基于波原子变换的语音信号去噪毕业论文.docx

Contents1. INTRODUCTION12. WAVE ATOM TRANSFORM42.1 Definition of Wave Atom Transform42.2 1-D Characteristics in Wave Atom Transform62.3 The Structure of 1-D filter-bank in Wave Atom.103. THRESHOLDING METHOD FOR NOISE REDUCTION173.1 Hard and Soft Thresholdings173.2 Determination of Threshold204. EXPERIMENTS AND DISCUSSIONS.274.1Experimental Conditions.274.2 Results and Discussions285.CONCLUSION37REFERENCE38ABSTRACT (IN KOREAN)40i 1. INTRODUCTIONSpeech is one of the most natural and convenient ways of intercommunication. Thus, speech signal processing techniques for a man-machine interface like speech recognition, speech synthesis and speech coding have drawn much attention. As speech processing moved from the laboratory to the field, it has become increasingly important to deal with the ambient noise. Since the quality and intelligibility are degraded by the ambient noise, the problem of reducing noise components of the noisy speech is still regarded as an important issue in the field of speech research 1-3. Most classical methods for handling ambient wideband noise were variants of an approach based on subtracting an estimate of the spectrum of the noise from that of the noisy speech 4-7. And then a novel approach for noise reduction using the wavelet transform has been proposed by D. L. Donoho 8. It employs the thresholding technique in the wavelet domain and has shown to have good properties for a wide class of signals corrupted by additive white Gaussian noise. Over the past two decades, numerous signal processing techniques with the wavelet transform have been developed to make use of its superiority to the conventional Fourier transform.Recently, wave atom transform has been proposed, and has shown its potential in denoising experiments on both seismic and fingerprint images 1. Wave atom was introduced as a variant of 2D wavelet packets obeying the parabolic scaling. In a nutshell, wave atoms interpolate exactly between directional wavelets and Gabor, in the sense that the period of oscillations of each wave packet (wavelength) is linked to the size of the essential support (diameter) by the parabolic scaling, i.e., wavelength (diameter)2 1,2.In this thesis, we apply the 1D wave atom transform to noisy speech signals for noise reduction. In the first place, we analyzed the algorithm of 1D wave atom transform in the signal processing point of view. Then, using thresholding methods for the wave atom coefficients of noisy speech signals, noise reduction experiments were performed for noisy speech having various signal-to-noise ratios. We also did the similar experiments using the wavelet transform to compare them with wave atom transform. Experimental results have shown that the wave atom transform gives somewhat better performance than the wavelet transform for noise reduction from noisy speech with additive Gaussian noise. This thesis is organized as follows. Chapter 2 provides some background information on the 1D wave atom transform relating to the wave packet tree and filter-bank. Then thresholding methods for noise reduction are explained in chapter 3 with some preliminary experiments to determine the threshold. In chapter 4, noise reduction experimental results using the wave atom transform and the wavelet transform are presented with discussions. Finally, the conclusion is given in chapter 5.2. WAVE ATOM TRANSFORMIn this chapter, we give an overview of the mathematical mechanism of wave atom. At the beginning, definition of wave atom transform will be represented. And then, the view of phase-space will be conveyed based on ordinary space and characteristics of wave atom transform will be explained. Before introducing the presentation of the filter-bank of wave atom, we start with the relationship of phase-space and frequency domain, an then an example of the wavelet packet tree and filter-bank structure for wave atom transform with signal length of 32 samples will be expounded. Finally, at the end of this chapter, some examples of the wave atom coefficients and wavelet coefficients for clean and noisy speech signals will be shown.2.1Definition of Wave Atom TransformLet us define 2D Fourier transform as:f=e-ixf(x)dx (2.1)fx=1(2)2eixf()d (2.2)wave atom is noted as (x), with subscript =j,m,n=(j,m1,m2,n1,n2), j is scale or level, m is the frequency index, n is the time index. All five quantities j,m1,m2,n1,n2 are inter-valued and index a point (x,) in phase-space. They are related together as follows 1.x=2-jn, =2jm, (2.3)where and are two parameters which are all for implementation of computing boundary. is the center coordinate of in phase-space. At the same time, presents each bump of () as . So wave atom should obey the relation from phase-space to frequency domain as the point (x,).Under the condition above, and in Eq. 2.3 with the parameter and to keep the boundary should build a frame of wave packets which supports the equations below.