BLIND SOURCE SEPARATION BASED ON SPACE-TIME-FREQUENCY DIVERSITY. Scott Rickard, Radu Balan, Justinian Rosca

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1 BLIND SOURCE SEPARATION BASED ON SPACE-TIME-FREQUENCY DIVERSITY Scott Rckard, Radu Balan, Justnan Rosca Semens Corporate Researc, 7 College Road East, Prnceton, NJ 8 ABSTRACT We nvestgate te assumpton tat sources ave dsjont support n te tme doman, tme-freuency doman, or freuency doman We call suc sgnals dsjont ortogonal Te class of sgnals tat appromately satsfes ts assumpton ncludes many syntetc sgnals, musc and speec, as well as some bologcal sgnals We measure te dsjont ortogonalty of te bencmark sgnals n te ICALAB Toolbo n te tme, tme-freuency, and freuency domans and sow tat most satsfy te assumpton n at least one representaton In order to compare ts assumpton wt oter common source assumptons, we derve a demng algortm for nosy nstantaneous mtures based on dsjont ortogonalty and compare ts performance to te algortms n te ICALAB Toolbo, all of wc rely on te second-order statstcs, non-statonarty, or gerorder statstcs of te sources Te results ndcate tat space-tmefreuency dversty s a useful assumpton for te desgn of BSS/ICA algortms INTRODUCTION Blnd source separaton and ndependent component analyss algortms leverage te knowledge tat te sources satsfy certan statstcal or determnstc condtons n order to perform te separaton Amar and Cock [] lst four common source property assumptons tat form te bass for most BSS/ICA algortms: Hger-order statstcs (HOS) Sources are statstcally ndependent Ts s usually practcally enforced by lookng to te t order moments or cumulants of te mtures 2 Second-order statstcs (SOS) Sources are decorrelated 3 Non-statonarty and SOS (NS) Sources are decorrelated and ave tme-varyng varances Space-tme-freuency dversty (STF) Sources are dsjont n te tme doman, tme-freuency doman, or freuency doman Ts s te assumpton we analyze n ts paper An alternatve STF assumpton s presented n [2] Most metods fall nto one of te frst tree categores, and few tecnues make use of space-tme-freuency dversty For eample, te ICALAB Toolbo [3], a software program tat allows one to compare te performance of BSS/ICA algortms, contans 9 BSS/ICA metods, none of wc can be classfed as a STF metod Surprsngly, owever, of te 7 bencmark sgnal famles contaned n te ICALAB Toolbo, we wll sow tat of tem possess a large degree of tme-freuency dversty Moreover, none of te tecnues (ncludng te one we present ere) are able to dem te two bencmark famles wc do not possess a g level of Scott Rckard s also wt te Program n Appled and Computatonal Matematcs, Prnceton Unversty In Proceedngs of te t ICA-BSS Conference, Aprl 23, Japan tme-freuency dversty So, all te practcally demable ffteen bencmarks are tme-freuency dverse Specfcally, n ts paper we analyze te assumpton tat te sources ave dsjont support n eter te tme doman, freuency doman, or te tme-freuency doman We call sgnals for wc tere ests an nvertble lnear transform suc tat n te transform doman te sgnals ave dsjont support dsjont ortogonal Wen te transform s te wndowed Fourer transform, we call te sgnals W-dsjont ortogonal For suc dsjont ortogonal sources, we derve a separaton algortm Wle te dsjont ortogonal source assumpton may seem too restrctng, we argue tat t s n practce appromately satsfed by many sgnals of nterest Specfcally, tme-dvson multpleed communcaton sgnals are by desgn tme doman dsjont, freuencydvson multpleed communcaton sgnals are by desgn freuency doman dsjont, and te goal of freuency opped CDMA sgnals s tat te sgnals are dsjont n te tme-freuency doman Addtonally, peraps surprsngly, speec sgnals are W-dsjont ortogonal enoug to allow for accurate mng parameter