A novel accelerated artificial bee colony algorithm for ...

Multidim Syst Sign Process DOI 10.1007/s11045-015-0352-5

A novel accelerated artificial bee colony algorithm for optimal design of two dimensional FIR filter Supriya Dhabal1 · Palaniandavar Venkateswaran2

Received: 17 November 2014 / Revised: 17 July 2015 / Accepted: 29 July 2015 © Springer Science+Business Media New York 2015

Abstract This paper presents a novel approach for the design of two-dimensional (2D) Finite Impulse Response (FIR) filters. The design of FIR filters is generally non-differentiable, multimodal and higher dimensional; especially for 2D filters. A large number of filter coefficients are optimized either using constrained or unconstrained optimization approach. Due to the large number of constraints, traditional design methods cannot produce optimal filters required for some crucial applications. This makes meta-heuristic algorithms as good alternatives for addressing such constraints more efficiently. In order to improve the performance of 2D filters, we propose an Accelerated Artificial Bee Colony algorithm, termed as AABC algorithm. The earlier reported ABC based methods perform the modification of a single parameter of the solution in each cycle. But in this proposed AABC algorithm, we have adopted multiple parameters change of search equation at each step. This in turn improves the convergence speed of the algorithm by three times than the classical ABC algorithm and two times with respect to recently developed CABC method. In order to achieve better exploration behaviour of abandoned bees, we have also introduced a change during the initialization strategy of scout bees in the proposed AABC algorithm. The efficiency and robustness of the proposed algorithm are demonstrated by comparing its performance with classical Genetic Algorithm (GA), Particle Swarm Optimization , ABC and CABC methods. Keywords

B

FIR filter · Mini-max design · 2D filter · ABC · CABC · GABC

Supriya Dhabal [email protected] Palaniandavar Venkateswaran [email protected]

1

Department of Electronics and Communication Engineering, Netaji Subhash Engineering College, Kolkata, West Bengal 700152, India

2

Department of Electronics and Tele-Communication Engineering, Jadavpur University, Kolkata, West Bengal 700032, India

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Multidim Syst Sign Process

1 Introduction Recently, two dimensional (2D) FIR filter design has received considerable attention in various signal processing systems due to its inherent stability, linear phase characteristics, and low coefficient sensitivity (Lu 2002; Lu and Hinamoto 2006; McClellan 1973; Pei and Shyu 1995). 2D filters have found extensive applications in the domain of biomedical imaging, seismic data processing, satellite image processing, and pattern recognition (Kockanat et al. 2012; Dhabal and Venkateswaran 2014). The design techniques of 2D filter are grouped into two categories: generalized design procedure, where arbitrary frequency responses are approximately evaluated without any structural constraints of the filter and specialized design techniques, where filters utilize some essential features to simplify the problem of implementation. The first approach includes mini-max design (Lu 2002; Lu and Hinamoto 2006), frequency sampling, windowing techniques and least pth norm FIR filter design proposed by Abatzoglou and Jaffer (1995). Other methods like separable approach of 2-D filter design, McClellan transformation based design of 2D filter from 1D prototype developed by McClellan (1973) and nonrectangular transformations falls into the category of specialized designs. Previously reported works effectively utilized different iterative methods to design 2D filter in mini-max sense. These are semi-definite programming (Lu 2002), cone programming (Lu and Hinamoto 2006) and least squares design methods (Zhao and Lai 2013; Lai and Cheng 2007). Most of these algorithms do not have accurate control on pass-band and stop-band regions and also produces sub-optimal impulse response of 2D filters. Recently, evolutionary algorithms have become the most effective alternative to design optimal 2D filters (Boudjelaba et al. 2014; Tzeng 2007). It is observed that the recently reported adaptive Genetic Algorithm (GA), by Boudjelaba et al. (2014), cannot be used effectively for the design of higher order filters because it leads to sub-optimal solution and also it requires a large computational time. Therefore, the design task of higher order 2D filter has become a challenging research area as large number of independent coefficients are need to be optimized in multimodal error surface of 2D filter. For example, to design a Type1 FIR filter of size N1 × N2 , the order of optimized variables is [(N1 + 1) × (N2 + 1)] /4 and large numbers of constraints are imposed on a dense set of frequency grid points over the baseband [− π, π] × [− π, π]. In this work, the design task of 2D FIR filter has been formulated as an unconstrained optimization problem and solved using different ABC based algorithms (Karaboga and Basturk 2007; Karaboga and Akay 2009; Zhang et al. 2013; Zhu and Kwong 2010; Gao et al. 2013). The efficiency and robustness of the proposed AABC algorithm are demonstrated by comparing its performance with classical GA, Particle Swarm Optimization (PSO), ABC and CABC methods. Numerical comparisons show that the proposed algorithm yields a better approximation of desired filter although a long time is required as compared to some traditional design approaches like frequency transformation, frequency sampling and window method, Semi-definite or cone programming. In comparison to other evolutionary approaches like GA (Boudjelaba et al. 2014) or PSO, the proposed AABC has the following advantages: (1) easy to implement with same number of tuning parameters as in ABC, (2) feasibility of higher order 2D filter design in dense set of frequency grid points with satisfactory tolerance, (3) the new initialization strategy of scout bees help in better exploration, (4) reduces the number of functional evaluations, and hence the computational time, to find an acceptable solution. Rest of the paper is organized as follows: The formulation of design problem to obtain 2D FIR filter is presented in Sect. 2. Sect. 3 presents a brief overview of ABC algorithm.

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Multidim Syst Sign Process

The novel approach and the design flow of proposed method are explained in Sect. 4. The detailed simulation results with three different design examples are given in Sect. 5. The tuning of different control parameters are also discussed in this section elaborately. Finally, conclusions and further research scopes are given in Sect. 6.

