An Adaptive Sensor Array Using an Affine Combination of Two Filters

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1 An Adaptive Sensor Array Using an Affine Combination of Two Filters Tõnu Trump Tallinn University of Technology Department of Radio and Telecommunication Engineering Ehitajate tee 5, Tallinn Estonia Abstract: We study an adaptive sensor array that uses a combination of two filters as the adaptive scheme to update the beamformer weights. The Generalized Sidelobe Canceller configuration is used for computing the adaptive weights of the beamformer. As the adaptive scheme we use the recently proposed affine combination of two adaptive filters. The filters are joined with help of an adaptive parameter. This way the scheme forms a two stage adaptive structure. The combination of two adaptive filters is a new interesting way of obtaining both fast initial convergence and small steady state error at the same time. We present the results of a steady state analysis of the scheme in a sensor array processing scenario. The theoretical results are verified in our simulation study. Key Words: Adaptive filtering, antenna arrays, combination of two adaptive filters. 1 Introduction In this paper we investigate an adaptive beamformer that can be used to combine the output signals of an array of sensors. The beamformer is constrained to filter out the signal form the desired source and suppress all the other signals impinging at the antenna array. The desired source is identified by its steering vector so that any signal incident on the array from the direction of interest appears as an identical replica of the source signal at the beamformer output. All other signals received by the array arriving from different angles are considered to be noise and/or interference. The purpose of the beamformer is thus to minimize the effects of noise and interference at the array output while preserving the desired signal. We are going to use so called Generalized Sidelobe Canceller [1] beamformer structure that has two branches. One of the branches determines the direction of interest and the other branch includes a blocking matrix followed by an adaptive algorithm who s purpose is to find the best possible rejection of noise and interferers based on the mean square criterion. As an adaptive algorithm we use the recently proposed combination of two adaptive filters. Usually one faces a trade off between the initial convergence speed and the mean square error in steady state when designing adaptive algorithms. In case of algorithms belonging to the Least Mean Square (LMS) family this trade off is controlled by the step-size parameter. Large step size leads to a fast initial convergence but the algorithm also exhibits a large mean square error in the steady state and in contrary, small step size slows down the convergence but results in a small steady state error [1, 2]. Recently there has been an interest in a combination scheme that is able to optimize the trade off between convergence speed and steady state error [3]. The scheme consists of two adaptive filters that are simultaneously applied to the same inputs as depicted in Figure 2. One of the filters has a large step size allowing fast initial convergence and the other one has a small step size for a small steady state error. The outputs of the filters are combined through a mixing parameter λ. The performance of this scheme has been studied for some parameter update schemes [4, 5, 6]. The reference [4] uses a convex combination i.e. λ is constrained to lie between 0 and 1. The reference [5] presents a transient analysis of a slightly modified version of this scheme. The parameter λ is in those papers found using an LMS type adaptive scheme and computing the sigmoidal function of the result. The reference [6] takes another approach computing the mixing parameter using an affine combination. This paper uses the ratio of time averages of the instantaneous errors of the filters. The error function of the ratio is then computed to obtain λ. Both of those approaches thus use nonlinear schemes to compute the combination parameter. In this paper we compute the mixing parameter λ ISBN:

from output signals of the individual filters in a linear fashion.

The scheme was independently proposed in [7] and [8] and the transient performance of it was investigated in [9]. In [10], the output signal based combination was used in adaptive line enhancer.

In the analysis part of the paper we investigate the behaviour of the adaptive combination scheme in steady state i.e. in the situation when discrete time n approaches infinity.

The italic, bold face lower case and bold face upper case letters will be used for scalars, column vectors and matrices respectively.

The operator E[ ] denotes mathematical expectation and Re{ } stands for the real part of a complex variable. The block diagram of the Generalized Sidelobe Canceller is shown in Figure 1.

