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IEEE COMMUNICATIONS LETTERS, VOL. 20, NO. 5, MAY 2016
On the Optimality of Spatial Repetition Coding for MIMO Optical Wireless Communications Yan-Yu Zhang, Hong-Yi Yu, Jian-Kang Zhang, Yi-Jun Zhu, Jin-Long Wang, and Xin-Sheng Ji
Abstract—As a spatial diversity transmission scheme, repetition code (RC) is conjectured to be optimal in the sense of error performance for an intensity modulated direct detection multi-inputmulti-output optical wireless communication (IM/DD MIMOOWC) over log-normal fading channels. Despite the fact that all the experimental evidences thus far have strongly demonstrated that this hypothesis is indeed true, its mathematical proof remains a long-standing open problem mainly due to the lack of an explicit signal design criterion like MIMO radio frequency communications. In this letter, subject to two commonly used power constraints, we prove the optimality of RC under a much weaker condition in the sense of maximizing both large-scale and small-scale diversity gains for any space signalling using the recently established pair-wise error probability design criterion for a maximum likelihood (ML) detector. Index Terms—Intensity modulation with direct detection (IM/DD), multiple-input-multiple-output (MIMO), optical wireless communication (OWC), log-normal fading channels, full diversity, repetition coding (RC), space code, maximum likelihood (ML) detector.
I. I NTRODUCTION
R
ECENTLY, intensity modulated direct detection optical wireless communication (IM/DD OWC) has drawn much attention [1]–[4] due to its advantages of low cost, high security, freedom from spectral licensing issues and etc. However, some challenges still remain, especially for atmospheric environments, where some factors, such as rain, snow, fog and temperature variation, affect link performance. To design reliable high date rate IM/DD OWC links, these impairments-induced fading [5] should be taken into consideration. This fading of the received intensity signal can be described by a log-normal (LN) statistical model [6]–[8]. To attain robust performance, multi-input-multi-output OWC (MIMO-OWC) systems provide diversity by designing transmitted symbols distributed over transmitting apertures (space) and (or) symbol periods (time). Contrasted with space-time diversity techniques for MIMO radio frequency communications [9] and coherent Manuscript received September 28, 2015; accepted February 15, 2016. Date of publication March 2, 2016; date of current version May 6, 2016. This work was supported in part by NSFC of China under Grant 61271253, in part by NHTRDP (863 Program) of China under Grant 2013AA013603, in part by open research fund of National Mobile Communications Research Laboratory under Grant 2013D09, and in part by the NSERC. The associate editor coordinating the review of this paper and approving it for publication was C. Zhong. Y.-Y. Zhang, H.-Y. Yu, Y.-J. Zhu, and X.-S. Ji are with the National Digital Switching System Engineering and Technological Research Center, Zhengzhou 450000, China (e-mail:
[email protected];
[email protected];
[email protected];
[email protected]). J.-K. Zhang is with the Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON L8S 4K1, Canada (e-mail:
[email protected]). J.-L. Wang is with the Department of Communication Engineering, University of Science and Technology, Nanjing, China (e-mail: wjl543@sina. com). Digital Object Identifier 10.1109/LCOMM.2016.2537322
MIMO-OWC [10]–[12], the diversity techniques for IM/DD MIMO-OWC have two major differences. The first major difference is that a space-only code can assure a full diversity gain for IM/DD MIMO-OWC [13]. A very simple and attractive scheme to provide this full diversity is repetition coding (RC), which transmits the same symbol across all the transmitter apertures [6], [14]–[16] and has a fast maximum likelihood detector. The second major difference is a nonnegative constraint on the design of signals in IM/DD MIMO-OWC. Because of this reason, the currently well-developed MIMO radio frequency techniques cannot be directly utilized for IM/DD MIMO-OWC. Despite the fact that by properly adding some direct-current components into transmitter designs, orthogonal space time block codes [17]–[21] can be modified for the use in IM/DD MIMO-OWC, its error performance is worse than RC [6], [14]–[16]. Thus far, all computer simulations have shown that RC is the best full diversity space code (FDSC) for IM/DD MIMO-OWC [6], [14]–[16]. Therefore, RC is conjectured to be optimal [14], [15]. However, the rigorous proof of its optimality still remains unsolved due to the lack of an explicit signal design criterion. Fortunately, a general error performance criterion for the design of FDSC has been established