Rev. Lett. 54, 566 (1985).,j. A. D. Matthew and M. G. Devey, J. Phys. C 9, L413 (1976). 18W. K. Ford, E. W. Plummer, J. L. Erskine, and D. Pease, Bull.9j F. Herbst, Phys. Rev. B 15, 3720 (1977). Am. Phys. Soc. 29, 523 (1984).'11 L Erskine, Phys. Rev. B 24, 2236 (1981). 19K. Jacobi and H. H. Rotermund, Surf. Sci. 116, 435 (1982).SK. Horn, M. Scheffmer, and A. M. Bradshaw, Phys. Rev. Lett. 41,

V. -

I

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PHYSICAL REVIEW B VOLUME 33, NUMBER 10 15 MAY 1986

Electronic structure and properties of epitaxial Fe on Cu(100): Theory and experiment

M. F. Onellion

Department of Physics, University of Texas, Austin, Texas 78 7)2

C. L. FuDepartment of Physics and Astronomy, Northwestern Universiy, Evanston, Illinois 60201

M. A. Thompson and J. L. ErskineDepartment of Physics. University of Texas, Austin, Texas 78712

.A. J. Freeman

Department of Physics and Astronomy, Northwestern University. Evanston, Illinois 60201(Received 27 February 1986)

Results of a combined experimental-theoretical study of the electronic structures and properties of epitax-ial Fe on Cu(l00) are reported. Angle-resolved photoelectron spectroscopy is used to determine the elec-tronic structure of one, two, and four layers of epitaxial Fe on Cu(100). The experimentally determinedtwo-dimensional energy bands of p (1 x I )Fe monolayers and bilayers verify predictions of local spin-densityfull-potential linearized augmented plane-wave calculations. Changes in electronic properties with coverageand the evolution of the bulk electronic structure of the substrate-stabilized fcc iron are described.

* Advances in (i) novel sample preparation techniques, in angle-resolving photoelectron spectrometer, described previ-particular epitaxial growth of layered structures, and (ii) in ously,3 7 was used to prepare the epitaxial crystals and to ob-

Slocal spin-density electronic-structure theory of surfaces and tam the photoelectron spectra. Our sample preparationinterfaces are stimulating new interest and excitement in the techniques and surface characterization methods were alsofield of thin-film magnetism. Taken together, they offer identical to those described previously.3

unique opportunities for developing new magnetic materialst The iheoretical electronic structures were determined,as well as advancing our understanding of fundamental from local spin-density functional theory by means of themagnetic interactions in solids.2 The opportunities for ad- highly precise all-electron full-potential linearized augment-

" 'ancing our fundamental understanding of magnetic phe- ed plane-wave (FLAPW) method 8 The surfaces are mod-nomena are particularly attractive in the subfield of solid- eled by a single-slab geometry with Fe layer(s) atop a five-state physics in which carefully characterized materials of layer Cu(001) substrate: The stacking has atoms in theknown structure are experimentally studied and the results fourfold hollow site of adjacent atomic planes. The Fe-Cucoordinated with first-principles calculations. interlayer spacing was determined from total energy calcula-

In a recent publication3 we reported experimental results tions. We find that Fe forms an ordered overlayer onfor epitaxial p( I xl I)Ni films on Cu(100). This study Cu(001) with an fe-Cu interlayer spacing which is very

• .' presented some of the first detailed experimental evidence close (within 0.05 A) to that of the substrate. This result isshowing that high-quality epitaxial magnetic films could be confirmed by low-energy electron-diffraction (LEED) stud-grown on metallic single-crystal surfaces,' and that the two- ies described later. For the case of two monolayersdimensional electronic structure and magnetic exchange p( I x I )Fe/Cu(001), the problem is complicated by thesplitting of these films could be determined with sufficient magnetic coupling between the Fe layers. The magneticaccuracy to provide meaningful tests of the predictions of ground state was therefore determined from a comparisonfirst-principles calculations.5 of spin-polarization energy between various magnetic states.

The present Rapid Communication continues to explore The ferromagnetic coupled bilayers are found to have thethe prospects of advancing our understanding of two-di- lowest total energy, whereas the antiferromagnetic couplingmensional magnetic structures based on the interplay between the Fe layers exists as a metastable state with a to-between ab initlo first-principles calculations and photoemis- tal energy 0.2 eV above the ferromagnetic state.sion studies of novel thin-film structures fabricated by Despite the presence of the nonmagnetic Cu substrate,molecular-beam epitaxy. Our experimental results confirm strongly enhanced magnetic moments localized at the Fe

" the predictions of the computational studies,5 and indicate site are found from these calculations: (1) 2.85Mto for Ithat the p( 1x I )Fe on Cu(100) system is a second suitable Fe/Cu(001); and (2) 2.83A. 8 and 2.58 gH for the surface andcandidate for detailed experiments in which electronic and interface Fe layers, respectively, for the two monolayer coy-magnetic properties can be probed by the rapidly increasing erage. The Fe-derived localized interface states and the nar-number of surface and spin sensitive spectroscopic tech- rowing of the d band appears to be the mechanism drivingniques.-6 the enhancement of the magnetic moments over the value

Our experiments were performed at the Synchrotron Ra- (2,12Ma) in bulk Fe.diation Facility in Stoughton, Wisconsin. The I-m stain- Previous experimental work9 has established that the ex-less-steel Seya-Namioka monochrometer was used to cellent bulk lattice constant match permits pseudomorphicdispense radiation from the Tantalus storage ring, and an growth of fcc Fe on Cu(001). Evidence for pseudomorphic

33 7322 01986 The American Physical Society

.. % - %., P. ?* e

4 J~ .. o 4-0

33 ELECTRONIC STRUCTURE AND PROPERTIES OF EPITAXIAL Fe ON Cu(100): ... 7323

growth is based on transmission electron microscopy studies (EDC's) for epitaxial layers were taken using s-polarizedof 1000 A Fe films grown on Cu(l00) films in high vacuum light with the A vector along a symmetry axis of the crystal(10-' torr). Unlike the Ni on Cu(100) system, which we and with the emitted electrons detected either in that mirror

* previously studied, 3 apparently, no structure studies of plane (even symmetry) or perpendicular to it (odd sym-1-5-monolayer films grown on well-characterized substrate metry). All spectra were taken at room temperature (300surfaces in 10-1 0-torr vacuum are currently available for the K) at an energy resolution (monochromator plus analyzer)Fe on Cu(100) system. Therefore, we have conducted an of approximately 100 meV. Approximately 200 spectra forextensive investigation of the growth properties and struc- one- and two-monolayer Fe films on Cu(100) were taken inture of thin Fe layers on Cu(100) surfaces. Our LEED and even and odd geometry for k, along F-X and F-M direc-Auger analysis of this system are reported elsewhere.' 0 The tions of the two-dimensional Brillouin zone. Several photonimportant results of this work relevant to the present paper energies were used. Except for some minor variations ofare (1) p(l xl )Fe grows on Cu(100) as an extension of binding energies with film thickness (discussed below),the substrate with identical ( ±0.1 A) lattice constant, (2) spectra corresponding to a given symmetry and k1l valuethicker films (up to 4 layers) appear to form excellent epi- were consistent.taxial layers of fcc iron stabilized by the Cu(100) substrate Figure 2 presents the two-dimensional electronic structureand having a lattice constant equal to the substrate, (3) in- of one- and two-layer films obtained from our photoemis-terdiffusion on Fe and Cu at the interface is not apparent sion data. Solid lines and dashed lines in Fig. 2 representfor substrate temperatures below 250'C, and (4) the epitax- calculated majority spin and minority spin bands for aial growth appears to be dominated by a layer-by-layer p( lx I)Fe film on a five-layer Cu(l00) slab which havemechanism at substrate temperatures of 150'C. over 50% of their wave function derived from Fe basis

Figure 1 displays representative angle-resolved photoemis- functions. These are the specific two-dimensional energysion spectra for one and two monolayers of p (1 x 1 )Fe on bands to which our experiments should be most sensitive.Cu(l00) along the F-X direction of the two-dimensional

'.p. Brillouin zone. All of our energy distribution curves

-_ _ _ _ _ __"_p 0 X 1) MONOLAYER Fe/Cu(IOO)EVEN SYMMETRY

p(IXI) LAYER k, .23V M r X~~~~Fe/Cu (100) P ' ,F/Cu.(100) . '150 100 050 0 050 1.00ODD SYMMETRY E,__________"" -x DIRECTION 0.0 -,

h, a 21.22eV 050I LAYER 1 00 spBAND

ha' - 16 65 Ov2 LAYERS I 50 h. - 2122ev

2,00

, . .I'- -- -- --------------

, '~o D SYMMETRYcrr

Cnz ,5o 1.00 0.50 0 0.50 1.00r..'." k.. =0'75A-'

,* ---t, 050 x , . ,,,.,..k,, =0 504-

,e1 , 50

- - ----.- ; 200"5, " . CALCULATED

k,, =025 ' 2ML IML h. IML 2ML

k,, O O " ' "' ... ' .. ... ... ... ... " "o • 16 8 V MAJORITY -- -

k,, 0 V a 2122 ev ---- MINORITY.

' FIG. 2. Two-dimensional electronic structure of p(lxl )Fe onS- Cu(100) The two broad curves indicate the regions of binding en-

- -6 -5 -4 -3 -2 -I E, ergy and k1 where a prominent structure resulting from the Cu spBINDING ENERGY (eV) band is observed Light solid and dashed curves represent calculat-

ed (Ref 5) surface Fe bands having over 50% surface character0_P FIG. I. Angle-resolved photoemission spectra for one- and two- Data are represented by empty (two-monolayer films) and solid

layer p(I x I )Fe films on Cu(100). Values of k,1 correspond to the (one-monolayer films) circles (h' - 16.85 eV) and rectanglesr- direction of the two-dimensional Brillouin zone. (hv - 2 1.I1 eV).

7324 M. F. ONELLION, C. L. FU, M. A. THOMPSON, J. L. ERSKINE, AND A. J. FREEMAN 33

The broad shaded areas in the even symmetry bands discriminate between film growth in which islands of therepresent regions of k1l where prominent features associated second layer begin to nucleate when there is less than 15%with the Cu sp band sweep through the angle-resolved of the first layer remaining to be completed.energy-distribution curves (EDC's) near EF. These struc- The calculated bands for two layers of Fe and Cu(00l)tures were identified by k1 scans on clean Cu(100) sur- are modified from those of the monolayer coverage. Thefaces; their presence precludes the study of even symmetry overall Fe surface layer derived features and dispersionsFe bands in several regions of the two-dimensional Brillouin remain similar to those shown plotted in Fig. I. However,zone at specific photon energies. Interference of the sp the localization of the surface state wave functions on theband in studies of p( I x I )Ni on Cu(100) (Ref. 3) and on surface Fe layer becomes less pronounced due to the hy-Ag(100) (Ref. 11) is less of a problem because the over- bridization between surface and subsurface Fe layers. Thislayer Ni d-band features are much narrower than corre- change of bonding also manifests itself in a reduced upwardsponding Fe features, and are therefore more easily dis- dispersion of odd symmetry bands (dominated by d, and d,tinguished from the sp-band features. character) along the F-Al direction (cf. Fig. 2). Further-

Some results of the theoretical studies are shown in Fig. more, the calculation predicts the existence of additional2. It is clear from the comparison that the calculations are majority spin bands having a strong admixture of wavein reasonably good agreement with the results shown in Fig. functions localized on the subsurface Fe layer, in particular,2 in the regions of the two-dimensional Brillouin zone effec- odd symmetry bands with binding energy - 2 eV along F-tively probed by our experiments. Based on the calcula- T, and an upward dispersion band along F-M (not shown intions, the most prominent features observed in even sym- Fig. 2). Interference from the Cu sp band precludes observ-metry geometry in the vicinity of EF should be minority ing the new even symmetry band for two monolayers, andspin bands. Clearly, spin-polarized photoemission studies of the broad peak widths associated with the Fe overlayersthis system will be informative. Our experimental results renders it very difficult to differentiate small differences inaround F, where Fe-derived features are not masked by the spectra for one- or two-monolayers. Since the size of theCu sp band, support the number of bands and their bindingenergies predicted by the calculations. Odd symmetry bandsobserved along F-X also appear to agree rather well with thecalculated results, although there are significant differences(0.25 eV) between measured and calculated binding ener- FOUR LAYERSgies. In general, we find that the measured bands lie closer fcc Fe/Cu(IOO)to the Fermi energy than the calculations predict for the NORMAL EMISSIONmajority and minority spin bands. This result could imply /that the magnetic exchange splitting is actually smaller than 29 eV

predicted theoretically, or could be a manifestation of theimportance of many-body effects (i.e., core-hole relaxationand correlation). Our previous experimental studies of the 27 eV /7

bulk 7 and surface 2 electronic properties of ferromagnetic Fe I-have shown, however, that correlation effects appear to be Zsmall; we therefore assume that these effects are also ofminor importance in the Fe overlayer system. Cr 25ev -

One source of the discrepancy between the experimentsand the theoretical results could be due to the fact that ourmeasurements were conducted at 300 K, and our calcula- I /tions correspond to T=0 K. We have not yet attempted z --2v -any low-temperature experiments or experiments above w//T=300 K to detect possible temperature dependences in - -

T 21 eVthe results. A second possible source of disagreement z Ibetween our experimental results and the calculations are 19evsome subtle layer dependencies we have observed in elec-ironic properties of the Fe layers. Our LEED studies re- 170Vvealed two domain (2 x I) LEED patterns in the coverage 16 1 "range between 0.5 and 1.0 monolayers. 1 These structureswere not observed above full monolayer coverage, but their 15evpresence at low coverage suggests a competition between 14evthe substrate stabilized fcc structure and the possible ten- 13eVdency for Fe to form some other structure, in particular thenormal bcc structure of bulk Fe. Displacement of ironatoms too small to detect by our LEED measurements is -not inconceivable, and such displacements would cause sig- , 6 5 4 3 -2 -1nificant changes in the electronic properties, particularly in BINDING ENERGY (eV)the r-Al portion of the two-dimensional Brillouin zone. As FIG. 3. Normal emission photoemission spectra for four-layer fccnoted in detail elsewhere.") while we have eliminated film Fe films stabilized on the Cu(100) surface. Peaks near /. resultgrowth models that assume the second layer begins before from the Fe film and exhibit clear evidence of direct bulk interbandthe first layer is more than 85/ complete, we cannot transitions.

.............. . . . .. . . - - . -

~**~-. :.-:-:--- . - - - -. -. '',.

33 ELECTRONIC STRUCTURE AND PROPERTIES OF EPITAXIAL Fe ON Cu(100): ... 7325

magnetic moments for the surface Fe layers remains almost ization, and in obtaining accurate electronic structure infor-identical for one- or two-monolayer coverage, the exchange mation. The p(l x l) Fe on Cu(100) appears to representsplitting (AE,,) is expected to be the same for both cases an additional excellent model system in which to explore the(AE, 2.65 ± 0.05 eV). relationship between magnetism and electronic structure

Figure 3 displays normal emission spectra for four layers from the point of view of thin-film magnetism (two-of Fe on Cu(100). Our LEED analysis' ° has shown that dimensional magnetism), and the magnetism of a new artifi-high-quality fcc films of epitaxial Fe form at this coverage. cially stabilized bulk phase (fcc Fe).Features in the spectra near E)- are definitely due to emis-sion from the Fe overlayer. This assignment was checkedby obtaining corresponding spectra for clean Cu(100).

S. Peaks near EF exhibit clear dispersion with photon energy We wish to thank the staff of the Synchrotron Radia~ionk k). indicating strong influence of direct bulk transitions. Center, Stoughton, Wisconsin, particularly Roger Otte, forDipole selection rules limit the symmetry of initial states their support. This work was supported at the University of

*-. probed in normal emission to A1 and A2 symmetry. The Texas by the National Science Foundation under Grant Nodispersion of the peaks near EF with ki is consistent with DMR-83-04368, and the University of Texas Joint Servicesthe calculated bulk bands 1 of fcc Fe along the V-X'direction Electronics Program A FOS R-F49620-8 2-C-0033. and atof the three-dimensional Brillouin zone. It is therefore pos- Northwestern University by the Office of Naval Researchsible to carry out detailed energy band measurements of the under Grant No N 00014-81-K-0438. and by a computingbulk band structure of the fcc phase of ferromagnetic iron grant from CRAY Research. Inc. The Tantalus storage ringstabilized on Cu(100).1 4 is also supported by the National Science Foundation. We

In summary. our results have identified a second thin- wish to thank Dr. Alex igna'iev for the loan of a spot pho-film magnctic system in which considerable success has tometer. Finally, we wish to thank J. C. Hermanson forbeen achieved in the synthesis of the film, in its character- making his calculations available to us prior to publication.

IR. M. White. Science 229, 11 (1985). York. 1985). Vol 1082U Gradmann, Appl. Phys. Lett. 3, 161 (1974), and references 'A M. Turner, A W Donoho. and J. L. Erskine. Phys. Rev. B 29,

therein. 2986 (1984), and references therein.3M. Thompson and J. L. Erskine, Phys. Rev. B 31, 6832 (1985) 8E Wimmer, H Krakauer. M Weinert, and A. J. Freeman, Phys.4Epitaxy of metals is not new. see, for example, the review article Rev B 24, 864 (1981). and references therein.

by E. Bauer, Appl. Surf. Sci. 11/12, 479 (1982). Our experiments 9K. W A Jesser and J. W. Matthews, Philos, Mag. 15, 1097 (1967).are among the first to explore the electronic properties and mag- 10M. F. Onellion, M. A Thompson. J. L. Erskine, C. B. Duke. andnetic exchange splitting of epitaxial magnetic films using photo- A. Paton (unpublished).emission. One previous study of a similar nature has been report- lM. A Thompson, M F Onellion, and J. L. Erskine (unpub-ed by R. Miranda, F. Yndurain, D. Chandesris, D. Lecante. and lished).Y. Petroff, Phys. Rev. B 25. 527 (1982). 12A. M. Turner and J. L. Erskine, Phys. Rev. B 30, 6675 (1984);

5C. L. Fu, A. J. Freeman, and T. Ogucht. Phys. Rev. Lett. 54, 2700 28, 5628 (1983).(1985). 13D. Bagayoko and J. Calaway, Phys. Rev. B 23, 5419 (1983).

6J. M. Kirschner. Springer Tracts in Modern Physics (Springer, New t4M. A. Thompson and J. L Erskine (unpublished).

.A

'-...

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w °.° ,o o-° • - '§. . .. 2: . .. .. .. . . . . .~

Reprinted from Applied Optics. Vol. 25, Page 3864, November 1, 1986Copyright @ 1986 by the Optical Society of America and reprinted by permission of the copyright owner.

Laser-induced damage and ion emission of GaAs at 1.06 Jim

Austin L. Huang, Michael F. Becker, and Rodger M. Walser

This study focused on the multipulse laser damage and the subdamage threshold ion emission of GaAs. Theinitial goals were to determine the pulse-dependent damage threshold and to correlate ion emission withsurface damage. A Q-switched Nd:YAG laser was used to irradiate the (100) GaAs samples. Using values oN from I to 100, we obtained accumulation curves based on 50% damage probability. Corresponding damagethreshold fluences were 0.4-0.8 J/cm2 for N > I and 1.5 J/cm2 for N - 1. We observed large site-to-sitefluctuations in ion emission and found the onset of emission at 0.2 J/cm 2 for all cases. Once surfa-e damageoccurred, ion emission increased greatly. The observed behavior supports a surface cleaning model for theion emission which precedes surface damage. Measurements of linear and nonlinear free carrier absorptionwere made, but no anomalous absorption was observed.

