December 2005
Intermediate to advanced
475 pages
12h 6m
English
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6.8 z-TRANSFORM OF DELAYED UNIT SAMPLE SEQUENCE
The delayed unit sample sequence is expressed by
x1(n) = δ (n − k) for right shift and also by x2(n) = δ (n + k) for left shift.
We have already shown that Z[δ (n)] = 1 where ROC is entire z-plane.
Using shifting property of z-transform we can write
Z [x(n − k)] = z−kX(z)
Applying this property to x1(n) = δ (n − k), we have
Z[x1(n)] = Z[δ (n − k)] = z−kZ[δ (n)] = z−k.1 = z−k (6.56)
Z[x1(n)] has infinite value at z = 0 whereas it has finite values for all remaining values of z
Therefore, the ROC of Z[x1(n)] is entire z-plane except at z = 0.
Applying this property to x2(n) = δ(n + k), we have
Z[x2(n)] = Z[δ (n + k)] = zkZ[δ (n)] = zk.1 = zk
Z[x2(n)] has infinite value at z = ∞ ...
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