CHAPTER 3 ROBUST ADAPTIVE BEAMFORMING

50 CHAPTER 3 ROBUST ADAPTIVE BEAMFORMING 3.1 INTRODUCTION Adaptive beamforming is used for enhancing a desired signal while suppressing noise and interference at the output of an array of sensors. It is well known that adaptive beamformers can suffer significant performance degradation, when the array response vector for the desired signal is not known exactly. This degradation is especially noticeable at high signal-tonoise ratio (SNR). Imperfect knowledge of the array response vector may be due to (i) (ii) uncertainty in the source direction-of-arrival (DOA) or sensor characteristics or (iii) improper modeling and variations in the propagation medium between the source and array. The array signal processing has been studied for some decades as an attractive method for signal detection and estimation in harsh environment. An array of sensors can be flexibly configured to exploit spatial and temporal characteristics of signal, noise and has many advantages over single sensor. There are two kinds of array beamformers: fixed beamformer and adaptive beamformer. The weight of fixed beamformer is pre-designed and it does not change in applications. The adaptive beamformer automatically

51 adjusts its weight according to some criteria. It significantly outperforms the fixed beamformer in noise and interference suppression. A typical representative is the linearly constrained minimum variance (LCMV) beamformer. A famous representative of LCMV is Capon beamformer (Capon, 1969). In ideal cases, the Capon beamformer has high performance, in interference and noise suppression, provided that the array steering vector (ASV) is known. However, the ideal assumptions of adaptive beamformer may be violated in practical applications. The performance of the adaptive beamformers highly degrades when there are array imperfections such as steering direction error, time delay error, phase errors of the array sensors, multipath propagation effects and wavefront distortions. This is known as target signal cancellation problem. To overcome the problem caused by steering direction error, multiple-point constraints (Hudson 1981) were introduced in adaptive array. The idea of this approach is intuitive. With multiple gain constraints at different directions in the vicinity of the assumed one, the array processor becomes robust in the region where constraints are imposed. However, the available number of constraints is limited because the constraints consume the degrees of freedom (DOFs) of array processor for interference suppression. Introducing the derivative constraints into the array processor leads to another class of solution. With the derivative constraints, the array response is almost flat in the vicinity of target direction. The beamformer has wide beam width in the target direction. With a small steering direction error, the beamformer does not cancel the target signal. However, the wide beam width is achieved at the cost of reduced capability in interference suppression because the additional derivative constraints consume the DOFs of beamformer. Derivative constraints can be used to obtain not only a flat

52 response of array processor, but also a flat null in the assumed signal direction in blocking matrix design. Quadratic constraints can be used to minimize the weighted mean square deviation between the desired array response and the response of the processor over the variations in parameters, such as the steering error, the phase errors and the array geometry error, etc., When array processor cancels target signal, the norm of the filter coefficients grows to a large value beyond the normal value, for noise and interference suppression. An inequality constraint is imposed on the coefficients norm of adaptive beamformer to limit the growth of tap coefficients. The excess coefficients growth problem can also be solved by using noise injection method. Artificially generated noise is added to reference signals of adaptive filters. Although the artificial noise causes estimation errors in the beamformer coefficients, it prevents tap coefficients from growing excessively, resulting in robustness against array imperfections. A similar approach called, the leaky least mean square (LMS) algorithm, can also be used for this purpose. The calibration based approaches can eliminate the inherent error of the array processor, such as geometry error, sensor response error, etc. However, it cannot eliminate dynamic errors, such as steering error when the source is moving in a vicinity of the assumed direction. In target tracking methods, the look direction is steered to the continuously estimated directionof-arrival (DOA). One problem is that, this method may mis-track to the interference, in the absence of target signal, unless some other methods are used to limit the tracking region. Robust beamformer, for real time applications, iteratively searches for the optimal direction. It maximizes the mean output power of Capon beamformer, using first-order Taylor series approximation, in terms of steering direction error. This method does not suffer from performance loss in interference/noise suppression. However, its

