Odd symmetry of weights vector in linearly-constrained adaptive arrays with desired signal
Keywords:adaptive array, recursive least mean squares, RLS, linear constraints, odd symmetry, adaptive filtering algorithm
AbstractThe paper discusses the conditions of an odd symmetry in linearly-constrained least squares criterion adaptive array. It is shown, that the vectors, which optimal weight vector of such adaptive array consists of, and the optimal vector itself, have an odd symmetry, i.e. pairs of symmetrical elements of the vectors are complex-conjugated. To ensure this property, the vector of constrains (radiation pattern values in directions of interest) has to be specified as a real-valued ones. The odd symmetry allows to calculate the weights of the adaptive array in real-valued arithmetic at the cost of two or four times less number of arithmetic operations, comparing with similar calculations, based on complex-valued arithmetics. Adaptive algorithms, based on real-valued arithmetics, provide a 1.5 … 2 times shorter transient response and a 2 … 3 dB deeper notches in the steady-state radiation pattern towards interference sources comparing with complex-valued algorithms.
O. L. Frost, “An algorithm for linearly constrained adaptive array processing,” Proc. IEEE, Vol. 60, No. 8, p. 926-935, 1972. DOI: https://doi.org/10.1109/PROC.1972.8817.
J. A. Apolinario, S. Werner, P. S. R. Diniz, and T. I. Laakso, “Constrained normalized adaptive filters for CDMA mobile communications,” Proc. of 9th European Signal Processing Conf., 8-11 Sept. 1998, Island of Rhodes, Greece. IEEE, 1998, 4 p. URL: http://ieeexplore.ieee.org/document/7089625/.
M. R. L. De Campos and J. A. Apolinario, “The constrained affine projection algorithm – development and convergence issues,” Proc. of the First Balkan Conf. on Signal Processing, Communications, Circuits, and Systems, May 2000, Istanbul, 4 p.
V. I. Djigan, “Joint use of constant modulus and least squares criteria in linearly-constrained communication arrays,” Radioengineering, Vol. 16, No. 4, p. 88-95, 2007. URL: http://www.radioeng.cz/fulltexts /2007/07_04_088_095.pdf.
K.-C. Huarng and C.-C. Yeh, “Adaptive beamforming with conjugate symmetric weights,” IEEE Trans. Antennas Propag., Vol. 39, No. 7, p. 926-932, 1991. DOI: https://doi.org/10.1109/8.86911.
V. I. Djigan, “Algorithms of linearly constrained blind adaptive signal processing in digital arrays with odd symmetry,” Digital Signal Processing, No. 2, p. 3-13, 2015. URL: http://www.dspa.ru/en/2015/dsp- 2015-2.htm.
V. I. Djigan, Adaptive Signal Filtering: Theory and Algorithms [in Russian]. Moscow: Technosphera, 2013, 528 p.
V. I. Djigan, “Adaptive filtering algorithms with quatratized cost function for linearly constrained arrays”, Proc. of IX Int. Conf. on Antennas Theory and Techniques, 16-20 Sept. 2013, Odessa, Ukraine. IEEE, 2013, p. 214-216. DOI: https://doi.org/10.1109/ICATT. 2013.6650729.
A. Cantoni and P. Butler, “Properties of the eigenvectors of persymmetric matrices with applications to communication theory,” IEEE Trans. Commun., Vol. 24, No. 8, p. 804-809, 1976. DOI: https://doi.org/10.1109/TCOM.1976.1093391.
R. Nitzberg, “Application of maximum likelihood estimation of persymmetric covariance matrices to adaptive processing,” IEEE Trans. Aerospace Electronic Systems, Vol. AES-16, No. 1, p. 124-127, 1980. DOI: https://doi.org/10.1109/TAES.1980.308887.
K.-C. Huarng and C. C. Yeh, “A unitary transformation method for angle-of-arrival estimation,” IEEE Trans. Signal Processing, Vol. 39, No. 4, p. 975-977, 1991. DOI: https://doi.org/10.1109/78.80927.
V. I. Djigan, “Multi-beam adaptive antenna array,” Proc. of Southern Federal University, No. 2, p. 23-29, 2012.