chapter-5 group search optimization for the design...

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CHAPTER-5

GROUP SEARCH OPTIMIZATION FOR THE DESIGN OF OPTIMAL IIR DIGITAL FILTER

5.1 Introduction

Optimization is a consortium of different methodologies that works concurrently and

provides flexible information processing capability for handling real-life ambiguous

situations. In the recent past among various swarm / animal intelligence techniques, the

group search optimizer (GSO) based on the phenomena of animal searching behavior,

has attracted much attention of the researchers to solve various complex optimization

problems. The aim of this chapter is to explore the capability of an enhanced variant of

GSO for the design of digital IIR filters.

Digital filter is an important constituent of digital signal processing systems and

is used to filter discrete time signals. The function of digital filter is to restrain certain

range of frequencies and to pass the other frequency range without alteration. Digital

filters on the basis of impulse response length and relation between input and output are

categorized as, infinite impulse response (IIR) filters and finite impulse response (FIR)

filters. FIR filters exhibit linear phase characteristics and are inherently stable.

However, in comparison to IIR filters a much higher filter order is required by FIR

filters in order to meet a given specifications. Due to the attribute of high selectivity,

digital IIR filter is frequently used as computation tool for numerous digital signal

processing applications such as: speech processing, telecommunications data

processing, image processing, military electronics, industrial applications, noise

suppression and radar. The design methods for IIR digital filters are principally

categorized as: 1) transformation methods 2) optimization methods. Filters designed

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with transformation method are not efficient in terms of phase response, filter structure,

coefficient quantization error and attainment of global optimal solution (Lu and

Antoniou, 2000). In optimization methodology, digital IIR filters are designed to meet

the specified frequency response by employing some optimization criterions to get the

best performance. Due to non-linear and multimodal error surface of IIR digital filter,

conventional design algorithms may get stuck at local minima, sometimes.

Considering the drawbacks of conventional optimization techniques, in the

recent past bio-inspired evolutionary algorithms (EA) have been used to optimize the

IIR filter design problem. Some of latest EA's applied for the design of stable digital IIR

filter are as follows: genetic algorithms (GA) (Liu, et al., 2004; Nilsson, et al., 2004;

Tsai, et al., 2006); hierarchical genetic algorithm (HGA) (Tang, et al., 1998), simulated

annealing (Chen, et al., 2001); ant colony optimization (ACO) (Karaboga, et al., 2004);

tabu search (Kalinli and Karaboga, 2005); taguchi immune algorithm (TIA) (Tsai and

Chou, 2006); particle swarm optimization (PSO) (Chen and Luk, 2010; Upadhyay, et

al., 2014); seeker-optimization-algorithm (Dai, et al., 2010); real structured genetic

algorithm (Tsai, et al., 2009); two-stage ensemble evolutionary algorithm (Li, et al.,

2013); gravitation search algorithm (Saha, et al., 2014); real-coded genetic algorithm

(Kaur et al., 2014) and many more.

The design of digital IIR filter in terms of magnitude approximation is generally

formulated in two design steps (Antoniou, 2005): (i) an objective function based on the

difference between the designed and ideal amplitude response is optimized (ii) the

ripple magnitude in pass-band and stop-band is minimized by optimizing the

coefficients of transfer function. Another important aspect is that the filter order needs

to be minimum in a bid to reduce the computational complexity and cost of the designed

filter. To realize a filter of low order is a cumbersome task and require special

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consideration in the design procedure of digital IIR filters. In the prior research work for

the design of digital IIR filter carried out by Tang et al. (1998), Tsai et al. (2006) and

Tsai and Chou (2006) the minimization of filter order was not considered in the

optimization procedure.

Swarm intelligence (SI) a recent category of optimization technique based upon

the behaviour of animals has attracted surmount attention of researchers. The most

prominent swarm intelligence algorithms are ACO and PSO. ACO is particularly

inspired by behaviour of ants, whereas bird flocking and schooling of fish forms the

backbone of PSO. Recently, one more feather has been added to the category of swarm

intelligence based algorithms called group search optimization (GSO) (He, et al., 2006).

GSO unlike other existing SI algorithms is motivated by general searching behaviour of

animals. It is based upon the general forging strategy namely producer-scrounger

(Barnard and Sibly, 1981). Unlike its other counterpart namely PSO, in GSO the

population is divided into three different type of members: producers, scroungers and

dispersed members (Rangers). Producers perform producing, the searching operation

and differs from the operation performed by scroungers and rangers. Whereas in PSO

all individuals follow the same criteria for searching. Moreover in GSO the dispersed

members perform ranging operation thus ensuring that GSO does not get stuck in local

minima. Most importantly GSO and EA's work on a common concept of fitness to

achieve the better solution. Further the scrounging strategy applied in GSO is analogous

to crossover operators (Mühlenbein and Schlierkamp-Voosen, 1993). Last but not least

the scanning operation performed by GSO is comparable to simplified direct search

method. Summing up GSO includes the heterogeneous attributes of direct search, EA's

and Swarm Intelligence (He et al., 2009). Due to complex multimodal nature of digital

IIR filter design problem the need arises for an optimization algorithm possessing

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properties of both population-based global search and that of standalone local search

algorithm. The need of alliance of exploration and exploitation motivates to hybridize

the local and global optimization techniques (Lin and Chen, 2011).

The intent of this chapter is to minimize the non-linear and multimodal error

surface of IIR digital filter by applying a swarm intelligence based global search

optimization technique namely enhanced group search optimization (EGSO) algorithm.

Further the exploratory search is employed to exploit the local search space. The

original GSO algorithm is retailored by blending the concept of opposition-based

learning for evolution of group members to initiate the search process with good

candidates and enhance the convergence rate. The performance of the EGSO algorithm

is investigated for solving global optimization numerical problems and design of digital

IIR filters in terms of magnitude approximation while employing the stability

constraints. Further, the optimized filter coefficients are obtained with EGSO in order to

achieve minimum magnitude response error along with optimal order of the filter

simultaneously. The digital IIR filter design problem is analyzed in two ways: In first

case order of the filter is taken as a variable in the design process, so that magnitude

response error can be minimized while achieving the optimal order. In second case,

magnitude response error is minimized for higher orders of the filter by varying the first

and second order blocks.

