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CN113078948B - Downlink transmission optimization method of LiFi-WiFi polymerization system - Google Patents

Downlink transmission optimization method of LiFi-WiFi polymerization system Download PDF

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CN113078948B
CN113078948B CN202110325208.0A CN202110325208A CN113078948B CN 113078948 B CN113078948 B CN 113078948B CN 202110325208 A CN202110325208 A CN 202110325208A CN 113078948 B CN113078948 B CN 113078948B
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CN113078948A (en
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马帅
秦莉莉
张凡
杨瑞鑫
李世银
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China University of Mining and Technology CUMT
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04B—TRANSMISSION
    • H04B10/00—Transmission systems employing electromagnetic waves other than radio-waves, e.g. infrared, visible or ultraviolet light, or employing corpuscular radiation, e.g. quantum communication
    • H04B10/11—Arrangements specific to free-space transmission, i.e. transmission through air or vacuum
    • H04B10/114—Indoor or close-range type systems
    • H04B10/116—Visible light communication
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04Q—SELECTING
    • H04Q11/00—Selecting arrangements for multiplex systems
    • H04Q11/0001—Selecting arrangements for multiplex systems using optical switching
    • H04Q11/0062—Network aspects
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04W—WIRELESS COMMUNICATION NETWORKS
    • H04W24/00—Supervisory, monitoring or testing arrangements
    • H04W24/02—Arrangements for optimising operational condition
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04W—WIRELESS COMMUNICATION NETWORKS
    • H04W28/00—Network traffic management; Network resource management
    • H04W28/16—Central resource management; Negotiation of resources or communication parameters, e.g. negotiating bandwidth or QoS [Quality of Service]
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04W—WIRELESS COMMUNICATION NETWORKS
    • H04W72/00—Local resource management
    • H04W72/04—Wireless resource allocation
    • H04W72/044—Wireless resource allocation based on the type of the allocated resource
    • H04W72/0473—Wireless resource allocation based on the type of the allocated resource the resource being transmission power
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04Q—SELECTING
    • H04Q11/00—Selecting arrangements for multiplex systems
    • H04Q11/0001—Selecting arrangements for multiplex systems using optical switching
    • H04Q11/0062—Network aspects
    • H04Q2011/0086—Network resource allocation, dimensioning or optimisation
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02D—CLIMATE CHANGE MITIGATION TECHNOLOGIES IN INFORMATION AND COMMUNICATION TECHNOLOGIES [ICT], I.E. INFORMATION AND COMMUNICATION TECHNOLOGIES AIMING AT THE REDUCTION OF THEIR OWN ENERGY USE
    • Y02D30/00—Reducing energy consumption in communication networks
    • Y02D30/70—Reducing energy consumption in communication networks in wireless communication networks

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Abstract

The invention provides a downlink transmission optimization method of a LiFi-WiFi polymerization system, which comprises the following steps: step 1, setting a LiFi-WiFi polymerization system; step 2, solving the reachable rate of the polymerized LiFi-WiFi system; step 3, solving the optimal discrete constellation input of the LiFi-WiFi polymerization system; and 4, solving the optimal discrete constellation input distribution based on the lower bound and the upper bound. The constellation distribution and power optimized by the method can obviously improve the reachable rate.

Description

LiFi-WiFi聚合系统的下行链路传输优化方法Downlink Transmission Optimization Method for LiFi-WiFi Aggregation System

技术领域technical field

本发明属于可见光通信领域，尤其涉及一种LiFi-WiFi聚合系统的下行链路传输优化方法。The invention belongs to the field of visible light communication, and in particular relates to a downlink transmission optimization method of a LiFi-WiFi aggregation system.

背景技术Background technique

物联网设备的数量不断增加，不断给无线网络带来巨大的带宽负担。考虑到在室内环境中产生的无线数据80％以上，可见光通信(VLC)或光保真度(LiFi)在400-790T Hz中具有巨大的免许可证带宽，支持高速数据传输和室内照明同时进行。LiFi利用现成的发光二极管(LED)和光电二极管(PD)作为收发器，可以集成到物联网设备中。虽然LiFi是下一代无线解决方案的竞争对手，但易受阻塞和小信号覆盖仍然给各种室内应用带来许多挑战。The ever-increasing number of IoT devices continues to place a huge bandwidth burden on wireless networks. Considering more than 80% of wireless data generated in indoor environment, visible light communication (VLC) or light fidelity (LiFi) has huge license-free bandwidth in 400-790T Hz, supporting high-speed data transmission and indoor lighting simultaneously . LiFi utilizes off-the-shelf light-emitting diodes (LEDs) and photodiodes (PDs) as transceivers that can be integrated into IoT devices. While LiFi is a competitor to next-generation wireless solutions, susceptibility to blocking and small-signal coverage still pose many challenges for a variety of indoor applications.

发明内容SUMMARY OF THE INVENTION

发明目的：为解决背景技术中存在的技术问题，本发明提出LiFi-WiFi聚合系统的下行链路传输优化方法，包括如下步骤：Purpose of the invention: In order to solve the technical problems existing in the background technology, the present invention proposes a downlink transmission optimization method for a LiFi-WiFi aggregation system, including the following steps:

步骤1，对LiFi-WiFi聚合系统进行设定；Step 1, set the LiFi-WiFi aggregation system;

步骤2，求解聚合LiFi-WiFi系统可达速率；Step 2, solve the reachable rate of the aggregated LiFi-WiFi system;

步骤3，求解LiFi-WiFi聚合系统的最优离散星座输入；Step 3, solve the optimal discrete constellation input of the LiFi-WiFi aggregation system;

步骤4，求解基于下界和上界的最优离散星座输入分布。Step 4, solve the optimal discrete constellation input distribution based on the lower bound and the upper bound.

步骤1包括：考虑一个LiFi-WiFi聚合系统的下行链路传输，其中发射机配备了一个发光二极管LED和一个WiFi天线，接收机配备一个单光子探测器PD和一个射频天线，发射机同时通过LiFi链路和WiFi链路传输信息，其中LiFi链路和WiFi链路的带宽分别为B1和B2；Step 1 includes: consider the downlink transmission of a LiFi-WiFi aggregation system, where the transmitter is equipped with a light-emitting diode LED and a WiFi antenna, the receiver is equipped with a single-photon detector PD and an RF antenna, and the transmitter is simultaneously connected through LiFi The link and the WiFi link transmit information, wherein the bandwidths of the LiFi link and the WiFi link are B 1 and B 2 respectively;

设

Figure BDA0002994354290000011
表示发送的信号向量，其中x1∈R和x2∈C分别表示LiFi链路的发送信号和WiFi链路的发送信号，R为实数集合，C为复数集合。Assume
Figure BDA0002994354290000011
represents the transmitted signal vector, where x 1 ∈ R and x 2 ∈ C represent the transmitted signal of the LiFi link and the transmitted signal of the WiFi link, respectively, R is the set of real numbers, and C is the set of complex numbers.

步骤1还包括：在LiFi-WiFi聚合系统中，传输的信号分布在离散星座上，设定LiFi链路信号通过M脉冲幅度调制发送，WiFi链路信号通过N-正交幅度调制发送，信号x1取自具有基数M的非负实离散星座集Ω1，表示为：Step 1 further includes: in the LiFi-WiFi aggregation system, the transmitted signals are distributed in discrete constellations, and it is set that the LiFi link signal is sent through M pulse amplitude modulation, the WiFi link signal is sent through N-quadrature amplitude modulation, and the signal x 1 is taken from the non-negative real discrete constellation set Ω 1 with cardinality M, expressed as:

Figure BDA0002994354290000012
Figure BDA0002994354290000012

其中Pr(·)表示求概率；x1,k表示星座点，取值为非负实数；k表示星座点的序号，p1,k表示x1＝x1,k的概率；参数A，

Figure BDA0002994354290000013
Pe,1分别表示x1的峰值光功率、平均光功率和电功率门限；where Pr( ) represents the probability; x 1,k represents the constellation point, which is a non-negative real number; k represents the serial number of the constellation point, p 1,k represents the probability of x 1 =x 1,k ; parameter A,
Figure BDA0002994354290000013
P e,1 represent the peak optical power, average optical power and electrical power threshold of x 1 , respectively;

WiFi信号x2取自一个具有基数N的复数离散星座集Ω2，表示为：The WiFi signal x 2 is taken from a complex discrete constellation set Ω 2 with base N, expressed as:

Figure BDA0002994354290000021
Figure BDA0002994354290000021

其中x2,l表示星座点，取值为复数，l表示星座点的序号，p2,l表示选择x2,l的概率，Pe,2表示x2的电功率门限。where x 2,l represents the constellation point, which is a complex number, l represents the sequence number of the constellation point, p 2,l represents the probability of selecting x 2,l , and P e,2 represents the electric power threshold of x 2 .

步骤1还包括：设q1∈R和q2∈C分别表示x1的功率放大因子和x2的功率放大因子，q1和q2需满足平均功率约束，即：Step 1 also includes: Let q 1 ∈ R and q 2 ∈ C represent the power amplification factor of x 1 and the power amplification factor of x 2 respectively, and q 1 and q 2 need to satisfy the average power constraint, namely:

Figure BDA0002994354290000022
Figure BDA0002994354290000022

其中η1和η2分别表示LiFi链路的功率放大器的效率和WiFi链路的功率放大器的效率，PT表示平均电功率门限，中间参数ε1和ε2分别为：where η 1 and η 2 represent the efficiency of the power amplifier of the LiFi link and the power amplifier of the WiFi link, respectively, P T represents the average electrical power threshold, and the intermediate parameters ε 1 and ε 2 are:

Figure BDA0002994354290000023
Figure BDA0002994354290000023

Figure BDA0002994354290000024
Figure BDA0002994354290000024

步骤1还包括：对LiFi信号的功率控制需要满足平均光功率和峰值光功率要求，如下所示：Step 1 also includes: the power control of the LiFi signal needs to meet the average optical power and peak optical power requirements, as follows:

Figure BDA0002994354290000025
Figure BDA0002994354290000025

q1A≤Pins，q 1 A≤P ins ,

其中，

Figure BDA0002994354290000026
表示求均值，
Figure BDA0002994354290000027
表示x1的均值；Po和Pins分别表示平均光功率和瞬时光功率门限。in,
Figure BDA0002994354290000026
represents the mean value,
Figure BDA0002994354290000027
Represents the mean value of x 1 ; P o and Pins represent the average optical power and instantaneous optical power threshold, respectively.

步骤1还包括：设

Figure BDA0002994354290000028
表示信道向量，其中g1和g2分别是LiFi链路的信道增益和WiFi链路的信道增益；设y1和y2分别表示来自LiFi链路的接收信号和来自WiFi链路的接收信号，写成如下矢量形式：Step 1 also includes: setting
Figure BDA0002994354290000028
represents the channel vector, where g 1 and g 2 are the channel gain of the LiFi link and the channel gain of the WiFi link, respectively; let y 1 and y 2 denote the received signal from the LiFi link and the received signal from the WiFi link, respectively, Written in vector form as follows:

Figure BDA0002994354290000029
Figure BDA0002994354290000029

其中

Figure BDA00029943542900000210
是来LiFi链路的实高斯噪声，
Figure BDA00029943542900000211
是来自WiFi链路的复高斯噪声，
Figure BDA0002994354290000031
表示均值为0、方差为
Figure BDA0002994354290000032
的高斯分布，
Figure BDA0002994354290000033
表示均值为0、方差为
Figure BDA0002994354290000034
的复高斯分布；in
Figure BDA00029943542900000210
is the real Gaussian noise from the LiFi link,
Figure BDA00029943542900000211
is the complex Gaussian noise from the WiFi link,
Figure BDA0002994354290000031
means that the mean is 0 and the variance is
Figure BDA0002994354290000032
the Gaussian distribution of ,
Figure BDA0002994354290000033
means that the mean is 0 and the variance is
Figure BDA0002994354290000034
The complex Gaussian distribution of ;

步骤2包括：Step 2 includes:

步骤2-1，将聚合LiFi-WiFi系统可达速率RLiFi-WiFi定义为：Step 2-1, define the achievable rate R LiFi-WiFi of the aggregated LiFi-WiFi system as:

Figure BDA0002994354290000035
Figure BDA0002994354290000035

其中

Figure BDA0002994354290000036
和
Figure BDA0002994354290000037
分别表示LiFi链路的可达速率和WiFi链路的可达速率，I(x；y)表示信道平均互信息；in
Figure BDA0002994354290000036
and
Figure BDA0002994354290000037
respectively represent the reachable rate of LiFi link and the reachable rate of WiFi link, I(x; y) represents the channel average mutual information;

步骤2-2，基于离散星座点输入的LiFi-WiFi聚合系统，给定LiFi链路和WiFi链路带宽B1和B2，则LiFi-WiFi聚合系统可达速率RLiFi和RWiFi分别为：Step 2-2, based on the LiFi-WiFi aggregation system input by discrete constellation points, given the LiFi link and WiFi link bandwidths B 1 and B 2 , the achievable rates R LiFi and R WiFi of the LiFi-WiFi aggregation system are respectively:

Figure BDA0002994354290000038
Figure BDA0002994354290000038

Figure BDA0002994354290000039
Figure BDA0002994354290000039

其中，

Figure BDA00029943542900000310
表示求关于z1函数的均值，
Figure BDA00029943542900000311
表示求关于z2函数的均值，
Figure BDA00029943542900000312
表示信道增益g2的共轭；in,
Figure BDA00029943542900000310
means to find the mean of the z 1 function,
Figure BDA00029943542900000311
means to find the mean of the z 2 function,
Figure BDA00029943542900000312
represents the conjugate of the channel gain g2 ;

步骤3包括：Step 3 includes:

步骤3-1：LiFi-WiFi聚合系统的最优离散星座输入问题表述为：Step 3-1: The optimal discrete constellation input problem for the LiFi-WiFi aggregation system is formulated as:

Figure BDA00029943542900000313
Figure BDA00029943542900000313

Figure BDA00029943542900000315
Figure BDA00029943542900000315

q1≤min(Po/μ,Pins/A),(7c)q 1 ≤min(P o /μ,P ins /A),(7c)

Figure BDA00029943542900000314
Figure BDA00029943542900000314

Figure BDA0002994354290000041
Figure BDA0002994354290000041

Figure BDA0002994354290000042
Figure BDA0002994354290000042

Figure BDA0002994354290000043
Figure BDA0002994354290000043

其中，

Figure BDA0002994354290000044
表示LiFi链路星座点的序号，
Figure BDA0002994354290000045
表示WiFi链路星座点的序号；in,
Figure BDA0002994354290000044
Indicates the sequence number of the LiFi link constellation point,
Figure BDA0002994354290000045
Indicates the serial number of the WiFi link constellation point;

步骤3-2：可达速率RLiFi-WiFi写为：Step 3-2: The achievable rate R LiFi-WiFi is written as:

Figure BDA0002994354290000046
Figure BDA0002994354290000046

其中，

Figure BDA0002994354290000047
和
Figure BDA0002994354290000048
分别表示LiFi链路的发射功率和WiFi链路的发射功率；in,
Figure BDA0002994354290000047
and
Figure BDA0002994354290000048
represent the transmit power of the LiFi link and the transmit power of the WiFi link, respectively;

步骤3-3：定义

Figure BDA0002994354290000049
约束条件(7e)和(7d)写为：Step 3-3: Definition
Figure BDA0002994354290000049
Constraints (7e) and (7d) are written as:

Figure BDA00029943542900000410
Figure BDA00029943542900000410

其中，

Figure BDA00029943542900000411
表示LiFi链路的星座点向量，x1,M表示LiFi链路的第M个星座点，Υ1表示关于向量p集合，p1表示LiFi链路的星座点概率向量，p1,M表示x1＝x1,M的概率，
Figure BDA00029943542900000412
表示元素全为1的1×M的行向量，p代表向量；in,
Figure BDA00029943542900000411
Represents the constellation point vector of the LiFi link, x 1, M represents the Mth constellation point of the LiFi link, Υ 1 represents the set of about vectors p, p 1 represents the constellation point probability vector of the LiFi link, p 1, M represents x 1 = probability of x 1,M ,
Figure BDA00029943542900000412
Represents a 1×M row vector with all 1 elements, p represents a vector;

定义

Figure BDA00029943542900000413
约束条件(7f)和(7g)写成：definition
Figure BDA00029943542900000413
Constraints (7f) and (7g) are written as:

Figure BDA00029943542900000414
Figure BDA00029943542900000414

其中，

Figure BDA0002994354290000051
表示WiFi链路的星座点向量，x2,N表示WiFi链路的第N个星座点，Υ2表示关于向量p集合，p2表示WiFi链路的星座点概率向量，p2,N表示x2＝x2,N的概率，
Figure BDA0002994354290000052
表示元素全为1的1×N的行向量；in,
Figure BDA0002994354290000051
Represents the constellation point vector of the WiFi link, x 2,N represents the Nth constellation point of the WiFi link, Υ 2 represents the set about the vector p, p 2 represents the constellation point probability vector of the WiFi link, p 2,N represents x 2 = probability of x 2,N ,
Figure BDA0002994354290000052
Represents a 1×N row vector whose elements are all 1s;

引入辅助变量w、

Figure BDA0002994354290000053
r、
Figure BDA0002994354290000054
Introduce auxiliary variable w,
Figure BDA0002994354290000053
r.
Figure BDA0002994354290000054

Figure BDA0002994354290000055
Figure BDA0002994354290000055

Figure BDA0002994354290000056
Figure BDA0002994354290000056

Figure BDA0002994354290000057
Figure BDA0002994354290000057

Figure BDA0002994354290000058
Figure BDA0002994354290000058

Figure BDA0002994354290000059
Figure BDA0002994354290000059

Figure BDA00029943542900000510
Figure BDA00029943542900000510

则可达速率RLiFi-WiFi重写为：Then the achievable rate R LiFi-WiFi can be rewritten as:

Figure BDA00029943542900000511
Figure BDA00029943542900000511

步骤3-4：问题(7)等效如下问题(14)：Steps 3-4: Problem (7) is equivalent to the following problem (14):

Figure BDA00029943542900000512
Figure BDA00029943542900000512

Figure BDA00029943542900000513
Figure BDA00029943542900000513

Figure BDA00029943542900000514
Figure BDA00029943542900000514

p1∈Υ1,p2∈Υ2,(14d)p 1 ∈Υ 1 ,p 2 ∈ Υ 2 ,(14d)

其中

Figure BDA00029943542900000515
τ表示Po/μ,Pins/A中的最小值；in
Figure BDA00029943542900000515
τ represents the minimum value in P o /μ, Pins /A;

问题(14)中，

Figure BDA00029943542900000516
和
Figure BDA00029943542900000517
的功率分配变量只包含在约束(14b)和(14c)中，而分布变量p1和p2只包含在约束(14d)中，问题(14)通过迭代求解以下两个子问题来处理，直到总体问题收敛：In question (14),
Figure BDA00029943542900000516
and
Figure BDA00029943542900000517
The power distribution variables of are only included in constraints (14b) and (14c), while the distribution variables p1 and p2 are only included in constraints (14d), problem ( 14 ) is addressed by iteratively solving the following two subproblems until the overall The problem converges:

功率分配子问题1：给定的p1和p2优化

Figure BDA0002994354290000061
和
Figure BDA0002994354290000062
Power distribution subproblem 1: optimization given p 1 and p 2
Figure BDA0002994354290000061
and
Figure BDA0002994354290000062

概率分布子问题2：给定的

Figure BDA0002994354290000063
和
Figure BDA0002994354290000064
优化p1和p2；Probability Distribution Subproblem 2: Given
Figure BDA0002994354290000063
and
Figure BDA0002994354290000064
optimize p 1 and p 2 ;

对于功率分配子问题1：当给出p1和p2时，问题(14)是一个最优的功率分配问题，如下问题(15)所示：For power allocation subproblem 1: when p 1 and p 2 are given, problem (14) is an optimal power allocation problem, as shown in problem (15) below:

Figure BDA0002994354290000065
Figure BDA0002994354290000065

Figure BDA0002994354290000066
Figure BDA0002994354290000066

Figure BDA0002994354290000067
Figure BDA0002994354290000067

其中，

Figure BDA0002994354290000068
h(·)表示关于
Figure BDA0002994354290000069
的函数；in,
Figure BDA0002994354290000068
h( ) means about
Figure BDA0002994354290000069
The function;

问题(15)对

Figure BDA00029943542900000610
和
Figure BDA00029943542900000611
是一个凸问题，采用注水法解决该问题，并得到最优功率分配
Figure BDA00029943542900000612
和
Figure BDA00029943542900000613
Question (15) to
Figure BDA00029943542900000610
and
Figure BDA00029943542900000611
is a convex problem, and the water injection method is used to solve the problem and obtain the optimal power distribution
Figure BDA00029943542900000612
and
Figure BDA00029943542900000613

对于概率分布子问题2：当给出

Figure BDA00029943542900000614
和
Figure BDA00029943542900000615
时，问题(14)表示为如下问题(16)：For the probability distribution subproblem 2: when given
Figure BDA00029943542900000614
and
Figure BDA00029943542900000615
, problem (14) is expressed as the following problem (16):

Figure BDA00029943542900000616
Figure BDA00029943542900000616

s.t.p1∈Υ1,(16b)stp 1 ∈ Υ 1 , (16b)

p2∈Υ2,(16c)p 2 ∈Υ 2 , (16c)

问题(16)是一个有两个变量p1和p2的凸优化问题，采用不精确梯度下降法，并得到LiFi链路的概率分布p1和WiFi链路的概率分布p2；Problem (16) is a convex optimization problem with two variables p 1 and p 2 , using inexact gradient descent method, and obtain the probability distribution p 1 of LiFi link and the probability distribution p 2 of WiFi link ;

综上所述，求解优化问题(7)，可以通过迭代求解功率分配子问题(15)和概率分布子问题(16)，可以得到最大可达速率RLiFi-WiFi。To sum up, to solve the optimization problem (7), the power distribution sub-problem (15) and the probability distribution sub-problem (16) can be solved iteratively, and the maximum achievable rate R LiFi-WiFi can be obtained.

步骤4包括：Step 4 includes:

步骤4-1：在离散星座点输入条件下，LiFi链路传输速率RLiFi的上界和下界的闭式表达式分别为Step 4-1: Under the condition of discrete constellation point input, the closed-form expressions of the upper and lower bounds of the LiFi link transmission rate R LiFi are respectively

Figure BDA00029943542900000617
Figure BDA00029943542900000617

Figure BDA0002994354290000071
Figure BDA0002994354290000071

步骤4-2：在离散星座点输入条件下，给出WiFi链路可达速率的上界和下界，如下所示：Step 4-2: Under the input condition of discrete constellation points, give the upper and lower bounds of the WiFi link achievable rate, as follows:

Figure BDA0002994354290000072
Figure BDA0002994354290000072

Figure BDA0002994354290000073
Figure BDA0002994354290000073

步骤4-3：让

Figure BDA0002994354290000074
和
Figure BDA0002994354290000075
分别表示RLiFi-WiFi的下界和上界，得到：Step 4-3: Let
Figure BDA0002994354290000074
and
Figure BDA0002994354290000075
Representing the lower and upper bounds of R LiFi-WiFi , respectively, we get:

Figure BDA0002994354290000076
Figure BDA0002994354290000076

Figure BDA0002994354290000077
Figure BDA0002994354290000077

步骤4-4：基于可达速率

Figure BDA0002994354290000078
的下界，优化LiFi链路和WiFi链路的输入星座点概率分布和功率分配，以获得最大的传输速率下界
Figure BDA0002994354290000079
该优化问题可表示如下：Step 4-4: Based on reachable rate
Figure BDA0002994354290000078
The lower bound of , optimizes the input constellation point probability distribution and power allocation of LiFi link and WiFi link to obtain the maximum transmission rate lower bound
Figure BDA0002994354290000079
The optimization problem can be expressed as follows:

Figure BDA00029943542900000710
Figure BDA00029943542900000710

此外，通过定义：Furthermore, by defining:

Figure BDA00029943542900000711
Figure BDA00029943542900000711

Figure BDA00029943542900000712
Figure BDA00029943542900000712

Figure BDA0002994354290000081
Figure BDA0002994354290000081

Figure BDA0002994354290000082
Figure BDA0002994354290000082

Figure BDA0002994354290000083
Figure BDA0002994354290000083

Figure BDA0002994354290000084
Figure BDA0002994354290000084

将

Figure BDA0002994354290000085
改写如下：Will
Figure BDA0002994354290000085
Rewritten as follows:

Figure BDA0002994354290000086
Figure BDA0002994354290000086

则问题(20)用如下公式表示：Then problem (20) is expressed by the following formula:

Figure BDA0002994354290000087
Figure BDA0002994354290000087

为了解决问题(23)，通过迭代求解以下两个子问题，功率分配子问题3和概率分布子问题4，直到整体问题达到收敛为止：To solve problem (23), the following two sub-problems, power distribution sub-problem 3 and probability distribution sub-problem 4, are solved iteratively until the overall problem reaches convergence:

功率分配子问题3：当p1和p2被固定时，优化LiFi链路和WiFi链路的功率分配

Figure BDA0002994354290000088
和
Figure BDA0002994354290000089
问题(23)表述如下问题(24)：Power allocation subproblem 3: Optimizing power allocation for LiFi links and WiFi links when p1 and p2 are fixed
Figure BDA0002994354290000088
and
Figure BDA0002994354290000089
Question (23) formulates the following question (24):

Figure BDA00029943542900000810
Figure BDA00029943542900000810

Figure BDA00029943542900000811
Figure BDA00029943542900000811

Figure BDA00029943542900000812
Figure BDA00029943542900000812

Figure BDA00029943542900000813
Figure BDA00029943542900000813

采用近似梯度投影法解决问题(24)，并得到LiFi和WiFi链路的最优功率分布

Figure BDA00029943542900000814
和
Figure BDA00029943542900000815
Approximate gradient projection method to solve problem (24) and obtain optimal power distribution for LiFi and WiFi links
Figure BDA00029943542900000814
and
Figure BDA00029943542900000815

