IEEE Signal Processing - May 2018 - 128

Magnitude (dB)

form of the filter is known
The reason for the signiTable 1. The mean absolute decibel errors
instead of the rational form.
ficantly better performance
of the converted transfer functions.
Therefore, this procedure is
compared to partial fraction
Error with Partial Fraction
Error with Leastnot recommended.
expansion is that the numerical
Filter order
Expansion conversion
Squares conversion
errors in finding the poles are
(50/50)
2.36 # 10 -10 dB
3.86 # 10 -10 dB
compensated by the numeraLeast-squares fit
-4
(100/100)
7.97 # 10 dB
5.52 # 10 -8 dB
tors of the second-order secOne possibility is obtaining
tions: the least-squares fit will
the delayed parallel form by
(200/200)
2.84 dB
6.78 # 10 -8 dB
give the best possible impulse
the least-squares fit as we have
(500/500)
4.45 dB
7.02 # 10 -8 dB
response match for the given
seen before for the direct-to1.70 # 10 -7 dB
(1,000/1,000)
NaN
(slightly inaccurate) denomiparallel conversion case. Of
(1,500/1,500)
NaN
1.34 # 10 -6 dB
nators. Besides simplicity, the
course, the target impulse
ability to correct the numeriresponse h(i) is computed by
cal errors of pole finding (even
running the series version
unstable poles) is a great benefit of the
reasons. Other examples can be series
of the filter, and we are not converting
least-squares design also compared to
graphic or parametric equalizers [10]
the series or pole-zero form to direct
other conversion methods proposed preand equalizer filters iteratively designed
form. Also, the numerically problematic
viously [4], [5].
directly in the series form [11] or obtained
root finding is avoided, since either the
from a warped IIR design [12].
poles, or the second-order denominaFor
strictly
proper
transfer
functions
tors, are known.
Conversion from series or
Such a conversion example is pre(
M
1
N
),
the partial fraction expanpole-zero form
sented
in Figure 4. The measured transsion
works
well
since
the
poles
are
either
In some situations, the transfer function
fer function of an average living room is
known (pole-zero form) or computed
we are converting to a parallel form is
the starting point to design a warped IIR
from second-order polynomials (series
given as a series of second-order secfilter. Warped filter design is one possecond-order form), and the numerator
tions, or, equivalently, in pole-zero form.
sible methodology to obtain filters with
can also be evaluated in its factored form.
Examples include classic low-pass, bandlogarithmic-like frequency resolution
On the other hand, when M $ N, we
pass filters such as Butterworth, Cheto fit the resolution in human hearing,
byshev, etc. [1]. The butter, cheby1, etc.
need to perform polynomial long divia desirable property in audio applicacommands in MATLAB/Octave can
sion to reduce the order of the numerations. However, warped filters require
give the pole-zero versions of the filters
tor and, for that, the denominator and
a complicated all-pass-based structure
making the implementation possible for
numerator have to be recombined from
for implementation. This can be avoided
such a high-order/low-cutoff-frequency
the poles and zeros. At this point, we
when they are converted series secondfilters where the direct form implemenwould lose all of the numerical benefits
order sections [12]. To show the robusttation is unfeasible due to numerical
coming from the fact that the pole-zero
ness of the conversion method, in the
example of Figure 4, a 1,000th-order
warped IIR filter is designed by the
0
prony function in MATLAB based on
Series
the warped room response (the warping
Implementation
−10
parameter is m = 0.8) . Next, the warped
IIR filter is converted to series second−20
order sections [12]. The net transfer
function of the 500 series second-order
sections is displayed by blue line. Finally,
−30
the filter is converted to the delayed parallel form by the least-squares method,
−40
displayed by a red line. The two transfer
functions in Figure 4 match perfectly: the
Parallel
−50
mean absolute decibel error from 20 Hz
Implementation
to 22.05 kHz is 5.7 # 10 -12 dB.
−60

102

103

104

Factoring out zeros before partial
fraction expansion

Frequency (Hz)

If we are willing to give up the full parallelization of the filter structure, there
is another possibility for converting the

Figure 4. The parallel implementation of a (1,000/1,000) transfer function. The blue line shows the
response of the 500 series second-order sections, which is then converted to 500 parallel second-order
sections by the least-squares fit, displayed by red line. The red curve is offset by −20 dB for clarity.

128

IEEE Signal Processing Magazine

May 2018

Table of Contents for the Digital Edition of IEEE Signal Processing - May 2018