A practical approach to real-time application of speaker recognition using wavelets and linear algebra

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1 A practical approach to real-tie application o speaker recognition using wavelets and linear algebra Duc Son Pha, Michael C. Orr, Brian Lithgow and Robert Mahony Departent o Electrical and Coputer Systes Engineering Monash University VIC 68 Eail: orr@ieee.org Abstract: A continuous wavelet transor, with Morlet wavelets as the basis unctions, is used to ap speech into the tie-requency doain. Forward and inverse FFT routines are used to ipleent the wavelet transors. A coeicient covariance atrix is deined and an Eigenvalue decoposition is used to optially deterine signiicant wavelet based ilters that accurately represent speech and potentially identiy dierent speakers. covariance atrix o the wavelet coeicients. The authors believe that the excellent tie-requency resolution o the Morlet will provide a good representation o huan speech. The abstract nature inoration obtained in the wavelet transor doain is overcoe by using covariance analysis on the wavelet coeicients to deterine the best representation o speaker characteristics. In this preliinary study, it is deonstrated that the proposed ethod can be used or eicient reconstruction using either a ull set or a signiicantly reduced set o coeicients. Key words: speaker recognition, wavelet transor, Morlet, Eigenvalue decoposition. Introduction: Huan perception o speech is, in part, reliant on detection o orant requencies and their transitions ties. Subects utilize orant transition length to categorise sounds []. Real tie speaker recognition is an active area o speech research. One possible explanation or the ailure o existing algoriths to identiy speakers is the accuracy o the signal analysis used, in characterising, or exaple, stop consonants []. A siple speaker recognition syste generally consists o two stages. Firstly, the purely tie series data iltered and apped into a doain ore suited to eature extraction. Many existing schees use a windowed FFT algorith to ap speech data into a tie-requency doain. Secondly, the transored data is copared with prior knowledge about a library o speakers to ind a atch. In order to obtain reasonable speech recognition the transored signal ust have both localised tie and requency resolution, i.e. the ability to capture non-stationary aspects o speech such as stop-constants. The windowed FFT process used by any speech recognition systes provides excellent inoration on orant requencies and slower transitions, however, it has serious deiciencies in representing stop consonants and other key sounds used by huans in speech recognition. In this paper we propose a wavelet-based analysis o speech using an Eigenvalue decoposition o the Figure : Proposed speech-processing syste. Selection o wavelet and ilter bank: Many speech-processing techniques have tried to iic the auditory ilter bank through the use o cepstral ilters []. These techniques are oten solely based on the spectru o a signal, unlike huan hearing which also utilises teporal inoration [7]. Wavelet-based analysis provides better localisation in both tie and requency doains. In speech processing, this is essential or tracking vowels and consonants. The Morlet wavelet provides best tie and requency localisation. / t / t ψ ( t) = (π ) e e To design o wavelet ilter bank, the ain considerations are the bandwidth (o individual wavelet ilters) and the nuber o wavelet ilters per octave. In this paper, the signal is sapled at 0kHz. The speech spectru considered is the requency range 00Hz 8kHz. To provide separation o orant requencies, each ilter in the ilter bank has a Q o approxiately 6. The table below shows how the spectru was split into octaves.

2 Octave Starting requency Ending requency 6.5 Hz 5 Hz 5 Hz 50 Hz 50 Hz 500 Hz 500 Hz khz 5 khz khz 6 khz khz 7 khz 8 khz Table : Frequency range o ilter bank Additionally, in order to get a reasonably lat response across the spectru, the ilters were designed to to overlap at db points. The wavelet ilter bank can be then described by the ilter coeicients β =[β, β,.. β ] in which β =, β < and the coeicient β i corresponds to the ilter / t / β t ψ ( t) = (π ) e e Experients on the response o individual ilters indicate that ilters per bank is a good choice. A discussion o the derivation o the ilter coeicient values is given in an appendix. The β values used are [ ]. Figures and show the requency response o the ilters in the octave between 50Hz 500 Hz 0 Pxx - X Power Spectral Density Figure :. Band pass ilter50hz- 500Hz ripple. Continuous wavelet transor: The continuous wavelet transor is t τ CWT ( a, τ ) = x( t) ψ ( ) dt () a a The wavelet unction ψ (t) used is the Morlet wavelet. The discrete version o the wavelet transor can be written as CWT (, n) = where k = 0 x[ k] / e ( αk / ) : requency scale = 0 6 n : tie shit n = 0-0 k : saple index k = 0-0 e π ( k n) and α= 0.00 is the scale actor to it the Morlet wavelet to the tie rae used. Direct ipleentation o () and () requires a large nuber o coputations. I () is viewed as the response o a signal to a linear syste, i.e. a siple convolution, coputationally it is advantageous to calculate () in the requency doain [, 5]. In addition, i the ter (k-n) in () is considered as the requency ter in the FFT, the wavelet transor or an entire octave can be evaluated as the inverse Fourier transor CWT (, n : 0... ) = IFFT FFT ( x( k))* FFT ( ψ ) () () { } Frequency Figure : Frequency response o wavelet ilters 50Hz-500Hz 0 Pxx - X Power Spectral Density where the scaled wavelet or a particular octave is ψ = ψ 0 ( k / / Even though the Morlet wavelet bases are not orthogonal, the reconstruction can be evaluated ro the ollowing orula [5] M δ CWT (, n) { x n : n = 0... } = () Cδψ (0) = 0 where Cδ is a constant (0.776) or Morlet wavelet and δ will be evaluated ro experients [5]. To soe extent, δ is the copensation actor because the continuous wavelets are not orthogonal. An algorith siilar to [5] is proposed or ast ipleentation o the wavelet transor. The algorith is to be hardware ipleentable: ) Frequency. Design the undaental wavelet bases to cover the lowest requency range. FFT the sapled signal on a 0 saple rae basis. At each scale and or each sub-band, deciate the undaental wavelet, padding zeros to have equal nuber o points as the signal and FFT the baby wavelets

