ARTICLE IN PRESS

Engineering Science and Technology, an International Journal ■■ (2015) I

Contents lists available at ScienceDirect

Engineering Science and Technology, an International Journal

journal homepage: http://www.elsevier.com/locate/jestch

Full Length Article

Design of multiplier-less sharp non-uniform cosine modulated filter banks for efficient channelizers in software defined radio

Shaeen Kalathil *, Elizabeth Elias

Department of Electronics and Communication, National Institute of Technology Calicut, Kerala, India

ARTICLE INFO

ABSTRACT

Article history: Received 13 April 2015 Received in revised form 6 June 2015 Accepted 30 June 2015 Available online

Keywords: Cosine modulation Non-uniform filter banks Frequency response masking approach Canonic signed digit representation Meta-heuristic algorithms Software defined radio Channelizer

Forthcoming software defined radios require filter banks which satisfy stringent specifications efficiently with low implementation complexity. Cosine modulated filter banks (CMFB) have simple and efficient design procedure. The different wireless standards have different channel spacing or bandwidths and hence demand non-uniform decomposition of subbands. The non-uniform CMFB can be obtained from a uniform CMFB in a simple and efficient approach by merging the adjacent channels of the uniform CMFB. Very narrow transition width filters with low complexity can be achieved using frequency response masking (FRM) filter as prototype filter. The complexity is further reduced by the multiplier-less realization of filter banks in which the least number of signed power of two (SPT) terms is achieved by representing the filter coefficients using canonic signed digit (CSD) representation and then optimizing using suitable modified meta-heuristic algorithms. Hybrid meta-heuristic algorithms are used in this paper. A hybrid algorithm combines the qualities of two meta-heuristic algorithms and results in improved performances with low implementation complexity. Highly frequency selective filter banks characterized by small passband ripple, narrow transition width and high stopband attenuation with non-uniform decomposition of subbands can be designed with least the implementation complexity, using this approach. A digital channelizer can be designed for SDR implementations, using the proposed approach. In this paper, the non-uniform CMFB is designed for various existing wireless standards.

Copyright © 2015 The Authors. Production and hosting by Elsevier B.V. on behalf of Karabuk University.

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Digital filter banks are largely used in different applications such as compression of speech, image, video and audio data, transmultiplexers, multi carrier modulators, adaptive and biosignal processing [1]. Filter banks decompose the spectrum of a given signal into different subbands and each subband is associated with a specific frequency interval. Most of the applications of filter banks demand good frequency response characteristics with reduced implementation complexity. The frequency selectivity of individual filters is due to small passband ripple, high stopband attenuation and narrow transition width. If the specifications are very stringent or a good frequency selective filter is required, then the complexity of the conventional FIR filter will be very high, since the order of the FIR filter is inversely proportional to the transition width. Frequency response masking (FRM) is a cost efficient way for the design of FIR filters with narrow transition width [2]. The upcoming software

* Corresponding author. Tel.: +91 9447100244, fax: +91 4952287250. E-mail address: shaeen_k@yahoo.com (S. Kalathil). Peer review under responsibility of Karabuk University.

defined radios (SDR) require efficient digital channelizers, which will select individual channels from the digitized wideband signal. Extremely frequency selective filter banks with excellent design flexibility and low implementation complexity are highly appreciated in SDR Channelizers. Hence the filter bank proposed in this paper is an outstanding choice for digital channelizer.

The digital channelizers in SDRs select individual channels from the digitized wideband signal. In order to extract equal bandwidth signals, polyphase DFT filter banks are efficiently used [3]. But when the individual subbands are from different communication standards, uniform filter banks will not be a feasible option. Different non-uniform filter banks are proposed, using tree structured filter banks, multi-stage filter banks, variable bandwidth filter banks and using FIR prototype in non-uniform CMFB [4-7]. The stringent specifications of wireless communication standards require filter banks with narrow transition widths and the least implementation complexity.

In perfect reconstruction (PR) filter banks, the output will be a weighted delayed replica of the input. In case of near perfect reconstruction (NPR) filter banks, a tolerable amount of aliasing and amplitude distortion errors are permitted. Cosine modulated filter banks are one popular class among the different M-Channel

http://dx.doi.org/10.1016/jjestch.2015.06.003

2215-0986/Copyright © 2015 The Authors. Production and hosting by Elsevier B.V. on behalf of Karabuk University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

ARTICLE IN PRESS

S. Kalathil, E. Elias/Engineering Science and Technology, an International Journal ■■ (2015) I

maximally decimated filter banks [1]. Design of NPR CMFB is easier and less time consuming compared to the corresponding PR CMFB. Even though small amounts of aliasing and amplitude distortion errors exist, these filter banks are widely used in different applications due to the design ease [1]. Also, it is difficult to attain high stopband attenuation with PR CMFB. Hence, as a compromise, NPR structures can be preferred in those applications, where some amount of aliasing can be tolerated. NPR CMFBs with narrow transition widths and good stopband attenuation and small passband ripple give very small amplitude distortion and aliasing distortion errors, which are comparable to that of PR CMFB.

In uniform filter banks, the spectrum is decomposed into equal bands. In many applications a non-uniform decomposition of subbands is preferred. The input signal is decomposed into signals having different bandwidths. A simple and efficient design of nonuniform filter bank is by the cosine modulation of the prototype filter and then merging appropriate filters of the resulting uniform filter bank [8]. The non-uniform CMFB design is derived from a uniform CMFB. Hence the attractive properties of a uniform CMFB are retained in the non-uniform CMFB. This is an easy and efficient design method. The prototype filter alone is required to be designed and optimized. All the other analysis and synthesis filters with unequal bandwidths are obtained from this filter, by merging the appropriate filters of the uniform filter bank. The prototype filter is designed using non-linear optimization in Reference 8. A modified approach, in which the prototype filter is designed using linear search technique is given in Reference 9. Multiplier-less design of non-uniform CMFB using different prototype filter design techniques for FIR filter such as Window method, weighted Chebyshev approach and weighted constrained least square is given in Reference 10.

NPR uniform CMFB design using FRM prototype filter has been proposed by Furtado et al. in References 11 and 12. The design involved non-linear optimization and their main aim was to reduce the number of optimization variables. This was achieved since the FRM prototype filter contains less number of distinct coefficients compared to the conventional FIR filter. NPR CMFB using a different class of prototype filter was proposed by Rosenbaum et al. [13], where the prototype filter is a non-linear phase filter with increased overall delay. Their main aim was to reduce the arithmetic complexity by replacing two cosine modulation blocks with a single sine modulation block. This design also involves non-linear optimization. A multiplier-less design of NPR uniform CMFB with sharp transition band using modified meta-heuristic algorithms is proposed in Reference 14.

