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Optimum design of group delay equalizers
Digital Signal Processing 21 (2011) 1–12
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Optimum design of group delay equalizers Mauricio F. Quélhas, Antonio Petraglia ∗ Program of Electrical Engineering, COPPE, EE – Federal University of Rio de Janeiro, CP 68504, CEP 21945-970, Rio de Janeiro, RJ, Brazil
a r t i c l e
i n f o
a b s t r a c t
Article history: Available online 13 July 2010
This paper introduces a novel optimization procedure for the design of group delay equalizers, which is simple to implement, robust and of fast convergence. A pole-zero placement technique is applied to ensure filter stability and direct updating equations. By cascading second-order allpass filter sections to the IIR filter to be equalized, the technique is able to produce group delay responses that are practically as close to a flat response as desired. The new optimization scheme is shown to avoid local minima often encountered by other algorithms used in the design of delay equalizers. Simulation results are presented to verify the effectiveness of the proposed approach. © 2010 Elsevier Inc. All rights reserved.
Keywords: Allpass filters Equiripple group delay response Optimization Phase linearity
1. Introduction It is well known that digital and sampled-data analog (switched-capacitor and switched-current, for instance) filters can be implemented by FIR and IIR networks. Having all poles at the origin of the z-plane, the FIR filters have been widely used, mainly because of their inherent stability, high-speed implementation using FFT and other algorithms, exact linear-phase frequency response and in this case symmetry in the filter coefficients. On the other hand, with their poles adequately positioned inside the unit circle, IIR filters are able to satisfy frequency selectivity requirements with considerably lower computational complexity, and hence employ substantially fewer components (multipliers and adders), than their FIR competitors. But, because their poles lie outside the origin, IIR filters produce nonlinear phase frequency responses. The choice for the class of filters for a specific application often depends on the evaluation of tradeoffs involving frequency selectivity, phase linearity and computational complexity. For example, to accommodate a number of narrow channels in the available frequency band, modern communication systems require sharp selective filters. Here, efficiently designed phase-equalized IIR elliptic filters with sufficiently linear phase response, such as to keep inter-symbolic interference (ISI) and amplitude signal distortion within pre-defined bounds, would replace with advantages computationally expensive linear-phase FIR filters. Various design techniques have been proposed with the objective of producing selective digital networks that exhibit approximately linear phase in the frequency band of interest [1–15]. A widely applied technique is phase or delay equalization, by cascading a selective IIR elliptic filter to an equalizer, generally composed by first- and second-order allpass filters [3,4]. The equalizer is designed with the purpose of achieving approximately linear phase response. Fig. 1(a) presents the block diagram of a phase-equalized IIR filter, and Fig. 1(b) depicts the equalizer as a cascade connection of N /2 second-order allpass sections. The most popular methods are based on finding the best coefficients of the equalizer transfer function that meet given phase/group delay specifications. The cost functions are defined in the frequency-domain [5–11]. In order to guarantee
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Corresponding author. E-mail address: [email protected] (A. Petraglia).
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2010 Elsevier Inc. All rights reserved.
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M.F. Quélhas, A. Petraglia / Digital Signal Processing 21 (2011) 1–12
Fig. 1. Schematic of group delay equalization: (a) cascade connection of the IIR filter and equalizer, and (b) equalizer structure employing N /2 cascaded second-order allpass sections.
filter stability, constrained optimization is evaluated in the minimax [2,11] or least pth [6–8,10] sense, using, for instance, eigenvalue [9,10] and second-order coning program (SOCP) [11] approaches. Time-domain optimization routines [12,13] have been proposed to ease phase constraints at higher attenuation frequency regions. Since constrained optimization becomes highly nonlinear, genetic algorithms have also been applied for allpass equalizer design [18,19]. For improved robustness, some procedures search for the optimal location of the equalizer poles and zeros [14–17], restricting the poles’ radii to be lower than 1, thereby assuring stability. In this work, a novel approach for the equalization of the group delay response is presented. The algorithm is designed with the purpose of dividing the search space, such that inside each part simpler monotonic cost functions are defined, and then quickly minimized. As a result, the algorithm converges to the global minimum very fast, even in designs of high-order allpass equalizers. The developed methodology is shown also to avoid the need for a good starting set of allpass equalizer coefficients. The following sections are organized as follows. In Section 2 the group delay response properties of allpass equalizers composed of cascaded second-order allpass sections are reviewed. Section 3 highlights difficulties often encountered by optimization routines that search the solution in the minimax sense. In Section 4 the proposed equiripple optimization procedure is described. In Section 5 simulation results verifying the effectiveness of the method are shown. In Section 6 concluding remarks are made. 2. Group delay response of allpass filters The transfer function of a second-order allpass filter can be written in terms of its complex conjugated pairs of poles and zeros as
A 2 ( z) =
( z − 1/r · e − j θ )( z − 1/r · e j θ ) . ( z − r · e j θ )( z − r · e − j θ )
(1)
The magnitude frequency response | A 2 (e j ω )| equals 0 dB for 0 ω π , and the phase frequency response A 2 (e j ω ) decreases monotonically from 0, at ω = 0, to −2 × π radians, at ω = π radians/s. Since the phases of the poles and zeros are ±θ in the above equation, the distortion introduced by the equalizer is characterized in the group delay response by a single peak at ω p = θ/π , whose height increases as the pole radius r increases. Therefore, each second-order allpass section has its group delay response completely defined by the pair (θ, r ). For example, a second-order group delay equalizer applied to a bandpass IIR filter having the group delay response shown in solid line in Fig. 2(a) produces the response indicated in dashed line. The result obtained with a fourth-order equalizer is also shown in Fig. 2(a). In general, if an Nth-order equalizer employing N /2 cascaded second-order allpass sections is applied to an IIR filter having N c cutoff frequencies, then the resulting equalized group delay response contains N c + N /2 peaks: one peak near each of the filter cutoff frequencies, ωC i , produced by the poles of the IIR filter closest to the unit circle, and N /2 peaks introduced by the allpass sections. In the bandpass example of Fig. 2(b), we have N c = 2, and hence the equalized responses (in dashed lines) employing 1 and 2 second-order sections have 3 and 4 peaks, respectively. Even if equiripple response is not the optimal one, as stated in [2,10], it is a sufficiently good approximation of the best approach for the group delay response of equalized IIR filters [5,7–9]. The methodology advanced in this paper searches for the allocation of poles and zeros of an equalizer that, when cascaded to the desired IIR filter, produces an equiripple response.
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Fig. 2. Group delay response of (a) an IIR filter (solid line), IIR filter cascaded to a 2nd- and 4th-order equalizer (dashed lines), and (b) of a generalized Nth-order equalizer with N /2 + 2 peaks.
