阅读说明：本技术 一种可见光通信多输入多输出系统信道容量可达方法 (Method for reaching channel capacity of visible light communication multi-input multi-output system ) 是由 马帅 刘栋 张蕴琪 杨瑞鑫 汪奇 李世银 于 2021-07-01 设计创作，主要内容包括：本发明提供了一种可见光通信多输入多输出系统信道容量可达方法,包括如下步骤：步骤1,设定MIMO VLC系统的参数；步骤2,计算得到最优的输入分布；步骤3,求解离散分布输入的上界和下界；步骤4,求解使可达速率最大化的最佳功率分配。仿真结果表明,随着信噪比的增加,需要更多的离散点来获得精确的信道容量。在低信噪比区域,精确的信道容量的上界和下界与紧密相关。此外,在高信噪比区域,信道容量的下界也很紧密。基于最优输入的功率分配方案比等概率输入具有更好的性能。(The invention provides a method for reaching channel capacity of a visible light communication multi-input multi-output system, which comprises the following steps: step 1, setting parameters of an MIMO VLC system; step 2, calculating to obtain optimal input distribution; step 3, solving an upper bound and a lower bound of discrete distribution input; and 4, solving the optimal power distribution for maximizing the reachable rate. Simulation results show that as the signal-to-noise ratio increases, more discrete points are required to achieve accurate channel capacity. In the low snr region, the upper and lower bounds of the exact channel capacity are closely related. In addition, in the high snr region, the lower bound of channel capacity is also very tight. The power allocation scheme based on the optimal input has better performance than the equal probability input.)

1. A method for reaching channel capacity of a visible light communication multi-input multi-output system is characterized by comprising the following steps:

step 1, setting parameters of an MIMO VLC system;

step 2, calculating to obtain optimal input distribution;

step 3, solving an upper bound and a lower bound of discrete distribution input;

and 4, solving the optimal power distribution for maximizing the reachable rate.

2. The method of claim 1, wherein step 1 comprises:

step 1-1, the transmitting end of the MIMO VLC system has N_TA Light Emitting Diode (LED) with a receiving end having N_RSingle photon detector PD, wherein N_T>1 and N_R>1；

Step 1-2, settingRepresenting a transmit signal vector for a MIMO VLC system, the elements of the transmit signal x being real and non-negative;

wherein the content of the first and second substances,denotes the Nth_TThe signal sent by each LED;represents N_TA real vector of dimensions;

steps 1-3, to protect the human eye and for practical circuit considerations, the peak and average optical and electrical power of x are limited, and therefore

Where A is the peak optical power of a single LED; x is the number of_jSignal representing the jth LED transmission; μ is the average optical power; ε represents an electrical power constraint;represents an integer of 1 to N_TA set of (a);averaging the 1 norm of x;averaging the squares of the 2 norms of x;

receiving signals in a channel of a MIMO VLC systemExpressed as:

y＝Hx+z (1)

wherein the content of the first and second substances,representing a MIMO VLC system channel matrix;is an independent Gaussian noise with mean of 0 and variance of σ²；Is the Nth_RThe signal received by the PD is a signal,represents N_RA real vector of dimensions;

step 1-4, decomposing the MIMO VLC system channel into parallel sub-channels by using Singular Value Decomposition (SVD);

step 1-5, the elements of the transmit signal vector x are represented.

3. The method of claim 2, wherein steps 1-4 comprise: the channel matrix H is decomposed into H ═ U Λ V^TWhereinAndis a unitary matrix, Λ ═ diag { λ₁,...,λ_NIs the singular matrix, λ_NIs the equivalent channel gain of the Nth parallel sub-channel, the ith parallel sub-channelExpressed as:

in the formula, λ_iIs the channel gain of the ith parallel subchannel;is the equivalent input signal of the ith parallel subchannel;is the gaussian noise of the ith parallel subchannel; n denotes the rank of the channel matrix H.

4. The method of claim 3, wherein steps 1-5 comprise: in steps 1-5, the elements of the transmit signal vector x are represented as:

in the formula (I), the compound is shown in the specification,is a unitary matrix V^TThe (j) th column vector of (a),which represents an equivalent input signal, is,is a unitary matrix V^TOf the jth column vector v_jN of (2)_TAn element of a row;

the peak power, optical power and electrical power constraints of a MIMO VLC system are equivalently redefined as:

in the formula (I), the compound is shown in the specification,is an equivalent input signalAn average optical power constraint of;equivalent input signalAn average electrical power constraint of;

the ith sub-channel according to the constraint of (4a)The amplitude of the input signal is bounded, and thereforeThe achievable distribution of channel capacity of (a) is discrete,satisfies the following expression:

wherein the content of the first and second substances,is an equivalent form to describing the input signal x with constellation points,pr (-) represents the probability of the solution,is an input signalOf the kth constellation point, p_i,kIs the kth constellation pointA corresponding probability;is an input signalOf the kth constellation point, p_N,kIs the kth constellation pointCorresponding toProbability; k represents the serial number of the constellation point;represents a set of integers 1 to N;represents an integer of 1 to K_iA set of (a);