Multipliers are the most expensive components for implementing a digital filter in hardware. The multipliers in filters can be implemented using shifters and adders, if the coefficients are represented by signed power of two (SPT) terms. Canonic signed digit (CSD) representation is a particular subset of SPT representation [15]. It contains minimum number of SPT terms and the adjacent digits will never be both non-zeros. As a result, efficient implementation of the multipliers using shifters/adders is possible [15]. But CSD representation of the coefficients may lead to deterioration of the filter performances. Hence suitable optimization techniques have to be deployed to improve the performances.

In this paper an approach for the design of multiplier-less NPR non-uniform CMFB is given, in which the prototype filter is designed using FRM approach. The continuous coefficient design of the FRM prototype filter does not require any non-linear optimization. The edge frequencies are iteratively adjusted to satisfy the 3-dB condition as given in Reference 16. The sub-filters in the FRM prototype filter are designed using weighted Chebyshev approximation. A highly frequency selective filter bank having nonuniform decomposition with less implementation complexity is designed. The coefficients are quantized using canonic signed digit

H0(z) —► «VI

M —* F0(z)

H^z) —» IM

1M -» Fj(z)

HM-i(z) -----, flvi Fm-i(z)

Fig. 1. M channel maximally decimated filter bank.

(CSD) representation. The finite precision performances of the filter bank in the minimal SPT space can be made at par with those with continuous coefficients. To improve the frequency response characteristics of the filters, optimization in the discrete domain is required. Conventional gradient based approaches cannot be deployed here, as the search space is discrete. Meta-heuristic algorithm is a proper choice for such problems and it is reported [17] to result in global solutions by properly tuning the parameters. Hybrid meta-heuristic algorithm, which combines the best qualities of two algorithms, further improves the performance with less number of adders. Multiplier-less design of NPR non-uniform CMFB with FRM filter as the prototype filter and the coefficients synthesized in the CSD form using modified meta-heuristic algorithms are hitherto not reported in the literature. The filter bank proposed in this paper is a suitable choice for digital channelizers in SDRs.

The remaining part of the paper is organized as follows: Section 2 gives an introduction of NPR CMFB. Section 3 briefly illustrates the design of non-uniform NPR CMFB. Section 4 gives a brief description of the frequency response masking approach. Section 5 explains the design of the proposed continuous coefficient CMFB. Different design examples and comparison with conventional FIR filter are also given. Section 6 outlines the design and optimization of the CSD coefficient filter bank using various modified meta-heuristic algorithms. Section 7 explores the applicability of the proposed filter banks as efficient channelizers in SDRs and also gives a brief review of existing digital channelizers. Result analysis is given in section 8 and the conclusion in section 9.

2. Cosine modulated uniform filter banks

The structure of an M-channel maximally decimated uniform CMFB is shown in Fig. 1. The input signal is decomposed into subband signals having equal bandwidths. A set of M analysis filters Hk(z), 0 < k < M - 1 decomposes the input signal into M subbands, which are in turn decimated by M fold down-samplers. A set of synthesis filters Fk(z), 0 < k < M - 1 combines the M subband signals after interpolation by a factor of M on each channel.

The reconstructed output, Y(z) is given by Equation (1) [1].

Y (z ) = To (z )X (z ) + X T (z )X (ze-j 2niM )

where T0(z) is the distortion transfer function and Tl(z) is the aliasing transfer function.

To (z) = M X Fk (z)Hk (z)

T (z) = M X Fk (z)Hk (ze- w )

I = 1,2.........M -1

ARTICLE IN PRESS

S. Kalathil, E. Elias/Engineering Science and Technology, an International Journal ■■ (2015) I

■1M0

Hi(z) \Mx --- tAfi

^M-lO2)

У(п)

Fig. 2. Cosine modulated non-uniform Alter bank.

Amplitude distortion error is given by Er = max [Щ (e>)|-1]

For the design of NPR CMFB, a linear phase FIR filter with good stopband attenuation and which provides a flat amplitude distortion function, is initially designed. All the analysis and synthesis filters are generated from this prototype filter by cosine modulation. All the coefficients are real. The coefficients of the analysis and synthesis filters are given by Equations (5) and (6) respectively [1]. po(n) is the impulse response coefficients of the prototype filter, with filter order N.

hk(n) = 2po (n)cosI M(k + 0.5)| n~ ) + (-!)'

fk (n) = 2po (n)cos | M (к + 0.5)) n - N2 ] - (-1)k П

к = 0,1,2.........M -1

n = 0,1,2.........N -1

Different techniques are available for the design of the prototype filter of the NPR CMFB using different objective functions and using different FIR filter approximations [18]. Since the prototype filter is cosine modulated to obtain the analysis and synthesis filters, the filter bank design is reduced to the design of the prototype filter. If the prototype filter has linear phase response, then the overall filter bank will have linear phase response. The adjacent channel aliasing cancellation is inherent in the filter bank design. Remaining is the aliasing between non-adjacent channels. Prototype filter with narrow transition width and good stopband attenuation reduces

aliasing between the non-adjacent channels [19]. The 3-dB cut-off

frequency of the prototype filter should be at ac,3dB = [16]. This condition will reduce the amplitude distortion around the transition frequencies ^M—, where k = 0,1,.....M-1 [1].

3. Cosine modulated non-uniform filter banks

The non-uniform filter banks decompose the input signal into subbands of unequal bandwidths. The structure of the MM channel cosine modulated non-uniform filter bank is shown in Fig. 2. A set of M analysis filters Hk (z), 0 < k < M -1 decomposes the input signal into MM subbands. A set of synthesis filters Fk (z), 0 < k < MM -1 combines the MM subband signals. The decimation ratios are not equal in all the subbands. In this paper, the MM -channel non-uniform design is obtained from the M-channel uniform CMFB by merging appropriate channels [8]. For maximally decimated filter banks, the decimation factors should satisfy the condition given below.

Fig. 3. Realization structure of frequency response masking Alter.

k=0 Mk

The non-uniform bands are obtained by merging the adjacent analysis and synthesis filters. Consider the analysis filters Hi (z), which are obtained by merging li adjacent analysis filters.

ni+!j-1

H (z)=7 Hk (z), i = 0,1.......M -1

Here, Ui is the upper band edge frequency (n0 = 0 < n1 < n2 <.....< nM = M) and k is the number of adjacent channels to be combined. The synthesis filters Fi (z) are obtained in a similar way.

F (z) = y'7'Fk (z), i = 0,1.......M -1

li k=ni

The corresponding decimation factors Mt are given by Mi = —.

The condition to be satisfied for alias cancellation is that lf and n are chosen such that n is an integral multiple of lt, for all

i = 0,1........Mm -1 [8].