by [13], [22]. This criterion reveals that FDSC is attained if and only if all the entries of a non-zero space coding matrix are positive. Particularly for 2 × 2 IM/DD MIMO-OWC with unipolar pulse amplitude modulation (PAM), it was proved that RC is the optimal linear FDSC [13], [22] that maximizes both the large-scale and small-scale diversity gains. However, the techniques using the Farey sequence for the proof developed in [13], [22] is hard to be extended into a general IM/DD MIMO-OWC. In this letter, our main task is to attack this long-standing open problem by providing an affirmative answer. Our contribution can be summarized as follows: 1) Unlike the numerical evidence on RC optimality demonstrated in [6], [14]–[16], we mathematically prove that RC is the optimal FDSC in the sense of maximizing both large-scale and small-scale diversity gains; 2) Contrasted with the specific linear design for a special 2 × 2 IM/DD MIMO-OWC in [13], [22], our proof of RC optimality in this letter is among all high-dimensional space constellations without any additional assumptions; 3) Subject to two commonly used power constraints, it will be proved that the optimal FDSC for any IM/DD MIMO-OWC system is exactly spatialrepetitional transmission of the PAM constellation; 4) The technique developed in this letter provides a new and useful tool to solve a class of the max-min optimization problems with continuous and discrete mixed variables, which are usually encountered in the design of transceivers for digital communication systems.
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ZHANG et al.: OPTIMALITY OF SPATIAL REPETITION CODING FOR MIMO OPTICAL WIRELESS COMMUNICATIONS
II. S YSTEM M ODEL Let us consider an M × N IM/DD MIMO-OWC system having M receiver apertures and N transmitter apertures transmitting a symbol vector x = [x1 , x2 , · · · , x N ]T ∈ {x0 , · · · , x2 K −1 }, denoted by X2 K , where xn for n = 1, 2, · · · , N is non-negative to satisfy the unipolarity requirement of an intensity modulator and represents the symbol to be transmitted from the n-th transmitter aperture. These symbols are then transmitted to the receivers through flat-fading path coefficients, which form the elements of an M × N channel matrix H. Hence, the received signal, denoted by an M × 1 column vector y, can be represented by y = Hx + n,
(1)
where the entries of the channel matrix H are assumed to be independent and LN distributed, i.e., h i j = e zi j , where z i j ∼ N(μi j , σi2j ), i = 1, · · · , M, j = 1, · · · , N . The probability density function (PDF) of h i j is f H (h i j ) = − √ 1 e 2π h i j σi j M N i=1 j=1
(ln h i j −μi j )2 2σi2j
.
The
PDF
of
H
is
f H (H) =
f H h i j . For noise vector n, the two primary sources at the receiver front end are due to noise from the receiver electronics and shot noise from the received direct current photocurrent induced by background radiation [23], [24]. By the central limit theorem, this high-intensity shot noise for the lightwave-based OWC is closely approximated as additive, signal-independent and white Gaussian noise [24] with zero mean and variance σn2 . According to the results of the log-amplitude fluctuation of 11/6 2 d plane waves in [23], [25], σi2j = 0.307 2π i j C h , where λ λ is the optical wavelength, di j is the distance between the j-th transmitter and i-th receiver apertures in meters and C h2 denotes the refractive index structure coefficients and depends on altitude and the wind speed. For a given system, λ and C h2 are fixed. Despite the fact that σi2j may be diverse resulted from different di j in the commonly used IM/DD MIMO-OWC systems, the receiver aperture array is usually symmetrical with respect to the transmitter array a satisfactory align Mto make σi2j is equal for different j. ment and as a result, j = i=1 Therefore, our work in this letter is under assumption that 1 = 2 = · · · = N . The following two commonly used power constraints are considered respectively. 1) Constraint 1: total average optical power. In most practical modulated optical sources, the average optical power N ≤ 1. x constraint is commonly used [24], [26], i.e., E n n=1 2) Constraint 2: total average electrical power. For a unipolar signal [x1 , · · · , x N ]T with the optical average power xi ≥ 0, N constraint depends on E n=1 x n ≤ 1, whereas the electri N 2 ≤ 1, cal average power constraint is given by E x n=1 n which is the standard power measure in digital and wireless communications [27] and thus, also considered for OWC systems [28]. Under the above assumptions, our primary task in this letter is to prove the optimality of RC for any space constellation
847
N , where notation R N denotes a X2 K = {x0 , · · · , x2 K −1 } ⊆ R+ + set consisting of all N -dimensional nonnegative vectors.
