A new improvement to the Powell and Chau linear phase IIR filters

Abstract

An improvement to the realization of the linear-phase IIR filters
is described. It is based on the rearrangement of the numerator
polynomials of the IIR filter functions that are used in the real-time
realizations proposed in literature. The new realization has better
total harmonic distortion when a sine input is used, and it has smaller
phase and group delay errors due to finite section length

Full text

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 6, JUNE 1998 1685
A New Improvement to the Powell
and Chau Linear Phase IIR Filters
Bojan Djoki´
c, Miodrag Popovi´
c, and Miroslav Lutovac
Abstract—In this correspondence, an improvement to the realization of
the linear-phase IIR filters is described. It is based on the rearrangement
of the numerator polynomials of the IIR filter functions that are used
in the real-time realizations recently proposed in literature. The new
realization has better total harmonic distortion when sine input is used,
and it has smaller phase and group delay errors due to finite section
length.
I. INTRODUCTION
Recently, a notable interest has been shown in real-time imple-
mentation of IIR filters having linear phase. Powell and Chau [1]
have devised an efficient method for the design and realization of the
real-time linear-phase IIR filters using suitable modification of the
well-known time reversing technique [2]. In their method, which is
shown in Fig. 1, the input signal is divided into -sample segments,
time-reversed, and twice filtered using two IIR filter blocks whose
transfer functions are the same, e.g. . The
transfer function is usually of elliptic type, giving the best
selectivity. It has been shown that the proposed procedure is much
faster than previous methods, having at the same time low distortions
of amplitude and phase characteristics.
The performance of this technique has been further improved
by Willson and Orchard in [3], where double zeros on the unit
circle have been separated, giving better amplitude response in the
stopband and slightly smaller amplitude and phase distortions. The
transfer functions and are different but have the same
denominator.
In this correspondence, a new improvement to the Powell–Chau
and Willson–Orchard methods is described, based on the reordering
of polynomials in the numerators of the filter transfer functions.
II. DESCRIPTION OF THE NEW METHOD
Let the transfer functions of the two IIR filter blocks in Fig. 1 be
(1)
(2)
where the polynomials and , whose zeros lie on the unit
circle, may be equal, as in [1], or different, as in [3]. In general, they
have no influence to the phase characteristic. However, the phase
error, which is caused by the segmentation of input sequence into
length- segments, may be influenced by the numerator polynomials,
as can be concluded from the results in [1] and [3]. In addition, it
may be noticed that the effects of the segmentation of the input signal
are concentrated into the time-reversed part of the block diagram,
Manuscript received June 8, 1995; revised August 8, 1997. The associate
editor coordinating the review of this paper and approving it for publication
was Dr. Victor E. DeBrunner.
B. Djoki´
c is with MOBTEL BK-PTT, Belgrade, Yugoslavia.
M. Popovi´c is with the Faculty of Electrical Engineering, University of
Belgrade, Belgrade, Yugoslavia.
M. Lutovac is with IRITEL, Belgrade, Yugoslavia.
Publisher Item Identifier S 1053-587X(98)03929-4.
where infinite impulse responses of the blocks are interrupted after
samples.
Following the results obtained by Willson and Orchard in [3],
it is interesting to examine whether some other rearrangements of
polynomials and into transfer functions and
may be better with respect to truncation noise, phase, and amplitude
errors.
First, truncation noise, which is noise caused by truncation to -
sample segments, of several realizations will be examined. Since
the truncation noise depends only on the realization of , the
spectrum of the signal (at the output of the time-reversing part of
the diagram in Fig. 1) will be analyzed. As in [1], the total harmonic
distortion (THD) of the signal is measured in response to a
single frequency input . The input
signal frequency is chosen so that an integer number of periods
is contained in the input samples, e.g., . In this case,
we have used the values and section length
. The THD of the input signal alone is approximately 150
dB. Some representative results are shown in Fig. 2, where effects
of placing the polynomial before or after time-reversed section
, or included in , are examined. In all examples, the
polynomials and are synthesized to satisfy the
requirements from [1, Example 6]
dB dB (3)
In Fig. 2, only the envelopes of spectra of all examined realiza-
tions are plotted. First, the curve represents the spectrum of the
time-reversed section [without direct section
] of the original solution [1]. If the FIR part of
the direct filter is implemented before the time-reversed section
, the input signal in is increased, and consequently, the
truncation noise at the output of is increased at all frequencies
(curve ). On the contrary, if is realized after the time-reversed
section , the truncation noise is filtered by , and the noise
at higher frequencies is more attenuated (curve ).
Finally, when is included in the time-reversed section ,
the noise is considerably decreased at lower frequencies (curve ). As
can be seen in Fig. 2, this realization produces the lowest truncation
noise in the passband and slightly larger noise in the stopband than
the solution represented by curve . It should be noticed that the
noise of the last realization in the stopband is still below the noise at
the passband edge frequency. Hence, from the truncation noise point
of view, the following choice of time-reversed and direct transfer
functions is optimal.
(4)
(5)
where the numerator polynomials and and denominator
polynomial are the same as in [1] or [3].
In Fig. 3(a) spectra of output signals [after the direct block
in Fig. 1] are shown, both for the original realization using
(1) and (2) and for the new realization using (4) and (5). The input
signal and the polynomials and are the
same as in the previous example. The spectrum of the Powell–Chau
technique is shown by the thin line, whereas the spectrum for the
new realization is shown by the thick line. It can be easily seen that
the harmonic distortion produced by the new realization is smaller by
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1686 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 6, JUNE 1998
Fig. 1. Block diagram of the Powell–Chau implementation of linear phase IIR filter.
Fig. 2. Envelopes of single frequency responses.
. Time-reversed sections are designated by , anddirect
section by . (a) . (b) . (c)
. (d) .
20–40 dB in most of the passband. This means that the finite section
length produces smaller harmonic distortion when both numerator
polynomials are concentrated in the time-reversed section, as in (4).
The new realization also compares favorably with the Will-
son–Orchard solution [3], as can be seen in Fig. 3(b), where the
comparison between the new solution and Willson–Orchard solution
is made. In this example, the polynomials and
are synthesized to satisfy the requirements from [1, Example 3].
dB dB (6)
that were used for comparison in [3].
This choice of the transfer functions and also has
positive influence on the decrease of the impulse response, as shown
in Fig. 4(a). As can be seen, after few samples of impulse response
that are larger than in the original realization [1], the remaining
samples become significantly smaller. This fact is a consequence of
the increased number of zeros in the time-reversed transfer function
. In order to better explain this behavior of the impulse
response, we can decompose the time-reversed transfer function
into FIR and IIR parts as
remainder (7)
Consequently, the impulse response is also the sum of two re-
sponses [finite impulse response and infinite impulse re-
(a)
(b)
Fig. 3. Spectra of output signals obtained for single frequency input.
. New realization—thick line, old realization—thin
line. (a) Comparison with Powell–Chau technique [1]. (b) Comparison with
Willson–Orchard technique [3].
sponse ], where and are impulse responses of
and , respectively. The first part of impulse
response is very short. The rates of decrease of the infinite
impulse responses of realizations according to (1) and (4) are the
same due to same denominators in both realizations. However,
the new solution has a smaller amplitude of impulse response ,

