# Comprehensive Performance Analysis of Non Blind LMS Beamforming Algorithm using a Prefilter

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2 them in another domain other than frequency domain so as to distinguish between the mixed signals that form the input signal. Therefore, the technique is based on the idea that the desired and interfering signals arrive at the antenna array from different directions. Thus, these differences between arriving signals can be exploited. Applying the FFT on the array propagation (Equation 1) gives, ( ) ( ( )) ( ) ( ). ( ) ( )/ (2) Assuming d= λ/2, and solving Equation (2) and equating the result to zero, the following formula gives the index KMSC (or the order) of the most significant coefficient as a function of the direction of arrival θ and the number of array elements M.As KMSC.must be an integer the equation takes the form of equation (3). K MSC, * ( ( ( ) )+ - (3) Fig.1 Adaptive beam former with Pre-filtering system The distinction is obtained by converting the input signal to the spectrum of the spatial domain (this domain is the sine of the direction of arrival, or sinθ domain).the desired signal is extracted from the input signals simply by making a band-pass filter in the spectrum of the spatial domain, i.e. in the sinθ spectrum. This filtering process is shown in Fig. 2, and is explained in (Abu-Ella O et al,2008; Abu-Ella O et al, 2010) as follows: The Most Significant Coefficient (MSC) of the transformed signal is selected. This is ranked as the largest sample of the transformed desired signal. The most significant coefficient is placed at its rank in the M zeros element vector (zero padding). The Inverse Fast Fourier Transform (IFFT) is applied to the filtered vector of the previous step to reconstruct an alternative input signal that contains a reduced amount of interference and noise. The reconstructed data vector is used as input signal to the conventional adaptive beam forming system. Where mod M is the modulus notation performed on M points. Equation (3) can be simplified to K MSC (* +) (4) This Kmsc is used then to reconstruct the modified input signal which has reduced interference and noise. Simulation results presented later in this paper show that the prefiltering technique significantly reduces the mean square (MSE). This prefiltered output is then used as input to any conventional beamforming algorithm to enhance its performance. The results obtained by applying this technique to the non blind Conventional LMS algorithm are discussed here. 3. LMS Algorithm In adaptive filtering applications for modeling, equalization, control, echo cancellation, and Beam forming, the widely used least-mean-square (LMS) algorithm has proven to be both a robust and easilyimplemented method for on-line estimation of Timevarying system parameters (S. C. Douglas et al, 1994). Fig.2 Pre-filtering Process Mathematically, assuming that the propagation vector for the θ direction of arrival, is given by ( ) ( ) Where M is the number of array elements, d is the spacing distance between any two adjacent elements, and λ is the wavelength of the operating carrier frequency. (1) Fig.3 LMS adaptive beamforming process Fig.3 shows a generic adaptive beamforming system which requires a reference signal. As shown in Fig.3, the outputs of the individual sensors are linearly combined after being scaled using corresponding weights such that 2025 International Journal of Current Engineering and Technology, Vol.4, No.3 (June 2014)

