# Introduction

or the Software Defined Radio (SDR) [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 [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 [Lim Y.C., 1986] had been proposed by Smitha K.G. et. al. (2011,2008) and Mahesh R. et. al (2011,2007), which offers complexity reduction over the conventional per channel (PC) approach [Hentschel T, 2002]. In the PC approach, a separate filter is used for each communication standard. The low complexity reconfigurable architecture proposed by [Smitha K.G. et. al. (2011[Smitha K.G. et. al. ( , 2008) ) and Mahesh R. et. al (2011,2007) 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. (2011) and Mahesh R. et. al (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 [Bellanger M., 1981] is very high, we can realize the prototype filter also as an FRM filter [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. (2011) and Mahesh R. et. al (2007). By suitably combining appropriate low pass channels, we can get sharp transition-width band-pass channels also.

The filters designed using Parks-McClellan approach [Parks T. et.al, 1972, 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. 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.

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 Mercier et.al, 2007], [Yu Y J and Y.C. Lim, 2002] and [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.

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.


# II.

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  


# 2012

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)

# 2012

May
f 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
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. 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.

The FRM approach can be extended to more number of channels to form the reconfigurable nonuniform bandwidth channels [Smitha K.G. et. al., 2011, 2008and Mahesh R. et. al, 2011, 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 [Smitha K.G. et. al., 2011, 2008and Mahesh R. et. al, 2011, 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.


# Genetic Algorithm

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 Lim, 2003.

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.

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.

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 [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. .

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 (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 (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.

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 (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.


# VI.


# Design Example and Simulation Results

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. 


# May

The architecture for this non-uniform bandwidth two channel implementation is shown in Fig. 6. Using equation ( 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 (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 (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. 5 and when it was implemented using FRM method as in Fig. 6, are shown in Table 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 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 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.

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

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 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 3 : Performance comparison of CSD represented filters with maximum 3 SPT terms before and after GA optimization .

If the number of non-zero bits is reduced to 2, further reduction in complexity is obtained. But from Table 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 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.


# Complexity Comparision

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 by
f(N) = (N + 1)/2 if N is odd (N/2) + 1 If N is even
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 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.


# VIII. Band-pass Channels From Low Pass Channels

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. 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 11(a-c    