2 Problem formulations The frequency response of a 2D FIR filter with impulse response h (n 1 , n 2 ), n 1 = 0, 1, . . . , (N1 − 1), n 2 = 0, 1, . . . , (N2 − 1) is given by Boudjelaba et al. (2014) H (ω1 , ω2 ) =

N 1 −1 N 2 −1 

h(n 1 , n 2 ) e− j (n 1 ω1 +n 2 ω2 )

n 1 =0 n 2 =0

= M(ω1 , ω2 )e

−j



N1 −1 2

ω1 +

N2 −1 2

ω2



(1)

where M(ω1 , ω2 ) is the magnitude response of 2D filter. For quadrantal-symmetric filter with N1 , N2 odd, the impulse response holds following relationship (Tzeng 2007):  h

N2 − 1 N1 − 1 − k1 , − k2 2 2





 N1 − 1 N2 − 1 − k1 , + k2 2 2   N1 − 1 N2 − 1 + k1 , − k2 =h 2 2   N1 − 1 N2 − 1 + k1 , + k2 for 1 ≤ k1 =h 2 2   ≤ (N1 −1) 2 , 1 ≤ k2 ≤(N2 −1) 2 =h

(2)

Consequently, the magnitude response of the filter is written as M(ω1 , ω2 ) =

N 1 −1 N 2 −1 

a(n 1 , n 2 ) cos(n 1 ω1 ) cos(n 2 ω2 ).

(3)

n 1 =0 n 2 =0

The coefficients a(n 1 , n 2 ) are related to the filter impulse response h (n 1 , n 2 ) by

   N1 − 1 N2 − 1 N1 − 1 N2 − 1 , a(0, n 2 ) = 2h , , − n2 , 2 2 2 2   N1 − 1 N2 − 1 − n1, , a(n 1 , 0) = 2h 2 2   N1 − 1 N2 − 1 N1 − 1 a(n 1 , n 2 ) = 2h − n1, − n2 , for n 1 = 1, 2, . . . , 2 2 2 N2 − 1 n 2 = 1, 2, . . . , . 2 

a(0, 0) = h

(4)

Further, the magnitude response of the filter given in (3) can be written as a closed form expression M(ω1 , ω2 ) = P T (ω1 , N1 )A P(ω2 , N2 ) (5)

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Multidim Syst Sign Process

where the superscript  T  denotes the matrix  transpose ⎤ operation, P(ω, N ) is a matrix given ⎡ 1 cos (ωi1 ) · · · cos  N 2−1 ωi1 ⎢ 1 cos (ωi2 ) · · · cos N −1 ωi2 ⎥ 2 ⎢ ⎥ by P(ωi , Ni ) = ⎢ . ⎥, i ∈ (1, 2) and A is a real matrix .. .. .. ⎣ .. ⎦ . . .  N −1 1 cos (ωim ) · · · cos 2 ωim related to h (n 1 , n 2 ) with an appropriate size. Let us consider Md (ω1 , ω2 )represents the desired magnitude response of the 2D filter as a function of the frequency variables ω1 and ω2 (ω1 , ω2 ∈ [− π, π]). Therefore, the design task is to find each coefficient a(n 1 , n 2 ) of matrix A such that the function M(ω1 , ω2 ) approximates the desired magnitude response Md (ω1 , ω2 ). This approximation can be achieved by minimizing the weighted square error between M(ω1 , ω2 ) and Md (ω1 , ω2 ), mathematically expressed as: m

E=

m

1  2 

w jk



2  2     Md (ω1 j , ω2k ) − M(ω1 j , ω2k ) =  Md − P T A P ◦ W 

j=1 k=1

F

(6)

  where W = w jk ≥ 0 ( j = 1, 2, . . . , m 1 ; k = 1, 2, . . . , m 2 ) are weight coefficients matrix, the notations ∗ F and  ◦ denote the matrix Frobenius norm and Hadamard matrix product respectively (Zhao and Lai 2013).

3 Overview of ABC algorithm The ABC algorithm is a recently introduced swarm based meta-heuristic algorithm proposed by Karaboga and Basturk (2007). Here, a colony consists of three sets of bees: employed bees, onlookers and scout bees. The role of employed bees is to collect nectar from the food sources, and convey the information about its own food source with onlooker bees. Onlooker bees wait inside the hive to collect the knowledge from the employed bees about their discovered food sources. The employed bees share their information about food sources by dancing in the designated dance area inside the hive. The Scout bees will always search for new food sources near the hive. Therefore, employed and onlooker bees perform the job of global exploitation and scout bees help in exploration (Karaboga and Basturk 2007; Karaboga and Akay 2009). In ABC algorithm, first half of the colony consists of employed bees and another half contains the onlookers. Here, the position of a food source corresponds to a possible solution of the problem and the nectar amount of each food source indicates the quality of the solution. The four main phases of this algorithm is presented below:

3.1 Initialization phase The algorithm starts with a population of randomly distributed S N solutions. Each solution  X i = xi,1 , xi,2 , . . . , xi,D is a D-dimensional vector, where D is the number of optimization parameters. The initial food sources are randomly generated using the equation (Karaboga and Basturk 2007)  xi j = xmin, j + rand(0, 1) xmax, j − xmin, j (7) where i = 1, 2, . . . , S N , j = 1, 2, . . . , D, rand(0, 1) ∈ [0, 1] is a random number and xmin, j , xmax, j are respectively the lower, upper boundaries of the solution space for j th dimension.

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Multidim Syst Sign Process

3.2 Employed bee phase All employed bees remember their previous best position and find a new food source within its neighbourhood. If the new food source has equal or better quality than the old source, then replace old source by new one; otherwise the old source is retained. The employed bees determine the new food source based on the following equation (Karaboga and Basturk 2007; Karaboga and Akay 2009):  vi, j = xi, j + φi, j xi, j − xk, j (8) where k is a randomly selected food source, i = k ∈ {1, 2, . . . , S N }, j ∈ {1, 2, . . . , D} and φi, j is a random number within the range [-1 1]. The quality of food source is calculated by f iti (X i ) using the formula (Zhang et al. 2013)   1 f (X i ) ≥ 0 1+ f (X i ) , f iti (X i ) = (9) 1 + | f (X i )| , f (X i ) < 0 where f (X i ) is the objective value of X i .

3.3 Onlooker bee phase An onlooker bee chooses a food source depending on the probability value pi associated with that food source SN  pi = f iti f it j (10) j=1

where f iti is the fitness value of solution i and update the source food using (8).

