![Date : 30-7-2020
Z-Transform Definition
• Definition of Z-Transform
The Z-transform of a sequence x[n], denoted X(z), is defined as
Chapter-3](https://image.slidesharecdn.com/chapter-3-231024185125-408357d4/85/Chapter-3-pptx-3-320.jpg)
![Date : 30-7-2020
Representation of DT systems using Difference Equation
The input /output relationship of a LTI discrete-time system is given by the linear
constant-coefficient difference equation (LCCDE).
Example:
y n = 2x n + 5x n − 1 − 3x n − 2 + 0.5y n − 1 − 4y[n − 2]
System function:
If 𝑥[𝑛]
𝑍.𝑇
𝑋(𝑧)
𝑦[𝑛]
𝑍.𝑇
𝑌(𝑧)
ℎ[𝑛]
𝑍.𝑇
𝐻(𝑧)
then
𝑆𝑦𝑠𝑡𝑒𝑚 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝐻 𝑧 =
𝑌(𝑧)
𝑋(𝑧)](https://image.slidesharecdn.com/chapter-3-231024185125-408357d4/85/Chapter-3-pptx-10-320.jpg)
![Date : 30-7-2020
The general form of a Linear Constant-Coefficient Difference Equation (LCCDE)
a0 y[n] + a1 y[n-1] + a2 y[n-2] + … aN y[n-N] = b0 x[n] + b1 x[n-1] + b2 x[n-2] +… bM x[n-M]](https://image.slidesharecdn.com/chapter-3-231024185125-408357d4/85/Chapter-3-pptx-11-320.jpg)
Chapter-3.pptx
- 1. Date : 30-7-2020 ECEN341 Signals and Systems Chapter-3 Z-Transform and System Representation using Difference Equations for DT systems Chapter-3
- 2. Date : 30-7-2020 Z-Transform What is the use of z-transform • Z-transform is a mapping of discrete-time sequences from the time domain into a complex frequency variable domain • Very useful in the analysis of discrete-time signals and systems
- 3. Date : 30-7-2020 Z-Transform Definition • Definition of Z-Transform The Z-transform of a sequence x[n], denoted X(z), is defined as Chapter-3
- 4. Date : 30-7-2020 Z-Transform of Elementary Signals Chapter-3 Z-Transform of an Impulse Sequence Z-Transform of a Unit Step Sequence Note that for an infinite geometric series, we have 1 + 𝑎 + 𝑎2 + 𝑎3 + 𝑎4 + ⋯ = 1 1−𝑎 when |a| < 1 X(z) = 1 1−𝑧−1 = 1 1− 1 𝑧 = 𝑧 𝑧− 1
- 6. Date : 30-7-2020 Chapter-3 Problem: Find the Z-Transform of the causal sequence
- 7. Date : 30-7-2020 Chapter-3 Problem: Find the Z-Transform of the DT sequence
- 8. Date : 30-7-2020 Chapter-3 Problem: Find the Z-Transform of the two sided DT sequence
- 9. Date : 30-7-2020 Problem: Find the Z-Transform of the sequence x(n) = u(n) – u(n-3). Chapter-3
- 10. Date : 30-7-2020 Representation of DT systems using Difference Equation The input /output relationship of a LTI discrete-time system is given by the linear constant-coefficient difference equation (LCCDE). Example: y n = 2x n + 5x n − 1 − 3x n − 2 + 0.5y n − 1 − 4y[n − 2] System function: If 𝑥[𝑛] 𝑍.𝑇 𝑋(𝑧) 𝑦[𝑛] 𝑍.𝑇 𝑌(𝑧) ℎ[𝑛] 𝑍.𝑇 𝐻(𝑧) then 𝑆𝑦𝑠𝑡𝑒𝑚 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝐻 𝑧 = 𝑌(𝑧) 𝑋(𝑧)
- 11. Date : 30-7-2020 The general form of a Linear Constant-Coefficient Difference Equation (LCCDE) a0 y[n] + a1 y[n-1] + a2 y[n-2] + … aN y[n-N] = b0 x[n] + b1 x[n-1] + b2 x[n-2] +… bM x[n-M]
- 12. Date : 30-7-2020 Chapter-2 Time Domain Analysis of systems Problem : A causal discrete-time LTI system is described by 𝑦 𝑛 − 3 4 𝑦 𝑛 − 1 + 1 8 𝑦 𝑛 − 2 = 𝑥 𝑛 where x(n) and y(n) are the input and the output of the system respectively. Determine the system function H(z).
- 13. Date : 30-7-2020 Problem : Given a transfer function determine the system difference equation H(z)= (𝑧+1)2 (𝑧− 1 2 )(𝑧+ 3 4 )
- 14. Date : 30-7-2020 14 References • “Signals and Systems” by Oppenhehim, Willsky and Hamid,, 2nd edition, Pearson Education India, 2015. • Signals and Systems :Principles and Applications by Shaila Dinkar Apte , Cambridge University Press Chapter-1
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