# Conclusion

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  
1![If H a (z) and H c (z) are interpolated with a factor M, H a (z M ) and H c (z M ) are obtained, whose transitionwidth is 1/M times the transition-width of H a (z) i.e. (f asf ap )/M. The filters H a (z M ) and H c (z M ) are cascaded to the masking filters H Ma (z) and H Mc (z) respectively, which suppress the unwanted images of H a (z M ) and H c (z M ). Thus, the transfer function of the overall FIR filter H(z) [Lim Y.C., 1986] is given by (2) The structure of the FRM FIR filter is given in Fig. 1 [Lim Y.C., 1986].](image-2.png "( 1 )")
1![Fig.1 : Basic FRM Filter Architecture](image-3.png "Fig. 1 :")
2![Fig.2 : Frequency response illustration of FRM approach](image-4.png "Fig. 2 :")
3![Fig.3 : Structure of two stage FRM filter III. Reconfigurable Filters Based on Frm a) Reconfigurable filters with single stage masking](image-5.png "Fig. 3 :")
![Fig 4.](image-6.png "")
45![Fig. 4 : Two-channel single-stage architecture](image-7.png "Fig. 4 :Fig. 5 :")
6![Fig. 6 : Reconfigurable non-uniform bandwidth filters with multistage masking and multistage FRM IV. Canonic Signed Digit Representation](image-8.png "Fig. 6 :")
![use of GA for the optimization of the frequency responses of the CSD represented filters is employed by Yu Y.J. and .C. Lim, 2002, Samadi P. and M. Ahmad, 2007, [Fuller A. et.al, 1999 and Uppalapati H. et.al., 2005. Genetic algorithm for the optimization of FRM filter in the SPT space is used by Patrick Mercier et.al, 2007, Yu Y J and Y.C.Lim, 2002 and Kilambi S. and B. Nowrouzian, 2006. a) Encoding of the optimization variables](image-9.png "")
![.al, 1999, Uppalapati H. et.al., 2005]. But these restoration algorithms increase the computational complexity of GA. A ternary coding based GA and a simple modified restoration algorithm is proposed by Manoj V.J. and E. Elias, 2009. We use this GA and restoration algorithm in our work. b) Objective function for the design of the CSD coefficients of filters When the filter coefficients are rounded to the Any FIR filter can be represented [Zhangwen Tang et.al, 2002] as representation of an n-bit number is floor(n/2), compared with n bits in the 2's complement representation. The number of adders/subtractors needed to realize CSD represented filter coefficient will be floor(n/2)-1.](image-10.png "")
1![Optimization of prototype filter: The basic steps in GA to optimize the filters are given in Fig 7 and are explained below [Randy L.H. and Sue E.H, 2004]. a) Initialization: The initial chromosome is generated by concatenating the first half of the continuous filter coefficients of the prototype filter. If the prototype filter is implemented by another FRM filter, the initial chromosome is generated by concatenating the first half of the continuous filter coefficients of all the sub-filters constituting the FRM prototype filter. This initial chromosome has been rounded to the nearest CSD representation with maximum number Global Journal of Researches in Engineering Volume with restricted number of SPT Max {|(|H(e j? )| -1)|, ? < ? p |H(e j? )|, ? > ? s }](image-11.png "P Phase 1 :")
7![Fig. 7 : GA design flow for filter optimization b) Fitness Evaluation of initial population : Each chromosome in the population is evaluated using the objective function given in (9) and they are ranked. c) Selection : The best chromosomes based on ranking are selected from the population and they form the mating pool. In this paper, Roulette wheel rank selection [Randy L.H. and Sue E.H, 2004] is used for selecting the mates for cross over and reproduction.d) Crossover : Offspring for the next generation are generated by exchanging the genes of parents. We use two-point crossover in our work, in which the genes are swapped between the parents between the two selected points. e) Mutation : Some of the best solutions from the current population are propagated to the next generation without any change and the remaining chromosomes are mutated with some new information and propagated to the next generation. Due to cross over and mutation, the canonic property of the filter coefficients may be violated. The canonic property is retained using a simple restoration algorithm in which when two consecutive non-zero terms come, one of them is made zero [Manoj V.J. and E. Elias 2009].](image-12.png "Fig. 7 :")
![Channel (CDMA) Pass band (PB) frequency: 1250 kHz Maximum pass band ripple: 0.1 dB Stop band (SB) frequency: 1251 kHz Minimum stop band attenuation: 40 dB Sampling Rate: 6 MHz Normalized PB frequency,f p1 = 1250/6000 = 0.2083 Normalized SB frequency,f s1 = 1251/6000 = 0.2085 Second Channel (PHS) Pass band frequency: 300 kHz Maximum pass band ripple: 0.1 dB Stop band frequency: 301 kHz Minimum stop band attenuation: 40 dB Sampling Rate: 6 MHz Normalized PB frequency, f = 300/6000 = 0.05 Normalized SB frequency, f = 301/6000 = 0](image-13.png "First")
8![Fig. 8 : Frequency response of channel 1 (M 1 = 6) and channel 2 (M 2 = 25) using infinite precision filters and CSD represented filters with maximum 3 SPT terms Next, the filters are represented using CSD with restricted number of bits. The restriction in the number of non-zero bits in the CSD representation is varied from 2 to 6 and the performance is compared. The frequency responses of the infinite precision filters and CSD represented filters with maximum three SPT terms, for channel 1 and channel 2 are shown in Fig 8a and 8b respectively.In Table2, 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 Table2, 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](image-14.png "Fig. 8 :")
![Number of Iterations = 500 Population Size = 50 Number of population members that survive each generation = 5 Mutation Rate = 0.2 Number of the best population which is kept without change during mutation (Elite Count) = 10 The frequency responses of channel 1 and channel 2 after GA optimization are given in Fig 9a and 9b respectively.](image-15.png "")
9![Fig. 9 : Frequency response of channel 1 (M 1 = 6) and channel 2 (M 2 = 25)with CSD represented filters with maximum 3 SPT terms before and after GA optimization](image-16.png "Fig. 9 :")
![2113 dB and the minimum stop band attenuation is degraded by 10.9 dB compared with the infinite precision filter responses. The different parameters used for the optimization of the prototype filter are given below: Number of Iterations = 500 Population Size = 50 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) = 5 The different parameters used for the optimization of the masking filters of channel 1 and channel 2 are given below: Number of Iterations = 500 Population Size = 50 Number of population members that survive each generation = 15 Mutation Rate = 0.1 Number of the best population which is kept without change during mutation (Elite Count) = 20 The frequency responses of channel 1 and channel 2 after GA optimization are given in Fig 10a and 10b respectively.](image-17.png "")
10![Fig. 10 : Frequency response of channel 1 (M = 6) and channel 2( M = 25)with CSD represented filters with maximum 2 SPT terms before and after GA optimization](image-18.png "Fig. 10 :")
![Pass band frequency: 900 kHz Maximum pass band ripple: 0.1 dB Stop band frequency: 901 kHz Minimum stop band attenuation: 40 dB Sampling Rate: 6 MHz Normalized PB frequency, fp2 = 900/6000 = 0.15 Normalized SB frequency, fs2 = 901/6000 = 0.1502 The band-pass responses are shown in Fig 11(d-f). The band-pass response shown in Fig. 11d is derived by combining the first and second channel low pass responses. The band-pass response shown in Fig. 11e is obtained by combining the first and third low pass channel responses. The band-pass response shown in Fig. 11f is obtained by combining the second and third channel responses. Journals Inc. (US)](image-19.png "")
11a11c![Fig.11a : First low pass channel response](image-20.png "Fig. 11a :Fig. 11c :")
1Channel 1Channel 2MethodsMax. Pass Band Ripple (dB)Min. Stop Band Attenuation (dB)Max. Pass Band Ripple (dB)Min. Stop Band Attenuation (dB)Reconfigurable two channel filters with two stage masking and direct implementation of0.21337.280.226837.98prototype filterReconfigurable two channel filters with two stage masking and FRM prototype filter0.17838.660.230639.14
2
4Channel 1Channel 2CoefficientsPeak Pass Band Ripple(dB)Minimum Stop Band Attenuation (dB)Peak Pass Band Ripple(dB)Minimum Stop Band Attenuation (dB)With Infinite precision0.17838.660.230639.14With CSD representation0.296532.07-0.441928.24With CSD representation and GA optimization0.213334.670.161334.58
5a Ma
201219Volume XII Issue v v v v VI Version ID D D D )(Global Journal of Researches in Engineering
			© 2012 Global Journals Inc. (US)
			F © 2012 Global Journals Inc. (US)
		
