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.



  • 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

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

04-Laplace-Transform (28)

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

04-Laplace-Transform (29)

Characteristic Families of Signals and Corresponding ROC

04-Laplace-Transform (30)

04-Laplace-Transform (31)

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:

04-Laplace-Transform (32)

04-Laplace-Transform (33)

04-Laplace-Transform (34)

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