## Abstract

In this paper, we consider the design of space code for an intensity modulated direct detection multi-input-multi-output optical wireless communication (IM/DD MIMO-OWC) system, in which channel coefficients are independent and non-identically log-normal distributed, with variances and means known at the transmitter and channel state information available at the receiver. Utilizing the existing space code design criterion for IM/DD MIMO-OWC with a maximum likelihood (ML) detector, we design a diversity-optimal space code (DOSC) that maximizes both large-scale diversity and small-scale diversity gains and prove that the spatial repetition code (RC) with a diversity-optimized power allocation is diversity-optimal among all the high dimensional nonnegative space code schemes under a commonly used optical power constraint. In addition, we show that one of significant advantages of the DOSC is to allow low-complexity ML detection. Simulation results indicate that in high signal-to-noise ratio (SNR) regimes, our proposed DOSC significantly outperforms RC, which is the best space code currently available for such system.

© 2016 Optical Society of America

## 1. Introduction

As an emerging technology, intensity modulated direct detection optical wireless communication (IM/DD OWC) [1–4
] has become an attractive area of research. To provide more robust error performance in presence of atmospheric turbulence, the design of diversity transmitters for IM/DD MIMO-OWC was proposed [5–13
]. Unfortunately, the currently well developed diversity techniques for MIMO radio frequency (MIMO-RF) [14–17
] and coherent MIMOOWC [18–22
] can not be straightforwardly generalized to IM/DD MIMO-OWC because of the following *two* reasons. The *first* reason is that the transmitter design of IM/DD MIMO-OWC must be nonnegative to satisfy the unipolarity requirement of intensity modulation. As a result, the existing MIMO-RF [14] and coherent MIMO-OWC diversity techniques [21, 22] are not directly applicable to IM/DD MIMO-OWC. Although a full diversity gain can be provided by adding proper direct-current components into the code designs for MIMO-RF and coherent MIMO-OWC [5, 8, 9], the numerical error performance of some modified orthogonal space time block codes is even worse than that of the space-only repetition code (RC) [6, 7, 10, 11]. The *second* reason is that the channel coefficients of IM/DD MIMO-OWC are also nonnegative [23]. In fact, the results in [12, 13] indicate that because of this characteristic of channels, a properly designed space-only transmission of IM/DD MIMO-OWC enables a full diversity gain. As a simple space-only scheme, RC, which repeats the same symbol across all the space dimensions [6, 7, 10, 11], is the best currently available full diversity space code [6, 7, 10–13
] for IM/DD MIMO-OWC, which does not need the transmitter to exactly know channel state information. Here, a natural question is whether there exists a full diversity spatial transmission scheme outperforming RC, which, as far as the best knowledge of authors, is still a longstanding open problem.

Therefore, in this paper, we consider the design of space code for IM/DD MIMO-OWC systems. For such system, it is known that when the transmitter has perfect channel state information, the transmitter aperture selection method was proposed in [24–27 ] and shown to outperform RC. Unfortunately, perfect channel state information at the transmitter is not easily attainable in practice. For this reason, we specifically consider an IM/DD MIMO-OWC system in presence of log-normal fading [28, 29] with a practical assumption that different channel coefficients are independent and have nonidentical variances resulted from diverse atmospheric factors [30, 31]. For this scenario, an error performance design criterion of a space code with an ML detector has been established in [12, 13]. Using this criterion, the design problem of a diversity-optimal space code (DOSC) maximizing both large-scale and small-scale diversity gains is formulated into a max-min optimization problem with continuous-discrete mixed design variables. In [12, 13], linear DOSC is designed for a specific 2 × 2 IM/DD MIMO-OWC with given unipolar pulse amplitude modulation (PAM). Unfortunately, the technique in [12,13] using the Farey sequence in number theory is very difficult to be extended to a general IM/DD MIMO-OWC. In addition, significantly different from the specific linear design for a 2 × 2 IM/DD MIMO-OWC in [12,13], our design in this paper is to consider the DOSC for a general IM/DD MIMO-OWC system within the nonnegative orthant of a multi-dimensional real space without any other additional assumptions on the signal set. By fully using the full large-scale diversity condition on the space code design [12, 13], we will arrange all the space signal vectors in an increasing order and develop a novel geometrically-weighted inequality to attain a closed-form solution to the design problem. It will be proved that RC with a diversity-optimal power allocation is the DOSC for a general IM/DD MIMO-OWC system among all the nonnegative high-dimensional space constellations. In addition, it will be shown that this diversity-optimal design has a fast ML detection algorithm.

