Discrete evolutionary transform for time–frequency signal analysis

In this paper, we introduce the discrete evolutionary transform (DET) capable of representing deterministic non-stationary signals. Besides the signal representation, the DET permits the computation of a kernel from which the evolutionary spectrum of the signal is obtained. The signal representation is modeled after the Wold–Cramer representation used for random non-stationary signals in Priestleyʹs evolutionary spectral theory. The proposed transform generalizes the short-time Fourier transform and the spectrogram. To illustrate how to define the windows used in the DET we consider the Gabor and the Malvar cases. The Gabor-based window is time dependent and uses the bi-orthogonal analysis and synthesis windows of the expansion. The Malvar-based window is a function of time and of frequency, and depends on the orthogonal functions used in the expansion. Two types of transforms are shown: sinusoidal and chirp DETs. The sinusoidal DET represents well signals with narrow-band components, while the chirp transformation is capable of representing well signals with wide-band components provided that the instantaneous frequency information of the signal components is estimated. Examples are used to illustrate the implementation of the DET. The examples show the capability of the transform in providing excellent representation for a signal and a spectrum with very good time–frequency resolution.