CN111431598A - New inner and outer boundary calculation method for MAC capacity area in V L C network - Google Patents

Abstract

The invention provides a new inner and outer boundary calculation method of MAC capacity region in V L C network, the invention establishes suggested inner boundary by using single user ability to realize input distribution of each user, and can obtain suggested outer boundary by determining single user capacity of each user and calculating total capacity upper limit by relaxing input constraint condition.

Description

New inner and outer boundary calculation method for MAC capacity area in V L C network

Technical Field

The invention relates to the field of visible light communication, in particular to a new inner and outer boundary calculation method for an MAC capacity area in a V L C network.

Background

By utilizing existing light emitting diodes (L ED) as transmitters, V L C systems can not only provide both illumination and Communication, but also have several other advantages, such as ultra-low electromagnetic radiation, strong transmission security and high energy efficiency.

The capacity region of a MAC can characterize the fundamental limitation of achievable rates and therefore can serve as a theoretical basis for other practical V L C network designs, the disadvantage of the prior art is that the exact capacity region and optimal input distribution of the V L C MAC are still unknown, without closed form expressions, and the complexity is relatively high.

Disclosure of Invention

The invention aims to solve the technical problems in the background art, and provides a new inner and outer boundary calculation method for an MAC (multiple access channels) capacity area in a V L C (visible light communication) network, which comprises the following steps:

step 1, establishing a V L C system, wherein the V L C system comprises two transmitters and a receiver, the two transmitters are respectively marked as a first transmitter and a second transmitter, each transmitter is provided with a light-emitting diode (L ED), and the receiver is provided with a single Photon Detector (PD);

step 2, limiting the peak light power and the average light power of the V L C system;

and 3, solving the channel capacity and the inner and outer boundaries of the V L C system.

In step 1, X is set₁And X₂Representing the transmitted signals of the first transmitter and the second transmitter, respectively. Since the information is embedded in the intensity of the optical signal, X₁And X₂Should be real, non-negative.

In step 2, according to eye safetyStandard and actual lighting requirements, limiting peak and average optical power: x is not less than 0₁≤A₁，E{X₁}≤μ₁，0≤X₂≤A₂，E{X₂}≤μ₂Wherein A is₁And A₂Respectively representing the maximum value of the transmission signal of the first transmitter and the maximum value of the transmission signal of the second transmitter; e { X }₁And E { X }₂The mathematical expectations for the signals transmitted by the first and second transmitters, respectively; while mu₁And mu₂The maximum value of the expected value of the signal transmitted by the first transmitter and the second transmitter respectively. In the MAC channel, the received signal Y is represented as:

Y＝X₁+X₂+Z (1)

where Z is the independent Gaussian noise with a mean of 0 and a variance of σ²。

The step 3 comprises the following steps:

step 3-1, the MAC capacity field in the V L C system is a convex hull

Wherein R (X)₁,X₂) Indicating, for a fixed product distribution satisfying a given input constraint

The set of rate pairs (R)₁,R₂) The following conditions are satisfied:

wherein R is₁And R₂Respectively represent x₁Transmission rate and x₂The transmission rate of (c);

I(X₁；Y|X₂) Indicating the channel capacity of information about X1 obtained from Y when X2 is known to be known.

I(X₂；Y|X₁) Indicating the channel capacity of information about X2 obtained from Y when X1 is known to be known.

I(X₁,X₂(ii) a Y) representsUnconditional mutual information, i.e., channel capacity of information on X1 and X2 obtained from Y.

Step 3-2, solving channel capacity I (X)_i；Y|X_i) Wherein i is 1,2,

due to the finite amplitude (peak optical power constraint), the optimal input distribution of equation (2) should be a finite set of discrete points. Therefore, an effective method is proposed to estimate the channel capacity I (X) under discrete input distribution_i；Y|X_i) And its inner and outer boundaries, where i is 1,2, and i is 3-i.

Step 3-3, solving channel capacity I (X)_i；Y|X_i) The inner boundary of (a);

step 3-4, solving channel capacity I (X)_i；Y|X_i) To the outer boundary of (a).

