Channel capacity and receiver deployment optimization for multi-input multi-output visible light communications

Jin-Yuan Wang; Jianxin Dai; Rui Guan; Linqiong Jia; Yongjin Wang; Ming Chen

doi:10.1364/OE.24.013060

1. Introduction

Recently, as the light emitting diode (LED) becomes more and more popular, much research has been focused on visible light communications (VLC) [1]. In point-to-point VLC, achieving high transmission rate is challenging simultaneously with illuminating consistently due to the small bandwidth of the LED [2], but the typical illumination levels in VLC offer very high signal-to-noise ratio (SNR) [3]. The availability of a large number of high SNR channels with sufficient bandwidth makes the multi-input multi-output (MIMO) technique be an efficient transmission scheme for VLC to improve system performance significantly.

In VLC, both illumination and communication are simultaneously implemented. Moreover, the input signal is non-negative and the average optical power cannot change with time. Therefore, the derived results in traditional wireless radio frequency communications are not directly applicable to VLC. So far, the channel capacity for the point-to-point VLC is investigated. In [4], the capacity for VLC is analyzed by using the inverse source coding. In [5], the capacity bounds are derived for VLC under non-negativity and average power constraints. Moreover, by adding a peak power constraint, the capacity bounds for VLC are derived in [6]. In [7], the capacity bounds for VLC with input-dependent noise is analyzed. To the best of our knowledge, the capacity bounds for the MIMO VLC have not been discussed yet in open literature.

In addition, the deployment of the transceivers in MIMO VLC is one of the most important problems. The performance of the system can be dramatically improved by planning the positions of the transceivers. For a MIMO VLC system, the received power is derived in [8], and the conditions that the optimal locations of the transmitters satisfied are given. In [9], the received SNR is analyzed, and proposed a deployment scheme for LEDs, i.e., 4 LEDs are deployed in the corners and 12 LEDs are deployed evenly on a circle. In [10], the received SNR and bit error rate under three deployment schemes for LEDs are discussed. Note that [8–10] investigate the deployment for LEDs. By reasonably planning the positions of the LEDs, the received power or SNR in each position on the receive plane can be as large as possible. However, such deployment may not achieve the maximum capacity. Moreover, in practical indoor environment, the receivers are not deployed in every positions of the receive plane, there are only several places are used for deploying receivers. Therefore, when given the positions of LEDs, how to deploy the receivers to achieve maximum capacity should be studied in future.

Motivated by the above literature, this paper investigates the channel capacity and the receiver deployment optimization for MIMO VLC systems. Under the non-negativity, peak power and dimmable average power constraints, the system model for the MIMO VLC is derived. Assume that the perfect channel state information at the transmitter (CSIT) is known, the channel matrix is converted into several parallel, non-interfering sub-channels by using the singular value decomposition (SVD). According to the principle of the entropy power inequality (EPI) and variational methods, the closed-form expression of the lower bound on the channel capacity for MIMO VLC is derived. Based on the theoretical expression of the lower bound of the capacity, the receiver deployment optimization problem is proposed, which has been shown to be a NP-hard problem without computational efficient algorithms to obtain the optimal solution [11]. From an implementation point of view, heuristic optimization algorithms are preferred. As a very promising heuristic optimization algorithm, the particle swarm optimization (PSO) is employed to solve the optimization problem. Numerical results verify the derived capacity bound and the proposed deployment optimization scheme.

The remainder of this paper is organized as follows. The system model is described in Section 2. In Section 3, the channel capacity for the MIMO VLC is analyzed. Section 4 formulates the receiver deployment optimization problem and proposes a scheme to solve the problem. Simulation results are presented in Section 5 before conclusions are drawn in Section 6.

2. System Model

2.1. Received Signal

Consider an indoor MIMO VLC system, as shown in Fig. 1, the length, width and height of the room are a, b and c, respectively. At the transmitter, N_T LEDs are uniformly deployed on the ceiling, and the coordinate of each LED is $(x_{j}^{T}, y_{j}^{T}, c)$ , j = 1,2,·⋯, NT. At the receiver, N_R phonodiodes (PDs) are located on a receive plane with the height c^R, and the coordinate of each PD is $(x_{j}^{R}, y_{j}^{R}, c^{R})$ , i = 1,2, ⋯, NR. Mathematically, the received N_R × 1 dimensional signal vector Y at the receiver can be written as

Y = r GX + Z

where

X = {(X_{1}, X_{2}, \dots, X_{N_{T}})}^{T}

denotes the input signal vector. Z is the additive white Gaussian noise vector, and each element in Z follows a Gaussian distribution with zero mean and variance σ². r is the optoelectronic conversion factor. G is the channel matrix, which is given by

G = [\begin{matrix} h_{1, 1} & h_{1, 2} & \dots & h_{1, N_{T}} \\ h_{2, 1} & h_{2, 2} & \dots & h_{2, N_{T}} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ h_{N_{R}, 1} & h_{N_{R}, 2} & \dots & h_{N_{R}, N_{T}} \end{matrix}]

where h_i,j represents the channel gain between the i-th PD and the j-th LED.