III. D IVERSITY G AIN OF S PACE C ODE FOR IM/DD MIMO-OWC AND D ESIGN P ROBLEM In this section, we first review the recently established error performance criterion for the FDSC design and then, formally state our design problem. A. General Design Criterion In this section, we first briefly review the recently established error performance criterion for the space signalling design and then, introduce the concept of large-scale diversity gain and small-scale diversity gain for the IM/DD MIMO-OWC system over log-normal fading channels [13], [22]. Proposition 1: [13], [22] For ∀x, xˆ ∈ X with x = xˆ , if e = xˆ − x is positive up to a scale, then, the average pair-wise error probability for M × N IM/DD MIMO-OWC with a space signalling is bounded by C L (ln ρ)
−
−M N
Dl 8
2 N ek2 ln ρ+ln −ln M k=1
e Dl PU (x → xˆ ) + O e− 8
σi−2 j , CL =
M N
j=1 σi j − MN M N e 2 (4π ) i=1
≤ P(x → xˆ ) ≤ M N , where Dl = = i=1 j=1
− 1 2 N 2 and Q 12 k=1 ek
ln2 ρ
ln(N /M)− 3 ln Ds (e) 4 4 PU x → xˆ = CU G (e) ρ/ln2 ρ × (ln ρ)−M N e where CU =
2 1
N MN M N
e
2 − 8 ln
−
i=1 j=1 σi j M N σi j 2 ) i=1 j=1 (ln |e j |
Dl 8
N M
ln2
ρ ln2 ρ
, Ds (e) =
N
j=1 |e j |
j
ln ln Ds (e)
and G (e) = e 2 (N /M) 2 . To design the optimal FDSC, the following three factors of PU (x → xˆ ) must be optimized: 1) Large-Scale Diversity Gain: The exponent Dl determines the dominant behaviour of PU (x → xˆ ) in a high SNR regime. Also, M N governs the decaying speed of ln ρ and thus, two scales of decaying can be optimized simultaneously. Therefore, Dl is named as the large-scale diversity gain. Full large-scale diversity is achieved when all the M N terms in is utilized. This condition should be satisfied first. M −2 N 2) Small-Scale Diversity Gain: Ds (e) = i=1 |ei | j=1 σi j dictates the polynomial decaying of ρ/ln2 ρ and thus, is called small-scale diversity gain. mine Ds (e) should be maximized after. 3) Coding Gain: G (e) is defined ascoding gain, which affects the horizontal shift of the error curve. The coding gain can be further optimized if there remain freedoms after both diversity gains have been maximized. B. Problem Formulation Without loss of generality, we assume 1 = · · · N = 1. Now, it is time for us to formally state our optimization problem. Consider an M × N IM/DD MIMO-OWC with K bits per channel use (pcu). By Proposition (1), a necessary and sufficient condition for a space code to provide full large-scale diversity
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IEEE COMMUNICATIONS LETTERS, VOL. 20, NO. 5, MAY 2016
is that each non-zero error vector e = xˆ − x is unipolar without zero-valued entries. This condition enables us to sort all the N such that transmitted signal vectors x0 , . . . , x2 K −1 ∈ R+ x0 < · · · < x2 K −1 (2) where notation “a < b” means that the entries of b − a are all positive. Under this condition, we only need to optimize the small-scale diversity gain. Hence, our goal in this letter is to solve the following optimization problem: N min |xnk2 − xnk1 | (3) max xk k1 ,k2 ∈{0,··· ,2 K −1},k2 >k1 n=1 subject to Constraint 1 or Constraint 2. Note that the objective function of (3) has N 2 K continuous variables and 2 K (2 K − 1) discrete variables. Here, the main challenge is how to deal with Nthe first minimization problem: |xnk2 − xnk1 |. Using (2), mink1 ,k2 ∈{0,··· ,2 K −1},k2 >k1 n=1 ∀k1 , k2 ∈ {0, 1, · · · , 2 K − 1} and k2 ≥ k1 + 1, we have N |x − xnk2 ≥ xn(k1 +1) > xnk1 , therefore, resulting in n=1 N N nk2 xnk1 | = n=1 (xnk2 − xn(k1 +1) + xn(k1 +1) − xnk1 ) ≥ n=1 (xn(k1 +1) − xnk1 ), where the equality holds