Step 3-2 comprises:

step 3-2-1 to obtain X₁And X₂Optimum distribution (X)₁And X₂Channel capacity distribution) of the channel, setting signal X_iGet K_iA non-negative real value to obey a discrete distribution

i is 1,2, as follows:

in the formula x_i,jIs X_iJ (th) point of (p)_i,jRepresenting its corresponding probability; pr { X_i＝x_i,jIs means for X_i＝x_i,jThe probability of (d); sigma²Is the variance of the Gaussian noise Z, K₁And K₂Are respectively taken as

Step 3-2-2, based on step 3-2-1, obtaining:

wherein, P (X)_i) Represents the probability density function of the signal Xi, h (Xi + Z) represents the entropy of Xi + Z,

since the noise Z follows a Gaussian distribution, its probability density (pdf)

Is written as:

step 3-2-3, obtaining the capacity expression of formula (4d) on the basis of step 3-2-1 and step 3-2-2, expressing as a mathematical problem subject to optimization, as follows:

s.t.(3a),(3b),(3c)

step 3-2-4, due to variable K_i，

And

to overcome this difficulty, the problem (6) is difficult to solve by applying an imprecise descent gradient method (S.Boyd and L Vandenberghe, Convexoptimization Cambridge, U.K.: Cambridge Univ.Press,2004.) to obtain an optimal input distribution

Namely, it is

Sum channel capacity C_i。

The number of non-negative real values taken when representing the optimal input distribution,

represents X_iAt the point of the (j) th,

indicating its corresponding probability.

Step 3-3 comprises:

step 3-3-1, to obtain channel capacity C_1,2Inner boundary of (3), define

Wherein the probability density function f_Y(y) is:

wherein x_1,mRepresents X₁M point of (1), p_1,mRepresenting its corresponding probability;

x_2,nrepresents X₂N th point of (p)_2,nRepresenting its corresponding probability;

y represents a received signal;

step 3-3-2, according to the parameters obtained in step 3-3-1, the capacity domain of V L C MAC comprises two variables

And

this makes the optimization problem difficult to solve. Therefore, a sub-optimal solution needs to be found. In particular, will distribute

And

put to the rightmost side of formula (7), the result obtained is C_1，2The inner boundary of (a).

The steps 3-4 comprise:

step 3-4-1, defining a signal

Then sum signal

The peak optical power and the average optical power are respectively constrained to be

Wherein

Wherein

Represents an upper bound for the peak optical power,

represents the maximum value of the average optical power.

Step 3-4-2, setting signal

Subject to a discrete distribution, take

A non-negative real value

This gives:

in the formula

Step 3-4-3, according to step 3-4-2, obtaining C_1,2Has an outer boundary of

To simplify it, C_1,2Is written as

s.t.(9a),(9b),(9c)

Step 3-4-4, which is a discrete non-convex problem. The problem (10) can be solved by an inaccurate gradient descent method, and the channel capacity C can be obtained_1,2To the outer boundary of (a). The method for analyzing the channel capacity of the two users can be directly extended to N users (N is more than or equal to 3) multiple access communication.

The present invention studies the capacity region of V L C MAC with peak and average power limitations, the optimal input follows a discrete distribution due to peak optical power limitations, formulating the exact channel capacity region of the found V L C MAC into a hybrid discrete optimization problem by assuming discrete inputs, which is non-convex because the objective function has no analytical expressions.

The method has the advantages that the obtained difference between the inner boundary and the outer boundary is smaller, and the system performance is better due to the new inner boundary and the new outer boundary of the capacity areas of the Multiple Access Channels (MAC) in the visible light communication (V L C) network.

Drawings

The foregoing and/or other advantages of the invention will become further apparent from the following detailed description of the invention when taken in conjunction with the accompanying drawings.

FIG. 1 shows the respective optimum input positions of V L C MAC under different SNRs

Schematic diagram of the variation curve of (2).

FIG. 2 shows the respective discrete distribution optimal inputs of V L C MAC under different SNR

Schematic diagram of the variation curve of (2).

Fig. 3 is a graph showing the inner and outer boundary variation curves of the V L C MAC capacity domain at low SNR.

Fig. 4 is a graph showing the inner and outer boundary variation curves of the V L C MAC capacity domain at high SNR.

FIG. 5 shows the rate R only under the constraint of peak optical power₁And R₂Schematic diagram of the variation curve of (2).

FIG. 6 shows the rate R only under the constraint of peak optical power₁And R₂Schematic diagram of the variation curve of (2).

