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 (inphase and quadrature), then the
AR parameters for a grand total of
parameters.
Let the data be written
where ranges from to ,
is autoregressive noise and
is the deterministic (tonal) component,
and
and are the timeseries window function
coefficients (Hanning, etc.).
In a more compact matrix notation,
we let
where is the diagonal matrix of window function
coefficients.
The loglikelihood function is given by

(8.11) 
where is the autocorrelation matrix
for an AR process (see Section 9.1.3).
We break down
into its
wellknown 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 lowertriangular
and has ones on the diagonal.
For the noncircular (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 lowerorder 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 rewrite (8.11) as:

(8.12) 
The frequencydomain 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 frequencydomain 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
20170519