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\title{Effect of Signal to Noise Ratio on Adaptive Beamforming Techniques}
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             \author[1]{Ved Vyas  Dwivedi}

             \author[2]{Smita  Banerjee}

             \affil[1]{  RK University}

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\date{\small \em Received: 10 December 2016 Accepted: 4 January 2017 Published: 15 January 2017}

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\begin{abstract}
        


The capability of adaptive antenna array lies in forming higher gain in the user directions and lower gain in the interferer directions. The technique used to produce such radiation pattern by calculating the excitation weights are called the adaptive beamforming techniques. It tries to minimize the error between the desired and actual signal and maximize the signal to noise ratio (SNR). But in severe interference environment when the actual signal is weak, the effect of SNR on the radiation pattern needs to be considered. This paper describes the effect of SNR on different adaptive beamforming techniques such as non- blind Least mean Square (LMS), blind Constant Modulus Algorithm (CMA) and evolutionary Particle Swarm Optimization (PSO). The performance and validation of beamforming algorithms are studied through MATLAB simulation by varying SNR parameter for different desired and interference direction. Different weights are obtained using this beamforming algorithm to optimize the radiation pattern. The parameters for comparison are the main beam and null placement for different angles of user and interferer. The mean SLL and directivity are also studied.

\end{abstract}


\keywords{adaptive antenna, beamforming, particle swarm optimization, least mean square, constant modulus algorithm, signal to noise ratio.}

