1. I ntroduction

The Power-inversionAdaptive Array:Concept and PerformanceR.T. COMPTON, JR.

The Ohio State University ElectroScience Laboratory

Abstract

The use of a power-inversion adaptive array to improve the signal-

to-interference ratio in a communication system is described.

"Power inversion" refers to the ability of an adaptive array to

invert the power ratio of two received signals. The power-inver-

sion technique is useful because it does not require detailed know-

ledge about desired-signal structure or arrival angle. The perform-

ance attainable with a power-inversion array is described and typi-

cal design curves are presented.

Manuscript received January 11, 1978; revised March 8,1979,and April 25, 1979.

The work was supported in part by Naval Air Systems Commandunder Contract N00019-79-C-0291 and in part by the Joint Ser-vices Electronics Program under Contract N00014-78-C-0049.

Author's address: Department of Electrical Engineering, The OhioState University, 2015 Neil Avenue, Columbus, OH 43210.

0018-9251/79/1100-803 $00.75 ©2979 IEEE

This paper discusses the use of a power-inversion adap-tive array to improve the signal-to-interference ratio (SIR)in a communication system. "Power inversion" refers tothe ability of an adaptive array to invert the power ratioof two received signals. It does this by nuiling the strongsignal in favor of the weak one. The power-inversion tech-nique is useful because it does not require detailed infor-mation about desired signal structure or arrival angle. Thepower-inversion technique has been discussed previouslyoy Compton [1 ,Scliwegman and Compton [21, andZahin [3].

The least mean square (LMS) adaptive array (due toWidrow et al. [4] ) requires an estimate of the desiredsignal for use as the reference signal [5] in the array.'When a desired-signal waveform can be estimated in thereceiver, the LMS algorithm is a very useful technique.For certain communication systems it is possible to obtaina suitable reference signal [6, 7] while for others it is not.When a reference signal can be obtained, the LMS adap-tive array automatically tracks the desired signal andnulls interference. When no reference signal is available,however, the LMS array cannot be used.

Tlie power-inversion adaptive array is basically thesame as the Howells-Applebaum array [8, 9]. For powerinversion, however, the array is used differently than inits radar application. First, the number of degrees offreedom in the array must equal the number of interfer-ing signals. Second, the steering vector is chosen differ-ently than for radar. (It is assumed that the desired-signalarrival angle is not known in advance.) Finally, the loopgain must be chosen to optimize the output signal-to-interference-plus-noise ratio (SINR) over the dynamicrange of the received signals.

The purpose of this paper is to examine the perform-ance of a power-inversion array in a form useful for systemdesign. Curves will be presented showing the performanceof a two-element power-inversion array as a function ofdesired-signal power, interference power, thermal-noisepower, feedback-loop gain, signal arrival angles, andsignal bandwidths. We will show that the power-inversionteclhnique is most useful in two situations: either 1) whenthe received desired-signal level is nearly constant, or2) when an SINR of less than 0 dB is required at thearray output (as in some spread spectrum systems.)

Section II discusses the number of degrees of freedomin an array pattern. Section III discusses the power-inver-sion concept, and the relation between the LMS algorithmand the Howells-Applebaum array. Section IV providesdetailed curves of the SINR performance of the power-inversion array as a function of various parameters, par-ticularly the loop gain.

1The reference signal is called the "desired response" inWidrow et al. [4 ].

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-15, NO. 6 NOVEMBER 1979 803

Ill. Power-inversion Feedback

S (t, 0)

Fig. 1. An N-element linear array.

11. Pattern Degrees of Freedom

Consider an N-element linear array as shown in Fig. 1.Assume for simplicity that each element is omnidirec-tional and that all mutual impedances are zero. Let a unitamplitude CW signal at frequency w0 propagate into thearray from angle 0 relative to broadside. The complexsignal produced on the ith element is

Xi= exp [-j(27rQi/X) sin 0 ] exp(fco t) (1)

where X is the free-space wavelength and Qi is the distancebetween element 1 and element i. (We arbitrarily definethe phase of the signal on element 1 to be zero.) Eachxi is multiplied by a complex constant wi and then addedto produce the array output S (t, 0)

S(t, 0) = w, {I + (w2/wl) exp[-j(27rQ2/X) sin 0]

+ ... + (wN/wI) exp[-j(27rQN/X) sin 0] } exp(jwo t). (2)

S (t, 0) describes the voltage pattern of the array forany given weights wi. We have factored one of the con-stants, w1, out of the expression, and in this form, theangular dependence of 13j(t, 0) is contained in thebracketed term. Since this term containsN -1 com-plex constants (w2/wl, w3/w1, ..., wN/wl), there areN - 1 degrees of freedom in the pattern of the array atfrequency co0. The Nth constant w1 does not affect thepattern (the relative response versus angle), but merelycontrols the overall scale constant (magnitude and abso-lute phase) of the antenna response.

Thus an N-element array hasN -1 degrees of free-dom in its pattern at a given frequency. Therefore, witha two-element array, we may place one null in an arbitrarydirection in space (for a given frequency). Once that nullis specified, however, there is no further flexibility in thepattern other than its overall amplitude and phase. Witha three-element array we may place nulls in two arbitrarydirections; or, we may place a null in one direction anda beam maximum (d Is IldO = 0) in another. In eithercase, the pattern flexibility will be "used up" once wehave specified two directions.

