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2.3 The z-transform

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The z -transform

We introduced the z -transform before as

H ( z ) = ∑ k = - ∞ ∞ h [ k ] z - k

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

Theorem

If y = h * x , then 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

Let m = n - k , and note that z - 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 )

This yields the “transfer function”

H ( z ) = Y ( z ) X ( z ) .
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Source:  OpenStax, Digital signal processing. OpenStax CNX. Dec 16, 2011 Download for free at http://cnx.org/content/col11172/1.4
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