Interference rejection by amplitude shading of sonar transducer arrays

Interference rejection by amplitude shading of sonar transducer arrays

H. S.C. Wang

Hughes Aircraft Company, Fullerton, California 92634 (Received 15 September 1975; revised 20 December 1975)

It is demonstrated that strong coherent interference incident from arbitrary directions can be effectively rejected by amplitude shading or real weighting of channel signals. For a given spatial distribution of interference field, the total interference power received by a transducer array is formulated and superimposed on the power received from an isotropic noise field. The total noise plus interference power is then minimized with respect to the shading coefficients and subject to the constraint that the mainbeam peak be maintained at a fixed level. As an illustrative example, we have calculated the optimum amplitude shading coefficients for a linear array with logarithmically spaced transducer elements to reject an interference field uniformly distributed over a 15øX36 ø solid angle with an intensity up to 70 dB above the noise field. In each case, the directivity pattern of the array adjusts itself by bunching several null points in the direction of interference incidence to achieve a nearly complete rejection. The merit of a narrow-band processor based on the principle of optimum amplitude shading is then examined by comparing its performance with those of a conventional processor and an optimum processor with both amplitude and phase adaptation. The optimum amplitude shading processor is clearly superior to the ocnventional processor under strong interference, but much simpler to implement than the optimum processor with amplitude and phase adjustment. Furthermore, the very little additional SNR0 due to complex weighting probably cannot be achieved in practice because of the extreme super-gain condition which often results.

PACS numbers: 43.60. Gk, 43.30.Vh

INTRODUCTION

In a narrow-band system, the operation of a sonar re- ceiver is likely to be limited by coherent interferences in the same frequency band as the target signal but incident from directions other than that of the target. Such interferences can only be rejected (or its throughput minimized) by spatial processing of signals received from an array of transducers. For example, by apply- ing suitable weight (i.e., amplitude weighting and phase shift) to each transducer signal, the sum. of all these signals will contain less interference power relative to the signal power than in a single transducer. In fact, from the well-known theory of adaptive antenna and beamforming, the optimum channel weights which yield the maximum signal-to-noise (interference) ratio at the processor output are complex and may be given in column matrix form as

al e7X• ! a 2 eJX2

_Co = = q-v, (t)

an e •

where V is a set of phase factors depending on array geometry and the intended target direction, and Q an NxN noise/interference cross-poKer spectral d•'nsity matrix. If the complex weight a• e•X• is viewed in rectangular form, the channel signal weighting can be ac- complished by the introduction of a quadrature channel with real weights in both the in-phase and quadrature channels. s Such an optimum processor has been considered as more costly to implement in practice because the N complex weights or 2N real weights must

be obtained by matrix inversion or some other iterative calculations and then realized with additional multiplica- tive and delaying circuits.

Actually complex weighting of channel signals is not necessary in order to effectively reject coherent interference in the same band as the target signal. This is due to the fact that in an array with many receiving elements there are usually too many degrees of freedom or variables for adjustment. For instance, it can easily be shown that in the case of a line array with 2m + 1 omnidirectional transducers, m simultaneous plane- wave interferences with different angles of incidence can be completely rejected by 2m + 1 amplitude shading weights alone. To maintain a useful array gain, rea- sonably narrow mainbeam width and low side-lobe levels, the number of elements N in the array is usually larger than the minimum required for exact rejection of a fore- seeable number of interferences. If a smaller number

of variables less than 22V will suffice to reject the interferences, a natural choice is the N amplitude shading coefficients since they are relatively easy to adjust in addition to the fact that the beamformer now only in- volves one-half as many independent variables as a conventional optimum processor.