The remarks above suggest a method of using an adap-tive array to obtain interference protection for a communi-cation system. Assume two signals are to be received, onea weak desired signal and the other a strong interferencesignal. We wish to suppress the interference signal. Sup-pose we arrange to receive these signals with a two-elementarray. Suppose further that the weights wi in this arrayare adjusted to minimize the array output power (withthe constraint that not all wi = 0). The array outputpower will be minimized when the array has directedits only available null at the strong interference signal.As a result, the interference will be attenuated by the nullwhile the desired signal will not be in a null. The SIRwill be improved by the pattern ratio in the two direc-tions. This is the essence of the power-inversion tech-nique in adaptive arrays.How do we adjust the weights so the array output

power is minimized? This may be done with the feed-back concept due to Howells [8] and Applebaum [9].This adaptive array concept is closely related to the LMSalgorithm of Widrow et al. [4] and for our purposes itwill be helpful to compare these two systems.

Fig. 2 shows a two-element adaptive array. The sig-nal from each element is split into quadrature componentswith a quadrature hybrid. Let x1j (t) and xI2 (t) be thein-phase components and xQ 1 (t) and XQ2 (t) be thequadrature components. Each xji(t) or xQi(t) is mul-tiplied by a real weight wji or WQi and then summed toproduce the array output s(t).

In the LMS algorithm [4] the weights are derivedfrom the control equation

dwp /dt = -k(a/awp [E2(t)]i i

= 2ke(t) x (t), i= 1, 2

where P denotes in-phase or quadrature and where e(t)is the error signal, defined as the difference between areference signal R(t) and the array output s(t),

26(t) =R(t)- s(t) =R(t) - iEl [wI X,I(t)

+WQ xQ (t)]i i

as shown in Fig. 2. The feedback loop corresponding to(3) is shown in Fig. 3.

The behavior of the weights in this array can mosteasily be studied by making use of analytic signal nota-tion [10-12]. If the quadrature hybrids are assumed tobe broadband quadrature hybrids, then

XQ (t) = Xs t)

(3)

(4)

(5)

where the caret denotes the Hilbert Transform [10]. Wedefine the analytic signal x'(t) associated with the ithantenna element as

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEEMS VOL. AES-15, NO. 6 NOVEMBER 1979804

(6)Xi(t) =XIY() + ix i(t) =XI (t) +"

I(t)

and the signal vector X as the vector with componentsXi(t)

(7)

(T denotes the transpose). Also, we define the complexweight for the ith element as

Wi WI -W

and the weight vector as

W=(Wl,W2,...5,WN)T.

The analytic signal at the array output is then given by2

(8)

Fit,. 2. A two-element adaptive array.

(9) Fig. 3. The LMS feedback loop.

SIGNALS FROMOTHER CHANNELS

s (t) =wT X=-Xlw

and the real-valued array output signal is

s(t) = Re {s(t) }-

Using this notation, the LMS algorithm in (3) may bewritten

dwi/dt = k ?(t) e (t) = k xl't) [R(t) - s(t)] (12)

where the asterisk denotes the complex conjugate, ande (t) and R(t) are the analytic signals associated with thereal-valued error signal e(t) and reference signal R(t),respectively3. Substituting (10) for s (t) and writing(12) in vector form, one finds that the weight vector wsatisfies the differential equation

* INTEGRATOR TRANSFER

FUNCTION - 2ks

W = (j 1 S

where CF is the covariance matrix

4) = E(X*XT)

and S is the vector

S = E [X*R(t)]Twdwldt + kX*X w = kX*R(t) . (13)

In general, the solution w to (13) consists of the sum ofa mean value and a random jitter component.4 In thispaper we are interested only in the mean value of w,

whose behavior can be approximated by replacing X*XTand X*R(t) in (13) by their expected values. When thisis done, (13) may be solved by well-known techniques.The steady-state weight solution is

2The weights Wi are complex random processes that, in gen-eral, are not analytic. However, the spectra of Wi are bandlimitedaround zero frequency and, in a typical communications appli-cation, the xi(t) are narrowband processes at a carrier frequencycoo. In this case s(t) is analytic as long as the highest frequencyin wi is less than the lowest frequency in x i(t).

3Equation (12) is equivalent to (3) in the sense that kx'(t)e (t)in (12) contains the baseband components of 2ke(t) xpi(t) in (3).The second harmonic terms in (3) are assumed too high in fre-quency to affect the solution for wpi.

4The random component of w, which results in misadjustment[4 1, occurs because of the ranlom noise components of X (andpossibly R(t) ).

The major difficulty in using the LMS algorithm in a

practical communication system is the problem of obtain-ing a suitable reference signal. For certain communicationsystems [6, 7] it is possible to obtain a reference signalby processing the array output. When a reference signalcan be obtained, the LMS adaptive array yields automaticbeam tracking of the desired signal and good interferenceprotection for the communication system [13]. On theother hand, for many types of communication systemsit appears impossible to obtain a reference signal. In this

situation, the power-inversion adaptive array may stillbe useful.

