AD-A116 S86 OHIO STATE tR4IV COLUJMBUS ELECTROSCIENCE LAB /9,ADAPTIVE ARRAY BEHAVIOR WITH SINUSOI DAL ENVELOPE MODULATED INTE--ETC(U)MAR 82 A S AL-RUWAIS. Rt T COMPTON N00019-81-C-0093

UNCLASSIFIED ESI-713603-5

mEEEE~E~hhh.MEEEM["EI".

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ADAPTIVE ARRAY BEHAVIOR WITH SINUSOIDAL

ENVELOPE MODULATEDI The Ohio State University

i Abdulaziz S. AI-Ruwais and R.T. Compton, Jr.

. The Ohio State University

S I :- ElectroScience LaboratoryI Department of Electrical Engineering

Columbus, Ohio 43212

Technical Report 713603-5

Contract N00019-81-C-0093

March 1982

I

Ii

APPROVED FOR PUBLIC Rftft[ .,, DISTRIBUTION UNLIMITED

J Naval Air Systems CommandWashington, D.C. 20361I * - JUL 6 1982

T A

. .:

50272 -101REPORT DOCUMENTATION 1. REPORT NO. 2. 3. Recai,nt's Accisiori No

__ PAGE ~ ~ .A/'~~4. Title and Subtitle S. Report Date

Adaptive Array Behavior with Sinusoidal Envelope Modulated March 1982Interference

7. Author(s) S. performing Organization Rept. NoAbdulaziz S. Al-Ruwais and R.T. Compton, Jr. 713603-5

9. Performing Organization Name and Address 10. Project/ Task /Work Unit No

The Ohio State University 11. ContractlC) or Grant(G) No

ElectroScience LaboratoryDepartment of Electrical Engineering i(c) NOOO9-81-C-OO93Columbus, Ohio 43212 IG

12. Sponsoring organization Name and Address 13. Type of Report & Period Covered

Naval Air Systems Command Technical ReportWashington, D.C. 20361

IS. Supplementary Notes

IS. Abstract (Limit: 200 words)

'The behavior of an LMS adaptive array with modul-ited interference isdescribed. An interference signal with sinusoidal, double-sideband,suppressed-carrier modulation is assumed. It is shown that such interferencecauses the array to modulate the desired signal envelope but not its phase.The amount of the desired signal modulation is determ~ined as a function ofsignal arrival angles and powers and the modulation frequency of theinterference.

Such interference also causes the array output s'gnal-to-interference-plus-noise ratio (SINR) to vary with time. However, it is shown that when thedesired signal is a digital communication signal, the averaged bit errorprobability is essentially the same as for CW interference.

17. Docuent Analysis a. Oeacptori dpieAry

Interference Rejection

Antennas

b. Idiantileire/Open-Ended Terms

c. COSATI Flield/Group

IL Availability Statement 19.Security Class(This Report) 71. No of Page%

APPROVED FOR PUBLIC RELF.Sr, Unclassified 40DISTRIBUTION UNLlN,1IcQ 2 ecrt Class (This Pact) 72 22

-Unclassified(See ANSIS-239. If) See Iftetruttlons on Rteverse OPTIONAL FORM 272 14-71j)

ii fDepartmntn of Comme.rce

TABLE OF CONTENTS

Page

LIST OF FIGURES

I. INTRODUCTION I

I. FORMULATION OF THE PROBLEM 2

III. RESULTS 17

A. Typical Waveforms 17

B. The Effect of Angle of Arrival 19

C. The Effect of Modulation Frequency 22

0. The Effect of Interference-to-NoisL Ratio 28

E. The Effect of Desired Signal-to-Noise Ratio 28

F. Bit Error Probability 28

V. REFERENCES 40

LK..