Fig. 1 An indoor MIMO VLC system

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Because r scales the SNR only, without loss of generality, the value of r is set to one. Therefore, (1) can be simplified as

Y = GX + Z

2.2. Channel Gain

In (3), the channel gain h_i,j in (2) can be written as [12]

h_{i, j} = {\begin{array}{l} \frac{(m + 1) A r}{2 π D_{i, j}^{2}} T_{s} g \cos^{m} (φ_{i, j}) \cos (ψ_{i, j}), & if 0 \leq ψ_{i, j} \leq Ψ_{c, i} \\ 0, & if ψ_{i, j} \geq Ψ_{c, i} \end{array}

where m is the order of the Lambertian emission, A_r is the receiver area of the PD, T_s and g are the optical filter gain and the concentrator gain of the PD, respectively. Ψ_c,_i is the field of view (FOV) of the i-th PD. D_i,j, φ_i,j and ψ_i,j are the distance, the angle of irradiance and the angle of incidence from the j-th LED to the i-th PD, as shown in Fig. 2.

Fig. 2 The distance and angle relationship between the j-th LED and the i-th PD

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According to Fig. 2, we have

\cos (φ_{i, j}) = \cos (ψ_{i, j}) = \frac{c - c^{R}}{\sqrt{{(x_{j}^{T} - x_{i}^{R})}^{2} + {(y_{j}^{T} - y_{i}^{R})}^{2} + {(c - c^{R})}^{2}}}

In this paper, each PD employs very large FOV. Assume that no matter where transceiver is located in, 0 ≤ ψ_i,j ≤ Ψ_c,_i always hold. Therefore, (4) can be rewritten as

h_{i, j} = \frac{(m + 1) A_{r} T_{s} g {(c - c^{R})}^{m + 1}}{2 π {[{(x_{j}^{T} - x_{i}^{R})}^{2} + {(x_{j}^{T} - x_{j}^{R})}^{2} + {(c - c^{R})}^{2}]}^{\frac{m + 3}{2}}}

2.3. Signal Constraints

In VLC, the information is modulated as the instantaneous optical intensity, and thus all signals in X should be non-negative. In addition, due to the physical limitation of the LED, the peak optical power constraint should also be considered. Therefore, we have

0_{N_{T} \times 1} \leq X \leq A I_{N_{T} \times 1}

where A is the peak optical power of each LED.

Considering the illumination requirement in VLC, the average optical power cannot be changed but can be adjusted according to the users’ requirement (dimming target). Thus, the dimmable average power constraint can be written as

E (X) = ξ P I_{N_{T} \times 1}

where 0 < ξ ≤ 1 is the dimming target. P is the nominal optical power of each LED, which satisfies P ≤ A.

3. MIMO Capacity Analysis

3.1. Channel Decomposition

Assume that perfect CSIT is known, and thus the N_R × N_T dimensional channel matrix G can be converted to min{N_R, N_T} parallel, non-interfering sub-channels by using SVD, i.e.,

G = UD V^{H}

where U and V are N_R × N_R dimensional and N_T × N_T dimensional unitary matrixes. N_R × N_T dimensional matrix D = diag(λ₁, ⋯, λ_Γ, 0, ⋯, 0) is a diagonal matrix with non-negative entries, where Γ ≤ min {N_R, N_T} is the rank of G.

Let $\tilde{X} = V^{H} X$ , $\tilde{Y} = U^{H} Y$ and $\tilde{Z} = U^{H} Z$ , and thus the MIMO channel is converted to parallel, non-interfering sub-channels, which can be described as [13]

\tilde{Y} = U^{H} Y = D \tilde{X} + \tilde{Z}

where

\tilde{X}

and

\tilde{Y}

are the input and output for the transformed system.

\tilde{Z}

is the noise vector for the transformed system, and the elements in

\tilde{Z}

will be i.i.d. and have the same variance as Z. Figure 3 shows the principle diagram of the SVD for MIMO VLC.