The power-inversion adaptive array is based on a modi-fied LMS algorithm requiring no reference signal. If weassume the reference signal R(t) is zero, (16) becomes

S =0 (17)

so the LMS algorithm will force all the weights to zero,

as may be seen from (14). To prevent this weight shut-down, in a power-inversion array we make two changes

COMPTON: THE POWER-INVERSION ADAPTIVE ARRAY: CONCEPT AND PERFORMANCE

(10)

(1 1)

(14)

(15)

(16)

x =1-11 T

lj. l(t), X2(t) I... I .N.(t)l

805

x lp

*LOWPASS FILTER TRANSFER

FUNCTION =2k

Fig. 4. Power-inversion array feedback.

in the feedback. First, we offset the ith weight in thearray from zero; second, we add a wi term to the leftside of (12). Offsetting the ith weight from zero pre-vents the weights from going to zero when R(t) = 0.The extra wi term added to (12) controls the weightbehavior when no signals are being received, as will beseen below. With these changes, the new weight controequation is

T (dwi/dt) + Wi = w.0 - k xl*t)s (t)

where wio = WIio - jWQio is the ith offset weight (acomplex constant) and where two gain constants r andk are now included. The loop-gain constant k controlsthe steady-state weights in the array and the constant Tin conjunction with k, controls the time constants ofthe array response. The choice of k will be extensivelydiscussed below. The choice of T is determined by thespeed of response requirements on the array, and byallowable weight jitter. In this paper we do not discussthe choice of r. The quadrature-feedback loop corres-ponding to (18) is shown in Fig. 4.

When (10) and (15) are substituted in (18) the weigivector is found to satisfy

7(dw/dt) + (I + kF) w=wo

where I is the identity matrix and wo is the offset weiglvector

/Wio\wo =

\W20

or, as it is usually called, the steering vector. From (19it is clear that the steady-state weights are given by

The feedback loop in (12) and Fig. 3 is the loop sug-gested by Widrow et al. [4], while the feedback loop in(18) and Fig. 4 is the loop originally proposed by Howellsand Applebaum [9]. In the radar application of theHowells-Applebaum array, the steering vector wo is usedto obtain a main beam of the quiescent pattern in a spec-ified direction (see discussion below). In the power-inversion application described here, the steering vectorwill be chosen to obtain coverage over a broad sectorwhere desired signals may be received. The loop gain kwill be chosen to obtain a maximum SINR at the array out-put for the range of signal levels expected. These ideasare discussed in the next section.

IV. Array Performance

Consider an array of two isotropic elements spaced adistance Q apart with adaptive weight control loops asshown in Fig. 4. Suppose two signals are incident on thearray, one desired and one interference. In addition,thermal noise is present on each element signal. Thusthe element signals are

x.-j() =d(t+i (+n )1-1, 2 (22)

where dj(t) is the desired signal component, i j(t) is theinterference signal component, and n1(t) is a noise com-

[8) ponent. nj(t) will be assumed to be bandlimited zeromean Gaussian noise with variance a2 (the same variancefor each element), and to be statistically independentbetween elements, i.e.,

E{n (t) n *(t)} - u26I i

(23)

where 6ij is the Kronecker delta. The desired signal isassumed to arrive from angle Od (measured from broad-side, see Fig. 1). Moreover, d 1(t) and d 2 (t) are assumedto differ only by an interelement propagation delay.Thus

d,(t) =d(t) (24a)

(24b)d2(t)=d(t- Td)

(19) where d (t) is the desired signal waveform and

Td = (Q/c) sin Od

is the spatial time delay between elements for arrival

(20) angle Od. (c is the velocity of propagation.) Similarly,the interference arrives from angle Oi, and

i I (t) = i (t)

i 2 (t) = i (t -Ti)

(24c)

(25a)

(25b)

w = [I+ kflf1 wo (21) where i1(t) is the interference waveform and

which is not zero except perhaps in the limit as k-*oo. T4= (Q/c) sin OVIEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-15, NO. 6 NOVEMBER 1979

(25c)

806

The desired and interference signals are assumed to bezero mean and statistically independent of each otherand the thermal noise.

With these assumptions, the covariance matrix (D in(15) is found to be

(Rd(O) + Ri(O)(F =a2i +

Rd(Td) + Ri(T1)

R*(Td) + Ri(Ti)

Rd(O) + Ri(m) /

where Rd(r) and Rj(r) are the autocorrelation functionsof d (t) and i (t), given by

Rd(T) = E [d (t + r)d *(t)] (27;

and

Rj(r) = E [i (t + r) i *(t)J (27

Since Rd(O) and Rj(O) are the desired signal and inter-ference power on each element, it is convenient to define

Rd(O) = Sd = desired-signal power per element

Rj(O) = Si = interference power per element

The inverse of this is easily found

(I + k)-' =(1/D)

(26)

1 +K+Ktd +Kti

-KtdPd I-Kp

-K1dpd-KKtipi

I + K+Ktd +Kti/

(32a)

where D is the determinant of I + k(F

a) D=(l +K+K&d+KjK)2 - IKdpd + KtiPi2. (32b)

The steady-state array weights may now be calculatedlb) from (21) for any given steering vector w0.

Our goal is to protect a desired signal from a strongerinterference signal. To see how this may be done, it will behelpful to proceed in steps. We first consider the arrayperformance with noise alone, then with one signal, andfinally with both signals incident on the array. In theprocess, the power-inversion behavior will be seen, andit will become clear how the steering vector w0 and thefeedback gain k (or K) should be chosen.

and alsoA. Noise Alone

Pd= Rd(Td)lRd(O) = Rd(Td)lSd

Pi = Ri(T1)/Ri(0) = Ri(Ti)/S.

