The variance of multitaper spectrum estimates for real Gaussian processes

Multitaper spectral estimation has proven very powerful as a spectral analysis method wherever the spectrum of interest is detailed and/or varies rapidly with a large dynamic range. In his original paper D.J. Thomson (1982) gave a simple approximation for the variance of a multitaper spectral estimate which is generally adequate when the spectrum is slowly varying over the taper bandwidth. The authors show that near zero or Nyquist frequency this approximation is poor even for white noise and derive the exact expression of the variance in the general case of a stationary real-valued time series. This expression is illustrated on an autoregressive time series and a convenient computational approach outlined. It is shown that this multitaper variance expression for real-valued processes is not derivable as a special case of the multitaper variance for complex-valued, circularly symmetric processes, as previously suggested in the literature.>