## 2. System model and design criterion for space code

In this section, we first introduce the system model considered in this paper, and then, review our recently proposed design criterion for the space code of IM/DD MIMO-OWC systems [12, 13].

#### 2.1. System model

Let us consider an *M* × *N* IM/DD MIMO-OWC system which has *M* receiver apertures and *N* transmitter apertures. These *N* transmitter apertures transmit a space signal **s** = [*s*
_{1}, *s*
_{2}, ··· , *s _{N}*]

*randomly and independently selected with equal probability from a constellation*

^{T}*𝒮*= {

**s**

*, 0 ≤*

_{k}*k*≤ 2

*− 1} to be designed. To meet the nonnegativity condition of intensity modulation, the symbol*

^{K}*s*for

_{n}*n*= 1, 2, ··· ,

*N*to be transmitted from the

*n*-th transmitter aperture is nonnegative, say, $\mathcal{S}\in {\mathcal{R}}_{+}^{N}$. These symbols are then transmitted to the

*M*receiver apertures through an

*M*×

*N*channel matrix

**H**, whose entries are flat-fading path coefficients and also nonnegative. Therefore, the received signal, denoted by an

*M*× 1 vector

**y**, can be represented by

**H**are assumed to be independent and log-normal distributed, i.e.,

*h*=

_{mn}*e*

^{zmn}, where ${z}_{mn}~\mathcal{N}\left({\mu}_{mn},{\sigma}_{mn}^{2}\right)$,

*m*= 1, ··· ,

*M*,

*n*= 1, ··· ,

*N*. The probability density function (PDF) of

*h*is ${f}_{H}({h}_{mn})=\frac{1}{\sqrt{2\pi}{h}_{mn}{\sigma}_{mn}}{e}^{-\frac{{\left(\text{ln}{h}_{mn}-{\mu}_{mn}\right)}^{2}}{2{\sigma}_{mn}^{2}}}$, where ${\sigma}_{mn}^{2}$ is decided by diverse factors such as altitude, the light wavelength, the link distance and the root mean square wind speed [30,31]. These factors usually lead to non-identical ${\sigma}_{mn}^{2}$ for different

_{mn}*m*and

*n*in practical scenarios. The PDF of

**H**is $f(\mathbf{H})={\mathrm{\Pi}}_{m=1}^{M}{\mathrm{\Pi}}_{n=1}^{N}{f}_{H}\left({h}_{mn}\right)$. From [23,30,32,33], the noise vector

**n**is modelled as signal-independent, additive, white and Gaussian noise with zero mean and co-variance matrix $\frac{{\sigma}_{\mathbf{n}}^{2}}{M}{\mathbf{I}}_{M\times M}$.

To facilitate our analysis, in this paper we make the following two assumptions:

*Channel information*. The receiver knows the values of channel coefficients. The transmitter knows*μ*, the mean values of the logarithm of channel coefficients, and the following vector in terms of the channel variances, $\frac{1}{\overline{\sigma}}{\left[{\overline{\sigma}}_{1},\cdots ,{\overline{\sigma}}_{N}\right]}^{T}$, where ${\overline{\sigma}}_{n}={\sum}_{m=1}^{M}{\sigma}_{mn}^{-2}$ and $\overline{\sigma}={\sum}_{n=1}^{N}{\overline{\sigma}}_{n}$._{mn}

#### 2.2. Design criterion for space code

For presentation clarity, let us now introduce our recently established space code design criterion for the IM/DD MIMO-OWC system [12, 13] below:

*Design criterion*: If **e** = (*e*
_{1}, *e*
_{2},··· ,*e _{N}*)

