### 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.

unit circle

value of H(z)

abs(fft(h))