Let $\hat{X} = D \tilde{X} = {({\hat{X}}_{1}, \dots, {\hat{X}}_{Γ}, 0, \dots 0)}^{T} = {(λ_{1} {\tilde{X}}_{1}, \dots, λ_{Γ} {\tilde{X}}_{Γ}, 0, \dots, 0)}^{T}$ , we have

\hat{X} = D V^{H} X = {(λ_{1} \sum_{j = 1}^{N_{T}} v_{j, 1} X_{j}, \dots, λ_{Γ} \sum_{j = 1}^{N_{T}} v_{j, Γ} X_{j}, 0, \dots, 0)}^{T}

where v_i,j, u_i,j and h_i,j are the i-th row and the j-th column element of matrixes V, U and G, respectively. Therefore, (10) can be transformed as

\tilde{Y} = \hat{X} + \tilde{Z}

Fig. 3 MIMO VLC channel decomposition

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3.2. Lower Capacity Bounds of Sub-channels

For the n-th sub-channel, (12) can be written as a scalar form

{\tilde{Y}}_{n} = {\hat{X}}_{n} + {\tilde{Z}}_{n}

According to (11), the relationship between ${\hat{X}}_{n}$ and X_j can be expressed as

{\hat{X}}_{n} = λ_{n} \sum_{j = 1}^{N_{T}} v_{j, n} X_{j}, n = 1, 2, \dots, Γ

From (7), we have 0 ≤ X_j ≤ A, ∀j ∈ {1,2, ⋯, N_T}, and thus

A \min {0, v_{j, n}} \leq v_{j, n} X_{j} \leq A \max {0, v_{j, n}}

Therefore, ${\hat{X}}_{n}$ should satisfy the following constraint

{\hat{A}}_{n} \leq {\hat{X}}_{n} \leq {\hat{B}}_{n}, n = 1, 2, \dots, Γ

where

{\hat{A}}_{n} = λ_{n} A \sum_{j = 1}^{N_{T}} \min {0, v_{j, n}}

,

{\hat{B}}_{n} = λ_{n} A \sum_{j = 1}^{N_{T}} \max {0, v_{j, n}}

.

According to (8), the average optical power of the n-th element ${\hat{X}}_{n}$ in $\hat{X}$ can be written as

E ({\hat{X}}_{n}) = ξ P λ_{n} \sum_{j = 1}^{N_{T}} v_{j, n}

Furthermore, the mutual information between ${\hat{X}}_{n}$ and ${\tilde{Y}}_{n}$ can be expressed as

\begin{array}{r} I ({\hat{X}}_{n}; {\tilde{Y}}_{n}) = H ({\tilde{Y}}_{n}) - H ({\tilde{Y}}_{n} | {\hat{X}}_{n}) \\ = H ({\hat{X}}_{n} + {\tilde{Z}}_{n}) - H ({\tilde{Z}}_{n}) \end{array}

where H(·) denotes the entropy.

For two mutually independent variables ${\hat{X}}_{n}$ and ${\tilde{Z}}_{n}$ , the following EPI always hold [14]

e^{2 H ({\hat{X}}_{n} + {\tilde{Z}}_{n})} \geq e^{2 H ({\hat{X}}_{n})} + e^{2 H ({\tilde{Z}}_{n})}

Therefore, (18) can be further written as

\begin{array}{r} I ({\hat{X}}_{n}; {\tilde{Y}}_{n}) \geq \frac{1}{2} \ln (e^{2 H ({\hat{X}}_{n})} + e^{2 H ({\tilde{Z}}_{n})}) - H ({\tilde{Z}}_{n}) \\ = \frac{1}{2} \ln (1 + \frac{\exp {- 2 \int_{{\hat{A}}_{n}}^{{\hat{B}}_{n}} f_{{\hat{X}}_{n}} (x) \ln [f_{{\hat{X}}_{n}} (x)] d x}}{2 π e σ^{2}}) \end{array}

Under the given constraints (16) and (17), maximizing the left hand side of (20) with respect to $f_{{\hat{X}}_{n}} (x)$ , the capacity for the n-th sub-channel can be derived. However, it is very hard or even impossible to obtain the actual channel capacity for the MIMO VLC. Fortunately, a lower bound on the capacity can be derived by maximizing the right hand side of (20) with respect to $f_{{\hat{X}}_{n}} (x)$ under constraints (16) and (17). To obtain a tight lower bound, we define a functional

J [f_{{\hat{X}}_{n}} (x)] = \int_{{\hat{A}}_{n}}^{{\hat{B}}_{n}} f_{{\hat{X}}_{n}} (x) \ln [f_{{\hat{X}}_{n}} (x)] d x

And then consider the following functional optimization problem

s . t . \begin{array}{l} \min_{f_{{\hat{X}}_{n}} (x)} J [f_{{\hat{X}}_{n}} (x)] \\ \int_{A_{n}}^{{\hat{B}}_{n}} f_{{\hat{X}}_{n}} (x) d x = 1 \\ \int_{A_{n}}^{{\hat{B}}_{n}} x f_{{\hat{X}}_{n}} (x) d x = ξ P λ_{n} \sum_{j = 1}^{N_{T}} v_{j, n} \end{array}

The functional optimization problem (22) can be solved by using the variational method [15]. Define the average-to-peak-power ratio (APPR) of the n-th sub-channel be α_n = ξ P/A, and the following theorem is derived.