MIMO VLC system channel capacity C_MIMOIs the sum of the channel capacities of the N subchannels,C_ithe capacity of the ith subchannel is represented by the following expression:

wherein σ²In order to be able to measure the power of the noise,representing signalsThe discrete distribution to be obeyed to,representing signalsA probability density function of;to representAndthe mutual information of (a) and (b),show aboutMaximum mutual information, conditional entropy ofRepresenting a given random variableThen, for random variableThe uncertainty that still exists;

due to noiseObeying Gaussian distribution, probability density functionExpression (2)Writing into:

5. the method of claim 4, wherein step 2 comprises:

step 2-1, replace equation (7) to (6C), capacity C of ith subchannel_iWriting into:

in the formula, p_i,mIs the probability of the mth discrete point correspondence;

the problem of channel capacity for MIMO VLC systems is expressed as:

s.t.(5a),(5b),(5c),(5d),(5e) (9)

step 2-2, restating the problem (9) as follows:

in the formula (I), the compound is shown in the specification,is a transmission signalAll the constellation points that are possible to obtain,is an input signalK of_iThe number of the constellation points is one,is a vector representing the square of a discrete point, p_iIs the probability value corresponding to the discrete point,is an intermediate variable, phi, for simplifying the objective function_i(p_i) Is the target function, gamma_iIs a set of all the constraints that are,is an input signalK of_iA discrete pointA corresponding probability;

problem (9) is equivalently re-expressed as the following problem (11):

s.t.p_i∈Υ_i (11b)；

question (11) about probability p_iIs convexBut with respect to points K_iAnd positionIs non-convex; to solve problem (11), an inaccurate gradient descent method is applied to obtain an optimal input distribution.

6. The method of claim 5, wherein step 3 comprises: due to C_iWithout closed form, approaching a lower bound and an upper bound to approximate the channel capacity of the ith subchannel;

step 3-1, solving the upper bound of the discrete input distribution:

and 3-2, solving the lower bound of the discrete distribution input.

7. The method of claim 6, wherein step 3-1 comprises: the upper bound on the channel capacity of a MIMO VLC system is written as:

in the formula, C_MIMORepresenting the channel capacity of a visible light Multiple Input Multiple Output (MIMO) communication system VLC;

obtaining an expression of an upper bound of channel capacity of the MIMO VLC system by solving the following optimization problems:

s.t.(5a)，(5b)，(5c)，(5d)，(5e) (13)

the following definitions are made:

in the formula (I), the compound is shown in the specification,and L_i(p_i) Are all intermediate variables, #_i(p_i) It is then a representation of the objective function,probability value p corresponding to discrete point_iThe transpose matrix of (a) is,is the K th_iIntermediate variables corresponding to the discrete points;

along with the definitions (10a), (10b) (10c), (10f), the question (13) is equivalently rewritten as:

s.t.p_i∈Υ_i (15b)

problem (15) is a non-convex problem, the objective function being related to probabilityIs non-convex, but is constrained with respect toIs convex and solves the problem (15) using the OP (best probability) -Frank-Wolfe method, which yields an upper bound on the distribution of the discrete inputs.

8. The method of claim 7, wherein step 3-2 comprises: the lower bound of the channel capacity of a MIMO VLC system is written as:

obtaining a lower bound of channel capacity of the MIMO VLC system based on the continuously distributed input:

when the input follows a continuous distribution, the following constraints need to be satisfied:

wherein f is_X(x) Is the probability density function corresponding to the input signal X,is N with elements all being 1_TThe vector of the dimension column is then calculated,is N_TReal space of dimension, in which all elements are [0, A ]]And is prepared from

Another lower bound on the channel capacity of a MIMO VLC system based on the continuously distributed input is given by:

in the formula, parameters alpha, beta and gamma are ABG distribution parameters, and the constraint of (17) is satisfied; det (H) is the determinant of the channel matrix H;

the parameters α, β and γ are solutions of the following equations:

wherein the intermediate parameterIntermediate parameterT(x)＝e^x(β+Aγ)Is an intermediate variable, an intermediate parameterGamma is an ABG distribution parameter;

wherein the distribution of the transmit signal vector x is given by:

9. the method of claim 8, wherein step 4 comprises:

step 4-1, setting input from discrete constellation setReceived signal of ith parallel channel after power distributionGiven by:

wherein the content of the first and second substances,is 0 as a mean and σ as a variance²Independent gaussian noise of (2); p_iA power allocation parameter representing an ith subchannel;

step 4-2, reachable Rate R of ith sub-channel_iComprises the following steps:

step 4-3, using M-QAM multi-system quadrature amplitude modulation method,the constraint of (2) is written as:

wherein the content of the first and second substances, represents the kth constellation point, P_oIs the total average optical power, P_eRepresenting a total electrical power constraint; taking into account the inequalityIn the inequality, a_iRepresenting the ith value, n represents the total number of values, and equation (23c) is approximately written as:

step 4-4, noiseObeying Gaussian distribution, the ith parallel channel can reach the rate R_iComprises the following steps:

achievable rate R for MIMO VLC systems_MIMOExpressed as:

the problem under consideration (27) is represented by the following mathematical formula:

s.t(23a),(23b),(24),(23d) (27)

constraint of optimization problem (27) for P_iIs convex, but the objective function is for P_iIs non-convex; next, solving a lower bound of the reachable rate based on minimum mean square error MMSE and an upper bound and a lower bound of the reachable rate based on a Jensen inequality;

step 4-5, solving the lower bound of the achievable rate based on MMSE: the optimal power distribution scheme is obtained by researching the relation between the mutual information and MMSE; the relationship between mutual information and MMSE is given by:

in the formula (I), the compound is shown in the specification,denotes x_iAndthe MMSE in between is carried out,is x_iMMSE estimation of conditional mean, I (x)_i；y_i) Representing channel average mutual information; SNR represents the signal-to-noise ratio;

the lagrangian function L of the problem (27) is given by:

wherein, ω is₁≥0，ω₂≧ 0 is the Lagrangian multiplier for constraints (23c) and (23 d);

then, the sufficient prerequisite KKT condition for the existence of the global optimal solution is expressed as:

ω₁≥0,ω₂≥0,

wherein the intermediate parameter

Power allocation parameter P_iThe expression of (c) is written as:

obtaining MIMO VLC systemUnified optimal power allocation P_i ^*Comprises the following steps:

MMSE_ithe upper bound of (A) is:

in the formula (I), the compound is shown in the specification,to representThe variance of (a);

obtaining a dual variable omega by adopting an OPA optimal power distribution dual method₁,ω₂；

And 4-6, solving an upper bound of the reachable rate based on the Jensen inequality: according to the Jensen inequality, (25b) converts to:

in the formula (I), the compound is shown in the specification,presentation pairCalculating an average value;

an upper bound for achievable rates is obtained:

an expression of the upper bound is obtained by solving the following optimization problem:

s.t.(23a),(23b),(23c),(23d) (35)

question (35) about P_iIs convex and solves it using an off-the-shelf convex optimization solver, such as CVX, to arrive at an achievable rate R_MIMOThe upper bound of (c);

and 4-7, solving the lower bound of the reachable rate based on the Jensen inequality:

according to the Jensen inequality, (25b) translates into:

thus, the lower bound on the achievable rates is:

obtaining a lower bound expression by solving the following optimization problem:

s.t.(23a),(23b),(23c),(23d) (38)

solving the problem (38) with a convex optimization solver to obtain an achievable rate R_MIMOThe lower bound of (c).

Technical Field

The invention relates to a method for reaching channel capacity of a visible light communication multi-input multi-output system.

Background

With the exponential growth of internet of things devices and applications, traditional Radio Frequency (RF) based indoor internet of things networks are facing the problem of spectrum scarcity, but they are unable to meet large-scale access and the increasing demand for high wireless data rates, while Visible Light Communication (VLC) with extensive license-free bandwidth is a promising complementary choice for radio frequency communication. In addition, VLC uses widely deployed LEDs, has the advantages of high data rate, no electromagnetic interference, low cost, high energy efficiency, high intrinsic safety, and the like, and has attracted extensive attention from the industrial and academic circles.

VLC Multiple Input Multiple Output (MIMO) systems may utilize spatial diversity to increase data rates using multiple Light Emitting Diodes (LEDs) at the transmitter and multiple Photodiodes (PDs) at the receiver. To date, the channel capacity of MIMO VLC is an open problem, and due to limited amplitude limitations, it has been demonstrated that the achievable distribution of channel capacity is a limited number of quality points.

Disclosure of Invention

The purpose of the invention is as follows: in order to solve the technical problems existing in the background technology, the method comprises the following steps:

step 1, setting parameters of an MIMO VLC system;

step 2, calculating to obtain optimal input distribution;

step 3, solving an upper bound and a lower bound of discrete distribution input;

and 4, solving the optimal power distribution for maximizing the reachable rate.

The step 1 comprises the following steps:

step 1-1, the transmitting end of the MIMO VLC system has N_TA Light Emitting Diode (LED) with a receiving end having N_RSingle photon detector PD, wherein N_T>1 and N_R>1；

Step 1-2, settingRepresenting a transmit signal vector for a MIMO VLC system, the elements of the transmit signal x being real and non-negative;

wherein the content of the first and second substances,denotes the Nth_TThe signal sent by each LED;represents N_TA real vector of dimensions;

steps 1-3, to protect the human eye and for practical circuit considerations, the peak and average optical and electrical power of x are limited, and therefore

Where A is the peak optical power of a single LED; x is the number of_jSignal representing the jth LED transmission; μ is the average optical power; ε represents an electrical power constraint;represents an integer of 1 to N_TA set of (a);averaging the 1 norm of x;is the square average of the 2 norm of x.

