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\title{Design of Low Power and Low Complexity Multiplier-less Reconfigurable Non-uniform Channel filters using Genetic Algorithm}
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\begin{document}

             \author[1]{V. Satish  Kumar}

             \author[2]{Elizabeth  Elias}

             \author[3]{Bindiya  T.S.}

             \affil[1]{  National Institute of Technology, Calicut, Kerala, India.}

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\date{\small \em Received: 5 February 2012 Accepted: 1 March 2012 Published: 15 March 2012}

\maketitle


\begin{abstract}
        


The key requirements of the channel filter in a software defined radio receiver are low power, low complexity and reconfigurability of the architecture used. An architecture based on frequency response masking (FRM) technique is recently reported which offers, reconfigurability at the filter and architecture levels, in addition to the inherent low complexity offered by the FRM technique. In this paper, we propose a modified architecture to reduce the overall complexity by realizing the prototype filter in the FRM technique by another FRM filter. The hardware implementation of the filter calls for the representation of the filter coefficients in the signed power of two (SPT) space. It is well known that if canonic signed digit (CSD) representation is employed in the SPT space, the hardware complexity can further be significantly reduced. Hence it is proposed in this paper to extend the CSD representation to the FRM based digital filters. The design of the FRM filter in the discrete space degrades the performance and this calls for the use of efficient non-linear optimization techniques. We use genetic algorithm (GA) based optimization which brings forth a near optimal solution. This results in very low power and low complexity FRM based multiplier-less reconfigurable non-uniform channel filters.

\end{abstract}


\keywords{Software Defined Radio, frequency response masking (FRM), Reconfigurability, canonic signed digit (CSD), genetic algorithm (GA), signed power of two (}

