Design of Highly Selective Quasi-Equiripple FIR Lowpass Filters ...

DESIGN OF HIGHLY SELECTIVE QUASI-EQUIRIPPLE FIR LOWPASS FILTERSWITH APPROXIMATELY LINEAR PHASE AND VERY LOW GROUP DELAYThomas Kurbiel, Daniel Alfsmann and Heinz G. GöcklerDigital Signal Processing Group (DISPO), Ruhr-Universität Bochum, 44780 Bochum, Germanyphone: +49 234 32 22106, fax: +49 234 32 02106, email: {kurbiel, alfsmann, goeckler}@nt.rub.de, web: www.dsv.rub.deABSTRACTIn this contribution, the design of approximately linearphase,low delay and quasi-equiripple FIR lowpass filters isresumed with particular emphasis on very low group delay.To this end, a novel unconstrained single-objective multicriteriondesign algorithm is proposed. Results show that themean group delay can be diminished down to roughly 20% ofthat of a corresponding linear-phase FIR filter. Moreover, theapplication of the proposed algorithm to the design of powercomplementary (square-root Nyquist) filters is demonstrated.1. INTRODUCTIONDigital linear-phase (LP) FIR filters have been widely usedin all fields of digital signal processing due to their robuststability, simplicity of implementation, and the absence ofphase distortions. For some specific applications, however,the (normalised) group delay τ g = (N − 1)/2 of standard LPFIR filters, where N represents the filter length, is unacceptablyhigh. Very low group delay is a challenge, for instance,in case of signal processing in digitally implemented hearingaids, where the difference between the direct and the processedsound signals must not exceed a critical value [1].Moreover, long delay is prohibitive, for instance, in two-waycommunication systems for intelligibility, and in feedbackcontrol systems for stability reasons.A first idea of approaching the design of highly selectiveLD digital filters is to use standard recursive or non-recursiveminimum-phase filters, respectively. However, the passbandgroup delay distortion of these filter classes is frequently unacceptable– and group delay equalisation additionally counteractsto our design goal of very low group delay. Moreover,it is suspected that recursive minimum-phase filters (especiallynarrow-band ones) possess a prohibitively high minimumgroup delay being caused by the positive group delaycontributions of the non-zero singularities of IIR filterswithin the z-plane unit circle [2]. Hence, in this investigation,the design of selective digital filters with very low group delayis focused on modified minimum-phase FIR filters, wherethe poles of the above IIR filters are replaced with zeros reducing,in contrast to the IIR counterpart, the passband groupdelay. Part of this group delay advantage is, however, lostagain, since FIR filter orders are known to generally exceedthose of corresponding IIR filters [2].In the past, many attempts to the design of FIR lowpassfilters have been made to approximate a desired magnituderesponse in conjunction with a linear or non-linear phase or,alternatively, with an approximately constant passband groupdelay response [3, 4, 5, 6]. In [7], a procedure for the designof FIR Nyquist filters with low group delay was proposedthat is based on the Remez exchange algorithm. Withthese approaches a filter group delay can always be obtainedthat ranges below the group delay of a corresponding LP FIRfilter. However, the minimisation of the minimum or meanvalue of the passband group delay was of no concern. In contrast,for instance, Lang [8] has shown that his algorithms forthe constrained design of digital filters with arbitrary magnitudeand phase responses have the potential to achieve a considerablereduction of group delay as compared to LP FIRfilters, even for high order FIR filters.The goal of this contribution is to present an alternativeunconstrained approach to the design of highly selective FIRfilters with very low and approximately constant group delay,and quasi-equiripple magnitude response. To this end, anovel multi-criterion objective function is proposed for optimisationallowing for individual weighting of the involvederror functions. This multi-criterion objective function considersthe deviation of the filter magnitude responses in passbandand stopband and, in passband only, both the deviationsof phase from linearity and of group delay from zero.Subsequently, the multi-criterion objective function is introducedand discussed in section II. In section III, three illustrativedesign examples are presented and discussed, while insection IV some conclusions are drawn.2. OPTIMISATION APPROACHThe design of FIR lowpass filters with approximately linearphase and very low group delay is first formulated as a multiobjectiveunconstrained optimisation problem. In practice,however, this problem is solved by combining all multipleobjectives into one scalar objective function and applyingstandard unconstrained optimisation techniques. The independentparameters of the objective functions are the FIR filtercoefficients.2.1 Multi-Objective FormulationThe frequency response of a real-valued digital FIR filter oflength N is characterized by the scalar product [2]:H(e jΩ N−1) =∑n=0where the N-dimensional coefficient vectorh n e −jnΩ = h T c, (1)h T = [h 0 , h 1 ,··· , h N−1 ] (2)comprises the FIR filter parameters to be optimised, and vectorc contains the corresponding exponential terms of (1).For the design of a lowpass filter we define the desiredfilter magnitude response:{M(e jΩ 1, 0 ≤ Ω ≤ Ωp) =(3)0, Ω s ≤ Ω ≤ π,

