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Generalized doubly complementary IIR digital filters
Signal Processing 29 (1992) 173-181 Elsevier
173
Generalized doubly complementary IIR digital filters Markku Renfors Department of Electrical Engineering, Tampere University of Technology, P.O. Box 527, 33101 Tampere, Finland Sanjit K. Mitra (Member EURASIP) Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106, USA P h i l i p A. R e g a l i a Department of Electronics & Communication, Institut National des Telecommunications, 9 Rue Charles-Fourier, 91011 Evry Cedex, France Yrj6 Neuvo (Member EURASIP) Department of Electrical Engineering, Tampere University of Technology, P.O. Box 527, 33101 Tampere, Finland Received 25 July 1991 Revised 2 March 1992
Abstract. A certain class of frequency selective digital IIR filters, including the classical Butterworth, Chebyshev and elliptic filters, can be realized as a parallel connection of two real or complex conjugate allpass filter. In this paper, a generalized structure where the allpass branches include arbitrary real or complex weighting constants is discussed. Applications of these generalized doubly complementary filters include gain equalizers, digital filters with multilevel magnitude respone, and sharpening the amplitude respone of digital filters realized as a cascade of identical filter stages. Zusammenfassung. Eine bestimmte Klasse von frequenzselektiven digitalen IIR-Filtern, einschlieBlich der klassischen Butterworth, Tschebyscheff und ellipischen Filter, kann als Parallelkombination von zwei reellen oder konjugiert komplexen AllpaB Filtern realisiert werden. In dieser Arbeit wird eine verallgemeinerte Struktur diskutiert, in der die AllpaB-Zweige reelle oder komplexe Gewichtsfaktoren enthalten. Anwendungen dieser verallgemeinerten doppel-komplement/iren Filter sind Entzerrer, digitale Filter mit mehrstufigen Frequenzg~ingen und die Flanken-Versteilerung bei Frequenzg~ingen von Filtern, die als Kaskade identischer Filterstufen realisiert sind. R6sum6. Une certaine classe de filtres num6riques IIR s~lectifs en fr6quence, dont les filtres classiques de Butterworth, Chebyshev et les filtres elliptiques peuvent ~tre obtenus par une connection en parall/~le de deux filtres passe-tout r~els ou conjugu6s complexes. On pr6sente dans cert article une structure g6n6rale compos6e de branches passe-tout pond6r6es par des constantes arbitraires r6elles ou conjugu6es complexes. Les applications de ces filtres g6n6ralis6s doublement compl6mentaires incluent les 6galisateurs de gain, les filtres num6riques gt r6ponse d'amplitude ~t plusieurs niveaux, et l'am61ioration de precision de la r6ponse en amplitude des filtres numbriques r6alisbe par la mise en cascade de plusieurs &ages de filtres identiques. Keywortls. Power complementary IIR filters; parallel allpass realizations; multilevel IIR filters; gain equalizers.
Correspondence to: Professor S.K. Mitra, Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106, USA. Fax: (805) 893-3262. 0165-1684/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved
M. Renfors et al. / Doubly complementary IlR digital filters
174
1. Introduction
2. Properties
Digital allpass filters are useful signal processing building blocks and are used in a variety of applications [6]. In particular, certain filter types, such as classical odd-order lowpass designs, can be realized as a parallel connection of two real or complex conjugate allpass filters [2, 3, 9-11]. These methods have significant advantages, such as modular structure, low roundoff noise power and reasonably low coefficient sensitivity. In addition, a complementary filter pair can be implemented at the cost of a single filter. This is particularly useful for implementing filter banks [ 1]. In this paper, we consider a generalization of the above concepts and define the weighted complementaryfilter pair, obtained by using different weights in the two allpass branches:
2.1. Complementary properties
G(z) = creAm(z)+ c2A2(z),
(la)
H(z) = emA m(z) - c2A2(z),
(1 b)
as illustrated in Fig. 1, where Al(z) and A2(z) are two real or complex allpass filters, and cl and e2 are two real or complex constants. In order to normalize the amplitude responses, we assume in most cases that the weighting constants satisfy
Ic, I + Ic21 = 1.
(2)
The classical lowpass designs mentioned are obtained as special cases of (1) with em=e2=0.5. We first derive the properties of the weighted complementary filter pair, including conditions for different types of complementary properties, and their corresponding amplitude response characteristics. Next, we outline some applications of the weighted complementary filters including multilevel designs, amplitude equalizers and weighted cascaded designs. c!
G(z) + H(z) = 2cmA. (z).
(3)
This shows that the filter pair is essentially allpass complementary, i.e. the frequency respone of the sum of the two transfer functions has a constant magnitude. In conventional filter designs, cm= 0.5 and the filter-pair is strictly allpass complementary. Adding up the squared-magnitude responses we get IG(eJ'°)lz + IH(eJ°')l 2= 2(le~l2 + Ic212),
(4)
implying that the filter pair is essentially power complementary. Again in conventional filter designs, the right-hand side of (4) is unity. From (4) it also follows that the transfer functions G2(z) and H:(z) are magnitude complementary, i.e. the sum of their magnitude responses is a constant.
2.2. Amplitude response characteristics From (1) is can be seen that the amplitude responses of the filter pair satisfy
Ile, I- Ic211~< pa(eJ~)l ~ le, I+ le21,
(5a)
IIc, I - Ic2ll ~< In(eJ'°)I ~< Ic,I + Ic21.
(5b)
Clearly, for highly selective filters, the magnitudes of Cmand c: must be very close to each other. For example, with cm= 0.51 and c2 = 0.49, the maximum value is unity and the minimum value corresponds to 34 dB of attenuation. In many cases, the filter pair of (1) is derived from a conventional (say classical lowpass) prototype design of the form G(z) = ½[Am(z) + A2(z)],
(6a)
/t(z) = ½[Am(z) - A2(z)].
(6b)
G(z) D
IN H(z) Fig. 1. Weighted complementary filter pair. Signal Processing
From (1) it follows that
Next we examine how the amplitude responses of the prototype design and the weighted complementary pair are related when real or complex conjugate weighting constants are used.
M. Renfors et al. / Doubly complementary HR digital filters
Real weights. In the case of real weighting con-
stants in (1) we obtain from (1) and (6) (7a)
H(z) = (c, - c2)G(z) + (c, + c2)H(z).
(7b)
Since the prototype filters are power complementary and allpass complementary and, consequently, their frequency responses are in phase quadrature at all frequencies, we can show that
c21Gl(eJ°')l 2,
(8a)
]H(eJ'°)[ 2 = (cl + c2) 2 - 4c,c21Gl(eJ'°)l 2.
