The Efect of Antenna Patterns on the Per-

formance of Dual-Antenna Radar Air-

borne Moving Target Indicators

HARRY URKOWITZ, SENIOR MEMBER, IEEE

Summary-A dual antenna radar airborne moving target indi-cation (AMTI) system is a scheme for eliminating or sharply reducingresidual ground clutter fluctuation and stationary target residue sothat moving target returns are easier to detect. Successive pulsetransmissions are alternately transmitted from the two antennas,each of which is connected to its own receiver. The two antennas arearranged longitudinally (in the flight path of the airplane) in such arelative position that the rear antenna occupies the same position asthe forward one after one repetition period. The pulse from the first(forward) antenna is delayed after square-law detection, and sub-tracted from the square law detected second pulse return. If the twoantennas have identical antenna patterns and transmitted pulseshapes, and if there are no position errors, the returns from stationaryobjects will be perfectly cancelled (except for a noise residue, neg-lected in this paper).

This paper derives clutter cancellation and moving target enhancement formulas in terms of the antenna patterns and position errors.The formulas are complicated, so certain approximations are sug-gested for rough-cut calculations.

INTRODUCTION

A IRBORNE MOVING TARGET indication(AM1TI) systems using a single antenna havebeen analyzed previously in the literature.1-3 The

type of system which was considered was a pulsed-radarAMTI system with a single antenna in which successivepulse returns are subtracted at video to cancel the re-flection from fixed objects. The motion of the aircraftcauses a fluctuation in the ground echo, leaving a residueafter cancellation. The fluctuation in the ground echocomes about from the following mechanism. The totalradar return at any instant is the resultant of the re-flections from a large number of scatterers on theground. Since the aircraft moves between the receptionof two successive pulse returns, the returns from thescatterers are rephased from pulse to pulse, so that thevideo returns will be slightly different.Now consider a system which is equipped with two

antennas arranged in the following way. Let the first

Manuscript received June 19, 1964. The work described here wassupported in part by the United States Air Force under Contract No.AF 33(038)-12473, Wright-Patterson Air Force Base, Ohio.

The author is with the General Atronics Corp., Philadelphia, Pa.I T. S. George, "Fluctuations of ground clutter return in airborne

radar equipment," Proc. IEE (London), vol. 99, pt. 4, pp. 92-99;April, 1952.

2 F. R. Dickey, Jr., "Theoretical performance of airborne movingtarget indicators," IRE TRANS. ON AIRBORNE ELECTRONICS, VOl.AE-8, pp. 12-23; June, 1953.

3 H. Urkowitz, "An extension to the theory of the performance ofairborne moving target indicators," IRE Trans. on Aeronautical andNavigational Electronics, vol. ANE-5, pp. 210-214; December, 1958.

antenna occupy a given position for one pulse trans-mission and let the second antenna be so located that,for the next pulse transmission, it occupies the samiieposition as the first antenna on the first pulse trans-mission. The antennas are connected to separate identi-cal (as far as possible) receivers; the output of the firstreceiver is delayed by the time between the pulse trans-missions from the first and second antennas. To preventinterference, each antenna transmits with a repetitionperiod 2T, so that the delay between the transmissionsfrom the two antennas is T, making T the effective repe-tition period of the system.A block diagram of a simple dual-antenna AMTI

system is shown in Fig. 1, and the timing diagram of thepulse transmissions is shown in Fig. 2. The duplexingswitch provides each antenna with a pulse train ofperiod 2T for transmission. The radar returns in thetwo receivers are out of line by a time interval T; thepurpose of the delay T is to bring them into line beforesubtraction. The process of delay and subtraction iscalled cancellation.

It is beyond the scope of this paper to consider thepractical aspects of design and construction of such adual-antenna system. However, it is apparent that thetwo receiver channels must be very closely matched.Also, the antenna patterns must be closely matched andit is important that the antenna spacing be proper forthe particular values of T and of the aircraft speed.

