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Module theory

comnum_mapping

Module theory 

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Closed-form error rates and information rates of constellations over AWGN.

The closed forms are functions of $E_{b}/N_{0}$ (ebn0_db) unless their name or their argument says otherwise: ser_esn0 and the orthogonal signals take $E_{s}/N_{0}$ (esn0_db).

They are properties of a constellation and its labelling, so they live with the constellations; comnum-metrics re-exports them as theory_ber, theory_ser, theory_mi and theory_gmi. The functions of a family of constellations take the family and the order first: (kind, m, …).

§Model

For BPSK over AWGN,

$$ P_{b} = Q\left(\sqrt{2 E_{b} / N_{0}}\right) $$

§Examples

use comnum_mapping::{theory, constellation::ConstellationKind};
let pb = theory::ber(ConstellationKind::Qam, 4, 6.0).unwrap();
assert!((pb - 2.388e-3).abs() < 1e-6);

§References

  1. J. G. Proakis and M. Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
  2. R. F. Pawula, S. O. Rice and J. H. Roberts, “Distribution of the phase angle between two vectors perturbed by Gaussian noise,” IEEE Trans. Commun., 30(8), 1828–1841, 1982.

Enums§

RateChannel
The channel of constellation_mi, constellation_gmi and of the cutoff rates of crate::bicm.

Constants§

DEFAULT_QUADRATURE_POINTS
The Gauss–Hermite nodes per dimension of the information rates when the caller does not choose (constellation_mi, constellation_gmi, the BICM cutoff rate): the default of the theory checks of the specifications and of the Python closed forms.