The subsequent structuring of the chapter is as follows: The design statement of

digital IIR filter is stated in Section 5.2; Section 5.3 explains the EGSO algorithm

implementation; The results obtained with EGSO algorithm are evaluated and compared

with the design results by Tang et al. (1998), Tsai et al. (2006) and Tsai and Chou

(2006) in Section 5.4; Finally, the conclusions and discussions are outlined in Section

5.5.

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5.2 Design Problem of Digital IIR Filter

Digital IIR filter design problem involves the optimizing the filter coefficient so that the

designed filter meet the desired specifications such as constant gain in pass-band,

minimum transition width and minimum value of ripples in pass-band and stop-band.

To design IIR digital filter the filter coefficients are evaluated to meet the desired design

specifications like minimum Lp-norm error, ripples in pass-band and stop-band,

minimum filter order and stability conditions. To avoid the coefficient quantization

problem which leads to instability of IIR digital filter, the cascading structure is the

mostly used for IIR filter implementation. The structure of cascading type digital IIR

filter is:

v

k ukuk

u

i i

i

zxzx

zxzx

zx

zxxzH ukuk

12

1241

24

21

11

12

12

1 1

1

1

1)( 124224 (5.1)

TMvuxxxX 114221 ],...,,[ is a decision variable vector of dimension M×1 with M = 2u +

4v + 1. x1 represents the gain, [x2, x3,..., xM] denotes the filter coefficients of first and

second order sections. u and v are the number of first and second order blocks,

respectively.

5.2.1 Magnitude response error in terms of Lp-norm approximation

The intention of designing digital IIR filter is that magnitude response of the designed

IIR digital filter should approach the ideal magnitude response. Ideal magnitude

response )( lIM of IIR digital filter is represented as:

stopbandforpassbandfor

Ml

llI

,0,1

)( (5.2)

where ωl denotes pass-band and stop-band frequency.

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The constraints are imposed in the design process on the denominator

coefficients given in Equation (5.1) to meet the stability condition of the designed IIR

filter. The frequency response is specified at K equally spaced discrete samples in pass-

band and stop-band. The magnitude response error in terms of Lp-norm approximation

is given as (Tsai and Chou, 2006):

pK

l

p

llI XMMxe/1

0

),()()(

(5.3)

For p=1, the magnitude response error becomes the L1-norm error and is defined as

given below:

K

lllI XMMxe

0

),()()( (5.4)

Mathematically, the constrained optimization problem for IIR digital filter design is

stated below:

Minimize )()( xexF (5.5a)

Subject to Stability Constraints:

)....,,2,1(01 12 uix i (5.5b)

)....,,2,1(01 12 uix i (5.5c)

)....,,2,1(01 124 vkx uk (5.5d)

)....,,2,1(01 12424 vkxx ukuk (5.5e)

)....,,2,1(01 12424 vkxx ukuk (5.5f)

5.2.2 Filter order design criterion

The design criterion for IIR filter involves obtaining of minimum magnitude

approximation error, lower pass-band and stop-band ripples by ensuring the lowest

order. The order of the IIR filter is determined as follows:

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v

kk

u

jj qpOrder

11

2 (5.6)

where pj and qk are jth and kth control genes, respectively, for corresponding first order

and second order sections. u and v are the number of first and second order blocks,

respectively. The maximum order of the filter is u + 2v.

5.2.3 Feasible solution

All the poles of designed digital IIR filter are forced to lie inside the unit circle by

imposing stability constraints obtained by applying the Jury method (Jury, 1964).

The stability constraints given in Equation (5.5b) to Equation (5.5f) have been

forced to satisfy by applying heuristics on the coefficients with random variation

described in Chapter 3, Section 3.2.3 Equations (3.7a) to (3.7c).

5.3 Enhanced Group Search Optimizer

EGSO is a nature inspired, population based algorithm that is stimulated by general

searching behaviour of animals. The behaviour of animals is primarily affected and

depicted in their search for resources i.e. shelter, food etc. In true spirit of a common

quote "Necessity is the need of invention" animals searching and surviving capabilities

are inspired by the availability of resources and their spatial and temporal distributions

in the environment. The most important survival strategy of animal world "group

living" has been analyzed intensively. Group living of animals leads to fast searching of

resources as patch finding rate is increased. The foraging techniques followed by group

of animals are primarily classified as: 1) producing, e.g., searching for resource; and 2)

joining (scrounging), e.g., joining resources unearthed by others. The member of a

group who is able to find a resource subsequently shares the resource with others.

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In order to analyze the optimal policy for joining, two models have been

proposed: information sharing (IS) and producer–scrounger (PS) (He, et al., 2009). The

IS model assumes that foragers search concurrently for their own resource while

searching for opportunities to join. On the other hand, foragers in the PS model are

assumed to use producing or joining strategies exclusively. Recent studies suggest that,

at least for the joining policy of ground-feeding birds, the PS model is more plausible

than IS model.

EGSO is based upon the general foraging strategy namely PS (Barnard and

Sibly, 1981). In EGSO population is analogous to a group and consist of individual

members representing different design variables. A set of members of a group are

initialized by applying opposition-based learning. Further an exploratory search is

exploited for fine tuning of local search space.

The population / group in EGSO consist of three types of members: producers,

scroungers and rangers. As stated earlier the framework adopted in EGSO is PS model

in which foragers exclusively follow the producer and joining strategies. Inheriting from

the basic PS model framework, EGSO assumes the best member as producer and rest of

members are divided in to scroungers and rangers. The joining / scrounging policy of

house sparrows is followed by the scroungers to join the resources found by the

producer. Further to assure that EGSO is not trapped in local minima some of the

members (rangers) perform random walk.

5.3.1 Group initialization

In a group consisting of NG members and M design variables, at any kth iteration, ith

member is identified by its position ),...,( 1tMi

ti

ti xxx and head angle

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),...,( )1(1t

Miti

ti angangang . The head direction ),...,( 1

tiM

ti

ti ddDir is compounded from

the head angle by a polar to cartesian transformation as depicted below:

1

11 )cos(

M

k

tik

ti angd (5.7)

)1,...,2()cos().sin(1

1)1(

Mjangangd

M

k

tik

tji

tij

(5.8)

)sin( )1(t

MitiM angd (5.9)

Each member of the group is initialized with the help of random search for value of all

filter coefficients. The starting point is initialized by exploring the global search space

and further the starting point is fine tuned in the local search space to find the best

starting point. The variable tjix is initialized and search process is initiated using

Equation (5.10):

)...,,2,1;...,,2,1()( minmaxmin MjNGixxRxx jjjt

ji (5.10)

where

R is a uniform random generated number between {0,1}.

