FSO Channel Capacity
Shannon capacity and the net rate of OOK, BPSK, QPSK, 8-PSK and 16-QAM after hard-decision FEC, gated on their exact AWGN bit error rates, with the required SNR and the gap to Shannon.
Shannon capacity
6.658 Gb/s
6.66 b/s/Hz
QPSK net rate
1.869 Gb/s
1.87 b/s/Hz after FEC
QPSK uncoded BER
7.62e-24
at Es/N0 = 20.0 dB
Required SNR
8.53 dB
Es/N0 for BER = 3.80e-3
Gap to Shannon
4.29 dB
required SNR − 10 log₁₀(2^η − 1) at the same net η
Shannon Eb/N0 limit
1.52 dB
(2^η − 1)/η at η = 1.87 b/s/Hz
Net rate vs SNR
Model
C = B log₂(1 + SNR), R = B·k/(1 + OH) if BER(Es/N0) ≤ limit, else 0
Complex AWGN channel with Nyquist signalling (symbol rate = B), so SNR = Es/N0 and k bits per symbol give k b/s/Hz. Bit error rates are exact for Gray mapping and ideal coherent detection: BPSK Q(√(2γ)), QPSK and coherent on-off keying Q(√γ), 16-QAM from Cho & Yoon (2002), 8-PSK from Craig's (1991) exact phase integral. The FEC is modelled by its input-BER limit; ≈ 3.8×10⁻³ is often quoted for 7 % hard-decision codes (ITU-T G.975.1 doesn't print it; its App. I.9 BCH code, 6.69 % overhead, takes 3.5×10⁻³ in to 2.1×10⁻¹⁴ out, Table I.22).
Scope: most terrestrial FSO links use intensity modulation with direct detection. Their channel is real and non-negative with peak and average power limits, its capacity has no closed form, and its SNR is defined on the photocurrent. Photon-counting links follow the Poisson channel instead; see the photon-counting BER page.
Model and references
Exact. The equations are exact within the stated assumptions, or are a standard's tabulated values. The results are as good as the inputs. About the tiers
Exact for coherent detection in complex AWGN: Gray mapping, Nyquist signalling (SNR = Es/N0). FEC is reduced to an input-BER limit. Direct-detection and photon-counting channels have other capacities.
- Shannon C. E. (1948). A mathematical theory of communication. Bell Syst. Tech. J. 27(3), 379–423. doi:10.1002/j.1538-7305.1948.tb01338.x
- Craig J. W. (1991). A new, simple and exact result for calculating the probability of error for two-dimensional signal constellations. MILCOM '91, 571–575. Exact M-PSK phase-error integral. doi:10.1109/MILCOM.1991.258319
- Cho K., Yoon D. (2002). On the general BER expression of one- and two-dimensional amplitude modulations. IEEE Trans. Commun. 50(7), 1074–1080. Gray 16-QAM bit error rate. doi:10.1109/TCOMM.2002.800818