Z transform Day 1

EE8591 Digital Signal Processing :
UNIT II DISCRETE TIME SYSTEM ANALYSIS
Z-transform and its properties, inverse z-transforms; difference equation – Solution by ztransform,
application to discrete systems - Stability analysis, frequency response –Convolution – Discrete Time Fourier transform , magnitude and phase representation

1. WEBINAR ON DISCRETE TIME SYSTEM
ANALYSIS


1
K.Vijay Anand - Associate Professor

Department of Electronics and Instrumentation Engineering

R.M.K Engineering College

2. Overview
Agenda:
•Z-transform and
•Z- transform its properties,
•Inverse z-transforms;
•Difference equation – Solution by z-transform,
•Application to discrete systems
- Stability analysis
-frequency response
•Convolution
•Discrete Time Fourier transform ,
•Magnitude and phase representation.

3. Recap:
•What is Digital Signal Processing?
•Application of DSP
•Signal:Continuous or analog signals,Discrete-time signals,Two-Dimensional Digital Signals
•Analog to Digital Conversion:Nyquist–Shannon Sampling Theorem
•Aliasing,Sampling Effect,Anti Aliasing Filter,Under-sampling
•Sampling of Band Limited Signals,Over-sampling
•Digital to Analog Conversion
•Classification of Signals : Causal signals
•Deterministic and Random signals,Types of Digital signal
•Digital Functions
•Types of Digital signal
Digital Functions (Impulse, Step,Ramp Exponential, Sine)
•Notation for Digital Signals

4. Introduction to Z transform


Definition
Region of convergence
Properties of z-transform
Inverse z-transform
Convolution
4

5. Introduction
•The Laplace Transform (s domain) is a valuable tool
for representing, analyzing & designing continuous-
time signals & systems.
•The z-transform is convenient yet invaluable tool for
representing, analyzing & designing discrete-time signals
& systems.
•The resulting transformation from s-domain to z-domain
is called z-transform.
•The relation between s-plane and z-plane is described
below : z = e
sT
•The z-transform maps any point s = σ + jω in the s-
plane to z- plane (r²θ).

7. The Z-Transform
For continuous-time signal,
Time
Domain
S‐Domain
For discrete-time signal,
Time Domain
Z‐Domain Ƶ
Ƶ
-1
Causal
System
where,

8. One Sided And Two Sided Z-Transform
8
One Sided Z-Transform
The one sided Z-Transform can be used to solve Differential
 Equations with initial conditions.

This Property makes it useful for use in Practical systems
It does not contain information about the signal x(n) for
 negative values of time.
 It is represented by the equation :

X

(z)  x (n) z
n
n0

9. Two Sided Z-Transform
9
•Two sided Z-transform requires that the signal be specified for the entire
range :
  

•It represented by the equation :

X ( z )   x ( n ) z
 n

n  

11. Properties of ROC
A ring or disk in the z-plane centered at the
origin.
The Fourier Transform of x(n) is converge
absolutely if the ROC includes the unit
circle.
The ROC cannot include any poles

12. REGION OF CONVERGENCE

16. Find the Z - Transform and mention the Region of Convergence
(ROC) for the following discrete time sequences.

1. x (n) = {2 1 2 3}

2. x (n) = { 2, 1, 2 3 }

3. x (n) = {1, 2 , 1,-2, 3, 1}
1
st
2
nd
3
rd
example Is of causal signal
example Is of anti-causal signal
example Is of non-causal signal
16

22. Z-Transform of Basic Signals
•Unit Impulse Function

x ( n )   ( n )
22
  n   n 
 n
n  0
X (z) 1
