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# z-Transform Examples

Module by: Mark A. Davenport. E-mail the author

## zz-transform examples

### Example 1

Consider the zz-transform given by H(z)=zH(z)=z, as illustrated below.

The corresponding DTFT has magnitude and phase given below.

What could the system HH be doing? It is a perfect all-pass, linear-phase system. But what does this mean?

Suppose h[n]=δ[n-n0]h[n]=δ[n-n0]. Then

H(z)=n=-h[n]z-n=n=-δ[n-n0]z-n=z-n0.H(z)=n=-h[n]z-n=n=-δ[n-n0]z-n=z-n0.
(1)

Thus, H(z)=z-n0H(z)=z-n0 is the zz-transform of a system that simply delays the input by n0n0. H(z)=z-1H(z)=z-1 is the zz-transform of a unit-delay.

### Example 2

Now consider x[n]=αnu[n]x[n]=αnu[n]

X(z)=n=-x[n]z-n=n=0αnz-n=n=0αzn=11-αz(if|α/z|<1)(GeometricSeries)=zz-αX(z)=n=-x[n]z-n=n=0αnz-n=n=0αzn=11-αz(if|α/z|<1)(GeometricSeries)=zz-α
(2)

What if az1?az1? Then n=0(αn)nn=0(αn)n does not converge! Therefore, whenever we compute a zz-transform, we must also specify the set of zz's for which the zz-transform exists. This is called the region of convergence (ROC). In the above example, the ROC={z:z>α}{z:z>α}.

### Example 3

What about the “evil twin” x[n]=-αnu[-1-n]x[n]=-αnu[-1-n]?

X(z)=n=--αnu[-1-n]z-n=n=--1-αnz-n=-n=--1zα-n=-n=1zαn=1-n=0zαn(convergesif|z/α|<1)=1-11-zα=α-z-αα-z=zz-αX(z)=n=--αnu[-1-n]z-n=n=--1-αnz-n=-n=--1zα-n=-n=1zαn=1-n=0zαn(convergesif|z/α|<1)=1-11-zα=α-z-αα-z=zz-α
(3)

We get the exact same result but with ROC={z:z<α}{z:z<α}.

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