Series Combination is Commutative Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search


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 [102].

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
\leftrightarrow
H_1\cdot H_2
=
H_2\cdot H_1
\leftrightarrow
h_2 \ast h_1
$


Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[How to cite this work] [Order a printed hardcopy]

``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (August 2006 Edition).
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA  [Automatic-links disclaimer]