Audio Engineering Society. Convention Paper. Presented at the 120th Convention 2006 May Paris, France

Transcription

1 Audio Engineeing Society Convention Pape Pesented at the 120th Convention 2006 May Pais, Fance This convention pape has been epoduced fom the autho's advance manuscipt, without editing, coections, o consideation by the Review Boad. The AES takes no esponsibility fo the contents. Additional papes may be obtained by sending equest and emittance to Audio Engineeing Society, 60 East 42 nd Steet, New Yok, New Yok , USA; also see All ights eseved. Repoduction of this pape, o any potion theeof, is not pemitted without diect pemission fom the Jounal of the Audio Engineeing Society. Adaptive Time-Fequency Resolution fo Analysis and Pocessing of Audio Alexey Lukin 1, AES Student Membe, Jeemy Todd 2, AES Membe 1 Moscow State Univesity, Moscow, Russia lukin@gaphics.cs.msu.u 2 izotope, Inc., Cambidge, MA jeemy@izotope.com ABSTRACT Filte banks with fixed time-fequency esolution, such as the Shot-Time Fouie Tansfom (STFT), ae a common tool fo many audio analysis and pocessing applications allowing effective implementation via the Fast Fouie Tansfom (FFT). The fixed time-fequency esolution of the STFT can lead to the undesiable smeaing of events in both time and fequency. In this pape, we suggest adaptively vaying STFT time-fequency esolution in ode to educe filte bank-specific atifacts while etaining adequate fequency esolution. Seveal stategies fo systematic adaptation of time-fequency esolution ae poposed. The intoduced appoach is demonstated as applied to spectogam displays, noise eduction, and spectal effects pocessing. 1. INTRODUCTION It is well known that signal pocessing algoithms dealing with multimedia infomation should account fo popeties of human peception in ode to achieve bette pocessing quality. Thee exist multiple studies of human auditoy and visual peception which ae extensively employed in image and audio compession algoithms. In this pape, we conside the timefequency esolution of filte banks commonly used in audio analysis and pocessing, and we popose a multiesolution appoach that impoves seveal existing algoithms. 2. SHORTCOMINGS OF STFT The Shot Time Fouie Tansfom (STFT) is a filte bank which is widely used in audio analysis and pocessing. The STFT can also be plotted on a 2D gaph as a function of both time and fequency, with colo epesenting magnitude, to fom a spectogam display. Spectogams ae becoming a popula tool among audio enginees as they ae much moe peceptually-oiented than a taditional wavefom display. Filte banks based on the STFT ae used in algoithms fo noise eduction and vaious spectal effects such as multiband delays, vocodes, and cente channel extaction.

2 Adaptive Time-Fequency Resolution It is known fom psychoacoustics that the fequency esolution of human heaing is not unifom. Instead, it follows a mel-scale which is appoximately linea below 500 Hz and logaithmic above it [1]. The fixed timefequency esolution of the STFT is puely linea, so it is not ideal fom a peceptual standpoint. The atifacts specific to STFT-based pocessing ae pe-echoes (time smeaing of tansient events) and insufficient fequency esolution at stationay pats (especially at low fequencies), leading to peceptually inadequate modeling of audio. leads to the multiesolution famewok fo signal pocessing algoithms depicted in Fig. 1. x[ A B Pocessing 1 Pocessing 2 Pe-echoes ae atifacts esulting fom the fact that any modification of time-fequency coefficients of a signal speads its effect along the entie window length of the filte bank in the time domain. Fo example, pocessing of tansfomed coefficients that captue an onset of a tansient event will esult in smeaing of tansient enegy in time within the filte bank window, both in the fowad and backwad diections. The backwad speading (pe-echo) is typically much moe audible due to popeties of tempoal masking of human heaing and the fact that ongoing tansient enegy will pobably mask the post-echo. Audibly this esults in swishy, non-focused sounding of tansients which include dums, pecussion and othe instuments with shap attacks. x 1 [ D Filte bank Mixe of coefficients Invese filte bank x 2 [ E Filte bank G F C Analysis contol Insufficient fequency esolution manifests itself diffeently in diffeent pocessing algoithms. Geneally, it pevents algoithms fom sepaating closely spaced tones. Fo example, in noise eduction this may lead to weake suppession of noise. In cente channel extaction o time stetching of audio this may lead to unwanted modulations in low fequencies. In this pape, we will popose a method fo educing these atifacts simultaneously. 3. ADAPTIVE TIME-FREQUENCY RESOLUTION Thee have been many attempts to build filte banks with vaiable time-fequency esolution fo audio compession puposes [2]. Howeve such attempts ae limited by the fact that compession equies the citical sampling popety of filte banks, to keep the amount of data in the signal at a minimum. This significantly esticts the feedom to vay time-fequency esolution. On the othe hand, image and audio pocessing methods allow edundancy in ovesampled filte banks which y[ Figue 1. Geneal scheme fo signal pocessing with adaptive time-fequency esolution. The same pocessing algoithm is unning seveal instances (only two ae depicted, but the famewok can be genealized to any numbe of instances), labeled A and B above, with diffeent fixed time-fequency esolutions that wok in paallel on the same input data steam. The esulting signals x 1 [ and x 2 [ ae pocessed signals which wee pocessed with diffeent time-fequency esolutions. Ou goal is to combine them in ode to achieve the desied esolution in evey aea of the time-fequency plane. This combination is pefomed by additional filte banks D and E, both with a single fixed time-fequency esolution that tansfoms these esulting signals into time-fequency coefficients on the same time-fequency gid. The esulting timefequency coefficients can be adaptively mixed by the mixe F to select desied coefficients in each aea of the time-fequency plane. The pocess of mixing can be contolled by some pio stategy (eflecting popeties of human peception) and/o depending on local signal AES 120th Convention, Pais, Fance, 2006 May Page 2 of 10

