Equalization of Audio Systems using Kautz Filters ... - TKK Acoustics

Audio Engineering SocietyConvention Paper 6767Presented at the 120th Convention2006 May 20–23 Paris, FranceThis convention paper has been reproduced from the author’s advance manuscript, without editing, corrections, orconsideration by the Review Board. The AES takes no responsibility for the contents. Additional papers may beobtained by sending request and remittance to Audio Engineering Society, 60 East 42 nd Street, New York, New York10165-2520, USA; also see www.aes.org. All rights reserved. Reproduction of this paper, or any portion thereof, isnot permitted without direct permission from the Journal of the Audio Engineering Society.Equalization of audio systems using Kautzfilters with log-like frequency resolutionTuomas Paatero and Matti KarjalainenHelsinki University of Technology, Laboratory of Acoustics and Audio Signal Processing, FinlandCorrespondence should be addressed to Tuomas Paatero (tuomas.paatero@tkk.fi)ABSTRACTThis paper presents a new digital filtering approach to the equalization of audio systems such as loudspeakerand room responses. The equalization scheme utilizes a particular infinite impulse response (IIR) filterconfiguration called Kautz filters, which can be seen as generalizations of finite impulse response (FIR) filtersand their warped counterparts. The desired frequency resolution allocation, in this case the logarithmic one,is attained by a chosen set of fixed pole positions that define the particular Kautz filter. The frequencyresolution mapping is characterized by the allpass part of the Kautz filter, which is interpreted as a formalgeneralization of the warping concept. The second step in the actual equalizer design consists of assigningthe Kautz filter tap-output weights, which is then in turn more or less a standard least-square configuration.The proposed method is demonstrated using measured loudspeaker and room responses.1. INTRODUCTIONThe utilization of digital signal processing (DSP)techniques for the equalization of audio systems,such as loudspeaker and room responses or combinedloudspeaker-room responses, have been studiedextensively for more than twenty years. A thoroughlist of references, an overview of methods, andconsiderations on known obstacles are provided byMourjopoulos [15] [16]. Some additional referencesmay also be cited [17, 13, 3, 7, 24, 8]. In particular,the problems associated with equalizer design byinverting an identified system function serve as themain motivation of this paper. The issue of nonminimumphase equalization can be considered asa fundamental question. On the other hand, it isalso important to understand what a chosen equalizerconfiguration tries to accomplish from a broaderpoint of view. The focus here is not particularly onnon-minimum phase design considerations, but morelikely on the undesired phenomenon that systemfunction zeros tend to transform into problematicprominent and ”long-ringing” resonances (poles) inan equalizer design by inverting the system function.Pre-processing, such as (complex) smoothing

PAATERO AND KARJALAINENEqualization using Kautz filters on log-scaleproduce orthonormal tap-output impulse responses.However, a Kautz filter is in fact more genuinelya generalization of the FIR filter and its warpedcounterparts, which will be demonstrated in termsof properties of the allpass filter that constitute thebackbone of the tapped transversal structure of Figure1.The time-domain counterpart of (1), the Kautz filterimpulse response, is given byĥ(n) =N∑w i g i (n), (2)i=0where functions {g i (n)} N i=0 are impulse responses orinverse z-transforms of functions {G i (z)} N i=0 . Themeaning of orthonormality is specified most economicallyby defining the time-domain inner product oftwo (causal) signals x(n) and y(n),(x, y) :=∞∑x(n)y ∗ (n). (3)n=0Now, impulse responses {g i (n)} N i=0 are orthogonal inthe sense that (g i , g k ) = 0 