The presence of the ambient noise field also calls for a set of weights which rejects most (but not all) of the interference power without letting through too much noise power at the processor output. In this paper, we shall study the feasibility of an "amplitude-shaded" optimum narrow-band processor by first deriving the power received by an array of transducers from the interference and ambient noise fields as a quadratic of the array-shading coefficients a•. This total interference and noise power at the beamformer output is then minimized with respect to the a• using the method of

1251 J. Acoust. Soc. Am., Vol. 61, No. 5, May 1977 Copyright ̧ 1977 by the Acoustical Society of America 1251

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1252 H.S.C. Wang: Interference rejection of sonar transducer arrays 1252

Lagrange multipliers subject to the constraint that the mainbeam-peak response be maintained at a fixed level. Using a 12-element logarithmically spaced linear array, it is demonstrated that an interference field of 15øx 36 ø

in angular extent and up to 70 dB above ambient noise field can be rejected almost completely by amplitude shading. To provide a quantitative comparison, per- formances of a conventional processor and an optimum processor with complex weighting under identical field conditions are also evaluated. The results indicate that

adaptive amplitude shading provides an improvement of signal-to-interference ratio by 27 dB as compared with a conventional or fix-parameter processor. However, in typical cases, the optimum complex weighting processor gives only a negligibly small additional gain, which requires weights unrealistic due to excessively high values of supergain index.

I. POWER RECEIVED BY A DISCRETE TRANSDUCER ARRAY IN A COHERENT INTERFERENCE FIELD SUPERIMPOSED ON AN ISOTROPIC NOISE FIELD

Consider an array of N discrete omnidirectional transducers located at (ri, 0i, •), i= 1, 2, ... ,N, where r, 8, and 05 denote the radius vector, colatitude, and longitude of a spherical coordinate system. The response of such an array after beamforming to a single frequency plane wave incident in the (8, 05) direction is specified by its directivity pattern function given in Ref. 4

[}r(cosa P(O, ½)= •-•.ai +2 Y•.•a,aj cos -cos ) i j>i

- krj(cosot• - cosot0•)] (2)

[throughout this paper, all summations over channel indices i, j are from 1 to N, and only the indices summed and their special restrictions (if any) will be explicitly written], where a• is the amplitude shading or weighting coefficient, a i the angie between the direc- tional vectors (0•, ½i) and (0, ½) with its cosine given by

eosa • = sin0 sin0 i eos(qb - qb i ) + cos0 cos0 i, (3)

and a0i is the angie between (0•, ½•) and (•0, ½0), the direction into which the array is steered. This pattern function varies with frequency or wavelength of the in- eident signal via the wave number k= 2•r/X.

If the intensity of the wave field is distributed in colatitude and longitude as g(0, ½) the total power re- eelveal by the array can be calculated as

P=•-• 1>(0, ½)g(O, ½)sinOdOd½. (4)

Although the technique to be described here can be ap- plied to any general three-dimensional array, we shall carry out the following development for the ease of linear arrays, which are of special interest and advantages in certain practical applications. Since a linear array of omnidirectional transducers (assumed to be on the z axis) possesses axial symmetry, the pattern function is independent of the longitude ½, and the ½ distribution of field intensity g(O, ½) becomes inconsequential. Thus, the received power can be expressed as

1 p(O)go(O)sinOdO (5) P , where go is the aggregate field distribution in latitude

go(O)= g(O, ½)d½ . (6)

For the purpose of this study, it suffices to assume that go for the interference field is uniformly distributed over the range (02, 0 z) so that ge(O) has the shape of a rectangular block of height 4vG as depicted in Fig. 1. The interference power received then becomes

Pt =G P(O)sinOdO . (7)

For a linear array on the z •is, the power pattern P(O, ½) in Eq. (2) is simplified into •

P(0) =• ai + 2••a•a• eos[y•(eos0 - cos00)] , (8) i

where

ytl = 2w(•zt-•zl) (9)

for the ith •d jth transducers located at z, and z•. Substituting Eq. (8) into Eq. (7) we obtain the total re- eelveal interference power by a shaded array as

P, =G(•a[(eosO 1 - cos0z) +2ZZ aiai {sin[yi i(eøs0• - cos00)] - sin[Ti i(eos• - cos00)] }) .