(8b)
- c2) 2 -~ 4c,
opposite. It can thus be shown that IG(eJ~°)l =211cr[ IG(eJ'°)l ± [c~lx/-i-L [G(eJo')[2 I,
G(z) = (c, + c2)G(z) + (cj - c2)H(z),
IG(eJ~°)[2 = (c,
175
The above shows that there is a one-to-one mapping between the amplitude responses of the prototype filter pair and those of the weighted complementary filter pair. This implies that if the prototype design has equiripple passband(s) and stopband(s), then the corresponding frequency bands of the weighted complementary pair are also equiripple. Assuming that both c~ and c2 are positive, the maximum and minimum amplitude response values 1 and 0 are mapped to the values cj + c2 and Icj- c2l, respectively. Notice that with the normalization of (2), the maximum amplitude response value is unity.
IH(eJ'°)l = 21 [c,[ IG(eJ'°)l ~= ]Cr[x/1 --[G(eJC°)[ 2 [. (1 lb)
3. Applications 3.1. Two-level filters [1]
We consider first the odd-order case. Let the amplitude response specifications for the two-level filter be given by DI-~j<~IG(eJ'°)I<~Dj
for 0~
(12a)
D2<~[G(eJ'°)I<~D2+82
for c02~
(12b)
Such a filter can be derived from an odd-order elliptic prototype filter by selecting the values of the real weighting constants appropriately. The specifications for the prototype are given in the form l-Sj~
c, = 0.5 e j* = c~+jci,
(9a)
c2 = 0.5 e -jo = cr -jci.
(9b)
G(z) = (cr + jci)Al(z) + (cr-jci)A2(z)
= 2crG(z) - 2c~/~(z). J
(10)
Since the frequency responses of G(z) and/4(z) are in phase quadrature at all frequencies, the phase responses of G(z) and H(z)/j must be the same or
(13b)
c2 = / ( D I - D 2 ) .
(14)
The passband and stopband ripples can also be determined from (8a):
~,=1~z = ~/
Now we can write
(13a)
Using (8a), the values of the weighting constants can be determined: I
that the weights in (1) are complex conjugates of each other and their magnitudes are 0.5, i.e.,
for0~
for co2~
Cl = ~ ( O l + D 2 ) ,
Complex conjugate weights. In this case we assume
(1 la)
~/fl
6,(2D, - ~,) D~_ D 2 ,
.~-- $ Dr-D2
•
(15a)
(15b)
Now the elliptic prototype filter is completely specified. The minimum-order elliptic filter satisfying these specifications is then designed and the allpass filters A j(z) and A2(z) are determined from the poles of the elliptic filter. Finally, the weighting constants are computed from (14). In a similar manner, two-level filters using a bandpass or multiband prototype filter can be designed. Vol. 29, No. 2, November1992
M. RenfiJrs et al. / Doubly complementary I1R digital filters
176
It is obvious that if the values of the weighting constants el and c2 are interchanged, the magnitude response remains the same but the phase response is changed. One of the choices may give considerably better phase response or group delay response than the other.
EXAMPLE 1. Consider the design of a two-level filter satisfying the specifications of (12) with c0~= 0.3n, o92= 0.33~, D~ =0.75, D2 =0.25 and 81 = &2= 0.01. The maximum passband and Stopband ripple values for the elliptic prototype filter, obtained from (15), are &~=0.015 and 32=0.101. The estimated minimum order for the elliptic filter is 4.7 so that a fifth-order filter is required. The amplitude response plot for the designed two-level filter are shown in Fig. 2(a). Figure 2(b) shows the group delay responses for both choices of the weighting constants.
I
auJ 0 . 7 5
'
I
t.-- 0 . 5 0 Z C9 < ~;
0.25
tr
0.01
n~ ¢.r u.i ,,,
In the even-order case, it is necessary to use complex conjugate weighting coefficients in order to obtain a real-coefficient overall transfer function. It is possible to determine the values cr and ci in (11 a), and the passband and stopband ripple values for the prototype design such that the specifications of (12) are satisfied. However, the behavior of the amplitude respone in this case is no longer monotonic in the transition band.
3.2. Amplitude equalizers The weighted filter can also be used as a gain equalizer [4]. In this case, the amplitude response of the filter is desired to approximate a certain desired function D(og) on given frequency band(s). To illustrate this application, consider the case where the desired function is monotonically increasing or decreasing, i.e. of lowpass or highpass type, and the frequency band is [0, rt]. We consider here only odd-order designs using real allpass filters. In this case, the weighting constants are computed as in (14) from the maximum and minimum values of the desired function:
c,=½[D(O)+D(n)], I.-
A
L
i " 1
I
i
e2=½[D(O)-D(x)].
(16)
Using (8a), the desired amplitude response for the prototype filter with weights c~ = 0.5 and c2 = -4-0.5 can be computed using
t
0
~ - -
D2(,~)
V ~-0.01 :~
I
~
0
I
0.2~
(a)
I
m
0.4n
I
m
0.6w
I
I
0.8w
I 1
FREQUENCY iii ._1 0,,.. ca} Z >,,a: _1 LLI Cb ¢,
(b)
i
0 rr r5
'
40
1
l ~ I ~ l ,
20
J
~b(~o)=2 cos-'(D(¢o)). I
0
0.2'n"
).4~
m
I
i
0.6~
L
(18)
L
O.8Tt
FREQUENCY
Fig. 2. Responses for the two-level design of Example 1. signal Processing
This desired amplitude function satisfies the condition 0~
This phase function is to be approximated by the allpass filter A(z)=Al(z)/A2(z). The phase approximation problem can be solved using a
M. Renfors et al. / Doubly complementary I1R digital fihers
Remez-type optimization algorithm proposed in [7]. In addition to the desired function, this algorithm utilizes a weight function W(o9). This weight function is computed from the specifications in such a way that an equiripple solution for the error function W(o9)(~b (o9) - arg[Al(eJ'°)/A2(eJ°')]) results in equiripple error for the amplitude response of the weighted filter. Finally, the resulting allpass filter A(z) splits into two parts Al(z) and 1/A2(z) in such a way that the allpass functions of the two branches, A~(z) and A2(z), are stable. There are again two choices for combining the weighting constants and the allpass subfilters which result in the same amplitude response but different phase responses. The above procedure can be extended in a straightfoward manner to the case where the desired amplitude response is a unimodal function with D ( 0 ) = D ( ~ ) . In this case, a bandpass-type prototype filter is used. The weighting constants are computed in a manner similar to that of (16) from the maximum and minimum values of the desired function. Then the desired phase difference can be computed using (17) and (18). The techniques described above can also be used in cases where the desired function is to be approximated only on a certain subset of [0, rt]. In such designs, the maximum and minimum values of the amplitude response may not always be determined by the specifications. In these cases, the optimum values for the weighting constants can be found by a search procedure.