3 the antenna array pattern is optimized to have maximum possible gain in the direction of the desired signal and nulls in direction of interferers (Lal.C.Godara et al, 1997; S. C. Douglas et al, 1994; R. S. Kawitkar et al 2005). LMS is nonblind algorithm which requires a training sequence of known symbols d(n), to train the adaptive weights. It uses the estimate of the gradient vector from the available data. This algorithm makes successive corrections to the weight vector in the direction of the negative of the gradient vector which finally concludes to minimum MSE (MMSE). This successive correction to the weight vector is the point at which optimum value w0 is obtained that relies on autocorrelation matrix R and cross correlation matrix p of the filter. LMS is an adaptive beamforming algorithm, defined by the following equations (Lal.C.Godara et al, 1997; S. C. Douglas et al, 1994; R. S. Kawitkar et al, 2005 ; B.Widrow et al, 2005;Simon Haykin et al, 2002) with input signal x(n) : capacity i.e number of users that can be served by the system. y(n) =W H (n) x (n) (5) e (n) = d(n) y (n) (6) w (n+1) = w(n) + µ x (n) e*(n) (7) where y (n) is the filter output,e(n) is the error signal between filter output and desired signal d(n) at step n. d(n) is the training sequence of known symbols (also called as a pilot signal), required to train the adaptive weights. Equation (7) is the weight w(n) update function for the LMS algorithm. μ is rate of adaption also called as a step size, controlled by the processing gain of the antenna. (R. S. Kawitkar et al, 2005) ; B.Widrow et al, 2005). Fig.4 Flowchart of the Hybrid (Prefiltered) adaptive beamforming technique Fig.5 a) Shows the beam pattern gain (magnitude response) of Conventional and Hybrid LMS algorithm for M =8, desired angle at=45 deg interference angles at 35, 50. µ=0.001 Fig.5 b) & c) show the Polar plot for the same in terms of the Antenna Array factor. 4. Hybrid (Prefiltered) Adaptive Beamforming Algorithm The complete hybrid system by applying prefiltering to the conventional adaptive beamforming algorithm discussed in sections II and III respectively is illustrated below with the help of flowchart in Fig 4. The technique aims at improving the performance of beam forming algorithm by reducing the interference and noise effects on the desired user signal. 5. Simulation Results Simulation of the technique is carried out using MATLAB software. The prefiltered signal is applied to the conventional LMS beam forming algorithm for a Uniform Linear array (ULA) with a distance between the elements d = λ/2. Results of magnitude response of Conventional and Hybrid LMS beamforming algorithms presented here are obtained by varying parameters like no of antenna elements (M) and step size parameter μ, for two or more interferes in random directions and Noise is assumed to be Gaussian. The performance analysis is done using following parameters such as Beam Pattern gain characteristics, Signal to Interference Ratio SIR with respect to number of iterations, Bit Error rate, Convergence speed and system Fig.5 a) Beam pattern gain of LMS and hybrid technique for M=8, µ=0.001 Fig.5 b) Polar plot for LMS c) Polar plot for Hybrid LMS 2026 International Journal of Current Engineering and Technology, Vol.4, No.3 (June 2014)

4 Fig.6 a) Shows the beam pattern gain (magnitude response) of Conventional and Hybrid LMS algorithm form =8, desired angle at=45 deg interference angles at 35, 50. μ=0.001 Fig.6 b) & c) show the Polar plot for the same in terms of the Antenna Array factor. Table 2. Results obtained for antenna elements M=12 and µ=0.001 Input DOA (Ө) in deg Beam Gain for Conventional Beam Gain for Prefiltered Total improvement Beam Gain (db) Table 3. Results obtained for antenna elements M=16 and µ=0.001 Input DOA (Ө) in deg Beam Gain for Conventional Beam Gain for Prefiltered Total improvement Beam Gain (db) Fig.6 a) Beam pattern gain of LMS and hybrid technique M=16, and μ= Fig.6 b) Polar plot LMS 330 c) Polar plot Hybrid LMS Plots obtained here do not give exact value of amplitude /gain response G (θ), hence for more accurate estimation we normally refer its computed value. Hence for the further analysis we refer its computed value in MATLAB. After observing Table 1, 2 & 3 show that the prefiltered technique improves antenna beam pattern gain than the conventional LMS algorithm for most of the DOA s. The technique works well even for close angular separation between desired user and interferers when the antenna elements are increased The Second parameter for comparison is Signal to interference Ratio SIR behavior with respect to number of iterations. For the same initial conditions set for the previous case graphs are plotted for LMS and Prefiltered (PF) LMS by increasing the number of iterations for DOA 45 deg & M=8. It is seen from the Fig.7 a) & b) that there is an improvement of 0 to 3 dbs in the SIR of the prefiltered algorithm than the conventional algorithm and also the improvement is achieved in the initial few iterations only and becomes steady with number of iterations from 200 to 800.The simulation is also carried out by varying the number of antenna elements 8, 12 and 16 for constant number of iterations which also shows improvement in SIR form 0 to 3dbs. 0 to 3dbs. Table 1. Results obtained for antenna elements M=8 and µ= Input DOA (Ө) deg Beam Gain for Conventional Beam gain for Prefiltered Total improvement Beam Gain (db) Fig.7 a) SIR versus the number of iterations 2027 International Journal of Current Engineering and Technology, Vol.4, No.3 (June 2014)

