Design of FIR Filter Using Biogeography Based ... - IEEE Xplore

2015 Second International Conference on Advances in Computing and Communication Engineering

Design of FIR Filter using Biogeography Based Optimization Mandeep Kaur

Urvinder Singh

Dilpal Singh

Chandigarh University, Gharuan Mohali, India [email protected]

Chandigarh University, Gharuan Mohali, India [email protected]

Chandigarh University, Gharuan Mohali, India [email protected]

particle swarm optimization (IPSO) and craziness based particle swarm optimization (CRPSO) whose results were much improved as compared to real code genetic algorithm (RGA), particle swarm optimization (PSO) and Park and McClellan algorithm (PM) [11, 12, 13]. The algorithms used to optimize problem function were basically adopted from nature and its habitants. Simon proposed another algorithm named biogeography based optimization (BBO) algorithm which has some unique features as compared to other biology based algorithms. This algorithm was demonstrated on real world sensor selection problem and it proved better than various other algorithms chosen for comparison [6]. The algorithm was also used for dynamic deployment of static and mobile sensors [7]. Du et al. implemented features of other heuristic algorithms i.e. evolutionary strategy and immigration refusal with BBO to modify the algorithm. The modified version of BBO performed better than original BBO when certain tests were implied over the new version BBO [8]. Ma and Simon proposed BBO with blended migration for constrained optimization problems and compared the modified BBO with GA and PSO [9]. BBO has been used to optimize the antenna and it provided superior results as compared to other optimization algorithms [18-24]. In another work same algorithm was compared with stud genetic algorithm (SGA) and standard particle swarm optimization 2007 (SPSO 07) where it was demonstrated that BBO performed better than other competing algorithms [14]. Another modification was proposed by Pattnaik et al. where clear duplicate operator was introduced into BBO and was tested over different benchmark functions. Enhanced BBO (EBBO) gave better results when compared to BBO and its other variants [10]. Recently, another modification of BBO was proposed as biogeography with chaos to improve the performance of BBO for multiobjective problems and it has been proven feasible [15].

Abstract—Digital signal processing (DSP) systems require filters to fulfill their needs for particular frequency characteristics. This paper presents designing and optimization of digital finite impulse response (FIR) highpass filter. Optimization is a process to bring out the best result for an objective function. BBO is the algorithm which is based on the biogeography of organisms in a habitat. The algorithm has certain modified variants such as BBO with chaos, enhanced BBO, blended BBO and BBO with immigration refusal (IR) which are used to optimize the filter and then compared with each other. The works concludes that out of various variants, blended BBO shines out with optimum results for a given set of parameters. The convergence profiles of these variants proved blended BBO as better converging variant of BBO. Keywords—FIR-finite impulse response;BBO-bio-geography based optimization; HSI- habitat suitability index;SIV-suitability index variables; IR-immigration refusal;

I.

INTRODUCTION

Filters are the electronic circuits which act as building blocks for all digital signal processing (DSP) system [1]. Digital filters are the filters which give output in digital form and are faster than analog filters when compared on the basis of performance, output and reliability. Digital filters can be classified into two types: Finite Impulse Response (FIR) and Infinite Impulse Response (IIR). FIR filters are more stable, free from phase distortion and less sensitive as compared to IIR filters [3]. Litwin explored the basics of digital filters and difference between FIR and IIR filters [1]. Further, Mastorakis et al. designed a two dimensional filter using Genetic Algorithm (GA) and compared it with some previous designs. The results showed that stability can be guaranteed in this method with simplification of the filter [2]. Karaboga and Cetinkaya designed digital FIR filter using differential evolution (DE) algorithm and its performance was compared with genetic algorithm and least squares method. It was found that DE calculated optimal results quicker than GA [3]. Ababneh et al. designed linear phase FIR filter with the help of particle swarm optimization (PSO) and GA where PSO outperformed GA [4]. Luitel designed linear phase FIR filter using PSO and differential evolution particle swarm optimization (DEPSO) and concluded that DEPSO proved to be a better choice for FIR filter in dynamic environment [5]. Mondal et al. designed linear phase highpass FIR filter by implementing improved 978-1-4799-1734-1/15 $31.00 © 2015 IEEE DOI 10.1109/ICACCE.2015.136