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 6, JUNE 1998 1687
(a)
(b)
Fig. 4. Impulse responses. (a) —thin line,
—thick line. (b) remainder
—thin line, remainder —thick
line.
which is caused by the numerator [the numerators of realization
according to (1) and (4) are different]. This is shown in Fig. 4(b).
The faster decrease of the impulse response of the new realization
produces less significant negative effects of the truncation to -
sample segments.
Having reduced truncation noise in the passband, we can also
expect that the phase response will be improved in the passband. In
the second experiment, the phase errors, which are produced by this
method, and methods in [1] and [3] are compared. The same filter,
which satisfies (3), was excited by a unit pulse sequence having 4096
samples. The results are presented in Fig. 5(a) for the new realization
(thick line) and old realization [1] (thin line). As can be seen, the
phase error in the passband is very small, having the maximum of
about rad. Compared with the previous realization, the new
realization produces 2–3 times smaller maximum phase error in the
passband. This experiment was performed again using the impulse
response of a high-order allpass filter as the input signal with very
similar results. In addition, the comparison with the Willson–Orchard
method [3], based on (3) and presented in Fig. 5(b), shows similar
results in favor of the new method.
(a)
(b)
Fig. 5. Comparison of the passband phase responses. .
New realization—thick line, old realization—thin line. (a) Comparison with
Powell–Chau technique [1]. (b) Comparison with Willson–Orchard technique
[3].
The new choice of the direct and time-reversed transfer functions
has some effect on the complexity of the realization. In order to
achieve the lowest possible number of multipliers, is realized
as a cascade connection of the FIR filter and the IIR filter
. The IIR filter is realized as in [1]. The
second IIR filter can be realized as a cascade connection of
the second-order sections without direct branches or even with the
same realizations as in [1] by proper selection of the output node.
The number of multipliers for and is the same
as in [1] for and , respectively. Since an
additional multiplier is required for every second-order section of
FIR filter , in the analyzed example, the number of multipliers
is increased from 21 in [1] to 27 in the new realization, as shown
in Table I. In addition, the use of the new technique that combines
several multipliers into multiplier blocks [4] can drastically reduce

