MIMO Zero-Forcing Performance Evaluation Using the Holonomic Gradient Method

For multiple-input-multiple-output (MIMO) spatial-multiplexing transmission, zero-forcing (ZF) detection is appealing because of its low complexity. Our recent MIMO ZF performance analysis for Rician-Rayleigh fading, which is relevant in heterogeneous networks, has yielded for the ZF outage probability and ergodic capacity infinite-series expressions. Because they arose from expanding the confluent hypergeometric function 1F1(·, ·, σ) around 0, they do not converge numerically at realistically high Rician K-factor values. Therefore, herein, we seek to take advantage of the fact that 1F1(·, ·, σ) satisfies a differential equation, i.e., it is a holonomic function. Holonomic functions can be computed by the holonomic gradient method (HGM), i.e., by numerically solving the satisfied differential equation. Thus, we first reveal that the moment generating function (m.g.f.) and probability density function (p.d.f.) of the ZF signal-to-noise ratio (SNR) are holonomic. Then, from the differential equation for 1F1(·, ·, σ), we deduce those satisfied by the SNR m.g.f. and p.d.f. and demonstrate that the HGM helps compute the p.d.f. accurately at practically relevant values of K. Finally, numerical integration of the SNR p.d.f. produced by HGM yields accurate ZF outage probability and ergodic capacity results.