practical supergain ypj

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IEEETRANSACTIONS ONACOUSTICS, SPEECH, A ND SIGNALPROCESSING, VOL. ASSP-34,NO.3, UNE 1986 39 3

Practical SupergainHENRY COX, FELLOW, IEEE, ROBERT M. ZESKIND, SENIOR MEMBER, IEEE,

A N D THEO KOOIJ, MEMBER, IEEE

Abstract-The problem considered is that of designing endfire linearray shadings which provide a useful amount of supergain withoutextreme sen sitivity to random errors. Optimum shading weights areobtained subject to a constraint n the gain against un correlated hitenoise. The results of optimum array gain versus white noise gain con-straint aye presented parametrically for arrays of different interele-mentspacings,an ddifferentnoise ields.Resultsar epresented orspherically and cylindrically sotropic noise, and other wavenumberlimited noise fields, used in modeling ocean am bient noise. It is foundthat nearly optimum performance can be obtained in a simple delayand umbeamform er by shading o educe idelobes and modestoversteering to redu ce m ainlobe width ithout too large a reductionnmainlobe sensitivity.

I INTRODUCTIONT has long been known that a ignificant increase in theendfire beam array gain of a linearray over that of con-ventional beamforming can be achieved in isotropic noisefields when adjacent elements are separated by less thanone-half wavelength 11-[4]. This phenomenon has beencalled supergain at endfire.Forexample, or N veryclosely spaced elements, endfire array gain approachingN 2 can be achieved in spherically isotropic noise [3] anda corresponding array gain o f 2 N 1 can be achieved fora cylindrically isotropic noise field 121.Early work involved steering Dolph-Chebyshev shadedarrays past endfire to narrow their mainlobes [6], 7]. Forclosely spaced elements, this can be done without a grat-ing lobe entering visible space. It soon became apparent[5] that the theoretical gains of optimum performance werenot obtainable in practice due to the extreme sensitivityof these highly tuned systems to small random errors en-countered in realworldapplications. mportanterrors,such as amplitude and phase errors in sensor channels, arenearly uncorrelated from sensor to sensor and affect thebeamformer in a similar manner to spatially white noise.Hence, he array gainagainst uncorrelated or spatiallywhite noise is a good measure the array processor'srobustness to errors.

Gilbert and Morgan [8] first addressed the problem ofmaximizing array gain against isotropic noise with a con-straint on gain against uncorrelated noise and showed amonotonicdependencebetween he gain and hecon-straint. A closely related problem was examined by Uz-soky and Solymar [lo]. We consider the generalized ver-Manuscript received April 22, 1985; revised October 14, 1985.H. and R . M. Zeskind are with he BBN Laboratories, Inc., Ar-T. Kooij is with theOffice of Naval Technology, Arlington, VA 222IEEE Log Number 8407205.

lington, VA 22209.

sion 113 of that problem; thatf finding the set of optimalcomplex weights which maximizes the array gain againstcorrelated noise subject to an inequality constraint on thegain against spatially white noise. In this problem, hewhite noise gain is specified for a desired level of insen-sitivity to random errors.The solution of the constrained optimization problemyields endfire array gain which is ess han he uncon-strained supergain value but greater than for conventionalbeamforming. Constrained optimal solutions represent agood tradeoff betweenncreased array gain and sensitivityto errors.' The complex.weightsmay be implemented in afrequency domain beamformer. These weights generallydo not have a linear phase elationship associated with thesimple delay and beamformer.In hispaperwe first consider he sensitivity con-strained optimization problem and present results for sev-eral types ofbackgroundnoise.Then heproblem ofachieving near optimum performance with a delay andsum beamformer is examined and a design techniquepre-sented. It is found that near optimum performance can beachieved using a simple delay and sum beamformer withproper delay selection and amplitude shading.

SENSITIVITY CONSTRAINEDPTIMALEAMFORMINGConsider a line array of N equally spaced sensors. TheHermitian cross-spectral density matrix of the sensor out-puts at a frequencyf s given by

where a: is the noise power spectral density averaged overthe N sensors, and is the average signal power pectraldensity of the N sensors. The signal ross-spectral densitymatrix is P and that for the noise 1s Q. Both P and Q arenormalized to have their traces equal to N . The quantityo:/aZ is the input signal-to-noise spectral ratio.The output spectral density of a beamformer z ( f ) canbe expressed as the Hermitian formz ( f ) w * ( f ) R(f) f f ) (2)

where w( ) is the vector of complex beamformer weightsat frequencyfand the asterisk denotes omplex conjugatetranspose. The array gain G is defined as the output sig-nal-to-noise ratio divided by the input signal-to-noise ra-tio and is given by