		
			

				


* 
	
		On computational complexity in digital filters
		
			MBellanger
		
	
	
		Proc. Of the European Conference on Circuit Theory and Design
				Of the European Conference on Circuit Theory and DesignHague, Netherlands
		
			August 1981
			
		
	




* 
	
		Optimization of FIR digital filters over the canonical Signed-digit coefficient space using genetic algorithms
		
			AFuller
		
		
			BNowrouzian
		
		
			FAshrafzadeh
		
	
	
		Proceedings of the 1998, Midwest Symposium on Circuits and Systems
				the 1998, Midwest Symposium on Circuits and Systems
		
			August 1999
			
		
	




* 
	
		Channelization for software defined base-stations
		
			THentschel
		
	
	
		Annales des Telecommunications
		
			57
			386420
			2002
		
	




* 
	
		The digital front-end of software radio terminals
		
			THentschel
		
		
			MHenker
		
		
			GFettweis
		
	
	
		IEEE Person. Commun. Magazine
		
			
			1999
		
	




* 
	
		A Personal History of the Parks McClellan Algorithm
		
			JamesHMcclellan
		
		
			ThomasWParks
		
	
	
		IEEE Signals Processing Magazine
		
			March 2005
		
	




* 
	
		A genetic algorithm employing correlative roulette selection for optimization of FRM digital filters over CSD multiplier coefficient space
		
			SKilambi
		
		
			BNowrouzian
		
	
	
		IEEE APCCAS
		
			Apr 2006
		
	
	in press




* 
	
		The optimum design of one-and two-dimensional FIR filters using the frequency response masking technique
		
			YCLim
		
		
			YLian
		
	
	
		IEEE Transactions on Circuits Systems II. Analog Digit Signal Process
		
			40
			2
			
			1993
		
	




* 
	
		Frequency-response masking approach for the synthesis of sharp linear phase digital filters
		
			YCLim
		
	
	
		IEEE Trans. Circuits. Syst
		
			33
			
			Apr. 1986
		
	




