Interference Cancellation when Direction-of-Arrival is Time-Varying

Koshy George† and Kiran S. Sajjanshetty‡

† ‡P.E.S. Centre for Intelligent Systems,†Dept. of Telecommunication Engineering, P.E.S. Institute of Technology,

100 Feet Ring Road, BSK III Stage, Bangalore 560 085, India. Email: kgeorge@pes.edu

Abstract—Rejecting interferences, whether inten-tional or otherwise, enhances the quality of the signal-of-interest at the receiver, and hence is importantin several applications. Our goal in this paper is toreject intentional interferences whose directions ofarrival are both unknown and time-varying. We showthat a simple adaptation technique with switching canachieve this objective.

Keywords-Adaptive arrays, interference cancella-tion, direction-of-arrival, time-variations.

I. Introduction

Spatial array processing techniques have played animportant role in several fields including radar, sonar,communications, and seismology. Typical applicationsinclude determining the direction-of-arrival (DOA) ofsignals, beamforming in specified directions, and reject-ing interferences. Canceling interferences resulting fromcurrent operating environments enhances the quality ofthe signal-of-interest (SOI). Examples of such interfer-ences include transmission by other users in communi-cation systems, and radar clutter. In this paper, however,we deal with the rejection of intentional interferences, orjamming signals, that are deliberately used to prevent auser from receiving the SOI. Rejecting such interferencesbecomes non-trivial when the sources of such signals areintentionally made mobile. We propose here an adaptivesolution to the problem of extracting a narrowbandsignal in the presence of an interference whose DOA isvarying with time.

Beamforming via spatial matched filtering is sufficientto maximize the visibility of a signal in the presence ofthermal noise of the receivers [1], [2]. However, this isinsufficient to reject an interference whose power maybe several orders of magnitude higher than the SOI.On the contrary, the minimum-variance distortionlessresponse (MVDR) beamformer [1] maximizes the signal-to-interference-plus-noise ratio (SINR), and hence notonly casts a beam in the direction of the SOI, but aswell places a null in the direction of the interference.The depth of this null is comparable to the interferencepower [3]. A generalization of this is the linearly con-strained minimum-variance (LCMV) beamformer [4], [5].

Optimal beamformers (e.g., MVDR beamformer) re-quire the knowledge of the second order statistics of the

interference and noise signals. Adaptive beamformers areused when such a priori knowledge is not available. Blockadaptive sample matrix inversion (SMI) [6], or sample-by-sample adaptive methods such as the least meansquares (LMS) algorithm [2], the recursive least squares(RLS) algorithm [7]–[10], and the Frost algorithm [11]have been proposed in the past. In these algorithms,information about the direction of the SOI is used inlieu of the ‘desired’ signal, since the latter is not directlyavailable. In general, these techniques are used when theDOA is time-invariant.

Canceling interferences due to moving sources havebeen considered in, for instance, [12]–[15]. While a vari-ant of the RLS algorithm is used in [12], post-correlationcancellation and adaptive cancelers are considered in [13]at the cost of an increase in the noise level. The objectivein [14] is to determine only the power spectral densityestimate of the output. A planar array of sensors isused in [15] and the weights adapted using variants ofthe LMS algorithm. Here, exact nulls are not placed inthe direction of the interferences. In these references, itappears that the problem of spatial coherence betweenthe SOI and the interference has not been meaningfullyaddressed. It is to be emphasized that an interferencethat is more powerful than the SOI is a cause forconcern when there is spatial coherence, even when theinterference has moved away from the SOI.

In this paper, we assume a possibly hostile environ-ment wherein the position of the source of the inter-ference is altered continuously to frustrate the effortsof the beamformer to cast a null in its direction. Wedemonstrate that an adaptive technique based on theoptimal MVDR beamformer together with switching isable to track the interference direction, nullify its effectand extract the desired signal even when there is spatialcoherence between the two signals. The novelty of theapproach lies in the use of both slow and fast adaptationfor spatial processing to achieve our objectives.

This paper is organized as follows: In Section II weintroduce briefly the fundamentals of array signal pro-cessing. The adaptive beamformer used in this paper isdescribed in Section III. The efficacy of the technique isdemonstrated through simulations. The results of theseare presented in Section IV.

