Utilization of a Unitary Transform for Efficient Computation in the ...

YILMAZER et al.: UTILIZATION OF A UNITARY TRANSFORM FOR EFFICIENT COMPUTATION 177
residues or complex amplitudes at ;
angular frequencies ;
damping factors.
Therefore, one can write the sampled signal as
One can also write
where
(19)
(20)
(13)
(14)
.
.
.
.
. ..
.
.
where, , for .
In this study, it has been assumed that the damping factor
is equal to zero. The objective is to find the best estimates
for , and from . Let us consider the matrix ,
which is obtained directly from . is a Hankel matrix,
and each column of is a windowed part of the original data
vector,
. It is assumed that, there
are data samples
.
.
.
.
. ..
.
.
Now, let us consider the MP
(21)
(22)
(23)
(24)
.
.
.
.
. ..
.
.
(15)
The parameter is called the pencil parameter. is chosen between
and for efficient noise filtering [7]. The variance
of the estimated values of and will be minimal, if the
values of are chosen in this range.
The matrix is obtained from (14) by writing it in the following
matrix form, shown in (16) at the bottom of the page.
From the matrix , we can define two submatrices, say
.
.
. .. .
(25)
Here is the identity matrix. One can show that the rank
of will be , provided that [7].
However, if , , the row of
is zero, then the rank of this matrix is . Therefore, the
parameters can be found as the generalized eigenvalues of
the matrix pair . So, the solution to the problem can be
reduced to an ordinary eigenvalue problem, and will be the
eigenvalues of
(26)
where is the Moore–Penrose pseudo inverse of , which is
defined as
(27)
.
.
. .. .
(17)
The DOA is obtained from , where
.
For the noisy data, the singular value decomposition (SVD)
is useful to reduce some of the noise effect [15]. The matrix
can be written as
(18)
(28)
.
.
. .. . .
(16)