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### Transcript of Z Transform (1) Hany Ferdinando Dept. of Electrical Eng. Petra Christian University.

• Slide 1
• Z Transform (1) Hany Ferdinando Dept. of Electrical Eng. Petra Christian University
• Slide 2
• Z Transform (1) - Hany Ferdinando2 Overview Introduction Basic calculation RoC Inverse Z Transform Properties of Z transform Exercise
• Slide 3
• Z Transform (1) - Hany Ferdinando3 Introduction For discrete-time, we have not only Fourier analysis, but also Z transform This is special for discrete-time only The main idea is to transform signal/system from time-domain to z- domain it means there is no time variable in the z-domain
• Slide 4
• Z Transform (1) - Hany Ferdinando4 Introduction One important consequence of transform-domain description of LTI system is that the convolution operation in the time domain is converted to a multiplication operation in the transform-domain
• Slide 5
• Z Transform (1) - Hany Ferdinando5 Introduction It simplifies the study of LTI system by: Providing intuition that is not evident in the time-domain solution Including initial conditions in the solution process automatically Reducing the solution process of many problems to a simple table look up, much as one did for logarithm before the advent of hand calculators
• Slide 6
• Z Transform (1) - Hany Ferdinando6 Basic Calculation They are general formula: Index k or n refer to time variable If k > 0 then k is from 1 to infinity Solve those equation with the geometrics series or
• Slide 7
• Z Transform (1) - Hany Ferdinando7 Basic Calculation Calculate:
• Slide 8
• Z Transform (1) - Hany Ferdinando8 Basic Calculation Different signals can have the same transform in the z-domain strange The problem is when we got the representation in z-domain, how we can know the original signal in the time domain
• Slide 9
• Z Transform (1) - Hany Ferdinando9 Region of Convergence (RoC) Geometrics series for infinite sum has special rule in order to solve it This is the ratio between adjacent values For those who forget this rule, please refer to geometrics series
• Slide 10
• Z Transform (1) - Hany Ferdinando10 Region of Convergence (RoC)
• Slide 11
• Z Transform (1) - Hany Ferdinando11 Region of Convergence (RoC)
• Slide 12
• Z Transform (1) - Hany Ferdinando12 Region of Convergence (RoC)
• Slide 13
• Z Transform (1) - Hany Ferdinando13 RoC Properties RoC of X(z) consists of a ring in the z- plane centered about the origin RoC does not contain any poles If x(n) is of finite duration then the RoC is the entire z-plane except possibly z = 0 and/or z =
• Slide 14
• Z Transform (1) - Hany Ferdinando14 RoC Properties If x(n) is right-sided sequence and if |z| = r o is in the RoC, then all finite values of z for which |z| > r o will also be in the RoC If x(n) is left-sided sequence and if |z| = r o is in the RoC, then all values for which 0 < |z| < r o will also be in the RoC
• Slide 15
• Z Transform (1) - Hany Ferdinando15 RoC Properties If x(n) is two-sided and if |z| = r o is in the RoC, then the RoC will consists of a ring in the z-plane which includes the |z| = r o
• Slide 16
• Z Transform (1) - Hany Ferdinando16 Inverse Z Transform Direct division Partial expansion Alternative partial expansion Use RoC information
• Slide 17
• Z Transform (1) - Hany Ferdinando17 Direct Division If the RoC is less than a, then expand it to positive power of z a is divided by (a+z) If the RoC is greater than a, then expand it to negative power of z a is divided by (z-a)
• Slide 18
• Z Transform (1) - Hany Ferdinando18 Partial Expansion If the z is in the power of two or more, then use partial expansion to reduce its order Then solve them with direct division
• Slide 19
• Z Transform (1) - Hany Ferdinando19 Properties of Z Transform General term and condition: For every x(n) in time domain, there is X(z) in z domain with R as RoC n is always from to
• Slide 20
• Z Transform (1) - Hany Ferdinando20 Linearity a x 1 (n) + b x 2 (n) a X 1 (z) + b X 2 (z) RoC is R 1 R 2 If a X 1 (z) + b X 2 (z) consist of all poles of X 1 (z) and X 2 (z) (there is no pole-zero cancellation), the RoC is exactly equal to the overlap of the individual RoC. Otherwise, it will be larger a n u(n) and a n u(n-1) has the same RoC, i.e. |z|>|a|, but the RoC of [a n u(n) a n u(n-1)] or (n) is the entire z-plane
• Slide 21
• Z Transform (1) - Hany Ferdinando21 Time Shifting x(n-m) z -m X(z) RoC of z -m X(z) is R, except for the possible addition or deletion of the origin of infinity For m>0, it introduces pole at z = 0 and the RoC may not include the origin For m