estmaton and blnd separaton [] Indeed, as we wll sow, musc and speec, as well as some bologcal sgnals, are appromately W-dsjont ortogonal For te separaton algortm, we consder an addtve nose mng model wt an arbtrary number of sensors and possbly more sources tan sensors (te degenerate separaton problem ) Te bass for our approac to nosy model estmaton by mamum lkelood, under te nstantaneous mng assumptons, s tat te sources are dsjont ortogonal Te mplementaton of te derved crteron nvolves teratng two steps: a parttonng of te tme-freuency plane for separaton followed by an optmzaton of te mng parameter estmates Te soluton s applcable to an arbtrary number of sensors and sources Tat s, one can dem by convertng te parttoned tme-freuency representatons back nto te tme doman However, n order to compare wt te oter metods n te ICALAB Toolbo, we wll use te estmate of te mng matr and perform standard nverse mng matr demng Epermentally, we sow te capablty of te tecnue on te ICALAB data Te organzaton of te paper s as follows Secton 2 presents te sgnal mng model and Secton 3 provdes epermental motvaton of te W-dsjont ortogonalty sgnal model Secton sows te dervaton of te ML estmator of mng parameters and source sgnals, and ts mplementaton by an teratve procedure Te algortm performance on te ICALAB bencmarks s compared to te performance of te oter ICALAB tecnues n Secton

2 C - j % X s X ˆ ˆ 2 MIXING MODEL AND SIGNAL ASSUMPTION 2 Te Mng Model Consder te measurements of source sgnals by nstantaneous mng model: sensors n an! &%' )( *+ () " #$ *, were -/3276 s te nstantaneous mng matr We assume - as 8 full8 rank, :9 8 wc 8; ensures space dversty We denote by <=>?$@!AB6, C3?D@EAB6, FGH?$@EAB6 te wndowed Fourer transform of sgnals I+>3JK6@ML>33JM6, and NO>3JK6, respectvely, wt respect to a wndow P3JM6, were? s te frame nde, and A te freuency nde Wen no danger of confuson, we sall drop te arguments?d@ea n <QH@>C and FG Te mng model () s tus <?$@EAB6+ or, more compactly, were <U@EF R 2 S C?D@EAB6TUF?$@VA)6W@OXZY\[GY& (2) <&?D@!AB6]&-^C_a`b_!?D@EAB6TQF&?D@!AB6 (3) are te -vectors of components and C _a`b_ s te -vector of components C; We sall denote by -c te d t column of -, -egf - KKMK Our problem s: gven measurements 3I 3JK6, kkk>k, I 3JM66 we want to determne te ML estmates of te mng parameters -no ml and te source sgnals L 3JM6, kkkkk, L 3JK66 In order to solve ts we rely on te W-dsjont ortogonalty 8 8lof te sources and te assumpton tat te sensor noses are ndependently dstrbuted and ave Gaussan dstrbutons wt zero mean and pr varance 22 Te W-Dsjont Ortogonal Sgnal Model In [] we called two sgnals L and L W-dsjont ortogonal (W DO), for a gven wndowng functon P3JM6, f te supports of te wndowed Fourer transforms of L and L are dsjont, tat s: For sources C?D@EAB6C?$@EAB6+\st@vu,?$@VA (),kkkmk,c te assumpton generalzes to: Cw!?D@EAB6CmK?D@EAB6]&s@^uyXY&zZ{ } cy\ ~@;u]?d@ea () Suc a determnstc constrant s not only rarely satsfed, but t also mples tat te sgnals are, n general, statstcally dependent, wc s easly proved by te fact tat te condtonal dstrbu- C s6)t $L { 6 s dfferent from te condtonal ton C C ƒl L C \s6 In [6] t as been notced, owever, tat relaton () s satsfed n an appromate sense by real speec sgnals Tus, () can be seen as te matematcal dealzaton of te condton tat eac mture tme-freuency pont wt sgnfcant power s most often domnated by a sngle source Te case of sngle source domnance of speec mtures n te tme-freuency doman as been notced and utlzed several tmes [7, 8, 9,,, 2] It was also sown n [3] tat () s te lmt of a stocastc source model 3 W-DISJOINT ORTHOGONALITY OF THE BENCHMARKS We proposed n [6] te normalzed dfference between te sgnal energy contaned n te domnant tme-freuency ponts of a sgnal n a mture and te nterference energy n tose ponts as a measure of W-dsjont ortogonalty In order to measure W-dsjont ortogonalty for a sgnal of nterest n a mture for a gven representaton, we partton te ponts (n te TD, TF, or FD representaton) of te mture nto tose domnated by te source of nterest and tose domnated by te nterference We defne te mask wc s te ndcator functon of te domnant tme-freuency ponts for source were ˆ H?D@EAB6 s te nterference, ˆ K?