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

By BYJU'S Exam Prep

Updated on: September 25th, 2023 The Z-transform is a fundamental tool used in the field of electrical engineering, particularly in the study of discrete-time systems and signal processing. It plays a crucial role in the analysis and representation of discrete signals, offering a powerful alternative to the continuous Laplace transform. For engineering students preparing for the Graduate Aptitude Test in Engineering (GATE) or other electrical engineering exams, a comprehensive understanding of the Z-transform is essential.

In these study notes, we delve into the principles and applications of the Z-transform, providing a clear and concise explanation of its concepts. We explore the conversion of discrete-time signals from the time domain to the complex Z-plane and learn how to obtain the Z-transform of various common signals and sequences. Moreover, we discuss the significance of the Z-transform in analyzing discrete-time systems, such as digital filters, and explore its applications in solving different equations and characterizing system stability. By mastering the Z-transform through these study notes, students can enhance their problem-solving abilities and tackle advanced topics in digital signal processing and control systems effectively.

Table of content ### 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 a 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 the causality of the system.
• ROC can be used to determine the 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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