# 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.

Download Formulas for GATE Electrical Engineering – Electrical Machines

### 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

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