for i ≠ k, and normal,since (g i , g i ) = 1 for i = 0, . . . , N.A reasonable presumption in modeling a real responseis that the poles should be real or occurin complex conjugate pairs. For complex conjugatepoles, an equivalent real Kautz filter formulation [1],depicted in Fig. 2, prevents dealing with complex(internal) signals and filter weights. The allpasscharacteristics of the transversal blocks is restoredby shifting the denominators in Fig. 2 one step tothe right and by compensating for the change in thetap-output blocks. A mixture of structures in Fig. 1and Fig. 2 is used in the case of both real and complexconjugate poles.3. MODELING AND EQUALIZATION USINGKAUTZ FILTERSOur previous proposals for utilizing Kautz filters inrelation to audio and acoustic systems have beenbased on approximative modeling of given (measuredof somehow identified) target responses. Thedesign procedure is then particularly straightforward:a desired pole set is generated with respectFig. 2: One possible realization of a real Kautzfilter, corresponding to a sequence of complexconjugatepole pairs [1]. The normalization termsare p i = √ √(1 − ρ i )(1 + ρ i − γ i )/2 and q i =(1 − ρi )(1 + ρ i + γ i )/2, where γ i = −2RE{z i }and ρ i = |z i | 2 are expanded polynomial coefficientsof the second-order blocks.to the target response h(n) and the approximation(or the model) is composed asĥ(n) =N∑c i g i (n), c i = (h, g i ). (4)i=0That is, the filter weights are the orthogonal expansioncoefficients (Kautz-Fourier coefficients) of h(n)with respect to the choice of basis functions. One ofthe favorable specialities of Kautz filter design comparedto other IIR or pole-zero filter configurationsis that the approximation is independent of rearrangementof the pole set, which implies means forreducing as well as extending the model by pruning,tuning, and appending poles, respectively. In addition,the use of orthogonal expansion coefficientscorrespond to least-square (LS) design with respectto the particular pole set, and as a consequence ofthe orthogonality, the approximation error (energy)is given simply asE = (h, h) −N∑c 2 i , (5)i=0where (h, h) is the energy of the target response.As an alternative to the evaluation of c i = (h, g i )using the inner product formula (3), the Kautz filtertap-output weights are also attained by feeding thesignal h(−n) to the Kautz filter and reading the tapoutputs x i (n) = G i [h(−n)] at n = 0: c i = x i (0).That is, all inner products in (4) are implementedAES 120 th Convention, Paris, France, 2006 May 20–23Page 3 of 12

PAATERO AND KARJALAINENEqualization using Kautz filters on log-scalesimultaneously using filtering. In the case of an FIRfilter this would equal design by truncation of h(n).3.1. Least-square equalization using Kautz filtersAs for any pole-zero model configuration, it is notin general a straightforward operation to invert or“flip” a Kautz model of a system into its inversesystem or equalizing counterpart. In fact, this kindof direct inversion schemes are particularly unattractivefrom the point of view of Kautz filters becauseof the numerator configuration in the transfer function.The art of equalization by direct inversiontechniques [17, 14, 3, 15] will not be considered further,but it is nevertheless interesting to notice thatmany of the consequential requirements posed uponan equalizer are inherently available in a Kautz filterconfiguration, namely,• genuine IIR nature of the equalizer• explicit control over the poles• pure delays and allpass filters as building blocks• linear in the weight-parameters, allowing a wellposedLS configurationThe equalization configuration that is more interestingfrom the Kautz filter point of view can becharacterized as direct equalization: the equalizer(with impulse response h eq (n)) is identified in cascadewith the system h(n) to be equalized, to meetor approximate the ideal equalization targetĥ(n) = h(n) ∗ h eq (n) ≈ δ(n − ∆), (6)where