Estimation of Exponential Power Spectra

Abstract

Many naturally occurring random processes, such as ambient sea noise, have the property that the logarithm of their power spectra is approximately a linear function of log frequency. It is difficult to estimate a power spectrum with a steep negative slope due to the leakage of low-frequency power into the higher-frequency bands by the digital filter of the estimator. This spectral estimation problem can be simplified by assuming a two-parameter model of the spectrum S (f). Let X (t) be a stationary Gaussian process with zero mean and lnS (f) = ln (σ2θ/2) −θ|f|, where θ and σ2 are unknown parameters. The total power of X (t) is σ2 and the mean frequency is θ−1. These parameters are estimated by simple functions of the number of zero crossings of X (t) and the number of fixed level crossings in a given finite record. The estimator of ln S (f) is ln (σ̂2θ̂/2) − θ̂f, where σ̂2 and θ̂ are the parameter estimates. For fixed σ2, the large sample variances of θ̂ and lnŜ are derived and computed for θ = 14, 12, 1, and 2. Monte Carlo methods are used to estimate the medium sample properties of θ̂ and σ̂2. The variance of lnŜ is of the order of T−1, whereas the variance of the standard digital estimator of lnS is of the order T−12, where T is the record length.