EXAMPLE 2. Consider the design of a gain equalizer approximating the desired function D(og) = (o9/2)/sin(o9/2). The requirements for the amplitude error is +0.001 or smaller. This equalizer compensates for the amplitude droop of a zeroorder hold circuit. For a full-band gain equalizer, we select D ( 0 ) = 1 and D(rc) = 1.571. Using (16) we obtain the values of the weighting constants c j = 1.2855 and c2 = -0.2855. In this case, a seventh-order filter is needed and the resulting amplitude error is
177
1.60 Q
1.40
k,.-
L9
1.2o
1.00
tv-
I.-au'ltu' n"0.0010 ~0~ <~-0.001
0 0.2r~ 0.4v 0.61r 0.8w 1 FREQUENCY
F i g . 3.
Responses for the gain equalizer of Example 2.
+0.00072. Figure 3 shows the amplitude response plot for the equiripple amplitude error solution obtained through phase appoximation as described above. All the poles of the allpass filters of this design are located on the real axis. If the approximation band is [0, 0.95rt], then a fifth-order filter is sufficient, and the resulting amplitude error is 4-0.00041. For a comparison, optimum IIR filters satisfying the same specifications were designed using the techniques proposed in [7]. In the fullband case, the specifications could just be satisfied using a filter with five poles and two zeros. For the approximation band [0, 0.957t], a filter with two poles and two zeros is sufficient. However, these filters cannot be realized using allpass subfilters. Also, at least in this case, the weighted complementary implementation is very favorable with respect to the finite wordlength properties because all of its poles are located on the real axis.
3.3. Weighted cascaded designs Now consider implementing a digital filter as a cascade of two filter stages. These filter stages are realized as a parallel connection of the same two allpass filters Al(z) and A2(z). The problem of determining the weighting constants optimally for both stages is considered here. Obviously, real weighting constants cannot be used for improving Vol. 29, No. 2, November 1992
M. RenJbrs et aL/ Doubly complementary HR digital filters
178
the selectivity of the filter. However, we shall see that optimally selected complex weighting constants improve the stopband attenuation considerably when compared to that obtained using the straightforward cascade structure. In the following we consider using both real allpass filters (oddorder lowpass prototype) and complex conjugate allpass filters (even-order lowpass prototype). We consider the following transfer function: G(z) = [cA,(z) + c*A2(z)][c*Al(z) + cAz(z)] = Icl2[AZ(z) + AZz(Z)]
(19)
+ 2 Re(c)2A ffz)A2(z).
The realization of above using real allpass subfilters and Icl = 0.5 is shown in Fig. 4. It is quite easy to see that, using real allpass filters and complex conjugate weighting constants, (19) is the only way to get a real coefficient overall transfer function. With complex conjugate allpass filters, different complex conjugate weights could be used for the two stages, but the best results are obtained by selecting the coefficients as in (19). The remaining problem is to determine the complex weighting constant c optimally. We can treat the overall filter as a cascade of two stages of the form of (10). As the stages have complex conjugate weights, the two components in (1 i a) are in phase in one of the stages and in opposite phases in the other stage. This means that the overall amplitude response is a product of two terms of the form of ( l l a ) and, for each ¢o, one of the terms has a '+' sign and the other one has a ' - ' sign. Assuming that Icl =0.5, we now obtain ]G(eJ°~)] = [IG(eJ'°)l2 - 4[Im(c)]21,
(20)
1
4
I
Fig. 4. Structure for the weighted cascaded filter using real allpass subfilters. Signal Processing
where G(z)=[Affz)+A2(z)]/2 is the prototype filter. From (20) it is easy to see that the optimum value for c is obtained by selecting 4[Im(c)] 2= ~ / 2 , where ~s is the stopband ripple of G(z). Therefore the optimum selection is Im(c) = 6s/2x/2.
(21)
The resulting stopband ripple of the overall filter is 6s = 6s2/2, i.e. the stopband attenuation is 6 dB higher than that of the direct cascade design. The passband maximum and minimum values are 1 - ~s and ( 1 - ~p)2_ ~s, respectively, where ~p is the passband ripple of the prototype filter G(z). Typically, the value of ~s is so small that its effect on the passband response can be ignored, and the passband amplitude response is practically the same as that of the cascade design. 3. An IIR filter, with passband and stopband edges at 0.1 x and 0.15x, passband ripple of at most 0.1 dB, and a minimum stopband attenuation of at least 80 dB, is to be designed. In the case of direct elliptic designs, the minimum filter order is eight which results in 87.5 dB stopband attenuation with 0.1 dB passband ripple. This eighth-order filter can be realized using complex allpass filters. A ninth-order elliptic filter gives 102 dB stopband attenuation with 0.1 dB passband ripple and it can be realized as a parallel connection of two allpass filters. The specifications can also be satisfied by using a cascade of two fifth-order elliptic subfilters. The direct cascade gives 82 dB stopband attenuation, whereas the attenuation increased to 88 dB with optimized weighting constants. Even though the required overall order is somewhat higher than in direct elliptic designs, the cascaded designs are useful because they have lower coefficient sensitivity if similar allpass filter structures are used in both cases. Other advantages of the cascaded design are lower passband group delay variation and slightly lower roundoff noise level. To illustate these points, we consider a ninthorder direct elliptic design and cascaded designs with fifth-order subfilters, in which both subfilters EXAMPLE
M. Renfors et al. / Doubly complementary HR digital filters
can be implemented using real allpass filters. We assume that the allpass filters are implemented using cascades of first- and second-order sections with direct-form coefficients [8]. Table 1 shows the passband ripple and stopband attenuation for different designs as function of the coefficient wordlength. We consider direct rounding of the coefficients, without any further optimization. It is well known that filter structures based on parallel connection of allpass filters have much better passband sensitivity than stopband sensitivity, especially when the stopband attenuation is high as in this example [6, 8]. Also in this example, the coefficient wordlength is limited mainly by the stopband behavior. Using the structure of Fig. 4, the weighting coefficient 2 Re(c 2) may need somewhat longer wordlength than the allpass filter coefficients. In this case we used the value 2 Re(c 2) =0.5-2 -14. Using first- and second-order direct-form I allpass filter sections [8], the estimated noise gains are 21.1 dB for the ninth-order filter and 18.6 dB for the cascaded designs. The passband group delay response is in the range [19.9, 93.6] for the ninth-order filter and [16.5, 64.7] for the cascaded designs.
179
Approximately magnitude complementary filter pair
We can also find the complementary lilter to the transfer function of (I 9) : H(z) = [dA,(z) - d*Ae(z)][d*Al(z) - dA2(z)]
A[(z)]
= Idl2[A?(z) +
- 2 Re(d)2Al(z)A2(z).