(28a) First, suppose no signals are present other than thermalnoise. Ifd .(t) = 0 and ii(t) = 0 in (22), then td = 0 andti = 0 and (21) and (32) yield

(28b)

w= [1/(1 +K)]wo = [1/(1 t ka2)] w0 -

Then (F may be written

Sd +si(D = GP2++

PdSd +piSi

PdSd + P ,

S +

d dI I

By also defining the normalized parameters

K = ko2 = normalized loop gain (30a)

td = Sdl/a2 = desired signal-to-noise ratio per element (30b)

ti = 5i/a2 = interference-to-noise ratio per element (30c)

the matrix I + k(F in (21) may be written

1+K+K#d + Kti KtdPd + Ktip7

I+k(F= (31)KtdPd +Ktipi I +K KKd +Kti/

Thus when dj(t) and i j(t) are absent, the steady-stateweight vector equals the steering vector scaled down bythe constant (1 + ku2)-1 Since this constant is the samefor every element of the array, the array pattern withoutsignals (which we call the quiescent pattern) is the same

as the pattern that would result from the steering vectorw0, except for an overall magnitude reduction.

Thus the steering vector should be chosen to obtainthe quiescent pattern desired from the array. In radarapplications w0 is chosen to obtain a quiescent patternwith a main beam at some desired look angle and withsuitable sidelobe performance. In communication appli-cations where the angle of arrival of the desired signalwill not be known in advance, the quiescent pattern shouldcover the sector of space from which desired signals mayarrive. Such coverage may be obtained by using elementswhose patterns cover this sector. Then by choosing thesteering vector to have only one nonzero component(so one element is "on" and the other is "off") the qui-escent pattern of the array will just be the element pat-tern of one element. By this choice, we can assure that

COMPTON: THE POWER-INVERSION ADAPTIVE ARRAY: CONCEPT AND PERFORMANCE

(33)

807

a desired signal has access to the receiving system fromany direction within the sector.5

In this paper we will assume the desired signal mayarrive from any angle within -7r < 0d < r. To providecoverage over this sector, we choose isotropic elementswith half-wavelength element spacing.6 Furthermore,to be specific, we will assume that element 1 is the oneturned on in the quiescent mode. Thus we choose thesteering vector

w= (A\O= I

From (21) and (32) the resulting weights are

w = (lID) (

1 + K + K#d +K \

-KtdPd-Ktipi

(35)

where D is given in (32b). With these weights the desiredsignal at the array output is

s j (t) = [d (t) d (t - Td)I] w

=-(1/D){f[I +K+Kd + Kti d(t)

(KtdPd + Ktjp,) d (t (36)

The array output desired-signal power is

d E[s d(t) S d*(t)]

- Sd/2D2){(I + K + K4d + Kti)[l+ K + K#d(l-22pd 12)

+ Kt,(1-2 Re {Pip*})]

(34)

and the array output interference power is

P 2= 2 E[s i(t) s (t)]

= (Si/2D2) {(l + K + K4d + Kk)

[l+K+K&d(l-2 Re {PdP*})+ Kti(l -2 p, 12 )]

+ PKdPd + Ktip, 2 } (39)

Finally, the thermal-noise signal at the array output is

(l) [(_ )Is n(t) = [n l(t) n2 (t) w (40)

and the output-noise power is

p =I E[s (t) s *(t)I

= (a2/2) ( lw 112 + IW2 12)

=(a2/2D2) [(I + K+KKd +Kf)2+ 1KVdKd+KipI12 ] (41)

Now let us apply these results to two cases: 1) desiredsignal and noise present in the array and 2) desired signal,itterference, and noise present in the array.

B. Desired Si jnal-PI us-Noise

If only desired signal and thermal noise are present inthe array, we may obtain the desired-signal output powerand noise power from (37) and (41) by letting ti =SIG2= 0. To be specific, let us first assume the desired signalis CW, with frequency wo. (We consider the case ofnonzero bandwidth below). We have

d (t) = VSd exp[(o t + 'hd)I

+ 1KVdpd + Kt,pj 12}2

Similarly, the interference signal at the array output is

s i(t) = [ i (t) i (t - T.)] w

=(I/D)[(I + K + Kd +IK) ,(t)

- (KVdPd + Ktjp,) i (t - T,)]

5In addition, we should choose the element spacing so thearray cannot have spurious nulls within the sector of interest(i.e., an interference null at one angle within the sector willnot create a grating null at another angle within the sector.)

6Half-wavelength spacing permits only one null in visiblespace.

(37)where 4'd is uniformly distributed on [-ir, 7T]. Then

-

Rd(Td) =E[d(t + Td) d*(t)] =Sd expLfcoTd]

and thus

Pd = Rd(Td)/Sd = exp(Jc0Td ) _ exp(jqd) (43)

where qd is the interelement phase shift at frequency cwo.(38) Substituting |Pd 12 = I into (37) and (41), we find the

output signal-to-noise ratio (SNR) to be

SNR = Pd/Pl = #d {(l + K)2/[( +K+Kd)2 K2I1d

(44)From this equation we may obtain an understanding

of the effect of the loop gain and the input SNR on the

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-15, NO. 6 NOVEMBER 1979

(42)

808

array performance. Fig. 5 shows a typical plot of outputSNR versus K for several values Of Ud and Fig. 7 showsthe output SNR versus td for several values of K.

From these curves we can see how the loop gain Kshould be chosen. Since the signal being received bythe array is the desired signal, we do not wish to null it.When the input SNR (td) is small, the weights in (35)are not much different than with noise alone and thearray does not null the desired signal. This result maybe seen in Fig. 5; for example, for td = -10 dB, K haslittle effect on the output SNR. With a larger input SNR,increasing K causes the array feedback to null the desiredsignal and the output SNR drops. For example, in Fig. 5the curve for td = +30 dB shows this behavior.

