ࡱ> Gjbjbjss Ğ: %!%!%!4Y!Y!Y!Y!"\Y!8^q"$$$$$VH%$l%\\\\\\\$`lc\A%!%$$%%\ $$]J2J2J2% 8$%!$\J2%\J2J2U !X$aY!|'vV(w\^08^Vd*dPXd%!X%%J2%%%%%\\.%%%8^%%%%d%%%%%%%%% :Signal Denoising using STFT with Bayes prediction and Ephraim-Malah estimation Zoran Brajevi1, Antonio Petoai2 1Croatian Radio, Prisavlje 3, Zagreb,Croatia,  HYPERLINK "mailto:zbrajevic@hrt.hr" zbrajevic@hrt.hr 2 Faculty of Electrical Engineering and Computing, Unska 3, Zagreb, Croatia Abstract - This paper introduces a new audio cleaning method which is composed of combination of stochastically and orthogonal frequency-based systems. This method can be implemented on signals which have been inherently contaminated with some degree of stationary noise. Beside Short Time Fourier Transform (STFT), this paper focuses on stochastically approach which is needed to ensure the information about minimum mean-square error (MMSE) of the spectral amplitude estimator (SAE). This type of estimation will is performed on a silence- or pause- interval via Bayes prediction method and Ephraim-Malah estimation. This procedure results in with significant reducing of spectral coefficients and therefore the elimination of redundant or noise data. After being performed on an arbitrary mathematical function, the described cleaning method is applied on a PCM (Pulse Code Modulation) wave (22.05 kHz, 8 bit). It is shown that described method is dealing very well with both, noise and discrete disturbance which are the most common problems in the daily work with audio material. The realization of mentioned signal denoising is achieved with MATLAB developing software. Keywords denoising, noise distrubances, audio material, Discrete Fourier Transform (DFT), (STFT), Bayes prediction, Ephraim-Malah estimation, Minimum Mean Square Error (MMSE), spectral amplitude Introduction There are several types of noise disturbances in audio signals. These are globally separated into two major groups: impulsive based disturbance and frequency based disturbance. 'Cliks' and 'crackels' are short impulsive disturbances typical for old records and Discrete Wavelet Transformation (DWT) can be applied [5] without additional statistical based approach. Removing of impulse based disturbance is presented in paper [1], by the same author. Broadband noise, on the contrary, is a common to all recording methods and it has been manifested through constant low or high present frequency usually perceived as 'hiss'. In this paper, the broadband noise is taken into further consideration and if its energy, in analyzed frequency bands, is constant over time, we are speaking about stationary noise. The degree of the noise energy could change with time and therefore a need for variable tresholding. The idea of signal denoising using STFT with variable tresholding, is realized through noise power estimation, is shown in general block diagram on Fig. 1. The threshold level corresponds to the output of noise power estimator and it is given with the factor gk(m). Factor gk(m) takes values between 0 and 1. If the useful part of the audio signal is strongly represented, factor gk(m) takes values close to 1. On the other hand, on a strongly noise presence, factor gk(m) takes values close to zero.  General Block Diagram The transition from gk(m) = 1 to gk(m) = 0 is supposed to be gradual but not abrupt. s(n) and n(n) denote the speech and the noise processes, respectively while w(n) represents correlated noise process which is taken from the part of the audio signal that contains only noise. It often happens, at the beginning of recording (on plate or tape or even digital born recordings), a small portion (up to 100 ms) of noise which can be used for noise spectral estimation [2]. The principle of STFT and DFT, because of shortness of this paper, will not be elaborated and it is assumed that the reader is familiar with it. Wiener Filtering, MMSE estimator and some theoretical observations The origin idea comes from basic Wiener Filter theory under consideration of the Least Mean Square (LMS) Algorithm [3]. The Wiener filter, depicted in Fig.2, adjusts its weight(s) (i.e. the filter response) to produce filter output y(n), in order to reduce the amount of noise present in a signal. This could be done under assumption that both, signal and noise, are stationary linear stochastic processes with known spectral characteristic or, what is more likely and what is relevant for this paper, known autocorrelation and cross-correlation. Hence, under previous assumption, at the subtracted output, the noise is canceled, and the output e(n) contains clean or noiseless signal.  