基于波原子变换的语音信号去噪毕业论文(编辑修改稿)

基于波原子变换的语音信号去噪毕业论文(编辑修改稿)内容摘要：

60 . 70 . 80 . 91F i l t e r b a n k p l o t f o r N = 3 2 w a v e a t o m c o e f f i c i e n t s x  [ r a d s ] c e n t e r f r e q . 13 Fig. The relationship of timefrequency (Heisenberg boxes) The wavelet packet tree and the relationship of timefrequency (Heisenberg boxes) [5] for wave atom coefficients with signal length of 32 samples are shown in Fig. . In Fig. , we know that each bump in frequency is supported on an interval of length 2𝜋2𝑗 and the center of the positive frequency bump is 𝜋2𝑗𝑚. The 𝜔𝑗,𝑚 is defined as the center of the positive frequency bump as follows. 𝜔𝑗,𝑚 = 𝜋2𝑗𝑚 () For each wave number 𝜔𝑗,𝑚 , the coefficients 𝑐𝑗,𝑚,𝑛 can be seen as a decimated convolution at scale 2−𝑗. 14 𝑐𝑗,𝑚,𝑛 = ∫𝜑𝑚𝑗 (𝑥 −2−𝑗𝑛)𝑢(𝑥)𝑑𝑥 () By Plancherel theorem [18], 𝑐𝑗,𝑚,𝑛 = 12𝜋∫𝑒𝑗2−𝑗𝑛𝜔𝜑̂𝑚𝑗 (𝜔)̅̅̅̅̅̅̅̅̅𝑢̂(𝜔)𝑑𝜔 () Assuming that the function u is accurately discretized at 𝑥𝑘 = 𝑘𝑕,𝑕 =1/𝑁,𝑘 = 1,…,𝑁, 𝑁 means signal length, so the discrete coefficient equation as follows. 𝑐𝑗,𝑚,𝑛 ≈ 𝑐𝑗,𝑚,𝑛𝐷 = ∑ 𝑒𝑗2−𝑗𝑛𝑘𝜑̂𝑚𝑗 (𝑘)̅̅̅̅̅̅̅̅̅𝑢̂(𝑘)𝑘=2𝜋(−𝑁2+1:1:𝑁2) () This equation makes sense for couples（ 𝑗,𝑚） for which the support of 𝜑̂𝑚𝑗 (𝑘) lies entirely inside the interval [−𝜋𝑁,𝜋𝑁], so we may write 𝑘 ∈ 2𝜋𝑍 [1]. Fig. shows examples of wave atom coefficients and wavelet coefficients with DB8 wavelet for a clean speech signal. Fig. shows examples of a noisy speech signal with SNR = 10dB and its coefficients. 15 (a) (b) (c) Fig. Examples of a clean speech and its transformation coefficients (a) clean speech signal (b) wave atom coefficients (c) wavelet coefficients 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5x 1 04 0 . 4 0 . 3 0 . 2 0 . 100 . 10 . 20 . 30 . 4c le a n s p e e c h0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5x 1 04 0 . 8 0 . 6 0 . 4 0 . 200 . 20 . 40 . 60 . 8w a v e a t o m c o e f f i c i e n t s o f cl e a n sp e e c h0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5x 1 04 0 . 5 0 . 4 0 . 3 0 . 2 0 . 100 . 10 . 20 . 30 . 4. 5w a v e le t c o e f f i ci e n t s o f cl e a n sp e e c h 16 (a) (b) (c) Fig. Examples of a noisy speech and its transformation coefficients (a) noisy speech signal (b) wave atom coefficients (c) wavelet coefficients 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5x 1 04 0 . 4 0 . 3 0 . 2 0 . 100 . 10 . 20 . 30 . 4n o i sy s p e e c h0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5x 1 04 0 . 8 0 . 6 0 . 4 0 . 200 . 20 . 40 . 60 . 8w a v e a t o m c o e f f i c i e n t s o f n o i sy sp e e c h0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5x 1 04 0 . 5 0 . 4 0 . 3 0 . 2 0 . 100 . 10 . 20 . 30 . 40 . 5w a v e l e t c o e f f i ci e n t s o f n o i sy sp e e c h 17 3. THRESHOLDING METHOD FOR NOISE REDUCTION The 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 parison 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 transform as well as wave atom transform is done using the tool box released in the inter [12]. Hard and Soft Thresholdings Let y be a finite length observation sequence of the signal x which is corrupted by . zero mean, white Gaussian noise n with a standard deviation 𝜀. 𝑦 = 𝑥+ 𝜀𝑛 () 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 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 () 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 pared 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 𝑥 ≥ 𝑡𝑕𝑟 and is 0 if x thr, where „thr‟ is a threshold value. Soft thresholding is an extension of hard thresholding, in other words, first setting to zero the 19 elements whose absolute values are lower than the threshold, and then shrinking the nonzero coefficients towards 0. If 𝑥 ≥ 𝑡𝑕𝑟, the soft threshold signal is (𝑠𝑔𝑛(𝑥)∙(𝑥 − 𝑡𝑕𝑟)) and if x thr, the soft threshold signal is 0. Given a transformed signal Y and threshold 𝜆 0 , two thresholding methods can be expressed as follows. The hard thresholding method is given as Eq. .   YYYYT H R h a r d ,0 ,)( () The soft thresholding method is given by    YYYYYT H R s o f t ,0 ),)(s g n ()( () where 𝑇𝐻𝑅(∙) represents the output value after thresholding. Fig. displays the hard and soft thresholding functions. 20 (a)。