$@EAB6+Š w wœ R R CmK?D@!A)6 ƒˆ >?D@EAB6 oterwse (6) CwE?D@EAB6k (7) Now we defne two mportant performance crtera: () ow well te domnant tme-freuency ponts preserve te source of nterest, and (2) ow well consderng only te domnant tme-freuency ponts suppresses te nterferng sources We defne te preservedsgnal-rato (PSR) of a source n a mture as PSR3 6OŽ >?$@VA)6C"K?$@EAB6 C"H?D@EAB6 wc measures te percentage of energy of source contaned n ts domnant tme-freuency ponts We defne te sgnal-to-nterference rato of te domnant tme-freuency ponts of a source n a mture, SIR3 6 Ž >?$@EAB6C">?D@EAB6?D@EAB6?$@EAB6 (8) k (9) Tese two crtera, te PSR and SIR, are combned to form te measure of W-DO WDO3 6]}?D@EAB6C">?D@EAB6 +?D@EAB6 M?$@EAB6 () C"H?D@EAB6 PSR3 6 PSR3 6 SIR3 6k () For sgnals wc ave dsjont support, we note tat PSR X, SIR, and tus WDO X Moreover, WDO X mples tat PSR X, SIR, and tat te sgnal as dsjont support compared to te nterferng sources In general, sey PSR Y X, SIR \s, and WDO YX (and can be negatve) In order to summarze te W-dsjont ortogonalty of a famly of sgnals, we look to te average WDO and te mnmum WDO, defned as follows, R WDO3 6 (2) mwdo \ c š WDO3 6 (3) awdo Fgure lsts te 7 bencmark sgnal famles from te ICALAB Sgnal Processng Toolbo (Verson ) [3] Bencmarks ACsnd, ACvsparse, and Speec are dsplayed n Fgure 2 We measure te W-dsjont ortogonalty of te bencmarks for tree wndow szes: one sample, 2 samples, and te sgnal lengt Tese tree szes correspond to te tme doman (TD), tme-freuency doman (TF), and freuency doman (FD) representatons of te sgnal In te TD case, te A freuency nde s meanngless and n

3 ¾ Ð ¾ ¾ - Ò Î te FD representaton te? tme nde s meanngless However, for ease of notaton and reference, we wll use?d@!a)6 and refer to tme-freuency ponts for all tree representatons Te average and mnmum WDO for te ICALAB bencmarks are lsted n Fgure 3 for te tree representatons of te sgnals Note tat all but 2 (AC-7sparse and EEG9) ave eual to or greater tan % average WDO For eac sgnal, te largest awdo and mwdo s glgted n bold ACsnd ACsnd ACsparse ACvsparse ABo7 Sergo7 AC-7sparse acspeec6 Speec Speec8 Speec Speec2 alo 2depspeec nband Gnband EEG sne waves sne waves sparse bell-saped sources very sparse spkng sgnals 7 typcal bologcal sgnals 7 random sources, some asymmetrcally dstrbuted sources mtures of 7 from ACsparse 6 typcal speec sgnals speec and musc sources 8 speec and musc sources speec and musc sources 2 speec and musc sources speakers sayng te same tng 2 speakers sayng te same tng narrow band sources fort order colored sources 9 EEG sgnals Fg ICALAB bencmarks Fg 2 Bencmarks ACsnd, ACvsparse, and Speec TD TF FD bencmark awdo mwdo awdo mwdo awdo mwdo ACsnd ACsnd ACsparse ACvsparse ABo Sergo AC-7sparse 3 2 acspeec Speec Speec Speec Speec alo depspeec nband Gnband EEG Fg 3 Average and mnmum WDO for te ICALAB bencmarks Most of te sources ebt a g level of dsjont ortogonalty, but we need to, gven only te mtures, determne n wc representaton (TD, TF, or FD) te sources are most W-DO One approac would be to run te algortm descrbed n te net secton tree tmes, once n eac doman and ten coose te soluton wt te most W-DO outputs Alternatvely, we can measure ow sparse te mtures are n eac representaton and ten