M is number of decision variables (filter coefficients).

NG is number of members in a group.

t is the iteration counter.

maxjx and min

jx are the maximum and minimum values of jth decision variable (filter

coefficient), respectively of vector X.

5.3.2 Evolution using Opposition-based learning

The evolution of group members is refined with the help of opposition-based learning

(OBL). In addition to refining the selection of good members, the convergence rate of

EGSO is also enhanced employing OBL. The main idea behind opposition-based

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learning is to consider current population and its opposite population at the same time in

order to select better current candidate solution.

OBL is applied in EGSO by adopting the methodology presented in Chapter 3,

Section 3.3.4.

5.3.3 Fitness function evaluation

The objective function defined by Equation (5.5a) is used to obtain the expected fitness

function f . For the design of IIR digital filter the expected fitness function of ith member

of the group in tth iteration is given below:

))((minimize xFf ti

ti )...,,2,1( NGi (5.11)

At the end of first iteration the function value of the fittest member is set as global best (

f global ) and corresponding value of variables / filter coefficients is set as global best

variables (Gj).

)...,,2,1(;min NGiff ti

global (5.12)

5.3.4 Diversification by producers

EGSO seeks motivation from animal searching behaviour especially the way animals

pursue their prey / resources. The most important aspect while seeking resources by

animals is scanning done through sensory receptors, bodily movements etc. EGSO

adopts the "vision" scanning media for searching as depicted in Figure 5.1.

The search space as depicted above in Figure 5.1 is represented by maximum

pursuit angle θmax and maximum pursuit distance lmax. The member having the best

(minimum) objective function value is designated as the producer.

The producer performs diversification of the search space by scanning three

different points in the search space randomly: at zero degree, through right side of the

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hypercube and through left side of the hypercube. The movement of producer tpx at tth

iteration is depicted as follows:

)(max1tt

ptp

ts angDirlRxx (5.13)

)2/( max2max1 RangDirlRxx ttp

tp

tr (5.14)

)2/( max2max1 RangDirlRxx ttp

tp

tl (5.15)

Where R1 is a random number having normal distribution whose mean is zero and

standard deviation is 1. R2 is a uniform random generated number between {0,1}.

Figure 5.1: Scanning field representation in 3D space

If any of three points explored by producer is better than present position in term

of function value, then producer will shift; else it will not move and will change its

direction to new random generated angle as follows:

max21 Rangang tt (5.16)

where αmax denotes maximum turning angle.

After ta iterations producer is not able to find better point, then it changes its

direction to zero degree as given below:

(Forward directed)

00

θmax

θmax

lmax

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ttt angang a (5.17)

5.3.5 Intensification by Scroungers

Once the resource (the best point in terms of fitness) is found by the producer most of

the other members act as scroungers and exploit the findings of the producer by

performing a random walk towards producer. There are many joining strategies

available in literature, EGSO follows the area copying mechanism. EGSO assures that

the search is not trapped in local minima by switching the role of producers and

scroungers among themselves. If during random walk a scrounger finds a better point

than the producer, then in next iteration the scrounger will act as producer and vice-

versa. The movement of scrounger at tth iteration is depicted as follows:

)(31 t

itp

ti

ti xxRxx (5.18)

where

R3 is a uniform sequence random generated number between {0,1}.

tix is the value of the ith member (scrounger) at tth iteration.

tpx is the value of the producer at tth iteration.

5.3.6 Rangers

Ranging is the process of searching of resources without any cue (Dusenbery, 1989). In

animal kingdom there exist many searching strategies for ranging animals like

systematic search, random walk etc (Higgins and Strauss, 2003). EGSO adopts the

random walk methodology for members designated as rangers. Random walk performed

by rangers also ensures that EGSO is not stuck at local minima point by exploring

unearthed promising regions. During each iteration rangers perform the random walk by

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generating random angle using Equation (5.16) and computing random distance l and

performing the walk as follows:

max1 lRtl ai (5.19)

)( 11 ttii

ti

ti angDirlxx (5.20)

where

R1 is a random number having normal distribution whose mean is zero and

standard deviation is 1.

ta is constant denoting fixed number of iterations.

lmax is maximum pursuit distance.

tix is the value of the ith member (ranger) at tth iteration.

ang t+1 new random generated angle.

tiDir is the new head direction.

5.3.7 Exploratory Move

EGSO refines the search in local search space by exploiting an exploratory move.

Further exploratory move also ensures the fast convergence speed of EGSO. The best

point obtained after performing the ranging move is further refined in the local search

space by perturbing each design variable in positive and negative directions of its range

respectively. At the end of each variable perturbation the current point is updated to the

improved solution. The perturbation of variable xi is performed as follows:

)...,,2,1;...,,2,1( MjMiuxoldxnew jiiii (5.21)

where

jiji

u ji 0

1 and M denotes number of members / variables / filter

coefficients.

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The objective function denoted by )( ixf is calculated as follows:

Otherwisexold

xoldfuxoldfuxoldxoldfuxoldfuxold

xnew

i

iiiiiii

iiiiiii

i

;)()(;)()(;

(5.22)

where ( i=1,2,...,M ) and ∆i is random for global search and fixed for local search.

In case the point discovered after the perturbation for all the member variables is

different from the original point, the exploratory move is considered as successful and

overall minimum is set as next iteration starting point.

5.3.8 Termination and calculation of best function values

At the end of each iteration if the function value obtained by the best group member is

better than the global best ( f global ) then it replaces the global best and corresponding

marks obtained by the best member are stored as the global best marks (Gj).

Otherwiseffff

f global

globalttglobal

;; minmin (5.23)

where )...,,2,1(minmin NGiff ti

t

A heuristic optimization algorithm is usually terminated based upon the value of

maximum number of iterations, tolerance or number of function evaluations. EGSO

employs maximum number of iterations as criteria to terminate. The above procedure is

repeated with incremented t value until the value of t reaches the maximum value of

iterations specified. Otherwise, jG is the best value of filter coefficients and bestf is the

best / minimum value of the objective function.

5.3.9 Pseudo-code for IIR Filter Design Using EGSO

The search procedure of the proposed EGSO method is as shown below:

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1. Read data; viz. Filter coefficient vector X, maximum allowed iterations, ITmax,

minjx and ),.....2,1(max Mjxj , etc.