This paper considers the effect of differences in an-tenna patterns upon ground clutter cancellation andupon moving target enhancement. Use will be made ofthe random ground model1'3 and of previous definitionsfor cancellation and enhancement.3

CANCELLATION AND ENHANCEMENT

Let V1(t) and V2(t) be, respectively, the video signalsbefore cancellation out of the first and second receivers.We take the viewpoint that each transmission has itsown origin, and that the delay of V1(t) simply brings thetwo origins into line. Thus, the process of cancellationobtains directly the difference between V1(t) and V2(t).Another viewpoint is to say that the second pulse returnis T seconds after the first so that we should writeV2(t- T). The delay after Receiver No. 1 in Fig. 1 de-lays V1(t) so that it becomes V1(t- T), which is equiv-alent to treating V2(t-T) as V2(t) and V1(t-T) as

218

Urkowitz: Antenna Patteterns

¶ DIRECTIONOF FLIGHT

CANCELLATIONRESIDUE

Fig. 1-Simplified dual-antenna AMTI system.

ANTENNA NO.

II ~~~~~~~tI ~~~~~ ~ ~~~~I I I

I: ' ANTENNA NO.2

Fig. 2- Timing diagram of pulse transmission.

V1(t). After cancellation, the residue signal will be

R(t) = V2(t)- V1(t)

The mean square value of the residue is

R2(t) = V22(t) + V12(t) - 2V2(t)VI(t),

Also of interest is the residue when a fixed or a movingtarget is present. The ratio of moving target meansquare residue to stationary target mean square residueis defined as the moving target enhancement (or simplyenhancement) E. Letting the subscript s stand for thestationary target and the subscript m stand for the sametarget when it is moving,

Rm2 1-lOmE= =

R82 1- 08(7)

It is clear that the problem is reduced to finding thecrosscorrelation functions 4)0, 4)m and 0,, in terms of the

1) radar parameters and target geometry. As before,3 re-(1) sults are computed for square-law detectors.

VIDEO CORRELATION OF GROUND CLUTTER(2)

where the bar indicates a statistical average taken oversuccessive signals. It is supposed that the mean squarevalues of V1(t) and V2(t) are the same,

V12(t) = V22(Q). (3)

The normalized, but not dc-suppressed, video cross-correlation function is designated by 4 and is given by

V2(_)V1(_ ) V2(t)Vl(t) (4)-[V(t) V22(t)] 1/2 V12(t)

Then

R2= 2V12(t)[1 - 4]. (5)

The cancellation ratio is defined in terms of the resi-due. Using the subscript c for clutter, the clutter cancel-lation Cc is defined as

V12 (t)cc -

RC2

1

2[t - oj~(6)

The definition is made this way so that Cc will be largerthan unity.

The geometry of the ground patch is shown in Fig. 3.The aircraft is presumed to be flying at a constant speed.The limits of the ground patch are determined by thepulse duration and the antenna patterns. The distanced is the distance between the proper and actual positionsof the second antenna when it transmits. If the antennais positioned exactly right, d=O. It has been shown3that if the patch were uniformly illuminated, the in-stantaneous RF return may be written as

N

i(t) = Cn Cos (oot -o ))n=l

(8)

where wo is the transmitted angular frequency, and I%is a random phase angle uniformly distributed in theinterval (0, 27r). The index n indexes the various scat-terers in the patch, and N is presumed to be large.i(t) will have a Gaussian distribution for large N. Letb2 be the (uniform) reflectivity of the patch (power perunit area). Then, except for an unimportant propor-tionality constant to give Cn the proper dimensions,

Cn = bV\A/N,where A is the area of the patch.

(9)

219

220 IEEE TRANSACTIONS ON AEROSPACE AND NAVIGATIONAL ELECTRONICS

Z AIRCRAFT

(dlo,z )

s2

x

-r2

Fig. 3 Geometry of ground patch

The antenna pattern and pulse shape can be broughtin by weighting Cn with their effect on the magnitudeand adding the proper phase angle to In. Thus, for any

particular return,

il(t) = E C,E(0)p(rj) cos (coot - (PI- ()), (10)n

where E(0,n) is the magnitude of the two-way antennapattern at an angle 0n, the azimuth of the nth scatterer;p(rn) is the pulse height at rn, the radial ground dis-tance of the nth scatterer, and t(6n) is the phase shiftof the two-way antenna pattern at azimuth angle On.