53 performance degrades when there exist multiple errors, such as the steering direction error, the array geometry error and the array sensor phase error. The array steering vector is assumed to be a vector function of steering direction only. When multiple imperfections exist, the assumed model of the ASV is violated. Recently, robust methods use uncertainty set of the ASV. The true ASV is assumed to be an ellipsoid centered at the nominal ASV. The designed beamformers are robust against arbitrary variation of true ASV within an assumed uncertainty set. These beamformers are equivalent and belong to diagonal loading approach. The diagonal loading factor can be calculated from the constraint equation. Traditional approaches for increasing robustness to DOA uncertainty include linearly constrained minimum variance (LCMV) beamforming, diagonal loading, quadratically constrained beamforming, and combinations of these. These techniques allow desired signal to arrive from a region in the DOA space rather than just from a single direction only. It relies implicitly on assumptions about the strength of the desired signal and the interval over which the DOA can vary. In these techniques, robustness to DOA uncertainty is increased at the expense of a reduction in noise and interference suppression. A different approach is to use samples of the sensor data to estimate the signal DOA or the signal subspace. Direction-finding (DF)-based techniques, estimate the DOA of the desired and interference signals and proceed as if they are known. Subspace techniques estimate the signal plus interference subspace to reduce mismatch. These data-driven techniques are more complex to implement but can have nearly optimal performance when the data is sufficient to yield good estimates of the DOA or subspace. However, they suffer significant performance degradation when these estimates are not reliable. Techniques that improve the robustness of

54 data-driven beamformers, in the presence of moving and spatially spread sources, by incorporating additional linear constraints have also been already proposed. An adaptive beamformer using a Bayesian approach balances the use of observed data and a priori knowledge about source DOA. In this approach, DOA is assumed to be a discrete random variable with a known a priori probability density function (pdf) that characterizes the level of uncertainty about source DOA. The resulting beamformer is a weighted sum of minimum variance distortionless response (MVDR) beamformers. It is pointed at a set of candidate DOA s, where the relative contribution of each MVDR beamformer is determined from a posteriori pdf of the DOA conditioned on observed data. A simple approximation to a posteriori pdf allows for a straightforward implementation that is somewhat more complex than LCMV beamformer but considerably less complex than data-driven beamformers. Performance of Bayesian beamformer is better when compared to both LCMV and data-driven beamformers, in a variety of scenarios. The worst-case (WC) performance optimization has been shown as a powerful technique which yields a beamformer with robustness against an arbitrary signal steering vector mismatch, data non-stationarity problems and small sample support. The WC approach explicitly models an arbitrary mismatch in the desired signal array response and uses WC performance optimization to improve the robustness of the minimum variance distortionless response (MVDR) beamformer. In addition, the closed-form expressions for the SINR are derived therein. Unfortunately, the natural formulation of the WC performance optimization involves the minimization of a quadratic function subject to infinity non-convex quadratic constraints. WC optimization is also modeled as a convex second-order cone program (SOCP) and solved efficiently via the well-established interior point method

55 (IPM). Regrettably, the SOCP method does not provide a closed-form solution for the beamformer weights and even it cannot be implemented online, whereas the weight vector needs to be recomputed completely with the arrival of a new array observation. Approaches based on eigen decomposition of sample covariance matrix developed a closed-form solution, for a WC robust detector using Lagrange method which incorporates the estimation of the norm of weight vector and/or the Lagrange multiplier. A binary search algorithm followed by a Newton-like algorithm can be used to estimate the norm of the weight vector after dropping the Lagrange multiplier. Although these approaches have provided closed-form solutions for the WC beamformer, they unfortunately, incorporate several difficulties. First, eigen decomposition for the sample covariance matrix is required with the arrival of a new array observation. Second, the inverse of diagonally loaded sample covariance matrix is required to estimate the weight vector. Third, some difficulties are encountered during algorithm initialization and a stopping criterion is necessary to prevent negative solution of the Newton-like algorithm. Two efficient ad hoc implementations of the WC performance optimization problem are, first, the robust MVDR beamformer with a single WC constraint implemented using an iterative gradient minimization algorithm with an ad hoc technique. It estimates the Lagrange multiplier instead of the Newton like algorithm. This algorithm exhibits several merits including simplicity, low computational load and no need for either samplematrix inversion or eigen decomposition. A geometric interpretation of the implementation has been introduced to supplement the theoretical analysis. Second, a robust linearly constrained minimum variance (LCMV) beamformer with multiple beam WC (MBWC) constraints is developed using a novel multiple WC constraints formulation. The Lagrange method is

