Module 11: Applications : Linear prediction, Speech Analysis and Speech Enhancement Prof. Eliathamby Ambikairajah Dr. Tharmarajah Thiruvaran School

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1 Module : Applicatios : Liear predictio, Speech Aalysis ad Speech Ehacemet Prof. Eliathamby Ambiairajah Dr. Tharmarajah Thiruvara School of Electrical Egieerig & Telecommuicatios The Uiversity of New South Wales Australia

2 Liear Predictio The degree of which a sigal ca be predicted from its past samples I time domai a predictable sigal has smooth ad correlated fuctios I frequecy domai the eergy of a predictable sigal is cocetrated i relatively arrow bads of frequecies I cotrast, the eergy of a upredictable sigal, such as a white oise, is spread i a wide bad of frequecies ELEC97

3 Liear predictio for a quassi periodic sigal s s p a a s s + + a s G u + a 3 s G u Curret speech sample Prevoius speech samples Betwee pitch pulses G u0. Therefore s ca be predicted from a liearly weighted summatio of past samples see above equatio However, if G u is icluded the we ca predict s oly approximately 3 T T t

4 Liear predictio models are extesively used i speech processig e.g. low bit rate speech coders Impulse Trai Geerator Radom Noise Geerator T T u t G Gai T Pitch period Vocal Tract Model p a s Speech Time model for speech productio S U p G a 4

5 A liear predictor ca be used to fid values of the coefficiets a, a, a 3, of the vocal tract filter, usig the speech samples. Speech G u sigal + e T T t + a - s ^ s Predictio error a - - a p - - p Vocal Tract model all pole model 5 Liear predictor all ero model

6 ŝ Liear Predictor is ofte referred to as iverse filterig, as its aim is to determie the all ero filter which is the INVERSE of the vocal tract model p e Liear Predictor s p a s Vocal Tract model G u t T T p a G U S p S E G u e G U E G S E U S U E a the If : Note 6

7 Suppose we process the speech sigals with a liear predictor ad the predictor coefficiets are ad the predictor output is: s Speech sigal Liear Predictor s p ŝ e sˆ sˆ p s T s Vector form The basic problem of liear predictio aalysis is to determie a set of predictor coefficiets directly from the speech sigal. How do we obtai the same as a? 7 The error betwee the actual sigal s ad the predicted ^ value s is give by: e e e s s s sˆ p s T s Vector form

8 Oe approach is based o miimisig the average squared predictio error over a short segmet e.g. frame 00 samples.5 ms; fs8000h of the waveform.the shorttime average predictio error is defied as: [ ] s s J s s e J p ˆ 8 [ ],,...p j for 0 j s s s j J j J p J is miimised whe :

9 p [ s s j ] [ s s j ] j p p equatios are obtaied. A matrix ca be formed from the past ad preset samples to 00 i all frames. If p, the simultaeous equatios ad with uows have to be solved to obtai the predictor coefficiets A recursio algorithm ow as Durbi s method is ormally used to calculate,, 3, p ad it offers a solutio without havig to solve the matrix. 9

10 This is similar to h opt R - P Wieer filter i time domai p p p p j N j s s j ss φ φ φ φ φ φ φ φ φ φ φ. ss 0.. ss ss 0 ss ss. ss ss 0 ss. ss ss p j 0 0 [ ] [ ] p j j s s p j s s

11 G u Vocal Tract model s Liear Predictor s e T T t p a Speech sigal p ŝ Speech productio Model If a the e Liear Predictor/Iverse filter G u A by product of the liear predictio aalysis is the geeratio of the error sigal e ad it is a good approximatio to the excitatio source Error sigal e It is expected that the predictio error will be large for voiced speech at the begiig of each pitch period.

12 Note: Iverse filterig Im Re Poles correspod to the vocal tract Zeros correspod to the iverse filter

13 Liear Predictio usig Steepest Descet Algorithm The diagram below shows a bloc diagram of a liear predictor. The predictor coefficiets are adjusted cotiually durig adaptatio to reduce the squared predictio error e towards its miimum vale s Speech sigal p Liear Predictor s ŝ e e Predictor coefficiets:,, 3, p 3

14 Liear Predictio usig Steepest Descet Algorithm [ ],,..p ] [ ˆ + + s ce s e e e e s s s s e p The updatig of the predictor coefficiets is carried out usig the steepest descet algorithm. The predictor coefficiets are up dated o a sample by sample basis e c + ] [ Curret Previous Gradiet c is the learig rate 0<c< 4

15 Iverse Filterig Let us cosider a model of a bad limited detector H m see below d Impulse y d d respos e h m Iverse filter H m H d If H m A/B the the iverse system fuctio is the give by H d B/A. The result idicates that the poles of the iverse filter system fuctio correspod to the eros of the model. The eros of the iverse filter are the poles of the model. For stable ad causal iverse filter, the poles of the iverse filter trasfer fuctio must lie withi the uit circle. 5

