Introduction to Z Transform with examples

Introduction to Z Transform:

Z transform of a signal provides a valuable technique for analysis and design of the discrete time signal and discrete time LTI system.

Z transform of a discrete time signal has both imaginary and real part. The plot of imaginary part versus real part is called as the z plane. The poles and zeros of the discrete time signals are plotted in the complex z plane. Pole-zero plot is the main characteristics of the discrete time signals. Using pole zero plot we can check the stability of the system which we will see in the upcoming posts.

Advantages of the Z transform:

Following are some of the main advantages of the Z transform:

  • We can simplify the solution of differential equation using z transform.
  • By the use of z transform we can completely characterize given discrete time signals and LTI systems.
  • The stability of the LTI system can be determined using z transform.
  • Mathematical calculations can be reduced by using z transform. For example convolution operation is transformed into simple multiplication operation.

Z transforms:

There are two types of z transform:

  1. Single sided Z transform.
  2. Double sided Z transform.

Single sided Z transform:

Single sided Z transform can be defined as,

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Double sided Z transform:

Double sided Z transform can be defined as,

2

 

In single sided Z transform only positive values of n are used (from 0 to 8) hence called single sided, whereas in the double sided values of n are ranging from -8 to +8.

Representation of the Z transforms:

Z transform of the signal x(n) is represented as,

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The relation between x(n) and X(Z) is denoted as follows:

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Where X(Z) is the Z transform of the signal x(n). The arrow is bidirectional which indicates that we can obtain x(n) from X(Z) also, which is called as inverse Z transform.

x(n) and X(Z) is called as Z transform pair.

Region of convergence (ROC):

“Region of convergence is defined as a set of all values of Z for which X(Z) has finite value. It is very important to denote ROC every time when we find Z transform.”

Significance of ROC:

  • ROC will decide whether the given system is stable or unstable.
  • ROC is also useful for determining the type of sequence. i.e. whether the system is causal or non-causal, finite or infinite.

Examples on Z Transform:

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