2 from output signals of the individual filters in a linear fashion. The way of calculating the mixing parameter is optimal in the sense that it results from minimization of the mean-squared error of the combined filter. The scheme was independently proposed in [7] and [8] and the transient performance of it was investigated in [9]. In [10], the output signal based combination was used in adaptive line enhancer. In this paper we join the Generalized Sidelobe Canceller beamformer structure with the combination of two adaptive filters. In the analysis part of the paper we investigate the behaviour of the adaptive combination scheme in steady state i.e. in the situation when discrete time n approaches infinity. We will assume throughout the paper that the signals are complex valued and that the combination scheme uses two LMS adaptive filters. The italic, bold face lower case and bold face upper case letters will be used for scalars, column vectors and matrices respectively. The superscript denotes complex conjugation and the superscript H Hermitian transposition of a matrix. The operator E[ ] denotes mathematical expectation and Re{ } stands for the real part of a complex variable. The block diagram of the Generalized Sidelobe Canceller is shown in Figure 1. The structure consists of two branches. The upper branch is the steering branch, that directs its beam toward the desired source. The lower branch is the blocking branch that blocks the signals impinging at the array from the direction of the desired source and includes an adaptive algorithm that minimizes the mean square error between the output signals of the branches. u(n) w s C b x(n) d(n) w b Σ + - e(n) 2 Generelized Sidelobe Canceller In this Section we describe the adaptive beamformer used in this paper. The beamformer is often termed as Generalized Sidelobe Canceller [1]. Let φ denote the the angle of incidence of a planar wave impinging a linear sensor array, measured with respect to the normal to the array. The electrical angle θ is related to the incidence angle as θ = 2πd λ sin φ, (1) where λ is the wavelength of the incident wave and d is the spacing between adjacent sensors of the linear array. Suppose that the signal impinging the array of M sensors is given by u(n) = A(Θ)s(n) + v(n), (2) where s(n) is the vector of emitter signals, Θ is a collection of directions of arrivals, A(Θ) is the array steering matrix with its columns a(θ) defined as responses toward the individual sources s(n) and v(n) is a vector of additive circularly symmetric Gaussian noise. The M vectors a(θ) are often called the steering vectors of the respective sources. Let us also suppose that the source of interest is located at the electrical angle θ 0. Figure 1: Block diagram of generelized sidelobe canceller The weights in steering branch w s are selected from the condition w s a(θ 0 ) = g (3) i.e. we require the response in the direction of the source of interest θ 0 to equal a constant g. The signal at the output of the upper branch is given by d(n) = w H s u(n). (4) In the lower branch we have a blocking matrix, that will block any signal coming from the direction θ 0. The columns of the M M 1 blocking matrix C b are defined as being the orthogonal complement of the steering vector a(θ 0 ) in the upper branch a H (θ 0 )C b = 0. (5) The vector valued signal x(n) at the output of the blocking matrix is formed as x(n) = C H b u(n). (6) The output of the algorithm is e(n) = d(n) w H b (n)x(n). (7) ISBN:

3 The signals x(n) and d(n) can be used as the input and desired signals respectively in an adaptive algorithm to select the blocking weights w b. In this paper we are going to use the combination of two adaptive filters that gives us fast initial convergence and low steady state misadjustment at the same time. The adaptive algorithm is described in the next Section. 3 Combination of Two Adaptive Filters x(n) e 1 (n) y 1 (n) w 1 (n) λ(n) y(n) + 1 λ(n) w 2 (n) y 2 (n) e 2 (n) Figure 2: The combined adaptive filter. d(n) Let us consider two adaptive filters, as shown in Figure 2, each of them updated using the LMS adaptation rule w i (n) = w i (n 1) + µ i e i (n)x(n), (8) e i (n) = d(n) w H i (n 1)x(n), (9) In the above w i (n) is the N vector of coefficients of the i-th adaptive filter, with i = 1, 2 and x(n) is the known N input vector, common for both of the adaptive filters. The input process is assumed to be a zero mean wide sense stationary Gaussian process. µ i is the step size of i th adaptive filter. We assume without loss of generality that µ 1 > µ 2. The case µ 1 = µ 2 is not interesting as in this case the two filters remain equal and the combination renders to a single filter. The desired signal in (4) can alternatively be expressed as d(n) = w H o x(n) + ζ(n)., (10) where the vector w o is the optimal Wiener filter coefficient vector for the problem at hands and the process ζ(n) is the irreducible error that is statistically independent of all the other signals. The outputs of the two adaptive filters are combined according to y(n) = λ(n)y 1 (n) + [1 λ(n)]y 2 (n), (11) where y i (n) = wi H (n 1)x(n) and the mixing parameter λ can be any real number. We define the a priori system error signal as difference between the output signal of the optimal Wiener filter at time n, given by y o (n) = wo H x(n) = d(n) ζ(n), and the output signal of our adaptive scheme y(n) e a (n) = y o (n) λ(n)y 1 (n) (1 λ(n))y 2 (n). (12) Let us now find λ(n) by minimizing the mean square of the a priori system error. The derivative of E[ e a (n) 2 ] with respect to λ(n) reads E[ e a (n) 2 ] λ(n) = 2E[Re{(y o (n) y 2 (n)) (y 2 (n) y 1 (n)) } (13) +λ(n) (y 2 (n) y 1 (n)) 2 ]. Setting the derivative to zero results in λ(n) = E[Re{(d(n) y 2(n))(y 1 (n) y 2 (n)) }] E[ (y 1 (n) y 2 (n)) 2, ] (14) where we have replaced the Wiener filter output signal y o (n) by its observable noisy version d(n). Note however, that because the input signal x(n) and irreducible error ζ(n) are independent random processes, this can be done without introducing any error into our calculations. The denominator of equation (14) comprises expectation of the squared difference of the two filter output signals. This quantity can be very small or even zero, particularly in the beginning of adaptation if the two step sizes are close to each other. Correspondingly λ computed directly from (14) may be large. To avoid this from happening we add a small regularization constant ɛ to the denominator of (14). The constant ɛ should be selected small compared to E[x T (n)x(n)] but large enough to prevent division by zero in given arithmetic. 4 Analysis In this Section we investigate the steady state error of the adaptive scheme described above. It can be deduced from the results of transient analysis presented in [9] that the excess mean square error (EMSE) of ISBN:

4 the combined filter in steady sate when n can be computed as EMSE = 1 M N 1 m=0 ω m E [ λ( )b k,m ( ) +(1 λ( ))b l,m ( ) 2], (15) where λ( ) is the steady state combination parameter, ω m is the m-th eigenvalue of the correlation matrix of x(n), R x = E[x(n)x H (n)] and b k,m ( ) is the m-th element of the transformed weight error vector of the k-th filter in steady state. The transformed weight error vector of k-th filter at time n is defined as b k (n) = Q H (w o w k (n)), (16) where Q is the unitary matrix whose columns are the orthogonal eigenvectors of R x. The correlation matrix of the input signal to the adaptive filter is given by R x = C H b E[u(n)u H (n)]c b (17) = C H b A(Θ)E[s(n)s H (n)]a H (Θ)C b +C H b E[v(n)v H (n)]c b, (14) have been replaced by exponential averaging of the type P u (n) = (1 γ)p u (n 1) + γp(n), (21) where p(n) is the quantity to be averaged, P u (n) is the averaged quantity and γ is the smoothing parameter. We have used γ = in this paper. The averaged quantities were then used in (14) to obtain λ. In our first simulation example we have used a 10 element linear array with half wave-length spacing. The noise power is 10 6 in this simulation example. The useful signal with signal to noise ratio (SNR) of 20 db arrives form the broadside of the array. There are two strong interferers at 10 and 15 with SNR 1 = 46 db and SNR 2 = 40 db respectively. The step sizes of the adaptive combination are µ 1 = 0.3 and µ 2 = The steady state antenna pattern is shown in Figure 3. One can see that the algorithm has formed deep nulls in the directions of the interferers while the response in the direction of the useful signal is equal to the number of antennas. where statistical independence of the emitter signals s(n) and the noise v(n) has been assumed. The components of type E[b k,m ( )b l,m ( )] can be computed as [9] E[b k,m ( )b l,m( )] = J min ω m µl + ωm µ k ωm 2, (18) where J min is the mean square error of the Wiener filter for the problem. For the steady state combination parameter λ we then have λ( ) = where EMSE 2,2 EMSE 2,1 EMSE 1,1 2EMSE 2,1 + EMSE 2,2 (19) Figure 3: The antenna patern. EMSE kl = 1 M N 1 i=0 5 Simulation Results [ ] ω i E b k,i( )b l,i ( ). (20) In this Section we present the results of our simulation study. The curves shown in the Figures to follow are averages over 100 independent trials. In order to obtain a practical algorithm, the expectation operators in both numerator and denominator of In figure 4 we show the evolution of EMSE in this simulation. The cyan line is the EMSE of the fast converging filter and the green line is the EMSE of the slow converging filter. The EMSE of the combination is shown in blue. One can see that the structure converges rapidly with the first filter and stabilizes then at a certain level. When the slow adapting filter catches the fast one a second convergence occurs and eventually we reach a converged state at a lower EMSE level. In addition there is a red line in the Figure indicating the theoretical steady state EMSE. ISBN:

In the second simulation example we use a 16 element linear array with half wavelength spacing. The noise power is 4 10 4.

