A new interpretation of Capon's maximum likelihood method of frequency-wavenumber spectral estimation

The maximum likelihood method of frequency-wavenumber spectral estimation (MLM) is a high-resolution alternative to conventional Fourier analysis. This paper provides a new interpretation for MLM. The point of view taken is that at each wavenumber we attempt to estimate the power of a discrete planewave at that wavenumber, a problem complicated by the presence of interfering energy at other wavenumbers (noise), with an unknown spectral density matrix. It is shown that, subject to some assumptions, the problem of choosing the variance of the planewave complex amplitude (that is, the plane-wave power) and the noise covariance matrix jointly, to maximize the likelihood function for the observations, has no unique solution. There is a continuous range of values for the estimates all of which have the same (maximum) likelihood, the lower bound on the planewave power estimate being zero, and the upper bound being the MLM power estimate. Some important differences between the problem of estimating the planewave complex amplitude for each independent data segment and the problem of estimating the variance of the planewave complex amplitude are indicated.