Design a digital filter approximation to each of these ideal analog filters by sampling a truncated version of the impulse response and using the specified window. In each case choose a sampling frequency which is 10 times the highest frequency passed by the analog filter. Choose the delays and truncation times such that no more than 1% of the signal energy of the impulse response is truncated. Graphically compare the magnitude frequency responses of the digital and ideal analog filters using a dB magnitude scale versus linear frequency.

What will be an ideal response?




fc = 1 ; type = 'LP' ; fs = 10*fc ;


%


%	Lowpass, Rectangular Window


%


hd = FIRDF(type,fc,fs,'RE',0.01) ;


N = length(h) ; Ts = 1/fs ; n = [0:N-1]' ;


F = [0:0.001:1/2]' ; [Hd,F] = DTFT(n,hd,F) ;


subplot(2,2,1) ;


p = xyplot(F*fs,20*log10(abs(Hd)),[0,fs/2,-120,0],'\itf ',...


'|H_d(\ite^{{\itj}2{\pi}{\itf}{\itT}_s} )|',...


'Times',18,'Times',14,...


'Lowpass - Rectangular Window','Times',24,'n','c') ;





subplot(2,2,2) ;


p = xyplot(F*fs,20*log10(abs(Hd)),[0,fs/10,-5,0],'\itf ',...


'|H_d(\ite^{{\itj}2{\pi}{\itf}{\itT}_s} )|',...


'Times',18,'Times',14,...


'Lowpass - Rectangular Window','Times',24,'n','c') ;


%


%	Lowpass, von Hann Window


%


hd = FIRDF(type,fc,fs,'VH',0.01) ;


N = length(h) ; Ts = 1/fs ; n = [0:N-1]' ;


F = [0:0.001:1/2]' ; [Hd,F] = DTFT(n,hd,F) ;


subplot(2,2,3) ;


p = xyplot(F*fs,20*log10(abs(Hd)),[0,fs/2,-120,20],'\itf ',...


'|H_d(\ite^{{\itj}2{\pi}{\itf}{\itT}_s} )|',...


'Times',18,'Times',14,...


'Lowpass - Von Hann Window','Times',24,'n','c') ;


subplot(2,2,4) ;


p = xyplot(F*fs,20*log10(abs(Hd)),[0,fs/10,-5,0],'\itf ',...


'|H_d(\ite^{{\itj}2{\pi}{\itf}{\itT}_s} )|',...


'Times',18,'Times',14,...


'Lowpass - Von Hann Window','Times',24,'n','c') ;





%	Function to design a digital filter using truncation of the 


%	impulse response to approximate the ideal impulse response.  The 


%	user specifies the type of ideal filter,


%


%	LP	-	lowpass


%	BP	-	bandpass


%


%	the cutoff frequency(s) fcs (a scalar for LP and HP a 2-vector


%	for BP and BS), the sampling rate, fs, the type of window,


%


%	RE	-	rectangular


%	VH	-	von Hann


%	BA	-	Bartlett


%	HA	-	Hamming


%	BL	-	Blackman


%


%	and the allowable truncation error,


%	err, as a fraction of the total impulse response signal energy.


%


%	The function returns the filter coefficients as a vector.


%


%	function hd = FIRDF(type,fcs,fs,window,err)


%


function hd = FIRDF(type,fcs,fs,window,err)





if fs == 0 | fs == inf | fs == -inf,


disp('Sampling rate is unusable') ;


else


fs


Ts = 1/fs ; 


type = upper(type) ; window = upper(window) ;


switch type


case 'LP',


fc = fcs(1) ; zc1 = 1/(2*fc) ;


N = 200*zc1/Ts ; n = [0:N]' ;


Etotal = sum(sinc(2*fc*n*Ts).^2)


E = 0 ; N = 1 ;


while abs(E-Etotal) > Etotal*err,


N = 2*N ;


n = [0:N]' ;


E = sum(sinc(2*fc*n*Ts).^2) ;


end


delN = floor(N/10) ;


while abs(E-Etotal)1,


N = N-delN ;


n = [0:N]' ;


E = sum(sinc(2*fc*n*Ts).^2) ;


delN = floor(delN/2) ;


end


N = ceil(N/2)*2 ;	%	Make number of pts even


nmid = (N/2-1) ;


w = makeWindow(window,n,N) ;


hd = w.*sinc(2*fc*(n-nmid)*Ts) ;


hd = hd/sum(hd) ;


case 'BP'


fl = fcs(1) ; fh = fcs(2) ; fmid = (fh + fl)/2 ; 


df = abs(fh-fl) ; zc1 = 1/df ;


N = 200*zc1/Ts ; n = [0:N]' ;


Etotal = sum((2*df*sinc(df*n*Ts).*...


cos(2*pi*fmid*n*Ts)).^2) ;


E = 0 ; N = 1 ;


while abs(E-Etotal) > Etotal*err,


N = 2*N ;


n = [0:N]' ;


E = sum((2*df*sinc(df*n*Ts).*...


cos(2*pi*fmid*n*Ts)).^2) ;


end


delN = floor(N/10) ;


while abs(E-Etotal)1,


N = N-delN ;


n = [0:N]' ;


E = sum((2*df*sinc(df*n*Ts).*...


cos(2*pi*fmid*n*Ts)).^2) ;


delN = floor(delN/2) ;


end


N = ceil(N/2)*2 ;	%	Make number of pts even


nmid = (N/2-1) ; n = [0:N-1]' ;


w = makeWindow(window,n,N) ;


hd = w.*2.*df.*sinc(df*(n-nmid)*Ts).*...


cos(2*pi*fmid*(n-nmid)*Ts) ;


hd = hd/sum(hd.*cos(2*pi*fmid*(n-nmid)*Ts)) ;


end


end





function w = makeWindow(window,n,N)


switch window


case 'RE'


w = ones(N,1) ;


case 'VH'


w = (1-cos(2*pi*n/(N-1)))/2 ;


case 'BA'


w = 2*n/(N-1).*(0<=n & n<=(N-1)/2) + ...


(2-2*n/(N-1)).*((N-1)/2<=n & n

case 'HA'


w = 0.54 - 0.46*cos(2*pi*n/(N-1)) ;


case 'BL'


w = 0.42 - 0.5*cos(2*pi*n/(N-1)) + ...


0.08*cos(4*pi*n/(N-1)) ;


otherwise


w = ones(N,1) ;


end


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