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Engineering Science and Technology, an International Journal

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

Full length article

Design of low complexity sharp MDFT filter banks with perfect reconstruction using hybrid harmony-gravitational search algorithm

V. Sakthivel*, Elizabeth Elias

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

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ARTICLE INFO

Article history: Received 14 January 2015 Received in revised form 24 March 2015 Accepted 30 March 2015 Available online 14 May 2015

Keywords:

MDFT filter banks with PR Frequency response masking Canonic signed digit Hybrid harmony-gravitational search algorithm

Artificial bee colony algorithm Gravitational search algorithm Harmony search algorithm Genetic algorithm

ABSTRACT

The design of low complexity sharp transition width Modified Discrete Fourier Transform (MDFT) filter bank with perfect reconstruction (PR) is proposed in this work. The current trends in technology require high data rates and speedy processing along with reduced power consumption, implementation complexity and chip area. Filters with sharp transition width are required for various applications in wireless communication. Frequency response masking (FRM) technique is used to reduce the implementation complexity of sharp MDFT filter banks with PR. Further, to reduce the implementation complexity, the continuous coefficients of the filters in the MDFT filter banks are represented in discrete space using canonic signed digit (CSD). The multipliers in the filters are replaced by shifters and adders. The number of non-zero bits is reduced in the conversion process to minimize the number of adders and shifters required for the filter implementation. Hence the performances of the MDFT filter bank with PR may degrade. In this work, the performances of the MDFT filter banks with PR are improved using a hybrid Harmony-Gravitational search algorithm.

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

Multirate filter banks are used in many applications such as wireless communication, image compression, speech processing, sub-band coding and adaptive signal processing [1]. Due to the simple design and easy realization, the modulated filter banks are more popular than the other filter bank structures. Mainly there are two types of modulated filter banks, Discrete Fourier Transform (DFT) polyphase filter bank and Cosine modulated filter bank. Using the same prototype filter, the analysis and synthesis filter banks are generated by exponential modulation in the DFT filter banks. This will lead to the easy implementation of DFT filter banks, but here, no inherent aliasing cancellation structure is available. Also the polyphase matrix of the analysis and synthesis filters need to be invertible for the M-channel PR uniform filter banks [2].

To overcome the disadvantage of DFT filter banks, the modified DFT filter banks can be used [3—5]. MDFT filter banks provide linear phase in both the analysis and synthesis filters, provided, the

* Corresponding author. Tel: +91 9995335962; fax: +91 4952287250. E-mail address: sakthi517@nitc.ac.in (V. Sakthivel). Peer review under responsibility of Karabuk University.

prototype filter is chosen to have linear phase. In MDFT filter banks, a structure inherent alias cancellation is available which will automatically cancel all the odd alias spectra. This leads to near perfect reconstruction(NPR) MDFT filter banks. In the NPR MDFT filter banks, the aliasing distortion and amplitude distortion are very low. In many applications like image processing, filter banks with PR are essential. Hence we need to design the MDFT filter banks with PR.

The non-critically sub-sampled DFT filter banks are considered for the design of MDFT filter banks with PR instead of NPR MDFT filter banks [6], because the output signals in the NPR MDFT filter banks and the non-critically decimated filter banks differ only in a scaling factor and an additional time delay. The non-critically sub-sampled DFT filter banks are chosen for PR to simplify the structure for PR [6]. To get PR, the aliasing, amplitude and phase distortions should be zero. In the MDFT filter banks, the phase distortion is zero, since all the filters are selected as linear phase filters. To achieve PR, the polyphase realization of the DFT filter banks is used [6]. When the continuous filter coefficients are converted into discrete filter coefficients in the signed power of two (SPT) space, the implementation complexity will be significantly reduced [7]. The shifters and adders in the SPT space will replace the multipliers in the filter implementation. Thus we can get multiplier-free MDFT

http://dx.doi.org/10.1016/jjestch.2015.03.012

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/).

filter banks with PR [8]. CSD representation is a special form of SPT representations and is a minimal one [9]. However, when the continuous filter coefficients are represented in the CSD space, the performances of the filter may degrade. Hence meta-heuristic algorithms are deployed which will optimize the filter coefficients [10,11], to improve the performances of the filter with the CSD represented coefficients.

Sharp transition width filters are required in the filter banks used in many applications such as image processing and wireless communication. This will increase the order of the prototype filter. Increase in the filter order will increase the implementation complexity. In order to reduce the complexity, frequency response masking(FRM) technique [12] can be used to design the prototype filter. Using the same prototype filter, the analysis and synthesis filters of the MDFT filter banks can be derived. In [13] FRM technique is used in the design of MDFT filter banks with NPR.

