O'Reilly logo

Signals and Systems by Smarajit Ghosh

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

6.8 z-TRANSFORM OF DELAYED UNIT SAMPLE SEQUENCE

The delayed unit sample sequence is expressed by

x1(n) = δ (nk) 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(nk)] = zkX(z)

 

Applying this property to x1(n) = δ (nk), we have

 

Z[x1(n)] = Z[δ (nk)] = zkZ[δ (n)] = zk.1 = zk     (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 = ∞ ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required