Notes on design of transformation-based reduced-rank generalized sidelobe canceller for matrix LCMV filter

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This paper is concerned with the design of transformation-based reduced-rank generalized sidelobe canceller (RRGSC) for the linearly constrained minimum variance (LCMV) filter. Most RRGSCs introduced in the previous literature were limited to vector form, and their performances were not compared with the LCMV filter. In order to overcome these drawbacks, we present a detailed discussion about the relationship between the general matrix LCMV filter and the transformation-based RRGSC, and obtain several new results. First, we study the effect of transformation matrix on the performance of RRGSC, and find the optimal design of the transformation matrix that can minimize the performance loss of RRGSC relative to the LCMV filter. Second, we obtain a sufficient and necessary condition for the performance equivalence between RRGSC and LCMV filter. Third, we obtain the minimum loss-free reduced-rank of RRGSC, and formulate the relationship between the reduced-rank and the minimum performance loss of RRGSC relative to LCMV filter. Application cases and computer simulations are provided to demonstrate the usefulness and correctness of the above-mentioned theoretical results.