Theorem 1 (1) When α_n = 0.5, the solution of optimization problem (22) can be derived as 1

f_{{\hat{X}}_{n}} (x) = \frac{1}{{\hat{B}}_{n} - {\hat{A}}_{n}}, {\hat{A}}_{n} \leq x \leq {\hat{B}}_{n}

(2) When α_n ≠ 0.5 and α_n ∈ (0,P/A], the solution of optimization problem (22) can be derived as

f_{{\hat{X}}_{n}} (x) = \frac{c_{0}^{'}}{e c_{0}^{'} {\hat{B}}_{n} - e c_{0}^{'} {\hat{A}}_{n}} e^{c_{0}^{'} x}, {\hat{A}}_{n} \leq x \leq {\hat{B}}_{n}

where c′₀ is the solution to

α_{n} = \frac{{\hat{B}}_{n} e^{c_{0}^{'} {\hat{B}}_{n}} - {\hat{A}}_{n} e^{c_{0}^{'} {\hat{A}}_{n}}}{(e^{c_{0}^{'} {\hat{B}}_{n}} - e^{c_{0}^{'} {\hat{A}}_{n}}) ({\hat{A}}_{n} + {\hat{B}}_{n})} - \frac{1}{c_{0}^{'} ({\hat{A}}_{n} + {\hat{B}}_{n})}

Proof 1 See Appendix A.

According to Theorem 1, substituting (23) or (24) into the right hand side of (18), the lower bound on the capacity for the n-th sub-channel can be derived, as shown in the following theorem.

Theorem 2 (1) When α_n = 0.5, the lower capacity bound for the n-th sub-channel is given by

C_{n} \geq \frac{1}{2} \ln [1 + \frac{{(λ_{n} A \sum_{j = 1}^{N_{T}} | v_{j, n} |)}^{2}}{2 π e σ^{2}}]

(2) When α_n ≠ 0.5 and α_n ∈ (0,P/A], the lower capacity bound for the n-th sub-channel is given by

\begin{matrix} C_{n} \geq \frac{1}{2} \ln {1 + [\frac{\exp (- c_{0}^{'} ξ P λ_{n} \sum_{j = 1}^{N_{T}} v_{j, n})}{c_{0}^{'} \sqrt{2 π e σ^{2}}} \\ {\times (\exp (c_{0}^{'} λ_{n} A \sum_{j = 1}^{N_{T}} \max {0, v_{j, n}}) - \exp (c_{0}^{'} λ_{n} A \sum_{j = 1}^{N_{T}} \min {0, v_{j, n}}))]}^{2}} \end{matrix}

Proof 2 See Appendix B.

3.3. Lower Capacity Bound of MIMO Channel

After using SVD, the channel capacity for MIMO VLC can be expressed as

C_{MIMO} = \sum_{n = 1}^{Γ} C_{n}

Substituting (26) or (27) into (28), the lower capacity bound of the MIMO channel is derived. i.e., when α_n = 0.5, the lower capacity bound for MIMO VLC can be written as

C_{MIMO} \geq C_{Low} = \frac{1}{2} \sum_{n = 1}^{Γ} \ln [1 + {(\frac{λ_{n} A \sum_{j = 1}^{N_{T}} | v_{j, n} |}{\sqrt{2 π e σ^{2}}})}^{2}]

When α_n ≠ 0.5 and α_n ∈ (0,P/A], the lower capacity bound for MIMO VLC is given by

\begin{array}{l} C_{MIMO} \geq C_{Low} = \frac{1}{2} \sum_{n = 1}^{Γ} \ln {1 + [1 + \frac{\exp (- c_{0}^{'} P λ_{n} \sum_{j = 1}^{N_{T}} v_{j, n})}{c_{0}^{'} \sqrt{2 π e σ^{2}}} \\ {\times (\exp (c_{0}^{'} λ_{n} A \sum_{j = 1}^{N_{T}} \max {0, v_{j, n}}) - \exp (c_{0}^{'} λ_{n} A \sum_{j = 1}^{N_{T}} \min {0, v_{j, n}}))]}^{2}} \end{array}

