Data Resynthesis
The estimation of the AR and sinusoid parameters
is an iterative feature extraction approach.
Despite this, given a set of
estimated or randomlyselected parameters,
we can create data that has the prescribed ML parameter
estimates. We use the approach of Section 4.5.4
and 4.4.
Given a set of SINAR parameters,
we first undo the SNR norlalization,
which was described in Section 8.2.1.
Then, we can construct the the vector and matrices
, then
using (8.15), we can require to meet the condition

(8.18) 
where
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

(8.19) 
where
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

(8.20) 

(8.21) 
We stack the matrices to produce the single constraint
where
Now, in order to approximately meet the AR constraints, the whitened
residual
,
where
must have variance .
Now
must meet
as well as the constraint
which can be written
where
To generate data that meets these constraints,
we we follow the approach of Section 4.5.4.
To generate this data, let
where
is the orthonormal matrix
of basis functions that span the orthogonal complement of
.
Since
is of dimension
, then
is of dimension
.
To meet
we need
vector to have norm
So, to create data
, generate
an
vector of independent Gaussian noise, then
normalize it to have norm
,
then let
This vector will meet conditions
as well
as
If we unwhiten
as follows:
which can be achieved in the frequency domain by dividing
each Fourier coefficient by ,
we will have an approximate solution to resynthesis.
But, although meets the linear constraints exactly
(for amplitude and derivative), it does not meet the
autocorrelation function (ACF) constraints. We must insure that
ACF computed from the residual
satisfies
the desired ACF, which is the ACF of the autoregressive spectrum
for
.
This can be achieved by making a slight
adjustment to within the linear subspace
orthogonal to the alreadysatisfied linear constraints.
The linear constraints (in the unwhitened world) are given by
where
Let matrix
be the orthonormal 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 resynthesis technique is implemented in
software/module_sinar_synth.m.
Baggenstoss
20170519