56 exploited to solve this optimization problem, which reveals that the solution of the robust LCMV beamformer with MBWC constraints entails solving a set of nonlinear equations. As a consequence, a Newton-like method is mandatory to solve the ensuing system of nonlinear equations which yields a vector of Lagrange multipliers. It is worthwhile to note that these approaches adopt ad hoc techniques to optimize the beamformer output power with spherical constraint on the steering vector. Unfortunately, the adaptive beamformer is sensitive to noise enhancement at low SNR and additional constraint is required to bear the ellipsoidal constraint. 3.2 ROBUST ADAPTIVE BEAMFORMING Adaptive beamforming is a complementary means for signal-tointerference-plus-noise-ratio (SINR) optimization (Van Trees 2002, Dimitris and Ingle 2005, Godara 1997). Our investigation starts with the formation of a lobe structure those results from the dynamic variation of an element-space processing. A weight vector is controlled by an adaptive algorithm, which is the MVDR-Sample Matrix Inversion algorithm (Jiang and Zhu 2004, Dimitris and Ingle 2005, Godara 1997). It minimizes cost function of a link s SINR by ideally directing beams toward the signal-of-interest (SOI) and nulls in the directions of interference. In optimum beamformers, optimality can be achieved in theory if perfect knowledge of the second order statistics of the interference is available. It involves calculation of interference plus noise correlation matrix R i n. In adaptive beamformer, the correlation matrix is estimated from collected data. In sample matrix Inversion technique, a block of data is used to estimate adaptive beamforming weight vector. The estimate Rˆi n is not really a substitute for true correlation matrix R i n. Hence there is degradation in performance. The SINR which is a measure of performance of the beamformer degrades as sample support (the number of data) is low.

57 3.3 SYSTEM MODEL An uniform linear array (ULA) of M elements or sensors is considered in this investigation. A desired signal S 0 from a point source from a known direction 0 with steering vector a 0 and L number of J (jammer or) interference signals from unknown directions 1, 2, 3... L, specified by the steering vectors a1, a2, a3,... a L, respectively impinges on the array. The white or sensor or thermal noise is considered as n. A single carrier modulated signal S ( ) 0 t is given by S0 ( t) S0( t)cos(2 Fc) t (3.1) It is arriving from an angle 0 and is received by the ith sensor. The signal S ( ) 0 t is a baseband signal having deterministic amplitude and random uniformly distributed phase with F c as the carrier frequency. The symbol is used to indicate that the signal is a pass band signal. Let X 1 (k) be the single observation or measurement of this signal made at time instant k, at sensor 1, which is given as T X 1(k)= a0s 0(k)+ [a 1,a 2...a L ][J 1(k),J 2(k) J L(k)] + n(k) (3.2) L a S ( k) a J *( k) n( k ) (3.3) 0 0 j 1 j j Hence the single observation or measurement made at the array of elements at the time instant k, called array snapshot is given as a vector with T as the transpose, X ( k) [ X ( k) X ( k) X ( k)... X ( k )] T (3.4) 1 2 3 M

58 The general model of the steering vector is given as a 2 2 d 2 d j d cos( ) j2 cos( ) j( M 1) cos( ) 1 e e... e M (3.5) It is assumed that the desired signal, interference signals and noise are mutually uncorrelated. 3.4 ADAPTIVE BEAMFORMING TECHNIQUES In optimum beamformer, a priori knowledge of true statistics of the array data is used to determine the correlation matrix which in turn is used to derive the beamformer weight vector. Adaptive Beamforming is a technique in which an array of antennas is exploited to achieve maximum reception in a specified direction by estimating the signal arriving from a desired direction while signals of the same frequency from other directions are rejected. This is achieved by varying the weights of each of the sensors used in the array. Though the signals emanating from different transmitters occupy the same frequency channel, they still arrive from different directions. This spatial separation is exploited to separate the desired signal from the interfering signals. In adaptive beamforming, the optimum weights are iteratively computed using complex algorithms based upon different criteria. For an adaptive beamformer, covariance or correlation matrix must be estimated from unknown statistics of array snapshots to get the optimum array weights. The optimality criterion is to maximize the signal-to-interference-plus-noise ratio to increase the visibility of the desired signal at the array output. In this investigation, it is assumed that the angle of arrival of the desired signal is known.