3 . Using linear vector space algorith, process the tie requency inoration. 5. Inverse FFT the processed result backs to tie doain to obtain the approxiate signal o a speaker. Figure depicts the syste response or a short speech saple by a ale speaker. The axiu agnitude o the error is within 5% o the peak agnitude o the original signal. Figure : Word one by a ale speaker and the reconstructed signal with ull set o wavelet coeicients. Best vector basis and speaker recognition: Matheatically, the wavelet coeicients associated with the requency response and localized in tie can be viewed as a vector in linear vector space: = {,,... K,} = { CWT (, n) : =.. M, n = : } This vector ay be thought o as a tie-sequence itsel that varies in agnitude and direction in the linear vector space according to the tie-varying characteristics o the processed speech. I the vector is randoly distributed, such as would be obtained i Gaussian noise were substituted or the speech signal, the vector will span the all directions with equal probability. The authors hypothesise that the characteristic sounds ade by individual speakers will lead to a unique subspace structure in the tievarying response o the wavelet ilter coeicients averaged over tie. The structure ay be the result o a nuber o physical attributes o the speaker, such as structure o the vocal tract, etc. To use this structure to design a speech recognition algorith it is suicient to characterise the subspace associated with a speciic speaker or class o speakers. To illustrate, suppose that Ω P ={ P } and Ω Q ={ Q } denote the subspaces in the linear vector space associated with the average ale and eale pitch tones. I a signal is processed and it is ound that the average wavelet ilter coeicients lie consistently in Ω Q and are signiicantly disparate ro Ω P then the speaker is ore likely to be eale than ale. I a vector tie-sequence has a structure that carries suicient inoration to be identiied and extracted, it should be possible to identiy the subspace associated with this structure by studying the covariance o this signal. Suppose this subspace has diension P where (0<P<K), then the subspace is represented by a sequence o P vectors that span the subspace directions ^ = {,... This is the best basis that can optially represent a given speaker. It is expected that or our speaker recognition, () the P value will be sall and () dierent speakers will have dierent sets o code-vector basis. O course, the nuber o speakers that can be dierentiated depends upon the total nuber o subspaces in the linear vector space and the nuber o subspaces needed to represent each speaker. To ind out the structure or the best basis, Eigenvalue decoposition (EVD) is used on the covariance atrix [, 6]. The covariance atrix Q is essentially the atrix that describes the correlation between the subspaces o a given vector. H * * * Q = = (... K )... K The structure o a vector will develop or a statistical averaging o the covariance atrix J Q = Q = The covariance atrix can be decoposed in to the diagonal and the Eigen vector atrices Q = VDV I the structure in the Q atrix is represented by a sall nuber o subspaces, the diagonal atrix should have a sall nuber o signiicant Eigenvalues and the rest are insigniicant λ λ P D = ε... ε K P The best basis will be the Eigen vectors that are associated with the P signiicant Eigenvalues {λ, λ,..., λ P ) Suppose the best basis is V P = {v, v,...v P }, the rectiied speech vector is the proection o the original vector on the subspace o the best basis P }