概率分布子问题4：用给定的

Figure BDA00029943542900000816
和
Figure BDA00029943542900000817
优化p1和p2的概率分布，当
Figure BDA00029943542900000818
和
Figure BDA00029943542900000819
固定时，问题(24)表示为如下问题(25)：Probability distribution subproblem 4: Using a given
Figure BDA00029943542900000816
and
Figure BDA00029943542900000817
optimize the probability distribution of p1 and p2 , when
Figure BDA00029943542900000818
and
Figure BDA00029943542900000819
When fixed, problem (24) is represented as problem (25) as follows:

Figure BDA0002994354290000091
Figure BDA0002994354290000091

s.t.p1∈Υ1,(25b)stp 1 ∈Υ 1 , (25b)

p2∈Υ2,(25c)p 2 ∈Υ 2 ,(25c)

其中，in,

Figure BDA0002994354290000092
Figure BDA0002994354290000092

Figure BDA0002994354290000093
Figure BDA0002994354290000093

则问题(25)分为两个独立的子问题，分别为问题(26)：Then problem (25) is divided into two independent sub-problems, namely problem (26):

Figure BDA0002994354290000094
Figure BDA0002994354290000094

s.t.p1∈Υ1,(26b)stp 1 ∈Υ 1 , (26b)

和问题(27)and questions (27)

Figure BDA0002994354290000095
Figure BDA0002994354290000095

s.t.p2∈Υ2(27b)stp 2 ∈ Υ 2 (27b)

其中，

Figure BDA0002994354290000096
表示关于p1的函数，
Figure BDA0002994354290000097
表示关于p2的函数；in,
Figure BDA0002994354290000096
represents a function with respect to p 1 ,
Figure BDA0002994354290000097
represents a function of p 2 ;

应用Frank-Wolfe方法来解决问题(26)和(27)，从而得到LiFi链路p1的概率分布和WiFi链路p2的最优概率分布。The Frank-Wolfe method is applied to solve problems (26) and (27), resulting in the probability distribution of LiFi link p 1 and the optimal probability distribution of WiFi link p 2 .

步骤4-4还包括：将得到的

Figure BDA0002994354290000098
p1,p2带入公式(22)，得到最大可达速率下界RL LiFi-WiFi。Steps 4-4 also include: the resulting
Figure BDA0002994354290000098
p 1 , p 2 are brought into formula (22) to obtain the lower bound R L LiFi-WiFi of the maximum achievable rate.

有益效果：本发明所提出的LiFi-WiFi聚合系统框架可以克服频繁的链路切换，从而提高系统数据速率，并提供可靠的通信。本发明提出的LiFi-WiFi聚合系统的下行链路优化方法，导出了具有任意离散分布的精确可达速率表达式，不仅方法计算精度高，求解速度快，而且能得到最优的输入分布和功率分配，从而显著地提高了系统的可达速率和能效。Beneficial effects: The LiFi-WiFi aggregation system framework proposed by the present invention can overcome frequent link switching, thereby increasing the system data rate and providing reliable communication. The downlink optimization method of the LiFi-WiFi aggregation system proposed by the present invention derives an accurate achievable rate expression with any discrete distribution. The method not only has high calculation accuracy and fast solution speed, but also can obtain the optimal input distribution and power distribution, thereby significantly improving the reachability and energy efficiency of the system.

附图说明Description of drawings

下面结合附图和具体实施方式对本发明做更进一步的具体说明，本发明的上述和/或其他方面的优点将会变得更加清楚。The present invention will be further described in detail below in conjunction with the accompanying drawings and specific embodiments, and the advantages of the above-mentioned and/or other aspects of the present invention will become clearer.

图1a为LiFi链路在不同SNR1下各自的最优概率分布{x1,k,p1,k}的变化曲线示意图。Figure 1a is a schematic diagram of the variation curve of the respective optimal probability distribution {x 1,k , p 1,k } of the LiFi link under different SNR 1 .

图1b为LiFi链路在不同SNR1下各自的最优输入位置{x1,k}的变化曲线示意图。Figure 1b is a schematic diagram of the variation curves of the respective optimal input positions {x 1, k } of the LiFi link under different SNR 1 .

图1c为LiFi链路在不同SNR1下各自的可实现速率RLiFi的变化曲线示意图。Figure 1c is a schematic diagram of the change curves of the respective achievable rates R LiFi of LiFi links under different SNR 1 .

图2a为WiFi链路在SNR2＝-6dB下的最优概率分布{x2,l,p2,l}的变化曲线示意图。Fig. 2a is a schematic diagram of the variation curve of the optimal probability distribution {x 2,l ,p 2,l } of the WiFi link under the condition of SNR 2 =-6dB.

图2b为WiFi链路在SNR2＝4dB下的最优概率分布{x2,l,p2,l}的变化曲线示意图。FIG. 2b is a schematic diagram of the variation curve of the optimal probability distribution {x 2,l , p 2,l } of the WiFi link at SNR 2 =4dB.

图2c为WiFi链路在SNR2＝12dB下的最优概率分布{x2,l,p2,l}的变化曲线示意图。FIG. 2c is a schematic diagram of the variation curve of the optimal probability distribution {x 2,l ,p 2,l } of the WiFi link under the condition of SNR 2 =12dB.

图2d为WiFi链路在不同SNR下各自的可达速率RWiFi的变化曲线示意图。FIG. 2d is a schematic diagram of the variation curves of the respective attainable rates R WiFi of WiFi links under different SNRs.

图3a为LiFi-WiFi系统在不同PT下各自可达速率RLiFi-WiFi的变化曲线示意图。Figure 3a is a schematic diagram of the change curve of the respective attainable rate R LiFi-WiFi of the LiFi-WiFi system under different P T.

图3b为LiFi-WiFi系统在不同Pins下各自可达速率RLiFi-WiFi的变化曲线示意图。Figure 3b is a schematic diagram of the variation curve of the respective attainable rate R LiFi-WiFi of the LiFi -WiFi system under different Pins.

图4a为LiFi-WiFi系统在不同PT下各自可达速率下界RL LiFi-WiFi的变化曲线示意图。Figure 4a is a schematic diagram of the variation curve of the lower bound R L LiFi -WiFi of the LiFi-WiFi system under different PTs.

图4b为LiFi-WiFi系统在不同Pins下各自可达速率下界RL LiFi-WiFi的变化曲线示意图。Figure 4b is a schematic diagram of the variation curve of the lower bound R L LiFi-WiFi of the respective achievable rate of the LiFi -WiFi system under different Pins.

图5a为LiFi-WiFi系统在带宽B1下各自可达速率下界RL LiFi-WiFi的变化曲线示意图。Figure 5a is a schematic diagram of the variation curve of the lower bound R L LiFi-WiFi of the respective achievable rates of the LiFi-WiFi system under the bandwidth B 1 .

图5b为LiFi-WiFi系统在带宽B2下各自可达速率下界RL LiFi-WiFi的变化曲线示意图。Figure 5b is a schematic diagram of the variation curve of the lower bound R L LiFi-WiFi of the respective achievable rates of the LiFi-WiFi system under the bandwidth B 2 .

图6a为在问题(7)的最优解下RLiFi-WiFi、RL LiFi-WiFi和RU LiFi-WiFi在不同PT下的变化曲线示意图。Figure 6a is a schematic diagram of the change curves of R LiFi-WiFi , R L LiFi-WiFi and R U LiFi-WiFi under different P T under the optimal solution of problem (7).

图6b为在问题(20)的最优解下RLiFi-WiFi、RL LiFi-WiFi和RU LiFi-WiFi在不同PT下的变化曲线示意图。Figure 6b is a schematic diagram of the change curves of R LiFi-WiFi , R L LiFi-WiFi and R U LiFi-WiFi under different P T under the optimal solution of problem (20).

具体实施方式Detailed ways

在本发明中，从实际的通信角度考虑了一个LiFi-WiFi聚合系统。首先，导出了具有任意离散分布的LiFi-WiFi聚合系统的精确可达速率表达式，鉴于这种速率表达式不是封闭形式的，进一步导出了下界和上界。然后，优化了离散星座输入分配和功率分配，以最大程度地提高导出的可达速率，为了解决这个非凸问题，利用了互信息和最小均方误差(MMSE)之间的关系，通过不精确梯度下降法计算出离散星座的最优概率分布。本发明的结果为LiFi-WiFi聚合系统提供了一个相对实用的设计框架。In the present invention, a LiFi-WiFi aggregation system is considered from the practical communication point of view. First, an accurate achievable rate expression for LiFi-WiFi aggregation systems with arbitrary discrete distributions is derived, and since this rate expression is not closed-form, lower and upper bounds are further derived. Then, the discrete constellation input allocation and power allocation are optimized to maximize the derived achievable rate, and to solve this non-convex problem, the relationship between mutual information and minimum mean square error (MMSE) is exploited by imprecise The gradient descent method computes the optimal probability distribution for discrete constellations. The results of the present invention provide a relatively practical design framework for the LiFi-WiFi aggregation system.

LiFi-WiFi聚合系统的模型：Model of LiFi-WiFi aggregation system:

考虑了一个LiFi-WiFi聚合系统的下行链路传输，其中发射机配备了一个发光二极管(LED)和一个WiFi天线，接收机配备一个单光子探测器(PD)和一个射频天线。发射机同时通过LiFi链路和WiFi链路传输信息，其中LiFi链路和WiFi链路的带宽分别为B1Hz和B2Hz。设

Figure BDA0002994354290000101
表示发送的信号向量，其中x1∈R和x2∈C分别表示LiFi链路和WiFi链路的发送信号，R为实数集合，C为复数集合。The downlink transmission of a LiFi-WiFi aggregation system is considered, where the transmitter is equipped with a light-emitting diode (LED) and a WiFi antenna, and the receiver is equipped with a single-photon detector (PD) and an RF antenna. The transmitter transmits information simultaneously through a LiFi link and a WiFi link, where the bandwidths of the LiFi link and WiFi link are B 1 Hz and B 2 Hz, respectively. Assume
Figure BDA0002994354290000101
represents the transmitted signal vector, where x 1 ∈ R and x 2 ∈ C represent the transmitted signals of the LiFi link and WiFi link, respectively, R is the set of real numbers, and C is the set of complex numbers.

在一个实用的LiFi-WiFi通信系统中，传输的信号分布在离散星座上，设定LiFi链路信号通过M脉冲幅度调制(PAM)发送，WiFi链路信号通过N-正交幅度调制(QAM)发送。更具体地说，信号x1取自具有基数M的非负实离散星座集Ω1，可表示为：In a practical LiFi-WiFi communication system, the transmitted signals are distributed in discrete constellations, and the LiFi link signal is set to be sent by M-pulse amplitude modulation (PAM), and the WiFi link signal is sent by N-quadrature amplitude modulation (QAM) send. More specifically, the signal x 1 is taken from a non-negative real discrete constellation set Ω 1 with base M, which can be expressed as:

Figure BDA0002994354290000111
Figure BDA0002994354290000111

WiFi信号x2取自一个具有基数N的复杂离散星座集Ω2，表示为：The WiFi signal x 2 is taken from a complex discrete constellation set Ω 2 with base N, expressed as:

Figure BDA0002994354290000112
Figure BDA0002994354290000112

设q1∈R和q2∈C分别表示x1和x2的功率放大因子，q1和q2需满足平均功率约束，即：Let q 1 ∈ R and q 2 ∈ C represent the power amplification factors of x 1 and x 2 , respectively, and q 1 and q 2 need to satisfy the average power constraint, namely:

Figure BDA0002994354290000113
Figure BDA0002994354290000113

其中η1和η2分别表示LiFi链路的功率放大器的效率和WiFi链路的功率放大器的效率，

Figure BDA0002994354290000114
PT表示平均电功率门限。where η 1 and η 2 represent the efficiency of the power amplifier of the LiFi link and the efficiency of the power amplifier of the WiFi link, respectively,
Figure BDA0002994354290000114
PT represents the average electrical power threshold.

此外，为了人的眼睛安全考虑，对LiFi信号的功率控制还需要满足平均光功率和峰值光功率要求：

Figure BDA0002994354290000115
和q1A≤Pins，其中，
Figure BDA0002994354290000116
表示求均值，
Figure BDA0002994354290000117
表示x1的均值
Figure BDA0002994354290000118
Po和Pins分别表示平均光功率和瞬时光功率门限。In addition, for the safety of human eyes, the power control of LiFi signals also needs to meet the requirements of average optical power and peak optical power:
Figure BDA0002994354290000115
and q 1 A≤P ins , where,
Figure BDA0002994354290000116
represents the mean value,
Figure BDA0002994354290000117
means the mean of x 1
Figure BDA0002994354290000118
P o and Pins represent the average optical power and instantaneous optical power thresholds, respectively.

