Time-Frequency Analysis Background

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The Time-Frequency Concept

A time-frequency representation (TFR) maps a 1-d signal to a 2-d time-frequency image that displays how the frequency content of the signal changes over time.  A bat echolocation chirp provides an excellent motivation for time-frequency-based signal processing.  Neither the time signal

Bat chirp signal

time

nor its Fourier spectrum

Bat chirp Fourier spectrum

frequency

reveal the true structure of the signal.  In contrast, a time-frequency image of the signal clearly exposes its nonstationary character

Optimal-kernel TFR

In the above TFR, time runs horizontally and frequency vertically, and the colors indicate the energy level.  (Click on any of these images to obtain a larger and higher resolution version.)

While each signal has a unique Fourier spectrum, a time-frequency analysis of a signal is nonunique.  In other words, many different TFRs can `explain' the same data.  For instance, here are two additional TFRs of the bat signal: the Wigner distribution at left and the spectrogram at right.

Wigner Distribution And Spectrogram

Since for any given signal some TFRs are `better than others,' TFR design has become an important research area.

TFRs can be characterized in terms of a 2-d kernel function. In order to find the `best' TFR for a given signal, our research at Rice focuses on optimization-based kernel design.