(22)
In this case, the optimum value for the weighting coefficient d is such that Im(d) = ~s/2x~, where ~ is the stopband ripple of the complementary prototype filter H ( z ) = [ A l ( z ) - A2(z)]/2. Denoting by es the stopband ripple of H(z), we can write IG(eJ~°)l + IHteJ'°)l = IIG(e~'°)t2 - ¢~sI+ tl/~(eJ')l 2 - E~I. (23) It can be seen that one or both of the following conditions: I(~(eJ~')l2/> 3s,
In(eJ°')l 2 ~>es
(24)
always holds. Since G(z) and H(z) are a power
Table 1 Magnitude response characteristics versus coefficient wordlength for Example 3 Filter structure
Ninth-order direct
Two fifth-order cascade
Two fifth order weighted cascade
Number of bits (including sign)
Passband ripple in dB
Stopband attenuation in dB
Passband ripple in dB
Stopband attenuation in dB
Passband ripple in dB
Stopband attenuation in dB
0.100 0,100 0.101 0.100 0.101 0.102 0.104
102.0 100.5 98.7 80.5 82.2 70.8 63.3
0.100
82.0
0.101
88.0
0.101 0.107 0.108 0.111 0.125
81.8 81.1 82.0 78.5 67.9
0,102 0.108 0.109 0.112 0.126
84.3 ~ 84.3 a 84.3 ~ 84.2" 69.3 ~
32 22 18 17 16 15 14 13 12 11 10
Weighting coefficient value was 0.5 2 ~4 corresponding to 16-bit wordlength. Vol. 29, No. 2, November1992
M. Renfors et al. / Doubly complementary IIR digital filters
180
complementary filter pair, we arrive at 1 - 6 s - es ~
(25)
Since the stopband ripple values are in practice very small, it can be seen that the weighted cascaded pair {G(z), H(z)} is practically magnitudecomplementary [5]. Comparing with straightforward cascade designs ( c = d = 0 . 5 ) which are exactly magnitude-complementary, the stopband attenuations of both filters have been increased by 6 dB. It can also be shown that the phase responses of G(z) and H(z) are identical except for some parts of the stop band region of H(z). This follows because in (19) and (22) the terms [Al(z)+ Az(Z)] 2 and A~(z)Az(z) have identical phase responses. Therefore, the phase difference of G(z) and H(z) is if I[At(e j°') + Az(eJ'°)]21< IA~(eJ~°)Az(eJ'°)l. Otherwise the phase responses are identical.
Implementation In the case of complex conjugate allpass filters, the implementation of weighted cascaded filters is very simple. Here the allpass filters Aj (z) and A2(z) include complex multipliers fl and /3* [11]. The complex weights can be combined with these multipliers, and the transfer function of (19) can be implemented just like a cascade of two filter stages, each realized as a parallel connection of complex conjugate allpass filters as described in [11]. With real allpass filters, the weights of the two stages have to be combined as in the latter form of (19) in order to find a real coefficient implementation. Figure 5 shows the implementation of an
l
Fig. 5. Structure for an approximately magnitude complementary filter pair using real allpass subfilters. signal proce~ing
approximately magnitude-complementary filter pair. Note that in comparison with the implementation of a single weighted cascaded filter, one of the allpass subfilter blocks has to be repeated. However, this is unavoidable when implementing magnitude-complementary filter pairs. The only hardware penalty when using the optimized structure proposed in this paper that, for the center taps, true multipliers have to be used instead of simple scaling by ½. The allpass transfer functions A l(z) and A2(z) are interchangeable in the structure, so it is advantageous to select them in such a way that the order of A2(2") is lower than that of A~(z).
4. Concluding remarks In this paper we have examined the properties of weighted complementary IIR filters utilizing either real or complex conjugate allpass subfilters. The most important applications of such filter-pairs included here are in the design and implementation of multilevel filters, gain equalizers, weighted cascaded filters and improved magnitude-complementary filter pairs. We have shown how to design two-level filters, approximating two arbitrary constant values in two disjoint frequency bands, using standard routines for elliptic filter design. We have also shown that certain types of gain equalizers can be realized using a parallel connection of two allpass filters with suitable weighting constants. As an example, we considered the design of a filter which compensates for the amplitude error of a zero-order hold circuit. We also considered realizing a digital filter or filter pair using a cascade of two identical (usually elliptic) filter stages. This makes it possible to realize a magnitude complementary filter pair. Another advantage of this approach is the improved finite wordlength performance, especially much lower coefficient sensitivity when compared with direct elliptic designs realized as parallel connection of allpass filters. Thus, cascaded realizations are attractive in many applications, and they are
M. Renfors et al. / Doubly complementary I1R digital filters
competitive with other low sensitivity low noise structures, even though the overall filter order is somewhat higher than that of the direct elliptic design. We have shown that the selectivity of such a cascaded filter or filter pair can be improved significantly at the cost of one or two simple weighting coefficients. The improvement is 6 dB in the stopband attenuation(s) of the filter(s).
Acknowledgments This work was supported in part by the Academy of Finland and in part by a University of California MICRO grant with matching supports from Granger-Telettra, Inc.
References [ 1] R. Ansari, "Multi-level IIR digital filters", IEEE Trans. Circuits and Systems, Vol. CAS-33, March 1986, pp. 337 341. [2] R. Ansari and B. Liu, "A class of low-noise computationally efficient recursive digital filters with applications to sampling rate alteration", IEEE Trans. Acoust. Speech Signal Process., Vol. ASSP-33, February 1985, pp. 90 97.
181
[3] A. Fettweis, H. Levin and A. Sedlmeyer, "Wave digital lattice filters", lnternat. J. Circuit Theory AppL, Vol. 2, June 1974, pp. 203 211. [4] P.A. Regalia and S.K. Mitra, "Tunable digital frequency response equalization filters", IEEE Trans. Acoust. Speech Signal Process., Vol. ASSP-35, January 1987, pp. 118 120. [5] P.A. Regalia and S.K. Mitra, "A class of magnitude complementary loudspeaker cross-overs", IEEE Trans. Acoust. Speech Signal Process., Vol. ASSP-35, November 1987, pp. 1509 1516. [6] P.A. Regalia, S.K. Mitra and P.P. Vaidyanathan, "The digital allpass filter: A versatile signal processing building block", Proc. IEEE, Vol. 76, January 1988, pp. 19 37. [7] M.K. Renfors and T. Saramfiki, "A class of approximately linear phase digital filters composed of allpass subfilters", Proc. IEEE Internat. Symp. on Circuits and Systems, San Jose, CA, May 1986, pp. 678 681. [8] M. Renfor and E. Zigouris, "Signal processor implementation of digital all-pass filters", IEEE Trans. A coust. Speech Signal Process., Vol. ASSP-36, May 1988, pp. 714 729. [9] T. Saram~iki, "On the design of digital filters as a sum of two allpass filters", IEEE Trans. Circuits and Systems, Vol. CAS-32, November 1985, pp. 1191 1193. [10] P.P. Vaidyanathan, S.K. Mitra and Y. Neuvo, "A new approach to the realization of low-sensitivity IIR digital filters", IEEE Trans. Acoust. Speech Signal Process., Vol. ASSP-34, April 1986, pp. 350 361. [11] P.P. Vaidyanathan, P.A. Regalia and S.K. Mitra, "Design of doubly-complementary IIR digital filters using a single complex allpass filters, with multirate applications", IEEE Trans. Circuits and Systems, Vol. CAS-34, April 1987, pp. 378 389.
