njit-etd2003-081 - New Jersey Institute of Technology

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3.1 Time-Frequency Distributions
From a given time series signal, x(t), one can easily see how the "energy" of the signal
is distributed in time. By performing a Fourier transform to obtain the spectrum, X(ω) ;
one can readily see how the "energy" of the signal is distributed in the frequency
domain. For a stationary signal, there is usually no need to go beyond the time or
frequency domain. Unfortunately, most real world signals have characteristics that
change over time, and the individual time domain or frequency domain does not provide
means for extracting this information. There is obvious need for creating a function,
W(t,ω), that represents the energy of the signal simultaneously in time and frequency.
This function, W(t, co), is commonly referred to as a time-frequency representation or
time-frequency distribution. The most well-known time-frequency distribution is the
spectrogram (squared magnitude of the short-time Fourier transform). However, there is
a rich theory behind time-frequency analysis of which the spectrogram is just a small
part. Two other well-known time-frequency distributions are the continuous wavelet
transform and the Wigner (or Wigner-Ville) distribution.
The areas of time- frequency and time-scale analysis have seen many exciting
developments in recent years [18-24]. The theory has been progressing rapidly and has
been successfully applied to a wide variety of signals including the following: speech,
biological, geophysical, machine monitoring, radar, and others. While there has been
rapid progress in recent years, there are still many open questions regarding the
construction of time-frequency distributions.
Since there are a wide variety of methods for constructing time-frequency
distributions, it is useful to classify them according to their structure and properties.