4. Receiver Deployment Optimization

4.1. Problem Formulation

Given the positions of the transmitters, this subsection will planning the positions of N_R receivers by maximizing the lower capacity bound for MIMO VLC. Mathematically, the optimization problem can be formulated as

s . t . \begin{array}{l} \max_{W} C_{Low} \\ 0 \leq x_{i}^{R} \leq a, i = 1, 2, \dots, N_{R} \\ 0 \leq y_{i}^{R} \leq b, i = 1, 2, \dots, N_{R} \end{array}

where

W = {(x_{1}^{R}, \dots, x_{N_{R}}^{R}, y_{1}^{R}, \dots, y_{N_{R}}^{R})}^{T}

denotes the position vector of N_R receivers. In problem (31), Γ, λ_n and v_j,n in the objective function C_Low can be determined by SVD, which are implicit functions of W. Therefore, problem (31) is a non-convex and non-linear optimization problem, which has been shown to be NP-hard [11].

4.2. Problem Solving

In problem (31), N_R receiver positions have infinite possibility, which make the problem hard to solve. Intuitively, the positions of the receivers can be discretized. Such discretized positions can be taken as the candidate positions of the receivers, the sub-optimal solutions can be obtained by using traversing search. However, the complexity of such method is too high, especially when the number of discrete points is large or when N_R is large. Therefore, such method is not practical. From an implementation point of view, computationally efficient algorithms are more preferred. In this subsection, a PSO based iterative searching algorithm is proposed to solve the problem (31).

PSO is one of the most important intelligence optimization algorithms, which was originally introduced by J. Eberhart and R. C. Kennedy [16]. A physical analogy might be a swarm of birds searching for a food source. In the analogy, by using its own flying experience and its companions’ flying experience, each particle (i.e. each bird) adjusts its flying to find the best available food source [17]. When employing PSO to solve problem (31), the first step is to map the problem into the structure of the PSO. In PSO, each particle contains two vectors, i.e., position vector and velocity vector. In problem (31), the coordinates of N_R receivers make up of the position vector, i.e., $W = {[x_{1}^{R}, x_{2}^{R}, \dots, x_{N_{R}}^{R}, y_{1}^{R} y_{2}^{R}, \dots, y_{N_{R}}^{R}]}^{T}$ , where $(x_{i}^{R} y_{i}^{R})$ is the coordinate of the i-th receiver. According to local best position, global best position and velocity, the current position vector of each particle is updated.

Assume that the swarm contains K particles, the velocity and position of the k-th particle in the t-th iteration can be written as

V_{k} (t + 1) = ω_{t} V_{k} (t) + c_{1} φ [Q_{j}^{L} - W_{k} (t)] + c_{2} ψ [Q * - W_{k} (t)]

W_{k} (t + 1) = W_{k} (t) + V_{k} (t + 1)

where

V_{k} (t) = {[v_{k, 1} (t), \dots, v_{k, 2 N_{R}} (t)]}^{T}

and

W_{k} (t) = {[x_{1, k}^{R} (t), \dots, x_{N_{R}, k}^{R} (t), y_{1, k}^{R} (t), \dots, y_{N_{R}, k}^{R} (t)]}^{T}

de-note the velocity vector and position vector of the k-th particle in the t-th iteration, respectively. c₁ and c₂ are learning factors, which are non-negative constant. φ and ψ are two independent random numbers uniformly distributed in the range of [0,1].

Q_{k}^{L}

denotes the current local best position vector of the k-th particle, and Q^∗ denotes the current global best position vector. ω_t is the inertia weight, which can be derived as

ω_{t} = ω_{\max} - \frac{ω_{\max} - ω_{\min}}{T_{\max}} \times t

where ω_max and ω_min denote the maximum and minimum weights. T_max represents the maximum iteration number.

To facilitate finding the solution of problem (31), a specific criterion named fitness will be used to evaluate the newly derived position vector of the swarm. Obviously, the particle with larger capacity bound should have a higher fitness value. Thus, the fitness function is chosen as

f_{k}^{fit} (t) = C_{Low} (W_{k} (t))

where C_Low(W_k(t)) is determined by (29) or (30).

After each evaluation, for the k-th particle, if the fitness of the newly obtained position vector W_k(t) outperforms that of the local best position vector $Q_{k}^{L}$ , $Q_{k}^{L}$ will be updated to W_k(t). Similarly, for the whole swarm, if the position vector of a particle W_l(t) outperforms that of the global best position vector Q^∗, Q^∗ will be updated to W_l(t). In most cases, a large number of iterations will be performed. For each iteration, every particle updates the velocity and the position vectors using (32) and (33), respectively, until the specified number of generations is exceeded. To facilitate the understanding, the proposed PSO-based iterative searching algorithm is illustrated as follows.

