Module correction 

Pilot-based gain correction in the presence of a cochannel interferer (Cimini 1985, Sec. II-C to II-F).

Model

Every sample of the desired signal $D$ (unit power) reaches the receiver through a flat Rayleigh gain $Z_{d} = A_{d} e^{j\theta_{d}}$, $\mathbb{E} A_{d}^{2} = 1$, with a cochannel interferer $I$ of independent gain $Z_{i}$, $\gamma = 1/\mathrm{SIR}$:

$$ R = Z_{d} D + \sqrt{\gamma}, Z_{i} I . $$

A pilot gives the complex envelope $Z_{p}$, equal to $Z_{d}$ in flat fading and correlated with it, $\mathbb{E}[Z_{d} Z_{p}^{*}] = \rho, e^{j\phi}$, when the pilot is $s$ apart in frequency. The correction $\hat{D} = R / Z_{c}$ either follows the fades only down to a limit $\varepsilon$ (amplitude relative to the RMS gain), $Z_{c} = \max(A_{p}, \varepsilon), e^{j\theta_{p}}$ (gain_limited, eq. (16)), or uses the optimum factor $Z_{c} = Z_{d}\left[1 + \gamma A_{i}^{2} / A_{d}^{2}\right]$ of eq. (15), which needs the amplitude of the interferer. Unlimited correction ($\varepsilon = 0$) boosts the interferer in the deep fades of the desired signal.

The signal-to-distortion ratio $\mathrm{SDR} = 1 / \mathbb{E}|\hat{D} - D|^{2}$ is, for gain-limited correction (eq. (25), and eq. (20) for $\rho = 1$, flat fading),

$$ \mathrm{SDR}^{-1} = \frac{\mathrm{SIR} + 1}{\mathrm{SIR}}, \frac{1 - e^{-\varepsilon^{2}}} {\varepsilon^{2}} + 1 - \frac{\rho \sqrt{\pi} \cos\phi}{\varepsilon} \operatorname{erf}(\varepsilon) + \left(1 - \rho^{2} + \frac{1}{\mathrm{SIR}}\right) E_{1}(\varepsilon^{2}), $$

sdr_gain_limited, and for the optimum correction (eq. (21)) $\mathrm{SDR}^{-1} = \frac{1}{\mathrm{SIR} - 1}\left(\frac{\mathrm{SIR}}{\mathrm{SIR} - 1} \ln \mathrm{SIR} - 1\right)$, sdr_optimum.

Three points of the paper do not hold as printed, and are corrected here:

• eq. (19) lacks the $-1$ of the first term, $\overline{(A_{d} \min(1/A_{d}, 1/\varepsilon) - 1)^{2}}$, without which eq. (20) does not follow;
• the coherence bandwidth of the two-ray model of eq. (23), where $\rho^{2} = 0.5$, is $s = \pi / (2\Delta)$ rad/s, $1/(4\Delta)$ Hz, not “$1/\Delta$”;
• the phase of the correlation of that model is $\phi = -s\Delta/2$ (a delay of $\Delta/2$ on average); $\arctan(-s\sigma)$, $\sigma = \Delta/2$, is that of an exponential delay profile. The two agree to $10^{-3}$ rad at the separations of Fig. 7 (500 Hz and 1 kHz with $\Delta = 50$ µs).

Examples

use comnum_waveform::ofdm::correction::{sdr_gain_limited, sdr_optimum};
use comnum_math::C64;
// SIR 20 dB: limiting the correction near -10 dB costs little against the optimum
let best = sdr_gain_limited(100.0, 10f64.powf(-0.5), C64::new(1.0, 0.0));
assert!(comnum_math::lin2db(best) > 13.0 && best < sdr_optimum(100.0));

References

1. L. J. Cimini, Jr., “Analysis and simulation of a digital mobile channel using orthogonal frequency division multiplexing,” IEEE Trans. Commun., 33(7), 665–675, 1985.

Functions

gain_limited

Gain-limited correction of eq. (16): $\hat{D} = R / (\max(|Z_{p}|, \varepsilon) e^{j \arg Z_{p}})$ for every sample of r and of the pilot envelope pilot (eps: the limit, an amplitude relative to the RMS gain; 0 for unlimited correction). A zero pilot with no limit gives a non-finite sample.

sdr_gain_limited

The SDR (linear) of gain-limited correction, eq. (25) of Cimini 1985 (see the module): sir_lin linear (infinite without interferer), eps the limit (amplitude relative to the RMS gain, positive), rho the complex correlation $\rho e^{j\phi} = \mathbb{E}[Z_{d} Z_{p}^{*}]$ of the data and pilot envelopes (1 in flat fading, eq. (20)).

sdr_optimum

The SDR (linear) of the optimum gain correction, eq. (21) of Cimini 1985: $1 / \mathbb{E}\left[\gamma A_{i}^{2} / (A_{d}^{2} + \gamma A_{i}^{2})\right]$ for independent Rayleigh amplitudes (sir_lin linear; sir_lin = 1 gives 2, the limit).