5 Fig.8.a) Plot for the BER Vs SINR for Antenna Elements (M) =12 Fig.7 b) SIR versus the number of iterations Iterations (N) = 800 The Third Parameter for the performance analysis is the Bit error rate for varied SNR. Fig.8 a) & b) shows the behavior of BER when the SINR is varied from -20 db to +20db for DOA 45º and antenna element M= 4 &12 respectively. It can be seen that BER of the proposed Prefiltered algorithm matches the Conventional algorithm and improves with the increase in SINR. Table 4 shows BER behavior for different DOA s and Table 5 shows BER when antenna elements are increased to 8, 12 and 16. It can be seen that BER of the proposed Prefiltered algorithm matches the Conventional algorithm and improves with the increase in SINR. It also indicates that for some DOA s conventional algorithm performs better while for certain DOA s Prefiltered algorithm performs better. Difference between the minimum BER achieved by Conventional and Prefiltered (Hybrid) algorithm is very less. Table 4. Comparison of minimum BER for different DOA s and With SINR variation from -20 db to 20 dbs. & M=8 Minimum BER using LMS Minimum BER using Prefiltered LMS Difference In BER Table 5. Comparison of minimum BER for different DOA s and antenna elements M=8, 12 &16 respectively. BER with M=8 BER with M= BER with M=16 Fig.8.a) Plot for the BER Vs SINR for Antenna Elements (M) = 4 Fig.9 a)system Capacity ( users) plot for DOA 45º, M= 4 and SNR Variation from -6 to 10 dbs 2028 International Journal of Current Engineering and Technology, Vol.4, No.3 (June 2014)

6 The Next parameter for Comparison is the Capacity i.e number of users that can be served by the system with respect to SNR and for different bit rates. Table 8. Comparison of Capacity ( Users) for different bit rates and antenna elements M=8 Bit Rate (bits/sec) Users using LMS Users using PF- LMS Difference in Users Table 9. Comparison of Capacity (Num.of Users) for different bit rates and antenna elements M=8, 12 &16 resp. Bit Rate (bits/sec) No. of users with M=8 No. of users with M=12 No. of users with M=16 Fig.9 b) System Capacity ( users) plot for DOA 45º, M= 16 and SNR Variation from -6 to 10 dbs Table 6. Comparison of Capacity ( Users) for different DOA s and With SINR variation from -20 db to 20 dbs & M=8 Users using LMS Users using PF- LMS Difference in Number of Users Table 7. Comparison of Capacity ( Users) for different DOA s and antenna elements M=8, 12 &16 resp. Fig.10 a) Convergence Speed for M= 8, µ=0.001 Difference in no.of Users with M=8 no.of Users with M=12 no.of Users with M= From Fig. 9 a) & b) it is observed that the number of user curves of both the systems almost overlap each other. The number of users increased with the increase in number of antenna elements. Also from Tables 6 & 7 and Tables 8 & 9 is clear that the system capacity increases with the increase in bit rate as well as SINR and the rate of increase in both conventional and prefiltered algorithm is identical. Fig.10 b) Convergence Speed for M= 12, µ=0.001 The Final parameter of comparison is the convergence speed determined by measuring the error behavior of the 2029 International Journal of Current Engineering and Technology, Vol.4, No.3 (June 2014)

7 algorithms versus the number of iterations. i.e. measuring the value of the cost function (the mean square error) at each sample time. Fig.10 a) to d) show Comparison of Convergence speed of LMS and Prefiltered (Hybrid) LMS in terms of number of iterations for number of antenna elements M= 8, 12, 16 & 32 & µ= 0.001respectively. Table 11.Convergence Speed for different angles of arrivals and number of antenna elements 12 & µ=0.001 for LMS Num. of iterations Hybrid Table 12.Convergence Speed for different angles of arrivals and number of antenna elements 16 & µ=0.001 Fig.10 c) Convergence Speed for M= 16, µ=0.001 for LMS PF-LMS Table 13. Convergence Speed for different angles of arrivals and number of antenna elements 32 & µ=0.001 for LMS PF-LMS Fig.10 d) Convergence Speed for M= 32, µ=0.001 As the graphs do not show the exact values, the approximate numbers of iterations for different angles of arrivals are tabulated in Tables 10 to 13 for number of antenna elements 8,12,16 32 respectively. With µ= Table 10.Convergence Speed for different angles of arrivals and number of antenna elements 8 & µ=0.001 for LMS PF-LMS From Fig 10.a) to d) and Tables 10 to 13 it can be seen that the convergence speed of the conventional algorithm is slightly faster than the Prefilterd technique, but the difference is very small. Both the algorithms converge at a faster rate with increase in number of antenna element. The difference between the numbers of iterations required to converge between Conventional and Hybrid algorithm reduces with increase in number of antenna elements. 6. Discussion and Conclusion The simulation results obtained for non blind LMS Beamforming algorithm by applying prefiltering technique are presented and discussed in the previous Chapter. These results obtained for the five performance measures can be observed and analyzed to conclude the following Points: 2030 International Journal of Current Engineering and Technology, Vol.4, No.3 (June 2014)

