122 IEEE SIGNAL PROCESSING LETTERS, VOL. 6, NO. 5, MAY 1999

Rejection and Tracking of an Unknown BroadbandSource in a Two-Element Array Through LeastSquare Approximation of Inter-Element Delay

C. C. Ko, Senior Member, IEEE,and C. S. Siddharth

Abstract—In this letter, a new algorithm to reject and track anunknown broadband source by approximating the inter-elementdelay in a two-element adaptive array is investigated. Essentially,by constraining the response of theL-tap processing filter toa delay in the least squares sense, the number of adaptiveparameters is reduced to merely an unknown gain and delay,which can then be adjusted to reject the unknown source byusing the LMS algorithm. Compared with the conventional LMSalgorithm, the new algorithm has the advantage that, at aboutthe same complexity, estimate for the source location is availableand the convergence speed is significantly improved.

Index Terms—Adaptive array, adaptive filter, DOA estimation,tracking.

I. INTRODUCTION

T HE USE OF adaptive array processing techniques toreject intentional or unintentional interference signals is

well known, with numerous techniques investigated [1] fornarrowband systems. In broadband applications, tapped delayline transversal filters, instead of a parallel bank of narrowbandsystems, are commonly used due to the smaller number ofadaptive weights needed for the same rejection performance[2]. One disadvantage with using transversal filters, however,is that the eigenvalue spread of the signal covariance matrixis increased due to temporal effects, leading to a deteriorationin convergence and tracking behavior [3].

To overcome this problem, this work proposes using con-straints for the processing weights so that the filter approxi-mates a delay in the least squares sense and the number ofadaptive parameters is reduced to an unknown gain and delay,which can then be adjusted to reject and track the unknownsource by using the LMS algorithm. Although the algorithmis derived in terms of a two-element power inversion array, itcan be readily extended to multi-element systems through theDavies beamformer [4].

II. I MPULSE RESPONSE ANDLEAST SQUARESAPPROXIMATION

Fig. 1 shows the broadband two-element array of interest,where the element spacing iselement zero has an unknowngain with respect to element one, and each element has

Manuscript received December 28, 1998. The associate editor coordinatingthe review of this manuscript and approving it for publication was Prof. T. S.Durrani.

The authors are with the Department of Electrical Engineering, NationalUniversity of Singapore, Singapore 0511 (e-mail: elekocc@nus.edu.sg).

Publisher Item Identifier S 1070-9908(99)03105-3.

independent receiver noise of power A far-field sourceor jammer of power arrives from a direction of Afterbandpass filtering to restrict the spectra to the frequencyband of interest from to where

is the center frequency and is the fractionalbandwidth, the element inputs are quadrature demodulated,lowpass filtered and sampled at a frequency of Theresulting complex samples are passed into a nonrecursive filterwith complex weights and these are adjusted to reject thebroadband source at the array output adaptively.

Using andthe array output is

(1)

where is the input sample from array element atthe th sampling instant, are the filter weightsemployed, and is the sample delay for before the latteris used to obtain Obviously, the array being implementedis a power inversion one and the rejection or tracking of thebroadband source can be achieved by adjusting the weightvector to minimize the output power.

Taking the sampling frequency to be one, without lossof generality, the spectra of and due to the sourcefrom can be shown from Fig. 1 to be related by

(2)

where is the inter-element delay in samplingperiod, is the inter-element phase shiftcorresponding to at and is the wavelength atFrom (2) and Fig. 1, the array response to the source at afrequency is

(3)

where Taking the inverse Fouriertransform, the impulse response corresponding to is

otherwise.(4)

From (4), if can be chosen such that is zerofor or is zero for all the source fromwill be totally eliminated at the output regardless of itsspectrum. This is equivalent to having approximated theresponse of a delay with complex

1070–9908/99$10.00 1999 IEEE

KO AND SIDDHARTH: UNKNOWN BROADBAND SOURCE 123

Fig. 1. Two-element broadband tapped delay line array.

gain by using a transversal filter with weight vectorperfectly. Although such a perfect approximation can be

easily seen from (9) to be impossible unlessis an integerand if and the weights are properly chosen, the broadbandsource will be well rejected if a good approximation can befound.

One approach to approximate by usingis to consider the average square error of the approx-

imation over :

(5)

From (4) and (5), the best weights which minimizes aregiven by

(6)

and the minimum value for is

(7)

Since is between and for ranging fromto and the maximum value for is two, will be

less than unity if the element spacing at the center frequency isnot more than half a wavelength and the direction of a sourceat this frequency is to be uniquely resolvable by the array.Using this and the versus characteristic, itwill be most appropriate to choose and such that

integer (8)

Equation (7) will then be symmetrical with respect toand

(9)

Since is an integer, the maximum value of foris approximately

(10)

when and is large. Approximating the summationby integration, this gives

(11)

the maximum square error in the case of very large bandwidth

For the more common scenario when the maximum valuefor is small compared with unity, the maximumsquare error is roughly

(12)

when Approximating the summation by inte-gration again, this becomes

(13)

By taking the unknown gain between the two array elementsto have a nominal value of 1, (11) or (13) can be used asdesign formulas for the determination ofto achieve a certainmaximum square error or jamming rejection performance.