FIG. 7 shows the rate and R for two users at a particular SNR (dB)₁+R₂Schematic diagram of the variation curve of (2).

Detailed Description

The invention provides a new inner and outer boundary calculation method of a MAC (multiple access channels) capacity area in a V L C (visible light communication) network, which comprises the following steps:

step 1, establishing a V L C system, wherein the V L C system comprises two transmitters and a receiver, the two transmitters are respectively marked as a first transmitter and a second transmitter, each transmitter is provided with a light-emitting diode (L ED), and the receiver is provided with a single Photon Detector (PD);

step 2, limiting the peak light power and the average light power of the V L C system;

and 3, solving the channel capacity and the inner and outer boundaries of the V L C system.

In step 1, X is set₁And X₂Representing the transmitted signals of the first transmitter and the second transmitter, respectively. Since the information is embedded in the intensity of the optical signal, X₁And X₂Should be real, non-negative.

In step 2, according to eye safety standard and actual lighting requirement, limiting peak light power and average light power: x is not less than 0₁≤A₁，E{X₁}≤μ₁，0≤X₂≤A₂，E{X₂}≤μ₂Wherein A is₁And A₂Respectively representing the maximum value of the transmission signal of the first transmitter and the maximum value of the transmission signal of the second transmitter; e { X }₁And E { X }₂The mathematical expectations for the signals transmitted by the first and second transmitters, respectively; while mu₁And mu₂The maximum value of the expected value of the signal transmitted by the first transmitter and the second transmitter respectively. In the MAC channel, the received signal Y is represented as:

Y＝X₁+X₂+Z (1)

where Z is the independent Gaussian noise with a mean of 0 and a variance of σ²。

The step 3 comprises the following steps:

step 3-1, the MAC capacity field in the V L C system is a convex hull

Wherein R (X)₁,X₂) Indicating, for a fixed product distribution satisfying a given input constraint

The set of rate pairs (R)₁,R₂) The following conditions are satisfied:

wherein R is₁And R₂Respectively represent x₁Transmission rate and x₂The transmission rate of (c);

I(X₁；Y|X₂) Indicating the channel capacity of information about X1 obtained from Y when X2 is known to be known.

I(X₂；Y|X₁) Indicating the channel capacity of information about X2 obtained from Y when X1 is known to be known.

I(X₁,X₂(ii) a Y) represents unconditional mutual information, i.e., channel capacity of information on X1 and X2 obtained from Y.

Step 3-2, solving channel capacity I (X)_i；Y|X_i) Wherein i is 1,2,

due to the finite amplitude (peak optical power constraint), the optimal input distribution of equation (2) should be a finite set of discrete points. Therefore, an effective method is proposed to estimate the channel capacity I (X) under discrete input distribution_i；Y|X_i) And its inner and outer boundaries, where i is 1,2,

step 3-3, solving channel capacity I (X)_i；Y|X_i) The inner boundary of (a);

step 3-4, solving channel capacity I (X)_i；Y|X_i) To the outer boundary of (a).

Step 3-2 comprises:

step 3-2-1 to obtain X₁And X₂Optimum distribution (X)₁And X₂Channel capacity distribution) of the channel, setting signal X_iGet K_iA non-negative real value to obey a discrete distribution

i is 1,2, as follows:

in the formula x_i,jIs X_iJ (th) point of (p)_i,jRepresenting its corresponding probability; pr { X_i＝x_i,jIs means for X_i＝x_i,jThe probability of (d); sigma²Is the variance of the Gaussian noise Z, K₁And K₂Are respectively taken as

Step 3-2-2, based on step 3-2-1, obtaining:

wherein, P (X)_i) Represents the probability density function of the signal Xi, h (Xi + Z) represents the entropy of Xi + Z,

since the noise Z follows a Gaussian distribution, its probability density (pdf)

(y_i) Is written as:

step 3-2-3, obtaining the capacity expression of formula (4d) on the basis of step 3-2-1 and step 3-2-2, expressing as a mathematical problem subject to optimization, as follows:

s.t.(3a),(3b),(3c)

step 3-2-4, due to variable K_i，

And

to overcome this difficulty, the problem (6) is difficult to solve by applying an imprecise descent gradient method (S.Boyd and L Vandenberghe, Convexoptimization Cambridge, U.K.: Cambridge Univ.Press,2004.) to obtain an optimal input distribution

Namely, it is

Sum channel capacity C_i。

The number of non-negative real values taken when representing the optimal input distribution,

represents X_iAt the point of the (j) th,

indicating its corresponding probability.

