## Statistical Model

Let's assume there are tonals in AR noise of order . There are parameters for each tonal: frequency, amplitide and phase, or equivalently frequency and complex amplitude with two components (in-phase and quadrature), then the AR parameters for a grand total of parameters. Let the data be written

where ranges from to , is auto-regressive noise and

is the deterministic (tonal) component, and

and are the time-series window function coefficients (Hanning, etc.).

In a more compact matrix notation, we let

where is the diagonal matrix of window function coefficients.

The log-likelihood function is given by

 (8.11)

where is the auto-correlation matrix for an AR process (see Section 9.1.3).

We break down into its well-known Cholesky factor form

where is the prewhitening matrix, which is a banded matrix formed from the prediction error filter. Each row of is formed from the prediction error filter , where the occupies the main diagonal. Matrix has a determinant of 1, as it is lower-triangular and has ones on the diagonal. For the non-circular (stationary process) case, on the rows less than row , the prediction error filter is truncated, and so the coefficients must be estimated from the corrsponding lower-order AR model. We can avoid the complexities of these end effects, especially since we are assuming that a window function is applied to the data, by assuming a circular AR process (See sections 9.1.2, 9.2.4). Then, matrix is circulant, with each row containing the full prediction error filter, wrapped around. Furthermore, the equation holds exactly. We can therefore re-write (8.11) as:

 (8.12)

The frequency-domain equivalent of (8.12) is

 (8.13)

where and are the Fourier coefficients of and , respectively. In obtaining (8.13), we used the fact that

This is just the frequency-domain equivalent of saying that the determinant of the circulant matric , with all ones on the diagonal, is 1. Although written in terms of Fourier coefficients, the expression is a PDF of . The equivalence of (8.13) and (8.12) can be readily seen.

Subsections
Baggenstoss 2017-05-19