59 3.4.1 Estimation of Correlation Matrix For p-dimensional data, the sample covariance matrix estimate becomes singular, and therefore unusable, if fewer than p+1 sample will be available, and it is a poor estimate of the true covariance matrix unless many more than p+1 sample are available. The correlation matrix can be estimated using different methods which would result in different performance and behavior of the algorithm. In block adaptive Sample Matrix Inversion technique, a block of snapshots are used to estimate the ensemble average of R x, a M M matrix and is written as H 1 K H Rx E{ x( k) x ( k)} x( k) x ( k) K 1 M (3.6) M a a R R (3.7) 2 H S 0 0 j n where M is the number of snapshots used and k is the time index, power of the desired signal and R j and 2 s is the R n are the jammer and noise correlation matrices, respectively and H is the complex conjugate transpose. The interference-plus-noise correlation matrix is the sum of these two matrices where R j n R j R n (3.8) 2 Rn Rn n I, and 2 n is the thermal noise power, I is the identity matrix. It is assumed that thermal noise is spatially uncorrelated. 3.4.2 Conventional Beamformer The expectation value at the antenna elements is written as E[ x( t)] [ X ( t) X ( t)... X ( t)][ X ( t) X ( t)... X ( t )] T (3.9) 1 2 M 1 2 M H where R E{ x( t) x( t ) }.

60 The output signal is H y( t) W x( t ) (3.10) This is the conventional beamformer s output signal with beamformer weight w which is shown in Figure 3.1. 0-10 -20-30 -40-50 -60-70 -80-90 -100-80 -60-40 -20 0 20 40 60 80 angle in Figure 3.1 Conventional beamforming showing the beam pattern Maximizing the beamformer output problem will result in Max 2 Max H w w P { y } ( w Rw ) (3.11) Solving this equation gives w a H a a (3.12) where a is the steering vector.

61 3.4.3 MVDR Beamforming If M number of sensors are used in a beamformer with spacing between them as d= /2, at any instant M 1 * jk 0 y( n) s0 ( n). Wk e (3.13) k 0 where 0 is the phase difference from the reference input and may be written as =(2 d/ ) sin = sin where is the angle of incidence. To protect all signals which are received from the wanted direction, a linear constraint may be defined as M 1 * jk 0 H. k ( ) ( 0) k 0 W e w n a g (3.14) The constraint g may be interpreted as gain at the look direction which is to be maintained as constant. A spatial filter that performs this function is called a linearly constraint minimum variance beamformer (LCMV). If the constraint is g =1 then the signal will be received at look direction with unity gain and the response at the look direction is distortionless. This special case of LCMV beamformer is known as minimum variance distortionless response (MVDR) beamformer which is shown in Figure 3.2. Mathematically, a weight vector w is to be calculated for this constrained optimization problem. w w * Rw Subject to w* a 0 1 (3.15) min Now the optimal weight vector may be written as w R a( ) / a ( ) R a ( ) (3.16) 1 H 1 x x

62 0-10 -20-30 -40-50 -60-70 -80-90 -100-80 -60-40 -20 0 20 40 60 80 angle in Figure 3.2 MVDR-the optimum beamformer beam pattern This beamforming method experiences the following drawbacks 1) Computational complexity in the order of 2 3 O( N ) too( N ). 2) In the case of large array, low sample support i.e (M>>k), Rx may result in singular matrix or ill-conditioned. 3.4.4 Sample Matrix Inversion (SMI) Sample matrix Inversion techniques solve the equation RxW0 directly by substituting the maximum likelihood estimates for the statistical quantities R x and r dx to obtain r dx W Rˆ rdx ˆ (3.17) 1 x cross correlation are The maximum likelihood estimates of the signal correlation and

63 M 1 H x k K k 0 R x x (3.18) and M 1 r dx ˆ (3.19) dx k 0 k When the input signal is stationary, the estimates only need to be computed once. However in cases where the signal statistics are time varying the estimates must be continuously updated. In SMI, the convergence performance is quantified in terms of number of statistically independent sample outer products that must be computed for the weight vector to be within 3dB of the optimum. 3.4.5 MVDR-SMI Beamformer MVDR is an optimal minimum variance distortionless response beamformer. It is also referred as the full rank solution as it uses all M adaptive degrees of freedom. It resembles the Wiener filter of the form 1 W R r (3.20) MVDR weight vector can be derived as w MVDR a H 2 i, n 2 i, n 1 a 1 a = s (3.21) w a (3.22) H where s is unit norm i.e a a 1 and is the Hadamard product. A standard method of estimating the covariance matrix is by constructing the sample covariance matrix

