TEMPORAL AND TIME-FREQUENCY CORRELATION-BASED BLIND SOURCE SEPARATION METHODS. Yannick DEVILLE

Transcription

1 TEMPORAL AND TIME-FREQUENCY CORRELATION-BASED BLIND SOURCE SEPARATION METHODS Yannick DEVILLE Université Paul Sabatier Laboratoire Acoustique, Métrologie, Instrumentation Bât. 3RB2, 8 Route e Narbonne, 362 Toulouse Ceex, FRANCE yeville@cict.fr ABSTRACT In this paper, we propose two versions of a correlation-base blin source separation (BSS) metho. hereas its basic version operates in the time omain, its extene form is base on the timefrequency (TF) representations of the observe signals an thus applies to much more general conitions. The latter approach consists in ientifying the columns of the (permute scale) mixing matrix in TF areas where this metho etects that a single source occurs. Both the etection an ientification stages of this approach use local correlation parameters of the TF transforms of the observe signals. This BSS metho, calle TIFCORR (for TIme- CORRelation-base BSS), is shown to yiel very accurate separation for linear instantaneous mixtures of real speech signals (output SNR s are above 6 B).. INTRODUCTION Blin source separation (BSS) methos aim at restoring a set of unknown source signals from a set of observe signals which are mixtures of these source signals []-[2]. Most of the approaches that have been evelope to this en are base on Inepenent Component Analysis (ICA). They assume the sources to be ranom stationary statistically inepenent signals, an they recombine the available observe signals so as to obtain statistically inepenent output signals. The latter signals are then equal to the sources, up to some ineterminacies an uner some conitions (especially, at most one source may be Gaussian for such methos to be applicable if no aitional constraints are set on the sources). hereas many ICA methos have been propose in the last 5 years, a few other concepts for BSS have also been consiere. Especially, several investigations base on time-frequency (TF) analysis have been reporte [3]-[]. Two main classes emerge from these TF-BSS methos. The first one is still significantly relate to classical BSS approaches, as it consists of TF aaptations of previously evelope joint-iagonalization methos, with subsequent moifications [3]-[5]. The other class inclues several methos base on the ratio of the TF transforms of the observe signals [6]-[]. Some of these methos require the sources to have no overlap in the TF omain [6]-[8]. On the contrary, only slight ifferences in the TF representations of the sources are requeste by the methos that we propose in []-[]. It shoul be note that, unlike ICA-base BSS methos, TFbase BSS approaches are intrinsically well-suite to non-stationary signals an set no restrictions on the gaussianity of the sources. They are therefore e.g. especially attractive for speech signals. This paper mainly introuces a TF-BSS approach base on a ifferent type of parameter than in the previously reporte TF- BSS methos, i.e. on the local correlation of the observe signals in the TF omain. Before escribing it, we present a purely temporal version of this approach. This aims at first introucing the propose concept in a simpler framework, but also yiels an aitional temporal BSS metho. The latter metho only applies to more restrictive conitions than its subsequent TF extension, as shown below. The remainer of this paper is therefore organize as follows. e efine the propose temporal BSS metho in Section 2 an then introuce its TF extension in Section 3. In Section 4, we escribe tests performe with artificial mixtures of real speech sources. Conclusions are rawn from this investigation in Section 5 an more specific topics are aresse in the Appenix. 2. PROPOSED TEMPORAL APPROACH 2.. Assumptions an efinitions In this section, evote to the propose statistical temporal BSS approach, we assume that unknown, ranom, possibly complexvalue, source signals are mixe in a linear instantaneous way, thus proviing a set of observe signals. This reas in matrix form: () where! " $#&% an ' ( )+*,.-( $#&% an where the matrix consists of all possibly complex-value mixing coefficients /, which is enote 2 / # below. e now introuce the only assumptions that we make in this section with respect to the sources an mixing stage, an the associate efinitions. Definition : a source is sai to occur alone in a time area if the variance of only this source is not equal to zero in this time area. The consiere time area may be restricte to a single time from a theoretical point of view, as we use a statistical approach in this section. However, in practice, all signal means are estimate over time winows an the consiere time area then consists of such a winow. This efinition correspons to the theoretical point of view. From a practical point of view, this means that the ranges of values taken by all other sources are negligible with respect to those of the source that occurs alone.