1. h ducltlon N. Samples and ApparatusThe interaction between laser radiation and solids The GaAs samples used in the experiment were sup-

has been a perplexing problem for many years. Often plied by the microwave integrated circuit productionthe lifetime of an optical device is determined by its group at Texas Instruments in Dallas, TX. The sam-susceptibility to optical damage. For example, it is pies had a (100) orientation and were very lightlystill a significant problem that the performance of doped with chromium (-1 X 10'r cm - 1) but otherwiseGaAs injection lasers degrades at different rates with undoped and unannealed. These samples were char-respect to power level thus implying an accumulation acterized by a resistivity of > 1 X 10 -1-cm and an etcheffect.' There has been much controversy and unex- pit density of 40-60 x 10.1 cm- 2 . Only the front face ofplained phenomena associated with the energy trans- the wafer was polished to optical quality.fer mechanism of normal laser damage as well as the To prepare the sample for the vacuum tank we sephysical nature of surface damage. quentially cleaned the GaAs wafers in boiling solvents

The objectives of the experiments reported here of trichlorethylene, acetone, and methanol to assurewere to characterize the statistical nature of surface that all contaminants had been removed. While indamage for GaAs, particularly for multiple pulses on each solvent, the wafer was ultrasonically cleaned be-one site (N-on-1), to observe the relationship of fore proceeding. After the final cleaning in methanol.charged particle emission to surface darrage, and to the wafer was rinsed in deionized water (>5 min atobserve the damage morphology of GaAs. In silicon it room temperature) and placed into the vacuum tankhas been reported that charged particle emission is To measure the linear and nonlinear absorption incoincident with surface damage. 2 A relation between the transmission tests, the GaAs wafers were given ancharged particle emission and surface damage has yet optical quality finish on both sides. The backside ofto be reported for GaAs. To explore the statistical the wafer was mechanically polished in a two-stepnature of surface damage of GaAs, we performed vari- process. Rough polishing with a grinding pad and 6-ous single-shot and N-on-1 subthreshold laser tests. um diamond ferrous paste was used to initially buff theWe measured the positive charged particles emitted wafer. The finishing pad used a liquid 0.05-pm alumi-during each laser pulse to correlate the charge emission na nonferrous suspension as a grinding media. Afterevents with surface damage. Theoretical calculations polishing, the wafer was chemically cleaned as de-

for a thermal model, as well as a plasma production scribed above.model, have been carried out to determine the mecha- The experimental system is shown schematically innism responsible for the surface damage. Fig. 1. It utilized a Q-switched Nd:YAG laser with a

full width at half-maximum (FWHM) pulse length of45 ns, TEM51 transverse mode, and wavelength of

The authors are with University of Texas at Austin, Austin, Texas 1.064 pm. Although the pulse envelope was Gaussain.7871?. the laser was not single longitudinal mode. A knife-

Received 18 April 1986 edge scanning technique was used to measure the fo-0003-6935/86/213864-07502.00/0. cused spot diameter on the sample surface, which was.C 1986 Optical Society of America. found to be 580 Am. The laser energy fluctuated from

3864 APPLIED OPTICS / Vol. 25, No 21 I 1 November 1986

% %" .% . ,% " € * J , # " • " ". . .. . . .. . 4 ..

• ., - , , . , , ..,.P,. . ... %... .. . .. . . . -.. . - . . .. .. .. .. . .' . . .'' . , ' . '

1-.1 U.. s '. . 6

Pet.i s e p.

0 o o CotoeCm~lo

• p. Fig I. Experimental setup.

* pulse to pulse resulting in a 2-5% standard deviation in X 10 at a voltage of -2 kV at the first dynode. A dataenergy for each data run. The laser was operated at a value of 1000 in the more sensitive A-D converter15-Hz repetition rate, and the incident energy was channel corresponded to a charge of n-1.5 X 10-17 C.

Sattenuated by rotating a halfwave plate in front of a This -2-k( bias potential not only determined thepolarizer, gain of the electron multiplier, but it created an attrac-

A computer was utilize red o tr ad count the tive potential sufficient to collect all emitted positivelaser pulses striking the GaAs sample and to reord the ions.charge collected. The computer had a Z80 micro-processor and supported several data acquisition ports il Experin Datafvia sample and hold amplifiers and a multichannel A- Three experiments were conducted on the cleaned

D converter. The delay for the sample and hold am- samples: (1) an emission scan of the GaAs sample at a-,, plifiers was set to 130 s in hardware. This assured fixed laser fluence; (2) an N-on-i emission/damage

that the integrated values read for the charge emission threshold test; and (3) a transmission test. All thewere near the peak value but were collected well after tests were performed in vacuum with the exception ofthe laser noise burst. The electronics were triggered the transmission test. The calibration and beam pro-by a vacuum photodiode. A shutter, controlled by one files were checked at the start and end of each experi-of the computer's digital output ports, was used to mental session to assure accurate beam fluence me -select pulses for sample irradiation. To allow time for surements.data computation and storage betwee pu e shut- The emission scan of the Ga s sample checked the

nter seleted every third pulse from the laser train, uniformity of charge emission under constant laserA 760-mm focal length lens focused the beam onto fluence. The scan spots were separated by 0.8 mm,

% lthe sample. The long focal length gave a region of and forty-two samples were taken. The sample wastconstant spot size i3 mm deep at the focal plane and irradiated

permitted greater error in the placement of the sample. The results in Fig. 2 show a bilevel emission contour.A system of 90° prisms was placed after the lens to scan A different pattern of emission variations was ob-

-e. the beam across the sample in the vacuum tank. served on each sample. Site-to-site fluctuations of the"""To detect charged particle emission the sample and defect density in GaAs have been previously noted 3

detection device were placed into the vacuum environ- and could disrupt the statistics of charged particlea ment. The system was dry pumped to achieve experi- emission and the emission threshold. To minimize

' mental pressures of <1 x 10-6 Torr. A Hamamatsu the effects of these site-to-site variations, we conduct-electron multiplier tube model R596 served as the ion ed the emission/damage tests over a small area of thedetector and was placed inside the vacuum system at a wafer.

• 36° angle to the laser beam path. The sample surface Other experiments were concerned with the multi-S was normal to the optical path. The output of the pulse and single-pulse irradiance of GaAs. In the mul-

electron multiplier was connected to a capacitive volt- tipulse experiments, fluences below the single-pulsee. .. age divider and amplifier. The dynamic range of the damage threshold were used to search for accumula-

-P . data acquisition system was increased by making one tion effects associated with either emission or damage.

channel 10OX less sensitive than the other. The elec- The objectives of these experiments were (1) to identi-tron multiplier current gain was estimated to be -0.75 fy accumulation effects associated with surface dam-

1 November 1986 / Vol. 95, No. 21 I APPLIED OPTICS 3865

I . . % , ,. ,.

*~0~ %

-10 1

9- 0.9-

w0.8

* .0 0.8-E

4- 0.8 •

.. : 0.4-

S0.3-

1..m~l0 ,, : i ,. 0.2-

0 1'0 2'0 30 0.2 i " ' Dmson

-Fit toPosition (mm) 0.Damage Data

Fig 2. 'harged partice emission vs position on wafer for a constant 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

laser 'luence of 0.63 J/cm2. Fluenco (J/cm2 )

Fig. 3. Damage and emission probabilities vs fluence for N = 1.

09-

08-

os7- , g/S05- 08

03 -

CL.. a a 0.6-0r., 02-05

0.1 "

' " 01- • -Fit to 04

/~~~ Dmage Data o03

2 00.37

0 0 2 04 0. 08 102 104 I' 1I

: "Fluence (J/cm 2 )

0 .2 - o a• o.

* Damage

-"mFig. 4. Damage and emission probabilities vs fluence for N - 10. 0.4 E..yo0.- it-Fit to

t Damage ata

,.. ~0 ,-" .: I:. . .. .0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.

age, (2) to measure a damage threshold based on 50% F0.2,c. (J/c,2)

probability statistics, (3) to search for correlations be- Fig. 5. Damage and emission probabilities vs fluence for N =100.

tween surface damage and charged particle emission,

and (4) to determine the damage morphology as afunction of fluence and number of pulses.

For each value of the number of pulses N used (N =1, 3, 10, 30, and 100) we irradiated 150 to 200 differentsites to obtain the statistics of surface damage. Sur- oo-•face damage was identified by searching for surfacechanges with a Normarski optical microscope at 200X.The damage data for N = 1, 10, and 100 are plotted in 30-Figs. 3-5, respectively. In each plot we used a linearcurve fitting program to obtain the damage probabili-ty. From the linear fit we obtained the 50% probabili-.':.-ty fluence for each value of N and used this value in the

accumulation plot in Fig. 6. z 3For all values of N, we observed the onset of charged

particle emission at an average fluence of 0.2 J/cm2

±-.07 / The values for the onset of emission 1% and the 50% damage threshold are given in Table I .0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

From these damage fluence values and their plot in FLUENCE (J/cm2 )j. Fig. 6, it is apparent that an accumulation effect was Fig. 6. Number of pulses required to reach 50% damage probability

present in that the threshold decreased after the first vs fluence. The line was hand drawn to aid the viewer.

,% 3866 APPLIED OPTICS / Vol. 25. No. 21 1 November 1986

@4 %-,%a%",

_Wo. %, .- .-1 . . , . ,, . .v .- . : .. . .. . . . .. . . , .. . .

.' Tabl I. On"! of Charge Emission and 50% Damage Threahd Fluence 100000 T-for Non-1 Experm to 's

Charged particleN emission onset 50% Damage 10 000

pulses fluence 1J/cm'2) fluence (J/cm2 )

1 0.18 1.52 1000 41

3 0.15 0.7810 0.17 0.64 ., ,30 0.312 0.79 loo,

100 0.17 0.68 E

pulse. As the number of laser pulses incident on thewafer increased, the 50% damage fluence decreased ' ... . .and then leveled off. It is not clear from this scattereddata whether the decrease continues slowly for N> 2Q 25 30

or the curve is level. Scatter in the threshold data andthe related observation of low slopes in the damage Fig. 7. 'ul-.. , ... .. p , ...... h, .... 9 30 at an

probability vs fluence curves have been associatedwith the presence of the local defects.' This possibili-ty will be correlated with damage mcrphology in a latersection. Similar curves to the one shown in Fig. 6, but

" showing a monotonic decrease in threshold, have beenreported for metals,5 polymethylmethacrylate, andmodified polymethylmethacrylate.6 The single-shot50% damage fluence for GaAs is in agreement with

* :values previously measured.-,

The positive charged particle emission data werecollected on a pulse-by-pulse basis. A typical profilefor N = 30 is shown in Fig. 7. The site from which thisemission profile was obtained was damaged. Fromthese data we note that the first few pulses (from twoup to ten depending on the case) induced a chargeemission that decreased as the pulse number in- ,

. creased. After this, the emission increased greatly.Since the microscopy and determination of damagewere done after termination of the experiment, we

.. were unable to fix accurately the pulse number at . ,,,which damage was initiated. Visual observation of the Fig."M d i tr. la)at hre.rI lses at an averageinitial flash was a somewhat insensitive measure ofdamage initiation. We interpret the decrease in thefirst few pulses as a surface cleaning effect. These firstpulses cleaned any residue either left from the chemi-cal cleaning process or ejected from previously laserirradiated sites. This emission could also be due inpart to the depletion of the more volatile specie, As,

* from the surface atomic layers. Figure 7 also showsthat, once the site experiences surface damage, chargeemission is greatly increased. The emission at non-damaged sites always decreased with increasing num- .5ber of pulses.

Figures 8 and 9 are SEM micrographs showing thedevelopment of laser-induced surface damage mor-phologies. The initial surface was featureless andwithout contrast. In the Normarski microscope, theinitial surface change appeared as a depression with anarea equal to that of the laser spot. Within this area,there were several very small pitted regions (similar tothose shown in Fig. 8 but not as well defined). As thefluence increased, or the number of pulses increased,the initial surface damage evolved into the melt pits Fig. !). SFI nmr, grih .t I 'an ,,u,a t,, thirt\ pulse, at an

shown in Fig. 8. Similar damage pits have been seen in ,,.: a;,, ,.,. ,I

1 November 198t V., ?' , APPLIED OPTICS 3867

04,".4%

... " " " " ."." .'." .r ., . . " " '.'_ ,," '." ,. '_",' ,.',." ." -. ' .. e - ,. - . ." .. .. .-. . . . . .... " " ,, .

3A dN/dt = ,,l/hf + ij'1/2hj - N1t, (2)

.3.0 " % dldz = -,,j - - aN1,

2.8 0 -where N = N(xy,z,t) is the time- and space-dependent"* 26 carrier density, I = I(x,y,z,t) is the time- and space-

2 4 -" , • 00 dependent optical intensity, hf is the photon energy, t,.E 2 2 is the free carrier lifetime, & is the two-photon absorp-

2o - 0 tion coefficient, and a is the free carrier optical cross1.8 - section. In Eq. (2), carrier relaxation due to Auger

Z. 1.6 recombination has been neglected due to the low ex-1.4 pected carrier densities. At 10 MW/cm 2 , for example,2 I 1 . . . .. two-photon absorption is negligible with respect to one

0 2 4 6 s 10 12 14 16 18 20 photon and free carrier absorption; Oc/ = 0.23 cm-'Inenit and jio = 0.023 cm/MW.10 For this reason, two-photon

Fig. Intensity (MW/cm 2 ) absorption will be neglected in both Eqs. (2) and (3).I.: Fig. I Inveise transmission vs intensity for a GaAs water polished Also in Eq. (2) we have assumed that the one-photon

o tn hoh sides. 'he solid line represents a least -squares linear fit to%. : the data. absorption is impurity dominated and that only onecharge carrier is generated per absorbed photon.

The solution to these equations will proceed much asdescribed by Boggess et al. I except that in our case the

GaAs at 1.064 pm.1- -' The damage evolution ends optical pulse length is much longer than the free carrierwith severe cratering as shown in Fig. 9. The theory of lifetime, and the solution to Eq. (2) may be taken to beFauchet and Siegman ,' would seem to apply in this quasi-steady state. The solution to Eq. (2) for instan-case. They postulate that laser initiated ripple pat- taneous carrier density is given byterns on the surface of silicon and GaAs are formed

* from the interaction of the incident laser wave front N = t, k,J/hf. (4)

, - with scattered optical surface waves. At 1.064 gm, This allows Eqs. (3) and (4) to be combined and solved.however, with a pulse length of 45 ns and spot diameter That is, Eq. (3) must be integrated over z as well as x,%,

- • ot 580 gin. no ripple patterns were seen in our experi- and t to obtain a solution in terms of toal transmittedinents. Accordingly, the longer pulse lengths may in- energy, which is an experimentally observable parame-hibit ripple formation. ter. These integrations may be performed as shown inIV. and M s Ref. 11 for Gaussian temporal and spatial profiles ifV Ds.l dthe free carrier absorption is not excessively large.

Thermal models have been used in the past to ex- This approximation is possible if the intensity is lessplain both laser annealing processes and laser damage than a critical intensity for free carrier absorption asphenomena. These models assumed uniform heating, given byand the possibility of inhomogeneities was not includ-ed. The necessity of inhomogeneous processes is I<. = h

made apparent by examining the laser absorption in at (I - R)[I - exp(-,.JI

the sample. From the slope and intercept of the linear For the paramaters in our experiments, I, 17.3 MW/fit of the inverse transmission vs intensity shown in cm 2 . The solution to Eq. (3) for this case is conven-

i Fig. 10, the linear and nonlinear absorption behavior tionally written in terms of the inverse total energyA,'- * may be determined. Although small values of linear transmission T, which linearizes the relationship

absorption coefficient aj are not measured accurately

by this method, a value for a, is a by-product of the = ll -

curve fitting procedure. The linear absorption coeffi-cient was found using the equation 2;r hf(I - R)expl-,,JAf ti ,

%.7,, 11 R)- expl-,,,L) where E is the total transmitted laser energy, tr is the

where T, is the zero fluence transmission intercept, R optical pulse width (FWHM), and u)() is the laser spot--'2,- is the reflectivity at the air-gallium arsenide wafer radius (1/e2 intensity). The term on the right in

interface, and L is the thickness of the GaAs sample. brackets may be thought of as an effective intensityIn our case T was 0.45 (from Fig. 10), R was calculated and is the quantity plotted on the abscissa in Fig. 10.to be 0.306 based on tabulated refractive-index data, A linear fit to the data in Fig. 10 gives a slope of 40.2and L = 0.0635 cm. Substituting these values into Eq. and a corresponding value of (at) = 2.0 X 10- 25, s cm 2 .(1) and solving yield a = 1.2 cm-I (±40%). The free This product of two experimentally measurable quan-carrier or nonlinear absorption can be determined tities may be checked against published data, but un-from a linearized equation for inverse transmission vs fortunately there is uncertainty and dispersion in theintensity as shown in the following analysis. The cou- published values. For a, values ranging from 5 X 10--"pled differential equations describing the carrier den- (Ref. 12) to 5 X 10- 1- cm2 (Ref. 6) extrapolated to 1.06sity and optical intensity are pm have been reported. For t,, one direct measure-

3868 APPLIED OPTICS I Vol 25, No 21 / 1 November 1986

% %

ment gave 32 ns,' but this value is known to be strong- peak carrier density of N = 1.7 X 10" cm . Forly dependent on impurities and defects which increase comparison, the compensated intrinsic carrier concen-the recombination rate. If we use a value of t, = 3.2 X trations were calculated. Accounting for the chromi-10 - 8 s (one of the largest values reported, presumably um dopant concentration (jt 1 X 101 cm -) and thefor relatively pure material), our experiments predict a resistivity of the sample (?> 1 X 107 0 cm), we calculatedfree carrier cross section of 6.3 X 10-11 cm 2 , which is an electron concentration of 7.35 X 10, cm - "and a holewell within the range of reported values given above concentration of 1.56 X 109 cm - 3. Mobilities of 5000even if the free carrier lifetime were a factor of 10 and 300 cm 2 /V s were used for the electrons and holes,smaller to account for a possibly larger deep level im- respectively. Although the photogenerated carrierpurity density. In addition, the functional depen- density is much greater than the compensated intrin-dence of T - I on intensity agrees with the theoretically sic density, it is still quite small compared to the densi-predicted form as shown in Fig. 10. From this, we ties where plasma effects become important, 1011 102'1

conclude that we have included the major homoge- cm-'. Therefore, we can disregard the plasma modelS. neous absorption mechanisms in our calculations and from further consideration as a primary damage mech-

that no gross anomalous absorption was involved. anism in GaAs.To determine the mechanism responsible for the The charged particle detector system monitored the

laser-induced damage, we examine the temperature pulse-by-pulse charge emitted from the GaAs sample.change of the surface. Both linear and free carrier The magnitude of the emitted charge has been plottedabsorption must be included in the heating model. Fig. 7 on a pulse-by-pulse basis for a site which exhibit-Assuming that all the absorbed energy is thermalized ed surface damage. By examining the emission pro-

.- without thermal or carrier diffusion, the peak surface files, the emitted charge was observed to decrease after* - -. temperature rise can be written the first few pulses and later increase at least 2 orders

AT= (dI/dx)p, t/(p. (7) of magnitude when damage occurred. The relativelysmall decreasing emission profile of the initial pulses

where t,. is the pulse length, C. is the heat capacity, and appears to be a surface cleaning effect. The later* 3 p is the mass density. In this approximate calculation increase of charge emission after the fifth pulse in this

we assume a rectangular pulse shape. Substituting case is attributable to surface damage. For GaAs, onlyEqs. (3) and (4) into Eq. (7) and taking the surface two types of charge emission profile were observed:reflectivity into account give either a cleaning effect without damage or a cleaning

= [ + (1 - R)t i/4hf(1 - R)lt/C~p (8) effect with damage. Damage in the absence of a clean-AT'-" + (--ing effect was never observed, although at very highwhere 1o is the incident intensity. For GaAs at room intensities the two profiles merged and became indis-temperature, we have C, = 0.327 J/g K and p = 5.32 g/ tinguishable. This differs greatly from the emissioncm". Using a typical damage intensity for the single- characteristics of silicon as reported in Ref. 2. Forshot case, Io = 19 MW/cm 2 and the measured value for silicon, emission of charged particles coincided with(ot), we obtain a maximum temperature rise of 6.1°C. surface damage initiation, and no cleaning emissionThis diffusionless model assumes homogeneous ener- was observed. Furthermore, the emission of chargegy absorption and temperature-independent reflec- prior to damage in GaAs did not affect the observedtion and linear and nonlinear absorption coefficients. damage behavior in any detectable way.

* From the results of the previous calculation we con- It appears that neither the uniform heating modelclude that uniform heating is not sufficient to explain nor the carrier pair production model can adequately

-. the melting effects observed on the GaAs surface. A describe the damage mechanism of GaAs. In addition,similar conclusion has been reported for GaAs by Gra- it does not seem possible to relate the nature of thesyuk and Zubarev and for silicon by Merkle et al.', damage to charge emission due to surface cleaning,and Becker et al.2 However, this does not exclude the although the latter appears to be a necessary, and

. possibility of an abnormal absorption or heteroge- possibly essential, precuisor to damage. From theseneous absorption in defects at or near the GaAs sur- results we are led to speculate on other possible influ-face. ences that may cause, or enhance, surface damage.

An alternative process that might lead to damage is Lattice defects near the surface of the GaAs wafer arelattice disruption caused by a high-density carrier possible energy absorption sites for nucleating dam-plasma produced during the laser-solid interaction, age. These lattice defects include anomalous vacan-The possible relation of this process to silicon anneal- cies, interstitials, and dislocations introduced duringing has been reported previously.' - To investigate the crystal growth or surface pieparation processes. A

"" this possibility, we calculated the peak number of car- single-point defect seems an improbable physical areariers produced by the impurity absorption. for absorbing sufficient energy to cause a melt spot. It

S.The peak carrier density produced is calculated is more probable that a cluster of point defects will actfrom Eq. (4) where ambipolar diffusion was neglected as an efficient absorption site. This would result in a

• because of the short pulse length. To obtain an upper surface damage morphology of random melt pointsbound for the carrier density, we used the incident within the beam diameter of the laser.