iiiL

LIST OF FIGURESPage

Figure 1. A Three-Element LMS Array 3

Figure 2. Sinusoidally Modulated Interference 5

Figure 3. adn(t') versus timeed07 , ei =T 5 d--Ml dB,d= -,,Ed1

Ei=20 dB, fm' 2 18

Figure 4. INR versus time-d= N-=57 ,-d=10 dB,ci:20 d8, fm'= 2 20

Figure 5. SINR versus timeBd= , ,i :d= 0 dB,i=20 dB, fm' =2 21W

Figure 6. m versus oi-ed--,d=O dB, fm'=2 23

Figure 7. amax versus 0iOd- ,U - O"B, fm'=2 24

Figure 8. m versus f-d---, =d-B, Ei= 20 dB 25

Figure 9. a versus fBd= Ed-- i=20 dB 26

Figure 10. adn(t') versus timeOd=O ° , 7 ?T dB, qi=20 dB 27

Figure 11. m versus f

80-, 6i=5 d=0 dB 29

Figure 12. a versus f

d i,° , d:O dB 30

Figure 13. a t versus time8d:13e, oi--T, d-dB, fm':l 31

Figure 14. m versus f

q,=20 dB 32

iv

I

Page

Figure 15. amax versus f. , ei=5c, i=20 dB 33

Figure 16. a{ j versus f 't'ed=U-, oT-, i20dB 34

Figure 17. Bit Error Probability versus fm'Od'-, =i3O', :6 dJ B 37

Figure 18. Bit Error Probability versus fIL"d-O ,&==TB, i=20 dB 38

Is

I. INTRODUCTION

Adaptive arrays based on the LMS algorithm of Widrow, et al[1] can be

used to protect communication systems from interference. These antennas

optimize reception of a desired signal while simultaneously nulling

interference. For a given set of incoming signals, the LMS array yields the

maximum attainable signal-to-interference-plus-noise ratio (SINR) at the array

output[2,3].

Most studies of adaptive array performance have assumed interference

signals that have constant power. However, a modulated interference signal

may be more difficult for the array to null. If the modulation rate of the

interference is close to the natural response rate of the array, such a signal

may keep the array in a continual transient state.

In a previous paper[4], one of the authors examined the performance of an

LMS array with pulsed interference. It was shown, for example, that when the

array is used to receive a differential phase-shift keyed (DPSK) signal,

pulsed interference can increase the bit error probability more than CW

interference does. However, the increase due to the pulse modulation is

small, and occurs only if the pulse repetition rate, duty cycle and

interference power are all properly chosen. Otherwise, pulsed interference

has less effect than CW interference.

In this report, we study the effect of another type of envelope modulated

interference on the LMS array. Specifically, we consider an interference

signal with double-sideband, suppressed carrier, sinusoidal envelope

modulation. This type of signal has been chosen for two reasons. First,

such interference contains a spectral line on each side of the desired signal

frequency; by changing the interference modulation frequency, we can explore

the frequency response of the LMS loop. (The LMS loop is nonlinear in the

kL1

input signals.) Second, such modulation yields a system of differential

equations for the array weights that can be solved. In general, an

interference signal with arbitrary modulation leads to an intractable

mathematical problem.

As we shall show, modulated interference of this type has two effects on

the array. First, it causes the array to modulate the desired signal envelope

(but not its phase!). Second, it makes the output SINR from the array vary

with time. However, it turns out that this SINR variation causes almost no

change in average bit error probability (as compared with CW interference)

when the array is used in a digital communication system.

In section II of the report we establish notation, define the interference

signal and solve for the weight response of the adaptive array. In section

III we present calculated results obtained from the equations in section II

and discuss the effect of each interference signal parameter on the array

performance. Section IV contains the conclusions.

II. FORMULATION OF THE PROBLEM

Consider an LMS adaptive array[i] consisting of three isotropic elements

a half wavelength apart, as shown in Figure 1. The analytic signal Zj(t) from

element j is multiplied by complex weight wj and then summed to produce the

array output (t). The error signal (t), which is the difference between the

array output signal and the reference signal F(t), forms the input to a

feedback system that adjusts the wj. In the LMS array, the weights satisfy

the system of equations[1,5]

dwdw + koW = kS, (I)

where W=(wl, w2 , w3 )T is the weight vector, € is the covariance matrix,

I = E[X*XT], (2)

2

x3( x

IN3 IN2 I

Figure 1.A Three-Element LMS Array

3

S is the reference correlation vector,

S = E[X*F(t)], (3)

and k is the LMS loop gain. In these equations, X is the signal vector,

X = [Wl(t), x2(t), x3(t)] T , (4)

T denotes transpose, * complex conjugate, and E[.] expectation.

We assume that a desired and an interference signal are incident on the

array and that thermal noise is also present in each element signal. The

signal vector then contains three terms,

X = Xd + Xi + Xn, (5)

where Xd, Xi and Xn are the desired, interference and thermal noise vectors,

respectively.

Specifically, we assume a CW desired signal is incident from angle Od

relative to broadside. (0 is defined in Figure 1.) The desired signal vector

is then

Xd = Ade jko t + ld )Ud, (6)

where Ad is the amplitude, wo is the carrier frequency, 'Pd is the carrier

phase angle, and Ud is a vector containing the interelement phase shifts,

Ud = (1, e - d, e- 2 d)T (7)

with

d = irsingd. (8)

We assume 4)d to be a random variable uniformly distributed on (0,2,).