3. Difficulties often encountered by minimax optimization of group delays As the order of the allpass equalizer increases, the surface described by the cost function becomes increasingly crowded with local minima. To illustrate the difficulties often arising in minimax optimization algorithms1 in such cases, even if the starting point is close to the global solution, we consider an 8th-order equalization of a 4th-order IIR elliptic lowpass filter with normalized cutoff frequency of 0.15. The optimum equalized group delay response is shown in Fig. 3(a), which is achieved with the following optimum set of equalizer poles:
Λopt = (θ1 , r1 ); (θ2 , r2 ); (θ3 , r3 ); (θ4 , r4 ) = (0.0574, 0.9187); (0.1734, 0.9174); (0.2930, 0.9193); (0.3998, 0.9309) . The differential group delay, which is the difference between the maximum and the minimum group delays inside the passband, is 2.1 samples. Next, we slightly modify the optimum set Λopt by keeping the phases and replacing the radii by r T , so that the modified set becomes
ΛT = (0.0574, r T ); (0.1734, r T ); (0.2930, r T ); (0.3998, r T ) . This set is then used as the starting point in the minimax optimization algorithm. With r T = 0.90 the algorithm converged to the global solution Λopt . However, when r T is decreased to 0.88, the best equalized group delay response produced by the minimax algorithm is the one presented in Fig. 3(b), which shows a differential group delay of 3.7 samples. With r T = 0.85 the result obtained is displayed in Fig. 3(c), where the differential group delay is 4.8 samples. This example shows that the minimax routine fails even when starting at a close neighborhood of the optimum solution, because of the large number of local minima of the cost function. 1 Widely applied in group delay equalizer designs [2,4,11], such optimization routines search for the optimal solution in the minimax sense, i.e., they minimize the maximum value of the error function.
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Fig. 3. Group delay response (a) with optimum parameters A opt and with rt = 0.90, (b) with rt = 0.88 and (c) with rt = 0.85.
3.1. Presence of a tail in the optimum group delay response For some filter specifications the optimum allpass equalizer may produce a “tail” at the end of the passband. As illustrated in Fig. 4(a) for a 6th-order equalization of an IIR bandpass filter, the resulting group delay response presents the five expected peaks and a tail near each of the two cutoff frequencies. Except for the small fraction of the passband, where the tail occurs, the differential group delay is 6 samples. By comparison, the group delay response achieved by the minimax algorithm is shown in Fig. 4(b), which is not equiripple, has only four peaks, and the differential group delay equals 12 samples. Similar behavior is observed in other filter types such as lowpass, highpass and bandstop. In applications where the tail
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Fig. 4. Equalized group delay responses: (a) carried out manually, showing “tails” close to the cutoff frequencies, (b) minimax solution, without tails, but larger differential delay in most of the passband.
can be tolerated, the optimum equalizer produces, in most of the passband, a differential group delay that is substantially smaller than the one generated by the minimax optimization. The novel optimization scheme described next allows the presence of tails and does not require starting points as close to the optimum solution as do the minimax approach. 4. Equiripple optimization To avoid the aforementioned difficulties of the minimax optimization procedure, the technique described next divides the search for the optimum group delay equalizer into smaller regions, so that simple cost functions can be formulated for each smaller part of the group delay response, thereby reducing the number of local minima in each search. In addition, by relaxing the requirement of equiripple group delay in the whole passband and tolerating, in some cases, the occurrence of a tail in a small fraction of the passband, the algorithm produces a group delay that is entirely or partially equiripple in the frequency band of interest. Parameter adjustments and updating procedures are described next. 4.1. Parameter adjustments Once the equiripple (optimum) group delay response is achieved, each peak introduced by the equalizer has the form shown in Fig. 5(a), where h p is the difference between the height of the last peak (produced by the outmost pole of the IIR filter to be equalized) and the height of the peak of the considered pair of poles, and h v is the difference between the heights of the two valleys closest to the peak under consideration. Hence, when all the N /2 peaks introduced by an Nth-order equalizer satisfy the conditions h p = 0 and h v = 0, the overall group delay response achieves the equiripple form. Before the optimization procedure succeeds, one of the four situations presented in Figs. 5(b), (c), (d) and (e), or a combination of them, may occur. For both Figs. 5(b) and (c) h p = 0, and adjustments of the radius of the corresponding pair of equalizer poles need to be evaluated in order to approach the equiripple response. Figs. 5(d) and (e) show cases in which h v = 0 and, therefore, the phase of the poles must be adjusted.
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Fig. 5. Individual group delay responses (a) equiripple and with anomalies, (b) h p > 0, (c) h p < 0, (d) h v > 0, (e) h v < 0.
Let us now introduce the following set
V p = {τmin, L ; ωmin, L ; τmin, R ; ωmin, R ; τmax, p ; τmax,IIR }
(2)
which contains all the information that can be used to access the status of the group delay response at any stage of the proposed optimization scheme, as well as for the adjustment of the equalizer poles. Depicted in Fig. 6, these parameters are defined as follows: – – – – – –
τmin,L : height of valley to the left; ωmin,L : frequency in which τmin,L occurs; τmin, R : height of valley to the right; ωmin, R : frequency in which τmin, R occurs; τmax, p : height of the considered peak; τmax,IIR : height of the last peak (caused by the filter pole) inside the passband. Both
ωmin,L and ωmin, R are in the interval [0, π ]. Hence, h p and h v may be written as
h p = τmax,IIR − τmax, p ,
(3)
h v = τmin, L − τmin, R .
(4)
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Fig. 6. Definition of parameters pertaining to set V p .
Table 1 Radius adjustment. Radius adjustment rule
τmax,IIR > τmax, p τmax,IIR < τmax, p
hp > 0 hp < 0
r A > r0 r A < r0
Table 2 Phase adjustment. Phase adjustment rule
τmin,L > τmin, R τmin,L < τmin, R
hv > 0 hv < 0
θ A > θ0 θ A < θ0
For an equiripple response we observe, from Figs. 5 and 6, that the two conditions must be satisfied:
Condition 1:
τmin,L = τmin, R , i.e., h v = 0;
Condition 2:
τmax,IIR = τmax, p , i.e., h p = 0.
In order to satisfy these two conditions, the optimization must proceed with the adjustments indicated in Tables 1 and 2 for the phases and radii, respectively, of the equalizer poles. It should be noticed at this point that a pair (h p , h v ) is assigned to each of the N /2 peaks of the group delay response. The radii of poles are adjusted to achieve h p = 0, whereas the phases are adjusted to achieve h v = 0. Therefore, the overall optimization is carried out through N simpler cost functions of one free variable each, defined in a small fraction of the whole search space. It can thus be inferred from the previous considerations together with Fig. 5 that there is only one solution for each h p and h v , and that the signs of h p and h v indicate the gradient direction. The radius r0 and phase θ0 determine the position of a given equalizer pair of poles before the adjustment iteration, and r A and θ A determine their adjusted position. Table 1 indicates that when the group delay peak introduced by a given equalizer pole is lower (higher) than the peak introduced by the outmost filter pole, the radius of the equalizer poles should be increased (decreased). Table 2 indicates that the phase should be changed such as to rotate the pole towards the frequency of the smaller of the two valleys located beside the peak under assessment. 4.2. Update equations The update equations of key parameters applied during the optimization process are derived next. 4.2.1. Phase update Let us first state two ineffective adjustment steps: (i) interchanging the positions of two consecutive pairs of poles only interchanges the positions of the respective peaks; (ii) adjusting the phases of the pair of poles to ωmin, L or πωmin, R merges two successive peaks into one, which always deteriorates the group delay response. We thus conclude that the updated phase θ A must lie between ωmax, p and ωmin,( L , R ) , where ωmax, p is the frequency at τmax, p , and ωmin,( L , R ) is defined as
ωmin,(L , R ) =
ωmin, R , if h v > 0, ωmin,L , if h v < 0.