- intensity of a typical 1-on- 1 damage pulse, I = 19 MW/ Other possible sites for the nucleation of damage arecm-. Accounting for the reflectivity, we obtained a physical surface defects caused by processing or physi-

, 1 November 1986 / Vol 25. No 21 / APPLIED OPTICS 3869

...

.'* *%

cal handling of the wafer that would scratch the sur- Referwic..face. Surface defects on an optical surface will in-crease absorption and scattering, possibly creating 1. H. Kressel and H. Mierop. "Catastrophic Degradation in GaAsabsorption nuclei which could lead to damage. The Injection Laser," J. Appl. Phys. 38,5419( 1967 ).potential for nucleation at physical defects have been 2. M. F. Becker. Y. K. Jhee, M. Bordelon. and R. M. Walser,

previously speculated by Smith." In our experiments, "Charged Particle Exoemission from Silicon during Multi-Pulsemicroscopic inspection before damage has eliminated Laser Induced )amage," in Fourteenth ASTM Laser Damage

Srmposium. NBS Spec. Publ. 669 (1983); Y. K. Jhee, M. F.uthe possibility of gross surface defects (>1 tm) as a Becker, and R. M Walser, "Charge Emission and Precursorcause of damage. One would not have expected these Accumulation in the Multiple-Pulse Damage Regime of Sili'types of defect in integrated circuit quality material, con," ,l. Opt. Soc Am. B 2, 1626 (1985).

3. S. Dannefaer, B. Hogg, and D. Kerr, "Defect Characterization inV. Ckil mo and Cofmment Gallium Arsenide By Positron Annihilation," in Thirteenth

We observed accumulation effects in GaAs with International ('onference on Defects in Semiconductors, L. C.multiple-pulse laser irradiance below the 1-on-1 dam- Kimerling and J. M. Parsey, Jr.. Eds. (American Institute ofage threshold fluence. These experiments also yield- Mining, Metallurgical. and Petroleum Engineers, New York.ed various N-on-I laser damage thresholds derived 1985), pp. 1029-1033.from the statistics of the surface damage. In the ex- 4. S. R. Foltyn. "Spotsize Effects in Laser Damage Testing," in

Fourteenth ASTM Laser Damage S.mposium, NBS Spec.periments we also measured the linear and nonlinear Publ. 669 11983).optical absorption constants for the GaAs sample. 5. C. S. Lee, N. Koumvakalis, and M. Bass "A Theoretical Model

With this information, calculations were performed to For Multiple-Pulse Laser-Induced damage to Metal Mirrors," J.determine the mechanism responsible for the surface Appl. Phys. 54, 5727 (1983)

damage. Results for the uniform heating and carrier 6. A. A. Manenkov, G A. Matvushin. V. S. Nechitailo. A M.

production models clearly show that these mecha- Prokhorov. and A. S. Tsaprilov, "On The Nature of Accumula-

nisms could not be responsible for damage. The possi- tiin Effect in the Laser-Induced Damage to Optical Materials."

bility remains that nonuniform localized carrier gener- at Fourteenth ASTM Laser Damage S mposium, NBS Spec.ation and heating may be involved in the damage Publ 669 19831o An inhomogeneous heating model is sup- J

R. Meyer, M. R. Kruer, and F J. Bartoli. "Optical Heating inprocess. Semiconductors: Laser Damage in Ge. Si, InSb. and GaAs." .ported by the observed damage morphologies. It is App. Phys. 51, 5513 1980i.

also possible that the sample contained absorbing de- 8. J L. Smith. "Surface Damage of GaAs from 0 694 and 1.06 mmfects, or defect clusters, that could grow and contribute Laser Radiation." I Appl. Phys. 43, 3399 11972).

to the damage process or surface defects (e.g., pits and 9. P, M. Fauchet and A. E. Siegman, "Surface Ripples on Silicon

scratches) that might cause a melt spot to nucleate, and Gallium Arsenide under Picosecond Laser Illumination."Although in our experiments the former is believed to Appl. Phys. Lett. 40, 924 11982).be more likely. 10. E. W. Van Stryland. NI. A. Woodall, H. Vanherzeele. and M. J.

By investigating the pulse-by-pulse history of the Soileau. 'Energy Band-Gap Dependence of Twi-Photon Ab-

charged particle emission, a cleaning-type profile was sorption," Opt. Lett. 10. 490 (1985.discovered. This cleaning emission proved to have no 11. T. F. Boggess, K M. Bohnert, K. Mansour. S C Moss, 1. W.Boyd, and A. ,. Smirl. "Simultaneous Measurement of the Two-photon Coefficient and Free-carrier Cross Section Above the

parently, adsorbates remaining on the wafer surface Baodgap of ('rystalline Silicon." IEEE J. Quantum Electrom.

after cleaning did not act as absorbing sites to cause QE-22, 360 (1986).surface damage. The charge emission proved to be 12. R. K. Willardson and A, C. Beer. Semiconductors and Semimt-very noisy from site to site. Although the emission did al.s. Optical Properties of Il- V 'Compounds. Vol. 3. tAcadem

not strongly correlate with the event of surface dam- ic. New York. 1967). p. 409.age, once surface damage commenced an increase in 13. D. S. Chemla, 1). A. B. Miller. P. W. Smith. A. C. Gossard, and

charge emission was observed. This behavior is con- W. Wiegmann. "Rioom Temperature Excitonic Nonlinear Ab-

sistent with the idea of a hot partially ionized vapor sorptiin and Refraction in GaAs/AIGaAs Multiple Quantumbeing released from melt pits as damage proceeds. Well Structures," IEEE J. Quantum. Electron. QE-20, 265urtherieastig tpi s eededamag to odersnd.t 11984).

Further investigation is needed to understand the 14. A. Z. Grasvuk and I. G. Zubarev. "Interaction of Semiconductorsmechanism of energy transfer that causes damage and with Intense Light Fluxes," Soy. Phys. Semicond. 3,576 (1969)

emission. The relationship of the accumulation effect 15. K. L. Merkle, R. H. Uebbing, H. Baumgart. and F. Phillipp.to lattice defects could prove to be a fundamental key "Picosecond Laser Pulse Induced Damage in Crystalline Sili-in understanding the damage process. Another pa- cn," in Laser and Electron-Bream Interactions with Solids, B

rameter which might influence the damage threshold R. Appleton and G. K. Celler, Eds. (Elsevier, New York, 1982)S

is the dopant level of the semiconductor material. pp. 337-:342.It). ,J. A. van Vechten and A. I). Compaan, "Plasma Annealing State

of Semiconductors Plasmon ('ondensation to a Superconduc-tivitv Like State at 10(W) K'"" Solid State Commun. 39. ,M7

We gratefully acknowledge support given by the (1981).DoD Joint Services Electronics Program under re- 17 D.M. Kim, R. H.Shah, 1). Von der Linde. and 1). L. 'rosthwait.search contract AFOSR F49620-82-C-0033. We also "Picosecond Dynamics of Laser Annealing," in Laser and El,'c

thank John Beall at Texas Instruments for providing tron-Beam Interaitions itith Solids, B. R. Appleton and G K.

the GaAs samples. ('eller, Eds. I Elsevier, New York. 1982) pp. 85 90).

3870 APPLIED OPTICS / Vol 25, No. 21 / 1 November 1986

I; . . - . .. . .. . , , . . .. . . ,, . . . + . . . .. . . . +. . - . . . . . . . .. . . . + . . . . .. . . .. . .

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208 IEEI ELECTRON DEVICE LEFT ERS. Vo. EDL 7. NO 4. APRIL 1986

Modeling of Ion-Implanted GaAs MESFET's bythe Finite-Element Method

N. SONG, DEAN P. NEIKIRK, MFMBER. IEEE, AND TATSUO ITOH, FELLOW, IE-E"

4bstract-We discuss the results of a new io-dimensional (2-1)) characteristics and equivalent circuit parameters have been -finite-element model for GaAs ME.FET's made b, ion implantation, calculated. Unlike the triangular elements preferred in manySeveral different devices are characteried b) s.aring gate recess and semiconductor simulations, the bilinear rectangular elementsdoping profile. The simulation, in qualitative agreement with experimen- a d uctohere make it easier to manipulate the input and output

al findings, shows that a FET with a shallow gale recess exhibits a similar apbehavior to a FET with a deep implantation, i.e., an improvement in data by block mesh generation. The boundary conditions arelinear.y. a higher pinch-off voltage, and a decrease in transconductance. applied easily by the so-called penalty method [8]. It is found

that satisfactory results are obtained with less than 500 nodes. p

1. IntRODUcTInO For a typical node number of 307, it takes less than 2 s ofexecution time for each iteration on a CDC Dual Cyber 170/

T HE PERFORMANCE of ion-implanted GaAs MESFET's 750 at the University of Texas at Austin.

is usually predicted [I , [21 using an analytical model suchas the "two region model" [31. To include diffusion current Ill. DEvicE STRUCTURE AND APPROXIMATIONSterms and transverse current terms moeadequately, w have

more a ,we h The basic device structure used in this simulation is shown -adopted a two-dimensional finite-element method (2-D FEM) in Fig. 1. We have assumed a finite active-layer depth,

• ,' to nunmerically simulate GaAs MESFET's with nonuniformneglecting carrier transport far from the implant peak. The

doping profiles and various gate recesses. In particular, we deep region is considered to be semi-insulating. This assump-have studied a model ion-implanted structure, varying both the tion is based on the reported observations that phenomena suchdepth of the implant relative to the source-drain surface and as deep traps [21. [91, 110] contribute to mobility degradation

, the depth of the gate recess. For these numerical simulations as deep as region ribte silty degradaio

the FEM has several advantages over the finite-difference in this deep active region. For all the simulations we have alsoassumed a Gaussian-like active-layer doping profile, with a

method, such as the flexibility of the mesh size used in thecalculation and the inclusion of current conservation without 10 " cm Firstcal o ad tthe effects of gate recess changes were studied by holding thethe need of phantom nodes at insulatory boundaries where depth of the doping profile constant (as shown in Fig. 1(b)).default Neumann boundary conditions are applied, with the peak located at 85 nm from the source-drain surface.

II. SIMULATION Next, to study the effects of implant depth, the gate recess wasset to zero and the surface location of the electrodes was varied

Using a finite-element algorithm from a previous work [41, instead, which results in a change in the gate-to-profile peakthe coupled Poisson equation and current continuity equations distance. Here the implanted doping profilis left unchae

,, in the device are solved to find the unknown potentials and Tcoite uc and• The doping concentration below the source and drain* electron concentrations under various bias conditions using a contacts is fixed at 2 x 10'' cm -, and below the Schottky- standard FEM formulation [5). Once the electric field is gate contact is set to zero to approximate the built-in depletion

' calculated by taking an average for all Gaussian point values in gate ieo olg ap x me t bilt-in deltian element, an effective mobility is obtained using the 0.8 V. It isasnume vel ocint y veru e ectri -field relationships [6]. Using assumed that the low field mobility is 3200 cm /V s (appro-hassumed velocity versus electric elocit , a tempeatur s priate for electrons near the peak of the doping profile), the

saturation velocity is 10' cm/s, and the threshold electric field. found from the energy transport equation 171 neglecting the is 4 kV/cm. In order to simplify the model we have taken these

time dependence of energy for a first-order approximation.*1' With the effective mobtltty and temperature, the diffusion p

coefficient is calculated from Einstein's relation. IV. RNstits AND DISCUSSIONThe time step used in the simulation is fixed at 0.01 ps, i.e., The I-V characteristics obtained by varying the gate recess

thee di/ctt reaato timeteisic forine maera dopedn toe aat levelssilt theric relaxation time for material doped to a level depth are shown in Fig. 2. The data for a gate depth of 55 nmsimilar to the peak of the doping profile. From these, I- V are comparable to previous experimental results Ill1. As the

Manuscript received November 4, 1985. revised January 16, 1986 This gate depth increases, the transconductance becomes larger, thework was supported by Joint. Srvicev Electronic Program under Contract F magnitude of pinch-eff voltage becomes smaller, and the49620 82-C-)033 linearity of transconductance variation becomes poorer. Simi-

The authors are with the Department of Electrical and Computer Engineer- lar phenomena are observed ftr the various implanton cassing. University of Texas at Austin. Austin, TX 78712

IEEE Log Number 8W07960 shown in Fig. 3. Here the I- Vcurves beha,'e in a fashion quite

0741-3106/86/0400-0208$01.00 . 986 IEEEI I

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V: -

SONG ee al.: MODELING OF ION-IMPLANTED GaAs MESFET's 209

Lgs Lg Lgd 50 red....

source drain 40 recessed

ggate aedph---- planar

Sgtdp 30 gate-to-peakactive layer ditance(nm)

(a) =U

'10 30

-2 0 1 2 (-V)

0 GATE-TO-SOURCE VOLTAGE

Fig. 4. Plot of drain-current variation versus gate-to-source voltage for Figs., 2 (recessed) and 3 (planar).

similar to the I- V curves for a gate recess with identical gate-. . . . 200 to-profile peak distances. These phenomena are explicitly

0 so 100 ISO 200 shown in Fig. 4 for one typical drain voltage. Here, the5.. DEPti(nm) variation of the slope is related to the linearity. Such results as

(b) an improvement in linearity with a corresponding decrease in

Fig. I. Two-dimensional GaAs MESFET model. (a) Dimensions of device intermodulation distortion for deeper implants (i.e., largersimulated. Drain and source length: 0.2 Am: gate length L.: 0.5 pm; gate- gate-to-peak distance) have been reported previously [121-to-source distance L,,: I gm; gate-to-drain distance Lir: 1.5 Am; gatewidth: 300 pm. (b) Carrier profile used in simulation. All the carrier [151. This particular characteristic is important in large-signalprofiles used have a peak concentration of 2.5 x 10' 7/cm'. For Fig. 2. the applications, and therefore a trade-off between transconduct-source and drain surface is located at 0 nm. and for Fig. 3 the source, drain. ance and linearity may be required in these applications.and gate surfaces are all located at 30. 55. and 70 nm for project ranges 55.30. and 15 nm, respectively. The effects of changes in the tail of the active-layer doping

profile have also been investigated. Current-voltage curvesfor devices with the same profile shown in Fig. I(b) but with

recessed gate depth Vg(IV/step) an additional tail 10-20 nm long (extending the active-layerS0 - 300 A" . doping concentration down to 3 x 10 cm' from the' "" .,----*55SSO40 7S00 original I x 10") have been calculated. It is found that the

1, pinch-off voltage increases somewhat, while the transconduct-ance decreases near pinch-off. These changes are thought to be

I related to the steepness of the doping profile near the substrateinterface. The steeper profiles appear to give more linearbehavior near pinch-off-an important consideration in thet z

I 0design of low noise devices.- Typical values of the small-signal transconductance and

0 .. gate-to-source capacitance at zero gate bias are shown in Fig.1 2 3 5. In the case of recessed gate device, it can be seen that the

DRAIN VOLTAGE(V) cutoff frequency fr(fr = g,/(2wC,)) increases with gate

Fig. 2. I- V characteristics simulated by varying gate depth in recessed gate recess depth. This effect can be explained as follows: as thestructure, distance between the gate and the active layer implant peak

decreases, a smaller gate voltage swing is required to affect a

gate-to-profile Vg(tV/step) given drain current change [2). thus increasing the transcon-S0 peak distance(Rp) ductance g,. At the same time gate-to-source capacitance C,,5s0 A" decreases because the gate depletion layer extends beyond the

--- 300-is 0 implant peak as the recess becomes deeper. These effects

1 combine to produce an increase in fT. For very deep recesses0 (i.e., very small gate-to-implant peak separation) the transcon-

ductance begins to decrease, and the cutoff frequency frIsaturates at about 24 GHz. Although no graphical data areJ ,shown here, somewhat different results are obtained for gate.1 bias voltages near pinch-off, where the depletion layer is

always deeper than the implant peak. It is found that as the

DRAIN VOLTAGE(V) recess increases, C, increases rather than decreases as it does

Fig. 3. 1-V characteristics simulated by varying carrier profile in nonre- in the low gate bias case.

ceased gate structure. The results for planar devices obtained by varying gate-to---4

210 IEEE ELECTRON DEUCEF LErr7TERS. SOf-)l NO 4. APRIL lY6

gate-to-profile peak distance in progress to determine bias-dependent parameter changes.

80 70 60 50 40 30 20 (w) which are especially important in large-signal applications.-C .2

REFERENCES: tJ III

J A Higgens. R 1. Kusas. F H tisen. and I) R ('h'en. "Lowo noise GaAs FET's prepared b) ion implanation.' IWEE Trans

Electron Devices. sol ED 25. pp 587 S96 1978U 121 J N1 M Golio and R J Tre . "Proile studies of ion implanted

- .MESFET's." IIl: Frans..lcrowaie Theor. Feth sol MTT'I.pp 1066-1071. 198I

o 0recessed 01 I! R A Pucel. H A Haus. and H Sia:t. *Signal and noise properties ofp0aH GaAs inierovsase FET',, in Advances in Electronics and Electronplanar Physics ,.,1 I8 Nc York Avademki 197. pp 195-265

141 N Song and T toh. ''Accurale imulaiin it MESVET by finiteU 10 20 30 40 50 60 7Qfirm) elemnent nwthcxl including energy transposn and substrate effects." in

Proc. 1985 European Microwave Con! pp 24s 250

gate surface location in Fig. lb 51l 1 J Barnes and R I Liomax. "Finite e emt mnehods in semiconductfr device iniulatiion. IEEE Trans. Electron Devices. vol ED24.

Fig Calculated result, of transconductance and gate-io-,our~e capac:- pp 1082 -I9O8) 197-

lance for Figs. 2 and 3 1I1 K Yamaguc'ht arni H Kidcra. "Two-dmicnisinal nunmerial analysis olstablht criteria of GaAs FET's.'" lEF Fran. [lletron l[),vces.sol ED-23. pp 128 1289. 1976

171 V, R Curlice and N Yun. ' A tcn'peralutc model for the GaAspeak distane show similar effects. The larger g, at high drain NIFSFFI ." IFEE Trans. Flectron E)'D,' , ) 28. pp 954

current Ior the recessed gate structure, compared It) the planar 2 M 9y u." Co4,,truct.ire. is caused h., the thicker aeti,,e regions under the 41,ret. ipp[/ U.n,(. Eig . sol 31. pp $()i II . I)X4 i

sourt.C and drain in the rcessed gate structure. 191 K Lcc. N1 S Shur. K Lee. T T \,u. P ( I Roberts. and M JHelix. A i-A nid mobility in GaAs ion implanied l'FT's." It E:Trans. F- :,-' Dc. ce.s. vol ED 31. pp IAI 19 1 1 984

) C(ON('I[cSIoNS 1101 R H Vlit., at.d P x Jay. "Deep level, and eleIron nobility near the

acti:e late' jsibscaic nterface in GaAs MESF-Ks. in .Sem-lnsulat-- A tvo-dinlle,( nal finite element method has been used to mng 3 5 Alatcria.s. S Makramni Fixid and T f)iin -I', Nantwich.

This simulation K Shisa. 9S,2. pp 144-351characterize on-implanted GaAs NIESFE- T s al . -eng. V K Eu. T Zielinski. H KRai r . Vs. B Henderson,

allossvs accurate tn'deing of changes in device gcometry. (;jA, MtSFFT,, made by ion implantaii.n vo, %1C('VD butt er* including the location of both the channel implant and the gate layers.: IEEE E ctron Device I et , .ol It . pp lN 20. 1984

recess. Results of the lcodel have shown that FET's with a 1121 R F Wilam, aid D W Shas. GaAs FF1 , I.r risd lnartyand noise figure. IEEE Trans. Fle(tron leviPe 'ool FD 25. pp'shallo. recess ,how ttiilar behavior to those , cith a deep (-) N(5. 1978

implant. i.e.. an improvement in Itnearity. a higher pinch-off [III 1 A Higgeris and R t. Kusas. Anal ',Nsis and imrprosenient of

voltage, a decrease in transconductance. and a corresponding itermodulai,.' distorliin in GaAs txol M- r I2 I It h E Tran n%

Microcsaie Theori T ech siil VITT' 28. pp 1) 1 1980) 6decrease in cutoff frequency. Depending on the device 1141 R A Pucel, "'Profile design for distornion reduction in rnicro, ae

* application, different choices for these parameters would be FEr's., Electron Lett.. vol 14. no 16. pp 21W 2(i. 1978apr t T1151 J I !1 fDekkers. 1 Ponse. and H 'encking. Buried channel G,-sappropriate. Thus this simulation could be useful as a guide in MSEFT's'' IEEE Trans. Electron Devces. %id FIt 28. pp 15.5

the fabrication of optimized devices. Further calculations are 1070. 1981

Z-,

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."5

Physica 20D (1986) 320-334 INorth-Holland, Amsterdam %

ESTIMATION OF NONLINEAR TRANSFER FUNCTIONSFOR FULLY DEVELOPED TURBULENCE

Ch.P. RITZ and E.J. POWERSUnverstir of Texas at Austin, Austin, TX 78712. USA

Received 13 March 1985Revised manuscript received 3 September 1985

A statistical method for modeling the linear and quadratically nonlinear relationship between fluctuations monitored at twopoints in space or time in a turbulent medium is presented. This relationship is described with the aid of linear and quadratictransfer functions and the concept of coherency is extended to quantify the goodness of the quadratic model. A unique feature

of the approach described in this paper is that it is valid for non-Gaussian "input" and "output" signals. The validity of the

%" approach is demonstrated with simulation data. The method is applied to experimental data taken in the turbulent edge plasmaSof the TEXT tokamak. The results indicate a three wave process with energy transfer to large scale fluctuations. The estimation

of transfer functions is a first step in quantitatively measuring coupling coefficients and the energy transfer.