δ(·) is the unit impulse, including a potentialdelay ∆. Also in this direct configuration it isstill possible to use an inverted target response asthe basis of design, directly as in (4) or for exampleto identify just the poles. These alternatives willbe demonstrated in the loudspeaker equalization example.It is noteworthy that some of the knownproblems associated to using an inverted response toconstruct the equalizer, such as “ringing poles” andtruncation effects, do not (necessarily) pass on to theKautz equalizer. This may by itself reduce the needfor complicated pre-processing operations. In addition,even this simple approximation setup is by nomeans constrained to minimum-phase or magnitudeonlyequalization: the target response may obviouslybe constructed to take into account phase as well asmagnitude equalization.A more genuine form of direct equalization is providedby the least-square configuration [14]: thesquare error in the approximation (6) is minimizedwith respect to the equalizer parameters (filter coefficients).In terms of the Kautz equalizer, the tapoutputweights {w i } are optimized according to( )∑min (ĥ(n) − δ(n −w ∆))2 , (7)inwhere the equalizer responseĥ(n) =N∑w i x i (n), x i (n) = g i (n) ∗ h(n), (8)i=0is the Kautz filter response to the input h(n). Usingsystem identification terminology, the equalizationsetup is an output-error configuration with respectto a special choice of model structure. It can evenbe considered as a generalized linear prediction formulation.Furthermore, it is a quadratic LS problemwith a well-defined and unique solution thatis obtained from the corresponding normal equations:if the Kautz equalizer tap-output responsesx i (n) = g i (n) ∗ h(n) are assembled into a “generalizedchannel convolution matrix”⎛⎞x 0 (0) · · · x N (0)x 0 (1) · · · x N (1)S = ⎜⎝.. .. .⎟⎠ , (9)x 0 (L) · · · x N (L)then the normal equations submit to the matrix formS T Sw = s, w = [w 0 · · · w N ] T , (10)where s is the (cross-)correlation vector between thetap-output responses and the desired response d(n),s i = (d, x i ). The matrix product S T S, where (·) Tdenotes transpose of a matrix, implements correlationanalysis of the tap-output responses, (x i , x j ), interm of the inner product (3), where it is presumedthat the Kautz filter responses are real-valued. Inthe case of an impulse as the desired response,d(n) = δ(n − ∆), the correlation vector simply picksthe (∆ + 1)th row of the matrix S,s = [x 0 (∆) · · · x N (∆)] T . (11)AES 120 th Convention, Paris, France, 2006 May 20–23Page 4 of 12

PAATERO AND KARJALAINENEqualization using Kautz filters on log-scale20a) poles16b) allpass block phasesMagnitude / dB100−10−20−30Imaginary part0.50−0.5−1−1 −0.5 0 0.5 1Real partc) tap phases and pole freqs.phase / rad4200 1 2 3angle / radd) ap phase der vs. mag resp−4010 −1 10 0Frequency / Hzphase / rad60402010.5Fig. 3: Magnitude responses of the Kautz filter tapoutputimpulse responses with respect to the proposedlogarithmic distribution of poles.00 1 2 3angle / rad01 2 3angle / radbetween the allpass operator and the correspondingorthonormal filter structure (the Kautz filter) is explainedmore thoroughly in [21]. The negated phasefunction of the allpass filter (solid line in Fig. 4(c))can be interpreted as the frequency scale mapping,whereas its derivative in Fig. 4(d) characterizes thefrequency resolution allocation introduced by thechoice of poles. The upper curve in Fig. 4(d) is thesum of magnitude responses of Fig. 3, which showsquite explicitly that the chosen resolution descriptionis somehow meaningful.The effect of pole radius tuning is demonstrated inFig. 