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

178 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 54, NO. 1, JANUARY 2006
Here, and are unitary matrices whose columns are the
eigenvectors of , and , respectively. is the singular
values of , which are located on the main diagonals in the descending
order
. If the data is noiseless,
the first singular values are nonzero, the rest is zero, where
(29)
(30)
If the data is noisy, needs to be estimated, and how to do it
is illustrated in [7]. The ratio of each of the singular value to the
largest one determines the value of . We choose
, where is the number of accurate significant decimal
digits of the data vector . One can rewrite the (25) as
with
Since,
Therefore
(39)
(40)
(41)
(42)
where , and , with
.
The matrices , and are defined as follows:
.
.
.
. .. .
.
(31)
. .. . .
(32)
For data contaminated with noise, it is better to take the
SVD of so that some of the noise effect can be reduced.
It can be shown that the singular vectors of
correspond to the signal space [9]. One can then solve for
as the generalized eigenvalue of the matrix
and , where, is the
largest singular vectors of . is estimated number of the
signals. That is, . Here, ,
and are orthogonal matrices. is a diagonal matrix
with its element . , such that
the corresponding singular values
satisfy
.Or
is the
eigenvalue of .
Since we do not know how many frequency components exist
in the signal, the number of the estimated frequencies should
be determined using some criteria. Typically the singular values
beyond are set equal to zero [7].
.
.
.
. .. . .
(33)
IV. SUMMARY OF THE ALGORITHM
Basically the algorithm can be summarized as follows:
From (31) and (33), one can write
(34)
(35)
Note that , and is real from the theorems
(1, 2, 3). Therefore
(36)
It can be shown that , and
, , and therefore, (36) can
be represented by
(37)
• Step 1: From the data vector ,
find the MP from (2);
• Step 2: Compute the real data matrix
, using theorem (3);
Or,
from theorem (2);
• Step 3: Evaluate and
;
• Step 4: Perform a SVD of and
calculate , which is the
largest singular vectors of ;
• Step 5: Calculate the generalized
eigenvalues
of
and ;
• Step 6: Calculate .
.
.
Hence, (36) converts to
(38)
In this algorithm all computations are made using real numbers
and therefore, no variable is complex including the eigenvalues
and the eigenvectors in this procedure.