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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
n satellite communication systems, the receiver receives extremely weak signals from the satellite. To enhance reception and radiation patterns dynamically in response to the signal environment, such technologies depend on adaptive array signal processing. An adaptive antenna is an array of antenna elements followed by a sophisticated signal processor that can adjust or adapt its own radiation pattern in order to focus the reception of the antenna array in a certain direction and rejects the signal from other direction. The necessity to remove the effect of the undesired signal to the desired one motivates advances in communication receiver antenna and hence synthesizing methods \hyperref[b0]{[1]}\hyperref[b1]{[2]}\hyperref[b3]{[3]}\hyperref[b4]{[4]}.\par
An adaptive antenna array combines the outputs of antenna elements. The directional gain of the antenna is controlled by adjusting phase or amplitude or both at each individual element. The weighted signals are summed and the output is fed to a controller. These weights are computed adaptively to adapt to the changes in the signal environment. Different adaptive beamforming algorithms are employed to minimize the error between the desired signal and the array output that adjusts the weights to satisfy an optimization criterion \hyperref[b5]{[5]}\hyperref[b6]{[6]}\hyperref[b7]{[7]} {\ref [8]}\hyperref[b8]{[9]}\hyperref[b9]{[10]}\hyperref[b10]{[11]}.\par
The capability of adaptive antenna array lies in forming higher gain in the user directions and lower gain in the interferer directions. There are different adaptive beamforming algorithms studied in literature which are used in the adaptive antenna array \hyperref[b11]{[12]}\hyperref[b12]{[13]}\hyperref[b13]{[14]}\hyperref[b14]{[15]}\hyperref[b15]{[16]}\hyperref[b16]{[17]}\hyperref[b17]{[18]}\hyperref[b18]{[19]}\hyperref[b19]{[20]}\hyperref[b20]{[21]}\hyperref[b21]{[22]}\hyperref[b22]{[23]}\hyperref[b23]{[24]}. Beamformers based upon statistically optimum blind and non-blind adaptive beamforming are analyzed and compared on the basis of beamforming capability and rate of convergence. It is observed that the convergence rate of Least Mean Square (LMS) is slowest where as Constant CGM is the fastest among all. SMI is found to have more computational complexity. Recursive Least Square (RLS) is found to have higher side lobe level (SLL) and null depths as compared to CGM \hyperref[b15]{[16]}. It was observed that the conventional Adaptive Beamforming (ABF) technique like Minimum Variance Distortionless Response (MVDR) improves the signal-to-interferenceplus-noise ratio (SINR) but unable to reduce the SLL \hyperref[b16]{[17]}. Hence to improve the SINR with reduced SLL, many optimization techniques have been used in ABF application. Adaptive Mutated Boolean Particle Swarm Optimization (AMBPSO) technique takes the uncorrelated desired and interferer signal directions and succeed in providing good SINR value with lower SLL as compared to conventional MVDR \hyperref[b17]{[18]}. Adaptive Dispersion Invasive Weed Optimization (ADIWO) shows improvement in steering ability regarding the main lobe and the nulls, faster as compared to PSO and achieves better SLL than the PSO and MVDR \hyperref[b18]{[19]}. Hybrid Particle Swarm Optimization with Gravitational Search Algorithm (Hybrid PSOGSA) shows its ability for optimization in beam-forming for a larger number of user signals and speedy computation using parallel GSA as compared to sequential stand alone algorithms but cannot maximise the gain along the user direction \hyperref[b19]{[20]}\hyperref[b20]{[21]}. Mementic algorithm shows optimal radiation pattern design to maximise the signal to interference ratio (SIR) by perturbing the phase-position \hyperref[b21]{[22]}. But, for the case of adaptive antennas, the position of the antenna elements cannot be changed so it should be kept fixed. As the required phase controls are available at no extra cost. Hence only phase weights are considered for optimal radiation pattern which shows good null depth along the undesired direction but the array factor (AF) gain along the main lobe is not satisfactory \hyperref[b22]{[23]}\hyperref[b23]{[24]} .\par
