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1 Sesors 0,, ; doi:0.3390/s OPE ACCESS sesors ISS Article A Iovatios-Based oise Cacellig Techique o Iverse Kepstrum Whiteig Filter ad Adaptive FIR Filter i Beamformig Structure Jisoo Jeog Faculty of Biomedical Egieerig ad Health Sciece, Uiversiti Tekologi Malaysia, 830 UTM Skudai, Johor, Malaysia; s: jisoojeog05@hamail.et; jeog@utm.my; Tel.: ; Fax: Received: 30 April 0; i revised form: 3 May 0 / Accepted: 5 May 0 / Published: 9 Jue 0 Abstract: This paper presets a acoustic oise cacellig techique usig a iverse kepstrum system as a iovatios-based whiteig applicatio for a adaptive fiite impulse respose (FIR) filter i beamformig structure. The iverse kepstrum method uses a iovatios-whiteed form from oe acoustic path trasfer fuctio betwee a referece microphoe sesor ad a oise source so that the rear-ed referece sigal will the be a whiteed sequece to a cascaded adaptive FIR filter i the beamformig structure. By usig a iverse kepstrum filter as a whiteig filter with the use of a delay filter, the cascaded adaptive FIR filter estimates oly the umerator of the polyomial part from the ratio of overall combied trasfer fuctios. The test results have show that the adaptive FIR filter is more effective i beamformig structure tha a adaptive oise cacellig (AC) structure i terms of sigal distortio i the desired sigal ad oise reductio i oise with omiimum phase compoets. I additio, the iverse kepstrum method shows almost the same covergece level i estimate of oise statistics with the use of a smaller amout of adaptive FIR filter weights tha the kepstrum method, hece it could provide better computatioal simplicity i processig. Furthermore, the rear-ed iverse kepstrum method i beamformig structure has show less sigal distortio i the desired sigal tha the frot-ed kepstrum method ad the frot-ed iverse kepstrum method i beamformig structure. Keywords: iovatios; whiteig; AC; beamformig; iverse kepstrum; adaptive FIR filter; system idetificatio; acoustic trasfer fuctio

2 Sesors 0, 687. Itroductio Durig the last five decades, oise cacellig ad sigal ehacig techiques have bee developed. The techiques are fudametally based o spectral subtractio, cepstrum ad complex cepstrum methods usig sigle-microphoe sesor, ad AC ad beamformig methods usig multiple-microphoe array sesors. Sice research o echo cacellatio usig adaptive filters ad two-microphoe sesors started i 965, the adaptive filterig techique has bee used as a solutio tool for sigal ehacig ad oise cacellig schemes []. I 975, Widrow et al. [] proposed a FIR least mea square (LMS) algorithm-based AC method usig two microphoes. This method requires that the primary microphoe takes sigal plus oise ad the referece microphoe takes oise aloe, whereby the adaptive filter estimates oise statistics from the referece microphoe ad the uses them to miimize the output power i a miimizig mea square error (MMSE) calculatio. From the theory, the applicatio shows a practical problem due to the difficulty of separatio betwee the period of the oise aloe ad the period of sigal plus oise. This causes sigal leakage ito the referece microphoe, which makes the adaptive filter estimate oise as well as the desired sigal, hece results i limitatios i the maximum cacellatio from the output sigal-to-oise ratio (SR) with sigal distortio. Pulsipher et al. [3] have aalyzed a ukow system used for the idetificatio of acoustic path trasfer fuctios betwee two-microphoe sesors ad oise sources from the AC method ad ivestigated the adaptive filter estimates of the ratio of acoustic path trasfer fuctios for the correlated oise ad foud out that the problem comes from the omiimum phase compoet of the ratio of acoustic trasfer fuctios. To resolve it, the applicatio uses a large amout of adaptive filter weights. Harriso et al. [4] have itroduced a ew approach i the AC method, whereby they use a oise estimatig techique of a small separatio betwee two-microphoe sesors with the use of a voice activity detector (VAD) durig the oise aloe period. The result has show that it could sigificatly reduce the amout of adaptive filter weights required for oise cacellatio while miimizig the presece of reverberatio. evertheless, the AC method shows a limitatio i the maximum cacellatio i SR because this maximum cacellatio is related to the coherece level of the oise betwee two-microphoe sesors [5], whereby it requires a sigificat coherece for eve modest oise cacellig performace. The applicatio of close ad direct applicatio of a desired sigal [6-8] i frot of two microphoes with the use of adaptive filter usig VAD durig the oise aloe period may reduce sigal distortio. The beamformig techique has bee itroduced to maximize sigal directivity, therefore it icreases the performace i SR. This method may require may microphoes with accurate phase aligmet amog the microphoes array ad hece, a computatioal complexity i processig is expected. To icrease the performace i SR, the techique has bee developed with the use of adaptive filter ad also VAD [9]. The cepstrum processig techique [0] may provide a solutio for sigal separatio, but with the practical limitatio due to miimum phase iformatio. The complex cepstrum method [] may give

