Gallery of classical results#

The textbook curves of digital and optical communications, simulated with comnum and compared with their closed forms. Every script runs as is (the figures of this page are produced when the documentation is built), and the same results are checked at a few points by tests/test_performance.py.

Against the literature#

The scripts are grouped by theme, in the order of a link. Some reproduce one named figure or table of a paper or a standard, with the same system parameters, and say what the library’s curve is expected to match: where the published points are available as numbers, they are drawn on the figure; otherwise the caption names the figure to open next to the plot. The others compare with closed forms and cite the equations they use. The table below gathers every comparison of this gallery.

Reference	What is reproduced	Where
Alamouti, IEEE J. Sel. Areas Commun. 16(8), 1998, Fig. 4	BER of coherent BPSK over Rayleigh fading: no diversity, MRRC with 2 and 4 receive antennas, the two-branch transmit diversity scheme with 1 and 2 receive antennas	Alamouti 1998, Fig. 4: transmit diversity against MRRC
X.-Y. Hu, E. Eleftheriou, D. M. Arnold, IEEE Trans. Inf. Theory 51(1), 2005, Fig. 5	Bit and block error rates of a regular (3, 6) PEG code of length 504 over AWGN with BPSK and sum-product decoding	Hu, Eleftheriou & Arnold 2005, Fig. 5: a regular PEG code of length 504
Thompson et al., IEEE Trans. Commun. 56(8), 2008, Fig. 2 and eq. (15)	BER of the CE-OFDM phase demodulator receiver for 8-PAM data and several modulation indices, against the high-CNR approximation	Thompson et al. 2008, Fig. 2: the CE-OFDM phase demodulator
Häger, H. D. Pfister, IEEE J. Sel. Areas Commun. 39(1), 2021, Fig. 5	Effective SNR versus launch power over 25 × 80 km, linear equalisation and digital back-propagation with one step per span; the published points are drawn	Häger & Pfister 2021, Fig. 5: linear equalisation and back-propagation
Proakis, M. Salehi, Digital Communications, 5th ed., 2008, Sec. 3.3-2	MSK detected by maximum-likelihood sequence detection has the BER of BPSK once precoded; without precoding, that of differentially decoded BPSK	Proakis & Salehi 2008, Sec. 3.3-2: MSK has the error rate of BPSK
J. Guerreiro et al., IEEE Commun. Lett. 20(11), 2016, Fig. 1 and eq. (8); M. J. López Morales et al., Globecom Workshops 2022, Fig. 3; A. U. Ahmed, J. R. Zeidler, IEEE Trans. Signal Process. 63(17), 2015, Fig. 6; R. A. Pacheco, D. Hatzinakos, ICASSP 2006, Fig. 1; J. Guerreiro et al., VTC Fall 2015, Fig. 3	CE-OFDM: the phase detector, GAMP and the optimum BER at \(2\pi h = 1.4\); the arctangent, basic linear and enhanced receivers at 0.7; the 99 % bandwidth; the spectrum at \(2\pi h = 2\)	CE-OFDM receivers, spectrum and bandwidth
O. Afisiadis et al., IEEE Trans. Wireless Commun. 19(2), 2020, Figs. 2 and 7; L. Vangelista, IEEE Signal Process. Lett. 24(12), 2017, Fig. 1; M. Chiani, A. Elzanaty, IEEE Internet Things J. 6(5), 2019, Fig. 3a	The LoRa modulation: the SER over AWGN for SF 7 to 12, FSCM against FSK over a frequency-selective channel, the continuous spectrum, a same-SF interferer	The LoRa modulation: error rates, spectrum and interference
ETSI EN 302 307-1 V1.4.1, Table 13	The \(E_s/N_0\) thresholds of the 28 DVB-S2 MODCODs against the simulated frame error rates	DVB-S2: MODCODs, constellations and frame error rates
Proakis & Salehi, 5th ed., Secs. 4.3 and 13.3; eq. (13.4-15)	Error rates of QAM, PSK and PAM over AWGN, BPSK over Rayleigh fading, maximal-ratio combining	Bit error rate of M-QAM over AWGN, M-PSK over AWGN with the Python runner, BPSK over flat Rayleigh fading, Transmit diversity: Alamouti and space-time block codes
Ochiai, H. Imai, IEEE Trans. Commun. 49(2), 2001, eqs. (11) and (27)	The distribution of the PAPR of OFDM at the Nyquist rate and oversampled	PAPR of OFDM: the CCDF and its closed forms
Polyanskiy, H. V. Poor, S. Verdú, IEEE Trans. Inf. Theory 56(5), 2010	The normal approximation of the finite-length limit, against polar and LDPC codes of length 256	Channel coding: from convolutional codes to the finite-length limit
P. Poggiolini, J. Lightw. Technol. 30(24), 2012; A. Carena et al., Opt. Express 22(13), 2014	The GN and EGN models of nonlinear interference against split-step simulations of a WDM link	WDM nonlinearity: the GN model and digital back-propagation
Böcherer, F. Steiner, P. Schulte, IEEE Trans. Commun. 63(12), 2015	Probabilistic amplitude shaping of 64-QAM at 3 bits per symbol	Probabilistic shaping: the rate gain and probabilistic amplitude shaping

The README of references/ in the repository lists every document the library is validated against, with its status, and specs/validation/ holds the campaigns that CI checks against these sources.

Foundations#

Constellations, error rates over AWGN and Rayleigh fading against their closed forms, information rates.

Constellations and Gray labels

Bit error rate of M-QAM over AWGN

M-PSK over AWGN with the Python runner

BPSK over flat Rayleigh fading

Information rates of QAM

Coding and shaping#

Channel codes against the finite-length limits and the literature, the codes of 5G NR, probabilistic amplitude shaping.

Channel coding: from convolutional codes to the finite-length limit

Hu, Eleftheriou & Arnold 2005, Fig. 5: a regular PEG code of length 504

5G NR channel codes: LDPC and polar

Probabilistic shaping: the rate gain and probabilistic amplitude shaping

Waveforms#

OFDM and its PAPR, continuous-phase modulation, constant-envelope OFDM, chirp spread spectrum, filtered multicarrier (FBMC/OQAM, UFMC) and OTFS.

PAPR of OFDM: the CCDF and its closed forms

PAPR reduction techniques

OFDM with impairments: imperfect channel estimates and nonlinear amplifiers

Proakis & Salehi 2008, Sec. 3.3-2: MSK has the error rate of BPSK

Thompson et al. 2008, Fig. 2: the CE-OFDM phase demodulator

CE-OFDM receivers, spectrum and bandwidth

The LoRa modulation: error rates, spectrum and interference

Filtered multicarrier and OTFS

MIMO#

Detection, space-time codes, capacity, message passing for large MIMO.

MIMO detection: linear, successive cancellation and maximum likelihood

Transmit diversity: Alamouti and space-time block codes

Alamouti 1998, Fig. 4: transmit diversity against MRRC

Ergodic capacity of MIMO channels

Large MIMO: message passing (LAMA and OAMP)

Biguesh & Gershman 2006, Fig. 4: LS, scaled LS, relaxed MMSE and LMMSE

Optical#

Fibre propagation and its compensation, nonlinear interference, Raman amplification, optical modulators; optical wireless links (visible light indoors, free-space optics).