* 
	
		Vinod Reconfigurable Low Area Complexity Filter Bank Architecture Based on Frequency Response Masking for Non-uniform Channelization in Software Radio Receivers
		
			MaheshR
		
		
			AP
		
	
	
		IEEE Transactions on Aerospace and Electronic Systems
		
			47
			2
		
	




* 
	
		Frequency response masking based reconfigurable channel filters for software radio receivers
		
			MaheshR
		
		
			APVinod
		
	
	
		Proc.IEEE Int Symp. On Circuits and Systems
				.IEEE Int Symp. On Circuits and SystemsNew Orleans
		
			May 2007
			
		
	




* 
	
		Design of multiplierless nonuniform filter bank transmultiplexer using genetic algorithm
		
			VJManoj
		
		
			EElias
		
	
	
		Signal Processing
		
			89
			11
			
			2009
		
	




* 
	
		Design of nonuniform filter bank transmultiplexer with canonic signed digit filter coefficients
		
			VJManoj
		
		
			EElias
		
	
	
		IET Signal Processing
		
			3
			3
			
			2009
		
	




* 
	
		Software radio architecture
		
			JMitola
		
		
			2000
			John Wiley
		
	




* 
	
		Cheyshev approximation for nonrecursive digital filters with linear phase
		
			TParks
		
		
			JMcclellan
		
	
	
		IEEE Trans. on Circuit Theory
		
			19
			
			1972
		
	




* 
	
		Sai Mohan Kilambi, and Behrouz Nowrouzian Optimization of FRM FIR Digital Filters Over CSD and CDBNS Multiplier Coefficient Spaces Employing a Novel Genetic Algorithm
		
			PatrickMercier
		
	
	
		Journal of Computers
		
			2
			7
			September 2007
		
	




* 
	
		
			LHRandy
		
		
			EHSue
		
		Practical genetic algorithms
				
			John Wiley and Sons, Inc., Publication
			2004
		
	
	2nd edn




* 
	
		Canonical Signed Representation for FIR Digital Filters
		
			ReidMHewitt
		
		
			EarlSSwartzlander
		
		
			Jr
		
	
	
		IEEE Transactions on Signal Processing Systems
		
			
			2000
		
	




* 
	
		Performance analysis of genetic algorithm for the design of linear phase digital filter banks with CSD coefficients
		
			PSamadi
		
		
			MAhmad
		
	
	
		Proceedings of the Third IEEE International Conference on Natural Computation
				the Third IEEE International Conference on Natural Computation
		
			August 2007
		
	




* 
	
		A Low Complexity Reconfigurable Multi-stage Channel Filter Architecture for Resource-Constrained Software Radio Handsets
		
			KG A PSmitha
		
		
			Vinod
		
	
	
		J Sign Process Systems
		
			
			2011
		
	




* 
	
		Vinod A Reconfigurable Multi-stage Frequency Response Masking Filter Bank Architecture for Software Defined Radio Receivers
		
			KGSmitha
		
		
			RMahesh
		
		
			AP
		
	
	
		IEEE International Symposium on Circuits and Systems
				
			2008
			
		
	




* 
	
		Use of the Remez Algorithm for Designing FRM Based FIR
		
			TapioSaramaki
		
		
			YongChingLim
		
		
	




* 
	
		Signed power-of-two term allocation scheme for the design of digital filters
		
			DLYong Ching Lim
		
		
			RuiYang
		
		
			JSong
		
	
	
		IEEE Transactions on Circuits and Systems -Analog and Digital Signal Processing
				
			May 1999
			II
			
		
	




* 
	
		Genetic algorithm approach for the optimization of multiplier less sub-filter generated by the frequency response masking technique Process
		
			Y JYu
		
		
			YCLim
		
	
	
		IEEE International conference in electronics, circuits and systems
				
			2002
			3
			
		
	




* 
	
		A novel genetic algorithm for the design of a signed power-of-two coefficient quadrature mirror filter lattice filter bank
		
			YJYu
		
		
			YCLim
		
	
	
		Circuit Syst. Signal Process.
		
			21
			3
			
			2002
		
	




* 
	
		
		
			ZhangwenTang
		
		
			JieZhang
		
		
			HaoMin
		
		
	
	A high Speed, programmable, CSD coefficient FIR filter