## Series and Parallel Transfer Functions

The transfer function conveniently captures the*algebraic structure*of a filtering operation with respect to

*series or parallel combination*. Specifically, we have the following cases:

*Transfer functions of filters in series multiply together.**Transfer functions of filters in parallel sum together.*

### Series Case

*series connection*of two filters and . The output from filter 1 is used as the input to filter 2. Therefore, the overall transfer function is

*multiply*together.

### 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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