Butterworth Lowpass IIR Filter Design

Source Code:

Wp=1500/4000;                     % normalised passband frequency
Ws=2000/4000;                     % normalised stopband frequency
Rp=0.03;
Rs=40;
[N,Wn]=buttord(Wp,Ws,Rp,Rs);

[num,den]=butter(N,Wn);         % computer the nominator and
                                                %  denominator of impulse response

[aaa,freq]=wavread('noisy2.wav');
n=length(aaa);                          %length of the original signal
signal=filter(num,den,aaa);         %generate the filtered signal

% plot original signal in time domain;
figure;
plot((1:n)/freq,signal);
title('Original signal in time domain');
xlabel('second');
grid;

% plot original signal in frequency domain.,
figure;
signal0=signal-mean(signal);
fsignal=fft(signal0);
 

plot((1:n/2)/n*freq,abs(fsignal(1:n/2)));
title('Original signal in frequency domain');
xlabel('Hz');
ylabel('magnitude');
grid;

figure;
plot((1:n/2)/n*freq,unwrap(angle(fsignal(1:n/2))));
%use help command to learn about unwarp function
title('original signal in frequency domain');
xlabel('Hz');
ylabel('phase');
grid;

origin=wavread('orig2.wav');
% calculate the MSE
total=0;
m=65536;% obtained from n (number of sample)
for i=1:m
   res=(signal(i)-origin(i))^2;
   total=total+res;

end
mse=(1/m)*total % MSE of filtered signal

tot2=0;

% calculate the SNR
for i=1:m
   tot2=tot2+origin(i)^2;
end
total2=(1/m)*tot2;
snr=10*log10(total2/mse) %SNR of filtered signal

tot3=0;
%SNR
for i=1:m
   tot3=tot3+signal(i)^2;
end
total3=(1/m)*tot3;
snr=10*log10(total3/mse) % SNR of original signal
 

Figure 1: Numerator

Figure 2: Denominator

Figure 3: Filtered signal in time domain

Figure 4: Filtered signal in frequency domain

Figure 5: Filtered signal(phase)  in frequency domain
 

SNR & MSE results:

MSE                  :0.1635
SNR(Filtered)    :17.3184
SNR(Original)   :0.089
 

Observation and Elaboration

    Figure 1 and figure 2 show the series of numerator and denominator of the IIR filter impulse response.
     whereas figure 3 to 5 show the characteristic of the filtered signal. the figures show that from the frequency range 1700 hz onwards, all of the signal is cutted down because signal above this range are all noise, cutting down these part will get rid of the noise at the same time maintain the content of the original signal (Orig2.wav).
      Besides, the characteristic of signal in time domain is closer to the original time domain signal compare with the one obtained in FIR  filter design, this show that the IIR filter design give a bette performance than the FIR filter design.