Signal Averaging in Wavelet Domain

We present a method to implement signal averaging using wavelet transforms. Signal averaging plays an essential role in reducing noise from physical experiments, but is currently performed outside wavelet transforms and removes the information of individual data points or experimental scans. We introduce the conditions that need to be satisfied to implement signal averaging using wavelet transforms, showing that only the Haar wavelet can be used to obtain signal-averaged information. Our method identifies the wavelet component that contains the signal-averaged value and develops a procedure to recover it. The method is tested on model data using varied amounts of Gaussian and Poisson noise at different data lengths as well as experimental data from the Electron Spin Resonance (ESR) spectroscopy. While the Haar wavelet is successful in obtaining the mean, other wavelets yield weighted averages. The incorporation of this method enables signal averaging and analysis to be carried out simultaneously in the wavelet domain, expanding the application of wavelet transforms and improving wavelet analysis. The code is available via GitHub and denoising.cornell.edu, as well as the corresponding author's group website (signalsciencelab.com).