Wiener Filter If a single weight case is y(n)=hx(n), the error signal is e(n)=d(n)-hx(n) and the most challenging task is to find the best weight factor h. This is exactly what we are trying to do in this paper using the STFT, realized with DFT filterbanks together with statistical approach. Before start, let us have a little better theoretical insight into the Wiener filtering depicted on the Fig.2. because that would be, at the same time, a theoretical insight into the procedure described in this paper. Taking the square of the output error leads to evaluating the e2(n). The very next step is to observe the statistical expectation of the e2(n) which is labeled as E(e2(n)). With short and simple mathematical calculation described in [3], the E(e2(n)) can be presented as quadratic equation as follows:  EMBED Equation.3  (1) (1) where  EMBED Equation.3  (2) (2) is a power of corrupted signal while  EMBED Equation.3 ,  EMBED Equation.3  (3) is the cross-correlation between the d(n) and x(n) and autocorrelation of x(n), respectively. The E(e2(n)) is plotted in Fig.3.  Calculation of the weight factor h The best (or optimal) weight factor h' could be found at the location of the MMSE of the E(e2(n)). Implementation and the evaluation of the parameter g(m) (Fig.1) begins under the consideration of Bayes-estimator or MMSE-estimator which we need to minimize the median square error:  EMBED Equation.3  (4) Let s(n) and n(n) denote the speech and the noise processes, respectively.(Fig.1). The next task is to find the best possible estimate of the amplitude of the useful signal from the STFT of the contaminated signal.  EMBED Equation.DSMT4  (5) , where k=0,1,& ,N-1 and Nk(m) is the contaminated part of the signal. In this approach only the amplitude Ak(m) will be changed and the phase (m) remains unchanged. This consideration comes from the fact that the amplitude information is decisive factor in our aurally perception. There are plenty statistical methods which can be used in order to estimate the k(m). Among them we emphasize the Bayes estimator (or, in this case, the MMSE estimator) which can be used to minimize the mean square error [1],[4];  EMBED Equation.DSMT4  (6) for every frame m and channel k. There is another precondition which, together with equations (1) (4) and (6), is necessary to be fulfilled for an accurate calculation of the MMSE estimator; we need to have an a priori information (or knowledge) in the form of Probability density function (PDF). This-a priori knowledge-can be, in praxis, determined only in few cases and for this reason we reach for statistical model with independent Gaussian random variable. Derivation of Amplitude estimator In this section the MMSE amplitude estimator is derived and that will be done under the assumed Gaussian statistical model. The observed signal is given by x(n)=s(n)+w(n), where, without loss of generality, we let the observation interval be n=0,1,2N-1. Further, formula (7) define the kth spectral component of the signal s(n), the noise w(n), and the noisy observations, respectively;  EMBED Equation.DSMT4  (7) The most commonly used Fourier analysis technique in the world is the so-called fast Fourier transform which is an efficient algorithm for computing the DFT [4]. After performing equation (7) we note that the signal x(n) and s(n) can be written in terms of its spectral components X(k) and A(k), where A(k) is a spectral component of x(n). Formula (8) represents the noiseless part of the audio signal written in terms of its spectral components:  EMBED Equation.DSMT4  (8) (The same formula is used for noise and composite signal). Moreover, on the basis of the Gaussian statistical model for the spectral components assumed here, the series (8) converges almost surely to s(n) for every n. Therefore, it can be shown that s(n) and X(k) bears the same information. It means that the MMSE estimation problem can be reduced