3.4 Scout bee phase If the productivity of a food source cannot be improved within a predefined number of cycles, then it is abandoned and the employed bee becomes a scout bee. The scout bee replaces abandoned source xi by randomly generated solution using (7).

4 The proposed accelerated ABC algorithm In order to improve the performance of classical ABC, frequently used strategy is to modify the search equation of employed and onlooker bee phase. Among the various search strategies suggested by researchers, the recently reported best performed methods are global-best guided ABC (GABC) proposed by Zhu and Kwong (2010) and CABC by Gao et al. (2013). In GABC, the exploitation of bees is improved by sharing the information of global best solution with a randomly selected candidate solution. As in the case of PSO, the GABC method also suffers from an oscillation as the direction of two terms in the search equation may be in opposite (Gao et al. 2013). During our simulation, it is also observed that CABC method outperformed the GABC for this particular design problem. Hence, the search equation for employed and onlooker bee phase is selected as in CABC in the proposed method:  vi, j = xr 1, j + φi, j xr 1, j − xr 2, j (11)

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Multidim Syst Sign Process

where r 1 = r 2 = i ∈ [1, S N ] and φi, j ∈ [−1, 1] is a random number. Here the new food source is found based on the difference between two randomly selected solutions which is also controlled by a random step size. Therefore, the searching for new solution is done by both employed and onlooker bees using (11) which is comparable to the self-adapting mutation process of Differential Evolution (Karaboga and Basturk 2008). In all the previously reported ABC algorithms like classical ABC, GABC, CABC, only one dimension of the food position is modified at each repeat. Therefore, the search process becomes too slow and it takes a long time to find optimal solution; especially for higher dimensional problem (Alizadegan et al. 2013). From the simulation results, as shown under Sect. 5.2, it is observed that multiple changes of parameter values have accelerated the search process during the implementation of higher dimensional filters. Hence, instead of single change in parameters, we proposed multiple changes that helped to accelerate the search procedure to a great extent, besides maintaining the robustness of the algorithm. However, the number of parameter changes in every step should be selected properly in order to ensure the true global minima. The experimental results reveal that upto eight simultaneous changes in parameter values, for most of the design examples, have produced significant improvements in convergence speed in addition to ensuring good quality of solutions. Generally, in most of the ABC algorithms, scout bees control the exploration process very effectively by selecting random food sources as in (7). However, in few cases, the employed bees may also get trapped in local optimum that actually lies closer to the global solution. Therefore, by simply introducing a random food source in the population, the employed bee may not reach true global solutions quickly. To overcome this issue, a new initialization strategy is imposed for scout bees as:   xmin,j + rand(−1, 1) xmax,j − xmin,j iter < max_iter/δ xij = (12) iter ≥ max_iter/δ xbest,j + rand(−1, 1) xworst,j − xbest,j where xbest, j and xwor st, j are two parameters taken from the best and worst solution vectors found so far assuming a random value of j ∈ [1, D], iter is the current iteration, max_iter is the maximum number of iterations and δ is a scale factor which helps to determine the scout bee as a purely random solution or based on best and worst candidate. Here the best and worst solutions are achieved by searching the bees with minimum and maximum fitness as in (6). Therefore, the proposed AABC algorithm modifies the search strategy of honey bee swarms in all the three phases of basic method, towards finding the better optimal solution for multimodal, nonlinear, higher dimensional practical problem. The design goal is to optimize the coefficients of matrix A i.e. a(n 1 , n 2 ), based on the flowchart presented in Fig. 1a, b, and subsequently to compare the effectiveness of ABC and AABC algorithms as illustrated in Fig. 1a, b respectively.

5 Experimental results In order to validate the performance of proposed AABC method, here we design different types of 2D FIR filters of various orders. The complete simulation work is carried out using MATLAB 2009 on Genuine Intel(R) Core 2 Duo CPU E7300 @2.66GHz, 2GB RAM. It is known to all that the maximum and average ripple components appear in pass-band and stopband regions are two important parameters to determine whether the designed filter is good or not in terms of quality of the filter. Therefore, all the associated algorithms are executed for 30 independent runs and the following performance parameters are considered: (a) maximum and average ripple in pass-band and stop-band regions, (b) average computational time and

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Multidim Syst Sign Process

Start

(a)

(b)

Assume N 1, N 2 , weight matrix response

Assume N 1, N 2 , weight matrix response

and desired

Initialize SN, limit, max_iter, SN/2 food sources using (7) Employed bee phase: update food sources using (8) and evaluate fiti

Start and desired

Initialize SN, limit, max_iter, SN/2 food sources using (7) and no of parameters to be optimized.

Employed bee phase: update food sources using (11) and evaluate fiti

Onlooker bee phase: calculate pi , select food sources based on pi and update using (8)

Onlooker bee phase: calculate pi , select food sources based on pi and update using (11)

Scout bee phase: create new food sources using (7) Scout bee phase: create new food sources using (12) No iter =max_iter?

No iter =max_iter?

Yes Design 2D FIR filter

Yes Design 2D FIR filter

Fig. 1 a Flowchart of existing ABC algorithm. b Flowchart of proposed AABC algorithm

Table 1 Control parameters of AABC, CABC, ABC, PSO and GA Algorithm

Parameters

AABC, CABC, ABC

NP = 50, limit = 50, maximum NOFE = 105 (for 7 × 7, 11 × 11 and 15 × 15) and NOFE = 1.25 × 105 (for 19 × 19 and 23 × 23)

PSO

NP = 500, c1 = c2 = 2.05, ω = 0.9 → 0.4, NOFE = 105

GA

NP=100, Roulette wheel selection(probability1/3), two points crossover(cr = 0.9), Gaussian mutation (rate 0.01), NOFE = 105

average number of functional evaluations NOFE to reach an acceptable threshold value of fitness function. The maximum and average ripple of the filter is obtained by assuming a dense set of frequency grid points over the baseband region [− π, π] × [− π, π] and the maximum and mean difference value from the desired specifications are measured. In this simulation work, the population size is chosen as NP = 2 × S N = 50 for ABC, CABC and AABC; 500 for PSO and 100 for GA; the limit for abandonment of scout bees is 50. The remaining control parameters for different algorithm are listed in Table 1. The maximum NOFE for each algorithm is selected as 105 for lower order filters, 1.25 × 105 for higher order filters and all the methods are executed for 30 independent runs. It is also verified in ABC based algorithms that, assuming larger values of NP and limit will not significantly improve the performance of designed filter.

