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
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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




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


Series Combination is Commutative
Since multiplication of complex numbers is commutative, we have

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

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