1. CT-321 Digital Signal Processing
Yash Vasavada
Autumn 2016
DA-IICT
Lecture 13
Z Transform
7th September 2016

2. Review and Preview Review of past lecture: Preview of this lecture:
Inverse Z Transform
Preview of this lecture:
Properties of Z Transform and their Applications
Linear Constant Coefficient Difference Equations
Reading Assignment
OS: Chapter 3 and Chapter 4
PM: Chapter 3 and Chapter 4 section 4.4

3. Properties of Z Transform
Refer to Section 4 of Chapter 3

4. Application of the Properties of Z Transform
We will make use of only the following Z Transform Pair:
Time-shifting property:
Time reversal property:
Differentiation property:
Exponential multiplication property:

5. Linear Constant-Coefficient Difference Equations
An important class of LSI systems are those for which the output 𝑦(𝑛) and the input 𝑥(𝑛) satisfy the 𝑁 𝑡ℎ order linear constant coefficient difference equation (LCCDE):
What are such types of LSI systems? Exactly those for which Z Transform of the impulse response is a rational function (i.e., a ratio of polynomials) in 𝑧:
Take Z Transform of both sides of LCCDE:

6. Accumulator Expressed as LCCDE
Representation of the accumulator in time domain:
…and in Z domain:

7. A Moving Sum Expressed as LCCDE
Therefore:
LCCDE representation of Moving Sum:
Z Transform Representation
Consider a moving sum of past 𝑁 samples:
This operation can be viewed as the sum of the output of two accumulators

8. Z Transform of Finite Duration Sequences
Consider the following sequence defined only over 𝑛=0,1,…,𝑀−1
Its Z Transform is as follows:
=0; otherwise
ROC of finite length exponential
𝑀=25
ROC of infinite length exponential
𝑀=∞
ROC
ROC
ROD

9. Frequency Response of LTI Systems
Consider the frequency response 𝐻 𝑓 = 𝐻 𝑓 exp 𝑗∠𝐻(𝑓) of LTI systems in the polar coordinates:
Here, 𝐻 𝑓 is called the magnitude response of the filter and it is the gain that the filter applies to a complex exponential at frequency 𝑓
∠𝐻(𝑓) is the phase response and it is the phase offset that the filter applies to a complex exponential at frequency 𝑓
As we have seen, for LTI systems, the following holds:
Therefore,