0.4 Time-frequency dictionaries

Motivated by quantum mechanics, in 1946 the physicist Gabor (Gabor:46)proposed decomposing signals over dictionaries of elementary waveforms which he calledtime-frequency atoms that have a minimal spread in atime-frequency plane. By showing that such decompositionsare closely related to our perception of sounds, and that they exhibit important structures in speech and music recordings,Gabor demonstrated the importance of localized time-frequency signal processing.Beyond sounds, large classes of signals have sparse decompositions as sums of time-frequency atoms selected from appropriate dictionaries.The key issue is to understand how to construct dictionaries with time-frequency atomsadapted to signal properties.

Heisenberg uncertainty

A time-frequency dictionary $\mathcal{D}={\left\{{\phi }_{\gamma }\right\}}_{\gamma \in \Gamma }$ is composed of waveforms of unit norm $\parallel {\phi }_{\gamma }\parallel =1$ , which have a narrow localization in time and frequency.The time localization u of φ _γ and its spread around u , are defined by

Similarly, the frequency localization and spread of ${\widehat{\phi }}_{\gamma }$ are defined by

The Fourier Parseval formula

shows that $⟨f,{\phi }_{\gamma }⟩$ depends mostly on the values $f\left(t\right)$ and $\widehat{f}\left(\omega \right)$ , where ${\phi }_{\gamma }\left(t\right)$ and ${\widehat{\phi }}_{\gamma }\left(\omega \right)$ are nonnegligible , and hence for $(t,\omega )$ in a rectangle centered at $(u,\xi )$ , of size ${\sigma }_{t,\gamma }\times {\sigma }_{\omega ,\gamma }$ . This rectangle is illustrated by [link] in this time-frequency plane $(t,\omega )$ . It can be interpretedas a “quantum of information” over an elementary resolution cell. The uncertainty principle theorem proves(see Chapter 2) that this rectangle has a minimum surface that limits thejoint time-frequency resolution:

Constructing a dictionary of time-frequency atoms can thus be thought of as covering the time-frequency plane with resolution cells having a timewidth ${\sigma }_{t,\gamma }$ anda frequency width ${\sigma }_{\omega ,\gamma }$ which may vary but with a surface larger than one-half. Windowed Fourier and wavelet transforms are two important examples.

Windowed fourier transform

A windowed Fourier dictionary is constructed by translating in time and frequency a time window $g\left(t\right)$ , of unit norm $\parallel g\parallel =1$ , centered at $t=0$ :

The atom $g_{u,\xi }$ is translated by u in time and by ξ in frequency. The time-and-frequency spread of $g_{u,\xi }$ is independent of u and ξ . This means that each atom $g_{u,\xi }$ corresponds to a Heisenberg rectanglethat has a size ${\sigma }_t\times {\sigma }_{\omega }$ independent of its position $(u,\xi )$ , as shown by [link] .

The windowed Fourier transform projects $f$ on each dictionary atom $g_{u,\xi }$ :