Design of Digital Band Pass FIR Filter Using Heuristic Optimization Technique

8/20/2019 Design of Digital Band Pass FIR Filter Using Heuristic Optimization Technique

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International Journal of Scientific Research Engineering & Technology (IJSRET), ISSN 2278 – 0882Volume 4, Issue 6, June 2015

Design of Digital Band Pass FIR Filter UsingHeuristic Optimization Technique

Shalika Sharma 1, Balraj Singh 2

1, 2(Department of Electronics and Communication Engineering (ECE),

Giani Zail Singh PTU Campus, Bathinda, Punjab, India)

ABSTRACT This paper presents the efficient way of designing FiniteImpulse Response (FIR) band pass filter usingDifferential Evolution algorithm. In this variousstrategies of differential evolution have been attemptedfor the designing of FIR band pass filter. Impulseresponse coefficients of designed FIR filters have beenrepresented as the sum or differences of powers of two

individuals. The performance of filters depends upon itsmagnitude response and its hardware cost. DE is theheuristic approach which helps in minimizing possiblynon-linear and non-differentiable functions. This methodconverges faster with more certainty. This includesinitialization, mutation, crossover and selection. Thisalso requires few control parameters like population size,mutation factor and crossover rate. DE is robust, easy touse but sometimes causes instability problems.

Keywords - Differential Evolution algorithm, FIR filters, Magnitude response, ripple magnitude.

I. INTRODUCTION

Digital signal processing has a wide range ofapplications like in communication fields, patternrecognition, image processing etc. It becomes more

popular application in electronic engineering because ofsome advantages like more flexibility, good

performance, better time response, environmentalstability and lesser cost of equipment production. Themain component of DSP is digital filters and it is definedas these are those systems on which mathematicaloperations are performed on sampled discrete timesignals to boost some aspects of that signal. This is thefilter which helps in performing all basic functions ofDSP like filtering, addition of signals or separation ofsignals etc.Processing of digital data is accomplished by widevarieties of digital filter and these are classified into twomain categories: one is Infinite Impulse Response (IIR)Filters and other is Finite Impulse Response (FIR)Filters. FIR filters are those filters whose impulse

response is of finite duration. These filters are non-recursive type filters because their output depends on

present and past input values whereas IIR filters arethose whose impulse response is of infinite duration andthey are recursive type filters because their outputdepends on past output values. FIR filters has followingadvantages as compared to IIR filters: (1) finite impulseresponse (2) easy to optimize (3) linear phase (4) more

stable.Optimization is a procedure of finding and comparingvarious solutions until no better solution can be found.To design FIR band pass filter, various optimizationtechniques have been developed. Optimizationtechniques are divided into two categories: one is basedon classical optimization and other is nature basedoptimization. Classical optimization includes directsearch method and gradient method whereas nature

based optimization includes Genetic Algorithm (GA),Particle Swarm Optimization (PSO) and DifferentialEvolution (DE). There are various drawbacks of gradientmethod such as function must be continuous,differentiable and completely defined and drawback ofdirect search method is that this method is valid for localoptimum point because validation of global optimum isdifficult. Hence nature based optimization is used

because it is based on random variables.GA is capable of searching multi-dimensional andmultimodal decision spaces. This has also been used forthe optimization of complex and discontinues functions.Sometimes GA attempts very low convergence and highdimensional optimization problems. Due to the existenceof various local optima, GA traps in the local optima ofobjective function.To overcome these disadvantages PSO has beendeveloped by Kennedy and Ebehart. PSO is simple,easy, fast and robust. It has some shortcomings also suchas convergence behaviour which depends upon its

parameters. If the parameters have been changedincorrectly which results in divergent particle trajectorieswhich traps into local minima due to which convergence

problem has been occurred which may lead to highdimensional optimization problem.



