Overview

We now turn to the problem of decomposing one-dimensional signals so as to illuminate two important properties: localization in time of transient phenomena, and presence of specific frequencies. Our starting point will be Heisenberg's inequality, which limits how precisely we can compute these properties.

The main tool will be expansion in orthonormal bases whose elements have good time-frequency localization. Features in this context are just the basis elements which contribute large components to the expansion; they are detectable from their size. Alternatively, we can look for combinations of large components, or of not-so-large components that share similar time or frequency location. The localization of the basis elements does most of our work for us; when we find a large component, we can mark the time-frequency location of its basis element to build a time-frequency picture of the analyzed signal.

10.1 The Time-Frequency Plane

The time- frequency plane is a two-dimensional space useful for idealizing two measurable quantities associated to transient signals. A signal may be represented in this plane in a number of ways.

10.1.1 Waveforms and Time-Frequency Atoms

The game is to choose the best representation for a given signal, and our strategy will be to decomposing a signal into pieces called time- frequency atoms, then drawing idealized representations of these atoms in the plane.

Suppose that ? is a modulated waveform of finite total energy, and suppose that both the position and momentum uncertainties of ? are finite:

(10.1)

These quantities...