In this paper, the FRM technique is proposed to be used in the design of MDFT filter banks with PR. The design of FRM filter involves the design of a band edge shaping filter, masking filter and complementary masking filter. The complexity of the FRM prototype filter is further reduced by converting the continuous filter coefficients into equivalent CSD representation. This may degrade the performance of the prototype filter and hence the MDFT filter banks with PR. Hence efficient optimization technique in the discrete space is required to be deployed. The classical gradient based techniques cannot be directly applied due to the fact that the search space contains integers. By properly selecting the parameters, a global solution can be reached using meta-heuristic algorithms [10]. The various meta-heuristic algorithms used in this paper are Harmony Search algorithm (HSA), Gravitational Search algorithm (GSA), Artificial bee colony (ABC) algorithm, Genetic algorithm (GA) and hybrid harmony-gravitational search algorithm. HSA is a music based optimization algorithm introduced by Geem and Lee for the optimization of mathematical problems [14,15]. A modified integer coded HSA algorithm is proposed in [16]. This modified HSA is used in this paper. GSA is a population based algorithm proposed by Rashedi in 2009 [17], based on Newtonian law of gravity and motion. A modified GS algorithm for the design of 2D sharp wideband filter was proposed in [18]. This modified GSA is used in this paper. The Artificial Bee Colony (ABC) algorithm [19—22] is introduced by Karaboga and Basturk. Integer coded ABC algorithm has been proposed for the design of non-uniform filter bank trans-multiplexer in [23]. A modified integer coded ABC algorithm is proposed for the optimization of FRM filter in [24]. This modified integer coded algorithm is used in this paper. An integer coded genetic algorithm (GA) is proposed in [25]. This integer coded GA is also used in this paper. A hybrid Harmony-Gravitational Search algorithm which combines the qualities of HSA and GSA is proposed in [26] to further reduce the complexity by reducing the number of adders due to SPT terms. This hybrid algorithm is proposed in this paper to be used for the design of multiplier-free MDFT filter banks with PR using FRM. Our design reduces the implementation complexity of the MDFT PR filter bank by first using the FRM technique which reduces the number of multipliers. Then it is made multiplier-free by replacing the multipliers by shift/add operations. The CSD method reduces the number of adders to a minimum. The structural adders and the reduced number of adders due to SPT terms, when realized on hardware will lead to low area, low power and high speed of operation [27—29]. To the best of our knowledge, neither the design of FRM based MDFT filter bank with PR, nor its multiplier-free design is proposed in the literature so far.

The rest of the paper is organized as follows: Section 2 gives details about the design of Modified DFT filter bank with PR, explains the FRM technique used in the proposed design and gives the

details of the prototype filter for the design of the MDFT with PR using FRM. Section 3 illustrates the design of CSD represented FRM filter coefficients in MDFT filter banks with PR. Section 4 explains the optimization of the CSD represented filter coefficients using hybrid HSA-GSA and other meta-heuristic algorithms. Section 5 gives the details of results and analysis. Section 6 concludes the paper.

2. Design of modified DFT filter banks with PR

2.1. Modified DFT filter banks

In the DFT filter banks [30], synthesis and analysis filters are derived from the prototype filter by exponential modulation. This will lead to easy implementation of DFT filter banks. But in the DFT filter banks, there is no inherent mechanism to cancel the aliasing. Hence DFT filter banks do not give perfect reconstruction due to aliasing and linear distortion. This disadvantage can be removed by introducing some modifications to the DFT filter banks which leads to the MDFT filter banks [3—5].

The MDFT filter bank is a complex modulated M-channel filter bank with a two step decimation of the sub-band signals as shown in Fig. 1. Initially the sampling rate is decimated by a factor M/2 and later decimated by the factor 2 with and without a delay of one sampling period, using either the real or the imaginary part in the sub-bands respectively as given in [6]. Due to the modification in the structure of the filter banks, all adjacent alias spectra and all odd alias spectra are canceled within the MDFT filter banks. Non-adjacent alias terms can be made small by selecting high stop-band attenuation in the design of the prototype filter. This will give NPR in MDFT filter banks. In many applications like image processing, filter banks with PR are required. Hence the design of MDFT filter banks with PR is essential.

The output signal X(z) in the MDFT filter banks is given as [6].

