MIMO Zero-Forcing Detection Performance Evaluation by Holonomic Gradient Method

We have recently derived infinite-series expressions for performance measures of multiple-input multipleoutput (MIMO) spatial multiplexing with zero-forcing detection (ZF) for Rician–Rayleigh fading, which is relevant in heterogeneous networks. These expressions ensue from a well-known infinite-series expansion around σ = 0 of the confluent hypergeometric function 1F1(·, ·, σ). Theoretically, this expansion converges for any σ. Numerically, convergence becomes slow with increasing σ. Consequently, our ZF performance-measure expressions diverge numerically at practically-relevant Rician K-factor values. Therefore, herein, we deploy instead the holonomic gradient method (HGM), which computes a function by numerically solving the differential equation it satisfies. HGM is applicable because 1F1(·, ·, σ) is holonomic, i.e., it satisfies a differential equation with polynomial coefficients with respect to σ. First, using properties of holonomic functions, we 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. HGM is shown to yield accurate p.d.f. computation for practically-relevant values of K (at which infinite-series truncation breaks down). Numerical integration of the SNR p.d.f. obtained from HGM yields accurate outage probability and ergodic capacity assessments for MIMO ZF under Rician–Rayleigh fading. Index Terms Confluent hypergeometric function, differential equation, holonomic function, infinite-series, holonomic gradient method, MIMO, numerical convergence, Rayleigh and Rician (Ricean) fading, spatial multiplexing, zero-forcing.