4. Frequency response masking approach

In conventional FIR filters, as the transition width reduces, the filter order increases, thereby increasing the implementation complexity. There are several equations available for determining the order of linear phase FIR filters. In one approach, the length of a filter, N, can be estimated using Bellanger's equation [20]:

N -21og (10g,g2 ) -1 3Af

where & and & are the peak pass-band and stop-band ripple magnitudes respectively, and Af is the normalized transition width. The high complexity cosine modulated transmultiplexers given in Reference 11 have a filter order of 32767. When the application demands highly frequency selective filters, the transition width will be very narrow and that will result in high order filters. Y. C. Lim [2] introduced the cost efficient way of designing FIR filters with very narrow transition width using FRM approach. In FRM technique, most of the coefficients in the sub-filters are sparse in nature. This leads to very low hardware implementation complexity.

The basic realization structure for the FRM filter is shown in Fig. 3 [2]. Fa(z) is the band-edge shaping filter. The complementary band-edge filter, Fc(z), can be easily implemented by subtracting the output of Fa(z) from the (N - 1)th delayed version of its input. N is the length of the filter Fa(z). The basic concept is that interpolation by a factor of L reduces the transition band by a factor of L. The interpolated band-edge shaping filter and its complementary filter jointly create the arbitrary passbands with narrow transition band. The unwanted passbands are removed with the cascade connections of the two masking filters Fma(z) and Fmc(z) with Fa(z) and Fc(z)

ARTICLE IN PRESS

S. Kalathil, E. Elias/Engineering Science and Technology, an International Journal ■■ (2015) I

Table 1

Performance comparison of prototype filter design using FRM and FIR approach.

MinSB attenuation

Max. PB ripple (dB)

Max. Ampl. distortion

Total no. of distinct coefficients

Direct method (FIR) FRM approach

60 dB 61.7 dB

2.4 x 10-3 10.9 x 10-3

2.4 x 10-3 3.9x 10-3

5130 Fa 183

respectively. Suppose H(z) is the desired overall transfer function of the FRM filter.

H (z ) = Fa (zL )Fma (z ) + Fc (zL )Fmc (z )

The design equations for case A and case B designs are given below [2]. Suppose op and o are the required passband and stopband edge frequencies respectively of the prototype filter and L is the interpolation factor. The interpolation factor L is related to the number of channels M as given by L = (4m + 1)M and L = (4m -1)M for case A and case B design repectively [13]. Here m is an integer, whose value should be less than L. mvMa and msMa are the passband edge and stopband edge frequencies respectively of the masking filter. copMc and cosMc are the passband edge and stopband edge frequencies respectively of the complementary masking filter.

For case A design,

0 = rapL - 2mn 2mn + 0 rap,Ma L 2mn-0 rap,Mc L (12) (13) (14) $ = rasL - 2mn r 2(m + 1)n-$ ras,Ma L 2mn + $ ®s,Mc L (15) (16) (17)

For case B design,

0 = 2mn-rarL 2(m - 1)n + $ rap,Ma L 2mn-$ rap,Mc L (18) (19) (20) $ = 2mn - rapL 2mn-0 ras,Ma L 2mn + 0 ®s,Mc L (21) (22) (23)

The passband edge frequency [21], cut-off frequency [22] or both edge frequencies simultaneously with fixed transition width, can be iteratively adjusted with small step size to satisfy the condition (25) within a given tolerance value.

5.1. Design specifications (8-channel CMFB)

Number of Channels: 8 Cut-off frequency: 0.0625ft Transition width: 0.0016ft Stopband Attenuation: 62 dB

The prototype filter of the uniform CMFB is designed using the FRM approach. The sub-filters of the FRM filters are designed using Parks McClellan algorithm [27]. For the specifications of 8 channel CMFB, the prototype filter is also designed as a conventional FIR filter. From Table 1, it is clear that the order of the conventional FIR filters are very high for narrow transition width filter. The total number of distinct coefficients of the FRM prototype filter is very less compared to the conventional FIR filter. The coefficients of the FRM filter can be easily quantized into signed power of two terms [2]. The less number of optimization variables for FRM filter results in reduced design time.

5.2. Performance analysis

5. Proposed design of continuous coefficient non-uniform CMFB using FRM approach

The prototype filter design using weighted Chebyshev approximation using a linear search technique is proposed in Reference 21. The prototype filter for cosine modulated filter bank using different types of windows and with different objective functions in an iterative manner is proposed in References 16 and 22. The prototype filter design using WCLS approximation is proposed in Reference 23.

In this paper, the prototype filter of the CMFB is designed using the FRM approach. Initial design of linear phase FIR filter with sharp transition band using FRM approach involved linear programming methods to design the different subfilters [2,24]. The design approaches proposed in References 25 and 26 involved non-linear optimizations. The FRM sub-filters synthesized using minimax or using weighted Chebyshev approximation are proposed in Reference 27. The passband and stopband edge frequencies are iteratively adjusted, with fixed transition width to satisfy the 3-dB condition [16]. To eliminate the amplitude distortion, the condition to be satisfied by the prototype filter P0(z) is given as [16]

= 1, for 0<m< — M

From the above relation it can be shown that

P0I e2M « 0.707

An 8 channel, 16 channel and 32 channel uniform CMFB with FRM prototype filters is initially designed. The frequency response plot of uniform 8 channel filter bank is shown in Fig. 4 and a zoomed plot showing first two channels is shown in Fig. 5. By appropriately merging different channels, a non-uniform 5 channel filter bank is obtained. The channels merged are channels 1 and 2, 3 and 4, 7 and 8 (2,2,1,1,2) resulting in decimation factors (4,4,8,8,4). Similarly, from the 16 channel uniform CMFB, seven channel filter bank with decimation factors (4,8,8,4,16,16,8) and from the 16 channel uniform CMFB, seven channel filter bank with decimation factors

-80 ! -1 ■

: ? i' ■■

0.4 0.6

Fig. 4. Frequency response of uniform 8-channel filter bank.

j n j »-TV

20 0 -20 -40 -60 -80

ARTICLE IN PRESS

S. Kalathil, E. Elias/Eugiueering Science and Technology, an International Journal ■■ (2015) I

/WW 1 1 ' ; ■ '„' V i ' 1 v " " » ---Channel 1 —Channel 2

0.05 0.1 0.15 0.2 0.25

Fig. 5. Zoomed plot of first two channels (X-X').

Fig. 7. Amplitude distortion function plot (4,4,8,8,4).

(8,16,16,8,8,4,32,32,16,8) are also designed. The different other nonuniform combinations that can be obtained from an 8-channel uniform CMFB are with decimation factors (2,4,8,8), (8,8,4,2), (4,4,2), (2,4,4), (8,8,4,4,4), (4,4,4,8,8) and (8,8,4,4,8,8). Whenever any number of channels is merged by satisfying the constraints, the outcome will be a single channel with an equivalent filter. The non-uniform channels will have the same number of coefficients as that of the prototype filter and also the transition width is the same as that of the prototype filter. The total number of channels in the nonuniform filter bank is always less than that of the initial uniform channel filter bank.