X (z)=-

1 M/2-1

E E k=0 l=0

Fk(z)Hk(zWM)X(zWM

The equation shows that all odd aliasing terms are canceled and only the even alias spectra are remaining which are the same as the output of a non-critically decimated M-channel DFT filter bank with a sub-sampling factor of M/2 given as [6].

2 M-1 M/2-1

XDFT (z)=MûE E

Fk(z)HklzWM) X(zWM

k=0 l=0

The only difference between the output signals X(z) and XDFT(z) are the delay and the scaling factor of the amplitude. Hence the non-critically sub-sampled M-channel DFT filter bank as shown in Fig. 2 [6] is considered for PR, instead of NPR MDFT filter banks. This will simplify the design of the MDFT filter banks with PR [6].

The filter bank should satisfy the conditions such as exact compensation of aliasing, no phase and amplitude distortion, to get PR. The polyphase realization of the DFT filter banks as shown in Fig. 2 [6] is essential to impose the PR conditions on the analysis and synthesis filters. The Type-1 polyphase filters of the low-pass filter h0(n) used in the analysis filter bank are given as [30,31].

Ho (z)= £ z-kGJzM k=0

Янй

Hk+i(z)

¿2 Rp

J, M/2

z"1 ¿2 j.im

Î2 z-

n TM/2

Fi-dz)

12 j.lm

4, M/2

z-' Re

12 Re

J, M/2

z"1 12 j.lm

Î M/2

t2 z-

n T M/2

F*+ i(z)

Fig. 1. Modified DFT filter bank.

H0(z) i M/2

Hi(z) i M/2

Ям-М J, M/2

YM-i(z)

Î M/2 Fo(z)

Î M/2 F\(z)

Î M/2 FM-I(z)

xdft(z) = zmI12x(z)

Fig. 2. Complex modulated M-channel filter bank with a sub-sampling factor of M/2.

Gk(z) = E gk(n)z-n

gk(n) = ho(Mn + k), k = 0,1, 2,......M - 1

where M is any integer.

The Type-3 polyphase filters used in the synthesis filter bank are given as [30,31].

Ho(z)=22 zkGjzM k=0

Gk(z) = Ê gk(n)z

gk(n) = ho(Mn - k), k = 0,1, 2,......M - 1

where M is any integer.

The MDFT filter banks guarantee PR as proved in [6], if the prototype filter has a length of N = rM + 1, where r is an integer and only if the following condition on the polyphase filters is satisfied [6].

Gk(z)Gk(z) + Gk+M/2(z)Gk+M/2(z) = M "

where a is the delay.

For general case, N = r.M + s, r is an integer and. 0 < s < M The condition for PR on the poyphase filters is given [6] as

Gk (z)Gk(z) + Gk+M/2(z)Gk+M/2(z) = m^P^)

In the MDFT filter banks, the amplitude distortion function is given as [1,30].

Tdist (z)=MJ2Fk(z)Hk (z)

The PR condition can be obtained for MDFT filter banks, if equation (5) or 6 is satisfied depending on the value of s.

2.2. Design of MDFT filter banks with PR using FIR

Linear phase prototype FIR filter using Parks McClellan method is used for the design of MDFT filter bank with PR [8]. The polyphase realization of the non-critically sub-sampled M-channel DFT filter bank is used to get PR in MDFT filter banks [6]. Due to the presence of filters with linear phase in the MDFT filter banks, the phase distortion is zero. The aliasing distortion is completely removed in the design of MDFT filter banks with PR.Very low amplitude distortion occurs due to the degree of overlap between the adjacent filter responses. The pass-band and the stop-band edge frequencies are selected carefully, so that the adjacent filter responses intersect at 3 dB level [8].

The specifications of the prototype filter with sharp transition bandwidth are selected as follows:

H(Z) = Fa(zL)FMa(z)+Fc(zLyMc(z)

Let fp and fs be the passband and stopband frequencies of the final filter H(z), L, the interpolation factor, |YJ, the largest integer which is less than Y, fap and fas be the passband and stopband frequencies of the prototype filter Ha(z), fmap and fmas be the pass-band and stop-band frequencies of the masking filter FMa (z), and fmcp and fmcs be the respective passband and stopband frequencies of FMc(z). Then the following are the design equations of the sub-filters [12]:

m = \fp*LJ fap = fpL - m fas = fsL -

fmap = fp fm

m - fc

m + 1 - fas

fmcp =

ap f f

fmcs = fs

Maximum pass-band ripple: 0.004dB Minimum stop-band attenuation: 60dB Pass-band edge frequency: 0.062p Stop-band edge frequency: 0.06336p Number of channels: 8

The design of the prototype filter is done as in [8] and the filter bank with PR is obtained. The number of multipliers required to realize FIR prototype filter is obtained as 1283.

