  # Z-Transform-1 Study Notes for GATE Electronics and Communication Engineering Exam

By BYJU'S Exam Prep

Updated on: September 25th, 2023 In this article, you will find the Z-Transform-1 which will cover the topic as Z-Transform, Region of Convergence (ROC), Summary of ROC of Discrete-Time Signals for the sequences, Characteristic Families of Signals and Corresponding ROC, Summary of 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.ejω

• The discrete-time Fourier Transform (DTFT) is obtained by evaluating Z-Transform at z = ejω.
• 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:   Candidates can practice 150+ Mock Tests with BYJU’S Exam Prep Test Series for exams like GATE, ESE, NIELIT by the following link:

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