Noise filtering and deconvolution of XPS data by wavelets and Fourier transform

In experimental sciences, the recorded data are often modelled as the noisy convolution product of an instrumental response with the 'true' signal to find. Different models have been used for interpreting x-ray photoelectron spectroscopy (XPS) spectra. This article suggests a method of estimate the 'true' XPS signal that relies upon the use of wavelets, which, because they exhibit simultaneous time and frequency localization, are well suited to signal analysis. First, a wavelet shrinkage algorithm is used to filter the noise. This is achieved by decomposing the noisy signal into an appropriate wavelet basis and then thresholding the wavelet coefficients that contain noise. This algorithm has a particular threshold related to frequency and time. Secondly, the broadening due to the instrumental response is eliminated through a deconvolution process similar to that developed in the previous paper in this series for the analysis of HREELS data. This step mainly rests on least-squares and on the existing relation between the Fourier transform, the wavelet transform and the convolution product.