1

19. Series Representation of Stochastic ProcessesGiven information about a stochastic process X(t) in

can this continuous information be represented in terms of a countable set of random variables whose relative importance decrease under some arrangement?

To appreciate this question it is best to start with the notion of a Mean-Square periodic process. A stochastic process X(t) is said to be mean square (M.S) periodic, if for some T > 0

i.e Suppose X(t) is a W.S.S process. Then

Proof: suppose X(t) is M.S. periodic. Then

,0 Tt

(19-1). allfor 0])()([2

ttXTtXE

( ) ( ) with 1 for all .X t X t T probability t

)(PILLAI

( ) is mean-square perodic ( ) is periodic in the

ordinary sense, where

X t R

* ( ) [ ( ) ( )] R E X t X t T

2

But from Schwarz’ inequality

Thus the left side equals

or

i.e.,

i.e., X(t) is mean square periodic.

. period withperiodic is )( TR

2 2 2*1 2 2 1 2 2

0

[ ( ){ ( ) ( )} ] [ ( ) ] [ ( ) ( ) ]E X t X t T X t E X t E X t T X t

* *1 2 1 2 2 1 2 1[ ( ) ( )] [ ( ) ( )] ( ) ( )E X t X t T E X t X t R t t T R t t

Then periodic. is )( Suppose )( R

0)()()0(2]|)()([| *2 RRRtXtXE

( ) ( ) for any R T R

.0])()([2 tXTtXE (19-2)

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(19-3)

*1 2 2[ ( ){ ( ) ( )} ] 0E X t X t T X t

3

Thus if X(t) is mean square periodic, then is periodic and let

represent its Fourier series expansion. Here

In a similar manner define

Notice that are random variables, and

0

0

1( ) .

T jnn R e d

T

0

0

1( )

T jk tkc X t e dt

T

kck ,

(19-5)

PILLAI

(19-6)

)(R

00

2( ) , jn

nR eT

(19-4)

0 1 0 2

0 1 0 2

0 2 1 0 1

* *1 1 2 22 0 0

2 1 1 22 0 0

( ) ( )2 1 2 1 1 0 0

1[ ] [ ( ) ( ) ]

1( )

1 1[ ( ) ( )]

m

T Tjk t jm tk m

T T jk t jm t

T T jm t t j m k t

E c c E X t e dt X t e dtT

R t t e e dt dtT

R t t e d t t e dtT T

4

i.e., form a sequence of uncorrelated random variables, and, further, consider the partial sum

We shall show that in the mean square sense as i.e.,

Proof:

But

0 1

,

1 ( )*1 0

0, [ ] { }

0 .m k

T mj m k tk m m T

k mE c c e dt

k m

(19-7)

nnnc }{

0( ) .N

jk tN k

K N

X t c e

)()(~ tXtX N

2 2 *

2

[ ( ) ( ) ] [ ( ) ] 2 Re[ ( ( ) ( )]

[ ( ) ].

N N

N

E X t X t E X t E X t X t

E X t

(19-8)

.N

. as 0])(~)([2

NtXtXE N(19-9)

(19-10)

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5

0

0

0

2

* *

( ) *

0

( )

0

[ ( ) ] (0) ,

[ ( ) ( )] [ ( )]

1 [ ( ) ( ) ]

1[ ( ) ( )] .

k

kk

Njk t

N kk N

N T jk t

k N

N NT jk tk

k N k N

E X t R

E X t X t E c e X t

E X e X t dT

R t e d tT

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(19-12)

Similarly

i.e.,

0 02 ( ) ( )* *

2

[ ( ) ] [ [ ] .

[ ( ) ( ) ] 2( ) 0 as

Nj k m t j k m t

N k m k m kk m k m k N

N

N k kk k N

E X t E c c e E c c e

E X t X t N

(19-13)

0( ) , .jk tk

k

X t c e t

(19-14)

and

6

Thus mean square periodic processes can be represented in the form of a series as in (19-14). The stochastic information is contained in the random variables Further these random variables are uncorrelated and their variances This follows by noticing that from (19-14)

Thus if the power P of the stochastic process is finite, then the positive sequence converges, and hence This implies that the random variables in (19-14) are of relatively less importance as and a finite approximation of the series in (19-14) is indeed meaningful. The following natural question then arises: What about a general stochastic process, that is not mean square periodic? Can it be represented in a similar series fashion as in (19-14), if not in the whole interval say in a finite support

Suppose that it is indeed possible to do so for any arbitrary process X(t) in terms of a certain sequence of orthonormal functions.