run te algortm on te most sparse representaton Te logc n ts s tat te dsjont ortogonalty comes from, n general, eac sgnal avng a sparse representaton and te few large coeffcents of eac source not overlappng wt one anoter Because of lnearty, sparse sgnal representatons sould lead to sparse mture representatons and we ope tat te most sparse mture representaton corresponds to te representaton wt te largest W-DO In Append A, we sow tat ts logc olds true for te speec and musc bencmarks and some of te syntetc bencmarks, but fals for some of te syntetc bencmarks THE MAXIMUM LIKELIHOOD ESTIMATOR OF SIGNAL AND MIXING PARAMETERS In ts secton we derve te jont mamum lkelood estmator of parameters and source sgnals under assumpton () Te source sgnals naturally partton te tme-freuency plane nto dsjont were eac source sgnal s non-zero (e actve) Tus te sgnals are gven by te collecton,kkkmk, and one comple varable C tat defnes te actve sgnal: C3?D@EAB6+tC]?D@EAB6Xœ"ž!?$@VA)6 () Let te model parameters Ÿ consst of te mng parameters - /3236, te partton Œ6 and C Its lkelood and mamum log-lkelood estmator are gven by: 8 8 r ; R & œ ž «M ±³²; >µ ¹aº M b µ^» ¼ ¹3º >½ argmnàdá 9R "Œª Á 3 œ ž b µc» ¼ () For any 9 $aª 6 we defne te selecton map ÂÄà TF-plane ~È, Â?D@!AB6ZÉd ff?$@!ab6êg Clearly  defnes a unue partton Optmzng over C n () we obtan ¾ ¼»:Ë» (6) were d tâ?d@!ab6 Insertng (6) nto (), te optmzaton problem reduces to:» ¾ ¹ ̾ argmaí Ï» ¹ Ì (7) were: 3-c@M 6 - Ë Î 9 Î <&?D@EAB6 9 $ 9 $ (8) Note te crteron to mamze 9 $ depends on a set of contnuous parameters -, and a selecton map  A typcal optmzaton algortm for suc a crteron works as follows Te optmzaton s done n two steps: frst te optmzaton over te contnuous parameters, and ten te optmzaton over te selecton map (or, euvalently, te partton) Suc a procedure s terated untl te crteron reaces a saturaton floor Because te crteron s bounded above, we are guaranteed t wll converge Net we descrbe solutons for te two optmzaton problems Optmal Partton, te op- Gven a set of mng parameters, -ÑÉ32 6 tmal selecton map s smply gven by 8 8 :9 8 8; Ì ¹3º» ¹Œº argma>ò Ë Ò» Te partton s ten mmedate: ƒæ³?d@!a)6 Â?D@EAB6+}dÈ (9) 2 Optmal Mng Parameters Now gven a partton Œ6, te optmal mng parameters are obtaned ndependently for eac d by: 8 8;» argmaí žôó œ ž 9 $aª» ¹3º Ë» > Ò (2)

4 Ö Ð Denote Ë (2) œ ž 9 $aª Ö -c argmaí - ž Ë - (22) -c Ten te optmzaton problem turns nto: wose soluton s te man egenvector of te symmetrc and nonnegatve matr, -cƒ 3 ML Algortm Ö v &ØcÙ³k Ú³k ŒÛÜ&ØOÛ@Ý7ÞßWÙÞ cà Ûe{ &s (23) Summng tese fndngs, te optmzaton algortm becomes:, and selec- Step Intalze Ð, -ãâbg32sä6 8 8 r9 8 8 Set LB\s, \s, and coose a stoppng Eç tresold æ ; Eç Step Fnd te optmal partton 6 ton map,  by solvng (9) wt -e&- Eç 8 ;!ç 8; Step 2 Fnd te optmal parameters - partton g, and all XZY\dOY\ Ð, ; Ð, ÐWè Step 3 Set L= L)TƒX Ð,, and compute 6 æ ten go to Step ; oterwse: Step Te et values are -ÑÜ-, and,é wt, for nstance random values (or wt te metod presented n Append B); from (23) for te 6 If, obtaned after L teratons Te source sgnal are ten computed by convertng te estmated tme-freuency representatons back nto te tme doman Te core of ts algortm s essentally te same as tat presented n [9] It can also be seen as a specfcaton to nstantaneous mtures of te anecoc mng metod presented n [3] BSS Algortm (STF-ER) In order to compare wt te tecnues n te ICALAB Toolbo, we do not dem va parttonng, but rater use te mng matr estmate and standard mng matr nverson demng Te overall BSS algortm based on space-tme-freuency dversty wc operates n te most effcent representaton (STF-ER) s as follows: Measure te 9% effcency as descrbed n Append A and select te most effcent representaton