In all of the above adaptive beamforming techniques proposed so far try to minimize the error between the desired and actual signal and maximize the signal to interference ratio (SIR). But in severe interference environment when the actual signal is weak, the effect of SNR on the radiation pattern needs to be considered.\par
The present study analyses different adaptive techniques such as non-blind LMS, blind CMA and evolutionary PSO. The performance of beamforming algorithms are studied through MATLAB simulation by varying SNR parameter for different desired and interference direction. Different weights are obtained using this beamforming algorithm to optimize the radiation pattern. The parameters for comparison are the main beam and null placement for different angles of user and interferer. The mean SLL and directivity are also studied.\par
The rest of the paper is arranged as follows: Section II describes the mathematical model of signal, Section III formulates the adaptive beamforming problem, Section IV, V and VI describes adaptive beamforming using PSO, LMS and CMA, Section VII compares the results and Section V concludes the whole study. 
\section[{II.}]{II.} 
\section[{Signal Model}]{Signal Model}\par
Consider a Uniform Linear Array (ULA) with N elements as shown in Figure \hyperref[fig_0]{1}. Let S(k) is the S X1 signal vector from the S th source with DOA equal to ? S.\par
[ ]) ( ... ... ) ( ) ( ) ( 2 1 k S k S k S k S s = (1)\par
We define the input signals as X 1 (k), X 2 (k),.........X N (k). As they reach the antenna elements, the N X 1 signal vector X(k) can be written as? = = s s s s SV k S k X 1 ) ( * ) ( ) ( ? (2)\par
Where SV (?) is the steering vector or array response vector of N X 1 which controls the direction of antenna beam.T N j j j SV ))] sin( ) 1 ( exp( .... .... )) sin( 2 exp( )) sin( exp( 1 [ ) ( ? ? ? ? ? ? ? ? ? ? = (3)\par
Now if the signal 1,2??S consist of U number of desired user arriving from ? 1 ,? 2 ,? 3 ,??? U , I number of interferences arriving from ? 1 ,? 2 ,? 3 ,??? I with variance ? i 2 and noise with variance ? n 2 , then the input signal consist of user signal S u , interferer signal S i and noise N. The received signal can be written as ) ( ) ( * ) ( ) ( * ) ( ) ( 1 1 k N SV k S SV k S k X I i i i U s u u + + = ? ? = = ? ? (4) 
\section[{Global}]{Global})) sin( ) 1 ( exp( ..... )) sin( exp( 1 ) ( u u u N j j SV ? ? ? ? ? ? ? ? =\par
is the steering vector of the desired signal along the user and [ ])) sin( ) 1 ( exp( ..... )) sin( exp( 1 ) ( i i i N j j SV ? ? ? ? ? ? ? ? =\par
is the steering vector along the interferer direction. 
\section[{III.}]{III.} 
\section[{Adaptive Beamforming Problem Formulations}]{Adaptive Beamforming Problem Formulations}\par
An ULA will receive the incoming signals which will be multiplied by the weights of antenna elements which are then summed to get the output in the form of received signal. The received signal will be graphical represented in the form of the radiation properties as a function of space coordinates known as radiation pattern. The radiation pattern of the linear array for far field is represented in terms of array factor (AF) by \hyperref[b14]{[15]},n N n w k X AF * ) ( 1 ? = = (5)\par
where N= number of elements, In adaptive antenna beamforming, the radiation pattern of ULA is controlled through various adaptive algorithms. Adaptive algorithm dynamically optimizes the radiation pattern according to the changing electromagnetic environment. The output or received signal is given to the adaptive algorithm where it checks the output radiation pattern with the desired radiation pattern. If the received actual radiation pattern does not meet the user demands, then adaptive algorithm will try to adjust the weights of the antenna array such that the actual and desired radiation pattern remains same. The antenna array pattern is optimized to have maximum possible gain in the direction of the desired signal and nulls in the direction of the interferers.\par
Figure 2 shows the block diagram of an adaptive antenna array. Particle Swarm Optimization (PSO) was developed by Eberhart and Shi \hyperref[b24]{[25]}. It is used as adaptive algorithm to search the optimized adaptive antenna radiation pattern. This is done using the algorithm summarized in the Table 1 \hyperref[b25]{[26]}. In every iteration, PSO algorithm will try to increase the AF gain of the desired user and decrease the AF gain of the interfering user as compared of the previous iteration. The converged value of weights produces an optimized adaptive antenna radiation pattern.\par