*,( { } )k m k k mE c c 0 as .k k

. , kck

2(0) [ ( ) ] .k

k

R E X t P

kk

0 as .k k

,k

, t 0 ?t T

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7

i.e.,

where

and in the mean square sense

Further, as before, we would like the ck s to be uncorrelated random variables. If that should be the case, then we must have

Now

1

)()(~

nkk tctX (19-15)

(19-16)

(19-17)

( ) ( ) in 0 .X t X t t T

*,[ ] .k m m k mE c c (19-18)

* * *1 1 1 2 2 2 0 0

* *1 1 2 2 2 1 0 0

*1 1 2 2 2 1 0 0

[ ] [ ( ) ( ) ( ) ( ) ]

( ) { ( ) ( )} ( )

( ){ ( , ) ( ) }

T T

k m k m

T T

k m

T T

k XX m

E c c E X t t dt X t t dt

t E X t X t t dt dt

t R t t t dt dt

(19-19)PILLAI

*

0

*, 0

( ) ( )

( ) ( ) ,

T

k k

T

k n k n

c X t t dt

t t dt

8

and

Substituting (19-19) and (19-20) into (19-18), we get

Since (19-21) should be true for every we must have

or

i.e., the desired uncorrelated condition in (19-18) gets translated into the integral equation in (19-22) and it is known as the Karhunen-Loeve orK-L. integral equation. The functions are not arbitrary and they must be obtained by solving the integral equation in (19-22).They are known as the eigenvectors of the autocorrelation

*1 1 2 2 2 1 1 0 0

( ){ ( , ) ( ) ( )} 0.XX

T T

k m m mt R t t t dt t dt

1 2 2 2 1 0( , ) ( ) ( ) 0,

XX

T

m m mR t t t dt t

( ), 1 ,k t k

*, 1 1 1 0

( ) ( ) .T

m k m m k mt t dt (19-20)

(19-21)

1 2 2 2 1 1 0( , ) ( ) ( ), 0 , 1 .

XX

T

m m mR t t t dt t t T m

1)}({ kk t

(19-22)

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9

function of Similarly the set represent the eigenvaluesof the autocorrelation function. From (19-18), the eigenvalues represent the variances of the uncorrelated random variables This also follows from Mercer’s theorem which allows the representation

where

Here and are known as the eigenfunctions and eigenvalues of A direct substitution and simplification of (19-23) into (19-22) shows that

Returning back to (19-15), once again the partial sum

1 2( , ).XX

R t t

*1 2 1 2 1 2

1

( , ) ( ) ( ), 0 , , XX k k k

k

R t t t t t t T

(19-23)

*, 0

( ) ( ) .T

k m k mt t dt

1 2( , ) respectively.XX

R t t)(tk ,k

k( ) ( ), , 1 .k k kt t k

1

( ) ( ) ( ), 0N

N

k k Nk

X t c t X t t T

PILLAI

(19-24)

(19-25)

1{ }k k

k,kc

1 .k

1k

10

in the mean square sense. To see this, consider

We have

Also

Similarly

* * *

1

* *

01

*

01

*

1

[ ( ) ( )] ( ) ( )

[ ( ) ( )] ( ) ( )

( ( , ) ( ) ) ( )

( ) ( )= | (

N

N k kk

N T

k kk

N T

k kk

N

k k k k kk

E X t X t X t c t

E X t X t d

R t d t

t t

2

1

) | .N

k

t

(19-26)

N

kkkN ttXtXE

1

2* |)(|)](~)([ PILLAI

2 2 *

* 2

[| ( ) ( ) | ] [| ( ) | ] [ ( ) ( )]

[ ( ) ( )] [| ( ) | ].

N N

N N

E X t X t E X t E X t X t

E X t X t E X t

2[| ( ) | ] ( , ).E X t R t t (19-27)

(19-28)

(19-29)

11

and

Hence (19-26) simplifies into

i.e.,

where the random variables are uncorrelated and faithfully represent the random process X(t) in provided satisfy the K-L. integral equation.

Example 19.1: If X(t) is a w.s.s white noise process, determine the sets in (19-22).