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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
or the Software Defined Radio (SDR) \hyperref[b12]{[Mitola J., 2000]}, we require a technology which will cater to multi-band, multi-standard and multi-service. This requires systems which are reconfigurable and reprogrammable by software. From the modulated input signal, depending on the communication standards, sub-bands with non-uniform bandwidths are to be extracted by the channel filters in the channelizer in a SDR receiver  {\ref [Hentschel T et.al, 1999]}. The channelizer operates at the highest sampling rate and hence requires low power and low complexity filters. Reconfigurability is an important requirement in an SDR handset. To facilitate this, a reconfigurable structure based on FRM  {\ref [Lim Y.C., 1986]} had been proposed by Smitha K.G. et. al.  {\ref (2011,} {\ref 2008)} and Mahesh R. et. al  {\ref (2011,} {\ref 2007)}, which offers complexity reduction over the conventional per channel (PC) approach \hyperref[b2]{[Hentschel T, 2002]}. In the PC approach, a separate filter is used for each communication standard. The low complexity reconfigurable architecture proposed by \hyperref[b18]{[Smitha K.G. et. al. (2011}\hyperref[b19]{[Smitha K.G. et. al. ( , 2008) )} and Mahesh R. et. al  {\ref (2011,} {\ref 2007)} is to realize channels with non-uniform bandwidths and is based on the frequency response masking (FRM) method. The same prototype filter but separate interpolation factors and masking filters are used to derive different non-uniform bandwidth channels from the wideband input. So this is a very useful method to get non-uniform sharp transition-width channels and can be extended to realize sharp non-uniform filter banks with low complexity. The complexity of these reconfigurable filter banks can still be reduced by employing multistage masking instead of single stage masking whenever the order of the masking filters is high as proposed by Smitha K.G. et. al.  {\ref (2011)} and Mahesh R. et. al  {\ref (2007)}. The problem with this method is that a high order prototype filter also increases the overall complexity. To address this problem, we use a modified architecture in which, if the order of the prototype filter estimated using Bellanger's equation \hyperref[b0]{[Bellanger M., 1981]} is very high, we can realize the prototype filter also as an FRM filter  {\ref [Lim Y.C., 1986]}. Design example shows that this method offers 98.77\% complexity reduction when compared to the PC approach and 53.01\% complexity reduction when compared to the approach proposed by Smitha K.G. et. al.  {\ref (2011)} and Mahesh R. et. al  {\ref (2007)}. By suitably combining appropriate low pass channels, we can get sharp transition-width band-pass channels also.\par
The filters designed using Parks-McClellan approach  {\ref [Parks T. et.al, 1972} {\ref , James H. et.al, 2005]} result in filter coefficients with infinite precision. But for hardware implementation, they have to be represented using finite number of bits. The signed power-of-two (SPT) system [Yong Ching Lim D.L. et.al, 1999] is a representation in which the multiplier coefficients are represented using few non-zero bits. Thus the multiplication operation can be done using a series of shift and add operations. FIR filters with SPT representation is named as multiplier-less digital filters. Canonic Signed Digit (CSD) representation [Reid M. \hyperref[b16]{Hewitt et. al, 2000}] is a special case of the SPT system. If we restrict the number of non-zero bits in the representation of the filter coefficients, the number of non-zero partial products needed to realize the filter will be reduced and hence the switching activity. Since the F power consumption is directly proportional to the switching activity, power consumption also reduces.\par
But there will be degradation in the magnitude response of the filter due to the restricted number of SPT terms in the representation of the filter coefficients. Degradation in the frequency response due to the truncated FIR filter coefficients can be improved using suitable optimization techniques resulting in low power and low complexity FRM based multiplier-less reconfigurable non-uniform channels. Genetic algorithm for optimizing the CSD represented FRM filter is reported in [Patrick  {\ref Mercier et.al, 2007]}, [Yu Y J and Y.C. \hyperref[b23]{Lim, 2002]} and \hyperref[b5]{[Kilambi S. and B. Nowrouzian, 2006]}. In this paper, we use ternary coded genetic algorithm [Manoj V.J. and E.  for improving the performance of the multiple channel CSD represented filter architecture.\par
The paper is organized as follows. Section II gives an overview of the single stage and multi stage FRM method. Section III gives an insight into the reconfigurable architecture based on FRM. Section IV explains the CSD technique. Section V gives an overview about genetic algorithm based optimization. A design example and MATLAB simulation results are presented in section VI. Section VII compares the number of multipliers in the proposed method with the existing methods. Section VIII illustrates the method of obtaining band pass channels from low pass channels. Section IX concludes the paper. 
\section[{II.}]{II.}\par
Overview of Frm Approach a) Single Stage FRM Let H(z) be the transfer function of the desired FIR low pass filter with pass band and stop band edge frequencies f p and f s respectively. In the FRM technique, the overall sharp transition width filter is composed of many sub-filters of wide transition width. If H a (z) represents the transfer function of a low pass linear phase filter, its complementary filter H c (z) can be expressed as given below   
\section[{2012}]{2012}\par
May  The masking filters for each channel can be designed using the equations given in (4). The structure of the two channel reconfigurable filters is given in  H c (z) = z -(N-1)/2 -H a (z) H(z) = H a (z M ) H Ma (z) + H c (z M ) H Mc (z) 
\section[{2012}]{2012}\par
Mayf ap = f p1 M 1 -f p1 M 1 = f p2 M 2 -f p2 M 2 f as = f s1 M 1 -f s1 M 1 = f s2 M 2 -f s2 M 2\par
