By Jack Xin, Yingyong Qi

The target of the booklet is to offer an obtainable advent of mathematical types 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 primary to various technological advances of the electronic international some time past decade, from song compression in MP3 to electronic listening to aids, from community dependent voice enabled prone to speech interplay with cell phones. arithmetic and computation are in detail on the topic of those leaps and limits. nonetheless, speech and listening to are strongly interdisciplinary components the place multiple medical and engineering guides and methods frequently coexist and make it tricky for newbies to enter.

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In Matlab, w = conv(x,y). Example: compute w =x*y, x w = [2,3],y conv(x,y) = = = [1, -4,S], [2, -S, -2, IS]. S3) Here NI = 2, N2 = 3, N = NI + N2 - 1 = 4, so zero-pad as: xa = [x, 0, 0], Ya = [y,0]. w = Xa 8Ya = real(ifft(fft(xa ). *fft([ya))) = [2, -S, -2, IS]. 53). 3 Circular Convolution and z-Transform Let x, y be N-dimensional vectors with complex components, w = x 8 y. Let X(z), Y(z), W(z) be their z-transforms. Then: W(z) =X(z)Y(z) modz- N . S4) Here mod z-N means that zn and zn-N are regarded as the same, for any integer n.

What is the windowing effect in Fourier domain? Multiplication in the time domain implies convolution in the Fourier (OTFT) domain: Xw(w) =X(w) HecW(w), where: rec W() w N-l = L e-2rriwl1 11=0 1 = - e 1- -2rriwN . e- 2rr1w . ( sm nwN) =. sm(nw) e-rriw(N-l) The function s~~~(rrww~) = D( w, N) is called Dirichlet kernel. A plot of Dirichlet kernel as function of w E ( -1 /2,1/2] is in Fig. 2. It is even in w E [-1/2,1/2], has maximum N as e ---+ o. The first two (symmetric) zeros are located at w = ±1/N, the region between the two zeros is called the main lobe.

The simplest window function is the rectangular function: recw(n) I nE [0,~-1] =] o otherwIse. What is the windowing effect in Fourier domain? Multiplication in the time domain implies convolution in the Fourier (OTFT) domain: Xw(w) =X(w) HecW(w), where: rec W() w N-l = L e-2rriwl1 11=0 1 = - e 1- -2rriwN . e- 2rr1w . ( sm nwN) =. sm(nw) e-rriw(N-l) The function s~~~(rrww~) = D( w, N) is called Dirichlet kernel. A plot of Dirichlet kernel as function of w E ( -1 /2,1/2] is in Fig. 2. It is even in w E [-1/2,1/2], has maximum N as e ---+ o.

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