Receiving signals in a channel of a MIMO VLC systemExpressed as:

y＝Hx+z (1)

wherein the content of the first and second substances,representing a MIMO VLC system channel matrix;is an independent Gaussian noise with mean of 0 and variance of σ²；Is the Nth_RThe signal received by the PD is a signal,represents N_RA real vector of dimensions;

step 1-4, decomposing the MIMO VLC system channel into parallel sub-channels by using Singular Value Decomposition (SVD);

step 1-5, the elements of the transmit signal vector x are represented.

The steps 1 to 4 comprise: the channel matrix H is decomposed into H ═ U Λ V^TWhereinAndis a unitary matrix, Λ ═ diag { λ₁,...,λ_NIs a singular matrix of the symbols,

λ_Nis the equivalent channel gain of the Nth parallel sub-channel, the ith parallel sub-channelExpressed as:

in the formula, λ_iIs the channel gain of the ith parallel subchannel;is the equivalent input signal of the ith parallel subchannel;is the gaussian noise of the ith parallel subchannel; n represents the rank of the channel matrix H;

the steps 1 to 5 comprise: in steps 1-5, the elements of the transmit signal vector x are represented as:

in the formula (I), the compound is shown in the specification,is a unitary matrix V^TThe (j) th column vector of (a),which represents an equivalent input signal, is,is a unitary matrix V^TOf the jth column vector v_jN of (2)_TAn element of a row;

the peak power, optical power and electrical power constraints of a MIMO VLC system are equivalently redefined as:

in the formula (I), the compound is shown in the specification,is an equivalent input signalAn average optical power constraint of;equivalent input signalAn average electrical power constraint of;

the ith sub-channel according to the constraint of (4a)The amplitude of the input signal is bounded, and thereforeThe achievable distribution of channel capacity of (a) is discrete,satisfies the following expression:

wherein the content of the first and second substances,is an equivalent form to describing the input signal x with constellation points,pr (-) represents the probability of the solution,is an input signalOf the kth constellation point, p_i,kIs the kth constellation pointA corresponding probability;is an input signalOf the kth constellation point, p_N,kIs the kth constellation pointA corresponding probability; k represents the serial number of the constellation point;represents a set of integers 1 to N;represents an integer of 1 to K_iA set of (a);

MIMO VLC system channel capacity C_MIMOIs the sum of the channel capacities of the N subchannels,C_ithe capacity of the ith subchannel is represented by the following expression:

wherein σ²In order to be able to measure the power of the noise,representing signalsThe discrete distribution to be obeyed to,representing signalsA probability density function of;to representAndthe mutual information of (a) and (b),show aboutMaximum mutual information, conditional entropy ofRepresenting a given random variableThen, for random variableThe uncertainty that still exists;

due to noiseObeying Gaussian distribution, probability density functionExpression (PDF)Writing into:

the step 2 comprises the following steps:

step 2-1, replace equation (7) to (6C), capacity C of ith subchannel_iWriting into:

in the formula, p_i,mIs the probability of the mth discrete point correspondence;

the problem of channel capacity for MIMO VLC systems is expressed as:

step 2-2, restating the problem (9) as follows:

in the formula (I), the compound is shown in the specification,is a transmission signalAll the constellation points that are possible to obtain,is an input signalK of_iThe number of the constellation points is one,is a vector representing the square of a discrete point, p_iIs the probability value corresponding to the discrete point,is an intermediate variable, phi, for simplifying the objective function_i(p_i) Is the target function, gamma_iIs a set of all constraints, p_i,KiIs an input signalK of_iA discrete pointA corresponding probability;

problem (9) is equivalently re-expressed as the following problem (11):

s.t.p_i∈Υ_i (11b)；

question (11) about probability p_iIs convex, but with respect to the number of points K_iAnd positionIs non-convex; to solve problem (11), an inaccurate gradient descent method is applied to obtain an optimal input distribution.

The step 3 comprises the following steps: due to C_iWithout closed form, approaching a lower bound and an upper bound to approximate the channel capacity of the ith subchannel;

step 3-1, solving the upper bound of the discrete input distribution:

and 3-2, solving the lower bound of the discrete distribution input.

Step 3-1 comprises: the upper bound on the channel capacity of a MIMO VLC system is written as:

in the formula, C_MIMORepresenting the channel capacity of a visible light Multiple Input Multiple Output (MIMO) communication system VLC;

obtaining an expression of an upper bound of channel capacity of the MIMO VLC system by solving the following optimization problems:

the following definitions are made:

in the formula (I), the compound is shown in the specification,and L_i(p_i) Are all intermediate variables, #_i(p_i) It is then a representation of the objective function,probability value p corresponding to discrete point_iThe transpose matrix of (a) is,is the K th_iIntermediate variables corresponding to the discrete points;

along with the definitions (10a), (10b) (10c), (10f), the question (13) is equivalently rewritten as:

s.t.p_i∈Υ_i (15b)

problem (15) is a non-convex problem, the objective function being related to probabilityIs non-convex, but is constrained with respect toIs convex and solves the problem (15) using the OP (best probability) -Frank-Wolfe method, which yields an upper bound on the distribution of the discrete inputs.