2.3. Review of FRM approach

Since the order of the FIR filter used in the above design is very high, the implementation complexity of the filter will also be high. To design the prototype filter with sharp transition width and reduced complexity, FRM approach is an efficient method [12]. The arbitrary bandwidth sharp transition width FIR filter can be implemented using FRM. The transition width is reduced by L times, when it is interpolated by L. The structure of the FRM filter is shown in Fig. 3 [12]. Here, Fa(z) is the band edge shaping filter, which is a linear phase FIR filter of even order N and Fc(z) is its complementary filter. Fc(z) can be easily obtained from Fa(z) by using the following equation [12].

-(N-1)

Fc(z)=Z— - Fa(z) (12)

Fa(zL) and Fc(zL) are the interpolated filters of Fa(z) and Fc(z) respectively and their transition bandwidths are L times smaller. FMa(z) and FMc(z) are the two masking filters, which are used to eliminate the undesired bands in the band edge shaping and the complementary filters respectively. The FRM filter transfer function H(z) is given as [12].

Fig. 3. Basic FRM filter architecture.

2.4. Design of FRM prototype filter

The initial phase of this work is the design of the continuous coefficient sharp transition width PR MDFT filter bank with all the analysis and synthesis filters having linear phase property. Using the same prototype filter, all the analysis and synthesis filters of the MDFT filter banks are derived using complex modulation. Hence, the problem of designing the MDFT filter bank with PR reduces to the problem of designing a single prototype filter. Also, the complexity of implementing the MDFT filter bank with PR is the same as the complexity of implementing the prototype filter. When the transition widths of the sub-channels need to be very narrow for various wireless communication applications, the prototype filter should have sharp transition width which results in a high order filter and hence high complexity MDFT filter bank with PR. Hence, the prototype filter of the MDFT filter bank is designed using the FRM approach [12,13], to reduce the implementation complexity. All the sub-filters of the FRM filter, Fa(z), FMa(z) and FMc(z) are realized as per the original work on FRM [12] using the Parks—McClellan method [32] which results in filters with linear phase property.

2.4.1. Design specifications

Maximum pass-band ripple: 0.004dB Minimum stop-band attenuation: 60dB Pass-band edge frequency: 0.062p Stop-band edge frequency: 0.06336p Number of channels: 8

When the prototype filter is designed using the minimax method for this set specifications, the filter order is found to be 2565. The number of multipliers required to implement the filter is found to be 1283. To reduce the complexity of the filter implementation, the prototype filter is realized as an FRM filter. The lengths of the sub-filters Fa(z), FMa(z) and FMc(z) are obtained as 221, 85 and 89 respectively. The total number of multipliers required to implement an FRM filter is the sum of the number of multipliers required to implement its sub-filters. This is obtained as 199. It is observed that the length of the sub-filter Fa(z) is high. In order to further reduce the complexity, the sub-filter Fa(z) is realized as an FRM filter. Hence Fa(z) is re-designed using two stage FRM as shown in Fig. 4 [33—36]. The number of multipliers to implement the two stage FRM prototype filter is now obtained as

Fig. 4. Structure of two stage FRM filter.

Table 1

Prototype filter of PR MDFT filter bank.

Max.PB ripple (dB) Min.SB attn. (dB) No. of multipliers

FIR 0.003952 -60 1283

FRM 0.007993 -59.38 147

complexity of the FIR and FRM filters for the same specifications. Fig. 6 shows the frequency response plot of the analysis filters of MDFT filter banks. The amplitude distortion function plot is shown in Fig. 7.

147. Hence the implementation complexity is reduced as compared to minimax method. The parameters of the FRM prototype filter with continuous coefficients are given in Table 1. The frequency response plot of the FRM prototype filter is shown in Fig. 5. p(z) in equation (10) which satisfies PR condition is also plotted in Fig. 5. Table 1 gives a comparison of the performances such as passband (PB) ripple and stopband (SB) attenuation and the implementation

3. Design of multiplier-free sharp MDFT filter banks with PR

The implementation complexity of the filter coefficients can be reduced by converting the continuous coefficients into discrete filter coefficients using SPT space. This leads to the replacement of multipliers by adders/subtractors and shifters in the filter implementation [7]. CSD representation is a special form of SPT representation and is a minimal one [9]. CSD is used for encoding a floating-point value into a two's complement representation. It is

MDFT Continuous Coefficients

MDFT after CSD

MDFT with Hybrid HSA-GSA

0.2 0.4 0.6 0.8

0 0.2 0.4 0.6 0.8

Fig. 5. Magnitude Response of FRM Prototype filter before and after optimization Hybrid HSA-GSA and Condition for PR (p(z)).