Step 3-3 comprises:

step 3-3-1, to obtain channel capacity C_1,2Inner boundary of (3), define

Wherein the probability density function f_Y(y) is:

wherein x_1,mRepresents X₁M point of (1), p_1,mRepresenting its corresponding probability;

x_2,nrepresents X₂N th point of (p)_2,nRepresenting its corresponding probability;

y represents a received signal;

step 3-3-2, according to the parameters obtained in step 3-3-1, the capacity domain of V L C MAC comprises two variables

And

this makes the optimization problem difficult to solve. Therefore, a sub-optimal solution needs to be found. In particular, will distribute

And

put to the rightmost side of formula (7), the result obtained is C_1，2The inner boundary of (a).

The steps 3-4 comprise:

step 3-4-1, defining a signal

Then sum signal

The peak optical power and the average optical power are respectively constrained to be

Wherein

Wherein

Represents an upper bound for the peak optical power,

represents the maximum value of the average optical power.

Step 3-4-2, setting signal

Subject to a discrete distribution, take

A non-negative real value

This gives:

in the formula

Step 3-4-3, according to step 3-4-2, obtaining C_1,2Has an outer boundary of

To simplify it, C_1,2Is written as

s.t.(9a),(9b),(9c)

Step 3-4-4, which is a discrete non-convex problem. Can be applied with an imprecise gradientThe down method solves the problem (10) and obtains the channel capacity C_1,2To the outer boundary of (a). The method for analyzing the channel capacity of the two users can be directly extended to N users (N is more than or equal to 3) multiple access communication.

Examples

FIGS. 1 and 2 illustrate the respective optimum input positions of V L C (V L C, Visible L illumination communication) MAC (MAC, Multiple Access Channel model) under different SNR (signal-to-NOISE RATIO, SIGNA L NOISE RATIO, SNR) when using the method of the present invention

(x_i,jJ-th point representing Xi) and a discrete distribution optimal input

(p_i,jRepresents p_iThe j point of (1), i.e. x_i,jProbability) of in which

For SNR ≦ 10dB, the optimal input position has two discrete points {0, A } for different probabilities, i.e., {0.8,0.2 }. therefore, OOK modulation system can also get the capacity of V L C MAC at low SNR>At 10dB, there are more than two discrete points at the optimal input position, which indicates that a PAM Modulation system (PAM) can obtain the capacity of the V L C MAC.

Fig. 3 and 4 of the method diagram according to the invention illustrate, respectively, that at low SNR, i.e. at low SNR

(A1 denotes the peak optical power of emitter 1) and

(A2 denotes the peak optical power of emitter 2, σ is the standard deviation of Gaussian noise Z), and at high SNR, i.e., at

And

inner and outer boundaries of the V L C MAC capacity domain, where

As can be seen from fig. 3, the proposed outer boundary is lower than the existing outer boundary, i.e. it is lower than the existing outer boundary

And

and the proposed inner boundary is larger than the maximum discrete entropy inner boundary.

Fig. 5 and 6 of the method diagram according to the invention illustrate the special case, i.e. only under the constraint of peak optical power, where

Figures 3, 4 and 5, 6 all show that the proposed outer boundary is lower than the existing outer boundary, whereas the proposed inner boundary is lower than the outer boundary except for

The other existing inner boundaries on the high SNR side are all high, i.e., in FIG. 6

Sum rate R₁+R₂(transmission rate).