Step 3-2 comprises: the lower bound of the channel capacity of a MIMO VLC system is written as:

obtaining a lower bound of channel capacity of the MIMO VLC system based on the continuously distributed input:

when the input follows a continuous distribution, the following constraints need to be satisfied:

wherein f is_X(x) Is the probability density function corresponding to the input signal X,is N with elements all being 1_TThe vector of the dimension column is then calculated,is N_TReal space of dimension, in which all elements are [0, A ]]And is prepared from

Another lower bound on the channel capacity of a MIMO VLC system based on the continuously distributed input is given by:

in the formula, parameters alpha, beta and gamma are ABG distribution parameters, and the constraint of (17) is satisfied; det (H) is the determinant of the channel matrix H;

the parameters α, β and γ are solutions of the following equations:

wherein the intermediate parameterIntermediate parameterT(x)＝e^{x (β+Aγ)}Is an intermediate variable, an intermediate parameterGamma is an ABG distribution parameter;

wherein the distribution of the transmit signal vector x is given by:

step 4 comprises the following steps:

step 4-1, setting input from discrete constellation setReceived signal of ith parallel channel after power distributionGiven by:

wherein the content of the first and second substances,is 0 as a mean and σ as a variance²Independent gaussian noise of (2); p_iA power allocation parameter representing an ith subchannel;

step 4-2, reachable Rate R of ith sub-channel_iComprises the following steps:

step 4-3, using M-QAM multi-system quadrature amplitude modulation method,is restricted byWriting into:

wherein the content of the first and second substances, represents the kth constellation point, P_oIs the total average optical power, P_eRepresenting a total electrical power constraint; taking into account the inequalityIn the inequality, a_iRepresenting the ith value, n represents the total number of values, and equation (23c) is approximately written as:

step 4-4, noiseObeying Gaussian distribution, the ith parallel channel can reach the rate R_iComprises the following steps:

achievable rate R for MIMO VLC systems_MIMOExpressed as:

the problem under consideration (27) is represented by the following mathematical formula:

constraint of optimization problem (27) for P_iIs convex, but the objective function is for P_iIs non-convex; next, solving a lower bound of the reachable rate based on minimum mean square error MMSE and an upper bound and a lower bound of the reachable rate based on a Jensen inequality;

step 4-5, solving the lower bound of the achievable rate based on MMSE: the optimal power distribution scheme is obtained by researching the relation between the mutual information and MMSE; the relationship between mutual information and MMSE is given by:

in the formula (I), the compound is shown in the specification,denotes x_iAndthe MMSE in between is carried out,is x_iMMSE estimation of conditional mean, I (x)_i；y_i) Representing channel average mutual information; SNRRepresenting the signal-to-noise ratio;

the lagrangian function L of the problem (27) is given by:

wherein, ω is₁≥0，ω₂≧ 0 is the Lagrangian multiplier for constraints (23c) and (23 d);

then, the sufficient prerequisite KKT condition for the existence of the global optimal solution is expressed as:

ω₁≥0,ω₂≥0,

wherein the intermediate parameter

Power allocation parameter P_iThe expression of (c) is written as:

obtaining optimal power distribution of MIMO VLC systemComprises the following steps:

MMSE_ithe upper bound of (A) is:

in the formula (I), the compound is shown in the specification,to representThe variance of (c).

Obtaining a dual variable omega by adopting an OPA optimal power distribution dual method₁,ω₂；

And 4-6, solving an upper bound of the reachable rate based on the Jensen inequality: according to the Jensen inequality, (25b) converts to:

in the formula (I), the compound is shown in the specification,presentation pairCalculating an average value;

an upper bound for achievable rates is obtained:

an expression of the upper bound is obtained by solving the following optimization problem:

question (35) about P_iIs convex and solves it using an off-the-shelf convex optimization solver, such as CVX, to arrive at an achievable rate R_MIMOThe upper bound of (c);

and 4-7, solving the lower bound of the reachable rate based on the Jensen inequality:

according to the Jensen inequality, (25b) translates into:

thus, the lower bound on the achievable rates is:

obtaining a lower bound expression by solving the following optimization problem:

solving the problem (38) with a convex optimization solver to obtain an achievable rate R_MIMOThe lower bound of (c).