0 0.1 0.2 0.3 0.4 0.5

(o/2i[

Fig. 6. Frequency Response of analysis filters of MDFT PR filter bank (Continuous Coefficients, CSD Rounded coefficients and Optimized using Hybrid HSA-GSA).

having fewer non-zeros as compared to the two's complement form. In the CSD no two adjacent digits are non zero.

A fractional number g is represented in CSD format as [9].

g = X£dj2R-j (17)

where P is the word length of the CSD number, dj = {1,0, -1} and the integer R represents a radix point in the range 0 < R < P.

For a given word length, if the filter coefficients are rounded to the nearest CSD representation without any restriction in the number of non-zero bits, it will result in the filter with maximum precision in the CSD space for that word length. The CSD representation of a decimal number using n-bits of word-length cannot have more than (n + 1)/2 non-zero bits, often it is fewer. For the word-lengths of 12, 14 and 16 bits respectively, the maximum number of non-zero bits that will be present in the CSD representation of a number are 6,7 and 8 respectively. Here all the bits are used for the fractional part. A look-up-table approach is used to

0 0.1 0.2 0.3 0.4

Fig. 7. Amplitude distortion function plot of MDFT PR filter bank (Continuous Coefficients, CSD Rounded coefficients and Optimized using Hybrid HSA-GSA).

Table 2

A typical look up table entry.

Index CSD Equivalent Decimal Equivalent Number of non-zeros

6196 10-1000010-1010-1 0.3781 6

convert the filter coefficients into their CSD equivalents [23,37]. A typical CSD look-up-table contains index, CSD equivalent, decimal equivalent and the number of non-zeros as shown in Table 2. Table 3 shows the frequency performance parameters of the continuous coefficients and the maximum precision parameters of the CSD representation for different word lengths used. In the 12 bit CSD representation the implementation complexity is less but the filter performances are bad. The 16 bit CSD representation gives good filter performances but the implementation complexity is high. A 14 bit CSD representation is used as a compromise. The frequency response plot of the CSD represented prototype filter is also shown in Fig. 5. Fig. 6 also shows the frequency response plot of the CSD represented analysis filters of MDFT filter banks with PR. The amplitude distortion function plot of the CSD represented MDFT filter banks with PR is also shown in Fig. 7.

Thus we will get a multiplier-free MDFT filter bank with PR. Table 4 shows that the performance of the filter degrades, when the continuous filter coefficients are represented using CSD space. Hence meta-heuristic algorithms which will optimize the filter coefficients are used, to improve the performances of the CSD represented filter coefficients.

3.1. Objective function formulation

The CSD represented filter coefficients should be optimized to design the optimal multiplier-free FRM prototype filter and hence the MDFT filter banks with PR. Hence an objective function is to be formulated. The optimization of the CSD represented MDFT filter banks with PR is to minimize the objective function.

The problem is formulated as a multi objective function. Let Fp and Fs be the passband and stopband errors of the prototype filter. The amplitude distortion error is defined by Fdist, where Tdist(z) is defined by equation (7).

Fdist = max I Tdist (u)-1|

0 < u < p

Fp = max IIH(u)I- 1|

0 < u < up

Fs = max jH(u)I

Us < u < p

The constraint to minimize the average number of SPT terms using the penalty method [38] is given as

p(bH, Bh) = max(0, bH - BH)

where bH is the average number of SPT terms and BH is the upper bound.

In this paper, the number of non-zero SPT terms in the optimized multiplier-less FRM filter coefficients is variable, keeping the average number of non-zero SPT terms fixed. This method gives more flexibility in the design of discrete filters [37]. Thus after optimization, we will get a better filter bank in terms of performance and with reduced number of non-zero SPT terms.

The final objective function is

Minimize ô = ß^Fp + b2Fs + ß3Fdist + ß4p(b-H, Bh)

where b1 , b2, b3 and b4 are the weights, which define the relative importance given to each term in the objective function.