4 ^ P = v < v, > i= i An algorith to approxiate a signal by best vector subspaces can be described as ollow:. Process the signal and ap the signal to a tierequency vector.. oralise the vector. Calculate the covariance atrix Q. Average the covariance atrix Q over J ties (J should be large enough to ensure the structure develops). 5. EVD the average Q atrix 6. Pick out the best P vector basis corresponding to the P largest Eigenvalues 7. Proect the speech vector onto the best basis subspace 8. Using inverse wavelet transor to bring back to tie doain to obtain the rectiied signal. We describe here our irst successul attept to ilter a real speech signal using best vector basis approach. In this experient, a structure between the audio bands o a speech signal is to be exained. It deines which sub-bands over seven octaves are signiicant in representing the unique characteristics o each speaker. The experient ais at the possibility o representing a speaker with a saller nuber o linear vector subspaces than the ull set o 8 as in our design. Figure shows the graphical representation o the diagonal atrix and the Eigen vector atrix. The diagonal atrix suggests that there are a sall nuber o signiicant Eigen values. Figure 5 depicts the signal as reconstructed ro the best 0 linear vector subspaces, hence it appears possible to represent speech with the reduced subspace. Figure. The diagonal atrix (let), Eigen vector atrix (iddle) and best vector basis atrix (right) i Figure 5. The signals original (top) and as rectiied ro best 0 (iddle) and ull subspace (botto) Another series o tests have also been conducted. In ost cases, as ar as speech intelligibility is concerned, the representation o speech by the best basis approach provides good intelligibility. The longer the saples over which the average statistical Q atrix is easured, the better the representation. Based on this success, it appears theoretically and practically possible to iprove speaker recognition. The critical point in uture extensions o the work is the deinition and estiation o structure necessary to set up a proper raework or vector subspace analysis. The next stage is to develop the covariance atrix algorith or short-ter apping o target speakers. It is envisaged that in doing this, an algorith or iltering overlapping speakers could be developed. Such analysis would require careul and detailed work and is beyond the scope o this paper. 5. Conclusion: A novel approach or speech characterization and reconstruction using wavelets and linear algebra was described. The design o wavelet ilters and the wavelet transor have been proven eicient and suicient. The authors believe that the algorith described in this paper could be adapted in the uture to be the ront end o a speaker recognition syste. Even though real tie requireents are not strictly iposed in this developent, the proposed approach is proising. 6. Reerences: [] Delas, J.P., "On adaptive EVD asyptotic distribution o centro-syetric covariance atrices," IEEE Transactions on Signal Processing, vol. 7, pp. 0-06, 999. [] Liberan, A. M., Harris, K. S., Hoan, H. S., and Griith, B. C., "The discriination o

5 speech sounds within and across phonee boundaries," Journal o Experiental Psychology, vol. 5, pp , 957. [] Sits, R., "Accuracy o quasistationary analysis o highly dynaic speech signals," Journal o the Acoustical Society o Aerica, vol. 96, pp. 0-5, 99. [] Tokuda, K., Kobayashi, T., and Iai, S., "Adaptive cepstral analysis o speech," IEEE Transactions on Speech and Audio Processing, vol., pp. 8-89, 995. [5] Torrence, C. and Copo, G. P., "A practical guide to wavelet analysis," Bulletin o the Aerican Meterological Society, vol. 79, pp. 6-78, 998. [6] Yu, K.-B., "Eicient, parallel adaptive eigen-based techniques or direction o arrival estiation and tracking," IEEE Transactions on Signal Processing, pp. 7-5, 990. [7] Zwicker, E. and Fastl, H., Psychoacoustics: Facts and odels, nd ed: Springer-Verlag, 999 Appendix Calculation o ilter coeicients The Q actor is deined as the ratio o requency and the bandwidth associated with that requency Q = () BW Assue we are designing two ilters centred at and with bandwidth BW and BW. Then in order to have and intersect at the db point, we would require BW = + BW () (assue that > ) In addition, we would like to have these two ilters have constant Q Q = = () BW BW Hence, ro () () and () + Q Q + = = Q Q ow i we want to design a bank o our ilters o our ilters at,,, such that the lower db o is at start-band, the high db o is at * start-band or end-band and they intersect each other at db points, then where is a unction o Q Hence =a =a =a =a =a Q + a = Q Q + = Q The start band requency and the end-o-band requency are ound ro start band = BW end band = + BW Fro the relation between BW, Q and as derived above and given that start-band = end-band We have ( + ) = ( ) Q Q Then ro the result or above Q + = Q This equation yields Q Q + =.89 = Q In ters o the start-band requency start band =.095 Q =.89 =.07 =.89 =.89 =.58 =.809 Thereore, the theoretical array o coeicients o ilters is [ ]