Algorithm 1. PSO-based iterative searching algorithm

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5. Simulation Results

In this section, the derived lower capacity bound and the proposed iterative searching algorithm will be verified through simulations. The main simulation parameters are shown in Table 1.

Table 1. Main simulation parameters

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5.1. Channel Capacity Simulations

In this subsection, when given the positions of the transmitters, the lower capacity bound will be evaluated by some representative simulations. In the simulation, the positions of the receivers are uniformly distributed, the coordinates are set to be (ja/(N_R + 1),b/2,c^R), ∀j = 1, ⋯, N_R.

Figure 4 illustrates the lower capacity bound versus the peak power of the LED A for different numbers of the receivers when A = 2P and ξ = 0.3. It can be seen that, with the increase of A, the lower capacity bound increases. Moreover, the lower capacity bound also increase with the increase of N_R. However, with the increase of the number of receivers, the increasing trend of the lower capacity bound becomes more and more slowly.

Fig. 4 Lower capacity bound versus peak power of the LED A for different numbers of the receivers when A = 2P and ξ = 0.3

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Figure 5 shows the lower capacity bound versus the dimming target ξ for different APPRs when A = 60 dB and N_R = 3. As can be seen, when A = P, the lower capacity bound increases and then decreases with the increase of ξ. Moreover, the curve in this case is completely symmetric with respect to ξ = 0.5. When A = 1.5P and A = 2P, the lower capacity bound also increases and then decreases with the increase of ξ. However, the curve in this case is asymmetric with respect to ξ = 0.5. Specifically, when ξ = 1 and A = P, the capacity bound is zero, which indicates that the VLC with full-illumination target (i.e., ξ = 1) cannot be used for data transmission. When ξ = 1 and A ≠ P, the capacity bound is not zero, which indicates that the VLC in this case can also be used for data transmission.

Fig. 5 Lower capacity bound versus dimming target ξ for different APPRs when A = 60 dB and N_R = 3

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5.2. Receiver Deployment Simulations

In this subsection, the performance of the proposed PSO-based iterative searching algorithm will be verified. The simulation parameters for the proposed algorithm are shown in Table 2.

Table 2. Simulation parameters for the PSO algorithm

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Figure 6 shows the lower capacity bound versus the iteration number in the proposed PSO-based iterative searching algorithm. As is illustrated in Fig. 6, with the increase of the iteration number, the value of the lower capacity bound becomes larger and larger, and tends to be stable finally. This indicates that the lower capacity bound converges to a good solution.

Fig. 6 Lower capacity bound versus the iteration number

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Figure 7 shows the vertical view of LEDs’ and PDs’ distributions in XY-plane when N_R =1, 2, 3 and 4, respectively. The XY-coordinates of 4 LEDs are fixed as (1.25,1), (1.25,3), (3.75,1) and (3.75,3), while the XY-coordinates of N_R PDs are determined by using the proposed algorithm. Obviously, when N_R = 1, to obtain the higher capacity, the PD is not located in the center of the receive plane, but located in (2, 2). According to the symmetry of the receive plane, the position (3,2) can also achieve the same capacity. When N_R = 2, the two PDs are placed on the center of Y-axis, the coordinates of the two PDs are (1.15, 2) and (3.85, 2). When N_R = 3, two PDs are almost located underneath the two LEDs, another PD is located on the center of Y-axis. When N_R = 4, the four PDs are almost located underneath the four LEDs.

Fig. 7 The distribution of the receivers when N_R=1, 2, 3 and 4

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Furthermore, Fig. 8 the vertical view of LEDs’ and PDs’ distributions in XY-plane when N_R =15, 20, 25 and 30, respectively. From Fig. 8, it can be observed that when the number of the PDs are large, the receivers are almost located underneath the LEDs to obtain a large capacity, which provides a certain reference for future location planning in VLC.

Fig. 8 The distribution of the receivers when N_R=15, 20, 15 and 30

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6. Conclusion

Considering the non-negativity, peak power and dimmable average power constraints, this paper investigates the channel capacity bound and the receiver deployment optimization for MIMO VLC. By using SVD, the MIMO channel is transformed to several parallel, non-interfering sub-channels. After that, the lower bound on the channel capacity is derived by employing the EPI and variational method. To obtain appropriate positions of the receivers, a receiver deployment optimization problem is proposed, and a PSO-based iterative searching algorithm is proposed to solve the problem. Simulation results show that the channel capacity increases with the increase of the number of receivers. Moreover, when the receivers are located underneath the transmitters, the MIMO system can achieve large capacity.