The returns can be labeled, the subscript 1 beingused for the return for the first antenna and the sub-script 2 for the return for the second antenna,

i(t) =, C,Ei(0,)p,(rn) cos (cot sn- V00)

n

i2(t) = C.E2(0tn)P2(rnz) COS (Wot- (D.- 42(0n^) an) >(11)

where a,,= a(rn, 09) is the extra phase shift due to thenth scatter caused by inexact positioning of the secondantenna. a(r, 0) is given by3

d24wr d cos 0 -

\ ~~2a (r, 0) =

XA/1 + z2 /r2(12)

where

Il = bV/AIN E Ei(O,)pl(rn) cos [in + tl(6n)Jn

12 = bV/AIN El(0n)pl(rn) sin [4). + (l(0)]n

13 = bV\ A N E E2(0n)p2(r,) cos [4b. + 42(0n) + an]n

I4 = bV\ A/N , En(On)p2(rn) sin [ctn + {2(0n) + a.]j. (14)n

In what follows, certain averages will be needed.

Ab212 = 122 = El(0n)p12(r2)

2Nn

Ab2I32 = I42 = _ E22(0)p22(rn), (15)

since J.,) is uniformly distributed in the interval (0, 27r).To pass to the limit for N-> cc, we write

A= zAA = rArAO-rdrdO.

N(16)

Then

b2r,2 = 122 = _ E 12E(0) p12(r)rdrdO

where X is the wavelength and z is the aircraft altitude.In (12), d is the error in position of the second antenna,and is presumed to be along the direction of flight. Eq.(11) may be written as

il(t) = Il cos coot + 12 sin coot

i2(t) = 13 cos coot + I4 sin coot

I32 = I42 = -ff E22(0)P22(r)rdrdO. (17)

Examination of (13) shows that _l2 is the mean square

value of ii(t) while 132 is the mean square value ofi2(t). We have assumed these to the same. Call thecommon valueg,u

I12 = I22 = I32 = I42 = AO. (18)

(o0o,z )

y

December

(13)

Urkowitz: Antenna Patterns

In a manner similar to that above, it is found that

b2rrI113 = I2I4 = P13 = f Ei () E2(0)pl(r)p2(r)

,cos [a(r, 0)-AS(0)]rdrdOb2rr

I114 = 12I3 = bL14 = ffEi(0)E2(0)pl(r)P2(r)

.sin [ac(r, 0)- A(O)]rdrd0,

where

AU(0) = 41(0) -42(0).

The video outputs of the square-law detector are

V1(t) = Il2 + I22

V2(t) = 132 + 142.

Then the video crosscorrelation is

Vl(t)V2(t) = (I12 + I22)(I32 + 142).

while the phase difference arises from 1) error in posi-tion of second antenna, 2) motion of target and 3)difference in the phases of the antenna patterns. Conse-quently, the second pulse radar return may be writtenas

i2(t) = P2 cos (wot - Y) + 13 cos cOIt + 14 sin coot, (28)

where P2 is the amplitude of the signal component of(19) i2(t) and y is the phase shift of this component from

that of i1(t). We write

7 = 71 + Y2,(20)

(21)

(22)

This may be evaluated in terms of the /i's of (18) and(19) by using a formula of Davenport and Root.4 Aftera little manipulation, the result is

Vl(t)V2(t) = 4,io2(1 + p32),

where

12 = (g132 + U142) /(o2.

Since

V12=8=O2it follows that

qc = (1 + I2)/2.

What this section has done so far is to showplicitly) through (19) how antenna pattern differeaffect random ground cancellation. Before the imptions for design are discussed, it would be useful tcformulas for moving target enhancement which invthe antenna patterns.

POINT TARGET PLUS GROUND CLUTTERThe total return for the first pulse may be writte

il(t) = P3 cos coot + 13 cos coot + I2 sin coot,

where P1 is the amplitude of the first radar return fthe target. The second radar return will have a clureturn which will be the same as i2(t) of (11). The sicomponent of the second return will differ, in genfrom that of i1(t) of (27) in both amplitude and plThe amplitude difference arises from the differencmagnitudes of the antenna patterns (and pulse ampliti

4V. B. Davenport, Jr., and W. L. Root, "An Introduction tTheory of Random Signals and Noise," McGraw-Hill Bookpany, Inc., New York, N. Y., Problem 2, p. 168; 1958.

