Algebraic Geometry Based Design for Generalized Sidelobe Canceler

Generalized sidelobe canceler (GSC) uses a two step procedure in order to produce a beampattern with a fixed mainlobe and suppressed sidelobes. In the first step, a beampattern with a fixed response in the look direction is produced by convolving a vector of constraints with a normalized beamforming vector with the desired mainlobe response. In the second step, the signals in the look direction are blocked out using so-called blocking matrix, while the output power is minimized. Observing that for Griffiths-Jim GSC the beamforming vector contains the coefficients of a polynomial with at least one root at 1, we find here that all rows of a blocking matrix should be the coefficients of polynomials from the polynomial ideal with a root at 1. This allows us to reveal and exploit the underlying algebraic structure for GSC blocking matrix design using methods from computational algebraic geometry. It also allows to arrive to and prove several generalized statements. For example, the necessary and sufficient condition for a signal to be blocked can be easily found. The condition to a row-space of blocking matrix for blocking multiple signals impinging upon the array from multiple directions can also be easily formulated. The linear independence of rows of blocking matrix implies that all the corresponding polynomial share a single root. In general, understanding the algebraic structure that GSC's blocking matrix has to satisfy makes the GSC's design simpler and more intuitive.