3 Sesors 0, 688 the solutio for the miimum phase problem, where this miimum phase iformatio ca be directly estimated from the power spectrum. The kepstrum [,3] is similar to the complex cepstrum due to the fact that the miimum phase spectral factor ca be directly obtaied from a power spectrum estimatio ad it is equivalet with logarithmic miimum phase trasfer fuctio. The kepstrum method has bee used as a system idetificatio techique for ukow systems from the acoustic path trasfer fuctios betwee microphoe sesors iput ad the oise source i beamformig structure [4,5], with a phase recoverig techique from the causal kepstrum domai, where the frot-ed kepstrum method has produced a improved SR performace i a differet iput SR for real-time processig i reverberat room eviromets. It has also bee tested i differet locatios o a oise source usig ostatioary music soud, tued to a radio statio. With a kepstrum method, a techique for prevetig sigal distortio has bee used with the modified applicatio [6-8] to a desired sigal, ad also modified applicatio [4] to a adaptive filter with the use of VAD for differetiatig the periods betwee sigal plus oise ad oise aloe. I additio, the kepstrum has a distictio i the case of sigal plus oise, where the logarithmic miimum phase trasfer fuctio becomes the miimum phase kepstrum spectral factor ad it ca be represeted as a Kolmogorov [6] power series expasio. Furthermore, the sigal ad oise may be implemeted i a iovatios-based form, where it was origially discovered by Kalma ad Bucy [7] ad it may the be applied to a ifiite impulse respose (IIR) Wieer filterig structure [8]. For the iovatios-based approach, it may be represeted as a output of ormalized miimum phase spectral factor from iovatios white oise iput ad has bee used by Moir ad Barrett [9]. By applyig iovatios-based whiteig applicatio i a AC structure, it has bee ivestigated i a simulatio test, where it has bee applied with the use of a FIR ormalized least mea square (LMS) algorithm for oise cacellatio [0] ad also with a FIR recursive least squares (RLS) algorithm [], where it has bee foud that the applicatio of iovatios-based iverse kepstrum to cascaded adaptive filter gives a stable ad causal system because all poles ad zeros of the system are coverted ito uit circles due to the fact that the whiteig applicatio i the referece iput works as all-pass filter so that it allows oly oe acoustic path trasfer fuctio to be cosidered as the ukow system. For the real-time processig usig adaptive RLS filters, the iovatios-based whiteig applicatio has bee applied as a frot-ed applicatio i beamformig structure with rear-ed zero-model FIR RLS filters ad it was foud that it gives better oise cacellig performace tha a pole-zero model IIR RLS filter i a AC structure []. From the previous studies [4,0-] based o modified applicatios [4,6-8], it has bee foud that there are importat features to be further ivestigated to verify the performace. I this paper, by cosiderig: () sigal distortio i a desired sigal o the istat applicatio of oise statistics i reverberat eviromet, () oise reductio o oise characteristics with omiimum phase compoets ad its cosistecy o its iverted acoustic path trasfer fuctio, ad (3) the use of small amout of adaptive FIR filter weights for the real-time processig, hece for the fast covergece i estimate o oise statistics, it is aalyzed i both AC ad beamformig structures. The iverse kepstrum method is the applied to the rear-ed (of sum-ad-subtract fuctio) i beamformig

4 Sesors 0, 689 structure to verify the performace by comparig sigal waveforms i the time domai, spectra i the frequecy domai as well as estimated coefficiets arrays of iverse kepstrum ad weights arrays of adaptive FIR filter ad its pole-zero placemets i oise statistics. Furthermore, the rear-ed iverse kepstrum method is also compared with the frot-ed kepstrum method [4], which uses idetificatio of two paths with the ratio of overall acoustic path trasfer fuctio, ad also with frot-ed iverse kepstrum method usig a whiteig applicatio [].. Aalysis of Iovatios-Based Iverse Kepstrum This sectio describes the aalysis of cepstrum ad complex cepstrum (kepstrum) with the relatio of miimum phase kepstrum, ad also for its whiteig applicatio (iverse kepstrum). It shows that the miimum phase kepstrum coefficiets may be obtaied from the logarithm of the miimum phase trasfer fuctio (Sectio.) ad also from the logarithm of the miimum phase spectral factor (Sectio.). For the sigal ad oise, it shows that it ca be represeted as a output of ormalized miimum phase spectral factor from iovatios white oise iput (Sectio.3). Based o this, it shows that logarithm of iverse miimum phase trasfer fuctio may be implemeted as the iovatios form of the ormalized miimum phase kepstrum spectral factor for the whiteig applicatio (Sectio.4)... Aalysis of the Miimum Phase Trasfer Fuctio It is kow that the causal trasfer fuctio ca be expressed by Schwarz s classical formula [3] as: H jλ e z z H λ dλ λ e z ( ) R ( ), z j () 0 jw where as the itegratio variable ad z re. This Equatio gives the causal trasfer fuctio H ( whose real part o the uit circle is H R (w). Based o this, the phase iformatio ca be recovered by Hilbert s trasform relatio. The logarithm of miimum phase trasfer fuctio log ca be writte as: H M j e z log H ( ) log ( ), M z H M d j e z z () where log H ( ) is the magitude part of miimum phase logarithmic trasfer fuctio. M This idicates that the miimum phase trasfer fuctio may be expressed i terms of Schwarz s formula ad hece the miimum phase iformatio ca be recovered from Hilbert s trasform relatio. jw By defiig that z e, it ca be described as magitude ad phase term: For the -poit discrete form: log H M = log H jw M ( e ) log ( jw ) arg[log ( jw H )] M e j H M e (3) log H M k c 0 c e j k ce j k j c e j k ce j k (4)

5 Sesors 0, 680 From Equatio (4), the magitude of logarithmic miimum phase trasfer fuctio is: log H M ( e j k ) c0 c j k e ce j k (5) ad the phase of the logarithmic miimum phase trasfer fuctio is: argh M ( e j k ) j c e j k ce j k (6) This shows that Equatio (5) shows a eve cepstrum fuctio, hece miimum phase kepstrum coefficiets ca be processed i the cepstrum domai by multiplyig by two () i the positive time series, except for the first zeroth coefficiet i the time series... Aalysis of Miimum Phase Spectral Factor From the power spectral desity (, it ca be represeted as causal spectral factor H ( aticausal couterpart H ( as: as: Let z z, the it follows that ( z ) H ( z ) H It follows that ( z ) ( z ). By defiig ad H H H H ( z ) (7) z e jw, we ow have a logarithmic power spectrum ( w) ( w) (8) ( w) H ( e jw ) (9) log ( w) log H ( e jw ) (0) For the -poit discrete form: K k log k log H ( e j k ) k0 k As a result of the symmetry property of k, it ca be expressed as: K k k0 k e j k e j k k j k e Furthermore, sice k are real, oly a half portio i legth ca be cosidered as: () () K ( / ) k j k k k e 0 (3) 0 From Equatio (3), it shows that the kepstrum coefficiets ca be processed i the causal kepstrum domai by halvig the first zeroth coefficiet with the remaiig coefficiets trucated i size to ( / ). By trucatig i size less tha ( / ), kepstrum ow becomes complex cepstrum, which is a approximatio of the theoretical mathematical costruct [9].