3 Adaptive Time-Fequency Resolution featues (e.g. on its stationaity) detemined by analysis at C of the oiginal input signal. Finally, the invese filte bank G etuns the pocessed signal to the time domain, foming the output y[. Since mixing of pocessed signals x 1 [ and x 2 [ is pefomed in the tansfom domain, the poposed famewok allows abitay time-fequency esolution in abitay aeas of the time-fequency plane. The numbe of individual pocessos opeating at diffeent timefequency esolutions contols the smoothness of vaiation of esolution in the combined signal. Also, by mixing togethe coefficients fom seveal esolutions, we can intepolate between given discete esolutions. The poposed famewok can also be modified to pefom signal analysis with abitay time-fequency esolution. As shown in Fig. 2, the input signal x[ is fed diectly to filte banks H and I opeating with diffeent fixed time-fequency esolutions. To simplify the mixe K, filte banks H and I should poduce outputs a f,t,1 and a f,t, 2 at the same gid of time-fequency locations. This can be accomplished in the case of the STFT by using diffeent time-domain window lengths in H and I but using the same analysis hop and FFT size (zeo-padding windowed data as necessay). The mixe K combines the outputs of the filte banks accoding to analysis pefomed in J, poducing outputs a f, t in the tansfom domain. x[ J Analysis 4. ADAPTATION STRATEGIES In this section, we descibe two stategies fo vaying the time-fequency esolution of an STFT filte bank. Both of them incopoate pio knowledge about the fequency esolution of ou heaing and adapt to the time-vaying popeties of a signal. The fist stategy is based on a signal tansience estimato. We descibe the estimato based on analysis of enegy evolution in citical bands. Then we descibe the stategy of vaying time-fequency esolution in ode to educe pe-echoes in tansient egions of the time-fequency plane and incease the fequency esolution in stationay egions. The second stategy is based on the pinciple of minimal desciption length (MDL) [3]. It estimates the optimality of diffeent time-fequency esolutions and selects the one that is locally optimal. Optimality is defined as minimal possible enegy smeaing both in time and fequency diections. This method is analogous to the geneal MDL paadigm of finding the tansfom with the most compact suppot fo tansfomed enegy Tansience adaptation One appoach to adaptive time-fequency esolution of a filte bank is to explicitly account fo signal stationaity. Stationaity means pesevation of signal popeties acoss time, including powe and spectal shape. We define tansience as the opposite of stationaity: vaiance of signal popeties in time. To educe the time smeaing of tansients we will incease the tempoal esolution of the filte bank at tansient signal segments. Duing stationay segments, we will use highe fequency esolution. H I Filte bank 1 Filte bank 2 a f,t,1 a f,t,2 K Mixe of coefficients a f,t contol Some simple detectos of tansience ae descibed in [4]; they estimate the vaiance of a shot-time spectum in adjacent time fames. Such a spectal similaity measue is susceptible to false detections of tansients at stationay noisy pats of a signal esulting fom statistical vaiance of shot time spectal estimates of noise. In this wok, we ae using an algoithm which integates signal enegy in citical bands [2] and detects fast enegy onsets on a pe-band basis. Figue 2. Geneal scheme fo signal analysis with adaptive time-fequency esolution. The signal is tansfomed into the STFT domain with a window size of 12 ms and an analysis hop of 6 ms. Fo each fame the signal powe is integated inside 24 AES 120th Convention, Pais, Fance, 2006 May Page 3 of 10