The amplitudes excitations are kept constant whereas the phase excitations are selected as the optimization parameters. Hence the AF can be written as  Step-1: Initialize population, number of iterations, tuning parameters ) ( Step-2: Step-3: Evaluate the fitness function for each particle b n (i). Compute FF (i, k) as per the equation \hyperref[b7]{(7)}.\par
Step-4: Compute pbest(i, k) = FF(i, k) and gbest(i) = max (pbest (i, k)) with its location pbest (k) and gbest.\par
Step-5: Update velocity v (i+1, k) and position b n (i+1, k) using) ( )) , ( ) ( ( 2 ) ( )) , ( ) ( ( 1 ) , ( ) , 1 ( i u k i b ib g i u k i b ik b p k i v w k i v n n n n ? + ? + * = + ? ? ) , 1 ( ) , ( ) , 1 ( k i v k i b k i b n n + + = +\par
Step-6: Update fitness function FF(i+1, k).\par
Step-7: If FF(i+1, k) > FF(i, k), then pbest(i+1, k)=FF(i+1, k).\par
Step-8: Update gbest (i+1, k) = max (pbest(i+1,k)).\par
Step-9: If i<i max then increment i and go to step-5, else stop.\par
V. 
\section[{Adaptive Beamforming Using Least Mean Square Algorithm}]{Adaptive Beamforming Using Least Mean Square Algorithm}\par
Least Mean Square (LMS) algorithm was first developed by Widrow and Hoff in 1960. The optimum weights can be estimated with LMS algorithm. The algorithm recursively computes and updates the weight vector. Successive corrections to the weight vector in the direction of the negative of the gradient vector eventually lead to the MMSE between the beamformer output and the reference signal. At this point the weight vector assumes to be its optimum value. The algorithm contains three steps in each recursion: the computation of the processed signal with the current set of weights, the generation of the error between the processed signal and the desired signal, and the adjustment of the weights with the new error information. The following Table  {\ref 2} summarize the above three steps \hyperref[b12]{[13]}.\par
Table  {\ref 2}: Algorithm for Adaptive Beamforming using LMS\par
Step-(1): Initialize number of iteration i max and the value of µ.\par
Step-(2): Initialize weight W LMS , error E LMS and output y LMS as 0.\par
Step-(3): Compute Output, y LMS (i, k)= W LMS (i, k) H x(k)\par
Step-(  {\ref 4}): Compute Error, E LMS (i, k) = S u (k)-y LMS (i, k)   Step-(  {\ref 5}): Compute Weight, W LMS (i+1, k) =W LMS (i,   k)+µx(k)E LMS *(i, k)\par
Step-(  {\ref 6}): If i>i max , then stop, otherwise go to step (3) to update output, error and weight.\par
VI. 
\section[{Adaptive Beamforming Using Constant Modulus Algorithm}]{Adaptive Beamforming Using Constant Modulus Algorithm}\par
The constant modulus algorithm (CMA) was first proposed by Godward. It is used for blind equalization of signals that have a constant modulus where reference signals are not available. The algorithm contains three major steps in each recursion: the computation of the output signal with the current set of weights, the generation of the error, and the adjustment of the weights with the new error information. The following Table \hyperref[tab_1]{3} summarize the above three steps \hyperref[b15]{[16]}. Step-( \hyperref[formula_11]{1}): Initialize number of iteration i max and the value of µ.\par
Step-(  {\ref 2}): Initialize weight W CMA , error E CMA and output y CMA as 0.\par
Step-(  {\ref 3}): Compute Output, y CMA (i, k)= W CMA (i, k) H x(k).\par
Step-(4): Compute Error, E CMA (i, k) = y CMA (i, k)/ |y CMA (i, k)|-y CMA (i, k).\par
Step- \hyperref[b5]{(5)}: Compute Weight, W CMA (i+1, k) =W CMA (i, k)+µx(k)E CMA *(i, k)\par
Step-(  {\ref 6}): If i>i max , then stop, otherwise go to step (3) to update output, error and weight.\par
VII. 
\section[{Numerical Simulation Results}]{Numerical Simulation Results}\par
A 16 element ULA with ?/2 interelement spacing is taken. PSO, LMS and CMA were applied on a 16-element ULA. Three algorithms were compared on the basis of the SNR. In order to compare the performance, the simulations are done using MATLAB. All the algorithms were executed for 200 iterations and the termination criterion is set for the number of iterations. For PSO, the population size is assumed as 100 and tuning parameter 1 ? and 2 ? are set to 2.0.\par