1. Introduction single input and single output which will be mod-eled in the spatial or temporal frequency domain

Many fluctuation phenomena in nature, as well by linear and quadratic elements of the formas in various technical problems, can be reducedto relatively simple systems which are describable Y = L .P + P X +(

by a set of source ("input") signals and the re- P p p 2 P P1 X )sponse ("output") signals of the system. The sys- P-P1 +P2

tern can then be considered as a "black box". Bymodeling this "black box" by an appropriate net- Such a system is presented schematically in fig. 1.work, consisting of linear, quadratic and higher- Lp and QP2 are usually called linear and

order nonlinear elements, it is possible to gain quadratic transfer functions and are generallyconsiderable insight into the dynamics of the sys- complex quantities. The Fourier transforms of the

tem under test. measurable input signal x(s) and of the output

The simplest such network consists of a single signal y(s) are XP and Y,, respectively. The sig-input, single output system which is purely linear, nals x(s) and y(s) are assumed to be zero mean

since the nonlinear contributions of the system are stationary random processes. The error term E is

assumed to be negligible. The determination of the the Fourier transform of a process which is as-

linear system model from the measured input and sumed to be statistically independent of the first

output signals is based upon cross-correlation two terms of eq. (1). It can be regarded as the

techniques. Applications of the linear network led error due to noise inherent in the measurement as

for example to the estimation of the dispersion well as systematic errors not described by linear

relation in plasmas 11 -41. It is also often used to and quadratic terms. The goal is to estimate the

test complex electronic circuits. In this work we linear and quadratic transfer functions from the

will discuss a system which can be described by a measured input and output signals.

0167-2789/86/$03.50 C Elsevier Science Publishers B.V.(North-Holland Physics Publishing Division)

% %

£2 2 ,;,'. Ir ' "-.-""- :'" """""%" ,%'''''-.-.- . . . " -. ". ,"""-"-"-"-" ". . -"": . . -- ,"..". " .. ' "-.-_..-.-.-.-

Ch. P. Rit: and E.J. Pouers / Estimation P1 nonlinear tran qer fun, itom 321%

reported preliminary experimental results whichestimate the coupling coefficient and energ-

S Lp cascading in such a turbulent plasma [17]. Theestimation of the nonlinear transfer function is a

X+ +Yp necessary first step to quantifying the couplingcotfcients and the energy transfer. The approach

0 PVP2 has not yet been published.P ENote that eq. (1) is the simplest variant of an

equation which can be related to wave-wave cou-pling, as we assume that four-wave coupling and %

SYSTEM MODEL higher order processes are much weaker thanthree-wave coupling. In some cases when

Fig I Schematic model of the nonlinear s'stem, given in eq. three-wave nonlinear coupling is forbidden by the

dispersion properties of the system, as for example pfor surface gravity waves in water [181. higher-orderterms must be included.

Indeed, we have motivation for modeling vai- The determination of the transfer functions Lous nonlinear physical systems with such a and QPIP: is straight forward and well known forquadratically nonlinear equation, e.g. eq. (1). Ex- the case of a Gaussian input signal x(s) [19-211.amples of applications include electronic net- Such a method can be used for analyzing a systemworks. electromagnetics [5, 61, and parametric that can be excited externally by a Gaussian noiseexcitation in mooring dynamics [7-91, in structural source [20] and for systems with input signalsvibrations [101 and in nonlinear optics [11]. Char- which can be assumed to be Gaussian [211. Man,acteristic of these examples is that the spectral systems such as turbulent fluids and plasmas, how-components X, and Ye stand for the temporal ever, do not allow such a restrictive assumption forfrequency spectra. On the other hand, many physi- the input signal. In order to obtain insight into thecal phenomena can be described by an equation physical mechanisms of turbulence in these media,similar to eq. (1) with Fourier components X, and one can monitor the fluctuations at two points inY, representing spatial frequencies, i.e. wave num- time or space and study the change of the spectrabers. When we consider the Navier-Stokes equa- between these points. In these cases and in generaltion for neutral fluids and Fourier transform in the input should not be considered to be a Gauss-space, we find a wave-coupling equation which ian process because of nonlinear history of thedescribes the temporal change of a spectral com- fluctuations.ponent of given wavenumber due to a linear mech- The main objective of this work is to present aanism (growth rate, dispersion) and due to three technique which enables one to estimate, in anwave coupling [e.g.. 12, 13]. The linear and qua- efficient way, the linear and quadratic transferdratic nonlinear terms in this equation can be functions from the measured fluctuation signalsmodeled by eq. (1). Similiar wave coupling equa- x(s) and y(s). These transfer functions serve astions describe the turbulent behavior in plasmas the fundamental quantities with which to estimate[14, 15] and in the solid state [16]. In several of the growth rate, the wave-wave coupling coeffi-these fields, competing theoretical models exist to cients and finally the energy transfer betweendescribe the formation of the turbulent spectra. different spectral components [171. This paperTherefore, it is useful to have an experimental focuses on the technique of estimating the transfermeasurement available with which the competing functions and qualitatively describing the physicaltheories can be compared directly. In the past we results. A subsequent paper will cover the applica-

JAC. .Mt R 4 e 4 : i

**** 4 .. ..- ~ 44~~4*!

322 Ch.P. Rit: and E.J. Poters,/ Estimation of nonlinear transfer funatns

tion of this concept to estimating quantitatively tically similar realizations, we obtainthe coupling coefficient and energy cascading.

In section 2 we present a new technique to KY,X;) - P Q'(XXX, - KX,)estimate transfer functions. For the case of non- L P1 'P'

Gaussian input signals the usual definition of thecoherency must be generalized. The influence of P =P1 +P:. (2)noise and of systematic errors will also be brieflydiscussed. As the approach involves rather exten- Similarly. by multiplying Y, by X*XA and en-sive computation, the convergence and accuracy of semble averaging, we havethe model must be tested prior to applying it toreal data. We present the results of such a test in KY p , X*) = LP( XX, X,,! X)section 3. In section 4. the method is applied todata from turbulence measured in the edge plasma + ' Q ,'P(XXX;X,+) + (rXx; *),of the TEXT tokamak. We estimate the linear and P1 'P,quadratic transfer functions between two points in (3)space and interpret them in physical terms. Nextwe compare the linear transfer function with the withone obtained by neglecting nonlinear contribu-tions. Such a comparison is of interest since most P = P1 + P2' = P', + P,,-present correlation methods are based on a linearassumption. To simplify eq. (3) and, therefore, reduce the

computation time we approximate the fourth-ordermoment (XX,, X X, ) with second-order mo-

X Pj12I P.'2. Method ments (IXgX,.I2 by neglecting components

(X, X,,X*,X,) with (p 1 , P2)*(p.,p2). How-

-5. The transfer functions for the quadratic nonlin- ever, we retain the third-order moment of theear system given by eq. (1) can be derived in a input signal (XP, P.,XpXpX,,), as we assume that

straightforward way when a sufficient number of the input signal is non-Gaussian. Such an ap-independent realizations of the spectra XP and Y proach was proposed by Millionshchikov (22] andare available. The variable p represents for exam- is used to close the system in weak turbulencepIe the wavenumber or temporal frequency. theories [12-15). This approach was also used in

We first rewrite eq. (1) by using the symmetry many strong turbulence theones [23. 24]. in whichrelation. QP: = QPiP, (Note that the spectral the linear mode structure is strongly altered by the

% components X,, and X are interchangeable in turbulence and for which A/a is large. Compan-eq. (1)). It is therefore sufficient to sum the son with simulations which do not use such aquadratic terms over the frequency components closure scheme confirmed the validity of this ap-with p, >p. proximation also in strongly turbulent cases

124, 25]. Approximating the fourth-order momentsrequires that the fluctuations must be close to" Y , LX, + QP'+XP, XP,.z + E,. (1a)

+PX" Gaussian distributed, a condition which is notP1 -'P2+PvP' P" generally valid for strong turbulence. In a future

Lwork it is planned to compute the complete set ofMultiplying eq. (Ia) with the complex conjugate of fourth-order moments and solve for the transferthe input signal X, and computing the expected functions. This will allow us also to test the valid-value by ensemble averaging ( ) over many statis- ity of the closure scheme on real data. Using the

,4,

% o."

% (h.P. Rit: and E.J Pov.ters / EAstimaton of hnoinear frun er uion % 323

Millionshchikov hypothesis we have well-known statistical quantities in the pointed

4brackets, the auto-power spectrum P X, X*),.y..QI pthe cross-power spectrum CP = KY,,XP*), the auto-

bispectrum B,( pl, p2) =(X,-, .X*Xp*,) and the,!, Y PX P) - L(XpX*X,) - (ePXX,*X) cross-bispectrum B,(p 1 , P2) KYp,,. X*).

SI X, XP,1 2) The cross-power spectrum measures the statistical. P , 3) dependence of amplitude and phase of the sameSp p+4 p,. (3a) spectral components in the input and output.

" .' Therefore the cross-power spectrum must play anThe second term in the numerator of equations important role in detecting the dispersion relation

12) and (3a) vanish for a Gaussian input because and the growth rate. The auto- and cross-bispectrathen the skewness ((x 3(s))/x(s)) 3 ) is zero, measure the statistical relationship of amplitudetherefore ( X, .XX,* = 0. The second terms and phase between the spectral components P1, P2can thus be interpreted physically as corrections and p =PI +P2. If the waves at P1. P2 and pdue to the nonlinear ,,istory of the fluctuation have statistically independent random phases, then

- .,-" detected at the input. As long as the error term e the resulting (bi-)phase of the polar representations zero mean and statistically independent with (OP, .,, - ,, -0P ) will be random and the ex-respect to the input signal, the error terms in eqs. pected value of the bispectra convcrges to zero. If,

* 12) and (3a) vanish, however, a coherent phase relationship exists dueThe set of dependent equations (2) and (3a) can to nonlinear coupling of these waves, the bispec-

be solved iteratively to find the transfer functions tra. averaged over many realizations, will reach aL, and Q~P:'. As an initial guess we neglect the finite value. The auto- and cross-bispectra play,quadratic contribution in eq. (2) and find for the therefore, an important role in detecting three-linear transfer function wave coupling effects. The mathematical and stat-

istical background of the estimation of these ex-pected values with the aid of digital signal

L1-" - ,' . (2a) processing are thoroughly discussed in [26-281.

The amount of information gained from thetransfer functions can be quite large. Therefore it

The number of iterations used for a given frequency is useful to introduce normalized quantities top depends on the accuracy of the estimated mo- provide an easier interpretation of the input-out-ments. It is also sensitive to the stationarity of the put relationship. A convenient normalization issignal to be analyzed and is thus larger for real the coherency, which gives the fraction of thedata than for simulated data. The number of itera- power in the output signal which can be accountedtions is also dependent on the magnitude of the for by the linear and the quadratic transfer func-

, quadratic transfer function with respect to the tion model. To define the coherencies in terms oflinear one. The typical number of iterations needed the transfer functions, we multiply eq. (la) by itsforL to change by 6 less than l% is of order 5 complex conjugate, take a statistical average andfor simulation data and of order 10 for measured divide the result by the output-power (YY*). Wedata (6 < 1%). Note that the special case of a findGaussian input signal can be solved directly

* without an iterative procedure as the first term onthe right-hand side of eq. (3) vanishes. 1= y ( p) + y( P) + YQLQ( P )

As indicated by equations (2) and (3a), L, and," ' can be obtained by determination of the + y,( p) + (error terms), (4a)

. "

324 Ch.P Rir: and L. Po r .. tiinw ,' n.nlmear fran.ler Iun tln

with zero and unity. The "'goodness of fit" of the modelcan then be characterized by the total coherenc"

L (VX, of the model.

Y, Xp Xt. 2) 1(p y

Y2( P) = 2( P) + P( p) + Y [Q( P). (5)Pi -. XX

S <)YPYO)P1 -,2 Note that the above definition of the coherency

2 Re ( L [Q ' ] ( x¥ X*,; . converges to the commonly used definition [19. 201.']( P I= P, P,,: when the input signal is Gaussian. In this case

SXX, .:X;,) --- 0 and the term yLQ( p) vanishes.We find

Yn 1 1 Y1 (4b) Y ,YL,()- <P)i ) =2 I<XP >l",

whereP 1+ 2P IYP XP*X l (6)

p. ,~r +'., : E •2)YPP

-he linear coherency y"(p) and the quadratic Y2Q( P) = 0.coherency y(p) denote the fraction of outputpower accounted for by the linear and quadratic where

transfer functions. The value under the summa- p =Pl +p2.tion in y,(p) is called the cross-bicoherencyb2,,( pl, p.) and measures the portion of the power The linear coherency L'(P) is bounded by zeroof the output signal YPp, .. , which is phase locked and unity as can be shown with the Schwartzwith and X,, in the input signal. The inequality. Because of eq. (4a). y2( p) must also becoherency -yLQ( P) gives the portion of the output bounded by unity. The application of eq. (6) topower, for which the response due to the linear experimental data with a non-Gaussian input sig-transfer and due to the quadratic transfer are nal will thus lead to an erroneous result and cancorrelated. Note that this term arises because of yield values for the total coherency which arethe non-Gaussian input x(s). The term y'(p) greater than unity.represents the ratio of the output power due tonoise, which can not be accounted for by thesystem. The error terms in eq. (4a) vanish as long 3. Simulation testas the error term E of the output signal is zero

* mean and independent of the input signal of the To test the validity of the approach described in

system. The usual definitions of the coherencies the previous section, we have carried out a com-1191 are bounded by zero and unity (as we shall puter simulation. We start with analytically de-

.-" show below for the case of a Gaussian input fined linear and quadratic transfer functions. Forsignal). However, the individual coherencies given non-Gaussian input signals we compute theY"( p). y ( p) and yLQ( p) defined above are not corresponding output signals. Next, we apply the

., necessarily bounded by unity. While y,(p) and method of section 2 to estimate the linear and'Y 2( p) must be larger than zero, the term -yQ( p) quadratic transfer functions from the input andis allowed to take on negative values as well. The output data, and then we compare the results wthnoise term y2( p) can take on any value between the expected values.

%-

Ch.P. Rit: and E.J. Powers/ Estimation o nonlinear translerfunt ]u,,m 321

For this example, we define L and QP as 1 - ........ . .................P PL,,, = 1.0 - 0.4 P_ + i0.8 p Re(L& ReLp)

P , '.4 P Nvq .e 81

PI{ PPA P22- 5pN q I +Pp 8* . C)

. . ............ ) ....... ........

where___ ~~a ---- ------.. v.---P =t, tp, 1= , .b 8 d)

12 lm )ImffLp)

The magnitude and shape of the quadratic trans- 44fer lunctions defined above are chosen arbitrarily. /

but are realistic in that the values are of the same / /order of magnitude as the ones predicted by the OL 0Hasegawa Mima equation 1151. The linear trans- 0 .5 1 0 .5 1

fer function is defined such that the input and P/Pq P/PNyq',. output spectra are similar in shape as would be Fig 2 Companson or the analytically defined linear transferexpected for a stationary state. The Nyquist function with the reconstructed ones: a) shows the real: and

frequenc-v shall be abbreviated as p ,q*. The real b) the imaginary component of the "true'" values; c) and d) theand imaginar parts of L. are illustrated in fig. 2a. estimated ones

2b and the absolute value of QP, is shown in acontour plot in fig. 3a. To approximate the situa- good agreement with the true values. The symbol,tion which actually occurs in a continuous medium, "', denotes an estimator. To save space, we com-we consider five identical " black boxes" of the pare here only the absolute value of the quadratictvpe shown in fig. I. which are connected in series, transfer function. The phase comparison wouldA Gaussian signal is applied to the input of the show a similar agreement. Note, however, that thefirst black box. The output, which is now phase information is also important for the inter-non-Gaussian because of the nonlinear nature of pretation of the nonlinear system. Because of thethe black box, becomes the input to the second symmetry properties possessed by the quadraticblack box and so on. For the simulation we utilize transfer function, it is not necessary to plot QP2,.,the input and output of the fifth black box. The over the entire two-dimensional planet. Fig. 3input signal, X,. for the estimation can thus be gives the value of IQP,'P21 at the frequency p =p,assumed to be approximately as "non-Gaussian" +P2 due to wave-wave coupling with p, and P2.as the output signal Y". The area, for which the transfer function is plotted,

In a second step we estimate the linear and can be subdivided into three regions with essentialquadratic transfer functions L1 and Q P2 using differences in the physical content (I, 1I, ii in fig.the approach of section 2. As shown in figs. 2c, 2d 3a). The triangular region (I) gives the quadraticand fig. 3b. the estimated transfer functions, are in transfer function at the highest frequency p in-

volved in the interaction (p > Pt, P2). Region (II)*The maximal frequency which can be resolved digitally is

g,en hs the Nquist theorem. The theorem indicates that the tit is sufficient to compute and plot the transfer function forsmallest detectahle period must contain at least two sampling pa > p, because QP- P2 - Q.2 Pl) and for p > 0 (becausepoints Frequencies larger than Psq produce an erroneous QP, P2 - (Q P,, P2I, as the Fourier transform for real data,pctrum below p,,, due to ahasing. ( a ) satisfies A ,, - X*P).

I',.5

.1*

".,

0,

326 Ch. P. Rit: and E.J. Povt'ers! Estimation of nonlinear transferfunctions

'5-.5 b)

.5 0 -0

... .. ..

%0 .5 1 0 .51

P,/PNyq Pl/PNyq

•% Fig 3 Contour plot of the amplitude of a) the analytically defined quadratic transfer function. Point A illustrates "m.I at.A%5'.% pI /rp s,, = 0.78 due to coupling with p2pq -0.22 and p/p, vq O 56 : b) reconstruction of I QP P21 by the iterative method. The

ontour interval is 0.05 in a) and 0.06 iu b).

gives the strength of the coupling at the inter- difference between the estimated total coherencymediate frequency p due to wave-wave coupling and unity can be regarded as due to systematicwith a spectral component of larger frequency Pi errors of the estimation approach and due to theand one with a smaller frequency P2(Pt > P > variance of the estimator of the statistical quanti-IP21). The triangular region (Ill) shows IQiP21 at ties (e.g. the cross-bispectrum). Fig. 4a showsthe lowest frequency component p due to cou- the contribution of the linear, quadratic and mixedpling with the two others (p <P1 , IP21). For a coherency to the total coherency of the modelconvenient graphical representation of the three y 2(p). We have chosen relatively small values of

Scoupling regions, we plot positive and negative the quadratic transfer function with respect to the

values of P2. Note, however, the positive and linear component in order to simulate a realisticnegative spectral components are related by X_ situation; consequently the contribution of 2,( p)= X. To facilitate the interpretation of fig. 3, we is small. Note that the total coherency of thearbitrarily pick out one point, which is indicated model y2(p) is close to unity indicating that theas A. The number of contours at this point gives systematic error of the approach is small. We

"V " the amplitude of the transfer function 1QiP21 at conclude that the iterative method is able to pro-

frequency p/p Nyq = 0.56 as a result of its coupling duce a good fit of the data.with a higher frequency component PI/PNyq = To illustrate the necessity of applying the itera-0.78 and with a lower frequency component tive method, we compute the transfer functionsI p21/P Nl=0.22- with the usual method by disregarding the

"- .-. To test the "goodness of fit" of this simulation, non-Gaussianity of the input signal, X.. The%- we compute the coherency from eq. (4b). The non-iterative method then leads to a quadratic

r"~pp .r '* . % --

% %

Ch.P. Rit: and EJ. Powers/ Estimation of nonlinear transfer functions 327

I 1.0 . _2 .)-

/y 2 (p)L. (,")I)P

1.0

3.4

5 .5

a) b)

yo 2(p) Q2(p)

0001 .5 1

Frequency P/PNyq

Fig. 4. Coherence spectra. Y2 ( p) linear component, y2( p) quadratic contribution, yv2o(p) mixed term. y2 ( p) summation over thethree components. The difference between y-(p) and 1.0 represents the error term y4 ( p): a) gives the coherence for the iterativeprocedure (eq. (4a. b); b) the one computed with the usual definition (eq.(6)).

transfer function which is much different from the cess. To illustrate this, we show in fig. 5 thetrue one. This erroneous result can easily be visual- convergence of both the power spectrum Pp andized by applying the usual definition of the the amplitude of the auto- and cross-bispectra to acoherency, given by eq. (6), which leads to values stable value as a function of an increasing amountfor the total coherency much larger than unity (fig. of realizations. We have chosen for this demon-

4b). The systematic error introduced by assuming stration the frequency components P/P Nyq = 0.47,a Gaussian input signal can thus be considerable. PI/PNyq = 0.31, and P2/PNyq = 0.16. While theNote, for the non-Gaussian case, the linear auto-power spectrum stabilizes after 500 reali-coherency is very close to unity. This is due to the zations, more than 1500 realizations for the auto-very small change of the signal between the input and cross-bispectra are needed. The large variance

and output as a result of the quadratic interaction of the higher-order spectra can be attributed to(which is of order y¥(p)in fig. 4a). the fact that all spectral components in a turbu-

It is important to recognize that a good estima- lent spectrum couple with each other and there-tion of the quadratic transfer function requires a fore the contribution of each triplet of waves islarge number of realizations. For this simulation relatively small.we have used 2000 realizations. This large number By testing the iterative method under different

of realizations is necessary in order to estimate the conditions, we have found another interesting fea-auto- and cross-bispectra accurately and thus to ture which we briefly report. For a purely Gauss-assure a proper convergence of the iterative pro- ian input, X., the iterativc nrocedure appears to

1.-.