if and only if k2 = k1 + 1. Hence, we obtain mink1 ,k2 ∈ 0, · · · , 2 K − 1 , k2 > N N k1 n=1 |xnk2 − xnk1 | = mink∈{0,··· ,2 K −2} n=1 (xn(k+1) − problem xnk ). This shows that the original optimization N (xn(k+1) − in (3) is equivalent to maxxk mink∈{0,··· ,2 K −2} n=1 N,x xnk ) subject to xk+1 and xk ∈ R+ k+1 − xk > 0 with a constrained power budget. This significant simplification is a key that allows us to prove the optimality of RC for a general IM/DD MIMO-OWC system. IV. M AIN R ESULTS : O PTIMALITY OF RC The main task in this section is to prove the optimality of RC as an FDSC maximizing the small-scale diversity gain. A. Optimal FDSC Under Constraint 1 Subject to Constraint 1, the optimization problem that maximizes the small scale diversity gain is formulated as N max min xn(k+1) − xnk (4) xk 0≤k≤2 K −2
subject 1 N 2K
to 2 K −1
Fig. 1. Specific scenarios for 1 = 2 .
Theorem 2: The solution to (6) is determined by X2 K = pk, k = 0, 1, · · · , 2 K − 1 .
(7)
√ where p = 61 N ×1 / N (2 K − 1)(2 K +1 − 1). The detailed proof of Theorem 2 is provided in Appendix B. Now, subject to the above two kinds of power constraints, the respective optimal space constellations have been successfully solved and the optimality of RC has been proved mathematically for the general M × N IM/DD MIMO-OWC with K bits pcu. To deeply appreciate the optimality of RC, we would like to make the following two remarks: 1) Optimality of RC: It should be emphasized here that the optimality of RC is proved without assuming whether the FDSC is linear or not. Hence, this optimality is for all FDSCs in the sense of maximizing both the small-scale and large-scale diversity gains. In addition, Theorems 1 and 2 reveal the optimality of RC under the condition of 1 = 2 = · · · = N , which is much weaker than that of all σi j being equal. Fig. 1 shows us two realistic cases for N = 2 that 1 = 2 holds assured by −11/6 −11/6 −11/6 −11/6 + d21 = d12 + d22 even if σi2j is unequal. d11 2) Optimal Space Constellation: It has been noticed that proving the optimality of RC is essentially finding highdimensional space constellation of maximizing the small scale diversity gain. Such an optimal space constellation, in fact, is the constellation that allows each aperture to repeatedly transmit the same point from an equally-spaced unipolar PAM constellation.
n=1
N,x xk+1 , xk ∈ R+ k+1 − xk > 0
n=1 k=0 x nk = 1. Theorem 1: The optimal solution to (4) is given by 2k1 N ×1 K , k = 0, 1, · · · , 2 − 1 X2 K = N 2K − 1
and
(5)
The proof of Theorem 1 is postponed into Appendix A.
V. C ONCLUSION AND D ISCUSSIONS In this letter, we have proved that RC is optimal among all high-dimensional space constellations under two kinds of commonly used power constraints in the sense of maximizing both the small-scale and large-scale diversity gains for a general IM/DD MIMO-OWC with the ML receiver. Our result in this letter is for the case with 1 = · · · = N . In addition, for the case with nonidentical i , the optimal space code design is under consideration.
