Orthonormal wavelet bases provide an alternative technique for the analysis of non-stationary signals. Unlike the Gabor representation, the basis functions in the wavelet representation all have the same band width on a logarithmic scale. This thesis develops a general framework for the time-scale analysis of signals. In this context, the ON wavelets form a subclass of DWT wavelets. Efficient algorithms for the computation of the wavelet transforms are also developed. As an application, we discuss the problem of detection of (wideband) signals subjected to scale-time perturbations. The probable unknown parameters for scale-time perturbed signals are the gain, and the scale and time perturbations. This problem is set in the context of classical composite hypothesis testing with unknown parameters, and depending on what the unknown parameters are, one of the wavelet transforms, developed is shown to naturally lead to a detector.