CN109150304A - A kind of calculation method of free space light intensity channel up to capacity - Google Patents

Abstract

The invention discloses a kind of free space light intensity channels up to the calculation method of capacity, for realizing the capacity of Free Space Optics (FSO) channel.Under peak value constraint and average light power constraint, the capacity for seeking FSO channel can be considered as the continuous discrete optimization problems of device of mixing, and wherein objective function is can not to accumulate.To overcome this difficulty, the present invention uses numerical integration method approximate objective function and its gradient.Then, it was demonstrated that the gap between original function and approximation can be arbitrarily small.Based on approximate function, the present invention proposes the inaccurate gradient descent method of one kind to solve to mix continuous discrete optimization problems of device, theoretically shows that optimal solution obtained converges on optimal discrete distribution, FSO channel capacity may be implemented.Finally, simulation results show the method that the performance of this method proposes, achievable rate are higher than existing method.

Description

A kind of calculation method of free space light intensity channel up to capacity

Technical field

The present invention relates to the communications field more particularly to a kind of free space light intensity channels up to the calculation method of capacity.

Background technique

Free space optics (FSO) widely exempts from licensed spectrum, low EMI, high security and high data speed due to it Rate (bibliography: [1], [2], [3], [4]), communication cause extensive research concern in academia and industry recently.No It is same as traditional radio communication, FSO communication detects (IM/DD) scheme using intensity modulated and directly.In order to meet eye-safe And the considerations of actual illumination, transmitting signal limitation should be controlled by meeting peak value and average light power.Under this configuration, (bibliography: [5], [6]) have been proven that FSO capacity realize distribution channel be on limited one group of point it is discrete, because And it realizes the Gaussian Profile of the classical channel capacity of RF and may not apply to FSO channel.Up to the present, for FSO channel how Effectively the reachable capacity of search channel it is discrete distribution or it is unknown, and current method only at each signal-to-noise ratio (SNR) Point carries out exhaustive search.

In order to evade inefficient calculating, the bound of FSO channel capacity has been obtained.Bibliography [7], [8], [9], [10] in, capacity is limited by average optical has studied power constraint.Based on sphere packing method, author is in bibliography [7] Upper and lower bound is exported, the gap between two boundaries is about 0.5 bit transmitted every time.In bibliography [8], by most Bigization source entropy proposes lower limit in the non-uniform Distribution of series of discrete, and the upper bound is also to pack to prove by sphere.With reference to text Offer capacity when two in [8] boundaries progressively describe FSO channel low signal-to-noise ratio.By utilizing a kind of new approximation method Obtain inherent volume simple form, the upper bound developed in bibliography [9] improves bibliography [7], the knot in bibliography [8] Fruit.Author is derived upper and lower bound in bibliography [10], and the average light power that goes to zero of the gap between boundary tends to It is infinitely great.Based on new recursion method, the upper limit proposes that it is further improved in bibliography [10] in bibliography [11] Sphere top filling.

In addition, bibliography [10], [11], [12], [13] have studied the volume marginal optical power under peak value and average value Limitation.For the fixed ratio of mean power and peak power, gap is when SNR becomes infinity, in bibliography [10] It goes to zero between upper and lower bound.In addition, by using truncation Gaussian Profile, it is shown that derived lower bound in bibliography [11] The top filling (bibliography [10]) close to spherical shape in high s/n ratio.It is derived in bibliography [12] by maximizing source The achievable rate of discrete non-homogeneous input distribution is provided, it is nervous in high s/n ratio.In peak optical powers, average light power Constrained with electric power, closing form lower bound (referred to as ABG lower bound) in the bibliography [13] development using entropy weight inequality and Lagrangian method.

In short, existing paper lays particular emphasis on the lower limit or the upper limit of FSO channel capacity, numerical result demonstrates channel boundary Compactness.However, existing work, which does not directly give how to be distributed with discrete input, realizes FSO channel capacity.Therefore, do not have Theoretical method can effectively obtain the channel capacity of FSO communication system.

Bibliography: [1] T.Komine and M.Nakagawa, " Fundamental analysis for visible-light communication system using led lights,”IEEE Trans.Consum.Electron,vol.50,no.1,pp.100–107,Feb.2004.

[2]H.Elgala,R.Mesleh,and H.Haasn,“Indoor optical wireless communication:potential and state-of-the-art,,”IEEE Commun.Mag.,vol.49,no.9, pp.56–62,Dec.2011.

