Musical Noise Generation Analysis for Noise Reduction Methods Based on Spectral Subtraction and MMSE STSA Estimation

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(1)MUSICAL NOISE GENERATION A NALYSIS FOR NOISE REDUCTION民1ETHODS BASED ON SPECTRAL SUBTRACTION A ND MMSE STSA ESTIお1ATION t Yoshihisa Uemura，↑Yu. Takahashí， tHiroshi Saruwatari， tKiyohiro Shikano， and t.Kazunobu Kondo. tNara Institute of Science and Technology， Nara， 630-0192， JAPAN t.SA Group， Center for Advanced Sound Technologies， Yamaha Corp.， Shizuoka， 438-0192， JAPAN. this paper， we realize the comparison using kurtosis ratio， and con sequ巴ntly we reveal a new抗ndings about the degree of generated musical noise in MMSE STSA vs， SS.. ABSTRACT. 1n this paper， we reveal new findings about the g巴nerated musícal noíse in minimum mean-squ紅'e error short-tÍlne spec住al amplitude O仏1SE STSA) processing， Recently we have proposed a objective metric of musical noise based on kurtosis change ratio on spec位al subtraction (SS). Also we found an interesting relationship among the degree of g巴nerated musical noise，the shapes of signal's proba bility density function， the s甘ength p紅ameter of SS processing. This paper IS包med to automatically 巴valuate the sound quality of various types of noise r巴duction metllods using kurtosis change ratio， We give a mathernatical analysis based on higher-order statistics view point， and lead to a valuable relation in that .tv仏1SE STSA has a weakness in speech period distortion rather than noise period， and vice versa in SS. 2. OVERVJEW OF NONLJNEAR PROCE SSlNG 2.1. SS At自民t， we introduce two kinds of representative nonlinear process ing， i.e.， SS and .tv仏1SE STSA. Although various types of SS metll ods are proposed， we address single-charUIel SS in出巴 power do mおn， which is used for any speech enhancement Let the coπupted speech signal o(t) be represent巴d as. [5].. bldex Terms- Musical noise，higher-order statistics， minimum mean-square e，汀or short-time spectral amplitude， spectral sub住ac tion，speωh enhancement 1. nぜTRODUCTION. [1]. Nonlinear proc巴ssing，e.g" spectral subtraction (SS) and min1m山n mean-square eηor short-time sp巴C位al ampli加de STSA) often generates p訂ticular distortion，th巴 so-called sical noise， It is one of the critical problems inherent in nonlín巴ar processing because musical noise is perceived as harsh and artificial tone. Thus a lot of countermeasures to handle the musical noise have been proposed. However musical noise can not be れraluated by 位aditional metric about sound quality， e.g.，cepstrum dist組ce，組d we just had to su均ectively assess tIle sound quality [3]. To begin with， we did not know so much thing and tIleory about subjective evaluation of musical noise， Thus we c組 not evaluate the actual performance of each of countermeasures ag出nst musical noise.. [2]，. k. (2). 2.2. l\-仏1SE STSA es“mator MMSE STSA is a method for estimating the clean speech spectral amplitude from co汀upted speech signal by mìnimization of m巴an square eπor. Tt is supposed that the statistical mod巴1 of noise ìs Gaussian mod巴1， and the rnodel is statistically independent and has zero m巴an. The given spec凶1 gain by.tv仏1SE is written by. G(k) = r(1.5)主主xplーさ) ， 1(1 叫ん(�)+ vkll(刊 (4 ) 、ん '1 Yk. 、 4レ， 1. 、ゐ，. where r(ー) denotes出巴E鉱山na functionんand 11 denotes the modi tìed Bessel functíons of zero and first order，resp古川;tively. Also functions in the equation ar巴 defined by. 