328 Ch.P. Ritz and EJ. Powers/Estimation of nonlinear transfer functions

6

- Ba(pi,p2)PP Bc(P1.p)

".1

000 2000 0 2000

a) Rafatios b)

Fig. 5. Convergence of the spectral estimates to a stable value as function of the number of realizations, a) Convergence of the powerspectrum at P/PNyq - 0.47; and b) of the amplitude of the auto- and cross-bispectra at P/P Nyq - 0.47 due to wave-wave couplingwith P2./PNq - 0.16 and PI/PNq - 0.31.

considerably reduce the number of realizations fusion, as the fluctuations in the edge can substan-

needed for a given statistical error of the quadratic tially affect the global plasma confinement. Thetransfer function. While the auto- and cross-power study of the linear and nonlinear behavior of thespectra are already estimated acceptably, the waves and instabilities in the edge plasma can add

estimate of the bispectra is significantly poorer for important information which, together with theo-the same amount of realizations. Because of a retical models, lead to optimized edge conditionssimilar behavior of the auto- and cross-bispectra with minimized particle and energy fluxes to the(see their convergence in fig. 5), the statistical wall.

errors therefore approximately cancel in eq. (3a) The linear features of fluctuations in the edge ofand the estimation of QPp2 is improved for the tokamak plasmas have been studied for some timesame amount of realizations. We conclude that the with probes [e.g. 1-4, 29-311. In the following weiterative procedure is valuable to increase the speed extend this observation and give information aboutof the calculation for systems with a purely Gauss- the nonlinear wave interaction by the estimationian input signal and also requires fewer realiza- of the nonlinear transfer functions.

tions. To make a direct comparison with theoreticalworks on turbulence in plasmas one is most inter-

4. Nonlinear wave coupling in the edge of a tokamak ested in the transfer function for wavenumberspectra. Such spectra are, however, not easily ob-

The plasma which connects the hot interior tainable in tokamak plasmas as this would neces-plasma of a tokamak with the cold wall is often sitate a large number of spatial samples withcalled edge plasma. This region of the tokamak is probes or a simultaneous measurement of many

characterized by steep density and temperature spectral components with scattering technique. Ingradients and by a high density fluctuation level, a tokamak the installation of a Langmuir probe

Asymmetries induced by the limiter and an array has recently been realized [4]. A large probeincreased impurity level due to the plasma wall array may substantially perturb the plasma. Multi-

contact make this region even more complex. A channel scattering experiments are able to mea-good understanding of the physics of the edge sure instantaneously different wave numbers. Forplasma in a tokamak can, however, be crucial for the analysis a heterodyne system would be needed

.: ::.

Ch. P. Ritz and E.J. Powers / Estimation of nonlinear transfer functions 329

and the influence due to different scattering tion in the ion diamagnetic drift direction while itvolumes has to be understood. To circumvent results in a rotation in the electron direction onthese technical questions we chose for this initial the inside. The measurements also exhibit a local-experiment a two probe technique. ized instability which occurs in the region of maxi-

Two spatially separated probes are capable of mum velocity shear and which is different frommeasuring temporal variations of the fluctuation, the turbulence structure outside of the velocityInstead of computing the bispectra for different shear. The measured phase velocity of the turbu-wave numbers which fulfill the selection rule k = lence can be described by an E, x B drift super-ki + k2 we compute them for the frequency com- imposed on a pressure gradient drift [29]. In thisponents f= f, + f2. This approach enables us to work we include a description of nonlinear effectsdetect only the resonantly coupled components, for the region outside the shear layer. Model equa-for which the frequency mismatch is zero. Those tions [23, 24, 32, 33] applicable in the edge regioncoupled spectral components with frequency mis- are characterized by quadratic nonlinearities,match, which are present in strong turbulence hence, the three wave interaction is the relevantcausing the broadening of the dispersion relation, wave coupling process. Model equations with cubiccontribute to the error term in our method. nonlinearities [34, 35] giving rise to four wave

We measured the signals with Langmuir probes interaction are generally not considered in thein the turbulent edge region of the TEXT tokamak edge plasma context.

* | (major radius 100 cm, minor radius a = 27 cm). Our goal is to estimate the linear and quadratic- -- To measure the density fluctuations, the probes transfer functions between the signals of two

were biased into ion saturation current and the poloidally separated points. The experimental" potential fluctuations were detected with floating setup is as follows: two radially movable Lang-

". probes. The signals are digitized with a 10 bit muir probes are located at the top of the toka-digitizer with 32k words storage capability per mak. They are separated by .x = 3.5 mm. For thechannel. Each time series is subdivided into 512 dominant power at low frequencies, this sep-time segments of 64 data points. To estimate the aration is small compared to the poloidal correla-transfer functions we have averaged over more tion length (several centimeters). In this paper. wethan 1500 realizations gathered from 3 identical analyze data taken at a radial position 1 cmshots. The sampling interval used in the following behind the limiter. The density at this location ispresentation is 1 ps, which defines a temporal approximately 5.0 x 101 cm ', the electron tem-Nyquist frequency of 500 kHz, which is well above perature T = 10 eV and Ih/nI = 40%. Th -owerthe dominant components of the turbulent power spectra of the density fluctuations monitored byspectrum. both probes located at this radial position are

We will consider data obtained with the follow- shown in fig. 6. Both probes yield nearly the sameing tokamak parameters. The toroidal magnetic spectrum.field is B, = I T, the plasma current is 10 = 100 The data, which have been digitized and storedkA, the chord averaged density is 1.0 x 1013 cm in the computer, are processed using the procedureand we have a peak electron temperature of 600 discussed in section 2. The propagation direction

% eV. In the region behind the limiter we observed a is first determined by observing the sign of the%, broad turbulent spectrum with fluctuation levels phase shift of the cross power spectrum between

ih/nJ of up to 50%. The edge plasma is char- the two probes. The signal of the probe which firstacterized by a nonuniform radial electric field samples the turbulent structures (i.e. the "up-which changes sign just inside of the outermost stream" probe) is treated as the input signal. Theclosed flux surface. Behind this flux surface the "downstream" probe signal is considered as theradial electric field causes an Er x B plasma rota- output.

'V.

330 Ch. P. Rit and E.J. Powers /Estimation of nonlinear tran /erfun, itmp

1 --- - -- - -- --- -

--. Xf a)-Yf

YI%].! .205E

%- - ------ 10----

: , . .. 0 -. . . . . .. . . . . . .. . . . . . .. . 5 02.0 250 5

l=w/2n (kHz)

.0 __________,_______Fig. 7 Amplitude a) and phasqe b) of the measured linear:-V" "" 0 250 500 transfer function The dotted line represents the transfer func-

tion assuming Gaussian input.Frequency (kHz)

Fig 6. Power spectrum of the density fluctuation 1.0 cm be- propagating a distance x. An amplitude smaller

hind the limiter for the two Langmuir probes, separated by 3.5 than one can be due to a linear damping mecha-mm in the poloidal direction. nism or a transfer of energy to other waves as a

result of nonlinear wave-wave coupling. OtherThe linear transfer function estimated from this reasons for the change of the amplitude also exist,

data is presented, for easier physical interpretation as for example, the neglect of higher-order nonlin-' , in terms of amplitude and phase of L1 = ear terms, an external source of noise or a multidi-

ILflexp(i0L(f)). Amplitude and phase are shown mensional behavior of the fluctuation. In fig. 7a wein figs. 7a. b. The phase of the linear transfer observe that I L1 is always smaller than unity and

-'-. function 0L(f) gives the phase relationship be- drops off rapidly for frequencies greater than 350_..- tween the input and the output signal due to kHz. We conclude that all spectral components are

linear effects. In our experiment, kL(f) can be damped; however, we would require an unphysi-related to the mean dispersion relation, ko(f), cally large damping coefficient to describe thebecause a wave which propagates with wavenum- damping at high frequencies. For more insight weber ke(f), will undergo a phase shift in propaga- look at the "goodness of fit" of the model byting between the two probes equal to ko(f)Ax. In plotting the coherency (fig. 8). The total coherencyfig. 7b we give the phase information as well as the of the model (eq. (5)), y2(f), is high for spectralrelated averaged wavenumber ke(f) = L( f)/ax. components up to the frequency where the ampli-We find an approximately linear dispersion rela- tude of the linear transfer function starts to de-tion and thus a nearly constant phase velocity of crease rapidly. The fast increase of ,y'(f) = 1.0 -the wave over the observed frequency range. At y2(f) for frequencies larger than 350 kHz demon-the radial position discussed here (1 cm behind the strate that systematic errors become important atlimiter), the propagation direction of the phase high frequencies. The effect of instrument noisevelocity is in the ion diamagnetic drift direction. can be neglected, as the power of the signal is

, V,,., The magnitude, ILI, gives the change of the significantly larger than the noise power for allsignal level between the input X/ and the output frequencies. We believe that the deviation of theY due to linear effects. An amplitude I L/ larger coherency from unity is mainly due to the one-than unity indicates a wave which grows while dimensional approach we have chosen. In our

% % %

C . P Rit: and ij. Pow ers stmatin ,l nonlinear rat er J, runt t, 331

1.0 -----------------------------------------------. interaction with two waves of sim ilar frequency.2 -M ,The strength of this wave-wave coupling increases

L ",when f, and f, approach the Nyquist frequency.2( M- We must keep in mind, however, that because of

'- the low power at high frequencies the absolutecontribution of the high frequency components to

5, the output signal is small, despite a large value of

the transfer function.When we examine the coherency due to qua-

dratic interaction of the waves y2(f) (fig. 8a) weS 2 2(f find very low values. The fraction of the total

0 M LOM power at frequency f which originates from inter-.. .action with all frequency pairs which satisfy the

0 250 500 selection rule f=f +f2 is therefore small for aFrfeqL*y (kHz) probe separation of 3.5 mm. The fact that the

magnitude of the mixed term -Y Q(f) is larger inFig. 8 Coherence between the two probes magnitude than Y2(f) is not surprising. For small

probe separations the turbulent structures reach-

experimental set up, we look only at the poloidal ing the first probe are still present when they reachcomponent of the fluctuation. When a turbulent the second probe (due to the linear transfer) andstructure propagates at an angle with respect to contribute even more to the output signal thanthe two probes it may be observed by only one of that generated due to quadratic interaction be-the probes and will cause a decorrelation between tween the two probes.the signals. This effect will dominate when the Having discussed the various sources of errorsscale length and thus the wave length of the struc- which affect the "goodness of fit" of the model, wetures becomes very small, as is the case for high have to ask if the measured quadratic transferfrequencies (see fig. 7b). function gives a meaningful result. This question

We now turn our attention to the quadratic is critical, as we know from the coherency of thetransfer function, for which the amplitude, I Qf, 1, quadratic term as well as from theoretical modelsis shown in a three-dimensional plot in fig. 9. that the quadratic interaction is small. Also, theDuring the propagation between the two probes, variance of the estimate is larger than that of theonly a relatively small amount of the spectral linear one, just as an estimate of a two-dimen-amplitude of the output signal, Y, is generated at sional surface has more variance than an estimatefrequency f=fi +1f2 >fl, 12 due to wave-wave of a one-dimensional curve, using the same num-

interaction with the frequency components jL and ber of data points. We have, however, good indi-f2 (see region (I) in Fig. 3a). The amount of the cation that the quadratic coefficient is qualitativelysignal which is transferred to intermediate fre- correct. When we look at the scaling of the coeffi-quencies due to coupling with a higher frequency cients with probe separation. we find: for an in-and a lower frequency component is larger. The creased spacing of the probes, y2(f) increases andlargest values of the transfer function occur in the It2 Q(f) decreases. We can expect such a result forshaded region in fig. 9, corresponding to region a correct measurement: for an increased probe

• (!1) in fig. 3a. We conclude from this that the separation, the interaction time for quadraticquadratic transfer function is strongest for the processes and linear damping is longer. This ob-dominant low frequency components of the spec- servation indicates that the quadratic transfer

'i,. trum in fig. 6 (around 30 kHz) as a result of their function is estimated correctly.

. . .". ., ..

* 332 Ch.P. Rit: andE.J. Powters/E, turnatotoftnon/litnurtraiferfuithtinon

500

'"250 f2 (kHz)

[O Fig 9. Three-dimensional plot of the amplitude of the quadratic transfer function. ohtined from the densmt, fluctuations measured

' at the tw.o probes.

- How do these results compare with different a dominant coupling of higher frequency spectral' -" ,..,theoretical predictions? Fundamental differences components in a way as to increase the amplitude

1 are predicted theoretically for two-dimensional and at the lowest frequency component of the triplet.• , "three-dimensional turbulence: in three-dimen- The strongest effect is found for coupling of two

' Y sional turbulence the energy cascades due to vortex spectral components of comparable frequency to a,f - istretching from large scale turbulent structures to very low frequency component. This result indi-., .'small scale structures. In two-dimensional turbu- cates an efficient "one step" process between rela-

lene heenergy ispredicted tocsae ncon- tively small scale turbulent structures (k > 10

,- trast to the three-dimensional model, to smaller cm - ) to large scales (k 5 cm- t). The experi-r, "wave numbers and frequencies, as the vorticity is ment indicates that the turbulence in the edge

tndepndentof the third direction. For tokamak pam ofthe TEXT tokamak is esnilyofplasmas with inherently strong. mainly toroidal, two-dimensional nature as expected theoretically

.r .imagnetic field a two-dimensional behavior of the for tokamak plasmas.-turbulence is generally expected. The essentially We mentioned in the introduction that much

• :'-,.'-two dimensional picture of the turbulence can be work dealing with turbulence is based on linear.-... ,justified by the strong magnetic field which con- transfer functions [e.g. 1 -4, 29]. The contribution' '-'+:fines the charged particles in the perpendicular of the nonlinear term in eq. (2) can then be. - "direction, while the electrons and ions can propa- neglected. The magnitude of the quadratic transfer

gate relatively freely along the field lines. There function and, thus, the coherency y,( f) we ob-le, qlare many different theoretical turbulence models served suggest that such an assumption is, indeed.

", -.. for tokamak plasmas because a variety of linear valid. The approximation can be confirmed quan-" €',instabilities and nonlinear mode coupling pro- titatively. When we neglect the quadratic terms

" ." cesses can be included 1361. Our experiment shows and cgmpute the linear transfer function (eq. (2a))

-,"-,

0~0250 f ,

2 (k4*~z)

Fi,9 he-iesoa lto h ampliude ofteqartc rnfrfnton bandfo tedn'fututosmaue

A

Ch.P Rt: and VJ Powers ji It mation ,l nonh near tra er a. Pe j trIn , r

we find %alues which are ver-v close to the ones for quencies and lead to an increase of the amplitude Ithe non-Gaussian input case (dashed lines in figs. at the low-frequenc,, component. This result indi-7a and b). While our results are obtained for cates an efficient "one step" process between smallturbulence observed in the edge plasma of the scale and large scale structures. The portion of ti,eTEXT tokamak with probes separated byAx = 3.5 signal which is generated between the tmo probes

mm, the, suggest that the quadratic terms can be due to quadratic interaction is small with respectneglected for similar experimental arrangements to the linear contributions, but proves that manNw hen one is interested only in the linear behavior waves couple with each other in a fullk turbulentof fully developed turbulence in the edge region of spectrum. Our observation is consistent with two-tokamaks. Note that the neglect of the nonlinear dimensional turbulence models which predict anterms in the estimation of the linear transfer func- energy transfer to smaller scale structures due totion can lead to an incorrect result for cases with three-wave coupling, which is in contrast to threesmall linear component l.p,. as in this case the dimensional models with energy cascading tocontribution of the quadratic term to eq. (2) can larger scales.be considerable. We demonstrated in this paper the usefulness of

the method to estimate the linear and quadratictransfer function for non-Gaussian inputs. Al-

5. Conclusion though we applied the method to plasma turbu-lence data, we emphasize that the method is equally

WVe hase found that the usual method of esti- applicable to analyze other data, as in fluid turbu-mating linear and quadratic transfer functions be- lence. for example. In a future work. we willtween two measurement points for Gaussian input extend this method to estimate the three-wavesignals can be extended to non-Gaussian input coupling coefficients and thus quantify energysignals. This extension leads to a model with which cascading in turbulence. Such experimental mea-a self-excited and fully turbulent system can be surements can be useful for a direct comparisonstudied experimentally. The model is tested with with theoretical models.simulated data and can reproduce the transferfunctions. A check of the "goodness of fit" showsthat the coherency is close to unity. The true Acknowledgementsoutput signal can thus be modeled nearly exactlyfrom the input signal by using the estimated linear We wish to thank Roger Bengtson for his en-and quadratic transfer functions. Analysis of the couragement and careful reading of the manu-same signals with the usual method, which as- script. We are pleased to thank Ken Gentle, Scottsumes a Gaussian input signal. gives nonphysical Levinson and Sung Bae Kim for useful discussionsvalues of the coherency with values larger than and comments on the manuscript. The consistentunit, theoretical motivation and help of Pat Diamond

Using two Langmuir probes in the edge plasma and Paul Terry is gratefulk acknowledged. Weof the TEXT tokamak we find: the linear transfer would further like to thank the TEXT group of

function yields a nearly linear dispersion relation the Fusion Research Center at U.T. Austin forof the waves with propagation in the ion diamag- their technical support.neic drift direction. All spectral components are This work was supported b, the U.S. Depart-at least marginally damped. The amplitude of the ment of Energy Contract No. DI--AC5-quadratic transfer function is largest at the do- 78ET53043. the Joint Services Electronics Pro-minant low frequencies around 30 kHz due to gram Contract No. F49020-92-C-(0)33. and in partcoupling with two waves having comparable fre- by DOE Contract I)E-AC05-85ER80251. and the

%A5

334 Ch.P. Ritz and EJ. Powers /Estimation of nonlinear transfer functions

Texas Atomic Energy Research Foundation. Ch.P. [171 Ch.P. Ritz. SiJ. Levinson, E.J. Powers, R.D. Bengtson and

Ritz was supported partially by the Swiss Na- K W. Gentle. Proc. Int. Conf. Plasma Phys. (1984) 73.tionl Sienc Fondaton.[181] 0. M. Phillips. in: Nonlinear Waves. Leibovich and Seebas.

* tinal ciene Fondaton.eds. (Cornell Univ. Press. Ithaca. 1974) 186.[191 Y.C. Kim. W F. Wong. E.J. Powers and J.R. Roth. Proc

IEEE 67 (1979) 428.*[201 J.Y. Hong. Y.C. Kim and E.J. Powers. Proc IEEE 68References (1980) 1026.

[211 D. Choi, R.W. Miksad. E.J Powers and F.J. Fischer. J ofSound and Vibr. 99 (1985) 309.

ill Si. Zweben. P.C. Liewer and R.W. Gould. J. Nucl. Mater. [22] M.D. Millionshchikov. Doki. Akad. Nauk SSSR 32 (1941)111-112 (1982) 39. 611.