Solution: Here

2 * * 2

1

[| ( ) | ] [ ] ( ) ( ) | ( ) | .N

N k m k m k kk m k

E X t E c c t t t

. as 0|)(|),(]|)(~)([|1

22

N

kkkN tttRtXtXE

1

( ) ( ), 0 ,k kk

X t c t t T

1}{ kkc

0 ,t T ( ),k t

(19-30)

)(),( 2121 ttqttRXX PILLAI

(19-31)

(19-32)

1{ , }k k k

(19-33)

1 ,k

12

and

can be arbitrary so long as they are orthonormal as in (19-17)and Then the power of the process

and in that sense white noise processes are unrealizable. However, if the received waveform is given by

and n(t) is a w.s.s white noise process, then since any set of orthonormal functions is sufficient for the white noise process representation, they can be chosen solely by considering the other signal s(t). Thus, in (19-35)

2

1 1

[| ( ) | ] (0) kk k

P E X t R q

)()()( 212121 ttqttRttR ssrr

.1 , kqk

( ) ( ) ( ), 0r t s t n t t T

)(tk

PILLAI

(19-34)

(19-35)

(19-36)

1 2 2 1 1 2 2 1 0 0

1 1

( , ) ( ) ( ) ( )

( ) ( )

XX

T T

k k

k k k

R t t t dt q t t t dt

q t t

13

and if

Then it follows that

Notice that the eigenvalues of get incremented by q. Example19.2: X(t) is a Wiener process with

In that case Eq. (19-22) simplifies to

and using (19-39) this simplifies to

)( 21 ttRss

*1 2 1 2

1

( ) ( ) ( ) ( ).rr k k kk

R t t q t t

2 1 21 2 1 2

1 1 2

( , ) min( , ) , 0

XX

t t tR t t t t

t t t

(19-39)

1

1

1 2 2 2 1 2 2 2 0 0

1 2 2 2 1

( , ) ( ) ( , ) ( )

( , ) ( ) ( ),

T t

XX k XX k

T

XX k k kt

R t t t dt R t t t dt

R t t t dt t

PILLAI

12

*121 )()()(

kkkkss ttttR (19-37)

(19-38)

0 1t T2dt

1

1

2 2 2 1 2 2 1 0

( ) ( ) ( ).t T

k k k ktt t dt t t dt t (19-40)

14

Derivative with respect to t1 gives [see Eqs. (8-5)-(8-6), Lecture 8]

or

Once again, taking derivative with respect to t1, we obtain

or

and its solution is given by

But from (19-40)

1

1 1 1 1 2 2 1

( ) ( 1) ( ) ( ) ( )T

k k k k ktt t t t t dt t

( ) cos sin .k kk k kt A t B t

.)()(

1221

T

t kkk tdtt

,0)0( k PILLAI

(19-41)

(19-42)

1 1( 1) ( ) ( )k k kt t

21

121

( )( ) 0,k

kk

d tt

dt

(19-43)

15

(19-45)

(19-47)

(0) 0, 1 ,

( ) cos ,k k

k k

k k

A k

t B t

PILLAI

and from (19-41)

This gives

and using (19-44) we obtain

Also

( ) 0.k T (19-44)

2

2 212

( ) cos 0

2 1 2

, 1 .( )

k k

k

k k

k

T B T

T k

Tk

k

(19-46)

16

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Further, orthonormalization gives

Hence

with as in (19-47) and as in (19-16),

( ) sin , 0 .kk kt B t t T (19-48)

2 1 cos2 2 2 22 0 0 0

sin 2 sin(2 1) 02 2 2

2 40

2

( ) sin

1 1

2 2 2

2 / .

k

k

k

k k

tT T T

k k k

Tt k

k k k

k

T

t dt B t dt B dt

T TB B B

B T

(19-49) 2 2 ,1

2( ) sin sinkk T T

tTt t k

k kc

17

is the desired series representation.

Example 19.3: Given

find the orthonormal functions for the series representation of the underlying stochastic process X(t) in 0 < t < T.

Solution: We need to solve the equation

Notice that (19-51) can be rewritten as,

Differentiating (19-52) once with respect to t1, we obtainPILLAI

1

)()(k

kk tctX

| |( ) , 0,XX

R e (19-50)

0 1t T2dt

2dt

1 2 | |

2 2 1 0( ) ( ).