This paper presents various migration variants of BBO. The variants are enhanced BBO, blended BBO, immigration refusal BBO and BBO with chaos. Each variant tries its best to find the most optimum result i.e. fitness. All four variants are compared on the various platforms and conclusions are brought out. The paper has been divided into various sections. Section I gave a brief introduction about filters and BBO. Section II explains about FIR filter and the objective function. The following section i.e. section III explores the algorithm

312

III.

implied with its variants. Further section IV discusses the results with graphs and tables. Meanwhile section V tells about conclusion withdrawn. II.

Biogeography based optimization is the algorithm which is based on concept of biogeography i.e. distribution of organisms over space and time. The geographical area is named as “island” or “habitat”. Each area is characterized by some features which include position, availability of vegetation, rainfall etc, and are known as habitat suitability index (HSI). The number of species which a habitat hosts also helps in determining the value of HSI because it is in direct proportion to the HSI’s value. The habitability is defined by certain variables referred as suitability index variables (SIVs). These can be termed as independent variables while HSI are to be termed as dependent variables. The habitats having high value of HSI resist changes but are ready to dispense their features with the habitat with low HSI. Similarly habitats with low HSI are more vulnerable to changes and are ready to accept features from habitats with high HSI. The entering of species into the habitat is termed as immigration and the exiting of species from the island is termed as emigration.

HIGHPASS FIR FILTER DESIGN

Finite impulse response (FIR) filter are the digital filter which always produce a stable output. FIR filter’s response can be given as given in (1).

H ( z )  h(0)  h(1) z 1  ...  h( N ) z  N or, H ( z ) 

N

 h( n) z

N

(1) (2)

n 0

where h(n) is the impulse response and N is order of the filter. Filter’s length can be determined from N+1. The designing process is chosen so as to obtain impulse response of filter. Filter’s frequency response can be formulated as, N

H (e

jwk

)   h( n)e

 jwk n

The algorithm works with two operators namely Migration operator and Mutation operator. Migration operator adjusts the habitat by moving the SIVs of better habitat into it which are chosen according to their immigration rate (λ) and emigration rate (μ) thus improving the habitat and further improving the generation. Mutation operator modifies the SIVs according to probability which is based on λ and μ parameters. Fig. 1 shows the variation of emigration rate with immigration rate with the number of species in single habitat. ‘I’ is the maximum immigration rate whereas ‘E’ is the maximum emigration rate, which are usually taken as one. ! is the maximum number of species a habitat can handle making the immigration rate minimum i.e. 0 and emigration rate to maximum i.e. 1.

(3)

n 0



where  = , H(  ) is the Fourier transform vector.  The frequency in [0, π] is sampled with N points. The error function refers to the difference between the magnitude of an ideal filter and the designed filter. Thus, it can be formulated as, N

Error  max{[|| H i (e jwi ) |  | H d (e jwi ) ||]}

(4)

n 0

E ()  G()[ Hi (e jwi )  H d (e jwi )]

BBO AND ITS VARIANTS

(5)

where G(ω) is the function which provide weights for different frequency bands and is termed as weighting function;  (  ) is the frequency response of ideal filter and it can be defined as, 1 ,1 ≤ ω ≤ 

H i (e jw )   0 , ℎ

(6)

where  is cut-off frequency of the designed filter and  (  ) is the frequency response of designed filter. The error function needs to be minimized so fitness function has been reformulated as

J  max p (| E () |  p )  max p (| E() |  s ) (7)

Fig. 1. Relationship of immigration and emigration rates with number of species in one habitat

where  and  are the pass band and the stop band ripple values;  and  are the normalized edge frequencies in the pass band and stop band respectively.

The values of λ and μ can be calculated as,



The error function in (7) gives a generalized error function which has to be minimized by proposed BBO. This function will be used to optimize the filter’s performance.