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Bengtson. NucI. Fusion 24 (1984) 527. [24] P.W. Terry. P.H. Diamond. K.C. Shaing. L. Garcia and[4] S.J Zweben and R.W. Gould. NucI. Fusion 25 (1985) 171. B.A. Carreras. "The Spectrum of Resistivity Gradient[5] M A Flemming. F.H. Mullins and A.W.D. Watson. Proc. Driven Turbulence". Institut for Fusion Studies. Report

lEE Int. Conf. RADAR-77. Oct. 25-28 (1977) 552. IFSR-174. Austin. Texas.[6] E.J. Powers. l.Y. Hong and Y.C. Kim. IEEE Trans. [25] R.E. Waltz and R.R. Dominguez. Phys. Fluids 26 (1983)

Aerospace and Electronic Systems AES-17 (1981) 602. 3338.. ~ ~ [7]1 JA. Pinkster. 1975 Society of Petroleum Engineering [26] D.R. Brillinger and M. Rosenblatt, in: Spectral Analysis

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Inlernaional Journal of Infrared and Milmeter 14ayes. Vol. 7. No. 3, 1986

THE APPLICATION OF HOMODYNE SPECTROSCOPYTO THE STUDY OF LOW-FREQUENCY

MICROTURBULENCE IN THE TEXT TOKAMAK

D. L. Brower, N. C. Luhmann, Jr., and W. A. PeeblesUniversity of California, Los Angeles

Los Angeles, CA 90024

and

Ch. P. Ritz and E. J. Powers

University of Texas, Austin

Austin, TX 78712

Received December 19, 1985

A new homodyne spectroscopy technique has been applied

for the first time to tokamak microturbulence measurements

in order to ascertain the frequency spectra and wave

propagation direction of low-frequency density

fluctuations. This method is employed in lieu of more

expensive and complicated heterodyne detection schemes

typically available for far-infrared laser scattering

systems.

Introduct ion

Laser and millimeter-wave scattering techniques are

commonly used to study the space-time statistics of

electron density fluctuations in tokamak and other plasmas

Of particular importance is the determination of the

direction of propagation of the fluctuations. Since the

scattering geometry fixes the direction of the scattering

wave vector k, the direction of propagation information is

carried by the sign of the fluctuation frequency w. The

fact that waves may be propagating both parallel and

447

0195.9271/l6/030-OU7 1O .00/O 1966 Piclum Pubbshmg Corpmalon

4 %

44 Brower el 61.

antiparallel to k is manifested by the presence of blue andred sidebands centered around the incident wave frequency0

To recover the propagation direction informationcontained in the blue and red sidebands. heterodynedetection techniques are typically employed. This approachrequires two coherent sources, with a frequency differenceAw. to be utilized as the incident and local oscillatorbeams. After mixing, the frequency range of the resultantsignal is A ± w, where w is the frequency associated withthe plasma fluctuations. As long as Aw >> w, the blue/redsidebands may be resolved. In contrast, if a classical

homodyne approach is used, A, = 0, and it is no longer Ipossible to unambiguously determine the wave propagationdirection.

Realization of a heterodyne detection system in thefar-infrared is expensive and technically nontrivial.Utilization of a rotating grating to frequency shift a

W,.. portion of the source beam is a feasible alternative

* although the frequency offset is limited to roughly Av <,150 kHz (insufficient for microturbulence measurementswhere fluctuations are observed up to I MHz). In addition,

there is noise associated with the grating which limitsresolution near zero frequency and fabrication can becostly.

ExRe entl Technque nd ARarUat.

A considerably simpler and inexpensive method proposedby Tsukishima I and Asada et al., permits detection of thewave propagation direction fr% the analysis of homodynesignals. The IF output of the scattered signal after themixer is a real quantity described by

v(t) = Re Jf_' d'N(k+,w)eit1,

where k+= k - k in the wave vector of the plasmafluctuation and Nk+ ) - n(k+,wd with i being the densityfluctuation level The sign of the frequency spectrumrepresents the propagation direction of the wave in the

Homody. Spentrocopy ia be TEXT Tokamak 409

laboratory frame of reference and thus N(k+.w) N(k+,-)However. from the real time signal v(t), one is restricted

to the reconstruction of the spectrum N'(k ,w) with thesyaetry property N'(k+.W) = N' (k+.-,), thereby losing

wave propagation direction information The key idea ofthe new homodyne spectroscopy technique proposed byTsukishimaI is to reconstruct the complex time signal w(t)

and recover the complete wave information by using twohomodyne IF signals vl(t) and v2 (t) which are phase shiftedby 900 with respect to each other permitting one to writew(t) = vl(t) + 3 v2(t)

The blue and red sidebands of the scattered radiation.S+(j) and S_(w) where S,(w) - [n.(w)] . can be readily

calculated from the two IF signals vTt) and v2 (t).

S,(w) = (GII(() + G2 2 (w)) M 2 lm(G 12 (w)), w > 0.

where GII(w ) and G2 2 (() are the auto-power and G12(w) the

cross-power spectral densities of the two signals vl(t) and

v2 (t), and are given by

Gik(a) = <V.()Vk(W)> ik = 1.2

The spectral component V,(w) = f(v (t) ) is the Fourier

transform of the IF signal v (t) and < '> denotes anensemble average over many statistically similar

realizations.

In the experimental results to be shown later, a timeseries of 32k data points (length of the time sample T a 16

ms at a sampling rate of 2 MHz) was subdivided into 128realizations of 256 data points (T' a 122 Ms By

employing a fast Fourier transform algorithm. the frequencyspectrum was obtained with a resolution of &./2n = I/T' - 8kJz Mixer and amplifier noise contributions could be

subtracted from the autopower spectra although-. signal-to-noise levels were sufficiently large so as to

'S..S make it unnecessary

In "A

'I.I

%' S.

4 Brower et l.

A schematic of the experimental arrangement emplSyedfor application of the homodyne spectroscopy technique tocollective far-infrared scattering is shown in Fig. I. Thesource beam utilized for the incident and local oscillatorradiation is a C I 3F far-infrared laser producing = 20 mWof power at 245 GHz (1.22 m). Detection is achieved by

1 Incident Beam

H- Plasma

*Scattered Beam

PhaseShifter

l ~ ~ o 'r , , (=/2)=' • I I r

'-,- Mixers

'4-------------B.S.

Local Oscillator

Source

Figure I Experimental arrangement for homodynespectroscopy measurement

o-p

do V "

::-

-,A g' ,-CC :: ' ' ' - ." ' " " . . " " : " " " " " ' ' ' , " , , ,- , , - ' " : " " " : " " ' " " " - , " " -, " " - . : ': i.,'I ,C,,, . , . . , , , ", .,',. "/ ., " K E .,, ' . ,. . . . , _, . .' .; . . .. .,'. . .L - ' ' ' - .'. . . " -, J ," . ' - " %

Homod)m Speciroscopy in Ih TEXT Tokamak 1

the use of quasi-optical biconnical Schottky barrier diodemixers. The frequency shifted scattered radiation may becollected over a range of wave vectors from 0 < kI < 15cm . Detailed information on the scattering syslem andcalibration procedures is described by Park et al . The

%" portion of the far-infrared laser utilized as the probe%- beam is weakly focussed along a vertical chord to a waist

of = 2 cm producing a measured wavenumber resolution 6k,

±1 cm -1 . The length of the scattering volume varies as afunction of wavenumber and ranies from ±8 cm (e- I point ofscattered power) at k = 12 cm to a chord average as k '0. Scattered radiation in the plane perpendicular to t he

"d toroidal magnetic field is examined.

At a particular wave vector k the scatteredradiation beam is divided equally into fwo components whichare coupled into detectors 1 and 2 by 50 % reflectivitybeam splitters. Similarly, the local oscillator beam is

equally divided to provide rf drive for each mixer. In oneleg of the local oscillator beam (det. 2), a phase shifteris inserted. This phase shifter consists of a piece ofhigh density polyethylene (excellent transmissioncharacteristics at 245 GHz) mounted on a rotation stage.

By tuning the rotation angle, the path of the LO beamthrough the polyethylene is altered thereby changing itsphase with respect to det. 1. This phase shifter is tunedsuch that there is a 900 phase difference between dets. Iand 2. The signal from each detector is then amplified anddigitized so that the cross- and auto- power spectra may be

computed.

* EXrimezLal ests

' The above described technique may now be applied to

density fluctuation measurements in a high temperaturetokwmak plasma. Microturbulence (low-frequency densityfluctuation) is driven by the free energy associated withplasma inhomogeneities such as density and temperature

-S gradients. For drift wave type fluctuations, it ispredicted that the turbulence will exhibit a phase velocity

%,- = Wje = [kBTe/eBTnel Vne x DOW , where iDe is the

electron diamagnetic drift velocity, k0 and w are thepoloidal wave vector and frequency of the fluctuation, BTis the toroidal magnetic field, Te is the electrontemperature, and ne is the electron density. The geometry

.... for scattering from electron drift waves is shown

-. % % %

I'. o Z 2 2',2 2 2""2 ] 2',2 / ~' 2''g' " , ,2 : ':'''2""''''',-£'". , . . .2". ""' ""

45~ Browfe c al.

schematically in Fig 2 The scattering system ispositioned such that the incident beam impinges upon theplasma from the top of the torus (also see Fig 1) along avertical chord at the major radius R = I m. This providesfor scattering from fluctuations with a poloidal wavevector, lie. Depending upon the orientation of thecollection optics with respect to the incident beam, thewavenumber matching condition (momentum conservation) gives

i= k ± 1i where ks. ko. ]i are the wave vectors of thescattered beam, incident beam, and plasma fluctuation,respectively. It is important to note that for aparticular scattering geometry, a sign change will occur(±1k) when one switches from the plasma top to bottom (seeFig. 2 (a) and (c) or (b) and (d)).

(a) ks -ko + (b ks =ko -k

.r..- k - kO

TOP

k kVn e

':-O-: k-o s -ko

BOTTOM

2-'.... (c ks ' k -W (d) _kS " [5 + 1i

Figure 2 Tokamak scattering geometry

Experimental results from the TEXT tokemak (majorradius R = Im and minor radius a = 27 cm) are shown in Fig3, for a scattering volume located at the plasma bottom

W

%

.- A.-,.A

..............................

." .

-. ,..one

Homody.- Spllrowopy II be TEXT Tokamak 453

".). a' b)

C

ionV~ e leto

C~'C

0 00 00 0 500 1000

1000,WU0[0 50 1000

d"[k az2 J (kzz]

Figure 3. Application of homodyne spectroscopy technique to

tokamak low-frequency microturbulence date. (a)J= homodyne signal from detect, or 1. (b) homodyne signal

spectroscopy, ks, = k° - k¢. and (d) frequency spectra. . .. using homodyne spectroscopy, kjs = Lo + k.

4- %%,%

S4 Om er et al.

with poloidal wave vector kQ = 7 cm- The dischartSparameters were I = 400 kA. BT = 28 kG, and ne = 2 x 10cm . In Figs 3 (a) and (b), the homodyne power spectraSk(w./2n) from the two mixers are illustrated. Each is

characterized by a broad spectra which falls-off in powerfor w/2n > 300 kHz. The homodyne spectra provide noinformation regarding wave propagation direction as the !wcomponents are detected as I±+II Now however, byimplementing the homodyne spectroscopy technique ofTsukishima . wave propagation direction information can beascertained as depicted in Fig. 3(c). Here, thefluctuations are observed to possess a clear peak at +w/2n250 ± 50 kHz in the electron diamagnetic drift direction

as measured in the laboratory frame of reference Thigindicates a fluctuation phase velocity vph(=/k) = 2 x 0cm/sec which is in the drift wave region of velocities.substantial component is also observed at w/2n < 0.corresponding to the ion drift direction. The scatteringgeometry for this measurement is oriented according to Fig.2(c), i.e s = -k. If we reverse the geometry tothat of Fig. 7d), i.e. 1 = Lo + k , one would expect achange in sign from the results of Fig. 3(c), which isindeed the case as shown in Fig. 3(d). The features ofthe scattered spectra are the same except that --w nowcorresponds to the electron drift direction.

The component of the frequency spectra correspondingto the ion drift direction may result from factors otherthan a true ion drift feature of the plasma On the TEXTtokamak. density fluctuations in the limiter shadow andscrape-off regions have been observed to propagate in theion drift direction due to a strong radial electric fieldinducing a plasma rotation effect Another possibility is

r that the interaction volume may extend to the opposite sideof the plasma thereby introducing components at tw althoughboth represent the same propagation direction This effectwill be described more thoroughly in the ensuing paragraph

- -. In Fig 4, the frequency spectra at wave vector k0 =

7 cm - are shown at three spatial positions along a

vertical chord through plasma center, scattering volume (L- ± 14 cm) positioned at the plasma top. midplane andbottlm The tokamak discharge conditions were I 300 kA.BT = 28 kG. and n = 3 1 I0 cm- At the plasma top (see

Fig 4(a)), the low frequency density fluctuations areobserved to be propagating largely in the electron

Pv?

% .V OL:' . .- *.- ..

LI' M:4:

Hadmew Setroacopy Is tie TWXT Tokamak 4SS

(a) TOP

k±.- 7cm1 electron

-i-on

14

-1000 -500 0 500 1000FREQUENCY kHz

(b)MIDPLANE• kIt - n I

7cm

'-. -1000 -500 0 500 1000FREQUENCY kHz

W(cBOTTOM-n1

k, 7cm;electron

Ion

"-S. -1000 -500 0 S00 1000

.% FREQUENCY kHz'..

Figure 4 Homodyne spectroscopy frequency spectra for

scattering volume centered at plasma (a) top, (b)

%.. mdplane. and (c) bottom

,%, *

J. .

% S

456 Brower teal.

diamagnetic drift direction with a peak at w/2n - -300 kHz.The fluctuations traveling in the ion drift direction are

typically at much lower frequency with no clear peak exceptat zero frequency. Likewise, at the plasma bottom (seeFig 4(c)). the fluctuations are again observed to bepropagating primarily in the electron drift direction withcj/2- = +300 kHz. The change in sign results from thereversal in direction of the fluctuations wave vector k0with respect to the fixed wave vectors k and k. of theincident and scattered radiation (see Fig. 2). When thescattering volume is centered on the midplane (see Fig.4(b)), fluctuatons are detected both above and below themidplane resulting in peaks at ± /2n. The wavepropagation direction can only be resolved if thescattering volume is situated completely above or below themidplane. Similar observations have been made at otherwavenumbers.

The wave propagation direction of microturbulence in atokamak plasma has been accurately measured by application

* of a new homodyne spectroscopy technique. This method hasbeen used in conjunction with a collective far-infraredlaser scattering experiment on TEXT. The low-frequencydensity fluctuations are observed to propagate primarily inthe electron diamagnetic drift direction, however, thebroadband spectra also possess an appreciable level offluctuations traveling in the ion drift direction.Application of the homodyne spectroscopy techniquerepresents an inexpensive and easily implementedalternative to the more technically demanding heterodyneschemes available in the far-infrared.

Acknowledgements

The authors wish to express their appreciation to theFusion Research Center, University of Texas at Austin, forgenerously providing machine time and operation for thesemeasurements. This work is supported by the U.SDepartment of Energy, Office of Fusion Energy undercontract No. DE-ACO5-78ET53043 and through subcontract no.UT-1-24120-52019 and the Department of Defense JointServices Electronics Program under contract no.F49620-82-C-0033.

04

% %. %

Hemedyme SPKrw"Y In The TEXT Tokamalk 47

)I] Tsukishima, T.. 0. Asada. Japan J. Appl. Phys.12.2059(1978).

(2) Asada. 0., A. Inoue, T. Tsukishima, Rev. Sea.Instrum. 51.1308(1980).

131 Tsukishima, T.. et al., Int. Conf. on Infrared andMillimeter Waves, Takarazuka. Japan, 1984.

[4] Park, H., CX. Yu. W A. Peebles, N.C. Luhmann, Jr.,R. Savage, Rev. Sci. Instrum. U3,1535(1982).

[5] Park, H., D.L. Brower, W.A. Peebles, N.C. Luhmann,Jr.. R.L. Savage, Jr., CX. Yu. Rev. Sci. Instrum.6,11055(1985).

[6] Brower, D.L.. W.A. Peebles. N.C. Luhmann, Jr., R.L.Savage, Jr, Phys. Rev. Lett. 5A,689(1985).

[7] Ritz, C.P., R.D. Bengtson, S.J. Levinson, E.J.Powers, Phys. Fluids ?a,2956(1984).

"'_1

".9

".4 4, Z ,".' . ., , . """"--""/ ,.,. '." / -. ., . ,,-'

Digital complex demodulation applied to interferometryD. W. Choi,' ) E. J. Powers, Roger D. Bengtson, and G. Joyceb)

The University of Texu. at Austin. Austin. Texas 78712

0. L. Brower, N. C. Luhmann, Jr., and W. A. Peebles

University of California. Los Angeles. California 90024

(Presented on II March 1986)

The objective of this paper is to describe the principles of digital complex demodulation, and tosummarize its advantages with respect to rapid time response and insensitivity to noise. Theseadvantages are demonstrated by application to interferometry data collected on the Texasexperimental tokamak (TEXT).

INTRODUCTION The objective of this paper is to describe the principlesInterferometry involves the measurement of the line-inte- and application of digital complex demodulation, which isgrnteideromrerctiolves t mesuemeInt ofy the-int me- the equivalent of digital heterodyning. The advantages of the

grated index of refraction of a medium. In any interferome- digital technique include an insensitivity to transient noise,ter a beam of coherent radiation is split, passed along two faster time response, and improved accuracy of the phasepaths (known as the reference and working arms), and re-mixed on a detector surface. Changes in the index of refrac- measurements.

tion of the medium along one of these paths, the working In the following sections, we describe the principles ofcomplex demodulation, our approach to avoiding 2n" phase

arm, alter the interference of the remixed beams, changing ambiguities, the effects of noise, and an application to inter-* ~the detector output. If' the variable medium is a plasma, mea- fe

surement of the detected signal leads to an estimate of the ometry data collected on the Texas experimental toka-mak (TEXT).line-averaged electron density along the working beam path.

In the simplest interferometer, the probe beam associat-ed with the working arm is mixed with a reference beamwhose phase is fixed. However, this simple scheme suffers I. DIGITAL COMPLEX DEMODULATION

from a severe limitation. It is not possible to distinguish Digital complex demodulation is a digital version ofbetween the cases of increasing and decreasing plasma den- analog heterodyne demodulation and allows the simulta-

sity. This shortcoming may be overcome by introducing a neous measurement of the amplitude and phase modulationssmall constant frequency shift w between the beams of the of a narrow-band modulated carrier signal as functions ofworking and reference arms. The detector output becomes a time. This technique has been shown to be effective and flexi-phase-modulated sine wave at frequency (o, with the relative ble compared with other demodulation techniques.' In thephase proportional to the line-integrated plasma density. following, we outline the key ideas of digital complex de-The phase of the detector output is obtained by comparison modulation.of this signal with a reference signal oscillating at the fre- A sampled narrow-band signal x(t, ) at a "carrier fre-quency too. In an electronic system, the relative phase of the quency" w,, whose amplitude and phase are modulated, maydetector signal with respect to the reference signal is ob- be described bytained via sine and cosine producing phase comparators.Electronic systems are attractive for the following reasons:the comparator output signals require only simple process- where A (t ) and 0(t. ) are the respective amplitude and

ing, allowing for real-time data reduction and the computer phase modulates at the time 1, = nAt. Equation ( I ) may be

signals need only be sampled at a low rate. There are also rewritten in the form

several disadvantages: the comparators are sensitive to var- x(t ) 1/2A (t, )(exp[ iw1 t + iO(t )]

iations in the amplitude of the detector and reference signals;the comparators may introduce large errors into the phase + exp{ - i[Wat. + O(t, ) ]}). (2)

calculation: and electronic systems have difficulty coping As in analog demodulation, we must downshift the frequen-with transient noise. As a result of this noise sensitivity, cy by an amount - w, In digital demodulation this down-fringes are often skipped or added. To counter the effects of shifting is accomplished by simply multiplying (in the com-noise, the time response of the electronic system is length- puter) the time series data representing Eq. ( 1 ) or (2) by aened (i.e., bandwidth reduced). Unfortunately, however, "local oscillator" component 2 exp( - iw t ). This results

this has the undesirable effect of"washing out" rapid fluctu- in difference and sum frequency terms. to,, - andations in plasma density associated with phenomena such as t,, + WoD, respectively. By choosing w , = o, the difference

sawtooth oscillations. frequency is set to zero,

19 Rev. Sct. Instrum. 57 (8), August 1986 0034-67481861081989-03S01.30 c 1986 American Institute of Physics 19to

.= .p= -% .% -.. - ]j. ,j -% . .%.% . .% = .%.% .% .. % -.. . . . . .,.... .•...% %" .;,'". " ,, ".".. ".". . , "% V.. ., _'. ,. ., .. ,. , ,. ,, . . .. . ,,.,.. . . . .. ,% .%, , ,., .. ,., ..

x(t. )2 exp( - iwtit ) the gain of the signal-to-noise ratio is f IfF In view of the

A (I. )exp[iO(t,)] + exp{ - i[ 2tot + 0(t)], fact that the low-pass filtering is being done digitally, it is arelatively simple matter to adjust the low-pass cutoff fre-

(3) quency to enhance the signal-to-noise ratio.Also, as in analog demodulation, a low-pass filter is aj. plied The error in the calculated phase modulate 0(t, ) de-to the signal. In this case a digital low-pass filter is applied pends on the signal-to-noise ratio SNR(t, ) of the filteredwhose passband to, is chosen to block the sum frequency signal,y(t,, ). To avoid errors in the calculated phase due to acomponent, wo, < to + (o, but to admit the inherent band- momentary low signal-to-noise ratio, a threshold value of

, .. width of the carrier due to phase modulation, w, > IdO(t)/ y(, ),v_.,,, is defined which depends on the amplitude of thedt I,,,. A linear digital low-pass filter, MAXFLAT, 2 is se- noise. When the signal amplitude drops so that ly(t, )Ilected for this purpose. This filter has a variable passband <y..., the calculated phase modulate 0(t, ) is ignored andand a variable transition region in which the filter response the previously calculated phase modulate is carried over. Ifdrops from 0.95 to 0.05. The amplitude modulation A(t, ) the noise is a transient, it is reasonable to neglect the phaseand phase modulation 0(t, ) are easily recovered from the measurement and assume the previous value of the phase asdigital filter output, y(t. ) =A(t, )exp[iO(t. )], the current value.

