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:

1. Transfer functions of filters in series multiply together.
2. Transfer functions of filters in parallel sum together.

Series Case Figure 6.1 illustrates the 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 In summary, if the output of filter is given as input to filter (a series combination), as shown in Fig.6.1, the overall transfer function is --transfer functions of filters connected in series 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 where we needed only linearity of the z transform to have that .

Series Combination is Commutative

Since multiplication of complex numbers is commutative, we have which implies that any ordering of filters in series results in the same overall transfer function. Note, however, that the numerical performance of the overall filter is usually affected by the ordering of filter stages in a series combination . 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: Next Section:
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