### Parallel Case

Figure 6.2 illustrates the *parallel combination* of two
filters. The filters and are driven by the
*same* input signal , and their respective outputs
and are *summed*. The transfer function of the parallel
combination is therefore

*z*transform to have that .

#### Series Combination is Commutative

Since multiplication of complex numbers is commutative, we have

*numerical*performance of the overall filter is usually affected by the ordering of filter stages in a series combination [103]. Chapter 9 further considers numerical performance of filter implementation structures.

By the convolution theorem for
*z* transforms, commutativity of a product of transfer functions implies that
*convolution is commutative*:

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Example

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Series Case