(5)
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Fig. 7. Weighted average adjustment.
Owing to the second restriction described above, the step size for updating the phase is chosen as a small percentage of this interval, that is
θ A = θ0 + v max,θ · |ωmax, p − ωmin,( L , R ) | · tanh f θ ( V p ) .
(6)
The factor v max,θ lies in the interval between 0 and 1 to increase the convergence speed, while keeping the routine robustness. In the simulations presented in Section 5, v max,θ was chosen as 0.05, and hence the phase change was limited to 5% of the difference between ωmin,( L , R ) and the initial phase ωmax, p . Once the step size limits are determined, the adjusted phase lies between ± v max,θ · |ωmax, p − ωmin,( L , R ) |, depending on the measured h v , and equals the initial value θ0 whenever h v = 0. The function tanh provides a smooth transiting between the two limits, crossing the origin when its argument is zero. The argument of the tanh function,
f θ ( V p ) = kθ ·
hv
τmax, p − min{τmin,L ; τmin, R }
,
(7)
establishes the update direction and speed depending, respectively, on the sign and magnitude of h v (see Figs. 5(d) and (e)), which is a measure of the deviation from the optimum equiripple group delay. The scaling factor kθ exploits the existing tradeoff between convergence rate and robustness, and was chosen as 5 in the illustrative examples shown in Section 5. Observe that the denominator of Eq. (7) is always positive, in accordance to Table 2. If h v = 0, satisfying one of the conditions for the optimum equiripple group delay, then f θ (·) = 0, and hence no update is made by Eq. (6). 4.2.2. Radius update The update equation for the pole radius is
r A = r0 + v max,r · r0 · tanh f r ( V p ) ,
(8)
where the factor v max,r lies between 0 and 1, restricting the step size to be lower than a small percentage of the initial value of the radius. Of course, the adjustment of the pole radius is such that r A < 1, for stability reasons. The function
f r ( V p ) = kr ·
hp
τmax, p − min{τmin,L ; τmin, R }
(9)
is similar to f θ (·) in Eq. (7). In the simulations presented below, the factors v max,r and kr were chosen as 0.01 and 2, respectively, to improve the algorithm’s robustness and convergence rate. 4.2.3. Weighted average adjustment In some cases the step size is too large, and consequently the updated parameter exceeds the optimum value in a given iteration. In other words, the cost function for the new parameter value produces an opposite gradient sign. Therefore, the best solution for this iteration is somewhere in between the initial parameter x0 and the adjusted one x A ,2 as illustrated in Fig. 7. In such cases, when an inversion of the gradient sign occurs, a weighted average of x0 and x A is applied as follows. Let us denote h p ,0 or h v ,0 as the values of h p and h v , respectively, computed at the beginning of a given iteration, and as h p , A and h v , A their values computed after the phase and radius update given by Eqs. (6) and (8), respectively. Two situations may then happen: (i) Either sign(h p , A ) = sign(h p ,0 ) or sign(h v , A ) = sign(h v ,0 ); (ii) Either sign(h p , A ) = sign(h p ,0 ) or sign(h v , A ) = sign(h v ,0 ).
2
x denotes the phase or radius of the considered pole. The sub-indexes 0 and A denote the initial and the adjusted parameters, respectively.
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Fig. 8. Optimization routine sequence.
In situation (i), when there is no change in the gradient sign, after the adjustment the response is closer to the equiripple case, and hence no further adjustment is made. In situation (ii) the sign of the gradient was changed, indicating that the step size should be reduced. In this case the phase is updated by a weighted average of θ0 and θ A , given by
θF =
|h v , A | · θ0 + |h v ,0 | · θ A . |h v , A | + |h v ,0 |
(10)
Similarly, the radius is updated by
rF =
|h p , A | · r0 + |h p ,0 | · r A . |h p , A | + |h p ,0 |
(11)
4.3. Optimization routine The optimization routine is summarized in the flow diagram of Fig. 8. The procedure starts by providing an initial guess. As discussed in Section 4.2.1, the poles must be kept apart from each other, in order to avoid merging two distinct peaks. Therefore, a good initial estimative for the phases is to equally space them inside the passband. The initial radii were all set to 0.9 in the simulations. This approach for the initial guess has proved robust in a large variety of simulations, even for very large equalization order, as shown through illustrative examples in Section 5. The optimization continues until the number of iterations exceeds a given limit, or the maximum deviation
p max = max |θi , F − θi ,0 |; |r i , F − r i ,0 | ,
(12)
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Fig. 9. Group delay equalization of the 5th-order elliptic filter with 2nd- up to 10th-order equalization.
Table 3 Equalization results for the 5th-order elliptic filter. Equalizer order
Number of iterations
Group delay deviation (in samples)
2 4 6 8 10
11 5 5 8 15
31.2 23.2 14.2 7.4 3.4
Fig. 10. Group delay equalization of the 7th-order Chebyshev filter with 2nd- up to 10th-order equalization.
for all parameters, that is, for i = 1, 2, . . . , N /2, is lower than a threshold. In most of the worked problems the algorithm reaches the equiripple response with p max < 2 × 10−5 , in less than 100 iterations. 5. Simulation results The effectiveness of the proposed approach has been observed in the group delay equalizations employing different numbers of equalizer sections. In the following paragraphs lowpass filter examples are shown. The technique can be immediately extended to bandpass, bandstop and highpass filters. The initial parameter estimates were obtained with the algorithm described in [17]. As a first example the proposed algorithm is applied to the group delay equalization of a 5th-order elliptic filter with normalized passband edge frequency of 0.12, passband ripple of 0.5 dB and stopband attenuation of 40 dB. The equalized group delay responses obtained with 1 up to 5 allpass sections are plotted in Fig. 9, and the main results are summarized in Table 3. It should be observed that the optimum (equiripple) response is achieved in all cases within a small number of iterations. By contrast, the FIR linear phase filter that meets the specifications requires a minimum order of 109 and has a group delay of 54.5 samples. A general canonical implementation of this filter performs 55 multiplications per sample, whereas the 6th-order equalized filter computes only 13 multiplications per sample, while producing an average delay whose value is about the same as the group delay of the FIR filter.
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Table 4 Equalization results for the 7th-order Chebyshev filter. Equalizer order
Number of iterations
Group delay deviation (in samples)
2 4 6 8 10
16 13 8 9 17
13.7 9 .9 6 .1 3 .3 1 .5
Fig. 11. Group delay equalization of a 7th-order elliptic filter with 40th-order equalization.