Next, we assume an envelope modulated interference signal as shown in

Figure 2, arriving from angle 3i. The interference signal vector is then

4

INTERFERENCESIGNAL

Figure 2. Sinusoidally Modulated Interference

5

ai(t)

Xi = e j (°t+¢i) ai (t-T i )e -jo ° Ti , (9)

ai (t-2Ti )e-J2woTi

where ai(t) is the envelope modulation as received on element 1, i is the

carrier phase angle, and Ti is the interelement time delay given by

Ti = sinei. (10)

We assume pi is a random variable uniformly distributed on (0,21T) and

statistically independent of 'd-

In this paper, we shall study the case where ai(t) is a simple sinusoid,

a i ( t ) = Aicosumt, (1

with Ai and u. the amplitude and frequency, respectively. I.e., the

interference is a double-sideband, suppressed carrier signal. As mentioned

above, we have chosen this type of modulation for two reasons. First, this

signal has two spectral lines, one on each side of the desired signal

frequency. By varying am we can change the separation between the desired and

interference spectral lines and examine the frequency response of the array

weights. Second, this type of modulation leads to a tractable mathematical

problem. More complicated modulations yield differential equations for the

weight vector that are more difficult to solve.*

We shall assume the frequency u in (11) is very small compared to the

carrier frequency wo. Under this condition the time delay Ti represents a

negligible delay in the modulation envelope, so the modulation on each element

signal is essentially the same,

a i ( t ) = a i ( t - Ti ) t a i ( t - 2 T i ) .

Specifically, the modulation in (11) yields a 3-term recursion relation inEg.(57) below. If, for example, ai(t) is assumed to contain a carrier term inaddition to the two sidebands, a 5-term recursion relation results instead of(57). The weight differential equation then cannot be solved by the methodspresented here. 6

* In this case Xi reduces to

Xi=Ai cosutej (wt+Ji )UI, (13)

where

Ui [1, e~jie ji2 if' (14)

with

-rnO 1.6 (15)

Finally, we assume the thermal noise vector is given by

nn=[ltK() 3(t)] ,(16)

where the Kj(t) are zero-mean, gaussian thermal noise voltages, all

statistically independent of each other, and each of power 02. Thus,

with S the Kronecker delta. Also, we assume the j(t) are statistically

independent of lpd and 'pi.

Uinder these assumptions, the covariance matrix in (2) becomes

D + i + 'Dn' 1

where

'= Ad 21Jd*UdT, (19)

Oi=Ai~cOs?uftJi *UiT

= (1/2)Ai2[1+cOs?0'Tt11)l*UlT, (20)

and

= a2=,((21)

where I is the identity matrix.7

To compute the reference correlation vector S in (3), we must first define

the reference signal F(t). In practice, the reference signal is usually derived

from the array output[6,71. It must be a signal correlated with the desired

signal and uncorrelated with the interference. Here we assume the reference

signal to be

F(t) = Arej(clot+ d). (22) j

Eq.(3) then yields

S = ArAdUd*. (23)

Eqs.(18)-(21) and (23) can now be inserted in (1) to give the differential

equation for the weight vector W,

S-+ k[a 2I+Ad 2Ud *UdT+(I/2)A (I+cos2umt)Ui*UiT]W = kArAdUd*. (24)

Before attempting to solve (24), it is helpful to put it in normalized form.

First, dividing by ko2 gives

+ [I+ dUd*UdT+(1/2) i(l+cos2m't')IJi*UiT]W(t-) =A (25)

where

d2 = input signal-to-noise ratio (SNR) per element,

.d 2

Ai = peak input interference-to-noise ratio (INR) per element.

and where we have also used normalized time and frequency variables,

4n'= wm/ko 2 = normalized modulation frequency,

t'= ko2t = normalized time.

8

L

Next we note that the constant Ar/o on the right side of (25) will just appear

as a scale factor in the solution for W. It has no effect on the array output

signal-to-noise ratios to be computed below. Hence we arbitrarily set Ar/c=1 to

eliminate it. Finally, by defining

Oo = I + dUd*UiT + (1/2)iUi*UiT, (26)

we can write (25) in the form,

+ [Oo+(1/2)Eicos(2cm't')Ui*uiT]W(tI) =vdUd*. (27)

Eq.(27) is a linear vector differential equation with a constant source

term on the right but with periodic coefficients. The general properties of

such differential equations have been described by D'Angelo[8]. He has shown

that the solution for W(t') will be a periodic function of time once the initial

transients have died out.

In this paper, we shall not consider the initial transient behavior, but

shall concentrate on the periodic steady-state solution for W(t'). Once W(t')

is periodic, it can be represented as a Fourier series

j ne2n (q 't 'W(t') Cne (28)

where Cn is a vector Fourier coefficient. Substituting (28) into (27),

replacing cos(2um't') with exponentials, and collecting terms with the same

frequency, we find that the coefficients Cn must satisfy

Y [(oo+j2ncm'I)Cn+(/4) iUi*iT(Cn~l+Cn+l)3ejnu nt= /dUd*. (29)

n=-

Enforcing this equation for each frequency component separately gives, for n=O,

OoCo+(1/4)&iUi*uiT(C_1+Cl) FdUd*, (30)

9

and for n*O,

(to+j2nLm'I)Cn + (1/4)qiUi*UiT(Cnl+Cn+l) = 0. (31)

