The bilateral Z transform is
where z = A e^{-j ω} for some real number A.
Motivation for using the Z transform in digital signal processing
The Z transform is a generalization of the discrete-time Fourier transform, which is defined as
As with other Fourier transforms, the discrete-time Fourier transform translates a complex valued signal x(k) as a function of time into a set of complex numbers that carry the frequency and phase of each simple wave in the signal. The discrete-time Fourier transform is used when x(k) is discrete, but not of finite duration and not periodic (alternatively, one can use the continuous or the discrete Fourier transform). The discrete-time Fourier transform is often thought of as the logical result of sampling the input to the continuous Fourier transform x(t) into discrete time x(n).
The bilateral Z transform generalizes the discrete-time Fourier transform by introducing the real number A, which represents the magnitude of the transform basis A e^{-j ω k}.
Alternatively, the discrete-time Fourier transform is simply the Z transform with A = 1. When A = 1, the transform basis becomes the numbers e^{-j ω k}, which in the complex plane form a circle of radius 1 around the origin. Thus, we can say that the discrete-time Fourier transform is the Z transform evaluated on the unit circle.
The motivation behind using the discrete-time Fourier transform in digital signal processing is that it allows us to use a discrete signal, but a continuous set of frequencies and phases. The motivation behind using the Z transform in digital signal processing rather than the discrete-time Fourier transform is only that it significantly reduces notation. In fact, in digital signal processing we will always evaluate the Z transform on the unit circle – with A = 1 – which means that we will only really use the discrete-time Fourier transform and we will never use its generalized form – the Z transform. We do however use the Z transform notation. It is much easier to write z^{-k} rather than e^{-j ω k}.
Some useful properties of the Z transform
One useful property of the Z transform is that it is linear, much like the Fourier transforms.
Second, time-shifting of the signal x(k) results in the following.
This means that, if the output of a system a(k) on the input x(k) is given by
which means a(k) could be a finite impulse response filter, then
and
which is the transfer function of the system a(k). This transfer function has nice properties. In linear time-invariant systems, the magnitude |H(z)| of H(z), z = A e^{-j ω}, when evaluated at A = 1 (on the unit circle), produces the magnitude response of the system
The phase response of the system is
The notation |H(e^{-j ω})| and Φ(e^{-j ω}) is used to show that the magnitude response and phase response are evaluated with e^{-j ω} and not with z = A e^{-j ω}.
Example transfer functions of a comb filter with the Z transform
A feedforward comb filter is just a simple delay. It has the form
where g is some gain, 1 – g is the decay of the delayed signal, and m is the delay in number of samples. The Z transform of both sides of the above equation produces
The transfer function of this comb filter then is
The magnitude response of the comb filter at a particular frequency ω is
This function of ω, when plotted, produces the characteristic comb like magnitude response of the comb filter shown below. This example feedforward comb filter was computed with the sampling frequency f_{s} = 2000 Hz, a delay of m = 50 samples, and with g = 0.8.
Other examples
For other examples of using the Z transform, see the topics on Comb filter, Transfer function, Gibbs phenomenon, Butterworth filter, All pass filter, and Shroeder-Moore filter.