Introduction
This paper is about signal processing in Matlab. IEEE Signal Processing Society organize a competition among universities all around the world. Beside, this year, the goal of the competition was to track the beat of a song in real-time. You are supposed to implement a simple tempo estimation algorithm. The details about Signal Processing are given in the problems.
Problem 1 Signal Processing
Sampling frequency is 44100 Hz. Design a digital IIR Butterworth filter
• Pass-band edge frequency is 200 Hz. The ripple at pass-band should be -1 dB.
• Stop-band edge frequency is 400 Hz. The minimum attenuation at stop-band is -40 dB.
Manipulate the Matlab Code at the bottom of the page to design your filter. Plot the resulting magnitude response in two
subplots. x axes are frequency in terms of Hertz, y axes are magnitudes in terms of decibels. • For the first subplot, x axis range is [0 − 22050] Hz.
• For the second subplot, x axis range is [0 − 600] Hz (Hint: xlim command).
Questions:
• The final Matlab code,
• Filter coefficients ai and bis,
• Transfer function H(z),
• Magnitude response plots (two subplots),
• Pole-zero map of the filter (Hint: pzmap command)
Firstly, Matlab Code:
Fs=44100;
Ts=1/Fs;
Fn=Fs/2;
wp=200/Fn;
ws=400/Fn;
Rp=1;
Rs=40;
a(0) = 1
a(1) = -7.83573
a(2) = 26.8636
a(3) = -52.6303
a(4) = 64.4488
a(5) = -50.5127,
a(6) = 24.7452
a(7) = -6.9274
a(8) = 0.848502
b(0) = 3.98943e-15
b(1) = 3.19154e-14
b(2) = 1.11704e-13
b(3) = 2.23408e-13
b(4) = 2.7926e-13
b(5) = 2.23408e-13
b(6) = 1.11704e-13
b(7) = 3.19154e-14
b(8) = 3.98943e-15
Question 1 :
[N,wc]=buttord(wp,ws,Rp,Rs,’s’); [z,p,k]=butter(N,wc,’s’); Hbanalog=zpk(z,p,k) [b,a]=butter(N,wc);
Hbdigital=tf(b,a,Ts)
[h,w] = freqz(b,a);
dB = mag2db(abs(h));
figure(1)
subplot(2,1,1)
plot((w/pi)*(Fs/2),dB)
xlabel(‘\omega / \pi’)
ylabel(‘Magnitude (dB)’)
xlim([0 22050])
grid;
subplot(2,1,2)
plot((w/pi)*(Fs/2),dB)
xlabel(‘\omega / \pi’)
ylabel(‘Magnitude (dB)’)
xlim([0 600])
grid;
for k = 1:N+1
fprintf(‘a(%d) = %8.6g, \t b(%d) = %8.6g \n’, k-1, a(k),k-1, b(k));
end
figure(2)
pzmap(Hbdigital)
grid;
The transfer function H(z)
1.1728e-16
—————————————————————————————————————-
(s^2 + 0.02001s + 0.0001041) (s^2 + 0.01696s + 0.0001041) (s^2 + 0.01134s + 0.0001041) (s^2 + 0.00398s + 0.0001041)
The magnitude response plots
Pole-zero map of the filter
Problem 2 of Signal Processing
In this paragraph, use the filter designed in Problem 1 to filter the input signal. Use the filtfilt command to avoid delay. Plot the input and output signals.
Questions:
• The final Matlab code,
• Input and output signals in two subplots,
• The input and output FFT plots in two subplots.
Firstly, Matlab Code:
load question2_signal;
load low_pass_coeff;
Fs=44100;
Ts=1/Fs;y=filtfilt(b,a,x);
yf=fft(y);
xf=fft(x);N = 4411;
dF = Fs/N;
f =0:dF:Fs-dF;
subplot(2,2,1)
plot(t,x);
xlabel(‘Time’);
ylabel(‘Magnitude’);
title(‘Input’);
grid;subplot(2,2,3)
plot(t,y);
xlabel(‘Time’);
ylabel(‘Magnitude’);
title(‘Output’);
ylim([-15 10])
grid;
subplot(2,2,2)
plot(f,abs(xf)/N);
xlabel(‘Frequency (in hertz)’);
ylabel(‘Magnitude’);
title(‘Input’);
grid;subplot(2,2,4)
plot(f,abs(yf)/N);
xlabel(‘Frequency (in hertz)’);
ylabel(‘Magnitude’);
title(‘Output’);
grid;
The input and output signals
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