These equations may be solved for the Cn by expressing each Cn in terms of

its components parallel and perpendicular to the vector Ui*. To do this, we

first form a set of three orthonormal basis vectors* as follows. Let el be a

unit vector parallel to Ui*,

el =(32)Ui TU *

and let e2 and e3 be unit vectors perpendicular to each other and to el, i.e.,

ejtek = 6jk, (33)

(where t is conjugate transpose). In particular, let e2 be in the plane defined

by Ud* and Ui*,

e2 = C[Ud*-Ui*], (34)

where and c are constants. Enforcing the orthogonality condition elte2=0

and choosing C so e2 has unit magnitude yields,

e2 = 1 [Ud* - ----d--i*]- (35)

V / UdTUd uiTUd* UiTUi*

uiTui*

e3 is easily found from el and e2 but will not be needed in the sequel, so we

shall not compute it explicitly.

Each coefficient Cn may be written in terms of the unit vectors ek as

3Cn = Yn,kek, (36)

k=1

where the ca,k are scalar coefficients. an,k is the component of Cn along the

unit vector ek.

* The array has 3 elements, so W(t') and Cn each have 3 components. Hence 3

basis vectors are needed to span the space for Cn.

10

Substituting Eq.(36) into (30) and (31) and utilizing the fact that

UjTej=0 for j*I, we obtain

-t a,kek + (14& = /dUd*, (37)k=1

and

I~ an,k to+j2num'I)ek+(1/4) iIi * ~ Ii+an+1,1)=O, n*0. (38)

Multiplying these two equations by ejt on the left gives

3

k=1

and

k cnk(8jk+j2nu~m'Sjk)+(1/4) iIUi*12 (an...,1+can1,1)6jI=O, n*01 (40)

where we have defined

ajk = ejtIoek. (41)

Eqs.(39) and (40) hold for 1<j<3. THe values of ajk are readily found by

substituting (26), (32) and (35) into (41) and making use of (33) for e3. The

results are

B = 1+Eu Iidi 1+(1/2)iUiTUi*, (42)

622 = 1+dP, (43)

B12 = 21 = d"Fujud (44)

813 = 831 823 = 32 =0, (45)

and

033 =, (46)

where

P= UdTUd IuiTud*lZ2. (47)UiTui

*

In addition, one finds

elti d t(jiTUd (48)

UiTui*

e2tUd* =P, (49)

= and

e3tUd* 0. (50)

Our problem now is to solve (39) and (40) for the an, k . We may do this as

follows. Writing out (39) and (40) for l-j<3 gives the six equations,

S1 IIO, 1+6j 2 a0 , 2 +(1/4) Fi I Ui*) 2 (i, 1+Q1, 1) =Vdeltd*, (51)

82 1co, 1+ B22MO, 2=de2tUd*, (52)

ao,3=0, (53)

( 611+j2n uin' )aen, j+ 12an,2=- (I/4)q rI Ui *i1 e.,ie1i ) (54)

21 an, 1+ ( 22+J 2n ' )c'n, 2=0, (55)

and

01n,3 0, (56)

where the last three equations hold for n*O. Solving (54) and (55) for Qn,j and

cn,2 yields

an,l: -Yn(an-1,1+-n+1,l (57)

12

and

- 612 (58)n,2 - (22+j 2n Lm,'

where

= F i (B22+j2nun') IUi* 2 (59)• Yn ~~4[ ( B1l+j2n~m' ) (22+j2nom') TM B1212 I5)

Eq.(58) allows one to calculate an,? from an,1, so the problem is reduced

to determining the an,, . To find the an1', we must solve (57).

Eq.(57) is a three-term recursion relation for the coefficients an,l-

Given cn,1 for two successive n, one could calculate all the remaining an,1 from

(57). However, if one starts from arbitrary initial values for the an,1, one

finds that in general an,1 will not approach zero as n+±-, so the series in (28)

will not converge. Only the correct initial values of an,1 will cause the

series to converge.*

To determine the correct initial values for the an, we manipulate (57)

into two forms,

an1 _ -+for n>0, (60)ani,1 I 1+ QI

Yn an,I

and

S -1 for n<O. (61)a n+l ,I + _-Yn °n,1

Consider first the behavior of an,1 for large positive n. Eq.(59) shows that

for large n,

1 8jn~m' (62)Yn r'i IU!* 2

*The situation here is similar to that encountered in solving the Mathieuequation[9].

