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(IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 3, March 2011 Analysis on Robust Adaptive Beamformers T.S.JEYALI LASEETHA 1, DR.(MRS) R.SUKANESH2 1. Professor Department of Electronics and Communication Engineering, Anna University of Technology Tirunelveli, Tamil Nadu INDIA email id: laseetha@gmail.com 2. Professor Department of Electronics and Communication Engineering, Madurai,Tamil Nadu INDIA Abstract: MVDR (minimum variance distortionless response) beamformer is the optimal beamformer which utilizes the second order statistics of the actual data for obtaining the Covariance matrix from which the weight vector of the antenna array is determined. In adaptive beamfomer which utilizes MVDR beamformer along with SMI(sample matrix inversion), actual data is not available to calculate the covariance matrix. Instead, covariance matrix is estimated from the available data. It includes finding the Matrix inversion. It may result in bad conditioning. To avoid this, some amount of loading is introduced to the diagonal elements, which is called diagonal loading. Diagonal loading can be inserted by adding a scaled version of identity matrix. Diagonal loading imparts Robustness to the adaptive beamformer against signal mismatch due to low sample support and helps to achieve desired sidelobe level and SINR improvement. Various methods in diagonal loading are analyzed in this paper with different loading levels and a novel hybrid algorithm for MVDR-SMI beamformer with colored adaptive diagonal loading is also proposed. The performance of the proposed methods is compared with other methods such as Conventional, MVDR, MVDR-SMI, MVDR-SMI-Diagonal Loading, MVDR-SMI-Colored –DL, MVDR-SMI-Adaptive DL by conducting simulations experiment. The proposed method shows the improvement in directivity and SINR compared to other methods. Key-words: Smart antennas, Adaptive beamforming, Uniform Linear Array, Minimum Variance Distortionless Response Beamformer (MVDR), Sample-Matrix Inversion(SMI), Adaptive colored diagonal loading 1.Introduction In Wireless Communications, smart technologies that results from the dynamic variation of an are not only being applied at the antenna level, but element-space processing weight vector as opposed also at the receiver for direction of arrival to a switched-beam or beam-space antennas, is estimation, detection, diversity combining and controlled by an adaptive algorithm, which is the equalization and at baseband processing software MVDR-Sample Matrix Inversion algorithm[2,7,10]. levels. The ultimate benefit of these techniques is to It minimizes cost function reduction of a link’s increase cellular capacity and range. Adaptive SINR by ideally directing beams toward the signal- beamforming reveals to be a complementary means of-interest (SOI) and nulls in the directions of for signal-to-interference-plus-noise-ratio (SINR) interference. Many algorithms differ largely in optimization [6,7,10]. In this paper, at antenna array complexity, correlation matrix, eigen value spread elements level, the formation of a lobe structure, dependence, inherent gradient noise, limited 171 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 3, March 2011 dynamic range and limited number of samples used "{ { " { {IJJ{ I{ . It is arriving from an [1], [2]. angle θ0 and is received by the ith sensor. The signal S0(t) is a baseband signal having a deterministic In optimum beamformers optimality can be amplitude and random uniformly distributed phase achieved in theory if perfect knowledge of the and Fc is the carrier frequency. The symbol is second order statistics of the interference is used to indicate that the signal is a pass band signal. available. It involves calculation of interference plus noise correlation matrix . For real world X1(k) is the single observation or measurement of scenarios, the adaptive methods are followed to this signal made at time instant k, at sensor 1, which obtain optimality. In adaptive beamformer, the is given as correlation matrix is estimated from the collected I# { { I" " { { - {I# I$ I {{H# { { H$ { { H { {{ - J{ { (1) data. In sample matrix Inversion technique a block I" "{ {- (# I H { { - J{ { of data is used to estimate the adaptive beamforming (2) weight vector. The estimate is not really a substitute for true correlation matrix . Hence Hence the single observation or measurement there is loss in performance. The SINR which is a made at the array of elements at the time instant k, measure of performance of the beamformer called array snapshot is given as a vector degrades as the sample support (the number of data) is low. The lower band on sidelobe levels of the I{ { {I# { { I$ { { I% { { I { {{ (3) beamformer when no interference sources were found at an angle is also to be calculated. Training The general model of the steering vector[13] is issues like the presence of desired signal in the given as correlation matrix is also dealt with. The {{ – { È { {{ % {{ % { È { {{ desired signal in the training set results in the # cancellation and subsequent lose in performance. (4) The paper is organized as follows. In Section 2, Also it is assumed that the desired signal, Problem formulation and general model is interference signals and noise are mutually presented. In Section 3 Adaptive beamforming with uncorrelated. various beamforming methods are presented along with the Novel Hybrid algorithm -Adaptive colored diagonal loading. In Section 4 simulation experiments are presented. Section 5 contains Results and discussions. Section 6 presents the 3. Adaptive Beamforming conclusions. 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 2. Problem Formulation And General is a technique in which an array of antennas is Model exploited to achieve maximum reception in a specified direction by estimating the signal arrival An uniform linear array (ULA) of M elements or from a desired direction while signals of the same sensors is considered. Let a desired signal S0 from a frequency from other directions are rejected. This is point source from a known direction θ0 with achieved by varying the weights of each of the steering vector ‘a0’ and L number of J(jammer or) sensors used in the array. Though the signals interference signals from unknown emanating from different transmitters occupy the directions{ # $ % {, specified by the same frequency channel, they still arrive from steering vectors {I# I$ I% I { respectively different directions. This spatial separation is impinge on the array. The white or sensor or thermal exploited to separate the desired signal from the noise is considered as ‘n’. interfering signals. In adaptive beamforming the optimum weights are iteratively computed using A single carrier modulated signal "{ { is given by complex algorithms based upon different criteria. For an adaptive beamformer, covariance or correlation matrix must be estimated from unknown 172 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 3, March 2011 statistics of the array snapshots to get the optimum This beamforming method experiences the array weights. The optimality criterion is to following drawbacks maximize the signal-to-interference-plus-noise ratio to increase the visibility of the desired signal at the 1) the computational complexity is more in the array output. The determination of the presence of order of { $ { J { % { . signals of interest is known as detection while the 2) In the case of large array, low sample inference of their parameters likes, the angle of support i.e(M>>k), may result in arrival θ0, is referred to as estimation. In this paper it singular matrix or ill-conditioned. is assumed that the angle of arrival of the desired signal is known conventional beamforming 0 3.1 Estimation Of Correlation Matrix -10 The correlation matrix can be estimated[6,7,8,9] -20 using different methods which would result in different performance and behavior of the algorithm. -30 beam response in dB In block adaptive Sample Matrix Inversion -40 technique, a block of snapshots are used to estimate -50 the ensemble average of and is written as[8] -60 # { { { { {{ (# { { { { (5) -70 $ I" I" - - -80 =H (6) -90 where N is the number of snapshots used and k is the time index, $ is the power of the desired signal -100 -80 -60 -40 -20 0 20 40 60 80 angle inθ and and are the jammer and noise correlation matrices, respectively. The interference-plus-noise