By Jack Xin, Yingyong Qi

The goal of the publication is to provide an obtainable creation of mathematical versions and sign processing tools in speech and listening to sciences for senior undergraduate and starting graduate scholars with simple wisdom of linear algebra, differential equations, numerical research, and chance. Speech and listening to sciences are basic to various technological advances of the electronic international long ago decade, from tune compression in MP3 to electronic listening to aids, from community dependent voice enabled providers to speech interplay with cell phones. arithmetic and computation are in detail concerning those leaps and limits. nonetheless, speech and listening to are strongly interdisciplinary components the place varied clinical and engineering guides and ways usually coexist and make it tricky for newbies to enter.

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N - I smce e N = e -N - for mteger m. Note that the approximation is better when k is much smaller than N. Letx E eN be defined by xn = s( "};), n = 0, 1, ... ,N -1. Then DFT(x) =X E eN with N-I Xk = ~ L. -2rrikn k=O,I, ... ,N-1. e-j\-J-Xn n=O We will see next that the Xk' s are related to the coefficients in representing x in terms ofthe orthogonal basis consisting ofthe vectors ek for k = 0, I, ... , N - 1. 5 Discrete Fourier Transform The N-point discrete Fourier transform is a linear map from N-J ~ -2nikn Xk = L.

37). 35) over I is replaced by these finite sums (l = 1,2,3,4 or 1= 1,2) accordingly. 36) for Hamming window by appending the following to the Matlab script file of Example 1. 35) end end figure; plot(su); % visualize variation in m (approximately constant). The lower and upper limits of the j loop are seen as follows. J is the integer part of the number dot. The above synthesis method is called overlap and add method. 16 1 Background Signal Processing, Statistical and Optimization Methods Spectrogram is the image display of the two dimensional function S( n, OJ) ~ for visualizing the relative energy concentration across frequencies as a function of time.

When trying to reconstruct continuous signals from their discrete samples, aliasing error occurs when these lower frequency aliases are recovered instead of the original higher frequency components. Even at a sampling rate of2v, sin(2nvt) ends up being sampled at sin(2nvn 21v ) = sin(nn) = 0 and is indistinguishable from the zero function. However, any higher sampling rate suffices to represent sin(2nvt) unambiguously. In general, the types of continuous signals that can be completely recovered from their sampled versions are band limited signals, namely those whose frequency content is bounded.