## 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

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

*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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