Fig 1 conventional beamforming showing the correlation matrix is the sum of these two matrices beampattern - (7) Where $ H, and $ is the thermal noise Mvdr beamforming power, I is the identity matrix. It is assumed that 0 thermal noise is spatially uncorrelated. -10 -20 -30 3.2 MVDR Beamformer beam response in dB -40 The MVDR beamformer whose pattern is shown in Fig(2) is an adaptive high resolution beamformer -50 that minimizes the output power while maintaining -60 unity response in the desired direction. Mathematically a weight vector ‘w’ is to be -70 calculated with the constrained optimization of -80 problem -90 c J I I J c I" (8) -100 -80 -60 -40 -20 0 20 40 60 80 Now the optimal weight vector may be written as angle inθ # { {È { { # { { (9) Fig 2 MVDR-the optimum beamformer- beampattern 173 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 3, March 2011 0 Mvdr-smi beamforming minimum loading level must be equal to noise power. Diagonal loading increases the variance of the artificial white noise by the amount $ . This -10 -20 modification forces the beamformer to put more -30 effort in suppressing white noise rather than beam response in dB -40 interference. When the SOI steering vector is -50 mismatched, the SOI is attenuated as one type of -60 interference as the beamformer puts less effort in suppressing the interferences and noise[17]. However when $ is too large, the beamformer -70 -80 -90 fails to suppress strong interference because it puts most effort to suppress the white noise. Hence, there -100 -80 -60 -40 -20 0 20 40 60 80 is a tradeoff between reducing signal cancellation angle inθ and effectively suppressing interference. For that reason, it is not clear how to choose a good diagonal loading factor $ in the traditional MVDR Fig 3 MVDR-SMI beamformer with beam response beamformer. 3.3 Diagonal Loading (Dl) This conventional diagonal loading can be thought of as a gradual morphing between two To overcome the above mentioned drawback no. 2, different behavior, a fully adaptive MVDR solution a small diagonal matrix is added to the covariance (H , no loading) and a conventional uniformly matrix. This process is called diagonal loading[15] weighted beampattern (H ∞, infinite loading)[5]. or white noise stabilization which is useful to The conventional DL weight vector can be provide robustness to adaptive array beamformers calculated as against a variety of conditions such as direction-of- arrival mismatch; element position, gain, and/or { - $ H{ # { { (10) phase mismatch; and statistical mismatch due to finite sample support[12,14]. Because of the where is the normalization constant robustness that diagonal loading provides it is given by { { { - $ H{ # { { always desirable to find ways to add diagonal (11) loading to beamforming algorithms and appropriate amount of loading. But little analytical information and $ reduces the sensitivity of the beampattern to is available in the technical literature regarding unknown uncertainities and interference sources at diagonal loading [11]. To achieve a desired sidelobe the expenses of slight beam broadening[3]. The level in MVDR-SMI beamformer sufficient sample choice of loading can be determined from L-Curve support ‘k’ must be available. However due to non- approach[12] or adaptive diagonal loading. Beam stationarity of the interference only low sample shape for Diogonal loading as shown in Fig(2) is support is available to train the adaptive better when compared to previously stated methods beamformer. We know that the beam response of an optimal beamformer can be written in terms of its MVDR-diagonal loading 0 eigen values and eigen vectors. The eigen values are random variables that vary according to the sample -10 support ‘k’. Hence the beam response suffers as the -20 eigen values vary. This results in higher sidelobe -30 beam response in dB level in adaptive beam pattern. A means of reducing -40 the variation of the eigen values is to add a weighted -50 identity matrix to the sample correlation matrix. -60 The result of diagonal loading of the correlation matrix is to add the loading level to all the eigen -70 values. This in turn produces the bias in these eigen -80 values in order to