4 Adaptive Time-Fequency Resolution citical bands coveing the entie audible spectum. The integated enegy is aised to the powe of 1/8 to povide bette sensitivity to elatively high enegy onsets at small absolute levels. Then we detect vaiation of enegy in time within each citical band by cosscoelating enegies e[b, with a filte h[ = {-1, -1, -1, 0, 1, 1, 1} (hee b is the citical band numbe, t is the index of the STFT fame): v[ b, = e[ b, h[ The tansience T[b, of the signal in each citical band is estimated as v[ b,, T [ b, = v[ b,, 10 v[ b, 0 v[ b, < 0 This povides 10 times bette sensitivity to enegy onsets than to enegy decays. When the tansience of a signal in each citical band is estimated, we can use it to contol the time-fequency esolution of a filte bank by educing fequency esolution aound tansients. This educes the smeaing of tansients in time while keeping good fequency esolution at stationay pats of the signal. The default behavio of the mixe of coefficients can eflect the peceptual popety of bette low-fequency esolution. At the same time, the suggested tansience detecto can alte the default mixing stategy towads bette time esolution aound tansients (see section 6.1 fo details) Maximal enegy compaction pinciple When plotting spectogams, the main poblem with fixed time-fequency esolutions is the smeaing of signal enegy. Smeaing in fequency causes hamonics to appea as thick lines and can pevent distinguishing closely spaced hamonics. Smeaing in time means loss of time esolution and can negatively affect estimation of positions and duations of tansient events in the signal. It would be desiable to jointly educe smeaing of enegy in both diections. Howeve this is not possible due to the uncetainty pinciple. What we popose in this section is to estimate the amount of enegy smeaing fo diffeent fixed timefequency esolutions and select the esolution that minimizes such smeaing in both tempoal and fequency diections. Let s conside a small ectangula aea Ω of the timefequency plane and shot-time Fouie tansfoms with diffeent time-fequency esolutions of the same signal in this aea. STFT coefficients fo diffeent esolutions can be obtained by calculating the STFT with timedomain windows of vaying length. Analysis hops of windows and fequency gids should be equal fo all STFT esolutions, just as they ae in the geneal analysis famewok depicted in Fig. 2. This ensues that squaed STFT magnitudes a f, t, at diffeent esolutions ae calculated in the same gid of time-fequency locations. Hee t and f ae time and fequency indices of STFT coefficients, and indexes available STFT esolutions. Ou task is to select the that minimizes enegy smeaing in the aea Ω. a, To achieve this, we sot the f, t inside Ω by thei magnitudes, in descending ode, fo each esolution. We name the soted esults a i,. Next we define enegy smeaing of evey paticula STFT esolution as: S i i = i a a i, i, + ε The numeato of the faction evaluates the fist moment of the statistical distibution of squaed magnitudes. The denominato nomalizes the numeato by the total enegy of the signal in aea Ω. The squae oot in the denominato assigns highe smeaing to esolutions with highe oveall enegy in the aea Ω. Since the enegy in Ω vaies only due to diffeences in amount of leakage (smeaing) fom adjacent egions at vaious esolutions, this penalizes esolutions whee excessive smeaed enegy comes fom suounding aeas of the timefequency plane. The small constant ε pevents division by zeo. When enegy smeaing measues S ae calculated fo evey esolution, we select the esolution 0 which minimizes enegy smeaing: agmins 0 = AES 120th Convention, Pais, Fance, 2006 May Page 4 of 10