The responses of the analysis filters and the amplitude distortion plot for the 5 channel CMFB (4,4,8,8,4) obtained from the 8-channel CMFB are shown in Figs. 6 and 7 respectively. The responses of the analysis filters and the amplitude distortion plots for the 7 channel (4,8,8,4,16,16,8) non-uniform CMFB obtained from the 16 channel uniform CMFB are shown in Figs. 8 and 9 respectively. The responses of the analysis filters and the amplitude distortion plots for the 10 channel (8,16,16,8,8,4,32,32,16,8) non-uniform CMFB obtained from the 32 channel uniform CMFB are shown in Figs. 10 and 11 respectively.

6. Proposed multiplier-less design of non-uniform CMFB

If the coefficients in the filters are represented using SPT terms, the multipliers can be implemented using shifters and adders [28]. CSD contains minimum number of SPT terms and results in reduced number of shifters and adders [28]. For any decimal number, the

corresponding CSD equivalent has a unique SPT representation. CSD is a radix-2 representation within the digit set {1, 0, -1}. CSD has a canonical property that the non-zero digits (1 and -1) will be never adjacent. The number of non-zero digits will be minimum. As a result, a minimum number of adders and shifters is required for the implementation. The filter coefficients of the 8-channel nonuniform CMFB are converted to finite word length CSD representation with restricted number of SPT terms.

The following example demonstrates how efficiently a CSD filter coefficient can be implemented with less number of adders. Suppose

0 0.2 0.4 0.6 0.8 1

Fig. 8. Frequency response of analysis filters (4,8,8,4,16,16,8).

i|U||i|y»,J|l i -60 H|M|ay||||i|||y|

0.2 0.4 0.6 0.8

Fig. 6. Frequency response of analysis filters (4,4,8,8,4).

Fig. 9. Amplitude distortion function plot (4,8,8,4,16,16,8).

ARTICLE IN PRESS

S. Kalathil, E. Elias/Engineering Science and Technology, an International Journal ■■ (2015) I

20 0 -20 -40 -60 -80 -100

Table 2

A typical look up table entry.

0 0.2 0.4 0.6

Fig. 10. Frequency response of analysis filters (8,16,16,8,8,4,32,32,16,8).

Index CSD equivalent Decimal Number of

equivalent non-zeros

2-1 + 2-5 + 2-7 - 2-10 - 2-12 + 2-14

8814 100010100-10-101 0.5379 6

6.1. Quantization of filter coefficients in CSD space

A look-up-table approach is used for the fast conversion of the filter coefficients to their corresponding CSD equivalents with prescribed number of non-zero terms [28]. A typical look-up-table entry for 16 bit CSD conversion is shown in Table 2. The look-up-table consists of four fields: an index, CSD equivalent, corresponding decimal and number of non-zeros present in the CSD equivalent. The coefficients can be converted to their nearest values in the minimal SPT space with specified number of non-zero terms using the look-up-table.

a filter coefficient h1 has a value of 0.4688. This can be represented in SPT terms as (2-2 + 2-3 + 2-4 + 2-5), same coefficients in CSD form is represented as (2-1 - 2-5). The implementation of multiplier for this coefficient using SPT form is shown in Fig. 12a and using CSD reprepresentation is shown in Fig. 12b. It can be observed that CSD coefficient requires just one adder, whereas direct representation requires 3 adders. The symbol ">>2" denotes a right

shift by 2.

6.2. Performance comparison

The filter coefficients are converted to finite precision CSD using a look-up-table [28]. The performances of the CMFB for different word-lengths are given in Table 3. The 12 bit CSD representation gives the worst performance with the lowest implementation complexity. The 16 bit CSD representation gives the best performance with the worst implementation complexity. Hence as a compromise between filter performances and implementation complexity, it is good to choose 14 bit CSD representation.

0 0.2 0.4 0.6 0.8 1

'ig. 11. Amplitude distortion function plot (8,16,16,8,8,4,32,32,16,8)

»1 »5 k

Fig. 12. A filter coefficient implemented using adders and shifters.

6.3. Objective function formulation

The optimization goal in the multiplier-less CMFB is to reduce the following objective functions.

F = max [TO (ejro)\-1] (26)

F2 = max I Po (eiro)\ (27)

F3 = max (0, g (x )-gb ) (28)

min^ = a-iF-i + a2F2 + a3F3 (29)

The design problem is formulated as a multi-objective optimization problem. The objective function given in (26) minimizes the overall amplitude distortion and (27) is to minimize the maximum error in the stopband of the filter and (28) is the constraint added to the objective function using the penalty method that lessens the number of non-zero terms [29]. Here g(x) denotes the average number of non-zero terms and gb is the required upper limit. The value of gb is chosen as 2.7. Eq. (29) combines the three objective functions, where a1, a2 and a3 are the trade-off parameters, which define the relative importance given to each term in the final objective function. The constants a1, a2 and a3 are chosen by trial and

Table 3

Performance comparison of CSD coefficient CMFB for different wordlengths.

12 bits CSD 14 bits CSD 16 bits CSD

Min. SB attn. 48.0 58.12 60.95

Max. PB ripple 2.2 x 10-2 1.04 x 10-2 1.15x 10-2

Max. amp. dist. 1.08 x 10-2 3.9 x 10-3 3.95x 10-3

Adders due to SPT terms 250 416 570

ARTICLE IN PRESS

S. Kalathil, E. Elias/Engineering Science and Technology, an International Journal ■■ (2015) I

Fig. 13. Flow chart of hybrid meta-heuristic algorithm.

error method. The optimization algorithms are terminated when F1 and F2 reach the specified limits.

6.4. Optimization of non-uniform CMFB (4,4,8,8,4) using modified meta-heuristic algorithms

The different modified meta heuristic algorithms used in this paper are Artificial Bee Colony (ABC) algorithm, Gravitational Search algorithm (GSA) and Harmony Search algorithm (HSA). The ABC algorithm is modified by Manoj and Elias in Reference 29, HSA and GSA algorithms are modified by Manuel and Elias in Refrence 30 and by Manuel et al. in Reference 31 respectively. The advantage of meta-heuristic algorithms is that the objective function need not be differentiable and continuous [32].

6.5. Optimization of CMFB using ABC algorithm

ABC Algorithm is a population based search technique introduced by D. Karaboga and B. Basturk [33]. Employed Bees, Onlooker Bees and Scout Bees constitute the artificial colony of honey bees. Possible solution of the problem is represented as the food source and the corresponding fitness is the amount of the nectar of the

Table 4

Parameters of hybrid GSA-HSA algorithm.