Phase excitation b n is chosen as the design variable in the PSO with lower and upper limit taken in the range of [-2?, 2?] with initial values of position and velocities are taken as random. For LMS and CMA, µ is taken as 0.001 and the initial weight and error are set to 0. Based upon the aims to maximize the AF gain of the desired user and minimize the AF gain of the interfering user. PSO will try to maximize the value of the AF gain along User1 while minimize the AF gain along interferer1 and interferer2. LMS will recursively computes and updates the weight vector between the output signal and the desired signal. CMA will update the information based upon the new error information.\par
To validate the study, two different scenarios are studied with different position of interferer. In scenario\#1, the ULA receives a desired signal arriving from angle 1 towards 1 i ? and 2 i ? . The AF gain along the main lobe is 0 dB whereas the AF gain towards the null is -20 dB to -50 dB as shown in Table \hyperref[tab_2]{4}. The maximum SLL is -15dB to -17dB with directivity of 7 dB as shown in Figure  {\ref 3} and Figure  {\ref 4}.\par
LMS algorithm also produces main lobe gain of 0 dB along the 1 s ? direction and null gain of -33 dB to -66dB for SNR=30dB to SNR=-10dB as shown in Table \hyperref[tab_2]{4}. As SNR reduces more than -10 dB, LMS fail to point the main beam and null along the user and the interferer direction in both the scenarios.\par
CMA algorithm works well for SNR=30 dB to SNR=10dB. As SNR starts deteriorating CMA does not produce main beam along the user and fails to point lower gain along the interferer as shown in Table \hyperref[tab_2]{4}. In both the scenarios, LMS and CMA gives reduced SLL.\par
The comparative Table \hyperref[tab_3]{5} for both the scenario shows that PSO is better as compared to LMS and CMA for every value of SNR. LMS and CMA fail to adapt for lower value of SNR. However LMS and CMA shows better SLL as compared to PSO. Table \hyperref[tab_4]{6} gives the optimized excitation weights for PSO, LMS and CMA for SNR=30dB. (W PSO )\#1 (W PSO )\#2 (W LMS )\#1 (W LMS )\#2 (W CMA )\#1 (W CMA )\#2\textbf{1}\par
1.00 + 0.00i 1.00 + 0.00i 1.00 + 0.00i 1.00 + 0.00i 1.00 + 0.00i 1.00 + 0.00i 2 0.84 -0.54i -0.28 + 0.95i 0.99 + 0.00i 0.99 + 0.00i 1.00 -0.02i 0.99 + 0.00i 3 0.59 + 0.80i 0.88 -0.46i 0.99 + 0.00i 0.98 + 0.00i 0.98 -0.00i 0.98 + 0.01i 4 0.99 + 0.02i -0.09 + 0.995i 0.99 + 0.00i 0.99 + 0.00i 0.97 + 0.00i 0.99 + 0.01i 5 0.53 -0.84 0.66 + 0.750i 0.99 + 0.00i 0.99 + 0.01i 0.99 + 0.02i 0.99 + 0.01i 6 -0.04 -0.99i -0.72 + 0.688i 0.99 + 0.00i 0.98 + 0.00i 0.99 -0.00i 0.99 + 0.01i 7 0.66 -0.74i 0.63 + 0.770i 1.00 + 0.00i 0.99 + 0.00i 0.97 -0.00i 0.99 -0.00i 8 0.54 + 0.83i -0.99 + 0.032i 1.00 + 0.00i 0.99 + 0.00i 0.98 -0.00i 0.98 + 0.01i 9 -0.29 -0.95i 0.29 -0.954i 0.99 + 0.00i 0.98 + 0.00i 0.98 + 0.00i 0.98 + 0.01i 10 -0.92 -0.37i -0.40 + 0.912i 0.99+ 0.00i 0.99 + 0.00i 0.98 -0.01i 1.00 + 0.00i 11 -0.14 + 0.98i 0.82 -0.571i 0.99 + 0.00i 0.99 + 0.00i 0.99 -0.01i 1.00 + 0.00i 12 -0.79 + 0.60i 0.03 -0.999i 0.99 + 0.00i 0.99 + 0.00i 0.98 + 0.00i 0.99 + 0.00i 13 0.17 + 0.98i -0.34 + 0.939i 0.99 -0.00i 0.99 -0.00i 0.99 + 0.01i 0.99-0.00i 14 -0.62 -0.78i 0.44 -0.896i 0.99 + 0.00i 0.99 + 0.00i 0.99 -0.00i 0.98 + 0.00i 15 0.82 -0.56i -0.44 + 0.896i -0.99 + 0.00i 0.99 + 0.00i 0.97 -0.01i 0.99 + 0.00i 16 0.21 -0.97i 0.90 -0.431i 1.00 + 0.00i 0.98 + 0.00i 0.97 + 0.00i 0.99 + 0.00i  
\section[{Conclusions}]{Conclusions}\par
In this paper, ABF based on PSO, LMS and CMA method have been simulated for 16 elements ULA. A performance analysis and validation is done by changing the values of SNR for two different positions of interferers. The main lobe gain and null depth are calculated to validity this approach. It is shown that the PSO-based beamformer provides accurate 0dB main beam gain and null depth of -20dB to -50dB with better SLL for each case of SNR. However, CMA fail to provide main beam and null placement for SNR< 0dB and LMS for SNR< -20dB. Therefore, the PSO method seems to be simple and appropriate in ABF applications based on the fitness function. ABF using PSO shows mean side lobe level (SLL) of -15 dB to -17 dB with a directivity of 7dB for each case of SNR. LMS and CMA show better SLL than PSO. It can be further studied with complex fitness functions in order to improve the value of SLL.\begin{figure}[htbp]
\noindent\textbf{1}\includegraphics[]{image-2.png}
\caption{\label{fig_0}Figure 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-3.png}
\caption{\label{fig_2}}\end{figure}
    \begin{figure}[htbp]
\noindent\textbf{1} \par 
\begin{longtable}{}
\end{longtable} \par
 
