Analysis of noisy coefficients in the discrete Hermite transform domain with application in signal denoising and sparse signal reconstruction

The influence of an additive Gaussian noise on the discrete Hermite transform based signal representation is analyzed. The Hermite coefficients of noisy signals are random Gaussian variables. Based on the derived respective mean values and the variances, an efficient nonlinear threshold for a simple signal denoising approach is introduced, suitable for signals well concentrated in this transform domain. Moreover, the results are easily incorporated into a coefficient thresholding based compressed sensing algorithm for the reconstruction of noisy signals with missing samples. These approaches and the theory behind are motivated by the signals concentrated in the Hermite transform domain, such as the QRS complexes and UWB signals. Numerical examples validate the presented theory.