T t tn n ne t dt t (19-51)

0 0

1 1 2 2 1

1

t ( ) ( )2 2 2 2 1 0 t

( ) ( ) ( )Tt t t t

n n n ne t dt e t dt t

(19-52)

18

Differentiating (19-53) again with respect to t1, we get

or

1 1 2 2 1

1

1 1 2 2 1

1

( ) ( )1 2 2 1 2 2 0

1

1

( ) ( ) 12 2 2 2 0

1

( ) ( ) ( ) ( ) ( )

( )

( ) ( ) ( )

t Tt t t tn n n nt

nn

t Tt t t t n nn nt

t e t dt t e t dt

d t

dt

d te t dt e t dt

dt

(19-53)

1 1 2

2 1

1

( )1 2 2 0

2 ( ) 1

1 2 2 2 1

( ) ( ) ( )

( )( ) ( )

t t tn n

T t t n nn nt

t e t dt

d tt e t dt

dt

1 1 2 2 1

1

1

( ) ( )1 2 2 2 2 0

( ) {use (19-52)}

21

21

2 ( ) ( ) ( )

( )

n n

t Tt t t tn n nt

t

n n

t e t dt e t dt

d t

dt

PILLAI

19

or

or

Eq.(19-54) represents a second order differential equation. The solution for depends on the value of the constant on theright side. We shall show that solutions exist in this case only if or

In that case Let

and (19-54) simplifies to

( )n t

21

1 21

( )( 2) ( ) n n

n n

d tt

dt

21

121

( ) ( 2)( ).n n

nn

d tt

dt

(19-54)

(19-55)

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( 2) /n n

2,n

( 2) / 0.n n

20 .n

(19-56)

221

121

( )( ).n

n n

d tt

dt

(19-57)

2 (2 )0,n

nn

20

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General solution of (19-57) is given by

From (19-52)

and

Similarly from (19-53)

and

Using (19-58) in (19-61) gives

1 1 1( ) cos sin .n n n n nt A t B t (19-58)

2

2 2 0

1(0) ( )T t

n nn

e t dt (19-59)

2

2 2 0

1( ) ( ) .T t

n nn

T e T t dt (19-60)

2

1

12 2 0

1 0

( )(0) ( ) (0)

T tnn n n

nt

d te t dt

dt

(19-61)

2

2 2 0( ) ( ) ( ).

T tn n n

n

T e T t dt T

(19-62)

21

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or

and using (19-58) in (19-62), we have

or

Thus are obtained as the solution of the transcendental equation

,n n

n

A

B

n n nB A

(19-63)

(19-64)

sin cos ( cos sin ),

( )cos ( )sinn n n n n n n n n n

n n n n n n n n

A T B T A T B T

A B T A B T

2

2 2 / 2 / 2( / )tan .

( ) 11 1

n n n n n nn

nn n n nn n nn n

A AB B

A A B A BT

A B

2

2( / )tan ,

( / ) 1n

nn

T

sn

22

which simplifies to

In terms of from (19-56) we get

Thus the eigenvalues are obtained as the solution of the transcendentalequation (19-65). (see Fig 19.1). For each such the corresponding eigenvector is given by (19-58). Thus

since from (19-65)

and cn is a suitable normalization constant.

2 (or ),n n

2

( ) cos sin

sin( ) sin ( ), 0

n n n n n

n n n n nT

t A t B t

c t c t t T

(19-66)2 2

20.n

n

1 1tan tan / 2,n nn n

n

AT

B

(19-68)

(19-67)

sn

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tan( / 2) .nnT

(19-65)

23

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Fig 19.1 Solution for Eq.(19-65).

2

1T

0

tan( / 2)T

/

24

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Karhunen – Loeve Expansion for Rational Spectra

[The following exposition is based on Youla’s classic paper “The solution of a Homogeneous Wiener-Hopf Integral Equation occurring in the expansion of Second-order Stationary Random Functions,” IRETrans. on Information Theory, vol. 9, 1957, pp 187-193. Youla istough. Here is a friendlier version. Even this may be skipped on a first reading. (Youla can be made only so much friendly.)]

Let X(t) represent a w.s.s zero mean real stochastic process with autocorrelation function so that its power spectrum

is nonnegative and an even function. If is rational, then the process X(t) is said to be rational as well. rational and evenimplies

( ) ( )XX XX

R R

( )XX

S ( )

XXS

0( ) ( ) 2 ( )cos

XX XX XX

j tS R e dt R d

(19-69)

(19-70)2

2

( )( ) 0.