设

Figure BDA0002994354290000119
表示信道向量，其中g1和g2分别是LiFi链路的信道增益和WiFi链路的信道增益；设y1和y2分别表示来自LiFi链路的接收信号和来自WiFi链路的接收信号，它们可以写成矢量形式Assume
Figure BDA0002994354290000119
represents the channel vector, where g 1 and g 2 are the channel gain of the LiFi link and the channel gain of the WiFi link, respectively; let y 1 and y 2 denote the received signal from the LiFi link and the received signal from the WiFi link, respectively, They can be written in vector form

Figure BDA00029943542900001110
Figure BDA00029943542900001110

其中

Figure BDA00029943542900001111
是来LiFi链路的实高斯噪声，
Figure BDA00029943542900001112
是来自WiFi链路的复高斯噪声，
Figure BDA00029943542900001113
表示均值为0，方差为
Figure BDA00029943542900001114
的高斯分布，
Figure BDA00029943542900001115
表示均值为0，方差为
Figure BDA0002994354290000121
的复高斯分布。in
Figure BDA00029943542900001111
is the real Gaussian noise from the LiFi link,
Figure BDA00029943542900001112
is the complex Gaussian noise from the WiFi link,
Figure BDA00029943542900001113
means that the mean is 0 and the variance is
Figure BDA00029943542900001114
the Gaussian distribution of ,
Figure BDA00029943542900001115
means that the mean is 0 and the variance is
Figure BDA0002994354290000121
complex Gaussian distribution.

A.LiFi-WiFi聚合系统的可达速率A. Achievable rate of LiFi-WiFi aggregation system

对于离散星座点的LiFi-WiFi聚合系统，可达速率仍然是未知的。为了解决这个问题，将可达速率RLiFi-WiFi定义为：For LiFi-WiFi aggregation systems with discrete constellation points, the achievable rate is still unknown. To solve this problem, the achievable rate R LiFi-WiFi is defined as:

Figure BDA0002994354290000122
Figure BDA0002994354290000122

Figure BDA0002994354290000123
Figure BDA0002994354290000123

其中

Figure BDA0002994354290000124
和
Figure BDA0002994354290000125
分别表示LiFi链路的可达速率和WiFi链路的可达速率，I(x；y)表示信道平均互信息；in
Figure BDA0002994354290000124
and
Figure BDA0002994354290000125
respectively represent the reachable rate of LiFi link and the reachable rate of WiFi link, I(x; y) represents the channel average mutual information;

引理1：基于离散星座点输入的LiFi-WiFi聚合系统，给定LiFi链路和WiFi链路带宽B1和B2，则LiFi-WiFi聚合系统可达速率RLiFi和RWiFi分别为：Lemma 1: LiFi-WiFi aggregation system based on discrete constellation point input, given LiFi link and WiFi link bandwidth B 1 and B 2 , the achievable rates of LiFi-WiFi aggregation system R LiFi and R WiFi are:

Figure BDA0002994354290000126
Figure BDA0002994354290000126

Figure BDA0002994354290000127
Figure BDA0002994354290000127

其中，

Figure BDA0002994354290000128
求关于z1函数的均值，
Figure BDA0002994354290000129
求关于z2函数的均值，
Figure BDA00029943542900001210
表示信道增益g2的共轭。in,
Figure BDA0002994354290000128
Find the mean with respect to the z 1 function,
Figure BDA0002994354290000129
Find the mean about the z 2 function,
Figure BDA00029943542900001210
represents the conjugate of the channel gain g2 .

Lifi-Wifi聚合系统的最优离散星座输入Optimal Discrete Constellation Input for Lifi-Wifi Aggregation System

基于RLiFi-WiFi的导出表达式，以下问题是优化LiF链路i和WiFi链路的信号分布和功率分配，以获得聚合系统的最大可达速率。数学上，这样的优化问题可以表述为Based on the derived expression of R LiFi-WiFi , the following problem is to optimize the signal distribution and power distribution of LiF link i and WiFi link to obtain the maximum achievable rate of the aggregated system. Mathematically, such an optimization problem can be formulated as

Figure BDA00029943542900001211
Figure BDA00029943542900001211

Figure BDA00029943542900001212
Figure BDA00029943542900001212

q1≤min(Po/μ,Pins/A), (7c)q 1 ≤min(P o /μ,P ins /A), (7c)

Figure BDA00029943542900001213
Figure BDA00029943542900001213

Figure BDA0002994354290000131
Figure BDA0002994354290000131

Figure BDA0002994354290000132
Figure BDA0002994354290000132

Figure BDA0002994354290000133
Figure BDA0002994354290000133

其中，

Figure BDA0002994354290000134
表示LiFi链路星座点的序号，
Figure BDA0002994354290000135
表示WiFi链路星座点的序号；in,
Figure BDA0002994354290000134
Indicates the sequence number of the LiFi link constellation point,
Figure BDA0002994354290000135
Indicates the serial number of the WiFi link constellation point;

对于问题(7)，可以用反证据法证明q2的最优相位与g2的最优相位相同。因此，最优q2可以写成：For problem (7), it can be proved by counter-evidence that the optimal phase of q 2 is the same as the optimal phase of g 2 . Therefore, the optimal q can be written as :

Figure BDA0002994354290000136
Figure BDA0002994354290000136

其中，

Figure BDA0002994354290000137
表示WiFi链路的发射功率；in,
Figure BDA0002994354290000137
Indicates the transmit power of the WiFi link;

通过将(8)替换到(5)，可以将可达速率RLiFi-WiFi可写为：By replacing (8) with (5), the achievable rate R LiFi-WiFi can be written as:

Figure BDA0002994354290000138
Figure BDA0002994354290000138

其中，

Figure BDA0002994354290000139
表示LiFi链路的发射功率；in,
Figure BDA0002994354290000139
Indicates the transmit power of the LiFi link;

通过定义

Figure BDA00029943542900001310
约束条件(7e)和(7d)可写为by definition
Figure BDA00029943542900001310
Constraints (7e) and (7d) can be written as

Figure BDA00029943542900001311
Figure BDA00029943542900001311

其中，

Figure BDA00029943542900001312
表示LiFi链路的星座点向量，x1,M表示LiFi链路的第M个星座点，Υ1表示关于向量p集合，p1表示LiFi链路的星座点概率向量，p1,M表示x1＝x1,M的概率，1TM表示元素全为1的1×M的行向量，p代表向量；in,
Figure BDA00029943542900001312
Represents the constellation point vector of the LiFi link, x 1, M represents the Mth constellation point of the LiFi link, Υ 1 represents the set of vectors p, p 1 represents the constellation point probability vector of the LiFi link, p 1, M represents x 1 = the probability of x 1, M , 1T M represents a 1×M row vector with all 1 elements, and p represents a vector;

定义

Figure BDA0002994354290000141
约束条件(7f)和(7g)可写成definition
Figure BDA0002994354290000141
Constraints (7f) and (7g) can be written as

Figure BDA0002994354290000142
Figure BDA0002994354290000142

其中，

Figure BDA0002994354290000143
表示WiFi链路的星座点向量，x2,N表示WiFi链路的第N个星座点，Υ2表示关于向量p集合，p2表示WiFi链路的星座点概率向量，p2,N表示x2＝x2,N的概率，
Figure BDA0002994354290000144
表示元素全为1的1×N的行向量；in,
Figure BDA0002994354290000143
Represents the constellation point vector of the WiFi link, x 2,N represents the Nth constellation point of the WiFi link, Υ 2 represents the set about the vector p, p 2 represents the constellation point probability vector of the WiFi link, p 2,N represents x 2 = probability of x 2,N ,
Figure BDA0002994354290000144
Represents a 1×N row vector whose elements are all 1s;

接下来，引入辅助变量：Next, introduce auxiliary variables:

Figure BDA0002994354290000145
Figure BDA0002994354290000145

Figure BDA0002994354290000146
Figure BDA0002994354290000146

Figure BDA0002994354290000147
Figure BDA0002994354290000147

Figure BDA0002994354290000148
Figure BDA0002994354290000148

Figure BDA0002994354290000149
Figure BDA0002994354290000149

Figure BDA00029943542900001410
Figure BDA00029943542900001410

则可达速率RLiFi-WiFi重写为：Then the achievable rate R LiFi-WiFi can be rewritten as:

Figure BDA00029943542900001411
Figure BDA00029943542900001411

基于上述定义，问题(7)可等效如下Based on the above definition, problem (7) can be equivalently as follows

Figure BDA00029943542900001412
Figure BDA00029943542900001412

Figure BDA00029943542900001413
Figure BDA00029943542900001413

Figure BDA00029943542900001414
Figure BDA00029943542900001414

p1∈Υ1,p2∈Υ2,(14d)p 1 ∈Υ 1 ,p 2 ∈ Υ 2 ,(14d)

其中

Figure BDA0002994354290000151
τ表示Po/μ,Pins/A中的最小值。in
Figure BDA0002994354290000151
τ represents the minimum value in P o /μ, Pins /A.

问题(14)中，

Figure BDA0002994354290000152
和
Figure BDA0002994354290000153
的功率分配变量只包含在约束(14b)和(14c)中，而分布变量p1和p2只包含在约束(14d)中。因此，考虑将优化过程分解为两个子问题。具体来说，问题(14)可以通过迭代求解以下两个子问题来处理，直到总体问题收敛：功率分配子问题1：给定的p1和p2优化
Figure BDA0002994354290000154
和
Figure BDA0002994354290000155
概率分布子问题2：给定的
Figure BDA0002994354290000156
和
Figure BDA0002994354290000157
优化p1和p2。接下来，将给出这两个子问题的解决方案。In question (14),
Figure BDA0002994354290000152
and
Figure BDA0002994354290000153
The power distribution variables of are only included in constraints (14b) and (14c), while the distribution variables p1 and p2 are only included in constraints (14d). Therefore, consider decomposing the optimization process into two subproblems. Specifically, problem (14) can be addressed by iteratively solving the following two sub-problems until the overall problem converges: Power distribution sub-problem 1: optimization given p 1 and p 2
Figure BDA0002994354290000154
and
Figure BDA0002994354290000155
Probability Distribution Subproblem 2: Given
Figure BDA0002994354290000156
and
Figure BDA0002994354290000157
Optimize p 1 and p 2 . Next, solutions to these two sub-problems will be given.

A.功率分配子问题1：A. Power distribution sub-problem 1:

当给出p1和p2时，问题(14)是一个最优的功率分配问题如下：Problem (14) is an optimal power allocation problem when p 1 and p 2 are given as follows:

Figure BDA0002994354290000158
Figure BDA0002994354290000158

Figure BDA0002994354290000159
Figure BDA0002994354290000159

Figure BDA00029943542900001510
Figure BDA00029943542900001510

其中，

Figure BDA00029943542900001511
h(·)表示关于
Figure BDA00029943542900001512
的函数。in,
Figure BDA00029943542900001511
h( ) means about
Figure BDA00029943542900001512
The function.

问题(15)对

Figure BDA00029943542900001513
和
Figure BDA00029943542900001514
是一个凸问题，采用注水法解决该问题，并得到最优功率分配
Figure BDA00029943542900001515
和
Figure BDA00029943542900001516
Question (15) to
Figure BDA00029943542900001513
and
Figure BDA00029943542900001514
is a convex problem, and the water injection method is used to solve the problem and obtain the optimal power distribution
Figure BDA00029943542900001515
and
Figure BDA00029943542900001516

B.概率分布子问题2：B. Probability distribution subproblem 2:

当给出

Figure BDA00029943542900001517
和
Figure BDA00029943542900001518
时，问题(14)表示如下：when given
Figure BDA00029943542900001517
and
Figure BDA00029943542900001518
, problem (14) is expressed as follows:

Figure BDA00029943542900001519
Figure BDA00029943542900001519

s.t.p1∈Υ1,(16b)stp 1 ∈ Υ 1 , (16b)

p2∈Υ2,(16c)p 2 ∈Υ 2 , (16c)

这是一个具有两个变量p1和p2的凸优化问题，然而，目标函数没有解析表达式，这阻碍了计算最优概率分布。为了克服这个困难，采用不精确梯度下降法，并得到LiFi和WiFi链路的概率分布p1和p2。This is a convex optimization problem with two variables p1 and p2 , however, there is no analytical expression for the objective function, which prevents computing the optimal probability distribution. To overcome this difficulty, an imprecise gradient descent method is employed, and the probability distributions p 1 and p 2 of LiFi and WiFi links are obtained.

综上所述，求解优化问题(7)，可以通过迭代求解功率分配子问题(15)和概率分布子问题(16)，可得到最大可达速率RLiFi-WiFi。To sum up, to solve the optimization problem (7), the power distribution sub-problem (15) and the probability distribution sub-problem (16) can be solved iteratively, and the maximum achievable rate R LiFi-WiFi can be obtained.

