A Signal Subspace Rotation Method for Localization of Multiple Wideband Sound Sources

A SIGNAL S UBSP A CE R OT A TION ME THOD FOR LOCALIZA TION OF MUL TIPLE WIDEB AND SO UND SOURCES Kainan Chen, 1 W enyu Jin, 1 , 2 Bharadwaj Desikan, 1 1 Huawei German Research Center , Munich, Germany { kainan.chen, bharadwaj.desikan } @huawei.com 2 Department of Algorithm, Starke y Hearing T echnologies, Eden Prairie, United States wenyu.jin@ieee.org ABSTRA CT In this paper, the problem of extending nar rowband mul- tichannel sound source localization algorithms to the wide- band case is addr essed. The DO A estimation of narrowband algorithm s is based on the estimate of inter-channel phase dif- ferences (IPD) between microph ones o f the so u nd sources. A new method for wideband soun d source DO A estimation based o n sign al subspa ce ro tatio n is present. The p roposed algorithm n ormalizes the narr owband signal statistics by ro- tating the estimated signal subspace to the wideban d coun- terpart in the eigenv ector domain. Then the wideban d DO A estimate can be obtained by estimatin g the no rmalized I PD from th ese wideband signal statistics. In additio n to requ iring less comp utational complexity compared to r epeating the n ar- rowband algo r ithms for all relev ant freq u encies of wideband signals, the pr oposed method also does not re quire any addi- tional prio r knowledge. The experimental results demon stra te the efficac y and the robustness o f th e pro posed method. Index T erms — Sou nd source localization, wideband, sig- nal subspace rotation 1. INTR O DUCTION Sound source loc a lization is an im portan t com ponen t in ma ny multichann el signal pro cessing systems aiming, e.g., at sou rce tracking, signal separation , enhanceme n t and noise suppres- sion [1, 2] . T ra d itional sou nd so u rce loc a lization alg o rithms such as GCC-PHA T [3] estimates the time delay of arriv a l (TDO A) between a pair of microphones to localize a sing le source. SRP-PHA T [4] as an extensio n can further local- ize multiple sources simultaneo usly . Anothe r grou p of algo- rithms th a t resolves th e simultan e ous multiple sou rce lo cal- ization pr oblem is based on h igh-reso lu tion subspace tech- niques, such as MUSIC [ 5] and ESPRIT [6] . For h ighly reverberant senarios, th e Indepen dent Com ponen t Analysis (ICA) based algorithms are prop osed [ 7, 8]. ESPRIT [6] exploits the algeb raic pro perties of the spatial covariance matrix. This m ethod feature s goo d performan c e for narrowband sign a ls. Howe ver , it is not directly applicab le to wideband signals. Hence, it pr ocess the wideband signals by estimatin g the individual narrowband results in th e Shor t T ime Fourier Transform (STFT) dom ain. Post-processing schemes such as the histogram meth od only exploit the nar- rowband estimatio n results while the mutual information between different frequency bins is not con sidered. Espe- cially for known typ e of the sou nd so u rces, su c h as speech, the process based on clustering STFT bins such as [9, 10] can further improve the accuracy . Many studies focus o n extension s of narrowband localiza- tion algorithms to wideband signals. The methods of [11, 12] transform signals to cylindr ical/spherical ha rmonics for c ir- cular/spher ic a l micro phon e arr ays. Then the rotational in vari- ances are n o rmalized through frequencies and the further es- timation via ESPRIT is b a sed o n wideband sign als. A Coher- ent Signal Subspace ( CSS) method that deri ves the so-called focusing m atrices based o n steering vectors w as introduced in [13]. The fo cusing matrices aim to adapt each narrowband signal covariance matrix to gen eralize to the wideband case. Theoretical analysis and exper iments show that the f ocus- ing matrices lead to a wideband MUSIC algorithm [13], and in [14, 15, 