to be that of estimating Xk. Nevertheless, since the spectral components are assumed to be statistically independent, the MMSE amplitude estimator can be derived from Xk only. The advance information is given in a form of probability density function (PDF) and is used, as noted before, a statistical approach with Gaussian distribution. The optimal amplitude estimator is given with (according to [2],[5]):  EMBED Equation.3  (9) Conditional PDF is given with:  EMBED Equation.3  (10) , where  EMBED Equation.3  (11) Pw(k) and Ps(k) denote the noise power density and uncontaminated signal power density, respectively.The appropriate signal processing (i.e. all necessary calculations) for evaluating above mentioned relations (8), (9) and (10) will be performed for every channel m and for every frame k. A prior And A posteriori SNR And Their Influence on Weight Factor The variables RPRIO and RPOST are defined as follows:  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  (12) They denote a priori SNR and a posteriori SNR, respectively. The MMSE amplitude estimator is derived under the implicit assumption that the a priori SNR and the noise variance E(w2(n)) are known [4],[5]. However, in the audio signal enhancement problem discussed here, these parameters are unknown in advance, as the audio signal alone is available. Therefore, they are replaced by their estimators in a practical system which will be described in the next chapter and which will be performed with MATLAB developing software. For this reason it is of interest to examine the sensitivity of the amplitude estimator to inaccuracy in these parameters. The a priori SNR is key parameter in the discussed problem, rather than the noise variance (expectation of w(n)) which is easier to estimate. Therefore we examine the ability and the sensitivity of MMSE amplitude estimator to the a priori SNR only. The a posteriori SNR will be the control variable which shows the actual SNR and which modifies the weight (control) factor in g(m) in the modified Wiener estimator. The sensitivity analysis provides also error information for the particular audio information. Further calculations are related to obtaining g(m) itself. First observation is the fact that g(m) is the function of a priori SNR and a posteriori SNR variable, respectively;  EMBED Equation.3  (13) which, again, both have to be calculated for every frame k and channel m. Increasing the number of channels means slowing down the process and more accurate frequency analysis at the same time. The formulas (7), (8), (9), (10) and (11) are taken into consideration to obtain:  EMBED Equation.DSMT4  (14) The Fig 4 shows us the relation between g(m) and RPRIO where all parameters are plotted in a logarithmic scale.  g(m) as the function of RPRIO with the constant RPOST So far we have defined three basic parameters that will give us the estimation of g(m). Beside the MMSE estimator, i.e X(k), there is the RPRIO and finally, there is the parameter RPOST which is given with 18 predictive values as follows [2],[3]:  EMBED Equation.3  This scale covers the range of values relevant for the practice. 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Put it in the simple way, if the net power (useful part of signal) gets smaller, the g(m) will be also smaller and vice versa. Analysis and Results First of all, we will apply the MMSE estimator (realized with STFT), on test function given as:  EMBED Equation.DSMT4  (15) Test function is rectangle function of width  with discontinuities at  /2 and /2 [1]. That function will be contaminated with Gaussian noise. (Fig.5) and after that we will apply the MMSE estimator on PCM wave with duration of 180 sec. (Fig.6) (AWGN 10 means Addition of White Gaussian Noise with 10 dB SNR per sample. It is assumed that the power of origin signal is 0 dB W in both cases, i.e. for test function and PCM wave, respectively.)  