EK P

  

K  P

(8) (9)

where E=maximum value of λ; I= maximum value of μ; P=population size; K=number of species of the K-th

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individual. The pseudo-code of the migration and mutation operators is given in following part.

the habitat [15]. The migration operator for this modified version can be given with the following pseudo-code.

Algorithm1:Pseudo-code for migration operator 1. For r =1 to P

Algorithm3:Pseudo-code for BBO with chaos For r =1 to P

2.

Choose " with probability proportional to λ"

1.

3.

If " is chosen

2.

Choose " with probability proportional to $"

4.

For s=1 to n

3.

If rand < $"

5.

Choose  with probability proportional to μ"

4.

6.

If  is selected

5.

For s =1 to n Choose H with probability proportional to μ" If rand < μ"

7.

Choose a random SIV σ from 

6.

8.

Substitute a random SIV in " with σ

7.

Choose random SIV σ from 

8.

" =   (&.'('*+- ))

9. 10.

End if

9.

End for End if

10.

12. End for

11.

11.

Use $" and μ" and calculate %"

4.

Choose SIV " (s) with probability similar to %"

5.

If " (s) is selected

6.

Substitute " (s) with randomly generated SIV

7.

End if

8. 9.

End if

 Enhanced BBO: Standard migration operator creates similar habitats which lead to duplication of habitats. The duplication ultimately reduces the diversity of the habitat. To avoid this harmful redundancy, a modified operator termed as ‘modified clear duplicate operator’ is introduced to clear the duplicates into basic code which increases the computation time of the algorithm but with a considerable increase in its performance. This operator clears the duplicate habitats by selecting a value between minimum and maximum value of SIVs [16]. The proposed migration operator can be coded in the pseudo-code as follows.

For s=1 to n

3.

End for

12. End for

Algorithm2:Pseudo-code for mutation operator 1. For r=1 to P 2.

End if

End for End for

Algorithm4:Pseudo-code for enhanced BBO

Here P=size of population; n=number of variables in a habitat; $" =immigration rate; μ" =emigration rate ; " =emigrating habitat;  =immigrating habitat; %" =immigration probability; σ=immigrating SIV.

1.

A. BBO Variants BBO’s migration operator exploits the habitats by finetuning the SIVs of that habitat. This means previous solutions are updated while mutation operator is the operator which generates new SIVs. In this paper, the variants have been modified in the migration operator of the algorithm because the major work in BBO is performed by the migration operator. So, mutation operator is not modified here to check the performance of migration operator variants. The variants analyzed in this paper are given below with their modified migration operator.  BBO with Chaos: In this variant of BBO, real coding is adopted. The migration operator is adopted from strength parent evolutionary algorithm (SPEA). The individuals possessing high HSI share their characteristics with chaos and improve the diversity of

For r = 1 to P

2.

For s = s+1 to P

3.

If " = 

4.

" = (min (" ) + (max ( ) - min (" ))*rand)

5.

End if

6.

End for

7.

End for

Here rand= any random integer between 0 and 1 

314

Blended BBO: In this variant of BBO, new solutions are made up of two parts: feature from other solution, and feature from itself. This technique has been adopted from blended crossover from GAs. Thus the upcoming generation consists of own parent’s feature as well as migrating SIV’s feature. A multiple ‘α’ has been introduced whose value is between 0 and 1 or it could be any random or deterministic value [14]. The pseudo-code for this BBO variant can be presented as given in following part.

The values of various parameters used for construction of filters are:

Algorithm5:Pseudo-code for blended BBO 1.

For r =1 to P

2.

Choose " with probability proportional to λ"

3.

If " is chosen

4.

For s=1 to n

5.

Choose  with probability proportional to μ"

6.

If  is chosen " = α (" ) + (1- α) 

7. 8. 9. 10.