1688 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 6, JUNE 1998
TABLE I
NUMBER OF MULTIPLIERS AND NOISE GAIN, . (a)
. (b) .
the complexity of the cascade realizations of and (a
single block can be used for each second-order section [4]). Thus, the
number of multiplier blocks of the new realization is approximately
the same as that corresponding in [1], and the complexity of the new
realization can be the same as in [1]. The number of multipliers in
allpass sections can be also reduced; half of the multipliers can be
implemented with a shifter and an adder or a shifter only [5].
Since is realized as two different filters [ and
], the quantization noise due to multiplication is increased,
as shown in Table I. The very high quantization noise of the filter
can be reduced by appropriate selection of transfer function
[6]. In addition, by increasing the wordlength in the last section only,
the quantization noise is reduced, and it can be made lower than the
noise caused by truncation to -sample segments.
III. CONCLUSION
In this correspondence, a new improvement to the realization of the
linear-phase IIR filters is described. It is based on the rearrangement
of the numerator polynomials of two IIR filter functions that are used
in the real-time realizations in [1] and [3]. The new realization has
better total harmonic distortion when sine input is used and smaller
phase error due to finite section length. It enables shorter sample delay
for the same phase error or lower phase error and THD improvement
for the same sample delay. The considerable improvement in phase
response and lower truncation noise are obtained at the expense of a
slightly increased number of multipliers and increased wordlength.
REFERENCES
[1] S. R. Powell and P. M. Chau, “A technique for realizing linear phase IIR
filters,” IEEE Trans. Signal Processing, vol. 39, pp. 2425–2435, Nov.
1991.
[2] J. J. Kormylo and V. K. Jain, “Two-pass recursive digital filter with
zero phase shift,” IEEE Trans. Acoust., Speech, Signal Processing, vol.
ASSP-30, pp. 384–387, Oct. 1974.
[3] A. N. Willson and H. J. Orchard, “An improvement to the Powell and
Chau linear phase IIR filters,” IEEE Trans. Signal Processing, vol. 42,
pp. 2842–2848, Oct. 1994.
[4] A. G. Dempster and M. D. Macleod, “Multiplier blocks and complexity
of IIR structures,” Electron. Lett., vol. 30, no. 22, pp. 1841–1842, Oct.
1994.
[5] M. D. Lutovac and L. D. Mili´
c, “Design of computationally efficient
elliptic IIR filters with a reduced number of shift-and-add operations
in multipliers,” IEEE Trans. Signal Processing, vol. 45, pp. 2422–2430,
Oct. 1997.
[6] B. Djoki´c, M. D. Lutovac, and M. Popovi´c, “A new approach to the
phase error and THD improvement in linear phase IIR filters,” in
Proc. 1997 IEEE Int. Conf. Acoust., Speech, Signal Process., Munich,
Germany, Apr. 21–24, 1997, pp. 2221–2224.
Generalized Digital Butterworth Filter Design
Ivan W. Selesnick and C. Sidney Burrus
Abstract—This correspondence introduces a new class of infinite im-
pulse response (IIR) digital filters that unifies the classical digital Butter-
worth filter and the well-known maximally flat FIR filter. New closed-
form expressions are provided, and a straightforward design technique is
described. The new IIR digital filters have more zeros than poles (away
from the origin), and their (monotonic) square magnitude frequency
responses are maximally flat at and at . Another result
of the correspondence is that for a specified cut-off frequency and a
specified number of zeros, there is only one valid way in which to split
the zeros between and the passband. This technique also permits
continuous variation of the cutoff frequency. IIR filters having more zeros
than poles are of interest because often, to obtain a good tradeoff between
performance and implementation complexity, just a few poles are best.
I. INTRODUCTION
The best known and most commonly used method for the design
of IIR digital filters is probably the bilinear transformation of the
classical analog filters (the Butterworth, Chebyshev I and II, and
Elliptic filters) [9]. One advantage of this technique is the existence
of formulas for these filters. However, the numerator and denominator
of such IIR filters have equal degree. It is sometimes desirable to be
able to design filters having more zeros than poles (away from the
origin) to obtain an improved compromise between performance and
implementation complexity.
The new formulas introduced in this correspondence unify the
classical digital Butterworth filter and the well-known maximally
flat FIR filter described by Herrmann [3]. The new maximally flat
lowpass IIR filters have an unequal number of zeros and poles and
possess a specified half-magnitude frequency. It is worth noting that
not all the zeros are restricted to lie on the unit circle, as is the case for
some previous design techniques for filters having an unequal number
of poles and zeros. The method consists of the use of a formula
and polynomial root finding. No phase approximation is done; the
approximation is in the magnitude squared, as are the classical IIR
filter types.
Another result of the correspondence is that for a specified number
of zeros and a specified half-magnitude frequency, there is only one
valid way to divide the number of zeros between and the
Manuscript received September 17, 1995; revised July 25, 1997. This work
was supported by BNR and by NSF Grant MIP-9316588. The associate editor
coordinating the review of this paper and approving it for publication was Dr.
Truong Q. Nguyen.
I. W. Selesnick is with Electrical Engineering, Polytechnic University,
Brooklyn, NY 11201-3840 USA (e-mail: selesi@radar.poly.edu).
C. S. Burrus is with the Department of Electrical and Computer Engineer-
ing, Rice University, Houston, TX 77251 USA.
Publisher Item Identifier S 1053-587X(98)03928-2.
1053–587X/98$10.00 1998 IEEE