Fig. 7, which is a graphical representation of the method according to the invention, shows the proposed inner and outer boundaries, maximizing the discrete entropy inner boundary,

and

rate sum of two users at a particular SNR (dB)R₁+R₂(bits/s/Hz), i.e. when only the peak optical power is constrained, wherein

It can be seen that at low SNR the proposed inner boundary is very close to the maximum discrete entropy inner boundary, whereas at high SNR the proposed inner boundary is higher than the maximum discrete entropy inner boundary. This is because, maximizing the discrete entropy inner boundary is to separate the input entropy H (X) into discrete entropy₁)+H(X₂) Maximized, this is only the difference entropy h (X)₁+X₂+ Z) is an approximation. At low and medium SNR, the proposed inner boundary ratio

The inner boundary is high. At the time of a high SNR, it is,

highest (only peak optical power constraint). Furthermore, the proposed outer boundary is higher than the existing outer boundary

And

the relevant english explanations in fig. 3 to 7 are as follows:

the deployed Outer represents the Outer boundary proposed by the method of the present invention;

the deployed inner means represents the inner boundary proposed by the method of the present invention;

max Entry denotes the maximum inner boundary of discrete Entropy proposed in the paper "Channel capacity and non-uniform signalling for free-space optical intensities channels" (A.A.F. and S.Hranilvic, "Channel capacity and non-uniform signalling for free-space optical intensities channels," IEEE J.Sel.areas Commun., vol.17, No.9, pp.1553-1563, Dec.2009.)

Outer

Denotes the outer boundary set forth in the paper "On the capacity of free space optical fibres

(A.Lapidoth,S.M.Moser,and M.Wigger,“On the capacityof freespace optical intensity channels,”IEEE Trans.Inf.Theory,vol.55,no.10,pp.4449–4461,Oct.2009.)

Outer

Denotes the outer boundary set forth in the paper "Free-space optical communications," capacity centers, approximations, and a new propagating property

(A.Chaaban,J.-M.Morvan,and M.-S.Alouini,“Free-space optical communications:capacity bounds,approximations,and a new spherepacking perspective,”IEEETrans.Commun.,vol.64,no.3,pp.1176–1191,Mar.2016.)

Outer

Denotes the outer boundary set forth in the paper "On the capacity of free space optical fibres

(A.Lapidoth,S.M.Moser,and M.Wigger,“On the capacityof freespace optical intensity channels,”IEEE Trans.Inf.Theory,vol.55,no.10,pp.4449–4461,Oct.2009.)

Outer

Denotes the outer boundary set forth in the paper "Bounds on the capacity region of the optical intensitymultiple access channel

(J.Zhou and W.Zhang,“Bounds on the capacity region of the optical intensity multiple accesschannel,”accepted by IEEE Trans.Commun.,2019.)

Inner

Denotes the outer boundary set forth in the paper "Bounds on the capacity region of the optical intensitymultiple access channel

(J.Zhou and W.Zhang,“Bounds on the capacity region of the optical intensity multiple accesschannel,”accepted by IEEE Trans.Commun.,2019.

It is noted that the outer boundary

And an inner boundary

Only for the specific case, i.e. with only peak optical power constraints, and the discrete Entropy maximizing inner boundary of Max entry is first extended to the V L C MAC scheme in the present invention.

In life, from the viewpoint of eye safety, instantaneous optical power, that is, amplitude needs to be considered. The average light power is also limited according to the actual lighting requirements, and

and

only the peak value constraint is considered, and the method is not applicable to actual life. Second, it is used for

And

the external environment of the method is worse than that of the method, the limit of the method is stricter, and the method is suitable for actual usersGrouping, power allocation, interference management and energy efficiency are all greatly facilitated, and are key to improving industrial efficiency.

The present invention provides a new method for calculating the inner and outer boundaries of the MAC capacity region in the V L C network, and a plurality of methods and approaches for implementing the method, and the above description is only a preferred embodiment of the present invention, and it should be noted that, for those skilled in the art, without departing from the principle of the present invention, several improvements and modifications may be made, and these improvements and modifications should be regarded as the protection scope of the present invention.

Claims (7)

1. A new inner and outer boundary calculation method of a MAC capacity area in a V L C network is characterized by comprising the following steps:

step 1, establishing a V L C system, wherein the V L C system comprises two transmitters and a receiver, the two transmitters are respectively marked as a first transmitter and a second transmitter, each transmitter is provided with a light emitting diode L ED, and the receiver is provided with a single photon detector PD;

step 2, limiting the peak light power and the average light power of the V L C system;

and 3, solving the channel capacity and the inner and outer boundaries of the V L C system.

2. The method of claim 1, wherein in step 1, X is set₁And X₂Representing the transmitted signals of the first transmitter and the second transmitter, respectively.