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Multidim Syst Sign Process

The selection of grid points plays an important role on the performance parameters of 2D filters. With the increase of grid points, frequency resolution of the filter increases but the cost function becomes highly multimodal. Therefore, to obtain the frequency response of the filter, a suitable set of uniform frequency points is selected over the area [− π, π] × [− π, π]. Throughout the experiment, for lower order filters (i.e. 7 × 7 and 11 × 11) 4N1 × 4N2 grid points are selected, whereas for higher order filters (i.e.15 × 15, 19 × 19 and 23 × 23) it is assumed as 2N1 × 2N2 . The weighting factor W is chosen as 1 in pass-band, 3/8 in transition-band and stop-band regions, so that the minimum of objective function (6) is attained.

5.1 Design examples of 2D filter and comparisons In this section, we experimentally analyze the performance of the proposed method using three different types of 2D filters; namely circularly symmetric, diamond-shaped and elliptical low-pass filters. Example 1 In this example, circular low-pass filter is designed with the desired magnitude response ⎧  ⎨ 1 P : 0 ≤ ω2 + ω2 ≤ ωp 2 1 D (ω1 , ω2 ) = (13) ⎩ 0 S : ω ≤ ω2 + ω2 , |ω | ≤ π, |ω | ≤ π s 1 2 1 2 where ωp = 0.425 π and ωs = 0.575 π, N1 × N2 = 7 × 7, 11 × 11, 15 × 15, 19 × 19, and 23 × 23. The magnitude response of circularly symmetric 2D filters, for 15 × 15 and 23 × 23, are shown in Fig. 2a, b respectively. For other sizes of circular filter, the magnitude response have been designed using the proposed AABC, and the results obtained are summarized in Table 2. From Fig. 2, it is clearly observed that the stop-band attenuation obtained for 15 × 15 is below −30 dB, while for 23 × 23 it is below −50 dB. In addition to maximum ripple in pass-band and stop-band, the average ripple is also reduced with the increase of filter order. In Table 3, performance of the proposed AABC method for the design of circularly symmetric filter is compared with frequency transformation, frequency sampling, window method, Pei and Shyu (1995), Lu et al. (1990), Lu and Hinamoto (2006), and other optimization based methods like GA and PSO. Here, we have listed the simulation results only for 7×7 and 23 × 23 filters. Although the optimization based methods require large computational time than non-optimization based approaches, they result in better pass-band and stop-band regions in terms of maximum and average ripple. Among the optimization techniques used here, all the ABC based methods such as classical ABC, CABC and the proposed AABC require less computational time than GA or PSO. For lower order filter i.e.7 × 7, the obtained results for ripple are almost same but for higher order filter like 23×23, the GA or PSO based methods cannot reach the global minimum due to the large number of filter coefficients. It is also observed that, for filters with order larger than 15 × 15 the GA or PSO cannot reduce the cost function significantly due to highly multimodal, nonlinear objective functions. But ABC based methods exhibit their ability to optimize the higher dimensional, multimodal problems. Another one crucial observation is that, the proposed AABC method produces better result with less simulation time and also reduces the average NOFE by approximately three folds than classical ABC.

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Multidim Syst Sign Process

(a) 0

Magnitude ( dB )

−20 −40 −60 −80 −100 1 1 0.5

0

0 −0.5

ω1

(b)

−1

−1

ω2

0

Magnitude ( dB )

−20 −40 −60 −80 −100 1 1 0 0

ω1

−1

−1

ω2

Fig. 2 a Magnitude response of 15 × 15 circularly symmetric 2D filter. b Magnitude response of 23 × 23 circularly symmetric 2D filter

Example 2 In this example, diamond-shaped filter is designed with the desired magnitude response  1 P:0 ≤ (|ω1 | + |ω2 |) ≤ ωp D (ω1 , ω2 ) = (14) 0 S : ωs ≤ (|ω1 | + |ω2 |) , |ω1 | ≤ π, |ω2 | ≤ π where ω p = 0.8 π and ωs = 0.96 π, N1 × N2 = 7×7, 11×11, 15×15, 19×19, and 23×23. The magnitude response for 15 × 15 and 23 × 23 is depicted in Fig. 3a, b respectively. For other sizes of diamond shaped filter, the results obtained are summarized in Table 4. From Fig. 3, it is clearly observed that the stop-band attenuation obtained for 15 × 15 is around −30 dB, while for 23 × 23 it is −40 dB approximately.

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Multidim Syst Sign Process Table 2 Best results for circular symmetric 2D FIR filter using proposed AABC algorithm; NP = 50; limit = 50; maximum NOFE = 105 (for 7 × 7, 11 × 11 and 15 × 15) and NOFE = 1.25 × 105 (for 19 × 19 and 23 × 23) N1 × N2

Minimum fitness (E)