The invention deduces the accurate channel capacity of the MIMO VLC system, and researches the optimal discrete distribution under the constraints of peak value, average light power and electric power by using an inaccurate gradient descent method. Since the precise solution of the channel capacity is a continuous-discrete hybrid optimization problem, the invention adopts a numerical integration method of an approximate objective function and the gradient thereof. Given herein are the upper and lower bounds of MIMO VLC channel capacity, taking into account the actual discrete constellation input. In addition, the invention also deduces the MIMO VLC channel capacity under the condition of continuous input, and obtains a closed expression of the lower bound of the capacity. On the basis, the invention researches the optimal power distribution problem which takes the maximum reachable speed as a target, and compares the power distribution scheme based on the optimal input and the equal probability input. Simulation results show that as the signal-to-noise ratio increases, more discrete points are required to achieve accurate channel capacity. In the low snr region, the upper and lower bounds of the exact channel capacity are closely related. In addition, in the high snr region, the lower bound of channel capacity is also very tight. The power allocation scheme based on the optimal input has better performance than the equal probability input.

Has the advantages that: the invention provides a non-precise gradient descent method for calculating the channel capacity of the MIMO VLC under the constraints of peak value, average optical power and electric power. In addition, the invention also deduces the upper bound and the lower bound of the capacity of the MIMO VLC channel and provides a closed expression. And, the present invention establishes for the first time a theoretical framework for MIMO VLC channel capacity with electrical power constraints. On the basis, the invention provides an optimal power distribution scheme suitable for any actual discrete constellation input, and compared with an equal power distribution scheme, the power distribution scheme provided by the invention can obviously improve the rate of low-order modulation input.

Drawings

The present invention will be described in further detail with reference to the accompanying drawings and specific embodiments.

Fig. 1 is a variation curve of respective exact channel capacities of the MIMO VLC at different SNRs when k is 4, 5.

Fig. 2 is a variation curve of respective precise channel capacities of MIMO VLCs at different points K under different SNRs.

Fig. 3 is a graph comparing the exact channel capacities of the upper and lower bounds of a discrete distributed input and the lower bound of a continuous distributed input.

Fig. 4 shows the curves of the respective capacities at different iteration numbers when k is 4 and SNR is-5 dB.

FIG. 5a shows the respective optimal input positions of MIMO VLC under different SNRThe change curve of (2).

FIG. 5b is the respective optimal input distribution of MIMO VLC under different SNRThe change curve of (2).

FIG. 6a shows the respective optimal input positions of MIMO VLC under different SNRThe change curve of (2).

FIG. 6b is the respective optimal input distribution of MIMO VLC under different SNRThe change curve of (2).

Fig. 7 is a graph showing the variation of the respective achievable rates at different electrical power thresholds when the optical power threshold μ is 3W.

FIG. 8 is a graph of the change in the respective achievable rates for different electrical power thresholds for different input profiles.

Detailed Description

The invention provides a visible light communication multi-input multi-output system MIMO VLC channel capacity reaching method, which is used for researching the MIMO VLC channel capacity with peak light power, average light power and electric power constraints. The accurate capacity and the optimal discrete distribution input of the MIMO VLC system channel are firstly obtained. And (4) taking the actual discrete constellation input into consideration, and acquiring the lower limit and the upper limit of the capacity of the MIMO VLC channel by using a closed expression based on a Jensen inequality. In addition, the capacity reachable rate of the MIMO VLC system with continuous input is deduced, and a closed expression is obtained; based on the optimal input distribution obtained above, the optimal power distribution that maximizes the achievable rate under the constraints of light power and electric power is studied. Then, by using the relationship between mutual information and MMSE, an optimal power allocation scheme is obtained using a bisection method.

Accurate channel capacity of MIMO VLC system:

considering IM/DD MIMO VLC system channel, the transmitting end has N_TEach LED having N receiving terminals_RA PD of which N_T>1 and N_R>1. Order toRepresenting the transmitted signal vector for a MIMO VLC system. The elements of the transmitted signal x are real and non-negative.

Furthermore, for human eye safety considerations, both the peak and average optical power of x are limited, so 0 ≦ x_j≤A,Where a represents the peak power of each LED and μ represents the maximum average optical power. Furthermore, the electrical power of x is also limited due to practical circuit considerations. Therefore, the temperature of the molten metal is controlled,where epsilon represents the maximum electrical power of the LED.

Order toRepresenting the VLC channel matrix. Receiving a signalIs shown as

y＝Hx+z (1)

WhereinIs an independent Gaussian noise with mean of 0 and variance of σ²。

Then, Singular Value Decomposition (SVD) is applied to decompose the MIMO VLC system channel into parallel subchannels. Wherein, H ═ U Λ V^T，Andis a unitary matrix. Considering N_T≥N_RThen rank (H) ≦ N_R. Wherein Λ ═ diag { λ ═ λ₁,...,λ_NIs a singular matrix, obtained by multiplying by the matrix U^TObtaining equivalent MIMO VLC system channel, and expressing the ith parallel subchannel as

WhereinIs an equivalent input signal, anIs a mean of 0 and a variance ofGaussian noise. Obviously, since the receiver does not receive, i>When the N is greater than the N value,has no impact on the channel capacity.

