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 the 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 a 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 a Z-Transform.
- Mathematical calculations can be reduced by using the Z-Transform. For example, the convolution operation is transformed into a simple multiplication operation.
Z-Transforms:
There are two types of Z-Transform:
- Single sided Z-Transform.
- Double sided Z-Transform.
Single sided Z-Transform:
Single sided Z-Transform can be defined as,
Double sided Z-Transform:
Double sided Z-Transform can be defined as,
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,
The relation between x(n) and X(Z) is denoted as follows:
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 a 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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- Addition of two images using MATLAB image processing.
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