5 using various displaying scales for the allpassphase mappings and corresponding derivatives. It isnoteworthy that the phase responses are quite insensitiveto relatively big changes in the radius profile.The pole distributions are chosen purposefully closeto the unit circle to produce spiking in the resolutionmapping: there is a clear transition from a localizedresolution to a smoother overall description whenthe radius parameter R in Eq. (16) is reduced. Anothernotable aspect is that the smoothness scaleswell (evenly for the whole frequency range) with thechosen rule for the pole radius mapping.4. EXAMPLES OF LS EQUALIZATION USINGKAUTZ FILTERSIn this section we apply the proposed principlesof Kautz filter design for the equalization of loudspeakerand room responses. The examination ofFig. 4: a) The poles set, b) negated phase functionsof 2nd order allpass blocks, c) (allpass) tapoutputphases – overall phase (solid line) is the frequencyscale mapping (circles indicate pole positionson this mapping), and d) phase derivative (lowercurve) w.r.t. sum of Kautz filter tap-output magnituderesponses (scaled).many practical issues would need a more thoroughinvestigation; here we take only some cases that illustratethe characteristics and capabilities of Kautzequalizers.4.1. Loudspeaker equalization, Case 1In this first example of Kautz filter equalization, aninverted minimum-phase target response (with respectto a measured loudspeaker response) is usedboth directly and indirectly to construct the equalizer.The Kautz filter poles are generated in bothcases using a warped counterpart of the BU-method[20] with respect to the inverted target response.The equalizer filter order is chosen to be 38 (18 complexconjugate pole pairs and two real poles). Thepurpose of this example is to demonstrate that twovery different equalizer parametrization schemes,corresponding to Equations (4) and (8), respectively,produce very similar magnitude response equalizationresults, as depicted in Fig. 6.The early part of the measured loudspeaker impulseresponse and the LS equalized response are displayedAES 120 th Convention, Paris, France, 2006 May 20–23Page 6 of 12

PAATERO AND KARJALAINENEqualization using Kautz filters on log-scale1a) pole distributionsb) allpass phases30Imaginary part0.50−0.5−1−1 −0.5 0 0.5 1Real partc) w.r.t. log−scalephase /rad604020phasederivative00 1 2 3angle / radd) log− and dB−scaleMagnitude / dB20100−10phase /rad604020010 0angle / rad (log)phase /rad (db)30201010 0angle / rad (log)−20−30−4010 2 10 3 10 4Frequency / HzFig. 5: Allpass filter characteristics for varying poleradius damping, a) pole sets, b) phase functions andphase derivatives, c) on log-scale, and d) in dBs onlog-scale.Fig. 6: Magnitude responses from bottom totop: measured loudspeaker response, LS equalized(method (8)), pole frequencies, equalized using (4)w.r.t. inverted target, corresponding equalizer responses.in panels (a) and (b) of Fig. 7. In Fig. 7(c) the LSequalizer is designed with respect to a delay ∆ = 12in the target of equalization. The pole set that isgenerated from a minimum phase target responseis not very good at producing pure delay components,which results also in inefficiency in magnitudeequalization (not shown). A somewhat trivial wayto attain better equalization is to include zeros inthe Kautz filter pole set: in Fig. 7(d) the equalizeris equipped with 12 additional poles at the origin,that is, part of the Kautz filter is implemented as anFIR filter substructure.4.2. Loudspeaker equalization, Case 2Another example of Kautz equalizer design is presentedin Fig. 8. It depicts the same loudspeakerresponse as in Case 1 having a relatively non-flatmagnitude response (curve (a)). The response iscorrected by a 24 th order (12 pole pairs) Kautz filterwith logaritmically positioned pole frequenciesbetween 80 Hz and 23 kHz (indicated by verticallines in the middle of the figure) and R = 0.03(see Eq. 16). After low- and high-frequency rolloffcompensations to avoid boosting off-bands of thespeaker, as shown by curve (c), the equalizer filterresulting from Kautz LS equalization shows its mag-nitude response in curve (d). The equalized responseis plotted in curve (e) and as a 1/3-octave smoothedversion in curve (f).Filter