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UNIT 3 | Signals and Systems

3.9 Unilateral Z Transform Concept | EC402 |

Welcome to Unit 3 of our Signals and Systems course, where we'll explore the powerful Z-transform and its various aspects. This unit is packed with essential concepts and properties that are crucial for understanding and analyzing discrete-time signals and systems.

What is Z-Transform?
We start by introducing the Z-transform, a key tool in discrete-time signal processing. You'll gain a fundamental understanding of what the Z-transform is and why it's essential in this field.

Region of Convergence (ROC) for Z-Transform
Understanding the Region of Convergence (ROC) is vital when working with Z-transforms. We'll dive into ROC and its significance, especially when dealing with various types of signals.

Z-Transform and ROC for Different Signal Types
Z-transform and ROC for Finite Duration Signal
Z-transform and ROC for Infinite Duration Signal:
Z-transform and ROC for Causal Infinite Duration Signal
Z-transform and ROC for Anti-Causal Infinite Duration Signal
Z-transform and ROC for Two-Sided (Non-Causal) Infinite Signals

We'll explore how the Z-transform and ROC apply to different signal categories, providing you with a comprehensive view of their behavior.

Properties of Z-Transform
Understanding the properties of the Z-transform is crucial for signal analysis. We'll cover several properties, including:
Linearity Property
Shifting Property
Time Scaling Property
Differentiation Property
Convolution Property
Correlation Property
Initial Value Theorem
Final Value Theorem

To solidify your understanding, we'll work through solved numerical examples based on these properties, giving you practical insights into their applications.

Inverse Z-Transform:
The inverse Z-transform is a critical operation when you need to recover a discrete-time signal from its Z-transform representation. We'll explore two methods:
Long Division Method
Solved numerical examples for causal and non-causal signals
Partial Fraction Method
Numerical examples for inverse Z-transform using partial fraction decomposition
Residue Method
Introduction to the residue method and numerical examples

Analysis of Discrete-Time LTI System using Z-Transform:
We'll delve into how the Z-transform is used to analyze discrete-time Linear Time-Invariant (LTI) systems, including techniques for estimating the transfer function and unit impulse response.

Relation between Discrete-Time Fourier Transform and Z-Transform:
Understanding the connection between these two transforms is crucial for frequency domain analysis in discrete-time signals and systems.

Z-Transform of Important Signals:
We'll conclude this unit by calculating the Z-transform of essential signals such as the unit impulse signal, unit step signal, sinusoidal signals, and more.

By the end of Unit 3, you'll have a strong foundation in Z-transform theory and its practical applications. These concepts are essential for anyone working with discrete-time signals and systems in various engineering and scientific fields.

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