Vol. 29, No. 2, November1992
173
Generalized doubly complementary IIR digital filters Markku Renfors Department of Electrical Engineering, Tampere University of Technology, P.O. Box 527, 33101 Tampere, Finland Sanjit K. Mitra (Member EURASIP) Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106, USA P h i l i p A. R e g a l i a Department of Electronics & Communication, Institut National des Telecommunications, 9 Rue Charles-Fourier, 91011 Evry Cedex, France Yrj6 Neuvo (Member EURASIP) Department of Electrical Engineering, Tampere University of Technology, P.O. Box 527, 33101 Tampere, Finland Received 25 July 1991 Revised 2 March 1992
Abstract. A certain class of frequency selective digital IIR filters, including the classical Butterworth, Chebyshev and elliptic filters, can be realized as a parallel connection of two real or complex conjugate allpass filter. In this paper, a generalized structure where the allpass branches include arbitrary real or complex weighting constants is discussed. Applications of these generalized doubly complementary filters include gain equalizers, digital filters with multilevel magnitude respone, and sharpening the amplitude respone of digital filters realized as a cascade of identical filter stages. Zusammenfassung. Eine bestimmte Klasse von frequenzselektiven digitalen IIR-Filtern, einschlieBlich der klassischen Butterworth, Tschebyscheff und ellipischen Filter, kann als Parallelkombination von zwei reellen oder konjugiert komplexen AllpaB Filtern realisiert werden. In dieser Arbeit wird eine verallgemeinerte Struktur diskutiert, in der die AllpaB-Zweige reelle oder komplexe Gewichtsfaktoren enthalten. Anwendungen dieser verallgemeinerten doppel-komplement/iren Filter sind Entzerrer, digitale Filter mit mehrstufigen Frequenzg~ingen und die Flanken-Versteilerung bei Frequenzg~ingen von Filtern, die als Kaskade identischer Filterstufen realisiert sind. R6sum6. Une certaine classe de filtres num6riques IIR s~lectifs en fr6quence, dont les filtres classiques de Butterworth, Chebyshev et les filtres elliptiques peuvent ~tre obtenus par une connection en parall/~le de deux filtres passe-tout r~els ou conjugu6s complexes. On pr6sente dans cert article une structure g6n6rale compos6e de branches passe-tout pond6r6es par des constantes arbitraires r6elles ou conjugu6es complexes. Les applications de ces filtres g6n6ralis6s doublement compl6mentaires incluent les 6galisateurs de gain, les filtres num6riques gt r6ponse d'amplitude ~t plusieurs niveaux, et l'am61ioration de precision de la r6ponse en amplitude des filtres numbriques r6alisbe par la mise en cascade de plusieurs &ages de filtres identiques. Keywortls. Power complementary IIR filters; parallel allpass realizations; multilevel IIR filters; gain equalizers.
Correspondence to: Professor S.K. Mitra, Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106, USA. Fax: (805) 893-3262. 0165-1684/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved
M. Renfors et al. / Doubly complementary IlR digital filters
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1. Introduction
2. Properties
Digital allpass filters are useful signal processing building blocks and are used in a variety of applications [6]. In particular, certain filter types, such as classical odd-order lowpass designs, can be realized as a parallel connection of two real or complex conjugate allpass filters [2, 3, 9-11]. These methods have significant advantages, such as modular structure, low roundoff noise power and reasonably low coefficient sensitivity. In addition, a complementary filter pair can be implemented at the cost of a single filter. This is particularly useful for implementing filter banks [ 1]. In this paper, we consider a generalization of the above concepts and define the weighted complementaryfilter pair, obtained by using different weights in the two allpass branches:
2.1. Complementary properties
G(z) = creAm(z)+ c2A2(z),
(la)
H(z) = emA m(z) - c2A2(z),
(1 b)
as illustrated in Fig. 1, where Al(z) and A2(z) are two real or complex allpass filters, and cl and e2 are two real or complex constants. In order to normalize the amplitude responses, we assume in most cases that the weighting constants satisfy
Ic, I + Ic21 = 1.
(2)
The classical lowpass designs mentioned are obtained as special cases of (1) with em=e2=0.5. We first derive the properties of the weighted complementary filter pair, including conditions for different types of complementary properties, and their corresponding amplitude response characteristics. Next, we outline some applications of the weighted complementary filters including multilevel designs, amplitude equalizers and weighted cascaded designs. c!
G(z) + H(z) = 2cmA. (z).
(3)
This shows that the filter pair is essentially allpass complementary, i.e. the frequency respone of the sum of the two transfer functions has a constant magnitude. In conventional filter designs, cm= 0.5 and the filter-pair is strictly allpass complementary. Adding up the squared-magnitude responses we get IG(eJ'°)lz + IH(eJ°')l 2= 2(le~l2 + Ic212),
(4)
implying that the filter pair is essentially power complementary. Again in conventional filter designs, the right-hand side of (4) is unity. From (4) it also follows that the transfer functions G2(z) and H:(z) are magnitude complementary, i.e. the sum of their magnitude responses is a constant.
2.2. Amplitude response characteristics From (1) is can be seen that the amplitude responses of the filter pair satisfy
Ile, I- Ic211~< pa(eJ~)l ~ le, I+ le21,
(5a)
IIc, I - Ic2ll ~< In(eJ'°)I ~< Ic,I + Ic21.
(5b)
Clearly, for highly selective filters, the magnitudes of Cmand c: must be very close to each other. For example, with cm= 0.51 and c2 = 0.49, the maximum value is unity and the minimum value corresponds to 34 dB of attenuation. In many cases, the filter pair of (1) is derived from a conventional (say classical lowpass) prototype design of the form G(z) = ½[Am(z) + A2(z)],
(6a)
/t(z) = ½[Am(z) - A2(z)].
(6b)
G(z) D
IN H(z) Fig. 1. Weighted complementary filter pair. Signal Processing
From (1) it follows that
Next we examine how the amplitude responses of the prototype design and the weighted complementary pair are related when real or complex conjugate weighting constants are used.
M. Renfors et al. / Doubly complementary HR digital filters
Real weights. In the case of real weighting con-
stants in (1) we obtain from (1) and (6) (7a)
H(z) = (c, - c2)G(z) + (c, + c2)H(z).
(7b)
Since the prototype filters are power complementary and allpass complementary and, consequently, their frequency responses are in phase quadrature at all frequencies, we can show that
c21Gl(eJ°')l 2,
(8a)
]H(eJ'°)[ 2 = (cl + c2) 2 - 4c,c21Gl(eJ'°)l 2.