orders from 8 up (4 pole pairs) give useful resultsin this case, although the selection of order andpole positions may introduce considerable variationin flatness of the result. Therefore full optimizationrequires a search over sets of poles and filter orders,in spite of the fact that the LS procedure itself alwaysgives optimal tap coefficients for a given fixedorder and pole set.Curve (g) in Fig. 8 demonstrates the effect of Kautzpole radii selection. In this case the poles are settoo close to the unit circle (R = 0.8), thus the frequencyranges around pole frequencies get too muchemphasis. Otherwise, in most cases, the selection ofpole radii is not critical at all. Even very small radii,such as R = 10 −5 , work well in this case.Comparison of curves (e) and (g) explains alsoclearly why LS equalization using the Kautz filterconfiguration behaves favorably with zeros in the responseto be equalized, while exact inversion of a responsewith deep dips results in undesirable peaksand long-ringing decay times in the equalizer [4]. InAES 120 th Convention, Paris, France, 2006 May 20–23Page 7 of 12

PAATERO AND KARJALAINENEqualization using Kautz filters on log-scale1a) loudspeaker response1b) EQ: poles from inv. target700.50.560(g)0050(f)−0.50 50 10010.5c) delay in EQ target−0.50 50 10010.5d) delay and zeroslevel [dB]4030(e)(d)0020(c)−0.50 50 100−0.50 50 100Fig. 7: Early part of time-domain responses: a) measuredloudspeaker, b) LS equalized (Kautz filter order38), c) using the same set of poles and includingdelay (∆ = 12) in target, and d) by including 12poles at the origin and delay (∆ = 12) in target.Kautz filters the pole radii determine the maximumQ values of resonances. If the pole radii are selectedconservatively, no excess peaking and ringing of resonancesappear in the equalizer response.4.3. Loudspeaker equalization, Case 3In this subsection we demonstrate further the use ofKautz modeling in loudspeaker equalization includingboth magnitude and phase correction. Whilein previous studies with warped and Kautz filters[7, 5, 21] we have applied magnitude-only equalization,it is interesting to investigate the ability ofKautz inverse models to correct also the phase behavior.For clarity of phase curves, it is preferable to demonstratethe phase correction by using a synthetic (simulated)loudspeaker response instead of a real measuredone. Figures 9 and 10 depict the magnitudeand group delay behaviors of an idealized two-wayloudspeaker. It consists of a low-frequency driver ina wented box (4 th order highpass at 80 Hz) and ahigh-frequency driver (up to 18 kHz), both with flatresponse between roll-off frequencies. They are combinedwith a second order Linkwitz-Riley crossover100(b)(a)10 2 10 3 10 4Frequency [Hz]Fig. 8: Basic example of loudspeaker response equalizationby Kautz filter inverse modeling. From bottomup: (a) measured magnitude response, (b) sameone 1/3-octave smoothed, (c) low and high roll-offcompensation, (d) magnitude response of 24 th order(12 pole pairs) Kautz equalizer, (e) equalized magnituderesponse, (f) same one 1/3-octave smoothed,and (g) Kautz equalized response with R = 0.03.Vertical lines at 35 dB level indicate the frequencypositions corresponding to logarithmically spacedpole angles.network [10, 11], which in an ideal case results in aflat magnitude response at the main axis.In this particular case we investigate a loudspeakerwhere the acoustic center of the high-frequencydriver is 17 cm behind the acoustic center of thelow-frequency unit. This means a temporal nonalignmentof about 0.5 ms, which results in rippleof the main axis magnitude response (‘Original’ inFig. 9) and similarly a non-flat group delay response(‘Original’ in Fig. 10). The magnitude response errorof this amount is easily audible. Although thegroup delay deviation remains within 1 ms above 300Hz, which is hardly noticeable in practice, it is interestingto check how the phase correction by KautzAES 120 th Convention, Paris, France, 2006 May 20–23Page 8 of 12