\caption{\label{tab_0}Table 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{3} \par 
\begin{longtable}{}
\end{longtable} \par
 
\caption{\label{tab_1}Table 3 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{4} \par 
\begin{longtable}{P{0.02437655860349127\textwidth}P{0.0529925187032419\textwidth}P{0.03497506234413965\textwidth}P{0.11764339152119702\textwidth}P{0.0646508728179551\textwidth}P{0.1367206982543641\textwidth}P{0.05511221945137157\textwidth}P{0.051932668329177055\textwidth}P{0.22998753117206983\textwidth}P{0.04239401496259352\textwidth}P{0.039214463840399\textwidth}}
\tabcellsep \tabcellsep \tabcellsep \multicolumn{5}{l}{scenario\#2 (*MB-Main Beam, * NP-Null Position)}\tabcellsep \tabcellsep \tabcellsep \\
SNR\tabcellsep Scenario\tabcellsep \tabcellsep PSO\tabcellsep \tabcellsep \tabcellsep LMS\tabcellsep \tabcellsep \tabcellsep CMA\tabcellsep \\
(dB)\tabcellsep \tabcellsep G\textunderscore S1\tabcellsep G\textunderscore I1\tabcellsep G\textunderscore I2\tabcellsep G\textunderscore S1\tabcellsep G\textunderscore I1\tabcellsep G\textunderscore I2\tabcellsep G\textunderscore S1\tabcellsep G\textunderscore I1\tabcellsep G\textunderscore I2\\
30\tabcellsep \#1\tabcellsep 0\tabcellsep -30\tabcellsep -23\tabcellsep 0\tabcellsep -33\tabcellsep -38\tabcellsep 0\tabcellsep -32\tabcellsep -37\\
\tabcellsep \#2\tabcellsep 0\tabcellsep -32\tabcellsep -42\tabcellsep 0\tabcellsep -48\tabcellsep -40\tabcellsep 0\tabcellsep -40\tabcellsep -47\\
20\tabcellsep \#1\tabcellsep 0\tabcellsep -25\tabcellsep -53\tabcellsep 0\tabcellsep -32\tabcellsep -50\tabcellsep 0\tabcellsep -37\tabcellsep -43\\
\tabcellsep \#2\tabcellsep 0\tabcellsep -22\tabcellsep -21\tabcellsep 0\tabcellsep -43\tabcellsep -36\tabcellsep 0\tabcellsep -37\tabcellsep -34\\
10\tabcellsep \#1\tabcellsep 0\tabcellsep -34\tabcellsep -45\tabcellsep 0\tabcellsep -48\tabcellsep -36\tabcellsep 0\tabcellsep -30\tabcellsep -28\\
\tabcellsep \#2\tabcellsep 0\tabcellsep -44\tabcellsep -30\tabcellsep 0\tabcellsep -35\tabcellsep -36\tabcellsep 0\tabcellsep -39\tabcellsep -26\\
0\tabcellsep \#1\tabcellsep 0\tabcellsep -32\tabcellsep -37\tabcellsep 0\tabcellsep -34\tabcellsep -40\tabcellsep \multicolumn{3}{l}{*MB and *NP are not exact}\\
\tabcellsep \#2\tabcellsep 0\tabcellsep -38\tabcellsep -45\tabcellsep 0\tabcellsep -39\tabcellsep -44\tabcellsep \multicolumn{3}{l}{*NP are not exact}\\
-10\tabcellsep \#1\tabcellsep 0\tabcellsep -34\tabcellsep -35\tabcellsep 0\tabcellsep -37\tabcellsep -39\tabcellsep \multicolumn{3}{l}{*MB and *NP are not exact}\\
\tabcellsep \#2\tabcellsep 0\tabcellsep -51\tabcellsep -48\tabcellsep 0\tabcellsep -66\tabcellsep -38\tabcellsep \multicolumn{3}{l}{*MB and *NP are not exact}\\
-20\tabcellsep \#1\tabcellsep 0\tabcellsep -41\tabcellsep -42\tabcellsep \multicolumn{3}{l}{*MB and *NP are not exact}\tabcellsep \multicolumn{3}{l}{*MB and *NP are not exact}\\
\tabcellsep \#2\tabcellsep 0\tabcellsep -50\tabcellsep -34\tabcellsep \multicolumn{3}{l}{*MB and *NP are not exact}\tabcellsep \multicolumn{3}{l}{*MB and *NP are not exact}\\
-30\tabcellsep \#1\tabcellsep 0\tabcellsep -35\tabcellsep -35\tabcellsep \multicolumn{3}{l}{*MB and *NP are not exact}\tabcellsep \multicolumn{3}{l}{*MB and *NP are not exact}\\
\tabcellsep \#2\tabcellsep 0\tabcellsep -36\tabcellsep -28\tabcellsep \multicolumn{3}{l}{*MB and *NP are not exact}\tabcellsep \multicolumn{3}{l}{*MB and *NP are not exact}\end{longtable} \par
 