The ambient noise field e• also be considered as a

propagating field with its source distributed throughout the volume of sea water or on a very large and dist•t surfaceß For simplicity, we shall assume that the noise field surrounding the array is homogeneous •d isotropic with • intensity (4•) '• per •it solid •gle. The total noise power received by the array with no re- gion of exclusion e• be obtained from Eq. (30) of Ref. 4 by setting the •gle of exclusion 6 = 0 therein:

=0)

• • a• a i = ai + 2•• • [sinTii eos(Ti• sine0)] , (11) '

gO (O)=f g(O, •)d• o

g8

4'rrG

0

4•r ß I0 0.1 INR (IN dB ) '

FIG. 1. Interference field distribution in colatitude O.

J. Acoust. Soc. Am., Vol. 61, No. 5, May 1977


1253 H.S.C. Wang' Interference rejection of sonar transducer arrays 1253

where •b 0 = 0 o -«v is the direction of steering in latitude. The total interference and noise power received at the beamformer output is the sum of Eqs. (10) and (11)

Pn,,•'=P• ' +Pn , (12)

which is seen to be a quadratic of the shading coefficients a i .

II. MINIMIZATION OF INTERFERENCE AND AMBIENT NOISE POWER RECEIVED BY THE ARRAY

In order to reject interference and achieve optimum array performance, we seek a set of shading coefficients {ai} such that P Na is a minimum with the constraint that the mainbeam peak be normalized to Unity:

R=•-•i - 1: O. (13) i

Method of Lagrange multiplier now yields the following equationsS:

OP•v+• •R aa i +n •-• =0, i= 1, 2, ... ,•, (14)

where • is a multiplier. Equations (14) and (13) consti- tute a set of N + 1 linear algebraic equations for the N+ 1 unknowns ax, az, ... a,, and •. Writing in matr•

l- htz." hln •

1

1

hnz". hnn 2

1..-1 0

form we have

hN1

1 --

[1

[2

(15)

where the general expression for the elements his is given by

1 aZPna i,j=l 2, ... N. (16) h•=2 Oai0a r' ' ' , For a linear array, the matrix elements h i• are obtained by inspection from Eqs. (10) and (11):

his= 1 +G(cos0x- cos0z) , i= 1, 2, . .. , N (17)

and for i •j,

his = (1/Yi •) sinyi • cos(yi • simb 0)

+ (G/y i •) {sin[y i •(cos0• - cos 00)1

, - sin[yij(eos0•. - eos00)] }. (18)

Thus, for a given array geometry in an arbitrary interference and noise field, all/h• can be numerically evaluated and Eq. (15) solved for the optimum shading coefficients

In order to interpret the meaning of the parameter G, we shall assume that the interference field intensity

g(0, ½) is also uniformly distributed over the longitude ½•.- ½•. The above formulation then implies that the interference-to-noise field intensity ratio is INR = 4vG/ I•.-•1 or

C:(I ½•.-½•1/4•r) 10 ø'l (19)

when INR is expressed in decibels and qb• and qb•. in radians. The performance of the array and beamformer for agivenlNR is measured by the amount of interference and noise power P•+• at the beamformer output, or, since the mainbeam peak is constrained at unity, by the output signal-to-noise (including interference) ratio

SNR0 = (p•+•) -1 (20) for'a unit amplitude plane-wave target signal in the steered direction. The optimum shading coefficients {ai}o• yield the minimum P•+• and therefore maximum SNR 0 under a given noise and interference condition.

III. ILLUSTRATIVE EXAMPLE

To illustrate the principle of interference rejection outlined above, we have calculated the performance of a 12-element logarithmically spaced linear array as described in Ref. 4 and shown in Fig. 2, with optimally adjusted amplitude-shading coefficients. The array is assumed to be operating in an isotropic ambient noise field and an interference field of uniform intensity over

the longitude ½•.-•b• = 36 ø and fr6m latitude • =- 75 ø to •. =- 60 ø. For incident INR varying from - • to + 70 dB, matrix elements /h• were calculated with the parameter G given by Eq. (19) in each case, and the optimum shading coefficients {ai}o• solved from Eq. (15). Fig- ures 3-6 show the typical directivity patterns corresponding to these {ai}o• at INR =- •, 30, 50, and 70 dB, respectively, when the array is steered broadside (the patterns are symmetric with respect to the broadside and plotted only between •=- 90 ø and •= 0ø). It is seen that as the interference field intensity increases, sev-

eral null points of the pattern are shifted into the direction of incident interference (- 75 ø < •< -60 ø) and grad- ually form a deep null zone when INR reaches 70 dB for a very strong interference.