() CM2-j(1+2-j|-|)-M+CM2-j(1+2-j|+|)-M for all (2.4)The parabolic scaling is applied in the definition of wave atom. When the scale is or 22j, both bumps are of the size of in frequency domain and in phase-space the size of each wave atom is . Some detailed mathematical proofs are given well in 2.2.2 1-D Characteristics in Wave Atom TransformBefore introducing the 1-D characteristics of wave atom transform, it is helpful to understand 2-D conditions. The origin of phase-space is from lots of complex mathematic derivation. The main purpose of the origin of phase-space is to extend the ordinary space domain. In order to decompose the signal with the feature of texture, more information of direction is needed in phase-space domain. So, around the concept of the “direction”, more bound conditions are defined to build the domain of phase-space 1. From the mathematical derivation based on ordinary space, we can see that the definition of phase-space is stricter than ordinary space because of the parameter of the “direction”. Fig. 2.1 conveys the relationship between the phase-space and frequency domain.From Fig. 2.1, it is clearly shown that each singularity in phase-space is corresponding to a pair of polar bump in the frequency domain. The left side of Fig.2.1 is the singularities in phase-space, and the right side is the frequency domain. Here, bumps are distributed as a surround status from outside-circle to inside-circle. Of course, no matter where the bump is, a polar bump also exists as its symmetry. In Fig. 2.1, we should note that there are two parameters namedto describe the size of the bump. The parameter describes the multiscale nature of the transform, from 0 (uniform) to 1 (dyadic). And the parameterindexes the wave packets directional selectivity, from 0 (best selectivity) to 1 (poor selectivity) 2. Wave atoms parameter for is . Parameters for other conventional transforms are shown in Fig. 2.2.Here in Fig.2.1, the sign “” presents a relationship of size other than a computing method of complex numbers. Meanwhile, its used for the definition of the size of each bump in the frequency domain.Fig. 2.1 Relationship of phase-space and frequency domainFig. 2.2 Various transforms as families of wave packetsFor computing the coefficients of wave atom, we must pay attention to the Fig. 2.1 and understand the symmetrical conditions. As a matter of fact, wave atom coefficients are symmetric to the origin in the frequency domain. The point of the center of the circle in Fig. 2.1 is zero point in frequency axis. Suppose that the bumps of upper 180 degree belong to the positive part of frequency axis, on the contrary, the other bumps of 180 degrees will be in the negative part of the frequency axis. So, the result of rebuilding the bumps in frequency axis is shown in Fig. 2.3. Where each bump in frequency of length is 22j and the center frequency of the positive frequency bump is 2jm.Fig. 2.3 Result of rebuilding the bumps in 1-D frequency axis2.3 The Structure of 1-D filter-bank in Wave AtomWave atom has a special parameter , and,are equal to 12, respectively. Therefore, there exists a g function given in Eq. 2.5 6 which can make symmetry of the bumps in the negative part of the frequency axis to the positive part. (2.5)Thus, based on wave atoms special definition and the parameters, coefficients of wave atom can be calculated along each bump by using Eq.2.6 and Eq. 2.7.(2.6) (2.7)where m=(-1)m and m=2(m+12).Next, the structure of filter-bank for wave atom transform will be explained. In wavelet transform, filter-bank can be represented like the form given in Fig. 2.4. (a) (b)Fig. 2.4 (a) Filter-bank representation of wavelet transform (b) wavelet tree representationLike the same way, a wavelet packet tree representing the filter-bank of wave atom coefficients can be expressed as shown in Fig. 2.5 for signal length is 32. In Fig. 2.5, the bump has a relationship with g function. In here, g() is defined as “left-handed”, whereas g(-) is defined as “right-handed”, and also, the “left-handed” and “right-handed” have relationships with m. When m is even, the bump is called left-handed; when m is odd, the bump is called right-handed. Therefore, the uniform partitioning of the frequency axis is obtained as an alternating sequence of staggered right-handed and left-handed bumps. On the positive part of the frequency axis, a scale doubling can be achieved by concatenating two right-handed bumps at scales differing by a factor 2. And corresponding to the figure Fig. 2.4, each point in the wavelet packet tree corresponds to a bump in the frequency axis, meanwhile, the distribution of bumps is determined by the relationship between time domain and frequency domain. (a) (b)Fig. 2.5 (a) Wave atom packet tree corresponding to bump in frequency axis (b) actual bump plotFig. 2.6 The relationship of time-frequency (Heisenberg boxes)The wavelet packet tree and the relationship of time-frequency (Heisenberg boxes) 5 for wave atom coefficients with signal length of 32 samples are shown in Fig. 2.6.In Fig. 2.5, we know that each bump in frequency is supported on an interval of length 22j and the center of the positive frequency bump is 2jm. The j,m is defined as the center of the positive frequency bump as follows.j,m=2jm (2.8)For each wave number j,m, the coefficients cj,m,n can be seen as a decimated convolution at scale 2-j.cj,m,n=mjx-2-jnu(x)dx (2.9)By Plancherel theorem 18,cj,m,n=12ej2-jnmju()d (2.10)Assuming that the function u is accurately discretized at xk=kh, h=1/N, k=1,N, N means signal length, so the discrete coefficient equation as follows.cj,m,ncj,m,nD=k=2(-N2+1:1:N2)ej2-jnkmjku(k) (2.11)This equation makes sense for couples（j,m）for which the support of mj(k) lies entirely inside the interval -N,N, so we may write k2Z 1.Fig. 2.7 shows examples of wave atom coefficients and wavelet coefficients with DB8 wavelet for a clean speech signal. Fig. 2.8 shows examples of a noisy speech signal with SNR=10dB and its coefficients.