[3]A.Jovicic,J.Li,and T.Richardson,“Visible light communication: opportunities,challenges and the path to market,”IEEE Commun.Mag.,vol.51, no.12,pp.26–32,Dec.2013.

[4]P.H.Pathak,X.Feng,P.Hu,and P.Mohapatra,“Visible light communication,networking,and sensing:a survey,potential and challenges,”IEEE Commun.Surveys Tuts.,vol.17,no.4,pp.2047–2077,Sept.2015.

[5]J.G.Smith,“The information capacity of amplitude-and varianceconstrained scalar gaussian channels,”Inf.Contr.,vol.18,no.3,pp.203– 219,Feb.1971.

[6]S.H.T.Chan and F.Kschischang,“Capacity-achieving probability measure for conditionally gaussian channels withbounded inputs,”IEEE Trans.Inf.Theory,vol.51,no.6,pp.2073–2088,Jun.2005.

[7]J.W.M.C.J.Wang Q.Hu.and J.Wang,“Tight bounds on channel capacity for dimmable visible light communications,”J.Lightwave Technol.,vol.31,no.23, pp.3771–3779,Dec.2013.

[8]A.A.Farid and S.Hranilovic,“Capacity bounds for wireless optical intensity channels with gaussian noise,”IEEE Trans.Inf.Theory,vol.56,no.12, pp.6066–6077,Dec.2010.

[9]Q.W.Rui Jiang Zhaocheng Wang.and L.Dai,“A tight upper bound on channel capacity for visible light communications,”IEEE Commun.lett.,vol.20, no.1,pp.1089–7798,Jan.2016.

[10]A.Lapidoth,S.M.Moser,and M.Wigger,“On the capacity of free-space optical intensity channels,”IEEE Trans.Inf.Theory,vol.55,no.10,pp.4449–4461, Oct.2009.

[11]A.Chaaban,J.-M.Morvan,and M.-S.Alouini,“Free-space optical communications:capacity bounds,approximations,and a new sphere-packing perspective,”IEEE J.Sel.Areas Comm.,vol.64,no.3,pp.1176–1191,Mar.2016.

[12]A.A.Farid and S.Hranilovic,“Channel capacity and non-uniform signaling for free-space optical intensity channels,”IEEE J.Sel.Areas Comm., vol.17,no.9,pp.1553–1563,Dec.2009.

[13]S.Ma,R.Yang,H.Li,Z.-L.Dong,H.Gu,and S.Li,“Achievable rate with closed-form for siso channel and broadcast channel in visible light communication networks,”J.Lightwave Technol.,vol.35,no.14,pp.2778–2787, Jul.2017.

Summary of the invention

The purpose of the present invention is developing a kind of effective method, is constrained in peak value and average light power and find and can reach To the Optimal Distribution of FSO channel capacity, specifically, the present invention provides a kind of free space light intensity channels up to the calculating of capacity Method includes the following steps:

Step 1, channel model is set；

Step 2, the capacity of channel model is solved；

Step 3, the optimal solution of the capacity of channel model is sought.

Step 1 includes: one typical IM/DD (intensity modulation and direct of setting Detection, intensity modulated/DC detecting) FSO (Free-space optical, free space light intensity) channel, it includes one A LED or laser diode LD are as transmitter, and a single-photon detector PD is as receiver；

The peak optical powers and average light power of input signal X are all constraints, so that 0≤X≤A, and Wherein, A is the amplitude of signal,For the mean value of signal, μ is electrical power；On FSO channel, receives signal Y and given by following formula Out:

Y=X+Z (1)

Wherein Z is independent Gaussian noise, mean value zero, variance σ²。

Since information is built-in in the intensity of optical signal, the signal X of transmission should be real non-negative.Further, since Eye-safe standard and practical lighting requirement, the peak optical powers and average light power of signal X all should be constraint, so that 0≤X ≤ A, and

Step 2 includes:

For the channel model that step 1 is set, capacity is defined as being distributed in all possible input provided defeated Enter the maximum mutual information C exported in channel^FSO:

Wherein I (X；It Y) is mutual information, H (Y) is the entropy of Y, and H (Y | X) is combination entropy, and P (X) indicates the distribution of X, f_Y(y) table Show Y probability density function (pdf, probability density function), it is clear that f_YIt (y) is the function of P (X).