Furthermore, the elements of the transmission signal x may be represented as

Wherein the content of the first and second substances,is V^TThen, the peak power, optical power, and electrical power constraints of the MIMO VLC system may be equivalently redefined as

The ith sub-channel according to the constraint of (4a)The amplitude of the input signal is bounded, and thereforeThe achievable distribution of channel capacity of (a) is discrete. Order toIs represented by K_iA non-negative mass pointOfIs distributed discretely. More specifically, the present invention is described in detail,the constraints (4a), (4b) and (4c) are satisfied. Therefore, the following expression is provided:

wherein the content of the first and second substances, is an input signalThe k-th discrete point of (1), p_i,kIs the corresponding probability. Obviously, i>When the N is greater than the N value,independent of the input signal. Thus, MIMO VLC system channel capacity C_MIMOIs the sum of the channel capacities of the N subchannels,C_irepresents the capacity of the ith subchannel, which may be given by the following expression:

due to noiseObeying Gaussian distributions, PDFsWriting into:

replace (7) to (6C), capacity C of ith subchannel_iWriting into:

therefore, the problem of channel capacity for MIMO VLC systems is expressed as:

however, problem (9) is a non-convex problem, and to solve problem (9), it is first restated in a compact form, given the following definitions:

wherein the content of the first and second substances,is K_iAll elements of x 1 are column vectors of 1. The problem (9) is equivalently re-expressed as:

s.t.p_i∈Υ_i (11b)

question (11) about probability p_iIs convex, but with respect to the number of points K_iAnd positionIs non-convex. The problem (11) is solved by using an inaccurate gradient descent method, and an optimal input distribution is obtained.

Lower and upper bounds for MIMO VLC channel capacity

Since the capacity of the ith sub-channel does not have a closed form, the channel capacity of the ith sub-channel is approximated in a manner of approximating a lower bound and an upper bound, thereby obtaining a lower bound and an upper bound of the capacity of the MIMO VLC channel.

A upper bound of discrete distributed input

Introduction 1: the upper bound of MIMO VLC channel capacity is written as:

by solving the following optimization problem, an expression of the upper bound of the capacity of the MIMO VLC channel is obtained.

To solve the problem (13), it is rewritten to a concise form. The following is defined:

along with the definitions (10a), (10b) (10c), (10f), the question (13) is equivalently rewritten as:

s.t.p_i∈Υ_i (15b)

problem (15) is a non-convex problem, the objective function being related to probabilityIs non-convex, but is constrained with respect toIs convex, applies OP (best probability) -Frank-Wolfe method to solve the problem (15), and gets the upper bound of MIMO VLC channel capacity.

B. Lower bound of discrete distributed input

2, leading: the lower bound of the MIMO VLC channel capacity is written as:

the OP (best probability) -Frank-Wolfe method is used for solving the problem (15), and the lower limit of the MIMO VLC channel capacity is obtained.

C. Lower bound of continuously distributed input

Considering the system model proposed in (1), a lower bound for MIMO VLC channel capacity having a closed form can be obtained based on continuously distributed inputs.

When the input follows a continuous distribution, the following constraints need to be satisfied:

wherein the content of the first and second substances,is N_TReal space of dimension, in which all elements are [0, A ]]And is prepared from

And 3, introduction: another lower bound on MIMO VLC channel capacity from a continuously distributed input is given by:

the parameters α, β and γ are solutions of the following equations.

Wherein the content of the first and second substances,

wherein the distribution of the input signal x is given by:

power allocation for capacity and rate maximization of MIMO VLC system

In practical applications, the input is typically based on a discrete signal constellation. Setting the input from a discrete constellation setAccording to the system model provided above, the MIMO VLC system is decomposed into N parallel sub-channels, and the ith parallel sub-channel is subjected to power distributionThe received signal of a channel may be given by:

wherein the content of the first and second substances,is 0 as a mean and σ as a variance²Independent gaussian noise. P_iIndicating the power allocation parameter for the ith subchannel. The achievable rate for the ith subchannel is written as:

since the M-QAM modulation method is used,the constraint of (2) is written as:

wherein the content of the first and second substances, representing the kth constellation point, p_i,kIs the correlation probability, P_oIs the total average optical power, P_eRepresenting the total electrical power constraint.

Because of the above obtained optimum input positionAnd optimal input probabilityFor power allocation, only the variable P needs to be considered_i。

Taking into account the inequality(23c) Written approximately as:

then, the ith parallel channel achievable rate is:

finally, the achievable rate for a MIMO VLC system is expressed as:

the problem considered is expressed mathematically as follows:

constraint of optimization problem (27) for P_iIs convex, but the objective function is for P_iIs non-convex.

A. MMSE-based lower bound on achievable rates

Due to (27) the lack of a closed form expression for the objective function, the relationship between mutual information and MMSE can be studied to arrive at an optimal power allocation scheme.

The relationship between mutual information and MMSE is given by:

the lagrange function of the problem (27) is given by:

wherein, ω is₁≥0，ω₂≧ 0 is the Lagrangian multiplier for constraints (23c) and (23 d).