2. Generate an group / array of (M×2NG).

3. for (i=1;i<=(2NG);i++) // group / population counter

4. for (j=1;j<=M;j++) // member / filter coefficient counter

5. Generate the initial value of filter coefficients tijx using Equation (5.10)

and tif using Equation (5.11)

6. Apply the opposite-based learning using Equation (3.9) and tNGif using

Equation (5.11)

endfor

endfor

7. for (i=1;i<=((2NG)-1);i++)

8. for (j=i+1;j<=(2NG);j++)

9. if )( ti

tj ff

10. Set ti

tj ff and ),.....,2,1( Mkxx t

kit

kj

endif

endfor

endfor

11. for (i=1;i<=(NG);i++)

12. for (j=1;j<=M-1;j++)

13. 4/, jiang // Set initial head angle

endfor

endfor

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14. Set constant 1 Mroundta , Maximum Pursuit angle 2max / at ,

Maximum Turning angle 2/maxmax and Maximum pursuit distance

M

jjj xxl

1

minmaxmax

15. Set tglobal ff 1 , ),.....,2,1( MjxG tjij and 1MIN

16. Set iteration counter, t=0

17. Increment iteration counter, t=t+1

18. for (i=1;i<=NG;i++)

19. if )( MINi then // Producer phase

20. Scan straight at zero degree using Equation (5.13) and evaluate function

value tSf based on new generated values of coefficients

),.....,2,1( Mjxstji

21. Scan right side using Equation (5.14) and evaluate function value tRf

based on new generated values of coefficients ),.....,2,1( Mjxr tji

22. Scan left side using Equation (5.15) and evaluate function value tLf

based on new generated values of coefficients ),.....,2,1( Mjxl tji

23. ),,(ofBest tL

tR

tS

new ffff ; ),.....,2,1()or,( Mjxlxrxsxnew tji

tji

tji

tji

24. if )( ti

new ff then

25. Set newti ff and ),.....,2,1( Mjxnewx t

jit

ji

26. FAIL = 0

else

27. FAIL = FAIL + 1

28. if (FAIL ≥ ta) then

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29. FAIL = 0 and change the producer angle to zero using

Equation (5.17)

else

30. Change the producer angle using Equation (5.16)

endif

endif

else

31. if (R1 ( )≥ 0.2)then // Scrounger phase

32. Model the area copying behaviour with the help of random walk

using Equation (5.18)

else // Dispersion

33. for (j=1;j<=M-1;j++)

34. max1,, )( Rangang jiji

endfor

35. Choose new random distance using Equation (5.19) and disperse

to new point using Equation (5.20)

endif // End of SCROUNDING / DISPERSION

endif

endfor

36. Generate the opposite population of learners using Equation (3.11) and select the

best NG groups out of (2×NG) groups.

37. Set i=1, tff 1min,

38. for (i=2;i<=NG;i++)

39. if )( minff ti

40. Set tiff min, and iimn

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endif

endfor

41. Perform the Exploratory move using Equation (5.21) and Equation (5.22)

42. if (t >= ITmax) goto 47

43. if )( min globalff

44. Set minff global and ),...,2,1(, MjxG tjimnj

endif

45. Store tif , t

jix and tjiang ),...,2,1(;),...,2,1( MjNGi for next iteration.

46. goto 17

47. stop

5.4 Evaluation of Design Results

The efficacy of designed digital IIR filters by employing EGSO is established by firstly

solving the global numerical optimization problems. The obtained results are further the

compared with GSO to establish the enhancement capability of EGSO.

Secondly EGSO is applied for the design of digital IIR filter and the obtained

results are compared with the GSO, ETLBO and other heuristic techniques used by

various researchers. To establish the versatility of the proposed EGSO method the

following design criteria's are adopted for the design of digital IIR filter:

L1 magnitude approximation while searching the order of filter

L1 magnitude approximation for higher orders of the filter

5.4.1 Application of EGSO and GSO for solving standard test optimization

problems

The enhancement in the performance of EGSO in comparison to GSO is validated by

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solving a group of seven benchmark functions each with different characteristics

presented in Chapter 3, Section 3.4.1, Table 3.1.

For the application of EGSO and GSO the dimension of benchmark test

functions presented in Chapter 3, Section 3.4.1, Table 3.1 is kept as 100 (i.e. M = 100).

SR denotes the search ranges of the variables (i.e. xmin , xmax ), fmin is the objective value

of the global optimum, and all the functions have an optimal value fmin= 0. The test

functions presented in Chapter 3, Section 3.4.1, Table 3.1 possess heterogeneous

characteristics such as F1 and F2 are unimodal, F3 is a discontinuous step function, F4 -

F7 are multimodal, whose number of local minima increase exponentially with increase

in dimension.

In order to make a fair comparison of EGSO and GSO the number of function

evaluations (NFE) is kept as 1×106. The NFE of GSO and EGSO are summarized by

Equations (5.24) and (5.25) respectively.

maxGSO 2123NFE ITMNGNG (5.24)

maxEGSO 21232NFE ITMNGNGNG (5.25)

where NG is the number of members, ITmax is maximum number of iterations and M is

number of variables / dimension size.

The parameter values employed for the execution of GSO and EGSO are

presented in Table 5.1.

For each benchmark test function 30 independent trial runs are performed. The

random seed number is used to generate random numbers giving different initial values

for each independent run. The maximum function value, minimum function value,

average function value, variance and the standard deviation of the function values are

all recorded for each benchmark test function and are shown in Table 5.2.

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Table 5.1: Parameters values employed for benchmark functions using GSO and EGSO

Control Parameter Value

Function evaluation size (FES) 1×106 Initial head angle (ang) / 4

Number of rangers 20% of NG Constant (ta) 1Mround

Maximum pursuit angle (θmax) 2/ at

Maximum pursuit distance (lmax)

M

jjj xx

1

2minmax )(

Maximum turning angle (αmax) θmax / 2

It can be seen from Table 5.2 that:

EGSO significantly outperforms GSO on all of seven benchmark test functions

and is able to achieve a more promising solution.

The comparison reflects that in case of unimodal functions F1 and F2 EGSO

possesses better local search ability and higher search accuracy. For the

multimodal functions, F4 - F7 EGSO possess a stronger ability to escape from

local minima and jump to a better near-global optimum. In case of F3 a

discontinuous step function EGSO and GSO attains the global minima.

