Series Combination is Commutative

Since multiplication of complex numbers is commutative, we have

$\displaystyle H_1(z)H_2(z)=H_2(z)H_1(z),

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 [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:

$\displaystyle h_1 \ast h_2
H_1\cdot H_2
H_2\cdot H_1
h_2 \ast h_1

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Example: The General Biquad PFE
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Real Linear Filtering of Complex Signals