2 " " " " L L a L Definition 2: a source is sai to be visible in the time omain if there exist as least one time area where it occurs alone. Assumption : i) each source is visible in the time omain an ii) there exist no time areas where the variances of all sources are equal to zero. It shoul be note that the consiere sources are therefore non-stationary 2. Assumption 2: the sources are mutually uncorrelate. Note that we o not require the complete inepenence of the sources, as the approach introuce below only uses their seconorer statistics. Assumption 3: the number of observations is equal to the number of sources. The matrix is then a square matrix. Assumption 4: i) is invertible an ii) none of its elements is equal to zero. For the sake of simplicity, the notations an introuce above irectly refer to the centere version of the signals in this section (whereas they correspon to the original, possibly non-centere, signals in Section 3) Problem statement an solution BSS woul ieally consist in eriving an estimate 3 then etermine the vector of, so as to (2) 3 6 (3) Each component 4 of the vector 4 woul then be equal to the source signal having the same inex, i.e. to (up to estimation errors). It is well known however that this can only be achieve up to two types of ineterminacies, which resp. concern the scale factors an orer with which the source signals appear in the output vector 4. e etail this phenomenon hereafter, as we will use this iscussion in the BSS metho propose below. Any of the mixe signals corresponing to the matrix form () reas exlicitly: 8 ;: / <>=?, (4) However, it may also be expresse in a ifferent way, by applying two transforms to it. The first one consists in changing the orer in which the terms (resp. associate to the sources ; " ) appear in the sum in (4). This is achieve by applying a ) to the inices A in this sum. The above mixe signal then reas:? /.B CED F CED ;F <(>=G;, (5) : The secon transform consists in normalizing the scales of the contributions of each source signal CD ;F with respect to the contribution of this signal in the first mixe signal 3. The latter contribu- 2 More precisely, they are long-term non-stationary, but they shoul be short-term stationary in practice in orer to make it possible to estimate the above-mentione signal means over short time winows. 3 The same principle may of course may applie to any other mixe signal instea. tion is equal to / B CED F CED ;F, so that we express as:? / HBCED F ;: / BCED F / B CD ;F CD ;F <I=? ;, (6) e therefore introuce the notations: J K / BCD ;F CED F (7) M /.BCD ;F / BCD ;F (8) where J L are the permute scale source signals an are the corresponing permute scale mixing coefficients. Eq. (6) then reas:? or in matrix form: : N J <>=?! () OP J () with J QR J J" $# % an OSR #. e thus get back to the initial mixture expression (), except that the mixe signals are now expresse with respect to the permute scale source signals J. The iscussion at the beginning of this subsection may then be reinterprete as follows: assume that we succee in eriving an estimate O 3 of O. Then, by computing the output vector 4 J K O 3 6 ( () OT6 3 4 OP JH (2) all components J of this vector are resp. equal to J (up to estimation errors), i.e. to the contributions of the (possibly) permute sources in the first mixe signal. The two BSS approaches propose in this paper precisely aim at estimating this matrix B, then proviing the corresponing vector 4 J of separate source signals. In the framework of this section, we propose a metho which takes avantage of the above Assumption -i, i.e. of the fact that there exist time areas where each source occurs alone. Such areas shoul first be etecte, so as to operate insie them. As all observe signals are proportional in any such area, an appealing approach for etecting these areas consists in checking the cross-correlation coefficients UVX VY between the observe signals ( an, efine as: ( ]_^ ` UVX)VY Z\[ ZP[ b ] ^ ` ZP[ ) ] ^ ` (3) where ZP[ c` stans for expectation. More precisely, we show in the Appenix that a necessary an sufficient conition for a source to occur alone at time is: U VX V Y >= e\< SfPgh<8ghQ (4) Now consier such an area where a single source occurs, say i. The observe signals (4) then become restricte to:?j/e i i <>=?, (5) Again using correlation parameters associate to these observe signals then makes it possible to ientify part of the matrix O. More precisely, when (5) is met, ZP[ ]_^ ` ZP[ ( ] ^ ` / i / i <kfl ;, (6)

