Wavelets, Detection, Estimation, and Sparsity

Abstract Questions concerning the detection and estimation of signals in noise using the discrete wavelet transform are considered. A theorem is presented that shows the discrete wavelet transform of white noise is spread out over all the transform plane. It is established that a signal which is represented by a small number of wavelet coefficients can be accurately estimated. Moreover, it is also shown that the number of wavelet coefficients needed to represent a signal has a dramatic effect on the ability to estimate the signal. Specifically, two estimation techniques are presented and analyzed. Probability of detection and probability of false alarm results are given for a wide variety of signals. An example is presented to demonstrate the results.