IEEE Signal Processing - May 2018 - 127
This least-squares procedure gives
the parameters of the delayed parallel
form directly and is robust even for large
(. 1, 000) filter orders. The method
is inspired by the fixed-pole design of
parallel filters [9], a methodology used
for obtaining IIR filters having uneven
(e.g., logarithmic)-frequency resolution. First, the roots of denominator
A (z -1) are found that are used to form
the denominator polynomials of the second-order sections A l (z -1) . Next, the
numerators of the sections are obtained
via a least-squares fit such that the difference between the impulse responses
of the original and parallel structures
is minimized. (Alternatively, it is also
possible to minimize the difference
between the complex transfer functions of the direct form and parallel
form filters, leading to practically the
same results.) This can be done easily because the transfer function (7)
becomes linear in its free parameters
ufk, bu l, 0, bu l, 1 once the denominator coefficients a l, 1, a l, 2 are determined.
While the procedure is also applicable to the traditional, nondelayed parallel
form, it will be illustrated for the numerically better-performing delayed version.
We can see in (7) for M 2 N that the
first K + 1 = M - N + 1 samples of the
impulse response are solely determined
by the FIR coefficients ufk, and thus the
problem reduces to finding the parameters of the second-order sections such
that the resulting impulse response
is the closest to the original impulse
response starting from sample K + 1.
The steps of the conversion are given
next. The procedure is outlined for the
L
hu (i) = / bu l, 0 u l (i) + bu l, 1 u l (i - 1) (9)
l =1
is closest to the impulse response of the
original filter h (i) starting from sample
i = K + 1.
Since (9) is linear in its free parameters bu l, 0, bu l, 1, it can be written in a
matrix form
hu = Ubu ,
(10)
where U contains the impulse responses
u l (i) and their delayed versions u l (i - 1)
0
Magnitude (dB)
Magnitude (dB)
0
-20
-40
−60
in its columns, and bu is a column vector
composed of the corresponding bu l, 0 and
bu l, 1 values. Now the resulting impulse
response vector hu should be the closest possible to the target h vector containing the samples h (i) from i = K + 1
in the least-squares sense. This is a
standard linear least-squares problem
(overdetermined set of equations) and
can be solved by the mldivide function in
MATLAB/Octave by using the syntax
bu = U\h.
Figure 3(b) shows the net transfer
function of the delayed parallel form
when the conversion is done by the aforementioned least-squares fit. Now the
conversion is much more accurate compared to the one obtained by using partial
fraction expansion shown in Figure 3(a).
As for the size of the least-squares pro blem, the impulse response fit was
made for I = 2M = 400 samples, to
have more equations than unknowns (we
have M = 200 free parameters).
Table 1 lists the mean absolute decibel
errors computed between the original
and converted transfer functions in
the range of 20 and 22.05 kHz for various filter orders, including the (200/200)
example of Figure 3. The significantly
better accuracy of the least-squares procedure is apparent starting from order
100. For the orders of 1,000 and 1,500,
some of the extracted poles p n are outside the unit circle, thus, the partial
fraction expansion method leads to an
unstable filter. On the other hand, the
proposed procedure still produces accurate results since it starts with stabilizing
the poles by flipping them inside the unit
circle in step 1.
case of no-pole multiplicity. In the case
of repeated poles, terms of higher
than second-order must also be included, similarly to the case of partial
fraction expansion.
1) Compute the roots p n of the denominator A (z -1), flip the unstable poles
| p n | 2 1 inside the unit circle by
replacing them with 1/p n, find the
complex-conjugate pairs and re combine the denominators of the
second-order sections A l (z -1) .
2) Compute the impulse response h (i)
of the filter H (z -1) = B (z -1) /A (z -1)
for samples i = 0fI.
3) For M $ N, the coefficients of
t h e FIR part equal to the fi r s t
K = M - N + 1 samples of the filter
impulse response, i.e., ufk = h (k)
for k = 0fK.
4) Compute the impulse responses u l (i)
of the denominators 1/A l (z -1) =
1/ (1 + a l, 1 z -1 + a l, 2 z -2) .
5) Find the numerator coefficients
bu l, 0, bu l, 1 by a least-squares fit such
that the resulting impulse response
102
103
Frequency (Hz)
(a)
104
-20
-40
−60
102
103
Frequency (Hz)
(b)
104
Figure 3. The delayed parallel implementation of a (200/200) transfer function: conversion done (a) by partial fraction expansion and (b) by a leastsquares fit. The thick blue line is the original transfer function, and the red dashed line is the net transfer function of the delayed parallel form. The thin
colored lines show the responses of the individual second-order sections, and the black line displays the transfer function of the constant gain.
IEEE Signal Processing Magazine
|
May 2018
|
127
Table of Contents for the Digital Edition of IEEE Signal Processing - May 2018
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