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
Let the data be written
where ranges from to ,
is auto-regressive noise and
is the deterministic (tonal) component,
and are the time-series window function
coefficients (Hanning, etc.).
In a more compact matrix notation,
where is the diagonal matrix of window function
The log-likelihood function is given by
where is the auto-correlation matrix
for an AR process (see Section 9.1.3).
We break down
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:
The frequency-domain equivalent of (8.12)
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.