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+j\omega is the parameter in the Laplace domain and T is the sampling period.

The j\omega 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))