8 1) Beam Pattern Characteristics: The beam pattern response of the prefiltered (hybrid) algorithm is better than the conventional algorithm. The improvement varies between 0 to 3 dbs for given experimental conditions. The amplitude response (beam pattern gain) of the hybrid technique for certain angles increases with increase in number of antenna elements even for the close spatial separation between desired user and interferers. The Spatial accuracy of the Prefiltered (Hybrid) algorithm is slightly less than the conventional algorithm.the difference varies from 0 to 5 degrees. This is because the Prefiltering is done using FFT and then IFFT which affects the spatial accuracy but this improves with increase in number of antenna elements. 2) Signal to Interference Ration (SIR) Vs iterations: The signal to interference ratio improves by about 1 to 3 db s in the Prefiltered algorithm than the conventional algorithm as the number of antenna elements are increased.the SIR improvement is achieved in the less number of iterations and stays constant as the number of iterations are increased. 3) Bit Error Rate (BER) with respect to SINR: The BER of the proposed Prefiltered algorithm matches the Conventional algorithm and improves with the number of antenna elements. For certain DOA s Prefiltered algorithm performs better than the conventional algorithm. Difference between the minimum BER achieved by Conventional and Prefiltered (Hybrid) algorithms is very less. 4) System Capacity ( Users in the system) Vs SNR: It is observed that the number of users serviced by both the systems is almost same. The number of users increased with the increase in number of antenna elements. Also system capacity increases with the increase in bit rate as well as SINR and the rate of increase in both conventional and prefiltered algorithm is identical. 5) Convergence Speed i.e. Mean Square Error with respect to Iterations: It can be inferred that the Prefiltered (Hybrid) algorithm is slow to converge as compared to the conventional but the difference is marginal. Both the algorithms converge at a faster rate with increase in number of antenna element. The numbers of iterations are reduced from maximum 250 to minimum 20 as number of antenna elements are increased from 4 to 32 for given experimental conditions. Moreover the difference between the numbers of iterations required to converge between Conventional and Hybrid algorithm reduce with increase in number of antenna elements. This analysis indicates that Prefiltering technique will be useful to enhance the performance of the Beamforming in systems which are corrupted with noise and interference and significantly especially when there are more number of antenna elements and convergence speed is not of much concern. References Lal.C.Godara, (July 1997), Applications of Antenna Arrays to Mobile Communications, Part I; Performance Improvement, Feasibility, and System Considerations, Proceeding of the IEEE, VOL. 85, NO. 7, pp Lal.C.Godara, (August 1997), Applications of Antenna Arrays to Mobile Communications, Part II; Beam- Forming and Directional of Arrival Considerations, Proceeding of the IEEE, VOL.85, NO. 8, pp Abu-Ella O, El-Jabu, (January 2010), Adaptive Beamforming Algorithm Using a Pre-filtering System. Source: Aerospace Technologies Advancements, Book edited by: Dr. Thawar T. Arif, ISBN , INTECH, Croatia, pp O. Ali Abu-Ella, B. El-Jabu, (2008), Increasing capacity of blind mobile system using pre-filtering technique, IET Microw. Antennas Propag, Vol. 2, No. 5, doi: /iet-map: , pp WINTZ P (1972), Transform Picture Coding, Proc.IEEE, 1972, 60 (7), pp S. C. Douglas and T. Meng,( June 1994), Normalized data Nonlinearities for LMS adaptation, IEEETransactions on Signal Processing, Vol. 42, No.6,pp , R. S. Kawitkar and D. G. Wakde, (2005), Smart antennaarray analysis using LMS algorithm, IEEE Int.Symposium on Microwave,Antenna, Propagation and EMC Technologies for Wireless Communications, pp B. Widrow and S.D. Stearns, (1985), Adaptive Signal Processing. Pearson Eduation, Inc. Simon Haykin, (2002), Adaptive Filter Theory, Fourth edition, Pearson Eduation, Inc. J.C.Liberti,T.S.Rappaport.(1999), Smart Antenna for wireless communication,prentice Hall India International Journal of Current Engineering and Technology, Vol.4, No.3 (June 2014)

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