Utilization of a Unitary Transform for Efficient Computation in the ...
Utilization of a Unitary Transform for Efficient Computation in the ... 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; ;
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)
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YILMAZER et al.: UTILIZATION OF A UNITARY TRANSFORM FOR EFFICIENT COMPUTATION 177<br />
residues or complex amplitudes at ;<br />
angular frequencies ;<br />
damp<strong>in</strong>g factors.<br />
There<strong>for</strong>e, one can write <strong>the</strong> sampled signal as<br />
One can also write<br />
where<br />
(19)<br />
(20)<br />
(13)<br />
(14)<br />
.<br />
.<br />
.<br />
.<br />
. ..<br />
.<br />
.<br />
where, , <strong>for</strong> .<br />
In this study, it has been assumed that <strong>the</strong> damp<strong>in</strong>g factor<br />
is equal to zero. The objective is to f<strong>in</strong>d <strong>the</strong> best estimates<br />
<strong>for</strong> , and from . Let us consider <strong>the</strong> matrix ,<br />
which is obta<strong>in</strong>ed directly from . is a Hankel matrix,<br />
and each column <strong>of</strong> is a w<strong>in</strong>dowed part <strong>of</strong> <strong>the</strong> orig<strong>in</strong>al data<br />
vector,<br />
. It is assumed that, <strong>the</strong>re<br />
are data samples<br />
.<br />
.<br />
.<br />
.<br />
. ..<br />
.<br />
.<br />
Now, let us consider <strong>the</strong> MP<br />
(21)<br />
(22)<br />
(23)<br />
(24)<br />
.<br />
.<br />
.<br />
.<br />
. ..<br />
.<br />
.<br />
(15)<br />
The parameter is called <strong>the</strong> pencil parameter. is chosen between<br />
and <strong>for</strong> efficient noise filter<strong>in</strong>g [7]. The variance<br />
<strong>of</strong> <strong>the</strong> estimated values <strong>of</strong> and will be m<strong>in</strong>imal, if <strong>the</strong><br />
values <strong>of</strong> are chosen <strong>in</strong> this range.<br />
The matrix is obta<strong>in</strong>ed from (14) by writ<strong>in</strong>g it <strong>in</strong> <strong>the</strong> follow<strong>in</strong>g<br />
matrix <strong>for</strong>m, shown <strong>in</strong> (16) at <strong>the</strong> bottom <strong>of</strong> <strong>the</strong> page.<br />
From <strong>the</strong> matrix , we can def<strong>in</strong>e two submatrices, say<br />
.<br />
.<br />
. .. .<br />
(25)<br />
Here is <strong>the</strong> identity matrix. One can show that <strong>the</strong> rank<br />
<strong>of</strong> will be , provided that [7].<br />
However, if , , <strong>the</strong> row <strong>of</strong><br />
is zero, <strong>the</strong>n <strong>the</strong> rank <strong>of</strong> this matrix is . There<strong>for</strong>e, <strong>the</strong><br />
parameters can be found as <strong>the</strong> generalized eigenvalues <strong>of</strong><br />
<strong>the</strong> matrix pair . So, <strong>the</strong> solution to <strong>the</strong> problem can be<br />
reduced to an ord<strong>in</strong>ary eigenvalue problem, and will be <strong>the</strong><br />
eigenvalues <strong>of</strong><br />
(26)<br />
where is <strong>the</strong> Moore–Penrose pseudo <strong>in</strong>verse <strong>of</strong> , which is<br />
def<strong>in</strong>ed as<br />
(27)<br />
.<br />
.<br />
. .. .<br />
(17)<br />
The DOA is obta<strong>in</strong>ed from , where<br />
.<br />
For <strong>the</strong> noisy data, <strong>the</strong> s<strong>in</strong>gular value decomposition (SVD)<br />
is useful to reduce some <strong>of</strong> <strong>the</strong> noise effect [15]. The matrix<br />
can be written as<br />
(18)<br />
(28)<br />
.<br />
.<br />
. .. . .<br />
(16)