A(t) = [y(t ), (4) In most interferometers the shift frequency w, (referredM [t to as the carrier frequency in this section) is generally not

0(t. Im[y(tr,)] + 2NI( known with perfect precision and may also be unstable. The" Re [y(t. ) ] application of a constant digital local oscillator frequency

where N is any integer, shift, wto' which is not precisely equal to oo will thereforeresult in an accumulated phase error which increases linear-ly in time, (w, - w,o )t,,. Other sources of phase error in-

II. PHASE AMBIGUITIES AND NOISE clude phase distortions introduced by the low-pass filter anddeviations from a uniform sampling rate. To cancel these

As a result of the multivalued nature of the arctangent errors a reference signal x, (t), which equals the carrier sig-function, there is a 2r ambiguity in the phase demodulation; nal without the plasma phase modulation 0(t, ), is also sam-

-? we make two assumptions: (1) The original phase demodu- pled,late 0(t) before sampling is continuous. This assumption isreasonable ifA () never crosses zero, since the bandlimited- x, (t) = A, (t, )exp(iot),ness of the modulated signal x(t) ensures the continuity of Digital complex demodulation is then applied to this signal0(t). (2) The change in the calculated phase modulate as well, to yield a reference phase modulate 0, (t ). The de-between any two consecutive samples is less than ir. Thus, if sired phase modulate Op (t, ), due to the plasma is simply the

- one obtains a true, unambiguous value of 0(t, ), this condi- difference of these calculated phases,-[ tion guarantees an unambiguous value of (t, + ) by re--*:.:. stricting the range of possible values in the interval

0lt. of 0(t .in(tev )s +2r. III. APPLICATION TO INTERFEROMETER DATA

The length of this interval is 21T.We now show that the proper choice of the sampling We have applied digital complex demodulation to dat.

rate ensures that the phase shift between two samples is less from a 1.2-mm far-infrared laser interferometer on TEXT.J'-'k than r. The function x(t, ) is a product of A(t, ) and The detector and reference signals had carrier frequencies of

exp{i[wot, + 0(i, )]}. According to the convolution 80 kHz. These signals were sampled at 200 kHz for 325 ms.theorem, the bandwidth B, of x(t, ) is greater than the The digital low-pass filter, MAXFLAT, was set to a band-

* , bandwidth of either A (t, ) or exp{i[ wt, + 0(t, ) ] }. Thus, pass frequency of 20% of the carrier, resulting in 200 filterwe have IAO(t )/2rAt J._., < B.. This inequality may bewritten in terms of the Nyquist frequency f,v = (2A) -as"AO(t ) i,*, <r(B /f), If the sampling frequency is suf 30

ficiently high to satisfy the sampling theorem (B, ,< f, ), A- OCTAL COM t E DEMODUATION 1 2

eel. then JA0(t, ) 1,, < ir, and the problem of 21T phase ambigu- , '

ities is removed. 20.

'.ro' A time series to be demodulated may be contaminatedby noise which includes the quanti7ation noise at the A/D z -

conversion stage as well as electronic pickup and plasma -.noise. The multiplication of x(t, ) by exp( - to,, ) trans-lates the frequency contents of x(t, ) by -- ,, in the fre- 1

quency domain, but has no effect on the signal-to-noise ratio.However, low-pass filtering increases the signal-to-noise ra- °0 00

tio when the signal A(1, )exp[iO(t, )I is within the paiss- TIME Imsec

%, . band. If the noise is assumed to be white up to the Nyquist Fl(. I }'Lilsm, den! , timerdedu rd frodigital, dn nlodulated interfer-

. frequency f, and the low-pass cutoff frequency is f, then onleter data taken ... the II XT

SI10 Rev. Scl. Instrum., Vol. 57, No. 8. August 1986 IR and FIR 1990

.. . -. .

Note also that the time response of the complex demodula-,ad ..... tion approach is sufficiently fast to recover the Mirnov oscii-

I-, lations which are superimposed on the sawteeth oscillations.2 WTETH These examples demonstrate that digital complex de-

r, modulation possesses a number of advantages with respeci,.- ,E L TI to insensitivity to noise, faster time response, and phase mea-l-,ELECTRONIC DE:MODULATION surement accuracy. Also, the digital approach is inherently

flexible, allowing relatit ely easy adjustment of the lo -passfilter cutoff frequency and the amplitude threshold.,... Thelimitations of digital complex demodulation stem from the

0a o ISO 170 memory and computation requirements We t)picallh sam-TIME (msec) pie the 80-kltz detector and reference sign'als at 200 klHz for

FIG 2 Comparisonofsawtoothoscillationsobsered by electronicdemodu- 325 ms, yielding 130 000 samples. In addition. a 16-kHt di-lation and digital complex demodulation gital low-pass filter (20% ) contains 200 coetli -ients repre-

senting the filter impulse response. The resultin8 -onsolu-tion integrals require 13 min on a VAX 11/780 with a

coefficients which characterize the filter impulse response.The amplitude thresholdy,_,n for phase interpolation was setto 5% of the maximum signal level when free of noise. Figure probably reduce this time requirement considerabl

1 shows the calculated density for a typical shot. Note thatthe digital complex demodulation result faithfully tracks the ACKNOWLEDGMENTSdensity including the dip during the current rise at - 25 ms. This A.ork is supported b) U. S. DOE Contract DE-

Figure 2 compares data reduced via complex demodula- ACO5-78ET53043. The phase-sensitive complex demodula-tion with data from a standard interferometer utilizing elec- tion technique was developed under the auspices of the DODtronic phase comparators. Both data are from a portion of a Joint Services Electronics Program Contract F4%-o20-82-C-discharge during which "sawtooth" activity was occurnng. 0033.The digitally produced traces display the sawtooth shape(generally observed via other faster diagnostics) while the "Pre.ent address AT&T" Bell Laboratorc-s. N Ando',er. Massachusetselectronically reduced data resemble a sine wave, due to the ",Present address GTE, Inc, Waltham, Massal,hu,,enssmoothing resulting from a derated time response, the latter 'P. Y Ktonas anJ N Papp. Signal 'rocess 2. 3I3 (190)being necessary to avoid skipped fringes due to noise effects. 'J. F Kaiser and W A Reed. Re, Set Instrum 48, 144' (l"')

1991 Rev. SO. Instrum., Vol. 57, No. S, August 1986 IR and FIR 1991

" _. .. ... . , .. . . ...- .. .-, . . , ... . .-. .- . . .- -.-. .-.., . .. .. ., - - . .. -.-.-..% S. . . ... .

* Resolving the propagation direction of tokamak microturbulence viahomodyne spectroscopy

0 L Brower, W. A. Peebles, and N. C. Luhmann, Jr.

* Lnt'rtv o -California. Los Angeles. C-iliforniw 9tKX)24

Oh. P. Ritz and E_ J. Powers

L'nivenity of Texras. .4uiian. Texas 78712

(Presented on I I March 1986)

A ri % homody ne spectroscopy technique IT. Tsukishima and 0. Asada. Jpn. J. Appi. Phys. 17.205S) (11)7S) I has been applied to tokamak microturbulence measurements in order to resolhe thefrequency spectra and wave propagation direction of low-frequency density fluctuationsApplication of this method provides a high -resolution, inexpensive, and easily implementedalternatise to the more technically demanding heterodyne detection schemes typically availableComparison of heterody ne and the new homodyne spectroscopy results will be made.

*INTRODUCTION 1. EXPERIMENTAL TECHNIQUE AND APPARA TUSi

Laser and millimeter-wase scattering techniques are com- Ti~e IF output of the scatt-red si,:;ia t!,morly used to study the space-time statistics of electron den- real quantity described bsity fluctuations in tokamak and other plasmas. Of particu---dlar importance is the determination of the direction of t, (t) Re V-"Nprpagation of the fluctuations. Since the scattering geome-

tyfixes the direction of the scattering wave vector k, the where k -. =k, k, is i hcAs~is

direction of propagation information is carried by the sign of ain tdNA~the fluctuation frequency w. The fact that waves may be fluctuation le\,el The ozri!rIpropagating both parallel and antiparallel to k is manifested sents the propagiry ii d-- J

by the presence of blue and red sidebands centered around frame of refeteri~e tni

the incident wave frequency t1)0 fiowke\ er. from t lit (a'

To recover the propagation direction information con- the re:onrsrru I'l 1.tamned in the blue and red sidebands. heterodyne detection rictr~ pr,,peristechniques are typically employed. This approach requires V63%11 r tA ''

two coherent sourc,_s, with a frequency difference Ato, to be ne, outilized as the incident and local oscillator beams. After minu\ m~ ..

ing, the frequency range of the resultant signal is .1,ji-Liwherew is the frequency associated with the plasmia fl~j-uu '

W:ations. As long as the blue/red sidcbands, mnix be* resolved. In contrast, if a classical hornod~ li approia Ii is,

* used. 1(w = 0, and it is no longer possible to unamiiUOUs,determine the wave propagation direLtlOl

-- Realization of a heterody ne detection sN stern ilil hr* ~~infrared is expensive (e~g., tw o lasers) arid t- r ~i

* - *trivial [e.g.. intermediate frequeni % I ( 1 stahil rii

e mentation of a rotating grating to r un hit tthe source beam is a feasible alrerrrarp, ,I*,

411 ~ quency offset IS limited to ro0Uglil A i -H.1 cient for microturhulCe1t Tr(1Tt 711rrWrt'A

are observed upto I Milli;* rxj

expensive methcod proposed h% I ii ~'mats resolution of the % a, r ['r 1j.i'1

analy sis of homod~ neosgtisi I h; ,copy technique utilize-s is.

while requiring niinor nnriirji, 1!tem

0 1977 Rev Sci insirum t &.j

-Al77 348 JOINT SERVICES ELECTRONICS PROGRAM! BASIC RESEARCH tN SAXELECRRONICS (JSEP) (U) TEXAS UNIV AT AUSTINELECTRONICS RESEARCH CENTER E J POSERS 31 DEC 8

UNCLASSIFIED AFOSR-TR-87-8891-RPP F49628-86-C-fl45 F/G 9/5 ULEEEEEEEEEEElllll

" i j

L 6

- 6L 111 .8

11111L.25

MICROCOPY RESOLUTION TEST CHART

NAIIONAL BUPEAU OF STANDARDS 19(1 A

-p

density fluctuations) is driven by the free energy associatedwith plasma inhomogeneities such as density and tempera-ture gradients. For drift-wave-type fluctuations, it is predict-

v Incident Beam ed that the turbulence will exhibit a-phase velocity of ordervc, = cu/k = (k 8 T/eBrn,)Vn. XBr/IBI, where v,, isthe electron diamagnetic drift velocity, ke and a are the

S ,Plasma poloidal wave vector and frequency of the fluctuation, B, isthe toroidal magnetic field, T, is the electron temperature,and n, is the electron density. The scattering system is posi-tioned such that the incident beam impinges upon the plas-

SctedBma from the top of the torus (see Fig. 1) along a verticalScattered Beam chord at the major radius R = Im. This provides for scatter-

Phase ing from fluctuations with a poloidal wave vector. ke. De-Shifter pending upon the orientation of the collection optics with

% I I -- ( -w/ 2 ) respect to the incident beam, the wave-number matchingI % condition (momentum conservation) gives k, = k, ± k,

where k,, ko, and k are the wave vectors of the scatteredI Mixers beam, incident beam, and plasma fluctuation, respectively.

StB.S. Similarly, energy conservation dictates w, = wo ± c± , whereLocal Oscillatr .the subscripts have the same meaning as above. In addition,ocOcloit is important to note that for a specific scattering geometry,

a sign change ( ± k) will occur in the detected signal whenone switches from the plasma top to bottom for fluctuations %traveling in a particular poloidal direction.

SExperimental results from the Texas Experimental To-Source mkamak (TEXT) for a scattering volume located at the plas-

FIG. I. Experimental arrangement for homodyne spectroscopy measure- na bottom with poloidal wave vector k, = 7 cm' arement. shown in Fig. 2. The discharge parameters were I, = 400

kA,Br = 28 kG, and i, = 2X 10" cm - . In Fig. 2(a), the

transform algorithm, the frequency spectrum was obtainedwith a resolution of &w/217 = I/T'".8 kHz. Mixer and am- .. a) -

plifier noise contributions could be subtracted from the auto- ion electronpower spectra although signal-to-noise levels were suffi- •ciently large so as to make it unnecessary.

A schematic of the experimental arrangement employedfor application of the new homodyne spectroscopy tech- -__-

nique to collective far-infrared scattering is shown in Fig. 1.Detailed information on the scattering system is described

* by Park et ai.!At a particular wave vector k, the scattered radiation ,t _-

beam is divided equally into two components which are cou- 0pled into the detectors by 50% reflectivity beam splitters. 1002 0 [kHz]1000Similarly, the local oscillator beam is equally divided to pro-vide rf drive for each mixer. In one leg of the local oscillator 1beam a phase shifter is inserted. This phase shifter consists of b)a piece of high-density polyethylene (excellent transmission e nicharacteristics at 245 GHz) mounted on a rotation stage. By 1 e 0 io*tuning the rotation angle, the path of the LO beam throughthe polyethylene is altered, thereby changing its phase. Thisphase shifter is tuned such that there is a 90" phase differencebetween the detected signals in the two channels. The signal orfrom each detector is then amplified and digitized so that thecross- and auto-power spectra may e computed.

0II. EXPERIMENTAL RESULTS -1000 0 O 1000

The new homodyne spectroscopy technique will now be -lx (zapplied to density fluctuation measurements in a high-tern- Fio. 2. Application ofhomodyne spectroscopy technique to tokamak low-perature tokamak plasma. Microturbulence (low-frequency frequc.y microturbulence data; (a) k, k,, - k. and (b) k, k. -,- k.

1978 Rev. Ski. Instrum., Vol. 57, No. 8, August 1981 IR and FIR 19.

r L rL'....(' , _' ,. r ,. -. .-. , (..- .- , , %,. .". . ,. . ,. ,.' .' % .", " 5" ."". • ". "" "". "'-'' "- .""." " -- % . " "k ."''." ." " -' %' ' 7 . .'• -. . - " € -" - 4 , ... , , ,"'' ,', % " """"

bution from the scattered spectra and defining it as the newelectron ion zero, one can plot the fluctuation spectrum. The resolution

7cm near zero frequency is limited by the IF bandwidth which isk = 7cm- 1 - 20 kHz. As with the homodyne spectroscopy technique,

moving the scattering volume from plasma top to bottomresults in a change of sign for the fluctuation spectrum. Incomparing the results of Figs. 2 and 3, it is very evident thatboth the new homodyne spectroscopy and heterodyne detec-tion methods produce similar spectra. Any differences can

-1000 0 1000 be attributed to plasma discharge conditions. The homodyne- 4 "/2W (kHz) spectroscopy method provides improved resolution near

zero frequency ( 8 kHz).

Ip= 300 kA Ill. SUMMARY

Bt 28 kG The wave propagation direction of microturbulence in aho 2 x 1013cm - 3

tokamak plasma is measured by application of a new homo-k= 7 cm- 1 dyne spectroscopy technique. The accuracy of this method is

established by comparison with results from a heterodyne

FiG. 3. Heterodyne detection frequency spectra. scattering system which are similar. Both techniques showthe low-frequency density fluctuations to be propagatingprimarily in the electron diamagnetic drift direction: how-

fluctuations are observed to possess a clear peak at + , / ever, the broadband spectra also possess an appreciable level2ut275 ±" 50 kHz in the electron diamagnetic drift direc- of fluctuations traveling in the ion drift direction. Applica-tion as measured in the laboratory frame of reference. This tion of the homodyne spcctroscopy technique represents anindicates a fluctuation phase velocity v ( = o/ inexpensive and easily implemented alternative to the morek )-2 x ct s cm/s, which is in the drift wave region of e- technically demanding heterodyne schemes available in the"e)t_ 0 ms hc sinteditwv eino e far infrared.locities.3 A substantial component is also observed at n/2ir < 0, corresponding to the ion drift direction. If we reversethe geometry from that of Fig. 1, one would expect to see achange in sign, which is indeed the case as shown in Fig. ACKNOWLEDGMENTS2(b). The features of the scattered spectra are the same ex- The authors wish to express their appreciation to thecept that - w now corresponds to the electron drift direc- Fusion Research Center, University of Texas at Austin, fortion. generously providing machine time and operation for these

The component of the frequency spectra corresponding measurements. This work is supported by the U.S. DOE,tothe ion drift direction may result from factors other than a Office of Fusion Energy under contract No. DE-AC05-true ion drift feature of the plasma. On the TEXT tokamak, 78ET53043 through subcontract No. UT-1-24120-52019density fluctuations in the limiter shadow and scrape-off re- and the Department of Defense Joint Services Electronicsgions have been observed to propagate in the ion drift direc- Program under Contract No. F49620-82-C-0033.tion because of a strong radial electric field inducing a plas-ma rotation effect.

Heterodyne scattering results for ke = 7 cm-' underplasma conditions of 1, = 300 kA, BT = 26 kG, andfi, = 2 X 10" cm- are shown in Fig. 3 for scattering vol- 'T. Tsukishima and 0. Asada, Jpn. J. Appl. Phys. 17. 2059 (1978).

2H. Park. D. L. Brower. W. A. Peebles, N. C. Luhmann, Jr., R. L. Savage,ume positioned at the plasma bottom. The frequency differ- Jr., and C. X. Yu, Rev. Sci. Instrum. 56. 1055 (1985).ence between the local oscillator and incident beams (IF) is 3D. L. Brower, W. A. Peebles, N. C. Luhmann, Jr.. and R. L. Savage. Jr.

Aw/2yrr-n 1100 kHz. By subtracting the IF frequency contri- Phys. Rev. Lett. 54. 689 (1985).

1979 Rev. Sol. Instrum., Vol. 57, No. 8, August 1986 IR and FIR 1979

DIGITAL ESTIMATION OF LINEAR/QUADRATIC TRANSFER FUNCTIONSWITH A GENERAL RANDOM INPUT I

Kyoung II Kim and Edward J. PowersDepartment of Electrical and Computer Engineering

and Electronics Research CenterThe University of Texas at Austin

Austin, Texas 78712 USA

Presented at

IEEE International Conference on Acoustics,Speech, and Signal Processing 86

Tokyo, Japan

April 8-11, 1986

0t* 5',* ~ .'.,~-,-, - S " '~. ;5~ - 'S' ~ ' S ~ ' S

DIGITAL ESTIMATION OF LINEAR/QUADRATIC TRANSFER FUNCTIONSWITH A GENERAL RANDOM INPUT

Kyoung 11 Kim and Edward J. Powers

Department of Electrical and Computer Engineeringand Electronics Research CenterThe University of Texas at Austin

Austin. Texas 78712, U.S.A.

ABSTRACT tions can be evaluated in the discrete frequency

A new digital method of estimating linear and domain by solving matrix equations. In section 3,

quadratic transfer functions of a quadratic system a known system is analyzed to show the feasibility

with a general random input is presented. The of the analysis results. In addition, the results

feasibility of the technique is demonstrated by are compared with the transfer functions estimated

analyzing simulated data. It is also shown that by the "Gaussian input method" In order to Illus-

considerable error occurs in estimating the trate the deleterious effects of assuming a Gauss-

transfer functions based on a Gaussian input ian input when in reality it is not.

assumption, when in fact the input is non- 2. ESTIMATION OF TRANSFER FUNCTIONSGaussian.