Appendix A Proof of Theorem 1

Assume that ${\tilde{f}}_{{\hat{X}}_{n}} (x)$ is the solution to problem (22), define a perturbation function ${\tilde{f}}_{{\hat{X}}_{n}} (x)$ as follows

{\tilde{f}}_{{\hat{X}}_{n}} (x) = f_{{\hat{X}}_{n}} (x) + ε η (x)

where ε is a variable. The perturbation function

{\tilde{f}}_{{\hat{X}}_{n}} (x)

should also satisfy the two constraints in problem (22). Equivalently, η(x) in (36) should satisfy

\begin{array}{l} \int_{{\hat{A}}_{n}}^{{\hat{B}}_{n}} η (x) d x = 0 \\ \int_{{\hat{A}}_{n}}^{{\hat{B}}_{n}} x η (x) d x = 0 \end{array}

Define a function ρ(ε) with respect to ε as

ρ (ε) = J [f_{{\hat{X}}_{n}} (x) + ε η (x)]

Because the minimum value of ρ(ε) is obtained when ε = 0, and we have

{\frac{d ρ (ε)}{d ε} |}_{ε = 0} = \int_{{\hat{A}}_{n}}^{{\hat{B}}_{n}} η (x) {\ln [f_{{\hat{X}}_{n}} (x)] + 1} d x = 0

Considering the two constraints in (37), the following two equations can be derived as

\ln [f_{{\hat{X}}_{n}} (x)] + 1 = c_{0}

\ln [f_{{\hat{X}}_{n}} (x)] + 1 = c_{0}^{'} x

where c₀ is an arbitrary constant, and

c_{0}^{'}

is a non-zero constant.

Therefore, there may be two solutions for $f_{{\hat{X}}_{n}} (x)$ , i.e.,

f_{{\hat{X}}_{n}} (x) = e^{c_{0} - 1}, {\hat{A}}_{n} \leq x \leq {\hat{B}}_{n}

or

f_{{\hat{X}}_{n}} (x) = e^{c_{0}^{'} x - 1}, {\hat{A}}_{n} \leq x \leq {\hat{B}}_{n}

If the PDF of ${\hat{X}}_{n}$ is (42), substituting (42) into the first constraint of (22), we have $e^{c_{0} - 1} = \frac{1}{{\hat{B}}_{n} - {\hat{A}}_{n}}$ Substituting (44) into (42), (23) is derived.
Then, substituting (23) into the second constraint of problem (22), and using the definitions of ${\hat{A}}_{n}$ and ${\hat{B}}_{n}$ , we have
$ξ P = \frac{A}{2}$ which can be equivalently expressed as α_n = 0.5.
If the PDF of ${\hat{X}}_{n}$ is (43), substituting (43) into the first constraint of (22), we have $e^{- 1} = \frac{c_{0}^{'}}{e^{c_{0}^{'} {\hat{B}}_{n}} - e^{c_{0}^{'} {\hat{A}}_{n}}}$ Substituting (46) into (43), (24) is derived.
Then, substituting (24) into the second constraint of problem (22), we have
$\frac{{\hat{B}}_{n} e^{c_{0}^{'} {\hat{B}}_{n}} - {\hat{A}}_{n} e^{c_{0}^{'} {\hat{A}}_{n}}}{e^{c_{0}^{'} {\hat{B}}_{n}} - e^{c_{0}^{'} {\hat{A}}_{n}}} - \frac{1}{c_{0}^{'}} = ξ P λ_{n} \sum_{j = 1}^{N_{T}} v_{j, n}$
According to the definitions of ${\hat{A}}_{n}$ and ${\hat{B}}_{n}$ , we have
${\hat{A}}_{n} + {\hat{B}}_{n} = λ_{n} A \sum_{j = 1}^{N_{T}} v_{j, n}$
By putting (47) and (48) together, (25) is derived.