Step 3 includes:

Step 3-1, setting input signal X is discrete stochastic variable, has K nonnegative real number { x_k}_1≤k≤K, meet:

Pr { X=x in formula_k}=p_kIndicate X=x_kWhen corresponding probability value be p_k, x_kIt is k-th point, p_kIt is x_kAccordingly Probability,For positive integer；

Step 3-2: noise Z follows Gaussian Profile, f_Y(y) pdf is converted into following form:

The capacity for solving channel model is equally written as following optimization problem:

Due to variable K, { p_k}_1≤k≤K{ x_k}_1≤k≤K, above-mentioned optimization problem is the discrete non-convex problem of mixing.In addition, Objective function (5a) can not accumulate, and without the analytical expression of objective function (5a).Therefore, above-mentioned optimization problem is difficult to It solves.The present invention will develop a kind of effective method to search for optimal input distribution.

Step 3-3, is such as given a definition:

φ (p) is objective function in formula, and Υ is constraint set, 1_KIt is the vector of K × 1, wherein all elements are equal to 1, then ask Optimization problem in topic (5a) i.e. step 3-2 equivalent is rewritten as following problem (7):

s.t.p∈Υ (7b)

There are three key variables, i.e. K, p and x in revised problem；When K and x are fixed, which is that the convex of variable p is asked Topic；This must to reduce design variable and passes through fixed variable x.

Step 3-4 uses equidistant fixed x:

K value { x is selected from [0, A] range equal intervals_k}_1≤k≤K, it may be assumed that

Set proposition 1: setting K^*WithIt is the optimal solution of problem (7), γ is indicatedMiddle any two points Between minimum range, i.e., Any two points are indicated, for given ε₀> 0, whenThere are a sequence { x_l}_1≤l≤KMeet:

For k^*Can value set；

It proves: without loss of generality, it is assumed thatIt is an ascending.Wherein k= 2,...,K^*, γ is allowed to indicate the smallest d_k, i.e.,

The precision ε given for one₀> 0 constructs a sequence { x_k}_1≤k≤K, one of them sufficiently large (generally 20) 'sMeet following formula

|x_k-x_k+1|≤ε₀ (10)

X in formula_kThe definition in (8).

Therefore, for any pointWhereinIn the presence of point x_lMeet:

Under proposition 1, optimal solutionIt is direct.It must be under given arbitrary accuracy comprising certain {x_k}_1≤k≤K.Since K is greater than K*, in { x_k}_1≤k≤KIn there may be many redundant points.But this redundancy will not influence Objective function realizes maximum value, because redundant points influence to reduce by optimizing the pdf of p.Therefore, given for some K, can determine one in { x_k}_1≤k≤KIn subset near-optimizationAs shown in proposition 1.

Step 3-5 solves the problems, such as (7) using gradient projection method.

Step 3-5 includes:

Step 3-5-1, allowsThe gradient for indicating objective function, that is, formula (7a), is given by:

Step 3-5-2, still, either objective function φ (p) or gradientAll none analytical expressions. In order to solve this problem, using numerical integration method respectively to φ (p) andIt carries out approximate: due to 0≤X≤A, and Z follows Gaussian Profile, [- τ₁,A+τ₁] and [- τ₂,A+τ₂] respectively indicate φ (p) andIntegrating range, τ₁、τ₂Indicate product Point slight gap, be two very littles taken for approximation value (can take the number between 0~1, for example, 0.4,0.5), wherein τ₁> 0 and τ₂> 0, allowsWithRespectively indicate objective function φ (p) approximation andApproximation, it may be assumed that

Allow p₀Indicate a feasible initial point, p_nIndicate n-th of iteration feasible point, wherein n=1,2 ..., inaccurate GradientGradient projection iteration p_nAnd p_n+1It is given by:

α in formula_n∈ (0,1] be nth iteration step-length,

In formula,(16) are defined according to projection, projection operation (15b) is to find a vectorSo that it withBetween distance it is minimum, projection (15b) constitutes following optimization problem:

p_n+1≥0 (17d)

Problem (17) is a convex quadratic programming problem, and can be by using ready-made convex optimization solver effectively It solves, such as CVX.

Symbol: bold-faced lowercase and capitalization respectively represent vector sum matrix.Transposition and Frobenius norm, Mark and the Kronecker product of order, matrix are expressed as ()^TWith | | | |, With The element of X is rounded up to immediate integer.

The utility model has the advantages that the present invention solves the problems, such as Channel of Free-space Optical Communication capacity, solution is original problem, Method before is all approximation, is not achieved relatively good effect (comparing with exhaustion), and the method for exhaustion is very time-consuming also accurately smart Degree can just provide accurate solution, and it is good as exhaustion that simulation result of the invention can achieve the effect that, propose a kind of inaccurate Gradient descent method solve to mix continuous discrete optimization problems of device, theoretically show that optimal solution obtained converges on optimal discrete Distribution, may be implemented FSO channel capacity.