*=*

^{T}**ŝ**−

**s**is positive up to a real-valued scale for any

**s**,

**ŝ**∈

*𝒮*satisfying

**s**≠

**ŝ**, then, in high SNR=

*ρ*regimes, the average pair-wise error probability

*P*(

**s**→

**ŝ**) for a space code of an

*M*×

*N*IM/DD MIMO-OWC system (1) is bounded by ${C}_{L}{\left(\text{ln}\rho \right)}^{-MN}{e}^{-\frac{{\mathcal{D}}_{\text{large}}}{8}{\left(\text{ln}\rho +\text{ln}\overline{\sigma}-\text{ln}\left(M{\sum}_{k=1}^{N}{e}_{k}^{2}\right)\right)}^{2}}\le P(\mathbf{s}\to \widehat{\mathbf{s}})\le {P}_{D}(\mathbf{s}\to \widehat{\mathbf{s}})+\mathcal{O}\left({e}^{-\frac{{\mathcal{D}}_{\text{large}}}{8}{\text{ln}}^{2}\rho}\right)$, where

*𝒟*

_{large}=

*σ̄*, ${C}_{L}=\frac{{\prod}_{m=1}^{M}{\prod}_{n=1}^{N}{\sigma}_{mn}}{{\left(4\pi \right)}^{MN}{e}^{-MN/2}}Q\left(\frac{1}{2}{\left({\sum}_{k=1}^{N}{e}_{k}^{2}\right)}^{-1/2}\right)$ with

*Q*being the standard Q-function and

It should be noted that the results in [12, 13] are for the case when *μ _{mn}* = 0 for any

*m*,

*n*. However, the channel fading not only attenuates the received optical power, but also attenuates the signal. In addition, different channel links may suffer from attenuations. Therefore, in the following, we provide a signal design criterion with

*μ*taken into consideration. By following the performance analysis techniques in [12,13], we can, without much difficulty, attain the corresponding dominant behavior of the average pair-wise error probability in high SNR regimes as follows:

_{mn}*P*(

_{D}**s**→

**ŝ**) is defined in (2).

From (2) and (3), we can observe that in high SNR regimes, the following three major factors significantly affect the decaying speed of *P̃ _{D}* (

**s**→

**ŝ**).

*Large-scale diversity gain*. In a high SNR regime, the first main factor that determines the exponential decaying speed of*P̃*(_{D}**s**→**ŝ**) is*𝒟*_{large}. Therefore,*𝒟*_{large}is named*large-scale diversity gain*[12, 13]. The maximum*𝒟*_{large}or full large-scale diversity gain*σ̄*is achieved when**e**=**ŝ**−**s**is positive up to a scale for any pair of distinct**s**,**ŝ**∈*𝒮*.*Small-scale diversity gain*. The second main factor is ${\mathcal{D}}_{\text{small}}(\mathbf{e})={\prod}_{n=1}^{N}{\left|{e}_{n}\right|}^{{\overline{\sigma}}_{n}}$, which controls the polynomial decaying speed of $\frac{\rho}{{\text{ln}}^{2}\rho}$, and thus, is defined as small-scale diversity gain [12, 13]. When a full large-scale diversity gain has been assured, the worst-case small-scale diversity gain min_{e}*𝒟*_{small}(**e**) should be further maximized.*Coding gain*. Since $\mathcal{G}=(\mathbf{e}){\prod}_{n=1}^{N}{\left|{e}_{n}\right|}^{-{\sum}_{m=1}^{M}{\mu}_{mn}{\sigma}_{mn}^{-2}}$ only affects the horizontal shift of the error curve, this term is defined as*coding gain*. Under the condition that both large-scale diversity and small-scale diversity gains have been optimized, ${\text{max}}_{\mathbf{e}}\mathcal{G}(\mathbf{e}){\prod}_{n=1}^{N}{\left|{e}_{n}\right|}^{-{\sum}_{m=1}^{M}{\mu}_{mn}{\sigma}_{mn}^{-2}}$ should be minimized if there are still design freedoms to be optimized.

