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

176 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 54, NO. 1, JANUARY 2006
II. THE UNITARY TRANSFORM
A square matrix, , is called unitary, if it satisfies
. The superscript denotes the complex conjugate
transpose of a matrix. Hence , where is the identity
matrix. For the matrix to be unitary, the columns of the matrix
must be orthonormal. This implies .
A matrix A, where
, is called centro-hermitian
[9]–[11], if it satisfies
is called the exchange matrix and defined as
.
. . . . .
.
.
.
.
.
(1)
. Here, is a square
matrix, and is conjugate of .
Theorem 1: If a vector
, where is an odd number, is centro-hermitian, that is
, then the following matrix :
.
.
.
.
. (6)
.
Therefore, . is centro-hermitian.
Theorem 3: If the matrix is centro-hermitian, then
is a real matrix. Here the matrix is unitary, whose
columns are conjugate symmetric and has a sparse structure
[8], [14]. For even, we have
Here, and are matrices that have the dimension of and
. When is odd, we have
(7)
(8)
.
.
.
.
. ..
.
.
is also centro hermitian.
Proof: Since
, then
for
. Therefore
(2)
Here and are matrices that have the dimension of ,
and 0 is a vector whose elements are 0.
Proof: Using , the conjugate of is
Since
(9)
(10)
(11)
.
.
.
.
. ..
.
.
(3)
Therefore,
is a real matrix.
.
.
. .. . .
(4)
Hence,
is a also centro-hermitian matrix.
Theorem 2: The following matrix
.
is centro hermitian for any matrix [9].
Proof: Using
.
. , and
.
. ,wehave
III. THE MP METHOD
The narrowband sources located in the far field of a uniformly
spaced array consisting of isotropic omnidirectional point sensors
radiating in free space is considered. This results in a uniform
linear array (ULA). The focus here is to use the unitary
transform to convert the complex matrices used in the MP formulation
to real matrices and use these matrices to estimate the
DOA of multiple signals simultaneously impinging on the ULA.
The vector is the set of voltages measured at the feed point
of the antenna elements of the ULA. Therefore, can be
modeled by a sum of complex exponentials, i.e.
(12)
.
.
. (5)
where
observed voltages at a specific instance ;
noise associated with the observation;
actual noise free signal;
;