Full LargeScale Diversity Space Codes for MIMO Optical Wireless Communications
Abstract
In this paper, we consider a multipleinputmultipleoutput optical wireless communication (MIMOOWC) system suffering from lognormal fading. In this scenario, a general criterion for the design of full largescale diversity space code (FLDSC) with the maximum likelihood (ML) detector is developed. Based on our criterion, FLDSC is attained if and only if all the entries of the space coding matrix are positive. Particularly for MIMOOWC with unipolar pulse amplitude modulation (PAM), a closedform linear FLDSC satisfying this criterion is attained by smartly taking advantage of some available properties as well as by developing some new interesting properties on Farey sequences in number theory to rigorously attack the continuous and discrete variables mixed maxmin problem. In fact, this specific design not only proves that a repetition code (RC) is the best linear FLDSC, but also uncovers a significant difference between MIMO radio frequency (RF) communications and MIMOOWC that spaceonly transmission is sufficient for a full diversity achievement. Computer simulations demonstrate that FLDSC substantially outperforms spatial multiplexing with the same total optical power and spectral efficiency and the latter obtains only the smallscale diversity gain.
Full largescale diversity, lognormal fading channels, multipleinputmultipleoutput (MIMO), optical wireless communications (OWC), space code.
I Introduction
Optical wireless communications (OWC), due to its potential for bandwidthhungry applications, has become a very important area of research [1, 2, 3, 4, 5, 6, 7]. However, some challenges remain, especially in atmospheric environments, where robustness is a key consideration. Therefore, in the design of high date rate OWC links, we need to consider the atmospheric impairmentsinduced fading which can be described by the lognormal (LN) statistical model [8, 9]. To combat fading, multiinputmultioutput (MIMO) OWC (MIMOOWC) systems introduce the design for the transmitted symbols distributed over transmitting apertures (space) and (or) symbol periods (time). Full largescale diversity is achieved when the total degrees of freedom (DoF) available in the MIMOOWC system is fully utilized.
Unfortunately, unlike MIMO techniques for radio frequency (MIMORF) communications with Rayleigh fading, there are two significant challenges in MIMOOWC communications. The first is that there does not exist any available mathematical tool that could be directly applied to the analysis of the average pairwise error probability (PEP) when LN is involved. Although there are really mathematical formulae in literature for numerically and accurately computing the integral involving LN [10, 11, 8], it can not be used for the theoretic analysis on diversity. The second is a nonnegative constraint on the design of transmission for MIMOOWC, which is a major difference between MIMO RF communications and MIMOOWC. It is because of this constraint that the currently available welldeveloped MIMO techniques for RF communications can not be directly utilized for MIMOOWC. Despite the fact that the nonnegative constraint can be satisfied by properly adding some directcurrent components (DC) into transmitter designs so that the existing advanced MIMO techniques [12] for RF communications such as orthogonal spacetime block code (OSTBC) [13, 14] could be used in MIMOOWC, the power loss arising from DC incurs the fact that these modified OSTBCs [15, 16] in a LN fading optical channel have worse error performance than the RC [11, 17, 18].
All the aforementioned factors greatly motivate us to develop a general criterion on the design of full largescale diversity transmission for MIMOOWC. As an initial exploration, we consider the spacealone code, and intend to uncover some unique characteristics of MIMOOWC by establishing a general criterion for the design of FLDSC and attaining an optimal analytical solution to a specific two by two linear FLDSC.
Ii Channel Model And Space Code
Iia Channel model with space code
Let us consider an MIMOOWC system having receiver apertures and transmitter apertures transmitting the symbol vector , , which are randomly, independently and equally likely, selected from a given constellation. To facilitate the transmission of these symbols through the transmitters in the one time slots (channel use), each symbol is mapped by a space encoder to an space code vector and then summed together, resulting in an space codeword given by , where the th element of represents the coded symbol to be transmitted from the th transmitter aperture. These coded symbols are then transmitted to the receivers through flatfading path coefficients, which form the elements of the channel matrix . The received spaceonly symbol, denoted by the vector , can be written as
(1) 
where is the average optical power of and, the entries of channel matrix are independent and LN distributed, i.e., , where . The probability density function (PDF) of is
(2) 
The PDF of is . The signalling scheme of is unipolar pulse amplitude modulation (PAM) to meet the unipolarity requirement of intensity modulator (IM), i.e., . As an example, the constellation of unipolar ary PAM is , where is a positive integer. Then, the equivalent constellation of is , i.e., .
Furthermore, for noise vector , the two primary sources at the receiver front end are due to noise from the receive electronics and shot noise from the received DC photocurrent induced by background radiation [19, 20]. By the central limit theorem, this highintensity shot noise for the lightwavebased OWC is closely approximated as additive, signalindependent, white, Gaussian noise (AWGN) [20] with zero mean and variance .
By rewriting the channel matrix as a vector and aligning the codechannel product to form a new channel vector, we can have , where denotes the Kronecker product operation and . For discussion convenience, we call a codeword matrix, denoted by . Then, the correlation matrix of the corresponding error coding matrix is given by
(3) 
where , is the error vector with and . All these nonzero form an error set, denoted by .
IiB Problem formulation
To formally state our problem, we make the following assumptions throughout this paper.