In the architecture given in section IIIB, the problem with higher order prototype filter is not addressed. If the order of the prototype filter is high, the overall complexity of the multiple channel filter structure also will be high. In the proposed method, we modify the architecture in section IIIB by implementing the prototype filter using FRM as discussed in section IIB. This is shown in Fig. \hyperref[fig_6]{6}. Using this method, sharp transition-width channels with very low complexity can be obtained. Also since the number of filters is increased, we get improved pass band and stop band characteristics. By selecting appropriate low pass channels for subtraction, sharp transition-width bandpass channels with very low complexity can also be obtained. This is illustrated in section VIII.\par
The FRM approach can be extended to more number of channels to form the reconfigurable nonuniform bandwidth channels \hyperref[b18]{[Smitha K.G. et. al., 2011}\hyperref[b19]{, 2008} {\ref and Mahesh R. et. al, 2011}\hyperref[b14]{, 2007]}. In this technique, the reconfigurability is achieved using the same prototype filter for all the channels. Different interpolation factors and masking filters are used to derive the different channels. For example, suppose we need two channels with pass band frequencies f p1 and f p2 and stop band frequencies f s1 and f s2 respectively. Let f ap and f as be the pass band and stop band frequencies respectively of the prototype filter which is the same for both channels. The prototype filter with specifications f ap and f as and interpolation factors M 1 and M 2 for channels 1 and 2 respectively can be designed by iterating the equation given below for different values of M 1 and M 2 \hyperref[b18]{[Smitha K.G. et. al., 2011}\hyperref[b19]{, 2008} {\ref and Mahesh R. et. al, 2011}\hyperref[b14]{, 2007]}. Where c i = \{-1, 1, 0\} and W is the word length of the CSD number. Since, this encoding uses -1, 0 and 1 digits, it is called ternary coding. No adjacent digits in the CSD representation can be non-zero i.e. c i *c i-1 = 0, where c i is the i th digit in the CSD representation. The maximum number of non-zero digits in the CSD V. 
\section[{Genetic Algorithm}]{Genetic Algorithm}\par
Several optimization methods are proposed for the optimization of infinite precision FRM filter in which either separate optimization of sub-filters is done or joint optimization of the filters is done. The sub-filters designed using linear programming in the paper [Lim Y. C., 1986] reduced the error in the pass band and stop band of the overall FRM filter. Another approach to reduce the error in the pass band and stop band of the overall FRM filter is to design the sub-filters using Remez algorithm as proposed by Tapio Saramaki and Yong Ching  {\ref Lim, 2003}.\par
The optimization of the FRM filter in the discrete space is a complicated process and so efficient nonlinear optimization techniques need to be used. The classical gradient based optimization techniques cannot be directly applied to this problem because, here the search space consists of integers. In this context, metaheuristic algorithm is a good optimization tool as the proper selection of the parameters with respect to a particular design problem can bring forth global solution.\par
Genetic algorithms (GA) have been established as a good alternative for the optimization of multimodal, multi dimensional problems. This is a population based evolutionary algorithm where, in each iteration, candidate solutions are generated using genetic operations like reproduction, crossover and mutation. The Once the infinite precision filters are designed, the coefficients of these filters have to be converted to the CSD representation. As we have discussed in section IV, one way to encode filters in the CSD space is to represent them using ternary coding as proposed by Manoj V.J. and E. Elias, 2009. A look up table is created which has four fields. The four fields are index, CSD numbers, decimal equivalents and the number of SPT terms. In our work, the maximum allowed precision is 12 bits.\par
When crossover and mutation operations are performed on these ternary coded coefficients, the canonical property of the CSD representation of the coefficients may be lost. To ensure the canonical property of the filter coefficient representation, many restoration algorithms are proposed  {\ref [Fuller A. et} terms, the stop band and pass band properties get degraded. So the filter coefficients in the CSD format have to be optimized to improve the pass band and stop band characteristics. In this paper, we use genetic algorithm based optimization technique to improve these characteristics. In our design, the CSD filter coefficients can use any number of SPT terms. But the total number of SPT terms used is restricted. This method gives more flexibility over restricting each coefficient with fixed number of SPT terms [Manoj V.J. and E. .\par
The pass band and stop band characteristics are taken care of by minimizing the pass band ripple and maximizing the stop band attenuation. So, the objective function used in our work is given by Minimize \hyperref[b8]{(9)} p and s are the pass band and stop band cut off frequencies respectively. So the objective function is to optimize the above values under the constraint that the total number of SPT terms used is restricted. c) Optimization of multiple channel filter architecture using ternary coded GA Since all the filters in our architecture have linear phase property, only the first half of the CSD represented filter coefficients are optimized and the other half of the filter coefficients are obtained using linear phase property. The genetic algorithms for optimizing the multiple channel filter architecture have two phases. Since we have multiple channels in our architecture, each channel has to be optimized separately. Also, since all the channels are derived from the same prototype filter outputs, we have to separately optimize the masking filters of each channel and the prototype filter. So, in the first phase, the prototype filter is optimized and in the second phase the masking filters of each channel are optimized. of non-zero bits, which is chosen as 6 in this work. By changing the initial chromosome by random perturbations, a population pool of N-1 chromosomes is generated, where N is the population size. The coefficients of all these N-1 filters are then converted to the CSD representation with restricted number of nonzero bits. These N-1 perturbed chromosomes along with the initial non-perturbed chromosome will make the initial population of size N. f) Fitness Evaluation of New Population: Each chromosome in the new population is evaluated using the objective function given in \hyperref[b8]{(9)} and they are ranked. The steps from c to f are repeated until a maximum number of iterations are reached and then GA is terminated. When it is terminated, the best chromosome is taken from the population and is decoded to get the optimum CSD represented filter. The above steps have to be repeated for the prototype filter and masking filters separately.\par