64 ˆ 1 R x ( k) x ( k) K H i, n i, n i, n k k 1 (3.23) H x i, n( k) is the k th training sample and k is the total number of training samples that are available. The sample covariance matrix R ˆi, n is the maximum likelihood estimate of the true covariance matrix R i, n. Now the approach is called sample matrix inversion with MVDR beamforming and the weights are calculated as W Rˆ a 1 i, n ( MVDR SMI ) H ˆ 1 a R i, n a (3.24) The beam response for MVDR_SMI beamformer is shown in Figure 3.3. Mvdr-smi beamforming 0-10 -20-30 -40-50 -60-70 -80-90 -100-80 -60-40 -20 0 20 40 60 80 angle in Figure 3.3 MVDR-SMI beamformer with beam response MVDR method may suffer from significant performance degradation when there are even small array steering vector errors. Several

65 approaches for increasing robustness to array steering vector errors have been proposed during the past few decades. Diagonal loading, linearly constrained minimum variance (LCMV) beamforming, quadratically constrained beamforming and second order cone programming (SOCP) are some of them. In this work, adaptive colored diagonal loading is proposed to improve the SINR and to eliminate the steering vector errors. 3.4.6 Diagonal Loading (DL) To overcome the above mentioned drawback no. 2 in Section 3.4.3, a small diagonal matrix is added to the covariance matrix. This process is called diagonal loading (Li 2003) or white noise stabilization which is useful to provide robustness to adaptive array beamformers against a variety of conditions such as direction-of-arrival mismatch; element position, gain, and/or phase mismatch; and statistical mismatch due to finite sample support (Hiemstra 2003, Li and Stoica 2006). Because of the robustness that diagonal loading provides it is always desirable to find ways to add diagonal loading to beamforming algorithms. But little analytical information is available in the technical literature regarding diagonal loading (Fertig 2000). To achieve a desired sidelobe level in MVDR-SMI beamformer, sufficient sample support k must be available. However due to nonstationarity of the interference, only low sample support is available to train the adaptive beamformer. The beam response of an optimal beamformer can be written in terms of its eigen values and eigen vectors. The eigen values are random variables that vary according to the sample support k. Hence the beam response suffers as the eigen values vary. This results in higher sidelobe level in adaptive beam pattern. A means of reducing the variation of the eigen values is to add a weighted identity matrix to the sample correlation matrix.

66 The result of diagonal loading of the correlation matrix is to add the loading level to all the eigen values. This in turn produces the bias in these eigen values in order to reduce their variation which in turn produces side bias in the adaptive weights that reduces the output SINR. Recommended loading levels of loading level. 2 2 < 2 n L 10 n where 2 is the noise power and 2 n L is the diagonal The minimum loading level must be equal to noise power. Diagonal loading increases the variance of the artificial white noise by an amount 2 L. This modification forces the beamformer to put more effort in suppressing white noise rather than interference. When the SOI steering vector is mismatched, the SOI is attenuated as one type of interference as the beamformer puts less effort in suppressing the interferences and noise. However when 2 L is too large, the beamformer fails to suppress strong interference because it puts more effort to suppress the white noise. Hence, there is a tradeoff between reducing signal cancellation and effectively suppressing interference. For that reason, it is not clear how to choose a good diagonal loading factor 2 L in the traditional MVDR beamformer. The conventional diagonal loading beamformer is shown in Figure 3.4. This beamformer can be thought of as a gradual morphing between two different behavior, a fully adaptive MVDR solution (L = 0, no loading) and a conventional uniformly weighted beam pattern (L =, infinite loading) (Hiemstra 2002).

67 0-10 -20-30 -40-50 -60-70 -80-90 -100-80 -60-40 -20 0 20 40 60 80 angle in Figure 3.4 MVDR-Diagonal Loading The conventional DL weight vector can be calculated as W Rˆ I a (3.25) 2 1 MVDR DL MVDR DL[ L ] ( ) where MVDR DL is the normalization constant given by MVDR DL a Rˆ I a (3.26) H 2 1 ( ) [ L ] ( ) and 2 L reduces the sensitivity of the beam pattern to unknown uncertainties and interference sources at the expenses of slight beam broadening. The choice of loading can be determined from L-Curve approach (Hiemstra 2003) or adaptive diagonal loading.