Fig. 6 illustrates the effect of the desired signal on thepattern. It shows the pattern for several values of td forthe case K = 0.1. It is seen that for td < 0 dB, the desiredsignal is not strong enough to cause a pattern null. Attd = +10 dB, however, the array has begun to null thedesired signal (which is at 00). The null depth increasesas td increases; it is -46.1-dB deep when td = +30 dB.(0 dB in Fig. 6 is defined as the array response when theweight vector equals the steering vector.)

Fig. 7 shows the output SNR as a function of theinput SNR Ud for several values of the loop gain K. Itis seen that for a given K, the output SNR at first increaseswith Ud (in the range where td is too small to affect theweights) and then decreases with td (when the desiredsignal is nulled by the array). For strong signals a 10-dBincrease in td produces a 20-dB increase in the null depth,so the output SNR drops 10 dB.

The problem faced by the designer is to prevent thearray from nulling the desired signal. The optimum choiceof the loop gain K depends on two factors: 1) the requiredminimum SNR out of the array (which depends on thereceiver and the type of modulation used in the communi-cation system), and 2) the dynamic range of signal levelthat must be accommodated by the array. For example,if the input signal level varies between 0 dB and 20 dB,and an output SNR of 0 dB is required, Fig. 7 shows thatK cannot be larger than about 0.07. If a higher value ofKis used, the output SNR will drop below 0 dB before tdreaches +20 dB. A lower value ofK than 0.07 shouldalso not be used, because lower values ofK yield poorerinterference protection, as will be seen below.

Several other effects may also be seen in Fig. 7. First,for td < -10 dB, the output SNR is independent of K. Thisbehavior results because such a weak desired signal hasno effect on the array weights. Increasing K simply lowersthe quiescent weights, which lowers both the signal andnoise powers out of the array by the same factor but doesnot affect the output SNR.

Second, the curve for K = 10 is essentially the same asthe curve for K = oo. For K = 10 the output SNR neverexceeds -6.5 dB regardless of the input SNR.

Third, the larger the dynamic range of the input signal,the smaller K must be to keep the output SNR above agiven minimum over the whole range. If the input SNR

LOOP GAIN K=2kcT2

Fig. 5. Output SNR versus loop gain K (no interference).

Fig. 6. Array pattern-no interference: K = 0.1, Od = 00.

DESIREDSIGNAL

dB

Fig. 7. Output SNR versus input SNR (no interference).

:O 30 40 50SNR C (dB)

varies from 0 dB to +60 dB, for example, we must haveK < 0.002 if the output SNR must exceed, say, 0 dB.As will be seen in the next section, such a low value ofK offers little interference protection. For this reason,the power-inversion technique is most effective when itis necessary to accommodate only a small dynamic rangefor the desired signal.

Moreover, the ideal application of the power-inversionarray is to a communication system where the outputSNR can be less than 0 dB. One such application is tospread-spectrum systems, which often operate with thesignal below the noise, because of the processing gain ofthe spread spectrum receiver. With the desired signalbelow noise, the array does not try to null the desired

COMPTON: THE POWER-INVERSION ADAPTIVE ARRAY: CONCEPT AND PERFORMANCE 809

and

Pi = exp(ico T.) A exp(qoi) . (46)

ej is the interelement phase shift at the interference arrivalangle.

The output SINR, given by

SINR = Pd (P-I + Pd)

Fig. 8. Output SINR versus input SNR . 0 =0° = 500,K = 0.01.

Fig. 9. Output SINR versus input SNR td. 6d = °0 6,= 500o

K = 0.1.

SNR Cd (dB)

Fig. 10. Output SINR versus input SNR d. Od=00° = 500,K = 1.0.

20

z

-tO LIMIT CURVE:- ~ 20

30-30 -20 -10 0 10 20 30 40 50 6C

SNR (dB)

signal. In this situation the designer has wide latitudein the choice of K, which can then be chosen for goodinterference suppression.

C. Desired Signal, Interference, and Noise

Now assume two signals are incident on the array,

one desired and one interference. (However, the array

has no way of knowing which is which!) Assume thatboth signals are CW. For the desired signal, Pd is givenin (43). For the interference we let

1-

i (t) = v/Si exp V(Wot + VPi)]

where ipi is uniformly distributed on [-rr, 7r]. Then

R1(Ti) = EI[i (t + T7) i *(t)] = Si exp(Qwo T.)

may now be calculated by substituting Pd and pi into (37),(39), and (41). These equations can be used to computea variety of curves that are helpful for understanding thebehavior of the array. Figs. 8, 9, and 10 are typical ex-

amples showing the SINR as a function of Ud for K = 0.01,0.1, and 1.0, respectively. These figures assume Od = 00,

Oi = 50° and half-wavelength element spacing. Each figureshows output SINR for several values of ti, the input inter-ference-to-noise ratio.