PAATERO AND KARJALAINENEqualization using Kautz filters on log-scale153Level / dB1050-5EQ excess-phaseEQ min-phaseOriginalGroup delay / ms2.521.510.5EQ excess-phase-10-15HP LP10 2 10 3 10 4Frequency / Hz0- - - EQ min-phase ––– Original-0.510 2 10 3 10 4Frequency / HzFig. 9: Magnitude responses of the simulated loudspeaker:LP = low-pass crossover; HP = highpasscrossover; original = response due to driverdistance misalignment; minimum-phase equalized;excess-phase equalized response.LS equalization works. This brings necessarily increasedlatency beyond the maximum group delayof the original response.Curves ‘EQ min-phase’ in Figs. 9 and 10 show themagnitude and group delay responses of the simulatedloudspeaker when a Kautz equalizer is designedbased on the minimum-phase part of the loudspeakerimpulse response. A Kautz filter of 18 pole pairs wasdesigned with logarithmic distribution of poles between80 Hz and 23 kHz and pole radius coefficientR = 0.1. High- and low-frequency roll-offs are compensatedin EQ design to remain as they were originally.After equalization the magnitude response isflat within ±1 dB, while the group delay (dashedline) is not essentially improved.Curves ‘EQ excess-phase’ in Figs. 9 and 10 illustratethe results of magnitude plus phase equalizationwith a Kautz LS equalizer. In this case thetarget response of the equalized system is given asa delayed impulse, with a latency higher than themaximum delay of the loudspeaker itself. The targetgroup delay was set here to 1.5 ms (66 samplesat 44.1 kHz sample rate). A Kautz equalizer was designedwith 8 logarithmically distributed pole pairswithin 80 Hz to 23 kHz, with R = 0.05, plus 96 polesat the origin. Notice that the latter ones correspondagain to FIR filter behavior, so that the equalizer isa mixture of an FIR and an IIR filter.Fig. 10: Group delay responses of the simulatedloudspeaker: original = ripple due to driver distancemisalignment; minimum-phase equalized; excessphaseequalized response with extra group delay.After applying excess-phase equalization the magnituderesponse in Fig. 9 is again within ±1 dB, whilethe group delay curve in Fig. 10 has ripple less than±0.1 ms. (The growth of low-frequency group delaycomes from the high-pass behavior of the loudspeaker,which is not compensated for.)4.4. Room response equalization, Case 4In this subsection we examine a basic example ofloudspeaker plus room response correction usingKautz LS equalization. The loudspeaker used hada lower roll-off frequency of about 80 Hz, compensatedin target response design. The room was alistening room of 33 m 2 with fairly well controlledacoustics. Figure 11 shows the first 5 ms of the measuredimpulse response in pane (a) and magnituderesponse in pane (c) in full resolution and 1/3-octavesmoothed.A Kautz equalizer of order 24 (12 pole pairs) wasdesigned with logarithmically positioned pole frequenciesbetween 50 Hz and 20 kHz, using the poleradius parameter value R = 0.5. The resulting impulseresponse and magnitude response are plottedin Fig. 11, panes (b) and (d). The magnitude responseis flattened as desired. In the impulse responsesome low-frequency oscillation is damped,but the peaks corresponding to reflections from surfacescannot of course be canceled out by such a loworderequalizer. However, even equalizer orders ofAES 120 th Convention, Paris, France, 2006 May 20–23Page 9 of 12