Summing up EGSO is able to obtain an excellent balance between exploitation and

exploration and gives smaller standard deviations of function values than GSO. Hence,

EGSO has a more stable solution quality.

5.4.2 L1 magnitude approximation while achieving the lowest order

The L1 stable digital IIR filter given by Equation (5.5a) are successfully designed by

employing EGSO while taking care of the stability constraints stated by Equation (5.5b)

to Equation (5.5f) under the prescribed design conditions presented in Chapter 3,

Section 3.4.2, Table 3.4.

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Table 5.2: Comparison of GSO and EGSO for seven 100 dimension functions over 30 independent runs with 1×106 NFE

Function Type GSO EGSO Sphere, F1

Maximum 5.66×10-03 6.48×10-06 Minimum 1.67×10-06 1.34×10-07 Average 6.53×10-04 1.26×10-06 Variance 1.62×10-06 2.91×10-12

Standard Deviation 1.27×10-03 1.71×10-06 Axis, F2


Standard Deviation 1.17×10-02 8.94×10-07 Step, F3

Maximum 0 0 Minimum 0 0 Average 0 0 Variance 0 0

Standard Deviation 0 0 Rastrigin, F4


Standard Deviation 3.07×10-01 1.12×10-02 Griwank, F5


Standard Deviation 2.78×10-03 6.50×10-06 Ackley, F6


Standard Deviation 2.03×10-01 8.32×10-05 Alpine, F7


Standard Deviation 2.17×10-01 3.48×10-04

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The overall performance of group search optimization method is not sensitive to

the values of different control parameters (He, et al., 2009). The point of concern is only

the percentage of group members to be designated as rangers. The value of different

control parameters used for the deployment of EGSO is presented in Table 5.3.

Table 5.3: Control parameters values for EGSO to design LP, HP, BP and BS filters

Control Parameter Value Maximum number of iterations (ITmax) 450 Number of members in a group (NG) 70

Initial head angle (ang) / 4 Number of rangers 20% of NG

Constant (ta) 1Mround Maximum pursuit angle (θmax)

2/ at


M

jjj xx

1

2minmax )(


The NFE needed for successful designing of LP, HP, BP and BS digital IIR

filters using EGSO is calculated using Equation (5.25) and are summarized in Table 5.4.

Table 5.4: NFE achieved by EGSO for L1 magnitude approximation error while searching order of the filter

Filter Type NFE

LP 9.689×104

HP 9.914×104

BP 9.731×104

BS 8.814×104

The examples of the IIR filters considered in Tang et al. (1998), Tsai et al.

(2006) and Tsai and Chou (2006) are referred to test and compare the L1 stable digital

IIR filter designed with the proposed EGSO algorithm. Its pertinent to mention, that in

the design of digital IIR filter carried out by Tang et al. (1998), Tsai et al. (2006) and

Tsai and Chou (2006) the minimization of filter order was not considered in the

optimization procedure i.e. the order of filter was fixed.

169

In order to optimize the order of filter, in this chapter a control gene is employed

to represent the structure of digital IIR filter. The order of the filter is included in the

optimization procedure as given in Chapter 4, Section 4.2.4, Equation (4.12). The

structure of digital IIR filter is represented by control gene presented in Chapter 3,

Section 3.4.4, Figure 3.6. The integer value of variable 242 vux is optimized along with

the filter coefficients to obtain optimum Order of designed IIR filter.

TMvuvu xxxxX 124214221 ],,...,,[ is the final decision variable where M = 2u + 4v + 2.

The designed LP, HP, BP and BS digital IIR filter models obtained by the

EGSO approach representing the value of filter coefficients are shown in Equations

(5.26), (5.27), (5.28) and (5.29) respectively.

)0.7395981.39537()0.67125(

)0.9909480.330330()0.999317(0.034417)(

2

2

zzz

zzzzHLP (5.26)

)0.7375751.374665)(0.655820(

)0.9979330.469350)(0.999400(0.041281)(

2

2

zzz

zzzzHHP (5.27)

)0.7786230.626148(

)0.9169230.059816(

)0.5603300.006753()0.7806520.622032(

)927485.0179689.0()063284.10.244515(0.025142)(

2

2

22

22

zz

zz

zzzz

zzzzzHBP

(5.28)

)0.5256010.81318)(0.5256030.813181(

)1.00200.4014707)(999569.00.401344(0.427592

22

22

zzzz

zzzzHBS (5.29)

The digital IIR filter is realized by setting 200 evenly distributed points within

the frequency span ,0 , such that the number of discrete frequency points are 182 for

the LP and HP filters, or 143 for the BP and BS filters in the union of prescribed pass-

band and stop-band shown in Chapter 3, Section 3.4.2, Table 3.4. The plot of obtained

magnitude response with respect to normalized frequency of L1 stable LP, HP BP and

BS digital IIR filters designed with the proposed EGSO are presented in Figure 5.2.

170

Figure 5.2: Magnitude response with respect to normalized frequency of L1 stable

digital IIR filters using EGSO

The scrutinizing of the magnitude response plots depicted in Figure 5.2 reveals

that the designed IIR digital filters employing EGSO posses flat pass-band and stop-

band response edges.

The stability of the designed LP, HP, BP and BS IIR digital filter is established

by drawing pole-zero diagrams presented in Figure 5.3. The magnitude and angle of

zeros is (0.9993, 3.1416), (0.9955, ± 1.4041) for LP digital IIR filter, (0.9994, 0),

(0.9990, ± 1.8079) for HP digital IIR filter, (1.1606, 0), (0.9879, 0), (0.8774, 0),

(1.0571, 3.1416), (0.9281, 3.1416), (0.9161, 3.1416) for BP digital IIR filter and

(1.0001, ± 1.3687), (0.9998, ± 1.7729) for BS digital IIR filter and the magnitude and

angle of poles is (0.8600, ± 0.6245), (0.6712, 0) for LP digital IIR filter, (0.8588, ±

2.4986), (0.6558, 3.1416) for HP digital IIR filter, (0.8835, ± 1.2111), (0.8824, ±

1.9335), (0.7486, ± 1.5753) for BP digital IIR filter and, (0.7250, ± 2.1662), (0.7250, ±

0.9754) for BS digital IIR filter. The maximum magnitude of pole is 0.8600 for LP

171

digital IIR filter, 0.8588 for HP digital IIR filter, 0.8835 for BP digital IIR filter and

0.7250 for BS digital IIR filter. By analyzing the Figure 5.3 and magnitude of poles and

zeros, it is concluded that the digital IIR filters designed with EGSO follow the stability

constraints as all poles lie inside the circle. The lying of zeroes outside the unit circle

has no effect on the stability of the designed filter. Further, as per the maximum

magnitude of pole of all types of IIR digital filter it can be concluded that the designed

IIR filter gives less quantization noise because the all the poles are not close to the unit

circle.

