Z Transform, Properties, Formula & Applications
Table of Contents
A Z Transform is essential for the analysis of discrete signals and systems. In the time domain, we are accustomed to continuous or analog signals. However, modern communication and systems are built on digital processing. As a result, we have to convert our analog signals to the digital domain. The analog signal is sampled as a discrete series of points at a rate above a threshold (known as the Nyquist sampling rate) in order to achieve this. Time has been discretized at these points. A sample is taken at time t=nTs, where Ts is the sampling interval.
These samples must first be sampled, quantized to one of M levels, and then encoded to binary before being stored, processed, or transmitted.
Z transforms are particularly useful for analyzing discretized-in-time signals. As a result, we are given a string of time-domain numbers. We investigate the stability, frequency response and other properties of these sequences in the frequency domain (also known as the z domain) by applying the z transform. Z transforms are therefore equivalent to Laplace transforms when applied to continuous signals.
Z Transform Formula
When studying discrete-time signals and systems, both the time-domain and the frequency domains are taken into account. We are given a third representation of the study by the z-transform. However, there are connections between the three domains. A unique feature of the z-transform is that all analysis will be expressed in terms of polynomial ratios for the signals and system that are important to us. And we are aware that using these polynomials is generally simple.
Definition
A discrete-time signal, which is a series of real or complex numbers, is transformed into a complex frequency-domain representation by the Z-transform in mathematics and signal processing.
Also, it can be considered as a discrete-time equivalent of the Laplace transform.
Where,
x[n]= Finite length signal[0, N] = Sequence support interval
z = Any complex number
N = Integer
Solved Problems
Example 1: Write the z-transform for a finite sequence given below.
x = {-2, -1, 1, 2, 3, 4, 5}
Solution:
Given sequence of sample numbers x[n]= is x = {-2, -1, 1, 2, 3, 4, 5}
z-transform of x[n] can be written as:
X(z) = -2z0 – z-1 + z-2 + 2z-3 + 3z-4 + 4z-5 + 5z-6
This can be further simplified as below.
X(z) = -2 – z-1 + z-2 + 2z-3 + 3z-4 + 4z-5 + 5z-6
Region of Convergence (ROC)
The range of variation of z for which z-transform converges is called region of convergence of z-transform.
Z transform properties
- ROC does not contain any poles.
- ROC of z-transform is indicated with circle in z-plane.
- If x(n) is a finite duration causal sequence or right sided sequence, then the ROC is entire z-plane except at z = 0.
- If x(n) is a infinite duration causal sequence, ROC is exterior of the circle with radius a. i.e. |z| > a.
- If x(n) is a finite duration anti-causal sequence or left sided sequence, then the ROC is entire z-plane except at z = ∞.
- If x(n) is a infinite duration anti-causal sequence, ROC is interior of the circle with radius a. i.e. |z| < a.
- If x(n) is a finite duration two sided sequence, then the ROC is entire z-plane except at z = 0 & z = ∞.
Applications
- Digital filters
- Mathematical and signal processing
- System design and analysis
- Linear discrete system
- Telecommunication automatic controls