Non-stationary Spectral Analysis

Abstract

A modern approach to spectral analysis of non-stationary signals is provided by the continuous wavelet transform (CWT), in which the signal in its entirety is not compared with infinitely-long sinusoids, but with waveforms called wavelets, which are concentrated in time and frequency. In this method, the concept of period (inverse of a frequency) is replaced by the concept of scale. Using the language of continuous time-signals that allows for avoiding some mathematical difficulties, we will describe what a wavelet is and how signals can be analyzed in time and scale; we will then establish a relation between scale and frequency and investigate CWT resolution in time and frequency, thus introducing the concept of multiresolution analysis (MRA). We will also define the conditions under which a signal can be reconstructed from its CWT coefficients. For practical applications, the CWT must be made discrete in time and scale; we will discuss the most popular discretization scheme. Next we will show how an average power spectrum estimate, the global power spectrum (GWS), can be derived from CWT by averaging over time, and how significance tests for the spectral features detected in CWT analysis can be devised. Real-world application examples will be provided.