68 3.4.7 Colored Diagonal Loading (CDL) In the presence of colored noise, DL can be applied which is termed as colored diagonal loading (CDL) and the morphing process may result in a beam pattern of our choosing. The colored diagonal loading is similar to W MVDR DL but the diagonal loading level of by the term (Hiemstra 2002) 2 L =, end point, can be altered W Rˆ R a (3.27) 2 1 MVDR CDL MVDR DL[ L dq ] ( ) where R dq is the covariance matrix that captures the desired quiescent structure. It may be determined directly based on 1) a priori information where R dq, need not be a diagonal or 2) desired weight vector where R dq must be diagonal. It is given as R diag diag w a (3.28) dq 1 ([ ( dq)] ( )) where w dq is the desired quiescent weight vector. The colored diagonal loading shows no improvement in pattern shape as shown in Figure 3.5. 0-10 -20-30 -40-50 -60-70 -80-90 -100-80 -60-40 -20 0 20 40 60 80 angle in Figure 3.5 MVDR-Colored Diagonal Loading

69 3.4.8 Adaptive Diagonal Loading (ADL) In this method the loading level is calculated assuming the a priori information about the SNR is available. The SNR can be estimated from link budget or using some SNR estimation algorithm. A variable loading MVDR.(VL-MVDR) is proposed in (Gu and Wolfe 2006) in which the loading level is chosen as ( 2 R) and the beam pattern is shown in Figure 3.6. 2 1 W MVDR-ADL MVDR ADL[ ADL ] ( ) R I a (3.29) where ADL M. SNR 0-10 -20-30 -40-50 -60-70 -80-90 -100-80 -60-40 -20 0 20 40 60 80 angle in Figure 3.6 MVDR- Adaptive Diagonal Loading beam pattern 3.4.9 MVDR-SMI Beamformer with Adaptive Colored Diagonal Loading In Adaptive Colored Diagonal Loading, which is our proposed method, the loading level is calculated assuming the a priori information about the Signal to Noise Ratio (SNR) is available.

70 The SNR can be estimated from link budget or using some SNR estimation algorithm. A variable loading MVDR (VL-MVDR) is proposed in (Gu and Wolfe 2006) which the loading level is chosen as 2 ( R ˆ ) W Rˆ I a (3.30) 2 1 MVDR ADL MVDR DL[ ADL ] ( ) where ADL M. SNR White noise stabilization is nothing but diagonal loading in which the adaptive colored loading technique is embedded to get a novel hybrid method which is proposed as W Rˆ R a (3.31) 1 MVDR ACDL MVDR DL[ dq ] ( ) The beam pattern for MVDR-ADCL is shown in Figure 3.7. 0-10 -20-30 -40-50 -60-70 -80-90 -100-80 -60-40 -20 0 20 40 60 80 angle in Figure 3.7 MVDR- Adaptive Colored Diagonal Loading beam pattern

71 3.5 COMPUTATION For the proposed hybrid algorithm, a 10 element Uniform Linear Array is considered with SNR of 20 db for the desired signal coming from s = 0 and INR of 70 db for two jammer signals coming from the directions i = -70, and 30. The element spacing is d = 0.5. The beam patterns for various methods of beamforming are obtained and compared with the performance of MVDR-Adaptive Colored Diagonal Loading. It is observed that the conventional beamformer performs well to get the maximum gain in the desired look direction of 0. But its performance is worst regarding the cancellation of interferences. Figure 3.5 shows the MVDR Colored Diagonal Loading beam pattern which performs much better than the conventional beamformer. This shows a greater improvement in SINR than the conventional. The null is placed properly with out any angle deviation. Figure 3.6 shows MVDR-ADL beam pattern. Figure 3.7 shows MVDR-ACDL beam pattern. This beam pattern gives improvement in SINR when compared to other diagonal loading methods. Figure 3.8 shows the beam patterns of the above mentioned techniques. The interferers angle and their corresponding beam responses are given below. Interferer 1 at angle -70 Interferer 2 at angle 30 : -60dB : -60dB 3.6 RESULTS 3.6.1 Number of Elements For the ULA which is considered for experimental work, the beam patterns are analyzed by changing the number of elements as 4, 8, 12, 16, 24, 50 and 100.

