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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
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Thesis outline
FIR Filter & Different design methods of it.Optimization TechniqueGenetic AlgorithmResults Conclusion & Future scope
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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
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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
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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
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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)
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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
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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
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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 )
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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
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The Objective For maximization problem For minimization problem Relative Fitness Function
Important steps in GA Encoding Selection Crossover Mutation
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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)
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Selection Roulette Wheel Selection Selected based on fitness
Rank Selection
Situation before ranking Situation after ranking
Less chance to be selected slow convergence problem
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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 %).
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• 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
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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
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FIR Filter using rectangular window method
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FIR Filter using Hanning window method
FIR Filter using Hamming window method
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FIR Filter using Kaiser window method with β = 5.8
FIR Filter using Frequency Sampling method
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FIR Filter using Least squared error method
FIR Filter using Roulette Wheel Selection method
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FIR Filter using Rank Selection method
FIR Filter using Tournament Selection method
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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
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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
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Any queries ?
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THANK YOU