基于下界和上界的最优离散星座输入分布：Optimal discrete constellation input distribution based on lower and upper bounds:

由于可实现速率(6a)和(6b)都不是封闭表达式，因此(5)的计算效率很低。为了降低计算复杂度，可以使用显式表达式作为目标函数。因此，可使用LiFi链路的可达速率RLiFi的容量约束。Since neither achievable rates (6a) nor (6b) are closed expressions, (5) is computationally inefficient. To reduce computational complexity, an explicit expression can be used as the objective function. Therefore, the LiFi link's achievable rate R LiFi 's capacity constraints can be used.

引理2：在离散星座点输入条件下，LiFi链路传输速率RLiFi的上界和下界的闭式表达式分别为Lemma 2: Under the input condition of discrete constellation points, the closed-form expressions of the upper and lower bounds of the LiFi link transmission rate R LiFi are respectively

Figure BDA0002994354290000161
Figure BDA0002994354290000161

Figure BDA0002994354290000162
Figure BDA0002994354290000162

此外，还推导了WiFi链路可达速率RWiFi的封闭形式上界和下界。In addition, closed-form upper and lower bounds for the WiFi link reachable rate R WiFi are also derived.

引理3：在离散星座点输入条件下，给出WiFi链路可达速率的上界和下界分别为Lemma 3: Under the input condition of discrete constellation points, the upper and lower bounds of the achievable rate of the WiFi link are given as

Figure BDA0002994354290000163
Figure BDA0002994354290000163

Figure BDA0002994354290000164
Figure BDA0002994354290000164

然后，让

Figure BDA0002994354290000165
和
Figure BDA0002994354290000166
分别表示RLiFi-WiFi的下界和上界，因此，Then, let
Figure BDA0002994354290000165
and
Figure BDA0002994354290000166
denote the lower and upper bounds of R LiFi-WiFi , respectively, therefore,

Figure BDA0002994354290000167
Figure BDA0002994354290000167

Figure BDA0002994354290000171
Figure BDA0002994354290000171

基于可达速率

Figure BDA0002994354290000172
的下界，优化LiFi链路和WiFi链路的输入星座点概率分布和功率分配，以获得最大的传输速率下界
Figure BDA0002994354290000173
数学上表示如下：Based on reachable rate
Figure BDA0002994354290000172
The lower bound of , optimizes the input constellation point probability distribution and power allocation of LiFi link and WiFi link to obtain the maximum transmission rate lower bound
Figure BDA0002994354290000173
Mathematically expressed as follows:

Figure BDA0002994354290000174
Figure BDA0002994354290000174

此外，通过定义Furthermore, by defining

Figure BDA0002994354290000175
Figure BDA0002994354290000175

Figure BDA0002994354290000176
Figure BDA0002994354290000176

Figure BDA0002994354290000177
Figure BDA0002994354290000177

Figure BDA0002994354290000178
Figure BDA0002994354290000178

Figure BDA0002994354290000179
Figure BDA0002994354290000179

Figure BDA00029943542900001710
Figure BDA00029943542900001710

可将

Figure BDA00029943542900001711
改写如下：can be
Figure BDA00029943542900001711
Rewritten as follows:

Figure BDA00029943542900001712
Figure BDA00029943542900001712

则，问题(20)用公式表示如下：Then, problem (20) is formulated as follows:

Figure BDA00029943542900001713
Figure BDA00029943542900001713

可以看出目标函数(23)相对于

Figure BDA00029943542900001714
和
Figure BDA00029943542900001715
是凸的。而问题(23)关于p1和p2是非凸的。为了解决问题(23)，应用与前文相似的思想，通过迭代求解以下两个子问题，直到整体问题达到收敛为止。It can be seen that the objective function (23) is relative to
Figure BDA00029943542900001714
and
Figure BDA00029943542900001715
is convex. And problem ( 23 ) is nonconvex with respect to p1 and p2 . To solve problem (23), applying similar ideas as before, the following two sub-problems are solved iteratively until the overall problem reaches convergence.

A.功率分配子问题3：A. Power distribution sub-problem 3:

首先，解决了功率分配子问题3：固定的p1和p2，优化LiFi链路和WiFi链路的功率分配

Figure BDA0002994354290000181
和
Figure BDA0002994354290000182
当p1和p2被固定时，问题(23)可以表述如下：First, the power allocation sub-problem 3 is solved: fixed p 1 and p 2 , optimizing power allocation for LiFi links and WiFi links
Figure BDA0002994354290000181
and
Figure BDA0002994354290000182
When p 1 and p 2 are fixed, problem (23) can be formulated as follows:

Figure BDA0002994354290000183
Figure BDA0002994354290000183

Figure BDA0002994354290000184
Figure BDA0002994354290000184

Figure BDA0002994354290000185
Figure BDA0002994354290000185

Figure BDA0002994354290000186
Figure BDA0002994354290000186

由于log2 x是一个凹函数，采用近似梯度投影法解决问题(24)，并得到LiFi和WiFi链路的最优功率分布

Figure BDA0002994354290000187
和
Figure BDA0002994354290000188
Since log 2 x is a concave function, the approximate gradient projection method is used to solve problem (24) and obtain the optimal power distribution for LiFi and WiFi links
Figure BDA0002994354290000187
and
Figure BDA0002994354290000188

B.概率分布子问题4：B. Probability distribution subproblem 4:

接下来，解决概率分布子问题4：用给定的

Figure BDA0002994354290000189
和
Figure BDA00029943542900001810
优化LiFi链接和Wi Fi链接p1和p2的概率分布。当
Figure BDA00029943542900001811
和
Figure BDA00029943542900001812
固定时，LiFi-WiFi聚合系统(24)的最大化问题表示为：Next, solve the probability distribution subproblem 4: with the given
Figure BDA0002994354290000189
and
Figure BDA00029943542900001810
Optimize the probability distribution of LiFi links and Wi Fi links p 1 and p 2 . when
Figure BDA00029943542900001811
and
Figure BDA00029943542900001812
When fixed, the maximization problem of the LiFi-WiFi aggregation system (24) is expressed as:

Figure BDA00029943542900001813
Figure BDA00029943542900001813

s.t.p1∈γ1， (25b)stp 1 ∈ γ 1 , (25b)

p2∈γ2， (25c)p 2 ∈ γ 2 , (25c)

其中，in,

Figure BDA00029943542900001816
Figure BDA00029943542900001816

Figure BDA00029943542900001817
Figure BDA00029943542900001817

然后，问题(25)分为如下两个独立的子问题：Then, problem (25) is divided into two independent subproblems as follows:

Figure BDA00029943542900001818
Figure BDA00029943542900001818

s.t.p1∈Υ1,(26b)stp 1 ∈Υ 1 , (26b)

和and

Figure BDA00029943542900001819
Figure BDA00029943542900001819

s.t.p2∈Υ2(27b)stp 2 ∈ Υ 2 (27b)

其中，

Figure BDA0002994354290000191
表示关于p1的函数，
Figure BDA0002994354290000192
表示关于p2的函数；in,
Figure BDA0002994354290000191
represents a function with respect to p 1 ,
Figure BDA0002994354290000192
represents a function of p 2 ;

应用Frank-Wolfe方法来解决问题(26)和(27)，从而得到LiFi链路p1的概率分布和WiFi链路p2的最优概率分布。The Frank-Wolfe method is applied to solve problems (26) and (27), resulting in the probability distribution of LiFi link p 1 and the optimal probability distribution of WiFi link p 2 .

进一步而言，将以上子问题得到的

Figure BDA0002994354290000193
带入公式(22)，可得到最大可达速率下界RL LiFi-WiFi。Further, the above sub-problems get
Figure BDA0002994354290000193
Bringing into formula (22), the lower bound R L LiFi-WiFi of the maximum achievable rate can be obtained.

注意，基于可达速率上界

Figure BDA0002994354290000194
也能找到信号幅度和功率分配的最优分布，以最大限度地提高LiFi-WiFi聚合系统的可达速率，这与下界情况相似。Note that based on the upper bound on the achievable rate
Figure BDA0002994354290000194
The optimal distribution of signal amplitude and power allocation can also be found to maximize the achievable rate of the LiFi-WiFi aggregation system, which is similar to the lower bound case.

本发明从以下两个角度研究LiFi-WiFi聚合系统的性能：输入的最优离散星座图和功率分配方案。对于离散星座输入，首先推导出LiFi-WiFi聚合系统的可达速率表达式。然后，研究了星座分布和功率分配的速率最大化问题，通过利用互信息和离散输入的最小均方误差之间的关系，提出了一种非精确梯度下降法来获得最优概率分布。为了在复杂性和性能之间取得平衡，最大限度地提高星座分布和功率优化方面的可达速率，降低了复杂度，其中，最优功率分配可以以封闭的形式得到，星座分布问题可以用Frank-Wolfe方法有效地解决。多个数值结果表明，与最新方案相比，优化的星座分布和功率可显著提高可达速率。The present invention studies the performance of the LiFi-WiFi aggregation system from the following two perspectives: the optimal discrete constellation diagram of the input and the power allocation scheme. For discrete constellation input, the achievable rate expression of the LiFi-WiFi aggregation system is first derived. Then, the rate maximization problem of constellation distribution and power allocation is studied, and an inexact gradient descent method is proposed to obtain the optimal probability distribution by exploiting the relationship between mutual information and the minimum mean square error of discrete inputs. In order to strike a balance between complexity and performance, maximize the achievable rate in terms of constellation distribution and power optimization, and reduce complexity, where the optimal power distribution can be obtained in closed form, the constellation distribution problem can be solved using Frank -Wolfe method solves efficiently. Multiple numerical results show that the optimized constellation distribution and power can significantly increase the achievable rate compared to the state-of-the-art scheme.

图1a和图1b说明了LiFi链路在不同信噪比SNR1下各自的最优概率分布{x1,k,p1,k}和最优输入位置{x1,k}，其中莱斯因子K1＝8。图中{x1,k}表示最优输入位置，SNR1表示LiFi链路的信噪比，{p1,k}表示概率分布。结果表明，对于低信噪比，最优输入位置包括两个概率相等的离散点；对于高信噪比，最优输入位置超过两个离散点，随着信噪比的增加，最优概率分布更接近等概率分布。Figure 1a and Figure 1b illustrate the respective optimal probability distributions {x 1,k ,p 1,k } and optimal input positions {x 1,k } of LiFi links at different SNR 1 , where Rice Factor K 1 =8. In the figure, {x 1, k } represents the optimal input position, SNR 1 represents the signal-to-noise ratio of the LiFi link, and {p 1, k } represents the probability distribution. The results show that for low SNR, the optimal input position includes two discrete points with equal probability; for high SNR, the optimal input position exceeds two discrete points, and with the increase of SNR, the optimal probability distribution closer to an equal probability distribution.

图1c显示了LiFi链路在不同信噪比SNR1下各自的可达速率RLiFi。图中SNR1表示LiFi链路的信噪比，Achievable rate RLiFi表示可达速率，Equiprobable表示等概分布，Proposed Method表示提出的方案。可以观察到，所提出方案的LiFi链路的可达速率RLiFi高于等概率分布。此外，随着信噪比的增加，所提出的方案与等概率分布之间的差距变小。Figure 1c shows the respective achievable rates R LiFi of LiFi links at different signal-to-noise ratios SNR 1 . In the figure, SNR 1 represents the signal-to-noise ratio of the LiFi link, Achievable rate R LiFi represents the achievable rate, Equiprobable represents the equal probability distribution, and Proposed Method represents the proposed scheme. It can be observed that the achievable rate R LiFi of the LiFi link of the proposed scheme is higher than the equal probability distribution. Furthermore, as the signal-to-noise ratio increases, the gap between the proposed scheme and the equal probability distribution becomes smaller.

图2a、图2b和图2c说明WiFi链路在信噪比SNR2＝-6dB，4dB和12dB下的最优概率分布{x2,l,p2,l}，其中莱斯因子K2＝16。图中{p2,l}表示概率分布，{Im(x2,l)}表示最优输入位置的虚部，{Re(x2,l)}表示最有输入位置的实部。结果表明，对于信噪比SNR2＝-6dB，最优输入位置包括四个概率相等的离散点；对于信噪比SNR2＝4dB，最优输入位置超过四个离散点，而最优概率分布不是等概率分布；此外，对于信噪比SNR2＝12dB，最优输入位置包括16个离散点，且每个输入点概率相等，即为等概率分布。Figures 2a, 2b and 2c illustrate the optimal probability distributions {x 2,l ,p 2,l } for WiFi links at signal-to-noise ratios SNR 2 =-6dB, 4dB and 12dB, where Rice factor K 2 = 16. In the figure, {p 2,l } represents the probability distribution, {Im(x 2,l )} represents the imaginary part of the optimal input position, and {Re(x 2,l )} represents the real part of the most input position. The results show that for SNR 2 =-6dB, the optimal input position includes four discrete points with equal probability; for SNR 2 =4dB, the optimal input position exceeds four discrete points, and the optimal probability distribution It is not an equal probability distribution; in addition, for a signal-to-noise ratio SNR 2 =12dB, the optimal input position includes 16 discrete points, and each input point has an equal probability, that is, an equal probability distribution.

