Wavelet Thresholding Techniques for Power Spectrum Estimation

by Pierre Moulin

Estimation of the power spectrum $S(f)$ of a stationary random process
can be viewed as a nonparametric statistical estimation problem.
We introduce a nonparametric approach based on a wavelet representation
for the logarithm of the unknown $S(f)$.
This approach offers the ability to capture statistically significant
components of $ln S(f)$ at different resolution levels and
guarantees nonnegativity of the spectrum estimator.
The spectrum estimation problem is set up as a problem of inference on
the wavelet coefficients of a signal corrupted by additive non-Gaussian noise.
We propose a wavelet thresholding technique to solve this problem
under specified noise/resolution tradeoffs and show that the wavelet
coefficients of the additive noise may be treated as
independent random variables.
The thresholds are computed using a saddle-point approximation to
the distribution of the noise coefficients.