(8b)
- c2) 2 -~ 4c,
opposite. It can thus be shown that IG(eJ~°)l =211cr[ IG(eJ'°)l ± [c~lx/-i-L [G(eJo')[2 I,
G(z) = (c, + c2)G(z) + (cj - c2)H(z),
IG(eJ~°)[2 = (c,
175
The above shows that there is a one-to-one mapping between the amplitude responses of the prototype filter pair and those of the weighted complementary filter pair. This implies that if the prototype design has equiripple passband(s) and stopband(s), then the corresponding frequency bands of the weighted complementary pair are also equiripple. Assuming that both c~ and c2 are positive, the maximum and minimum amplitude response values 1 and 0 are mapped to the values cj + c2 and Icj- c2l, respectively. Notice that with the normalization of (2), the maximum amplitude response value is unity.
IH(eJ'°)l = 21 [c,[ IG(eJ'°)l ~= ]Cr[x/1 --[G(eJC°)[ 2 [. (1 lb)
3. Applications 3.1. Two-level filters [1]
We consider first the odd-order case. Let the amplitude response specifications for the two-level filter be given by DI-~j<~IG(eJ'°)I<~Dj
for 0~
(12a)
D2<~[G(eJ'°)I<~D2+82
for c02~
(12b)
Such a filter can be derived from an odd-order elliptic prototype filter by selecting the values of the real weighting constants appropriately. The specifications for the prototype are given in the form l-Sj~
c, = 0.5 e j* = c~+jci,
(9a)
c2 = 0.5 e -jo = cr -jci.
(9b)
G(z) = (cr + jci)Al(z) + (cr-jci)A2(z)
= 2crG(z) - 2c~/~(z). J
(10)
Since the frequency responses of G(z) and/4(z) are in phase quadrature at all frequencies, the phase responses of G(z) and H(z)/j must be the same or
(13b)
c2 = / ( D I - D 2 ) .
(14)
The passband and stopband ripples can also be determined from (8a):
~,=1~z = ~/
Now we can write
(13a)
Using (8a), the values of the weighting constants can be determined: I
that the weights in (1) are complex conjugates of each other and their magnitudes are 0.5, i.e.,
for0~
for co2~
Cl = ~ ( O l + D 2 ) ,
Complex conjugate weights. In this case we assume
(1 la)
~/fl
6,(2D, - ~,) D~_ D 2 ,
.~-- $ Dr-D2
•
(15a)
(15b)
Now the elliptic prototype filter is completely specified. The minimum-order elliptic filter satisfying these specifications is then designed and the allpass filters A j(z) and A2(z) are determined from the poles of the elliptic filter. Finally, the weighting constants are computed from (14). In a similar manner, two-level filters using a bandpass or multiband prototype filter can be designed. Vol. 29, No. 2, November1992
M. RenfiJrs et al. / Doubly complementary I1R digital filters
176
It is obvious that if the values of the weighting constants el and c2 are interchanged, the magnitude response remains the same but the phase response is changed. One of the choices may give considerably better phase response or group delay response than the other.
EXAMPLE 1. Consider the design of a two-level filter satisfying the specifications of (12) with c0~= 0.3n, o92= 0.33~, D~ =0.75, D2 =0.25 and 81 = &2= 0.01. The maximum passband and Stopband ripple values for the elliptic prototype filter, obtained from (15), are &~=0.015 and 32=0.101. The estimated minimum order for the elliptic filter is 4.7 so that a fifth-order filter is required. The amplitude response plot for the designed two-level filter are shown in Fig. 2(a). Figure 2(b) shows the group delay responses for both choices of the weighting constants.
I
auJ 0 . 7 5
'
I
t.-- 0 . 5 0 Z C9 < ~;
0.25
tr
0.01
n~ ¢.r u.i ,,,
In the even-order case, it is necessary to use complex conjugate weighting coefficients in order to obtain a real-coefficient overall transfer function. It is possible to determine the values cr and ci in (11 a), and the passband and stopband ripple values for the prototype design such that the specifications of (12) are satisfied. However, the behavior of the amplitude respone in this case is no longer monotonic in the transition band.
3.2. Amplitude equalizers The weighted filter can also be used as a gain equalizer [4]. In this case, the amplitude response of the filter is desired to approximate a certain desired function D(og) on given frequency band(s). To illustrate this application, consider the case where the desired function is monotonically increasing or decreasing, i.e. of lowpass or highpass type, and the frequency band is [0, rt]. We consider here only odd-order designs using real allpass filters. In this case, the weighting constants are computed as in (14) from the maximum and minimum values of the desired function:
c,=½[D(O)+D(n)], I.-
A
L
i " 1
I
i
e2=½[D(O)-D(x)].
(16)
Using (8a), the desired amplitude response for the prototype filter with weights c~ = 0.5 and c2 = -4-0.5 can be computed using
t
0
~ - -
D2(,~)
V ~-0.01 :~
I
~
0
I
0.2~
(a)
I
m
0.4n
I
m
0.6w
I
I
0.8w
I 1
FREQUENCY iii ._1 0,,.. ca} Z >,,a: _1 LLI Cb ¢,
(b)
i
0 rr r5
'
40
1
l ~ I ~ l ,
20
J
~b(~o)=2 cos-'(D(¢o)). I
0
0.2'n"
).4~
m
I
i
0.6~
L
(18)
L
O.8Tt
FREQUENCY
Fig. 2. Responses for the two-level design of Example 1. signal Processing
This desired amplitude function satisfies the condition 0~
This phase function is to be approximated by the allpass filter A(z)=Al(z)/A2(z). The phase approximation problem can be solved using a
M. Renfors et al. / Doubly complementary I1R digital fihers
Remez-type optimization algorithm proposed in [7]. In addition to the desired function, this algorithm utilizes a weight function W(o9). This weight function is computed from the specifications in such a way that an equiripple solution for the error function W(o9)(~b (o9) - arg[Al(eJ'°)/A2(eJ°')]) results in equiripple error for the amplitude response of the weighted filter. Finally, the resulting allpass filter A(z) splits into two parts Al(z) and 1/A2(z) in such a way that the allpass functions of the two branches, A~(z) and A2(z), are stable. There are again two choices for combining the weighting constants and the allpass subfilters which result in the same amplitude response but different phase responses. The above procedure can be extended in a straightfoward manner to the case where the desired amplitude response is a unimodal function with D ( 0 ) = D ( ~ ) . In this case, a bandpass-type prototype filter is used. The weighting constants are computed in a manner similar to that of (16) from the maximum and minimum values of the desired function. Then the desired phase difference can be computed using (17) and (18). The techniques described above can also be used in cases where the desired function is to be approximated only on a certain subset of [0, rt]. In such designs, the maximum and minimum values of the amplitude response may not always be determined by the specifications. In these cases, the optimum values for the weighting constants can be found by a search procedure.
EXAMPLE 2. Consider the design of a gain equalizer approximating the desired function D(og) = (o9/2)/sin(o9/2). The requirements for the amplitude error is +0.001 or smaller. This equalizer compensates for the amplitude droop of a zeroorder hold circuit. For a full-band gain equalizer, we select D ( 0 ) = 1 and D(rc) = 1.571. Using (16) we obtain the values of the weighting constants c j = 1.2855 and c2 = -0.2855. In this case, a seventh-order filter is needed and the resulting amplitude error is
177
1.60 Q
1.40
k,.-
L9
1.2o
1.00
tv-
I.-au'ltu' n"0.0010 ~0~ <~-0.001
0 0.2r~ 0.4v 0.61r 0.8w 1 FREQUENCY
F i g . 3.
