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

Abstract

The invention discloses a method for calculating the reachable capacity of a free space optical intensity channel, which is used to realize the capacity of a free space optical (FSO) channel. Under the peak constraint and the average optical power constraint, seeking the capacity of the FSO channel can be regarded as a hybrid continuous discrete optimization problem, where the objective function is non-integrable. To overcome this difficulty, the present invention adopts the numerical integration method to approximate the objective function and its gradient. Then, it is shown that the gap between the original function and the approximation can be arbitrarily small. Based on the approximation function, the present invention proposes an imprecise gradient descent method to solve the mixed continuous discrete optimization problem, which theoretically shows that the obtained optimal solution converges to the optimal discrete distribution and can realize the FSO channel capacity. Finally, the simulation results verify the performance of the proposed method, and its achievable rate is higher than the existing methods.

Description

A method for calculating the reachable capacity of a free-space optical intensity channel

technical field

The invention relates to the field of communications, in particular to a method for calculating the reachable capacity of a free-space optical intensity channel.

Background technique

Free Space Optics (FSO), due to its wide license-exempt spectrum, low electromagnetic interference, high security and high data rates (refs: [1], [2], [3], [4]), has recently gained a lot of attention in academic communication It has aroused extensive research attention in the world and industry. Different from traditional radio frequency communication, FSO communication adopts intensity modulation and direct detection (IM/DD) scheme. To meet eye safety and practical lighting considerations, emission signal limits should be controlled by meeting peak and average optical power. In this setting, (refs: [5], [6]) it has been shown that the capacity of the FSO realizes that the distributed channel is discrete over a limited set of points, and thus the Gaussian distribution of the classical channel capacity that realizes the RF cannot be applied on the FSO channel. So far, it is unknown how to efficiently search the discrete distribution of the reachable capacity of the channel for FSO channels, and current methods only perform an exhaustive search at each signal-to-noise ratio (SNR) point.

In order to avoid inefficient calculations, upper and lower bounds for the FSO channel capacity have been derived. In refs [7], [8], [9], [10], the capacity is limited by the average optical limit and the power constraint is studied. Based on the sphere packing method, the authors derive upper and lower bounds in Ref. [7], the gap between the two bounds is about 0.5 bits per transmission. In Ref. [8], the lower bound is proposed by maximizing the source entropy over a series of discrete non-uniform distributions, and the upper bound is also demonstrated by sphere packing. Two boundaries in Ref. [8] asymptotically describe the capacity of an FSO channel at low signal-to-noise ratio. The upper bound developed in ref. [9] improves the results in ref. [7], ref. [8] by exploiting a new approximation method to obtain the intrinsic volume simplex. The authors derived upper and lower bounds in Ref. [10], and the gap between the boundaries tends to zero and the average optical power tends to infinity. Based on a new recursive method, the upper bound is proposed in Ref. [11], which further improves the sphere-filling upper bound in Ref. [10].

Furthermore, references [10], [11], [12], [13] investigate the capacity bound optical power limit at both peak and average values. For a fixed ratio of average power to peak power, the gap between the upper and lower bounds in Ref. [10] tends to zero as the SNR becomes infinite. Furthermore, by using a truncated Gaussian distribution, it was shown that the lower bound derived in Ref. [11] is close to the upper bound of spherical padding at high SNR (Ref. [10]). The achievable rates with discrete non-uniform input distributions are derived in Ref. [12] by maximizing the source, straining at high signal-to-noise ratios. Under peak optical power, average optical power and electrical power constraints, closed-form lower bounds (called ABG lower bounds) are developed in Ref. [13] using entropy weight inequality and Lagrangian function methods.

In conclusion, existing papers focus on the lower or upper bound of the FSO channel capacity, and the numerical results demonstrate the tightness of the channel bounds. However, existing work does not directly show how to achieve FSO channel capacity with discrete input distributions. Therefore, there is no theoretical method to efficiently obtain the channel capacity of an FSO communication system.

References: [1] T.Komine and M.Nakagawa, "Fundamental analysis forvisible-light communication system using led lights," IEEETrans.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 lightcommunication, networking, and sensing: a survey, potential and challenges," IEEECommun.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 with bounded 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 opticalintensity 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 onchannel 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-spaceoptical 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 opticalcommunications: 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-uniformsignaling 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 lightcommunication networks,”J .Lightwave Technol., vol. 35, no. 14, pp. 2778–2787, Jul. 2017.

SUMMARY OF THE INVENTION

The purpose of the present invention is to develop an effective method to find the optimal distribution that can reach the FSO channel capacity under the constraints of peak and average optical power. Specifically, the present invention provides a calculation of the reachable capacity of a free-space optical intensity channel method, including the following steps:

Step 1, set the channel model;

Step 2, solve the capacity of the channel model;

Step 3: Obtain the optimal solution of the capacity of the channel model.

Step 1 includes: setting a typical IM/DD (intensity modulation and directDetection) FSO (Free-space optical, free-space optical intensity) channel, which contains an LED or a laser diode LD as a transmitter, A single photon detector PD acts as a receiver;

Both the peak optical power and the average optical power of the input signal X are constrained such that 0≤X≤A, and Among them, A is the amplitude of the signal, is the mean value of the signal, μ is the electrical power; on the FSO channel, the received signal Y is given by:

Y=X+Z (1)

where Z is an independent Gaussian noise with zero mean and variance σ ² .