Maximum ripple Pass-band

Average ripple Stop-band

Pass-band

Stop-band

7×7

1.7378

0.13416

0.14679

0.006546

0.025412

11 × 11

1.1178

0.07961

0.08896

0.004903

0.012803

15 × 15

0.13414

0.06114

0.05480

0.004013

0.00651

19 × 19

0.076389

0.02529

0.02935

0.001189

0.00310

23 × 23

0.057065

0.015526

0.016897

0.000457

0.001527

Table 5 shows a detailed comparison study of the proposed AABC for the design of diamond shaped 2D filter. For 7 × 7, the obtained results for ripple are almost same but for 23 × 23, the GA or PSO based methods cannot reach the global minimum. The proposed AABC method produces much better result with less simulation time and also reduces the average NOFE by approximately two folds than CABC and by three folds than classical ABC. Example 3 This example deals with the design  of elliptic filter with the similar specifications as in Zhao and Lai (2013). The pass-band ωp and stop-band frequency (ωs ) are assumed as follows:   ω21 ω22 ωp = (ω1 , ω2 )| + ≤1 (0.4 π)2 (0.5 π)2   ω21 ω22 ωs = (ω1 , ω2 )| + ≥ 1, |ω1 | ≤ π, |ω2 | ≤ π (15) (0.45 π)2 (0.55 π)2 The weights on the pass-band, stop-band and transition-band are selected as 1, 0.25 and 0 respectively. The magnitude response, for N1 × N2 = 19 × 21, is shown in Fig. 4. From Fig. 4, it is clearly observed that the stop-band attenuation obtained is around −20 dB. Simulation results of elliptic filter obtained using the proposed AABC is compared with CABC, classical ABC and recently reported matrix iterative methods (Zhao and Lai 2013) in Table 6. The proposed method produces much better optimal filter with less computational time and average NOFE than CABC and ABC. The other approaches like GA or standard PSO cannot reach the global minimum due to higher dimensional and multimodal problem space. Eventually the matrix iterative algorithms I and II presented by Zhao and Lai (2013) take very less simulation time and fewer functional evaluations to optimize the filter but the maximum ripple in pass-band and stop-band shows that the results exhibited by these methods are suboptimal solutions. Therefore, the proposed AABC method leads to much better result with less simulation time and it also reduces the average NOFE by approximately three folds than CABC and by four folds than the classical ABC.

5.2 Effects of simultaneous parameter changes on the performance of AABC The improvement in convergence profile for selecting different number of simultaneous changes of parameters values have been illustrated in Fig. 5a, b and the corresponding simu-

123

0.13416 0.13416 0.13359 0.13416 0.16182

CABC

ABC

GA

PSO

Frequency transformation

23 × 23

0.13416

AABC

7×7

0.028165 0.015688 0.085760 0.01930

Pei and Shyu (1995)

Lu et al. (1990)

Lu and Hinamoto (2006)

0.015894

Frequency transformation 0.106400

–

GA,PSO

Frequency sampling

0.015898

Window method

0.015835

Lu and Hinamoto (2006)

ABC

0.1769

Lu et al. (1990)

CABC

0.43054

Pei and Shyu (1995)

0.015526

0.16117

Window method

AABC

0.21092 0.43439

Frequency sampling

0.02270

0.11266

0.035096

0.090588

0.051664

0.016763

–

0.02132

0.01691

0.016897

0.1787

0.27218

0.18439

0.27011

0.21416

0.16146

0.14679

0.147272

0.14679

0.14679

0.14679

–

0.00168

0.0012231

0.002294

0.000521

0.001241

–

0.000532

0.000497

0.000457

–

0.033979

0.012339

0.032445

0.008595

0.01187

0.006545

0.00656

0.006546

0.006546

0.006546

Pass-band

Pass-band

Stop-band

Average ripple

Maximum ripple

Method

N1 × N2

–

0.004551

0.006791

0.00404

0.00445

0.007318

–

0.00291

0.00153

0.001527

–

0.046145

0.074295

0.040943

0.035495

0.074722

0.025412

0.025498

0.025412

0.025412

0.025412

Stop-band

– – –

< 0.1 < 0.1 < 0.1

20.02

–

–

– –

< 0.1 < 0.1 < 0.1

– –

–

89,050

60,300

41,310

< 0.1

23.95

17.44

13.05

–

–

< 0.1

0.36

53,558 –

5.281

124,164

16,440

8410

6040

Average NOFE

< 0.1

12.59

1.777

0.937

0.812

Time (S)

Table 3 Comparisons with different algorithms for circular symmetric 2D FIR filter;—means that no feasible solutions were found; bold cases indicate best results

Multidim Syst Sign Process

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Multidim Syst Sign Process

(a) 0

Magnitude ( dB )

−20

−40

−60

−80 1 1 0.5

0 0 −0.5

ω1

−1

−1

ω

2

(b) 0

Magnitude ( dB )

−20 −40 −60 −80 −100 1 0.5

1 0.5

0

0

−0.5

ω

1

−0.5 −1

−1

ω

2

Fig. 3 a Magnitude response of 15 × 15diamond shaped 2D filter. b Magnitude response of 23 × 23 diamond shaped 2D filter

lation results are summarized in Table 7. Figure 5a indicates the change in fitness values for circularly symmetric filter with N1 × N2 = 23×23 and Fig. 5b represents the plot for 15×15 diamond shaped filter. From these figures it is obvious that, with the increase in number of parameter changes, the initial convergence process is accelerated significantly up to a certain extent; faster convergence in initial searching is observed for maximum eight parameters change at the same time. Beyond these, i.e. the simultaneous changes of ten parameters in each repeat will slow down the initial convergence process because the higher number of parameter changes jump out of true global minima at the early stages of optimization. But for higher order filters, during the last phase of searching, faster convergence may happen till 14 parameters change as shown in Table 7. Here the simulation time and NOFE are listed

123

Multidim Syst Sign Process Table 4 Best results for diamond shaped 2D FIR filter using proposed AABC algorithm; NP = 50; limit = 50; maximum NOFE = 105 (for 7 × 7,11 × 11 and 15 × 15) and NOFE = 1.25 × 105 (for 19 × 19 and 23 × 23) N1 × N2

Minimum fitness (E)

Maximum ripple Pass-band

Average ripple Stop-band

Pass-band

Stop-band

7×7

3.487

0.16486

0.22337

0.012568

0.025732

11 × 11

2.4714

0.07971

0.12328

0.005888

0.017325

15 × 15

1.6919

0.061375

0.08982

0.004662

0.011502

19 × 19

0.23094

0.041446

0.05780

0.00235

0.005302

23 × 23

0.19347

0.00694562

0.047571

0.000465

0.002986

based on the final stages of optimization for both circularly symmetric (15 × 15 and 23 × 23) and diamond shaped filters (7 × 7 and 23 × 23). For each experiment AABC algorithm was executed for 30 times and it was terminated when it reached the global minima within a gap of 10−3 . For lower order filters, like 7 × 7, significant improvement in computational time and NOFE is achieved up to eight parameters (randomly selected) at the same time. But for higher order filters, due to large number of unknown parameters, very small amount of improvement in simulation time is observed at the last few hundred iterations also. Therefore, the optimal number of simultaneous parameters update completely depends upon the order of filter.