PAATERO AND KARJALAINENEqualization using Kautz filters on log-scale0.5(a)0.5(b)Amplitude0Amplitude0-0.5-0.50 1 2 3 4 5Time [ms]0 1 2 3 4 5Time [ms]level [dB]20100-10-20level [dB]20100-10-2010 2 10 3 10 4Frequency [Hz]10 2 10 3 10 4Frequency [Hz]Fig. 11: Kautz equalization of a room response: (a) first 5 ms of the impulse response, (b) first 5 ms of theequalized impulse response, (c) original magnitude response, and (d) Kautz equalized magnitude response.In magnitude responses the lifted upper curves are computed through 1/3-octave smoothing.8–12 (4–6 pole pairs) seen to provide useable equalizationresults in this particular case.In principle the Kautz equalization is capable to approachperfect inversion or any resolution of magnitudeand phase equalization. In practice there arehowever limitations, such as ability to do major correctionsin phase behavior, e.g., proper temporal envelopemanipulation as done in [4]. This remains atopic for future research.5. DISCUSSION AND CONCLUSIONSMany DSP techniques have been proposed earlierfor the equalization of audio reproduction channels,either for loudspeaker only or for combined loudspeakerand room responses. We have earlier investigateddifferent frequency warping techniques,including Kautz filters, which allow for maximumflexibility of frequency resolution control in equalizerdesign [21]. The previous Kautz filter equalizerswere designed as models for responses that wereseparately inverted for given reproduction system responses.For simplicity, the inversion was done onlyfor the minimum-phase part.In the present study we have extended the use ofKautz filter equalizers to more general cases by systematicdesign of inverse filter models directly fromgiven impulse responses of audio reproduction systems.The novelty is to apply least square optimaldesign of Kautz filter tap coefficients for a fixed setof system poles. Logarithmic frequency resolution isapproximated by setting the pole angle distributionlogarithmically and by controlling the pole radii toapproximate constant-Q behavior. This is conceptuallysimilar to inverse (FIR) filter design throughlinear prediction for uniform frequency resolution.An advantage of the new method is that it canAES 120 th Convention, Paris, France, 2006 May 20–23Page 10 of 12

PAATERO AND KARJALAINENEqualization using Kautz filters on log-scalebe applied to non-minimum phase EQ design, asdemonstrated for loudspeaker equalization. Anotherfavorable feature is the inherent control of sharpmagnitude response dips to avoid sharp peaks andlong-ringing temporal decays in the equalizer design.This is done simply by limiting the Q-values of theKautz system pole pairs.After introducing Kautz filters in Section 2 and thenew principle of Kautz equalization in Section 3, aset of case studies are presented in Section 4. Theseinclude different loudspeaker response corrections,including non-minimum phase equalization. A typicallistening room case is also studied for combinedloudspeaker and room response equalization. Thisis primarily magnitude-only equalization; more detailedcontrol of phase behavior is left to future research.Low-order Kautz equalizers (4 to 12 pole pairs) arefound in practice to yield good magnitude responsecorrection. Phase correction may need higher orderswith a set of system poles at the origin and excessdelay required for the target response.Kautz filters are know to be numerically robust dueto the allpass type of elements used in the implementation.For low orders it is possible to map thetransfer function to direct form IIR structures formaximal efficiency on DSP processors. This andmany other practical questions remain however outof the scope of this paper, which is the first stepin introducing this new concept of audio equalizerdesign.6. ACKNOWLEDGMENTThe work of Tuomas Paatero has been funded by theAcademy of Finland, project No. 205787 (VirtualAcoustics and Audio 2015).7. REFERENCES[1] P. W. Broome, “Discrete Orthonormal Sequences”,Journal of the Association for ComputingMachinery, vol. 12, no. 2, pp. 151–168,1965.[2] J. C. Brown, “Calculation of a constant Q spectraltransform”, Journal of the Acoustical Societyof America, vol. 89, pp. 425–434, January1991.[3] R. Greenfield and M. Hawksford, “Efficient filterdesign for loudspeaker equalization”, Journalof the Audio Engineering Society, vol. 39,no. 10, pp 739–751, November 1991.