Since we will concentrate on frequency domain1. INTRODUCTION analysis and the objective is to find a digital

method that can be practically implemented, weA difficulty encountered when one attempts to will start from the input-output relationship in

apply the Volterra functional series to nonlinear the discrete frequency domain. In the following,

problem I the measurement of the Volterra ker- we assume the unknown nonlinear system is of

nels [see, e.g. 13. So far, a fundamental assump- second order (i.e., quadratic) thus, higher order

tion underlying many approaches involves the fact terms may be safely neglected. Then the model to

that the "input" is assumed to be a stationary be studied can be expressed as follows;

random process which possesses Gaussian statis- I() - H (f )X(f ) (1)tics, an assumption which allows a substantialsimplification of the relevant mathematics. In M-1many practical cases, however, the input excita-tion Is not under the direct control of the exper- here X(f ) and Y(f rspectively represent theimentalist which precludes the use of the so- wherete and frespcivl reprsentecalled "probing' method, thus one must use the discrete miourier transforms (DFT for a finite

number (N) of observations of the input and theoutput signals of the nonlinear system described

It has been shown In [2] that, for a zero- by Volterra series up to second order. On theItan asben snput, iepre a for ae zner other hand, H (f ) and H (f f ) are linear and

mean Gaussian input, expressions for the linear quadratic transermfunction wkicA, aro given byand quadratic Lransfer functions are respectively the Fourier transforms of Volterra kernels at agiven in terms of various spectral moments up to discrete set of frequenciesthird order (i.e., the bispectrum). However. when (fn-n/N; n--(N-1)/2 .... ,-1,,1,...,N/2. It willa general random input is applied to the system, be assumed that the quadratic transfer functionit is extremely difficult to find such closed form H (f ,f ) is a symetric function of its argu-expressions for the transfer functions. Katzenel- Jntk, I.e., H (f fl)-H (flf)son and Gould (3) described an iterative method to 2 fk#)1 (ffk).

solve this problem, and Eykhoff [43 considered a Detemination of the linear and quadraticdiscrete time version. Also Ritz and Powers 5 transeer functions in terms of the input and out-

showed that when the input Is weakly non'Gausslan,transfer functions of a quadratic system can be put characteristics can be carried out by solving

the following set of equgtions which Ire ob&ainedestimated by an iterative approach in the discrete by multiplying (1) by X (f ) and X (fi)X (f ),frequency domin, respectively, and then takint an expected valueof

each side.It is the purpose of this paper to describe a

now digital method of processing input and output E[X*(f )Y(f )] - Hl(f.)£[IX(f, )12

signals in order to quantitatively measure thelinear and the quadratic transfer functions even + I t H2(f fl)E[X((f )X fl) (2)when we cannot amume a particular characteristic 2 fksfI a k (of the input. In the next section, it is shown

*## that the linear and the quadratic transfer fune-

2

EEX*(f I)X*(f YtMfm)] H1 (r )E[X' Cf I X(f )X ))lfI I H 2 f k'fl E[X (f I X (f Xi f k Mf 03 (3)

Note that (3) Is meaningful only whenDf*-f of olf f~ because of the properties at higher

ordO slecralmoments [6). X

When the system Input is zero-mean Gaussian, fthe terms containing the third order moment at the -fM fminput In (2) and (3) vanish. In this case, theDlinear and the quadratic transfer functions can bedetermined separately and expressed by the variousspectra up to third order [2]. More specifically,they are given by Xn

H(M) E(X (f B)Y(fa)]() f

E(IX(f.)12JEDO~ )X(f f fFig. 1 Two .dimensional frequency

H ff)k I k'I k10 5 domain.H2 (fl ko I~.O (5)~f)"ECXf

However. for the case of a general Input, we have H (C*t -to solve (2) and (3) simultaneously so that It Is H E(Ct) IE(XY(fa)) (8)extremely difficult to find the closed form solu-- m

Stions like (4) and (5). If E[XXItJ Is not singular. The solution given by(8) can also be considered a~s a result of mul-

or Next, we will describe a method of solving tivariate linear regression analysis and thus the*(2) and (3) which can be digitally Implemented, transfer functions obtained In this manner are

Due to the symetricity assumed for H (f of ). we optimum In the man jsusare sense when there Is anycan express the output only In te o~ t e per- additive noise present In the output. Note thattions of the quadratic transfer function which are ED X I Is a Hermitian matrix consisting of varl-In the sun and difference Interaction regions of cus -spectral moments of the Input signal. Exceptthe two dimensional frequency domain (regions S for the first element, the first row and column ofand D In Fig. 1). Using this fact and expanding this matrix represent the bispectrum Wiile* thethe summation term In (1). we can rewrite (1) In first element Is the auto power spectrum of thethe following vector form; Input. The remaining elements are fourth order

where t denotes transposition, and

(HI(fm)p 2H2(f +,f 01),...,2H 2(fa f) .. ,2H 2(fmgfal],friod

CH IH(fS), H 2(f m fma) 2H 2(fm 0~ 1 fS1)..2 1rl .,H2(Nf ] for mnevn

tXf),Xf )X(f m-) .. X(rmf 0 '~ Xf- ,fri d

-* (~~1 X(fm) X m . ui ' ),o~ .X~ M Xf o d

t2 2X Mf n) X(f m)XfmIs) *X(fm0 )X(fm1 )e*..9X(fm)X(fo)* ...,X(fM)X(fM)N]. for m even.

S7 rl 7-1In (6), f signifies the Nyquist frequency assiated Wigh the sampling of the Input and outputsignals. Then solvine simultaneously (2) and (3) spectral moiet of the input. The size of theis equivalent to solving the following matrix matrix E[X X I to be inverted depends on theequation; number of data points (N-2t4 taken for the Dr's

and the frequency Index m. For example, It Is aErx~Y(fa)i *E(XX)H. (7 (N+21v/1 )K(N.2- 1 matrix If m is even, and It Is

(N .)X(N*2- 1-) when a Is odd. Therefore theEquation ()Is linear in te transfer function size decreases ai m Increases. The largest ane Isvector N, and so Hi Is given by (1*2)x(fl.2) when a-0, and the Smals one Is

(y#)x(r2)when a-l- or M.

If the Input is zero-m"ay Gaussian, It can be two methods. In this case, the mean square errorsshown that the matrix E(X X I becomes a diagonal have been calculated as follows; by the Gaussianmatrix and thus the transfer- functions given by Input method, MSE19.5. MSE -4.0, and by the gen-(8) have the same expressions for the linear and eral input method, MSE1 -O. 06q SE -0.04. Compar-quadratic transfer functions as those given by (4) Ing the mean square error values as well as theand (5). Consequently the solution given by (8) plots, one can clearly see the obvious differencesis a general one for the transfer functions of a between the transfer functions estimated by thequadratically nonlinear system with an arbitrary two methods. This indicates that, in order torandom Input, i.e., It Includes the zero-mean obtain useful estimates, one must use the newGaussian input as a special case. method developed in this paper when there Is not

any good a priori knowledge about the input signal3. ANALYSIS OF SIMULATED DATA statistics or when the input is not a zero-mean

Gaussian signal.FORTRAN programs have been written on the

basis of the analysis results described in the 4. CONCLUSION" previous section and tested by analyzing simulated

data generated by a known system. In this paper. we have discussed the problemof estimating system transfer functions by pro-

The known system is formed as follows; cesing random input and output data, and2 2 described a new digital estimation method whichy(t) - -O.64x(t) * x(t-2) * 0.9x (t) + x (t1).(9) can be successfully applied to the modeling of a

quadratically nonlinear system excited by a non-Therefore the transfer functions are given by Gaussian input. Analyzing simulated data, we have

14wf also demonstrated the feasibility of the tech-H (f) - -0.64 + e-1 (10) nique. Considering the computation time and theIi21r(f1+) size of the processor memory available for practi-H2(r1,f2) - 0.9 + e 2 cal applications, It Is desirable that the number

of data points in each segment of record data bejnot too large since it determines the dimensionsThese actual transfer functions of the given y- of the matrices that must be inverted In order to

term are shown in Fig. 2 and Fig. 3. solve the matrix equations.

The approach developed in this paper has been ACKNOWLEDGEMENTtested by applying a zero-mean exponentially dis-tributed Input signal which is generated by IMSL This work was supported by the Department ofoutine. For both the Input and output signals, Defense's Joint Services Electronics Program128000 sample points of data record have been gen- through the Air Force Office of Scientificerated, and they were divided up into 2000 seg- Research Contract F49620-82-C-0033.ment of 64 data points each.

For a quantitative measure of the quality of REFERENCES

the estimates, the normalized mean square errors [1] G. Hung and L. Stark, "The Kernel Identifica-involved in the computed transfer functions are tion Method - Review of Theory, Calculationdefined as follows; to ehd -Rve fTerCluain

n aApplication, and Interpretation," Math Biosci.,N |H (f )-A (f )12 37, pp. 135-190, 1977.

MSE 1 1 1 m (12) (2) L.J. Tick, "The Estimation of Transfer Func-1 M-1 IHl(f )12 tions of Quadratic Systems," Technometrics, Vol.

1 M 3, No. 4, pp. 563-567, Nov. 1961.H(f ))2 (31 J. Katzenelson and L.A. Gould, "The Design of

1 k' 1) 2(fkf Nonlinear Filters and Control Systems, Part I,"* ,' MSEq . k 1 I f 12 (13) Inf. and Control 5, pp. 108-143, 1962.

2Ck'r [4 P. Eykhorf, "Some Fundamental Aspects of Pro-. . cess Parameter Estimation," IEEE Trans. Auto.

'a. (kl D Cont., AC-8, pp. 347-357. Oct. 1963.[5 Ch.P. Ritz and E.J. Powers, "Estimation of

whe sNonlinear Transfer Functions for Fully Developed,'.where "signifies an estimated quantity, and N~i Tublne"FC Ns7,Fso Rsac etr: )the number of frequency points In the regioRS S Turbulence." FRCR No. 273, Fusion Research Center.

then mbr oThe University of Texas at Austin; accepted forand D. publication in Physica D, 'Nonlinear Phenomena',

By using the input and output signals gen- 1985.%: erated above, the estimation of the linear and [6) D.R. Brillinger and M. Rosenblatt, "Asymptotic

quadratic transfer functions have been carried out Theory of Estimates of K-th Order Spectra," infirst by the Gaussian input method (Eqs. () and pectral Analysis of Time Series Ed. by B. Harris,(5)), and then the results are compared with the pp. 189-232, Wiley, New York, 1967.estimates that have been obtained by the generalinput methoddeveloped in this paper (Eq. (8)).Fig. 4 and Fig. 5, respectively, show the linearand quadratic transfer functions estimated by the

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162 IEEE TRANSACTIONS ON FIFCTRoMAGNFTIC COMPATIBILITY, VOL FMC-28. NO. 3. AUGUST 1986

Correspondence

Identification of Nonlinear Systems in the Walsh II. DEVELOPMENT

Sequency Domain Now we consider the space of real nonnegative-coordinate signals.

SIL K AThus the input %If) is defined for t a 0. Following Barrett'sKOUNGIEDKIM. JAE OUNGS. HO. EM R. IEEE. ANDw Iorthogonal representation, we can express the output y(t) of an

EDWARD J. POWERS. JR.. FELLOW. IEEE unknown NLDI system for a zero-mean dyadic stationary 131. 181Gaussian input process x(t) as follows:

Abstract-A method of measuring transfer functions for nonlinear

dyadic-invariant (NLDI) systems described by Barrett's orthogonal modelis discussed. In particular, this paper develops the expressions for the y(1)=ho+3 h1(tj)x(t * ti) di,transfer functions up to third order, and the results show that the transfer

%. functions can be obtained from the raw input and output data bycomputing the appropriate Walsh sequenc power spectra. + 2 , , )(x(f * f)x(t a 1)

Key (lords-Nonlinear systems, Walsh sequency domain, transfer 0 0h

functions.Index Code-P2d. K2d. -Elx(, * 1)x(t * t2)]} d 1 dr,

,. INTRODUCTION + ( ) x

Ever since J. L. Walsh published a complete set of orthogonalfunctions I 1. Walsh functions have been one of the most importantexamples of nonsinusoidal functions in engineering applications. The - x(t 6 t)Elx(t e t)x~t t3)]computational efficiency of the fast Walsh transform (FWT) and the -x(t t2)Ejx(t O 10)feasibility of its software and hardware implementation provide theincentive to find useful applications. On the other hand. motivated by -x(ts 13)E[x(14 tt)x(tD 12M)} dI dt2 d1+ (A)the fact that the Walsh functions form the natural basis forrepresenting dyadic-invariant systems just as trigonometric functions where h is the nth order kernel. El'1 denotes the expectationdo for time-invariant systems. some studies have been carried out to operator, and the operator a signifies modulo-two addition (withoutfind useful properties of the dyadic-invariant systems. For example, carry) of the two real numbers involved. In the following context, andan optimal linear dyadic- ivariant (LDI) system was developed 121. without the loss of generality, we assume that the kernels arethe problem of modeling a multiple input/output LDI system in terms symmetric functions in their arguments since it can be seen that theof sequency transfer functions was considered 131. and Walsh series output y(t) would be identical for any permutation of the arguments.expansions were utilized to develop a method of measuring the The Walsh transform of (i)is given bykernels up to order two in Wiener's nonlinear system model 141.

In this study, we present a method of measuring kernels for theidentification of a class of nonlinear systems which can be described Y(o) = WI y(t)) = y(t)*(o, 1) dtby Barrett's model. Since we let the model have dyadic convolutionoperations, we refer to these systems as nonlinear dyadic-invariant(NLDI systems. In particular, we will obtain the expressions for =h06(o) + 3 H(o)X(o)6(o e a,) do,kernels up to third order using Walsh transform techniques. Thus thisstudy can be considered as an extension (through a different T ( C oapproach) of the results in 141. and also a parallel study to that which + H0(oi, o){ X(oi)X(o0)has been carried out by Hong et al. in the Fourier frequency domain15.-EIX(o)X(o 2)I}5(o 0 a,0 2 do, do2J151+O 2

%" , By orthogonalizing the Volterra functional series for a whiteGaussian input signal. Wiener provided a representation for nonlinear + H3(o, 02, 03){ X(o)X(o2)X(0 3 )systems 161. In many practical situations, however, the input to thesystem is usually nonwhite. In such cases, Barrett's orthogonal model X(oOE[ X(o2)X(a3)j - X(02)E[X(o3)X(vj]171 for nonlinear systems is a useful one since it is valid for Gaussianinputs with arbitrary spectral densities. -X(ov 3)E[X(oi)X(o 2 )

Manuscript received September 6. 1985. revised February 28. 1986. This 6(o o 01 4 0 2 62 o3) do do 2 do(3 + (2)work was supported by the Department of Defense Joint Services ElectronicsProgram through the Air Force Office of Scientific Research Contract where W[. signifies the Walsh transform operator, I(o, t)denotesF4%620-82-C-OO33.

K I Kim and E J Powers. Jr are with the Electronics Research Center a generalized Walsh function 181, and H.(oi, a2, , o) and X(o)and the Department of Electrical and Computer Engineering, University of are Walsh transforms of the nth order kernel and the input,Texas at Austin. Austin. TX 78712. (512) 471-6179. respectively. We refer to H,,(a, a2, • • •, a,) as the nth order transfer

J Y Hong was with the Electr(mics Research Center and the Department of function in the Walsh sequency domain. Note that the nth orderElectrical and Computer Engineering. University of Texas at Austin. Austin. transfer function has the same symmetry property as that of the nthTX 78712 He is now with Intelligent Signal Processing, Austin. TX 78752.(512) 495-6729. order kernel. In (2), the delta function 6(a) is the Walsh transform of

IEEE Log Number 86011940 the unit step function, i.e., 6(a) = J*(a, 1) dt, and it has the

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IIEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY. VOL EMC-28. NO. 3. AUGUST 1986 163

following properties 181 which are similar to those of the Dirac delta obtain (3)-(5) is equivalent to the approach in which we minimize thefunction: mean-square error between the output of the actual system and the

f(t)b(t * t,) dt=f(t,) model 112], 1131. Therefore, considering y(l) in (I) as the estimate ofthe desired signal and x(t) as the signal immersed in noise, thetransfer functions developed in this paper would specify the nonlinear

1 filter which is optimum in the mean-square sense 113). 114].

To •n) dt=l. for > 0. Although, due to the lack of a simple relationship between arithmeticconvolution and logical convolution, the existence of natural systems

To get (2) from (I). we have utilized the product rules of generalized that can he modeled by (I) is open to question, we can extend theWalsh functions which are expressed as results developed in this paper so as to design an optimal NLDI filter

which can be used as an alternative to a time-invariant filter, Thei(a-, 1)*(o:. ) *(a] 0 a implementation as well as the design of such filters in the Walsh

'(o. ti)*I(o )= */(0, * 0 12). sequency domain would be very efficient.

By averaging (2) and then integrating, h(I is readily obtained such REFERENCESthat II I J. L. Walsh. "A closed set oforthogonal functions." Amer. J. Math.,

vol. 44, pp. 5-24, 1923.h0 = El Y(o)I do. 121 F. Pichler, "Walsh functions and optimal linear systems." in Proc.Jo 1970 Symp. Applications of Walsh Functions (Washington, DC),

Mar. 31-Apr. 3. 1970. pp. 17-22.Since we assume that the transfer functions arc symmetric 131 S. Cohn-Sfelcu and S. T. Nichols. "On the identification of linearfunctions of their arguments. other transfer functions up to order n, dyadic invariant systems." IEEE Trans. Electromagn. Compat.. vol.i.e.. HnOl, 02,"', o) may be obtained by multiplying (2) by EMC-17. pp. Ill -117. May 1975.X(o )X(o ) "" X(a ). respectively. and then taking an expected 141 A. S. French and E. G. Butz. "The use of Walsh funclions in theXclefach s aue oWiener analysis of nonlinear systems." IEEE Trans. Comput., vol.value of each side. Because of the Gaussian nature of the input C23 .222.Mr194C-23, pp. 225-232, Mar. 1974.process and the orthogonality of the functional series in (2), only one 151 J. Y. Hong. Y. C. Kim, and E. J. Powers. "On modeling the nonlinearterm containing H,(o, . . ., a') remains in calculating El Y(o)X(o I) relationship between fluctuations with nonlinear transfer functions,"

X(o )J. In particular, the transfer functions up to third order are Proc. IEEE. vol. 68. pp. 1026-1027, Aug. 1980.given as follows: 161 N. Wiener, Nonlinear Problems in Random Theory. Cambridge.

MA: MIT Press. 1958.H.,5 (,a-) (3) 171 J. F. Barrett. "The use of functionals in the analysis of nonlinearH I(O r(a) (3) physical systems." J. Electron. Contr., vol. 15. pp. 567-615, 1963.

181 M. Maqusi. Applied Walsh Analysis. London, U.K.: Heyden,I (f' ,..,(a, az)l-h0Fx(aa)6(o ) ) 1981.

HAU, a2)= - 0()'()(4) 191 G. S. Robinson, "Discrete Walsh and Fourier power spectra." in2)Proc. 1972 Symp. Applications of Walsh Functions (Washington,

DC), Mar. 27-29. 1972, pp. 298-309.1 (a2, 03) H 1) 1101 N. Ahmed and T. Natarajan. "On logical and arithmetic autocorrela-Hilo.( or', [13 =) (02 * o0) tion functions," IEEE Trans. Electromagn. Compat.. vol. EMC-16.3, .( ) (a)r,(a2)Fr(o,) ((73) pp. 177-183. Aug. 1974.

11 J. S. Bendat and A. G. Piersol, Engineering Applications ofHl(° 2 ) 6 )Ha( ° ) '] Correlation and Spectral Analysis. New York: Wiley. 1980.

+ 40(o * 3) + [ Mal * 02) (5) (121 P. Eykoff, "'Some fundamental aspects of process-parameter estima-r.(02) Jtion," IEEE Trans. Automat. Contr., vol. AC-8, pp. 347-357, Oct.

1963.In (3)-(5), 1T(a) is the Walsh sequency power spectral density 113] T. Koh and E. J. Powers, "Second-order Volterra filtering and its

function of x(t), and 1y7, .,(ot, U2, "'a,,) is the nth order Walsh application to nonlinear system identification," IEEE Trans. Acoust.,cross spectrum such that r,(o)6(o a ,n) = E[X(o)X(O1 )1. and Speech, Signal Processing. vol. ASSP-33, pp. 1445-1455. Dec. 1985.Iy7 x...x(VI, '', o,)00(O 0 02 * ' • a. 0 o) = 1141 J. Katzenelson and L. A. Gould, "The design of nonlinear filters and

E[ Y()X(00X(02) ''' X(o,)]. Note that the nth order Walsh control systems, Part I." Inform. Contr. 5, pp. 108-143. 1962.

spectrum can be defined as the Walsh transform of the nth orderlogical correlation function, and, in general, it is not directly relatedto the Fourier spectrum. In the case of the second-order moment, therelationship between discrete Walsh and Fourier power spectra andtheir computation was discussed in 191, and the algorithmic propertiesof logical and arithmetic autocorrelation functions were investigated~in [101.

We can see that the resulting expressions for the kernels up to third

order have much the same form as those obtained in the Fourierfrequency domain (4), except that they are expressed in terms ofWalsh instead of Fourier spectra. This result implies that we may takeadvantage of the computation procedure with which we are familiarin Fourier analysis [Il]; that is, the nth order transfer functionH.(oa, 02, ''", a,,) can be obtained from the raw input and outputtime-series data by computing the appropriate power spectra usingthe fast Walsh transform algorithm.

Finally, we note that the statistical approach used in this paper to

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