5.3 Effects of N P on the performance of AABC In this section, circular and diamond shaped 2D filters with different dimensions are used to investigate the impact of number of populations (NP). AABC is executed for 30 different trials on each of these examples, and the mean values of the final results are listed in Table 8. From Table 8, we can observe that NP can influence the NOFE and hence simulation times. When NP is less or equal to 30, due to the poor explorations of employed and onlooker bees, we obtain a slower convergence or no feasible solutions in most of the runs. For almost all examples considered here, the best computational performance is achieved when it lies in the range of 40–60. Thereafter, further increase in NP leads to high explorations behavior of bees and hence lower convergence speed and more number of functional evaluations. Thus, throughout the simulations, the value of NP is roughly fixed at 50 for all design examples.

5.4 Effects of δ in scout bee phase on the performance of AABC When only one parameter in employed/onlooker bee phase is modified in each repeat like basic ABC algorithm or the number of parameter changes is very small, the selection of δ in equation (12) can improve the performance of AABC significantly. Table 9 shows the simulation results for different values of δ; assuming two parameters change in each iterations. The maximum number of iterations is fixed at 2000 for 7 × 7 and 2500 for 23 × 23 filters and the average values of 30 different simulations with random seeds are exposed here. The constant δ is varied from 1 to 40 for both circular symmetric and diamond shaped filters with different orders. The average execution time and NOFE is produced to select a

123

123

23 × 23

–

0.0072415

0.11484

0.052076

GA,PSO

Frequency sampling

Pei and Shyu (1995)

Lu et al. (1990)

0.39241

Lu et al. (1990)

0.0071958

0.32976

Pei and Shyu (1995)

0.0069583

0.15912

Frequency sampling

ABC

0.16486

PSO

CABC

0.164207

GA

0.0069456

0.16486

ABC

AABC

0.16486

0.16486

AABC

7×7

0.16464

0.17835

0.07934

–

0.047782

0.048027

0.047571

0.39972

0.3554

0.32493

0.22337

0.22366

0.22337

0.22337

0.22337

0.001078

0.001046

0.000551

–

0.000476

0.000465

0.000465

0.031221

0.001885

0.010053

0.012568

0.012575

0.012568

0.012568

0.012568

Pass-band

Pass-band

Stop-band

Average ripple

Maximum ripple

CABC

Method

N1 × N2

0.010119

0.029484

0.004478

–

0.004790

0.003115

0.002986

0.084952

0.084218

0.06668

0.025736

0.025829

0.025736

0.025732

0.025732

Stop-band

–

< 0.1

– –

0.87 < 0.1

– –

–

97,890

58,740

< 0.1

26.056

15.187

43,880

< 0.1 12.906

– –

< 0.1

42,690

129,813

10,380

6090

4440

Average NOFE

4.2376

12.813

1.1126

0.6532

0.5782

Time (S)

Table 5 Comparisons with different algorithms for diamond shaped 2D FIR filter;—means that no feasible solutions were found; bold cases indicate best results

Multidim Syst Sign Process

Multidim Syst Sign Process

0

Magnitude ( dB )

−20 −40 −60 −80 −100 1 0.5

1 0.5

0 0

−0.5

ω

−0.5 −1

1

−1

ω2

Fig. 4 Magnitude response of 19 × 21elliptic filter

Table 6 Comparisons with different algorithms for elliptic filter with N1 × N2 = 19 × 21, N P = 50 ; limit = 50; maximum NOFE = 1.25 × 105 ;—means that no feasible solutions were found; bold cases indicate best results Method

Maximum ripple Pass-band

Time (sec)

Average NOFE

Stop-band(dB)

AABC

0.12018

20.228

8.206

21765

CABC

0.12321

20.213

14.162

53751 88590

ABC

0.12605

19.692

20.145

GA,PSO

–

–

–

–

Zhao and Lai (2013) Algorithm I

0.2039

8.498

< 0.1

39

Zhao and Lai (2013) Algorithm II

0.2039

8.497

< 0.1

37

suitable value of δ. It is obvious from Table 9 that when number of parameter modifications is very small, approximately δ ≤ 4, leads to slower convergence whereas selecting higher values δ ≥ 5 accelerates the searching procedure significantly; therefore the number of cost functional evaluations is also reduced gradually with the increase in δ. But when the number of parameter changes was fixed at optimal position, like 8 parameters for 7 × 7 diamond shaped filter, the variation of δ from 1 to 40 did not show any significant improvement in performance of the algorithm. It is also noted that, for a fixed value of δ if we increase the max_iter then execution time and hence NOFE, is increased because with the increase in max_iter the step size is reduced gradually which leads to slower convergence of the algorithm.

123

Multidim Syst Sign Process Fig. 5 a Effect of different number of parameters changes for 23 × 23 circularly symmetric filter. b Effect of different number of parameters changes for 15 × 15 diamond shaped filter

(a)

5

10

2 Param 4 Param

4

10

6 Param 8 Param

3

10

10 Param 2

Fitness

10

1

10

0

10

−1

10

−2

10

0

1

2

3

4

5

6 4

x 10

No of functional evaluations

(b) 103 2 Param 4 Param 6 Param 8 Param 10 Param

2

Fitness

10

10

1

0

0.5

1

1.5

No of functional evaluations

123

2

2.5 4

x 10

3.6386

3.714

4.0522

4.1625

14

16

18

20

3.9658

8

3.5375

3.9908

6

3.6195

4.1094

4

10

4.9002

12

6.2126

2

28,700

28,120

26,147

25,602

25,192

24,625

26,230

27,460

29,160

36,180

47,370

13.6726

13.5071

13.2454

13.1227

13.2928

13.2662

15.3090

16.1218

17.3656

19.5126

24.9844

44,293

44,797

44,785

44212

46,762

47,540

49,260

51,650

57,050

68,200

88,460

–

–

0.7232

0.6857

0.65

0.5758

0.5266

0.5290

0.5288

0.6078

1.2543

Time (S)

–

–

4400

4051

3920

3792

3722

3962

4185

5090

8753

NOFE

Time (S)

