����PRESENTATION ON �PROPERTIES OF Z TRANSFORM

• BRANCH-E & TC ENGG
• SUBJECT- DIGITAL SIGNAL PROCESSING
• CHAPTER – 3 – THE Z TRANSFORM & ITS APPLICATION TO THE ANALYSIS OF LTI SYSTEM
• TOPIC- PROPERTIES OF Z TRANSFORM
• SEM-6^TH
• FACULTY – Er. ARADHANA DAS (Sr. LECTURER E & TC ENGG DEPARTMENT)
• AY-2021-2022, SUMMER-2022

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Introduction

• The z-transform is the most general concept for the transformation of discrete-time series.
• The Laplace transform is the more general concept for the transformation of continuous time processes.

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• For example, the Laplace transform allows you to transform a differential equation, and its corresponding initial and boundary value problems, into a space in which the equation can be solved by ordinary algebra.

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The Transforms

The Laplace transform of a function x(t):

The one-sided z-transform of a function x(n):

The two-sided z-transform of a function x(n):

Relationship to Fourier Transform

Note that expressing the complex variable z in polar form reveals the relationship to the Fourier transform:

which is the Fourier transform of x(n).

The Direct Z-Transform

The z-transform of a discrete time signal is defined as the power series

(1)

��Where z is a complex variable. For convenience, the z-transform of a signal x[n] is denoted by

� X(z) = Z{x[n]}

�Since the z-transform is an infinite series, it exists only for those values of z for which this series converges. The Region of Convergence (ROC) of X(z) is the set of all values of z for which this series converges.

Properties of z-transform

1. Linearity :

It states that when two or more individual discrete signals are multiplied by constants, their respective Z-transforms will also be multiplied by the same constants. �

If x₁[n] ↔ X₁(z)

and x₂[[n] ↔ X₂(z)

�then�a₁x₁[n] + a₂x₂[n] ↔ a₁X₁(z) + a₂X₂(z)

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a₁x₁[n] + a₂x₂[n] ↔ a₁X₁(z) + a₂X₂(z)�

���2) Time Shifting Property:�Time shifting property depicts how the change in the time domain in the discrete signal will affect the Z-domain� If x[n] ↔ X(z) � then x[n-k] ↔ z^-k X(z)��

Proof :

Since

�then the change of variable m = n-k produces

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3) Scaling

Time Scaling property tells us, what will be the Z- domain of the signal when the time is scaled in its discrete form

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If x[n] ↔ X(z)

Then aⁿ x[n] ↔ X(a^-1z) or X(z/a)

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Proof:

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4) Time Reversal

If x[n] ↔ X(z)

then x[-n] ↔ X(z^-1)

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Proof :

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5) Differentiation Property

If x[n] ↔ X(z)

then n x[n] = -z(d/dz X(z))

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Convolution Property (Proof 1)�

Convolution Property (Proof 2)

z Transforms of Elementary Functions

Z transform of Shifted Unit Step

Z transform of Shifted Unit Step

Z transform of Scaled Unit Step

z Transforms of Elementary Functions

Z transform of Shifted Impulse

Z transform of Shifted Impulse

z Transforms of Elementary Functions

Bottom into derivative of Top minus Top into derivative of Bottom upon Bottom Square

Revision

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