INV MNI a.1 Cl2 a.3

50 500 1 4 0.1

ARTICLE IN PRESS

S. Kalathil, E. Elias/Engineering Science and Technology, an International Journal ■■ (2015) I

Wideband Mixer

Filter ADC

Variable

Digital

Baseband Processing

Baseband Processing

Gaseband Processing

Fig. 14. Generic wide-band receiver [3].

food source. An employed bee is the bee who goes to the previously visited food source. Employed bees choose a food source within the neighbourhood of the food source in their memory. The new solution vector is formed adjacent to the existing vectors. Onlooker bee is the bee waiting in the dance area for taking the decision to choose a food source. Onlooker bees take the information provided by the employed bees regarding the fitness function. Onlooker bee selects the food source based on the fitness function. As a result, the food source with a high fitness value will get more onlookers. If the nectar quality of a food source is not improved after a certain number of iterations called the limit cycles, it is abandoned. The employed bee associated with the abandoned food source becomes a scout. The scout bee randomly finds a food source. Termination is reached either, when the stopband attenuation and error in amplitude distortion function reach the limits stated or when given number of iterations is reached.

6.6. Optimization ofCMFB using HSA algorithm

Motivated by the music improvisation scheme, the harmony search algorithm (HSA) was developed by Z.W. Geem for the optimization of mathematical problems. By adjusting the pitches, the musician searches for a better state of memory. The decision variables are represented as musicians and solutions are represented as harmonics. Aesthetics is equal to the fitness function and the pitch range denotes the range of values of the optimization variables.

A harmony Memory (HM) is initialized, in which the solution variables resemble different musical notes. Musicians improve the harmonies for getting better aesthetics. Similarly the harmony search algorithm explores the search space for finding the candidate solutions with good fitness value. In this algorithm a new solution is formed by the following three rules [30].

1. Memory Consideration: Selects any one value from the harmony memory.

2. Pitch adjustment: Selects an adjacent value from harmony memory.

3. Random Selection: Selects a random value from the possible range.

The fitness function of the new harmony vector is evaluated and if it is found better, then the worst harmony vector is replaced with the new vector. Termination is reached either, when the stopband attenuation and error in amplitude distortion function reach the limits stated or when a given number of iterations is reached.

6.7. Optimization ofCMFB using GSA algorithm

GSA is a population based heuristic algorithm proposed by Rashedi in 2009 [34]. GSA is based on Newtonian law of gravity and motion [34]. A modified GSA algorithm for the design of 2D sharp wideband filter is proposed in [31]. In GSA, solutions are represented as masses. A mass or agent is formed by the CSD encoded filter coefficients. The position of mass, inertial mass, active gravitational mass and passive gravitational mass constitutes the four specifications of each mass. The position of mass is equivalent to the solution and the corresponding gravitational and inertial masses are determined by the fitness function. Masses attract each other by force of gravity and the masses will be attracted by the heaviest mass which gives an optimum solution. The positions of the masses are updated in each iteration. Termination is reached either, when the stopband attenuation and error in amplitude distortion function reach the limits stated or when given number of iterations is reached.

Ho(z) JM0 Baseband Processing

H,(z) JM! Baseband Processing

Hm-I(Z) ——* |MM-i Baseband Processing

Fig. 15. Proposed filter bank view of wide-band channelizer.

6.8. Optimization ofCMFB using hybrid GSA-HSA algorithm

The hybrid meta-heuristic algorithms combine the qualities of different algorithms for preventing the premature convergence to a sub-optimal solution [35]. The steps involved in a general hybrid algorithm are given as a flowchart in Fig. 13. In this paper, the hybrid optimization algorithm is formed by combining the algorithms GSA and HSA. In this technique both HSA and GSA are run in parallel and the algorithms are coupled at regular intervals. The different parameters are initialised as shown in Table 4.

Table 5

Performance parameters of the non-uniform CMFB (4,4,8,8,4).

Max. PB ripplea Min. SB attnb Max. amp. dist.c Adders due to SPT terms

Continuous coefficients 1.09 x 10-2 61.7 3.9 x 10-3

CSD rounded (4 SPTs) 1.3x 10-2 57 4.2 x 10-3 395

Max. precision (7 SPTs) 1.04 x 10-2 58.12 3.9 x 10-3 416

GA 1.3x 10-2 57 3.98x 10-3 395

ABC 0.99 x 10-2 58.5 3.7x 10-3 400

GSA 0.90 x 10-2 58.55 3.4 x 10-3 392

HSA 0.97 x 10-2 58.15 3.5x 10-3 399

Hybrid GSA-HSA 1.02 x 10-2 58 2.9 x 10-3 383

a Maximum passband ripple (dB). b Minimum stopband attenuation (dB). c Amplitude distortion.

ARTICLE IN PRESS

S. Kalathil, E. Elias/Engineering Science and Technology, an International Journal ■■ (2015) I 0.02

0.03 0.04 0.05 0.06 0.07 Normalized Frequency (ra/n)

Fig. 16. Zoomed amplitude distortion plot of ABC optimized non-uniform CMFB (4,4,8,8,4).

6.8.1. Initialization

The initial solution vector is randomly perturbed to obtain an initial set of populations for HSA and GSA. Maximum number of iterations MNI, control parameter INV and other control parameters used in the HSA and GSA algorithms of the hybrid

algorithm are initialized. To obtain an expanded search space, the initial number of solutions is taken as the integer multiple of the total population size. The fitness function is evaluated for each solution vector and a set of best solutions is passed on to the next stage.

CSD Rounded HSA Algorithm Genetic Algorithm

0.01 0.02

0.03 0.04 0.05 0.06 0.07 Normalized Frequency (ra/rc)

0.08 0.09 0.1

Fig. 17. Zoomed amplitude distortion plot of HSA optimized non-uniform CMFB (4,4,8,8,4).

0.03 0.04 0.05 0.06 0.07 Normalized Frequency (ra/n)

Fig. 18. Zoomed amplitude distortion plot of GSA optimized non-uniform CMFB (4,4,8,8,4).

ARTICLE IN PRESS

Normalized Frequency (ra/n)

Fig. 19. Zoomed amplitude distortion plot of Hybrid GSA-HSA optimized non-uniform CMFB.

6.8.2. Check the condition for mixing

If the populations of the two algorithms are not combined in the previous INV number of iterations, go to step 3. Otherwise go to step 6.

6.8.3. Grouping

Group the populations of the GSA and HSA algorithms.

6.8.4. Splitting

A set of different filters are randomly selected from the group to form the population of HSA algorithm. The remaining filters in the group are used to form the population of GSA algorithm.

6.8.5. Update the population

The populations of the HSA and GSA algorithms are updated using the update procedure of the corresponding algorithms.

6.8.6. Update the best solution

If the best solution in the current populations of the two algorithms is better than the best solution obtained so far by the hybrid algorithm, the best solution is replaced by the best solution in the current populations.