Appendix B Proof of Theorem 2

When α_n = 0.5, the PDF of ${\hat{X}}_{n}$ is (23). Therefore, $H ({\hat{X}}_{n})$ can be derived as $H ({\hat{X}}_{n}) = \ln ({\hat{B}}_{n} - {\hat{A}}_{n})$
Furthermore, the lower bound on the capacity for the n-th channel can be expressed as
$C_{n} \geq \frac{1}{2} \ln (1 + \frac{{({\hat{B}}_{n} - {\hat{A}}_{n})}^{2}}{2 π e σ^{2}})$
According to the definitions of ${\hat{A}}_{n}$ and ${\hat{B}}_{n}$ , we have
${\hat{B}}_{n} - {\hat{A}}_{n} = λ_{n} A \sum_{j = 1}^{N_{T}} | v_{j, n} |$
Therefore, by substituting (51) into (50), (26) can be derived.
When α_n ≠ 0.5 and α_n ∈ (0, P/A], the PDF of ${\hat{X}}_{n}$ is (24). Therefore, $H ({\hat{X}}_{n})$ is given by $H ({\hat{X}}_{n}) = \ln (\frac{e^{c_{0}^{'} {\hat{B}}_{n}} - e^{c_{0}^{'} {\hat{A}}_{n}}}{c_{0}^{'}}) - c_{0}^{'} (\frac{{\hat{B}}_{n} e^{c_{0}^{'} {\hat{B}}_{n}} - {\hat{A}}_{n} e^{c_{0}^{'} {\hat{A}}_{n}}}{e^{c_{0}^{'} {\hat{B}}_{n}} - e^{c_{0}^{'} {\hat{A}}_{n}}} - \frac{1}{c_{0}^{'}})$ Substituting (25) and (48) into (52), we have $H ({\hat{X}}_{n}) = \ln [\frac{\exp (- c_{0}^{'} ξ P λ_{n} \sum_{j = 1}^{N_{T}} v_{j, n}) (e^{c_{0}^{'} {\hat{B}}_{n}} - e^{c_{0}^{'} {\hat{A}}_{n}})}{c_{0}^{'}}]$
According to the definitions of ${\hat{A}}_{n}$ and ${\hat{B}}_{n}$ , (53) can be further written as
$\begin{array}{l} H ({\hat{X}}_{n}) = \ln [(\exp (- c_{0}^{'} ξ P λ_{n} \sum_{j = 1}^{N_{T}} v_{j, n}) \\ \times \frac{(\exp (- c_{0}^{'} λ_{n} A \sum_{j = 1}^{N_{T}} \max {0, v_{j, n}}) (- c_{0}^{'} λ_{n} A \sum_{j = 1}^{N_{T}} \max {0, v_{j, n}}))}{c_{0}^{'}} \end{array}$
Therefore, the lower bound on the capacity for the n-th channel can be derived as (27).

Acknowledgments

This work is supported by National 863 High Technology Development Project (2014AA01A701), National Natural Science Foundation of China (61372106, 61223001, 61322112 & 61531166004), Science and Technology Project of Guangdong Province (20140119), and Research Project (2014CB360507).

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Parameters	Symbol	Value
Room size	a×b × c	5 m×4 m × 3 m
Number of LEDs	N _T	4
Positions of LEDs	$(x_{j}^{T}, y_{j}^{T}, c)$	(0.25a,0.25b,c),(0.25a,0.75b,c), (0.75a,0.25b,c),(0.75a,0.75b,c)
Height of receiver	c ^R	0.5 m
Order of Lambertian emission	m	1
Area of PD	A_r	1 cm²
Optical filter gain	T_s	2
Concentrator gain of PD	g	2
Variance of noise	σ ²	0 dB

Parameters	Symbol	Value
Maximum particle number	K	200
Maximum iteration number	T _max	100
Learning factors	c₁, c₂	1.4962
Maximum inertia weight	ω _max	0.9
Minimum inertia weight	ω _min	0.3

Parameters	Symbol	Value
Room size	a×b × c	5 m×4 m × 3 m
Number of LEDs	N _T	4
Positions of LEDs	$(x_{j}^{T}, y_{j}^{T}, c)$	(0.25a,0.25b,c),(0.25a,0.75b,c), (0.75a,0.25b,c),(0.75a,0.75b,c)
Height of receiver	c ^R	0.5 m
Order of Lambertian emission	m	1
Area of PD	A_r	1 cm²
Optical filter gain	T_s	2
Concentrator gain of PD	g	2
Variance of noise	σ ²	0 dB

Parameters	Symbol	Value
Maximum particle number	K	200
Maximum iteration number	T _max	100
Learning factors	c₁, c₂	1.4962
Maximum inertia weight	ω _max	0.9
Minimum inertia weight	ω _min	0.3

Channel capacity and receiver deployment optimization for multi-input multi-output visible light communications

Abstract

1. Introduction

2. System Model

2.1. Received Signal

2.2. Channel Gain

2.3. Signal Constraints

3. MIMO Capacity Analysis

3.1. Channel Decomposition

3.2. Lower Capacity Bounds of Sub-channels

3.3. Lower Capacity Bound of MIMO Channel

4. Receiver Deployment Optimization

4.1. Problem Formulation

4.2. Problem Solving

5. Simulation Results

5.1. Channel Capacity Simulations

5.2. Receiver Deployment Simulations

6. Conclusion

Appendix A Proof of Theorem 1

Appendix B Proof of Theorem 2

Acknowledgments

References and links

Cited By

Figures (8)

Tables (3)

Equations (54)

Optics Express