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II. Array Signal Processing

Consider a uniform linear array (ULA) of M sensorswith an inter-element spacing of d = λ

2 , where λ isthe wavelength of the carrier signal. We assume thatthe signal at the jth element xj(t) has been demodu-lated to baseband, filtered to receiver bandwidth, andsampled to yield xj(n). The array snapshot of theULA at any instant is then defined as follows: x(n) =(

x1(n) x2(n) · · · xM (n))T

, where AT denotes thetranspose of a matrix A. For a ULA, it is well-known [3]that

x(n) =√

Mv(φs)s(n) + η(n) + i(n) (1)

where s(n) is the signal-of-interest impinging on theULA at an angle φs with respect to the normal to theULA, η(n) is the vector of sensor thermal noises acrossthe bandwidth of the receiver of each of the sensors, i(n)is the vector of interference at the sensors, and

v(u) =1

√

M

(

1 e−j2πu· · · e−j2πu(M−1)

)T(2)

is the steering vector. Here, u = d sin φs

λis the normalized

spatial frequency. We note that the delay in the arrivalof the signal at the jth sensor element in the array ismeasured relative to the first sensor element. The angleφs is the look-direction, or the DOA of the signal-of-interest. In this paper, we assume that this DOA is eitherknown a priori or determined using any of the DOAestimation techniques (see, for instance, [3]).

To extract s(n), the M sensor signals are linearlycombined, with the output of the array given by y(n) =w

Hx(n), where AH represents the Hermitian transpose

of the matrix A. The beam response depends on theweights w, and is chosen appropriately to cast the beamin specified directions. The weight vector that phase-aligns a signal from direction φs is w = v(φs); thisbeamformer is known as the spatial matched filter. If thenoise is spatially white this filter is optimal in the sensethat it maximizes the signal-to-noise ratio (SNR) at theoutput of the beamformer. However, the sidelobes of thisfilter are large enough to allow strong interferences topass through to the output.

The optimal beamformer, on the contrary, maximizesthe signal-to-interference-plus-noise ratio (SINR), and isgiven by

wo =R

−1i+ηv(φs)

vH (φs)R−1i+ηv(φs)

, (3)

where Ri+η is the interference-plus-noise correlation ma-trix, and is equal to Ri +Rη, the sum of the correlationmatrices of the interference i(n) and the noise η(n). Here,we assume that the signal s(n), the interference, and thenoise are mutually uncorrelated. It may be noted thatmaximizing the SINR optimally improves the visibility

of the desired signal in the presence of interferences.This beamformer is also the solution to the follow-ing constrained optimization problem: minw w

HRi+ηw,

subject to the constraint that the gain in the look-direction is unity (i.e., w

Hv(φs) = 1). Therefore, thisis also known as the minimum-variance distortionlessresponse (MVDR) beamformer. The linearly constrainedminimum-variance (LCMV) beamformer is obtained byplacing additional constraints.

An important advantage of the MVDR beamformeris that a priori knowledge of the directions of arrivalof the interferences are not required. Such informationis embedded within the correlation matrix, and makesunnecessary their explicit estimation. This has a directbearing on the problem considered in this paper as theDOA of the interference is assumed to be changingcontinuously.

III. Adaptive Array Processing

The optimal (MVDR or LCMV) beamformer is basedon the assumption that the interference-plus-noise cor-relation matrix Ri+η is known a priori. We note thatRi+η in (3) may be replaced by Rx, the autocorrelationmatrix of the signal x as defined in (1) [3]. Under theassumptions on the signals s(n), i(n), and η(n), it followsthat Rx = σ2

sv(φs)vH (φs)+Ri+η , where σ2

s is the powerof the signal s(n). Using the matrix inversion lemma, itcan easily be shown that the MVDR beamformer is givenby

wo =R

−1x v(φs)

vH (φs)R−1x v(φs)

. (4)

Thus, it is sufficient to know the second order statisticsof the signal Rx. However, since a priori knowledge ofthis is not available, we estimate as follows:

Rx(n) =

K−1∑

k=0

x(n − k)xH(n − k), (5)

where K is the number of input samples used. Alterna-tively, we can express (5) as a recursion:

Rx(n) = Rx(n − 1) + x(n)xH (n). (6)

Thus, if the sample correlation matrix is updated usingthe above recursion, one can adapt the weights everyinstant as follows:

w(n) =R

−1x (n)v(φs)

vH (φs)R−1x (n)v(φs)

.

However, since the inverse of Rx(n) is more useful, thefollowing is obvious from (6) using the matrix inversionlemma:

P(n) = P(n − 1) −P(n − 1)x(n)xH (n)P(n − 1)

1 + xH(n)P(n − 1)x(n),

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where P(n)∆= R

−1x (n). The weights are then updated as

follows:

w(n) =P(n)v(φs)

vH (φs)P(n)v(φs). (7)

Similar to the adaptive algorithms for array signal pro-cessing available in the literature, we require the DOAof the signal-of-interest. (We assume that either thisis known a priori, or has been estimated.) Further, weobserve that the advantages of using the weight updatein (7) are three-fold: (i) This is an adaptive version ofthe MVDR beamformer, and it does not require the apriori knowledge of the second order statistics of the in-terference and the noise. Further, the explicit estimationof the DOA of the interference is not required. (ii) Theweights are updated at every instant of time. Therefore,the algorithm has the potential to track faster changesin the DOA of the interference. (iii) The computationalburden is considerably reduced since the inverse of thecorrelation matrix is updated at every instant of time.

