Module estimation 

Training-based estimation of a flat MIMO channel: the least-squares, scaled least-squares, LMMSE and relaxed MMSE estimators, their mean square errors, the optimal training matrices and the combination of several training periods (Biguesh & Gershman 2006).

Model

An $r \times t$ channel $H$ is probed by $N \ge t$ training vectors, the columns of the $t \times N$ matrix $P$ of power $\lVert P \rVert_{F}^{2} = \mathcal{P}$:

$$ S = H P + V, $$

$V$ white of variance $\sigma_{n}^{2}$ per entry (eq. (2)). The prior of $H$ is its correlation $R_{H} = E\{H^{H} H\}$ ($t \times t$, footnote 2: not that of $\mathrm{vec}(H)$), the rows of $H$ uncorrelated. Every estimator is linear, $\hat{H} = S A$ with an $N \times t$ matrix $A$ (ChannelEstimator):

Estimator	$A$	Needs	Eq.
LS	$P^{\dagger} = P^{H} (P P^{H})^{-1}$	—	(4)
scaled LS	$\gamma_{o} P^{\dagger}$, $\gamma_{o} = \frac{\mathrm{tr}\{R_{H}\}}{J_{\mathrm{LS}} + \mathrm{tr}\{R_{H}\}}$	$\mathrm{tr}\{R_{H}\}$, $\sigma_{n}^{2}$	(14), (16)
LMMSE	$(P^{H} R_{H} P + \sigma_{n}^{2} r I)^{-1} P^{H} R_{H}$	$R_{H}$, $\sigma_{n}^{2}$	(25), (26)
relaxed MMSE	the LMMSE of $R_{H} = \frac{\mathrm{tr}\{R_{H}\}}{t} I$	$\mathrm{tr}\{R_{H}\}$, $\sigma_{n}^{2}$	(47), (48)

Scaled LS and relaxed MMSE take $\mathrm{tr}\{R_{H}\}$ either known or estimated by $\mathrm{tr}\{\hat{H}{\mathrm{LS}}^{H} \hat{H}{\mathrm{LS}}\}$ (eq. (17), the LS-SLS and LS-RMMSE estimators; TracePrior).

Mean square error

With $B = I - P A$, the error $H - \hat{H} = H B - V A$ has

$$ J = E\{\lVert H - \hat{H} \rVert_{F}^{2}\} = \mathrm{tr}\{B^{H} R_{H} B\} + \sigma_{n}^{2} r \, \mathrm{tr}\{A^{H} A\}, $$

the closed form Training::mse for every estimator whose prior is known, also when the prior of the estimator is not the true $R_{H}$ (a mismatch). It gives $J_{\mathrm{LS}} = \sigma_{n}^{2} r \, \mathrm{tr}\{(P P^{H})^{-1}\}$ (7), $J_{\mathrm{SLS}} = \frac{J_{\mathrm{LS}} \mathrm{tr}\{R_{H}\}}{J_{\mathrm{LS}} + \mathrm{tr}\{R_{H}\}}$ (15) and $J_{\mathrm{MMSE}} = \mathrm{tr}\{(R_{H}^{-1} + \frac{1}{\sigma_{n}^{2} r} P P^{H})^{-1}\}$ (28). LS-SLS and LS-RMMSE have no closed form (a Monte Carlo).

Training

Orthogonal rows, $P P^{H} = \frac{\mathcal{P}}{t} I$ (9), minimise the MSE of LS, scaled LS and relaxed MMSE; Training::orthogonal is the DFT matrix of eq. (10), of entries of equal magnitude. For the LMMSE, the optimal training Training::mmse_optimal water-fills the power on the eigenvectors of $R_{H} = Q \Lambda Q^{H}$: $P = \sqrt{\sigma_{n}^{2} r}\, Q \,[\mathrm{diag}(p_{i})^{1/2}, 0]$, $p_{i} = (\mu_{o} - 1/\lambda_{i})^{+}$, $\sum_{i} p_{i} = \mathcal{P}/(\sigma_{n}^{2} r)$ (eqs. (41)–(44), the unitary $U$ of (44) the identity).

Several training periods

The LS estimates $\hat{H}{k}$ of $K$ periods of training $P{k}$, the channel unchanged, are combined by the best linear unbiased estimator blue_combine of weights blue_weights (Lemma 4, eq. (54)):

$$ \alpha_{k} = \frac{1 / \kappa_{k}}{\sum_{l} 1 / \kappa_{l}}, \qquad \kappa_{k} = \mathrm{tr}\{(P_{k} P_{k}^{H})^{-1}\}, $$

$\alpha_{k} = \mathcal{P}{k}/\mathcal{P}{\mathrm{tot}}$ for orthogonal training (58), of MSE blue_mse $\sigma_{n}^{2} r / \sum_{l} 1/\kappa_{l}$ (eqs. (59), (61)).

Examples

use comnum_mimo::estimation::{ChannelEstimator, Training};
use comnum_math::linalg::{identity, matmul};
let (t, r, power, noise) = (4, 2, 10.0, 0.5);
let p = Training::orthogonal(t, t, power)?;
let r_h = identity(t) * comnum_math::C64::from(r as f64);
// LS: sigma^2 t^2 r / P, eq. (12)
let j = p.mse(&ChannelEstimator::Ls, &r_h.view(), noise, r)?;
assert!((j - noise * (t * t * r) as f64 / power).abs() < 1e-12);
// a noiseless observation is estimated exactly by LS
let h = comnum_math::linalg::CMat::from_elem((r, t), comnum_math::C64::new(0.3, -1.0));
let s = matmul(&h.view(), &p.symbols().view());
let h_ls = p.estimate(&s.view(), &ChannelEstimator::Ls, noise)?;
assert!((&h_ls - &h).iter().all(|e| e.norm() < 1e-12));

References

1. M. Biguesh and A. B. Gershman, “Training-based MIMO channel estimation: a study of estimator tradeoffs and optimal training signals,” IEEE Trans. Signal Process., 54(3), 884–893, 2006.

Structs

Training

A training matrix $P$ ($t \times N$, see the module).

Enums

ChannelEstimator

A linear estimator $\hat{H} = S A$ of the channel (see the module).

TracePrior

The trace $\mathrm{tr}\{R_{H}\}$ used by the scaled LS and relaxed MMSE estimators.

Functions

blue_combine

The combination $\sum_{k} \alpha_{k} \hat{H}_{k}$ of the estimates estimates with the weights weights (eq. (52)).

blue_mse

The MSE $\sigma_{n}^{2} r \sum_{k} \alpha_{k}^{2} \kappa_{k} = \sigma_{n}^{2} r / \sum_{k} 1/\kappa_{k}$ of the BLUE combination of the LS estimates of trainings (eqs. (59), (61)); $\sigma_{n}^{2} t^{2} r / \mathcal{P}_{\mathrm{tot}}$ for orthogonal trainings.

blue_weights

The weights $\alpha_{k}$ of the best linear unbiased combination of the LS estimates of the training periods trainings (Lemma 4, eq. (54)); $\mathcal{P}{k} / \mathcal{P}{\mathrm{tot}}$ for orthogonal trainings (58).