End if End for End if

Here α=0.5.  BBO with immigration refusal (IR): Migration operator improves the SIVs by immigrating high HSI habitats to low HSI habitat. But there are chances that there is immigration from low-fitness solution to high fitness solution thus ruining the fitness of island. To tackle this problem, fitness of immigrating island is taken into consideration and is rejected if it is lower than the fitness of emigrating island [18]. The code for this variant is as follows: Algorithm6:Pseudo-code for BBO with IR For r= 1 to P

2.

Choose " with probability proportional to λ"

3.

If " is chosen

4.

For s=1 to n

5.

Choose  with probability proportional to μ"

6.

If fitness (" ) < fitness ( )

7.

Substitute " with 

8.

End if

9.

End for

10.



stopband frequency ( ) = 0.65;



passband ripples ( ) =0.1;



stop band ripples ( ) = 0.01;



Population size in BBO = 40;



Maximum number of generations = 200;



Dimension = 11;



Probability of mutation = 0.01;



Elitism parameter=2.

B. Comparison on basis of convergence curves To compare each modified variant on the basis of convergence profile, convergence curves have been drawn in Fig. 3 by plotting the average fitness values for a given number of iterations. The iteration number is selected as 200 for each modified variant.

End if

Fig. 3 concludes that blended BBO converges at a much faster pace as compared to other variants of BBO. Also, it attains a much lower value of fitness function than other modified variants in less number of iterations. Meanwhile, enhanced BBO produces a sub-optimal value for the fitness function and converges at a slower pace than blended BBO. The reason for this is that enhanced BBO remove the habitats which result in reduction of diversity. BBO with IR attained a low value which may be due to the refusal of SIVs. Furthermore, BBO with chaos did not come up with good results which might be due to the intake of parent SIVs but in the exponential form as given in Algorithm 3. Thus, blended BBO brought out better results as compared to other variants of BBO i.e. enhanced BBO, BBO with immigration refusal and BBO with chaos.

11. End for IV.

passband frequency ( ) = 0.75;

The code has been evaluated and run on MATLAB 2012b version on core (TM) i3 processor, 3.00 GHz with 3 GB RAM. Fig. 2 shows the magnitude response of all the four variants i.e. BBO with chaos, enhanced BBO, blended BBO and BBO with IR over same objective function of high pass filter. The frequency parameter has been normalized in the figure. Table I shows the optimized coefficients for the designed FIR filter having order 20. Meanwhile, Table II presents the best, worst and average fitness value of all the modified variants. The values from table and the graph suggest that best frequency response has been produced by the modified variant named blended BBO. The frequency response graph suggests that other variants have same response in transition band but there is an improved response in stop band region by blended BBO. This is because it makes use of two features: the parents and the immigrating habitat’s features which results in least fitness value and least ripples as seen in Fig. 2. BBO with chaos also uses both features same as blended BBO but with exponent which makes the difference in computation thus making blended BBO the best variant of BBO in optimization of FIR high pass filter.

11. End for

1.



RESULTS AND COMPARISONS

A. Study of magnitude response of FIR highpass filter In order to bring out the results and compare the effectiveness of all modified variants, FIR filter has been designed using MATLAB. The designed filter has an order of value 20 thus the number of coefficients is 21. The number of sampling points is taken as 256 and the sampling frequency is chosen as 1 Hz. Each modified variant of BBO has been run 40 times to get optimum solution.

315

TABLE I.

OPTIMUM COEFFICIENTS OF BBO VARIANTS

h(N)

BBO with chaos

Enhanced BBO

Blended BBO

BBO with IR

h(1)=h(21)

- 0.0172191591766668

0.0000000000000000

0.000808732857412956

-0.00135941944083007

h(2)=h(20)

0.0000000000000000

0.0200000000000000

0.0188318492348605

0.0000000000000000

h(3)=h(19)

0.0181391428940318

-0.0442181142862795

-0.0407884508250713

0.0368361088292808

h(4)=h(18)

0.0000000000000000

0.00403604349295295

0.00852423732438976

-0.0138256096358770

h(5)=h(17)

0.0000000000000000

0.0244260303745714

0.0380592939106986

0.0000000000000000

h(6)=h(16)