6.8.7. Termination

The algorithm comes to an end, when the total number of iterations reaches the number specified in MNI. If not steps indicated in sections 6.8.2 to 6.8.6 are repeated. When the algorithm is terminated, the best solution in the memory is taken as the optimum solution.

7. Proposed filter bank as efficient channelizer in software defined radio

Fig. 14 shows the generic wide-band receiver for software defined radio. The wideband mixer converts the frequency of the radio frequency signal to the intermediate frequency signal. This wideband signal is digitized by ADC. The digital channelizer which follows the ADC extracts the individual channels of different communication standards. The channelizer is a set of bandpass filters and the output of bandpass filters after decimations are fed to the baseband processors. Digital filter banks are commonly used digital channelizers [3]. Wireless communication specifications highly appreciate filters with narrow transition bands. But for FIR filters narrow transition widths result in high order. FRM approach is a cost efficient way of designing narrow transition filters, in which most of the coefficients are sparse. Multiplier-less realizations further reduce the

hardware complexity. A brief review of the digital channelizers available in the literature is given below.

Digital down converter chips are used to extract the different channels individually. Each filter in the channelizer is separately designed, hence the complexity increases when the number of received channels increases [3,36]. Digital down converter using variable bandwidth filters is proposed in Reference 6, which has a fixed FIR filter and two arbitrary sampling rate converters. Polyphase DFT filter bank is an efficient channelizer with less computational complexity compared to digital down converters [37,38]. When the received channels have uniform bandwidth and for more number of received channels, polyphase DFT filter banks are a good choice. An opened filter bank channelizer using a new DFT filter bank is given in Reference 3. For uniform bandwidth channelizers, Goertzel algorithm filter banks are proposed in Reference 39.

For the channels with unequal bandwidth filters, tree structured filter banks are proposed in Reference 5. Tree structured filter banks have certain limitations such as restricted non-uniform band-widths and high signal delay. Channelizers with non-uniform bandwidths are proposed using perfect reconstruction exponentially modulated filter banks in Reference 4. The outputs of the analysis section are added in the synthesis section, to get the nonuniform channels. But the output of synthesis section will always be integer multiples of the bandwidths of the analysis section.

The channelizers proposed in Reference 7 design a uniform CMFB and a non-uniform filter bank is obtained by suitably merging the adjacent channels of the uniform filter bank. Channelizers based on frequency response masking and coefficient decimation filters are proposed in Reference 36. In coefficient decimation filter approach, an FIR filter is initially designed and by decimating the coefficients by different factors, different multiband channels are obtained. The required bands are obtained by using masking filters which need to be separately designed. The non-uniform filters using frequency response masking also requires to design the prototype filter and different masking filters.

Fig. 15 shows the block diagram of the proposed non-uniform filter banks for efficient realization of digital channelizers in SDR.

Table 6

Hardware complexity comparison in terms of required number of LUTs.

Continuous coefficients Max. precision (7 SPT terms) Hybrid GSA-HSA

No. of multipliers 258 0 0

No. of adders 509 925 892

No. of LUTs required 25,486 7400 7136

ARTICLE IN PRESS

Table 7

Common world wide wireless technologies.

Service Technology Launch Channel

year spacing (MHz)

1G Cellular AMPS 1983 0.03

2G Cellular GSM 1991 0.2

3G Cellular CDMA2000 1xEV-DO 2002 1.25

3G Cellular CDMA2000 1xEV-DO 2010 3.75,5,7.5,10,

Rev.B 11.5,15,18.75

3G Cellular WCDMA FDD 2001 5,10

3.5G Cellular HSDPA 2007 5

4G MBWA i Burst HC-SDMA 2005 0.625

Fixed WiMAX IEEE802.16d 2004 1.75,3.5,7,20

4G Cellular LTE 2009 1.4,3,5,10,15,20

Digital radio DAB 1995 1.715

Digital TV 2007 8

Digital Cable TV 6

Personal Area Networks IEEE 802.15.4a (Zigbee) 2003 2,5

Personal Area Networks ANT 2006 1

Personal Area Networks IEEE 802.15.1 2002 1

(Bluetooth)

8. Results and discussion

All the simulations are done using a Dual Core AMD Opteron processor operating at 2.17GHz using MATLAB 7.12.0. The performances of the non-uniform CMFB after optimization in the CSD space are compared in terms of the error in amplitude distortion, stopband attenuation, passband ripple and also the implementation complexity in terms of adders. For the joint optimization of different sub-filters, the coefficients of Fa, Fma and Fmc are concatenated together and are taken as the design vector for the optimization algorithm. All the three sub-filters of FRM filter have linear phase. Hence, only half of the symmetrical coefficients of each sub-filter needs to be extracted and concatenated to form the solution vector. This reduces the computation time and the dimension of the optimization variable. The results shown in the manuscript are the best possible results obtained with 500 iterations and average of 10 simulations with the weights specified.

8.1. Performance of multiplier-less non-uniform CMFB using modified meta-heuristic algorithms

The prototype filter alone is required to be designed and optimized. All the other non-uniform bands are obtained by appropriately merging the adjacent subbands of uniform filter bank. After channelizing into corresponding subbands, the subbands can be moved to the baseband and the baseband processing is done. The proposed channelizers are suitable when signal bandwidths and band locations are given beforehand. The advantages of the proposed filter banks are listed below.

• Narrow transition filters with very low implementation complexity is achieved by using FRM filter.

• The prototype filter alone is required to be designed and all the other filters are obtained from this filter by cosine modulation.

• Multiplier-less realization using various modified meta-heuristic algorithms reduces the implementation complexity.

• Easier for configuring the different communication standards with differing bandwidths.

The CSD rounded filter coefficients in finite word length are optimized for the combined objective function given in (29), using various modified meta heuristic algorithms. Table 5 compares the performances of the prototype filter in terms of minimum stopband attenuation and maximum passband ripple achieved and also compares the CMFB for the maximum error in amplitude distortion and the run time required for the design. The zoomed amplitude distortion function plots for ABC algorithm, HS algorithm and GS algorithm are shown in Figs. 16-18 respectively. The zoomed amplitude distortion function plot fot hybrid GSA-HSA algorithm is shown in Fig. 19.

From Table 5, it can be observed that among the four optimization algorithms GSA algorithm has got the lowest error in amplitude distortion, minimum passband ripple and maximum stopband attenuation. HSA algorithm also gives comparable performance. Hence hybrid GSA-HSA algorithm gives good performance with less number of adders. The reduction obtained in hardware

CD TD ID

10 0 -10 -20 -30 -40 -50 -60 -70

Normalized Frequency (ra/rc)

Fig. 20. Uniform 16-channel CMFB.

