A Z-transform is a mathematical tool used to analyze and transform discrete-time signals into their frequency-domain representations. In a PowerPoint presentation about Z-transform, one might cover the following topics:
Introduction to Z-transform: This section provides a brief overview of what Z-transform is, and how it differs from Fourier Transform.
Discrete-time signals: This section explains what discrete-time signals are, and how they relate to continuous-time signals.
Z-transform equation: This section describes the Z-transform equation, which is used to transform a discrete-time signal into its frequency-domain representation.
Region of convergence (ROC): This section explains the Region of convergence (ROC), which is the set of values for which the Z-transform converges.
Inverse Z-transform: This section covers the inverse Z-transform, which is used to transform a frequency-domain representation of a signal back into its time-domain representation.
Properties of Z-transform: This section discusses the various properties of Z-transform, such as linearity, time-shifting, convolution, and others.
Applications of Z-transform: This section discusses the various applications of Z-transform in digital signal processing, control systems, and other fields.
Conclusion: This section summarizes the key points covered in the presentation and emphasizes the importance of Z-transform in understanding and analyzing discrete-time signals.
Overall, a PowerPoint presentation on Z-transform would help the audience to understand the mathematical concepts and practical applications of this important signal processing tool.
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11- Z Transform.pdf
1. Signals and Systems
17EL (Section-I and II), Fall-2019
Lecture-11
Z-Transform
Dr. Shoaib R. Soomro
2. ▪ Introduction
▪ ROC
▪ Z Transform Pairs
▪ Example Problems
▪ Inverse Z Transform
▪ Properties of Z Transform
▪ Solution Of Linear Difference Equations
▪ Stablity
2
Contents
Dr. Shoaib R. Soomro, Fall 2019
3. ▪ The counterpart of the Laplace transform for discrete-time systems is the z-transform.
▪ The Laplace transform converts integrodifferential equations into algebraic equations.
▪ In the same way, the z-transforms changes difference equations into algebraic equations, thereby
simplifying the analysis of discrete-time systems.
▪ where z is a complex variable. The signal x [n], which is the inverse z-transform of X [z], can be
obtained from X[z] by using the following inverse z-transformation:
▪ The symbol ∮ indicates an integration in counterclockwise direction around a closed path in the
complex plane
3
Introduction
Dr. Shoaib R. Soomro, Fall 2019
4. ▪ The direct z-transform X [z] may not converge (exist) for all values of z.
▪ The values of z (the region in the complex plane) for which the sum converges (or exists) is
called the region of existence, or more commonly the region of convergence (ROC), for X[z].
▪ Example Problem:
▪ Find the z-transform and the corresponding ROC for the signal γnu[n].
4
Region of Convergence
Dr. Shoaib R. Soomro, Fall 2019
8. ▪ As in the Laplace transform, we shall avoid the integration in the complex plane required to find
the inverse z-transform by using the (unilateral) transform table.
▪ Many of the transforms X[z] of practical interest are rational functions (ratio of polynomials in z),
which can be expressed as a sum of partial fractions, whose inverse transforms can be readily
found in a table of transform.
▪ The partial fraction method works because for every transformable x[n] defined for n ≥ 0, there
is a corresponding unique X[z] defined for |z| > r0 (where r0 is some constant), and vice versa.
8
Inverse Z Transform
Dr. Shoaib R. Soomro, Fall 2019
9. ▪ Find the inverse z-transform of
1.
2.
3.
9
Example Problems
Dr. Shoaib R. Soomro, Fall 2019
11. ▪ RIGHT SHIFT (DELAY)
▪ LEFT SHIFT (ADVANCE)
▪ CONVOLUTION
▪ MULTIPLICATION BY γn (SCALING IN THE z-DOMAIN)
▪ MULTIPLICATION BY n
▪ TIME REVERSAL
11
Some Properties of The Z-transform
Dr. Shoaib R. Soomro, Fall 2019
12. ▪ Find Z Transforms
▪ Show that the z transform od u[−n] is −1/(z − 1) with the ROC |z| < 1.
12
Example Problems
Dr. Shoaib R. Soomro, Fall 2019
14. ▪ the z-transform converts difference equations into algebraic equations
▪ that are readily solved to find the solution in the z domain. Taking the inverse z-transform of the
z-domain solution yields the desired time-domain-solution.
▪ Example Problem:
▪ if the initial conditions are y[−1] = 11/6, y[−2] = 37/36, and the input x[n] = (2)−n u[n].
14
Z-Transform Solution Of Linear Difference Equations
Dr. Shoaib R. Soomro, Fall 2019
16. ▪ An LTID system is asymptotically stable if and only if all the poles of its transfer function H[z]
are within the unit circle. The poles may be repeated or simple.
▪ An LTID system is unstable if and only if either one or both of the following conditions exist: (i)
at least one pole of H[z] is outside the unit circle; (ii) there are repeated poles of H[z] on the unit
circle.
▪ An LTID system is marginally stable if and only if there are no poles of H[z] outside the unit
circle, and there are some simple poles on the unit circle.
16
Stability
Dr. Shoaib R. Soomro, Fall 2019
