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CT-321 Digital Signal Processing

Learn about Properties and Applications of Z-Transform, Linear Constant-Coefficient Difference Equations, and Frequency Response of LTI Systems. Explore practical examples and equations in this insightful lecture.

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CT-321 Digital Signal Processing

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

  2. Review and Preview • Review of past 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 and the input satisfy the 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 : 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 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 • Its Z Transform is as follows: otherwise ROC of finite length exponential ROC of infinite length exponential ROC ROC ROD

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

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