Time (S)

NOFE

N1 × N2 = 7 × 7

N1 × N2 = 23 × 23

N1 × N2 = 15 × 15

NOFE

Diamond

Circular

1

No. of parameter changes

13.7844

12.5404

12.3741

12.2290

12.3007

12.3286

12.603

12.8782

13.9466

16.8344

24.1842

Time (S)

44,630

43,570

42,807

42,107

43,120

44,150

45,990

47,770

52,970

65,030

95,000

NOFE

N1 × N2 = 23 × 23

Table 7 Effect of parameter changes simultaneously for circular and diamond filter; N P = 50; limit = 50; maximum NOFE = 105 (for 7 × 7) and NOFE = 1.25 × 105 (for 23 × 23);—means that not applicable; bold cases indicate best results

Multidim Syst Sign Process

123

Multidim Syst Sign Process Table 8 Simulation results for varying number of populations (NP); limit = 50; maximum NOFE = 105 (for 7 × 7 and 15 × 15) and NOFE = 1.25 × 105 (for 23 × 23); —means that no feasible solutions were found; bold cases indicate best results NP

Circular

Diamond

N1 × N2 = 7 × 7

N1 × N2 = 15 × 15

N1 × N2 = 23 × 23

N1 × N2 = 23 × 23

Time (S)

NOFE

Time (S)

NOFE

Time (S)

NOFE

Time (S)

NOFE

10

–

–

–

–

–

–

–

–

20

–

–

–

–

–

–

–

–

30

1.704

11,910

6.688

37,560

21.381

62,064

17.04

63,480

40

0.6906

4928

3.7808

21,184

13.178

38,648

10.1626

37,888

50

0.7222

5170

4.1062

23,110

14.987

44,870

12.0466

44,720

60

0.7814

5616

4.8658

27,528

18.578

53,724

14.5346

53,892

70

0.875

6286

5.7906

32,732

22.409

63,616

16.6624

62,048

80

1.0248

7360

6.4812

36,640

23.271

72,720

19.5378

72,624

90

1.097

7902

7.3032

41,004

28.318

81,396

21.9906

81,900

100

1.2718

9140

8.2372

45,920

31.781

90,680

24.1656

90,240

Table 9 Simulation results for varying δ; no. of parameter changes=2; NP = 50; limit = 50; maximum NOFE = 105 (for 7 × 7) and NOFE = 1.25 × 105 (for 23 × 23);—means that no feasible solutions were found; bold cases indicate best results δ

Circular

Diamond

N1 × N2 = 7 × 7

N1 × N2 = 23 × 23

N1 × N2 = 7 × 7

N1 × N2 = 23 × 23

Time (S)

Time (S)

Time (S)

NOFE

Time (S)

NOFE -

NOFE

NOFE

1

-

-

-

-

2.5002

20850

-

2

5.5096

45,110

–

–

2.3686

20,090

–

–

3

3.7124

34,320

–

–

2.2158

18,430

29.8906

105,340

4

2.8096

23,400

25.156

97,570

1.9592

16,290

30.0936

100,690

5

1.7124

14,290

24.7408

95530

0.7094

5890

23.1688

89,260

10

1.706

14,280

23.119

83,670

0.9314

7740

20.4096

73,340

15

1.2282

10,210

23.4436

82,000

0.9122

7530

23.4812

81,580

20

1.0904

9150

20.900

77,990

0.8718

7210

19.9906

75,200

25

0.975

8080

22.0218

73820

0.6906

5830

22.0874

74,990

30

0.9534

8020

22.131

74040

0.6906

5810

21.2468

78,180

40

0.8466

7140

20.1906

74,150

0.728

6100

20.2374

69970

5.5 Effects of limit on the performance of AABC In this section, four different sizes of 2D filters are used to investigate the impact of limit. They are 7 × 7 and 23 × 23 circular symmetric and diamond shaped filter. Keeping other parameters as a constant, the AABC algorithm is executed on each of these design examples and the mean values of the final results are presented in Table 10. From Table 10, we can observe that limit can influence the results. When limit is fixed at 80, we obtain a faster

123

Multidim Syst Sign Process Table 10 Simulation results for different setting of limit; no. of parameter changes=2; NP = 50; δ = 20; maximum NOFE = 105 (for 7 × 7) and NOFE = 1.25 × 105 (for 23 × 23); bold cases indicate best results limit

Circular

Diamond

N1 × N2 = 7 × 7

N1 × N2 = 23 × 23

N1 × N2 = 7 × 7

N1 × N2 = 23 × 23

Time (S)

NOFE

Time (S)

NOFE

Time (S)

NOFE

Time (S)

NOFE

10

1.1314

9250

21.8282

83,510

0.9562

7970

18.5312

71,480

20

0.7874

6480

17.6656

67,750

0.6278

5220

17.0654

65,800

30

0.8124

6680

17.881

68,390

0.6158

5150

16.859

651,40

40

0.8032

6520

17.925

68,680

0.6502

5470

17.1624

66,210

50

0.8064

6610

17.6718

67,100

0.6312

5310

17.1406

66,210

60

0.7722

6320

17.644

67,620

0.6186

5180

17.2062

66,390

70

0.803

6540

17.7092

67,480

0.5968

4980

17.2126

66,500

80

0.7656

6300

17.5158

66910

0.6312

5270

16.9282

65,380

90

0.7874

6470

17.775

68,060

0.653

5450

17.0812

65,830

100

0.7968

6510

17.6936

67,960

0.628

5230

17.3378

65,890

150

0.7968

6510

17.8152

68,110

0.631

5300

17.0406

65,820

200

0.8156

6720

17.5154

67,110

0.6376

5320

19.2346

64,960

250

0.8062

6620

17.781

67,990

0.6626

5530

19.3438

65,730

300

0.8188

6740

17.6748

67,740

0.6562

5460

19.5282

65,680

convergence speed on the circular symmetric filters. For the diamond, the faster convergence speed is achieved for 70–80. Beyond this range of limit, the algorithm does not provide any significant improvement in convergence speed. Therefore, in our simulations, the control factor limit is set in the range of 50–100 for all design examples.