3 s s The set of values thus obtaine for all observations inexe by < ientifies one of the columns of O, as shown by (8) (which also inicates that the first row of O consists of ). By repeately performing such a column ientification for time areas associate to all sources, we eventually ientify the overall matrix O, which completes the propose approach. The overall structure of the BSS metho thus introuce may be summarize as follows. This approach consists of 3 stages, i.e:. The etection stage consists in etecting the time areas where a single source occurs. This is achieve by fining the times is very close (or practical time winows) where UVX VY to for all < SfPgm<8gn. 2. The ientification stage consists in ientifying the columns of O. This is achieve by successively consiering each of the single-source areas etecte in the first stage as follows. The correlation parameters on the left-han sie of (6) yiel a column of O. This column is kept only if its istance with respect to all previously ientifie columns is above a user-efine threshol, showing that the consiere time area oes not contain the same source as the previous ones. The ientification proceure ens when the number of columns of O thus kept becomes equal to the number of sources (this is guarantee to occur because all sources are assume to be visible in the consiere ata) The combination stage consists in recombining the mixe signals accoring to (), in orer to obtain the extracte source signals. 3. TIME-FREQUENCY EXTENSION OF THE PROPOSED APPROACH 3.. Motivations an basic principles The approach that we introuce in the previous section has the avantage of being simple. It may be consiere to be of limite practical applicability however, because it assumes all sources to occur alone in associate time areas, which is a restrictive conition. But it opens the way to a much more powerful metho if we now think in terms of the time-frequency (TF) istributions of the signals, instea of their plain time istributions consiere up to this point. Inee, the TF counterpart of the above approach may be briefly efine as follows: assume that each source occurs alone in a TF area, then it may be expecte from the above presentation that corresponing columns of O may be ientifie in such areas, thus allowing one to eventually perform BSS. The remainer of this section presents this approach in a more formal way. It shoul be stresse again that the TF version of the propose approach that we will thus introuce has a much broaer scope of application that its above temporal version. This is ue to the fact that it only requests each source to occur alone in a small boune TF area, which is a much less restrictive conition that having it alone in a time area, i.e. at all frequencies for that time area. For instance, mixtures of continuous speech signals meet the assumptions consiere here, because most of the energy of these signals is concentrate in a few boune time-varying frequency regions, corresponing to formants. 4 A possibly more robust version of this approach consists in keeping the ientifie columns corresponing to all available single-source areas, an then clustering them, so as to eventually erive each column of o as an average of all its ientifie occurences. It shoul also be note that the consiere time-frequency tools, which are presente in Subsection 3.2, are originally efine in a eterministic framework. This paper uses them in such a framework an concerns two cases i.e: i) either the consiere original sources are eterministic or ii) they are ranom processes, but in the latter case only a single realization of these processes is consiere (this is what is actually available in practice). The following iscussion then concerns this single, eterministic, realization an the tools an properties that we use only require such a realization frequency tools Many TF representations have