The KKT condition is then expressed as:

ω₁≥0,ω₂≥0,

wherein the content of the first and second substances,

thus, the power allocation parameter P_iThe expression of (c) is written as:

obtaining the optimal power distribution of the MIMO VLC system as follows:

for MMSE_iWithout closed form expression, but MMSE_iThe upper bound of (A) is:

then, the dual variable ω is derived by using the OPA (optimal power allocation) dual method₁,ω₂.

B. Upper and lower bounds of achievable rates based on the Jensen inequality

Next, the objective function in (26) is approximated in another way, according to the Jensen inequality, (25b) converted to:

thus, an upper bound on the achievable rate is obtained:

an expression of the upper bound is obtained by solving the following optimization problem:

question (35) about P_iIs convex, solving the problem (35) using a convex optimization solver such as CVX, which yields an upper bound on the achievable rate based on the Jensen inequality.

According to the Jensen inequality, (25b) translates into:

thus, the lower bound on the achievable rates is:

obtaining a lower bound expression by solving the following optimization problem:

problem (38) for P_iIs convex, so a convex optimization solver is used to solve the problem (38), resulting in a lower bound for the achievable rate based on the Jensen inequality.

Examples

For convenience, parameters are definedWhich represents the ratio of the amplitude to the average value of the optical power. Transmitting antenna N_T2, receiving antenna N_RThe peak power a of each LED is set to 1, 2. The channel matrix is given by:

fig. 1 illustrates the exact channel capacity of a MIMO VLC channel plotted against SNR when k is 4 and k is 5, whereIt is observed that the exact channel capacity increases with increasing snr, and that the exact channel capacity of k-5 is larger than that of k-4 over the entire snr region, with minimal differences in the low snr region. In addition, since subchannel 1 has better channel gain, the capacity of subchannel 1 is greater than that of subchannel 2.

Fig. 2 shows the variation of the respective exact channel capacities of the MIMO VLCs at different points K under different SNRs when K is 4. Obviously, only two points are needed in the low signal-to-noise ratio region, in which case OOK modulation can be used. Further, in the case where the parameter K is fixed, the capacity tends to be constant although the signal-to-noise ratio increases. Therefore, it is concluded that as the signal-to-noise ratio increases, more points are needed to achieve accurate channel capacity.

Fig. 3 is a graph comparing the upper and lower bounds of the discrete distribution input and the lower bound of the continuous distribution input with the accurate channel capacity, and it is apparent that the difference between the accurate channel capacity and the lower bound capacity is small in the low snr and high snr regions, however, the difference between the accurate channel capacity and the upper bound is increasingly large. Clearly, because discrete distribution is the optimal distribution for a VLC system, the lower bound of the discrete distribution input better approximates the exact channel capacity than the lower bound of the input after continuous distribution.

Fig. 4 illustrates the relationship between the capacity and the number of iterations for different points K of 2,4,8,16 when K is 4 and SNR-5 dB, and when K is greater than the optimal number K^*By optimizing the probability vector p_iThe effect of the redundant points on capacity is eliminated. Thus, the capacity can converge to a capacity with the best number of points.

Fig. 5a, 5b and 6a, 6b illustrate the optimal positions of two sub-channelsAnd optimal input distributionRelation to SNR. From fig. 5a, it can be understood that the discrete points increase with increasing SNR. Furthermore, the additional points tend to be distributed in the central area. Fig. 5a also shows that the optimal input positions tend to be distributed in both terminals in the low SNR region. In this case, since the optimum input position includes only one discrete point 0 in fig. 6a and the distributed electric power is zero, the sub-channel 2 is not effective. Thus, OOK modulation can be used to obtain the capacity of the low signal-to-noise ratio region MIMO VLC channel. As the signal-to-noise ratio increases, the optimal input position has more discrete points, and PAM modulation can be used to obtain accurate channel capacity for MIMO VLC systems. Furthermore, the optimal input positions in fig. 5b are more discrete points than those in fig. 6b, and since sub-channel 1 has better channel gain, it is possible to increase the number of input positionsSubchannel 1 is allocated more electrical power than subchannel 2.

FIG. 7 illustrates the change in achievable rate as a function of the electrical power threshold. The MMSE-based lower bound power distribution has better performance than the Jensen-inequality-based lower bound low electric power threshold power distribution. Furthermore, the lower bound based on the Jensen inequality performs better when the electrical power threshold power is greater than 4W, while the lower bound based on MMSE tends to be constant due to the constraint of the optical power threshold. However, because the difference between the upper and lower bounds becomes larger as the electric power threshold increases, the upper bound effect based on the Jensen inequality is not ideal. It can be seen from fig. 8 that MMSE-based lower power allocation has better performance at low power thresholds.

FIG. 8 plots achievable rates versus electrical power threshold for different input profiles. Clearly, the optimal input is better than the equiprobable input and the difference between the two forms of input distribution becomes smaller as the electrical power threshold increases. When the power threshold rises to 3.8W, the rate of attainment tends to be constant since the optical power threshold has been reached.