Module wiener 

Two-dimensional Wiener interpolation of pilots in time and frequency (Hoeher, Kaiser & Robertson 1997).

The channel $h_{l,k}$ of symbol $l$ and subcarrier $k$ is observed at the pilots $\mathcal{P}$, $\tilde{h}{l’,k’} = h{l’,k’} + n_{l’,k’}$ (the LS estimates, noise of variance $\sigma^{2}/E_{p}$), and every position is estimated by a linear filter of the $N_{\mathrm{tap}}$ pilots $\mathcal{T}(l, k) \subset \mathcal{P}$ nearest to it, their eqs. (2) and (9),

$$ \hat{h}{l,k} = \underline{w}^{T} \underline{\tilde{h}}, \qquad \underline{w}^{T} = \underline{\theta}^{T}, \Phi^{-1}, \qquad J{o}(l, k) = 1 - \underline{\theta}^{T} \Phi^{-1} \underline{\theta}^{*}, $$

with $\theta = E[h_{l,k} \tilde{h}^{*}]$ over the taps and $\Phi = E[\tilde{h} \tilde{h}^{H}] = R + \frac{\sigma^{2}}{E_{p}} I$ (eqs. (6)–(7), (12)): the linear MMSE of comnum_dsp::estimation::lmmse on the taps. For a WSSUS channel the correlation is separable (eq. (15)),

$$ R(\Delta l, \Delta k) = r_{t}(\Delta l), r_{f}(\Delta k), $$

the time correlation $r_{t}$ of the Doppler spectrum at the symbol period $T_{s}$ (uniform: $\mathrm{si}(2\pi f_{D} T_{s} \Delta l)$; Jakes: $J_{0}(2\pi f_{D} T_{s} \Delta l)$) and the frequency correlation $r_{f}$ of the delay power spectrum at the subcarrier spacing $\Delta F$ (uniform on $[-\tau_{\max}, \tau_{\max}]$: $\mathrm{si}(2\pi \tau_{\max} \Delta F \Delta k)$, or a PowerDelayProfile). The taps are the pilots nearest in the weighted distance $|\Delta l| f_{D} T_{s} + |\Delta k| \tau \Delta F$ (their Sec. 5, method B), ties broken by the smaller time distance then the pilot order. With pilots in one symbol only, the filter is the 1-D Wiener filter in frequency of their Fig. 2.

The sampling theorem of the grid (their eq. (17)): pilots every $N_{t}$ symbols and $N_{f}$ subcarriers resolve the channel if $f_{D} T_{s} N_{t} \le 1/2$ and $\tau_{\max} \Delta F N_{f} \le 1/2$ (two-sided spreads), $1/4$ for a twofold oversampling.

Examples

use comnum_waveform::ofdm::wiener::{Correlation2d, DelaySpectrum, DopplerSpectrum, Wiener2d};
// pilots every 3 symbols and 6 subcarriers on a block of 12 x 24
let pilots: Vec<(usize, usize)> =
    (0..12).step_by(3).flat_map(|l| (0..24).step_by(6).map(move |k| (l, k))).collect();
let corr = Correlation2d { fd_ts: 0.0832, doppler: DopplerSpectrum::Uniform, delay: DelaySpectrum::Uniform { tau_df: 0.0417 } };
let w = Wiener2d::new(pilots, corr, 0.01, 10)?;
let mse = w.mse(&[(1, 2), (0, 0)])?;
assert!(mse[1] < 0.01 && mse[0] < 0.05);

The robust estimator of Li, Cimini & Sollenberger (1998)

An estimator designed for the worst channel of maximum Doppler $f_{\max}$ and delay spread $t_{\max}$, a flat band-limited Doppler spectrum and a uniform delay profile on the $K_{o} = \lfloor K t_{\max}/T_{s} \rfloor + 1$ taps of the DFT grid between 0 and $t_{\max}$ (their eq. (56) prints $\lceil K t_{\max}/T_{s} \rceil$, one tap fewer when $K t_{\max}/T_{s}$ is an integer; their figures follow the count of the taps), filters every kept tap in time with the temporal Wiener filter of the correlation $\mathrm{sinc}(n\omega_{d}/\pi)$, $\omega_{d} = 2\pi f_{\max} T_{f}$ (block period $T_{f}$). Its MSE, normalised to the channel power, is for an infinite filter (eq. (57))

$$ \overline{\mathrm{MMSE}} = \frac{K_{o}\rho}{K}\left(1 - \left[\frac{\pi K}{K_{o} \rho \omega_{d}} + 1\right]^{-\omega_{d}/\pi}\right), $$

$\rho$ the noise to signal ratio, and for $L$ taps $\overline{\mathrm{MSE}} = K_{o} c_{0} \rho / K$ with $(c_{0}, \dots, c_{L-1})^{T} = (\bar{R}{t} + \frac{K{o}\rho}{K} I)^{-1} \bar{r}_{t}$ (eqs. (69)–(74)): RobustDesign::mse and RobustDesign::fir. It bounds the MSE of every channel within the design (their eqs. (34)–(36)).

References

1. P. Hoeher, S. Kaiser and P. Robertson, “Two-dimensional pilot-symbol-aided channel estimation by Wiener filtering,” Proc. IEEE ICASSP, vol. 3, 1845–1848, 1997.
2. Y. Li, L. J. Cimini and N. R. Sollenberger, “Robust channel estimation for OFDM systems with rapid dispersive fading channels,” IEEE Trans. Commun., 46(7), 902–915, 1998.

Structs

Correlation2d

The separable time-frequency correlation $r_{t}(\Delta l), r_{f}(\Delta k)$ of a WSSUS channel of unit power (see the module).

RobustDesign

The robust estimator of Li, Cimini & Sollenberger (1998) for $K$ tones (see the module): the design delay spread spans $K_{o}$ taps of the DFT grid, the design Doppler is $\omega_{d} = 2\pi f_{\max} T_{f}$.

Wiener2d

The 2-D Wiener interpolator of a set of pilots (see the module).

WienerPlan

The Wiener2d estimator of a set of positions, one factorisation per set of taps (Wiener2d::plan).

Enums

DelaySpectrum

The delay power spectrum of the channel.

DopplerSpectrum

The Doppler spectrum of every path, of maximum Doppler $f_{D}$.