Phase 2: Optimization of masking filters of each channel: After the prototype filter is optimized, the masking filters of each channel need to be optimized separately using the same objective function given in \hyperref[b8]{(9)}. In this phase, we adopt the joint optimization of the masking filters in which the initial chromosome is generated by concatenating the first half of the continuous filter coefficients of all the masking filters of a channel. The steps of GA algorithm are repeated until the maximum numbers of iterations are reached. This same optimization process has to be repeated for the masking filters of all the channels. 
\section[{VI.}]{VI.} 
\section[{Design Example and Simulation Results}]{Design Example and Simulation Results}\par
The design example shown here compares the proposed method discussed in section IIIC with the methods discussed in section IIIA and IIIB. In this example, two channels of non-uniform bandwidth are designed for CDMA (Code Division Multiple Access) and PHS (Personal Handy Phone System) standards. The specifications for each channel are given below.  
\section[{May}]{May}\par
The architecture for this non-uniform bandwidth two channel implementation is shown in Fig. \hyperref[fig_6]{6}. Using equation (  {\ref 6}) and iterating for different values of M 1 and M 2 , we obtain M 1 = 6, M 2 = 25, f ap = 0.25 and f as = 0.255. When these values are substituted in Bellanger's equation given in \hyperref[b4]{(5)}, the length of the prototype filter is obtained as 391 and the number of multipliers needed is 196. Since the length of the prototype filter is high, it is implemented as FRM filter and the number of multipliers needed for FRM implementation is then found to be 60. The CDMA filter response is implemented using single stage masking since its interpolation factor is small (M 1 = 6). For PHS filter response, the interpolation factor is 25 and can be factorized into 5 and 5. So, PHS filter is implemented using two stages masking with interpolation factor 5 in each stage. The length of the masking and masking complementary filters H Ma1 and H Mc1 of channel 1 is 24 and 23 respectively. The length of the first stage masking and masking complementary filters H Ma21 and H Mc21 of channel 2 is 20 and 19 respectively. The length of the second stage masking filter H Ma22 of channel 2 is 20. The total number of multipliers needed for the proposed method is 117 compared to the method proposed by Smitha K.G. et. al. (2011) and Mahesh R. et. al  {\ref (2007)} where 249 multipliers are needed. Thus, there is a 53.01\% reduction in the number of multipliers. The peak pass band ripple and minimum stop band attenuation obtained for channel 1 and channel 2 when the prototype filter was implemented directly using Parks -McClellan method as in Fig.  {\ref 5} and when it was implemented using FRM method as in Fig. \hyperref[fig_6]{6}, are shown in Table \hyperref[tab_0]{1}. From this table, it is seen that when the prototype filter was implemented using FRM method, the stop band characteristics are also improved compared to the direct implementation using Parks-McClellan method.  In Table \hyperref[tab_2]{2}, the peak pass band ripple and minimum stop band attenuation of the 2 channels for the infinite precision filters and CSD represented filters using different number of non-zero bits are shown. From Table \hyperref[tab_2]{2}, it can be seen that when the number of nonzero bits for the CSD representation is restricted to 3, for the first channel, the peak pass band ripple is increased by 0.0215 dB and minimum stop band attenuation is degraded by 3.22 dB. For the second channel, the stop band attenuation is degraded by 1.33 dB compared to the infinite precision filter response. So we have to optimize the CSD represented filter responses to get better responses for channel 1 and 2. Here, we have employed GA optimization to improve the pass band   and stop band responses.\par
For the multiple channel filter architecture, optimization includes two phases. In the first phase, the coefficients of the prototype filter are optimized which is implemented as an FRM filter. The FRM prototype filter is made up of sub-filters and each of these continuous filter coefficients has linear phase characteristics. So for optimizing the coefficients of a prototype filter, the genes corresponding to the first 30 genes of the FRM sub-filter, the first 16 genes of the masking filter and the first 16 genes of the masking complementary filter are concatenated, to generate the chromosomes in the initial population. The maximum allowed number of SPT terms is taken as 186 so that the average number of SPT terms in a filter coefficient is 3. The different parameters used for the optimization of the prototype filter are given below: Number of population members that survive each generation = 1 Mutation Rate = 0.02 Number of the best population which is kept without change during mutation (Elite Count) = 10\par
In the second phase, the coefficients of the masking filters of each channel are optimized. Each of these continuous filter coefficients has linear phase characteristics. Therefore, for optimizing the coefficients of the masking filters of channel 1, the genes corresponding to the first 12 genes of the masking filter and the first 12 genes of the masking complementary filter are concatenated to generate the chromosomes in the initial population. The maximum allowed number of SPT terms is taken as 72 so that the average number of SPT in a filter coefficient is 3. Similarly for optimizing the coefficients of the masking filters of channel 2, the genes corresponding to the first 10 genes of the first stage masking filter, the first 10 genes of the first stage masking complementary filter and the first 10 genes of the second stage masking filter are concatenated to generate the chromosomes in the initial population. The maximum allowed number of SPT terms are taken as 90 so that the average number of SPT terms in a filter coefficient is 3. The different parameters used for the optimization of the masking filters of both channels are given below: The peak pass band ripple and minimum stop band attenuation obtained for channel 1 and channel 2 outputs after GA optimization of the prototype filter and the masking filters with maximum 3 SPT terms are given in Table  {\ref 3}. From the table it is clear that, when we employed optimization, the pass band responses of channel 1 and channel 2 are improved by 0.0233 dB and 0.0723 dB respectively and the stop band responses of channel 1 and channel 2 are improved by 4.92 dB and 0.83 dB respectively.   Table  {\ref 3} : Performance comparison of CSD represented filters with maximum 3 SPT terms before and after GA optimization .\par
