Mixed time-frequency signal transformations

Mixed time-frequency representations are transformations of time-varying signals that depict how the spectral content of a signal is changing with time. They are multidimensional, timevarying extensions to conventional Fourier transform spectral analysis of signals and systems. Most time-frequency representations (TFRs) transform a one-dimensional signal, x(t), into a twodimensional function of time and frequency, Tx(t, f). These transformations represent a surface above the time-frequency “plane” which gives an indication as to which spectral components of the signal are present at a given time and their relative amplitude. They are conceptually similar to a musical score with time running along one axis and frequency along the other.37,50 Just as the location and the shape of the notes on a musical score represent the pitch, time of occurrence, and duration of each sound in a piece of music, so too does the location of the local maximum and the shape of the surface of the TFR give an indication as to the frequency content, onset, and duration of various dominant signal components. Such representations are useful for the analysis, modification, synthesis, and detection of a variety of nonstationary signals with time-varying spectral content.1-4,30,91 The purpose of this chapter is to give an overview of many of the linear and quadratic time-frequency representations that have been developed over the past 60 years. The chapter first reviews various one-dimensional spectral representations, such as the Fourier transform, the instantaneous frequency, and the group delay. A brief discussion follows of a few commonly used TFRs such as the short-time Fourier transform (STFT), the Wigner distribution (WD), the Altes Q distribution, and the Bertrand unitary P0 distribution. These TFRs will be used as examples in the subsequent section, which describes many useful properties that an ideal TFR should satisfy. Unfortunately, no one TFR exists which satisfies all of these desirable properties. The relative merits of these TFRs can be understood by grouping them into “classes” of TFRs that share two or more properties. The remainder of the chapter is devoted to defining and understanding these classes. Important insights into these classes of TFRs can be gained by examining a set of five two-dimensional kernel functions that are unique to each TFR. These kernels greatly simplify the analysis and application of TFRs