Test functions with AWGN 10 and denoising  180 seconds of 'Eine kleine Nachtmusik' with AWGN 10 and denoising MATLAB function y=kizodenoise(x,w,N,apri,gain) performs the denoising process described in this paper and gives the best result for relatively stationary signals (i.e. music). Input variables are: x is input signal and has to be row vector. In praxis it can be test function or PCM wave, w is the noise sequence used to estimate noise spectral power density, N is time window length, apri is a priori SNR and gain stands for gain of noise spectral power density. Single output variable is y and represents the resulting function (or resulting PCM wave). The distortion of resulting function will be measured with MSE (Mean Square Error) ([4, 5]):  EMBED Equation.DSMT4  (16) and with total SNR given in eq. 7.  EMBED Equation.DSMT4  (17) fi and f are input and output function, respectively, with N samples. TABLE  SEQ TABLE \* ARABIC 1. Comparison of methods SNR TABLE (all values in dB)STFTWIENER ADAPTIVE FILTERINGy=kizodenoise(x,w,N,apri,gain)TEST 34,633,538,9MUSIC44,035,849,7SPEECH29,331,332,5 TABLE  SEQ TABLE \* ARABIC 2. Comparison of SNRPRIOR and MSE SNRPRIOR (all values in dB)10 dB20 dB30 dBMSE605244Conclusion As it can be seen from the Table 1, that presented method gives the best result for relatively stationary signals. Even though, with the carefully choice of parameter (e.g. a priori SNR between 10 and 12 and minimum 70 ms of noise sequence) the results achieved for relatively non stationary signals (i.e. speech) are little better than those achieved with pure STFT or even with 'classic' Wiener filtering. The Figures 5 and 6 shows significant signal improvement and are correlated with the results shown in a Table 2. (Most left column of a priori SNR of 10 dB). References l2lmlqllllllmm m m m mm%m&m'm(mnn nn(n*nhnnnnnn*o0oPoopppp,ppppppp㲥ߡ㙡ߕו׍׍׍}v hA=khA=khkjuhVxUh*h+0hwgWh|)hh6h&CAjhihHpEHUj$>Q hihUVjhihUh~/h:2hM&hK'h ?h hihhhrb6 hrbhrb hrbh .ll m npppp4qs tPtt"uuuuuuv $IfgdK $Ifgd:|3-$gdK`gd,`gd<2gd ? $`a$gdHpgdv&gdHpgdgd~/gdgdppppppp"q%q3q4qEqcqqqqqqqqr2rSrUrVrrrrrrrrrrsss s2sJsTs]s^s_ssssssû÷ڻڻڳëóóó{jVS hih?UVjhihUh~/ hihhyheh:|36h:|3hyheh+6h+h?hyheh ?6h<) h hKhKh}Uh ?6h ? hcsChcsCjlhHpU hHphHphHphK'hcsC0ssstt tt(tNtPtRtTtttttttttttttttttt uu u"u.u0uZu\u^u`uؾحؙؕؑttbt"h}]gh}]g5CJaJmHnHu jh}]gh}]g5CJUaJh}]gh}]g5CJaJhhKhih6 hK6H* hK6h<)j6h?hoEHUjXS hih?UVh~/h<2hHp hihh6h?jhihUj1h?hoEHU&`ubuuuuuuuuuuuvv#v$v%v*v+v/v0v4v5v9v;v@vAvEvƺ|qfWOfOfOfWOhQ3mHsHhKhKCJaJmHsHhkhwgWmHsHhKhwgWmHsH hKhKhKhKmHsHhKmHsHhKhwgWCJaJmHsHh_h_5mHsHh_5mHsHhwgWCJaJmHsHh:|3h:|3CJaJmHsHh:|3CJaJmHsHh}]gh}]g5CJaJhK5CJaJh w5CJaJv$v%v+v0v5v:vRIII $Ifgd Zkd1;$$Ifl\-` W3R t044 layt}U $IfgdK:v;vAvFvKvPv[RIII $Ifgd Z $IfgdKkd;$$Iflj\-` W3R t044 layt}UEvFvJvKvOvQvWvXv\v]vavbvfvhvivovpvvvvvvvvvvvⷫsj_jShKh}]g5CJaJhK5CJH*aJhK5CJaJhKhK5CJaJh w5CJaJ"h}]gh}]g5CJaJmHnHu jh}]gh}]g5CJUaJh}]gh}]g5CJaJhyhehTmHnHsHhKhwgWCJaJmHsHhKhKCJaJmHsHhkhwgWmHsHhQ3mHsHh@hwgWmHsHPvQvXv]vbvgv[RIII $Ifgd Z $IfgdKkd<$$Ifl\-` W3R t044 layt}Ugvhvivvvvvv[PJAAAA $IfgdV-$gd}]g $ & Fa$gd}]gkd=$$Ifl\-` W3R t044 layt}Uvvvvvvvvvvvvvvvvvvvvvvvvvvvv˿˯vjjbTMIh<2 hih)nhkhT5\mHsHhQ3CJaJhQ3CJaJmHsHhkhTmHsHhTCJaJhOhTCJaJhKhTCJaJmHsHhTCJaJmHsHhOh_CJH*aJmHsHh_h_5mHsHh_5mHsHhTCJH*aJmHsHhOhTCJH*aJmHsHhOhTCJaJmHsHvvvvvvvRkdE>$$Ifl\*[ O1O t0 44 layt}U $IfgdVvvv(y3yܖ^[VQLGGGgd)ngdzgd)ngd)nkd ?$$Ifl\*[ O1O t0 44 layt}Uvvvwwww*wEwPwcww/x6xxxxxxx'yzĖږܖė2\^$j"`bdhjϸզhn hih h  hih@h@h@6 hih)nh~h~6h@U hihhVxh~h#'hKhQ3h}Uh:2h<2ht)Z.Brajevi, "Elimination of unwanted signals using Wavelet transform", Speech and Audio Processing,ELMAR 2011 Y.Ephraim, D.Malah, "Speech enhancement using a minimum mean square error short time spectral amplitude estimator", IEEE Trans. Acoustic, Signal Processing, vol.ASSP-32,pp. 1109-1121, Dec.1984 O.Cappe, "Elimination of the musical noise phenomenon with the Ephraim and Malah noise supressor", IEEE, Speech and Audio Prrocessing, vol 2, April 1994. Christoph Musialik and Urlich Hatje  Frequency-domain processors for efficient removal of noise and unwanted audio events , Algoritmix GmbH, AES 26th conference 2005 M. J. Roberts, Signals and Systems , McGraw-Hill, New York, 2004 ޚbdfhj$a$gd gd gd@= 0P&P 1h. A!"#8$ % DpD 0P&P 1h. A!"#8$ % P h*2Dp= 0P&P 1h. A!"#8$ % DpgDd-"F  C "A uvodbgn8Ĥdk(fD nfn8Ĥdk(PNG  IHDRT-֌sRGBgAMA a cHRMz&u0`:pQ<fYIDATx^?'5~ב"ҖۜD)[@'\ICnR`%BRB4H4HH4tW!mP!VT_-(ϙ?؞S/c->P^/In/4ݟ?y<97 ^z@ ;gϞ :k,] wֹ:Ksݻw䓏ojSCH@sf直K]u5q<mt\Az|'й]@ УGnc:~rP@#%C@@J D)BP1hPA#%C@*:x1Pvd @ETo#@;ʎ !@m4s({ @ّ! PQfe(;R2c*eGJTt @E69cH@(F3Gw )BP1hPA#%C@*:x1Pvd @ETo#@;ʎ !@m4s({ @ّ! 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