16], the f ocusing matrice s extend ESPRIT to wide- band signals. As pointed out in [ 17, 18], the focusing matrices method ad a pts the covariance matrices via linear tran sfo rma- tions that are depend e nt on fr equency and DO As. Therefor e, these extensions require a prio r i kn owledge of DOA estimates and o n the arra y manifo ld for initializing the transform ation matrices. T o av oid th is req u irement, the algorith m in [17] estimates an AR model for the transmission ch annel of the wideband signals as a priori knowledge for wideban d local- ization. The perfor mance of this algorithm cruc ia lly depend on how well the AR model is estimated. Howev er , the model that they proposed and the conv entional m odels are not suit- able for non-station ary scenarios. A novel metho d tha t adapts the n arrowband ESPRIT to wideband signals is intro duced in this work. The key idea is the rotatio n of the eigen vectors that span the signal sub- space by th e c orrespon ding frequ ency , and to reconstruct th e covariance m atrices for each fr e quency bin b ased on the ro- tated eigenvectors. In comparison with the focu sing matrices approa c h , the pro posed method utilize the wid eband signal second-o rder statistics wh ile d o es not require a priori kno wl- edge of the DO A wh ich is typically obtaine d by repeating nar- rowband MUSIC [ 16]. Experimen ts with simulated and real recordin gs show the advantages of the new method regarding perfor mance an d computation a l complexity . 2. SIGN AL MODEL W e assume a micro p hone array which fits the requiremen ts for using ESPRIT , i.e ., to loc alize Q sources. W e use P m i- cropho nes ( P > Q ) that for m a linear array with a unif orm spacing ∆ d an d with mutually uncorr e la te d zero- mean sensor noise. T h e en vir onmen t is assumed with free-field and far - field conditions. The sou rce-micr o phon e model in the STFT domain is defined as X p ( f i ) = Q X q =1 A p,q ( f i ) S q ( f i ) + N p ( f i ) , (1) where X p ( f i ) and N p ( f i ) denote th e obser vation and the noise at the p -th m icropho ne in the STFT doma in at f r e- quency bin f i , respectively , S q ( f i ) denotes the q - th source signal, and A p,q ( f i ) denotes the f requen cy re sponse of the propag ation path which are gi ven by the rows of A ( f i ) for q -th sour ce arri ving at the p -th microph one. The po wer spectral density matr ix for all Q sou rce signal compon ents at the i -th fr equency bin f i is den oted as R s ( f i ) and the power spectr al density matrix for the observed noise is denoted a s R n ( f i ) . The narrowband compo nent covariance matrix R ( f i ) of the micro phon e signals can be expressed as R ( f i ) = A ( f i ) R s ( f i ) A H ( f i ) + R n ( f i ) , (2) where ∗ H denotes the Hermitian transp o se, A ( f i ) denotes a P × Q matrix that captures the steerin g vector s at the fre- quency bin f i . The elem ent of A ( f i ) at the p − th row and q − th column, A ( f i ) p,q , can be expressed as A ( f i ) p,q = e − j 2 π f i ∆ t p,q , ∆ t p,q def = ( p − 1)∆ dc − 1 sin θ q . (3) Therefo re, the steering vectors are frequen cy-depend ent and the relatio nship between dif ferent freque n cy bins f 1 , f 2 can be expressed as, A ( f 1 ) ◦ f − 1 1 = A ( f 2 ) ◦ f − 1 2 , (4) where {∗} ◦ f − 1 describes elem ent-wise exponen tiation by f − 1 and f 1 , f 2 denote any tw o frequen cies below the spa tial aliasing frequ e ncy f a , f a = ⌊ c 2∆ d sin θ ⌋ , (5) where ⌊∗⌋ denotes a function that returns the next lower fre- quency bin. For our proposed ev olution o f ESPRIT to wideb and source localization, the first step is to estimate the narrowband steering vector . Then th e IPDs of each source are obtain ed from the estima tio n result. Therefo re, the IPDs are d epend- ing on f r equency and ESPRIT has to b e repeated f or each narrowband to localize wid eband sou rces, a s it is also the case for other narrowband localization algor ithms [5, 19]. A solution to estimate a unique I PD for each source thr ough the frequen cy b ins is to rotate the estimated steerin g vector based on (4) as it is shown in the following. 