IEEE Signal Processing - May 2018 - 124

Tips & Tricks
Balázs Bank

converting Infinite Impulse response Filters
to Parallel Form

D

iscrete-time rational transfer
functions are often converted to
parallel second-order sections
due to better numerical performance
compared to direct form infinite impulse
response (IIR) implementations. This
is usually done by performing partial
fraction expansion over the original
transfer function. When the order of
the numerator polynomial is greater
or equal to that of the denominator,
polynomial long division is applied
before partial fraction expansion re sulting in a parallel finite impulse
response (FIR) path.
This article shows that applying
this common procedure can cause a
severe dynamic range limitation in the
filter because the individual responses
can be much larger than the net transfer function. This can be avoided by
applying a delayed parallel form where
the response of the second-order sections is delayed in such a way that
there is no overlap between the IIR
and FIR parts. In addition, a simple
least-squares procedure is presented
to perform the conversion that is numerically more robust than the usual
Heaviside partial fraction expansion.
Finally, the possibilities of converting
series second-order sections to the delayed parallel form are discussed.

Introduction
IIR digital filters are part of most signal processing algorithms. They are
used not only in classic filtering applications (low-pass, high-pass, and so on) but
also as tools for approximating any given
transfer function, e.g., a measured frequency response that we wish to model
in discrete time. Compared to FIR filters,
IIR filters typically require lower computational resources for the same modeling
accuracy. However, care has to be taken
to assure their stability: a theoretically
stable IIR filter might become unstable
when implemented with finite coefficient
precision. The problem becomes pronounced when the filter has high order
and/or has poles near the unit circle. As a
remedy, IIR filters are often implemented
as a series or parallel combination of (typically, second-order) subfilters [1].
The conversion to series second-order
sections starts with finding the poles p n
and zeros z m of the transfer function
H (z -1) =

124

(1)
resulting in
H (z -1)
(1 - z -1 z 1) (1 - z -1 z 2) f (1 - z -1 z M)
.
(1 - z -1 p 1) (1 - z -1 p 2) f (1 - z -1 p N )

(2)
IEEE SIgnal ProcESSIng MagazInE

r1
1 - p 1 z -1
rN
r2
,
+
+g +
1 - p 2 z -1
1 - p N z -1
(3)

H (z -1) =

-1
-2
+
+
+ g + z -M
= b 0 b 1 z-1 b 2 z-2
,
1 + a 1 z + a 2 z + g + z -N

=K
Digital Object Identifier 10.1109/MSP.2018.2805358
Date of publication: 26 April 2018

B (z -1)
A (z -1)

Finally, the complex-conjugate pairs of
poles and zeros are recombined to form
second-order sections. Carefully pairing those poles and zeros is of utmost
importance, and the ordering of the sections is also critical as it influences the
roundoff noise and dynamic range of
the filter [1].
Today, the parallel implementation
is gaining more interest since it provides several advantages compared to
series biquads: it has lower quantization
noise [2], and, even more importantly, it
leads to a significant speedup in modern multicore processors that can take
advantage of the fully parallel filter
structure [3].
While alternative methods are available for direct-to-parallel conversion
[4], [5], by far the most common way
of converting filters to parallel form is
based on partial fraction expansion [1].
Here, the first step is converting the
transfer function (1) to the residue form

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May 2018

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where rn are the residues corresponding to the poles p n. The usual way of
determining rn is the Heaviside coverup method, which can be formulated
mathematically as
rn = (1 - z -1 p n) H (z)

z = pn .

(4)

1053-5888/18©2018IEEE
Table of Contents for the Digital Edition of IEEE Signal Processing - May 2018