图2d说明WiFi链路在不同信噪比SNR2下各自的可达速率RWiFi。图中SNR2表示WiFi链路的信噪比，Achievable rate RWiFi表示可达速率，Equiprobable表示等概分布，Proposed Method表示提出的方案。可以观察到，采用所提出的方案的WiFi链路的可达速率RWiFi高于等概率分布。此外，随着信噪比的增加，所提出的方案与等概率分布之间的差距变小。Figure 2d illustrates the respective achievable rates RWiFi of WiFi links at different signal-to-noise ratios SNR2 . In the figure, SNR 2 represents the signal-to-noise ratio of the WiFi link, Achievable rate R WiFi represents the achievable rate, Equiprobable represents the equal probability distribution, and Proposed Method represents the proposed scheme. It can be observed that the achievable rate RWiFi of the WiFi link with the proposed scheme is higher than the equal probability distribution. Furthermore, as the signal-to-noise ratio increases, the gap between the proposed scheme and the equal probability distribution becomes smaller.

图3a说明LiFi-WiFi系统在不同平均电功率门限PT下各自的可达速率RLiFi-WiFi。图中Total power PT表示平均电功率门限，Achievable rate RLiFi-WiFi表示可达速率，Proposed Method表示提出的方案，Equiprobable表示等概分布，Power of LiFi Link

Figure BDA0002994354290000201
表示LiFi链路发射功率，Power of WiFi Link
Figure BDA0002994354290000202
表示WiFi链路发射功率，Link transmitpowwer表示链路分配功率。观察到，在所提出的方案和等概率分布的情况下，随着平均电功率门限PT的增加，可达速率RLiFi-WiFi增加。此外，随着平均电功率门限PT的增加，LiFi和WiFi链路的发射功率
Figure BDA0002994354290000203
和
Figure BDA0002994354290000204
增加。Figure 3a illustrates the respective attainable rates R LiFi-WiFi of the LiFi-WiFi system under different average electric power thresholds P T . In the figure, Total power P T represents the average electrical power threshold, Achievable rate R LiFi-WiFi represents the reachable rate, Proposed Method represents the proposed solution, Equiprobable represents the equal probability distribution, and Power of LiFi Link
Figure BDA0002994354290000201
Indicates the transmit power of the LiFi link, Power of WiFi Link
Figure BDA0002994354290000202
Indicates the WiFi link transmit power, and Link transmitpowwer indicates the link distribution power. It is observed that the achievable rate R of LiFi-WiFi increases with the increase of the average electrical power threshold P T under the proposed scheme and the equal probability distribution. Furthermore, as the average electrical power threshold P T increases, the transmit power of LiFi and WiFi links
Figure BDA0002994354290000203
and
Figure BDA0002994354290000204
Increase.

图3b说明LiFi-WiFi系统在不同瞬时光功率门限Pins下各自的可达速率RLiFi-WiFi。图中Instant optical power threshold Pins表示瞬时光功率门限，Achievable rateRLiFi-WiFi表示可达速率，Link transmit powwer表示链路分配功率，Proposed Method表示提出的方案，Power of LiFi Link

Figure BDA0002994354290000205
表示LiFi链路发射功率，Power of WiFi Link
Figure BDA0002994354290000206
表示WiFi链路发射功率。观察到，所提出的方案的可达速率RLiFi-WiFi随着瞬时光功率门限Pins的增加先增加，然后保持不变。此外，随着瞬时光功率门限Pins的增加，LiFi链路的发射功率
Figure BDA0002994354290000211
首先增加，然后保持不变；WiFi链路的发射功率
Figure BDA0002994354290000212
首先减小，然后保持恒定，这是因为
Figure BDA0002994354290000213
假设平均光功率Po＝0.8Pins，均值μ为0.5A。对于较低的瞬时光功率门限Pins，发射功率
Figure BDA0002994354290000214
受τ约束，而对于高瞬时光功率门限Pins，发射功率
Figure BDA0002994354290000215
受平均电功率门限PT约束。Figure 3b illustrates the respective attainable rates R LiFi-WiFi of the LiFi -WiFi system under different instantaneous optical power thresholds Pins. In the figure, Instant optical power threshold Pins represents the instantaneous optical power threshold, Achievable rateR LiFi -WiFi represents the achievable rate, Link transmit powwer represents the link distribution power, Proposed Method represents the proposed solution, Power of LiFi Link
Figure BDA0002994354290000205
Indicates the transmit power of the LiFi link, Power of WiFi Link
Figure BDA0002994354290000206
Indicates the transmit power of the WiFi link. It is observed that the achievable rate R of the proposed scheme R LiFi-WiFi first increases with the increase of the instantaneous optical power threshold Pins and then remains constant. In addition, with the increase of the instantaneous optical power threshold Pins, the transmit power of the LiFi link
Figure BDA0002994354290000211
First increase, then stay the same; transmit power of WiFi link
Figure BDA0002994354290000212
It first decreases and then remains constant because
Figure BDA0002994354290000213
Assuming that the average optical power P o =0.8P ins , the average value μ is 0.5A. For lower instantaneous optical power threshold Pins , the transmit power
Figure BDA0002994354290000214
Constrained by τ, while for high instantaneous optical power threshold Pins , the transmit power
Figure BDA0002994354290000215
Constrained by the average electrical power threshold PT .

图4a说明LiFi-WiFi系统在不同平均电功率门限PT下各自的可达速率下界RL LiFi-WiFi。图中Total power PT表示平均电功率门限，Link transmit powwer表示链路分配功率，Proposed Method表示提出的方案，Equiprobable表示等概分布，Power of LiFiLink

Figure BDA0002994354290000216
表示LiFi链路发射功率，Power of WiFi Link
Figure BDA0002994354290000217
表示WiFi链路发射功率，Lowerbound RL LiFi-WiFi表示可达速率下界。观察到，当平均电功率门限PT增加时，所提出的方法和等概率分布的可达速率下界RL LiFi-WiFi都会增加。此外，随着平均电功率门限PT的增加，WiFi链路的发射功率
Figure BDA0002994354290000218
首先增加，并保持不变，发射功率
Figure BDA0002994354290000219
的传输功率一直增加。原因是发射功率
Figure BDA00029943542900002110
还受到光学功率约束的限制。Figure 4a illustrates the respective lower bounds R L LiFi-WiFi of the achievable rate of the LiFi-WiFi system under different average electric power thresholds P T . In the figure, Total power P T represents the average electrical power threshold, Link transmit powwer represents the link distribution power, Proposed Method represents the proposed solution, Equiprobable represents the equal probability distribution, Power of LiFiLink
Figure BDA0002994354290000216
Indicates the transmit power of the LiFi link, Power of WiFi Link
Figure BDA0002994354290000217
Indicates the transmit power of the WiFi link, and Lowerbound R L LiFi-WiFi indicates the lower bound of the achievable rate. It is observed that both the proposed method and the lower bound on the achievable rate of the equal probability distribution, R L LiFi-WiFi , increase when the average electric power threshold P T increases. In addition, as the average electrical power threshold P T increases, the transmit power of the WiFi link
Figure BDA0002994354290000218
First increase, and keep constant, the transmit power
Figure BDA0002994354290000219
The transmission power has been increasing. The reason is the transmit power
Figure BDA00029943542900002110
Also limited by optical power constraints.

图4b说明LiFi-WiFi系统在不同瞬时光功率门限Pins下各自的可达速率下界RL LiFi-WiFi。图中Instant optical power threshold Pins表示瞬时光功率门限，Lowerbound RL LiFi-WiFi表示可达速率下界，Link transmit powwer表示链路分配功率，ProposedMethod表示提出的方案，Power of LiFi Link

Figure BDA00029943542900002111
表示LiFi链路发射功率，Power of WiFiLink
Figure BDA00029943542900002112
表示WiFi链路发射功率。假设平均光功率Po＝0.8Pins，均值μ≤0.5A。观察到，随着瞬时光功率门限Pins的增加可达速率下界RL LiFi-WiFi先增加，然后保持不变。此外，随着瞬时光功率门限Pins的增加，LiFi链路的发射功率
Figure BDA00029943542900002113
先增大，然后保持不变；WiFi链路的发射功率
Figure BDA00029943542900002114
首先减小，然后保持恒定，这是因为
Figure BDA00029943542900002115
对于较低的瞬时光功率门限Pins，发射功率
Figure BDA00029943542900002116
受τ约束，而对于高瞬时光功率门限Pins，发射功率
Figure BDA00029943542900002117
受平均电功率门限PT约束。Figure 4b illustrates the lower bound R L LiFi -WiFi of the respective achievable rates of the LiFi-WiFi system under different instantaneous optical power thresholds Pins. In the figure, Instant optical power threshold Pins represents the instantaneous optical power threshold, Lowerbound R L LiFi -WiFi represents the lower bound of the achievable rate, Link transmit powwer represents the link allocation power, ProposedMethod represents the proposed solution, Power of LiFi Link
Figure BDA00029943542900002111
Indicates the transmit power of the LiFi link, Power of WiFiLink
Figure BDA00029943542900002112
Indicates the transmit power of the WiFi link. It is assumed that the average optical power P o =0.8P ins , and the average value μ≤0.5A. It is observed that with the increase of the instantaneous optical power threshold Pins, the achievable rate lower bound R L LiFi -WiFi first increases and then remains unchanged. In addition, with the increase of the instantaneous optical power threshold Pins, the transmit power of the LiFi link
Figure BDA00029943542900002113
Increase first, then stay the same; transmit power of WiFi link
Figure BDA00029943542900002114
It first decreases and then remains constant because
Figure BDA00029943542900002115
For lower instantaneous optical power threshold Pins , the transmit power
Figure BDA00029943542900002116
Constrained by τ, while for high instantaneous optical power threshold Pins , the transmit power
Figure BDA00029943542900002117
Constrained by the average electrical power threshold PT .

图5a说明LiFi-WiFi系统在不同带宽B1下各自的可达速率下界RL LiFi-WiFi。图中Lower bound RL LiFi-WiFi表示可达速率下界，Link transmit powwer表示链路分配功率，Proposed Method表示提出的方案，Power of LiFi Link

Figure BDA0002994354290000221
表示LiFi链路发射功率，Powerof WiFi Link
Figure BDA0002994354290000222
表示WiFi链路发射功率，Bandwidth of LiFi link B1表示LiFi链路带宽。观察到，随着带宽B1的增加，所提出的方法的可达速率下界RL LiFi-WiFi增加。此外，随着带宽B1的增加，LiFi链路的发射功率
Figure BDA0002994354290000223
增大，WiFi链路的发射功率
Figure BDA0002994354290000224
减小。Figure 5a illustrates the lower bounds R L LiFi-WiFi of the respective achievable rates of the LiFi-WiFi system under different bandwidths B 1 . In the figure, Lower bound R L LiFi-WiFi represents the lower bound of the achievable rate, Link transmit powwer represents the link distribution power, Proposed Method represents the proposed solution, Power of LiFi Link
Figure BDA0002994354290000221
Indicates the transmit power of the LiFi link, Powerof WiFi Link
Figure BDA0002994354290000222
Indicates the transmit power of the WiFi link, and Bandwidth of LiFi link B 1 indicates the bandwidth of the LiFi link. It is observed that as the bandwidth B1 increases, the achievable rate lower bound R L LiFi -WiFi of the proposed method increases. Furthermore, as the bandwidth B1 increases, the transmit power of the LiFi link
Figure BDA0002994354290000223
increase, the transmit power of the WiFi link
Figure BDA0002994354290000224
decrease.

图5b说明LiFi-WiFi系统在不同带宽B2下各自的可实现速率下界RL LiFi-WiFi。图中Lower bound RL LiFi-WiFi表示可达速率下界，Link transmit powwer表示链路分配功率，Proposed Method表示提出的方案，Power of LiFi Link

Figure BDA0002994354290000225
表示LiFi链路发射功率，Powerof WiFi Link
Figure BDA0002994354290000226
表示WiFi链路发射功率，Bandwidth of WiFi link B2表示WiFi链路带宽。观察到，随着带宽B2的增加，所提出的方法的可达速率下界RL LiFi-WiFi增加。此外，随着带宽B2的增加，LiFi链路的发射功率
Figure BDA0002994354290000227
减小，WiFi链路的发射功率
Figure BDA0002994354290000228
增大。Figure 5b illustrates the respective lower bounds R L LiFi-WiFi of the achievable rate of the LiFi-WiFi system under different bandwidths B 2 . In the figure, Lower bound R L LiFi-WiFi represents the lower bound of the achievable rate, Link transmit powwer represents the link distribution power, Proposed Method represents the proposed solution, Power of LiFi Link
Figure BDA0002994354290000225
Indicates the transmit power of the LiFi link, Powerof WiFi Link
Figure BDA0002994354290000226
Indicates the WiFi link transmit power, Bandwidth of WiFi link B 2 represents the WiFi link bandwidth. It is observed that as the bandwidth B2 increases, the achievable rate lower bound R L LiFi -WiFi of the proposed method increases. Furthermore, as the bandwidth B2 increases, the transmit power of the LiFi link
Figure BDA0002994354290000227
reduce, the transmit power of the WiFi link
Figure BDA0002994354290000228
increase.