For te mtures n te most effcent representaton, ntalze te mng matr estmate by clusterng a random selecton of te tme-freuency ponts wc make up te 9% effcency as descrbed n Append B Loop troug Steps 3, measure te awdo and mwdo of te estmated outputs after eac mng matr reestmaton After convergence of te crteron (or a fed number of loops), nvert te mng matr estmate correspondng to te largest sum of awdo and mwdo, and apply t to te mtures to produce te orgnal source estmates EXPERIMENTAL RESULTS We frst tested te STF-ER algortm on suare mtures of te sgnals n of te 7 ICALAB bencmarks No algortm was able to dem eter AC-7sparse or EEG9 because tey bot contan nearly dentcal source sgnals, so tese bencmarks were elmnated from te test set ACsnd, SpeecÆ,8,2È, and 2depspeec were also not consdered because oter sgnals contaned n te test set were very smlar For te remanng sgnals, n order to sow tat te metod presented ere as te possblty of workng, we ntalzed te metod wt te partton assgnng eac tme-freuency pont to te correspondng largest magntude orgnal source at tat tme-freuency pont For W-DO sources, ts s te optmal partton We ten performed one Step 2 mng matr estmaton and stopped Ts non-blnd algortm was run n te tme doman (STF- TD-OP), tme-freuency doman (STF-TF-OP), and freuency doman (STF-FD-OP) STF-ER-OP selects te most effcent representaton and ten runs te approprate metod Te results, sown n Fgure demonstrate tat parttons do est wc allow for demng More detals concernng te Performance Inde (PI) measure of demng performance can be found n [] and [3] Lower PI scores are better, zero mples perfect demng, and, for our purposes, a PI score less tan ndcates good demng performance For te comparson eperments we tested STF-ER aganst te 9 algortms n ICALAB, wc are lsted n Fgure For te tests, all algortms were run wt te default parameter settngs For a detaled descrpton and dscusson of te algortms consult [] and [3] For STF-ER, Steps 3 were looped tmes and te entre algortm was run 9 tmes wt te demng matr producng te outputs wt te largest sum of awdo and mwdo beng te one selected We frst tested te algortms usng te dentty matr as te mng matr Results are presented n Fgure 6 Te purpose of tese tests was to epose te algortms wt source assumptons nconsstent wt te propertes of te bencmarks As te mtures are already demed, te correct algortm beavor would be to leave tem unaltered, but as te results ndcate, ts rarely appens In fact, EVD2 s te only algortm tat as a PI of less tan (good demng performance) for all bencmark fles SOBI, SOBI-RO, JADETD, SANG, and STF-ER all dem seven of te ten bencmarks NG-FICA faled to dem (PI ) any of te bencmark fles We also tested te algortms on random suare mtures; Te results are presented n Fgure 7 STF-ER-OP demed 9 of te bencmarks consstently wt ACsparse beng te one bencmark t faled to dem STF-ER-OP demed ACsparse nto a seres of snsods nstead of a number of tme dsjont bell-saped bumps In fact, several of te tecnues proposed ts alternatve demng On te oter and, several of te tecnues wc faled on ACsnd dd so because tey proposed demtures tat looked lke ACsparse JADETD and SANG bot consstently dem 7 of te bencmarks STF-ER consstently demes 6 of te bencmarks, and s te only metod to dem te tme dsjont ACvsparse and te freuency dsjont ACsnd EVD2 (wc demed all bencmarks n te dentty case) and NG-FICA bot faled n ts demng test on all of te bencmark fles AMUSE BSS SVD EVD2 SOBI SOBI-RO SOBI-BPF SONS EVD2 JADEop JADETD FPICA Pearson opt SANG NG-FICA NG-OL ERICA SIMBEC UNICA FOBI-E Algortm for Multple Unknown Source Etracton based on EVD BSS SOS algortm based on SVD BSS SOS algortm based on symmetrc EVD Second Order Blnd Identfcaton Robust SOBI wt Robust Ortogonalzaton