Detailed description of the invention

The present invention is done with reference to the accompanying drawings and detailed description and is further illustrated, it is of the invention above-mentioned or Otherwise advantage will become apparent.

Fig. 1 a is to give under the conditions of φ=2, relationship under the conditions of different SNR, between points and achievable rate.

Fig. 1 b is to give under the conditions of φ=3, relationship under the conditions of different SNR, between points and achievable rate.

Fig. 1 c is to give under the conditions of φ=4, relationship under the conditions of different SNR, between points and achievable rate.

Fig. 2 is in φ=4, and under the conditions of different points K, discrete maximum entropy and optimal channel proposed by the present invention hold Measure the variation diagram with SNR.

Fig. 3 is the channel capacity that the method for exhaustion obtains and the mentioned preferred channels capacity of the present invention in φ=4, discrete maximum entropy With the variation diagram of SNR.

Specific embodiment

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

The present invention provides a kind of free space light intensity channels up to the calculation method of capacity, includes the following steps:

Step 1, channel model is set；

Step 2, the capacity of channel model is solved；

Step 3, the optimal solution of the capacity of channel model is sought.

Step 1 includes: one typical IM/DD (intensity modulation and direct of setting Detection, intensity modulated/DC detecting) FSO (Free-space optical, free space light intensity) channel, it includes one A LED or laser diode LD are as transmitter, and a single-photon detector PD is as receiver；

The peak optical powers and average light power of input signal X are all constraints, so that 0≤X≤A, and Wherein, A is the amplitude of signal,For the mean value of signal, μ is electrical power；On FSO channel, receives signal Y and given by following formula Out:

Y=X+Z (1)

Wherein Z is independent Gaussian noise, mean value zero, variance σ²。

Since information is built-in in the intensity of optical signal, the signal X of transmission should be real non-negative.Further, since Eye-safe standard and practical lighting requirement, the peak optical powers and average light power of signal X all should be constraint, so that 0≤X ≤ A, and

Step 2 includes:

For the channel model that step 1 is set, capacity is defined as being distributed in all possible input provided defeated Enter the maximum mutual information C exported in channel^FSO:

Wherein I (X；It Y) is mutual information, H (Y) is the entropy of Y, and H (Y | X) is combination entropy, and P (X) indicates the distribution of X, f_Y(y) table Show Y probability density function (pdf, probability density function), it is clear that f_YIt (y) is the function of P (X).

Step 3 includes:

Step 3-1, setting input signal X is discrete stochastic variable, has K nonnegative real number { x_k}_1≤k≤K, meet:

Pr { X=x in formula_k}=p_kIndicate X=x_kWhen corresponding probability value be p_k, x_kIt is k-th point, p_kIt is x_kAccordingly Probability,For positive integer；

Step 3-2: noise Z follows Gaussian Profile, f_Y(y) pdf is converted into following form:

The capacity for solving channel model is equally written as following optimization problem:

Due to variable K, { p_k}_1≤k≤K{ x_k}_1≤k≤K, above-mentioned optimization problem is the discrete non-convex problem of mixing.In addition, Objective function (5a) can not accumulate, and without the analytical expression of objective function (5a).Therefore, above-mentioned optimization problem is difficult to It solves.The present invention will develop a kind of effective method to search for optimal input distribution.

Step 3-3, is such as given a definition:

φ (p) is objective function in formula, and Υ is constraint set, 1 in formula_KIt is the vector of K × 1, wherein all elements are equal to 1, Optimization problem then in problem (5a) i.e. step 3-2 is equivalent to be rewritten as following problem (7):

s.t.p∈Υ (7b)

There are three key variables, i.e. K, p and x in revised problem；When K and x are fixed, which is that the convex of variable p is asked Topic；This must to reduce design variable and passes through fixed variable x.

Step 3-4 uses equidistant fixed x:

K value { x is selected from [0, A] range equal intervals_k}_1≤k≤K, it may be assumed that

Set proposition 1: setting K^*WithIt is the optimal solution of problem (7), γ is indicatedMiddle any two points Between minimum range, i.e., Any two points are indicated, for given ε₀> 0, whenThere are a sequence { x_l}_1≤l≤KMeet:

For k^*Can value set；

It proves: without loss of generality, it is assumed thatIt is an ascending.Wherein k=2 ..., K^*, γ is allowed to indicate the smallest d_k, i.e.,

The precision ε given for one₀> 0 constructs a sequence { x_k}_1≤k≤K, one of them sufficiently large (generally 20) 'sMeet following formula

|x_k-x_k+1|≤ε₀ (10)

X in formula_kThe definition in (8).

