, July 6-8, 011, London, U.K. Formanic Analysis of Speech Signal by Wavele Transform L.Falek, A.Amrouche, L.Fergani, H.Teffahi, A.Djeradi Absrac The goal of his sudy is o consider he insananeous frequencies corresponding o he speech signal forman using he wavele ransform. The developed mehod is based on an analysis of derivaive phase of he coninuous Morle wavele ransform coefficiens. Using synhesized signals produced by a forman model made i possible o adap his mehod o real speech signals and o deerminae he insananeous formanic frequencies for all he ypes of speech sounds (isolaed vowels, syllables and words). Resuls are saisfacory and join hose obained by oher researchers in he field (who were ineresed especially in isolaed vowels). Index Terms speech signal, insananeous frequencies, formans, complex wavele, phase derivaive. I. INTRODUCTION Formans are he maximum energy of insananeous frequencies of speech signal. They are basic componens of coding, recogniion or synhesis sysems of signals. They can also be useful for specialized applicaions as assisance o medical diagnosic (for larynx pahologies : by speech signal analysis ). Considering heir growing imporance, he formans are he subjec of many works. The difficuly in formans calculaion is relaed mainly o none saionnariy of he speech signal. Timefrequency represenaions (like coninuous wavele ransform) made formidable grea srides hese 30 las years wih he very fas evoluion of compuers calculaion capaciies. These represenaions are adaped o signals presening frequenial conens which vary during ime (wha is he case of he speech signal). Manuscrip received March 3, 011; revised April 8, 011. This work was suppored in par by he U.S. Deparmen of Commerce under Gran BS13456 (sponsor and financial suppor acknowledgmen goes here). L. Falek is wih he Universiy of science and echnology Houari 3, El Alia, Algiers, ALGERIA. (e-mail: lfalek@homail.fr). Aissa Amrouche is wih he Universiy of science and echnology Houari Boumediene (USTHB), Elecronic and Compuer Science Faculy, BP 3, El Alia, Algiers, ALGERIA. (e-mail: amrouche_a@yahoo.com). L.Fergani is wih he Universiy of science and echnology Houari 3, El Alia, Algiers, ALGERIA. (e-mail: lamifer@msn.com). H.Teffahi is wih he Universiy of science and echnology Houari 3, El Alia, Algiers, ALGERIA. (e-mail: heffahi@gmail.com). A.Djeradi is wih he Universiy of science and echnology Houari 3, El Alia, Algiers, ALGERIA. (e-mail: adjeradi05@yahoo.com). They provide a join represenaion in ime and frequency, conrary o he Fourier ransform who represens in only frequenial form he conained informaion in signal emporal, from where he disadvanage of loss of evens chronology. Waveles ransform akes up same idea as he Fourier ransform by adoping a muli-resoluion approach: if we look a a signal wih a broad window, we will be able o disinguish from he coarse deails. In a similar way, increasingly small deails could be observed by shorening he window size. Waveles analysis objecive is hus o carry ou a kind of adjusable mahemaical microscope. In his sudy, we developed a deerminaion mehod of insananeous variaion of a speech signal formanic frequencies based on complex coninuous wavele ransform. The mehod principle is he phase exploiaion of coefficiens ransformaion for he insananeous frequency exracion by using an analyical complex coninuous wavele ransform. The applicaion of his mehod o he speech signal is carried ou by aking accoun of he acousic characerisics of his signal. We proceeded in firs o he adjusmen of mehod parameers saring from hree vowels (/a/, /i/, /u/) obained using a formans synhesizer, hen we widened he applicaion o real speech signals (isolaed vowels, syllables and words). The resuls represened on a specrogram were compared wih hose obained using a radiional mehod (LPC). They were considered o be saisfacory. II. ANALYTICAL SIGNAL AND INSTANTANEOUS FREQUENCY The analyical signal concep was posed by Ville in 1948 [13] wih an aim of defining he insananeous frequency. The analyical signal is a complex signal associaed wih a real signal. I has ineresing properies, in paricular wih regard o is Fourier ransform, which is null for he negaive frequencies. An analyical signal z() can be calculaed saring from he Hilber ransform [11] of a real signal x() such as: (1) zx() = x() + ih(x()) wih H(x()) he Hilber ransform of x() : I is possible o define [6], saring from he analyical signal an insananeous frequency fx() [4], where zx() is an analyical signal