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To overcome convergence property, DE has beendeveloped by Storn and Price in 1995. DE is capable forfinding global minimum regardless of the initial

parameter values. It is simple and robust. It has fewcontrol parameters. It has parallel processing nature andfast convergence. DE has the ability to provide multiplesolutions in a single run. DE is also capable of findingoptimal solution for non-linear optimization problems.The objective function for the design of digital filtersinvolves various control parameters which are highlynon-uniform, non-linear, non-differentiable, multimodalin nature. These objective functions cannot be optimized

by classical optimization because of somedisadvantages: (1) highly sensitive when no. of variablesincrease which leads to increase the size of space (2)frequent convergence to local optimal solution ordivergence or revisiting the same sub optimal solution(3) requires continues and differentiable objective cost

function (4) problem of convergence and complexity. Toovercome these drawbacks, traditional and classicalmethods of optimization have been developed in whichDE is one of them which is meta heuristic approachwhich helps in minimizing non-linear and non-differentiable functions and this has been developed byStorn and Price in 1995.

II. FIR FILTER DESIGN PROBLEMS

FIR filters are known as non-recursive type filters whichare described by the difference equation expressed as:

= ( −)−1

=0 (1)Where is the set of filter coefficients. The output y(n)is the function of only input signal x(n). N is the order offilter. FIR filter specifications include the maximumtolerable pass band ripple, maximum tolerable stop bandripple, pass band edge frequency and stop band edgefrequency. The difference equation can be expanded as:y(n)= 0 +⋯+ +1 ( −+ 1) (2)The transfer function of FIR filter is given as:H(z)= −1=0 − (3)The unit sample response of the FIR system is identicalto the coefficients { }, that is,

h(n)= , 0 ≤ ≤ −10, otherwise (4)

FIR filters have symmetric and anti-symmetric properties, which are related to their h(n) undersymmetric and asymmetric conditions as described

below by equations:h(n) = h(N -1 - n) for even (symmetric) (5)h(n) = - h(N -1 - n) for odd (anti-symmetric) (6)For such a system the number of multiplications isreduced from N to N/2 for N even and to (N-1)/2 for

odd. Then for linear phase FIR filter, the followingcondition should be satisfied:h(n) = ( −1 −)−+ ; n=0,1… N – 1 (7)Errors: The FIR filter is designed by optimizing thecoefficients in such a way that the approximation errorfunction in -norm for magnitude is to be keptminimal. The magnitude response is specified at Kequally spaced discrete frequency points in pass-bandand stopband.err1 x = Hd ωi −H( ωi , x)N

i=0 (8)

err2 (x) = ( Hd ωi −H( ωi , x) )Ni=0

212 (9)

where 1 – absolute error 1 -norm of magnituderesponse, 2( ) – squared error 2 -norm ofmagnitude response and N is number of samples.Desired magnitude response Hd (w i) of FIR filter isdefines as:

Hd ωi =1, for ωi

∈

passband 0, for ωi

∈stopband (10)

The ripple magnitudes of pass-band and stop-band haveto be minimized which are given by andrespectively.

( ) = max ( , ) – min ( , ) (11)( ) = max ( , ) (12)

Aggregating all objectives, the multi-criterionconstrained optimization problem is stated as:Minimize O1 (x)= 1( ) (13)Minimize O2 (x)= 2( ) (14)Minimize O3 (x)= ( ) (15)

Minimize O4 (x)= ( ) (16)In multiple- criterion constrained optimization problemfor the design of digital FIR filter, a single optimaltradeoff point can be found by solving followingequations:-Minimize O(x) =

4=1 ( ) (17)

where denotes the weights.

III. DIFFERENTIAL EVOLUTION

It is a population based stochastic method applied tominimize performance index. DE is the combination of

arithmetic and classical operations of recombination,mutation and selection to evolve from a randomlygenerated starting population to a final solution. Thedifferent variants of DE are classified using: DE/ α/β/δα indicates the method for selecting the parentchromosome that will form the base of the mutatedvector. β indicates the number of difference vectors to

perturb the base chromosome. δ indicates recombinationmechanism used to create offspring population. The binacronym indicates that the recombination is controlled



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by a series of independent binomial experiments. This isthe method which is based on principles of GA’s butwith crossover and mutation operations. The majordifference is GA’s relies on crossover and DE relies onmutation. DE has few control parameters which made it

popular. DE algorithm is classified into four steps: (1)initialization (2) mutation (3) crossover (4) selection andcontrolling parameters are (1) size of population (2)dimension (3) mutation factor (4) crossover rate.

