Methods-MP

A general family of time-frequency atoms can be generated by scaling,translating and modulating a single window function $g(t)$. For any scale $s > 0$, frequency modulation $\psi$ and translation $u$, we denote $\gamma =(s, u, \psi)$ and define

$$g_{\gamma}(t)=\frac{1}{\sqrt{s}}\,g\left(\frac{t-u}{s}\right)e^{i\xi t}.$$ (6)

If $g(t)$ is even, which is generally the case, $g_{\gamma}(t)$ is centered at the abscissa $u$. Its energy is mostly concentrated in a neighborhood of $u$, whose size is proportional to $s$. The Fourier transform of $g_{\gamma}(t)$ can be expressed by

$$\widehat{g}_{\gamma}(\omega)=\sqrt{s}\,\widehat{g}\left(s\left(\omega-\xi \right)\right)e^{-i(\omega-\xi)u}.$$ (7)

Since $\vert\widehat{g}(\omega)\vert$ is even, $\vert\widehat{g}_{\gamma}(\omega)\vert$ is centered at the frequency $\omega=\xi$ . Its energy is concentrated in a neighborhood of $\xi$,whose size is proportional to $1/s$.

In this study we use software implementation of matching pursuit algorithm developed by Mallat and Zhang (1993). We use dictionary composed of Gabor functions supplemented with a canonical basis of discrete Dirac functions and the discrete Fourier basis of cosine and sine functions. The discrete Gabor time-frequency atom is defined by

$$g_{\gamma}(n)=g_{s}(n-p)\cos{(\frac{2\pi k}{N}n+\phi)},$$ (8)

where

$$g_{s}(n)=\frac{K_s}{\sqrt{s}}\sum^N_{p=0}2^{1/4}e^{-\pi\left(\frac{n-pN}{s}\right)^{2}},$$ (9)

constant $K_s$ normalizes the discrete norm of $g_s$ to $1$, and $\gamma=(s,p,\frac{2\pi k}{N},\phi)$ for $0\leq \phi<2\pi$, $0\leq p<N$ and $0\leq k<N$ . This dictionary provides a richer collection of atomic waveforms which are located on a much finer grid in time-frequency space than wavelet and cosine packet tables.

We compute a linear expansion of signal $f$ over a set of atoms selected from the dictionary, in order to best match its inner structures. After $m$ iterations, a matching pursuit decomposes a signal into

$$f=\sum^{m-1}_{n=0}\langle R^{n}f,g_{\gamma_n}\rangle g_{\gamma_n}+R^{m}f,$$ (10)

where $R^{m}f$ is the residual vector after $m$ iterations, and $\langle f, g \rangle$ denotes inner product of functions $f$ and $g$. The Matching Pursuit algorithm at each step selects atom $g_{\gamma_n}$, for which inner product $\langle R^{n}f,g_{\gamma_n} \rangle$ is largest.

To illustrate the decomposition into the time-frequency atoms we compute energy density defined by

$$Ef(t,\omega)=\sum^{m-1}_{n=0}\vert\langle R^{n}f,g_{\gamma_n}\rangle\vert^{2}Wg_{\gamma_n}(t,\omega),$$ (11)

where $Wg_{\gamma_n}(t,\omega)$ is the Wigner distribution of $g_{\gamma_n}(t,\omega)$ atom . Unlike the Wigner and the Cohen class distributions, it does not include cross terms. Energy distribution of atoms from our dictionary are visible as horizontal lines for cosine functions, vertical lines for Dirac $\delta$, or ellipses with axes proportional to time and frequency spread for Gabor functions.