6 Sesors 0, Aalysis of Miimum Phase Kepstrum Spectral Factor o Sigal ad oise I the case of radom sigal plus oise, it ca also be represeted as iovatios-based form. From Figure (upper part), it shows a equivalet relatio of output from the iputs betwee white oise ad iovatios white oise as Equatio (4): where v, ( z ) S S ( z ) xx ad i ww ss M ( M v M M H ( z ) H i (4) are the variaces of the additive white oise, white oise iput ad white oise iovatios process, respectively. M ( ad ( are coloured miimum phase trasfer fuctios S M ad M ( z ) ad S M ( z ) are maximum phase couterparts. H ( is the ormalized miimum phase spectral factor, which has all its zeros iside z whilst H ( outside z. is the couterpart, which has its zeros Figure. Equivalece of outputs based o iputs of: (a) white oise model ad (b) iovatios white oise model [upper part: sigal plus oise, lower part: oise aloe]. White oise v White oise S M ( M ( Sigal s oise x s x i v Iovatios white oise i v Iovatios white oise H ( x s x (a) (b) I the case of sigal plus oise, the logarithm of each positive- ad egative-sided trasfer fuctio becomes the kepstrum spectral factors of the z- trasform spectral desity ad these are represeted as a power series expasio. For the iovatios-based iverse kepstrum approach, sigal plus oise are represeted as a output of ormalized miimum phase spectral factor from the iovatios white oise iput. It may be applied to a optimum IIR Wieer filterig structure, where it has bee defied by Kailath [8] as combiatio of two cascaded filters, a frot-ed whiteig filter to geerate the white iovatios process ad cascaded spectral shapig filter to provide spectral shapig fuctio for the iput sigal..4. Aalysis of Iovatios-Based Iverse Kepstrum ad its Applicatio to oise Sigal Oly For the applicatio of oise aloe, it ca be estimated durig the absece of desired sigal (Figure (lower part)). Therefore, the additive oise ow ca be represeted as iovatios-based form, as show i Figure. Figure. Represetatio of oise as iovatios-based whiteig form. x oise [ H ] - i v Iovatios white oise

7 Sesors 0, 68 From the miimum phase kepstrum spectral factor of Equatio (5): K log H k k z k z... (5) 0 its expoetiatio becomes a causal spectral factor as: H exp[ K ] exp( k0 kz kz...) (6) By usig the fact that miimum phase spectral factor allows its ivertibility, it ca be expressed as: [ H ] exp[ K ] exp[ ( k 0 k z k z...)] (7) This shows that the iverse of the miimum phase spectral factor ca be obtaied from the kepstrum expoetial by multiplyig by mius oe ( ). It ca also be represeted i a ormalized form: H ( exp[ k0]exp[ K ] exp[ k0] H (8) log H k0 log H (9) I the case of the applicatio of additive oise aloe, i becomes v. From the fact that x HM v ad x H i, the relatioship betwee miimum phase trasfer fuctio H M ( ad ormalized miimum phase spectral factor H ( is described as: where v, i ad ( H M H i H v H σ ε (0) are white oise, iovatios process ad stadard deviatio of ormalized iovatios sequece respectively. By takig logarithm of Equatio (0): log H M log H log () From Equatios (9) ad () ad with the fact that log H log H where M : k log 0 () exp [ k 0 ] ad exp [k0] ad are stadard deviatio ad variace of ormalized iovatios sequece respectively. A ormalized iovatios-based iverse kepstrum is the represeted from logarithmic miimum phase trasfer fuctio or logarithmic miimum phase spectral factor, which described as: log [ H M ] log [ H ] (3) log K log (4) Therefore, it shows that logarithm of iverse miimum phase trasfer fuctio may be implemeted as iovatios form of ormalized miimum phase kepstrum spectral factor ( log [ H M ] = K (, assumig that stadard deviatio ).

8 Sesors 0, Iverse Kepstrum Processig ad Adaptive FIR LMS Algorithm For the whiteig applicatio, the iverse kepstrum is processed from the referece microphoe x. Therefore oly the estimate from a sigle acoustic path trasfer fuctio is required. This may be compared with kepstrum processig, where the estimate of two acoustic path trasfer fuctios from both the primary microphoe d ad referece microphoe x is required. From each iput microphoe sesor, periodograms are obtaied from Haig-widowed fast Fourier trasforms (FFTs) from the two-microphoe iputs as show i Figure 3. Figure 3. Periodogram estimate from each iput microphoe sesors. d Haig Widow FFT D x Periodogram () dd x Haig Widow FFT X x Periodogram () xx As a discrete estimate of the cotiuous power spectral desity, it uses a modified weighted overlapped segmet averagig (WOSA) algorithm ad the auto-periodograms are processed from 50% overlappig Haig-widowed FFTs i,048 frame size by usig smoothig method (5), with the forgettig factor = 0.8 [0]: * * dd( i) βdd( i ) ( β) X d ( i) X d ( i) xx( i) β xx( i ) ( β) X x( i) X x ( i) (5) From (5), its logarithm eeds to add Euler s costat to be ubiased due to bias i magitude [4]. Hece, each kepstrum coefficiet from the two-microphoe iputs are foud from the iverse FFT (IFFT) of the ubiased logarithmic auto-periodograms ad the by subtractig two kepstrum coefficiet vectors ( k k ), we ca get the kepstrum coefficiets ( k ) for kepstrum processig. O the other had, iverse kepstrum coefficiets ( k ) are foud by egatig the kepstrum coefficiets ( k ) from the referece microphoe. The processig differece betwee iverse kepstrum method ad kepstrum method ca be foud i Figure 4. Figure 4. Block diagram for the compariso betwee iverse kepstrum processig ( ad kepstrum processig ( k ). k ) k dd log( ) Euler costat xx : IFFT k k Average k k k k k This idicates that iverse kepstrum processig requires oly a egative sig of the kepstrum coefficiets, which ca be obtaied for the iverse of acoustic path trasfer fuctio from the referece microphoe iput Equatio (6). The egated kepstrum coefficiets are the ormalized by dividig the zeroth kepstrum coefficiet value:

9 Sesors 0, 684 log{ / H( } K K( (6) O the other had, it is compared with the kepstrum method, which uses the ratio of the acoustic trasfer fuctio betwee two-microphoe iputs as Equatio (7). log{ H( / H( } K( K( K( (7) Both the kepstrum ad iverse kepstrum coefficiets are trasformed ito the correspodig impulse respose usig recursive formula [] ad the it is covolved with referece iput sigal to get a refied ew iput sigal as show i Figure 5. Figure 5. The coversio procedure from iverse kepstrum ( k ) or kepstrum ( k ) to impulse respose ( h ), ad its covolutio with referece iput sigal ( x ). k k k k k x l 0 k0, g0 k0 g Impulse respose h Covolutio x For the cascaded adaptive FIR filter, the LMS algorithm [5] has bee used ad the weights are updated as: μxe w w.000 X (8) where μ is step size ( 0 ), divisio. 0 X is iput power ad the value of is used to prevet zero Table. Compariso of computatioal complexity i FLOPS o (i) iverse kepstrum processig, (ii) kepstrum processig ad (iii) LMS algorithm [0]. Algorithm Required processig FLOPS (i) Iverse kepstrum WOSA Periodogram log (5./ f ) 4 FFT / IFFT 4( / )log /3 Logarithm /Expoetial (log ) Total Computatio (*) (ii) Kepstrum WOSA Periodogram log (5./ f 5 FFT / IFFT 5( / )log /3 3 Logarithm /Expoetial 3 (log ) Total Computatio (*) (iii) LMS Real multiplicatio (A) Iteratios (B) (A) Total Computatio (**) (A) (B) ote: Total computatio: (*) is based o : 048 frame size, (**) is based o 00 LMS weights. For the compariso of processig amog: (i) iverse kepstrum coefficiets, (ii) kepstrum coefficiets ad (iii) LMS algorithm based adaptive FIR filter weights, computatioal complexity i floatig poit operatios per secod (FLOPS) ca be compared as show i Table, where it shows )

10 Sesors 0, 685 that iverse kepstrum processig gives the least computatioal complexity, ad kepstrum processig shows less computatioal complexity tha the ordiary adaptive FIR LMS algorithm [0]. 4. Iverse Kepstrum Method The iverse kepstrum method uses the whiteig applicatio from oe acoustic path trasfer fuctio, H ( z ) from the referece microphoe durig the oise aloe period as show i Figure 6(a), where the iverse kepstrum is to be aalyzed as the frot-ed applicatio to the adaptive FIR filter from the AC structure, ad the rear-ed applicatio to the sum-ad-subtract fuctio from the beamformig structure accordigly. It is also to be compared with the kepstrum method, which is based o idetificatio of the ratio of acoustic path trasfer fuctios, H ( z ) ad H ( z ). Durig the sigal ad oise period, the oise estimate is applied to obtai the desired sigal with a o-distortio. For the purpose, the method uses direct applicatio of desired sigal i frot of two microphoes as show i Figure 6(b). Direct applicatio of the desired sigal ca be foud i the AC structure [6] ad also i beamformig structure [7,8]. Figure 6. (a) Acoustic path trasfer fuctios as oise estimate ad (b) applicatio of desired sigal to the two microphoes. Primary MIC Primary MIC oise H H ( z ) (a) Referece MIC oise H H ( z ) Sigal Referece MIC (b) 4.. Iverse Kepstrum Method as Frot-Ed Applicatio to AC Structure I the AC structure, idetificatio as a ukow system is represeted as the ratio of acoustic trasfer fuctios, H H( / H( as show i Figure 7, where a ordiary adaptive FIR filter may be used to estimate the oise statistics durig oise aloe period. Its estimate is the applied to the sigal ad oise period to cacel the oise by usig the estimated oise statistics with the coditio that the desired sigal should be oly retaied with o distortio. Figure 7. Aalysis of idetificatio of acoustic path trasfer fuctios by a adaptive FIR filter i the AC structure durig the periods of (a) oise aloe ad (b) sigal plus oise. Primary MIC L D Primary MIC L D oise H H ( ) z Referece MIC H (/ H ( Adaptive FIR filter (a) Output e oise H Freeze Sigal Referece MIC H ( z H (/ H ( ) Adaptive FIR filter (b) Output e

11 Sesors 0, 686 Secodly, the iverse kepstrum is applied i frot of the adaptive FIR filter as show i Figure 8(a), where the iverse kepstrum filter estimates a deomiator part, / H ( z ) ad the cascaded adaptive FIR filter estimates a umerator part, H ( z ) from the ratio of overall trasfer fuctio. It is compared with the kepstrum method as show i Figure 8(b), where the kepstrum filter estimates the miimum phase term oly from the ratio of overall trasfer fuctio ad the cascaded adaptive FIR filter estimates remaiig term ad hece it works as a all-pass filter. Figure 8. Applicatio of (a) iverse kepstrum filter ad (b) kepstrum filter to the AC structure. Primary MIC d L D Primary MIC d L D Referece MIC x K ( x Iverse kepstrum filter (a) W Adaptive FIR filter Output e Referece MIC x K( x W Kepstrum filter Adaptive FIR filter (b) Output e Assumig that each acoustic path trasfer fuctios betwee two microphoes ad oise source are as give i (9), it is represeted as the ukow system of (30). Its estimates are to be aalyzed as show i Figure 9, where the operatio of the iverse kepstrum is to be compared with the kepstrum filter as a frot-ed applicatio to the cascaded adaptive FIR filter. Figure 9. Block diagram for the operatio compariso betwee a iverse kepstrum filter ad a kepstrum filter as frot-ed applicatio to a AC structure. White oise v Ukow system H H H Sigal oise D ff L K Iverse kepstrum filter K ( Kepstrum filter W ( Adaptive FIR filter W Adaptive FIR filter e Error e Error As a simple example, we assume that oe trasfer fuctio from acoustic path trasfer fuctios is omiimum phase term, such as: H ( z ad where H ( ) is omiimum phase trasfer fuctio. z H 0. z (9) The ukow system is the described as the ratio of trasfer fuctios ad it is represeted as:

12 Sesors 0, 687 H( ( z )/( 0.z 3 4 ).8z 0.36z 0.07 z z... (30) where it ca be estimated by ordiary adaptive FIR filter. For the operatio of the iverse kepstrum filter K ad adaptive FIR filter W ( ), each oe is estimated as: K /( 0.z ) z W ( z (3) This idicates that the frot-ed iverse kepstrum K estimates the deomiator part ad the cascaded adaptive FIR filter W ( z ) estimates the umerator part from the ratio of overall trasfer fuctios. O the other had, for the operatio of the kepstrum filter K ( ad adaptive FIR filter W ( z ), each oe is estimated as: K ( 0.5z )/( 0.z ) W ( z )/( 0.5z ) (3) This idicates that the frot-ed kepstrum estimates the miimum phase term oly from the ratio of overall trasfer fuctio, where omiimum phase term is reflected to the miimum phase term by a reciprocal polyomial as z H( z ). The cascaded adaptive FIR filter works the as a all-pass filter. From (3) ad (3), we ow have foud that both methods show the same result as (30). ow let us check with iverse of overall trasfer fuctio, such that: H ( 0. z ad where H ( ) is omiimum phase trasfer fuctio. z H ( z (33) This idicates that the trasfer fuctio ow has a omiimum phase term i the deomiator polyomial from the ratio of overall trasfer fuctios. The ukow system ca be estimated from the ratio of trasfer fuctios ad it is represeted as: H( ( 0.z ( 0.z )/( z ) [( 0.z )/( 0.5z )][( 0.5z )/( z )/( 0.5z ) 0.3z 0.5z... where it ca also be estimated by adaptive FIR filter. The Equatio (34) idicates that the omiimum phase term of deomiator part is coverted to a miimum phase term for the operatio of a adaptive FIR filter so that the cascaded remaiig part of Equatio (34) always works as a all-pass filter. The iverse kepstrum ad cascaded adaptive FIR filter are estimated as: )] (34) K /( 0.5z ) W ( 0. z (35) ad the kepstrum ad cascaded adaptive FIR filter are estimated as: K( ( 0.z ) /( 0.5z ) W (36) This idicates that the estimates, Equatios (35) ad (36), from both methods are the same as Equatio (34), which is the iverse of the overall trasfer fuctio. The omiimum phase compoet from the polyomial umerator ad deomiator of the overall trasfer fuctio may frequetly occur i reverberat eviromets [6] ad causes a fluctuatio i the spectrum due to the differece betwee Equatios (30) ad (34). To deal with this problem, it idicates that either the kepstrum or the iverse kepstrum methods provide a solutio i the AC structure. Alteratively, we may solve

13 Sesors 0, 688 this problem by swappig the positio of the two microphoes, but it is ot a practical solutio i a realistic eviromet. 4.. Iverse Kepstrum Method as a Rear-Ed Applicatio i Beamformig Structure I a beamformig structure, idetificatio as a ukow system is represeted as the ratio of combied acoustic trasfer fuctios, H 0.5( H H ) / 0.5( H H ( )) as show i Figure 0, ( z where a ordiary adaptive FIR filter may also be used to estimate the ratio of combied overall trasfer fuctios. Its estimate may the be applied to the sigal ad oise period to cacel the oise by usig estimated oise statistics. Figure 0. Aalysis of idetificatio of acoustic path trasfer fuctios by adaptive FIR filter i beamformig structure durig the periods of (a) oise aloe ad (b) sigal plus oise. oise H H ( ) z Primary MIC Referece MIC (a) 0.5 L D 0.5 H ( z ) H ( z ) H ( z ) H ( z ) Adaptive FIR filter Output e oise H H ( ) z Primary MIC Sigal Referece MIC 0.5 L D Freeze 0.5 H ( z ) H ( z ) H ( z ) H ( z ) Adaptive FIR filter (b) Output e Based o this, a iverse kepstrum filter is applied i frot of the adaptive filter as a rear-ed applicatio from the sum-ad-subtract fuctio from the beamformig structure as show i Figure (a), where the iverse kepstrum filter estimates the polyomial deomiator part, / 0.5( H( H( ) ad the cascaded adaptive FIR filter estimates the polyomial umerator part, 0.5( H( H( ) from the ratio of combied overall trasfer fuctios. It is compared with the kepstrum method as show i Figure (b), where the kepstrum filter estimates the miimum phase term oly from the umerator polyomial part, 0.5( H( H( ) ad the cascaded adaptive FIR filter estimates the remaiig part from the umerator polyomial part, 0.5( H( H( ), where this umerator polyomial part is to be a overall trasfer fuctio with a delay filter i the rear-ed primary iput, d. Figure. Rear-ed applicatio of (a) iverse kepstrum filter ad (b) kepstrum filter to beamformig structure. Primary MIC d 0.5 d L D Primary MIC d d 0.5 L D Referece MIC x 0.5 x K Output W ( ) z e Iverse kepstrum filter Adaptive FIR filter (a) Referece MIC x 0.5 x K ( W ( Output Kepstrum filter Adaptive FIR filter (b) e

14 Sesors 0, 689 To compare with the operatio i a AC structure, the same compoets of the acoustic trasfer fuctios are used as Equatio (9) ad this is represeted as a ukow system as i Equatio (37). It is to be aalyzed as show i Figure, where the operatio of the iverse kepstrum is to be compared with the kepstrum filter as a rear-ed applicatio to the sum-ad-subtract fuctio of the beamformig structure. From the ukow system, we have the umerator polyomial part from the ratio of overall combied trasfer fuctios ad it is represeted as a overall trasfer fuctio ad the deomiator polyomial part works as a delay filter, described i Equatio (37): H( 0.5[( z ) ( 0.z )].z with oe sample delay, D (37) Figure. Block diagram for the operatio compariso of a iverse kepstrum filter ad a kepstrum filter as rear-ed applicatios to a beamformig structure. Sigal White oise v Ukow system H( H H( H H oise D ff L K Iverse kepstrum filter K ( Kepstrum filter W Adaptive FIR filter ( W Adaptive FIR filter e Error e Error For the operatio of iverse kepstrum filter K ad adaptive FIR filter W ( ) is estimated as: K ad W. z with oe sample delay, z, each oe D (38) This idicates that the rear-ed iverse kepstrum K works as a whiteig filter with oe sample delay ad a cascaded adaptive FIR filter W ( z ) estimated a umerator part from the ratio of combied overall trasfer fuctios. O the other had, for the operatio of the kepstrum filter K ( ad the adaptive FIR filter W ( ), each oe is estimated as: z K 0.9z ad ( 0. z W with oe sample delay, D (39) This idicates that the rear-ed kepstrum estimates the miimum phase term oly from the polyomial umerator part from the ratio of combied overall trasfer fuctios, where the omiimum phase term is reflected to the miimum phase term by the reciprocal polyomial as z H( z ). The cascaded adaptive FIR filter estimates the remaiig term from the polyomial umerator of the ratio of combied overall trasfer fuctios. Based o this, the ukow system Equatio (37) may be estimated by the operatios (38) ad (39) as rear-ed applicatios of the iverse kepstrum ad kepstrum methods to the sum-ad-subtract fuctio of the beamformig structure, respectively.

