IEEE Signal Processing - May 2018 - 129

0
Magnitude (dB)

series form to the parallel form: factoring out zeros from the numerator until
we reach M 1 N. For the typical scenario of M = N, this means factoring
out one real zero, or, if there are no real
zeros, a complex conjugate pair. Now the
filter structure is a set of parallel secondorder sections (without additional delay)
and a first- or second-order FIR filter in
series. For the case of factoring out one
real zero, we obtain

−50

−100

-1

H (z )
= (1 - z -1 z 1) e /
L

l =1

bt 0, l + bt 1, l z -1
o.
1 + a 1, l z -1 + a 2, l z -2

−150
101

102

(11)
The conversion is illustrated in Figure 5 for an eighth-order high-pass Butterworth filter designed by the butter
command in MATLAB. The thick blue
line shows the magnitude response of
the series second-order implementation
obtained from the pole-zero form (the
direct-form implementation is unstable,
thus, not shown). Since all of the zeros
of the Butterworth filter equal to unity
z m = 1 for m = 1f8, we simply factor
out one of them before performing partial fraction expansion. Thus, the resulting filter structure is composed of four
second-order IIR filters in parallel and
an FIR filter Ft (z -1) = 1 - z -1 in series.
Obviously, the four second-order sections can be computed in parallel, but
the first-order FIR filter cannot, which
might lead to a slight increase in computational time in parallel architectures.
The benefit of the factored conversion is the significantly reduced design
time. Since no root finding is required,
the partial fraction expansion requires
very few (though complex) operations.
This allows computing the parallel form
on the fly when the user is manipulating the parameters of the series transfer
function in real time, e.g., by changing
the cutoff frequency of the high-pass filter. Another typical example would be
varying the parameters of a parametric
or graphic equalizer in series form and
converting it to the more efficient parallel form in real time.
We note that, for high-order filters
such as in the example of Figure 4,
partial fraction expansion can lead to

103

104

Frequency (Hz)

Figure 5. The factored parallel implementation of an eighth-order Butterworth high-pass filter. The
cutoff frequency is fc = 100 Hz, while the sampling rate is fs = 44.1 kHz. The thick blue line shows
the response of the 4 series second-order sections, and the red dashed line displays the response of
the factored parallel implementation. The thin-colored lines display the magnitude responses of the
parallel second-order sections in series with the response of the first-order FIR filter that has been
factored out.

numerical errors even when converting
from the series form. Indeed, the example of Figure 4 cannot be converted by
the factored partial fraction expansion
method discussed in the this section,
and the least-squares method should
be used.

Conclusions
In this article, the common approach
of converting direct form IIR filters
to parallel form has been revisited: the
method based on partial fraction expansion. If the order of the numerator M is
greater or equal to that of the denominator N, the responses of the individual
transfer functions can be significantly
larger than the net transfer function
leading to reduced dynamic range and
increased quantization noise. The use of
an alternative parallel form is suggested
to avoid the dynamic range problem: in
the delayed parallel filter the response
of the second-order sections is delayed
such that there is no overlap with the
parallel FIR path. The parameters of the
delayed parallel form can also be computed by the usual partial fraction expansion; the only difference is that the order
of the numerator polynomial is reduced
by performing polynomial long division
over the reversed numerator polynomial.
IEEE Signal Processing Magazine

May 2018

For high (>100) filter orders, conversion via partial fraction expansion can be
numerically sensitive. Therefore, a simple least-squares procedure is proposed
to obtain the delayed parallel form directly. The denominators of the second-order
sections are computed by recombining
the roots of the original denominator.
The coefficients of the parallel FIR part
simply equal to the first M - N + 1 samples of the original impulse response, and
the numerators of the second-order sections are estimated by a least-squares fit
such that the resulting impulse response
is the closest possible to the impulse
response of the original filter. Since we
are directly optimizing the error of the
impulse response, this guarantees optimal filter performance (the frequency
response error will be also minimal due
to Parseval's theorem). Indeed, filters in
the order of 1,000 can be factored with
very high accuracy.
Finally, the case in which the original
transfer function is available in a pole-zero
form or, equivalently, in a series combination of second-order sections, has been
tackled. In addition to the least-squares
fit, an alternative method has been presented that starts with factoring out
zeros from the numerator until M 1 N is
reached so that partial fraction expansion
129

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