Several things may be seen from these figures. First,for (i = -100 dB the interference is virtually not present,so the SINR curves for this case are identical to the resultsin Fig. 7 for the appropriate values of K. Second, as theinterference power is increased, we find that the SINRat first drops and then rises. For example in Fig. 8, with

U = 10 dB, we have SINR = 9.2 dB for ti = -100 dB, thenSINR = -3 dB for ti = +17 dB, and finally SINR -+ 12.2 dBas ti * Comparing Fig. 8 with Figs. 9 and 10 shows thatas K is increased, there is a less significant drop in the SINRfor intermediate values of ti. In fact, for K = 1 there is es-

sentially no drop in SINR as ti is increased. In this respectK = 1 represents the best choice of loop gain. However,K = 1 also results in a narrow range of acceptable desired-signal levels when no interference is present, as was seen

in Fig. 7, and also results in higher desired signal atten-uation, as will be seen below. Third, for any given tithere is a fimite range of td over which the output SINR isabove any given value. For interference powers close tothe desired-signal power, this range is least. For inter-ference power substantially less than or greater than thedesired signal power, the acceptable range for Ud is wider.Finally, it is interesting to note that for stronger inter-ference signals, the SINR can be substantially better thanit would be without the interference. For example, con-

sider Fig. 9. At td = 20 dB if ti = -100 dB we have SINR= -2.6 dB, but if ti = +40 dB we have SINR = +17.7 dB.The reason for this behavior is that, without interference,the adaptive array devotes its single null to the desiredsignal. However, with strong interference the array isforced to use its null on the interference. The desiredsignal is then not in a null.7

Fig. 11 shows some patterns that illustrate these re-

marks. It shows the array patterns for K = 0.1 and fori -100 dB, +7 dB, +20 dB, and +40 dB. The shift of

7This result is achieved because we are using a two-elementarray, so there is only one null. With a three-element array, one

fmds no such improvement as tj is increased, because the array

nulls both signals at once.

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-15, NO. 6 NOVEMBER 1979

Ma:zU,

(47)

810

DESIREDSIGNAL

Fig. 11. Array patterns. td = +20 dB, K = 0.1, d =0 i= 50 .

Fig. 12. Desired signal attenuation versus input SNRd= 0j 50 . d6d =o0 60 =500

=0

7060 ejs 01o

VT / S s 20dB100d

z, 2040 -

zw

30(-d(20 dBz020

0

X10

0

-30 -20 -10 0 10 20 30 40 50 60

ed(dB)the null from the desired signal to the interference as ti in-creases may be seen. It is interesting to note that duringthis change the array response in the desired signal direc-tion is nearly independent of the interference power.We may also compute the attenuation of the desired

signal. If the array weight vector equals the steering vec-tor, the desired-signal output power is8

Pod =Sd/2.From (37) the desired-signal output power with the arrayoperating Pd will be equal to Pod multiplied by factor q

n2= {(I +K + Ktd +Kts)[l +K+Ktd(1-2 lpd 12)

+ Kti(1-2 Re pip* })] + IKpdP + Kt,p, 12

[(1 + K + Ktd + Kt)2 - IKtdPd + Ktip. 12 ] 2 (48)

which we define as the attenuation of the desired signal.When R7 = 1, the array desired-signal output power is thesame as it would be if the weights are given by the steer-ing vector wo.

8Sd is the power behind one element of the array. One halfof this power is available in each quadrature channel. The weightvector w = w0 turns on only one of the quadrature channels behindelement 1.

0

ei (DEGREES)

Fig. 13. Output SINR versus interference angle 0. 6 =0,tp=0dB,tj=30dB.

F ig. 14. Output S I NR versus i nterference angle 60 00

td =-20 dB, t + 20 dB d

ei (DEGREES)

Fig. 15. Output SINR versus interference angle 6. 6 = 50,

td =-20 dB, t; = +20 dB. i'

61 (DEGREES)

Fig. 12 shows 71 in decibels versus the input SNR tdfor several values ofK and ti. In all cases, as td increases,the attenuation 7r increases. It is seen that, for a fixed K,the desired signal attenuation is almost independent ofti. The reason for this was noted in Fig. 11; although thepattern changes drastically with ti, the desired-signalresponse of the array is almost independent of ti.

Finally, we note that the array performance dependson the spatial separation of the desired and interferingsignals. All of the curves presented so far have been forOd =00 and Oi = 500. Figs. 13, 14, and 15 show typicalplots of the output SINR as a function of Oi for fixedOd . In Figs. 13 and 14 the desired signal arrives frombroadside (Od = 00). (In Fig. 13 td =0 dB and fi =30dB. In Fig. 14 td = -20 dB and ti = + 20 dB.) In Fig.15,Od =500, td = -20 dB, and ti = +20 dB. In all casesthe SINR drops when the interference signal is near the

COMPTON: THE POWER-INVERSION ADAPTIVE ARRAY: CONCEPT AND PERFORMANCE 811

desired signal, because the desired signal falls in the inter-ference null. In general, when interference is too closeto the desired signal, the performance of the array willbe unsatisfactory. How close the two signals may bedepends on the minimum SINR the receiver can accept.9

D. Power Inversion

Let us now assume the noise power in the signalsx (t) is very small compared to the signal powers, i.e.,(d > 1 and ti > 1. Also, let us assume the array loopgain k is large enough that kSd > 1 and kSi > 1. (Notefrom (30) that kSd = KUd and kSi = Kti.) Then we maydrop the 1 + K terms in (37) and (39) in comparisonwith the other terms. The output desired signal-to-inter-ference ratio (SIR) is then

SIR = Sd {(d + ti) IV1l2 lPd 1)

+ .( 1-2 Re {p,p* })j

+ tdPd + ipi }/

s -+i)[4d(l2 Re {PdPi + ti(l-2 lpi 12)]

+ dPd + hipi 2 } .(49)

When the signals are CW (Pd- exp(pkd), Pi = exp(jqi)),this simplifies to

the power-inversion array is not as good as with CW signals.A performance degradation occurs with nonzero band-width because the antenna pattern is frequency depen-dent, so its response varies over the signal bandwidth.In particular, the pattern varies much more rapidly withfrequency in the nulls than elsewhere, so it is primarilyinterference bandwidth that affects the performance.Desired signal bandwidth has only a negligible effect onthe results. In this section we briefly illustrate the effectsof bandwidth on array performance.1"

Let us assume the interference signal i (t) is a band-limited stochastic signal whose power spectral density isconstant over a band of width Awi rad/s centered atfrequency wO. Then the autocorrelation function ofi (t) is

Ri(T) = Si [sin(Aw1r/2)/(AwT/2)] exp(jcor) .