Therefore, for any pointWhereinIn the presence of point x_lMeet:

Under proposition 1, optimal solutionIt is direct.It must be under given arbitrary accuracy comprising certain {x_k}_1≤k≤K.Since K is greater than K^*, therefore in { x_k}_1≤k≤KIn there may be many redundant points.But this redundancy will not influence Objective function realizes maximum value, because redundant points influence to reduce by optimizing the pdf of p.Therefore, given for some K, can determine one in { x_k}_1≤k≤KIn subset near-optimizationAs shown in proposition 1.

Step 3-5 solves the problems, such as (7) using gradient projection method.

Step 3-5 includes:

Step 3-5-1, allowsThe gradient for indicating objective function, that is, formula (7a), is given by:

Step 3-5-2, still, either objective function φ (p) or gradientAll none analytical expressions. In order to solve this problem, using numerical integration method respectively to φ (p) andIt carries out approximate: due to 0≤X≤A, and Z follows Gaussian Profile, [- τ₁,A+τ₁] and [- τ₂,A+τ₂] respectively indicate φ (p) andIntegrating range, τ₁、τ₂Indicate product The slight gap divided is the value of two very littles taken for approximation, wherein τ₁> 0 and τ₂> 0, allowsWithTable respectively Show objective function φ (p) approximation andApproximation, it may be assumed that

Allow p₀Indicate a feasible initial point, p_nIndicate n-th of iteration feasible point, wherein n=1,2 ..., inaccurate GradientGradient projection iteration p_nAnd p_n+1It is given by:

α in formula_n∈ (0,1] be nth iteration step-length,

In formula,According to projection define (16), projection operation (15b) be find one to Measure pn+1 ∈ Υ so that it withBetween distance it is minimum, projection (15b) constitutes following optimization problem:

p_n+1≥0 (17d)

Problem (17) is a convex quadratic programming problem, and can be by using ready-made convex optimization solver effectively It solves, such as CVX.

In step 3-5-2, uses straight line backtracking line suitable step-length of selection in (15a) to successively decrease to reach, specifically includes:

Step 3-5-2-1, initialization: selection K >=2, λ_K-1≤ 0, c is set₂, c₃For iteration stopping parameter；

Step 3-5-2-2 selects a feasible initial point p if n=0₀∈Y；

Step 3-5-2-3, n=n+1, then calculateWith

Step 3-5-2-4, material calculation α_n；

Step 3-5-2-5 is calculated

Step 3-5-2-6, if | | p_n-p_n-1||≤c₂, then stop, thenOtherwise, step 3- is turned to 5-2-3；

Step 3-5-2-7, if | λ_K-λ_K-1|≤c₃, then stop, then exporting p^opt=p_n, K^opt=K, otherwise K=K+1, Then turn to step 3-5-2-2, wherein K^optIndicate discrete point { x_kOptimal number, λ_K-1For the initial value of objective function, p^optFor the optimal probability value for meeting condition.

Due to optimal discrete distribution { K^*,x^*,p^*It is a unique discrete random variable, Ke Yitong in limited numerical value It crosses simple linear search and obtains optimal number K^opt, K=K+1 is for next iteration, and wherein the initialization of K is not less than 2.Always It, lists the inaccurate gradient descent method proposed in the above method.

Step 3-5-2-4 includes:

Step 3-5-2-4-1, selection For initial step length, ρ is the stride length shrinks factor, and c is a ginseng Number, generally takes the number between 0 and 1；

Step 3-5-2-4-2 is repeated until meeting:

Wherein For the target function value of next iteration,For this The value of secondary iteration,For projection；

Step 3-5-2-4-3,← indicate assignment, i.e., the step-length of this time iteration multiplied by the stride length shrinks factor It is assigned to step-length next time；

Step 3-5-2-4-4 terminates to repeat；

Step 3-5-2-4-5, whenWhen terminate；

The optimality of inaccurate gradient descent method: it is worth noting that, if τ₁Value it is sufficiently large, [- τ₁,A+τ₁] to appoint Small gap of anticipating is close to [- ∞, ∞].In addition, following theorem shows approximationIt can be with arbitrarily small error close to φ (p)。

Theorem 1: for given precision ε₁> 0, there are a sufficiently large parameters to meet τ₁> σ

