Reliable MIMO Optical Wireless Communications Through Super-Rectangular Cover

In this paper, we consider an intensity modulated direct detection multiple-input-multiple-output optical wireless communication (IM/DD MIMO-OWC) system, where the channel coefficients are assumed to be completely known at the receiver. For such a system, since both the transmitted signals and the channel coefficients are nonnegative, the full rank condition on the unique identification of the signals for a noise-free channel and fully reliable (diversity) estimation of the signals for a noisy channel for MIMO radio frequency (MIMO-RF) communications is sufficient but not necessary. For this reason, from the viewpoint of detection theory, a novel so-called super-rectangular cover theory is developed to characterize both the unique identifiability and full reliability for IM/DD MIMO-OWC. In this theory, two important concepts, cover order and cover length are introduced. Among all the super-rectangles covering the feasible domain in the first quadrate determined by a semidefinite positive quadratic form smaller than any given positive constant, the largest superrectangular cover is the one with the maximum number of finitely lengthy sides, while the smallest super-rectangular cover is the largest super-rectangular cover with the length of each finitely lengthy side being the minimum. Cover order and cover length are defined, respectively, as the number of the finitely lengthy sides of the largest super-rectangle cover and the side length of the smallest super-rectangular cover. This theory states that a transmitted matrix signal can be uniquely identified if and only if the cover order is equal to the transmitter aperture number (maximum cover order), i.e., full cover. In particular, when this theory is applied to the diversity analysis for spacetime block coded IM/DD MIMO-OWC over commonly used log-normal fading channels, we prove that full reliability is guaranteed with a maximum likelihood (ML) detector if and only if the space-time block code (STBC) enables full cover. In addition, for this system, the diversity gain can be geometrically interpreted as the cover order of the super-rectangle, which should be maximized, and the volume of this super-rectangle, as the diversity loss, should be minimized. Using this established error performance criterion, the optimal linear STBC for block fading channels is proved to be spatial repetition code with an optimal power allocation. The design of the optimal non-linear STBC is shown to be equivalent to constructing the optimal multi-dimensional constellation, which is a classic and longstanding topic. Specifically, a multi-dimensional constellation from Diophantine equations is proposed and then, shown to be more energy-efficient than the commonly used nonnegative pulse amplitude modulation (PAM) constellation. Furthermore, for fast fading channels, a linear STBC is constructed by collaborating the signals of two successive channel uses and proved to be related This work was supported by NHTRDP (863 Program) of China (Grant No.2013AA013603). The work of Jian-Kang Zhang was partially supported by NSERC. Yan-Yu Zhang, Hong-Yi Yu and Jin-Long Wang are with National Digital Switching System Engineering and Technology Center, Henan Province (450000), China. Emails: [email protected], [email protected] and [email protected] Jian-Kang Zhang is with the Department of Electrical and Computer Engineering, McMaster University, 1280 Main Street West, L8S 4K1, Hamilton, Ontario, Canada. Email: [email protected]. to the Golden number √ 5+1 2 , say, Golden code. This Golden code has an important property of non-increasing diversity loss as constellation size increases and thus, presents encouraging error performances. These observations obtained in this paper reveal useful insights into how the space-time block coded IM/DD MIMO-OWC systems are remarkably different from, rather than “mimic”, the conventional space-time coded MIMO-RF systems (Tarokh, Seshadri, and Calderbank, 1998) from the perspectives of detection theory, error performance criterion and code design techniques.