Non-Linear Weighting Function for Non-stationary Signal Denoising

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1 Non-Linear Weighting Function for Non-stationary Signal Denoising Farès Abda, David Brie, Radu Ranta To cite this version: Farès Abda, David Brie, Radu Ranta. Non-Linear Weighting Function for Non-stationary Signal Denoising. IEEE. May 26, IEEE, pp.cdrom, 26. <hal > HAL Id: hal Subitted on 21 Dec 26 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 NON-LINEAR WEIGHTING FUNCTION FOR NON-STATIONARY SIGNAL DENOISING Farès Abda, David Brie, Radu Ranta CRAN UMR 739 CNRS-UHP-INPL, B.P. 239, 5456 Vandœuvre-lès-Nancy, France ABSTRACT We propose in this paper a new strategy for non-stationary signals denoising based on designing a tie-varying filter adapted to the signal short ter spectral characteristics. The basic idea leading us to use a new paraetric nonlinear weighting of the easured signal short ter spectral aplitude (STSA) is exposed. The overall syste consists in cobining the estiated STSA and the coplex exponential of the noisy phase. The proposed technique results in a significant reduction of the noise for a variety of non-stationary signals including speech signals. 1. INTRODUCTION The denoising proble consisting in enhancing a signal degraded by uncorrelated additive noise when only the noisy easure is available, is a fundaental tas in signal processing. Because of the non-stationarity of the original signal, the short-tie Fourier analysis and synthesis concepts have been widely used [1 4]. The noisy signal is decoposed into spectral coponents by eans of the short-tie Fourier transfor. The advantages of the spectral decoposition results fro a good separation of the original signal and noise as well as the decorrelation of spectral coponents. Therefore, it is possible to treat the frequency bins independently and the estiation proble is siplified since the tie varying filtering procedure can be ipleented in the frequency doain where ultiplicative odifications are applied on the short-tie spectru and the enhanced signal is synthesized fro the odified short-tie spectru using the OverLap Add ethod (OLA). In general, it is significantly easier to estiate the STSA of the original signal than to estiate both aplitude and phase. There are a variety of denoising techniques that capitalize on this aspect by focusing on enhancing only the STSA of the noisy signal. We propose in this article a new strategy for the restoration of the STSA. We introduce a paraetric spectral gain forulation based on a continuous non-linear function, naely, the arctangent. The paper is organized as follows. In section 2 we introduce the necessary concepts of analysis and synthesis using the short-tie Fourier transfor, with a special attention to the OLA ethod and the effects of spectral odifications on the reconstructed signal. In section 3, we expose the overall schee of the denoising syste using the nonlinear spectral weighting ethod. In section 4, we discuss the epirical paraeters deterination. In section 5, we suarize the paper and draw conclusions. 2. SHORT-TIME FOURIER ANALYSIS AND SYNTHESIS Let x(n) and d(n) denote the original signal and the statistically independent additive noise respectively. The set of observations expressed using the odel y x(n) + d(n) is analysed using the short-tie discrete Fourier transfor defined as Y w(n )y()e j2π/l (1) where w(n ) is an appropriate sliding window (e.g. Haing window) of size N, and L N is the nuber of analysis frequencies ω = 2π/L for =,...,L 1. The denoising procedure ay be viewed as the application of a weighting rule, or nonnegative real-valued spectral gain G (n), to each bin of the observed short-tie spectru Y (n), in order to for an estiate ˆX (n) of the original signal shorttie spectru. We recall in the following the OLA ethod [1] for reconstructing the estiated signal ˆx(n) fro the odified short-tie spectru ˆX (n). Expressing a tie-varying ultiplicative odification to the short-tie spectru of y(n) as ˆX Y (n)g (n), (2) the synthesized signal is obtained by inverse Fourier transfor of ˆX (n), and suing over all the windowed fraes ˆx [ ] 1 Y ()G ()e j2πn/l. (3) L By defining the tie-varying ipulse response corresponding to G (n) as g n () = 1 G ()e j2πn/l, (4) L and using the definition of Y (n), we obtain fro (3) ˆx [ ] y(l) w( l)g (n l). (5) l