124 IEEE SIGNAL PROCESSING LETTERS, VOL. 6, NO. 5, MAY 1999

III. W EIGHT ADJUSTMENT

To cancel the unknown broadband source, the weightscan be adjusted directly by using an algorithm

to minimize the output power continuously. However, aswill be large for good broadband rejection, the convergencebehavior of the system will deteriorate considerably if simplesteepest descent algorithms with complexity are em-ployed. Also, it will be difficult to deduce the source locationfrom the values of the weights. To improve these systemperformance, a “constrained weight adaptation” scheme isproposed. Specifically, instead of having“fully” adjustableweights, only and on the relative element gain andsource location, will be adjusted. The weights will beobtained from these two parameters using (6) so that abroadband null will be formed toward the source in a leastsquares sense in the frequency domain.

From (6), the changes and in and willresult in a change in the complex weight vector givenby

(14)

where

(15)

(16)

and

(17)

Based on these, one simple method to adjustand totrack and cancel the broadband source is to first estimate thegradients of the output power with respect to the weights fromthe most recent input samples by using (18), shown at thebottom of the page. Then, updating the weights in the steepestdescent manner would require

(19)

where is the feedback factor. This would imply adjustmentof and by

(20)

where the pseudoinverse, is given by

(21)

Combining, once the gradient is estimated, and canbe adjusted according to

(22)

TABLE ICOMPUTATIONAL COMPLEXITY FOR IMPLEMENTING THE NEW ALGORITHM

Finally, the weights corresponding to these new values ofand can be calculated using (6) and the whole process ofgradient estimation, parameter and weight updating can berepeated indefinitely to minimize the output power to rejectthe broadband source.

The above derivation depends on (14) which relates toand Obtained from differentiation, this relationship is

exact if the changes involved are small andhas full rank.The former will be true under the usual situation when thefeedback factor is small or steady state has been reached.While the latter is difficult to prove mathematically (due to thecomplexity of the various functions), it is feasible to calculateoff-line the rank of over all possible values of andor can then be changed to ensure thathas full rank. Infact, as is usually much smaller than one (its maximumvalue is 0.2 for 40% bandwidth at the variousfunctions in (16) and (17) will in general be nonzero andwill generally have full rank.

Table I summarizes the number of multiplications and di-visions needed to implement the new methd per output sam-ple, assuming that and whichdepend on only, have been precalculated and stored. Ob-viously, the complexity is , which is comparable to theconventional LMS algorithm.

IV. SIMULATION RESULTS

Some simulation results on the performance of the algorithmwill now be presented. Note that the primary performancemeasures are on how well and fast a broadband jammer orinterference can be rejected. This is because although it ispossible to obtain an estimate of the delay in the new algorithmand the problem of delay estimation [5]–[7] is related toadaptive broadband jammer rejection, the two problems areactually quite different. Specifically, in delay estimation, theimplementation of an adaptive structure that can reject the

(18)

KO AND SIDDHARTH: UNKNOWN BROADBAND SOURCE 125

Fig. 2. Output power trajectory of using the new algorithm and the LMSalgorithm directly in a five-tap array with an element spacing of half awavelength at the centre frequency. The environment consist of a far-fieldsource of power 30 dB, with a flat power spectrum density, arriving from 15�;

element zero has a gain of 1.05 with respect to element one, and each elementcontributes independent receiver noise of 0 dB. The fractional bandwidth is10%, the number of input samples used for the estimation of the output powergradients,N; is five, while the feedback factor is chosen for a misadjustmentof 1%.

jammer continuously is not needed and the variance of theestimate is an important performance indicator. In jammingrejection, the efficient implementation of an adaptive structureis crucial and the output power, which measures how well andfast the jammer can be rejected, is most important.

Figs. 2 and 3 show the variation of the output power andjamming rejection (ratio of the jammer power at element zeroto that at the array output) for a five-tap array with an elementspacing of half a wavelength at the centre frequency. Theenvironment consist of a far-field jammer of power 30 dBand flat power spectral density arriving from a direction of15 with respect to the normal of the array. Element zero hasa gain of 1.05 with respect to element one, and each elementcontributes independent receiver noise of 0 dB. The fractionalbandwidth is 10%, while the number of input samples used toestimate the gradients of the output power with respect to theweights, is chosen to be five. Also shown in the figures forcomparison is the convergence curve when the LMS algorithmis used directly for adjusting the five tap weights. Note thatboth algorithms have the same initial starting weights andhave feedback factors chosen to result in the same measuredmisadjustment of about 1% in the steady state.

With all other conditions remaining the same, Fig. 4 com-pares the rejection performance of the two algorithms whenthe jammer’s power is reduced to 5 dB. Clearly, in additionto the availability of the source direction in terms of thenew algorithm has a better convergence behavior than whenusing the LMS algorithm, directly due to the smaller numberof adaptive parameters that have to be adjusted.

V. CONCLUSIONS

A new algorithm to reject a broadband source by approx-imating the inter-element delay in a two-element adaptivearray is studied. By constraining the response of the-tap processing filter to a delay in the least squares sense,

Fig. 3. Jamming rejection trajectory in the scenario of Fig. 2.

Fig. 4. Jamming rejection trajectory in the scenario of Fig. 3 when thejammer power is reduced to 5 dB.

the algorithm reduces the adaptive parameters in the systemto an unknown gain and delay, which is then adjusted totrack the unknown source by using the LMS algorithm. Thenew algorithm has the advantage that it has about the samecomplexity as the LMS algorithm, estimate for the sourcelocation is available, and convergence speed is significantlyimproved.

REFERENCES

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[4] R. Khanna and B. B. Madan, “Adaptive beamforming using a cas-cade configuration environment,”IEEE Trans. Acoust., Speech, SignalProcessing, vol. ASSP-31, pp. 940–945, 1983.

[5] A. Zeira and P. M. Schultheiss, “Threshold and related problems in timedelay estimation,”IEEE ICASSP, 1991, pp. 1261–1264.

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