( )XX

NS

D

25

PILLAI

The total power of the process is given by

and for P to be finite, we must have (i) The degree of the denominator polynomial must exceed the degree of the numerator polynomial by at least two,and(ii) must not have any zeros on the real-frequency axis. The s-plane extension of is given by

Thus

and the Laplace inverse transform of is given by

2

2

( )

( )

1 12 2( )

XX

N

DP S d d

(19-71)

( ) 2D n 2( )D

2( )N ( ) 2N m

2( )D ( )s j

( )XX

S

(19-72)

2 2( ) ks

2 2 2( ) ( ) ikk

k

D s s (19-73)

( )s j 2

22

( )( ) | ( ) .

( )XX s j

N sS S s

D s

26

PILLAI

Let represent the roots of D(– s2) . Then

Let D+(s) and D–(s) represent the left half plane (LHP) and the right half plane (RHP) products of these roots respectively. Thus

where

This gives

Notice that has poles only on the LHP and its inverse (for all t > 0)converges only if the strip of convergence is to the right

1 2, , , n

1 20 Re Re Re n

(19-74)

(19-75)

| | 2 2 1

1

1 ( 1) ( 2)! | |

( 1)! ( 1)!( )!( ) (2 )

k k jk

k k jj

k je

k j k js

2( ) ( ) ( ),D s D s D s (19-76)

1 ( )

( )

C s

D s

*

0

( ) ( )( ) ( ).n

kk k k

k k

D s s s d s D s

(19-77)

22 1 2

2

( ) ( )( )( )

( ) ( ) ( )

C s C sN sS s

D s D s D s

(19-78)

27

PILLAI

of all its poles. Similarly C2(s) /D–(s) has poles only on the RHP and its inverse will converge only if the strip is to the left of all those poles. In that case, the inverse exists for t < 0. In the case of from(19-78) its transform N(s2) /D(–s2) is defined only for (see Fig 19.2). In particular,for from the above discussion it follows that is given by the inverse transform of C1(s) /D+(s). We need the solution to the integral equation

that is valid only for 0 < t < T. (Notice that in (19-79) is thereciprocal of the eigenvalues in (19-22)). On the other hand, the rightside (19-79) can be defined for every t. Thus, let

( ),XX

R

1 1Re Re Res 0,

0( ) ( ) ( ) , 0

XX

Tt R t d t T (19-79)

(19-80)

( )XX

R

Fig 19.2

Re

n

1Re Re

n

1Re

strip of convergencefor ( )

XXR

s j

0( ) ( ) ( ) ,

XX

Tg t R t d t

28

PILLAI

and to confirm with the same limits, define

This gives

and let

Clearly

and for t > T

since RXX(t) is a sum of exponentials Hence it follows that for t > T, the function f (t) must be a sum of exponentials Similarly for t < 0

, for 0.k tkk

a e t

.k tkk

a e

( ) 0( ) .

0 otherwise

t t Tt

(19-81)

(19-82) +

( ) ( ) ( )

XXg t R t d

+

( ) ( ) ( ) ( ) ( ) ( ) .

XXf t t g t t R t d

(19-83)

( ) 0, 0f t t T (19-84)

( ) 0

+

( ) { ( )} ( ) 0,

k

XX

D

d ddt dtD f t D R t d

(19-85)

29

PILLAI

and hence f (t) must be a sum of exponentials Thus the overall Laplace transform of f (t) has the form

where P(s) and Q(s) are polynomials of degree n – 1 at most. Also from(19-83), the bilateral Laplace transform of f (t) is given by

Equating (19-86) and (19-87) and simplifying, Youla obtains the key identity

Youla argues as follows: The function is an entire function of s, and hence it is free of poles on the entire

( ) 0

+

( ) { ( )} ( ) 0,

k

XX

D

d ddt dtD f t D R t d

, for 0.k tkk

b e t

contributes to 0contributions

in < 0contributions in

( ) ( )( )

( ) ( )sT

tt

t T

P s Q sF s e

D s D s

(19-86)

2

2 1 1( )( )( ) ( ) 1 , Re Re ReN s

D sF s s s

(19-87)

(19-88)2 2

( ) ( ) ( ) ( )( ) .

