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

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

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