As a result of this null shifting, the residual interference power at the beamformer output is always smaller than the total noise power and the noise and interference power PN+• remains nearly constant. In Fig. 7, the curve labeled NIL shows the noise and interference level of the optimally shaded array referred to the minimum noise power in the absence of interference and the curve labeled REINR, the residual interference- to-noise power ratio in decibels. These results sug- gest that, as the array now essentially operates under noise-limited conditions, further improvement of per-

4. 40X

I 2 3 II

, . , . .

,4 - o o c; oo 6 c; o -

_3

FIG. 2. An array of 12 transducers logarithmically spaced along a straight line.




ß

ß

ß

.

.

ß

,

.

-40- ß

ß

,

,

.

,

_

ß

.

,

ß

-60 ' ß

ß

ß

,

ß

.

_

ß

ß

ß

ß

ß

-80- ,

ß

ß

ß

,

ß

_

ß

ß

.

ß

-100 -•

-90

NO INTERFERENCE

N=12

L = 4.4X

T ......... '1' ......... '1'

#

#

# #

#

#

#

#

#

# #

# #

#

#

#

SHADING COEFFICIENTS

(1, 12) 0.1317 (00! (2, 11! 0•.261 (00) (3, 10) -0•608 (00) (4, 9) 0,9623 (00) (5, 8) -0.1366 (01) (6, 7) 0.8070 (00)

T T T T

BEARING IN DEG.

FIG. 3. Directivity pattern of optimally shaded array in isotropic noise fieldß

-30

# #

#

o

ß I INR = 3o dB I : I I

-20 ' : I I ' I I .

.

• I I -40 - I

-

ß I I

ß I I

: I i • ß I - I I . I.U I.U ' • I ' I.M

-80- ,', "' I ß 0. I ß

: ,, I " I -- • I •1 : • I ß I I ß

-100 ' T T .... / .... T T I

.

.

#

.

N=12

L = 4.4•,

#

.


(1, 12) 0.1243 (00)

(2, 11 ) 0•.550 (00)

(3, 10) -0.3864 (00)

(4, 9) 0.1348 (01)

(5, 8) -0.1998 (01)

(6, 7) 0.1157 (01)

1' T r T T

-90 -60 -30 BEARING IN DEG.

FIG. 4. Directivity pattern of optimally shaded array rejecting moderate interœerence.

#

#

# #

# #

#




0

-20' INR = 50dB ß

.

'

ß

-80 -

.

--

: ' [ I -100 T T ..... -90

.#

N = 12

L = 4.4X

#

#

. .

. #

.

#


(1, 12) 0.1161 (00) (2, 11) 0.2354 (00) (3, 10) -0.3403 (00) (4, 9) 0.1253 (01) (5, 8) -0.1834 (01) (6, 7) 0.1069 (01)

-• T T 'r T

-60 -30 0

BEARING IN DEG.

FIG. 5. Directivity pattern of optimally shaded array rejecting strong interference.

•o: T '1' ß

: I j ,

• I ß I I : I I

-20: I I ,

- I : I I ' I I

ß I

ß

.

ß

- I I ß

ß t,• I ß u.I I u.I

-80 - o j ß 0. I 0. I : •' I ß II j II ' •'- JN J : '-)'1 '-)' : I

-100 T ......... T ..... L T 'r -90

#

# #

# #

.

#

#

.

.

#

# .

.

