Module massive 

The uplink of multi-user massive MIMO: $K$ single-antenna users to a base station of $M$ antennas, with linear receivers (Ngo, Larsson and Marzetta 2013).

Model

The base station receives $y = \sqrt{p_{u}}\, G x + n$, with $G = H D^{1/2}$ the $M \times K$ channel: $H$ the small-scale fading (i.i.d. $\mathcal{CN}(0, 1)$, drawn by comnum_channel::fading::Mimo), $D = \mathrm{diag}(\beta_{k})$ the large-scale fading of the users, $x$ the symbols of unit energy, $n \sim \mathcal{CN}(0, I_{M})$ and $p_{u}$ the normalised transmit SNR of every user. With perfect channel state information (CSI), the receiver knows $G$. With estimated CSI, the users send orthogonal pilots of $\tau \ge K$ symbols at the power $p_{u}$: the MMSE estimate $\hat G$ has independent columns of variance $\tau p_{u} \beta_{k}^{2} / (\tau p_{u} \beta_{k} + 1)$ per entry (Uplink::estimate_variance), and its error, independent of $\hat G$, the variance $\varepsilon_{k} = \beta_{k} / (\tau p_{u} \beta_{k} + 1)$ (Uplink::error_variance). Then $y = \sqrt{p_{u}} \hat G x + n’$ with the noise and estimation error $n’$ of variance $1 + p_{u} \sum_{i} \varepsilon_{i}$ (Uplink::effective_noise_variance). The achievable rate of user $k$ is $\mathbb{E}\, \log_{2}(1 + \mathrm{SINR}{k})$ (eqs. (13), (37)): that of the unbiased estimates of crate::detection::Linear on the channel $\sqrt{p{u}} \hat G$ with the noise variance $1 + p_{u} \sum_{i} \varepsilon_{i}$.

Lower bounds

Uplink::rate_bound gives the closed-form lower bounds on the achievable rate of user $k$ (Propositions 2, 3, 6 and 7), with $p_{p} = \tau p_{u}$:

Receiver	Perfect CSI	Estimated CSI
MR	$\log_{2}\Bigl(1 + \frac{p_{u} (M - 1) \beta_{k}}{p_{u} \sum_{i \ne k} \beta_{i} + 1}\Bigr)$	$\log_{2}\Bigl(1 + \frac{\tau p_{u}^{2} (M - 1) \beta_{k}^{2}}{p_{u} (\tau p_{u} \beta_{k} + 1) \sum_{i \ne k} \beta_{i} + (\tau + 1) p_{u} \beta_{k} + 1}\Bigr)$
ZF	$\log_{2}(1 + p_{u} (M - K) \beta_{k})$	$\log_{2}\Bigl(1 + \frac{\tau p_{u}^{2} (M - K) \beta_{k}^{2}}{(\tau p_{u} \beta_{k} + 1) \sum_{i} \frac{p_{u} \beta_{i}}{\tau p_{u} \beta_{i} + 1} + \tau p_{u} \beta_{k} + 1}\Bigr)$

The bound of MR needs $M \ge 2$, that of ZF $M \ge K + 1$. As $M \to \infty$, with $p_{u} = E_{u} / M$ and perfect CSI the rate tends to $\log_{2}(1 + \beta_{k} E_{u})$, and with $p_{u} = E_{u} / \sqrt{M}$ and estimated CSI to $\log_{2}(1 + \tau \beta_{k}^{2} E_{u}^{2})$ (Propositions 1 and 5): doubling $M$ saves 3 dB of transmit power with perfect CSI and 1.5 dB with estimated CSI.

Examples

use comnum_mimo::detection::Criterion;
use comnum_mimo::massive::Uplink;
// one user, perfect CSI: the MR and ZF bounds are log2(1 + p_u (M - 1) β)
let up = Uplink::new(8, vec![0.5], 2.0)?;
assert!((up.rate_bound(Criterion::Mr, 0)? - 8f64.log2()).abs() < 1e-12);
assert!((up.rate_bound(Criterion::Zf, 0)? - 8f64.log2()).abs() < 1e-12);

References

1. H. Q. Ngo, E. G. Larsson and T. L. Marzetta, “Energy and spectral efficiency of very large multiuser MIMO systems,” IEEE Trans. Commun., 2013 (arXiv 1112.3810).

Structs

Uplink

The uplink of $K$ users of large-scale fading large_scale to antennas antennas, at the normalised transmit SNR snr_lin of every user (see the module).