5 Adaptive Time-Fequency Resolution This esolution is selected as optimal in the aea Ω and is used to build a spectogam o pocess the audio signal. 5. APPLICATION TO DISPLAY OF SPECTROGRAMS The audio spectogam has become an impotant tool in audio engineeing. Many common opeations, such as content analysis, emoval of atifacts, and basic editing opeations ae suppoted by spectogams in popula sound editos. The main factos limiting the usefulness of a typical STFT-based spectogam view ae a linea fequency scale obscuing many low-fequency details, and the fixed time-fequency esolution of the STFT leading to time o fequency smeaing of audio events. Figue 4. STFT spectogam, window size is 93 ms. If a typical STFT spectogam with fixed timefequency esolution is displayed with a peceptually meaningful fequency scale (e.g. the mel-scale) the lack of low-fequency esolution becomes obvious (Fig. 3). Howeve inceasing the fequency esolution of the STFT will poduce time smeaing of tansients (Fig. 4). It is possible to combine spectogams taking lowfequency spectogam data fom the STFT with high fequency esolution and high-fequency data fom the STFT with bette tempoal esolution (Fig. 5). Figue 5. Spectogam with combined STFT esolutions. Figue 3. STFT spectogam, window size is 12 ms. Figue 6. Spectogam with adaptive esolution. AES 120th Convention, Pais, Fance, 2006 May Page 5 of 10

6 Adaptive Time-Fequency Resolution In section 5.1 we intoduce an adaptive spectogam that povides futhe impovements in the adaptation of time-fequency esolution (Fig. 6) Resolution selection algoithm To futhe educe smeaing of enegy in spectogams, we apply the esolution selection stategy descibed in section 4.2. We calculate the STFT with 4 diffeent window sizes: 12, 24, 48, and 96 ms. Time-fequency magnitudes ae calculated on the same gid by zeo padding windowed signals and using equal STFT analysis hops fo evey esolution. To obtain the optimal esolution at evey point (f, t) of the time-fequency plane we conside a ectangula aea Ω aound this point that is 1 citical band wide and 48 ms long. The tadeoff hee is that small Ωs will not allow us to fom a obust estimate of enegy smeaing as thee will be too few STFT coefficients inside, and lage Ωs will not be local enough fo fine contol of esolution. We select citical bands because they have some peceptual meaningfulness, and also because they poject to equal-height aeas on ou mel-scale spectogam. Ou 48 ms width is chosen afte expeimental evaluation of the look and meaningfulness of spectogams with vaious widths fo Ω. Hee k is the nomalization constant selected so that the sum of all w is 1, and ε is a small constant peventing division by Simulation esults We have conducted simulations to compae the look and usefulness of conventional STFT spectogams and adaptive-esolution spectogams. The fist test signal consisted of an atificially geneated 1 khz tone with a shap fade-in lasting 2 ms and a smooth decay lasting 600 ms (Fig. 7). Since the tone onset is abupt, it contains a tansient enegy bust speading outside of the 1 khz band, as shown by the conventional spectogam (Fig. 8). By vaying the esolution of the spectogam, we can make eithe the hoizontal line (the decaying tone) o the vetical line (the tansient attack) thinne, but not both at once. Howeve ou adaptive spectogam is able to locally select the time-fequency esolution which minimizes smea both in time and fequency (Fig. 9) leading to less fequency speading of the slow tone decay and bette time localization of the tone onset. Next, accoding to section 4.2, we calculate the best esolution choice 0 of the 4 available esolutions. The STFT magnitude coefficient a f, t, is used to fom the 0 spectogam view at point (f, t). In ode to pevent had switching fom one esolution to anothe we have updated the algoithm to mix magnitude coefficients instead of switching between them. In this manne we ae able to intepolate between 4 available fequency esolutions. The mixing is pefomed accoding to espective enegy smeaing measues of each esolution: Figue 7. Wavefom of a tone onset. a f, t = 4 = 1 w a f, t, Mixing weights w ae calculated as follows: w = S 8 k + ε AES 120th Convention, Pais, Fance, 2006 May Page 6 of 10