, July 6-8, 011, London, U.K. 1 ω σ ψ() = expjω exp( ) exp c exp( ) C c σ 4 σ () III. MORLET CONTINUOUS WAVELET TRANSFORM A. The coninuous Morle wavele The complex coninuous Morle wavele ransform also makes i possible o define he concep of he insananeous frequency, when one employs an analyical wavele [7]. This is why we choose he complex coninuous Morle wavele like moher wavele. The laer is an analyzing wavele for small oscillaions (a cenre frequency FC: around 1Hz). Moreover, i is very quie localized in ime (beween -4 and 4s) and in frequencies (a peak around 0.8 Hz). Wha makes of i very a good candidae for he analysis of he speech signal. The Morle wavele is inspired by he elemenary Gabor signal. I is obained by modulaion of a Gaussian. I is given by he following relaion [3] (3) Where: C: facor which makes i possible o sandardize energy. ω c =πf c (F c : is he wavele cenre frequency ) σ =(1/(πσ f ) (σ σ is he sandard deviaion of Gaussian and 4σ is he effecive wavele duraion and 4σ f is band-widh). The produc ω c σ fixes he bond beween he widh of he wavele Gaussian envelope and is oscillaion frequency F C [8]. To have a wavele family, he produc mus be consan. For he Morle wavele, his las mus ake raher large values (ω c σ 5 in pracice). For low oscillaions: ω c = πfc=5.486 rad/s were Fc 0.8Hz. One usually uses for ωc values: 5 ω c 6. Then for Fc : (0.8 Fc 1) Hz [3] B. The wavele ransform Compared o he Fourier ransform, he basic idea of he wavele ransform is o break up a signal x() according o anoher base ha of he sinusoids, each wavele basis having paricular properies which guide is use for he ype of problem arising. Signal x() hus will be broken up on a funcions family relocaed and dilaed saring of a single funcion ψ () called wavele moher. The family pus iself in he form [3 9 1] (4) and is called wavele analyzing, wih "a" parameer of dilaion or scale parameer, defining he widh of he analysis window. The variable a cheek he role of he opposie frequency: more "a" is weak less, he wavele analyzing is wide emporally, and herefore more he cenre frequency of is specrum is high. The parameer b is he ranslaion parameer locaing he wavele analyzing in he emporal field. The Modificaion of "a" and "b" allow o have wavele a he desired frequency and he desired momen. By noing ψ * () complex combined of ψ (), he wavele Transform of signal x () is defined by [3 9 1] (5) This analysis makes i possible o locally describe he conens of x() in he viciniy of (a, b) in he ime-scale plan. I indicaes o us relaive imporance of he frequency 1/a around he poin b (or a he momen b) for a signal x(). Thus, if x() vibraes a a frequency definiely less raised or, on he conrary, much higher han 1/a, he module of he wavele ransform coefficien will be very small and almos negligible. I becomes consequen only if he signal conains a componen of his frequency a he poin considered. The coefficiens of he wavele ransform are hus a way of locaing wih precision he appearance of a frequency given o one momen given in a signal. This decomposiion is funcion of wo variables "a" and "b" and evaluaes he relevance of he use of he wavele in he descripion of x() IV. METHODOLOGY The implemenaion of he mehod was made in Malab language, by aking accoun of he moher Morle wavele parameers quoed in par III.A. We sared by validaing he mehod for a heoreical signal hen we passed o he applicaion on a speech signal A. Applicaion on a heoreical signal We chose a signal x() composed of 3 sinusoids such as: x()= 5sin (π*300) + 10sin (π*000) + 15sin (π*3500) The scales were seleced so ha hey cover he required frequencies: wavebands: [00, 1500]; [1400, 5000]. Produc values ωcσ 5 like known in he lieraure. We calculaed he module of he wavele ransform coefficiens (ampliude of he module according o he scales). The resul obained is illusraed in fig.1. The insananeous frequencies are esimaed by he maximum of he module of he coefficiens of he wavele ransform as specified of Fig.1. We locae hen he scales which correspond o maximum of he module of he coefficiens. We calculae he frequencies by deriving coefficiens phase of he wavele ransform which corresponds o hese scales compared o ime: (figure). These las correspond well o he frequencies of signal x() (relaive o each peaks of he module: max1, max3, max3) : F1=500Hz, F=000Hz e F3=3500Hz. Those can hus be used for a firs esimae of insananeous frequencies signal. Fig.3 represens he energy of one of he frequencies of Fig.. In fig.4, we have maximum energies of he frequencies obained of Fig.3 (represened on he specrogram of signal x()), correspondens well a he frequencies of signal x().