13

Hence, we conclude from (60) that anli/o._1 will approach zero as n+-'w, as long

as the next term, an+,i/an,, also approaches zero. Therefore, the approach we

shall take for finding the an,1 is to assume oN+1,1/oN,1=O for some suitably

large N and then use (60) to compute an,i/an_1,1 iteratively, starting at n=N

and working down. This procedure yields c~l i/al as a continued fraction:

2 , = - 1 _am

+ - -+ -1Y-2-I + - (63)

Y3+ -1

1YN

where the continued fraction terminates at YN. A similar continued fraction

may be found for a_1,1/co,I from (61). However, sinceY,' (64)

Y-n Yn*,(4

the result is just

a-) = t(65)

Hence it is sufficient to evaluate (63) for cI,I/aOI and then use (65) to find

a-I I //' ,I"

Suppose the continued fraction in (63) yields a value r:

r. (66)aO,1

We may then substitute

1,I ro0 ,l, (67)

and

•, (68)

14

into (51) to obtain

[B1l+(I/4)Ei Ui *12 (r+r*)]ao,l+R12a0,2=i/deltUd*. (69)

Solving (69) and (52) simultaneously yields

-O,?1 = / ?( 2e, tUd*' aetUd*) (70)aO I -[B11+(1/2)Ei IUi,*1Re(r)]B22IB12 12"

Once co),1 has been computed, aO,2 may be found from (52):

00,2 = frje2tUd* -1 oIn1 (71)822

We have now developed all the equations needed to compute the periodic

array weights in Eq.(28). To summarize, the procedure is as follows. Starting

from the signal parameters ed, Cd, ei, Fi, um', we first compute the vectors Ud

and Ui in (7) and (14), then the 8jk in (42)-(47) and the Yn in (59). From the

Yn, we evaluate r in (66) using the continued fraction in (63). (The number of

terms used in the continued fraction, N, is chosen large enough that the value

of r obtained is insensitive to small changes in N.) After r is found, we use

(70) to find a0,I, (71) to find a0,2, (67) to find a1,1, (57) to find Qn,l for

n*0,1, and finally (58) to find an,2 for n*O. From the cn,1 and an,2 (and using

1an,3=O), the Cn may be evaluated from (36) and then W(t') from (28).

The time-varying weights in (28) have two effects on array performance.

First, they cause the array to modulate the desired signal. (The array becomes

a time-varying, or frequency dispersive, channel[lO].) Second, the array output

signal-to-interference-plus-noise ratio (SINR) varies periodically with time.

Our major purpose in this paper is to evaluate these two effects.

Given a time-varying weight vector W(t'), the desired signal component of

the array output is

Sd(t') = AdWT(t,)UdeJ( olt ' +Pd), (72)

15

(where wo'=wo/ko 2). To study the modulation on d(t), we define

ad(t')e J nd( ) = AdWT(t')Ud. (73)

Then ad(t')=AdIWT(t')UdI is the envelope modulation and nd(t')=<WT(t')Ud is the

phase modulation. Furthermore, we define adn(t') to be the envelope normalized

to its value in the absence of interference, i.e.,

adn(t') a(t'))AdlWoTUdI

where Wo is the steady-state weight vector that would occur without

interference,

Wo = (Od + On) - S. (75)

(Od, On and S are given in (19),(21) and (23).) We present our results in terms

of adn(t') rather than ad(t'), because the effect of the interference can be

determined most easily by comparing the value of adn(t') with unity.

The output desired signal power is

Pd(t') =(1/2)E{Id(t')i21 = (1/2)Ad 2IWT(t')Udl2. (76)

The output interference signal is

si(t') = Aicos(m't')ej(wo't '+ i )WT(t ')Ui, (77)

and the output interference power is

Pi(t') = (1/2)E{Is'i(t')1 21 = (1/2)Ai2cos2(Um't')IWT(t')Ud12. (78)

The output thermal noise power is

2

Pn(t') : - Wt(t')W(t'). (79)

From these quantities, the output interference-to-noise ratio (INR),

NR Pi(t')INR= Pn(t, )' (80)

16

and the output signal-to-Interference-plus-noise ratio (SINR),

SINR = t'(81)Pl~t, ) + Pn(t, ) , 81

may be computed.

With this background, we now discuss the numerical results obtained from

these equations.

Il1. RESULTS

In this section, we describe the effect of the amplitude modulated

interference signal on the array. In Part A, we show typical curves of desired

signal modulation, output INR and SINR as functions of time. In Parts B-E, we

describe the effect of each signal parameter on the desired signal modulation.

In Part F, we assume the array is used in a digital communication system and

show how the bit error probability is affected by the interference.

A. Typical Waveforms

Figure 3 shows a typical curve of the normalized envelope modulation

adn(t') for the case Od=0 0 , ei=50 , d=10 dB, Ei=20 dB and fm'=um'/2w=2. * As

may be seen, there is substantial envelope modulation produced by the

interference for this set of parameters.

When the phase modulation nd(t') is calculated, one discovers an

interesting result: nd(t') is constant, not only for the parameters used in

Figure 3, but for all values of the signal parameters. Thus, this

interference does not produce phase modulation on the desired signal. A

similar result was also found previously for a pulsed interference signal[4].

