Time-Frequency Signal Analysis

Abstract

This chapter explained the need for defining time-frequency distributions when dealing with nonstationary signals. An attempt was made to provide a comparison between all available TFDs and a clear basis for choice. The criterion of choice is based on one's intuition of how a TFD should perform under some particular condition, for a practical and suitable analysis. The Wigner-Ville distribution has been preferred because it is the closest fit to the ideal time-frequency analysis tool that most signal analysts envisage; it verifies most of the desirable properties and is easily calculated. The notions of instantaneous frequency and time delay of a signal were also reviewed, and it was shown how these parameters are related to time-frequency signal analysis based on the Wigner-Ville distribution. Other properties of the WVD were reviewed and their usefulness for a practical analysis emphasized. The condition of positivity that seems intuitively basic to the concept of a TFD is shown not to be essential in many applications, but it is stressed that what is important is to find a relation between some physical parameters, and some features of the time-frequency distribution, (e.g., energy dissipation and instantaneous frequency [36J, [22]. The limitation of the WVD with regard to cross-terms was addressed. An experienced user of the WVD should be able to interpret correctly these time-frequency representations by discriminating these interferences when they are not too close together. Alternatively, a smoothing procedure will reduce their effect, at the expense of decreased resolution. Potential users of the WD should incorporate the analytic signal (Le., in fact use the WVD) if they want to avoid artifacts and aliasing created by the Wigner distribution (which uses the real signal). Implementation procedures of WVD analysis were shown to be relatively straightforward, in the case of both deterministic and random signals. The potential importance of the method for time-varying filtering of nonstationary signals was demonstrated, and examples were provided. As is the case in standard spectral analYSiS, there is no "best" method in timefrequency signal analysis. The correct approach is to select a TFD that will be optimal for the particular class of signals under consideration. In practice the most useful TFDs, in the author's opinion, are the STFT and the WVD - the STFT because of its negligible cross-terms and the WVD because it gives the least blurred time-frequency representation in the case of monocomponent signals. Other techniques are being currently developed which cater specifically for multicomponent signals. In some specific applications, for example speech, it could be advantageous to use methods such as those developed by Williams and co-workers [20], and Atlas and co-workers [35b]. There are still a number of open questions with regards to time-frequency analysis: positivity, cross-terms, instantaneous frequency, and so on. For further reading it is recommended that a recent tutorial by Mecklenbrauker [95] and a recent review paper by Cohen [32] be consulted.New developments in this field are reported at the annual session on TimeFrequency Signal Analysis organized by SPIE as part of the international conference on advanced signal processing algorithms and architectures, and at the "time-varying spectral analYSis" session of the IEEE International Conference on Acoustics, Speech, and Signal Processing, as well as other conferences.