Then have:

Wherein, φ (p) andIt is provided in (6d) and (14a) respectively, in addition,

It proves as follows:

ε_totalIt representsError between φ (p), is given by

Wherein,

In addition, ε_totalThe upper limit of absolute value is given by:

The upper bound are as follows:

Due to (y-x_k)²≥y², inequality (25b) is set up, for y≤- τ₁≤0。

Hereinafter, it will show with τ₁Increase, two, the right side of (25c) becomes zero.Specifically, the integral of the formulaIt is given by:

In addition, the formulaIntegral be given by:

(26b) and (27b) is substituted into (25c), is obtained:

Wherein,

Further, sinceHave:

The formulaThe upper limit are as follows:

Due to (y-x_k)²≥(y-A)², inequality (30b) is set up, for y >=A+ τ₁。

Hereinafter, it will show with τ₁Increase, two, the right side of (25c) becomes zero.

The integral of the formulaIt is given by:

In addition, the integral of the termIt is given by:

By obtaining (31b) and (32b) substitution (30c):

Equally, becauseWhen, have:

By combination (22a), (29) and (34) have

Work as τ₁When >=σ,WithBoth non-negative monotonic decreasing functions,

Then

Because erfc (x) be non-negative monotonic decreasing function andHave

Therefore, for any given precision ε₁> 0, there are parameter τ one big₁> σ meets:

Then have:

Equally, the τ sufficiently large for one₂,It can be close with arbitrarily small gap

Theorem 2: for given precision ε₂> 0, there are τ₂>=σ meets:

Then, have:

WithIt is provided in (13) and (14b) respectively.

It proves as follows:

IfIt indicatesWithBetween phasor difference, be given by:

In addition,Element can be write as:

Wherein,

In addition,The upper limit of the absolute value of element:

SoWithThe upper bound of norm difference are as follows:

Hereinafter, it will indicate that with τ₂Increase, the right item of (42) will become zero.Specifically, termBy following formula It provides:

Then, the integral of the termIt is given by:

In addition, the integral of this formulaIt is given by:

Finally, the integral of this formulaIt is given by:

By substituting into (43c), obtaining (44b), (45b) and (46):

Moreover, the termThe upper limit:

Then, the integral of the termIt is given by:

In addition, the integral of this formulaIt is given by:

Finally, the integral of this formulaIt is given by:

By substituting into (48c), obtaining (49b), (50b) and (51):

By having in conjunction with (47) and (52):

Equally, work as τ₂When >=σ,WithBoth non-negative monotonic decreasing functions, then

Due to erfc (x) be non-negative monotonic decreasing function andHave

Therefore, for any given precision ε₂> 0, there are a big parameter τ₂> σ meets:

Then have:

Therefore, according to inaccurate gradient descent method, optimal discrete input distribution can be calculated effectively.Moreover, logical Cross the sequence { p of the probability distribution of the acquisition of algorithm 2_nOptimal Distribution is converged to, this is by following proof theorem:

Theorem 3:(convergence) K, { p given for one_nThe optimal solutions of problem (12) is converged on, correspondingly, {φ(p_n) the corresponding optimal value for converging to problem (12).

It proves as follows:

Assuming that { p_nConverge on a non-stationary pointIt needs to prove this point:

According to bibliography [14] D.P.Bertsekas and D.P.Bertsekas., Nonlinear Programming., Athena Scientific, the proof of proposition 2.3.1 in 1999, condition (56a) are set up, and it is following not Equation is set up,

Wherein,

Then,Have:

It is also set up since (57) set up inequality (58c)；Inequality (58d) due to WhenWhen, equation is set up, wherein | κ |=| | e_n||。

According to theorem 2, | | e_n| | value can be arbitrarily small.Accordingly, there exist a parameter κ to meetThis Outside, due toIt is non-stable, condition (56b) establishment.

According to bibliography [14] D.P.Bertsekas and D.P.Bertsekas., Nonlinear Programming., Athena Scientific, proposition 2.3.1 is it is found that { p in 1999_nEach limit point be stable 's.In addition, because problem (12) there is fixed K be it is convex, stable point is globe optimum.

Theorem 3 ensure that the solution that method proposed by the present invention obtains is distributed up to capacity, i.e., discrete point of optimal input Cloth.In embodiment, numerical simulation is verified to notional result of the invention.