\caption{\label{tab_2}Table 4 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{5} \par 
\begin{longtable}{P{0.10419354838709677\textwidth}P{0.09322580645161291\textwidth}P{0.15903225806451612\textwidth}P{0.11516129032258066\textwidth}P{0.09322580645161291\textwidth}P{0.17\textwidth}P{0.11516129032258066\textwidth}}
SNR\tabcellsep PSO\tabcellsep Scenario\#1 LMS\tabcellsep CMA\tabcellsep PSO\tabcellsep Scenario\#2 LMS\tabcellsep CMA\\
30\tabcellsep *C\tabcellsep *C\tabcellsep *C\tabcellsep *C\tabcellsep *C\tabcellsep *C\\
20\tabcellsep *C\tabcellsep *C\tabcellsep *C\tabcellsep *C\tabcellsep *C\tabcellsep *C\\
10\tabcellsep *C\tabcellsep *C\tabcellsep *C\tabcellsep *C\tabcellsep *C\tabcellsep *C\\
0\tabcellsep *C\tabcellsep *C\tabcellsep *NC\tabcellsep *C\tabcellsep *C\tabcellsep *NC\\
-10\tabcellsep *C\tabcellsep *C\tabcellsep *NC\tabcellsep *C\tabcellsep *C\tabcellsep *NC\\
-20\tabcellsep *C\tabcellsep NC\tabcellsep *NC\tabcellsep *C\tabcellsep *NC\tabcellsep *NC\\
-30\tabcellsep *C\tabcellsep NC\tabcellsep *NC\tabcellsep *C\tabcellsep *NC\tabcellsep *NC\end{longtable} \par
 
\caption{\label{tab_3}Table 5 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{6} \par 
\begin{longtable}{}
\end{longtable} \par
 
\caption{\label{tab_4}Table 6 :}\end{figure}
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