Module sync 

Timing and carrier-frequency synchronisation of OFDM (Schmidl & Cox 1997).

The preamble is an OFDM symbol whose odd subcarriers are empty: its useful part is two identical halves of $L = LN/2$ samples. With the received samples $r$, the correlation and the energy of the second half,

$$ P(d) = \sum_{m=0}^{L-1} r^{*}[d + m], r[d + m + L], \qquad R(d) = \sum_{m=0}^{L-1} |r[d + m + L]|^{2}, \qquad M(d) = \frac{|P(d)|^{2}}{R(d)^{2}}, $$

the timing metric $M(d)$ reaches a plateau of the length of the cyclic prefix at the start of the preamble; the phase of $P$ there gives the fractional carrier frequency offset, in units of the subcarrier spacing, $\hat\varepsilon = \angle P(\hat d) / \pi$ (an offset of $\varepsilon$ rotates the second half by $\pi\varepsilon$).

schmidl_cox returns the metric; synchronize the timing (the middle of the points above PLATEAU_FRACTION of the peak, which lies in the cyclic prefix when data follow the preamble) and the offset, the phase of the correlation $P$ that the metric was computed from.

A repeated symbol (Moose 1994)

When a block of $N$ = Ofdm::fft_len samples is sent twice in a row (repeated_preamble), the DFTs $Y_{1k}$, $Y_{2k}$ of the two received blocks differ by the phase $2\pi\varepsilon$ of the offset alone, and the maximum-likelihood estimate is (Moose, eq. (21))

$$ \hat\varepsilon = \frac{1}{2\pi} \angle \sum_{k} Y_{2k} Y_{1k}^{*}, \qquad |\varepsilon| < \tfrac{1}{2}, $$

over the active subcarriers (moose); for small errors it is unbiased, of variance $1/((2\pi)^{2} E_{s}/N_{0})$ with $E_{s}$ the energy of one block (eq. (26), moose_std). An offset leaves on subcarrier $k$ the fraction $\operatorname{diric}(2\pi\varepsilon/N, N)^{2}$ of its power and adds the interference of the others (eq. (10), ici_power):

$$ \frac{E|I_{k}|^{2}}{|X|^{2}|H|^{2}} = \sum_{l \ne k} \left(\frac{\sin \pi\varepsilon}{N \sin(\pi(l - k + \varepsilon)/N)}\right)^{2} \le 0.5947 \sin^{2}\pi\varepsilon \quad (|\varepsilon| \le \tfrac{1}{2}), $$

hence the lower bound of the SNR after the DFT (eq. (15), cfo_snr_bound), $\mathrm{SNR}(\varepsilon) \ge \mathrm{SNR}, (\sin\pi\varepsilon/\pi\varepsilon)^{2} / (1 + 0.5947, \mathrm{SNR} \sin^{2}\pi\varepsilon)$.

The cyclic prefix (van de Beek, Sandell & Börjesson 1997)

Without a preamble, the $L$ samples of the prefix and their copies $N$ samples later carry the timing and the offset. Over a window of $2N + L$ samples that contains one whole symbol, the joint maximum-likelihood estimates are (their eqs. (6)–(8), (12)–(13))

$$ \hat\theta = \arg\max_{\theta} \left{ |\gamma(\theta)| - \rho, \Phi(\theta) \right}, \qquad \hat\varepsilon = -\frac{1}{2\pi} \angle \gamma(\hat\theta), $$

with $\gamma(\theta) = \sum_{k=\theta}^{\theta+L-1} r(k), r^{*}(k + N)$, the energy $\Phi(\theta) = \frac{1}{2} \sum_{k=\theta}^{\theta+L-1} |r(k)|^{2} + |r(k+N)|^{2}$ and $\rho = \mathrm{SNR}/(\mathrm{SNR} + 1)$ (cp_ml: $\hat\theta$ is the first sample of the prefix).