been propose over the last fifty years an are e.g. presente in [2]-[3]. e here use the shorttime Fourier transform (STFT), esp. because it yiels low computional loa thanks to FFT algorithms an it oes not introuce interference terms, thus keeping the linear instantaneous mixing structure when applie to the consiere observe signals. The STFT of a signal p is obtaine by first multiplying the signal p J by a shifte winowing function q+^ Jr, centere at time. This yiels the winowe signal p J )q ^ J r. This signal epens on two time variables, i.e. the selecte time where the local spectrum of p J is analyze an the running time J. The STFT of p( is then efine as the Fourier transform of the above winowe signal, i.e: ]t, kumv_w p ]J )q ^ ]J r )x6!yz { ]J} (7) 6 w ;$tq is then the contribution of the consiere signal p in the TF area corresponing to the short time winow associate with an to the angular frequency t Assumptions an efinitions The same assumptions an efinitions as in Subsection 2. are consiere hereafter, except that i) we eventually use a eterministic framework, as explaine above, an ii) the temporal concepts use in Subsection 2. are here replace by their TF counterparts, i.e: Definition -TF: a source is sai to occur alone in a TF area if only this source is such that its centere TF transform is not equal to zero everywhere in this TF area. Definition 2-TF: a source is sai to be visible in the TF omain if there exist as least one TF area where it occurs alone. Assumption -TF: i) each source is visible in the TF omain an ii) there exist no TF areas where the centere TF transforms of all sources are equal to zero everywhere. More precisely, each value of a TF transform correspons to a single TF point, associate to a single angular frequency t an to a complete time winow efine by the selecte analysis time an consiere finite-length winowing function q ^ ), as shown in Subsection 3.2. The BSS metho that we propose below then uses means associate to these TF transforms, compute over analysis zones which consist of ajacent TF points. An analysis zone may have any shape in the TF omain. e here focus on the case when it forms a temporal line, i.e. when all its points correspon to the same frequency t an to ajacent (half-overlapping) time winows. The latter winows resp. correspon to a iscrete set of ~ time positions $, with =? ~. This set is enote as hereafter. Each analysis zone is then specifie in terms of the couple ƒ]t,, which completely efines the part of the TF omain associate to this analysis zone.

4 Y Š Š The propose BSS metho then uses the following parameters, associate to the above-efine analysis zones. For any signal, whose TF transform is enote T ;]t,, the mean of its TF transform over the consiere analysis zone is: T $tq? = ~ˆ T $ ]t, (8) : Similarly, for any couple of signals an, whose TF transforms are enote ]t, an ;]t,, the cross-correlation of the (centere versions of the) TF transforms of these signals over the consiere analysis zone is measure either by Š,!Œ ƒ]t, = ~Ž b ]t, r b $tq $# :+ ]t, r ƒ)t, $# ^ () or by the corresponing TF local cross-correlation coefficient: Œ ƒ]t, a Š Š Œ $tq ƒ)t, Š Œ! Œ ƒ]t, (2) The source uncorrelation assumption of Section 2 is then replace by: Assumption 2-TF: over each analysis zone $tq, the (centere) TF transforms of the sources are uncorrelate, i.e: Š, ] ƒ]t, (ep< ma Propose approach As we progressively introuce most require concepts in the previous sections an subsections, the resulting propose TF-BSS metho may now be irectly presente, as the TF aaptation of the overall temporal approach that we escribe at the en of Section 2. It therefore consists of the same 3 stages as the latter approach, aapte to the TF context however, an therefore precee by a pre-processing stage, i.e:. The pre-processing stage consists in eriving the STFT transforms ]t, of the mixe signals, accoring to (7). 