Responses for the gain equalizer of Example 2.
+0.00072. Figure 3 shows the amplitude response plot for the equiripple amplitude error solution obtained through phase appoximation as described above. All the poles of the allpass filters of this design are located on the real axis. If the approximation band is [0, 0.95rt], then a fifth-order filter is sufficient, and the resulting amplitude error is 4-0.00041. For a comparison, optimum IIR filters satisfying the same specifications were designed using the techniques proposed in [7]. In the fullband case, the specifications could just be satisfied using a filter with five poles and two zeros. For the approximation band [0, 0.957t], a filter with two poles and two zeros is sufficient. However, these filters cannot be realized using allpass subfilters. Also, at least in this case, the weighted complementary implementation is very favorable with respect to the finite wordlength properties because all of its poles are located on the real axis.
3.3. Weighted cascaded designs Now consider implementing a digital filter as a cascade of two filter stages. These filter stages are realized as a parallel connection of the same two allpass filters Al(z) and A2(z). The problem of determining the weighting constants optimally for both stages is considered here. Obviously, real weighting constants cannot be used for improving Vol. 29, No. 2, November 1992
M. RenJbrs et aL/ Doubly complementary HR digital filters
178
the selectivity of the filter. However, we shall see that optimally selected complex weighting constants improve the stopband attenuation considerably when compared to that obtained using the straightforward cascade structure. In the following we consider using both real allpass filters (oddorder lowpass prototype) and complex conjugate allpass filters (even-order lowpass prototype). We consider the following transfer function: G(z) = [cA,(z) + c*A2(z)][c*Al(z) + cAz(z)] = Icl2[AZ(z) + AZz(Z)]
(19)
+ 2 Re(c)2A ffz)A2(z).
The realization of above using real allpass subfilters and Icl = 0.5 is shown in Fig. 4. It is quite easy to see that, using real allpass filters and complex conjugate weighting constants, (19) is the only way to get a real coefficient overall transfer function. With complex conjugate allpass filters, different complex conjugate weights could be used for the two stages, but the best results are obtained by selecting the coefficients as in (19). The remaining problem is to determine the complex weighting constant c optimally. We can treat the overall filter as a cascade of two stages of the form of (10). As the stages have complex conjugate weights, the two components in (1 i a) are in phase in one of the stages and in opposite phases in the other stage. This means that the overall amplitude response is a product of two terms of the form of ( l l a ) and, for each ¢o, one of the terms has a '+' sign and the other one has a ' - ' sign. Assuming that Icl =0.5, we now obtain ]G(eJ°~)] = [IG(eJ'°)l2 - 4[Im(c)]21,
(20)
1
4
I
Fig. 4. Structure for the weighted cascaded filter using real allpass subfilters. Signal Processing
where G(z)=[Affz)+A2(z)]/2 is the prototype filter. From (20) it is easy to see that the optimum value for c is obtained by selecting 4[Im(c)] 2= ~ / 2 , where ~s is the stopband ripple of G(z). Therefore the optimum selection is Im(c) = 6s/2x/2.
(21)
The resulting stopband ripple of the overall filter is 6s = 6s2/2, i.e. the stopband attenuation is 6 dB higher than that of the direct cascade design. The passband maximum and minimum values are 1 - ~s and ( 1 - ~p)2_ ~s, respectively, where ~p is the passband ripple of the prototype filter G(z). Typically, the value of ~s is so small that its effect on the passband response can be ignored, and the passband amplitude response is practically the same as that of the cascade design. 3. An IIR filter, with passband and stopband edges at 0.1 x and 0.15x, passband ripple of at most 0.1 dB, and a minimum stopband attenuation of at least 80 dB, is to be designed. In the case of direct elliptic designs, the minimum filter order is eight which results in 87.5 dB stopband attenuation with 0.1 dB passband ripple. This eighth-order filter can be realized using complex allpass filters. A ninth-order elliptic filter gives 102 dB stopband attenuation with 0.1 dB passband ripple and it can be realized as a parallel connection of two allpass filters. The specifications can also be satisfied by using a cascade of two fifth-order elliptic subfilters. The direct cascade gives 82 dB stopband attenuation, whereas the attenuation increased to 88 dB with optimized weighting constants. Even though the required overall order is somewhat higher than in direct elliptic designs, the cascaded designs are useful because they have lower coefficient sensitivity if similar allpass filter structures are used in both cases. Other advantages of the cascaded design are lower passband group delay variation and slightly lower roundoff noise level. To illustate these points, we consider a ninthorder direct elliptic design and cascaded designs with fifth-order subfilters, in which both subfilters EXAMPLE
M. Renfors et al. / Doubly complementary HR digital filters
can be implemented using real allpass filters. We assume that the allpass filters are implemented using cascades of first- and second-order sections with direct-form coefficients [8]. Table 1 shows the passband ripple and stopband attenuation for different designs as function of the coefficient wordlength. We consider direct rounding of the coefficients, without any further optimization. It is well known that filter structures based on parallel connection of allpass filters have much better passband sensitivity than stopband sensitivity, especially when the stopband attenuation is high as in this example [6, 8]. Also in this example, the coefficient wordlength is limited mainly by the stopband behavior. Using the structure of Fig. 4, the weighting coefficient 2 Re(c 2) may need somewhat longer wordlength than the allpass filter coefficients. In this case we used the value 2 Re(c 2) =0.5-2 -14. Using first- and second-order direct-form I allpass filter sections [8], the estimated noise gains are 21.1 dB for the ninth-order filter and 18.6 dB for the cascaded designs. The passband group delay response is in the range [19.9, 93.6] for the ninth-order filter and [16.5, 64.7] for the cascaded designs.
179
Approximately magnitude complementary filter pair
We can also find the complementary lilter to the transfer function of (I 9) : H(z) = [dA,(z) - d*Ae(z)][d*Al(z) - dA2(z)]
A[(z)]
= Idl2[A?(z) +
- 2 Re(d)2Al(z)A2(z).