Functions§

ber
Theoretical bit error rate over AWGN with Gray mapping (OptiCommPy theoryBER).
ber_dbpsk_coherent
Bit error rate of DBPSK with coherent detection and differential decoding over AWGN (Proakis & Salehi, §3.2-2): an isolated symbol error spoils two differences, so $P_{b} = 2p(1 - p)$ with $p = Q(\sqrt{2 E_{b}/N_{0}})$.
ber_differential
Bit error rate of $M$-DPSK over AWGN with the detection detection: ber_dpsk for the noncoherent detection, ber_dbpsk_coherent for the coherent detection with differential decoding (defined for $M = 2$).
ber_dpsk
Bit error rate of Gray-labelled M-DPSK with noncoherent detection over AWGN, as a function of $E_{b}/N_{0}$ (dB): $\frac{1}{2} e^{-E_{b}/N_{0}}$ for DBPSK (Proakis & Salehi, §4.5), and for $M \ge 4$ the average number of wrong bits over the decision sectors of the differential phase, whose probabilities come from differential_phase_tail at $\gamma_{s} = E_{b}/N_{0} \log_{2} M$:
ber_exact
The exact bit error rate of a Gray-labelled constellation over AWGN, as a function of $E_{b}/N_{0}$ (dB), where a closed form exists: ber_gray_pam for PAM, ber_gray_qam for square QAM (Cho and Yoon 2002), and ber for BPSK and QPSK, whose nearest-neighbour expression is exact.
ber_gray_pam
Exact BER of Gray-mapped m-PAM over AWGN (K. Cho and D. Yoon, “On the general BER expression of one- and two-dimensional amplitude modulations,” IEEE Trans. Commun. 50(7), 1074–1080, 2002, eqs. (9)–(10)), as a function of $E_{b}/N_{0}$ in dB:
ber_gray_qam
Exact BER of Gray-mapped square m-QAM over AWGN (Cho and Yoon 2002, eq. (16)), as a function of E_b/N_0 in dB: ber_gray_rect_qam with I = J = √m. Unlike the nearest-neighbour approximation of ber, it tends to 1/2 at low SNR, the BER of random decisions; for 16-QAM, P_b = (3Q(x) + 2Q(3x) - Q(5x))/4 with x = √(E_s/(5N_0)).
ber_gray_rect_qam
Exact BER of Gray-mapped mi × mj rectangular QAM over AWGN, two Gray-mapped PAM of mi and mj levels in quadrature (Cho and Yoon 2002, eqs. (19)–(22)):
ber_orthogonal_noncoherent
Bit error rate of $M$ orthogonal signals detected noncoherently over AWGN, as a function of $E_{s}/N_{0}$ (dB): $P_{b} = P_{s} M / (2(M - 1))$, every wrong symbol being equally likely (ser_orthogonal_noncoherent; Proakis & Salehi, §4.5).
ber_rayleigh_mrc
The bit error rate of BPSK (or Gray QPSK, per bit) with maximal-ratio combining of branches independent Rayleigh-faded branches of mean SNR per bit mean_ebn0 ($\bar\gamma$, linear) each (Proakis & Salehi, Digital Communications, 5th ed., eq. (13.4-15)):
constellation_gmi
The generalized mutual information (bits/symbol) of the bit-wise decoder of the labelled constellation c (point i carries the bits of i) with the probabilities px over the channel fading: gmi of any labelling, averaged over the fading. With uniform inputs it is the capacity of BICM (Caire, Taricco & Biglieri 1998, eq. (14)).
constellation_mi
The mutual information (bits/symbol) of the constellation c with the probabilities px (uniform if None) over the channel fading: mi of any constellation, averaged over the fading. With uniform inputs it is the capacity of coded modulation (Caire, Taricco & Biglieri 1998, eq. (10)).
differential_phase_tail
The probability that the phase angle between two vectors of the same SNR esn0_lin (linear, $E_{s}/N_{0}$ each) perturbed by independent Gaussian noise exceeds psi in absolute value, $0 \le \psi \le \pi$ (Pawula, Rice & Roberts, 1982, the integral form for equal SNRs):
ebn0_for_ber
The $E_{b}/N_{0}$ (dB) at which the BER of ber equals target: the bisection (200 steps) of $[-10, 60]$ dB, where the BER decreases.
ebn0_for_ser
The $E_{b}/N_{0}$ (dB) at which the SER of ser equals target, as ebn0_for_ber.
gmi
Generalised mutual information (bits/symbol) of the bit-wise decoder of a Gray-labelled constellation over AWGN, by Gauss–Hermite quadrature (OptiCommPy theoryGMI).
mi
Mutual information (bits/symbol) of a constellation over AWGN, by Gauss–Hermite quadrature (OptiCommPy theoryMI).
phase_pdf
Probability density of the phase error $\theta = \arg r - \varphi$ of a carrier $r = A e^{j\varphi} + w$ in circular Gaussian noise, at the SNR $\rho = A^{2}/\mathbb{E}|w|^{2}$ (linear) (S. O. Rice, Bell Syst. Tech. J. 27(1), 1948; as eq. (8) of A. U. Ahmed, S. C. Thompson and J. R. Zeidler, “Threshold extending receiver structures for CE-OFDM,” Proc. IEEE MILCOM, 2007):
ser
Theoretical symbol error rate over AWGN as a function of E_b/N_0.
ser_esn0
ser as a function of $E_{s}/N_{0}$ (dB), the SNR per symbol: the entry point of the closed forms whose SNR is per symbol (CE-OFDM, a symbol per subcarrier).
ser_mgf
The average SER of m-PSK or square m-QAM over a fading channel whose SNR per symbol $\gamma$ has the moment generating function mgf $(s) = \mathbb{E}, e^{s\gamma}$, $s \le 0$, by the MGF approach with Craig’s form of the $Q$ function (E. Basar et al., “Wireless communications through reconfigurable intelligent surfaces,” IEEE Access, 2019, eqs. (16) and (21)): with $g = \sin^{2}(\pi/M)$,
ser_orthogonal_noncoherent
Symbol error rate of $M$ orthogonal signals of equal energy detected noncoherently over AWGN (noncoherent M-FSK, chirp spread spectrum), as a function of $E_{s}/N_{0}$ (dB): the magnitude of the matched filter of the sent signal is Rice of location $v = \sqrt{2 E_{s}/N_{0}}$, the $M - 1$ others Rayleigh, all of unit scale, and the largest wins (Proakis & Salehi, §4.5; Afisiadis et al. 2020, eq. (16)):