图6a说明在问题(7)的最优解下，可达速率RLiFi-WiFi、可达速率下界RL LiFi-WiFi和可达速率上界RU LiFi-WiFi在不同平均电功率门限PT下的变化曲线；图6b说明在问题(20)的最优解下，可达速率RLiFi-WiFi、可达速率下界RL LiFi-WiFi和可达速率上界RU LiFi-WiFi在不同平均电功率门限PT下的变化曲线。图中Achievable rate表示可达速率，RLiFi-WiFi为可达速率，RL LiFi-WiFi为可达速率下界，RU LiFi-WiFi为咔哒速率上界，Total power PT表示平均电功率门限。图6a和图6b显示，对于平均电功率门限PT，可达速率RLiFi-WiFi和可达速率下界RL LiFi-WiFi的差距大于可达速率RLiFi-WiFi和可达速率上界RU LiFi-WiFi的差距；对于高平均电功率门限PT，可达速率下界RL LiFi-WiFi和可达速率RLiFi-WiFi的差距低于可达速率RLiFi-WiFi和可达速率上界RU LiFi-WiFi的差距。Figure 6a illustrates the achievable rate R LiFi-WiFi , the lower achievable rate R L LiFi-WiFi and the upper achievable rate R U LiFi-WiFi under different average electric power thresholds P T under the optimal solution of problem (7). Figure 6b shows that under the optimal solution of problem (20), the achievable rate R LiFi-WiFi , the lower bound of the achievable rate R L LiFi-WiFi and the upper bound of the achievable rate R U LiFi-WiFi at different average electrical power Variation curve under threshold P T. In the figure, Achievable rate represents the achievable rate, R LiFi-WiFi represents the achievable rate, R L LiFi-WiFi represents the lower bound of the achievable rate, R U LiFi-WiFi represents the upper bound of the click rate, and Total power P T represents the average electric power threshold. Figures 6a and 6b show that for the average electrical power threshold P T , the difference between the achievable rate R LiFi-WiFi and the lower bound RL LiFi-WiFi is greater than the achievable rate R LiFi-WiFi and the upper bound R U LiFi - the gap of WiFi ; for high average electrical power threshold P T , the gap between the lower achievable rate RL LiFi- WiFi and the achievable rate R LiFi-WiFi is lower than the achievable rate R LiFi-WiFi and the upper achievable rate R U LiFi - WiFi gap.

Claims (1)