With pulsed interference, it is possible to prove mathematically that there is

no phase modulation. For the interference considered here, we have not been

able to obtain such a mathematical proof. However, calculations of nd(t')

based on the equations in Section II consistently show that there is no phase

*Because the frequency of the periodic term in (27) is 2um', the envelope

modulation is periodic with period 1/2fm'=. 2 5. Figure 3 shows one period of themodulation.

17

0-

C5

-6o

0 0.05 0.10 0.15 0.20 0. 25 -t

T IME

Figure 3. ad,~(tl) versus time

q=2 dB, fm'=2

18

modulation, regardless of the values of ei, Or fm'. This result is of

considerable importance for adaptive arrays to be used in communication

systems with phase modulation.

Next, Figures 4 and 5 show the output INR and SINR as functions of time,

over one period, for the same signal parameters as in Figure 3. The sharp

drop in INR in Figure 4 at t'=.125 occurs when the incoming interference power

is zero. The variation in the SINR in Figure 5 is primarily due to the

variation in desired signal amplitude as shown in Figure 3. However, the peak

SINR occurs slightly earlier than the peak of adn(t'), reflecting the

influence of the time-varying interference and noise powers on the SINR.

Figures 3, 4 and 5 are intended merely to illustrate typical array behavior

with the interference modulation in Eq.(11). In general, the desired signal

modulation and the SINR variation change substantially as the signal parameters

ed, Ed, 0i, i and fm' are varied. In Sections B-E, we describe in detail the

effect of each signal parameter on the desired signal modulation. In Section F,

we show how the SINR variations affect bit error probability when the array is

used in a digital communication system.

To describe the desired signal modulation, we shall define three

quantities. First, we let amin and amax be the minimum and maximum values of

adn(t') during the modulation period. Then, we define

M= amax - amin. (82)

amax

m is the envelope variation normalized to its peak. It may be thought of as

"fractional modulation", analogous to percentage modulation in AM. In Parts

B-E below, we describe the effect of each signal parameter on amax and m.

B. The Effect of Angle of Arrival

Desired signal modulation effects are small unless 9j is close to Od"

When 1 is far from d, the envelope variation m is small and the peak amax is

19

TIME0 0.05 0.10 0.15 0.20 0. 25V

I0i l 1 1 1 0

0-

0

0-

0

01

02

6

5-

z

00.05 0.10 0.15 02 .25-t

TIME

Figure 5. SINR versus time

&i=20 dB, fm'=2

21

I

close to unity.*

Figures 6 and 7 show typical curves of m and amax as functions of 6i for

Od= 00 , d=10 dB and fm'=2. Four different curves are shown, for i=5, 10, 15

and 20 dB. It is seen that m is large and amax is small only when ei is close

to Od. Similar curves for od*O show that the modulation is always small

unless 6i-Od.

C. The Effect of Modulation Frequency

The variation m and the peak amax are large at low fm' and drop as fm'

increases. This result is illustrated in Figures 8 and 9, which show m and

amax as a function of frequency fm' for Od=O 0 , d=0 dB, i20 dB, and for

several values of ei .

This behavior is due to the array speed of response. For low fm', the

array is fast enough to follow the interference modulation. The array pattern

changes continuously as the interference power varies. At high fm', however,

the array weights are too slow to keep up with the modulation. Hence, as fm

is increased, the array weights settle into steady-state values reflecting the

time-averaged interference power. As they do so, the desired signal

modulation drops.

Figure 8 shows a small dip in m for intermediate values of fm' for some

Oi . This effect may be seen in the ei=15 ° curve, for example. Such behavior

is due to the way the shape of the modulation envelope changes as fm' varies.

Figure 10 illustrates this effect. It shows adn(t') as a function of time for

oi15° and for two values of fm', .015 and .04. (In order to plot the two

curves on the same graph, we have plotted both versuis fm't'.) Figure 10 shows

that amax-amin is essentially the same for f,,'=.04 as for f,'=.015. However,

m is lower at fm '=.015 than at fl'=.04 because amax is higher.

*An exception to this occurs if the interference causes a grating null in the

desired signal direction[I1]. For isotropic elements a half wavelength apart asconsidered here, there are no grating nulls.

22

m1.0

0.8 L --- Ci=20dB

15 dB

0.6 i

0.41 10 dB

,04fiO I

0.2 5dB

I. / ..

-90 -60 -30 0 30 60 90

e~ (DEGREES)

Figure 6. m versus e

Wd-=Uf--7td~O dB, fm'=2

23

-90~~~ -6m-00 a0 60 9

1.(DGRES

0.4

m

C5

1525

CD

0

07

0 L 1 I 111111 11114111 - I 11tH11 0 I 1111111 1 1 1111111 z I 1111111 3 1 1,1111114

10 10 10 10 10 10 10 10

Figure 9. ama versusfFi=2f dR

26

co, d

0

z

¢0- I0

l- i l J ,i i~ l ,,i I i . ,,00 0.1 0. 2 0.3 0.4 0.5

Figure 10. a rLt' versus time

27

D. The Effect of Interference-to-Noise Ratio

For low fm, the variation in is largest for high INR. For intermediate

values of fn, m peaks at intermediate INR. amax is unity at low fm' and drops

to a minimum value as fm' increases. The higher the INR, the farther amax drops

at high fm'. Figures 11 and 12 illustrate these results for the case Od= 00 ,

ei=50 , ~d=0 dB and Ei between 0 and 30 dB.