'k圭一生-:- Yk・ 1+6 '. H巴re Çk and 'yk ar巴 defined by. �nc s甘ategic Information and Com R&D Promotion Progr出百mein Japan. 978-1・4244・2354-5/09/$25.00 \Þ2009 IEEE. O(k， m) = S (k.l1l) + D(k， 111)，. Y(k，m) =ゾIO(い)12βEm [ID(k， 111)12]円(W.m)) (3) where Y(k， m) is an estimated speech signal‘βis an over-subtraction coefficient (i.e.， s位ength parameter) and E[.] is an expectation oper ator of . with respect to m. This work was partJy suppo口ed by munications. (1). where denotes the frequency subband and 111 is出e frame index. In SS， noise reduction is achieved by subtracting the power spec位um of th巴邸timated noise合om the power spec佐um of the noisy obser vation， This procedure is given by. (MMSE mu. We have propos巴d a novel mathematícal metric of musical noise [4]. The metric based on change of kurtosis，4th-order statis tícs， through nonlin巴ar processing， has high correlation with the amount of perceived musical noise by human. Also we found出at the degree of generated musical noise in processing is s仕ongly re lated wìth kurtosis nitio， Therefore now we can objectively evaluate the degree of generated musical noise in nonlinear processing wìth our kurtosis ratio Recently it is widely accepted for speech-e出ancement sωdies that 1仏1SE STSA sets up less musical noise tIlan SS， and we can obtain白e high-quality (Iess degraded) output signal. However no one confirrned ìt from theoretical and anal)1ical aspects because it is so difficult and unrealistic that both of MMSE STSA and SS are compared via subj巴ctive evaluation in everγparameters of itself. In. 。(t) = s(f) + d(t)，. where s(t) is a clean speech signal and d(t) is a noise signal. This processing is conducted on a frame-by-frame basis. The short-time Fourier回nsform (STFT) is used and血e previous model can be rewntten as. &λ，(k ) 一一一、. 日一. '!d(k). (5) (6). ，，2. 刊 'yk 圭で全日 Æd �K) wh巴re '!，(k)土E[IS k12] and ，(d(k)土E[lDkI2]. Çk and Yk are called a. 4433. ICASSP 2009. nHd n，白吋『目..

(2) (a) ぎ守. ‘竺. 会= 2悶ao』仏. い九時， ‘J wo 羽 R L 3 Y6 』U ，a，、鳴 e IBIIFlIωlML F ヨ ω g hu ] Z N {. { N Zpc sz￡. 〉、. (a). 正コ偲. ê t\ P(xl = c a.. 1 \ /. Xo-1. e-í. (b) ， a αe P(x) =c.! xα le→d�' JO (x = 0). て. Fα，6. P(z)=Ci(z+β由。)α4ピヰ争Z ー/ (x > 0). E. 出三ιー. Power. Power. Fig. 2. Shapes of p.d.f. (a) Original signal. (b) Processed signal. artificial signal pl∞essing. Hence， we turn our attention to l.:urtosis change ratio (kurtosis ratio) between before/after signal processing.. priori and a posterioげsignal-to-noise ratios (SNR)， res戸C臥!ely. h品1SE STSA can estimate the clean speech spectral anlpl山de as above‘ideally. However， in actual case， we can not know a p円on and a posterior SNR， and thus we estimate白em by the following equatlon，. 《. λ;(m ー 1). 会(m) =ηーよ一一一一 +(1- η)P[れ(m)ー1]，. Àd(k， m - 1 ). 0<ワ< 1，. 3.2. Kurtosis ratio in SS. We deriv巴 the relationship hetween kurtosis and the strength of SS Moreover， the relationship between kurtosis of processed signal and kurtosis of unprocessed signal are r巴vealed.. (8). 3.2.1. Gamma distribution lIlodeling. where Ak(川一1) i s the amplitud巴巴stimator o f the k出 signal spec汀al component in the (111 - l)th analysis企arne， and P[.] is an operator which is defined by. l. P[x] = <. _ 0. I. 江x> O.. We utilize山e garnma dis凶bution as a model of speech or noise sig nal [6). The ganuna dis凶bution have a lot of nseful mathematically a町ibutes which 紅e derived合om白e gamma function The p.d.f. of the garnma distribution is written as. (9). otherwise.. 