[4] P. Hatziantoniou and J. Mourjopoulos, “GeneralizedFractional-Octave Smoothing of Audioand Acoustic Responses”, Journal of the AudioEngineering Society, vol. 48, no. 4, pp. 259–280,April 2000.[5] A. Härmä, M. Karjalainen, L. Savioja, V.Välimäki, U. K. Laine and J. Huopaniemi, “Frequencywarped signal processing for audio applications”,J. Audio Engineering Society, vol.48, no. 11, 2000.[6] A. Härmä and T. Paatero, “Discrete Representationof Signals on a Logarithmic FrequencyScale”, in Proc. WASPAA’01, pp. 39–42, NewPaltz, NY, USA, October 2001.[7] M. Karjalainen, E. Piirilä, A. Järvinen, andJ. Huopaniemi, “Comparison of LoudspeakerEqualization Methods Based on DSP Techniques”,Journal of the Audio Engineering Society,vol. 47, no. 1/2, pp. 15-31, Jan/Feb 1999.[8] M. Karjalainen, T. Paatero, J. N. Mourjopoulos,and P. D. Hatziantoniou, “About room responseequalization and dereverberation ”, inProc. WASPAA’05, pp. 183–186, New Paltz,NY, USA, October 2005.[9] W. H. Kautz, “Transient Synthesis in the TimeDomain”, IRE Transactions on Circuit Theory,vol. CT–1, pp. 29–39, 1954.[10] S. H. Linkwitz, “Active Crossover Networks forNoncoincident Drivers,” J. Audio Eng. Soc.,Vol. 24, Number 1, pp. 2-8. Jan./Feb. 1976.[11] S. H. Linkwitz, “Passive Crossover Networks forNoncoincident Drivers,” J. Audio Eng. Society,Volume 26, Number 3, pp. 149-150. March 1978.[12] A. Marques and D. Freitas, “Infinite impulse response(IIR) inverse filter design for the equalizationof non-minimum phase loudspeaker systems”,in Proc. WASPAA’05, pp. 170–173, NewPaltz, NY, USA, October 2005.AES 120 th Convention, Paris, France, 2006 May 20–23Page 11 of 12

PAATERO AND KARJALAINENEqualization using Kautz filters on log-scale[13] M. Miyoshi and Y. Kaneda, “Inverse filtering ofroom acoustics”, IEEE Transactions on Acoustics,Speech, and Signal Processing, vol. 36, no.2, pp.145–152, February 1988.[14] J. Mourjopoulos, P. Clarkson, and J. Hammond,“A comparative study of least-squaresand homomorphic techniques for the inversionof mixed phase signals”, in Proc. ICASSP’82,vol. 7, pp. 1858–1861, May 1982.[15] J. Mourjopoulos, “Digital Equalization ofRoom Acoustics”, Journal of the Audio EngineeringSociety, vol. 42, no. 11, pp. 884–900,November 1994.[16] J. Mourjopoulos, “Comments on ”Analysis ofTraditional and Reverberation-Reducing Methodsof Room Equalization””, Journal of the AudioEngineering Society, vol. 51, no. 12, pp.1186–1188, December 2003.[17] S. Neely and J. Allen, “Invertibility of a roomimpulse response”, Journal of the AcousticalSociety of America, vol. 66, pp. 165–169, July1979.[18] T. Paatero, M. Karjalainen, and A. Härmä,“Modeling and Equalization of Audio SystemsUsing Kautz Filters”, in Proc. ICASSP’01, Vol.5, pp. 3313–3316, Salt Lake City, Utah, USA,May 2001.[20] T. Paatero, “An Audio Motivated Hybrid ofWarping and Kautz Filter Techniques”, in Proc.EUSIPCO’02, pp. 627–630, Toulouse, France,September 2002.[21] T. Paatero and M. Karjalainen, “Kautz Filtersand Generalized Frequency Resolution - Theoryand Audio Applications”, Journal of the AudioEngineering Society, vol. 51, pp. 27–44, 2003.[22] T. Paatero, ”Efficient Pole-zero Modeling ofResonant Systems using Complex Warpingand Kautz Filter Techniques,” in Proc. WAS-PAA’03, pp. 9–12, New Paltz, NY, USA, October2003.[23] T. Paatero, “Modeling of Long and ComplexResponses Using Kautz Filters and Time-Domain Partitions”, in Proc. EUSIPCO’04, pp.313–316, Vienna, Austria, September 2004.[24] B. Radlović and R. Kennedy, “Nonminimumphaseequalization and its subjective importancein room acoustics”, IEEE Transactionson Speech and Audio Processing, vol. 8, no. 6,pp. 728–737, November 2000.[25] J. L. Walsh, Interpolation and Approximationby Rational Functions in the Complex Domain,2nd Ed., Am. Math. Soc., Rhode Island, 1969.[19] T. Paatero and M. Karjalainen, ”New DigitalFilter Techniques for Room Response Modeling”,in Proc. AES 21st Int. Conf., St. Petersburg,Russia, June 1-3, 2002.AES 120 th Convention, Paris, France, 2006 May 20–23Page 12 of 12