B. Optimal FDSC Under Constraint 2 Subject to Constraint 2, the corresponding max-min design problem is given by N max xn(k+1) − xnk min xk k∈{0,··· ,2 K −2} n=1 N,x xk+1 , xk ∈ R+ k+1 − xk > 0, s.t. 1 2 K −1 N (6) 2 n=1 x nk = 1 k=0 2K
A PPENDIX A. Proof of Theorem 1 First, it can be verified that with (5), we have N by computation xn(k+1) − xnk = 2 N /(N (2 K − 1)) N . mink∈{0,··· ,2 K −2} n=1 In the following, we will prove that this is indeed the maximum by contradiction. Suppose that there k=2 K −1 N satisfying ⊆ R+ exists a space constellation {˜xk }k=0
ZHANG et al.: OPTIMALITY OF SPATIAL REPETITION CODING FOR MIMO OPTICAL WIRELESS COMMUNICATIONS
N x˜ k+1 − x˜ k > 0, x˜n(k+1) − x˜nk > mink∈{0,1,··· ,2 K −2} n=1 N 2 K −1 N and 21K k=0 2N / N 2K − 1 n=1 x˜ nk = 1. Then, N K N < ∀k ∈ {0, · · · , 2 − 2}, we have 2 / N 2 K − 1 N mean and geon=1 ( x˜ n(k+1) − x˜ nk ). Using the arithmetic N metrical mean inequality produces x ˜ n=1 n(k+1) − x˜ nk ≤ N N N 1 x ˜ − x ˜ . By substituting this n(k+1) nk N i=1 i=1 N N N x˜n(k+1) − x˜nk , < n=1 inequality into 2 N / N 2 K − 1 K we N arrive at Nthe fact 2that ∀k ∈ {0, · · · , 2 − 2}, n=1 x˜ n(k+1) − n=1 x˜ nk > 2 K −1 , which leads us to N N K x˜nk > 2 K2k−1 + k n=1 x˜n0 . Now, ∀k ∈ {1, . . . , 2 − 1}, n=1 2 K −1 N summing all these inequalities yields that k=0 n=1 x˜ nk ≥ N 2 2 K −1 K −1 2 K − 1 K + (1+ k + 1+2 x ˜ = 2 n=1 n0 2 K −1 k=0 N K −1 K K 2 −1 2 n=1 x˜ n0 ≥ 2 , which contradicts with the 1 2 K −1 N power constraint 2 K k=0 n=1 x˜ nk = 1. Therefore, space constellation (5) is indeed the optimal solution to (4). This completes the proof of Theorem 1. B. Proof of Theorem 2 Note that with the solution (7), the corresponding objective function of (6) is valued at mink∈{0,··· ,2 K −2} K K +1 N N N 2 − 1 2 . We n=1 x n(k+1) − x nk = 6 / N 2 − 1 2 prove this theorem by contradiction. Suppose that there exists k=2 K −1 N such that x ˘ k+1 − x˘ k > 0, some constellation {˘xk }k=0 ⊆ R+ N N mink∈{0,··· ,2 K −2} n=1 x˘n(k+1) − x˘nk > 6 2 / N 2 K − 1 K +1 N 2 K −1 N 2 2 − 1 2 and 21K k=0 n=1 x˘ nk = 1. Hence, ∀k ∈
N N 0, · · · , 2 K − 2 , we have 6 2 / N 2 K − 1 2 K +1 − 1 2 < N N N − x ˘ − x ˘ x ˘ ≤ x ˘ /N , where n(k+1) nk n(k+1) nk n=1 n=1 the last inequality is resulted from arithmetic mean and geometrical mean equality. This inequality gives us √ N x˘n(k+1) − x˘nk . Then, 6N / 2 K − 1 2 K +1 − 1 < n=1 √ N we get x ˘ ≥ k 6N / 2 K − 1 2 K +1 − 1 + n(k+1) n=1 √ N k 6N K n=1 x˘ n0 ≥ √ 2 K −1 2 K +1 −1 for 0 ≤ k ≤ 2 − 2. Based on ( )( ) this result, using Cauchy-Schwarz Sum Inequality produces 2 N N N 2 2 x ˘ ≤ N n=1 x˘n(k+1) and thus, n=1 x˘n(k) > n(k+1) n=1 6k for 1 ≤ k ≤ 2 K − 1. Then, summing up this (2 K −1)(2 K +1 −1) 2 K −1 N 1 2 K −1 2 inequalities yields that 21K k=0 n=1 x˘ n(k+1) > 2 K k=1 2
6k = 1, contradicting the power constraint 21K (2 K −1)(2 K +1 −1) 2 K −1 N 2 n=1 x˘ nk = 1. Thus, (7) is indeed the optimal solution k=0 to (6). This completes the proof of Theorem 2.
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