ARTICLE IN PRESS

0.3 0.4 0.5 0.6 0.7 Normalized Frequency (ra/n)

Fig. 21. Non-uniform CMFB for various wireless standards (WCDMA, Fixed WiMAX, HSDPA and CDMA2000).

complexity using the proposed multiplier-less approach is analysed by implementing the filters on Xilinx ISE, by selecting the device as SPARTAN 3E. In FPGA, configurable logic blocks are used to implement logic, which consists of look-up tables (LUTs) and flipflops. Table 6 shows the comparisons of number of LUTs exploited by the coefficients when implemented using multipliers and adders and for multiplier-less realization. The dedicated multiplier block inside the FPGA is not used. From Table 6, it can be verified that a tremendous reduction in terms of LUTs is achieved when the multipliers are implemented using CSD representation and optimized using hybrid GSA-HSA algorithm.

8.2. Digital channelizers for SDR implementations using proposed non-uniform CMFB

The proposed non-uniform sharp transition width CMFB can be applied for efficiently implementing the digital channelizer block of SDR. Table 7 shows the common world-wide wireless technologies and their corresponding channel bandwidths. It is clear from the table that different communication standards have different channel spacing and hence non-uniform filter banks are highly preferred. Fig. 20 shows the frequency response plot of 16-channel uniform CMFB. Figs. 21-24 show the frequency responses of the nonuniform CMFB designed for different wireless standards. Fig. 21 is obtained by suitably combining the channels of a 16 channel uniform CMFB for 4 wireless standards namely WCDMA-FDD, IEEE 802.16d, HSDPA and CDMA2000. Fig. 22 is obtained by suitably combining

the channels of a 12 channel uniform CMFB for 4 wireless standards namely HSDPA, WCDMA-FDD, CDMA2000 1x and CDMA2000 3x. Fig. 23 is obtained by suitably combining the channels of a 10 channel uniform CMFB for 4 wireless personal area network standards namely IEEE 802.15.4a, ANT, IEEE 802.15.4a and IEEE 802.15.1 (Bluetooth). Fig. 25 is obtained by suitably combining the channels of a 6 channel uniform CMFB for 3 different CDMA 2000 standards.

8.3. Complexity comparison with existing method

The hardware complexity of the proposed method is compared with the down-converter design using variable bandwidth filter, which consists of a fixed FIR filter and two arbitrary sampling rate converters [6]. The fixed FIR filter is designed using FRM technique. The filter orders for Fa, Fma and Fmc are shown in Table 8 and the design requires an additional complexity for an interpolation filter which has high order. Hence the method proposed in this paper has simple design procedure and low implementation complexity.

9. Conclusion

Upcoming digital filter bank applications demand low implementation complexity along with good performance charateristics. Channelizers in SDRs is one such application which demand highly frequency selective channels with non-uniform decompositions. To

5G-HSDPA (5 MHz) ■—■WCDMA (5 MHz) —CDMA2000 1x (1.25 MHz) CDMA2000 3x (3.75 MHz)

0.3 0.4 0.5 0.6 0.7 Normalized Frequency (ra/n)

0.8 0.9

Fig. 22. Non-uniform CMFB for various wireless standards (HSDPA, WCDMA, and CDMA2000).

ARTICLE IN PRESS

с та to

IEEE 802.15.4a (5 MHz) —ANT (1 MHz)

IEEE 802.15.4a (2 MHz) ---IEEE 802.15.1 (1 MHz)

"0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Normalized Frequency (ю/п)

0.8 0.9 1

Fig. 23. Non-uniform CMFB for various wireless PAN standards (Zigbee, ANT and Bluetooth).

meet such requirements, a totally multiplier-less NPR non-uniform cosine modulated filter bank is designed using frequency response masking approach and then optimized in the discrete space using various modified meta-heuristic algorithms. Highly frequency selective filter bank with non-uniform subband decomposition is achieved with the least implementation complexity. The individual

filters are characterized by small passband ripple, high stopband attenuation and narrow transition width. A comparative study of the non-uniform NPR CMFB in the finite precision space, optimized using various modified meta-heuristic algorithms, has been done in this paper. Hybrid GSA-HSA algorithm is observed to be the most suitable one for the optimization of multiplier-less

Normalized Frequency (ю/п)

Fig. 24. Non-uniform CMFB for various wireless standards (LTE, Digital Cable TV and CDMA 2000).

та to

800 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Frequency (ю/п)

Fig. 25. Non-uniform CMFB for different CDMA 2000 standards.

ARTICLE IN PRESS

S. Kalathil, E. Elias/Engineering Science and Technology, an International Journal ■■ (2015) I

Table 8

Complexity comparison with existing method.

Filter order

Interpolation filter

Total no. of

non-zero

coefficients

Proposed method prototype filter (6 channel) Method in Reference 6 variable bandwidth filter

63 111

303 900

non-uniform CMFB with sharp transition width. These multiplier-less non uniform CMFBs are very useful in filter bank applications where low implementation complexity and improved performances are required. The compatibility of the proposed filter bank as an efficient channelizer in a software defined radio is investigated in this paper. Examples of the non-uniform CMFB, designed for various wireless standards, are also given.

References

[1] P.P. Vaidyanathan, Multirate Systems and Filter Banks, Prentice-Hall, Englewood Cliffs, NJ, 1993.

[2] Y.C. Lim, Frequency-response masking approach forthe synthesis of sharp linear phase digital filters, IEEE Trans. Circ. Syst. 33 (4) (1986) 357-364.

[3] K.C. Zangi, R.D. Koilpillai, Software radio issues in cellular base stations, IEEE Trans. Selected areas Commn. 17 (4) (1999) 561-573.

[4] W.A. Abu-Al-Saud, G.L. Stuber, Efficient wideband channelizer for software radio systems using modulated PR filterbanks, IEEE Trans. Sig. Proc. 52 (10) (2004) 2807-2820.

[5] M. Narendar, A.P. Vinod, A. Madhukumar, A.K Krishna, A tree-structured DFT filter bank based spectrum sensor for estimation of radio channel edge frequencies in military wideband receivers, in: Information Sciences Signal Processing and their Applications (ISSPA), 2010 10th International Conference on, IEEE, pp. 534-537, 2010.

[6] J.T. George, E. Elias, Reconfigurable channel filtering and digital down conversion in optimal CSD space for software defined radio, AEU-Int. J. Electron. Commun. 68 (4) (2014) 312-321.

[7] B. Li, L. Ge, J. Zheng, Efficient dynamic channeliser based on nonuniform filter banks, Eur. Trans. Telecommun. 19 (3) (2008) 273-283.

[8] J.-J. Lee, B.G. Lee, A design of nonuniform cosine modulated filter banks, IEEE Trans. Circ. Syst. II: Anal. Dig. Signal Process. 42 (11) (1995) 732-737.