3. The method according to claim 2, wherein in step 2, the peak and average optical powers are limited according to eye safety standards and actual lighting requirements: x is not less than 0₁≤A₁，E{X₁}≤μ₁，0≤X₂≤A₂，E{X₂}≤μ₂Wherein A is₁And A₂Representing the maximum value of the first transmitter transmission signal and the maximum value of the second transmitter transmission signal, respectivelyA large value; e { X }₁And E { X }₂The mathematical expectations for the signals transmitted by the first and second transmitters, respectively; while mu₁And mu₂The maximum value of the expected value of the signals transmitted by the first transmitter and the second transmitter respectively;

in the MAC channel, the received signal Y is represented as:

Y＝X₁+X₂+Z (1)

where Z is the independent Gaussian noise with a mean of 0 and a variance of σ²。

4. The method of claim 3, wherein step 3 comprises:

step 3-1, the MAC capacity field in the V L C system is a convex hull

Wherein R (X)₁,X₂) Indicating, for a fixed product distribution satisfying a given input constraint

The set of rate pairs (R)₁,R₂) The following conditions are satisfied:

wherein R is₁And R₂Respectively represent x₁Transmission rate and x₂The transmission rate of (c);

I(X₁；Y|X₂) Channel capacity representing information about X1 obtained from Y when X2 is known to be known;

I(X₂；Y|X₁) Channel capacity representing information about X2 obtained from Y when X1 is known to be known;

I(X₁,X₂(ii) a Y) represents unconditional mutual information, i.e., channel capacity of information on X1 and X2 obtained from Y;

step 3-2, solving channel capacity

Wherein the ratio of i to 1,2,

step 3-3, solving channel capacity

The inner boundary of (a);

step 3-4, solving channel capacity

To the outer boundary of (a).

5. The method of claim 4, wherein step 3-2 comprises:

step 3-2-1 to obtain X₁And X₂Setting the signal X to an optimum distribution_iGet K_iA non-negative real value to obey a discrete distribution

As follows:

in the formula x_i,jIs X_iJ (th) point of (p)_i,jRepresenting its corresponding probability; pr { X_i＝x_i,jIs means for X_i＝x_i,jThe probability of (d); sigma²Is the variance of the gaussian noise Z; k₁And K₂Are respectively taken as

Step 3-2-2, based on step 3-2-1, obtaining:

wherein, P (X)_i) Represents the probability density function of the signal Xi, h (Xi + Z) represents the entropy of Xi + Z,

since the noise Z follows a Gaussian distribution, its probability density

Is written as:

step 3-2-3, obtaining the capacity expression of formula (4d) on the basis of step 3-2-1 and step 3-2-2, expressing as a mathematical problem subject to optimization, as follows:

s.t.(3a),(3b),(3c)

3-2-4, obtaining the optimal input distribution x by applying an inaccurate descending gradient method_iIs/are as follows

Namely, it is

And x_iChannel capacity C of_i，

The number of non-negative real values taken when representing the optimal input distribution,

represents X_iAt the point of the (j) th,

indicating its corresponding probability.

6. The method of claim 5, wherein step 3-3 comprises:

step 3-3-1, in order to obtain the inner boundary of the channel capacity, defining the channel capacity

Obtaining:

wherein the probability density function f_Y(y) is:

wherein x_1,mRepresents X₁M point of (1), p_1,mRepresenting its corresponding probability;

x_2,nrepresents X₂N th point of (p)_2,nRepresenting its corresponding probability;

y represents a received signal;

step 3-3-2, distribution

And

put to the rightmost side of formula (7), the result obtained is C_1，2The inner boundary of (a).

7. The method of claim 6, wherein steps 3-4 comprise:

step 3-4-1, defining a signal

Then sum signal

The peak optical power and the average optical power are respectively constrained to be

Wherein

Wherein

Represents an upper bound for the peak optical power,

represents the maximum value of the average optical power;

step 3-4-2, setting signal

Subject to a discrete distribution, take

A non-negative real value

This gives:

in the formula

Step 3-4-3, according to step 3-4-2, obtaining C_1,2Has an outer boundary of

To simplify it, C_1,2The outer boundary of (a) is written as:

s.t.(9a),(9b),(9c)

step 3-4-4, solving the problem (10) by an inaccurate gradient descent method and obtaining the channel capacity C_1,2To the outer boundary of (a).

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