3. MlJSICAL NOISE METRIC VL主KURTOSIS RATIO. J.I�. (10). μ3 where kurt denotes kurtosis and J.ln is the nth order moment which is given by. E [P(x)] =α(J，. α-. 12y. E[x]. α where y = log( E[x] ) - E[ log x] (see Refs. [7]).. (14). (15) (16). 3.2.2. Kurtosis ofmodeling signal. h出is way of modeling by the g釘凶na distribution. k."Urtosis is deter mined by the shape and the scale p紅白neters as below. At fi.rst， we rt"present the n白ーorder moment as (17) (岨 X"P(x) d x = C. (J"+n . r加的 JO Using (17) and useful relation，r紛=(σ-1)・。. (α-j)T(αーj)， we c組 obtain kurtosis of the mod巴led raw signal by the garnma dis往ibution as follows [4).. μn =. JO. Here， P(x) is p.d.f. of白e signal. We consider the SS in power spectral domain， so the integral range is only positive Although we can measure the number of the tonal compon巴nts by kurtosis， note that kurtosis itself is not enough to measure the rnusical noise. This is obvious in that kurtosis of some unpr∞essed signals， e.g.， speech signals， is also high， but we do not recogniz巴 speech as musical noise. Jn order to set aside the genuine tonal com ponents， we focus on the fact th.at musical noise is generated only in. 3-γ+、I(y-3)2 +24γ ô=. (. I )." P(x)d x.. (13). σ. where E[.] is an expectation operaωr. Th巴 garnma distribution mod eling is the estimation of the shape and the scale param巴ters from the raw input signal. In出is paper， we use the maximum likelihood estimation rnethod for estimating two pararnetersαand (J， as follows，. ) 1. μn =. r(め= fflf dx. H己reafter， in出is papeにl巴t C = 1 /[T(α) (J ]. lf ()' = 1，血is is白e exponential distribution. It is well known that the average of the g出nma distribution is given by. 3.1. Rel ationsbip between kurtosis ratio and musical noise. kuロ=. (12). JO. In this section， we introduce a basic idea of musical noise evaluation and musical noise metric for SS [4]. Hereinafter we give 組以場 planation of出e adequacy applying the metric to gen巴ral norùinear prω巴ssmg Nonlinear processing， including SS釦d恥仏1SE STSA， often gener ates characteristic isolated power spectral components (see Fig. 1 (a) and (b)). We define the musical noise as the generated audible iso笥 lated sp巴ctral components血rough nonlinear processing τbus we speculate that the arnount of musical noise is highly relat巴d to th巴 number of isolated components and the isolated level of them. Con sequently we realize the evaluation of the isolated components by kurtosis. We could say也at kurtosis can evaluate也e percentage of tonal cornponents in total cornponents. Bigger value indicates a signal with heavy skirt in its probability density function (p.d.fよit means that a signal has a lot of tonal components. Kurtosis is defined as. P(x) =ー土-:::- . xo-1 e-j，. T(，α)(Jcr where x三0，α> 0 and (J > O. Alsoσdenot巴s the shape par組leter and (J is the scale paranleter. The Garnma function is defined by. Consequently MMSE STSA is managed by estimating白e mth frarne spectral gain using previous frarne spectral g出n.. 一向一一一一 ku広ー日一可岡. μJ. ( α+ 2)(α+ 3) α(α+1). (18). 3ユ3. Change ofmodeling sigllal's kurtosis Í11 processing. SS is regarded as p.d.f. deforrning processing， i.e.， lateral shift of p.d.f. (See Fig. 2). Thus processed signal's kunosis depends on the S官'ength of processing. We formulated the deforrned signal's kurtosis. 4434. ハHV 内〈U.