( ) ( )

sTP s D s e Q s D ss

D s N s

0( ) ( )

T sts t e dt

30

PILLAI

finite s-plane However, the denominator on the rightside of (19-88) is a polynomial and its roots contribute to poles of Hence all such poles must be cancelled by the numerator. As a result the numerator of in (19-88) must possess exactly the same set of zeros as its denominator to the respective order at least.Let be the (distinct) zeros of the denominator polynomial Here we assume that is an eigenvalue for which all are distinct. We have

These also represent the zeros of the numerator polynomial Hence

and

which simplifies into

From (19-90) and (19-92) we get

).Re( s

)(s

1 2( ), ( ), , ( )n 2 2( ) ( ).D s N s

'k s

'k s( ) ( ) ( ) ( ).sTP s D s e Q s D s

1 20 Re ( ) Re ( ) Re ( ) .n (19-89)

( ) ( ) ( ) ( )kTk k k kD P e D Q (19-90)

(19-91)( ) ( ) ( ) ( )kTk k k kD P e D Q

( ) ( ) ( ) ( ).kTk k k kD P e D Q (19-92)

( ).s

31

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i.e., the polynomial

which is at most of degree n – 1 in s2 vanishes at (for n distinct values of s2). Hence

or

Using the linear relationship among the coefficients of P(s) and Q(s)in (19-90)-(19-91) it follows that

are the only solutions that are consistent with each of those equations,and together we obtain

( ) ( ) ( ) ( ), 1, 2, , k k k kP P Q Q k n (19-93)

( ) ( ) ( ) ( ) ( )L s P s P s Q s Q s (19-94)2 2 2

1 2, , ,

n

2( ) 0L s (19-95)

( ) ( ) ( ) ( ).P s P s Q s Q s (19-96)

( ) ( ) or ( ) ( )P s Q s P s Q s (19-97)

32

PILLAI

as the only solution satisfying both (19-90) and (19-91). Let

In that case (19-90)-(19-91) simplify to (use (19-98))

where

For a nontrivial solution to exist for in (19-100), wemust have

( ) ( )P s Q s (19-98)

1

0

( ) .n

ii

i

P s p s

(19-99)

0 1 1, , , np p p

1

0

( ) ( ) ( ) ( )

{1 ( 1) } 0, 1,2, ,

kTk k k k

ni i

k k ii

P D e D P

a p k n

(19-100)

( ) ( ).

( ) ( )k kT Tk k

kk k

D Da e e

D D

(19-101)

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The two determinant conditions in (19-102) must be solved together to obtain the eigenvalues that are implicitly contained in the and (Easily said than done!).

To further simplify (19-102), one may express ak in (19-101) as

so that

'i s 'ia s'i s

2 , 1, 2, ,kka e k n

1 11 1 1 1 1

1 12 2 2 2 2

1,2

1 1

(1 ) (1 ) (1 ( 1) )

(1 ) (1 ) (1 ( 1) )0.

(1 ) (1 ) (1 ( 1) )

n n

n n

n nn n n n n

a a a

a a a

a a a

(19-102)

(19-104)

(19-103)

/ 2 / 2

/ 2 / 2

1 ( ) ( )tan

1 ( ) ( )

( ) ( )

( ) ( )

kk k

k k k

k k

k k

Tk k k

k Tk k k

T Tk k

T Tk k

a D e De e

ae e D e D

e D e D

e D e D

h

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PILLAI

Let

and substituting these known coefficients into (19-104) and simplifyingwe get

and in terms of in (19-102) simplifies to

if n is even (if n is odd the last column in (19-107) is simply Similarly in (19-102) can be obtained by replacing with in (19-107).

0 1( ) nnD s d d s d s (19-105)

2tan , kh

2 3 0 2 1 3

2 3 0 2 1 3

( ) tan ( / 2) ( )tan

( ) ( ) tan ( / 2)k k k k

kk k k k

d d T d d

d d d d T

(19-106)

1 1 1

1 2[ , , , ] ).n n n

n

T 1 cot k htan kh

2 3 1 1 1 1 1 1 1 1

2 3 1 2 2 2 2 2 2 2

1 tan tan tan

1 tan tan tan

1 tan

n

n

n

2 3 1

0

tan tannn n n n n n

(19-107)

h

h

h

h

h

h

h

h

h

hh

h

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To summarize determine the roots with that satisfy

in terms of and for every such determine using (19-106).Finally using these and in (19-107) and its companionequation , the eigenvalues are determined. Once are obtained, can be solved using (19-100), and using that canbe obtained from (19-88).Thus

and

Since is an entire function in (19-110), the inverse Laplace transform in (19-109) can be performed through any strip of convergence in the s-plane, and in particular if we use the strip