For intermediate fm', it may be seen in Figure 11 that m peaks at

intermediate INR. This behavior is due to the way the modulation envelope

changes with INR and to the way m is defined. To illustrate this, Figure 13

shows the modulation envelope adn(t') over one period for fm'=l and several

INR. It may be seen that amax-amin at first increases with INR but then

decreases, because amin, the smallest value of adn(t') during the cycle, does

not drop below about 0.17.

E. The Effect of Desired Signal-to-Noise Ratio

The variation m is largest and the peak amax is smallest for low rd. As

d is increased, m decreases and amax increases.

Figures 14 and 15 illustrate this behavior. Figure 14 shows m and Figure

15 shows amnax, both versus fm' for Od= 0 0 , ei=50, i=20 dB and for five values

of d between 0 and 20 dB. It is seen that for a given SNR, m peaks slightly

at intermediate values of fm'. This behavior is due to the changes in the

shape of the modulation envelope. As an example, Figure 16 shows adn(t')

versus t' for d=O dB and 10 dB and for f1 '=.f1,.1? and 10. These values of

fm' span the peak of the curve for Fd= 10 dB in Figure 14 and show how the

shape of adn(t') changes with fin'. For d=0 dR, the curve of in vs. fm' in

Figure 14 shows no peak, and this behavior is seen in Figure 16.

F. Bit Error Probability

The effect of a time-varying SINR can he evaluated most meaningfully by

computing the bit error probability when the array is used in a digital

28

6-10 dB

D5 5dB

m 6-

0 dB

c'Ji

Figure 11. m versus fej dQ0 dB

29

0

% %"CD

6 -

\

Eoj

6 _- , 0l dB

c - 20 dB30 dB

0 - I0- 0-1I i I I 0 10liitti I Iiii 1011i~

f'

fm

Figure 12. a versus f- 30 dB

30

0-

0dB

0

30 dB

0 0.1 0.2 0.3 0.4 0.5

Figure 13. adnjt) versus time

31

0

m

10 d0Bt

Figur 1. dBesu

2O32

f I.

0

15dB

io10 d0' 02

= 20 dB

033

0~/~ ' -Cd= lOd B fm =0.01

10dB, 0.12o /OdB, 10

0d 0 f00

0- , 0 1

OdB, f 1-0.O

o -- '' d , ./

0 0.1 0.2 03 0.4 0.5

- - I t lf ' -

m

Figure 16. adin(t') versus f t

34

communication system. For this purpose we arbitrarily assume the desired signal

to be a differential phase-shift keyed (DPSK) biphase modulated signal[12].

Also, we assume the reference signal is a replica of the desired signal.*

Because the interference does not produce phase modulation on the desired

signal, the only effect of the interference on system performance will be to

vary the array output SINR and hence change the bit error probability. For

a DPSK signal in white noise, the bit error probability Pe is given by[14]:

(1/2 )e-Eb/No, (83)

where Eb is the signal energy per bit and No is the one-sided thermal noise

spectral density. For our purposes, it is convenient to express Eb/No in terms

of signal-to-noise ratio. Since Eb=PdTb, where Pd is signal power and Tb is the

bit duration, and since I/Tb is the effective noise bandwidth, NO/Tb is the

received noise power. Hence,

Eb Pd SNR. (84)- TNTT T = SR

In addition, for this analysis we shall assume the interference power at the

array output has the same effect on detector performance as thermal noise

power. With this assumption, the bit error probability may be written

-SINRPe = (1/2)e -S (85)

Finally, since the interference causes the SINR to vary periodically, we obtain

the effective bit error probability Te by averaging Pe over one period of the

interference modulation:

f(I/ 2fm' )

(e 2fm' (1/2)e -S INR(t ')dt'. (86)0

As long as the desired signal bandwidth is not excessive, these assumptionsproduce the same Dd as in (19) and the same S as in (23), and hence will yieldthe same weight behavior as the CW signal assumed above. Moreover, it has beenshown that desired signal bandwidth has almost no effect on array performanceeven if the bandwidth is large[13].

This calculation is valid as long as the bit rate is large compared with the

interference modulation period, which we assume to be the case.

