An Inhomogeneous Plane-Wave Based Single-Level Fast Direct Solver for the Scattering Analysis of Extremely Large Antenna Arrays

Abstract

A semianalytic single-level fast direct solver (FDS) is presented for the method-of-moments (MoM) analysis of large planar antenna arrays with up to few thousands of complex elements on current desktop computers. Assuming a partition of the impedance matrix, the proposed scheme factorizes the near-field blocks via a singular value decomposition (SVD)-based low-rank approximation and the intermediate-field blocks via an analytical inhomogeneous plane-wave (IPW) expansion. The near- and intermediate-field bases are then unified to leverage the Woodbury formula for the inversion. The solutions for the excitation of each array ports are obtained at once with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(N^{15/7} \ln ^{3} N)$ </tex-math></inline-formula> computational cost and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(N^{11/7} \ln N)$ </tex-math></inline-formula> memory for array problems with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> basis functions. The complexities are traced precisely and the error is controlled by means of analytical formulas for the required number of IPWs. The FDS is also combined with a macrobasis function (MBF) approach to deal with antenna geometries containing many subwavelength details. This article includes simulations of regular arrays of about <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$169^{2}~3$ </tex-math></inline-formula> -D dipoles and an irregular array composed of seven stations of the low-frequency square kilometer array (SKA) telescope comprising each 256 log-periodic (LP) antennas. The proposed FDS allows the analysis of the coupling effect both within any one station but also between the stations of the telescope, a size and complexity of problem which has not been possible with existing published techniques.