CT-321 Digital Signal Processing

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Presentation transcript:

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

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

Properties of Z Transform Refer to Section 4 of Chapter 3

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:

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:

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

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

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

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,