5.6 Improvement in convergence and robustness of AABC Figure 6a shows the convergence process of the five algorithms for circularly symmetric filter with 15 × 15, while for diamond shaped filter it is illustrated in Fig. 6b with 23 × 23. We obtained the similar characteristic curves for other dimensions, but for simplicity only two graphs have been demonstrated here. During the first few thousands functional evaluations, AABC converges faster than CABC, classical ABC, GA and PSO. In comparison to GA and PSO, ABC based methods find out best optimal solution; during the complete search process, the graphs of AABC, CABC and ABC is under the GA, and PSO indicating the performance are better. Figure 6b also clearly demonstrates that for higher order filters, GA or PSO cannot escape from local minimum once it is trapped. Finally, in order to demonstrate the robustness of proposed AABC than that of CABC and ABC, the comparison results between these three algorithms are presented in Tables 11, 12. The results for circularly symmetric filter are given in Table 11 whereas Table 12 shows the results for diamond shaped filter. Here the NOFE for different design cases are fixed beforehand and the average values of pass-band and stop-band ripple are analysed. For low order filter i.e. 7 × 7, each run is repeated for 25000 NOFE and for 15 × 15 and 23 × 23, it is selected as 50,000 and 75,000 respectively. It can be clearly observed from Tables 11, 12 that AABC performs better than ABC and CABC in terms of best, worst and mean fitness values

123

Multidim Syst Sign Process Fig. 6 a Convergence curves for 15 × 15 circularly symmetric filter. b Convergence curves for 23 × 23 diamond shaped filter

(a)

PSO 6

GA

10

ABC CABC AABC

4

Fitness

10

2

10

0

10

−2

10

0

2

4

6

8

10 4

No of functional evaluations

x 10

(b) 105 PSO

GA 4

10

ABC CABC AABC

3

Fitness

10

2

10

1

10

0

10

0

2

4

6

No of functional evaluations

8

10

x 10

4

as well as standard deviation is also reduced extensively. Therefore the proposed AABC algorithm resulted in a significant improvement during the design of 2D filters in terms of convergence, robustness and also solution quality.

123

Multidim Syst Sign Process Table 11 Comparison of different algorithms on the average values of pass-band ripple for circularly symmetric 2D filters in terms of best, worst, mean and standard deviation (SD); NP = 50; limit = 50; bold cases indicate best results N1 × N2

Algorithm

Best

Worst

Mean

SD

NOFE

7×7

AABC

0.006546

0.006546

0.006546

0

25,000

CABC

0.006546

0.006546

0.006546

0

15 × 15

23 × 23

ABC

0.006547

0.006549

0.00648

6.4072e−07

AABC

0.0040138

0.0041141

0.0041139

8.7509E−09

CABC

0.0041107

0.0041182

0.0041149

1.9165e−06

ABC

0.0041329

0.0042729

0.004222

5.6557e−05

AABC

0.0004577

0.0004834

0.0004679

6.2004E−06

CABC

0.0004976

0.0007351

0.0006176

5.9657e−05

ABC

0.0009562

0.0045163

0.0022229

7.8811e−04

50,000

75,000

Table 12 Comparison of different algorithms on the average values of stop-band ripple for diamond shaped 2D filters in terms of best, worst, mean and standard deviation (SD); NP = 50; limit = 50; bold cases indicate best results N1 × N2 7×7

15 × 15

23 × 23

Algorithm

Best

Worst

Mean

SD

NOFE 25,000

AABC

0.025732

0.025736

0.025734

7.1192e−18

CABC

0.025732

0.025736

0.025734

7.1192e−18

ABC

0.025736

0.025739

0.025736

1.5381e−06

AABC

0.011502

0.011704

0.011703

4.8936e−007

CABC

0.011695

0.011711

0.011704

3.4853e−006

ABC

0.011753

0.012247

0.011812

1.6225e−04

AABC

0.0029864

0.003186

0.0030503

5.0325e−005

CABC

0.0031159

0.003773

0.0034645

2.0384e−04

ABC

0.0047901

0.020069

0.0086514

3.3731e−03

50,000

75,000

6 Conclusions In this paper, a new swarm-based algorithm, namely AABC is proposed for solving higher dimensionality, multimodal problem more efficiently. The proposed modification suggested here can accelerate the convergence speed of the classical ABC by three times without introducing any new control parameter. From the simulation results, it was verified that the recently reported CABC method performs better than other ABC based methods available till now. In order to improve the performance of CABC further, the proposed method is presented. One more tuning is performed in scout bee phase to improve the exploration behaviour of the swarm. To demonstrate the performance of the proposed AABC, some traditional methods, and few optimization based methods like GA, PSO and classical ABC were tested for three different types of 2D filter design problems. The experimental results show that the proposed AABC performs better with respect to robustness, ability to jump out of the local optimum and also to accelerate the convergence process of both classical ABC and CABC. Hence, the proposed optimization approach can be used more effectively for the solution of many real world higher dimensional and multimodal problems. The realization of multiplier-less 2D FIR filters using the proposed hybrid method is left as a future work.

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Multidim Syst Sign Process

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Multidim Syst Sign Process Supriya Dhabal received the B.Tech degree in Electronics and Communication Engineering from Kalyani Govt. Engineering College, Kolkata, West Bengal, India in 2006 and ME degree in Electronics and Tele-Communication Engineering from Jadavpur University, Kolkata, West Bengal, India in 2008. In 2008 he joined Tata Consultancy Services, Kolkata as an Assistant System Engineer and at the mid of 2009 he joined Netaji Subhash Engineering College, Kolkata as an Assistant Professor. His research interests include efficient design of digital filter and applications of digital filter in several fields of Signal Processing and Communications.

Palaniandavar Venkateswaran has been working as an Associate Professor in the Dept. of Electronics & Tele-Communication Engg. (ETCE), Jadavpur University (JU), Kolkata, India since October 2001 and has been promoted as a Professor in 2009. He has published over 70 papers in various National / International, Journal / Conference Proceedings. His fields of interest are Computer Communication, Microcomputer Systems and Digital Signal Processing (DSP). He is a Senior Member of IEEE (USA).

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