The document contains MATLAB code for digital signal processing programs including: 1) Bandpass filters, Kaiser window functions, time domain windows, DFT of square waves with different duties, notch filters, and resonators. 2) Comb filters and the Welch method for calculating the power spectral density of a noisy signal. 3) A discrete Fourier transform program that calculates the forward and inverse DFT using twiddle factors.

1. MATLAB DSP Programs
Raja Rajasakeran
Venkat Peddigari
Pete Bernardin

2. Bandpass Filter
close all;
clear all;
r1 = 0.9;
r2 = 0.88;
r3=0.9;
theta1 = pi/5;
theta2 = 3*pi/10;
theta3=4*pi/10;
p1 = r1*cos(theta1) + i*r1*sin(theta1);
p2 = r1*cos(theta2) + i*r2*sin(theta2);
p3 = r3*cos(theta3) + i*r3*sin(theta3);
p2 = r2*cos(theta2) + i*r2*sin(theta2);
!aroots = [p1,conj(p1),p2,conj(p2), p3, conj(p3)];
aroots = [p2,conj(p2), p3, conj(p3)];
a = poly(aroots);
b = [1 zeros(1,18) -1];
b1 = [1];
[h,w] = freqz(b1,a);
freqzplot(h,w, 'mag');
figure;
freqzplot(h,w,'squared');
( )( ) ( )( )[ ]
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21
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21
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cos219.0cos219.0/1
119.0119.0/1)(
32
3322
−−−−
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−−
=
−−
−−−+−−−+
+−+−=
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+−+−=
−−−−=
zzzz
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zezezezezH jjjj
ππ
ωω
πωπω
ωωωω

3. Kaiser Bessel Window
clear all;
close all;
N=61;
w31 = window(@kaiser,N,1);
w33 = window(@kaiser,N,3);
w35 = window(@kaiser,N,5);
w37 = window(@kaiser,N,7);
plot(1:N,[w31, w33, w35, w37]); axis([1 N 0 1]);
legend('beta=1','beta = 3', 'beta=5', 'beta=7');
figure;
a=[1];
[fw31,f] = freqz(w31,a);
[fw33,f] = freqz(w33,a);
[fw35,f] = freqz(w35,a);
[fw37,f] = freqz(w37,a);
freqzplot([fw31,fw33,fw35,fw37],f, 'mag');
legend('beta=1','beta = 3', 'beta=5', 'beta=7');

4. Time Domain Windows
clear all;
close all;
N=61;
w = window(@blackmanharris,N);
w1 = window(@hamming,N);
w2 = window(@gausswin,N,2.5);
w3 = window(@kaiser,N,5);
w4 = window(@tukeywin,N,0.5);
plot(1:N,[w1,w3,w4]); axis([1 N 0 1]);
legend('Hamming','Kaiser', 'Tukey');
figure;
a=[1];
[fw,f] = freqz(w,a);
[fw1,f] = freqz(w1,a);
[fw2,f] = freqz(w2,a);
[fw3,f] = freqz(w3,a);
[fw4,f] = freqz(w4,a);
freqzplot([fw1,fw3,fw4],f, 'mag');
legend('Hamming','Kaiser', 'Tukey');

5. DFT of Square Waves, 3 Duties
close all;
clear all;
x = ones(1,16);
y = fft(x,16);
my = abs(y)/16;
ay = angle(y);
stem(my);
figure;
y1 = fft (x,32)/16;
my1 = abs(y1);
stem(my1);
y2= fft (x,128);
figure;
stem (abs(y2));
100% Duty 50% Duty
12.5% Duty

6. Notch Filters
clear all;
close all;
a = [1 -1.2726 0.81];
b = [1 -1.414 1];
[H, W] = freqz(b,a);
figure;
freqzplot(H,W,'linear');
title ('Pole-zero Notch Filter - Mag');
figure;
freqzplot(H,W);
title ('Pole-zero Notch Filter - dB');
a1 = [1];
b1 = [1 -1.414 1];
[H1, W] = freqz(b1,a1);
figure;
freqzplot(H1,W,'linear');
title ('All zero Notch Filter - Mag');
figure;
freqzplot(H1,W);
title ('All zero Notch Filter - dB');
1 2
3 4