3 If we let r = n l or l = n r, the latter relation becoes ˆx [ ] y(n r) w( n + r)g (r). (6) r If we define ĝ(r n,r) = ĝ(q,r) = g (r)w( q) (7) w(n) y(n) F F T Y (λ). I F F T G (λ) Noise PSD estiation Y (λ) OverLap Add ˆx(n) then (5) becoes ˆx r y(n r)ĝ(r n,r). (8) We see fro equation (7) that the resulting tie-varying filter ĝ(r n,r) is a filtered version of g (n) by eans of the low-pass filter w(n). However, in our case the spectral tievarying filter G (n) is a nonlinearly obtained spectral weighting fro Y (n). Thus, tie-varying odifications applied to the easured short-tie spectru Y (n) are only dependent on the latter. Therefore, since Y (n) is a slowly tie-varying function (by taing into account the band liiting effect of the analysis window), the analysis window will not have any significant effect on the designed tie-varying filter. We do not discuss here the selection of the rate at which Y (n) should be sapled in both tie and frequency to provide an unaliased representation of Y (n). This point is detailed in [1] fro which it turns out that a properly sapled short-tie transfor using a Haing window requires on the order of four ties ore inforation as would be required relative to the original signal. In turn to this redundancy one obtains a very flexible signal representation for which extensive odifications can be ade. 3. NONLINEAR WEIGHTING OF THE SHORT-TIME SPECTRAL AMPLITUDE Most of the denoising techniques operate in the frequency doain by applying a frequency dependent gain function to the spectral coponents of the noisy signal, in an attept to heavily attenuate the noise only spectral coponents, while preserving those corresponding to the original signal. These denoising algoriths coonly consists of three ajor coponents: a spectral analysis/synthesis syste which was described in section 2, a noise estiation algorith, and a spectral gain coputation. The gain odifies only the Fourier agnitudes of an input frae Miniu statistics noise PSD estiation Noise estiation usually involves soe ind of SNR based signal activity detection which restrict the update of the noise estiate to periods of signal absence. However, these traditional noise estiation approaches are difficult to tune and their application to low SNR signals results often in distorted Fig. 1: Bloc diagra of a single channel signal denoising algorith. signals. Due to these reasons, we used a useful noise estiation ethod, nown as the iniu statistics [5 7], which consists in tracing the inia values of a soothed power estiate of the noisy signal The spectral gain calculation The tie-varying filter G (n) is specified by a pass region (G (n) 1) which covers the effective support region of the original signal x(n), a stop region (G (n) ) which has to suppress the undesired spectral coponents representative of the noise and an interediate region ( < G (n) < 1) where both signal and noise are present with values depending on the relative spectral values of the original signal and noise. Based on this intuitively appealing interpretation of the spectral weighting function G (n), and given an estiate of the noise PSD ˆσ 2 (n), we propose to design the tie-varying filter as G π arctan [β ( Y (n) αˆσ (n))] + 1 }, (9) 2 where > are design paraeters that have to be chosen in order to adjust at best the shape of G (n). The values of α and β control the suppression level of the noise as well as the resulting signal distorsion. This nonlinear gain function is depicted in figure (2). We see that this function taes very low values corresponding to sall spectral aplitudes of the easured signal copared to the estiated noise PSD, and taes values close to the unity for large values of the spectral aplitudes. Consequently, this function perfors a selective weighting of spectral coponents based on the noise PSD estiate. However, since the gain function (9) is sensitive to the dynaical range of the spectral aplitudes, the input signal has to be noralised with its axial value in order to obtain the sae variation range for all signals. An other solution to this proble consists in using the odified gain function expressed as G where [ ( )] π arctan β γ (n) α + 1 }, (1) 2 γ Y (n) 2 ˆσ 2 (n) (11)

4 is the a posteriori SNR as defined in [2]. The proposed short-ter spectral aplitude estiator is therefore given by ˆX (n) = = π arctan [ β ( γ (n) α )] } Y (n) G (n) Y (n). (12) Finally, by cobining this spectral aplitude estiate with the noisy phase of the easured signal, we obtain the shortter spectral estiate as ˆX G (n)y (n), (13) and the corresponding synthesized signal ˆx (n) is obtained using (3). Input SNR db 5dB 1dB 15dB 2dB Signal 1 Signal 2 Signal 3 Signal 4 α β α β α β α β Table 1: Optial values of paraeters α and β obtained over 1 siulations for four different speech signals 1 and negative values indicates that there is a signal degradation. G (n).5 ασ (n) Fig. 2: Weighting function (9) for β = EMPIRICAL PARAMETER DETERMINATION We consider in this section the experiental deterination of the paraeters α and β using the Nelder-Mead algorith for the iniisation of the MSE between the noise free and the estiated signal calculated as 1 M [ M n=1 x(n) ˆx (n) ] 2, where M the length of the input signal. We used in a first study four speech signals corrupted by coputer-generated white Gaussian noises in order to have different SNRs. The obtained optial values of α and β with the corresponding output SNRs are given in table (1). Several siulations have been conducted and we found that reasonable values in the case of speech signals ust be chosen such that 2 < α 5 and 