# #

N=12

L = 4.4•,


(1, 12) 0.4367 (-02) (2, 11 ) 0.9659 (-01) (3, 10) 0.1348 (00) (4, 9) 0.1896 (00) (5, 8) -0.5527 (-01) (6, 7) 0.1298 (00)

'1' T T T

-60 -30 0

BEARING IN DEG.

FIG. 6. Directivity pattern of optimally shaded array rejecting very strong interference.




40

-60

N:12, l:4.4X -- INTERFERENCE DISTRIBUTION

8. 15 ø TO $0 ø

-- 4' 56 •

NIL---•

REINR--•/

/-

MAINBEAM V•IDTH • ./' -- _

-80-(0 I0 20 30 40 50 60 70 INTERFERENCE/NOISE FIELD INTENSITY RATIO [NR IN dB

FIG. 7. Characteristics of an optimally shaded linear array.

formance to that predicted by the theoretical maximum corresponding to the optimum complex weighting given in Eq. (1) is not necessary. A more-detailed comparison and discussion will be given in Sec. IV. Under all optimized conditions studied, the maximum side level is more than 13.5 dB below the mainbeam peak, and the mainbeam width at 3-dB points is essentially constant except in the case of a very strong interference. These data are also displayed in Fig. 7.

When the array is steered to an arbitrary direction •b 0 other than the broadside', the same optimization pro- cedure applies but the m•/trix elements hij in Eq. (15) will be different resulting in an asymmetric directivity pattern. Figt•re 8 shows the optimized pattern in the absence of interference and Fig. 9 that for INR = 70 dB

with the array steered to •b 0- -30 ø off the broadside.

IV. COMPARISON OF OPTIMUM AMPLITUDE

SHADING AND OPTIMUM COMPLEX WEIGHTING PROCESSORS

o

It is interesting to compare the performance of an array optimally amplitude shaded with that of the same

array with optimum amplitude and phase shading given by Eq. (1). To this end, we shall write the complex noise and interference correlation matrix in component form

where.the superscripts R and I on the matrix Q denote the real and imaginary components, respectively. For an interference field specified in Fig. 1 superimposed on an isotropic noise field, the real and imaginary parts of the elements of Q are evaluated as follows:

q•=l+G(cosO•-cosO•.), q•=O, i=l, 2, ...,N, (22)

q• sin•'/1 G = +•[sin()qj cose•) - sin()q• cose•.)] i •j, (23)

q• =(G/•'ij)[-cos(•/il cose•.) +cos()q• cose[)], i•j. (24) This complex matrix Q is inverted and then premulti- plied to the vector

_

exp(•.,(,•./x) •o.,e0) ß

. exp(•.,(•v/x )

(25)

to give the unnormalized optimum weighting vector C_o • as indicated in Eq. (1). in order to compare weighting coefficients and noise/interference power on the same footing, we divide C._•p by the square root of the signal power to obtain the normalized weighting vector

- 1

C__o, = (C_o*,A C__o,) y•' C__o, , (26) where the matrix used to calculate the signal power • is A = V Vt. Thus the normalized (•._o= not only yields the maximum SNR 0 but also unity target signal power as was in the calculation of optimum amplitude-shading coefficients {a•}•. The optimum processor output SNR 0 is then simply

-20

NO INTERFERENCE

-8O N = 12

L = 4.4•

-100-, .............. • .............. T .............. T .............. r .............. T .............. r ............. , .............. , .............. T .............. [ .............. ] .............. r- -90 -60 -30 0 30 60 90

BEARING IN DEG.

FIG. 8. Directivity pattern of optimally shaded array steered -30 ø from broadside in isotropic noise field.




-20

-100 -', ......................

INR = 70 dB

- 90 -60 -30 0

N=12

L = 4.4/• -

......................... T ............................. T-

30 60 90

BEARING IN DEG.

FIG. 9. Directivity pattern of optimally shaded array steered -30 ø from broadside and rejecting very strong interference.

1 1

SNRø = P•+, (C__= •__o,) = •__o*,Q •__o• ' (27) We have calculated this optimum SNR 0 for the logarith- mic array described in Sec. III steered broadside in an interference and noise field with INR from -oo to 70 dB.