1. InitializationTo initialize an individual population firstly setgeneration g=0 then initialize a population individualswith random values generated in search space accordingto the uniform probability distribution. This vector

population is initialized within the upper and lowerlimits of the search space.

= +rand() ( - ) (18)where = 1,2, …, ; = 1,2, …. . , The population vector may violate constraints and theseviolations are corrected by fixing them either at lower orat upper limit.

2. Mutation OperationMutation is an operation that adds a vector differential toa population vector of individuals according to thefollowing mutation strategies:Various Differential mutation strategies:There are several mutation strategies that can beemployed for optimization.Mutation Strategy 1:

= + ( - ) (19)where (j = 1,2,...NG; i =1,2,...,L)Mutation Strategy 2:

= + ( - -1 -

2 ) (20)where (j = 1,2,…,NG; i = 1,2,. .,L)Mutation Strategy 3:

= + ( - )+ (1 -

2 ) (21)where (j = 1,2,..,NG; i = 1,2,..,L)Mutation Strategy 4:

= +

+

−1 −

2 (22)

where ( = 1,2, …. . ; = 1,2, …. , ) Mutation Strategy 5:

= + 1 +2 −3 −4 (23)

where = 1,2, …. . ; = 1,2, …. , 1 , 2 3 , 4 are the random variables, g is generation,

is mutation factor, is the mutant or donor vector, is the parent vector, is the best population

vector,1 ,

2 ,3 ,

4 are the randomly generated population vectors.

3. Crossover operationFollowing the mutation operation, crossover is applied tothe population. Crossover is employed to generate a trialvector by replacing certain parameters of the targetvector by the corresponding parameters of a randomlygenerated donor vector.For each vector, +1 , an index 5 (i)∈ {1,2,…,NG} israndomly chosen using a uniform distribution, and a trialvector, +1 =[ 1

+1 , 2+1 ,…, +1 ]

+1 =4 ≤ = 5

4 > ≠5 (24)

Where ( = 1,2, …, ; = 1,2, …. . , ) is the donor vector or mutant vector, is the target

vector and +1 is the trial vector. 4 (j) is the jthevaluation of a uniform random number generation with[0,1]. 5 is the randomly chosen variable usinguniform distribution. CR is the crossover rate in [0,1].Usually, the performance of a DE algorithm depends onthree variables: the population size, the mutation factor (t) and the CR.

4. Selection OperationSelection is the procedure whereby better offspring are

produced. To decide whether the vector +1 should be

a member of the population comprising the nextgeneration, it is compared with the corresponding vector. Thus, if denotes the cost function under

minimization, then

+1 = +1 ; ( +1 ) < ( )

; (25)

where ( = 1,2, ……. , ; = 1,2 …. . , ) In this case, the cost of each trial vector +1 iscompared with that of its parent target vector . If the

cost function , of the target vector is lower than thatof the trial vector, the target is allowed to advance to thenext generation. Otherwise, a trial vector replaces thetarget vector in the next generation.

5. Control ParametersThe control parameters which controls DE that are

population size, mutation factor, crossover rate and thestopping criteria ( ).



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6. Algorithm for DEDesigning of FIR filter using DE algorithm:The search procedure of the proposed differentialevolution method has been discussed below:1. Read data: population size, mutation factor, crossoverrate, and maximum number of iterations.2. Generate an array of uniform random numbers.3. Generate the initial population individually andcompute augmented objective function.4. Arrange calculated objective function in ascendingorder and select first half of the population members.5. Set iteration counter.6. Increment the iteration counter.7. Select best member.8. Apply mutation, crossover and selection operations.9. If g< Tmax then go to 6.10. Write GBEST.11. Stop.

IV. DESIGN OF FIR FILTERS ANDRESULTS:

The band-pass digital FIR filter has been designed usingfive different mutation strategies of DifferentialEvolution (DE). The DE algorithm has beenimplemented by varying the filter order from 12 to 34and other DE parameters such as population size,mutation factor and crossover rate. The DE algorithm formutation strategy 4 has been implemented for filter orderfrom 12 to 34 and the filter order from achieved value of

Objective Function has been decided with optimumvalue of Objective Function. Five mutation strategiesgiven in Eq.19, Eq.20, Eq.21, Eq.22 and Eq.23 have

been implemented on the selected filter order. Then theDE algorithm parameter such as population size,mutation factor and crossover rate have been varied onselected mutation strategy. The magnitude and phaseresponse of designed digital FIR filters have been

plotted. The results have been shown below:-

1. Selection of OrderThe filter order has been varied from 12 to 34 usingmutation strategy 4 of DE algorithm. The achievedObjective Function for each order is given in Table 1.

