What is Z-transform?

Last Updated : 28 Apr, 2022

A z-Transform is important for analyzing discrete signals and systems. We know analog signals or signals that are continuous in the time domain. But modern-day communication and system are based on digital processing. This forces us to change our analog signals to the digital domain. The first step in doing this is to sample the analog signal at a rate above a threshold (known as Nyquist sampling rate) as a discrete sequence of points. These points are discretized in time. Each sample occurs at t=nT_s, where T_s is the sampling time.

After sampling, we need to quantize these samples to one out of M given levels and then encode the quantized samples to binary for further storing, analyzing, or transmitting.

z transforms are particularly useful to analyze the signal discretized in time. Hence, we are given a sequence of numbers in the time domain. z transform takes these sequences to the frequency domain (or the z domain), where we can check for their stability, frequency response, etc. Hence, taking z transforms is analogous to taking Laplace transforms for continuous signals.

Definition:

Suppose we have a sequence given to us as follows:

$y[n] = {y0, y1, y2, ..... }$

Here each point in the sequence is a sample of an analog signal.

The z transform of this sequence is defined as:

$y[z]= y_{0} + y_{1}z^{-1} + y_{2}z^{-2} + ........ = \sum_{\infty }^{n=0}y_{n}z^{-n}$

The infinite series must converge for Y(z) to be defined as a precise function of z.

A z-transform is the same as a Laplace transform, where s is simply a complex variable, z here is again a complex variable and, unlike n, it’s continuous. However, the z-transform does not converge for all sequences or for all values of z. The set of values of z for which the z-transform converges is called the region of convergence (ROC).

Now, we will see some transforms of popular signals.

Unit impulse

This is a simple but important sequence defined as:

$\delta_{n} = \begin{cases} & \text{ 1 } n = 0\\ & \text{ 0 } n = \pm 1,\pm 2,....... \end{cases}$