There are several different methods available the most popular are
We will just consider the window method.
38 Window Method
First stage of this method is to calculate the coefficients of the ideal filter.
This is calculated as follows
39 Window Method
Second stage of this method is to select a window function based on the passband or attenuation specifications then determine the filter length based on the required width of the transition band.
Using the Hamming Window 40 Window Method
The third stage is to calculate the set of truncated or windowed impulse response coefficients hn
for Where for 41 Window Method
Matlab code for calculating coefficients
close all clear all fc 8000/44100 cut-off frequency N 133 number of taps n -((N-1)/2)((N-1)/2) n n(n0)eps avoiding division by zero h sin(n2pifc)./(npi) generate sequence of ideal coefficients w 0.54 0.46cos(2pin/N) generate window function d h.w window the ideal coefficients gf freqz(d151244100) transform into frequency domain for plotting figure(1) plot(f20 log10(abs(g))) plot transfer function axis(0 2104 -70 10) figure(2) ste m(d) plot coefficient values xlabel(Coeffic ient number) ylabel (Value) title(Truncated Impulse Response) figure(3) freqz(d15124410 0) use freqz to plot magnitude and phase response axis(0 2104 -70 10) 42 Window Method 43 Equiripple Design
Starting point is minimax criterion e.g.
Based on theory of Chebyshev approximation and the alternation theorem which (roughly) states that the optimal ais are such that the max (maximum weighted approximation error) is obtained at L2 extremal frequencies
that hence will exhibit the same maximum ripple (equiripple)
Iterative procedure for computing extremal frequencies etc. (Remez exchange algorithm Parks-McClellan algorithm)
Very flexible etc. available in many software packages
Details omitted here (see textbooks)
FIR Filter design abundantly available in commercial software
bfir1(nWntypewindow) windowed linear-phase FIR design n is filter order Wn defines band-edges type is highstop
bfir2(nfmwindow) windowed FIR design based on inverse fourier transform with frequency points f and corresponding magnitude response m
bremez(nfm) equiripple linear-phase FIR design with Parks-McClellan (Remez exchange) algorithm
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