7. Resonators
clear all;
close all;
r = 0.99;
theta = pi/4;
a = [1 -2*r*theta r^2];
b = 1;
[H, W] = freqz(b,a);
figure;
freqzplot(H,W,'linear');
title ('Resonator r=0.99, theta = pi/4 - Mag');
figure;
freqzplot(H,W);
title (' Resonator r=0.99, theta = pi/4- dB');
r = 0.95;
theta = pi/4;
a = [1 -2*r*theta r^2];
b = 1;
[H, W] = freqz(b,a);
figure;
freqzplot(H,W,'linear');
title ('Resonator r=0.95, theta = pi/4 - Mag');
figure;
freqzplot(H,W);
title (' Resonator r=0.95, theta = pi/4- dB');
r = 0.99;
theta = pi/4;
a = [1 -2*r*theta r^2];
b = [1 0 -1]
[H, W] = freqz(b,a);
figure;
freqzplot(H,W,'linear');
title ('Pole-zero Resonator r=0.99, theta = pi/4 - Mag');
figure;
freqzplot(H,W);
title ('Pole-zero Resonator r=0.99, theta = pi/4- dB');
r = 0.9;
theta = pi/4;
a = [1 -2*r*theta r^2];
b = 1;
[H, W] = freqz(b,a);
figure;
freqzplot(H,W,'linear');
title ('Resonator r=0.9, theta = pi/4 - Mag');
figure;
freqzplot(H,W);
title (' Resonator r=0.9, theta = pi/4- dB');
r = 1.0;
theta = pi/4;
a = [1 -2*r*theta r^2];
b = 1;
[H, W] = freqz(b,a);
figure;
freqzplot(H,W,'linear');
title ('Oscillator, theta = pi/4 - Mag');
figure;
freqzplot(H,W);
title ('Oscillator, theta = pi/4- dB');
1

8. Resonators (continued)
2 43
98
765
10

9. Comb Filters
clear all;
close all;
a = [1 -1];
b = [1 0 0 0 0 0 0 0 0 0 0 -1];
b = b/11;
[H, W] = freqz(b,a);
figure;
freqzplot(H,W,'linear');
title ('Comb Filter M=10 - Mag');
figure;
freqzplot(H,W);
title ('Comb Filter M=10 - dB');
bz = zeros (1,54);
a1 = [1 0 0 0 0 -1];
b1 = [1 [bz] -1];
[H, W] = freqz(b1,a1);
figure;
freqzplot(H,W,'linear');
title ('LM Comb Filter L = 5 M=10 - Mag');
figure;
freqzplot(H,W);
title ('LM Comb Filter L = 5 M=10 - dB');
1 2
3 4

10. Welch Periodogram PSD
close all;
clear all;
t = 0:.001:4.096;
x = sin(2*pi*50*t) + sin(2*pi*120*t);
stdev = 2;
y = x + stdev*randn(size(t));
figure;
plot(y(1:50))
title('Noisy time domain signal')
FS = 1000.; % Sampling Rate
% 1 section
NFFT = 4096;
Noverlap = 0
figure;
pwelch(y, 4096, Noverlap, NFFT, FS)
% 16 overlapping sections
NFFT = 512;
Noverlap = 256
figure;
pwelch(y, 512, Noverlap, NFFT, FS)
% 32 overlapping sections
NFFT = 256;
Noverlap = 128
figure;
pwelch(y, 256, Noverlap, NFFT, FS)
¼ Hz Res
2 Hz Res 4 Hz Res
Time
1 2
43

11. Discrete Fourier Transform (DFT)
Forward Transform: for k=0,1,2,…,N-1
Inverse Transform: for n=0,1,2,…,N-1
-----------------------------------------------------------------------------------------------------------
N=8; % MATLAB “DISCRETE FOURIER TRANSFORM” PROGRAM
w0 = 2*pi/N;
K0 =3;
w=w0*K0*(0:N-1);
Data = complex(cos(w),sin(w)); % Data = ej2πnKo/N
for k =1:N
accum=complex(0,0);
for n=1:N
A = w0*(n-1)*(k-1);
Twiddle=complex(cos(A),-sin(A)); %Twiddle = e-j2πnk/N
accum = accum+Data(n)*Twiddle; % X(k)= Σej2πnKo/N
e-j2πnk/N
end % "n" is Time index
MAGXk=abs(accum) % Output the Magnitude of the DFT =N @ k=K0
end % "k" is the Frequency index