Embodiment

The performance and information source entropy maximization approach for the inaccurate gradient descent method that numerical result is used to illustrate to be proposed are led to It crosses maximization information source entropy and carrys out approximated channel capacity, be set to pedestal method.In addition, exhaustive search method is also compared, with Points increase computation complexity, defined parameters its

Fig. 1 a, Fig. 1 b and Fig. 1 c are set forth under the conditions of φ=2, φ=3 and φ=4, under the conditions of different SNR, Relationship between points and achievable rate.In fig 1 a, it can be observed that the achievable rate that inaccurate gradient algorithm obtains is higher than The method of maximum information source entropy.In addition, with the increase of K, the achievable rate of inaccurate gradient method is also increasing, however, maximum Information source entropy method is to start to be incremented by then to successively decrease.This is because the objective function H (X) of maximum information source entropy, instead of mutual information I (X；Y).With being incremented by for SNR, the achievable rate of inaccurate GRADIENT PROJECTION METHODS and maximum information source entropy method increases, but the two Gap successively decreasing.Identical result compares Fig. 1 a, Fig. 1 b and Fig. 1 c in Fig. 1 b and Fig. 1 c, it can be seen that with the increase of φ, The achievable rate of two methods is all increasing, however the gap of the two is being successively decreased.

Fig. 2 is depicted under φ=4, the relationship between different points K and achievable rate.Fig. 2 shows inaccurate ladder The achievable rate of degree method is higher than maximum information source entropy.With the increase of SNR, optimal channel, which may be implemented, in a biggish K holds Amount.

It in Fig. 3, gives under φ=4, at different SNR, reaches the optimal points K of achievable rate, exhaustion Method is also compared.In Fig. 3, the achievable rate of inaccurate gradient is higher than maximum information source entropy, especially in high s/n ratio condition.This Outside, the achievable rate of inaccurate gradient is identical with the method for exhaustion, this can also prove the optimality of inaccurate gradient.

Finally, the calculating time of three kinds of methods, it is as shown in the table.Table 1 is the difference points under the conditions of φ=4, SNR=0dB Complete the CPU time of three kinds of methods.All simulations use MATLAB (R2016b), have 3.4GHz CPU and 16GB RAM. In addition, maximizing information source entropy method solves nonlinear equation using 1stOpt software.As shown in table 1.The increasing counted with K Add, the CPU time that exhaustive method expends increases sharply, and the CPU time of the method proposed is slowly increased.Note that maximizing letter The CPU time of source entropy method has almost no change.This is because nonlinear equation is calculated by 1stOpt software, this is maximum Change the step of information source entropy method key.Table 2 is preferred channels capacity parameter.

The calculating time of table 1 compares (φ=4, SNR=0dB)

2 preferred channels capacity parameter of table

Parameter	Value
		Noise power σ2	1
φ, i.e. PAR	[2,3,4]
		Peak optical powers	sqrt(NOISE_VARIANCE)*10.^(SNR/10)
Peak value	SOURCE_VARIANCE*PAR

The present invention proposes inaccurate gradient descent method to solve to mix continuous discrete optimization problems of device, theoretically shows to be obtained The optimal solution obtained converges on optimal discrete distribution, and FSO channel capacity may be implemented.

The present invention provides a kind of free space light intensity channels up to the calculation method of capacity, implements the technical solution Method and approach it is very much, the above is only a preferred embodiment of the present invention, it is noted that for the general of the art For logical technical staff, various improvements and modifications may be made without departing from the principle of the present invention, these improve and Retouching also should be regarded as protection scope of the present invention.The available prior art of each component part being not known in the present embodiment is subject to reality It is existing.

Claims (7)

1. a kind of FSO channel is up to the calculation method of capacity, which comprises the steps of:

Step 1, channel model is set；

Step 2, the capacity of channel model is solved；

Step 3, the optimal solution of the capacity of channel model is sought.

2. the method according to claim 1, wherein step 1 includes: to set a typical IM/DD FSO certainly By spatial light intensity channel, it includes a LED or laser diode LD as transmitter, and a single-photon detector PD, which is used as, to be connect Receive device；

The peak optical powers and average light power of input signal X are all constraints, so that 0≤X≤A, andWherein, A is the amplitude of signal,For the mean value of signal, μ is electrical power；On FSO channel, receives signal Y and is given by:

Y=X+Z (1)

Wherein Z is independent Gaussian noise, mean value zero, variance σ²。

3. according to the method described in claim 2, it is characterized in that, step 2 includes:

For step 1 set channel model, capacity be defined as all possible input provided be distributed in input it is defeated Maximum mutual information C in channel out^FSO:

Wherein I (X；It Y) is mutual information, H (Y) is the entropy of Y, and H (Y | X) is combination entropy, and P (X) indicates the distribution of X, f_Y(y) indicate that Y is general Rate density function pdf, f_YIt (y) is the function of P (X).