0096-3518/86/0600-0393$01.00O 986 IEEE

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394 lEEE TRANSACTIONS ON ACOUSTICS,PEECH, AN D SIGNALROCESSING, VOL. ASSP-34, NO . 3, J UNE 1

The maximum gain is achieved by the complex weightvector w which maximizes ( 3 ) .The problem of maximizing the array gain ( 3 ) subjectto a constraint on the white noise gain, defined as

is most easily formulated as that of finding the vector wwhich minimizes 1/G with a constraint on l/G,,,, That is,minimizing

where E is a Lagrange multiplier. Finding the w whichminimizes ( 5 ) is equivalent to finding the w which maxi-mizes the reciprocal of ( 3 , which in turn is an eigenvalueproblem. The optimum w is chosen to be proportional tothe eigenvector corresponding to the largest eigenvalue of[Q E Z ] - ' P .When there is a single coherent wavefrontfrom direction 0, P is a dyad or rank one matrix whichmay be written as P rn (f, ) m* (f, ). Then the sen-sitivity constrained optimum w is any nonzero'scalar rnul-tiple of

[Q ~11-I 0) . (6 )The Lagrange multiplier provides a continuous param-eterizationbetween the array gainand the white noisegain. The relationship between these gains is monotonicin that increasing he white noise gain from ts uncon-strained optimum value at E 0 to its maximum value,10 log N , E causes the constrained optimum arraygain to decreasemonotonically rom its unconstrained

maximum to the array gain of a uniformly shaded con-ventional beamformer.As anexample,consideraneight-element uniformlyspaced endfire line array [ 1 2 ] . Figs. 1-4 present curvesof constrained optimal array gain versus the white noisegain for several values of the ratio of element spacing sto plane wave wavelengthX. The Lagrange multiplier E isa parameter along the curves. Fig. 1 shows the results fora spherically isotropic noise field, and Fig. 2 shows re-sults for a cylindrically isotropic noise field. Figs. 3 andshow similar results for other models of ocean ambientnoise.For very large E , the sensitivity constrained weight vec-tor of (6) will tend to m, ince the matrix I will dominateQ . But m is just conventional endfire bearnforming withuniform shading and plane wave phasing. Therefore, asthe desired white noise gain constraint is increased, theweight vector of (6) and the array gain approach that ofconventionalbeamforming. The upper imiton G,, forthis example, is 10 log 8 9 dB, the conventional arraygain against uncorrelated noise. As the minimum accept-ablevalueof G, is decreased, the constrainedoptimalarray gain ncreases pproaching the asymptotic iimitfor closelypacedlementsf N 2 for sphericallyisotropic noise field and 2N for cylindrically isotropic

G1

-50 -30 -2 0 70

Fig. 1. Arraygainversus G,, for aneight-elementendfire inearray spherically isotropic noise.

1'01-50 -40 -2 0 -10 0 70

Fig . 2. Arraygainversus G, fo ran eight-elementendfire inearray cylindrically isotropic noise.

0-50 -40 - 3 0 - 2 0 -70 0 10Idis]Fig. 3 . Arraygainversus G, for aneight-elementendfire inearray surface noise.noise field. Notice that there is an initial very sharp in the curves hat occurs as G, is allowed to decrease f9 dB. We will use 3 dB as a designconstraint forwhich permits a significant increased endfire array g



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COX et al.: PRACTICAL SUPERGAIN 395

G

3

-SO

7.36.2t

-40 -30 0WHITE

Fig. 4. Arraygain versus G , fo r aneight-elementendfire inearray ndistant shipping noise .

1

8.8

Fig. 5. Constrainedoptimumweights for cylindrically sotropicnoisewhere SIX 0.3, G , 3 dB. (a) l W , 2 1 . (b) Phase perturbation.S IA

Fig. 6 . Optimum beam pattern (cylindrically sotropic noise)

G 6.9 =9Fig. 7. Conventional beampattern (cylindrically sotropic no ise ).

against isotropic noise or ocean ambient noise withoutx-cessive sensitivity to random errors.Fig. presents thew which is the solution to (6) whenG, 3 dB, 0.3, in a cylindrically isotropic noisefield. The upper curve in Fig. is the magnitude of theweights, while the lower curve s the phase departure fromplanewavephasing. The endfire beampattern resultingfrom these constrained optimal weights is shown in Fig.6. Comparing Fig. 6 to he conventional endfire beam-pattern in Fig. 7 , we observe that the weights f (6 ) resultin a much narrower mainlobe and reduced sidelobe level