1.LiFi-WiFi聚合系统的下行链路传输优化方法，其特征在于，包括如下步骤：1. The downlink transmission optimization method of LiFi-WiFi aggregation system is characterized in that, comprises the following steps: 步骤1，对LiFi-WiFi聚合系统进行设定；Step 1, set the LiFi-WiFi aggregation system; 步骤2，求解聚合LiFi-WiFi系统可达速率；Step 2, solve the reachable rate of the aggregated LiFi-WiFi system; 步骤3，求解LiFi-WiFi聚合系统的最优离散星座输入；Step 3, solve the optimal discrete constellation input of the LiFi-WiFi aggregation system; 步骤4，求解基于下界和上界的最优离散星座输入分布；Step 4, solve the optimal discrete constellation input distribution based on the lower bound and the upper bound; 步骤1包括：考虑一个LiFi-WiFi聚合系统的下行链路传输，其中发射机配备了一个发光二极管LED和一个WiFi天线，接收机配备一个单光子探测器PD和一个射频天线，发射机同时通过LiFi链路和WiFi链路传输信息，其中LiFi链路和WiFi链路的带宽分别为B1和B2；Step 1 includes: consider the downlink transmission of a LiFi-WiFi aggregation system, where the transmitter is equipped with a light-emitting diode LED and a WiFi antenna, the receiver is equipped with a single-photon detector PD and an RF antenna, and the transmitter is simultaneously connected through LiFi The link and the WiFi link transmit information, wherein the bandwidths of the LiFi link and the WiFi link are B 1 and B 2 respectively; 设
Figure FDA0003407125930000011
表示发送的信号向量，其中x1∈R和x2∈C分别表示LiFi链路的发送信号和WiFi链路的发送信号，R为实数集合，C为复数集合；Assume
Figure FDA0003407125930000011
Represents the transmitted signal vector, where x 1 ∈ R and x 2 ∈ C represent the transmitted signal of the LiFi link and the transmitted signal of the WiFi link, respectively, R is the set of real numbers, and C is the set of complex numbers; 步骤1还包括：在LiFi-WiFi聚合系统中，传输的信号分布在离散星座上，设定LiFi链路信号通过M脉冲幅度调制发送，WiFi链路信号通过N-正交幅度调制发送，信号x1取自具有基数M的非负实离散星座集Ω1，表示为：Step 1 further includes: in the LiFi-WiFi aggregation system, the transmitted signals are distributed in discrete constellations, and it is set that the LiFi link signal is sent through M pulse amplitude modulation, the WiFi link signal is sent through N-quadrature amplitude modulation, and the signal x 1 is taken from the non-negative real discrete constellation set Ω 1 with cardinality M, expressed as:
Figure FDA0003407125930000012
Figure FDA0003407125930000012
 其中，Pr(·)表示求概率；x1,k表示星座点，取值为非负实数；k表示星座点的序号，p1,k表示x1＝x1,k的概率；参数A，
Figure FDA0003407125930000013
Pe,1分别表示x1的峰值光功率、平均光功率和电功率门限；Among them, Pr( ) represents the probability; x 1,k represents the constellation point, which is a non-negative real number; k represents the serial number of the constellation point, p 1,k represents the probability of x 1 =x 1,k ; parameter A,
Figure FDA0003407125930000013
P e,1 represent the peak optical power, average optical power and electrical power threshold of x 1 , respectively; WiFi信号x2取自一个具有基数N的复数离散星座集Ω2，表示为：The WiFi signal x 2 is taken from a complex discrete constellation set Ω 2 with base N, expressed as:
Figure FDA0003407125930000014
Figure FDA0003407125930000014
 其中，x2,l表示星座点，取值为复数，l表示星座点的序号，p2,l表示选择x2,l的概率，Pe,2表示x2的电功率门限；Among them, x 2,l represents the constellation point, which is a complex number, l represents the sequence number of the constellation point, p 2,l represents the probability of selecting x 2,l , and P e,2 represents the electric power threshold of x 2 ; 步骤1还包括：设q1∈R和q2∈C分别表示x1的功率放大因子和x2的功率放大因子，q1和q2需满足平均功率约束，即：Step 1 also includes: Let q 1 ∈ R and q 2 ∈ C represent the power amplification factor of x 1 and the power amplification factor of x 2 respectively, and q 1 and q 2 need to satisfy the average power constraint, namely:
Figure FDA0003407125930000021
Figure FDA0003407125930000021
 其中η1和η2分别表示LiFi链路的功率放大器的效率和WiFi链路的功率放大器的效率，PT表示平均电功率门限，中间参数ε1和ε2分别为：where η 1 and η 2 represent the efficiency of the power amplifier of the LiFi link and the power amplifier of the WiFi link, respectively, P T represents the average electrical power threshold, and the intermediate parameters ε 1 and ε 2 are:
Figure FDA0003407125930000022
Figure FDA0003407125930000022
Figure FDA0003407125930000023
Figure FDA0003407125930000023
 步骤1还包括：对LiFi信号的功率控制需要满足平均光功率和峰值光功率要求，如下所示：Step 1 also includes: the power control of the LiFi signal needs to meet the average optical power and peak optical power requirements, as follows:
Figure FDA0003407125930000024
Figure FDA0003407125930000024
 q1A≤Pins，q 1 A≤P ins , 其中
Figure FDA0003407125930000025
表示求均值，
Figure FDA0003407125930000026
表示x1的均值；Po和Pins分别表示平均光功率和瞬时光功率门限；in
Figure FDA0003407125930000025
represents the mean value,
Figure FDA0003407125930000026
Represents the mean value of x 1 ; P o and Pins represent the average optical power and instantaneous optical power threshold, respectively; 步骤1还包括：设
Figure FDA0003407125930000027
表示信道向量，其中g1和g2分别是LiFi链路的信道增益和WiFi链路的信道增益；设y1和y2分别表示来自LiFi链路的接收信号和来自WiFi链路的接收信号，写成如下矢量形式：Step 1 also includes: setting
Figure FDA0003407125930000027
represents the channel vector, where g 1 and g 2 are the channel gain of the LiFi link and the channel gain of the WiFi link, respectively; let y 1 and y 2 denote the received signal from the LiFi link and the received signal from the WiFi link, respectively, Written in vector form as follows:
Figure FDA0003407125930000028
Figure FDA0003407125930000028
 其中
Figure FDA0003407125930000029
是来LiFi链路的实高斯噪声，
Figure FDA00034071259300000210
是来自WiFi链路的复高斯噪声，
Figure FDA00034071259300000211
表示均值为0、方差为
Figure FDA00034071259300000212
的高斯分布，
Figure FDA00034071259300000213
表示均值为0、方差为
Figure FDA00034071259300000214
的复高斯分布；in
Figure FDA0003407125930000029
is the real Gaussian noise from the LiFi link,
Figure FDA00034071259300000210
is the complex Gaussian noise from the WiFi link,
Figure FDA00034071259300000211
means that the mean is 0 and the variance is
Figure FDA00034071259300000212
the Gaussian distribution of ,
Figure FDA00034071259300000213
means that the mean is 0 and the variance is
Figure FDA00034071259300000214
The complex Gaussian distribution of ; 步骤2包括：Step 2 includes: 步骤2-1，将聚合LiFi-WiFi系统可达速率RLiFi-WiFi定义为：Step 2-1, define the achievable rate R LiFi-WiFi of the aggregated LiFi-WiFi system as:
Figure FDA0003407125930000031
Figure FDA0003407125930000031
 其中
Figure FDA0003407125930000032
和
Figure FDA0003407125930000033
分别表示LiFi链路的可达速率和WiFi链路的可达速率，I(x；y)表示信道平均互信息；in
Figure FDA0003407125930000032
and
Figure FDA0003407125930000033
respectively represent the reachable rate of LiFi link and the reachable rate of WiFi link, I(x; y) represents the channel average mutual information; 步骤2-2，基于离散星座点输入的LiFi-WiFi聚合系统，给定LiFi链路和WiFi链路带宽B1和B2，则LiFi-WiFi聚合系统可达速率RLiFi和RWiFi分别为：Step 2-2, based on the LiFi-WiFi aggregation system input by discrete constellation points, given the LiFi link and WiFi link bandwidths B 1 and B 2 , the achievable rates R LiFi and R WiFi of the LiFi-WiFi aggregation system are respectively:
Figure FDA0003407125930000034
Figure FDA0003407125930000034
Figure FDA0003407125930000035
Figure FDA0003407125930000035
 其中，
Figure FDA0003407125930000036
表示求关于z1函数的均值，
Figure FDA0003407125930000037
表示求关于z2函数的均值，
Figure FDA0003407125930000038
表示信道增益g2的共轭；in,
Figure FDA0003407125930000036
means to find the mean of the z 1 function,
Figure FDA0003407125930000037
means to find the mean of the z 2 function,
Figure FDA0003407125930000038
represents the conjugate of the channel gain g2 ; 步骤3包括：Step 3 includes: 步骤3-1：LiFi-WiFi聚合系统的最优离散星座输入问题表述为：Step 3-1: The optimal discrete constellation input problem for the LiFi-WiFi aggregation system is formulated as:
Figure FDA0003407125930000039
Figure FDA0003407125930000039
Figure FDA00034071259300000310
Figure FDA00034071259300000310
 q1≤min(Po/μ,Pins/A), (7c)q 1 ≤min(P o /μ,P ins /A), (7c)
Figure FDA00034071259300000311
Figure FDA00034071259300000311
Figure FDA00034071259300000312
Figure FDA00034071259300000312
Figure FDA00034071259300000313
Figure FDA00034071259300000313
Figure FDA0003407125930000041
Figure FDA0003407125930000041
 其中，
Figure FDA0003407125930000042
表示LiFi链路星座点的序号，
Figure FDA0003407125930000043
表示WiFi链路星座点的序号；in,
Figure FDA0003407125930000042
Indicates the sequence number of the LiFi link constellation point,
Figure FDA0003407125930000043
Indicates the serial number of the WiFi link constellation point; 步骤3-2：可达速率RLiFi-WiFi写为：Step 3-2: The achievable rate R LiFi-WiFi is written as:
Figure FDA0003407125930000044
Figure FDA0003407125930000044
 其中，
Figure FDA0003407125930000045
Figure FDA0003407125930000046
和
Figure FDA0003407125930000047
分别表示LiFi链路和WiFi链路的发射功率；in,
Figure FDA0003407125930000045
Figure FDA0003407125930000046
and
Figure FDA0003407125930000047
represent the transmit power of LiFi link and WiFi link, respectively; 步骤3-3：定义
Figure FDA0003407125930000048
约束条件(7e)和(7d)写为：Step 3-3: Definition
Figure FDA0003407125930000048
Constraints (7e) and (7d) are written as:
Figure FDA0003407125930000049
Figure FDA0003407125930000049
 其中，
Figure FDA00034071259300000410
表示LiFi链路的星座点向量，x1,M表示LiFi链路的第M个星座点，
Figure FDA00034071259300000411
表示关于向量p集合，p1表示LiFi链路的星座点概率向量，p1,M表示x1＝x1,M的概率，
Figure FDA00034071259300000412
表示元素全为1的1×M的行向量，p代表向量；in,
Figure FDA00034071259300000410
represents the constellation point vector of the LiFi link, x 1, M represents the Mth constellation point of the LiFi link,
Figure FDA00034071259300000411
represents the set of vectors p, p 1 represents the probability vector of constellation points of the LiFi link, p 1,M represents the probability of x 1 =x 1,M ,
Figure FDA00034071259300000412
Represents a 1×M row vector with all 1 elements, p represents a vector; 定义
Figure FDA00034071259300000413
约束条件(7f)和(7g)写成：definition
Figure FDA00034071259300000413
Constraints (7f) and (7g) are written as:
Figure FDA00034071259300000414
Figure FDA00034071259300000414
 其中，
Figure FDA00034071259300000415
表示WiFi链路的星座点向量，x2,N表示WiFi链路的第N个星座点，
Figure FDA00034071259300000416
表示关于向量p集合，p2表示WiFi链路的星座点概率向量，p2,N表示x2＝x2,N的概率，
Figure FDA0003407125930000051
表示元素全为1的1×N的行向量；in,
Figure FDA00034071259300000415
represents the constellation point vector of the WiFi link, x 2, N represents the Nth constellation point of the WiFi link,
Figure FDA00034071259300000416
represents the set of vectors p, p 2 represents the constellation point probability vector of the WiFi link, p 2,N represents the probability of x 2 =x 2,N ,
Figure FDA0003407125930000051
Represents a 1×N row vector whose elements are all 1s; 引入辅助变量：Introduce auxiliary variables:
Figure FDA0003407125930000052
Figure FDA0003407125930000052
Figure FDA0003407125930000053
Figure FDA0003407125930000053
Figure FDA0003407125930000054
Figure FDA0003407125930000054
Figure FDA0003407125930000055
Figure FDA0003407125930000055
Figure FDA0003407125930000056
Figure FDA0003407125930000056
Figure FDA0003407125930000057
Figure FDA0003407125930000057
 则可达速率RLiFi-WiFi重写为：Then the achievable rate R LiFi-WiFi can be rewritten as:
Figure FDA0003407125930000058
Figure FDA0003407125930000058
 步骤3-4：问题(7)等效如下问题(14)：Steps 3-4: Problem (7) is equivalent to the following problem (14):
Figure FDA0003407125930000059
Figure FDA0003407125930000059
Figure FDA00034071259300000510
Figure FDA00034071259300000510
Figure FDA00034071259300000511
Figure FDA00034071259300000511
Figure FDA00034071259300000512
Figure FDA00034071259300000512
 其中
Figure FDA00034071259300000513
τ表示Po/μ,Pins/A中的最小值；in
Figure FDA00034071259300000513
τ represents the minimum value in P o /μ, Pins /A; 问题(14)中，
Figure FDA00034071259300000514
和
Figure FDA00034071259300000515
的功率分配变量只包含在约束(14b)和(14c)中，而分布变量p1和p2只包含在约束(14d)中，问题(14)通过迭代求解以下两个子问题来处理，直到总体问题收敛：In question (14),
Figure FDA00034071259300000514
and
Figure FDA00034071259300000515
The power distribution variables of are only included in constraints (14b) and (14c), while the distribution variables p1 and p2 are only included in constraints (14d), problem ( 14 ) is addressed by iteratively solving the following two subproblems until the overall The problem converges: 功率分配子问题1：给定的p1和p2优化
Figure FDA00034071259300000516
和
Figure FDA00034071259300000517
Power distribution subproblem 1: optimization given p 1 and p 2
Figure FDA00034071259300000516
and
Figure FDA00034071259300000517
 概率分布子问题2：给定的
Figure FDA0003407125930000061
和
Figure FDA0003407125930000062
优化p1和p2；Probability Distribution Subproblem 2: Given
Figure FDA0003407125930000061
and
Figure FDA0003407125930000062
optimize p 1 and p 2 ; 对于功率分配子问题1：当给出p1和p2时，问题(14)是一个最优的功率分配问题，如下问题(15)所示：For power allocation subproblem 1: when p 1 and p 2 are given, problem (14) is an optimal power allocation problem, as shown in problem (15) below:
Figure FDA0003407125930000063
Figure FDA0003407125930000063
Figure FDA0003407125930000064
Figure FDA0003407125930000064
Figure FDA0003407125930000065
Figure FDA0003407125930000065
 其中，
Figure FDA0003407125930000066
h(·)表示关于
Figure FDA0003407125930000067
的函数；in,
Figure FDA0003407125930000066
h( ) means about
Figure FDA0003407125930000067
The function; 问题(15)对
Figure FDA0003407125930000068
和
Figure FDA0003407125930000069
是一个凸问题，采用注水法解决该问题，并得到最优功率分配
Figure FDA00034071259300000610
和
Figure FDA00034071259300000611
Question (15) to
Figure FDA0003407125930000068
and
Figure FDA0003407125930000069
is a convex problem, and the water injection method is used to solve the problem and obtain the optimal power distribution
Figure FDA00034071259300000610
and
Figure FDA00034071259300000611
 对于概率分布子问题2：当给出
Figure FDA00034071259300000612
和
Figure FDA00034071259300000613
时，问题(14)表示为如下问题(16)：For the probability distribution subproblem 2: when given
Figure FDA00034071259300000612
and
Figure FDA00034071259300000613
, problem (14) is expressed as the following problem (16):
Figure FDA00034071259300000614
Figure FDA00034071259300000614
Figure FDA00034071259300000615
Figure FDA00034071259300000615
Figure FDA00034071259300000616
Figure FDA00034071259300000616
 问题(16)是一个有两个变量p1和p2的凸优化问题，采用不精确梯度下降法，并得到LiFi链路的概率分布p1和WiFi链路的概率分布p2；Problem (16) is a convex optimization problem with two variables p 1 and p 2 , using inexact gradient descent method, and obtain the probability distribution p 1 of LiFi link and the probability distribution p 2 of WiFi link ; 综上所述，求解优化问题(7)，通过迭代求解功率分配子问题(15)和概率分布子问题(16)，得到最大可达速率RLiFi-WiFi；In summary, the optimization problem (7) is solved, and the maximum achievable rate R LiFi-WiFi is obtained by iteratively solving the power distribution sub-problem (15) and the probability distribution sub-problem (16); 步骤4包括：Step 4 includes: 步骤4-1：在离散星座点输入条件下，LiFi链路传输速率RVLC的上界和下界的闭式表达式分别为Step 4-1: Under the condition of discrete constellation point input, the closed-form expressions of the upper and lower bounds of the LiFi link transmission rate R VLC are respectively
Figure FDA00034071259300000617
Figure FDA00034071259300000617
Figure FDA0003407125930000071
Figure FDA0003407125930000071
 步骤4-2：在离散星座点输入条件下，给出WiFi链路可达速率的上界和下界，如下所示：Step 4-2: Under the input condition of discrete constellation points, give the upper and lower bounds of the WiFi link achievable rate, as follows:
Figure FDA0003407125930000072
Figure FDA0003407125930000072
Figure FDA0003407125930000073
Figure FDA0003407125930000073
 步骤4-3：让
Figure FDA0003407125930000074
和
Figure FDA0003407125930000075
分别表示RLiFi-WiFi的下界和上界，得到：Step 4-3: Let
Figure FDA0003407125930000074
and
Figure FDA0003407125930000075
Representing the lower and upper bounds of R LiFi-WiFi , respectively, we get:
Figure FDA0003407125930000076
Figure FDA0003407125930000076
Figure FDA0003407125930000077
Figure FDA0003407125930000077
 步骤4-4：基于可达速率
Figure FDA0003407125930000078
的下界，优化LiFi链路和WiFi链路的输入星座点概率分布和功率分配，以获得最大的传输速率下界
Figure FDA0003407125930000079
优化问题表示如下：Step 4-4: Based on reachable rate
Figure FDA0003407125930000078
The lower bound of the input constellation point probability distribution and power allocation of the LiFi link and WiFi link is optimized to obtain the maximum transmission rate lower bound
Figure FDA0003407125930000079
The optimization problem is expressed as follows:
Figure FDA00034071259300000710
Figure FDA00034071259300000710
 此外，通过定义：Furthermore, by defining:
Figure FDA00034071259300000711
Figure FDA00034071259300000711
Figure FDA0003407125930000081
Figure FDA0003407125930000081
Figure FDA0003407125930000082
Figure FDA0003407125930000082
Figure FDA0003407125930000083
Figure FDA0003407125930000083
Figure FDA0003407125930000084
Figure FDA0003407125930000084
Figure FDA0003407125930000085
Figure FDA0003407125930000085
 将
Figure FDA0003407125930000086
改写如下：Will
Figure FDA0003407125930000086
Rewritten as follows:
Figure FDA0003407125930000087
Figure FDA0003407125930000087
 则问题(20)用如下公式表示：Then problem (20) is expressed by the following formula:
Figure FDA0003407125930000088
Figure FDA0003407125930000088
 为了解决问题(23)，通过迭代求解以下两个子问题，功率分配子问题3和概率分布子问题4，直到整体问题达到收敛为止：To solve problem (23), the following two sub-problems, power distribution sub-problem 3 and probability distribution sub-problem 4, are solved iteratively until the overall problem reaches convergence: 功率分配子问题3：当p1和p2被固定时，优化LiFi链路和WiFi链路的功率分配
Figure FDA0003407125930000089
和
Figure FDA00034071259300000810
问题(23)表述如下问题(24)：Power allocation subproblem 3: Optimizing power allocation for LiFi links and WiFi links when p1 and p2 are fixed
Figure FDA0003407125930000089
and
Figure FDA00034071259300000810
Question (23) formulates the following question (24):
Figure FDA00034071259300000811
Figure FDA00034071259300000811
Figure FDA00034071259300000812
Figure FDA00034071259300000812
Figure FDA00034071259300000813
Figure FDA00034071259300000813
Figure FDA00034071259300000814
Figure FDA00034071259300000814
 采用近似梯度投影法解决问题(24)，并得到LiFi和WiFi链路的最优功率分布
Figure FDA00034071259300000815
和
Figure FDA00034071259300000816
Approximate gradient projection method to solve problem (24) and obtain optimal power distribution for LiFi and WiFi links
Figure FDA00034071259300000815
and
Figure FDA00034071259300000816
 概率分布子问题4：用给定的
Figure FDA0003407125930000091
和
Figure FDA0003407125930000092
优化p1和p2的概率分布，当
Figure FDA0003407125930000093
和
Figure FDA0003407125930000094
固定时，问题(24)表示为如下问题(25)：Probability distribution subproblem 4: Using a given
Figure FDA0003407125930000091
and
Figure FDA0003407125930000092
optimize the probability distributions of p 1 and p 2 when
Figure FDA0003407125930000093
and
Figure FDA0003407125930000094
When fixed, problem (24) is represented as problem (25) as follows:
Figure FDA0003407125930000095
Figure FDA0003407125930000095
Figure FDA0003407125930000096
Figure FDA0003407125930000096
Figure FDA0003407125930000097
Figure FDA0003407125930000097
 其中，in,
Figure FDA0003407125930000098
Figure FDA0003407125930000098
Figure FDA0003407125930000099
Figure FDA0003407125930000099
 则问题(25)分为两个独立的子问题，分别为问题(26)：Then problem (25) is divided into two independent sub-problems, namely problem (26):
Figure FDA00034071259300000910
Figure FDA00034071259300000910
Figure FDA00034071259300000911
Figure FDA00034071259300000911
 和问题(27)and questions (27)
Figure FDA00034071259300000912
Figure FDA00034071259300000912
Figure FDA00034071259300000913
Figure FDA00034071259300000913
 其中，
Figure FDA00034071259300000914
表示关于p1的函数，
Figure FDA00034071259300000915
表示关于p2的函数；in,
Figure FDA00034071259300000914
represents a function with respect to p 1 ,
Figure FDA00034071259300000915
represents a function of p 2 ; 应用Frank-Wolfe方法来解决问题(26)和(27)，从而得到LiFi链路p1的概率分布和WiFi链路p2的最优概率分布；Apply the Frank-Wolfe method to solve problems (26) and (27), thereby obtaining the probability distribution of LiFi link p 1 and the optimal probability distribution of WiFi link p 2 ; 步骤4-4还包括：将得到的
Figure FDA00034071259300000916
p1,p2带入公式(22)，得到最大可达速率下界RL LiFi-WiFi。Steps 4-4 also include: the resulting
Figure FDA00034071259300000916
p 1 , p 2 are brought into formula (22) to obtain the lower bound R L LiFi-WiFi of the maximum achievable rate.
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