A covariance approach to spectral moment estimation

We are interested in estimating the moments of the spectral density of a comp[ex Gaussian signal process <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{ q^{(1)} (t) \}</tex> when the signal process is immersed in independent additive complex Gaussian noise <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{q^{(2)} (t) \}</tex> . Using vector samples <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q = \{ q(t_1),\cdots ,q(t_m)\}</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q(t) = q^{(1)}(t) + q^{(2)}(t)</tex> , estimators for determining the spectral moments or parameters of the signal-process power spectrum may be constructed. These estimators depend upon estimates of the covariance function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_1 (h)</tex> of the signal process at only one value of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h \neq 0</tex> . In particular, if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m = 2</tex> , these estimators are maximum-likelihood solutions. (The explicit solution of the likelihood equations for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m &gt; 2</tex> is still an unsolved problem.) using these solutions, asymptotic (with sample size) formulas for the means and variances of the spectral mean frequency and spectral width are derived. It is shown that the leading term in the variance computations is identical with the Cramér-Rao lower bound calculated using the Fisher information matrix. Also considered is the case Where the data set consists of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> samples Of continuous data, each of finite duration. In this case asymptotic (with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> ) formulas are also derived for the means and variances of the spectral mean frequency and spectral width.