In the above paper,1 the authors proposed an algorithm for the estimation of the two-dimensional (2-D) direction and polarization of incoming waves using loops ...
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 46, NO. 3, MARCH 1998
461
Comments (1), we know that 1) when sin i = 0 or tan i = e0i ; L1 E3 + j 2S E6 does not have the full column rank (K) and, hence, dj (j = 2; 3; 4; 5; k = 1; 2; 1 1 1 ; K ) are no longer the generalized eigenvalues of the matrix pencils defined by (22)–(25)1 and 2) when sin i = 0 or 0j ; 1 E3 0 E6 does not have the full column tan i = 0e L j 2S rank (K) and, hence, dj (j = 6; 7; 8; 9; k = 1; 2; 1 1 1 ; K ) are no longer the generalized eigenvalues of the matrix pencils defined by (26)–(29).1 .
Comment on “Direction and Polarization Estimation Using Arrays with Small Loops and Short Dipoles” Qi Cheng and Yingbo Hua
Index Terms— Antenna arrays.
In the above paper,1 the authors proposed an algorithm for the estimation of the two-dimensional (2-D) direction and polarization of incoming waves using loops and dipoles. The algorithm does not require sensors to be uniformly spaced. However, if sin i = 0, 0i , or tan i = 0e0j , the algorithm does not tan i = e necessarily provide correct estimates, even when the exact data covariance matrix (given by (1)1 ) is available. The reason is as shown in the following paragraph. The direction and polarization estimates in the above paper1 are computed from dj for j = 1; 2; 3; 4; 5 (or j = 1; 6; 7; 8; 9) and k = 1; 2; 1 1 1 ; K (K is the number of incoming waves). dj (j = 2; 3; 1 1 1 ; 9; k = 1; 2; 1 1 1 ; K ) are solved as the generalized eigenvalues of the matrix pencils given by (22)–(29).1 Since s = where T is a K 2 K nonsingular matrix, then
E
AT
1
Ld
E3 6 j 2Sl E6 = A
q
q
1
Ld
_ A3 6 j 2Sl A6 T = APT
P
(1)
Author’s Reply Jian Li
Index Terms—Antenna arrays.
The special cases pointed out by Cheng and Hua occur with zero probability. They correspond to k = 0 or or ( k ; k ) = (=4; 0) or (=4; 0 ). It can be easily checked to see if these rare cases have indeed occurred. It can be seen from the third and sixth elements of the vector (6)1 that for these special cases we have
q
= diag[p1 ; 1 1 1 ; pI ] with i = where _ = [ 1 ; 1 1 1 ; I ] and T j 6 cos i ). From [q1 ; q2 ; 1 1 1 ; qL ] and pi = sin i (0sin i e
Manuscript received March 11, 1996; revised August 14, 1997. This work was supported by the Australian Research Council and the Center for Sensor Signal and Information Processing. Q. Cheng is with the School of Electrical Engineering, Faculty of Technology, The Northern Territory University, Darwin, NT 0909, Australia. Y. Hua is with the Department of Electrical Engineering, The University of Melbourne, Parkville, VIC, 3052 Australia. Publisher Item Identifier S 0018-926X(98)01502-6. 1 J. Li, IEEE Trans. Antennas Propagat., vol. 41, pp. 379–387, Mar. 1993.
sin k sin k ej
6 cos k sin k = 0
which causes E6 and [(1=Lsd )E3 6 (=(j 2Asl )E6 )] to be singular. Hence, one could avoid using the proposed approach1 by simply checking to see if these matrices are ill conditioned. Manuscript received June 24, 1997. The author is with the Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL 32611 USA. Publisher Item Identifier S 0018-926X(98)01503-8. 1 J. Li, IEEE Trans. Antennas Propagat., vol. 41, pp. 379–387, Mar. 1993.
0018–926X/98$10.00 1998 IEEE