If the number of non-zero bits is reduced to 2, further reduction in complexity is obtained. But from Table \hyperref[tab_2]{2}, it can be seen that when the number of nonzero bits for the CSD representation is restricted to 2, the peak pass band ripple is increased by 0.1185 dB and minimum stop band attenuation is degraded by 6.59 dB for the first channel. For the second channel, the peak pass band ripple is increased by 0. The peak pass band ripple and minimum stop band attenuation achieved for channel 1 and channel 2 outputs after GA optimization of the prototype filter and masking filters are given in Table \hyperref[tab_4]{4}. From the table it is clear that, the pass band responses of channel 1 and channel 2 are improved by 0.0832 dB and 0.2806 dB respectively and stop band response of channel 1 and channel 2 are improved by 2.6 dB and 6.34 dB respectively, after optimization.   VII. 
\section[{Complexity Comparision}]{Complexity Comparision}\par
Here, we compare the number of multipliers used to design channel 1 and channel 2 filters. The number of multipliers used to design any filter is given byf(N) = (N + 1)/2 if N is odd (N/2) + 1 If N is even\par
Where f(N) denotes the total number of multipliers needed for the implementation of an FIR filter with order N. If N is the order of the prototype filter H a (z), N is the order of the masking filter H Ma (z), and N mc is the order of the masking complementary filter H Mc (z), the number of multiplications, implement the overall FIR filter is given by a ) + f(N ma ) + f(N mb ) In Table \hyperref[tab_6]{5}, the complexity comparison of the proposed method with the existing method in terms of multipliers is shown. We can see that the proposed method offers a 53.01\% reduction over the method discussed in section IIIB. 
\section[{VIII. Band-pass Channels From Low Pass Channels}]{VIII. Band-pass Channels From Low Pass Channels}\par
In this section, the method of extracting band pass channels from the designed low pass channels is shown. To illustrate this, we have modified the structure shown in Fig. \hyperref[fig_6]{6} to a three channel structure by adding one more channel with the following specifications. The frequency responses of all the three low pass channels are given in Fig \hyperref[fig_10]{11(a-c}     
\section[{Conclusion}]{Conclusion}\par
The existing architecture with low power, low complexity and reconfigurability for software defined radio is modified in this paper. The prototype filter in the FRM structure is replaced by another FRM structure. The filter coefficients are represented in the signed power of two spaces, where CSD representation is employed. Thus we get multiple channel architecture with very low complexity and low power which is ready for hardware implementation. Since the design of the filter in the discrete space degrades the performance, the response is optimized by a modified genetic algorithm which results in near optimal solutions. This leads to the implementation of low power, low  \begin{figure}[htbp]
\noindent\textbf{1}\includegraphics[]{image-2.png}
\caption{\label{fig_0}( 1 )}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{1}\includegraphics[]{image-3.png}
\caption{\label{fig_1}Fig. 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2}\includegraphics[]{image-4.png}
\caption{\label{fig_2}Fig. 2 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{3}\includegraphics[]{image-5.png}
\caption{\label{fig_3}Fig. 3 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-6.png}
\caption{\label{fig_4}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{45}\includegraphics[]{image-7.png}
\caption{\label{fig_5}Fig. 4 :Fig. 5 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{6}\includegraphics[]{image-8.png}
\caption{\label{fig_6}Fig. 6 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-9.png}
\caption{\label{fig_8}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-10.png}
\caption{\label{fig_9}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{1}\includegraphics[]{image-11.png}
\caption{\label{fig_10}P Phase 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{7}\includegraphics[]{image-12.png}
\caption{\label{fig_11}Fig. 7 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-13.png}
\caption{\label{fig_12}First}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{8}\includegraphics[]{image-14.png}
\caption{\label{fig_13}Fig. 8 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-15.png}
\caption{\label{fig_15}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{9}\includegraphics[]{image-16.png}
\caption{\label{fig_16}Fig. 9 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-17.png}
\caption{\label{fig_18}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{10}\includegraphics[]{image-18.png}
\caption{\label{fig_19}Fig. 10 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-19.png}
\caption{\label{fig_21}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{11a11c}\includegraphics[]{image-20.png}
\caption{\label{fig_22}Fig. 11a :Fig. 11c :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{1} \par 
\begin{longtable}{P{0.44479452054794516\textwidth}P{0.10479452054794519\textwidth}P{0.09547945205479451\textwidth}P{0.10945205479452054\textwidth}P{0.09547945205479451\textwidth}}
\tabcellsep \multicolumn{2}{l}{Channel 1}\tabcellsep \multicolumn{2}{l}{Channel 2}\\
Methods\tabcellsep Max. Pass Band Ripple (dB)\tabcellsep Min. Stop Band Attenuation (dB)\tabcellsep Max. Pass Band Ripple (dB)\tabcellsep Min. Stop Band Attenuation (dB)\\
Reconfigurable two channel filters with two stage masking and direct implementation of\tabcellsep 0.213\tabcellsep 37.28\tabcellsep 0.2268\tabcellsep 37.98\\
prototype filter\tabcellsep \tabcellsep \tabcellsep \tabcellsep \\
Reconfigurable two channel filters with two stage masking and FRM prototype filter\tabcellsep 0.178\tabcellsep 38.66\tabcellsep 0.2306\tabcellsep 39.14\end{longtable} \par
 
