Holonomic Gradient Method-Based CDF Evaluation for the Largest Eigenvalue of a Complex Noncentral Wishart Matrix

Abstract

The outage probability of maximal-ratio combining (MRC) for a multiple-input multiple-output (MIMO) wireless communications system under Rician fading is given by the cumulative distribution function (CDF) for the largest eigenvalue of a complex noncentral Wishart matrix. This CDF has previously been expressed as a determinant whose elements are integrals of a confluent hypergeometric function. For the determinant elements, conventional evaluation approaches, e.g., truncation of infinite series ensuing from the hypergeometric function or numerical integration, can be unreliable and slow even for moderate antenna numbers and Rician $ K $-factor values. Therefore, herein, we derive by hand and by computer algebra also differential equations that are then solved from initial conditions computed by conventional approaches. This is the holonomic gradient method (HGM). Previous HGM-based evaluations of MIMO relied on differential equations that were not theoretically guaranteed to converge, and, thus, yielded reliable results only for few antennas or moderate $ K $. Herein, we reveal that gauge transformations can yield differential equations that are {\emph{stabile}}, i.e., guarantee HGM convergence. The ensuing HGM-based CDF evaluation is demonstrated reliable, accurate, and expeditious in computing the MRC outage probability even for very large antenna numbers and values of $ K $.