(a)(b)(c)Fig. 2.7 Examples of a clean speech and its transformation coefficients(a) clean speech signal (b) wave atom coefficients (c) wavelet coefficients(a)(b)(c)Fig. 2.8 Examples of a noisy speech and its transformation coefficients(a) noisy speech signal (b) wave atom coefficients (c) wavelet coefficients3. THRESHOLDING METHOD FOR NOISE REDUCTIONThe content in this chapter is mainly around the threshold setting. The main subject is how to set the parameter of threshold under the condition of hard threshlolding and soft thresholding. Before introducing the presentation of the parameter setting, this chapter starts with the comparison of wave atom coefficients of the clean speech signal and noisy speech signal. Then, explain the figures about parameter setting under the condition of hard thresholding and soft thresholding.Wavelet transform as well as wave atom transform is done using the tool box released in the internet 12.3.1 Hard and Soft ThresholdingsLet be a finite length observation sequence of the signal which is corrupted by i.i.d. zero mean, white Gaussian noise with a standard deviation .y=x+n (3.1)The goal is to recover the signal from the noisy observations y. Let denote a wavelet transform matrix for discrete wave atom transform. Then equation can be written in the wave atom domain as (3.2)where capital letters indicate variables in the transformed domain, i.e., where W denotes a wave atom transform matrix. Let Xest be an estimate of the clean signal X based on the noisy observation Y in the wave atom domain. The clean signal x can be estimated by x=W-1Xest=W-1Ythr (3.3)where Ythr denotes the wave atom coefficients after thresholding. Before the setting the parameter, thresholding functions 16 will be introduced. The method is based on thresholding in the signal that each transformed signal is compared to a given threshold; if the coefficient is smaller than the threshold, then it is set to zero, otherwise it is kept or slightly reduced in amplitude. Hard and soft thresholding are used for denoising the signals. Hard thresholding can be described as the usual process of setting to zero the elements whose absolute values are lower than the threshold. The Hard threshold signal is x if xthr and is 0 if x thr, the soft threshold signal is 0. Given a transformed signal Y and threshold 0, two thresholding methods can be expressed as follows. The hard thresholding method is given as Eq. 3.4.(3.4)The soft thresholding method is given by(3.5)where THR() represents the output value after thresholding. Fig. 3.1 displays the hard and soft thresholding functions.(a) (b)Fig. 3.1 Thresholding function (a) hard threshold (b) soft threshold3.2 Determination of ThresholdRemoving noise components by thresholding the wave atom coefficient is based on the observation that in a speech signal, energy is mostly concentrated in a small number of wave atom dimensions. The coefficients of these dimensions are relatively large compared to other dimensions or to any other signal that has its energy spread over a large number of coefficients. Hence, by setting smaller coefficients to zero, one can eliminate noise while preserving the important information of the original signal.In our work, we pay attention to determine the value of threshold using the standard deviation of wave atom coefficients in the silence region of the noisy speech. The threshold is obtained using Eq. 3.6, and the parameter k is determined empirically. THR=k(3.6)where is the standard deviation of wave atom coefficients in the silence region of noisy speech. The block diagram is shown in Fig. 3.2.Fig. 3.2 The procedure of threshold calculationIn order to get a better denoising result, the setting of the threshold is very important. To select an appropriate threshold value, we did some preliminary experiments with noisy speech having SNR of -5dB, 0dB, 5dB, respectively. The speech signals corrupted by white Gaussian noise at -5dB, 0dB, and 5dB SNR are generated with sampling frequency of Fs = 8000Hz. Then each enhanced signal is obtained using wave atom transform by varying the parameter k from 0.5 to 10 with incremental step of 0.5. The noisy speech signals are obtained from one male speaker and one female speaker. Fig. 3.3 shown the relationship between parameter k and SNR by wave atom transform. According to Fig.3.3, under the condition of different SNRs, the ascent part was presented in quick-up, then slows down gradually after arriving peak and eventually reached a plateau. We can see that the peak value is around 3 and 1.5 in hard thresholding and soft thresholding, respectively. So the thresholding parameter is set to 3 and 1.5 for hard and soft thresholdings, respectively. Under the same experimental conditions, Fig. 3.4 shows the relationship between parameter and SNR using the wavelet transform.(a) (b)(c) (d)Fig. 3.3 Determination of parameter and output SNR (a) hard thresholding with male speech(b) hard thresholding with female speech(c) soft thresholding with male speech(d) soft thresholding with female speech(a) (b)(c) (d)Fig. 3.4 Determination of parameter and output SNR (a) hard thresholding with male speech(b) hard thresholding with female speech(c) soft thresholding with male speech(d) soft thresholding with female speechComparing with the parameter value by wave atom transform, Fig.3.4 shows similar results. Therefore, the same threshold parameter 3 and 1.5 are chosen for hard and soft thresholdings, respectively. So the wave atom speech enhancement algorithm is shown in Fig. 3.5, and is summarized as follows.1. Determine the threshold using the wave atom coefficients at the beginning part, i.e.,