Objectives:
Scope of this lecture:
We have already seen the implementation of Fourier Transform and Laplace Transform for the study of Continuous Time (C.T.) signals and systems. Now our interest lies in frequency domain analysis and design of Discrete Time (D.T.) signals and systems. The ZTransform provides a valuable technique for frequency domain analysis of D.T. signals and design of DT-LTI systems. Further ZTransform offers an extremely convenient and compact way to describe digital signals and processors. Numerical problems are presented for a better understanding of the relevant concepts involved.
Z-transform
The response of a linear time-invariant system with impulse response h[n] to a complex exponential input of the form can be represented in the following way :
In the complex z-plane , we take a circle with unit radius centered at the origin.
H(w) is periodic with period with respect to ' w ' .
When we replace z by ejω ,we get periodicity of in the form of a circle.
Nature of Region of Convergence
Laplace Transform:
The ROC of the Laplace transform X(s) of a two-sided signal lies between two vertical lines in the s-plane.
depend only on real part of s. For a right-sided signal and the corresponding ROC is referred to as right-half plane. Similarly for a left-sided signal This ROC is referred to as left-half plane. When x(t) is two-sided i.e; of infinite extent for both t > 0 and t < 0 ; both and are finite and the ROC thus turns out to be a vertical strip in the s-plane.
Z-transform:
The ROC of X(z) of a two sided signal consists of a ring in the z-plane centered about the origin.
depend only on magnitude of z. As in the case of Laplace transform fora right-sided sequence and for aleft -sided sequence. If x[n] is two-sided ;the ROC will consist of a ring with both and finite and non-zero.
41 videos|52 docs|33 tests
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1. What is the Z-transform and how is it related to the region of convergence? |
2. Why is the region of convergence important in Z-transform analysis? |
3. What are the different types of region of convergence in Z-transform analysis? |
4. How does the region of convergence affect the frequency response of a system? |
5. Can the region of convergence change for different signals in the same system? |
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