(22)
In this case, the optimum value for the weighting coefficient d is such that Im(d) = ~s/2x~, where ~ is the stopband ripple of the complementary prototype filter H ( z ) = [ A l ( z ) - A2(z)]/2. Denoting by es the stopband ripple of H(z), we can write IG(eJ~°)l + IHteJ'°)l = IIG(e~'°)t2 - ¢~sI+ tl/~(eJ')l 2 - E~I. (23) It can be seen that one or both of the following conditions: I(~(eJ~')l2/> 3s,
In(eJ°')l 2 ~>es
(24)
always holds. Since G(z) and H(z) are a power
Table 1 Magnitude response characteristics versus coefficient wordlength for Example 3 Filter structure
Ninth-order direct
Two fifth-order cascade
Two fifth order weighted cascade
Number of bits (including sign)
Passband ripple in dB
Stopband attenuation in dB
Passband ripple in dB
Stopband attenuation in dB
Passband ripple in dB
Stopband attenuation in dB
0.100 0,100 0.101 0.100 0.101 0.102 0.104
102.0 100.5 98.7 80.5 82.2 70.8 63.3
0.100
82.0
0.101
88.0
0.101 0.107 0.108 0.111 0.125
81.8 81.1 82.0 78.5 67.9
0,102 0.108 0.109 0.112 0.126
84.3 ~ 84.3 a 84.3 ~ 84.2" 69.3 ~
32 22 18 17 16 15 14 13 12 11 10
Weighting coefficient value was 0.5 2 ~4 corresponding to 16-bit wordlength. Vol. 29, No. 2, November1992
M. Renfors et al. / Doubly complementary IIR digital filters
180
complementary filter pair, we arrive at 1 - 6 s - es ~
(25)
Since the stopband ripple values are in practice very small, it can be seen that the weighted cascaded pair {G(z), H(z)} is practically magnitudecomplementary [5]. Comparing with straightforward cascade designs ( c = d = 0 . 5 ) which are exactly magnitude-complementary, the stopband attenuations of both filters have been increased by 6 dB. It can also be shown that the phase responses of G(z) and H(z) are identical except for some parts of the stop band region of H(z). This follows because in (19) and (22) the terms [Al(z)+ Az(Z)] 2 and A~(z)Az(z) have identical phase responses. Therefore, the phase difference of G(z) and H(z) is if I[At(e j°') + Az(eJ'°)]21< IA~(eJ~°)Az(eJ'°)l. Otherwise the phase responses are identical.
Implementation In the case of complex conjugate allpass filters, the implementation of weighted cascaded filters is very simple. Here the allpass filters Aj (z) and A2(z) include complex multipliers fl and /3* [11]. The complex weights can be combined with these multipliers, and the transfer function of (19) can be implemented just like a cascade of two filter stages, each realized as a parallel connection of complex conjugate allpass filters as described in [11]. With real allpass filters, the weights of the two stages have to be combined as in the latter form of (19) in order to find a real coefficient implementation. Figure 5 shows the implementation of an
l
Fig. 5. Structure for an approximately magnitude complementary filter pair using real allpass subfilters. signal proce~ing
approximately magnitude-complementary filter pair. Note that in comparison with the implementation of a single weighted cascaded filter, one of the allpass subfilter blocks has to be repeated. However, this is unavoidable when implementing magnitude-complementary filter pairs. The only hardware penalty when using the optimized structure proposed in this paper that, for the center taps, true multipliers have to be used instead of simple scaling by ½. The allpass transfer functions A l(z) and A2(z) are interchangeable in the structure, so it is advantageous to select them in such a way that the order of A2(2") is lower than that of A~(z).
4. Concluding remarks In this paper we have examined the properties of weighted complementary IIR filters utilizing either real or complex conjugate allpass subfilters. The most important applications of such filter-pairs included here are in the design and implementation of multilevel filters, gain equalizers, weighted cascaded filters and improved magnitude-complementary filter pairs. We have shown how to design two-level filters, approximating two arbitrary constant values in two disjoint frequency bands, using standard routines for elliptic filter design. We have also shown that certain types of gain equalizers can be realized using a parallel connection of two allpass filters with suitable weighting constants. As an example, we considered the design of a filter which compensates for the amplitude error of a zero-order hold circuit. We also considered realizing a digital filter or filter pair using a cascade of two identical (usually elliptic) filter stages. This makes it possible to realize a magnitude complementary filter pair. Another advantage of this approach is the improved finite wordlength performance, especially much lower coefficient sensitivity when compared with direct elliptic designs realized as parallel connection of allpass filters. Thus, cascaded realizations are attractive in many applications, and they are
M. Renfors et al. / Doubly complementary I1R digital filters
competitive with other low sensitivity low noise structures, even though the overall filter order is somewhat higher than that of the direct elliptic design. We have shown that the selectivity of such a cascaded filter or filter pair can be improved significantly at the cost of one or two simple weighting coefficients. The improvement is 6 dB in the stopband attenuation(s) of the filter(s).
Acknowledgments This work was supported in part by the Academy of Finland and in part by a University of California MICRO grant with matching supports from Granger-Telettra, Inc.
References [ 1] R. Ansari, "Multi-level IIR digital filters", IEEE Trans. Circuits and Systems, Vol. CAS-33, March 1986, pp. 337 341. [2] R. Ansari and B. Liu, "A class of low-noise computationally efficient recursive digital filters with applications to sampling rate alteration", IEEE Trans. Acoust. Speech Signal Process., Vol. ASSP-33, February 1985, pp. 90 97.
181
[3] A. Fettweis, H. Levin and A. Sedlmeyer, "Wave digital lattice filters", lnternat. J. Circuit Theory AppL, Vol. 2, June 1974, pp. 203 211. [4] P.A. Regalia and S.K. Mitra, "Tunable digital frequency response equalization filters", IEEE Trans. Acoust. Speech Signal Process., Vol. ASSP-35, January 1987, pp. 118 120. [5] P.A. Regalia and S.K. Mitra, "A class of magnitude complementary loudspeaker cross-overs", IEEE Trans. Acoust. Speech Signal Process., Vol. ASSP-35, November 1987, pp. 1509 1516. [6] P.A. Regalia, S.K. Mitra and P.P. Vaidyanathan, "The digital allpass filter: A versatile signal processing building block", Proc. IEEE, Vol. 76, January 1988, pp. 19 37. [7] M.K. Renfors and T. Saramfiki, "A class of approximately linear phase digital filters composed of allpass subfilters", Proc. IEEE Internat. Symp. on Circuits and Systems, San Jose, CA, May 1986, pp. 678 681. [8] M. Renfor and E. Zigouris, "Signal processor implementation of digital all-pass filters", IEEE Trans. A coust. Speech Signal Process., Vol. ASSP-36, May 1988, pp. 714 729. [9] T. Saram~iki, "On the design of digital filters as a sum of two allpass filters", IEEE Trans. Circuits and Systems, Vol. CAS-32, November 1985, pp. 1191 1193. [10] P.P. Vaidyanathan, S.K. Mitra and Y. Neuvo, "A new approach to the realization of low-sensitivity IIR digital filters", IEEE Trans. Acoust. Speech Signal Process., Vol. ASSP-34, April 1986, pp. 350 361. [11] P.P. Vaidyanathan, P.A. Regalia and S.K. Mitra, "Design of doubly-complementary IIR digital filters using a single complex allpass filters, with multirate applications", IEEE Trans. Circuits and Systems, Vol. CAS-34, April 1987, pp. 378 389.
Vol. 29, No. 2, November1992