15 Sesors 0, 6830 Let us ow check with the iverse of the overall trasfer fuctio, (33), where omiimum phase term i deomiator polyomial is o loger exist ad overall trasfer fuctio is ow obtaied from umerator polyomial part of the ratio of overall combied trasfer fuctio. Iverse kepstrum ad cascaded adaptive FIR filter estimates as: K ad W. z with oe sample delay, D (40) where iverse kepstrum filter works as whiteig filter ad adaptive FIR filter estimates umerator polyomial part of the ratio of overall combied trasfer fuctio. O the other had, kepstrum ad cascaded adaptive FIR filter estimates as: K 0.9z ad W ( 0. z with oe sample delay, D (4) where kepstrum filter works as miimum phase filter ad adaptive FIR filter estimate remaiig term from the umerator polyomial part of the ratio of overall combied trasfer fuctio. It shows that the estimates, Equatios (40) ad (4) by both methods o the iverted trasfer fuctio are same as Equatios (38) ad (39), which are the estimates by the both methods o the direct trasfer fuctio. It idicates that both kepstrum ad iverse kepstrum methods do provide a solutio i beamformig structure, which may give a practical solutio i a reverberat oise with omiimum phase compoet from overall trasfer fuctio because it does ot eed to swap the two microphoes positio. The detailed aalysis o omiimum phase trasfer fuctio has bee ivestigated betwee AC ad beamformig structures [7]. 5. Experimets Experimets were implemeted i both simulatio tests o pc software ad real tests usig real ostatioary oise i a room eviromet. Accordig to the mai three cosideratios (sigal distortio i the desired sigal, oise reductio performace i oise with omiimum phase compoets ad covergece level i estimates of oise statistics with the use of a small amout of adaptive FIR filter weights), the performaces achieved whe usig a iverse kepstrum filter were verified i both the AC ad beamformig structures. Furthermore, the rear-ed applicatio of the iverse kepstrum method i beamformig structure was also compared with two frot-ed applicatios of the kepstrum method [4] ad the iverse kepstrum method [] i beamformig structure. The methodology is based o the fact that the coefficiets (kepstrum ad iverse kepstrum) ad weights (adaptive FIR filter) are cotiuously updated durig oise aloe period to estimate oise statistics. Whe the desired sigal is applied to oise, the last updated coefficiets ad weights are froze ad applied to the desired sigal ad oise. For a precise test, it is programmed to stop updatig coefficiets ad weights, ad the these are applied to the desired sigal ad oise period. To check the stregth i amplitude ad distortio status i desired sigal, simple three sie waveforms are added ad used as the desired sigal for both simulatio ad real tests. For the test usig real oise, we use a ostatioary music soud, tued to a certai radio statio. For the use of a adaptive FIR filter, a LMS algorithm has bee used with the use of step size for the simulatio test ad 0. 5 for the real test. For the processig,,048 frame size, samplig frequecy of,050 Hz ad yquist frequecy of aroud,000 Hz have bee chose. Two preamplifiers ad two microphoes of uidirectioal electret codeser type are used, placed 7 cm distace apart i broadside cofiguratio

16 Sesors 0, 683 for the real test i a room [3.8 m(d) 3 m(w).8 m(h)]. The performace is to be verified by comparig sigal waveforms i time domai, spectra i frequecy domai, estimated coefficiets ad weights arrays ad its pole-zero placemets. 5.. Simulatio Test For the simulatio test, the acoustic trasfer fuctios of (9) are used as the ukow system, which has omiimum phase compoet i oise. The desired sigal, cosistig of three frequecies (500 Hz, 550 Hz ad 700 H, has arbitrarily bee used as show i Figure 4(a) Adaptive FIR filter i AC ad beamformig structures The first test is to verify the oise cacellig performace i the AC structure by applyig three adaptive FIR filter weights for the oise characteristic with omiimum phase compoet i the polyomial umerator (9) ad omiimum phase compoet i the polyomial deomiator (33) i the acoustic trasfer fuctio. It is also verified i the beamformig structure. From the simulatio test based o the block diagram (Figure 9) of the AC structure [Figure 7(a)], it is foud that the oise spectrum with omiimum phase term i the polyomial deomiator (33) shows worse ad much differet performace tha oe of omiimum phase term i the umerator (9), as show i Figure 3(a). O the other had, from the simulatio test based o the block diagram (Figure ) of the beamformig structure [Figure 0(a)], spectra of both Equatios (9) ad (33) show good ad almost same spectral performace, as show i Figure 3(b). From the test result, it is show that the adaptive FIR filter works well with the use of a small amout of weights ad shows cosistecy i spectra i the beamformig structure for the oise with both omiimum phase cases, Equatios (9) ad (33). O the other had, based o the test result i Figure 3(a), a fluctuatio i spectrum is expected i AC structure i the case that oise statistics is frequetly chaged i a reverberat eviromet. Figure 3. Compariso i spectra betwee direct trasfer fuctio (9) ad its iverse trasfer fuctio (33) o applicatio of adaptive FIR filter i (a) AC structure ad (b) beamformig structure. (9) (33) (9) (33) (a) (b) For the secod test, the acoustic trasfer fuctio Equatio (9) is estimated by usig three adaptive filter weights ad the it is applied to sigal ad oise. To get the best result, it is foud that it is 3 D eeded to set the three sample delay as to reduce sigal distortio i the AC structure. O the other had, it is foud that there is o eed to set the delay i the beamformig structure. From the test

17 Sesors 0, 683 result, by usig a small amout of three adaptive FIR filter weights, it is show that the performace i the beamformig structure provides o sigal distortio i the desired sigal without ay delay adjustmet ad also better oise reductio i oise with omiimum phase term tha the performace i the AC structure, as show i Figure 4. Figure 4. Simulatio test based o 3 adaptive FIR filter weights: (a) waveform of origial desired sigal; (b) waveform of desired sigal with three samples delayed i the AC structure; (c) waveform of desired sigal with o sample delay i the beamformig structure; (d) correspodig spectra of (b) ad (c). (a) (b) (b) (c) (c) (d) From the above two test results i terms of oise reductio performace i reverberat oise with omiimum phase (Figure 3) ad sigal distortio i the desired sigal (Figure 4), it is foud that a adaptive FIR filter works better i a beamformig structure tha i a AC structure Applicatio of iverse kepstrum method to two structures ad compariso with kepstrum The objective is to verify the performace of the iverse kepstrum method i AC ad beamformig structures. For the test i the AC structure, the acoustic trasfer fuctios of (9) have bee used, hece the ukow system is expected to be estimated as (30). As verified i Table (a) ad Figure 5 (a), a adaptive FIR filter usig oe zero does ot approximate to (30). It ca be approximated by icreasig the adaptive FIR filter weights to four from (a) so that it gives almost the same performace as Equatio (30), as show i Table (b) ad Figure 5(b). O the other had, by applyig two iverse kepstrum coefficiets to a cascaded adaptive FIR filter usig oe zero, it also gives almost the same performace as Equatio (30) as show i Table (c) (iii) ad Figure 5(c). For the compariso, with the use of same size of two kepstrum coefficiets, it idicates that four adaptive filter weights are eeded to approximate (30) as show i Table (d) (iii) ad Figure 5(d). From the test result, it is aalyzed that i the AC structure, the applicatio of a iverse kepstrum filter works well with a adaptive FIR filter i terms of covergece with a smaller amout i adaptive filter weights, rather tha whe the adaptive FIR filter is used with applicatio of a kepstrum filter.