RELATIONSHIPO OVERSTEERINGThe 3 dB beamwidth of a uniformly spaced line array

steered to endfire is approximately 2 J o 8 8 6 x I L . If theelements are spaced t one-half wavelength,a grating lobeappears at the opposite endfire.y decreasing the elementspacing slightly, he grating obe vanishes from visiblespace.There s ittlecorrespondingbroadening of themainlobe since, at endfire, the eamwidth depends on thesquare root of array length. The net result is a nearly 3dB ncrease in endfire array gainwhich is sometimescalled the endfire anomaly.It is possible to oversteer or steer past endfire byusing time delays phases) matched to a propagationvelocity which is less than the speed propagation c .The delay r , applied to the nth element at position is7 , (1

which is matched to a wave propagating at a velocity(1 where 6 is a small positive constant which deter-mines the amount of oversteering. Oversteering has theeffects of reducing the mainlobe width in visible space,reducing mainlobe sensitivity, and bringing nto visiblespace new lobes at the oppositendfire. The sidelobe sen-sitivities are not changed but their levels relative to thereduced mainlobe are increased. The closenessof the ele-ment spacing determines he amount of oversteering whichcan be introduced without bringing in a grating lobet the



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396 IEEE TRANSACTIONSNCOUSTICS, SPEECH,NDIGNALROCESSING,OL.SSP-34,O ., JUNE 1opposite endfire. Early work in superdirective arrays in-volved extreme shading for reatly reduced sidelobes, andextreme oversteering of arrays with very closely spacedelements resulting in a very narrow mainlobe of greatlyreduced sensitivity. These designs were soon found to beextremelysensitive to various errorsmaking hem m-practical. In our parlance, he extreme oversteering re-sulted in an unsatisfactorily low gain against uncorrelatednoise.The question naturally arises: Can one achieve nearconstrained optimum performance using mild oversteer-ing? An examination of Fig. 5 is encouraging since thephases of the optimum weights are nearly linear in ele-ment position. Fig. 6 shows that the optimum beam pat-tern of an eight-element array with s /h 0.3 and havinga 3 dB white noise gain constraint has a peak sidelobelevel of about 13 dB. This 13 dB plus a 6 dB loss inmainlobe sensitivity which reduced the white noise gainfrom 9 dB (10 log8) to 3 dB suggests that a design whichhas 19 dB sidelobe before oversteering might give similarperformance. As a candidate design, we will place slightlymore emphasis on sidelobe control. We proceed as fol-lows.1) Shade he array for 25 dB sidelobes beforeoversteering. This should result in 19 dB sidelobes after over-steering. For simplicity we use 25 dB Taylor [9] shading.

2) Oversteer the array a limited amount such that theresulting white noise gain is 3 dB.Thisapproximatelycorresponds to oversteering by an amount such that the 6dB down point on the mainlobe is at endfire.Since we oversteer only a fraction of the mainlobe, onlya fraction of an additional lobe enters visible spaceat theopposite endfire. The appearanceof a grating lobe will beavoided if

N - 22 N

so that is slightly less than X/2.The design procedure is illustrated in Figs. 7, 8 , and 9.Fig. 7 shows the beampattern of an unshaded rray steeredto endfire. Fig. 8 shows he corresponding beampatternwhen 25 dB shading is used. The shading has cost about0.4 dB loss in white noise gain. Fig. 9 shows the beam-pattern of the shaded array with oversteering such that G,is 3 dB. The gain of this array against two-dimensionalisotropic noise is only 0. 2 dB lower than the constrainedoptimum of Fig. 6 . Nearly uniform 19 dB down sidelobeshave been achieved. Thus, the simple shaded delay andsum oversteered beamformer is nearly optimum for thisexample.The time delays of (7) were adjusted in the previousexample to meet a 3 dB white noise gain constraint for acylindrically isotropic noise field. The question naturallyarises as to the performance of this simple delay and sumbeamformer when the noise field is not the one assumedin adjusting the time delay, i.e., the effect of model mis-match. For the uniformly spaced eight-element line arraywith G, 3 dB and s/X 0.3, the constrained optimal

SIX ~ 0 . 3

=6.7

F i g . 8. Beampattern with 25 dB shading (cylindrically isotr opic nois

=9.3

Fig. 9 . Shaded and oversteered beampattern (cylindrically isotropic noise)

array gain for spherically isotropic noise is found froFig. 1 to be 14.1 dB. Using the Taylor shadings and timdelays of the previous example, t was found that the arrgain achieved by the simple delay and sum beamformwas 13.7 dB, only 0.4 dB lower than the constrained otimum.Thus, the shaded and oversteered design gives neaoptimumperformance nboth cylindrically and sphecally isotropic noise fields. The reason for this is faiobvious. In both cases oversteering exploits the fact tthe noise fields are limited in wavenumber to those wavnumberscorresponding to real angles.The oversteerbeamformer exploits in a slightly different way the samproperties of heband-limited noise as does the costrained optimum.A s a second example, consider a 32-element line arrwith one-quarter wavelength spacing and a white noconstraint of 8 dB in a spherically isotropic noise fieThe interelement spacing is 0.25 X. Conventional endfsteering with uniform weights yields the beampattern



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COX er ai.: PRACTICAL SUPERGAIN 397

Fig. 10. Conventional beampattern (spherically sotropic noise).

