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The z-Transform

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

The zz-transform

We introduced the zz-transform before as

H ( z ) = k = - h [ k ] z - k H ( z ) = k = - h [ k ] z - k
(1)

where zz is a complex number. When H(z)H(z) exists (the sum converges), it can be interpreted as the “response” of an LSI system with impulse response h[n]h[n] to the input of znzn. The zz-transform is useful mostly due to its ability to simplify system analysis via the following result.

Theorem

If y=h*xy=h*x, then Y(z)=H(z)X(z)Y(z)=H(z)X(z).

Proof

First observe that

n = - y [ n ] z - n = n = - k = - x [ k ] h [ n - k ] z - n = k = - x [ k ] n = - h [ n - k ] z - n n = - y [ n ] z - n = n = - k = - x [ k ] h [ n - k ] z - n = k = - x [ k ] n = - h [ n - k ] z - n
(2)

Let m=n-km=n-k, and note that z-n=z-m·z-kz-n=z-m·z-k. Thus we have

n = - y [ n ] z - n = k = - x [ k ] n = - h [ m ] z - m z - k = k = - x [ k ] H ( z ) z - k = H ( z ) k = - x [ k ] z - k = H ( z ) X ( z ) n = - y [ n ] z - n = k = - x [ k ] n = - h [ m ] z - m z - k = k = - x [ k ] H ( z ) z - k = H ( z ) k = - x [ k ] z - k = H ( z ) X ( z )
(3)

This yields the “transfer function”

H ( z ) = Y ( z ) X ( z ) . H ( z ) = Y ( z ) X ( z ) .
(4)

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