Module noma 

Downlink non-orthogonal multiple access (NOMA) with users dropped at random in a disc (Ding, Yang, Fan & Poor 2014): superposition coding at the base station, successive interference cancellation (SIC) at the users, outage probabilities in closed form.

$M$ users lie uniformly in a disc of radius $\mathcal{R}{D}$ around the base station; the channel of a user at the distance $d$ is $h = g/\sqrt{1 + d^{\alpha}}$ with $g \sim \mathcal{CN}(0, 1)$. The unordered gain $|h|^{2}$ has the CDF (eq. (3)) $$F(y) = \frac{2}{\mathcal{R}{D}^{2}}\int_{0}^{\mathcal{R}{D}}\left(1 - e^{-(1 + z^{\alpha})y}\right) z, dz$$ (RandomDisc::gain_cdf, by Gauss–Legendre with 256 nodes), approximated in the paper by Gauss–Chebyshev with $N$ nodes (eq. (4), RandomDisc::gain_cdf_chebyshev). With the gains ordered $|h{1}|^{2} \le \dots \le |h_{M}|^{2}$, the power coefficients $a_{1} \ge \dots \ge a_{M}$ and the target rates $\tilde R_{m}$, $\phi_{j} = 2^{\tilde R_{j}} - 1$, user $m$ decodes the messages $1..m$ when $|h_{m}|^{2} \ge \psi_{m}^{} = \max_{j \le m}\psi_{j}$, $\psi_{j} = \phi_{j}/(\rho(a_{j} - \phi_{j}\sum_{i > j} a_{i}))$ and $\psi_{M} = \phi_{M}/(\rho a_{M})$ (eqs. (7)–(8); no outage threshold when $a_{j} \le \phi_{j}\sum_{i > j}a_{i}$: outage 1). The outage of user $m$ is that of the $m$-th order statistic (eq. (9)), $P_{m} = I_{F(\psi_{m}^{})}(m, M - m + 1)$ (NomaDownlink::outage), and at high SNR (eq. (11)) $P_{m} \approx \frac{\tau_{m}}{m}\eta^{m}(\psi_{m}^{*})^{m}$, $\tau_{m} = \frac{M!}{(m-1)!(M-m)!}$, $\eta = \frac{1}{\mathcal{R}{D}}\sum{n}\beta_{n}$ (NomaDownlink::outage_high_snr): user $m$ has the diversity order $m$.

Examples

use comnum_mimo::noma::{NomaDownlink, RandomDisc};
// Fig. 3 of Ding et al., case 1: 80 m, alpha = 3, rates 0.1 and 0.5, rho = 80 dB
let disc = RandomDisc { radius: 80.0, alpha: 3.0 };
let noma = NomaDownlink::new(vec![0.8, 0.2], vec![0.1, 0.5])?;
let p = noma.outage_high_snr(&disc, 1e8, 10)?;
assert!((p[0] / 3.781e-4 - 1.0).abs() < 1e-3 && (p[1] / 0.1837e-4 - 1.0).abs() < 1e-3, "{p:?}");

References

1. Z. Ding, Z. Yang, P. Fan, H. V. Poor, “On the performance of non-orthogonal multiple access in 5G systems with randomly deployed users,” IEEE Signal Process. Lett. 21(12), 1501–1505, 2014 (arXiv:1406.1516): eqs. (3)–(11), Figs. 1, 3.

Structs

NomaDownlink

A downlink NOMA policy: the power coefficients $a_{1} \ge \dots \ge a_{M}$ of the users ordered by increasing gain and their target rates $\tilde R_{m}$ (bit per channel use).

RandomDisc

Users dropped uniformly in a disc around the base station, path loss $1 + d^{\alpha}$, Rayleigh fading.

Functions

orthogonal_outage

The outage probability of a user scheduled at random on an orthogonal resource at the transmit SNR snr (linear) and the rate rate (bit per channel use): $F((2^{R} - 1)/\rho)$ (the “conventional MA” of Ding et al. 2014, Figs. 1 and 3).