16 Example Let a trasmissio chael exhibit oly oe echo, havig amplitude a { a <} ad occurig K samples after the trasmitted sigal is received. The the uit-sample respose of the chael is h m 0, h m Ka ad h m 0 otherwise Therefore H m +a -K The iverse filter is: H iv H m + a K 6 ELEC97

17 Example: Iverse filter for a o-miimum phase system Let us cosider a physical system whose model system fuctio is give by H m +a - a >. This system model is a o-miimum phase because it has a ero outside the uit circle -a. The iverse filter system fuctio is: H iv H m To avoid the above problem: + H m H m a Sice the pole of H iv is outside the uit circle, the iverse filter is ustable. H. H mi allpass + a + a + a Ay o miimum phase filter ca be expressed as a miimum phase system multiplied by a all-pass filter. 7

18 Example: Cosider the followig o-miimum system H m. Fid the trasfer fuctio of the iverse filter. Your iverse filter must be stable. H m

19 Example: Cosider the followig o-miimum system H m. Fid the trasfer fuctio of the iverse filter. Your iverse filter must be stable H m Solutio: 0.5. mi We ca write H m H m allpass H H H m as iv H Allpass filter 9

20 Speech Sigal quasi-statioary sigal I speech processig, speech souds are divided ito TWO broad classes which deped o the role of the vocal chords o the speech productio mechaism VOICED speech is produced whe the vocal chords play a active role i.e. vibrate i the productio of a soud UNVOICED speech is produced whe vocal chords are iactive 0 ELEC97

21 Vocal Chords The vocal chords vibrate at a particular frequecy, which is called the pitch of the soud 50 to 00 H for male speaers 00 to 400 H for female ad child speaers Vocal tract chages shape slowly Fixed characteristics over a short iterval of about 0ms Allows assumptio that speech is quasi-statioary sigal ELEC97

22 Resoat Frequecies of Vocal Tract Vocal Tract is a o-uiform acoustic tube Cross-sectioal area depeds o lips, togue, jaw ad velum Spectrum of vocal tract respose shows a umber of resoat frequecies These frequecies are called Formats Four formats preset below 4 H ELEC97

23 Speech Productio Model S U G p a Impulse Trai Geerator Radom Noise Geerator T T u t G Gai T Pitch period Vocal Tract Model p a s Speech 3 ELEC97

24 Pitch period T 4 ELEC97

25 Format Frequecies Speech ormally exhibits oe format frequecy i every H For VOICED speech, the magitude of the lower format frequecies is successively larger tha the magitude of the higher format frequecies For UNVOICED speech, the magitude of the higher format frequecies is successively larger tha the magitude of the lower format frequecies 5 ELEC97

26 Voiced Speech 6

27 Uvoiced Speech 7

28 Speech Sigal Aalysis Short-time Eergy The short-time eergy of a o-statioary sigal may be computed by dividig the sigal ito frames of N samples ad computig the total squared value of the sigal samples i each frame. 8 ELEC97

29 Speech sigal x Frame Frame Frame Splittig the sigal ito frames ca be achieved by multiplyig the sigal by a suitable widow fuctio w N E [ ] m s w m 0 0 N w m 0 otherwise m frame umber 9

30 E m N [ s w m 0 This equatio ca thus be iterpreted as: ] This equatio ca be writte as: N E m [ s ] h m 0 where h [ w ] s s h E m The sigal [s] is filtered by a liear filter with impulse respose h i the above equatio. 30

31 w Let w be a rectagular widow W + 0 N - 0 otherwise - [ ] N N - N θ j H θ e Nθ si θ si Hθ π/n π/n To see how the choice of widow affects the short-time eergy, let h is very log BadwidthBWπ/N is very small. Such widow would be equivalet of a very arrowbad lowpass filter. E m would chage very little with time. 3

32 Choice of Widow What is desired is some lowpass filterig, so that the short-time eergy reflects the amplitude variatios of the sigal. That is, we wish to have a short duratio widow to be resposive to rapid amplitude chages A widow that is too short will ot provide sufficiet averagig to produce a smooth eergy fuctio 3 ELEC97

33 UV V UV N00 samples V- voiced speech ; UV uvoiced speech; Speech sigal Six As ca be see i the above figure, the values of short-time eergy for uvoiced segmets are sigificatly smaller tha voiced segmets. 33

34 N00 samples The effect of the duratio of the rectagular widow o the eergy computatio for the uttrace oe spoe by a male speaer. If N is icreased, the eergy cotour would become smoother 34

