Objective:
Objective of this topic is to Design filter and to apply different type of IIR filters on audio signal.
TASK:
Design a discrete time low pass filter by applying the appropriate
type of following filters:
1.
Butterworth
2.
Chebyshev Type I
3.
Elliptic
The
specifications of the desired filters are:-
a.
b
.
1.
Design all three low pass filters on above
specifications observe poles, zeros and magnitude response and comment which
one is better and why?
2.
Now Add a noise using awgn command in audio
signal and apply all the 6 filters to remove noise, listen, observe and write
down which filter has performed well and why?
Part (A)
1. Butterworth:
2. Chebyshev Type 1:
3. Elliptic:
Part (B)
1. Butterworth:
2. Chebyshev Type 1:
Observations:
Part (A)
All the three filters are designed for the same specifications. The Butterworth filter was of order 10 with all the poles placed on imaginary axis. There are no ripples in the passband neither the stop band. The Chebyshev type 1 filter showed a decrease in the order and it was reduced to 6 with the change in position of the poles. However, no ripples were observed in the pass band due to the specifications. The elliptic filter showed extremely good response for the low pass filter reducing the order to 4. The stop band ripples are clearly visible but the pass band ripples are not. It is computationally fast as compared to the other two filters. Hence the elliptic filter is the best among these three.
Part (B)
Just like part a, all the three filters are designed for the another set of specifications. The Butterworth filter was of order 5 with all the poles placed on the right half in a curvy shape. There are no ripples in the passband neither the stop band. The Chebyshev type 1 filter showed a decrease in the order and it was reduced to 3 with the change in position of the poles. All the 3 poles are still on the right half, vertically. However, no ripples were observed in the pass band due to the specifications. The elliptic filter showed extremely good response for the low pass filter reducing the order to 2. The stop band ripples are clearly visible due to motion of the zeros on the circle, producing the dips/ripples. But the pass band ripples are not there. It is computationally fast as compared to the other two filters as it has the smallest number of coefficients and require less computations. Hence the elliptic filter is the best among these three.
02:
Code:
rec =
audiorecorder(8000,8,1); %create a
recorder object
record(rec,3); %record
audio for 3 seconds
x = getaudiodata(rec); %get the auio data array in x
y = awgn(x,10,'measured'); %adding gaussian noise to input
signal x);
T = 1/Fs; %increment
in time axis
%applying filters to the
noisy audio
y1 = filter(Ba,y); %Part a : Butterworth
y2 = filter(Ca,y); %Part a : Chebyshev Type 1
y3 = filter(Ea,y); %Part a : Elliptic
y4 = filter(Bb,y); %Part b : Butterworth
y5 = filter(Cb,y); %Part b : Chebyshev Type 1
y6 = filter(Eb,y); %Part b : Elliptic
sound(y,Fs)
%plot original and noisy
signals in time and frequency domain
figure
subplot(2,2,1)
tx = 0:T:(length(x)*T)-T;
plot(tx,x)
title('Original
Signal in Time Domain')
xlabel('Time')
ylabel('Amplitude')
subplot(2,2,2)
FFTx = abs(fft(x));
fx =
(0:length(FFTx)-1)*Fs/length(FFTx);
plot(fx,FFTx,'r')
title('Original
Signal in Frequency Domain')
xlabel('Frequency')
ylabel('Magnitude')