4. according to the method described in claim 3, it is characterized in that, step 3 includes:

Step 3-1, setting input signal X is discrete stochastic variable, has K nonnegative real number { x_k}_1≤k≤K, meet:

Pr { X=x in formula_k}=p_kIndicate X=x_kWhen corresponding probability value be p_k, x_kIt is k-th point, p_kIt is x_kCorresponding probability,For positive integer；

Step 3-2: noise Z follows Gaussian Profile, f_Y(y) pdf is converted into following form:

The capacity for solving channel model is equally written as following optimization problem:

Step 3-3, is such as given a definition:

φ (p) is objective function in formula, and Υ is constraint set, 1_KIt is the vector of K × 1, wherein all elements are equal to 1, then problem (5a) That is optimization problem in step 3-2 is equivalent to be rewritten as following problem (7):

s.t.p∈Υ (7b)

There are three key variables, i.e. K, p and x in revised problem；When K and x are fixed, which is the convex problem of variable p；

Step 3-4 uses equidistant fixed x:

K value { x is selected from [0, A] range equal intervals_k}_1≤k≤K, it may be assumed that

Set proposition 1: setting K^*WithIt is the optimal solution of problem (7), γ is indicatedMiddle any two points it Between minimum range, i.e., Any two points are indicated, for given ε₀> 0, whenThere are a sequence { x_l}_1≤l≤KMeet:

For k^*Can value set；

Step 3-5 solves the problems, such as (7) using gradient projection method.

5. according to the method described in claim 4, it is characterized in that, step 3-5 includes:

Step 3-5-1, allowsThe gradient for indicating objective function, that is, formula (7a), is given by:

Step 3-5-2, using numerical integration method respectively to φ (p) andCarry out approximate: due to 0≤X≤A, and Z is abided by Gaussian Profile is followed, [- τ₁,A+τ₁] and [- τ₂,A+τ₂] respectively indicate φ (p) andIntegrating range, wherein τ₁> 0 and τ₂ > 0, τ₁、τ₂The slight gap for indicating integral is the value of two very littles taken for approximation, allowsWithIt respectively indicates The approximation of objective function φ (p) andApproximation, it may be assumed that

Allow p₀Indicate a feasible initial point, p_nIndicate n-th of iteration feasible point, wherein n=1,2 ..., inaccurate gradientGradient projection iteration p_nAnd p_n+1It is given by:

α in formula_n∈ (0,1] be nth iteration step-length,

In formula,(16) are defined according to projection, projection operation (15b) is to find a vector p_n+1 ∈ Υ so that it withBetween distance it is minimum, projection (15b) constitutes following optimization problem:

p_n+1≥0 (17d)。

6. according to the method described in claim 5, it is characterized in that, recalling line in (15a) using straight line in step 3-5-2 It selects suitable step-length to successively decrease to reach, specifically includes:

Step 3-5-2-1, initialization: selection K >=2, λ_K-1≤ 0, c is set₂, c₃For iteration stopping parameter；

Step 3-5-2-2 selects a feasible initial point p if n=0₀∈Y；

Step 3-5-2-3, n=n+1, then calculateWith

Step 3-5-2-4, material calculation α_n；

Step 3-5-2-5 is calculated

Step 3-5-2-6, if | | p_n-p_n-1||≤c₂, then stop, thenOtherwise, step 3-5-2-3 is turned to；

Step 3-5-2-7, if | λ_K-λ_K-1|≤c₃, then stop, then exporting p^opt=p_n, K^opt=K, otherwise K=K+1, then Turn to step 3-5-2-2, wherein K^optIndicate discrete point { x_kOptimal number, λ_K-1For the initial value of objective function, p^optFor Meet the optimal probability value of condition.

7. according to the method described in claim 6, it is characterized in that, step 3-5-2-4 includes:

Step 3-5-2-4-1, selectionρ, c ∈ (0,1),For initial step length, ρ is the stride length shrinks factor, and c is a parameter, Generally take the number between 0 and 1；

Step 3-5-2-4-2 is repeated until meeting:

Wherein For the target function value of next iteration,Specifically to change The value in generation,For projection；

Step 3-5-2-4-3,← indicate assignment；

Step 3-5-2-4-4 terminates to repeat；

Step 3-5-2-4-5, whenWhen terminate.