, ,k k

'k s2 2( ) ( ) 0, 1, 2, ,k kD N k n (19-108)

k s tanh k s k s k s

kp s ( )i s

( )i s

2 2

( ) ( , ) ( ) ( , )( )

( ) ( )

sTi i

ii

D s P s e D s Q ss

D s N s

(19-109)

1( ) { ( )}.i it L s (19-110)

1

Re( ) 0i

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then the two inverses

obtained from (19-109) will be causal. As a result

will be nonzero only for t > T and using this in (19-109)-(19-110) we conclude that for 0 < t < T has contributions only from the first term in (19-111). Together with (19-81), finally we obtain the desired eigenfunctions to be

that are orthogonal by design. Notice that in general (19-112) corresponds to a sum of modulated exponentials.

Re Re( ) (to the right of all Re( )),n is

1 12 2 2 2

( ) ( ) ( ) ( ),

( ) ( ) ( ) ( )

D s P s D s Q sL L

D s N s D s N s

(19-111)

(19-112)

2 2

1 ( ) ( )

( ) ( ) sT D s Q s

eD s N s

L

( )i t

12 2

( ) ( , )( ) , 0 ,

( ) ( )

Re Re 0, 1,2, ,

kk

k

n

D s P st L t T

D s N s

s k n

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PILLAI

Next, we shall illustrate this procedure through some examples. First,we shall re-do Example 19.3 using the method described above.

Example 19.4: Given we have

This gives and P(s), Q(s) are constants here. Moreover since n = 1, (19-102) reduces to and from (19-101), satisfies

or is the solution of the s-plane equation

But |esT| >1 on the RHP, whereas on the RHP. Similarly|esT| <1 on the LHP, whereas on the LHP.

| |( ) ,XX

R e

( ) , ( )D s s D s s

1 11 0, or 1a a 1

1sT s

es

2

2 2 2

2 ( )( ) .

( )XX

NS

D

(19-113)

(19-114)

1 1 1

11

( )

( )T D

eD

1ss

1s

s

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Thus in (19-114) the solution s must be purely imaginary, and hence in (19-113) is purely imaginary. Thus with in (19-114)we get

or

which agrees with the transcendental equation (19-65). Further from(19-108), the satisfy

or

Notice that the in (19-66) is the inverse of (19-116) because as noted earlier in (19-79) is the inverse of that in (19-22).

1 1s j

n

2 2

0.2

nn

(19-116)

(19-115)

2 2 2 2( ) ( ) 2 0n

n n ns jD s N s

11tan( / 2)T

1 1

1

j T je

j

s

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Finally from (19-112)

which agrees with the solution obtained in (19-67). We conclude this section with a less trivial example.

Example 19.5

In this case

This gives With n = 2,(19-107) and its companion determinant reduce to

12 2( ) cos sin , 0n n n n n

n

st L A t B t t T

s

(19-117)

| | | |( ) .XX

R e e (19-118)

2

2 2 2 2 2 2 2 2

2 2 2( )( )( ) .

( )( )XXS

(19-119)

2( ) ( )( ) ( ) .D s s s s s

2 2 1 1

2 2 1 1

tan tan

cot cot

h h

h h

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or

From (19-106)

Finally can be parametrically expressed in terms of using (19-108) and it simplifies to

This gives

and

(19-120)

(19-121)

2 21 2 and

221

( ) ( ) 4 ( )

2

b b c

1 2tan tan . h h

2

2

( ) tan ( / 2) ( )tan , 1,2

( ) ( ) tan ( / 2)i i i

ii i i

Ti

T

hh

h

2 2 4 2 2 2

2 2

4 2

( ) ( ) ( 2 ( ))

2 ( )

0.

D s N s s s

s bs c

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PILLAI

and

and substituting these into (19-120)-(19-121) the corresponding transcendental equation for can be obtained. Similarly the eigenfunctions can be obtained from (19-112).

22 2 22 1

( ) ( ) 4 ( )( ) 4 ( )

2

b b cb c

i s