(3) 1ra ble 1. Expected situation of musical noise generation. |. SS. I. 剛SE STSA estimaTor. 14. I. 12f. _1011 国 3:!.8 � 6 cn. (Unclear) Moderate. noise: wb.ite Gaussian noise:. Mixing. e'quivale泊t SNR mixing. I I. Evaluation noml. placed by zero witb 0.01 increments in between SNR 釦d kurtosis ratio of clean speech初d cle組 noise signal. 5. コ出. Noise. 10. +. ，-. .. o. i. 1.25. ( 1 9). _.. (x=O).. we have. .;- 1 J，.1円一 le-_ �dxl. αの2 .") ，_ ， S脚〈向吋 �一加-I + 2. (rr+:!)- l. ，. o. 側. ( 2 尺r 小ケ柑 f} a門+叫肘叶 CαE一均吻仙州w伊le判 dx 壬d. 2. 2. 01 SS (日). 2.25. 2.5. 0.5. 町田・h・田4四，_，田・. ・_，_. 0.6. 0.7. 0.8. Strength parameter 01 MMSE STSA. 0.9. (η). Fig. 5. Relationshíp between noise's and speech's kurtosis ratio and. (， . � " . � .，. "n ， . �"" I)+ � " "Î' " . (β叫2 kurt括〉一一一{(α+2)(肘3)+向(a+2)(av �' (α-3)(σ一叫 '' 2 ' 一α(α+ 1) l' J (22) Here，αdenotes tbe shape par創neter of modeled garnma distribution， andβís tbe p紅四leter of SS processing strengtb. Thus ku巾.SIS ralto ín SS can be given as. th巴S釘巴ngth parameter MMSE STSA (.，，).. apply tbe above-mentioned tbeoη! to speech-noise mixed signal， and obtain the following theoretical prediction about tbe gen巴rat巴d mu sical noise when SS is applied to speechjnoíse-dominant íntervals. Speech signal has higher kurtosís than noise signal， in general. kurt.s. Therefore generated musical noise in speech-domínant intervals is. = l.'Urtorg. バ βα (α ー 1) Gβα)2(α-2)(a - 1) Î e吋{l +一一一一一+ = ト (α+ 3) 2 (α+ 2)(σ+ 3) J 1. Speech・ Noise 田市. F"・由畠幅・岨唱帽. 。. Thus p叩c∞芯s鉛sed s凱19I伊】al's kur式tos釘:is is est“imat犯.ed as. M凶s ralto. 1.75. the s田ngth parameter of SS (，β). Aω I the 2nd-order moment ís estimated as belo哨，. x+リβ. 1.5. Strength parameter. Fig. 4. Relationship between noise's and speech's kurtosís ratio and. (X〉O).. Here we appro啄imate (x+ β αw - I in (19) by Tay lor exp組sion， and. ρω心[fx(a門α. _，_. 5. 。。，，戸OEa凋骨内。 0 2 偲』 ω一的 o t コリL. (c. (x βー αe)αーl eーヰd ，:s lC._ tcr�9 x"._ 九- �ûdx. P(x) = {. (η). Speech -. s田ngth parameter: configure from 0.5ω0.99. signal is w吋tten as. 子2 [ドO:. 0.9. 201. negative power compollellts訂e re. and change ofkmtosis in SS[4]. The resultant p.d.f. of the processed. μ2. 0.8. 0.7 S廿ength parameter of MMSE STSA. Fig.3. Processed s氾nal's SNR on SS and MMSE STSA.. S仕engtb parameter: configure仕om0 to 2.5 with. MMSE STSA. 0.6. 0.5. 0.05 increments in between floorillg:. ___. 2. speech: Japan News Article Sentences SS. 2.5. MMSE噛由 SS. .. 時• r ---_.. --__-f .. .. .. __--0..4... Table 2. Su句巴ctive evaluation conditions Database. Stren g th parameterol SS (ß) 1.5 1. 0.5. 。. relatively less. Conversely， a lot of musical nO.ises generate in noise. (23). Also we found that tbe musical noise metric based 0司kurtosis ra tio is highly related with the amount of perceived musical noise by hum釦[4). Kurtosís ratio is strongly related witb tbe generated isolated. components in nonlinear processing， Thus we can evaluate the de gree of generated musical noise according to magnitude of kurtosís ratio value，. only intervals because noíse signal ís low kurtosís signal， e.g.