【正文】 te Gaussian noise n with a standard deviation ??. ?? = ??+ ???? () The goal is to recover the signal x from the noisy observations y. Let W 18 denote a wavelet transform matrix for discrete wave atom transform. Then equation can be written in the wave atom domain as NXY ?? () where capital letters indicate variables in the transformed domain, ., WyY? , where W denotes a wave atom transform matrix. Let ???????? 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 ?? = ???1???????? = ???1??????? () where ??????? 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 pared to a given threshold。 i Contents 1. INTRODUCTION .................................................................................................. i 2. WAVE ATOM TRANSFORM ................................................................................ 4 Definition of Wave Atom Transform ................................................................. 4 1D Characteristics in Wave Atom Transform .................................................... 6 The Structure of 1D filterbank in Wave Atom……………………………… .… 10 3. THRESHOLDING METHOD FOR NOISE REDUCTION .................................... 17 Hard and Soft Thresholdings .......................................................................... 17 Determination of Threshold ............................................................................ 20 4. EXPERIMENTS AND DISCUSSIONS……… ………………… ………………… ..27 Conditions………………………………………… ……… ……… ..27 Results and Discussions ................................................................................. 28 .................................................................................................... 37 REFERENCE ......................................................................................................... 38 ABSTRACT (IN KOREAN) .................................................................................... 40 1 1. INTRODUCTION Speech is one of the most natural and convenient ways of intermunication. Thus, speech signal processing techniques for a manmachine 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 bee increasingly important to deal with the ambient noise. Since the quality and intelligibility are degraded by the ambient noise, the problem of reducing noise ponents of the noisy speech is still regarded as an important issue in the field of speech research [13]. 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 [47]. 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 2 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, ., 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 signaltonoise ratios. We also did the similar experiments using the wavelet transform to pare 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 anized as follows. Chapter 2 provides some background information on the 1D wave atom transform relating to the wave packet tree and filterbank. 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 3 and the wavelet transform are presented with discussions. Finally, the conclusion is given in chapter 5. 4 2. WAVE ATOM TRANSFORM In 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 phasespace will be conveyed based on ordinary space and characteristics of wave atom transform will be explained. Before introducing the presentation of the filterbank of wave atom, we start with the relationship of phasespace and frequency domain, an then an example of the wavelet packet tree and filterbank structure for wave atom transform with signal length of 32 samples will be expounded. Finally, at the e