IEEE Signal Processing - May 2018 - 125

Note that (3) and (4) are general only
if poles p n are distinct. In the case of
pole multiplicity, higher-order terms
also appear [6].
The partial fraction expansion re quires that the transfer function H (z)
is strictly proper, i.e., the order of the
denominator is larger than the order of
the numerator (N 2 M) . If this is not
the case, polynomial long division has
to be performed to result in a FIR part
F (z -1) and a strictly proper IIR part
Bl (z -1) /A (z -1) as
H (z -1) =

B (z -1)
Bl (z -1)
-1 =
A (z )
A (z -1)
+ f0 + f1 z -1 + g + fK z -K ,
(5)

where K = M - N is the order of the
FIR part. Partial fraction expansion,
including the polynomial long division,
is performed by the MATLAB/Octave
function residuez included in the Signal
Processing Toolbox. The last step of the
conversion is combining the complexconjugate pairs to second-order sections
having real coefficients:
H (z)
L

=/

l =1

K
b 0, l + b 1, l z -1
-k
-1
-2 + / fk z .
1 + a 1, l z + a 2 , l z
k =0
(6)

30
20
10
0
−10
−20
−30

The dynamic range problem
As shown in (6), both the IIR part
Bl (z -1) /A (z -1) and the FIR part F (z -1)
have an impulse response starting at
sample zero, meaning that the first K + 1
samples are determined by the combination of the FIR and IIR parts. This overlap leads to the fact that the individual
transfer functions of the filter sections
can be significantly larger than the
net transfer function, leading to the limitation of practical dynamic range [7].
This is illustrated in Figure 1(a) for
an IIR filter designed to model a measured loudspeaker response. The sampling rate is fs = 44.1 kHz in all of the
examples in this article. First, a directform IIR filter is designed by the Steiglitz-
McBride method [8] (the stmcb command
in MATLAB) shown by a thick blue line
and then converted to a parallel set of
second-order sections plus an FIR part
as described in the "Introduction" section. The orders of the numerator and the
denominator are both 20, with the short
notation: (20/20). This results in ten second-order sections plus a constant gain sec-

Magnitude (dB)

Magnitude (dB)

To my knowledge, the method discussed so far is the one that is covered in
all digital signal processing textbooks.
However, this common method, when
implemented in real-world processors,
may produce several problems. First,
the overlap between the FIR and IIR
parts lead to a dynamic range problem
and, thus, increased quantization noise.

102

103
Frequency (Hz)
(a)

tion in parallel. The red dashed line shows
the net transfer function, overlapping
the direct-form transfer function (thick
blue line) perfectly. The black dashed
line displays the transfer function of the
FIR part F (z -1), which is now a constant gain, while the thin colored lines correspond to the magnitude responses of the
individual second-order sections.
Figure 1(a) shows that some of the
individual transfer functions (in this
case, the constant gain part displayed
by the black dashed line and one second-order transfer function displayed
by a thin purple line) are significantly
larger than the net transfer function.
Figure 1(a) also shows only magnitude
responses; therefore, it cannot be seen
that these two upper curves have almost
opposite phase, and the required net
response is a result of the phase cancellation of these individual responses. In
theory, this does not lead to any difficulties. However, when the filter is
implemented with finite word length,
this requires the downscaling of the
input signal to avoid overload, which,
in turn, will lead to increased quantization noise. (No direct downscaling is
required in floating-point implementation, since it is done "automatically" by
the number format. However, the dynamic range reduction remains the same.)
This dynamic range problem be comes even more pronounced if the
order of the numerator M is larger than
that of the denominator N. This is illustrated in Figure 2(a) for numerator
and denominator orders of 25 and 20,
respectively (25/20). Figure 2(a) shows
that the fifth-order FIR part (black

Second, the expansion can be numerically sensitive leading to problems
when factoring high-order (N 2 100)
transfer functions. These problems are
addressed in this article in addition to
covering the case of converting filters
given in series or pole-zero form.
All of the MATLAB codes used to
create the figures in this article are
available for download, which can help
the interested reader gain more insight
to the presented algorithms (see the
"Supplementary Material" section).

104

30
20
10
0
−10
−20
−30

102

103
Frequency (Hz)
(b)

104

Figure 1. The parallel implementation of a (20/20) transfer function: (a) The traditional parallel form and (b) the delayed parallel form. The thick blue line
shows the original transfer function, and the red dashed line displays the net response of the parallel implementation. The thin colored lines display the
magnitude responses of the second-order sections, and the black dashed line shows the magnitude response of the constant gain path.

IEEE SIgnal ProcESSIng MagazInE

|

May 2018

|

125



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

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