4. Optimization of multiplier-free sharp MDFT filter banks with PR using hybrid harmony-gravitational search algorithm

Meta-heuristic algorithms are used in this paper [10]. In the FRM prototype filter, the sub-filters are optimized jointly. All the subfilters are optimized simultaneously to get an optimal FRM filter response and hence an optimal MDFT filter bank with PR. The fitness function of all the algorithms are evaluated using the objective function. The various meta-heuristic algorithms used in this paper are Harmony Search algorithm (HSA), Gravitational Search algorithm (GSA), Artificial bee colony (ABC) algorithm, Genetic algorithm (GA) and Hybrid Harmony-Gravitational Search algorithm.

4.1. Optimization of MDFT filter banks with PR using HSA algorithm

Inspired by the music improvisation scheme, the Harmony Search algorithm (HSA) [14,15] was introduced by Geem and Lee for the optimization of mathematical problems. HSA is a music based optimization algorithm. The aim of the music is to search for a perfect state of harmony. The harmony in music is analogous to find the optimality in the optimization process. Each musician in the music performance plays a musical note at a time, and those musical notes together make a harmony. Similarly each variable in

Table 3

Frequency performances and complexity of the maximum precision CSD represented protype filter and MDFT filter bank for different word lengths.

Word-length

Prototype filter of PR MDFT FB

PR MDFT FB

Max.PB ripple (dB)

Min.SB attn. (dB)

Max.Amp.dist. (dB)

No. of adders due to SPT terms

Continuous coefficients 12 bits 14 bits 16 bits

0.007993 0.01677 0.009118 0.007872

-59.38 -52.08 -57.77 58.99

0.007986 0.01679 0.00902 0.007866

0 200 344 478

Table 4

Frequency performance and complexity of the CSD rounded FRM filter and MDFT filter bank with PR.

Prototype filter of PR MDFT FB PR MDFT FB Complexity

Max.PB ripple (dB) Min.SB attn. (dB) Max.Amp.dist. (dB) No. of structural adders No. of adders due to SPT terms Total no. of adders

Max.precision (7 SPTs) 0.009118 -57.77 0.00902 142 344 486

CSD rounded (3 SPTs) 0.02214 -44.45 0.02235 142 283 425

the optimization has a value at a time and those values will form a solution vector. The HSA explores the search space for finding the candidate solutions with good fitness value. The various phases involved in HSA are initialization of harmony memory, harmony improvisation, memory consideration, pitch adjustment, random selection, memory updates and termination [16,39,40]. The weights of the objective function for HSA are obtained as, b1 = 0.6, b2 = 2, b3 = 1 and b4 = 0.03 by trial and error method, in order to get the desired specifications. Termination is reached when the specified number of iterations are reached, otherwise steps 'Harmony improvisation' to 'Update the harmony memory' are repeated. Fitness function is evaluated after termination, for all harmony vectors which are retained and the best solution is taken to get the optimized filter coefficients.

The parameters of the HSA are taken as given below:

Harmony memory size: 50 Harmony memory considering rate: 0.95 Pitch adjusting rate: 0.01 Number of iterations: 1000

4.2. Optimization ofMDFTfilter banks with PR using GSA algorithm

GSA is a population based algorithm proposed by Rashedi in 2009 [17], based on Newtonian law of gravity and motion. The position of every mass represents a solution to the problem and the corresponding gravitational and inertial masses are determined by the fitness function. A mass or agent is formed by the CSD encoded filter coefficients. Each mass has four specifications: position, in-ertial mass, active gravitational mass and passive gravitational mass. Masses attract each other by the 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. The various steps involved in GSA are initialization of agents, fitness evaluation, gravitational constant update, acceleration of agents calculation, update the velocity and position of agents and termination [17,39,40]. The weights of the objective function for GSA are obtained as, b1 = 1, b2 = 2, b3 = 1 and b4 = 0.05 by trial and error method, in order to get the desired specifications. The program will be terminated when the maximum number of iterations is reached, otherwise steps 'Fitness evaluation' to 'Update the velocity and position of agents' will be repeated.

