Digital Signal Processing

Lab 2

Exercise 1

Fourier transform and the inverse Fourier transorm

1. Frequency response of the moving-average system (Page 44)

H = 0;

w=-pi:pi/5000:pi;

h = [1 1 1 1 1]/5;

for k=1:length(h)                 % transform

      H = H + h(k)*exp(-j*(k-1)*w);                              

end                              

mag=abs(H);

phase=angle(H);

figure(1)

plot(w,mag)

axis([-pi pi 0 max(mag)])

figure(2)

plot(w,phase)

axis([-pi pi -pi pi])

h2 = zeros(size(h));  

dw = 1/10000;

for k=1:length(h2)                 % inverse transform

      h2(k) = sum(H.* exp(j*(k-1)*w) )*dw;

end  

2. Ideal lowpass filter (Page 43)

wc = pi/3;

w=-pi:pi/5000:pi;

H = zeros(size(w));

t1 = round((-wc-(-pi))/(pi/5000))+1;

t2 = round((wc-(-pi))/(pi/5000))+1;

H(t1:t2) = ones(1, t2-t1+1);

mag=abs(H);

phase=angle(H);

figure(1)

plot(w,mag)

axis([-pi pi 0 max(mag)])

figure(2)

plot(w,phase)

axis([-pi pi -pi pi])

h2 = zeros(1, 30);  

dw = 1/10000;

for k=1:length(h2)                 % sum(hk exp(-jkw)

      h2(k) = sum(H.* exp(j*(k-1)*w) )*dw;

end

figure(3)

plot(1:length(h2),h2)

3. Square-summability for the ideal lowpass filter (Page 52)

h = h2;

H = 0;

for k=1:length(h)                 % sum(hk exp(-jkw)

      H = H + h(k)*exp(-j*(k-1)*w);

      if k > 1,

          H = H + h(k)*exp(-j*(-(k-1))*w);

      end

end                              

mag=abs(H);

phase=angle(H);

figure(4)

plot(w,mag)

axis([-pi pi 0 max(mag)])

figure(5)

plot(w,phase)

axis([-pi pi -pi pi])

Sources:

Fundamentals of Digital Signal Processing, by Joyce Van de Vegte, Prentice Hall, 2002.