Power constraint. The average optical power is constrained, i.e., . Although limits are placed on both the average and peak optical power transmitted, in the case of most practical modulated optical sources, it is the average optical power constraint that dominates [21].

SNR definition. The optical SNR is defined by , since the noise variance per dimension is assumed to be . Thus, in expressions on error performance involved in the squared Euclidean distance, the term , in fact, is equal to
(4) with optical power being normalized by . Unless stated otherwise, is referred to as the squared optical SNR thereafter.
Under the above assumptions, our primary task in this paper is to establish a general criterion on the design of FLDSC and solve the following problem.
Problem 1
Design the space encoder subject to the total optical power such that 1) meets the unipolarity requirement of IM; 2) Full largescale diversity is enabled for the ML receiver.
Iii Design Criteria for Space Code
This subsection aims at deriving the PEP of MIMOOWC and then, establishing a general design criterion for the linear space coded system.
Iiia PEP of MIMOOWC
Given a channel realization and a transmitted signal vector , the probability of transmitting and deciding in favor of with the ML receiver is given by [22]
(5) 
where with . Averaging (5) over yields
(6) 
To extract the dominant term of (6), we make an assumption for time being. Later on, we will prove that this condition is actually necessary and sufficient for to render full diversity.
Assumption 1
Any is unipolar without zero entry.
Theorem 1
Now, we can see that in (1), and have the same exponential term, , whereas the exponential term of is , which decays slower than against high SNR. That being said, we have successfully attained the dominant term, , of the upperbound of . Thus, our selection of is reasonable to capture the dominant behaviour of .
With all the aforementioned preparations, we enable to give the general design criterion for FLDSC of MIMOOWC in the following subsection.
IiiB Design Criterion for FLDSC
The discussions in Subsection IIIA tells us that is the dominant term of the upperbound of in (1). With this, we will provide a guideline on the space code design in this subsection. To define the performance parameters to be optimized, we rewrite as follows.
(8) 
where and .
Here, the following three factors dictate the minimization of :

Largescale diversity gain. The exponent with respect to governs the behavior of . For this reason, is named as the largescale diversity gain. The full largescale diversity achievement is equivalent to the event that all the terms in offered by the MIMOOWC are fully utilized. Thus, when we design space code, full largescale diversity must be assured in the first place.

Smallscale diversity gain. is called smallscale diversity gain, which affects the polynomial decaying in terms of . should be maximized to optimize the error performance of the worst error event. Since the smallscale diversity gain will affect the average PEP via the polynomially decaying speed of the error curve, the smallscale diversity gain of the space code is what to be optimized in the second place.

Coding gain. is defined as coding gain. On condition that both diversity gain are maximized, if there still exists DoF for further optimization of the coding gain, should be minimized as the last step for the systematical design of space code.
In what follows, we will give a sufficient and necessary condition on a full largescale diversity achievement. Hence, Assumption 1 is sufficient and necessary for FLDSC, which is summarized as the following theorem:
Theorem 2
A space code enables full largescale diversity if and only , is unipolar without zerovalued entries or equivalently, , is positive.
With these results, we can proceed to design FLDSC systematically in the following section.
Iv Optimal Design of Specific Linear FLDSC
In this section, we will exemplify our established criterion in (IIIB) by designing a specific linear FLDSC for MIMOOWC with unipolar pulse amplitude modulation (PAM). For this particular design, a closedform space code optimizing both diversity gains will be obtained by smartly taking advantage of some available properties as well as by developing some new interesting properties on Farey sequences in number theory.
Iva Design Problem Formulation
Consider a MIMOOWC system with , where and . By Theorem 2, should be positive to maximize the largescale diversity gain. On the other hand, from the structure of and (IIIB), the smallscale diversity gain is under the assumption that CSIT is unknown. Therefore, to optimize the worst case over , FLDSC design is formulated as follows:
(9) 
IvB Equivalent Simplification of Design Problem
For PAM, all the possible nonzero values of are
(10) 
IvB1 Preliminary simplification
After observations over (10), we have the following facts.