The parameters used for GSA are:

Gravitational constant: 100

Number of iterations (T): 1000

Number of agents: 50

4.3. Optimization ofMDFT filter banks with PR using ABC algorithm

The Artificial Bee Colony (ABC) algorithm [19—22] is introduced by Karaboga and Basturk. This algorithm contains employed bees, onlooker bees and scout bees. Employed bees will find a food source within the neighborhood of the food source in their memory. Onlooker bees will wait on the dance area to collect information from employed bees to select a food source. Scout bees will search randomly the environment surrounding the nest for new food sources. The possible solution to a problem to be optimized is the position of the food source. The quality of the solution represented by the food source is based on the amount of nectar of that food source. The nectar quality of a food source represents the fitness value of the corresponding solution. The main steps of the

algorithm [8] are initialization, employed bee phase, onlooker bee phase, scout bee phase and termination. The weights of the objective function for ABC algorithm are obtained as, b1 = 0.6, b2 = 2, b3 = 1 and b4 = 0.1 by trial and error method, in order to get the desired specifications. Termination is achieved after a maximum number of iterations is reached, otherwise steps 'Employed bee phase' to 'Scout Bee phase' are repeated. After the termination condition is satisfied, the food source with the best nectar quality is decoded using the look-up-table and the optimal filter coefficients are obtained.

The parameters of ABC algorithm are given below:

Population Size: 50 Limit cycles: 495 Number of iterations: 1000

4.4. Optimization of MDFT filter banks with PR using genetic algorithm

Genetic algorithm (GA) is a population based stochastic search algorithm which imitates the evolution of biological systems and is capable of finding global optimum in high dimensional multimodal search space [41]. GA models the evolution process of natural selection, where, in each generation, candidates are modified by genetic operations like crossover, selection and mutation. An integer coded genetic algorithm (GA) is proposed in [25]. The integer coded GA is used in this paper. The weights of the objective function for GA are obtained as, b1 = 1, b2 = 2, b3 = 1 and b4 = 0.1 by trial and error method, in order to get the desired specifications. The parameters of GA are given below:

Population Size: 50 Mutation rate: 0.01 Popkeep fraction: 0.5 Number of elite chromosomes: 5 Number of iterations: 1000

4.5. Proposed optimization of MDFT filter banks with PR using hybrid harmony-gravitational search algorithm

HSA and GSA give good performances with reasonable number of adders. Hence, a new hybrid algorithm to combine the qualities of these two algorithms is proposed in this work. The algorithms run in parallel and they are coupled in regular intervals. To regulate the interaction between the algorithms, a control parameter INV is used. The solutions in the population of both algorithms are grouped after INV number of iterations. For the next iteration the solutions to form the population are randomly selected from the group. The various steps involved in this algorithm are given below:

4.5.1. Initialization

The maximum number of iterations, HMS of HSA, PSHSA, harmony memory considering rate, pitch adjusting rate, number of agents of GSA, PSGSA, gravitational constant and control parameter, a, are set.

The continuous coefficient FRM prototype filter is designed and the initial solution vector of the hybrid algorithm is obtained by concatenating the CSD encoded coefficients of the model filter, masking filter and complementary masking filter. The initial solution vector is randomly perturbed to obtain the different solutions for the hybrid HSA-GSA algorithm. To have a wider search space, the initial number of solutions is taken as the integer multiple of the chosen population size of PSHSA + PSGSA.

Table 5

Hardware complexity in implementing MDFT filter bank with PR.

MDFT PR FB designed Multiplier-free MDFT PR FB with multipliers design (hybrid HSA_GSA)

and adders[6] [Proposed]

No. of multipliers 147 0

No. of adders 142 448

No. of LUTs required 13337 3584

for hardware implementation

4.5.2. Prioritised enlisting of solution vectors using the fitness function

The fitness function is evaluated for each solution vector, and the PSHSA + PSGSA number of best solutions will be passed to the next stage.

4.5.3. Initial population of HSA and GSA

Population of HSA is initialized by selecting PSHSA number of random filters from the group of filters generated in step 2. Population of GSA algorithm is initialized using the remaining PSGSA number of filters from the group.

4.5.4. 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 5. Otherwise go to step 7.

4.5.5. Grouping

Group the populations of the two algorithms.

4.5.6. Splitting

Split the grouped populations randomly into two groups of size N1 and N2, respectively.

4.5.7. Update the population

Update the populations using GSA and HSA algorithms in parallel. A new population of GSA is generated from the current population corresponding to GSA algorithm using steps 'Fitness evaluation' to 'Update the velocity and position of agents' of GSA. New populations of HSA are generated from the current population corresponding to HSA algorithm using steps 'Harmony improvisation' to 'Update the harmony memory' of HSA.

4.5.8. Update the best solution

The best solution of the hybrid algorithm can be replaced by the best solution in the current populations, 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.