where Ω p and Ω s represent the filter passband and stopbandcut-off frequencies. Using (3), we set up two related errorfunctions on a dense frequency grid to control the disjointL r -norms of the filter magnitude responses in passband andstopband during optimisation by using FFT techniques:e p (h) =e s (h) =[Kp∑k=0](∣ ( )∣ (1/r∣∣He jΩ k ∣∣ r− M e k)) jΩ , (4)[ ] K/2 (∣ ( )∣ (1/r∣∣H∑ e jΩ k ∣∣ r− M e k)) jΩ . (5)k=K sHence, the frequency grid is defined equidistantly accordingto Ω k = 2πk/K, k = 0,1,...,K − 1, with K even. Moreover,index p refers to passband, where Ω Kp ≈ Ω p , and indexs denotes stopband with Ω Ks ≈ Ω s . The exponent r, restrictedto an even integer number, is set to r = 2 for leastsquares (L 2 ) optimisation, whilst r = 20 is chosen for quasiequiripple(L 20 ≈ L ∞ ) magnitude design [9].Using the group delay frequency response of H(e jΩ ) asgiven by [2]:τ g (e jΩ ,h) = −∂ arg(ejΩ ,h)∂Ω= Re{ h T }Dch T , (6)cwhere Re{.} denotes the real part of a complex-valued quantity,and D = diag[0, 1,··· , N − 1] represents an N × N diagonalmatrix, a third objective function to be used for theminimisation of the maximum absolute value (L ∞ -norm) ofthe passband group delay is readily formulated:∣ ∣∣τge τ (h) = max (e jΩ k ∥,h) ∣ = ∥τ g (e jΩ k ,h) ∥ . (7)0≤Ω k ≤Ω p ∞Obviously, in (7), the underlying passband group delay desiredfunction is zero for all frequencies.As an error function to control phase linearity during theoptimisation process, we use the L r -norm of the deviation ofthe phase curvature from zero. Hence, this fourth objectivefunction is based on the first derivate of the group delay w.r.t.the normalised frequency Ω as follows:e φ (h) =∼[Kp∑k=0[Kp∑k=1∣∂τ g (e jΩ ,h)∂Ω∣∣τ g(e jΩ k∣]r 1/r|Ω=Ω k) ( )∣− τ g e jΩ k−1 ∣∣r] 1/r, (8)where the frequency-continuous definition of e φ (h) is approximatedon the dense grid, as defined in (8). It shouldbe noted that the grid density can be adapted to any value,as required. The (minimum) density we use throughout alldesigns is represented by K = 1024.Finally, to extend the range of applications of our optimisationapproach, we introduce a fifth error function forthe specific design of power complementary (square-rootNyquist) FIR filters potentially with low group delay, beingused for data transmission and in standard quadrature mirrorfilter banks. To this end, we control the L ∞ -norm of the filtermagnitude response at the 3dB-point in the centre of thetransition band, Ω t = (Ω p + Ω s )/2, leading to:e t (h) =∣ |H(ejΩ t)| − √ 1 ∣ ∣∣∣ =2∥ |H(ejΩ t)| − √ 1 ∥ ∥∥∥∞. (9)2Combining the five error functions (4), (5), (7) , (8) and (9)to an objective vector function:e(h) = [ e p (h), e s (h), e τ (h), e φ (h), e t (h) ] T , (10)our design problem is readily formulated as an unconstrainedmulti-objective least squares (L 2 ) optimization problem [10]:minh∈R N ‖e(h)‖2 2 = minh∈R N|e(h)|2 = minh∈R NeT (h)e(h). (11)As a consequence of the non-linear nature of the objectivevector function e(h) in dependence of the coefficient vectorh to be optimised, the minimisation problem (11) is generallymulti-modal and, hence, is expected to possess different(possibly many) local minima. Nevertheless, each optimumor sub-optimum solution of the minimisation process (11)shall represent a balanced compromise between the finallyremaining error contributions of the five individual objectivefunctions of (10) to the squared Euclidean norm of e(h). Theimprovement of an individual objective is always at the expenseof some or all other errors of (10).2.2 Single-Objective FormulationThe multi-objective L 2 /L r /L ∞ -norm based formulation ofour optimisation problem given in section 2.1 is, in compliancewith common practice, transformed to a singleobjectiveform, which is generally better suitable for the applicationof standard optimisation procedures. To this end,the 5-dimensional error vector (10) is pre-multiplied by a 5-dimensional weighting vector according to:e(h) = w T e(h), (12)where w T = [α, β, γ, δ, η], yielding the scalar objectivefunction:e(h)=αe p (h)+β e s (h)+γe τ (h)+δe φ (h)+ηe t (h). (13)Obviously, the individual weights of vector w have to beselected very carefully to obtain a balanced optimum resultof the minimisation procedure. Needless to say that this requiresa lot of procedural experience and sure instinct. Moreover,alternate specific selection of w allows for trading-offindividual objectives against others.The scalar (single-objective) form (13) of the error functionis amenable to differentiation w.r.t. the coefficients hand, hence, to gradient-based optimisation techniques. As aresult, for the optimisation of the vector h of N unknown coefficientssubject to the minimisation of (13), we have usedthe gradient-based problem solver fminunc, which is offeredby the MATLAB Optimization Toolbox. Due to the multimodalnature of our non-linear optimisation problem, we arebound to start each optimisation from a suitable initial set h.Details on finding these initial estimates will be discussed inconjunction with the presentation of the design examples inthe subsequent section.

