By Rene Carmona, Wen-Liang Hwang, Bruno Torresani
Time frequency research has been the thing of excessive learn task within the final decade. This publication supplies a self-contained account of tools lately brought to research mathematical features and indications at the same time by way of time and frequency variables. The publication offers a close presentation of the purposes of those transforms to sign processing, emphasizing the continual transforms and their functions to sign research difficulties, together with estimation, denoising, detection, and synthesis. to assist the reader practice those analyses, sensible Time-Frequency research offers a collection of helpful instruments within the type of a library of S features, downloadable from the authors' websites within the usa and France. Key positive aspects* targeted presentation of the Wavelet and Gabor transforms* functions to deterministic and random sign idea* Spectral research of nonstationary signs and methods* a variety of sensible examples starting from speech research to underwater acoustics, earthquake engineering, web site visitors, radar sign denoising, clinical information interpretation, and so forth* Accompanying software program and information units, freely downloadable from the book's website
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Extra resources for Practical Time-Frequency Analysis: Gabor and Wavelet Transforms with an Implementation in S
Example text
Since both the continuous time signal f and the sequence of samples have a Fourier transform (recall the definitions of the first section of the chapter), it is natural to investigate the possible relations between these two transforms. 24) k---c~ This formula tells us that ](d)(~), the Fourier transform at frequency ~ of the sampled sequence f(A) = { f j } j , is the sum of the contributions from the Fourier transform f of the continuous signal f, not only at ~ but also at the countably infinite set of frequencies ~ + 2kTc/Ax for k = +1, + 2 , .
37) j=O where we now use the notation oak for the natural frequency a~k = k / N . For notation purposes one should think of the frequency variable w as being linked to the Fourier variable ( by the scaling relation ( = 27rw. Recall also that the original sequence f can be recovered from its transform by the inverse Fourier transform 1 N-1 fJ - -N E ]ke2i'j~k' k- 0, 1,... , N - 1. 38) k=0 This inversion formula shows that any finite series f can be written as the linear combination of complex exponentials.
The Hilbert transform is conveniently expressed in the Fourier domain as ~ " - - f ( { ) - - i sgn(~)](~) , ~ e IR. 19) Notice that the Hilbert transform maps sine waves into cosine waves, and vice versa. Given a real-valued signal of finite energy f, the associated analytic signal is defined (up to a factor 2) as its orthogonal projection Zy onto the subspace H2(IR) of L2(II~). 20) where I denotes the identity operator. 21) 36 Chapter 1 Time-Frequency where O(~) denotes the Heaviside step function which is equal to I when ~ _ 0 and to 0 otherwise.