Dissertation Report On

Optimal design of FIR filter using different selection methods in Genetic Algorithm

GUIDE

Prof. P.K. SHAH

Associate Professor

ECED, SVNIT

SUBMITTED BYMr. RAKESH PATIDAR(P13VL010)

ELECTRONICS ENGINEERING DEPARTMENTSARDAR VALLABHBHAI NATIONAL INSTITUTE OF TECHNOLOGY

Thesis outline

FIR Filter & Different design methods of it.Optimization TechniqueGenetic AlgorithmResults Conclusion & Future scope

Digital Filters It operates on the digital samples of the signals. These kinds of filters are defined using linear difference equations. Analog Filters It operates on the analog(actual) signal. It is defined by linear differential equations. Applications of Filters: in Audio signal processing speech processing image enhancement Communication Systems

FIR vs IIR Filters FIR Filters are linear phase & always stable. FIR requires higher order(N). FIR generally Non-Recursive. Filter design methods: window method

w(n) = 1 0≤ n ≤ M-1 = 0 otherwise

h(n) = hd(n) w(n)   = hd(n) 0 ≤ n ≤ M-1

=0 otherwise

Bartlett triangular window Rectangular(Gibbs phenomenon effect ), Hanning, Hamming and

Blackman windows Kaiser window Advantage of window method Simplicity to use As compared to other methods. Disadvantages of window method This method is applicable only if Hd(w) is absolutely integrable.

Only in design of prototype filters. The Frequency Sampling Technique

11

2 1

0

( ) Sin2

( )Sin

2

j kj NT N

N Nj T

k

NTH k e

eH e

T kNN

j Tz e Put

• The approximation error would then be exactly zero at the sampling frequencies and would be finite in frequencies between them. • In order to reduce this error increase no. of sample points (N).

• Merit of Frequency Sampling Technique• Any given magnitude response• Non prototype filter • Demerit of Frequency Sampling Technique• Frequency response obtained = Desired Frequency response (Only at sampling points)

Optimal Filter Design Methods Least squared error frequency domain design

Weighted Chebyshev Approximation• The weighting function enables the designer to choose the relative size of

the error in different frequency bands.

When L=0 h(n) will be Symmetric. When L=1 h(n) will be AntiSymmetric.

Error(E) = |(H(wk)-Hd(wk)|2  

21

0

1( ) ( )

j kLN

k

h n H k eN

1

0

( ) ( )N

j n

n

H w h n e

L>N

E( ) = W( )[ ( )dH - ( )H ]

E( )= W( )[ ( )dH -P( )Q( )]

E( ) = W( )Q( )[ ( )dH /Q( ) - P( )]

W^( ) = W ( )Q( )

( )dH = Hd( )/Q( )

E( ) = W^( ) [ ( )dH – P( ) ]

[( ]) MinE EMax

Optimization Statement of an optimization problem Objective Function f ( X )

X= [x1 x 2 x3 ........ x n ]T

g j( X ) ≤ 0 j = 1, 2............................m

h j ( X ) = 0 j = 1, 2............................ p

g j( X ) & h j ( X ) are inequality &equality constraint. Singal variable optimization algorithm These algorithms optimize only single objective Multi variable optimization algorithm These algorithms optimize more than one objective f ( X ) = a1 f1 ( X ) + a 2 f 2 ( X )

Optimallity criteria Local optimal point Global optimal point Inflaction point

Genetic algorithm Darwin's evolution theory A set of solutions (represented by chromosomes)

called population. Solutions which are selected to form new solutions

(offspring) are selected according to their fitness. Population size

The Objective For maximization problem For minimization problem Relative Fitness Function

Important steps in GA Encoding Selection Crossover Mutation

Encoding Binary Encoding

Permutation Encoding

Value Encoding

Chromosome A 101100101100101011100101

Chromosome B 111111100000110000011111

Chromosome A 1  5  3  2  6  4  7  9  8

Chromosome B 8  5  6  7  2  3  1  4  9

Chromosome A 1.2324  5.3243  0.4556  2.3293  2.4545

Chromosome B ABDJEIFJDHDIERJFDLDFLFEGT

Chromosome C (back), (back), (right), (forward), (left)

Selection Roulette Wheel Selection Selected based on fitness

Rank Selection

Situation before ranking Situation after ranking

Less chance to be selected slow convergence problem

Tournament several tournaments among a few individuals chosen at random from the

population. If the tournament size is larger, weak individuals have a smaller chance to

be selected. Elitism Copy few best chromosome to new population. Crossover

Parent 1: 11001|010 Parent 2: 00100|111

Offspring1: 11001|111 Offspring2: 00100|010 Crossover probability all offspring are made by crossover (100%). offspring are exact copies of parents (0 %).

• Mutation Probability• Whole chromosome is changed (100%).• No change (0 %).• Multiobjective Optimization• Does not exist one solution that is best with respect to all objectives• In MOOP problem the decision maker articulates its preference concerning the different objectives: never, before, during or after optimization procedure.

Classifications of Methods of Multi Objective Optimization

Results FIR Low pass filter designed with following parameters for all

methods Passband ripple(rp) = 0.02 Stopband ripple(rs ) = 0.01 Passband frequency(fp) = 1500 Hz Stopband frequency(fs) = 2000 Hz Sampling frequency(fs) = 6000 Hz

physical realizable low pass filter

FIR Filter using rectangular window method

FIR Filter using Hanning window method

FIR Filter using Hamming window method

FIR Filter using Kaiser window method with β = 5.8

FIR Filter using Frequency Sampling method

FIR Filter using Least squared error method

FIR Filter using Roulette Wheel Selection method

FIR Filter using Rank Selection method

FIR Filter using Tournament Selection method

Filter design method Transition width Peak side lobe(in dB)

Rectangular window 0.0975 -21.09

Hanning window 0.2977 -44.03

Hamming window 0.3487 -55.21

Kaiser window 0.3742 -60.93

Frequency Sampling 0.4403 -59.94

Least squared error 0.3123 -37.46

Roulette Wheel Selection 0.13879 -50

Rank Selection 0.10604 -54.67

Tournament Selection 0.1572 -58

Comparison of different filter design methods

Conclusion & Future ScopeResults obtained through different selection

methods are more optimal than other methods.Different new heuristic optimization methods:-Reduce complexity of the implementation of the

algorithmsReduce design error increase convergence speed Improve performance of the design methods

Any queries ?

THANK YOU