The results were plotted as the curve labeled "optimum processing" in Fig. 10. This curve is essentially flat and indicates the theoretical ability of the optimum processor to reject almost all the interference field even at INR- 70 dB (with a degradation of SNR0 by 0.5 dB). The output SNR 0 of the optimum amplitude-shading processor as calculated from Eq. (20) with Ps+x corresponding to the optimum shading coefficients {a•op is also plotted in Fig. 10. It is seen that the performance of these processors are essentially identical except in the case of very strong interference (INR = 70 dB), the degradation of SNR 0 of the latter is 2.1 dB more. If the amplitude shading of the array is optimized in the absence of interference and then fixed, the performance of this array in the presence of the interference field of various INR's is plotted as a dotted curve in Fig. 10, and labeled as conventional amplitude shading.' We see that, without adaptive optimum amplitude shading, the performance of the array is degraded by approximately 30 dB.

It is interesting to observe that this is the only case where the two optimum processors tend to be identical in the limit of small interference field. We first note

that the inverse of the complex matrix Q is also complex and has the following real and imaginary components:

Q•_ Q•_•Q• . _ •=-_ _ (Q_•)'x (29)

The noise and interference power received under optimum complex weighting is

= C_o*, c_o, =_V'(Q_-b'v. (30) If the array is steered broadside, only the real part of Q'• contributes and Eq. (30) becomes

Ps+, = V*Q_•V . (31) In general, this is different from and slightly smaller than the noise and interference power when the array is steered on target with optimum amplitude shading normalized to the same signal power. To establish a comparison, we express the unknown column vector in Eq. (15) as

H1, •+ 1

=D 't ' (32)

•N, N+ 1

HN+I,N+I

where H•,•+• denotes the cofactor corresponding to the ith row and (N + 1)th column element of the square matrix in Eq. (15) and D is its determinant. By expanding this colactor along its last row, we see that

20 $1

0 -

-I0 - N=12

L=4.4

70=0

OPTIMUM PROCESSING 7 •'•• PTIMU• • 0

\ AMPLITUDE \ SHADING

\ CONVENTIONAL•

AMPLITUDE \ SHADING \

-2o // I I I I I I - w I0 20 ::50 40 50 60

[NR IN dB

FIG. 10. Broadside array performance with optimum amplitude shading compared with optimum processing and conventional amplitude shading.

\ \

\ \

7O



1258 • H.S.C. Wang: Interference rejection of sonar transducer arrays 1258

20--

OPTIMUM PROCESSING

OPTIMUM AMPLITUDE SHADING CONVENTIONAL '•"• \

AMPLITUDE SHADING

N=12

L=4.4X

tO = -$0 o

I0 I I I I I

ZO :50 40 50 60 70 [NR IN dB

FIG. 11. Steered array performance with optimum amplitude shading compared with optimum processing and conventional amplitude shading.

a, =- D'Z(Q• +Q• +-.. + Q•N) (33) or

a~ (•R)-z V . (34)

Only when the noise field is isotropic (INR =- oo dB) and hence Q_r= O, will Q.•= (QR)'• and optimum amplitude shading leads to the optimum solution, i.e., one yield- ing minimum PN+r for a given received target signal level.

We have also calculated the output noise and interference power and then SNR 0 of the same array under the same field conditions with the array steered -30 ø off the broadside. The performance data when the array signals are processed by optimum amplitude and phase shading, optimum amplitude shading and conventional amplitude shading are shown in Fig. 11. It is again seen that the performance of the two optimum processors is essentially identical to within about 0.7 dB.