The next example considers a 7th-order Chebyshev filter having passband edge frequency of 0.3, a passband ripple of 0.1 dB and a stopband attenuation of 30 dB. Equalizers having 1 up to 5 allpass sections were designed, and produced the group delay responses displayed in Fig. 10. As in the first example, the equiripple group delay response was obtained after a small number of iterations, as shown in Table 4. To illustrate the ability of the proposed method in escaping local minima, a 40th-order equalizer was designed for a 7th-order elliptic filter with normalized cutoff frequency of 0.2 rad/s, passband ripple of 1 dB, and stopband attenuation of 50 dB. The differential group delay of this filter is rather large (about 100 samples), which complicates the equalization procedure. Shown in Fig. 11, the equalized differential group delay was reduced after 65 iterations to only 6 samples. 6. Concluding remarks This paper presented a novel optimization approach for the design of group delay equalizers. The basic properties of allpass filter group delay responses were discussed, as well as the limitations of the well-known minimax optimization procedure. The method advanced in this paper does not employ global cost functions, and hence avoids the occurrence of local minima. The equalizer is optimized by exploiting the knowledge of the group delay response close to each peak generated by the allpass sections throughout the minimization procedure. Details for the adjustment of the equalizer parameters – phases and radii of the equalizer poles – were presented. As indicated by the illustrative examples, the proposed equiripple optimization approach shows excellent performance in terms of convergence rate and group delay in the filter passband. The technique produces entirely or, at least, partially equiripple responses. In latter cases a tail occurs in a small part of the passband. In these cases the tolerance for the tail may be adjusted, to satisfy the application requirements. The designer has the control over all the steps of the optimization procedure, which is an additional advantage of the proposed method. References [1] L.R. Rabiner, J.F. Kaiser, O. Herrmann, M.T. Dolan, Some comparisons between FIR and IIR digital filters, Bell Syst. Tech. J. 53 (2) (1974) 305–331. [2] A.G. Deczky, Equiripple and minimax (Chebyshev) approximations for recursive digital filters, IEEE Trans. Acoust. Speech Signal Process. ASSP-22 (1974) 98–111. [3] P.A. Regalia, S.K. Mitra, P.P. Vaidyananathan, The digital allpass filter: A versatile signal processing building block, Proc. IEEE 76 (1) (1988) 19–37. [4] A. Antoniou, Digital Filters: Analysis, Design and Applications, McGraw–Hill, 1993. [5] H.W. Schuessler, P. Steffen, On the design of allpasses with prescribed group delay, in: International Conference on Acoustics Speech Signal Process., Apr. 1990, pp. 1313–1316. [6] N. Ko, D. Shpak, A. Antoniou, Design of delay equalizers using constrained optimization, in: Proc. IEEE Pacific Rim Conf. on Comm. Comp. and Signal Process., Victoria, BC, 1997, pp. 173–177. [7] M. Lang, T.I. Laakso, Simple and robust method for the design of allpass filters using least-squares phase error criterion, IEEE Trans. Circuits Syst. II: Analog Digital Signal Process. 41 (1) (1994) 40–48.
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[8] C.C. Tseng, Design of IIR digital all-pass filters using least pth phase error criterion, IEEE Trans. Circuits Syst. II: Analog Digital Signal Process. 50 (9) (September 2003) 653–656. [9] X. Zhang, H. Iwakura, Design of IIR digital allpass filters based on eigenvalue problem, IEEE Trans. Signal Process. 47 (2) (February 1999) 554–559. [10] T.Q. Nguyen, T.I. Laakso, R.D. Koilpillai, Eigenfilter approach for the design of allpass filters approximating a given phase response, IEEE Trans. Signal Process. 42 (9) (September 1994) 2257–2263. [11] S.C. Chan, H.H. Chen, C.K.S. Pun, The design of digital all-pass filters using second-order cone programming (SOCP), IEEE Trans. Circuits Syst. II: Express Briefs 52 (2) (February 2005) 66–70. [12] M. Vucic, G. Molnar, Time-domain synthesis of IIR phase equalizers, in: ICECS IEEE Int. Conf. on Electronics, Circuits and Systems, December 2006, pp. 236–239. [13] M. Vucic, H. Babic, IIR equalizer design based on the impulse response symmetry criterion, in: Proceedings of ISCAS 2003 – IEEE Int. Symp. on Circuits and Systems, vol. 4, Bangkok, Thailand, May 25–28, 2003, pp. 245–248. [14] P. Bernhardt, Simplified design of high-order recursive group-delay filters, IEEE Trans. Acoust. Speech Signal Process. ASSP-28 (1980) 498–503. [15] K. Umino, J. Andersen, R.G. Hove, A novel IIR filter delay equalizer design approach using a personal computer, in: IEEE International Symposium on Circuits and Systems, vol. 1, May 1990, pp. 137–140. [16] M.F. Quelhas, A. Petraglia, M.R. Petraglia, Efficient group delay equalization of discrete-time IIR filters, in: European Signal Processing Conference, Sep. 2004, pp. 125–128. [17] M.F. Quelhas, A. Petraglia, Initial solution for the optimum design of delay equalizers, in: IEEE International Symposium on Circuits and Systems, vol. 4, May 2005, pp. 3587–3590. [18] V. Hegde, S. Pai, W.K. Jenkins, T.B. Wilborn, Genetic algorithms for adaptive phase equalization of minimum phase SAW filters, in: Thirty-Fourth Asilomar Conference on Signals, Systems and Computers, vol. 2, Oct. 2000, pp. 1649–1652. [19] S.U. Ahmad, A. Antoniou, A genetic-algorithm based approach for the design of delay equalizers, in: CCECE, Canadian Conf. on Elect. and Computer Eng., May 2006, pp. 763–766.
Mauricio F. Quélhas received the Electronic Engineer and M.S. degrees in electrical engineering from the Federal University of Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil, in 2003 and 2005, respectively. He is currently working toward the Ph.D. degree at the Program for Post-Graduate Engineering, COPPE, UFRJ. Since 2005 he has been with Petrobras S.A., the oil company owned by the Brazilian government, where he works as an Equipment Engineer in automation and instrumentation. His current research interests are in the design of analog and digital filter approximation functions, optimization, adaptive filtering and system identification applied in the control of chemical processes. He was a visiting researcher at Tampere University of Technology in 2001, 2004 and 2009. Antonio Petraglia received the Engineer and M.Sc. degrees from the Federal University of Rio de Janeiro (UFRJ), Brazil, in 1977 and 1982, respectively, and the Ph.D. degree from the University of California, Santa Barbara (UCSB), in 1991, all in electrical engineering. In 1979, he joined the Faculty of UFRJ as an Associate Professor of electrical engineering, where he served as a Co-Chair in the Department of Electronic Engineering from 1982 to 1984. During the second semester of 1991, he was a Post-Doctoral researcher with the Department of Electrical and Computer Engineering at UCSB. From March 2001 through March 2002 he was a Visiting Scholar with the Electrical Engineering Department at the University of California, Los Angeles. He has been involved in teaching and research activities in the areas of analog and digital signal processing, and in mixed analog–digital integrated circuit design. Dr. Petraglia served as an Associate Editor for the IEEE Transactions on Circuits and Systems-II: Analog and Digital Signal Processing in 2002–2003 and the Analog Integrated Circuits and Signal Processing in 2007–2008. He is currently serving as an Associate Editor for the Circuits, Systems and Signal Processing Journal.
