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

More documents

Recommendations

Info

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
Page 1: Audio Engineering SocietyConvention
Page 4: PAATERO AND KARJALAINENEqualization
Page 9 and 10: PAATERO AND KARJALAINENEqualization
Page 11 and 12: PAATERO AND KARJALAINENEqualization

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?