Except for the special case of broadside array in an isotropic noise field mentioned earlier, there is a sig- nificant difference in the weights of the two optimum

processors. For the same target signal power, the ab- solute value of the complex weights of the optimum processor is usually much larger than the optimized amplitude shading coefficients, sometimes by several orders of magnitude. For example, when the array is steered - 30 ø off broadside in an isotropic noise field, the amplitude and phase of the weights in channels 1-6 for the two optimum processors are plotted in Fig. 12. The weights are symmetrical with respect to the center of the array and those in channels 7-12 are therefore not shown. The phase for the optimum amplitude-shading processor are those needed to steer the array on target. It is seen that, in order to achieve an additional gain of 0.7 dB, the optimum processor with both amplitude and phase shading has gone to a high degree of supergain condition, as is evidenced by the large amplitude of channel weights and considerable target signal cancellation on the mainbeam axis. As was pointed out in an earlier paper, 4 the degree of supergain for a sonar receiving array can be measured by the supergain index

SGI = E a•/(target signal power) i

=lc, I (35)

for the optimum processor. Thus, for the case of =- 30 ø INR =- oo this index is evaluated to be

= 1. 089x 104, whereas for the optimum ampiRude-shading processors SGI = 0. 1150 with no target signal cancellation on the mainbeam axis.

Since excessively high values of supergain index represent an instable array susceptible to loss of array gain and side-lobe degradation, this index is calculated in each case of optimization with various intensities of interference field. The results are summarized in Fig.

13. Here one can see that optimum amplitude-shading results in SGI values within the order of magnitude 10, but if we allow both the amplitude and the phase of the channel weights to vary, SGI can soar to 103 or 104. The contrast is especially drastic when the array is steered away from the broadside.

lOO

0.1

0.05

50 --

0.01 I I -120

SGI= 1.089x 104

N -12

L -4.4

•i•-o - _30 INR =

z •,= I__• •-- I 5 3

OPTIMUM AMPLITUDE SHADING SGI = 0.1150

-80 -40 0 40 80 120 160

)•i IN DEGREES


FIG. 12. Shading coefficients of two optimum processors.



lO 6

105 •

104

103 •

102 •

10

1 m

10 -1

10 -2 m

10 -3 •

10 -4

x•r 0 = _ 30 o

N=12

L = 4.4 X.

-30 o

OPTIMUM AMPLITUDE SHADING

OPTIMUM PROCESSING

10 20 30 40 50 60

• INR IN dB

FIG. 13. Supergain,indexes of two optimum processors.

V. SUMMARY OF CONCLUSIONS

By minimizing the interference and noise power received by an array of transducers with respect to its amplitude shading coefficients, we have demonstrated the following in a narrow-band system:.

(a) It is possible to reject coherent interference incident from arbitrary directions by adjusting amplitude shading without resorting to phase shading or complex weighting.

(b) The rejection of interference can be visualized from the shifting or bunching of null points of array di- rectlytry pattern in the direction of the interference.

(c) The rejection of interference by optimum amplitude shading is nearly complete in the sense that the operation of the array becomes ambient noise limited and the increase of total noise and interference power is very small even in the presence of strong interference.

(d) A quantitative comparison of processing array signals with optimum amplitude shading and with optimum complex weighting reveals that the latter method

can only achieve an additional gain of less than 1 dB on the average. It is highly doubtful that such an additional gain can actually be realized due to the extreme supergain condition these optimum complex weights represent.

(e) It therefore appears that optimum amplitude shading or real weighting of channel signals is already suf- ficient for the improvement of sonar signal-to-interference ratio without involving significantly more in- vestment in hardware or computation as needed for complex weighting.

ACKNOWLEDGMENTS

Part of the work reported in this paper was done while the author was with EDO Corporation, College Point, NY.

1D. K. Cheng, Proc. IEEE 59, 1664 (1971). 2D. J. Edelblute, J. M. Fisk, and G. L. Kinnison, J. Acoust.

Soc. Am. 41, 199 (1967). 3S. W. W. Shor, J. Acoust. Soc. Am. 39, 74 (1966). 4H. S.C. Wang, J. Acoust. Soc. Am. 57, 1076 (1975). 5See, for example, I. S. Sokolnikoff and R. M. Redheifer,

Mathematics of Physics and Modern Engineering (McGraw- Hill, New York, 1958), p. 254.



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Interference rejection by amplitude shading of sonar transducer arrays