Module link_adaptation 

Variable-rate variable-power MQAM over Rayleigh fading (Goldsmith & Chua 1997): the Shannon capacity with power adapted in time, the spectral efficiency of MQAM whose size and power follow the channel gain at a target bit error rate, with a continuum or a finite set of constellations, and the average time spent in every fading region.

The bit error rate of MQAM at the SNR $\gamma$ is bounded by eq. (17), $\mathrm{BER} \le 0.2, e^{-1.5\gamma/(M-1)}$ (mqam_ber_bound), which fixes the constellation of a target BER at $M(\gamma) = 1 + K\gamma S(\gamma)/\bar S$, with the power gap $K = -1.5/\ln(5,\mathrm{BER})$ (mqam_power_gap, eqs. (20), (24)). Over Rayleigh fading of mean SNR $\bar\gamma$, $p(\gamma) = e^{-\gamma/\bar\gamma}/\bar\gamma$, every integral is an exponential integral $E_{1}$:

• capacity with water-filling in time, eqs. (5)–(7): the cutoff $\gamma_{0} = x\bar\gamma$ solves $e^{-x}/x - E_{1}(x) = \bar\gamma$ and $C/B = E_{1}(x)/\ln 2$ (rayleigh_water_filling_cutoff, rayleigh_capacity_water_filling);
• continuous-rate adaptive MQAM, eqs. (23)–(25): the same water-filling at the mean SNR $K\bar\gamma$ (rayleigh_adaptive_mqam_efficiency);
• discrete rates, eqs. (28)–(32): the constellations $M_{1} < \dots < M_{N}$ (none in region 0) are used for $M_{j} \le \gamma/\gamma_{K}^{} < M_{j+1}$ with the power $(M_{j} - 1)/(\gamma K)$; the power constraint (32) fixes $\gamma_{K}^{}$, $$\sum_{j=1}^{N} \frac{M_{j} - 1}{K\bar\gamma}\left[E_{1}!\left(\frac{\gamma_{K}^{}M_{j}}{\bar\gamma}\right) - E_{1}!\left(\frac{\gamma_{K}^{}M_{j+1}}{\bar\gamma}\right)\right] = 1,$$ and $R/B = \sum_{j} \log_{2}M_{j}, p(M_{j} \le \gamma/\gamma_{K}^{*} < M_{j+1})$ (rayleigh_discrete_mqam);
• the average fade region duration, eqs. (34)–(35): $\bar\tau_{j} = \pi_{j}/(N_{j+1} + N_{j})$ with $\pi_{j}$ the probability of region $j$ and $N_{j}$ the level-crossing rate of its lower edge (rayleigh_region_durations);
• a constant error $\epsilon = \hat\gamma/\gamma$ of the SNR estimate raises the BER of the target $\mathrm{BER}{0}$ to $0.2,(5,\mathrm{BER}{0})^{1/\epsilon}$, eq. (42) (estimation_error_ber).

Examples

use comnum_mapping::link_adaptation::{rayleigh_adaptive_mqam_efficiency, rayleigh_capacity_water_filling};
// 20 dB of mean SNR: 5.90 bit/s/Hz of capacity, 4.22 for adaptive MQAM at 1e-3 (Fig. 3)
let c = rayleigh_capacity_water_filling(100.0)?;
let r = rayleigh_adaptive_mqam_efficiency(100.0, 1e-3)?;
assert!((c - 5.90).abs() < 0.01 && (r - 4.22).abs() < 0.01, "{c} {r}");

References

1. A. J. Goldsmith, S.-G. Chua, “Variable-rate variable-power MQAM for fading channels,” IEEE Trans. Commun. 45(10), 1218–1230, 1997: eqs. (5)–(7), (17), (20)–(25), (28)–(35), (42), Figs. 3, 5, 10, Table II.

Structs

DiscreteRate

MQAM with a finite set of constellations adapted to Rayleigh fading (eqs. (28)–(32)).

Functions

estimation_error_ber

The bit error rate of adaptive MQAM designed for ber0 when the SNR is estimated as $\epsilon$ = epsilon times its value (perfect AGC), eq. (42): $0.2,(5,\mathrm{BER}_{0})^{1/\epsilon}$.

mqam_ber_bound

The bound (17) of the bit error rate of MQAM of size m at the SNR snr (linear): $0.2, e^{-1.5,\gamma/(M-1)}$, within 1 dB for $M \ge 4$ and SNRs of 0 to 30 dB.

mqam_power_gap

The power gap $K = -1.5/\ln(5,\mathrm{BER})$ of MQAM at the target bit error rate ber (eq. (24)): $-5.48$ dB at $10^{-3}$.

rayleigh_adaptive_mqam_efficiency

The spectral efficiency (bit/s/Hz) of MQAM with rate and power adapted continuously to Rayleigh fading of mean SNR mean_snr (linear) at the target bit error rate ber, eq. (25): the water-filling capacity at the mean SNR $K\bar\gamma$.

rayleigh_capacity_water_filling

The capacity (bit/s/Hz) of Rayleigh fading of mean SNR mean_snr (linear) with the power adapted by water-filling in time, eq. (7): $E_{1}(\gamma_{0}/\bar\gamma)/\ln 2$.

rayleigh_discrete_mqam

The discrete-rate adaptive MQAM of the constellation sizes sizes ($M_{1} < \dots < M_{N}$, no transmission below $M_{1}$) over Rayleigh fading of mean SNR mean_snr (linear) at the target BER ber: $\gamma_{K}^{*}$ from the power constraint (32), by bisection on its logarithm to $10^{-15}$, and the efficiency (31).

rayleigh_region_durations

The average time (s) spent in every region $0..N$ of a DiscreteRate policy over Rayleigh fading of mean SNR mean_snr (linear) and maximum Doppler shift doppler_hz (eqs. (34)–(35)): region 0 below $A_{1}$ (no transmission), region $j$ on $[A_{j}, A_{j+1})$; $\bar\tau_{j} = \pi_{j}/(N_{j} + N_{j+1})$ with $N_{j} = \sqrt{2\pi A_{j}/\bar\gamma}, f_{D}, e^{-A_{j}/\bar\gamma}$ ($N_{0} = 0$, $N_{N+1} = 0$).

rayleigh_water_filling_cutoff

The cutoff $\gamma_{0}$ (linear) of water-filling in time over Rayleigh fading of mean SNR mean_snr (linear), eq. (6): $x = \gamma_{0}/\bar\gamma$ solves $e^{-x}/x - E_{1}(x) = \bar\gamma$ (bisection on $\ln x$ to $10^{-15}$).