We show that the distribution of the scalar Schur complement in a noncentral Wishart matrix is a mixture of central chi-square distributions with different degrees of freedom. For the case of a rank-1 noncentrality matrix, the weights of the mixture
representation arise from a noncentral beta mixture of Poisson distributions.
We study zero-forcing detection (ZF) for multiple-input/multiple-output (MIMO) spatial multiplexing under transmit-correlated Rician fading for an N_R X N_T channel matrix with rank-1 line-of-sight (LoS) component. By using matrix transformations and
multivariate statistics, our exact analysis yields the signal-to-noise ratio moment generating function (m.g.f.) as an infinite series of gamma distribution m.g.f.s and analogous series for ZF performance measures, e.g., outage probability and ergodic capacity. However, their numerical convergence is inherently problematic with increasing Rician K-factor, N_R , and N_T. We circumvent this limitation as follows. First, we derive differential equations satisfied by the performance measures with a novel automated approach employing a computer-algebra tool which implements Groebner basis computation and creative telescoping. These differential equations are then solved with the holonomic gradient method (HGM) from initial conditions computed with the infinite series. We demonstrate that HGM yields more reliable performance evaluation than by infinite series alone and more expeditious than by simulation, for realistic values of K , and even for N_R and N_T relevant to large MIMO systems. We envision extending the proposed approaches for exact analysis and reliable evaluation to more general Rician fading and other transceiver methods.
For multiple-input multiple-output (MIMO) spatial-multiplexing transmission, zero-forcing detection (ZF) is appealing because of its low complexity. Our recent MIMO ZF performance analysis for Rician--Rayleigh fading, which is relevant in heterogeneo
us networks, has yielded for the ZF outage probability and ergodic capacity infinite-series expressions. Because they arose from expanding the confluent hypergeometric function $ {_1! F_1} (cdot, cdot, sigma) $ 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 $ {_1! F_1} (cdot, cdot, sigma) $ satisfies a differential equation, i.e., it is a textit{holonomic} function. Holonomic functions can be computed by the textit{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 $ {_1! F_1} (cdot, cdot, sigma) $, 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.
For multiple-input/multiple-output (MIMO) spatial multiplexing with zero-forcing detection (ZF), signal-to-noise ratio (SNR) analysis for Rician fading involves the cumbersome noncentral-Wishart distribution (NCWD) of the transmit sample-correlation
(Gramian) matrix. An textsl{approximation} with a textsl{virtual} CWD previously yielded for the ZF SNR an approximate (virtual) Gamma distribution. However, analytical conditions qualifying the accuracy of the SNR-distribution approximation were unknown. Therefore, we have been attempting to exactly characterize ZF SNR for Rician fading. Our previous attempts succeeded only for the sole Rician-fading stream under Rician--Rayleigh fading, by writing it as scalar Schur complement (SC) in the Gramian. Herein, we pursue a more general, matrix-SC-based analysis to characterize SNRs when several streams may undergo Rician fading. On one hand, for full-Rician fading, the SC distribution is found to be exactly a CWD if and only if a channel-mean--correlation textsl{condition} holds. Interestingly, this CWD then coincides with the textsl{virtual} CWD ensuing from the textsl{approximation}. Thus, under the textsl{condition}, the actual and virtual SNR-distributions coincide. On the other hand, for Rician--Rayleigh fading, the matrix-SC distribution is characterized in terms of determinant of matrix with elementary-function entries, which also yields a new characterization of the ZF SNR. Average error probability results validate our analysis vs.~simulation.
We analyze the performance of multiple input/multiple output (MIMO) communications systems employing spatial multiplexing and zero-forcing detection (ZF). The distribution of the ZF signal-to-noise ratio (SNR) is characterized when either the intende
d stream or interfering streams experience Rician fading, and when the fading may be correlated on the transmit side. Previously, exact ZF analysis based on a well-known SNR expression has been hindered by the noncentrality of the Wishart distribution involved. In addition, approximation with a central-Wishart distribution has not proved consistently accurate. In contrast, the following exact ZF study proceeds from a lesser-known SNR expression that separates the intended and interfering channel-gain vectors. By first conditioning on, and then averaging over the interference, the ZF SNR distribution for Rician-Rayleigh fading is shown to be an infinite linear combination of gamma distributions. On the other hand, for Rayleigh-Rician fading, the ZF SNR is shown to be gamma-distributed. Based on the SNR distribution, we derive new series expressions for the ZF average error probability, outage probability, and ergodic capacity. Numerical results confirm the accuracy of our new expressions, and reveal effects of interference and channel statistics on performance.
