Using a rectangular window of width 50 and a sampling rate of 10,000 samples/second, design an FIR digital filter to approximate the analog filter whose transfer function is
Compare the frequency responses of the analog and digital filters.
Inverse Laplace transforming we get the analog impulse response
Sampling at 10,000 samples/second and truncating to 50 samples we get the impulse response of the digital FIR filter as
The easiest way to find the frequency response of the digital filter is to use the FFT in MATLAB.
% Program to graph the frequency response of a windowed FIR filter with
% 50 samples sampled at 10 kHz to approximate an analog filter with
% transfer function
%
% Ha(s) = 2000s/(s^2+2000s+2e6)
%
close all ; figure('Position',[20,20,1500,800],'PaperPosition',[0.5,0.5,16,9.5]) ;
f = [0:20:5000]' ; % Vector of cyclic frequencies
w = 2*pi*f ; % Vector of radian frequencies
s = j*w ; % Vector of s's
Ha = 2000*s./(s.^2+2000*s+2e6) ; % Analog filter freq resp
fs = 10000 ; % Sampling rate
n = [0:49]' ; % Discrete-time vector for graphing
N = 50 ; % Length of impulse response
% Generate the digital filter's impulse response
hd = 2000*sqrt(2)*(0.9048.^n).*cos(0.1*n+0.7854).*(uD(n) - uD(n-50)) ;
% Extend the impulse response by a factor of padFac by zero padding
padFac = 8 ; % Zero padding factor
n = [0:padFac*N-1]' ; % Extend the impulse response time
hd = [hd;zeros((padFac-1)*N,1)] ; % Extend the impulse response
Hd = fft(hd) ; % Find the DTFT using the DFT
F = n/(padFac*N) ; % Vector of cyclic discrete-time frequencies
W = 2*pi*F ; % Vector of radian discrete-time frequencies
% Graph the results
subplot(2,2,1) ;
ptr = plot(f,abs(Ha),'k') ; set(ptr,'LineWidth',2) ; grid on ;
xlabel('Frequency, {\itf} (Hz)','FontName','Times','FontSize',24) ;
ylabel('|H_{\ita}( {\itj}2\pi{\itf} )|','FontName','Times','FontSize',24) ;
title('Analog','FontName','Times','FontSize',36) ;
set(gca,'FontName','Times','FontSize',18) ;
subplot(2,2,3) ;
ptr = plot(f,angle(Ha),'k') ; set(ptr,'LineWidth',2) ; grid on ;
xlabel('Frequency, {\itf} (Hz)','FontName','Times','FontSize',24) ;
ylabel('Phase of H_{\ita} ( {\itj}2\pi{\itf} )','FontName','Times','FontSize',24) ;
set(gca,'FontName','Times','FontSize',18) ;
W = W(1:padFac*N/2) ; Hd = Hd(1:padFac*N/2) ;
subplot(2,2,2) ;
ptr = plot(W,abs(Hd),'k') ; set(ptr,'LineWidth',2) ; grid on ;
xlabel('Frequency, {\Omega}','FontName','Times','FontSize',24) ;
ylabel('|H_{\itd} ({\ite}^{{\itj}\Omega})|','FontName','Times','FontSize',24) ;
title('Digital','FontName','Times','FontSize',36) ;
set(gca,'FontName','Times','FontSize',18) ;
subplot(2,2,4) ;
ptr = plot(W,angle(Hd),'k') ; set(ptr,'LineWidth',2) ; grid on ;
xlabel('Frequency, {\Omega}','FontName','Times','FontSize',24) ;
ylabel('Phase of H_{\itd} ({\ite}^{{\itj}\Omega})','FontName','Times','FontSize',24) ;
set(gca,'FontName','Times','FontSize',18) ;
print -depsc -adobecset 'temp' ;
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