The L1-norm approximation errors in terms of magnitude response and ripples in

both pass-band and stop-band obtained for LP, HP, BP and BS filters with EGSO are

summarized and compared with results obtained with GSO, ETLBO, HGA (Tang, et al.,

1998), HTGA (Tsai, et al., 2006) and TIA (Tsai and Chou, 2006) in Table 5.5.

The scrutinizing of results presented in Table 5.5 proves that the proposed

EGSO algorithm enhances the performance of GSO and outperforms HGA (Tang, et al.,

1998), HTGA (Tang, et al., 1998) and TIA (Tsai and Chou, 2006) algorithms while

achieving the lowest order. The detailed comparison is summarized below:

The digital filter designed employing the proposed EGSO algorithm offers best

L1-norm error magnitude response for LP, HP, BP and BS IIR digital filter in

comparison to GSO, TIA, HTGA and HGA.

EGSO offers smaller ripple magnitude in pass-band for LP, HP, BP and BS IIR

digital filter in comparison to GSO, TIA, HTGA and HGA.

The ripples in stop-band for LP, HP, BP, BS IIR digital filter are less as

compared to GSO, TIA, HTGA and HGA.

Beside it the LP, HP, BP, BS IIR digital filters designed with EGSO are stable in

nature as all poles lie inside the unit circle.

172

Moreover the scrutinizing of data presented in Table 5.3, Table 5.2 and Chapter

3, Table 3.3 it is concluded that ETLBO design gives better L1-approximated

digital IIR filter in less NFE in comparison to EGSO.

Figure 5.3: Pole-zero plots of L1 stable digital IIR filters using EGSO

Table 5.5: Design results for L1 stable digital IIR filter applying EGSO

Method OrderL1-norm

error Pass-band performance

(Ripple magnitude)

Stop-band performance

(Ripple magnitude) LP Filter

EGSO 3 3.7002 0.9040 ≤│H(e jω)│≤ 1.010

(0.1067) │H(e jω)│≤ 0.1478

(0.1478)

GSO 3 3.7810 0.9028 ≤│H(e jω)│≤ 1.013

(0.1103) │H(e jω)│≤ 0.1324

(0.1324)

ETLBO 3 3.6722 0.9134 ≤│H(e jω)│≤ 1.012

(0.0986) │H(e jω)│≤ 0.1564

(0.1564)

TIA 3 3.8157 0.8914 ≤│H(e jω)│≤ 1.000

(0.1086) │H(e jω)│≤ 0.1638

(0.1638)

HTGA 3 3.8916 0.8994 ≤│H(e jω)│≤ 1.000

(0.1006) │H(e jω)│≤ 0.1695

(0.1695)

HGA 3 4.3395 0.8870 ≤│H(e jω)│≤ 1.009

(0.1139) │H(e jω)│≤ 0.1802

(0.1802)

173

Table 5.5: Continued

Method OrderL1-norm

error Pass-band performance

(Ripple magnitude)

Stop-band performance

(Ripple magnitude) HP Filter

EGSO 3 3.9497 0.9420 ≤│H(e jω)│≤ 1.010

(0.0680) │H(e jω)│≤ 0.1549

(0.1549)

GSO 3 3.9814 0.9430 ≤│H(e jω)│≤ 1.010

(0.0669) │H(e jω)│≤ 0.1544

(0.1544)

ETLBO 3 3.9603 0.9372 ≤│H(e jω)│≤ 1.012

(0.0755) │H(e jω)│≤ 0.1414

(0.1414)

TIA 3 4.1819 0.9229 ≤│H(e jω)│≤ 1.000

(0.0771) │H(e jω)│≤ 0.1424

(0.1424)

HTGA 3 4.3413 0.9403 ≤│H(e jω)│≤ 1.000

(0.0597) │H(e jω)│≤ 0.1668

(0.1668)

HGA 3 14.5078 0.9224 ≤│H(e jω)│≤ 1.003

(0.0779) │H(e jω)│≤ 0.1819

(0.1819) BP Filter

EGSO 6 1.2400 0.9874 ≤│H(e jω)│≤ 1.006

(0.0195) │H(e jω)│≤ 0.0492

(0.0492)

GSO 6 1.3067 0.9892 ≤│H(e jω)│≤ 1.007

(0.0179) │H(e jω)│≤ 0.0529

(0.0529)

ETLBO 6 1.1616 0.9810 ≤│H(e jω)│≤ 1.010

(0.0299) │H(e jω)│≤ 0.0430

(0.0430)

TIA 6 1.5204 0.9681 ≤│H(e jω)│≤ 1.000

(0.0319) │H(e jω)│≤ 0.1638

(0.0679)

HTGA 6 1.5367 0.9677 ≤│H(e jω)│≤ 1.000

(0.0323) │H(e jω)│≤ 0.0680

(0.0680)

HGA 6 5.2165 0.8956 ≤│H(e jω)│≤ 1.000

(0.1044) │H(e jω)│≤ 0.1772

(0.1772) BS Filter

EGSO 4 3.2572 0.9563 ≤│H(e jω)│≤ 1.008

(0.0525) │H(e jω)│≤ 0.1225

(0.1225)

GSO 4 3.3215 0.9596≤│H(e jω)│≤ 1.009

(0.0499) │H(e jω)│≤ 0.1207

(0.1207)

ETLBO 4 2.9902 0.9384 ≤│H(e jω)│≤ 1.012

(0.0738) │H(e jω)│≤ 0.1282

(0.1282)

TIA 4 3.4750 0.9259 ≤│H(e jω)│≤ 1.000

(0.0741) │H(e jω)│≤ 0.1278

(0.1278)

HTGA 4 3.6334 0.9425 ≤│H(e jω)│≤ 1.000

(0.0575) │H(e jω)│≤ 0.1294

(0.1294)

HGA 4 6.6072 0.8920 ≤│H(e jω)│≤ 1.009

(0.1080) │H(e jω)│≤ 0.1726

(0.1726)

174

EGSO is tested for robustness by performing 30 random runs and the best results

are reported in this chapter. The statistical data thus obtained presented in Table 5.6

establishes that EGSO is a robust algorithm for the design of digital IIR filters being

small value of standard deviation.

