The estimation of the AR and sinusoid parameters
is an iterative feature extraction approach.
Despite this, given a set of
estimated or randomly-selected parameters,
we can create data that has the prescribed ML parameter
estimates. We use the approach of Section 4.5.4
Given a set of SINAR parameters,
we first un-do the SNR norlalization,
which was described in Section 8.2.1.
Then, we can construct the the vector and matrices
using (8.15), we can require to meet the condition
Then, using the fact that the derivative w/r to
frequency must be zero and using (8.14),
we can also require that meet the condition
Any generated data must therefore meet
the two linear constraints (8.18), (8.19).
These two constraints
insure that the ML sinusoidal parameter
estimates will coincide with the desired ones and
can be written as one linear constraints
by stacking the matrices.
The constraints can be also implemented in the ``whitened" domain.
That is to say that the whitened data
must meet the modified constraints
We stack the matrices to produce the single constraint
Now, in order to approximately meet the AR constraints, the whitened
must have variance .
as well as the constraint
which can be written
To generate data that meets these constraints,
we we follow the approach of Section 4.5.4.
To generate this data, let
is the ortho-normal matrix
of basis functions that span the orthogonal complement of
is of dimension
is of dimension
vector to have norm
So, to create data
vector of independent Gaussian noise, then
normalize it to have norm
This vector will meet conditions
If we un-whiten
which can be achieved in the frequency domain by dividing
each Fourier coefficient by ,
we will have an approximate solution to re-synthesis.
But, although meets the linear constraints exactly
(for amplitude and derivative), it does not meet the
auto-correlation function (ACF) constraints. We must insure that
ACF computed from the residual
the desired ACF, which is the ACF of the autoregressive spectrum
This can be achieved by making a slight
adjustment to within the linear subspace
orthogonal to the already-satisfied linear constraints.
The linear constraints (in the un-whitened world) are given by
be the ortho-normal matrix
spanning orthogonal complement space of matrix .
Then, if we let
, and select to meet the
above circular ACF constraints, we have a solution that
meets all constraints.
This re-synthesis technique is implemented in