(23)

(24)

(25)

where yi arises from causes 1) and 2) and 72 arises fromcause 3). Then

47r(s, - S2)71 =

Si = V/ra2 + Z2

S2 = [ (ra cos 0a + vaT cos -d)+ (ra sin 0a + vaT sin -q)2 + z2]1/2, (30)

where

si and S2=first and second slant ranges to the targetr,=ground range to the target (first pulse)da= azimuth of target (first pulse)va =speed of target= direction of target's motion measured fromtrack of airplane

T = repetition period.

Also

7Y2 = A{(O3a).

(26) From (27) and (28), the video quantities are

(im- Vl(t) = (P3 + I1)2 + I22:nces V2(t) = (P2 cos Y + I3)2 + (P2 sin y + I4)2.

We must now find V1(t) V2(t).

Vl(t) V2(t)

= [(Pl+l)2+I22] [(P2 cos Y+I3)2-+(P2 sin +I4)2]. (33)

It is shown in Appendix I that

V1(t) V2(t)

= 4pto2[(X1+ l)(X2+ 1) +2s\VXlX2 COS (Oo-y) +121, (34)

where tan Oo=I14/13 and x1 and X2 are signal-to-noiseratios defined by

X1 = P32/(2go)

X2 = P22/(2,uo) .

Some simplification can be obtained if we demandthat V12(t) = V22(t) for the target plus clutter case aswell as the case of clutter alone. Then xl=x2=x and

V1(t)V2(t) = 4,uo2[(x + 1)2 + 2x1 cos (O - y) + ,B2]. (35)

(29)

(31)

(32)

1964 221

222 IEEE TRANSACTIONS ON AEROSPACE AND NAVIGATIONAL ELECTRONICS December

The normalized video correlation function, with a targetpresent is

(x + 1)2 + 2xo3cos (6o - y) +11wa ~ (X + 1)2- 2x -1-t

Using (6), (7) and (26), it is found that the enE is given by

2x[1 - , cos (do - m)]+ /C2x[1 - 1 cos (06 - y)] + 1/Cc

where, as before, the subscripts m and s repring and stationary targets, respectively.5

USEFUL APPROXIMATIONS

The important results of this paper are (6)(37), by means of which clutter cancellation atarget enhancement may be calculated. Thesare very involved and it would be helpful if rhandled approximations could be used.The appearance of (19) leads one to belie-

phase discrepancy is much more importanidifference in magnitudes of the antenna patthfore, one useful approximation to make is I

E2(0), pl(r) and p2(r) are uniform over the groThe computation of 13 and 414 iS still rathebecause of the complicated expression for a(rThis may be replaced by a simpler approxim,it is realized that the major problem with groicancellation occurs when 0 = 900.1 In the viciithe following approximation holds:

cos 0 r/2 - 0.

Furthermore, d will be different from zero onof ground speed error so that it will be smad/r negligible for reasonable r and a reasonabO around 90°. Similarly, for altitudes whicgreat, zlr is negligible; the result of all this is

a(r, 0) 2wod(r/2 - 0)1c.

With all these assumptions, it is found that

Ab22-

A1I3 =02 0

cos [4rd(r/2 - 0)/X -()]dO602 -01 l s

1114 = sin [4rd(lr 2 - )/X - t(6)]d6,

5 At this point, the author would like to correct a typographicalerror in his earlier paper, Ibid. WXherever the factor cos (o-ay)appears, with or without a subscript on 'y, the factor should bemultiplied by ,B, as in (34)-(37) of this paper.

where 62-01 iS the angular extent of the ground patch.The formula for moving target enhancement may also

be simplified by assuming that Qa =90', and thatq = 90((the case of most interest). Then (30) becomes

'yi 4 [ra- (d2 + { ra + vaT } 2)l1 2] /X. (42)

For the sake of a more rapid calculation, it may beassumed that A(Oa.) -0, so that 'Y2=0 and =zy.The computation of An and A14 in approximate form