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Digital Signal Processing www.elsevier.com/locate/dsp
Optimum design of group delay equalizers Mauricio F. Quélhas, Antonio Petraglia ∗ Program of Electrical Engineering, COPPE, EE – Federal University of Rio de Janeiro, CP 68504, CEP 21945-970, Rio de Janeiro, RJ, Brazil
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a b s t r a c t
Article history: Available online 13 July 2010
This paper introduces a novel optimization procedure for the design of group delay equalizers, which is simple to implement, robust and of fast convergence. A pole-zero placement technique is applied to ensure filter stability and direct updating equations. By cascading second-order allpass filter sections to the IIR filter to be equalized, the technique is able to produce group delay responses that are practically as close to a flat response as desired. The new optimization scheme is shown to avoid local minima often encountered by other algorithms used in the design of delay equalizers. Simulation results are presented to verify the effectiveness of the proposed approach. © 2010 Elsevier Inc. All rights reserved.
Keywords: Allpass filters Equiripple group delay response Optimization Phase linearity
1. Introduction It is well known that digital and sampled-data analog (switched-capacitor and switched-current, for instance) filters can be implemented by FIR and IIR networks. Having all poles at the origin of the z-plane, the FIR filters have been widely used, mainly because of their inherent stability, high-speed implementation using FFT and other algorithms, exact linear-phase frequency response and in this case symmetry in the filter coefficients. On the other hand, with their poles adequately positioned inside the unit circle, IIR filters are able to satisfy frequency selectivity requirements with considerably lower computational complexity, and hence employ substantially fewer components (multipliers and adders), than their FIR competitors. But, because their poles lie outside the origin, IIR filters produce nonlinear phase frequency responses. The choice for the class of filters for a specific application often depends on the evaluation of tradeoffs involving frequency selectivity, phase linearity and computational complexity. For example, to accommodate a number of narrow channels in the available frequency band, modern communication systems require sharp selective filters. Here, efficiently designed phase-equalized IIR elliptic filters with sufficiently linear phase response, such as to keep inter-symbolic interference (ISI) and amplitude signal distortion within pre-defined bounds, would replace with advantages computationally expensive linear-phase FIR filters. Various design techniques have been proposed with the objective of producing selective digital networks that exhibit approximately linear phase in the frequency band of interest [1–15]. A widely applied technique is phase or delay equalization, by cascading a selective IIR elliptic filter to an equalizer, generally composed by first- and second-order allpass filters [3,4]. The equalizer is designed with the purpose of achieving approximately linear phase response. Fig. 1(a) presents the block diagram of a phase-equalized IIR filter, and Fig. 1(b) depicts the equalizer as a cascade connection of N /2 second-order allpass sections. The most popular methods are based on finding the best coefficients of the equalizer transfer function that meet given phase/group delay specifications. The cost functions are defined in the frequency-domain [5–11]. In order to guarantee
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Corresponding author. E-mail address: [email protected] (A. Petraglia).
1051-2004/$ – see front matter doi:10.1016/j.dsp.2010.06.001
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2010 Elsevier Inc. All rights reserved.
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Fig. 1. Schematic of group delay equalization: (a) cascade connection of the IIR filter and equalizer, and (b) equalizer structure employing N /2 cascaded second-order allpass sections.
filter stability, constrained optimization is evaluated in the minimax [2,11] or least pth [6–8,10] sense, using, for instance, eigenvalue [9,10] and second-order coning program (SOCP) [11] approaches. Time-domain optimization routines [12,13] have been proposed to ease phase constraints at higher attenuation frequency regions. Since constrained optimization becomes highly nonlinear, genetic algorithms have also been applied for allpass equalizer design [18,19]. For improved robustness, some procedures search for the optimal location of the equalizer poles and zeros [14–17], restricting the poles’ radii to be lower than 1, thereby assuring stability. In this work, a novel approach for the equalization of the group delay response is presented. The algorithm is designed with the purpose of dividing the search space, such that inside each part simpler monotonic cost functions are defined, and then quickly minimized. As a result, the algorithm converges to the global minimum very fast, even in designs of high-order allpass equalizers. The developed methodology is shown also to avoid the need for a good starting set of allpass equalizer coefficients. The following sections are organized as follows. In Section 2 the group delay response properties of allpass equalizers composed of cascaded second-order allpass sections are reviewed. Section 3 highlights difficulties often encountered by optimization routines that search the solution in the minimax sense. In Section 4 the proposed equiripple optimization procedure is described. In Section 5 simulation results verifying the effectiveness of the method are shown. In Section 6 concluding remarks are made. 2. Group delay response of allpass filters The transfer function of a second-order allpass filter can be written in terms of its complex conjugated pairs of poles and zeros as
A 2 ( z) =
( z − 1/r · e − j θ )( z − 1/r · e j θ ) . ( z − r · e j θ )( z − r · e − j θ )
(1)
The magnitude frequency response | A 2 (e j ω )| equals 0 dB for 0 ω π , and the phase frequency response A 2 (e j ω ) decreases monotonically from 0, at ω = 0, to −2 × π radians, at ω = π radians/s. Since the phases of the poles and zeros are ±θ in the above equation, the distortion introduced by the equalizer is characterized in the group delay response by a single peak at ω p = θ/π , whose height increases as the pole radius r increases. Therefore, each second-order allpass section has its group delay response completely defined by the pair (θ, r ). For example, a second-order group delay equalizer applied to a bandpass IIR filter having the group delay response shown in solid line in Fig. 2(a) produces the response indicated in dashed line. The result obtained with a fourth-order equalizer is also shown in Fig. 2(a). In general, if an Nth-order equalizer employing N /2 cascaded second-order allpass sections is applied to an IIR filter having N c cutoff frequencies, then the resulting equalized group delay response contains N c + N /2 peaks: one peak near each of the filter cutoff frequencies, ωC i , produced by the poles of the IIR filter closest to the unit circle, and N /2 peaks introduced by the allpass sections. In the bandpass example of Fig. 2(b), we have N c = 2, and hence the equalized responses (in dashed lines) employing 1 and 2 second-order sections have 3 and 4 peaks, respectively. Even if equiripple response is not the optimal one, as stated in [2,10], it is a sufficiently good approximation of the best approach for the group delay response of equalized IIR filters [5,7–9]. The methodology advanced in this paper searches for the allocation of poles and zeros of an equalizer that, when cascaded to the desired IIR filter, produces an equiripple response.
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Fig. 2. Group delay response of (a) an IIR filter (solid line), IIR filter cascaded to a 2nd- and 4th-order equalizer (dashed lines), and (b) of a generalized Nth-order equalizer with N /2 + 2 peaks.