2. The etection stage is base on the following property, shown in the Appenix: a necessary an sufficient conition for a source to occur alone in the TF analysis zone ƒ]t,, is: VX VY ƒ]t, >= ep< RfPgm<?gn, (2) In this stage, we therefore etect the analysis zones where a single source occurs by checking in which zones VX VY )t, is very close to for all < SfPgm<Ggh. 3. In the ientification stage, we then ientify the columns of O in single-source analysis zones. This is base on the same approach as in the temporal BSS metho, except that the parameter in (6) is here replace by 5 : VY}VX ƒ]t, VX VX $tq (22) which is equal to / in X/ i when only source i occurs in the consiere TF analysis zone. 5 In the ientification stage, the non-centere version of the crosscorrelation parameters Gš š œgžhÿ( may be use instea an is simpler. 4. In the combination stage, we eventually recombine the mixe signals accoring to (), in orer to obtain the extracte source signals. It shoul be note that the overall structure of this approach has some similarities with the TF approach that we propose in []- []. However, these two approaches use completely ifferent etection an ientification parameters. As the etection an ientification stages of any such approach are inepenent one from the other, we may also erive mixe approaches by using the etection metho of one of the two approaches resp. propose in this paper an in []-[]. an the ientification stage of the other approach. 4. TEST RESULTS e now present tests performe with two artificial linear instantaneous mixtures of two real speech signals, sample at 225 Hz. The mixing matrix is set to:.. (23) The corresponing matrix O is therefore equal to.. or.. (24) epening whether it correspons to a non-permute or permute version of the source signals. The consiere sources signals are shown in Fig.. Although the TF transforms of these source signals have significant ifferences (see Fig. 2 an 3), the TF transforms of the resulting mixe signals are quite similar (see Fig. 4 an 5), ue to the consiere har mixing conitions. Nevertheless, the TF BSS metho propose in this paper succees in separating these signals with a high accuracy, as will now be shown. As an example, the propose approach has been applie with 256 samples per STFT winow an 8 such winows per analysis zone. The estimate matrix O 3 thus obtaine is equal to...8. (25) which is extremely close to the first expression in (24). Consequently, the estimate output signals are almost equal to the (scale an non-permute) sources, as confirme by Fig. 6. More precisely, the Signal/Noise Ratios (SNR s) obtaine for these two estimate sources are resp. equal to 68. an 6.2 B, while the SNR s in the observe mixe signals are equal to 3.6 an B. This clearly emonstrates the high separation capability of the propose approach. 5. CONCLUSION In this paper, we introuce two versions of a correlation-base BSS approach. hereas its first version operates in the time omain, its secon version is base on TF analysis an thus applies to much more general conitions. The latter approach, calle TIFCORR for TIme- CORRelation-base BSS, consists in ientifying the columns of the (permute scale) mixing matrix in TF areas where this metho etects that a single source occurs. e experimentally showe that it yiels very goo performance

5 for linear instantaneous mixtures of real speech sources. Our future investigations will first consist in a more etaile characterization of the experimental performance of the two approaches that we propose in this paper. e will also compare, both from the theoretical an experimental points of view, the TIFCORR metho to the TF ratio-base approach that we introuce in []- [], which has the same structure by uses completely ifferent parameters, as explaine in Section 3. e will also aim at extening the approach propose in this paper to convolutive mixtures. 6. REFERENCES [] A. Hyvarinen, J. Karhunen, E. Oja, Inepenent Component Analysis, iley, New York, 2. [2] J.