3. P ROPOSED APPR OA CH The proposed approach co nsiders a novel way for adaptin g ESPRIT to a single or multip le wideband sources. In ca se of a single sou rce scenario, for each nar rowband componen t, the pr oposed appr oach rotates th e eigenv ectors that span the signal subspaces by normalizing the IPD. In case of multiple sources, it r econstructs covariance matrices from the rotated eigenv ectors. T o obtain an estimate of the sign al subspace, the least- squares (LS) criterion is conventionally employed to fin d Q vectors that describe the signal subspa c e an d P − Q noise vectors to represent the n oise subspace in an LS sense (see, e.g., [5, 6 ]). For Q indepen dent target sources, the P × P matrix A ( f i ) R s ( f i ) A H ( f i ) is at least of rank Q and posi- ti ve semidefinite by co nstruction . Therefore, by taking th e eigenv alu e decomposition of R ( f i ) , the eigenvectors corre- sponding to the largest Q eigenv alues are assumed to be opti- mum to span the signal subspace at the frequency b in f i (see, e.g., [6]). The eigenv alue decomp osition of R ( f i ) can be expressed as R ( f i ) U ( f i ) = U ( f i ) Λ Λ Λ( f i ) , (6) where U denotes the eigen vector matrix and Λ Λ Λ is a diago- nal matr ix that contain s the correspo nding eigenv alu e s. The eigenv ector spans the signal su b space is den oted as U s , a nd the eigenv ector for th e noise subspace is den oted as U n , U ( f i ) = [ U s ( f i ) | U n ( f i )] . (7) By the relationship defined in (4), to rotate the estimated subspaces such that th ey b ecome frequency-in depend ent for all f requen cy subband s, the estimated sign al subspa c e rota tio n is defined as U ′ ( f i ) = U ( f i ) ◦ f − 1 i . (8) Then the frequency comp onent of the IPDs are assumed to cancelled. ESPRIT is based on sou r ce subspac e analysis [6]. For the single - source ESPRIT , the estimated source subspace is described by the vector U ′ s ( f i ) . By weighting and summing the ro tated eig en vectors that span th e sour ce sub space U ′ s ( f i ) , the estimate d wideband source sub sp ace U ′ ss can be ob tained as U ′ ss = X i β ( f i ) U ′ s ( f i ) , (9) where β ( f i ) den otes a frequ ency-depen dent weighting fac- tor . The e igenv alue s of th e sources are re levant to th e signal power at a certain frequency and can be assumed to reflect the reliab ility of the signal subspaces estimates. Therefor e, the weighting func tio n is chosen as β ( f i ) = tra c e { Λ Λ Λ s ( f i ) } , (10) where trace {∗} d enotes a fu nction which retu rns the trac e of the matrix a nd Λ Λ Λ s ( f i ) denotes the eigenvalue matrix o f the signal subspace. Follo wing E SPRIT [6], the submatrices satisfy the in v ari- ance relation U ′ ss 2 = U ′ ss 1 Φ , (11) with Φ = diag { e − j 2 π ∆ t 1 , ..., e − j 2 π ∆ t P } (12) where U ′ ss 1 and U ′ ss 2 denote the vectors contain the first and last P − 1 elements o f U ′ ss , respectively . Since the combined vector U ′ ss is assumed to span th e wid eband sign a l subspace E s , it holds that E s = U ′ ss T , (13) where T is a no n -singular matrix . Th erefor e , the subspaces of the two subarrays can be defined as E s 1 = U ′ ss 1 T , E s 2 = U ′ ss 2 T = U ′ ss 1 ΦT , E s 2 = E s 1 Ψ , (14) where Ψ = TΦT − 1 . Ψ can then be obtain ed from (14) by applying a standard least-squ ares o r total least-squares solver . By realizing that the eigenvalues of Ψ are the diago nal ele- ments of Φ the locations of th e sou rces can be estimated. For the multi-sour c e ESPRIT , e ach estimated na r rowband source subspace is described