Robust SOBI wt bank of Band-Pass Flters Second Order Nonstatonary Source Separaton BSS SOS+FOS algortm based on symmetrc EVD Robust Jont Appro Dagonalzaton of Egenmatrces wt optmzed numercal procedures HOS Jont Appromate Dagonalzaton of Egen matrces wt Tme Delays Fed-Pont ICA Pearson system optmzed Self Adaptve Natural Gradent algortm wt nonolonomc constrants Natural Gradent - Fleble ICA On-lne adaptve Natural Gradent Euvarant Robust ICA - based on Cumulants SIMultaneous Blnd Etracton usng Cumulants Unbased uas Newton algortm for ICA Fourt Order Blnd Identfcaton wt Transformaton matr E Fg BSS/ICA Algortms n ICALAB [3] EVD = egenvector decomposton FOS = fort order statstcs

5 6 SUMMARY We ave nvestgated te assumpton tat sources ave dsjont support n te tme doman, tme-freuency doman, or freuency doman as a bass for blnd source separaton Tests on te bencmarks sgnals n te ICALAB Toolbo reveal tat, peraps surprsngly, most of tem ebt a large degree of W-dsjont ortogonalty n at least one doman Based on ts assumpton, we derved a blnd separaton algortm and tested t usng te ICALAB bencmarks Te results sow tat tere est parttons of te doman wc result n near perfect mng matr estmaton Iteratve blnd estmaton of te partton results n performance comparable to oter establsed BSS/ICA metods Ts dsparty n potental performance and actual performance suggests tat future work n space-tme-freuency metods may produce etremely powerful blnd source separaton metods

6 alg ACsnd ACsparse ACvsparse ABo7 Sergo acspeec6 ACSpeec alo nband Gnband STF-TD-OP STF-TF-OP STF-FD-OP STF-ER-OP Fg Demng Performance Inde (PI) for ICALAB bencmarks for dentty matr mng gven W-DO optmal partton Ë Ë alg ACsnd ACsparse ACvsparse ABo7 Sergo acspeec6 ACSpeec alo nband Gnband AMUSE BSS SVD EVD EVD SOBI SOBI-RO SOBI-BPF SONS skéê³ëê JADEop JADETD FPICA Pearson opt SANG NG-FICA 6 skéé³éé NG-OL ERICA SIMBEC UNICA FOBI-E STF-ER Fg 6 Demng Performance Inde (PI) for ICALAB bencmarks for dentty matr mng A * ndcates tat nose was added to avod program eecuton error (Gaussan nose 2 db SNR) Tose wt PI less tan are n bold to sgnfy good demng performance alg ACsnd ACsparse ACvsparse ABo7 Sergo acspeec6 ACSpeec alo nband Gnband STF-TD-OP STF-TF-OP STF-FD-OP STF-ER-OP AMUSE BSS SVD EVD2 EVD2 SOBI SOBI-RO SOBI-BPF SONS JADEop JADETD FPICA Pearson opt SANG NG-FICA NG-OL ERICA SIMBEC UNICA FOBI-E STF-ER Fg 7 Algortm demng performance for random mtures Eac metod was tested on fve mtures med wt a randomly generated non-sngular mng matr A ndcates tat te PI score was less tan for all fve tests A ndcates te metod faled to dem wt a PI score less tan at least once n te fve tests

7 î ø ù ù Append A - Sparseness Measure We measure sparseness, te property tat a small percentage of te sgnal coeffcents (n eter TD, TF, or FD) captures a large percentage of te sgnal energy, as follows For a fed ì, í Ë 3ìr6 argmaî 9 $aï ð ñ Ten te ì effcency level s eff3ìo6+ <&?D@EAB6 \ì 9 $að ò Ƴ?D@EAB6+à 9 " <&?D@EAB6 $ó í Ë 3ìO6È Æ?$@!AB6È <&?D@!AB6 (2) (2) For eample, te 9% effcency would be te mamum percentage of components (clearly te smallest magntude ones) tat we can trow away wle stll mantanng at least 9% of te sgnal energy A tresold ndependent measure of effcent s, sumeff â eff3ìo63['ì (26) Fgure 8 sows te 9% effcency level and sumeff for te ICALAB bencmarks Eac bencmark was med usng te dentty matr before te effcences were calculated Comparng Fgure 8 to Fgure 3, we note tat te most effcent representaton corresponds to te mamum average WDO representaton n out of te demable bencmarks Te remanng four; ACsnd, ACsnd, Sergo7, and ACsparse are more effcently represented n one doman but more W-DO n anoter Ts dfference s most pronounced n te case of