7 Adaptive Time-Fequency Resolution Figue 8. STFT spectogam of a tone onset, window size is 46 ms. Figue 10. STFT spectogam of folk music, window size is 46 ms. Figue 9. Adaptive esolution spectogam of a tone onset. Ou next example is a piece of folk music with flute, cello, guita and pecussive dums. The conventional STFT spectogam (Fig. 10) lacks low-fequency esolution and is unable to sepaate bass notes of the cello and guita. At the same time the tempoal esolution at high fequencies is not enough to shaply localize onsets of the dums. Ou adaptive spectogam (Fig. 11) fixes both of these poblems: the lowfequency esolution is inceased, and dum onsets ae displayed moe shaply due to bette local time esolution. The adaptiveness of the time-fequency esolution also enables us to use bette fequency esolution at high fequencies to esolve closely spaced guita ovetones above 3 khz. Figue 11. Adaptive esolution spectogam of folk music. Anothe example of ou adaptive spectogams is given in Fig. 6 displaying a piece of ock music with vocal, bass, dums, guitas, flute and violin. Again, the adaptive spectogam is able to esolve low-fequency hamonics, and it avoids the smeaing of bass dum hits (at 0.3 and 1.0 seconds). In the high-fequency aea we ae able to peseve the shapness of dum onsets and esolve closely spaced guita hamonics. These tests show that the poposed adaptive appoach to the calculation of spectogams allows a spectogam to display moe useful details and musical events with bette pecision. AES 120th Convention, Pais, Fance, 2006 May Page 7 of 10

8 Adaptive Time-Fequency Resolution 6. APPLICATION TO AUDIO PROCESSING In this section, we show how filte banks with adaptive time-fequency esolution can be applied to impove the quality of seveal audio pocessing algoithms: the spectal subtaction algoithm fo noise eduction [5] and the cente channel extaction algoithm. We un seveal instances of single-esolution pocessos and adaptively combine thei esults in the time-fequency plane using one of the suggested stategies. The esulting audio signal shows significant eduction of time smeaing of tansients and at the same time good fequency esolution allowing effective suppession of tonal noise o extaction of the cente channel. In spite of the inceased computational complexity compaed to a single-esolution STFT, both algoithms allow eal-time implementation. Figue 12. Noisy sample of guita and castanets Noise eduction Most noise eduction methods fo additive stationay noises in audio ae based upon the spectal subtaction algoithm [5, 6]. This algoithm tansfoms the noisy signal with a filte bank and attenuates coefficients that ae supposedly pat of the noise, using a-pioi knowledge of the noise spectum. Then the invese filte bank econstucts the cleaned signal. In this pape, we will not discuss details of spectal subtaction methods, but athe show how modification of a filte bank can impove the quality of the esult by educing atifacts specific to filte banks. A typical filte bank fo spectal subtaction is based on the STFT. Good fequency esolution of the STFT filte bank allows sepaation of closely spaced noise and signal hamonics. Good fequency esolution also leads to stonge possible noise attenuation due to lowe noise powe pe STFT bin. Howeve good fequency esolution equies long STFT windows leading to poo time esolution. Spectal subtaction with poo time esolution is not able to suppess noise befoe tansient onsets since the pat of the tansient falls into the window and aises the coefficient magnitude peventing attenuation (Figs. 12, 13). Anothe poblem is geneal pe-echo associated with the modification of STFT coefficients at poo time esolution. Figue 13. Result of noise eduction with a 46 ms STFT window. Figue 14. Result of noise eduction with adaptive timefequency esolution. AES 120th Convention, Pais, Fance, 2006 May Page 8 of 10