Table 5.6: Statistical data of IIR digital filter employing EGSO for 30 independent trial runs

Parameter Maximum Minimum Average Variance Standard Deviation

LP filter L1-error 3.714653 3.700185 3.703390 1.457×10-5 3.818×10-3 Mean square error 0.002843 0.002416 0.002542 1.62×10-8 1.27×10-4 Pass-band Ripples 0.121055 0.105986 0.110788 2.217×10-5 4.709×10-3 Stop-band Ripples 0.148416 0.139931 0.145321 7.696×10-6 2.774×10-3 HP filter L1-error 4.026051 3.918558 3.952062 3.56×10-4 1.889×10-2 Mean square error 0.002438 0.002104 0.002349 7.448×10-9 8.631×10-5 Pass-band Ripples 0.076560 0.065671 0.073618 1.016×10-5 3.187×10-3 Stop-band Ripples 0.156168 0.140445 0.146972 1.57×10-5 3.962×10-3 BP filter L1-error 1.609513 1.249070 1.456390 6.751×10-3 8.216×10-2 Mean square error 0.000455 0.000244 0.000357 2.338×10-9 9.37×10-6 Pass-band Ripples 0.023337 0.009711 0.012896 1.539×10-5 3.923×10-3 Stop-band Ripples 0.075698 0.048265 0.065902 3.281×10-5 5.728×10-3 BS filter L1-error 3.347578 3.165303 3.254794 1.32×10-3 3.633×10-2 Mean square error 0.004604 0.003592 0.004114 7.693×10-8 2.77×10-4 Pass-band Ripples 0.060833 0.047547 0.053357 1.187×10-5 3.446×10-3 Stop-band Ripples 0.133672 0.110798 0.122808 3.283×10-5 7.693×10-8

5.4.3 L1 - magnitude approximation for higher orders of the filter

The robustness of EGSO for the design of digital IIR filter design has been further

explored by obtaining L1 - magnitude approximation for higher orders of the filter by

varying u and v the first and second order blocks, respectively.

The higher orders filters are designed under the prescribed design conditions of

normalized frequencies for pass-band and stop-band and maximum filter order given in

Chapter 3, Section 3.4.3, Table 3.8.

175

The parameter values such as NG, ITmax, θmax employed for the EGSO algorithm

for designing higher order filters are summarized in Table 5.7.

Table 5.7: Control Parameters values for L1- magnitude approximation for higher orders of the LP, HP, BP and BS filters employing EGSO

Control Parameter Value Maximum number of iterations (ITmax) 500 Number of members in a group (NG) 70

Initial head angle (ang) / 4 Number of rangers 20% of NG

Constant (ta) 1Mround Maximum pursuit angle (θmax)

2/ at


M

jjj xx

1

2minmax )(


The NFE needed for the execution of EGSO for realizing higher order filters vary

with the value of number of filter coefficients (M) and is denoted by Equation (5.25).

The L1-norm approximation errors in terms of magnitude response and ripples in both

pass-band and stop-band obtained for LP digital IIR filters with EGSO by varying first

and second order blocks are summarized in Table 5.8.

Table 5.8: Design results for L1 stable higher order LP digital IIR filter applying EGSO

Order L1-norm

error Pass-band

ripples

Stop-band

ripples Order

L1-norm error

Pass-band

Ripples

Stop-band

ripples 3 3.700226 0.10671 0.147850 16 0.460518 0.024423 0.0538674 2.170055 0.094832 0.082623 17 0.469727 0.021068 0.0701905 1.625763 0.082512 0.057534 18 0.486240 0.011719 0.0603726 1.119918 0.030339 0.069928 19 0.490024 0.027512 0.0515447 0.7886238 0.022901 0.073934 20 0.499660 0.019709 0.0560668 0.4240741 0.009932 0.047061 21 0.479121 0.038154 0.0552349 0.540658 0.040674 0.058908 22 0.506583 0.019070 0.04554010 0.581800 0.012956 0.062706 23 0.477549 0.027107 0.06975711 0.594372 0.012602 0.066475 24 0.449052 0.009891 0.03726112 0.419057 0.016638 0.055140 25 0.585000 0.061484 0.04075813 0.604512 0.020563 0.066476 26 0.422061 0.019338 0.05235214 0.408095 0.024896 0.032182 27 0.404354 0.016591 0.04474115 0.483535 0.009348 0.049601 28 0.602325 0.045322 0.058321

176

The best LP digital IIR filter obtained employing EGSO as depicted in Table 5.8

is for the 8th order as it gives the better performance in all aspects of magnitude

response. The L1- approximation error of magnitude response for the 8th order LP

digital IIR filter is 0.4240741 and ripples in pass-band and stop-band are 9.932×10-3 and

4.7061×10-2 respectively.

The L1-norm approximation errors in terms of magnitude response and ripples in

both pass-band and stop-band obtained for HP digital IIR filter with EGSO by varying

first and second order blocks is presented in Table 5.9. As depicted in Table 5.9, the

best performance of HP digital IIR filter is for the 9th order. The L1- approximation

error for magnitude response is 0.341489 and ripples in pass-band and stop-band are

2.505×10-2 and 3.4751×10-2 respectively.

The performance for the higher order BP and BS digital IIR filter in terms of

magnitude response and ripples in pass-band and stop-band is presented in Table 5.10.

The best performance of BP digital IIR filter is for the 14th order. The L1-

approximation error for magnitude response is 0.324744 and ripples in pass-band and

stop-band are 1.1366×10-2 and 1.7764×10-2 respectively.