-F. Caroso, Blin signal separation: statistical principles, Proceeings of the IEEE, vol. 86, no., pp , Oct. 8. [3] A. Belouchrani, M.G. Amin, Blin source separation using time-frequency istributions: algorithm an asymptotic performance, Proc. ICASSP 7, pp , Munich, Germany, April 2-24, 7. [4] A. Belouchrani, M.G. Amin, Blin source separation base on time-frequency signal representations, IEEE Transactions on Signal Processing, vol. 46, no., pp , Nov. 8. [5] A. Holobar, C. Févotte, C. Doncarli, D. Zazula, Single autoterms selection for blin source separation in timefrequency plane, Proc. EUSIPCO 22, Toulouse, France, Sept 3-6, 22. [6] A. Jourjine, S. Rickar, O. Yilmaz, Blin separation of isjoint orthogonal signals: emixing N sources from 2 mixtures, Proc. ICASSP 2, vol. 5, pp , Istanbul, Turkey, June 5-, 2. [7] R. Balan, J. Rosca, Statistical properties of STFT ratios for two channel systems an applications to blin source separation, Proc.ICA 2, Helsinki, Finlan, June 8-22, 2. [8] S. Rickar, R. Balan, J. Rosca, Real-time time-frequency base blin source separation, Proc.ICA 2, pp , San Diego, California, Dec. -3, 2. [] F. Abrar, Y. Deville, P. hite, A new source separation approach base on time-frequency analysis for instantaneous mixtures, Proc. ECM2S 2, pp , Toulouse, France, May 3 - June, 2. [] F. Abrar, Y. Deville, P. hite, From blin source separation to blin source cancellation in the uneretermine case: a new approach base on time-frequency analysis, Proc. ICA 2, pp , San Diego, California, Dec. -3, 2. [] L. Giulieri, N. Thirion-Moreau, P.-Y. Arquès, Blin sources separation using bilinear an quaratic time-frequency representations, Proc. ICA 2, pp , San Diego, California, Dec. -3, 2. [2] F. Hlawatsch, G.F. Boureaux-Bartels, Linear an quaratic time-frequency signal representations, IEEE Signal Processing Magazine, pp. 2-67, April 2. [3] L. Cohen, -frequency istributions - a review, Proceeings of the IEEE, vol. 77, no. 7, pp. 4-8, July 8. A. APPENDIX e here show the valiity of the etection criteria (4) an (2) resp. use in the temporal an TF versions of the propose BSS approach. In the frame of the temporal version of this approach escribe in Section 2, by using the corresponing assumptions it may be shown that, for the mixe signals efine in (4), the cross-correlation coefficient efine in (3) may be expresse as: UVX VY?'!, (26) an resp. stan for the inner a -imensional vector is equal to /, with:? ZP[! ^ ` (27) where the notations prouct an vector norm, an where the A -th component of each Applying Schwarz inequality to (26) then shows that UVX VY g = e\< =gh<8ghq (28) with equality if an only if an are epenent. Let us now analyze this conition at a given time, epening. Due on the values of the source parameters baª«=n to Assumption -ii, at least one of the values is not equal to zero. If only one is not equal to zero, then all vectors are clearly epenent, so that equality hols for all of them in (28) an therefore the etection conition (4) is fulfille. The only case that remains to be consiere is then the situation when at least two values an i are not equal to zero. It may then be shown that if b an were epenent for all < f g < g, then the columns with inices A an ± of the mixing matrix woul be epenent, which is not true ue to Assumption 4-i. Therefore, in the consiere case, at least one pair of vectors } b oes not consist of epenent vectors, so that U VX V Y = an the etection conition (4) is not fulfille. As an overall result, this conition is fulfille if an only if exactly one of the values is not equal to zero at the consiere time, i.e. if only one source occurs at that time. Now, as for the TF version of this approach escribe in Section 3, by using the corresponing assumptions it may be shown that, for the mixe signals efine in (4), the cross-correlation coefficient over an analysis zone efine in (2) may again be expresse accoring to the right-han sie of (26) except that the vectors in (26) here epen on the consiere analysis zone ƒ]t,, an with the same notations except that is replace by ƒ]t,? Š, ƒ)t, (2) with Š ] $ ƒ]t, efine accoring to (). This leas to the same iscussion as above, except that the consiere time is replace by the consiere analysis zone $tq. This eventually yiels the etection criterion (2).

6 2 value of source 2 8 value of source sample inex x sample inex x 4 Fig.. Sample values of both sources Fig. 4. -frequency transform of first mixe signal Fig. 2. -frequency transform of first source. Fig. 5. -frequency transform of secon mixe signal. 2 8 value of output sample inex x Fig. 3. -frequency transform of secon source. value of output sample inex x 4 Fig. 6. Sample values of both estimate source signals.