DSP Lab – Week 1

Constructing the Complex Plane

Suppose we have a sampled signal defined by the sequence h(n), n=0,1,2,...,N-1

Its Z- transform is given by H(z) = sum_{n=0}^{N-1} h(n)z^{-n} .

It maps the original sequence into a new domain, which is the complex plane z=e^{sT} where s=sigma+jomega is the parameter in the Laplace domain and T is the sampling period.

The jomega axis in the s-plane maps onto the unit circle with centre at the origin in the z-plane. So the value of H(z) at different points on the unit circle actually gives the contribution of the frequency component given by angle z, in the original signal.

This, in effect, gives the Discrete Fourier Transform of the sequence. Consider the following example:

%original sequence
h = [1,2,3,4];

%number of chosen points on the unit circle
N = 64;

%define the chosen points
z = complex(cos(2*pi/N*(0:N-1)),sin(2*pi/N*(0:N-1)));

%evaluate H(z) at each point
for i = 1:N
H(i) = 1+2*z(i)^-1+3*z(i)^-2+4*z(i)^-3;
end

%plot the unit circle
plot(z)

%plot the value of H(z) along the unit circle
figure
plot(abs(H))

%plot the N-point DFT of h(n)
figure
plot(abs(fft(h,64)))

This example computes the value of H(z) at 64 uniformly spaced points on the unit circle and compares it with the 64 point DFT. We can see that both (fig. b & c) are identical.

lab1_a
unit circle
lab1_b
value of H(z)
abs(fft(h))
abs(fft(h))

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