by the c olumns o f the m atrix U ′ s ( f i ) . T he order of the colum ns of the matrix is g ener- ally not known and m ay be different for different frequency bins. Therefo re, the estimated signal sub spaces cannot be combined using (9). A simp ler and more robust solution than the sou rce sub- space identificatio n is to reco nstruct estimated signal sub- spaces back to the fo rm of a P × P cov ariance matrix like R ′ ( f i ) . This can be a chieved by the inverse pro c ess of eigenv alu e decom p osition (6) u sing the frequency compon ent cancelled matrix U ′ ( f i ) , R ′ ( f i ) = U ′ ( f i ) Λ Λ Λ( f i ) U ′ ( f i ) − 1 . (15) W ith the same weighting factor definitio n in (1 0), the re- constructed covariance matrices are calculated by R ′′ = X i β ( f i ) R ′ ( f i ) . (16) Therefo re, the eigen vectors that span the estimated sign al sub- spaces are r e mixed and accumu lated through fr e q uencies in R ′′ . By usin g the conventional narrowband ESPRIT , wide- band sign al subspace can b e separated b ack and wid eband DO A s can then be estimated. 4. IMPLEMENT A TION In the estimated signal subspace ro tation step (8), whe n f gets larger , a finer quan tization is required to limit the ef fect of quantization erro r s o n the DO A estimatio n . An iterative accu- mulation method is p roposed to solve this numerical sensitiv- ity problem . I n each iteration, th e e stimated signal subspac e from the i -th ( i ∈ N + ) fr equency bin is ro tated to a dapt the next f requen cy b in ( i + 1 ) to recon stru ct the covariance matrix R ′′ i +1 . After weigh ting and summing to R ′′ i , the ne w accu mu- lated covariance m atrix R ′′′ i +1 is rotated to the next f requen cy bin ( R ′′ i +1 ) f or the next iteration. The rotation step is then as small as the po wer of f i +2 f − 1 i +1 . The iteration process ca n be expressed as Initialization: R ′′ 1 = U ′ f 1 f − 1 0 s ( f 0 ) Λ Λ Λ( f 0 ) U ′ f 1 f − 1 0 s ( f 0 ) − 1 (17) In each iteration : R ′′′ i +1 = β ( f i ) U ′ s ( f i ) Λ Λ Λ( f i ) U ′ s ( f i ) − 1 + R ′′ i R ′′′ i +1 def = U ′′′ s Λ ′′′ Λ ′′′ Λ ′′′ U ′′′ − 1 s R ′′ i +1 = U ′′′ f i +2 f − 1 i +1 s Λ Λ Λ ′′′ U ′′′ f i +2 f − 1 i +1 s (18) Because of the spatial aliasing problem, the iteration con- tinues until the lowest aliasing frequen cy , d enoted by f a 0 , f a 0 def = ⌊ argmin θ f a ⌋ = ⌊ c 2∆ d ⌋ . (19) T o utilize high er f requen cies, the r e plication method [20] can potentially b e useful f or detectin g the aliasin g frequ ency f or each sou rce. Similar to multi-sour ce localization, the DO A can be obtaine d using the matrix R ′′ ( f a ) . 5. EV ALU A TION A set of experiments was perf ormed in ord er to ev aluate the perfor mance of the algo rithm using real-world record ings. The ev aluation includ es comparison s to the narrowband ES- PRIT with the histogr am meth od (hist-ESPRIT) [21] and to the CSS method [13]. 5.1. Experimenta l setup The re c ording s were captured by a uniform linear array (ULA) with five microph o nes in a low-re verberation lab (T60 ≈ 0 . 2 s o f size 9 m × 8 m × 3 m ). The micr o phon e model SNR=10dB SNR=0dB Algorithm MAE SDE MAE SDE hist-ESPRIT 2 . 10 ◦ 2 . 47 ◦ 3 . 44 ◦ 2 . 47 ◦ CSS 3 . 81 ◦ 5 . 58 ◦ 5 . 52 ◦ 7 . 34 ◦ Proposed 1 . 41 ◦ 1 . 62 ◦ 1 . 68 ◦ 1 . 75 ◦ T able 1 . Sing le white noise source was AKG C562CM. The spacing between micro p hones ∆ d was 0 . 