# Z-Transform Study Notes for GATE & Other Electrical Engineering Exams

By Yash Bansal|Updated : September 24th, 2021

In this article, you will find the Z-Transform  which will cover the topic as Z-Transform, Inverse Z-transform, Region of Convergence of Z-Transform, Properties of Z-Transform.

In this article, you will find the Z-Transform  which will cover the topic as Z-Transform, Inverse Z-transform, Region of Convergence of Z-Transform, Properties of Z-Transform.

### Z-Transform

• Computation of the Z-transform for discrete-time signals.
• Enables analysis of the signal in the frequency domain.
• Z-Transform takes the form of a polynomial.
• Enables interpretation of the signal in terms of the roots of the polynomial.
• z−1 corresponds to a delay of one unit in the signal.

The Z - Transform of a discrete time signal x[n] is deﬁned as ,  where z = r.e

• The discrete-time Fourier Transform (DTFT) is obtained by evaluating Z-Transform at z = e
• The z-transform defined above has both sided summation. It is called bilateral or both sided Z-transform.

Unilateral (one-sided) z-transform

• The unilateral z-transform of a sequence x[n] is defined as ### Region of Convergence (ROC):

• ROC is the region where z-transform converges. It is clear that z-transform is an infinite power series. The series is not convergent for all values of z.

Significance of ROC

• ROC gives an idea about values of z for which z-transform can be calculated.
• ROC can be used to determine causality of the system.
• ROC can be used to determine stability of the system.

Summary of ROC of Discrete Time Signals for the sequences Characteristic Families of Signals and Corresponding ROC  Note: X(z) = z{x(n)} ; X1 (z) = Z {xl (n)} ; X2(z) = z{x2 (n)}; Y(z) =z (y (n))

Summary of Properties of z- Transform:   Impulse Response and Location of Poles               If you are preparing for GATE and ESE, avail Online Classroom Program to get unlimited access to all the live structured courses and mock tests from the following link :

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