9 Adaptive Time-Fequency Resolution We suggest using one of the stategies descibed in section 4 to adapt the time-fequency esolution of the filte bank. Pope selection of time-fequency esolution will esult in bette enegy compaction in the tansfom domain, which is always desiable fo noise eduction. Adaptive esolution will also allow good fequency esolution in stationay signal pats and good time esolution aound tansients leading to less time smeaing atifacts. To test ou appoach, we selected the tansient detection stategy descibed in section 4.1. We used spectal subtaction with 3 STFT filte banks with window sizes of 24, 48, and 96 ms and combined thei esults using anothe STFT filte bank with a window size of 12 ms (we equie good time esolution when combining esults, but the fequency esolution is not as impotant since all of the noise eduction pocessing has aleady been done). The tansience detecto also opeates with a window size of 12 ms. The combination of esults is pefomed accoding to the following fomula: X f, t αx = αx f, t,2 f, t,1 + (1 α) X + (1 α) X f, t,3 f, t,2,, f 4000Hz f > 4000Hz Hee α depends on tansience fo a given bin of the STFT: 0, T[ f, T1 α =, T2 T1 1, 1 T[ f, < T T T[ f, < T T[ f, T Hee T 1 and T 2 ae use-defined thesholds, and we have selected T 2 = 2T 1. Such a mixing stategy uses 2 times bette fequency esolution below 4 khz (appoximating the popety of bette low-fequency esolution of ou heaing) and adapts the esolution to the local tansience of the signal inside each citical band. As a esult of such adaptation of esolution (Fig. 14), we have achieved eduction of time-smeaing atifacts without compomise in depth of noise eduction. The conclusions of ou infomal listening tests wee also confimed by an incease of S/N atio fo ecodings estoed with adaptive time-fequency esolution (Table 1). Filte bank algoithm S/N atio, db Noisy ecoding STFT, window 12 ms STFT, window 25 ms STFT, window 50 ms Adaptive esolution Table 1. S/N atios afte noise eduction of guita and castanets sample Cente channel extaction A multiband cente channel extaction algoithm is an impovement upon a widely used kaaoke featue that subtacts left and ight steeo channels to cancel the inphase signals compising the cente of a steeo field. Some sound editos adopt an STFT-based algoithm fo this task that attenuates those STFT coefficients whose magnitudes and phases ae close in the left and ight channels. This allows fo a steeo esult, which is impossible with a single-band algoithm. Again, without going into details of the attenuation of STFT coefficients, we will descibe the effects of STFT esolution on the esulting sound and popose an impovement with adaptive time-fequency esolution. Good fequency esolution allows deepe suppession of hamonic signals (such as vocals) in the cente channel. Howeve poo time esolution esults in smeaing of tansients in the econstucted wavefom afte modification of STFT coefficients. We popose that the time-fequency esolution is adapted accoding to tansience of the signal as descibed in section 6.1. We use 3 paallel STFT-based cente channel extactos and combine thei esults using a local tansience estimate in citical bands. As a esult, we ae able to get good fequency esolution on hamonic pats of a signal (which ae typically to be AES 120th Convention, Pais, Fance, 2006 May Page 9 of 10

10 Adaptive Time-Fequency Resolution suppessed in the cente channel) and good time esolution aound tansients (which pevents timesmeaing). Ou infomal listening expeiments veify the eduction of these atifacts. 7. CONCLUSION We have demonstated a geneal famewok fo effective multiesolution signal pocessing and analysis. This famewok avoids seveal undesiable side effects of the STFT's fixed time-fequency esolution such as the smeaing of events in both time and fequency. It allows signal pocessing and analysis to adapt its esolution accoding to a pedetemined stategy o the analysis of local signal featues. We have shown how this famewok can be applied to the display of spectogams, spectal subtaction algoithms fo noise eduction and cente channel extaction algoithms. Reseach wok fo PhD pogam, Univesitat Pompeu Faba, Bacelona, [5] J. Thiemann Acoustic Noise Suppession fo Speech Signals Using Auditoy Masking Effects // Ph.D. thesis, Depatment of Electical & Compute Engineeing, McGill Univesity, Mont-eal, Canada, July [6] S. Canazza, G. De Poli, G.A. Mian, A. Scapa Real Time Compaison Of Audio Restoation Methods Based On Shot Time Spectal Attenuation // Poceedings of Confeence on Digital Audio Effects (DAFx01), Decembe , Limeick, Ieland. [7] Demo web page fo poposed algoithms: Fo moe examples of these algoithms and applications, please see the demo web page established fo this pape [7]. 8. ACKNOWLEDGEMENTS The authos would like to thank izotope, Inc. fo suppoting the poject, and D. Y.M. Bayakovski, supeviso of Alexey s eseach in the Gaphics & Media Lab of Moscow State Univesity. 9. REFERENCES [1] B. Logan Mel Fequency Cepstal Coefficients fo Music Modeling // Poceedings of Intenational Symposium on Music Infomation Retieval, [2] T. Painte, A. Spanias A Review of Algoithms fo Peceptual Coding of Digital Audio Signals // Poceedings of 13th Intenational Confeence on Digital Signal Pocessing, 1997, vol. 1, 2-4 July 1997, pp [3] P. Gunwald A Tutoial Intoduction to the Minimum Desciption Length Pinciple // Chaptes 1 and 2 of Advances in Minimum Desciption Length: Theoy and Applications, MIT Pess, Apil 2005, ISBN [4] J. Bonada Audio Time-Scale Modification in the Context of Pofessional Audio Post-poduction // AES 120th Convention, Pais, Fance, 2006 May Page 10 of 10