Constructing the Complex Plane
Suppose we have a sampled signal defined by the sequence ,
Its Z- transform is given by .
It maps the original sequence into a new domain, which is the complex plane where is the parameter in the Laplace domain and is the sampling period.
The axis in the -plane maps onto the unit circle with centre at the origin in the -plane. So the value of at different points on the unit circle actually gives the contribution of the frequency component given by , 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 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.