72 0-10 -20-30 -40-50 -60 MVDR-diagonal loding Mvdr-adaptive DL Mvdr-adapt-col-DL -70-80 -90-100 -80-60 -40-20 0 20 40 60 80 angle in Figure 3.8 Beam pattern of various diagonal loading methods As the number of elements increases, the beam pattern shows higher resolution i.e the 3 db beam width becomes much narrower from to 26 to 1 for conventional beamformer and 17 to 1 for adaptive diagonal loading beamformer. Finer or sharper beams are obtained when more number of elements is used. Sharper the beam, the beamformer is not susceptible to jammers. But the numbers of sidelobes are also increased. The 3-dB beam width of different beamformers is tabulated in Table 3.1. A trade off can be obtained to reduce the cost and to have a compact size. Hence a maximum of 16 elements are chosen for further analysis. 3.6.2 Noise Effect An ULA with 16 elements is considered for analyzing the effect of noise on the peaks of the signal power. Signal to noise ratio (SNR) is varied in steps of 10 db starting from 10 db till 60 db. As SNR increases the peak becomes sharper. It shows that the interference sources are suppressed to a maximum extent, so that it will not be a disturbance while extracting the signal even in the presence of strong interferers.

73 Table 3.1 Effect of changing number of antenna elements 3-dB beam width 4 26.2 19.5 17.1 17 17 17 16 8 12.8 15.47 13.3 14.8 14.8 14.8 25.5 12 8.4 8.7 6.9 8.4 8.5 8.7 8.5 16 6.25 6.4 6.4 6.4 6.4 6.6 13.2 20 5.1 5.2 6 5.3 5.3 5.2 6.8 24 4.4 4.5 4.5 4.3 4.3 4.3 4.3 50 2 2 2 2 2 2 2 100 1 1 1 1 1 1 1 3.6.3 Training Issues with the Number of Array Snapshots Increasing the number of array snapshots lead to complexity and computational cost but the performance of the beamformer increases. It is a trade off between the cost and the performance. This is shown in Figure 3.9. 3.6.4 Element Spacing The spacing between the elements for an 16 element ULA was varied as /4, /2, 3 /4 and which in turn vary the effective aperture length of the array. Among the four choices /2 showed the best performance for the particular frequency used for expermiments. When the distance between the elements is increased beyond /2, it resulted in spatial aliasing i.e a lot of spurious peaks were obtained which correspond to different frequencies. Below /2 the resolution of the beams was not satisfactory.

74 0-5 -10 SINR-CDL SINR-ADL SINR-ACDL -15-20 -25-30 -35-40 -45-50 0 50 100 150 200 250 300 number of snapshots k Figure 3.9 Training issues with the number of snapshots The analytical results of the response of different beamforming methods are tabulated in Table 3.2. Table 3.2 Beam response of the signals - desired and jammers using various methods Beamforming method Desired signal =0 Beam response Power (in db) Jammer1 = 20 Beam response Power (in db) Jammer2 =-20 Beam response Power (in db) Jammer3 =-70 Beam response Power(in db) conventional 0-20 -20-26.5 MVDR 0-91 -66-91 MVDR-SMI 0-58 -61-72 DL 0-72.5-72.5-85 CDL -6-50 -57-66.5 ADL 0-72.5-72.5-85 ACDL 0-52 -56.5-62

75 3.7 CONCLUSION Our investigation deals with adaptive array beamforming in the presence of errors due to steering vector mismatch and finite sample effect. Diagonal loading (DL) is one of the widely used techniques for dealing with these errors. The diagonal loading techniques has the drawback that it is not clear how to get the optimal value of diagonal loading level based on the recognized level of uncertainty constraint. Recently, several DL approaches proposed, the so-called automatic scheme, on computing the required loading factor. In our investigation, this drawback is tackled while the diagonal loading technique is integrated into the adaptive update scheme by means of variable loading technique rather than fixed diagonal loading or ad hoc techniques. The novelty is that the proposed method does not require any additional sophisticated scheme to choose the required loading. We propose a fully data-dependent loading to overcome the difficulties. The loading factor can be completely obtained from the received array data. Analytical formulas for evaluating the performance of the proposed method under random steering vector error are further derived. Experimental results are proved the validity of the proposed method and make comparison with the existing DL methods.