Module budget 

The SNR-loss budget of an OFDM receiver (Speth, Fechtel, Fock & Meyr 1999).

In tracking mode every residual imperfection of the inner receiver, a noisy channel estimate, a timing offset, a carrier frequency offset, fading within a symbol, is a rotation of the subcarriers plus an additive noise of power $\sigma^{2}$ (relative to the power of a subcarrier, $E|a_{k}|^{2} = E|H_{k}|^{2} = 1$), and its cost is the loss of SNR it causes, their eq. (14),

$$ \Delta\gamma = \frac{\gamma_{\mathrm{ideal}}}{\gamma_{\mathrm{real}}}, \qquad \sigma_{\mathrm{res}}^{2} = \sigma_{\varepsilon}^{2} + \sigma_{\Omega}^{2} + \sigma_{\mathrm{fad}}^{2} < \frac{1}{\gamma}\left(1 - \frac{1}{\Delta\gamma_{\max}}\right) $$

(eqs. (50)–(51), $\gamma$ the SNR per subcarrier): the noises add, and a loss $\Delta\gamma_{\max}$ allows the total noise_budget. The noises of the imperfections:

• a channel estimate of gain $G$ (its error is the noise divided by $G$, eq. (19)): $\Delta\gamma = 1 + 1/G$, eq. (20), estimator_loss_db and estimator_gain;
• a timing offset of $n_{\varepsilon}$ samples beyond the part of the guard interval that the echoes leave free, eqs. (30)–(31), timing_noise_power;
• a carrier frequency offset $\Delta f T_{u}$ (in subcarrier spacings), eq. (42), cfo_noise_power, $\frac{\pi^{2}}{3}(\Delta f T_{u})^{2}$, and the largest one for a loss, eq. (43), max_cfo;
• fading of maximum Doppler $f_{D}$ within a symbol (Jakes), eq. (47), doppler_noise_power, $\frac{\pi^{2}}{6}(f_{D} T_{u})^{2}$, and the largest Doppler for a loss, max_doppler (the curves of their Fig. 7).

The approximations are those of the paper: one imperfection at a time, small errors (the attenuation of the wanted subcarrier neglected), every interference Gaussian; super::sync::ici_power gives the exact interference of a frequency offset.

Examples

use comnum_waveform::ofdm::budget::{estimator_gain, estimator_loss_db, max_doppler};
// Sec. IV-A: an estimator of gain 10 costs 0.41 dB; 0.1 dB needs a gain of 43
assert!((estimator_loss_db(10.0) - 0.414).abs() < 1e-3);
assert!((estimator_gain(0.1) - 42.9).abs() < 0.05);
// Fig. 7: at 20 dB of SNR, 1 dB of loss tolerates f_D T_u = 0.035
assert!((max_doppler(100.0, 1.0) - 0.0354).abs() < 1e-4);

References

1. M. Speth, S. A. Fechtel, G. Fock and H. Meyr, “Optimum receiver design for wireless broad-band systems using OFDM — Part I,” IEEE Trans. Commun., 47(11), 1668–1677, 1999.

Functions

cfo_noise_power

The power of the intercarrier interference of a small carrier frequency offset eps (in subcarrier spacings), $\frac{\pi^{2}}{3}\varepsilon^{2}$ (eq. (42)): the second-order term of super::sync::ici_power when every subcarrier is active.

doppler_noise_power

The power of the intercarrier interference of fading within a symbol, for a Jakes spectrum of maximum Doppler fd_tu (normalised to the subcarrier spacing, $f_{D} T_{u}$), $\frac{\pi^{2}}{6}(f_{D} T_{u})^{2}$ (eq. (47); taps of unit total power).

estimator_gain

The gain $G = 1/(\Delta\gamma - 1)$ a channel estimator needs for a loss of loss_db (dB): the inverse of estimator_loss_db (their Fig. 2).

estimator_loss_db

The SNR loss (dB) of a receiver that equalises with a channel estimate of gain $G$, $10 \log_{10}(1 + 1/G)$ (eq. (20)).

jakes_ici_sir

The signal-to-interference ratio of CP-OFDM under Jakes’ fading of maximum Doppler fd_tu (normalised to the subcarrier spacing, $f_{D} T_{u}$), the prefix covering the delay spread (no ISI), exact (Robertson & Kaiser 1999; Nissel, Schwarz & Rupp, IEEE J. Sel. Areas Commun. 35(8), 2017, eq. (42)):

max_cfo

The largest carrier frequency offset (in subcarrier spacings) for a loss of loss_db (dB) at the SNR snr_lin (linear, per subcarrier), eq. (43): cfo_noise_power equal to the noise_budget.

max_doppler

The largest maximum Doppler $f_{D} T_{u}$ (normalised to the subcarrier spacing) for a loss of loss_db (dB) at the SNR snr_lin (linear, per subcarrier): the bound of Fig. 7 of Speth et al., doppler_noise_power equal to the noise_budget (max_cfo times $\sqrt{2}$).

noise_budget

The largest total power $\sigma_{\mathrm{res}}^{2}$ of the residual noises (relative to the power of a subcarrier) for a loss of loss_db (dB) at the SNR snr_lin (linear, per subcarrier): $(1 - 1/\Delta\gamma_{\max})/\gamma$ (eqs. (32), (51)).

timing_noise_power

The power of the intersymbol and intercarrier interference of a timing offset (eqs. (30)–(31)): over the taps (delay, power) of the channel (delays in samples, powers of unit sum), $\sum_{i} p_{i},(2\Delta_{i}/N - (\Delta_{i}/N)^{2})$, with $\Delta_{i}$ the samples of the FFT window of $N$ = n_fft points that fall outside the symbol of tap $i$ for a window offset samples late (positive) or early (negative) of the end of the guard interval of guard samples: $\Delta_{i} = n_{\varepsilon} - \tau_{i}$ if $n_{\varepsilon} > \tau_{i}$, $\tau_{i} - N_{g} - n_{\varepsilon}$ if $n_{\varepsilon} < \tau_{i} - N_{g}$, 0 otherwise (the second condition is printed $0 < n_{\varepsilon}T < -(T_{g} - \tau_{i})$ in the paper, which cannot hold for an echo inside the guard interval).