2. The Time-Frequency Analysis

Signals are in general nonstationary. A complete representation of nonstationary signals requires frequency analysis that is local in time, resulting in the time-frequency analysis of signals. The Fourier transform analysis has long been recognized as the great tool for the study of stationary signals and processes where the properties are statistically invariant over time. However, it cannot be used for the frequency analysis that is local in time because it requires all previous as well as future information about the signal to evaluate its spectral density at a single frequency ω. Although time-frequency analysis of signals had its origin almost 60 years ago, there has been major development of the time-frequency distributions approach in the last three decades. The basic idea of the method is to develop a joint function of time and frequency, known as a time-frequency distribution, that can describe the energy density of a signal simultaneously in both time and frequency. In principle, the time-frequency distributions characterize phenomena in a two-dimensional time-frequency plane. Basically, there are two kinds of time-frequency representations. One is the quadratic method covering the time-frequency distributions, and the other is the linear approach including the Gabor transform, the Zak transform, the linear canonical transform, and the wavelet transform analysis. So, the time-frequency signal analysis deals with time-frequency representations of signals and with problems related to their definition, estimation, and interpretation, and it has evolved into a widely recognized applied discipline of signal processing.