ACsparse, wc as sgnals consstng of tme dsjont bell-saped bumps Mtures of ACsparse appear snusodal Tus, te sgnals of ACsparse are sgnfcantly more W-DO n te tme doman but te mtures of ACsparse are more effcently represented n te tme-freuency doman TD TF FD bencmark 9% sumeff 9% sumeff 9% sumeff ACsnd ACsnd ACsparse ACvsparse ABo Sergo AC-7sparse acspeec Speec Speec Speec Speec alo depspeec nband Gnband EEG Fg 8 Te 9% effcency level and sumeff for te ICALAB bencmarks Tus, as we reure te W-DO assumpton, and we ws to select te best wndow for te sources n terms of W-DO gven te mture, we measure te effcency of te mture representatons and select te wndow sze tat s most effcent as we ope, based on te epermental results, tat n tat representaton te sources wll be mamally W-DO Append B - Algortm Intalzaton One way to ntalze te mng matr n Secton nstead of usng random values s to cluster a small random selecton of tmefreuency ponts from tose wc make up te 9% effcency For eac of tese ponts, we consder te dmensonal vector <&?$@VA)6 Under te W-DO assumpton, te <&?D@EAB6 sould be ôm-ã for some ôõê=ö for some È We construct a dstance matr between all pars of <&?D@EAB6 usng te followng metrc Te dstance between vector ø and ù s, [!ø@sù+6+ (27) wc as te mportant property tat [!ø@ù+6q s ff ùú í ø, for some 2 d3û³2ƒêö Tat s, pars of observatons wc le on a lne troug te orgn are consstent wt te W-DO assumpton Clusters are formed usng te parwse dstance matr and MAT- LAB s cluster functon [] We cluster on a small random selecton of ponts nstead of all te ponts because of tme constrants In practce, we randomly select 3 ponts from tose tme-freuency ponts makng up te 9% effcency (nstead of smply selectng te largest 3 tme-freuency components) because often te largest components wll be domnated by a subset of te sources Consderng ponts makng up te 9% effcency elps to ensure tat even te lower power sources ave representaton n te ntalzaton 7 REFERENCES [] A Ccock and S Amar Adaptve Blnd Sgnal and Image Processng: Learnng Algortms and Applcatons Wley, Aprl 22 [2] A Beloucran and M Amn Blnd source separaton based on tmefreuency sgnal representatons IEEE Trans on Sgnal Processng, 6(): , November 998 [3] A Ccock, S Amar, and K Swek et al ICALAB Toolboes, ttp://wwwbspbranrkengojp/icalab [] O Ylmaz and S Rckard Blnd separaton of speec mtures va tme-freuency maskng IEEE Trans on Sgnal Processng Submtted November 22 [] S Rckard, R Balan, and J Rosca Real-tme tme-freuency based blnd source separaton In Proc ICA, pages 6 66, 2 [6] S Rckard and O Ylmaz On te W-dsjont ortogonalty of speec In Proc ICASSP, volume, pages 29 32, 22 [7] P Bofll and M Zbulevsky Blnd separaton of more sources tan mtures usng sparsty of ter sort-tme Fourer transform In Proc ICA, pages 87 92, Helsnk, Fnland, June [8] M Aok, M Okamoto, S Aok, and H Matsu Sound source segregaton based on estmatng ncdent angle of eac freuency component of nput sgnals acured by multple mcropones Acoust Sc & Tec, 22(2):9 7, 2 [9] L-T Nguyen, A Beloucran, K Abed-Meram, and B Boasas Separatng more sources tan sensors usng tme-freuency dstrbutons In Int Symp on Sg Proc and ts Applcatons (ISSPA), pages 83 86, Kuala Lumpur, Malaysa, August [] S T Rowes One mcropone source separaton In Neural Informaton Processng Systems 3 (NIPS), pages , 2 [] B Berdugo, J Rosenouse, and H Azar Speakers drecton fndng usng estmated tme delays n te freuency doman Sgnal Processng, 82:9 3, 22 [2] P Bofll Underdetermned blnd separaton of delayed sound sources n te freuency doman preprnt, 22 [3] R Balan, J Rosca, and S Rckard Scalable non-suare blnd source separaton n te presence of nose In sent to ICASSP23, Hong- Kong, Cna, Aprl 23 [] MATLAB Statstcs Toolbo, ttp://wwwmatworkscom ø