Table 1: Specification and performance of Ex. A versus ref.Performance Design Ref.Passband Ripple 0.5dBStopband Atten. 50dBFilter Length N 25 20Passband min. τ g 5.9 9.5Passband mean τ g 6.0 9.5Passband max. τ g 6.1 9.53. DESIGN EXAMPLESSubsequently, we present and investigate three different designresults of an FIR lowpass filter of length N = 25 obtainedwith the single-objective multi-criterion optimisationprocedure as described in section 2.2. The passband andstopband cut-off frequencies are fixed to Ω p = 0.4π andΩ s = 0.6π, respectively. To obtain quasi-equiripple magnitudeand phase responses, the exponent r of (4), (5) and (8) isset to r = 20. In addition, the impact of the weighting vectorw T of (12) and (13) on the frequency responses is studied.Each design is compared with a LP or minimum-phase (MP)FIR reference filter with the same magnitude response.3.1 Example A: Approximately Linear PhaseFor Ex. A, the weights are chosen as follows: w T =[1, 15, 0.01, 3, 0]. As a result, we expect a high stopbandattenuation, forced by weight β = 15 of (5), and a highlylinear phase controlled by weight δ = 3 of (8).The design results of Ex. A are presented in Tab. 1 anddepicted in Fig. 1, respectively, along with a LP FIR referencefilter that meets the same magnitude specifications. Despitethe fact that our design requires a higher filter length(excess order of 5), its group delay ranges completely belowthe constant value of τgref = 9.5 of the LP reference filter. Especiallyin passband, the approximately constant group delayamounts to roughly 60% of that of the reference filter only.As to be seen from Figs. 1(a),(c), the zeros effectuatingthe stopband of the transfer function H(z) of our design arelocated slightly off (i.e. inside) the z-plane unit circle. Hence,all zeros within the unit circle (there are no more group delayneutral zeros on the periphery of the unit circle) contributeto the overall group delay by at least a small negativeamount [2]. Obviously, to maintain the specified stopbandrejection, extra zeros inside but close to the unit circle areneeded, which explains the excess filter order of our designcompared to that of the reference filter.Finally, it is somewhat unexpected that the mean groupdelay of τgmean = 6.0 of our design is that low despite thevery low weight of γ = 0.01 put on the group delay errorfunction (7). Hence, we suspect that this error function hasa high overall impact on the scalar objective function (13).Furthermore, according to our experience gained from manydesigns with the above fixed weight vector, it should be notedthat an increase of the filter length predominantly increasesthe stopband attenuation of our design, whereas all other filterproperties remain essentially unchanged.3.2 Example B: Very Low Group DelayFor Ex. B, the weights are chosen as follows: w T =[1, 15, 0.01, 0, 0]. Here, we expect a lowpass filter with20⋅log 10|H(e jΩ )|Im z 0Im z 0τ g(Ω)0−20−40−60−80−1001098765|H(e jΩ )||H ref (e jΩ )|0 0.2 0.4 0.6 0.8 1Ω/π(a) Magnitude responsesτ g (Ω)τ g,ref (Ω)40 0.5 1 1.5Ω/Ω p10.50−0.5(b) Group delay−1−1 0 1 2 3 4 5 6Re z 010.50−0.5(c) Zero plot (design)−1−1 0 1 2 3 4 5 6Re z 0(d) Zero plot (reference)Figure 1: Design of Ex. A (solid) versus reference (dashed)high stopband attenuation, forced by weight β = 15 of (5),and very low group delay in the passband controlled byweight γ = 0.01 of (7).The design results are presented in Tab. 2 and Fig. 2, respectively,along with the corresponding LP FIR referencefilter. The very small magnitude deviation in passband isdeemed the main reason for the excess filter length of theLP reference by 9, although the stopband zeros of our designare again located inside the z-plane unit circle; cf. Fig. 2(a).Comparing group delay, we have obtained τgmean = 0.14·τg ref ,however, at the expense of the loss of phase linearity!Finally we report that, like in this example, very lowgroup delay is only achievable in connection with very small