3. Difficulties often encountered by minimax optimization of group delays As the order of the allpass equalizer increases, the surface described by the cost function becomes increasingly crowded with local minima. To illustrate the difficulties often arising in minimax optimization algorithms1 in such cases, even if the starting point is close to the global solution, we consider an 8th-order equalization of a 4th-order IIR elliptic lowpass filter with normalized cutoff frequency of 0.15. The optimum equalized group delay response is shown in Fig. 3(a), which is achieved with the following optimum set of equalizer poles:
Λopt = (θ1 , r1 ); (θ2 , r2 ); (θ3 , r3 ); (θ4 , r4 ) = (0.0574, 0.9187); (0.1734, 0.9174); (0.2930, 0.9193); (0.3998, 0.9309) . The differential group delay, which is the difference between the maximum and the minimum group delays inside the passband, is 2.1 samples. Next, we slightly modify the optimum set Λopt by keeping the phases and replacing the radii by r T , so that the modified set becomes
ΛT = (0.0574, r T ); (0.1734, r T ); (0.2930, r T ); (0.3998, r T ) . This set is then used as the starting point in the minimax optimization algorithm. With r T = 0.90 the algorithm converged to the global solution Λopt . However, when r T is decreased to 0.88, the best equalized group delay response produced by the minimax algorithm is the one presented in Fig. 3(b), which shows a differential group delay of 3.7 samples. With r T = 0.85 the result obtained is displayed in Fig. 3(c), where the differential group delay is 4.8 samples. This example shows that the minimax routine fails even when starting at a close neighborhood of the optimum solution, because of the large number of local minima of the cost function. 1 Widely applied in group delay equalizer designs [2,4,11], such optimization routines search for the optimal solution in the minimax sense, i.e., they minimize the maximum value of the error function.
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Fig. 3. Group delay response (a) with optimum parameters A opt and with rt = 0.90, (b) with rt = 0.88 and (c) with rt = 0.85.
3.1. Presence of a tail in the optimum group delay response For some filter specifications the optimum allpass equalizer may produce a “tail” at the end of the passband. As illustrated in Fig. 4(a) for a 6th-order equalization of an IIR bandpass filter, the resulting group delay response presents the five expected peaks and a tail near each of the two cutoff frequencies. Except for the small fraction of the passband, where the tail occurs, the differential group delay is 6 samples. By comparison, the group delay response achieved by the minimax algorithm is shown in Fig. 4(b), which is not equiripple, has only four peaks, and the differential group delay equals 12 samples. Similar behavior is observed in other filter types such as lowpass, highpass and bandstop. In applications where the tail
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Fig. 4. Equalized group delay responses: (a) carried out manually, showing “tails” close to the cutoff frequencies, (b) minimax solution, without tails, but larger differential delay in most of the passband.
can be tolerated, the optimum equalizer produces, in most of the passband, a differential group delay that is substantially smaller than the one generated by the minimax optimization. The novel optimization scheme described next allows the presence of tails and does not require starting points as close to the optimum solution as do the minimax approach. 4. Equiripple optimization To avoid the aforementioned difficulties of the minimax optimization procedure, the technique described next divides the search for the optimum group delay equalizer into smaller regions, so that simple cost functions can be formulated for each smaller part of the group delay response, thereby reducing the number of local minima in each search. In addition, by relaxing the requirement of equiripple group delay in the whole passband and tolerating, in some cases, the occurrence of a tail in a small fraction of the passband, the algorithm produces a group delay that is entirely or partially equiripple in the frequency band of interest. Parameter adjustments and updating procedures are described next. 4.1. Parameter adjustments Once the equiripple (optimum) group delay response is achieved, each peak introduced by the equalizer has the form shown in Fig. 5(a), where h p is the difference between the height of the last peak (produced by the outmost pole of the IIR filter to be equalized) and the height of the peak of the considered pair of poles, and h v is the difference between the heights of the two valleys closest to the peak under consideration. Hence, when all the N /2 peaks introduced by an Nth-order equalizer satisfy the conditions h p = 0 and h v = 0, the overall group delay response achieves the equiripple form. Before the optimization procedure succeeds, one of the four situations presented in Figs. 5(b), (c), (d) and (e), or a combination of them, may occur. For both Figs. 5(b) and (c) h p = 0, and adjustments of the radius of the corresponding pair of equalizer poles need to be evaluated in order to approach the equiripple response. Figs. 5(d) and (e) show cases in which h v = 0 and, therefore, the phase of the poles must be adjusted.
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Fig. 5. Individual group delay responses (a) equiripple and with anomalies, (b) h p > 0, (c) h p < 0, (d) h v > 0, (e) h v < 0.
Let us now introduce the following set
V p = {τmin, L ; ωmin, L ; τmin, R ; ωmin, R ; τmax, p ; τmax,IIR }
(2)
which contains all the information that can be used to access the status of the group delay response at any stage of the proposed optimization scheme, as well as for the adjustment of the equalizer poles. Depicted in Fig. 6, these parameters are defined as follows: – – – – – –
τmin,L : height of valley to the left; ωmin,L : frequency in which τmin,L occurs; τmin, R : height of valley to the right; ωmin, R : frequency in which τmin, R occurs; τmax, p : height of the considered peak; τmax,IIR : height of the last peak (caused by the filter pole) inside the passband. Both
ωmin,L and ωmin, R are in the interval [0, π ]. Hence, h p and h v may be written as
h p = τmax,IIR − τmax, p ,
(3)
h v = τmin, L − τmin, R .
(4)
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Fig. 6. Definition of parameters pertaining to set V p .
Table 1 Radius adjustment. Radius adjustment rule
τmax,IIR > τmax, p τmax,IIR < τmax, p
hp > 0 hp < 0
r A > r0 r A < r0
Table 2 Phase adjustment. Phase adjustment rule
τmin,L > τmin, R τmin,L < τmin, R
hv > 0 hv < 0
θ A > θ0 θ A < θ0
For an equiripple response we observe, from Figs. 5 and 6, that the two conditions must be satisfied:
Condition 1:
τmin,L = τmin, R , i.e., h v = 0;
Condition 2:
τmax,IIR = τmax, p , i.e., h p = 0.
In order to satisfy these two conditions, the optimization must proceed with the adjustments indicated in Tables 1 and 2 for the phases and radii, respectively, of the equalizer poles. It should be noticed at this point that a pair (h p , h v ) is assigned to each of the N /2 peaks of the group delay response. The radii of poles are adjusted to achieve h p = 0, whereas the phases are adjusted to achieve h v = 0. Therefore, the overall optimization is carried out through N simpler cost functions of one free variable each, defined in a small fraction of the whole search space. It can thus be inferred from the previous considerations together with Fig. 5 that there is only one solution for each h p and h v , and that the signs of h p and h v indicate the gradient direction. The radius r0 and phase θ0 determine the position of a given equalizer pair of poles before the adjustment iteration, and r A and θ A determine their adjusted position. Table 1 indicates that when the group delay peak introduced by a given equalizer pole is lower (higher) than the peak introduced by the outmost filter pole, the radius of the equalizer poles should be increased (decreased). Table 2 indicates that the phase should be changed such as to rotate the pole towards the frequency of the smaller of the two valleys located beside the peak under assessment. 4.2. Update equations The update equations of key parameters applied during the optimization process are derived next. 4.2.1. Phase update Let us first state two ineffective adjustment steps: (i) interchanging the positions of two consecutive pairs of poles only interchanges the positions of the respective peaks; (ii) adjusting the phases of the pair of poles to ωmin, L or πωmin, R merges two successive peaks into one, which always deteriorates the group delay response. We thus conclude that the updated phase θ A must lie between ωmax, p and ωmin,( L , R ) , where ωmax, p is the frequency at τmax, p , and ωmin,( L , R ) is defined as
ωmin,(L , R ) =
ωmin, R , if h v > 0, ωmin,L , if h v < 0.