0.0531976107751360

-0.0521174300633237

-0.0592740463931042

0.0450974662849897

h(7)=h(15)

-0.0768191037339498

0.0544200854784211

0.0464247337218026

-0.0867299707114245

h(8)=h(14)

0.0000000000000000

0.0300000000000000

0.0325544425941253

0.000000000000000

h(9)=h(13)

0.207746275696171

-0.160498653596845

-0.134532387021371

0.191909339460792

h(10)=h(12)

-0.275936628556418

0.247392588790788

0.270756153116677

-0.237300806497940

h(11)

0.286528896803798

-0.307971789732571

-0.278649888479125

0.313559044851423

TABLE II.

COMPARISON OF BEST WORST AND MEAN VALUES

BBO Variant

Best Value

Worst Value

Mean Value

BBO with chaos

0.2498

2.6187

0.6551

Enhanced BBO

0.0621

2.6796

0.2456

Blended BBO

0.0616

2.5451

0.2175

BBO with IR

0.1877

2.4374

0.4633

Fig. 2. Magnitude and frequency graph for different variants of BBO.

Fig. 3. Convergence profile of different variants of BBO.

316

V.

CONCLUSION

[14] H. Ma and D. Simon, “Blended biogeography-based optimization for constrained optimization,” Engineering Applications of Artificial Intelligence, vol. 24 , pp. 517525, 2011. [15] X. Wang and Z. Xu, “Multi-Objective Algorithm based on Biogeography with Chaos,” International Journal of Hybrid Information Technology, vol. 7 , pp. 225-234, 2014. [16] S. Singh and G. Sachdeva, “Yagi-Uda Antenna Design Optimization for Maximum Gain using different BBO Migration Variants,” International Journal of Computer Applications, vol. 58 , pp. 86-96, 2012. [17] S. Singh, Shivangna and S. Tayal, “Analysis of Different Ranges for Wireless Sensor Node Localization using PSO and BBO and its variants,” International Journal of Computer Applications, vol. 63, pp. 31-37, 2013. [18] U. Singh, H. Kumar and T.S. Kamal, “Design of Yagi-Uda Antenna Using Biogeography Based Optimization,” IEEE Transactions on Antennas and Propagation, vol. 58, pp. 3375-3379, 2010. [19] U. Singh, H. Kumar and T.S. Kamal, “Linear Array Synthesis using Biogeography Based Optimization,” Progress In Electromagnetics Research M, vol. 11, pp. 2536, 2010. [20] U. Singh and T.S. Kamal, “Design of non-uniform circular antenna arrays using biogeography based optimization,” Microwaves, Antennas & Propagation, IET, vol. 5, pp. 1365-1370, 2011. [21] U. Singh and T.S. Kamal, “Optimal synthesis of thinned array using biogeography based optimization,” Progress In Electromagnetics Research M, vol. 24, pp. 141-155, 2012. [22] U. Singh and T.S. Kamal, “Synthesis of thinned planar circular array antennas using biogeography based optimization”, Emerging Technology Trends in Electronics, Communication and Networking (ET2ECN),2012, IEEE, pp. 1-5, 2012. [23] U. Singh and T.S. Kamal, “Concentric Circular Antenna Array Synthesis using Biogeography Based Optimization,” Majlesi Journal of Electrical Engineering, vol. 6, pp. 4855, 2012. [24] U. Singh, “Design of Concentric Circular Antenna Array using Biogeography Based Optimization,” International Journal of Enhanced Research in Science Technology & Engineering, vol. 2, pp. 119-125, 2013.

In this paper, FIR high pass filter has been optimized with various modified variants namely, BBO with chaos; enhanced BBO; blended BBO; BBO with IR. The simulation results showed that out of all the variants, blended BBO gives out better results. The convergence profile also proved blended BBO as a better option for the proposed objective. For future work, modifications can also be brought out in the mutation operator of BBO and these variants can be applied to problems other than filters.

VI. [1] [2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

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