Examples

use comnum_waveform::ofdm::Ofdm;
use comnum_waveform::ofdm::sync::{preamble, synchronize};
use comnum_core::rng::Rng;
use comnum_math::{cis, C64};
let ofdm = Ofdm { n_cp: 16, ..Ofdm::new(64, 52)? };
let p = preamble(&ofdm, 1)?;                               // with its prefix
// silence, the preamble, then a data symbol (random QPSK)
let mut r = Rng::new(4);
let data: Vec<C64> = (0..52).map(|_| cis(std::f64::consts::FRAC_PI_2 * r.below(4) as f64)).collect();
let mut rx = vec![C64::new(0.0, 0.0); 100];
rx.extend(&p);
rx.extend(ofdm.modulate(&data)?);
// an offset of 0.2 subcarrier spacing
let mut rx = rx;
comnum_dsp::signal::rotate(&mut rx, std::f64::consts::TAU * 0.2 / ofdm.fft_len() as f64, 0.0);
let e = synchronize(&rx, &ofdm)?;
assert!((e.cfo - 0.2).abs() < 1e-6);
assert!(e.timing >= 100 && e.timing <= 100 + 4 * 16);       // within the prefix

References

1. T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun., 45(12), 1613–1621, 1997.
2. P. H. Moose, “A technique for orthogonal frequency division multiplexing frequency offset correction,” IEEE Trans. Commun., 42(10), 2908–2914, 1994.
3. J.-J. van de Beek, M. Sandell and P. O. Börjesson, “ML estimation of time and frequency offset in OFDM systems,” IEEE Trans. Signal Process., 45(7), 1800–1805, 1997.
4. J. Armstrong, “Analysis of new and existing methods of reducing intercarrier interference due to carrier frequency offset in OFDM,” IEEE Trans. Commun., 47(3), 365–369, 1999.

Structs

Estimate

A synchronisation estimate.

Constants

MOOSE_ICI_BOUND

The constant of Moose’s bound of the interference, eq. (13): $E|I_{k}|^{2} \le 0.5947 |X|^{2}|H|^{2} \sin^{2}\pi\varepsilon$ for $|\varepsilon| \le 1/2$.

PLATEAU_FRACTION

The fraction of the peak of the metric above which synchronize counts the points of its plateau (Schmidl & Cox 1997 average the two times, on either side of the maximum, at 90 % of it).

Functions

cfo_coefficient

The weight $c_{d}$ with which, under a frequency offset eps (in subcarrier spacings), the symbol of the subcarrier $d$ positions away reaches a subcarrier after the DFT of $N$ = n points (Armstrong 1999, eq. (8); Moose 1994, eq. (8)):

cfo_snr_bound

Moose’s lower bound of the SNR of a subcarrier after the DFT under a frequency offset eps (in subcarrier spacings), for an SNR snr_lin (linear, $E_{c}/N_{0}$; infinite for the interference alone), eq. (15): $\mathrm{SNR},(\sin\pi\varepsilon/\pi\varepsilon)^{2} / (1 + 0.5947, \mathrm{SNR} \sin^{2}\pi\varepsilon)$ (linear).

cp_ml

The joint maximum-likelihood timing and frequency offset of van de Beek, Sandell & Börjesson (1997) from the cyclic prefix, for an SNR snr_lin (linear) per sample: over every start $\theta$ such that the prefix and its copy lie in r (a window of $2N + L$ samples that contains one whole symbol), the maximum of $|\gamma(\theta)| - \rho\Phi(\theta)$ (delayed_correlation and windowed_energy); the timing is the first sample of the prefix.

ici_power

The power of the interference that a frequency offset eps (in subcarrier spacings) brings onto the active subcarrier i from the other active subcarriers, relative to the power of one subcarrier: Moose’s eq. (10), the sum of $|c_{l-k}|^{2}$ (cfo_coefficient) over the other active subcarriers $l$.

moose

The frequency offset (in subcarrier spacings, $|\varepsilon| < 1/2$) of a block repeated at start and start + N in r, Moose’s eq. (21) over the active subcarriers (Ofdm::to_active).

moose_std

The standard deviation of Moose’s estimate for small errors, $1/(2\pi\sqrt{E_{s}/N_{0}})$, with $E_{s}/N_{0}$ (linear) the energy of one block over the noise density (eq. (26)).

preamble

The preamble of Schmidl & Cox (with its cyclic prefix): QPSK on the even active subcarriers (scaled by $\sqrt{2}$ to keep the unit power), nothing on the odd ones.

repeated_preamble

The preamble of Moose (with its cyclic prefix): one block of QPSK on every active subcarrier, sent twice in a row, $L(N_{\mathrm{cp}} + 2N)$ samples.

schmidl_cox

The timing metric $M(d)$ for every start $d$ (halves of half samples), on delayed_correlation and windowed_energy.

synchronize

Estimate the timing and the frequency offset of the preamble of ofdm in r.