[9] Z. Zijing, Y. Yun, A simple design method for nonuniform cosine modulated filter banks, in: Microwave, Antenna, Propagation and EMC Technologies for Wireless Communications, 2007 International Symposium on, IEEE, pp. 1052-1055,2007.

[10] K. Shaeen, E. Elias, Non-uniform cosine modulated filter banks using meta-heuristic algorithms in CSD space, J. Adv. Res. (2014) doi:10.1016/ j.jare.2014.06.008.

[11] M. FurtadoJr., P. Diniz, S. Netto, T. Saramaki, On the design of high-complexity cosine-modulated transmultiplexers based on the frequency-response masking approach, Circuits and Systems I: Regular Papers, IEEE Transactions 52 (11) (2005)2413-2426.

[12] M.B. FurtadoJr., P.S. Diniz, S.L. Netto, Optimized prototype filter based on the FRM approach for cosine-modulated filter banks, Circ. Syst. Sign. Process. 22 (2) (2003) 193-210.

[13] L. Rosenbaum, P. Lowenborg, M. Johansson, An approach for synthesis of modulated M-Channel FIR filterbanks utilizing the frequency-response masking technique, EURASIP J. Adv. Signal Process. 2007 (2007) 068285.

[14] K. Shaeen, E. Elias, Design of multiplier-less cosine modulated filterbanks with sharp transition using evolutionary algorithms, Int. J. Comput. Appl. 68 (25) (2013) 1-9.

[15] A.T. Fuller, B. Nowrouzian, F. Ashrafzadeh, Optimization of FIR digital filters over the canonical signed-digit coefficient space using genetic algorithms, in: Circuits and Systems, 1998. Proceedings. 1998 Midwest Symposium on, IEEE, pp. 456-459, 1998.

[16] F. Cruz-Roldan, P. Amo-Lopez, S. Maldonado-Bascon, S.S. Lawson, An efficient and simple method for designing prototype filters for cosine-modulated pseudo-qmf banks, IEEE Let. Sig. Proc. 9 (1) (2002) 29-31.

[17] X.-S. Yang, Nature-Inspired Metaheuristic Algorithms, Luniver Press, 2011.

[18] K. Shaeen, E. Elias, Prototype filter design approaches for near perfect reconstruction cosine modulated filter banks - a review, Signal Process. Syst. 81 (2) (2015) 183-195. doi:10.1007/s11265-014-0929-5.

[19] T.Q. Nguyen, Near-perfect-reconstruction pseudo-QMF banks, IEEE Trans. Sig. Proc. 42(1) (1994) 65-76.

[20] M. Bellanger, On computational complexity in digital filters, in: Proc. of the European conference on circuit theory and design, pp. 58-63,1981.

[21] C.D. Creusere, S.K. Mitra, A simple method for designing high-quality prototype filters for m-band pseudo QMF banks, IEEE Trans. Sig. Proc. 43 (4) (1995) 1005-1007.

[22] Y.-P. Lin, P. Vaidyanathan, A Kaiser window approach forthe design of prototype filters of cosine modulated filterbanks, IEEE Let. Sig. Proc. 5 (6) (1998) 132134.

[23] A. Kumar, G. Singh, R. Anand, A simple design method forthe cosine-modulated filterbanks using weighted constrained least square technique, J. Frank. Instit. 348(4)(2011)606-621.

[24] Y. Lim, Y. Lian, The optimum design of one-and two-dimensional FIR filters using the frequency response masking technique, IEEE Trans. Circ. Syst. II: Anal. Dig. Signal Process. 40 (2) (1993) 88-95.

[25] T. Saramaki, H. Johansson, Optimization of FIR filters using the frequency-response masking approach, in: Circuits and Systems, 2001. ISCAS 2001. The 2001 IEEE International Symposium on, Vol. 2, IEEE, pp. 177-180, 2001.

[26] W.-S. Lu, T. Hinamoto, Optimal design of frequency-response-masking filters using semidefinite programming, IEEE Trans. Circ. Syst. I Fundam. Theory Appl. 50 (4) (2003) 557-568.

[27] T. Saramaki, Y.C. Lim, Use of the Remez algorithm for designing FIR filters utilizing the frequency-response masking approach, in: Circuits and Systems, 1999. ISCAS'99. Proceedings of the 1999 IEEE International Symposium on, Vol. 3, IEEE, pp. 449-455,1999.

[28] Y.J. Yu, Y.C. Lim, A novel genetic algorithm forthe design of a signed power-of-two coefficient quadrature mirror filter lattice filter bank, Circ. Syst. Sign. Process. 21 (3) (2002) 263-276.

[29] V.J. Manoj, E. Elias, Artificial bee colony algorithm forthe design of multiplier-less non uniform filter bank transmultiplexer, Inf. Sci. (Ny) 192 (2012) 193203.

[30] M. Manuel, E. Elias, Design of sharp 2D multiplier-less circularly symmetric FIR filter using harmony search algorithm and frequency transformation, J. Sig. Inf. Process. 3 (3) (2012) 344-351.

[31] M. Manuel, R. Krishnan, E. Elias, Design of multiplierless 2-D sharp wideband filters using FRM and GSA, Global J. Res. Eng. 12 (5-F) (2012).

[32] X.-S. Yang, Music-Inspired Harmony Search Algorithm, Springer, 2009.

[33] D. Karaboga, B. Basturk, On the performance of artificial bee colony (ABC) algorithm, Appl. Soft Comput. 8 (1) (2008) 687-697.

[34] E. Rashedi, H. Nezamabadi-pour, S. Saryazdi, GSA: a gravitational search algorithm, Inf. Sci. (Ny) 179 (13) (2009) 2232-2248.

[35] T. Bindiya, E. Elias, Design of FRM based MDFT filterbanks in the canonic signed digit space using modified metaheuristic algorithms, Int. J. Signal Imag. Syst. Eng. (2015) (in press).

[36] R. Mahesh, A.P. Vinod, E.M. Lai, A. Omondi, Filter bank channelizers for multi-standard software defined radio receivers, J. Signal Process. Syst. 62 (2) (2011)157-171.

[37] K.C. Zangi, R.D. Koilpillai, Efficient filterbank channelizers for software radio receivers, in: Communications, 1998. ICC 98. Conference Record. 1998 IEEE International Conference on, Vol. 3, IEEE, pp. 1566-1570,1998.

[38] W.H. Yung, M.Jian, Y.W. Ho, Polyphase decomposition channelizers for software radios, in: Circuits and Systems, 2000. Proceedings. ISCAS 2000 Geneva. The 2000 IEEE International Symposium on, Vol. 2, IEEE, pp. 353-356, 2000.

[39] T. Hentschel, Channelization for software defined base-stations, in: Annales des Telecommunications, vol. 57, Springer, 2002, pp. 386-420.