\caption{\label{tab_0}Table 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2} \par 
\begin{longtable}{}
\end{longtable} \par
 
\caption{\label{tab_2}Table 2 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{4} \par 
\begin{longtable}{P{0.2833333333333333\textwidth}P{0.14306930693069309\textwidth}P{0.13745874587458745\textwidth}P{0.1486798679867987\textwidth}P{0.13745874587458745\textwidth}}
\tabcellsep \multicolumn{2}{l}{Channel 1}\tabcellsep \multicolumn{2}{l}{Channel 2}\\
Coefficients\tabcellsep Peak Pass Band Ripple(dB)\tabcellsep Minimum Stop Band Attenuation (dB)\tabcellsep Peak Pass Band Ripple(dB)\tabcellsep Minimum Stop Band Attenuation (dB)\\
With Infinite precision\tabcellsep 0.178\tabcellsep 38.66\tabcellsep 0.2306\tabcellsep 39.14\\
With CSD representation\tabcellsep 0.2965\tabcellsep 32.07\tabcellsep -0.4419\tabcellsep 28.24\\
With CSD representation and GA optimization\tabcellsep 0.2133\tabcellsep 34.67\tabcellsep 0.1613\tabcellsep 34.58\end{longtable} \par
 
\caption{\label{tab_4}Table 4 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{5} \par 
\begin{longtable}{}
\end{longtable} \par
  {\small\itshape [Note: a Ma]} 
\caption{\label{tab_6}Table 5 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{} \par 
\begin{longtable}{P{0.85\textwidth}}
2012\\
19\\
Volume XII Issue v v v v VI Version I\\
D D D D )\\
(\\
Global Journal of Researches in Engineering\end{longtable} \par
 
\caption{\label{tab_7}May}\end{figure}
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