5 In the second simulation example we use a 16 element linear array with half wavelength spacing. The noise power is The useful signal arrives again from the broadside of the array with signal to noise ratio (SNR) equal to 0 db. There are three strong interferers at 3, 6 and 35 with respective SNR-s SNR 1 = 23.5dB, SNR 2 = 20dB and SNR 3 = 20dB. The step sizes of the adaptive combination are µ 1 = 0.3 and µ 2 = The steady state antenna response is shown in Figure 6. One can see three deep notches in the directions of the interferers. Figure 4: Evolution of EMSE in time. Figure 5 depicts the time evolution of the parameter λ in this simulation example. The blue line is the simulation result and the red line is the theoretically computed steady state λ. We can see that at the beginning of the simulation λ is close to one so that only the output signal of the fast adapting filter is passed to the output of the structure. Eventually the slowly adapting filter catches up with the fast one, λ starts to decrease and obtains a small negative value at the end of the simulation example. Figure 6: The antenna patern. In Figure 7 we show the EMSE evolution in this simulation example. The noisy blue line is the simulation result and the red curve indicates the theoretical steady state EMSE value. One can again see a two stage convergence curve with rapid initial convergence due to the fast adapting filter followed by another convergence due to the more accurate filter. The simulation result matches the theoretical steady state EMSE value. Finally the Figure 8 shows evolution of λ in this simulation example. Again the parameter is close to one in the beginning of the simulation example and decreases later on toward a small negative number. The final value of λ is close to that predicted by the performance analysis. 6 Conclusions Figure 5: Evolution of λ in thime. Herein we have studied an adaptive sensor array that uses affine combination of two adaptive filters as its filter weight update scheme. The sensor array is used in the Generalised Sidelobe Canceller setup. It was demonstrated that the adaptive combination of two filters can well be used for sensor array adaptation al- ISBN:

6 Figure 7: Evolution of EMSE in time. Figure 8: Evolution of λ in time. lowing fast initial convergence followed by a refinement of the estimates to a small mean square error level. The results of steady state analysis of the adaptive scheme were presented. Finally it was demonstrated in our simulation study that there is a good match between the theoretical and simulation results. References: [1] S. Haykin, Adaptive Filter Theory, Fourth Edition, Prentice Hall, New Jersey 2002 [2] A. Sayed, Adaptive Filters, John Wiley and sons, New Jersey 2008 [3] M. Martinez Ramon, J. Arenas-Garcia, A. Navia Vazquez, A. R. Figueiras-Vidal, An Adaptive Combination of Adaptive Filters for Plant Identification, Proc. 14th International Conference on Digital Signal Processing, Santorini, Greece, 2002, pp [4] J. Arenas-Garcia, A. R. Figueiras-Vidal, A. H. Sayed, Mean-Square Performance of Convex Combination of Two Adaptive Filters, IEEE Transactions on Signal Processing 54, 2006, pp [5] M Silva and V. H. Nascimento and J. Arenas- Garcia, A Transient Analysis for the Convex Combination of Two Adaptive Filters with Transfer of Coefficients, Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing, Dallas, TX, USA, 2010, pp [6] N. J. Bershad and J. C. Bermudez and J. H. Tourneret, An Affine Combination of Two LMS Adaptive Filters Transient Mean Square Analysis, IEEE Transactions on Signal Processing 56, 2006, pp [7] T. Trump, An output signal based combination of two NLMS adaptive algorithms, Proc. 16th International Conference on Digital Signal Processing, Santorini, Greece, [8] L. A. Azpicueta Ruiz, A. R. Figueiras-Vidal, J. Arenas-Garcia, A New Least Squares Adaptation Scheme for the Affine Combination of Two Adaptive Filters, Proc. IEEE International Workshop on Machine Learning for Signal Processing, Cancun, Mexico, 2008, pp [9] T. Trump, Output signal based combination of two NLMS adaptive filters - transient analysis, Proceedings of the Estonian Academy of Sciences 60, 2011, pp [10] T. Trump, Output statistics of a line enhancer based on a combination of two adaptive filters, Central European Journal of Engineering 1, 2011, pp ISBN:

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