## 3. Diversity-optimal power loading for space code

Our primary purpose in this section is to devise a diversity-optimal constellation *𝒮* that maximizes both the large-scale diversity and the small-scale diversity gains. Then, this proposed design will be shown to admit ML detection with low-complexity.

#### 3.1. Diversity-optimal space code

In this subsection, we specifically aim at solving the following optimization problem:

*Design problem*: For any given positive integers *N* and *K*, design a space constellation
$\mathcal{S}=\left\{{\mathbf{s}}_{k},0\le k\le {2}^{K}-1\right\}\subseteq {\mathbb{R}}_{+}^{N}$ such that
${\text{min}}_{0\le {k}_{2}\le {k}_{1}\le {2}^{K-1}}{\prod}_{n=1}^{N}{\left|{s}_{{\mathit{nk}}_{2}}-{s}_{{\mathit{nk}}_{1}}\right|}^{{\overline{\sigma}}_{n}}$ is maximized under conditions that ∀*k*
_{2} > *k*
_{1},
${\mathbf{s}}_{{k}_{2}}-{\mathbf{s}}_{{k}_{1}}\in {\mathbb{R}}_{+}^{N}$, and
$\frac{1}{{2}^{K}}{\sum}_{k=0}^{{2}^{K-1}}{\sum}_{n=1}^{N}{s}_{\mathit{nk}}=1$.

Here, it is noticed that
${\text{min}}_{0\le {k}_{1}\le {k}_{2}\le {2}^{K}-1}{\prod}_{n=1}^{N}{\left|{s}_{{\mathit{nk}}_{2}}-{s}_{{\mathit{nk}}_{1}}\right|}^{{\overline{\sigma}}_{n}}$ has continuous and discrete mixed design variables. Thus, one of the main technical challenges is to analytically solve the inner minimization problem with respect to the discrete variables, i.e.,
${\text{min}}_{0\le {k}_{1}\le {k}_{2}\le {2}^{K}-1}{\prod}_{n=1}^{N}{\left|{s}_{{\mathit{nk}}_{2}}-{s}_{{\mathit{nk}}_{1}}\right|}^{{\overline{\sigma}}_{n}}$. From the above-mentioned design criterion, we know that a full large-scale diversity gain is assured by the condition that **ŝ** − **s** is positive up to a scale for any pair of distinct **ŝ** and **s**. This condition enables us to rearrange all the elements of *𝒮*,
${\mathbf{s}}_{0},\cdots ,{\mathbf{s}}_{{2}^{K}-1}\in {\mathbb{R}}_{+}^{N}$, such that **s**
_{0} < ··· < **s**
_{2K−1}, where notation “**x** < **y**” denotes that all the entries of **y** − **x** are positive. Therefore, for 0 ≤ *k*
_{1} < *k*
_{2} ≤ 2* ^{K}* − 1, it holds that

*s*

_{nk2}≥

*s*

_{n(k1+1)}>

*s*

_{nk1}, giving us ${\prod}_{n=1}^{N}{\left|{s}_{{\mathit{nk}}_{2}}-{s}_{{\mathit{nk}}_{1}}\right|}^{{\overline{\sigma}}_{n}}={\prod}_{n=1}^{N}{\left({s}_{{\mathit{nk}}_{2}}-{s}_{n({k}_{1}+1)}+{s}_{n({k}_{1}+1)}-{s}_{{\mathit{nk}}_{1}}\right)}^{{\overline{\sigma}}_{n}}\ge {\prod}_{n=1}^{N}{\left({s}_{n({k}_{1}+1)}-{s}_{{\mathit{nk}}_{1}}\right)}^{{\overline{\sigma}}_{n}}$, where the equality holds if and only if

*k*

_{2}=

*k*

_{1}+ 1. Therefore, we can attain that ${\text{min}}_{0\le {k}_{1}<{k}_{2}\le {2}^{K-1}}{\prod}_{n=1}^{N}{\left|{s}_{{\mathit{nk}}_{2}}-{s}_{{\mathit{nk}}_{1}}\right|}^{{\overline{\sigma}}_{n}}={\text{min}}_{0\le k\le {2}^{K}-1}{\prod}_{n=1}^{N}{\left({s}_{n(k+1)}-{s}_{\mathit{nk}}\right)}^{{\overline{\sigma}}_{n}}$. This observation leads us to the equivalent form of our original design problem:

*Problem* 1: For arbitrarily fixed positive integers *N* and *K*, find an *N*-dimensional constellation
$\mathcal{S}=\left\{{\mathbf{s}}_{k},0\le k\le {2}^{K-1}\right\}\subseteq {\mathbb{R}}_{+}^{N}$ to maximize
${\text{min}}_{0\le k\le {2}^{K}-2}{\prod}_{n=1}^{N}{\left({s}_{n(k+1)}-{s}_{\mathit{nk}}\right)}^{{\overline{\sigma}}_{n}}$ subject to constraints that **s**
_{k+1} − **s**
* _{k}* >

**0**and $\frac{1}{{2}^{K}}{\sum}_{k=0}^{{2}^{K}-1}{\sum}_{n=1}^{N}{s}_{\mathit{nk}}=1$.

Now, it can be seen that the positivity of the error vectors (up to a scale) not only assures a full large-scale diversity gain, but also plays an important role in equivalently simplifying the inner minimization problem with respect to the discrete variables in the design problem.

For the purpose of attaining a closed-form solution to Problem 1, we first establish a power-product inequality. It is known that log*t* is concave with respect to positive *t*. In addition, the Jensen’s inequality [38] tells us that for *λ _{n}* ≥ 0 with
${\sum}_{n=1}^{N}{\lambda}_{n}=1$, log

*t*satisfies ${\sum}_{n=1}^{N}{\lambda}_{n}\text{log}{t}_{n}\le \text{log}{\sum}_{n=1}^{N}{\lambda}_{n}{t}_{n}$, where the equality holds if and only if

*t*

_{1}= ··· =

*t*. Now, let

_{N}*t*= (

_{n}*s*

_{n(k+1)}−

*s*)/

_{nk}*σ̄*and ${\lambda}_{n}=\frac{{\overline{\sigma}}_{n}}{\overline{\sigma}}$. Then, we have ${\sum}_{n=1}^{N}{\lambda}_{n}\text{log}{t}_{n}={\sum}_{n=1}^{N}\frac{{\overline{\sigma}}_{n}}{\overline{\sigma}}\text{log}\left({s}_{n(k+1)}-{s}_{\mathit{nk}}\right)-{\sum}_{n=1}^{N}\frac{{\overline{\sigma}}_{n}}{\overline{\sigma}}\text{log}{\overline{\sigma}}_{n}=\frac{1}{\overline{\sigma}}\text{log}{\prod}_{n=1}^{N}{\left({s}_{n(k+1)}-{s}_{\mathit{nk}}\right)}^{{\overline{\sigma}}_{n}}-\frac{1}{\overline{\sigma}}\text{log}{\prod}_{n=1}^{N}{\overline{\sigma}}_{n}^{{\overline{\sigma}}_{n}}$ and $\text{log}{\sum}_{n=1}^{N}{\lambda}_{n}{t}_{n}=\text{log}{\sum}_{n=1}^{N}\frac{\left({s}_{n(k+1)}-{s}_{\mathit{nk}}\right)}{\overline{\sigma}}$. Then, combining these two inequalities with ${\sum}_{n=1}^{N}{\lambda}_{n}\text{log}({t}_{n})\le \text{log}{\sum}_{n=1}^{N}{\lambda}_{n}{t}_{n}$ produces $\text{log}{\prod}_{n=1}^{N}{\left({s}_{n(k+1)}-{s}_{\mathit{nk}}\right)}^{{\overline{\sigma}}_{n}}\le \text{log}{\prod}_{n=1}^{N}{\overline{\sigma}}_{n}^{{\overline{\sigma}}_{n}}+\text{log}{\left({\sum}_{n=1}^{N}\left({s}_{n(k+1)}-{s}_{\mathit{nk}}\right)/\overline{\sigma}\right)}^{\overline{\sigma}}$, yielding

