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Professor A G Constantinides 1 Z - transform Defined as power series Examples:

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Professor A G Constantinides 2 Z - transform And since We get

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Professor A G Constantinides 3 Z - transform Define +ve and = 1 +ve and > 1 +ve and < 1

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Professor A G Constantinides 4 Z - transform We have ie Note that has a pole at on the z-plane.

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Professor A G Constantinides 5 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 the Unit circle

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Professor A G Constantinides 6 The unit circle 1 1

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Professor A G Constantinides 7 Z – transform properties (i) Linearity The z-transform operation is linear Z Where Z, i = 1, 2 (ii) Shift Theorem Z

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Professor A G Constantinides 8 Z - transform Let Z But for negative i.....

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Professor A G Constantinides 9 Z - transform Examples: (i) Consider generation of new discrete time signal from via Recall linearity and shift (ii) Z write

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Professor A G Constantinides 10 Z - transform From With from earlier result We obtain Z Z

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Professor A G Constantinides 11 Inverse Z - transform Given F(z) to determine. Basic relationship is may be obtained by power series expansion. It suffers from cumulative errors

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Professor A G Constantinides 12 Inverse Z - transform Alternatively Use for m = -1 otherwise where closed contour encloses origin

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Professor A G Constantinides 13 Inverse Z - transform Integrate to yield Examples (i) write

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Professor A G Constantinides 14 Inverse Z - transform And hence Pole at of Residue (ii)Let where and To determine

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Professor A G Constantinides 15 Inverse Z - transform From inversion formula But

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Professor A G Constantinides 16 Inverse Z - transform Hence Thus

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Professor A G Constantinides 17 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.)

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MULT. INTEGERS 1. IF THE SIGNS ARE THE SAME THE ANSWER IS POSITIVE 2. IF THE SIGNS ARE DIFFERENT THE ANSWER IS NEGATIVE.

MULT. INTEGERS 1. IF THE SIGNS ARE THE SAME THE ANSWER IS POSITIVE 2. IF THE SIGNS ARE DIFFERENT THE ANSWER IS NEGATIVE.

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