044 m. The backg r ound no ise was wh ite noise, and it was played back via 22 surround speakers to emulate dif- fuse backgr ound no ise. The set of sou r ces contains white sources (independ ent from the backgro und noise) and a set of speech re c ording s selected from th e GRID Corpu s [22]. There were two sourc es located at an angle of 45 ◦ and at − 45 ◦ at a distance of 3 m from the c e nter of the micr opho ne array . In order to evaluate the robustness of the pr oposed localization method, thre e test con ditions wer e u sed: (1) sin- gle white source (2 ) two competing white sources, (3) two competing talkers. The total length of the recording s is 800 s. The sampling rate was 1600 0 sam p les per second. The proposed method, hist-ESPRIT [ 21] and CSS [13] were processed block-wise with 50% overlaps. T he length of the block was 1024 sam p les. The fr equency band u sed for these algorithms was up to 3800 Hz , b elow the aliasing frequen cy . 5.2. Results The scenario of the first experimen ts con sisted of a sing le white no ise a s the source under 10 dB and 0 dB back groun d diffuse white n oise. The sou rce signal was play ed in an al- ternating f ashion from two speakers lo cated at 45 ◦ and − 45 ◦ . The resu lt is shown in T able 1. The pe rforma n ce of the alg o- rithm is e v aluated by the mean absolute error (MAE) an d the standard deviation of the erro r ( SDE). It can be seen from the results that the pr oposed algorithm is m o re accurate and mo re stable under this strong back groun d no ise scenar io. W ith the same backg round noise conditions, the secon d experiment features two wh ite noise sources being played by both sp eakers in a co mpeting fashion . Th e result is shown in T able 2. The MAE of hist-ESPRIT is slightly lo wer than the propo sed algo rithm as the erro r influen ce of estimatio n in low frequen cies is slightly lower , while the SDE fo r the proposed algorithm is clearly superio r . The final experimen t used the same scen ario as th e second experimen t, but two simultaneo usly active speech sources were to be localized. The le vel of the spe e ch signals changed over time, and the estimated SNR was in th e rang e of [ − 5 , 10] d B. The result is shown in T able 3. Th e perf ormanc e of the propo sed alg o rithm is better th an CSS algor ithm, but worse than the hist-ESPRIT algorithm, espec ia lly on SDE. The speec h signal is spectrally sparse a n d the energy is com- SNR=10dB SNR=0dB Algorithm MAE SDE MAE SDE hist-ESPRIT 2 . 57 ◦ 5 . 64 ◦ 3 . 28 ◦ 7 . 47 ◦ CSS 6 . 8 ◦ 9 . 67 ◦ 6 . 5 ◦ 12 . 22 ◦ Proposed 2 . 82 ◦ 3 . 55 ◦ 2 . 75 ◦ 5 . 62 ◦ T able 2 . T wo simultaneous white noise sources Algorithm MAE SDE hist-ESPRIT 4 . 28 ◦ 6 . 47 ◦ CSS 6 . 85 ◦ 15 . 37 ◦ Proposed 5 . 75 ◦ 10 . 62 ◦ T able 3 . T wo simultaneous talkers, SNR ∈ [ − 5 , 10] dB pacted in the fundamen tal freq uency and its harmo nics. In contrast, under the white noise backgrou nd, the inter vals be- tween harmonics in the spec tr um are n o isier ( lower SNR). W ith the inspection of th e narrowband processing, the esti- mated DO A results fro m the intervals had large errors (up to 90 ◦ ). The reason behind the r e duction in performan ce of the prop osed algorithm is the inf erior ro bustness of the least squares criterio n in co mparison to the histog ram method . In the white sources experiments above, the SNR is constant for the entire fr equency range. Ther efore, those experimen ts had better results. Throu g h the above experiments, comp a red to th e co mput- ing time of hist-ESPRIT algorith m , CSS algorithm (with 0 . 1 ◦ resolution of the spatial spectrum) was 13% faster, and the propo sed alg orithm was 22 . 6% faster . 6. CO NCLUSION A wideband signal subspace DO A estimation approach is presented. The pr oposed signal subsp ace r otation method and the narrowband signal covariance matrix reconstru ction method are high and ou tperfo r m the existing co n ventional approa c h es in com putation al c omplexity . Add itionally , the propo sed ap proach av oid s the ne cessity of additio nal prior knowledge for extending the narrowband ESPRIT to a wide- band scheme. Exp eriments b ased on real rec o rding s validate the effectiv eness and low computation al comp lexity of the propo sed meth od. 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