IV. Simulation Results

In this section, we demonstrate through simulationexperiments the efficacy of the weight update techniquepresented in the previous section. In these experimentswe assume that the DOA of the interference is changingcontinuously. For brevity, we present the results for onlytwo cases: In Case A, the DOA of the interference isinitially 30, and subsequently increases linearly at arate of 0.01 per sampling instant. In Case B , the DOAfollows a sinusoidal pattern between −30 and 30. Thevariations for both cases are illustrated in Fig. 1.

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Figure 2. Case A: Output of beamformer when signal power isgreater than interference power.

Tracking an interference, and canceling it, presents nodifficulty as long as the power of the SOI is larger thanthat of the interference, even when the arrival-directionsof both the signal and interference are the same. Thesimulation results for Case A illustrating this are shownin Figures 2 and 3. Here we assume that the signal-of-interest with DOA 30 is s(t) = α sin Ωt, where Ω is10 radians/sec, and the signal power is 6 dB. Further,we assume that the interference power is −63 dB. Thenumber of elements of the ULA M is assumed to be 16.From the array output depicted in the figures it is quiteobvious that the interference is rejected completely evenwhen there is a spatial coherence between the s(t) andthe interference. This is quite obvious from the fact thatadaptive systems generally tend to track the strongersignal [2]. The results for Case B are similar, and areomitted here for brevity.

The fact that adaptive systems tend to track thestronger signal presents a difficulty when the interferencepower is much larger than the signal power. (This situa-tion is of more interest in this paper.) The consequence ofthis is that when the DOA of the interference approachesthat of the signal, making the two signals spatially coher-ent, the adaptive algorithm champions the interferencesignal, and subsequently tracks it. The effect of thison the weight vector is depicted in Fig. 4 for Case Awhere we assume that the power of the interference is52 dB, which is considerably more than the power ofs(t). (Here, we assume that the DOA of the signal isknown a priori to be at 42.) Evidently, the weight vectorbecomes rather large when the DOA of the interference

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Figure 4. Case A: Changes in the norm of the weight vector.

approaches that of the signal-of-interest, and remainslarge even beyond.

We overcome this problem as follows: As long as therate of change of the power of the output is larger thana specified threshold, the weight vector is re-initialized.That is, the current weight vector is switched to theinitial weight vector. The output of the beamformer forCase A is shown in Figures 5 and 6 when the weightvector is not switched. We can observe from these figuresthat as long as the DOA of the signal and interference aredifferent, the beamformer rejects the interference, and

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Figure 5. Case A: Beamformer output without weight switching.

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Figure 6. The same output as in Fig. 5 on an expanded scale.

maximizes the visibility of the SOI. As the DOA of theinterference approaches that of the signal, the output ofthe beamformer deteriorates rapidly, and the adaptivealgorithm begins to track the interference. When theproposed change is adopted and the weights switched,the adaptive algorithm recovers from the interference,and the output of the beamformer once again tracks theSOI after the two signals are no longer spatially coherent.This is evident from Fig. 7.

The output of the beamformer for Case B is shown inFigures 8–10 with weight switching. We assume that the

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Figure 7. Output of beamformer for Case A with weight switching.

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Figure 8. Output of beamformer for Case B with weight switching.

DOA of the signal is 30. Since the changes in the DOAof the interference is periodic, it matches the DOA ofthe signal-of-interest periodically. Consequently, weightswitching is done periodically. From the figures it isevident that the output recovers from the effect of astronger interference periodically.

On the contrary, suppose that the changes in the DOAof the interference are as in Case B, but that the DOAof the signal-of-interest is 40. Since the interferenceand the SOI are never spatially coherent, the adaptivealgorithm places an effective null in the direction of the

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interference, and no weight switching is required. This isevident from the beamformer output shown in Figures 11and 12.

It is clear from these simulation experiments that theperformance of the strategy of adaptation presented inthis paper is quite satisfactory. We further note that thisis true up to interference-to-signal ratio of 80 dB. Beyondthis, the output of the beamformer begins to deteriorate.

V. Conclusions

In this paper we presented a two-level adaptive strat-egy to place a null in the direction of an interference in

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Figure 11. Output of beamformer for Case B when the signalsare always spatially incoherent.

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Figure 12. The same output as in Fig. 11 on an expanded scale.

situations wherein this direction changes continuously.At the first level, the optimal weights are determined atevery instant after propagating the inverse of the correla-tion matrix of the output of the sensors in the array. Thiscorresponds to slow adaptation. At the second level, theweights are switched whenever there is spatial coherencebetween the interference and the signal-of-interest. Thiscorresponds to fast adaptation. Simulation experimentsdemonstrate the efficacy of this chosen adaptive strategy.

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