(5)
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Fig. 7. Weighted average adjustment.
Owing to the second restriction described above, the step size for updating the phase is chosen as a small percentage of this interval, that is
θ A = θ0 + v max,θ · |ωmax, p − ωmin,( L , R ) | · tanh f θ ( V p ) .
(6)
The factor v max,θ lies in the interval between 0 and 1 to increase the convergence speed, while keeping the routine robustness. In the simulations presented in Section 5, v max,θ was chosen as 0.05, and hence the phase change was limited to 5% of the difference between ωmin,( L , R ) and the initial phase ωmax, p . Once the step size limits are determined, the adjusted phase lies between ± v max,θ · |ωmax, p − ωmin,( L , R ) |, depending on the measured h v , and equals the initial value θ0 whenever h v = 0. The function tanh provides a smooth transiting between the two limits, crossing the origin when its argument is zero. The argument of the tanh function,
f θ ( V p ) = kθ ·
hv
τmax, p − min{τmin,L ; τmin, R }
,
(7)
establishes the update direction and speed depending, respectively, on the sign and magnitude of h v (see Figs. 5(d) and (e)), which is a measure of the deviation from the optimum equiripple group delay. The scaling factor kθ exploits the existing tradeoff between convergence rate and robustness, and was chosen as 5 in the illustrative examples shown in Section 5. Observe that the denominator of Eq. (7) is always positive, in accordance to Table 2. If h v = 0, satisfying one of the conditions for the optimum equiripple group delay, then f θ (·) = 0, and hence no update is made by Eq. (6). 4.2.2. Radius update The update equation for the pole radius is
r A = r0 + v max,r · r0 · tanh f r ( V p ) ,
(8)
where the factor v max,r lies between 0 and 1, restricting the step size to be lower than a small percentage of the initial value of the radius. Of course, the adjustment of the pole radius is such that r A < 1, for stability reasons. The function
f r ( V p ) = kr ·
hp
τmax, p − min{τmin,L ; τmin, R }
(9)
is similar to f θ (·) in Eq. (7). In the simulations presented below, the factors v max,r and kr were chosen as 0.01 and 2, respectively, to improve the algorithm’s robustness and convergence rate. 4.2.3. Weighted average adjustment In some cases the step size is too large, and consequently the updated parameter exceeds the optimum value in a given iteration. In other words, the cost function for the new parameter value produces an opposite gradient sign. Therefore, the best solution for this iteration is somewhere in between the initial parameter x0 and the adjusted one x A ,2 as illustrated in Fig. 7. In such cases, when an inversion of the gradient sign occurs, a weighted average of x0 and x A is applied as follows. Let us denote h p ,0 or h v ,0 as the values of h p and h v , respectively, computed at the beginning of a given iteration, and as h p , A and h v , A their values computed after the phase and radius update given by Eqs. (6) and (8), respectively. Two situations may then happen: (i) Either sign(h p , A ) = sign(h p ,0 ) or sign(h v , A ) = sign(h v ,0 ); (ii) Either sign(h p , A ) = sign(h p ,0 ) or sign(h v , A ) = sign(h v ,0 ).
2
x denotes the phase or radius of the considered pole. The sub-indexes 0 and A denote the initial and the adjusted parameters, respectively.
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Fig. 8. Optimization routine sequence.
In situation (i), when there is no change in the gradient sign, after the adjustment the response is closer to the equiripple case, and hence no further adjustment is made. In situation (ii) the sign of the gradient was changed, indicating that the step size should be reduced. In this case the phase is updated by a weighted average of θ0 and θ A , given by
θF =
|h v , A | · θ0 + |h v ,0 | · θ A . |h v , A | + |h v ,0 |
(10)
Similarly, the radius is updated by
rF =
|h p , A | · r0 + |h p ,0 | · r A . |h p , A | + |h p ,0 |
(11)
4.3. Optimization routine The optimization routine is summarized in the flow diagram of Fig. 8. The procedure starts by providing an initial guess. As discussed in Section 4.2.1, the poles must be kept apart from each other, in order to avoid merging two distinct peaks. Therefore, a good initial estimative for the phases is to equally space them inside the passband. The initial radii were all set to 0.9 in the simulations. This approach for the initial guess has proved robust in a large variety of simulations, even for very large equalization order, as shown through illustrative examples in Section 5. The optimization continues until the number of iterations exceeds a given limit, or the maximum deviation
p max = max |θi , F − θi ,0 |; |r i , F − r i ,0 | ,
(12)
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Fig. 9. Group delay equalization of the 5th-order elliptic filter with 2nd- up to 10th-order equalization.
Table 3 Equalization results for the 5th-order elliptic filter. Equalizer order
Number of iterations
Group delay deviation (in samples)
2 4 6 8 10
11 5 5 8 15
31.2 23.2 14.2 7.4 3.4
Fig. 10. Group delay equalization of the 7th-order Chebyshev filter with 2nd- up to 10th-order equalization.
for all parameters, that is, for i = 1, 2, . . . , N /2, is lower than a threshold. In most of the worked problems the algorithm reaches the equiripple response with p max < 2 × 10−5 , in less than 100 iterations. 5. Simulation results The effectiveness of the proposed approach has been observed in the group delay equalizations employing different numbers of equalizer sections. In the following paragraphs lowpass filter examples are shown. The technique can be immediately extended to bandpass, bandstop and highpass filters. The initial parameter estimates were obtained with the algorithm described in [17]. As a first example the proposed algorithm is applied to the group delay equalization of a 5th-order elliptic filter with normalized passband edge frequency of 0.12, passband ripple of 0.5 dB and stopband attenuation of 40 dB. The equalized group delay responses obtained with 1 up to 5 allpass sections are plotted in Fig. 9, and the main results are summarized in Table 3. It should be observed that the optimum (equiripple) response is achieved in all cases within a small number of iterations. By contrast, the FIR linear phase filter that meets the specifications requires a minimum order of 109 and has a group delay of 54.5 samples. A general canonical implementation of this filter performs 55 multiplications per sample, whereas the 6th-order equalized filter computes only 13 multiplications per sample, while producing an average delay whose value is about the same as the group delay of the FIR filter.
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Table 4 Equalization results for the 7th-order Chebyshev filter. Equalizer order
Number of iterations
Group delay deviation (in samples)
2 4 6 8 10
16 13 8 9 17
13.7 9 .9 6 .1 3 .3 1 .5
Fig. 11. Group delay equalization of a 7th-order elliptic filter with 40th-order equalization.