2 < β 5. However, if we choose α = 2.8 and β = 3, the algorith perfors high noise reduction in the ost cases. Furtherore, we found that this algorith, using the proposed values outperfors (in ter of SNR iproveents) the Wiener [8] and the MMSE-LSA [3, 9] algoriths both using the decision-directed approach for estiating the a priori SNR. Table (2) shows the obtained SNR iproveents for four different speech signals using the proposed algorith, the Wiener and the MMSE-LSA estiators for several input SNRs. The SNR iproveent is calculated as [ M ] n=1 (x(n) y(n))2 SNR = 1log 1 M n=1 (x(n) ˆx(n))2 (14) Input SNR db 5dB 1dB 15dB 2dB Signal 1 Signal 2 Signal 3 Signal 4 A W M A W M A W M A W M Table 2: SNR iproveents (in db) for the proposed algorith (A), the Wiener (W) and the MMSE-LSA (M) filters for four speech signals and different noise levels. As we can see fro table (2), the proposed ethod gives the best results. Moreover, the residual noise sounds very siilarly to that obtained using the MMSE-LSA algorith which is nown to provide less usical tones phenoenon [1]. We also tested the proposed algorith for denoising siulated sparse signals, i.e., a few large coefficients doinate the signal tie-frequency representation. In this case, we found it necessary to use a larger value for the paraeter α while aintaining the sae value for β, and the best results were obtained using the fixed value α = 5. Table (3) gives the SNR iproveents calculated using (14) for different input SNRs obtained for three test signals, naely, the linear chirp, the quadratic chirp and the two chirps signals corrupted with coputer-generated white Gaussian noise. The corresponding noise-free spectrogras are represented in Figure (3). The obtained results confir the efficiency of the pro-

5 posed ethod in denoising such signals Tie (a) Linear Chirp Tie (b) Quadratic Chirp Fig. 3: Sparse test signals Tie (c) Two Chirps Input SNR db 5dB 1dB 15dB 2dB Linear Chirp Quadratic Chirp Two Chirps A W M A W M A W M Table 3: SNR iproveents (in db) for the proposed algorith (A), the Wiener (W) and the MMSE-LSA (M) filters for three sparse test signals and different noise levels [2] R. J. McAulay and M. L. Malpass, Speech enhanceent using a soft-decision noise suppression filter, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 28, no. 2, pp , April 198. [3] Y. Ephrai and D. Malah, Speech enhanceent using a iniu ean-square error short-tie spectral aplitude estiator, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 32, no. 6, pp , Deceber [4] P. J. Wolfe and S. J. Godsill, Siple alternatives to the Ephrai and Malah suppression rule for speech enhanceent, in Proc. 11th IEEE Worshop on Statistical Signal Processing, vol. II, pp , 21. [5] Douglas B. Paul, The spectral envelope estiation vocoder, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 29, no. 4, pp , August [6] R. Martin, Spectral subtraction based on iniu statistics, in Proc. Eur. Signal Processing Conf., pp , [7] R. Martin, Noise power spectral density estiation based on optial soothing and iniu statistics, IEEE Transactions on Speech and Audio Processing, vol. 9, no. 5, pp , July CONCLUSION We proposed a siple and intuitively appealing ethod for non-stationary signal denoising consisting in a non-linear weighting of the STSA coefficients. We believe that the full potential of the proposed approach is not yet fully exploited, although very encouraging results were obtained. A future wor will concern the developent of regularization-based paraeters estiation fro only the noisy easure by taing into account a priori nowledge about the original signal. The sae non-linear function can be applied to develop a shrinage function in a wavelets denoising context [11]. In that respect, the derivation of a SureShrin-type threshold for the wavelet based denoising approach is also thinable since the corresponding shrining function have a bounded wea derivative [12]. These issues are now being investigated. 6. REFERENCES [1] J. B. Allen and L. R. Rabiner, A unified approach to short-tie fourier analysis and synthesis, Proceedings of the IEEE, vol. 65, no. 11, pp , Noveber [8] J. S. Li and A. V. Oppehei, Enhanceent and bandwidth copression of noisy speech, Proceedings of the IEEE, vol. 67, no. 12, pp , Deceber [9] Y. Ephrai and D. Malah, Speech enhanceent using a iniu ean-square error log-spectral aplitude estiator, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 33, no. 2, pp , April [1] O. Cappé, Eliination of the usical noise phenoenon with the Ephrai and Malah noise suppressor, IEEE transactions on Speech and Audio Processing, vol. 2, pp , April [11] F. Abda, Débruitage de signaux non-stationnaires : approches teps-fréquence et teps-échelle, Tech. Rep., Master degree, Université Henri Poincaré, Nancy 1, October 25. [12] J. Bigot A. Antoniadis and T. Sapatinas, Wavelets estiators in nonparaetric regression : A coparative siulation study, Journal of Statistical Software, vol. 6, pp. 1 83, 21.