4.5.9. Termination

The steps from 4 to 8 are repeated until the specified number of iterations are reached. The best solution in the memory of the

hybrid algorithm is taken as the optimum solution, if the algorithm is terminated.

The parameters of hybrid HSA-GSA algorithm are given below: PShsa : 25

Harmony memory considering rate: 0.97 Pitch adjusting rate: 0.01 PSgsa: 25

Gravitational constant: 100 a: 20 INV: 50

Maximum number of iterations: 1000

The frequency response plot of the FRM prototype filter optimized using hybrid HSA-GSA is also shown in Figs. 5 and 6 also shows the frequency response plot of analysis filters of MDFT filter banks optimized using hybrid HSA-GSA and Fig. 7 also shows the amplitude distortion function plot of MDFT filter banks optimized using hybrid HSA-GSA.

5. Results and discussion

Initially, for the given specifications, the prototype filter with linear phase is designed using both FIR and FRM techniques. The implementation complexity using FRM technique is found to be very low compared to that using FIR technique. The coefficients of the sub-filters Fa, Fma and Fmc of the FRM prototype filter are converted into CSD equivalent representation using a 14 bit look-up-table. The coefficients of Fa, Fma and Fmc are concatenated together and this is taken as the initial design vector for the optimization problem. The number of look-up-table (LUT) logic in the Field Programmable Gate Array (FPGA), for the design given in this paper, is calculated by implementing the filters on Xilinx ISE, by selecting the device as Spartan 3E. The total number of LUTs is found to be 13,337, which is mainly due to the multipliers and adders, when dedicated multiplier blocks are not used. Average number of SPT terms is the ratio of total number of SPT terms to represent all the filter coefficients to the number of filter coefficients. For the word-lengths of 12,14 and 16 bits respectively, the maximum number of non-zero bits that will be present in the CSD representation of a number are 6, 7 and 8 respectively. If the average number of SPT terms reduced, then the total number of SPT terms and hence the number of adders will be reduced. One adder requires 8 LUTs. Hence if number of adders is reduced then the number of LUTs will also be reduced. One multiplier requires 83 LUTs. Because of the large number of LUTs needed, all the filters are made multiplier-free by converting the filter coefficients into canonic signed digit (CSD) space. To improve the results of the CSD represented FRM prototype filter, meta-heuristic algorithms are used. It is observed that HSA and GSA give good performances with reasonable number of adders. Hence a hybrid HSA-GSA method is proposed in this paper to further reduce the number of adders. This makes the filters multiplier-free. The LUTs required reduces to 3584

Table 6

Performance parameters of the MDFT filter banks with PR using FRM.

Continuous coefficients CSD rounded cofficients HSA GSA ABC GA Hybrid HSA-GSA

Amplitude distortion (dB) 0.007986 0.02235 0.00895 0.00805 0.0142 0.015 0.00861

Passband ripple (dB) 0.007993 0.02214 0.0089 0.00806 0.0141 0.0151 0.00853

Stopband attenuation (dB) -59.38 -44.35 -58 -56.2 -56.6 -56.3 -57.35

Number of multipliers 147 0 0 0 0 0 0

No. of structural adders 142 142 142 142 142 142 142

Adders due to SPT terms 283 319 330 332 338 306

Total number of adders 425 461 472 474 480 448

as given in Table 5. The huge reduction in the number of LUTs naturally leads to lower power consumption and low chip area. Table 6 shows the performances of the MDFT filter banks with PR, which is optimized using hybrid harmony-gravitational search algorithm and other meta-heuristic algorithms. The hybrid HSA-GSA gives better performance in terms of number of adders required for the implementation of MDFT filter banks with PR. With hybrid HSA-GSA the number of adders reduces from 344 to 306. In this paper, all the simulations are performed on an Intel(R) core(TM) i5 processor operating at 2.4 GHz using MATLAB 7.10.0(R2010a).

6. Conclusion

In this paper, the design of a low complexity sharp MDFT filter bank with PR, is proposed. The continuous FRM prototype filter coefficients are represented in the discrete space using CSD. This degrades the performance of the filter banks. Hence to improve the performances of the filter bank, meta-heuristic algorithms are used. The hybrid HSA-GSA is observed to be better than other algorithms, to obtain the multiplier-free MDFT filter bank with PR using multistage FRM and with the least hardware complexity. The proposed design of low complexity sharp MDFT filter bank with PR using hybrid HSA-GSA, can lead to low power consumption, low chip area and high speed of operation, which are the major requirements in the upcoming applications such as software defined radio, wireless communication and portable computing systems.

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