， Gaus SJan nOlse. On the otber h組d， we can not formulate generated musical noise amoUl1t in }，仏1SE STSA estimator， but we can s凶speculate出e de gree of generated musical noise in only instance of noise intervals Here， we suppose that noise signal is stationary and has low kurto sis. In血is inst組ce， the obtained spectral gain by r-.仏1SE STSA es timator is stationary and small value in noise intervals. Thus output signal's p.d.f. does not change so rnuch frorn original one. Conse quently， the amount of generated musical noise is less白組曲e case. 4. THEORETlCAL ANALYSIS As indicated in. of using SS. Howev巴r， in speech intervals. we can not forecast be. (23)， we can now白nd a relatíonship between the. amount of generated musícal noíse in SS and the orígínal sígnal's. cause the estimation of spec仕al gain using a. prior and a posterior. SNR depends on previous合出ne and is ex出mely complicated. k.'Urtosís. That ís， SS for hígh k'Urt . osis sígnal (αís small) results in. Table 1 lists the summary of the points about musical noise gen. less musical noise than SS for low kurtosís signal (αis laτge) even. eratíon. We will confirm the predictioTls and bring out the unclear. if we set tbe fixed subtraction parameter β.百lUS in tbis paper， w巴. points with 巴xperiment in the next section.. 4435. 司ペU 114.

(4) 20. 16. ，g �. �. 14. F-_ .. ...・p 噌p ・・・・. 一 � 一一一一・..---・・-一ー. 一一一・・司h・・・・・・・・・一ー一 -�----ー・. i. SS -←. 12. 10 6 6. 4. 2. 0. 6. 5. 7. B. SNR. [dBl. 9. 11. 10. Fig.6Compariωn of noise's k.-uれosis ratio on equivalent SNR 61. ozm』ωm。亡コ￥. au aa守内，‘ 0. (a) Original. 否6�. MMSE .... ←. 4. nu noco a匂内4 比 @』廿コ ω t hu uz UN工 {. '5. ピ. 16. MMSE-骨・ SS・←. -・・・・・・・・砂・・・・・・・・・司p--------�--------司þo --------咽・-. 5. 6. 7. B SNR. [dBl. 9. 10. 1.2. 11. Fig. 8. Spectrogram (a) Original speecb signal. (b) SS processed. Fig. 7. Comparison of speecb's kurtosis ratio on equivalent SNR.. speecb signal (SNR signal (SNR. 5. EXPERIMENT. = 11. =. 1 1 dB). (c) r.仏，fSE STSA processed speech. dB). mo凶y accepted theOlγthat �仏1SE STSA is superior to SS in terms. 5.1. Conditions We conduct an experiment and 0同巴ctive evaluation for musical noise on SS and MMSE STSA. One of our great inter巴st is cornparative merits and dem巴rits of SS and 1仏1SE STSA. Particularly. we are interested in the difference on the degree of gen巴rat巴d rnusical noise between both m巴thod，ωd betw巴en both signal of speech or noise. Conditions of experiment are listed in Table 2.. 1.4. Time(s). The s町ength. of musical noise. It is stiU仕ue凶noise-only p紅t. but， in speech. dominant part. rhis is misconceprion.τrus new tìndi.ng is con且nned by spectrogram (Fig. 8). Spectrogram of SS processed signal shows the degradation of speech signal but we can not detect the isolated component. On the other hand， spec甘ogram of 1仏1SE sbow the apparent isolated components. Th i. s is consist巴nt with our subjec tive impressions. Consequently 1ぬ1SE STSA gen巴rat巴s the isolated. parameter of MMSE STSA is set to comrno叫y-used value and the. components in speech signal and we have consci.ous access to mus】. strength parameter of SS is controlled as to 巴qual SNR perfoπnance. cal noise.. of noise r巴duction.. 