Capacity Bounds and Achievable rates of Visible Light Communications for Future Smart City

The Shannon expression derived for a power-limited linear time-invariant (LTI) channel with additive white Gaussian noise (AWGN) is often quoted as a capacity for visible light communications (VLC). However, the use of the Shannon expression, i.e., bandwidth times the logarithm of unity plus the signal to noise ratio (SNR), is challenged in intensity modulation with direct detection (IM/DD) VLC channel. In this paper, we overview some literature on capacity bounds and achievable rates of VLC. We classify the Shannon capacity as the upper bound for VLC with optical or electrical power constraints. A tight closed-form lower bound can be also derived with the peak optical power, average optical power and electrical power constraints. Moreover, with a specific modulation, the achievable rate can be derived by inverting the bit error rate (BER) error function to obtain the SNR needed to carry a certain constellation with for instance one-dimension pulse amplitude modulation (PAM) or two-dimension quadrature amplitude modulation (QAM) at an adequate BER or symbol error rate (SER). These yield Shannon-like expressions for the achievable rates, but with a correction factor for the SNR which is a function of the required uncoded BER. Since QAM is commonly utilized in each subcarrier of multi-carrier orthogonal frequency division multiplexing (OFDM), this suggests the achievable rates of QAM can be extended for the throughput calculation of DC-biased optical OFDM (DCO-OFDM). Numerical results indicate a gap between the lower bound and the achievable rates by the PAM and DCO-OFDM signaling, which means that neither PAM nor DCO-OFDM is the optimal modulation scheme for VLC channels and other capacity-achieving modulations can be further explored.