reduce their variation which in -90 term produces side bias in the adaptive weights that -100 -80 -60 -40 -20 0 20 40 60 80 reduces the output SINR. Recommended loading angle in θ levels of $ ≤ $ $ where $ is the noise Fig 4 MVDR-Diagonal Loading $ power and is the diagonal loading level. The 174 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 3, March 2011 { - $ H{ # { { (14) 3.4 Colored Diagonal Loading(Cdl) where [4] In the presence of colored noise, DL can be applied 0 MVDR-adaptive diagonal loading which is termed as colored diagonal loading (CDL) and the morphing process may result in a -10 beampattern of our choosing. The colored diagonal -20 loading is similar to but the diagonal -30 loading level of $ = ∞ , end point, can be altered beam response in dB -40 by the term[5] -50 { - $ { # { { (12) -60 -70 where is the covariance matrix that captures -80 the desired quiescent structure. It may be -90 determined directly 1)based on apriori information – -100 -80 -60 -40 -20 0 20 40 60 80 where , need not be diagonal or 2) desired angle in θ weight vector –where must be diagonal. It is given as Fig 6 MVDR- Adaptive Diagonal Loading beampattern # I {? I C { {{ (13) 3.6 Adaptive coloured White Noise where is the desired quiescent weight vector. Stabilization (ACDL) The colored diagoinal loading shows no As already discussed, white noise stabilization is improvement in pattern shape as shown in Fig.5 nothing but diagonal loading in which the adaptive MVDR -colored-Diagonal loading colored loading technique is imbedded to get a 0 novel hybrid method proposed as # -10 -20 ? - C { { (15) -30 beam response in dB -40 -50 MVDR-adaptive colored diagonal loading- the proposed algorithm 0 -60 -10 -70 -20 -80 -30 beam response in dB -90 -40 -100 -80 -60 -40 -20 0 20 40 60 80 -50 angle in θ -60 Fig 5 MVDR-Colored Diagonal Loading -70 -80 -90 -100 3.5 Adaptive Diagonal Loading (ADL) -80 -60 -40 -20 0 20 40 60 80 angle in θ In this method the loading level is calculated assuming the apriori information about the SNR is Fig 7 MVDR- Adaptive Colored Diagonal available. The SNR can be estimated from link Loading beampattern budget or using some SNR estimated algorithm.Gu and Wolf proposed a variable loading MVDR.(VL- MVDR) in which the loading level is chosen as( $ {[16] 175 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 3, March 2011 have a compact size. Hence a maximum of 16 elements are chosen for further analysis 4. Simulations and Experiments Table1. Effect of Changing number of antenna For the proposed hybrid algorithm ,a 16 element elements Uniform Linear Array is considered with SNR of 20 Desired Beam forming method dB for the desired signal coming from θs = 0° and signal Jammer1 Jammer2 Jammer3 INR of 70 dB for three jammer signals coming from θ=0° θ = 20° θ=-20° θ =-70° the directions θi = -20°, 20° and 70°. The element Beam spacing is d = 0.5 λ. Beam Beam Beam response The various methods of beamforming are obtained response response response Power(in to compare them with the performance of the Mvdr- Power Power Power dB) Adaptive colored Diagonal Loading. (in dB) (in dB) (in dB) conventional 0 -20 -20 -26.5 comparison of various diagonal loading methods 0 MVDR 0 -91 -66 -91 -10 -20 MVDR-SMI 0 -58 -61 -72 -30 DL 0 -72.5 -72.5 -85 beam response in dB -40 -50 CDL -6 -50 -57 -66.5 -60 -70 ADL 0 -72.5 -72.5 -85 MVDR-diagonal loding -80 MVDR-coloured DL Mvdr-adaptive DL ACDL 0 -52 -56.5 -62 -90 Mvdr-adapt-col-DL -100 -80 -60 -40 -20 0 20 40 60 80 angle in θ Table2: Beam response of the signals - desired Fig 8 Beampattern of various diagonal loading and jammers – using various methods methods 3dB beamwidth elements 5. Results and Discussion MVDR MVDR Diagon conven ve DL Adapti Adapti Colore - SMI No.of tional d DL ve al 5.1 Number of elements For the ULA which is considered for simulation 4 26.2 19.5 17.1 17 17 17 16 work , the beampatterns were analyzed by changing the number of elements as 4, 8, 12,16, 24, 50 and 8 12.8 15.4 13.3 14.8 14.8 14.8 25.5 100. As the number of elements increases, the 7 beampattern shows higher resolution i.e the 3 dB beamwidth becomes much narrower from to 26° to 12 8.4 8.7 6.9 8.4 8.5 8.7 8.5 1° for conventional beamformer and 17° to 