35 Calculatig the Short-time Eergy Recursively Note that a recursive lowpass filter H ca also be used to calculate the short-time eergy: H 0 < a < a It ca be easily verified that the frequecy respose Hθ has the desired lowpass property. Such a filter ca be implemeted by a simple differece equatio: E a E- + [x] 35 ELEC97

36 The Structure for Calculatig the Short-time Eergy Recursively S x [S] + Lowpass filter - E p The quatity E must be computed at each sample of the iput speech sigal, eve though a much lower samplig rate suffice. The value a ca be calculated as: a e f c π f s f c is the cut-off frequecy ad f s is the samplig frequecy. 36 a

37 Speech Ehacemet Degradatio of speech quality caused by acoustic bacgroud oise is commo i most speech processig applicatios, icludig mobile commuicatios ad speech recogitio. Therefore, the problem of removig ucorrelated oise compoets from oisy speech has bee widely studied i the past, ad still remais a importat issue i the field of speech research. 37 ELEC97

38 Speech Ehacemet usig Spectral Subtractio Spectral subtractio has bee widely used for removig additive oise, ad has resulted i may variatios which have ejoyed reasoable success. The spectral subtractio techique proceeds as follows. Cosider a sigal corrupted by oise, which is described by: y s + η where s represets the ucorrupted speech sigal, η is the ucorrelated oise ad y is the oisy speech. 38 ELEC97

39 Fourier Trasform of y s + η Taig the Fourier trasform yields Yθ Sθ + Nθ where Y is the spectrum of the oisy speech sigal. Similarly Sθ ad Nθ represet the spectra of the clea speech ad the oise, respectively. I spectral subtractio, the modulus of the speech spectrum Sθ is estimated by subtractig a estimate of the oise magitude spectrum, E[ Nθ ] from Yθ. 39 ELEC97

40 Estimatio of Noise Magitude Spectrum The oise spectrum is estimated durig the bacgroud silece or durig pauses betwee words. The estimated oise magitude spectrum is the subtracted from the oisy speech magitude spectrum. I the spectral subtractio method, some residual oise is observed i the processed speech due to iaccurate estimates of the oise magitude spectrum at all frequecies. This error produces oise i the recostructed speech, referred to as 'musical oise'. 40 ELEC97

41 Spectral Subtractio The diagram below illustrates a bloc diagram cofiguratio of spectral subtractio: Yθ Sθ + Nθ y DFT/FFT Yθ - Post subtractio processig ^ Sθ IDFT/IFFT ^ s ^ Nθ Noise Estimate 4 ELEC97

42 The parameter i the spectrum equatio cotrols the amout of oise subtracted from the oisy sigal. Sˆ θ Y θ Nˆ θ For full subtractio ad for over subtarctio >. The time averaged oise spectrum Nθ ^ Obtaied from the periods whe the sigal is abset ad oly the oise is preset as: N ^ θ Nˆ θ M M N θ is the spectrum of the th oise frame ad it is assumed that there are M frames i the oise-oly period M is variable 4

43 Alteratively the averaged oise spectrum ca be obtaied as the output of a first-order digital lowpass filter as: N θ N θ + N θ Lowpass filter reduces the oise variace For restoratio of a time domai sigal the magitude spectrum estimate Sθ ^ is combied with the phase of the oisy sigal ad the trasformed ito the time via iverse Fourier Trasform as: sˆ M M 0 S e jφ e jπ M 43 Phase of the oisy sigal y

44 Due to variatios of the oise spectrum, spectral subtarctio may produce egative estimates of the magitude spectrum. To avoid egative estimates the spectral subtractio output is post processed see below. For example we may choose a rule such that if: Sˆ θ Sˆ θ Sˆ θ < 0 0 Y θ Nˆ θ o chage i.e. Y θ < Nˆ θ the set Sˆ θ 0 or 0.0 y DFT/FFT Yθ ^ Nθ Noise Estimate - Post subtractio processig Sθ ^ IDFT/IFF T ^ s 44

45 Musical Noise If the spectral subtractio method, some residual oise is observed i the processed speech due to iaccurate estimates of the oise magitude spectrum at all frequecies This error produces oise i the recostructed speech, referred to as Musical Noise. 45 ELEC97

46 Speech Ehacemet usig Spectral Subtractio Method 8 poit FFT Speech +oise f s 8 H Magitude at each frequecy 0 to 7 Noise frames N Nˆ M M N Y Y S + N Speech frames Y Oe value for each frequecy 0 to 7 Φ Phase value at each frequecy Φ0, Φ, Φ,.. ^ N - Sˆ Y Nˆ Oe value for each frequecy 0 to 7 Post processig ^ S 46 8 poit IFFT -.5 ^ s Ehaced speech

47 Summary Liear predictio Speech aalysis Format frequecies STFT of speech Speech ehacemet usig spectral subtractio 47 ELEC97