18 Sesors 0, 6833 From the test results betwee (c) (ii) ad (d) (ii) i Table, it idicates that iverse kepstrum method eeds oly 50% of the adaptive FIR filter weights eeded by the kepstrum method. Table. Weights ad coefficiets arrays showig each estimate output from the simulatio test based o the block diagram i Figure 9. (a) Adaptive filter weights (oe zero).7 (b) Adaptive filter weights (three zeroes) (c) (i) Iverse kepstrum coefficiets (two poles) ad (ii) Adaptive filter weights (oe zeroes) (i) (ii).99 (iii) (d) (i) Kepstrum coefficiets (two poles) ad (ii) Adaptive filter weights (three zeroes) (i) (ii) (iii) ote: iii=i*ii, where * idicates covolutio. Figure 5. Sapshot of pole-zero placemet: (a) oe zero from adaptive FIR filter (b) icreased to three zeros from (a) (c) two poles from iverse kepstrum coefficiets ad oe zero from the adaptive FIR filter (d) two poles from kepstrum coefficiets ad icrease to three zeros from the adaptive FIR filter. (a) (b) (c) (d)

19 Sesors 0, 6834 For the test i beamformig structure, acoustic trasfer fuctios of Equatio (9) has bee used, hece ukow system is expected to estimate as Equatio (37). As verified i Table 3 (a) ad Figure 6(a), adaptive filter usig oe zero is well approximated to Equatio (37). Table 3. Weights ad coefficiets arrays showig each estimate output from the simulatio test based o the block diagram i Figure. (a) Adaptive filter weights (oe zero).0 (b) (i) Iverse kepstrum coefficiets (two poles) ad (ii) Adaptive filter weights (oe zero) (i) (ii).09 (iii) (c) (i) Kepstrum coefficiets (two poles) ad (ii) Adaptive filter weights (oe zero) (i) (ii) 0.50 (iii) (d) (i) Kepstrum coefficiets (two poles) ad (ii) Adaptive filter weights (eight zeroes) (i) (ii) (iii) ~0.000 ~0.000 ~0.000 ote: (A) iii=i*ii, where * idicates covolutio. (B) (d) (ii) ad (iii) shows oly seve weights estimates i arrays. Figure 6. Sap shot of pole-zero placemet: (a) oe zero from adaptive FIR filter (b) two poles from iverse kepstrum ad oe zero from adaptive FIR filter (c) two poles from kepstrum filter ad oe zero from adaptive FIR filter (d) two poles from kepstrum filter ad icreased to eight zeros from adaptive FIR filter. (a) (b)

20 Sesors 0, 6835 Figure 6. Cot. (c) (d) Based o this, by usig two iverse kepstrum coefficiets, it ca be verified that it ca also be approximated to Equatio (37) as show i Table 3 (b) (iii) ad Figure 6(b). O the other had, by applyig two kepstrum coefficiets to the cascaded adaptive FIR filter usig oe zero, it caot be approximated to (37), as show i Table 3 (c) (iii) ad Figure 6(c). For the compariso, with the use of same size of two kepstrum coefficiets, it idicates that eight adaptive filter weights are eeded to approximate (37), as show i Table 3 (d) (iii) ad Figure 6(d). From the test result, it is also aalyzed i a beamformig structure ad shows that applicatio of iverse kepstrum filter works well with the adaptive FIR filter i terms of covergece with much smaller amout i adaptive filter weights, rather tha whe the adaptive FIR filter is used with applicatio of a kepstrum filter. From the test results betwee (b) (ii) ad (d) (ii) i Table 3, it idicates that iverse kepstrum method eeds oly 5% (75% less) of the adaptive FIR filters tha the kepstrum method. From the compariso results betwee iverse kepstrum filter ad kepstrum filter to adaptive FIR filter i terms of covergece i oise statistics ad its pole-zero placemets, it is foud that the applicatio of the iverse kepstrum filter could give a covergece beefit with the use of a small amout i adaptive FIR filter weights i the AC structure [Table (c) (ii)], ad is much more effective i a beamformig structure [Table 3 (b) (ii)]. 5.. Real Tests The simulatio test results suggest that the iverse kepstrum should achieve more oise reductio without sigal distortio i the desired sigal by usig small amout of adaptive FIR filter for the real tests i a realistic reverberat eviromet. Therefore, the iverse kepstrum has bee tested i a beamformig structure [Figure (a)] i a room. To verify the performace i sigal, the desired sigal [Figure 7 I-(a)] cosistig of three frequecies (500 Hz, 550 Hz ad 700 H has bee used, ad music souds tued to a radio statio [Figure 7 I-(b)] has bee used as real ostatioary oise i a room. From the test o the AC structure, it is show that the shape of the sigal waveform has bee distorted i time domai, as show i Figure 7 I-(c). O the other had, it is show that almost the same shape of the sigal waveform has bee maitaied i the beamformig structure as show i Figures 7 I-(d). The performace has also bee compared i spectra i frequecy domai as show i Figure 7 II-(e). I the beamformig structure, the performace has bee compared betwee 00 adaptive FIR filter weights aloe [Figures 7 II-(f)-(i)] ad 3 iverse kepstrum coefficiets with 50 cascaded adaptive FIR filter weights [Figure 7 II-(f)-(ii)], where the test results show that iverse the