1° for adaptive diagonal loading beamformer. The detailed 16 6.25 6.4 6.4 6.4 6.4 6.6 13.2 results are tabulated in Table1. Finer or sharper beams are obtained when more number of elements 20 5.1 5.2 6 5.3 5.3 5.2 6.8 are used. Sharper the beam, the beamformer is not 24 4.4 4.5 4.5 4.3 4.3 4.3 4.3 susceptible to jammers. But the number of side lobes also increased. The 3-dB beamwidth of the 50 2 2 2 2 2 2 2 different beamformers are tabulated in Table 2. A trade off can be obtained to reduce the cost and to 100 1 1 1 1 1 1 1 176 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 3, March 2011 5.2 Noise effect 6. Conclusion An ULA with 16 elements is considered for A new Hybrid Robust adaptive beamforming analyzing the effect of noise on the peaks of the algorithm is proposed with Adaptive Colored signal power. Signal to noise ratio (SNR) was varied diagonal loading based on data dependent approach. in steps of 10 dB starting from 10 dB till 60 dB. As This method is computationally efficient. SNR increases the peaks becomes sharper. It also Simulation results show that the proposed method showed that the interference sources were provide robustness against steering vector errors and suppressed to a maximum extent, so that it will not random array position perturbations by comparing it be a disturbance while extracting the signal even in with various diagonal loading methods the presence of strong interferers References 5.3 Training issues with the number of array [1] Viktor V.Zaharov, Marvi Teixeira, “SMI- MVDR Beamformer Implementations for Large snapshots Antenna Array and Small Sample Size”, IEEE Increasing the number of array snapshots lead to Transactions on Circuits and Systems- I Regular complexity and computational cost but the Papers Vol.55 No.10, November 2008. performance of the beamformer increases. It is a trade off between the cost and the performance [2]. Biao Jiang, Ye Zhu, “A New Robust Quadratic Constraint Beamforming against Array Steering Vector Errors”, International Conference on 5.4 Element Spacing Communications Circiuts and Systems 2004, Vol 2, The spacing between the elements for an 16 element 29-29 June 2004. ULA was varied as λ/4, λ/2, 3λ/4 and λ which in turn vary the effective aperture length of [3]Y.X.Zou,S.C.Chan ,et.al, “ Recursive Robust the array. Among the four choices λ/2 showed the Variable Loading MVDR Beamforming in best performance for the particular frequency used Implusive Noise Environment”, IEEE Asia Pacific for simulation. When the distance between the Conference on Circuits and Systems 2008, DOI: elements is increased beyond λ/2, it resulted in /10.1109/APCCAS.2008.4746190. spatial aliasing i.e a lot of spurious peaks were obtained which correspond to different frequencies. [4] Pekka Lilja, Harri Saarnisaari, “Robust Adaptive Below λ/2 the resolution of the beams was not Beamforming in Software Defined Radio with satisfactory. Adaptive Diagonal Loading”, IEEE Military Communications Conference 2005, DOI: Number of elements = 16, loading level= 100 10.1109/MILCOM.2005.1606058. 0 -5 [5]John D. Hiemstra, “Colored Diagonal Loading”, SINR-CDL -10 SINR-ADL Proceedings of the 2002 IEEE Radar SINR-ACDL conference,DOI: 10.1109/NRC.2002.999682. -15 -20 [6]H.L.Van Trees, “Detection, Estimation , and SINR in dB -25 Modulation Theory”, Part IV, Optimum Array -30 Processing, Wiley , NY,2002. -35 [7]Dimitris G.Manolakis, Vinay.K.Ingle, “Statistical -40 and Adaptive Signal Processing”, Artech House, -45 2005 -50 0 50 100 150 200 250 300 number of snapshots k [8]Simon Haykin, “Adaptive Filter Theory”, Prentice Hall of India, 1996. Fig 9 Training issues with the number of snapshots [9]Frank Gross, Smart Antennas for Wireless Communications with Matlab, McGraw-Hill, 2005 177 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 3, March 2011 [10]Lal.C.Godara, “Applications of Antenna Arrays Communications, Circuits and Systems, 2004, PP to Mobile communications, Part I: Performance 765-768 Improvement, Feasibility, and system considerations”, Proceedings of IEEE Vol.85, No.7 [14]Jian Li and Petre Stoica, “Robust Adaptive July 1997 Beamforming”,John Wiley & Sons Publications, 2006. 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