6. CONCLUSION. 5.2. Results. We analyze the degree of generated musical noise in SS and MMSE. Figure 3 depicts processed signal's SNR on SS and �仏1SE STSA. As we can see from Fig. 3， both出巴 upper 1凶út performance of MMSE STSA and SS are about 1 1 d.B. Figur，巴s 4 and 5 show the re. STSA. Also we came up with the novel fact about bow to character isti.c generate musical noise in 1仏1SE STSA.. 7. REFERENCES. lationship between k.-urtosis ratio and the strength p紅ameter of each method. In SS. as we expected， kurtosis ratio is smaller value in. [ 1 ] S. F. Boll， "Suppression of acoustic . noise in spe巴:ch using spec 住al sub甘acti.on，" IEEE Trans. Acoustics. Speech， Signal Proc.， voI.ASSP-27， no.2， pp. I 1 3-1 20 ， 1979. [2] Y. Ephraim and D. Malah，“Speech Enhancement Using a Min・. speech int巴rvals than noise int巴rvals. This is because kurtosis ratio in SS depends on unprocessed signal's original kurtosis On the other band， in r.仏1SE STSA， noise's kurtosis rati.o is very small. but speecb's kurtosis is very high and changes rapidly. It is particular note around commonly-used pararneter.. 1m凶n Mean-Square Error Short-Time Spec仕al Arnplitude Esti. This phe. mator，" IEEE Trans. Acoustics. Speech and Signal Proc.， vo1 . 32，. 出e sむengtb parameter of 1仏1SE STSA estimator and musical noise. no.6， pp . I I0 9- 1 1 2 1 ， December 1984 [3] M. Kato， et a1.，“Noise Suppression with High Speech Quality. via the sllbjective evaluation [2]. Consequently， in SS， musical noise. Based on Weigbted Noise Estimation and r.仏1SE STSA (Digi. nomenon has already contìrrned as白e sensitive relationship between. mainly arises in noise interval， on another front，出払仏1SE STSA， tbe. tal Signal Processing)“IEICE Trans. FlIndamentals， vo1.E85 -A，. problem of musical noise generation is mainly boiled up in speech. no目7， July 2002. [4] Y. Uemura， et al.，“Automatic Optimization Scb巴me of Spectral. interval.. Subtraction Based on Musical Noise Assessment via Higher. We compar巴 MMSE STSA with SS in t巴rms of the degree of generated musical noise. Figures 6 and 7 present the kurtosis ratio. Order Statistics"ハμ4.ENC. 2008 [5] J. Li， et al.，“ nOls巴 r巴duction based on adaptiveβ-order g巴n. sults are v巴ry mt巴resting. Figure 6 shows the familiar phenomenon. eralized spectral subtraction for speech enhancement，"刀VTER・. on SS and MMSE STSA in condition of 巴quivalent SNR. These re. SPEECH， pp，802-805， 2007. [6] T. H. Dat， et al..“Gamrna modeling of speech power組d its. of SS tbat th巴 amount of generat巴d musical noise in SS is very much and gradually increasing as bigg巴r the s廿ength p紅白n巴ter， on the. on-li.ne estimati.on for statistical speech enhancement，" IEICE. other hand， it is less in MMSE STSA. Figure 7 presents the interest. Tmlls. INF & SYST. vol.E89 -D， no.3， 2006 [7] M. Evans， et al.， Statistical Distriblltions， 2nd ed. Wiley. 1 993. ing result that恥仏l.SE STSA generates more musical noise in speech signal than SS. This is a new findi.ng. We have believed tbe corn-. 4436. 円，l副作、u 句14.

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