21 Sesors 0, 6836 applicatio of the kepstrum with reduced adaptive FIR filter weights gives eve better oise reductio performace while maitaiig the origial desired sigal. It idicates that the applicatio of the iverse kepstrum could provide a beefit of more computatioal simplicity tha a kepstrum i processig as well as the ordiary adaptive FIR filter (Table ). To reduce the computatioal complexity, the alterative method could be cosidered by decreasig the widow size but the problem has bee foud with the wideig sidelobe o the desired sigal. By usig the iverse kepstrum, it has bee foud that icreasig to 4096 the widow size is acceptable for the real-time processig with the arrowig sidelobe o the desired sigal []. Figure 7. Real test i a room eviromet: (I) waveforms of (a) desired sigal (b) real ostatioary oise (c) processig with three samples delayed i AC structure (d) processig with o delay i beamformig structure, (II) correspodig spectra from (e) I-(c) ad I-(d) (f) compariso betwee (i) processig with 00 adaptive FIR filter weights aloe ad (ii) processig with 3 iverse kepstrum coefficiets ad 50 adaptive FIR filter weights i beamformig structure. I-(a) I-(b) I-(c) I-(d) I-(c) I-(d) (i) (ii) ote: II-(e) i (c) ad (d) idicates a distorted part of desired sigal (a). II-(f) Furthermore, the performace [Figure 7 II-(f)-(ii)] of the iverse kepstrum i rear-ed beamformig [Figure (a)] has bee compared with the frot-ed applicatio of the iverse kepstrum [Figure 8(a)] ad kepstrum [Figure 8(b) i a beamformig structure. With the use of a kepstrum (or iverse kepstrum) coefficiets to 3 ad a reduced size of adaptive filter weights to 50, it is ow to verify the performace o sigal distortio i a desired sigal whe the desired sigal is applied to the oise estimates of coefficiets ad weights, which are abruptly froze o the rapid chage of oise

22 Sesors 0, 6837 statistics. To verify the performace, the average discrepacy betwee stregths of three frequecies after processig has bee measured sice each iput frequecy is applied with equal stregths i db. For a trial test, the same desired sigal is applied at radom itervals to the frequetly chagig real oise eviromet, where the estimate of oise trasfer fuctio is radomly froze accordigly. Figure 8 shows sapshot of a performace amog the applicatio of frot-ed kepstrum, frot-ed iverse kepstrum with compariso of applicatio of rear-ed iverse kepstrum i time ad frequecy domais. Figure 8. Frot-ed applicatio of: (a) iverse kepstrum ad (b) kepstrum i beamformig structure; (c) waveform i the time domai by iverse kepstrum; (d) waveform i time domai by kepstrum; (e) spectra i frequecy domai betwee Figure 7 II-(f)-(ii) ad (c); (f) its spectra i frequecy domai betwee Figure 7 II-(f)-(ii) ad (d); [commoly based o 3 iverse kepstrum coefficiets (or 3 kepstrum coefficiets) ad 50 adaptive FIR filter weights]. Primary MIC d 0.5 Primary MIC d 0.5 Referece MIC x K ( x Iverse kepstrum filter (a) 0.5 W ( z Output ) e Adaptive FIR filter Referece MIC K( x x Kepstrum filter (b) 0.5 W ( z ) Adaptive FIR filter Output e Sigal stregth differece (c) (d) (Figure 7 II-(f)-(ii) Sigal stregth discrepacy (Figure 7 II-(f)-(ii) Sigal stregth discrepacy (d) (c) ote: (e) i (c) ad (d) idicates a distorted part from desired sigal (Figure 7 I-(a)) i time domai. i (e) ad (f) idicates that distorted part i desired sigal from the applicatio of rear-ed iverse kepstrum (Figure 7 II-(f)-(ii)) is compared with oe of applicatio from (c) frot-ed iverse kepstrum ad (d) froted kepstrum i frequecy domai. (f) As show i Figure 8(c,e), the test results show that frot-ed iverse kepstrum provides better oise reductio but with atteuated amplitude i a desired sigal. O the other had, the frot-ed kepstrum [4] shows more stregth i sigal amplitude with almost same oise reductio as show i Figure 8(d,f). However, it has bee foud from the test that both frot-ed methods [Figure 8(a,b)]

23 Sesors 0, 6838 are more vulerable to sigal distortio tha the rear-ed iverse kepstrum method i beamformig structure. As show i Table 4, the level of sigal distortio has bee compared by measurig a average discrepacy of sigal stregth i db from frot-ed kepstrum [4], frot-ed iverse kepstrum [] ad rear-ed iverse kepstrum, where it has bee calculated as 0.5 db, 0.45 db ad 0.5 db, respectively. Table 4. Compariso of average discrepacy o applicatio of frot-ed kepstrum [4], frot-ed iverse kepstrum [] ad rear-ed iverse kepstrum. Applicatio Frot-ed kepstrum Frot-ed iverse Rear-ed iverse kepstrum kepstrum Frequecy (H D D D Trial Trial Trial Trial Trial Average D (db) Summary ote: D idicates discrepacy i db from sigal stregth of desired sigal i frequecy domai. For real-time processig i a realistic reverberat eviromet, the kepstrum method ad iverse kepstrum method have bee applied to AC ad beamformig structures, hece its performace o modified applicatio [4,6-8] to desired sigal ad adaptive filter has bee ivestigated i detail i this paper. Firstly, the reverberat ature of most rooms gives rise to omiimum phase compoets i the acoustic trasfer fuctio [8] ad its iverse is ofte required i a realistic reverberat eviromet [6], hece it might produce istability i processig. Therefore, the acoustic trasfer fuctio ad its iverted trasfer fuctio have bee tested i AC ad beamformig structures. Secodly, the acoustic oise trasfer fuctio chages rapidly ad frequetly i a realistic reverberat eviromet ad the estimated oise statistics for the acoustic trasfer fuctio are abruptly froze. It is the istatly applied durig the desired sigal period, which might cause a sigal distortio. To verify the performace, a discrepacy i db has bee compared amog the applicatio of frot-ed kepstrum, frot-ed iverse kepstrum ad rear-ed iverse kepstrum i a beamformig structure. Thirdly, a fast covergece i adaptive filter applicatio is essetial i real-time processig so that a fast covergece with small amout of adaptive filter weights has bee tested i both kepstrum method ad iverse kepstrum method. For a accurate discrepacy measuremet i a desired sigal, three added simple sie waveforms (disregardig whether it is arrowbad or widebad i this test) have arbitrarily bee used as a sample of a desired sigal so that the distorted amout i a desired sigal could be measured by calculatig the average discrepacy i db to check the cosistecy level of the sigal stregths i db from the origial desired sigal. For the precise testig i estimate of coefficiets ad weights, istead of usig automatic VAD, it has bee programmed to stop ad make the last updated coefficiets ad weights to be froze o demad of the applicatio, ad the it is applied to the desired sigal ad real oise period.