The next example considers a 7th-order Chebyshev filter having passband edge frequency of 0.3, a passband ripple of 0.1 dB and a stopband attenuation of 30 dB. Equalizers having 1 up to 5 allpass sections were designed, and produced the group delay responses displayed in Fig. 10. As in the first example, the equiripple group delay response was obtained after a small number of iterations, as shown in Table 4. To illustrate the ability of the proposed method in escaping local minima, a 40th-order equalizer was designed for a 7th-order elliptic filter with normalized cutoff frequency of 0.2 rad/s, passband ripple of 1 dB, and stopband attenuation of 50 dB. The differential group delay of this filter is rather large (about 100 samples), which complicates the equalization procedure. Shown in Fig. 11, the equalized differential group delay was reduced after 65 iterations to only 6 samples. 6. Concluding remarks This paper presented a novel optimization approach for the design of group delay equalizers. The basic properties of allpass filter group delay responses were discussed, as well as the limitations of the well-known minimax optimization procedure. The method advanced in this paper does not employ global cost functions, and hence avoids the occurrence of local minima. The equalizer is optimized by exploiting the knowledge of the group delay response close to each peak generated by the allpass sections throughout the minimization procedure. Details for the adjustment of the equalizer parameters – phases and radii of the equalizer poles – were presented. As indicated by the illustrative examples, the proposed equiripple optimization approach shows excellent performance in terms of convergence rate and group delay in the filter passband. The technique produces entirely or, at least, partially equiripple responses. In latter cases a tail occurs in a small part of the passband. In these cases the tolerance for the tail may be adjusted, to satisfy the application requirements. The designer has the control over all the steps of the optimization procedure, which is an additional advantage of the proposed method. References [1] L.R. Rabiner, J.F. Kaiser, O. Herrmann, M.T. Dolan, Some comparisons between FIR and IIR digital filters, Bell Syst. Tech. J. 53 (2) (1974) 305–331. [2] A.G. Deczky, Equiripple and minimax (Chebyshev) approximations for recursive digital filters, IEEE Trans. Acoust. Speech Signal Process. ASSP-22 (1974) 98–111. [3] P.A. Regalia, S.K. Mitra, P.P. Vaidyananathan, The digital allpass filter: A versatile signal processing building block, Proc. IEEE 76 (1) (1988) 19–37. [4] A. Antoniou, Digital Filters: Analysis, Design and Applications, McGraw–Hill, 1993. [5] H.W. Schuessler, P. Steffen, On the design of allpasses with prescribed group delay, in: International Conference on Acoustics Speech Signal Process., Apr. 1990, pp. 1313–1316. [6] N. Ko, D. Shpak, A. Antoniou, Design of delay equalizers using constrained optimization, in: Proc. IEEE Pacific Rim Conf. on Comm. Comp. and Signal Process., Victoria, BC, 1997, pp. 173–177. [7] M. Lang, T.I. Laakso, Simple and robust method for the design of allpass filters using least-squares phase error criterion, IEEE Trans. Circuits Syst. II: Analog Digital Signal Process. 41 (1) (1994) 40–48.
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[8] C.C. Tseng, Design of IIR digital all-pass filters using least pth phase error criterion, IEEE Trans. Circuits Syst. II: Analog Digital Signal Process. 50 (9) (September 2003) 653–656. [9] X. Zhang, H. Iwakura, Design of IIR digital allpass filters based on eigenvalue problem, IEEE Trans. Signal Process. 47 (2) (February 1999) 554–559. [10] T.Q. Nguyen, T.I. Laakso, R.D. Koilpillai, Eigenfilter approach for the design of allpass filters approximating a given phase response, IEEE Trans. Signal Process. 42 (9) (September 1994) 2257–2263. [11] S.C. Chan, H.H. Chen, C.K.S. Pun, The design of digital all-pass filters using second-order cone programming (SOCP), IEEE Trans. Circuits Syst. II: Express Briefs 52 (2) (February 2005) 66–70. [12] M. Vucic, G. Molnar, Time-domain synthesis of IIR phase equalizers, in: ICECS IEEE Int. Conf. on Electronics, Circuits and Systems, December 2006, pp. 236–239. [13] M. Vucic, H. Babic, IIR equalizer design based on the impulse response symmetry criterion, in: Proceedings of ISCAS 2003 – IEEE Int. Symp. on Circuits and Systems, vol. 4, Bangkok, Thailand, May 25–28, 2003, pp. 245–248. [14] P. Bernhardt, Simplified design of high-order recursive group-delay filters, IEEE Trans. Acoust. Speech Signal Process. ASSP-28 (1980) 498–503. [15] K. Umino, J. Andersen, R.G. Hove, A novel IIR filter delay equalizer design approach using a personal computer, in: IEEE International Symposium on Circuits and Systems, vol. 1, May 1990, pp. 137–140. [16] M.F. Quelhas, A. Petraglia, M.R. Petraglia, Efficient group delay equalization of discrete-time IIR filters, in: European Signal Processing Conference, Sep. 2004, pp. 125–128. [17] M.F. Quelhas, A. Petraglia, Initial solution for the optimum design of delay equalizers, in: IEEE International Symposium on Circuits and Systems, vol. 4, May 2005, pp. 3587–3590. [18] V. Hegde, S. Pai, W.K. Jenkins, T.B. Wilborn, Genetic algorithms for adaptive phase equalization of minimum phase SAW filters, in: Thirty-Fourth Asilomar Conference on Signals, Systems and Computers, vol. 2, Oct. 2000, pp. 1649–1652. [19] S.U. Ahmad, A. Antoniou, A genetic-algorithm based approach for the design of delay equalizers, in: CCECE, Canadian Conf. on Elect. and Computer Eng., May 2006, pp. 763–766.
Mauricio F. Quélhas received the Electronic Engineer and M.S. degrees in electrical engineering from the Federal University of Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil, in 2003 and 2005, respectively. He is currently working toward the Ph.D. degree at the Program for Post-Graduate Engineering, COPPE, UFRJ. Since 2005 he has been with Petrobras S.A., the oil company owned by the Brazilian government, where he works as an Equipment Engineer in automation and instrumentation. His current research interests are in the design of analog and digital filter approximation functions, optimization, adaptive filtering and system identification applied in the control of chemical processes. He was a visiting researcher at Tampere University of Technology in 2001, 2004 and 2009. Antonio Petraglia received the Engineer and M.Sc. degrees from the Federal University of Rio de Janeiro (UFRJ), Brazil, in 1977 and 1982, respectively, and the Ph.D. degree from the University of California, Santa Barbara (UCSB), in 1991, all in electrical engineering. In 1979, he joined the Faculty of UFRJ as an Associate Professor of electrical engineering, where he served as a Co-Chair in the Department of Electronic Engineering from 1982 to 1984. During the second semester of 1991, he was a Post-Doctoral researcher with the Department of Electrical and Computer Engineering at UCSB. From March 2001 through March 2002 he was a Visiting Scholar with the Electrical Engineering Department at the University of California, Los Angeles. He has been involved in teaching and research activities in the areas of analog and digital signal processing, and in mixed analog–digital integrated circuit design. Dr. Petraglia served as an Associate Editor for the IEEE Transactions on Circuits and Systems-II: Analog and Digital Signal Processing in 2002–2003 and the Analog Integrated Circuits and Signal Processing in 2007–2008. He is currently serving as an Associate Editor for the Circuits, Systems and Signal Processing Journal.