Wavelets: A comparison with the spectrogram and other methods for time-frequency analysis.

Abstract

Many natural signals exhibit spectral content that changes over time. Methods for time-varying spectral analysis first emerged in the 1940s with the development of the “sound spectrograph” at AT&T Bell Laboratories. Since then, the spectrogram has become the primary method for time-frequency analysis. Originally implemented as a bank of band-pass filters, today the spectrogram is typically computed digitally via the short-time Fourier transform. Recently, wavelets have been proposed as a superior method for time-frequency analysis. The usual argument is that the spectrogram uses a fixed window length, whereas the wavelet approach uses windows that are longer for lower frequencies and shorter for higher frequencies. While the benefits of this approach are usually taken as self-evident, we explore in critical detail the aims of time-frequency analysis, and the benefits afforded by wavelets versus the spectrogram and modern methods such as the Choi–Williams distribution. In particular, since a primary aim of time-frequency analysis is to study the local spectral and temporal characteristics of signals, we examine the local moments of the various methods. Local moments are related to important signal features such as the instantaneous frequency and bandwidth. We show the effect of fixed windowing versus variable wavelet windowing on these features.