, it holds holds that
(11a)
So, all the possible minimum of in (IVA) are , and , where are irreducible, i.e., . These terms are denoted by and . After putting aside the common term, , we can see that is the piecewise linear function of and , respectively. So, (IVA) can be solved by fragmenting interval into disjoint subintervals. This fragmentation can be done by the breakpoints where . To characterize this sequence, there exists an elegant mathematical tool in number theory presented below.
IvB2 Farey sequences
First, we observe some specific examples of the breakpoint sequences. For OOK, the breakpoints . For 4PAM, they are . For 8PAM, we have the breakpoint sequence with the former part being
(12a)  
and the remaining being  
(12b) 
Through these special examples, we find that the series of breakpoints before (such as the sequence in (12a)) is the one which is called the Farey sequence [23]. The Farey sequence for any positive integer is the set of irreducible rational numbers with arranged in an increasing order. The series of breakpoints after (such as the sequence in (12b)) is the reciprocal version of the Farey sequence. Thus, our focus is on the sequence before .
The Farey sequence has many interesting properties [23], some of which closely relevant to our problem are given as follows.
Lemma 1
If , and are three successive terms of and , then,

and .

and .
However, having only Lemma 1 is not enough to solve our design problem in (IVA). We need to develop the other new properties of Farey sequences, concluded by Properties 1, 2 and 3.
Property 1
Given , assume and . If and are successive, then, and .
Property 2
Assume and . Then,

holds.

If , then, .

If , then, .

If , then, .
Property 3
If and are successive in and , then, and are the two worst cases.
IvC Techniques to Solve The Maxmin Problem
Thanks to Farey sequences, (IVA) is transformed into a piecewise maxmin problem with two objective functions. By solving this kind of problem, our code construction results can be presented as the following theorem.
Theorem 3
V Computer Simulations
In this section, we carry out computer simulations to verify our newly developed criterion in (IIIB). In light of our work being initiative, the only spaceonly transmission scheme available in the literature is spatial multiplexing (SM). Accordingly, we compare the performance of spatial multiplexing and FLDSC specifically designed for MIMOOWC in Section IV. In addition, we suppose that are independently and identically distributed and let . These schemes are as follows:

FLDSC. The optical power is normalized in such a way that yields . From (13), the coding matrix is .

SM. We fix the modulation formats to be OOK and vary . So the rate is 2 bits per channel use (pcu). The transmitted symbols are chosen from equally likely. The average optical power is .
We can see that both schemes have the same spectrum efficiency, i.e., 2 bits pcu and the same optical power. Through numerical results, we have following observations.
Substantial enhancement from FLDSC is achieved, as shown in Fig. 1. For , the improvement is almost 16 dB at the target bit error rate (BER) of . For , the improvement is almost 6 dB at the target BER of . Note that the smallscale gain also governs the negative slope of error curve. The decaying speed of the error curve of FLDSC is exponential in terms of , whereas that of SM is polynomial with respect to , even worse than singleinputsingaloutput (SISO).
SM presents only smallscale diversity gain illustrated in Fig. 2. By varying the variance of , we find that in the high SNR regimes, the error curve decays as as long as the SNR is high enough . From to , the error curve has a horizonal shift, which is the typical style of RF MIMO [12]. The reason is given as follows. The equivalent space coding matrix is with . It should be noted that there exists two typical error events: and . From the necessity proof of Theorem 2, for , the attained largescale diversity gain is zero, and at the same time, if with , then, the attained largescale diversity gain is only two for MIMOOWC. Therefore, the overall largescale diversity gain of SM is zero with smallscale diversity gain being attained.
Vi Conclusion and Discussions
In this paper, we have established a general criterion on the fulldiversity space coded transmission of MIMOOWC for the ML receiver, which is, to our best knowledge, the first design criterion for the full largescale diversity transmission of optical wireless communications with IM/DD over lognormal fading channels. Particularly for a case, we have attained an optimal closedform FLDSC, rigorously proving that RC is the best among all the linear space codes. Our results clearly indicate that the transmission design is indeed necessary and essential for significantly improving the overall error performance for MIMOOWC.
Vii Acknowledgements
This work was supported in part by Key Laboratory of Universal Wireless Communications (Beijing University of Posts and Telecommunications), Ministry of Education of P. R. China under Grant No. KFKT2012103, in part by NNSF of China (Grant No. 61271253) and in part by NHTRDP of China (“863” Program) (Grant No. 2013AA013603).
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