Substituting X = Ti and noting that

Aw T-/2 = X (AcolAo)(wo T.) = ' BiOi

where Bi is the fractional bandwidth,

Bi = Awilwo

we find from (28a) that

P== [sinI (B,O5)/' (B,O,)] exp(qi) -

SIR = (Sd/Si)(t2/02)(l -Re {dPI })/(l - Re XdPi*})As long as Xi :$ kd (mod 2ir), this is1o

Similarly, we may assume the desired signal to have aflat spectral density over a fractional bandwidth Bd, so

= [sin : (BdOd)I' (Bd4d)] exp(4id).SIR = SlSd (50)

which is the reciprocal of the SIR coming into the array.

I.e., an interfering signal 20 dB above the desired signalat the array input comes out 20 dB below the desiredsignal. This property is the reason we refer to this array

as a power-inversion adaptive array.

For finite gain and nonzero noise, the array approxi-mately inverts the power ratio of two signals, as long as

the noise is small and the loop gain is large. For example,Fig. 10 (for K = 1) shows that for td = 20 dB and ti =40 dB (so the input SIR is -20 dB) the output SINR is+18 dB.

E. Bandwidth Effects

The results presented above were all for CW signals.For signals with nonzero bandwidth, the performance of

9This result assumes we take no advantage of electromagneticpolarization differences between the two signals. If the two sig-nals have different polarizations, we may improve the perform-ance at close separations by including cross-polarized elementsin the adapative array.

loAs 4 approaches tr, higher and higher values of A arerequired to make the approximation in (49) hold.

For given bandwidths and arrival angles, these Pd and Pi

may be substituted into (37), (39), and (41) and thearray output SINR may be calculated from (47). Note

that the nonzero bandwidth case differs from the CW

case only in that IPd < I or jpij< 1.Fig. 16 shows a typical plot of the output SINR for

Od = 00, Oi = 500, K = 0.1, i=40 dB, 0. Bd <0.2, and

for several values of interference bandwidth in the range

0 < Bi < 0.2. (The curve for Bi = 0 is the same as the

-i= 40 dB curve in Fig. 9.) It may be seen that as Biincreases from zero, the output SINR drops for lowervalues of input SNR.

The reason for this behavior may be understood from

the array patterns, shown in Fig. 17. We find that in-creasing the interference bandwidth causes the magni-tude of the pattern to drop. This behavior occurs becausethe null depth varies with frequency over the interfer-ence bandwidth. As the bandwidth increases, more andmore interference power appears at the array output;

11 The method used here to analyze the effect of bandwidthwas originally suggested by D.M. DiCarlo [ 141. See also Baird,Martin, Rassweiler, and Zahm [ 15 ] and Rodgers and Compton[ 161 for additional results on bandwidth.

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-15, NO. 6 NOVEMBER 1979

(51)

(52)

(53)

812

0 10 20 30 40 50 60INPUT SNR (dB)

Fig. 16. Effect of interference bandwidth on output SINR.

d=0 i=50 ,K=0.1,t.=40dB,0.B .0.2.

to compensate for this, the feedback lowers the valuesof the array weights. The result is to lower the array

response in all directions, including the desired-signaldirection.On the other hand, we note that all curves in Fig. 16

coalesce for higher values of input SNR. The reason isthat with high input SNR, the array weights are alreadyreduced more by the presence of both desired signal andinterference than they are by the bandwidth. I.e., withboth strong desired signal and strong interference, sincethe array cannot null both signals, it turns down the weightsand reduces the overall pattern magnitude. At high inputSNR, this effect is stronger than the pattern reduction due

to bandwidth.We also remark that the output SINR is more sensitive

to interference bandwidth the higher the input interfer-ence power. Fig. 18 shows a plot of output SINR similarto Fig. 16 except that the input interference-to-noiseratio is now +60 dB instead of +40 dB. It is seen thatmuch smaller bandwidths are required to produce a givenSINR degradation when ti = 60 dB than when ti = 40 dB.

The curves in Figs. 16-18 have been computed for a

desired signal bandwidth Bd = 0. However, it is foundthat Bd has no noticeable effect on these curves over therange 0 S Bd < 0.2, even if the desired signal and interfer-ence arrival angles are interchanged (see discussion belowabout the effect of arrival angle on bandwidth degradation).The reason that Bd has little effect is that the pattern ismuch less frequency sensitive in the desired signal direc-tion than in the null, as discussed earlier.

Finally, we note that it is the product of the inter-ference bandwidth Bi and the interelement phase shiftki that affects pi (see (52) ). For this reason, for inter-ference at broadside (/j = 0), bandwidth has no effecton array performance, but for interference at endfire

(oi = 7r), bandwidth has its greatest effect. For applica-tions where interference bandwidth is significant, thedesigner may wish to minimize bandwidth effects bypositioning the array so its broadside direction is closeto the interference arrival angle, if that is possible.