Figure 17 shows typical curves of Pe as a function of fm' for Od=O0,

Gi=3 0o, Cd= 6 dB and for several values of Fi between -20 dB and 30 dB. Figure

18 shows 7e versus fm' for 6d=0 0 , Fd= 6 dB, Fi=20 dB and for several 6i between

50 and 200. Both these figures show that fm' has almost no effect on bit error

probability. At low fm', where the weights can track the interference

modulation, the SINR varies over a wider range than at high f,,', where the

weights are too slow to follow the modulation. Nevertheless, the average hit

error probability is essentially the same for all fm'.

The curves in Figures 17 and 18 show a slight increase in Te as fm'

increases. At high fffl', where the array is too slow to track the interference

power variation, the weights are constant. Their values are determined by the

time-average interference power, Ai2 /2 (see Eq.(20)), or equivalently by the

time-average input INR, qi/2. In this case one can show that Pe has the same

value as the bit error probability for an array receiving CW interference with

input INR of Fi/2. 7e is slightly lower at low f,' because in this case the

weights track the modulation and yield maximum available SINR for each

instantaneous value of interference power. However, the difference between Te

at low and high fm is so small that, for practical purposes, Te may be

considered constant, equal to that for CW interference with INR = Fi/ 2 .

IV. CONCLUSIONS

We have examined the effect of a sinusoidally modulated (double-sidehand,

suppressed-carrier) interference signal on the performance of an adaptive array.

The major effect of such interference is to cause envelope modulation on the

desired signal. This envelope modulation is large only when the interference

arrival angle is close to that of the desired signal and when the interference

modulation frequency is low relative to the array speed of response. The

36

IC4

30dB

20dB

100d

0 0s

-IOdB

-20dB

10 10 10 CF ICp 10 10 10 C

f!'1

Figure 17. Bit Error Probbility versus f.VO 8j'O d=1

37

i0o -

ej= 50

I0-'100

10i150

I0-3 __

200

4

10 - I II itil t i , t! I I I I I I 11 11 I I l iii. I I 3 111111 t 1iii i24

I0- I0- 1 0- 10 I0 102 I0

f m

Figure 18. Bit Error Probability versus f

Fue -d-O , = 6 , i=20 T - 3

38

envelope modulation is greatest when the interference power is high and the

desired signal power is low.

The modulated interference produces no phase modulation on the desired

signal. Moreover, if the desired signal is a digital communication signal, such

interference results in a bit error probability essentially the same as that for

CW interference (from the same angle and with power equal to the time-average

power of the modulated interference). Interference modulation frequency has

almost no effect on bit error probability.

39

V. REFERENCES

1. B. Widrow, P. E. Mantey, L. J. Griffiths and B. B. Goode, "AdaptiveAntenna Systems," PRoc. IEEE, vol. 55, p. 2143, Dec. 1967.

2. S. P. Applebaum, "Adaptive Arrays," IEEE Trans. Antennas Propagat.,vol.AP-24, p.585-598, Sept. 1976.

3. C. A. Baird, Jr. and C. L. Zahm, "Performance Criteria for Narrowband ArrayProcessing," 1971 IEEE Conference on Decision and Control, Miami Beach,Florida, Dec. 15-17, 1971.

4. R. T. Compton, Jr., "The Effect of a Pulsed Interference Signal on anAdaptive Array," to be published in IEEE Transactions on Aerospace andElectronic Systems, vol. 18, no. 3, May 1982.

5. R. L. Riegler and R. T. Compton, Jr., "An Adaptive Array for InterferenceRejection," Proc. IEEE, vol. 61, p. 748, June 1973.

6. R. T. Compton, Jr., R. J. Huff, W. G. Swarner and A. A. Ksienski, "AdaptiveArrays for Communication Systems: An Overview of Research at the Ohio StateUniversity," IEEE Trans. Antennas Propagat., vol. AP-24, p.599-607, Sept.1978.

7. R. T. Compton, Jr., "An Adaptive Array in a Spread Sptectrum CommunicationSystem," Proc. IEEE, vol. 66, p. 289-298, March 1978.

8. H. D'Angelo, Linear Time-Varying Systems: Analysis and Synthesis, Allyn andBacon, BostonT197.

9. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill BookCo., New York, 1953; Section 5.2, p,. S f.

10.R. S. Kennedy, Fading Dispersive Communication Channels, John Wiley and Sons,New York, 1969.

11.A. Ishide and R. T. Compton, Jr., "On Grating Nulls in Adaptive Arrays," IEEETrans. Antennas Propagat., vol. AP-28, p. 467-475, July 1980.

12. R. E. Ziemer and W. H. Tranter, Principles of Communication, HoughtonMifflin Co., Boston, 1976.

13. W. E. Rodgers and R. T. Compton, JR., "Adaptive Array Bandwidth with TappedDelay-Line Processing," IEEE Trans. on Aerospace and Electronic Systems,vol. AES-15, p. 21-28, January 1979.

14. W. C. Lindsey and M. K. Simon, Telecommunication Systems Engineering,Prentice-Hall Inc., Englewood Cliffs, N. J., 1973.

40

IA