_{n}*𝒮*satisfying the power constraint, the following inequality is true,

*𝒮*=

*𝒮̃*= {

**ŝ**

*, 0 ≤*

_{k}*k*≤ 2

*− 1} such that ${\text{min}}_{0\le k\le {2}^{K}-2}{\prod}_{n=1}^{N}{\left({\tilde{s}}_{n(k+1)}-{\tilde{s}}_{\mathit{nk}}\right)}^{{\overline{\sigma}}_{n}}\ge \frac{2\overline{\sigma}{\prod}_{n=1}^{N}{\overline{\sigma}}_{n}^{{\overline{\sigma}}_{n}}}{{\left(\left({2}^{K}-1\right)\overline{\sigma}\right)}^{\overline{\sigma}}}$, then, ∀*

^{K}*k*∈ {0, 1, ··· , 2

*− 2}, it holds*

^{K}*k*∈ {0, 1,...,2

*− 2}, ${\sum}_{n=1}^{N}{\tilde{s}}_{n(k+1)}-{\sum}_{n=1}^{N}{\tilde{s}}_{\mathit{nk}}>2/({2}^{K}-1)$. It follows that ${\sum}_{n=1}^{N}{\tilde{s}}_{\mathit{nk}}>2k/\left({2}^{K}-1\right)+k{\sum}_{n=1}^{N}{\tilde{s}}_{n0}$ for*

^{K}*k*= 1, 2,··· ,2

*− 1. Now, summing all these inequalities yields that ${\sum}_{k=0}^{{2}^{K}-1}{\sum}_{n=1}^{N}{\tilde{s}}_{\mathit{nk}}>2{\sum}_{k=0}^{{2}^{K}-1}k/\left({2}^{K}-1\right)+\left(1+{2}^{K}({2}^{K}-1)/2\right){\sum}_{n=1}^{N}{\tilde{s}}_{n0}={2}^{K}+\left(1+{2}^{K}({2}^{K}-1)/2\right){\sum}_{n=1}^{N}{\tilde{s}}_{n0}\ge {2}^{K}$, which contradicts with our power constraint $\frac{1}{{2}^{K}}{\sum}_{k=0}^{{2}^{K}-1}{\sum}_{n=1}^{N}{s}_{\mathit{nk}}=1$. Thus, inequality (5) is indeed true. By computations, we can have that the upper-bound in (5) is achieved by where $\mathbf{w}=\frac{2{\left[{\overline{\sigma}}_{1},\cdots ,{\overline{\sigma}}_{N}\right]}^{T}}{\overline{\sigma}\left({2}^{K}-1\right)}$. Therefore, the optimal solution to Problem 1 is determined by*

^{K}*𝒮*

_{opt}. Now, for a general IM/DD MIMO-OWC system, the DOSC has been designed. To further appreciate the diversity-optimal design, we make the following three comments:

- Diversity-optimal constellation. The proposed diversity-optimal design reveals the fact that the diversity-optimal space constellation is actually a diversity-optimal power-loaded version of the one-dimensional PAM constellation {
*k*: 0 ≤*k*≤ 2− 1}. Here, it should be emphasized that the DOSC is attained among all the full large-scale diversity space code schemes. Hence, this diversity-optimality of our design is for all space code schemes in the sense of maximizing both the small-scale and large-scale diversity gains. It can be seen that after both diversity gains have been optimized, there are no design freedoms left. Therefore, the diversity-optimality of our proposed result is irrelevant to the values of^{K}*μ*._{mn} - RC diversity-optimality. The existing experimental evidences [7,10] have strongly demonstrated that RC as a spatial diversity transmission scheme is diversity-optimal in the sense of error performance for IM/DD MIMO-OWC over log-normal fading channels with equal variances. However, its mathematical proof remains open for a long time due to the lack of an explicit signal design criterion. As a specific case of the diversity-optimal design, when
*σ̄*_{1}= ··· =*σ̄*, the resulting DOSC is exactly RC. Hence, we actually solve this long-standing open problem under a much weaker condition._{N} - Power loading. The diversity-optimal design only requires the knowledge of $\frac{{\overline{\sigma}}_{n}}{\overline{\sigma}}$ for
*n*= 1, 2, ··· ,*N*rather than each variance ${\sigma}_{mn}^{2}$,*m*= 1, ··· ,*M*,*n*= 1, ··· ,*N*itself. Essentially, the diversity-optimal structure of space code is the diversity-optimal power-loaded space-repetition of the symbols selected from {*k*: 0 ≤*k*≤ 2− 1}.^{K}

