Time-Frequency Analysis Background
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
timenor its Fourier spectrum
Bat chirp Fourier spectrum
frequencyreveal the true structure of the signal. In contrast, a time-frequency image of the signal clearly exposes its nonstationary character
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.