Fig. 1 1 . Optimum beampattern (spherically sotropic noise).

Fig. 12 . Beampattern with 25 dB shading (spherically isotropic noise).Fig. 10 with an array gain of 15 dB and coincidently G,15 dB. The constrained optimum beampattern for G,,,8 dB is shown in Fig. 11. It achieves an array gain of19.8 dB, a nearly 5 dB increase in array gain over con-

Fig. 13. Shaded and oversteered beampattern (spherically isotropic noise).ventional beamformingwhile retaining reasonable ro-bustness against random errors. Fig. 12 shows the effectof 25 dB Taylor shading on the array steered to endfire.The value ofG, is reduced by about 0.4 B. Finally, Fig.13 shows the shaded array oversteered such that G,,, 8dB. Thearray gain of 19.3 dB,within dB of optimum,is achieved with a simple delay and sum beamformer.

CONCLUSIONUsing constrained optimum beamforming,t is possibleto achieve a significant increase in endfire array gain fora variety of noise fields consisting of plane waves. Anessential feature of these noise fields is that they are bandlimited in wavenumber corresponding to waves propagat-ing at a fixed phase velocity. Thus, they have no energy

corresponding to slow waves. The constrainton the whitenoise gain limits the sensitivity of the array gain to smalldeviations from idealized assumptions. The use of realshadings plus mild oversteering n a simple delay and sumbeamformer has been found to give nearly optimum per-formance.ACKNOWLEDGMENT

The authors express their appreciation to. M . Owen,BBN Laboratories, Incorporated, who developed optimi-zation and simulation software which has been central toour recent work in optimum and adaptive beamforming.REFERENCES[ l ] W. W. Hansen and J . R. Wood yard, A new principle in directionalantenna design, Proc. IRE, vol. 26, Mar. 1938.[2 ] S . A. Schelkunoff, A mathematical heory of l inear arrays, BellTech. J . , vol. 2 , Jan. 1943.[3 ] A. I. Uzkov, An approach to the problem of optimum directive an-tenna design, Comptes Rendus (Doklady) de rlcademic desences del LURSS, vol. LIII, no . 1946.H . J . Riblet, Note on the maximum directivity of an antenna,IRE, vol. 36 , M ay 1948.[5] T. T. Taylor, A discussion of the maximum directivity of an an-tenna, Proc. IRE, vol . 36 , Sept. 1948.[6] R . H. Duhamel, Optimum patterns for endfire arrays, Proc. IRE,vol. 4 1 , M ay 1953.[7] R . L. ritchard, Optimum directivity for linear point arrays, Proc.J . Acoust. Amer., vol. 25 , Sept. 1953.



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39 8 IEEERANSACTIONSNCOUSTICS,PEECH,NDIGNALROCESSING,OL .SSP-34, NO. 3, JUNE 1[8]E. N. Gilbert and S . P. Morgan, Optimum design of antenna arrayssubject to random varia tions, Bell Syst. Tech. J . , vol. 34 , May 1955.[9] T. T. Taylor, Design of line-source antennas for narrow beamwidthand low side lobes, IR E Trans. Antennas Propag., vol. AP-3, Jan.1955.[lo] M . Uzsokyan d L . Solymar,Theory of super-direc tive inear ar-rays, Acta Phys. Acad. Sci. Hungary, vol. 6 , no . 2, 1956 .[!I] H . Cox,Sensitivity considerations inadaptivebeamforming, nSignalProcess ing, J . W. R. Griffiths, P. L.S tock l in ,andC. VanSchooneveld,Eds.Ne w York an dLondon:Academic,1973.[ I21 T. Kooij, Adap tive array processors for sensitivity constrained op -t imization, Ph.D. dissertation, Catholic Univ. America, Washing-ton , DC, June 1977.[ I 31 H. Cox , Spatial correlation in arbitrary noise fields with applicationto ambient sea noise, J . Acoust. SOC. A m er . , vol. 54, Nov. 1973.

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