Table 1: Filter Order v/s Objective Function

Sr. No. Filter Order Objective Function1. 12 11.6931702. 14 11.6932003. 16 4.0259824. 18 4.026045

5. 20 2.0343616. 22 2.0345307. 24 2.0217908. 26 2.0259359. 28 0.745541

10. 30 1.59987311. 32 4.06467212. 34 7.216461

Fig.1: Filter Order v/s Objective Function

From the Fig.1 it has been observed that objectivefunction is minimum when filter order is 28. As filterorder is increased beyond 28, the objective functionincreases rapidly. Order 28 exhibits the minimum valueof objective function. Now all five mutation strategieshave been applied at filter order 28.

2. Different Mutation Strategies for Order 28As filter order 28 is selected now. Then differentmutation strategies as described in Eq.19, Eq.20, Eq.21,Eq.22, Eq.23 are applied at filter order 28.

Table 2: Mutation Strategies v/s Objective Function

Sr.No. Mutation Strategy Objective Function

1. Mutation Strategy 1 0.752657

2. Mutation Strategy 2 1.3741783. Mutation Strategy 3 0.750166



Table 2 shows that mutation strategy 4 gives minimumobjective function.



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3. Selection of Population SizeAs filter order 28 and mutation strategy 4 is selected.

Now different population varying from 50 to 150 in thesteps of 10 on mutation strategy 4 whose order is 28 andthe results are given in Table3.

Table3: Population Size v/s Objective Function

Sr. No. Population Size Objective Function

1. 50 0.754350

2. 60 0.756531

3. 70 0.753228

4. 80 0.745625

5. 90 0.749732

6. 100 0.745541

7. 110 0.744351

8. 120 0.744594

9. 130 0.744478

10. 140 0.744405

11. 150 0.744151

Table 3 shows that population 150 gives minimumobjective function so that population 150 is selected andgraph of population versus objective function is shownin Fig.2.

Figure 2: Population Size v/s Objective Function

This graph shows that population 150 gives theminimum objective function when DE algorithm appliedon mutation strategy 4 at order 28.

4. Selection of Mutation FactorAs mutation strategy 4, filter order 28 and population150 is selected. Keeping filter order as 28 and populationsize as 150, the mutation factor has been varied from 0.6to 0.95 in the steps of 0.05 and the achieved value ofobjective function has been depicted in Table 4.

Table 4: Mutation Factor v/s Objective Function

Sr. No. Mutation factor( ) Objective Function

1. 0.60 0.742710

2. 0.65 0.769261

3. 0.70 0.748500

4. 0.75 0.774302

5. 0.80 0.744151

6. 0.85 0.746278

7. 0.90 0.745664

8. 0.95 0.753231

Table 4 shows that mutation factor 0.60 gives minimumobjective function when applied on mutation strategy 4whose order is 28 and population is 150 and the graph ofmutation factor versus objective function is shown infig.3.

Fig. 3: Mutation Factor v/s Objective Function



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Filter

order

Minimum

value

Maximum

value

Average

value

Standard

deviation

28 0.74271 1.338521 0.905296 0.204094

This table shows the minimum value, maximum value,average value and standard deviation of order 28.

V. CONCLUSION

In this thesis, filter order 28 has been selected for thedigital band-pass FIR filter with 29 filter coefficients.Five different mutation strategies of DE have been areapplied at selected filter order 28. The mutation strategy4 gives best objective function. After this, values of DEalgorithm parameters have been varied for achieving theoptimum value of objective function. It is concluded that

population size 150, mutation factor 0.6, crossover ratevalue 0.20 gives the optimum value of objective functionfor the designed algorithm of digital band-pass FIRfilter. The magnitude and phase responses also have

been studied.

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Design of Digital Band Pass FIR Filter Using Heuristic Optimization Technique

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