, July 6-8, 011, London, U.K. resuls are illusraed in fig 5. These resuls join hose obained by L.Cnockaer [8] while reasoning on he cenre frequency (F0, F1, F, F3) of he wavele signal. Table 1: Values vowels fixed for he formans for he synhesis of he 3 Figure 1: modulae coefficiens of he wavele ransform according o he scales. Figure. Phase derivaive of wavele ransform corresponding for each maximum F0 F1 F F3 F4 F5 /a/ 00 700 100 700 3900 5700 /i/ 00 70 400 3370 3800 4900 /u/ 300 350 900 3300 4000 500 Ampliude de l'énergie 10.31 10.31 10.308 10.306 10.304 10.30 10.3 0 10 0 30 40 50 60 70 80 échanillons Figure 3. Coefficiens energy of he wavele ransform, for frequencies one frequency These resuls show ha he mehod is applicable on a heoreical signal. Frequency (Hz) 5000 4500 4000 3500 3000 500 000 1500 1000 500 Fréquences insanannées du signal composé de la somme de 3 sinusoïdes 0 0 500 1000 1500 000 500 3000 35000 4000 4500 5000 échanillons N du signal Figure 4: Maximum energy for he 3 insananeous Table. Couples of parameers (F C, ω c σ ) valid for he 3 synheics vowels Frequencies bands Fc ω c σ [00,900] 0.8 4.5 [900,1500] 1 8 [1500,3000] 0.9 8 [3000,4500] 1 10 [[4500,fe/] 1 10 The insananeous formanic frequencies found on fig.5 correspond well o he frequencies fixed in able 1. These resuls will be applied for he calculaion of he formans in he case of a real speech signal. B. Applicaion o a speech signal The speech signal is a non saionary signal, being able o conain a big number of very variable insananeous frequencies. Wha makes is analysis delicaee by his mehod. For ha, we sared by analyzing a signal of word of which we know all frequencies preliminary in order o ry o fix he parameers of he mehod. 1) Speech Signals Analysis obained using formans synhesizer (Kla) We synhesized 3 vowels (/a/, /i/ and /u/) whose Fi formans are consan (see able 1). On he basis of he assumpion ha he produc ω c σ fixes he bond beween he widh of he Gaussian envelope of he wavele and is oscillaion frequency FC and ha o have a family of waveles, he produc mus be consan (for he Morle wavele: 0.8 FC 1e ω c σ 5), i hen acs o find he couples of parameers (FC, ωcσ ) which make i possible o deec formans values inroduced ino he synhesizer wih precision. These couples of values willl be hen used and adjused for real speech signals. The mehod used is he following one: - Choice of 5 wavebands allowing covering all he formanic frequencies (5 formans) for he hreee sounds. These las will be used o fix he scales for he wavele ransform: [00,900], [900,1500], [1500,3000], [3000,4500], [4500, fe/] in Hz; - We hen varied cenre frequency F C of he wavele moher by sep of 0.5Hz wih 0.8 F C 1 - For each value of FC we varied he produc ω c σ by sep of 0.5 while saring of 5 like ciy in he lieraure. Couples of parameers (F C, ω c σ ) obained for he 3 sounds are illusraed by able. The corresponding Figures 5 : Resuls obained for he vowels [a], [I], [u] (on he lef wih wavele ransform and on he righ, wih he sofware Winsnoori using he LPC)
, July 6-8, 011, London, U.K. ) Applicaion mehod for real speech signals On he basis of couple of parameers obained for synheic speech signals, we applied he mehod o real speech signals (vowels, syllables and words) and we represened he resuls on he specrogram relaive o each sound (Fig 6, 7, 8) /men/ /aʕila/ Figures 8. Case of Arabic words. On he lef, wih wavele ransform and on he righ, wih he LPC (sofware Winsnoori) V. CONCLUSION Figures 6. Case of he real isolaed vowels: /a/, /i/, /u/. On he lef, wih wavele ransform and on he righ, wih he LPC (sofware Winsnoori) /sa/ /ka/ This sudy showed ha he coninuous complex wavele ransform of a speech signal can make i possible o esimae he frequencies of he formans, when he parameers of he ransform are quie seleced. Indeed, he values of he produc ω c σ obained for he analyzed sounds (isolaed vowels, consonans, syllables and words) made i possible o obain saisfacory resuls and join hose obained by anoher researcher for isolaed vowels. The difficuly of his mehod lies in he choice of hese parameers like all he oher mehods of formans deecion; however, i has he advanage be applied direcly o non-saionary signals. In addiion, we noiced he appearance of addiional poins o hose of he exising formans on he various figures of he resuls presened for he case of real signals. We hink ha his problem can be relaed o he presence of oher frequencies in he speech signal (harmonic noises and frequencies, aspiraion, ) which energy is imporan and deeced by our mehod. /yu/ Figures 7. Case of real syllables On he lef, wih wavele ransform and on he righ, wih he LPC (sofware Winsnoori) REFERENCES [1] Bilezikian JP, Raisz LG, R. G. (00). Principles of Bone Biology- Second Ediion. Aca-demic Press. 95, 96 [] Casellengo, M. and Dubois, M. (005). Timbre ou imbres? propriéé du signal, de l insrumen ou consrucion cogniive. 88 [3] C. Chui, An inroducion o waveles, Academic Press,1995. [4] Cohen, L. and Lee, C. (1989). Insananeous frequency and ime-frequency disribuions. IEEE Inernaional Symposium on Circuis and Sysems, 1989., pages 131 134. 11. [5] Emiya, V. (004). Specrogramme d ampliude e de fréquences insananées (safi). Maser s hesis, Aix-Marseille II. 5 [6] Flandrin, P. (1993). Temps-fréquence. Hermès. 11, 7, 9, 3.
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