Table 2: Specification and performance of Ex. B versus ref.Performance Design Ref.Passband Ripple 0.01dBStopband Atten. 52dBFilter Length N 25 34Passband min. τ g 1.7 16.5Passband mean τ g 2.3 16.5Passband max. τ g 4.7 16.520⋅log 10|H(e jΩ )|τ g(Ω)0−20−40−60−80−100181614|H(e jΩ )||H ref (e jΩ )|0 0.2 0.4 0.6 0.8 1Ω/π(a) Magnitude responsesτ g (Ω)τ g,ref (Ω)1210864200 0.5Ω/Ω p1 1.5(b) Group delayFigure 2: Design of Ex. B (solid) versus reference (dashed)magnitude ripple in passband, i.e. reduction of group delayconcurrently decreases passband magnitude deviation.3.3 Example C: Square-Root Nyquist FilterFor Ex. C, the weights are chosen as follows: w T =[1, 1, 0.1, 0, 1]. Hence, we expect a nearly power complementaryFIR square-root Nyquist filter as a result of weightη = 1 of (9) with low group delay controlled by weightγ = 0.1 of (7). To support magnitude symmetry w.r.t. Ω = π,the same weights α = β = 1 are put on (4) and (5).The design results are presented in Tab. 3 and Fig. 3, respectively,along with a MP FIR reference filter that meetsthe same magnitude specifications [11]. Power complementarityis assessed by the logarithmic distortion function [11]∑a dist (e jΩ ) = 20 · log 10K/2k=0[ ∣∣∣H ( )∣e jΩ k ∣∣2 ∣ ( )∣ ∣∣H + e j(Ω k−π) ∣ ] 2 ,as depicted in Fig. 3(c). The reasonably close similarity ofboth designs of this example confirms that our approach alsoTable 3: Specification and performance of Ex. C versus ref.Performance Design Ref.Passband Ripple 0.01dBStopband Atten. 43dBFilter Length N 25 28FB-Distortion max. 0.45dB 0.2dBPassband min. τ g 1.4 1.5Passband mean τ g 1.9 2.0Passband max. τ g 3.5 3.7has the potential to design standard quadrature mirror filters(SQMF) with common quality. Moreover, our designapproach overcomes the restriction to even filter lengths incontrast to the approach by Herrmann & Schüssler [11].3.4 Numerical Properties of the Design AlgorithmDue to the non-linear nature of the single-objective errorfunction e(h), as defined by (12) and (13), the minimisationproblem (11) is multi-modal and, hence, there may existmany local minima, where each (sub-)minimum obtainedhighly depends on the choice of the initial coefficient vectorh 0 . To overcome this deficiency, we adopt a heuristicapproach: We perform each filter design M times, alwaysstarting from a different, randomly selected initial coefficientvector h 0 m , m = 1,...,M, where only the best (sub-)optimumsolution is retained (M = 30...60). In future, this multistartprocedure will be combined with more intelligent strategiesof genetic algorithms.Experience with our design algorithm has shown that,too, the choice of the weight vector w T has a high impact onthe quality of the final optimum solution. On the one hand,there are weight ratios that make the algorithm diverge since,most probably, no (sub-)optimum solution exists for this veryweight vector. This is, for instance, the case, if the group delayrequirement is too tight by imposing a too high weighton (7). On the other hand, minor changes of some individualweights may lead to completely different final solutions,or even to divergence, respectively. All these issues call forfurther detailed investigations in the future.4. CONCLUSIONIn this contribution, an effective and flexible approach to thedesign of highly selective quasi-equiripple FIR filters withvery low group delay and/or approximately linear phase hasbeen proposed. This method is based on the solution of amulti-objective unconstrained optimisation problem allowingfor individual weighting of the involved error functions.To this end, three L 20 -norm and up to two L ∞ -norm errorfunctions are combined to a weighted least squares (L 2 ) optimisationapproach. The impact of both the individual objectivefunctions and the associated weights have been investigated.Furthermore, we have presented a method to solve themulti-objective optimisation problem by using a MATLABroutine, and provided three examples. Finally, we have discussedthe properties of the design algorithm with referenceto the examples.Future investigations will be devoted to the design ofnarrow-band FIR lowpass filters with very low group delay,and their application in uniform, complex-modulated low delayfilter banks. Furthermore, we will compare the poten-