SHORT TIME FOURIER TRANSFORM PROBABILITY DISTRIBUTION FOR TIME-FREQUENCY SEGMENTATION

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1 SHORT TIME FOURIER TRANSFORM PROBABILITY DISTRIBUTION FOR TIME-FREQUENCY SEGMENTATION Fabien Millioz, Julien Huillery, Nadine Martin To cite this version: Fabien Millioz, Julien Huillery, Nadine Martin. SHORT TIME FOURIER TRANSFORM PROBA- BILITY DISTRIBUTION FOR TIME-FREQUENCY SEGMENTATION. Francis CASTANIE. 6, IEEE Signal Processing Society, pp. III , 6. <hal-8554> HAL Id: hal Subitted on Jul 6 HAL is a ulti-disciplinary open access archive for the deposit and disseination of scientific research docuents, whether they are published or not. The docuents ay coe fro teaching and research institutions in France or abroad, or fro public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de docuents scientifiques de niveau recherche, publiés ou non, éanant des établisseents d enseigneent et de recherche français ou étrangers, des laboratoires publics ou privés.

2 SHORT TIME FOURIER TRANSFORM PROBABILITY DISTRIBUTION FOR TIME-FREQUENCY SEGMENTATION Fabien Millioz, Julien Huillery, and Nadine Martin Laboratory of Iage and Signal (LIS) 96 rue de la Houille Blanche, BP Saint Martin d Hères Cedex, France eail: stnae.ndnae@lis.inpg.fr ABSTRACT Taking as signal odel the su of a non-stationnary deterinistic part ebedded in a white Gaussian noise, this paper presents the distribution of the coefficients of the Short Tie Fourier Transfor (STFT), which is used to deterine the axiu likelihood estiator of the noise level. We then propose an autoatic segentation algorith of the real and iaginary parts of the STFT based on statistical features, which is an alternative to the spectrogra segentations considered as iage segentations. Exaples of segented tiefrequency space are presented on a siulated signal and on a dolphin whistle.. INTRODUCTION Tie-Frequency Representations (TFR) are useful tools for nonstationnary signal analysis by deterining the tie-frequency patterns, which are tie-varying areas containing energetic signal. A segentation task is a helpful step in such a signal characterization by highlighting these patterns. With a general odel of signal considered as a nonstationnary deterinistic signal d[] ebedded in a white Gaussian noise n[] of variance σ x[] = d[] + n[], () we proposed ([], []) a spectrogra segentation based on statistical features. Given that spectrogra coefficients of signal described in equation () have a non-central χ distribution while white Gaussian noise have a central χ distribution, segentation task consists in discriinating non-central fro central χ distribution. In this paper, we propose a new way of segentation, by considering real and iaginary part of Short Tie Fourier Transfor (STFT). Instead of having a χ distribution, TFR coefficients have a Gaussian distribution, which allows a sipler segentation ethod. In section we deterine the real and iaginary part of STFT distribution and show that their respective variance are not always equal, in order to deterine an efficient noise level estiator in section 3. In a second part, we use these results to propose a new segentation algorith based on local statistical features of the STFT, and its application on a siulated signal and a dolphin whistle.. DIFFERENCE OF VARIANCE BETWEEN PROBABILITY DISTRIBUTION OF STFT REAL AND IMAGINARY PARTS The STFT of a discrete signal x[] is deterined by coputing the discrete Fourier transfor on N overlapping segents centered on n, which describes the spectral contents of x around the instant n. The STFT is defined by X φ [n,k] = x[]φ[ n]e iπk M φ +Z, () where k is the frequency index, φ is the M φ -length window function and Z the zero padding. We will consider an energynoralized window, so φ[] =. (3) For the signal defined in (), real and iaginary parts of the STFT, Xφ r[n,k] and Xi φ [n,k], are sus of M φ independant Gaussian variables. Xφ r[n,k] and Xi φ [n,k] thus are Gaussian variables, with ean respectively given by E(X r φ[n,k]) = E(X i φ[n,k]) = d[]φ[ n] cos( πk M φ + Z ) (4) d[]φ[ n] sin( πk ), (5) M φ + Z and variance by Var(Xφ[n,k]) r = σ φ[ n] cos( πk M φ + Z ) (6) Var(Xφ[n,k]) i = σ φ[ n] sin( πk. M φ + Z ) (7)

