The continuous wavelet transform: a tool for signal investigation and understanding

In this article, the continuous wavelet transform is introduced as a signal processing tool for investigating time-varying frequency spectrum characteristics of nonstationary signals. The transform is discussed within the context of Fourier methods and the general problem of time-frequency representation and is compared with the more traditional Gabor method of windowed Fourier transforms. To take full advantage of the potential benefit of wavelet characterization, a computer algorithm for generating wavelet images from data is needed. Such an efficient algorithm, the algorithme a trous, was invented and published several years ago by an important group of engineering researchers. A detailed derivation and analysis of their algorithm is presented.