(37) will be made easier by the use of tables of the "radarscatter integrals."16

esent mov- DISCUSSIONThe important results of this paper are (6), (26) and

(37), by means of which clutter cancellation and mov-ing target enhancement may be calculated. Eq. (20)

d(26), and gives the necessary averages for the computation of q5,.ued moving Normalization with /to removes the dependence uponre formulas the parameter b, leaving the result dependent on themore easily forms of the pulse packets which strike the ground and

upon the geometric parameters of the ground patchte that the through a (r, 0). Eqs. (19) show (implicitly) the effects oft than the nonidentity of antenna patterns. The difficulty in de-ern. There- termining explicitly these effects is the difficulty inthat E1(0), evaluating the double integrals. In general, this willund patch. have to be done numerically. In actual practice, these~r involved formulas cannot be used directly in a design procedure60iin (13). to determine the required identity of antenna patterns;

ation when the integrals will have to be evaluated for a variety ofund clutter antenna patterns, and the design choice made on thenity of 900, basis of the results. The trouble here lies in what choices

to assume for the antenna patterns.

(38) For rough calculations, the approximations of (41)(38) may be used, together with (42) for a quicker calculation

ly because of enhancement. Of course, the magnitudes of the an-

ill, making tenna patterns have been assumed uniform and identicalAle range of so that magnitude discrepancies do not enter into the-h are not approximation.

The formulation in this paper can encompass otherperturbations besides those treated. In particular, errors

(39) in the prescribed position of the second antenna can beincluded by finding their effect upon the phase differencebetween first and second pulse returns; that is, the posi-tion discrepancy would be used to find the actual value

(40) of s2, the second slant range, to substitute into (70).VI-'-'

APPENDIX IDERIVATION OF (34)

In the expansion of the right-hand side of (33), therewill be terms of the form Il132. Appendix II shows that

(41) I1132 = 0. In a similar manner it could be shown that allsuch terms are zero. Therefore

hI,2 = 0, i 7j. (43)

6 "The Radar Scatter Integrals," Computation Lab., HarvardUniv., Cambridge, Mass., Problem Rept. No. 64; August 1, 1952.

li .:i n ri-t-n i-n t

Urkowitz: Antenna Patterns

The expansion of (33) then yields

V1(t)V2(t) = P12P22 + 2Ao(P12 + P22)+ 4PIP2(Y13 cos y + 1P14 sin y)

+ 122I3 + I22I42 + _12132 + 112I42 (44)

Now,4

112142 = 122132 = I22132 + 2(1213)2 = Ao2 + 2A142

By adding and subtracting p132I12 to the terms inside thesquare brackets, (49) becomes

132= f 1I1J32-= K IJ132 exp -

( 12)co _ 2,u(14 1

[(13 - A131)2 + (1 - 132)1121} dIldl3. (50)

Let

112132= I22142 = Po2 + 2guN 2.

Define Go by

tan 6o = P14/AI3;and the signal-to-noise ratios x1 and x2 by

xi = P12/(2go), X2 = P22/(2yo).

(45)

(46)

(47)Then (33) follows.

APPENDIX IIDERIVATION OF (43)

It will suffice to show that

J-J}I32exp:2 (1 I- 5112

Then

I3I2 = K I,J exp (- -) dl.

In (51), let v=13-1131l. Then

K v2 1J = J (v + g,3Il)2exp L 2go(i- 132)j dv

- L:(v2 ± /1132112) exp -

2,o(l -1-131) dv.

(52)

(53)1I32 = 0. (48)

Let W(11, 13) be the joint probability density functionof the zero-mean Gaussian random variables I1 and I3,and let /u13 be their correlation coefficient. Their commonvariance is 1o. Then

00 cox

11112 = I0 f I1I32W(I11 13)dlldl3

= K f I1132 exp0000 ~ .2guo(l - 13')

Eq (53) results after the expansion of the expression inthe squared parentheses. The middle term results in anodd function of v for the integrand which results in zeroafter integration. When (53) is substituted into (52) onegets

I1132 = K f f I(2 + 1I)00 00

v-2 + (1 - A1132)I12]*exp L 2go(l - Y132) dvdIl. (54)

[I32- 2A113I113 + I]2B d1ldI3. (49) It is clear that the integrand is an odd function of h1 andit follows that 11132 = 0.

1964 223