3 3 α 3. NOISE ESTIMATION FROM REAL PART OF STFT frequency index k In order to segent the tie-frequency representation, we need to know the noise level of x[]. In this section we deterine a variance estiator using real part of STFT coefficients considering first that the deterinistic part is null. Then we study the effect of the unknown deterinistic part on the estiator. 3 4 tie index n Fig.. Variations of α[n,k], for a STFT with a Blackan window of 3 points, with a zero padding of 33. White points are values of α between.485 and.55. We notice that eans of Xφ r[n,k] and Xi φ [n,k] are respectively real and iaginary part of the STFT of d[]. To copare the variances, we then define α[n,k] as the ratio of the variance of the real part of the STFT to the su of the two variances Var(Xφ r α[n, k] = [n,k]) Var(Xφ r[n,k]) + Var(Xi φ [n,k]) (8) = Relation (3) induces φ[ n] cos(πk M φ + Z ). (9) Var(X r φ[n,k]) + Var(X i φ[n,k]) = σ. () Using trigonoetric identities, equation (9) writes α[n,k] = + M φ p= M φ φ[p] cos(4πk p + n ). () M φ + Z When the frequency value is far enough fro and M φ+z, the frequency of the cosine function will be high enough copared to the window variations to cancel the second ter, so the value of α[n,k] will be. Otherwise, α[n,k] will discriinate variances of Xφ r[n,k] and Xi φ [n,k]. Consequently, a bias in the variance estiation appears and the spectrogra coefficients do not have a χ distribution anyore. Fig. shows the variations of α along the tie and frequency indexes for a STFT coputed with a Blackan window of length M φ = 3 and with a zero padding of Z = 33. This paraeter extends the works of L.H. Koopans [3], N.L. Johnson and D.G. Long [4], who only deterined that the frequency bins where the spectrogra distributions coputed without zero padding do not atch a χ distribution are k = and k = M φ for a rectangular window and k =, k =, k = M φ and k = M φ for a Hanning window. 3.. Centered white Gaussian noise We consider a signal x[] () of length N, where the deterinistic part d[] is null. The Maxiu Likelihood (ML) estiator of the variance σ is unbiased and optial, and writes σ = N N = x[]. () We want to estiate σ with the real part of the STFT coefficients. In section we saw that Xφ r [n,k] has a non constant variance equals to α[n,k]σ, where α[n,k] is deterinistic. We thus define a new rando variable of constant variance X r φ[n,k] = Xr φ [n,k] α[n,k]. (3) X r φ [n,k] is a centered white Gaussian noise of variance σ. The variance can now be estiated as equation. The sobuild estiator fro the real part of STFT coefficients is a ML estiator, which is optial and reains unbiased. 3.. Deterinistic signal ebedded in a white Gaussian noise We now consider a general case where x[] is a white Gaussian noise of unknown ean d[] (). Xφ r[n,k] is a Gaussian variable of ean Dr φ [n,k], the real part of the STFT of d[]. The ean of the rectified rando variable (3) is consequently E ( Xφ[n,k] r ) = Dr φ [n,k] (4) α[n,k] which is unknown. In the context of a TFR segentation of an unknown deterinistic signal, we cannot specify which points are centered and which are not. When we estiate the noise level, we will take non-centered points. ML estiator thus overestiate σ as ) (X r E ( σ = E( NK φ [n,k] ) ) (5) n,k = σ + NK Dφ r[n,k], (6) α[n, k] where N and K are the nuber of tie and frequency indexes. n,k

