Z Transform Formula

What is Z Transform?

In order to define z transform, it is simply a mathematical mechanism of going from the discrete time domain to the z domain also referred to as the complex frequency domain. Under the discrete time domain, a signal is generally described as a series of real or complex numbers which is then converted to the z-domain by the mechanism of z-transform.


Z Transform Formula

The formula applied in order to convert a discrete time signal x[n] to X[z] is as below:

X(Z)|z = e = F. T[x(n)]

Where,

x[n] represents Finite length signal

[0, N] represents Sequence support interval

z  represents Any complex number

N represents Integer

X[z] represents the z-transform of the discrete time signal.

Now here is to notify that the z-transform comes in two parts. The first part is the formula as given above while the second part is to describe a region of convergence (ROC) for the z-transform. Both parts are required for a complete z-transform as a z-transform unaccompanied by ROC would not be of much assistance in signal processing. Moreover, though the z-transform is acquired by directly using this formula, the inverse z-transform needs some mathematical manipulations which are linked to the power series and geometric series.


Z Transformation Relation To Discrete-Time Fourier Transform (DTFT)

There is a close connection between DTFT and z-transform. In fact, each of their respective formulas are also pretty similar, which is most commonly overlooked. Thus, let’s take a look at the formulas for both the DTFT and the z-transform for a signal x[n].

A.   DTFT Formula:

ω = nω0 and X(ω)=Tcn

B.   Z Transform Formula List

1. Bilateral Z Transform Formula

\[X(z)=Z\left \{ x[n] \right \}=\sum_{n=-\infty}^{\infty }x[n]z^{-n}\] 


2. Unilateral Z Transform Formula

\[X(z)=Z\left \{ x[n] \right \}=\sum_{n=0}^{\infty }x[n]z^{-n}\]


Solved Z Transform Examples On How to Find Z Transform

Example: Express the z-transform for a finite series as given below.

x = {-2, -1, 1, 2, 3, 4, 5, 6, 7, 8}

Solution:

Known sequence of sample numbers x[n] = is x = {-2, -1, 1, 2, 3, 4, 5, 6, 7, 8}

Z-transform of x[n] can be mathematically expressed as:

X (z) = -2z0 – z-1 + z-2 + 2z-3 + 3z-4 + 4z-5 + 5z-6+ 6z-7+ 7z-8+ 8z-9

This can be further simplified as below.

X (z) = -2 – z-1 + z-2 + 2z-3 + 3z-4 + 4z-5 + 5z-6+ 6z-7+ 7z-8+ 8z-9

FAQs (Frequently Asked Questions)

1. What is Meant By Region of Convergence for a Z-transform?

Answer: As mentioned earlier, the z-transform comes in two parts. One is the region definition and other is a mathematical formula which is referred to as the region of convergence. This region of convergence requires to be described for all z-transforms, for values of z as it will describe the points on the z-plane where the z-transform converges and simply to say the z-transform exists.


Generally for systems we consist of transfer functions that have the forms:

H(z) = N(z)/D(z)

2. What are the Uses of Z Transformation?

Answer: The z-transform is quite a useful and a significant technique, applicable in areas of system designing, signal processing, and evaluation and control theory.


Moreover, frequency and the Time-domain are considered in the study of discrete-time signals and systems. The z-transform provides us a third representation for the study. But, all the three domains are linked with each other. A unique feature of the z-transform is that in reference to the signals and system of interest to us, all of the evaluations will be in terms of ratios of polynomials.