Fig. 17. Effect of interference bandwidth on array patterns.

od =0 i=50',d=OdB,K=0.1,Bd=0,ti=40dB.Fig. 18. Effect of interference bandwidth on output SINR.

=0 6, =50 ,K =0.1, =60 dB,0 <0.05.d d

3c

20

10

U)

-10

-20

-30-30 -20 -10 0 10 20 30 40 50 60

INPUT SNR (dB)

V. Conclusions

This paper has discussed the concept of the power-

inversion adaptive array. Such an array is based on theuse of the Howells-Applebaum feedback to minimize thearray output power. When two signals are received by a

two-element array with omnidirectional element patterns

and the proper weight feedback, the array will improvethe ratio of the weak to the strong signal by nulling thestrong signal. This technique is useful so long as the strong

signal is interference.The problem for the designer wishing to use this idea

is to control the situation so the array does not null thedesired signal. In situations where the desired-signal powerremains near thermal-noise level, suitable performancecan be obtained by appropriate choice of K. The widerthe dynamic range of the desired signal above thermalnoise, however, the more difficult it becomes to deter-mine a loop-gain constant suitable for all situations. Incases where no single value of loop gain is satisfactory,the designer may wish to consider a system where thepower-inversion feedback is switched on or off depend-ing on the presence of interference. Also, schemes involv-ing the continuous adjustment or control of the loopgain K are possible.

COMPTON: THE POWER-INVERSION ADAPTIVE ARRAY: CONCEPI AND PERFORMANCE

20

-10

0.05

2

z(

DESI REDSIGNAL

dB

-30 -20 -10

813

References

[1] R.T. Compton, Jr., "Adaptive Arrays: On power equal-ization with proportional control," Ohio State Univ. Electro-Science Lab., Dept. Electrical Eng., Rep. 3234-1, Dec. 1971.

[21 C.W. Schwegman and R.T. Compton, Jr., "Power inversionin a two-element adaptive array," Ohio State Univ. Electro-Science Lab., Dept. Electrical Eng., Rep. 3433-3, Dec. 1972,ASTIA document AD 758690.

[3] C.L. Zahm, "Application of adaptive arrays to suppressstrong jammers in the presence of weak signals," IEEETrans. Aerosp. Electron. Syst., vol. AES-9, p. 260, Mar. 1973.

[4] B. Widrow, P.E. Mantey, L.J. Griffiths, and B.B. Goode,"Adaptive antenna systems," Proc. IEEE, vol. 55, p. 2143,Dec. 1967.

[51 R.L. Riegler and R.T. Compton, Jr., "An adaptive arrayfor interference rejection," Proc. IEEE, vol. 61, p. 748,June 1973.

[6] R.T. Compton, Jr., R.J. Huff, W.G. Swarner, and A.A.Ksienski, "Adaptive arrays for communication systems:An overview of research at the Ohio State University,"IEEE Trans. Antennas Propagat., vol. AP-24, p. 599,Sept. 1976.

[7] R.T. Compton, Jr., "An adaptive array in a spread spec-trum communication system," Proc. IEEE, vol. 66, p. 289,Mar. 1978.

[8] P.W. Howells, "Explorations in fixed and adaptive resolu-tion at GE and SURC," IEEE Trans. Antennas Propagat.,vol. AP-24, p. 575, Sept. 1976.

[9] S.P. Applebaum, "Adaptive arrays," IEEE Trans. AntennasPropagat., vol. AP-24, p. 585, Sept. 1976.

[101 J. Dugundji, "Envelopes and preenvelopes of real wave-forms," IRE Trans. Inform. Theory, vol. IT-4, p. 53, Mar.1958.

[11] E. Bedrosian, "The analytic signal representation of modu-lated waveforms," Proc. IRE, vol. 50, p. 2071, Oct. 1962.

[12] T.G. Kincaid, "The complex representation of signals,"General Electric Co., Heavy Military Electronics Dept.,Syracuse, N.Y. 13201, Tech. Rep. R67EMH5, Oct. 1966.

[13] R.T. Compton, Jr., "An experimental four-element adap-tive array," IEEE Trans. Antennas Propagat., vol. AP-24,p. 697, Sept. 1976.

[14] D.M. DiCarlo, "The effect of interference bandwidth onthe performance of quadrature weighted adaptive arrays,"private communication.

[15] C.A. Baird, G.P. Martin, G.G. Rassweiler, and C.L. Zahm,"Adaptive processing for antenna arrays," RadiationSystems Division, Harris Intertype Corp., Melbourne, Fla. 32901,Final Rep., June 1972.

[16] W.E. Rodgers and R.T. Compton, Jr., "Adaptive arraybandwidth with tapped delay-line processing," IEEE Trans.Aerosp. Electron. Syst., vol. AES-15, p. 21, Jan. 1979.

R.T. Compton, Jr., was born in St. Louis, Missouri, on July 26, 1935. He received theS.B. degree from M.I.T. in 1958 and the M.Sc. and Ph.D. degrees from The Ohio StateUniversity, Columbus, in 1961 and 1964, respectively, all in electrical engineering.

He is a Professor of electrical engineering at The Ohio State University. From 1965 to1967 he was an Assistant Professor of Engineering at Case Institute of Technology,Cleveland, Ohio, and from 1967 to 1968 he was a National Science Foundation Post-doctoral Fellow at the Technische Hochschule, Munich, Germany.

Dr. Compton is a member of Sigma Xi and Pi Mu Epsilon.

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-15, NO. 6 NOVEMBER 1979814