4 4. TFR SEGMENTATION For the odel of signal (), we showed in section that real and iaginary parts of STFT coefficients have a Gaussian distribution, where the ean depends on the deterinistic part. The segentation task consists in identifying coefficients with non-zero ean, which are points containing deterinistic signal, in order to reconstruct tie-frequency regions called spectral patterns. As seen in equation (6), non-zero eans overestiate the value of the noise variance. The idea is to estiate local variances of (n,k) sites, and select with a threshold depending on the estiated noise level the points of highest variance. 4.. Local variance distribution and threshold As in [], we consider a sall cell of P points C n,k, centered on the (n,k) site of the real part of STFT. Local variance estiator () of the rectified rando variable (3) writes σ [n,k] = P ( X r φ [n,k] ). (7) C n,k The knowledge of the local variance distribution of points without deterinistic part allows us to propose a suitable threshold to discriinate (n, k) sites without deterinistic part fro others with a given false alar probability p fa. For (n,k) sites without deterinistic part, (7) is a su of P squared centered Gaussian variables. If they are independant, σ [n,k] have a central σ P χ P distribution. Due to STFT construction, {Xφ r[n,k] } are correlated, we thus have σ [n,k] σ δ χ δ (8) where δ is an unknown degree of freedo, verifying δ P. This distribution has two unknown paraeters, σ which depends on the analyzed signal, and δ which depends on the STFT construction. By coputing the STFT of a centered white Gaussian noise of known variance, the only unknown paraeter of the χ distribution is δ, which can be estiated with a axiu likelihood approach []. The second unknown paraeter σ is estiated by ML with equation (6). When the distribution (8) is fully estiated, we define a threshold t σ t σ / Prob{ σ [n,k] > t σ } = p fa, (9) where p fa is a given false alar probability. The use of this threshold in a segentation algorith is described in the next subsection. 4.. Segentation algorith The proposed algorith is a region growing algorith, applied to the TFR. We first overestiate the noise variance over all the STFT real part coefficients, which give the last unknown paraeter of the local variance distribution (8), and enable us to copute the threshold t σ (9). We then select (n,k) sites whose local variance is higher than the threshold t σ to be candidates to the segentation. These sites are supposed to contain deterinistic ean due to equations (6) and (8). Then, a "seed" with the highest local variance is choosen aong the candidates, associated with a given label l. If soe of its neighbours in the TFR are candidates, they becoe new seeds of sae label, which containate then their own neighbours. Iteratively, we create so a spectral pattern of label l. Once ost of the candidates have been segented, we estiate noise variance again with only the unlabelized coefficients. We thus obtain a less overestiated value. A new threshold is then coputed on the new estiated σ [n,k] distribution, which gives new candidates to the segentation. Consequently, at each iteration the estiated noise level coes closer to σ, which allow to segent ore points containing deterinistic part Segentation control We use two criterions in this algorith in order to supervise its perforance. The first one is the Kologorov distance d k [5] defined as d k = sup F n(x) F(x), () where F(x) represents the theoretical cuulative distribution function and Fn(x) the epirical cuulative distribution function. The Kologorov distances on the unlabelized points before and after containation are copared to validate a seed containation. If the algorith has effectively segented (n, k) sites containing deterinistic signal, the unlabelized points will converge to a Gaussian distribution and d k will decrease. Secondly, the kurtosis [6] defined as µ 4 K = (σ 3, () ) where µ 4 is the fourth centered oent, is estiated on the unlabelized points at each iteration, in order to stop the segentation when it reaches. Indeed, when the algorith does not have anyore deterinistic signal to segent, the unlabelized points have a zero ean Gaussian distribution, with a null kurtosis. Moreover, it provides an indicator of execution of the algorith. If the algorith ends before the kurtosis reaches zero, we know that all spectral patterns do not have been segented Segentation results Fig. shows the result of a synthetic signal TFR segentation. It s a su of a filtered noise of variance and a frequency-

5 .5 Spectrogra db.34 Spectrogra db (a) Spectrogra of a siulated signal Segentation result 4 6 (b) Segentation result of the siulated signal x 4 (a) Spectrogra of a dolphin whistle Segentation result Labels x 4 (b) Segentation result of the dolphin whistle Fig.. Su of a frequency-varying signal and a large band signal. The spectrogra (a) is segented (b) in three regions fro, set of points without deterinistic signal, fro. Fig. 3. Whistle of a dolphin. The spectrogra (a) is liited to the noralized frequency [.,.34] to have a white Gaussian noise on the coefficients. Nine patterns are segented (b). varying cosine function of aplitude.5, ebedded in a white Gaussian noise of variance σ = 6. The two spectral patterns are correctly segented, assigning label "" to the frequency-varying signal and label "" to the large band signal. Fig. 3 presents a dolphin whistle segentation. Given that the recording noise is not white, we liited the TFR to a frequency band of [.,.34] in order to have approxiatively a white noise. Six patterns are segented, three of the having ore than one label. 5. CONCLUSION We showed that STFT real and iaginary parts of a deterinistic signal ebedded in a white Gaussian noise have two different Gaussian distributions. The variances depend on the (n,k) point of the STFT. An efficient estiator of the noise variance in the real part was proposed. This estiator is used in a new non-stationnary signal TFR segentation, based on local statistics of the STFT. Exeples with a siulated signal and a dolphin whistle prove the efficiency of this approach. Current works shows that this new algorith provides less false alar patterns than []. 6. REFERENCES [] C. Hory, N. Martin and A. Chehikian, Spectrogra segentation by eans of statistical features of nonstationnary signal interpretation, IEEE Transactions on Signal Processing, vol. 5, no., pp , deceber. [] C. Hory and N. Martin, Maxiu likelihood noise estiation for spectrogra segentation control, in Proceedings of ICASSP, Orlando, USA,, pp [3] L.H. Koopans, The Spectral Analysis of Tie Series, Acadeic Press, 974. [4] P.E. Johnson and D.G. Long, The Probability Density of Spectral Estiates Based on Modified Periododra Avergares, IEEE Transactions on Signal Processing, vol. 47, ay 999. [5] R.O. Duda and P.E. Hart, Pattern Classification and Scene Analysis, Wiley, 973. [6] J. F. Kenney and E. S. Keeping, Matheatics of Statistics, Van Nostrand, 95.