#### 3.2. Fast ML detection for diversity-optimal space code

Notice that for the DOSC *𝒮*
_{opt}, the resulting MIMO channel model becomes

### 3.2.1. Classical ML receiver of MIMO channels

For a general MIMO channel model **y** = **Hs** + **n** with **s** ∈ *𝒮*, the output of the classical ML receiver is attained as
$\widehat{\mathbf{s}}={\text{argmin}}_{\mathbf{s}\in \mathcal{S}}{\Vert \mathbf{y}-\mathbf{Hs}\Vert}_{2}^{2}$. To determine the optimal estimate, the receiver has to exhaustively search all the 2* ^{K}* elements in

*𝒮*. Therefore, the complexity of the classical ML receiver is

*O*(2

*).*

^{K}### 3.2.2. Fast ML detector for proposed diversity-optimal design

For our proposed diversity-optimal design *𝒮 _{opt}*, the optimal estimate of

**s**=

**w**

*k*for the model (8) with the ML detector is equivalent to finding a nonnegative integer

*k*such that

**y**and

**Hw**are independent of

*k*. The observation tells us that the optimal ML receiver for our diversity-optimal design is equivalent to a linear zero-forcing receiver. Hence, the optimal estimate of the transmitted signal can be efficiently obtained by using the following algorithm:

**Algorithm 1 (Fast ML detection)**: Given the received signal **y** and the nonzero channel matrix **H**, the output of the ML detector for the DOSC *𝒮*
_{opt} is determined as follows:

From Algorithm 1, we can see that the complexity of the ML detector for our proposed transmission scheme is just *O*(*MN*). Therefore, the ML detector for our proposed diversity-optimal design has much lower complexity.

## 4. Simulation results and discussions

In this section, we carry our computer simulations to examine the performance of our diversity-optimal design by comparing with RC [6, 7, 10–13
], which is the best space code presently available in the literature for this system. Since the average optical power is normalized, the SNR is defined by
$\frac{1}{{\sigma}_{\mathbf{n}}^{2}}$. The channel realizations are generated by producing Gaussian random variables
${z}_{mn}~\mathcal{N}\left({\mu}_{mn},{\sigma}_{mn}^{2}\right)$ first for *m* = 1, ··· , *M*, *n* = 1, ··· , *N* and then, letting *h _{mn}* = exp(

*z*). The channel realizations are independently generated from time slot to time slot.

_{mn}All the schemes to compare are described as follows:

- Repetition code (RC). The space vector of RC for
*M*×*N*IM/DD MIMO-OWC with*K*bits per channel use is given by ${\mathbf{s}}_{k}=\frac{2\times {\mathbf{1}}_{N\times 1}}{N\left({2}^{K}-1\right)}$ with 0 ≤*k*≤ 2− 1.^{K} - Diversity-optimal space code (DOSC). The signal vector of the proposed design in (7) is determined as ${\mathbf{s}}_{k}=\frac{2k{\left[{\overline{\sigma}}_{1},\cdots ,{\overline{\sigma}}_{N}\right]}^{T}}{\overline{\sigma}\left({2}^{K}-1\right)}$, for 0 ≤
*k*≤ 2− 1.^{K}

It can be seen that the above two schemes have the same bit rate, i.e., *K* bits per channel use and normalized average optical power, i.e.,
$\frac{1}{{2}^{K}}{\sum}_{k=0}^{{2}^{K}-1}{\sum}_{n=1}^{N}{s}_{\mathit{nk}}=1$. To make the comparison as fair as possible, all the receivers are based on ML detection, which is equivalent to the zero-forcing receiver. More details of the simulations shown by Figs. 1
–4 are described as follows:

As illustrated in Fig. 1, we find that if max_{n1≠n2,σ̄n1≥σ̄n2}
*σ̄*
_{n1}/*σ̄*
_{n2} is sufficiently large, then, the more SNR gain of our diversity-optimal design over RC is attained. Furthermore, the gap between two error curves becomes larger against increasing SNR. When *σ̄*
_{2}/*σ̄*
_{1} = 0.5/0.001 = 500, as illustrated by Fig. 1, substantial SNR gains can be observed. For example, when looking at the target codeword symbol error rate of 10^{−6}, we can see that our DOSC obtains about the 4 dB SNR gain over RC, which will be expected to become larger if SNR is sufficiently high. This is, again, for the reason that the DOSC provides larger small-scale diversity gain than RC, and the small-scale diversity gain governs the polynomial decaying speed of the error curves. In addition, for *M* = 1, *N* = 2 and
${\sigma}_{1}^{2}=1$,
${\sigma}_{2}^{2}=0.001$, the attained gain, about 4 dB for target error rate of 10^{−6}, is robust to the bit rate as illustrated by Fig. 2. Furthermore, from Fig. 3, when *μ _{n}* is varied and the other parameters are fixed, substantial gains can still be attained. However, when

*μ*

_{1}= 0 and

*μ*

_{2}becomes smaller, the corresponding error curves of RC and our diversity-optimal design will have a horizontal shift to the right and the performance advantage of our proposed design will become noticeable in a higher SNR regime. For example, when

*μ*

_{1}= 0 and

*μ*

_{2}= −1.5, the path long-term attenuation factors exp(

*μ*

_{1}) and exp(

*μ*

_{2}) are far from different. Recall that the small-scale diversity gain of an IM/DD MIMO-OWC system is defined at high SNRs and here the upper end of the SNR range is not sufficiently high. Hence, even when

*μ*

_{1}= 0 and

*μ*

_{2}becomes very small with fixed ${\sigma}_{1}^{2}$ and ${\sigma}_{2}^{2}$, the attained small-scale diversity advantage of our diversity-optimal design is expected to be more noticeable against increasing SNRs.

To put the influence of aperture number on the error performance into perspective, we vary the receiver aperture number *M* and fix the transmitter number to be *N* = 2. The performance comparison is illustrated by Fig. 4. From Fig. 4, we can see that for fixed
${\sigma}_{mn}^{2}$, *μ _{mn}*,

*N*and

*K*, the performance advantage of our diversity-optimal design over RC depends on

*M*and decreases against increasing

*M*. For instance, when

*M*= 1, the attained gain is about 3 dB at the target error rate of 10

^{−8}. However, for the case with

*M*= 8, the attained gain by our diversity-optimal design is smaller than 0.1 dB for the same target error rate. The main reason is that our proposed design is only diversity-optimal and improves the error performance in the high SNR regime. When

*M*is large, the diversity gain provided by our diversity-optimal design will be noticeable when SNR is sufficiently high.

## 5. Conclusion and future work

In this paper, we have designed the DOSC in the sense of maximizing both the large-scale and small-scale diversity gains for a general IM/DD MIMO-OWC system in presence of independent and non-identical log-normal fading. By fully taking advantage of the full large-scale diversity condition and developing a novel power-product inequality, we have attained a closed-form solution to the DOSC design problem. It has been shown that RC with a diversity-optimal power allocation is the DOSC. Simulation results have indicated that substantial performance gains are attained by our diversity-optimal design over RC, which is the best space code currently available in literature for IM/DD MIMO-OWC systems. In future, we will consider the following two research directions: 1) the error performance of our proposed DOSC will be verified via experimental approaches and 2) an asymptotic error performance formula will be established, addressing the overall influence of
${\sigma}_{mn}^{2}$ and *μ _{mn}* on the optimal design of space code.

## Acknowledgments

This work was supported in part by Grant No. 2013AA013603 from China National “863” Program, by Grant 61271253 from China NSFC and by NSERC.

## References and links

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