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Z - transform. Defined as power series Examples:. Z - transform. And since We get. +ve and > 1. +ve and = 1. +ve and < 1. Z - transform. Define. Z - transform. We have ie Note that has a pole at on the z-plane. Z - transform. Note:
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Z - transform • Defined as power series • Examples: 1
Z - transform • And since • We get 2
+ve and > 1 +ve and = 1 +ve and < 1 Z - transform • Define 3
Z - transform • We have • ie • Note that has a pole at on the z-plane. 4
Z - transform Note: • (i) If magnitude of pole is > 1 then increases without bound • (ii) If magnitude of pole is < 1 then has a bounded variation • i.e. the contour on the z-plane is of crucial significance. • It is called theUnit circle 5
1 1 The unit circle 6
Z – transform properties (i)Linearity • The z-transform operation is linear • Z • Where Z , i = 1, 2 (ii) Shift Theorem Z 7
.... Z - transform • Let • Z • But for negative i. 8
Z - transform Examples: • (i) Consider generation of new discrete time signal from via • Recall linearity and shift • (ii) Z write 9
Z - transform • From • With from earlier result • We obtain Z Z 10
Inverse Z - transform • Given F(z) to determine . • Basic relationship is • may be obtained by power series expansion. It suffers from cumulative errors 11
Inverse Z - transform • Alternatively • Use form = -1 • otherwise • where closed contour encloses origin 12
Inverse Z - transform • Integrate to yield Examples • (i) write 13
Inverse Z - transform • And hence Pole at of Residue (ii) Let where and • To determine 14
Inverse Z - transform • From inversion formula • But 15
Inverse Z - transform • Hence • Thus 16
Inverse Z - transform Note: • (i) For causal signals for negative i. Thus upper convolution summation limit is in this case equal to k. • (ii) Frequency representation of a discrete-time signal is obtained from its z-transform by replacing where T is the sampling period of interest. (Justification will be given later.) 17