subplot(2,2,3)
t = 0:T:(length(y)*T)-T;
plot(t,y)
title('Noisy Signal
in Time Domain')
xlabel('Time')
ylabel('Amplitude')
subplot(2,2,4)
FFT = abs(fft(y));
f =
(0:length(FFT)-1)*Fs/length(FFT);
plot(f,FFT,'r')
title('Noisy Signal
in Frequency Domain')
xlabel('Frequency')
ylabel('Magnitude')
%plot outputs of all three
filters of part a in time and frequency domain
figure
subplot(3,2,1)
t1 = 0:T:(length(y1)*T)-T;
plot(t1,y1)
title('Butterworth
Filter')
xlabel('Time')
ylabel('Amplitude')
subplot(3,2,2)
FFT1 = abs(fft(y1));
f1 = (0:length(FFT1)-1)*Fs/length(FFT1);
plot(f1,FFT1,'r')
title('Butterworth
Filter')
xlabel('Frequency')
ylabel('Magnitude')
subplot(3,2,3)
t2 = 0:T:(length(y2)*T)-T;
plot(t2,y2)
title('Chebyshev
Type 1 Filter')
xlabel('Time')
ylabel('Amplitude')
subplot(3,2,4)
FFT2 = abs(fft(y2));
f2 =
(0:length(FFT2)-1)*Fs/length(FFT2);
plot(f2,FFT2,'r')
title('Chebyshev
Type 1 Filter')
xlabel('Frequency')
ylabel('Magnitude')
subplot(3,2,5)
t3 = 0:T:(length(y3)*T)-T;
plot(t3,y3)
title('Elliptic
Filter')
xlabel('Time')
ylabel('Amplitude')
subplot(3,2,6)
FFT3 = abs(fft(y3));
f3 =
(0:length(FFT3)-1)*Fs/length(FFT3);
plot(f3,FFT3,'r')
title('Elliptic
Filter')
xlabel('Frequency')
ylabel('Magnitude')
%plot outputs of all three
filters of part a in time and frequency domain
figure
subplot(3,2,1)
t4 = 0:T:(length(y4)*T)-T;
plot(t4,y4)
title('Butterworth
Filter')
xlabel('Time')
ylabel('Amplitude')
subplot(3,2,2)
FFT4 = abs(fft(y4));
f4 =
(0:length(FFT4)-1)*Fs/length(FFT4);
plot(f4,FFT4,'r')
title('Butterworth
Filter')
xlabel('Frequency')
ylabel('Magnitude')
subplot(3,2,3)
t5 = 0:T:(length(y5)*T)-T;
plot(t5,y5)
title('Chebyshev
Type 1 Filter')
xlabel('Time')
ylabel('Amplitude')
subplot(3,2,4)
FFT5 = abs(fft(y5));
f5 =
(0:length(FFT5)-1)*Fs/length(FFT5);
plot(f5,FFT5,'r')
title('Chebyshev
Type 1 Filter')
xlabel('Frequency')
ylabel('Magnitude')
subplot(3,2,5)
t6 = 0:T:(length(y6)*T)-T;
plot(t6,y6)
title('Elliptic
Filter')
xlabel('Time')
ylabel('Amplitude')
subplot(3,2,6)
FFT6 = abs(fft(y6));
f6 = (0:length(FFT6)-1)*Fs/length(FFT6);
plot(f6,FFT6,'r')
title('Elliptic
Filter')
xlabel('Frequency')
ylabel('Magnitude')
Output:
Original and noisy signals in time
and frequency domain
Outputs of Filters designed in Part A:
Outputs of Filters designed in Part B:
Observations:
All
the Outputs of the filters designed in part a still had a hollow noise in the
background. Elliptic was better than Chebyshev type 1 and it was better than
Butterworth.
Filters
in part b produced comparatively good results as compared to those in part a.
In Butterworth, the noisy effect was still there but it was quite diminished.
Same was the case with the Chebyshev where the noise was very minute but also
the actual audio signal got diminished. Overall the result was quite good
considering the reduction of noise to a large extent. In case of Elliptic filter, the output had
some noise but the audio was quite loud and clear. But it was not able to
remove the noise completely too. Considering the noise reduction aspect, Chebyshev
Type 1 in part b produced the best results. This is as the pass band
was quite small and it attenuated the higher frequency very smoothly. However, elliptic
in part b worked great in bringing back the original audio signal as
the amplitude of the lower frequencies did not attenuate.
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