Table 5.9: Design results for L1 stable higher order HP digital IIR filter applying EGSO

Order L1-norm

error

Pass-band

ripples

Stop-band

ripples Order

L1-norm error

Pass-band ripples

Stop-band

ripples 3 3.949719 0.068018 0.15491 16 0.393850 0.012649 0.0374824 1.842311 0.078228 0.084231 17 0.474495 0.024383 0.0249905 1.198188 0.008794 0.099297 18 0.429605 0.039840 0.0181806 0.520009 0.011006 0.050839 19 0.424068 0.044378 0.0484887 0.508187 0.042767 0.026279 20 0.477039 0.003728 0.0454858 0.514193 0.004043 0.053554 21 0.441467 0.018523 0.0305259 0.341489 0.025050 0.034751 22 0.543254 0.019285 0.03262910 0.548138 0.021311 0.052498 23 0.619128 0.027663 0.04076511 0.561177 0.037134 0.040218 24 0.540321 0.043163 0.04921012 0.521872 0.016681 0.041883 25 0.511425 0.008556 0.04349813 0.506887 0.009076 0.038359 26 0.683219 0.0145262 0.03824014 0.509748 0.011655 0.058877 27 0.650800 0.0222417 0.03659215 0.352576 0.016204 0.050369 28 0.565851 0.041262 0.039624

177

Table 5.10: Design results for L1 stable higher order BP and BS digital IIR filter applying EGSO

BP Filter BS Filter

Order L1-norm

error Pass-band

ripples

Stop-band

Ripples Order

L1-norm error

Pass-band

ripples

Stop-band

ripples 6 1.436763 0.024771 0.064959 4 3.257271 0.052548 0.1225108 0.616825 0.031593 0.023950 6 1.046763 0.050126 0.07075310 0.446327 0.038281 0.018932 8 0.789466 0.052476 0.02126412 0.426560 0.005602 0.029684 10 0.507505 0.013009 0.01735914 0.324744 0.011366 0.017764 12 0.448160 0.008858 0.00659916 0.377144 0.036867 0.013680 14 0.503693 0.024060 0.02941618 0.386877 0.053771 0.016827 16 0.595255 0.015370 0.04028220 0.514219 0.048077 0.029248 18 0.637822 0.012596 0.03926422 0.401016 0.007674 0.044240 20 0.647827 0.050133 0.01973124 0.312474 0.014639 0.026925 22 0.824932 0.027834 0.02591926 0.347862 0.010152 0.028720 24 0.698424 0.036561 0.03025228 0.3361478 0.0195313 0.016806 26 0.403542 0.018922 0.03741730 0.392150 0.026036 0.022316 28 0.863827 0.048961 0.05035532 0.429510 0.028108 0.020712 30 0.475322 0.024134 0.04785634 0.362953 0.016755 0.040270 32 0.652312 0.024345 0.03346236 0.376211 0.018926 0.032161 34 0.607825 0.014631 0.032746

The best performance of BS digital IIR filter is for the 12th order filter. The L1-

approximation error for magnitude response is 0.448160 and ripples in pass-band and

stop-band are 8.858×10-3 and 6.599×10-3 respectively.

The plot of magnitude response with respect to normalized frequency of best L1

stable digital IIR filters obtained using EGSO presented in Figure 5.4.

The trend of magnitude response with respect to normalized frequency depicted

in Figure 5.4 undoubtedly validate that the obtained best L1 stable digital IIR filters

using EGSO perform like an ideal digital IIR filters.

Moreover the pole zero diagrams shown in Figure 5.5 justify that the obtained

best L1 stable digital IIR filters with EGSO follow the stability constraints imposed in

the design procedure as all the poles lie inside the unit circle. The poles magnitude and

angles in radian are given by (0.8671, ± 0.6341), (0.6630, 0) for LP digital IIR filter,

(0.8544, ± 2.4802), (0.6164, 3.1416) for HP digital IIR filter, (0.8731, ± 1.1972),

(0.8746, ± 1.9459), (0.7282, ± 1.5773) for BP digital IIR filter and (0.7358, ± 0.9697),

178

(0.7355, ± 2.1714) for BS digital IIR filter. The first number in the parentheses is the

magnitude of pole and the second number is the angle in radians. The stability of filter

is not influenced by the zeros lying outside the unit circle. The maximum magnitude of

poles are 0.8671 for LP digital IIR filter, 0.8574 for HP digital IIR filter, 0.8746 for BP

digital IIR filter and 0.7358 for BS digital IIR filter.

Figure 5.4: Magnitude response with respect to normalized frequency of best L1

stable digital IIR filters using EGSO

The poles close to unit circles are sensitive to coefficient quantization but the

poles of all the designed filters with proposed method under different design criterion

are not close to unit circle. So the designed LP, HP, BP and BS filters are not prone to

quantization noise which is the advantage of the designed filters.

The plot of L1-norm error with respect to order of LP, HP, BP and BS filters

obtained with EGSO by varying first and second order blocks is shown in Figure 5.6.

The trend of L1-norm error v/s order of filter presented in Figure 5.6 establishes the fact

that magnitude response of digital IIR filters equals to that of an ideal filter with the

increase of order.

Further the trend of fitness function obtained with EGSO with respect to

179

iterations for all types of digital IIR filters presented in Figure 5.7, clearly emphasize

that EGSO attains global solution in fix number of iterations.

Figure 5.5: Pole-zero plots of best L1 stable digital IIR filters using EGSO

Figure 5.6: Variation of L1-norm error with order for L1-approximated digital IIR

filter applying EGSO

180

Figure 5.7: Variation of L1-norm error with iterations for L1-approximated digital

IIR filter applying EGSO

5.5 Conclusion

In this chapter a robust methodology for the design of digital IIR filter is proposed.

EGSO a swarm intelligence based algorithm is used for global search and further the

exploratory search is employed to exploit the local search space. The unique

combination of broad exploration and further exploitation yields a powerful option to

solve multimodal optimization problems that designs IIR filters. The original GSO

algorithm is retailored by blending the concept of opposition-based learning for

evolution of group members and to enhance the convergence rate.

The proposed EGSO successfully solves the standard test numerical functions

and provides an enhanced performance as objective function is better in comparison to

GSO. EGSO is able to obtain an excellent balance between exploitation and exploration

and gives smaller standard deviations of function values than GSO. EGSO satisfies the

prescribed amplitude specifications consistently and is a useful tool for the constrained

181

IIR filter design problem. As depicted by the experimental results obtained for the

design of IIR digital filter, it can be concluded that the proposed method is efficient

algorithm in terms of number of function evaluations with higher accuracy in achieving

the small magnitude error in pass-band and stop-band ripples while maintaining the

stability.

chapter-5 group search optimization for the design...

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