Since the information is embedded in the intensity of the optical signal, the transmitted signal X should be real non-negative. Furthermore, due to eye safety standards and practical lighting requirements, both the peak optical power and the average optical power of the signal X should be constrained such that 0≤X≤A, and

Step 2 includes:

For the channel model set in step 1, its capacity is defined as the maximum mutual information C ^FSO distributed over the input and output channels given all possible inputs:

where I(X; Y) is the mutual information, H(Y) is the entropy of Y, H(Y|X) is the joint entropy, P(X) is the distribution of X, and f _Y (y) is the probability density function of Y ( pdf, probability density function), obviously, f _Y (y) is a function of P(X).

Step 3 includes:

Step 3-1, set the input signal X to be a discrete random variable with K non-negative real numbers {x _k } _1≤k≤K , satisfying:

In the formula, Pr{X=x _k }=p _k indicates that the corresponding probability value is p _k when X=x _k , x _k is the kth point, and p _k is the corresponding probability of x _k , is a positive integer;

Step 3-2: The noise Z follows a Gaussian distribution, and the pdf of f _Y (y) is transformed into the following form:

Solving the capacity of the channel model is equivalently written as the following optimization problem:

Due to the variables K, {p _k } _1≤k≤K and {x _k } _1≤k≤K , the above optimization problem is a mixed discrete non-convex problem. Furthermore, the objective function (5a) is not integrable, and there is no analytical expression for the objective function (5a). Therefore, the above optimization problem is difficult to solve. The present invention will develop an efficient method to search for the optimal input distribution.

Step 3-3, define as follows:

where φ(p) is the objective function, Υ is the constraint set, 1 _K is the K×1 vector, where all elements are equal to 1, then the problem (5a), the optimization problem in step 3-2, is equivalently rewritten as the following problem (7):

s.t.p∈Υ (7b)

There are three key variables in the rephrased problem, namely K, p and x; when K and x are fixed, the problem is a convex problem of variable p; this makes it necessary to reduce the design variables by fixing the variable x.

Steps 3-4, use equal spacing to fix x:

Select K values {x _k } _1≤k≤K from the range [0,A] at medium intervals, namely:

Set Proposition 1: Set K ^* and is the optimal solution of problem (7), γ represents The minimum distance between any two points in represents any two points, for a given ε ₀ > 0, when There exists a sequence {x _l } _1≤l≤K satisfying:

is the set of values that k ^* can take;

Proof: Without loss of generality, suppose is an ascending order. where k=2,...,K ^* , let γ denote the smallest _dk , namely

For a given precision ε ₀ > 0, construct a sequence {x _k } _1≤k≤K , one of which is large enough (usually 20) satisfy the following formula

|x _k -x _k+1 |≤ε ₀ (10)

where x _k is defined in (8).

Therefore, for any point in There exists a point x _l that satisfies:

Under Proposition 1, the optimal solution is direct. must contain some {x _k } _1≤k≤K at a given arbitrary precision. Since K is greater than K*, there may be many redundant points in {x _k } _1≤k≤K . However, this redundancy does not affect the objective function to achieve the maximum value, because the influence of redundant points can be reduced by optimizing the pdf of p. Therefore, for a given K, a subset of {x _k } _1≤k≤K can be determined to be approximately optimal As Proposition 1 shows.

Steps 3-5, use gradient projection to solve problem (7).

Steps 3-5 include:

Step 3-5-1, let represents the gradient of the objective function, Eq. (7a), given by:

Step 3-5-2, however, whether it is the objective function φ(p) or the gradient Neither has an analytic expression. In order to solve this problem, the numerical integration method is used for φ(p) and Approximation: Since 0≤X≤A, and Z follows a Gaussian distribution, [-τ ₁ ,A+τ ₁ ] and [-τ ₂ ,A+τ ₂ ] denote φ(p) and The integral interval of , τ ₁ and τ ₂ represent the small interval of the integral, which are two very small values to be approximated (numbers between 0 and 1 can be taken, such as 0.4 and 0.5), where τ ₁ >0 and τ ₂ > 0, let and respectively represent the approximate value of the objective function φ(p) and approximation, namely:

Let p ₀ denote a feasible initial point and p _n the nth iteration feasible point, where n = 1, 2, ..., the inexact gradient The gradient projection iterations _pn and _pn+1 are given by:

where α _n ∈(0,1] is the step size of the nth iteration,

In the formula, According to the projection definition (16), the projection operation (15b) is to find a vector make it with The distance between them is the smallest, and the projection (15b) constitutes the following optimization problem:

p _n+1 ≥ 0 (17d)

Problem (17) is a convex quadratic programming problem and can be solved efficiently by using off-the-shelf convex optimization solvers such as CVX.

Symbols: Lowercase and uppercase letters in bold font represent vectors and matrices, respectively. The transpose and Frobenius norm, rank, trace and Kronecker product of a matrix are denoted as ( ) ^T and || |||, respectively, and Round the elements of X to the nearest whole number.

Beneficial effects: The present invention solves the problem of free space optical communication channel capacity, and solves the original problem. The previous methods are all approximate and cannot achieve better results (compared with the exhaustive method), and the exhaustive method is very expensive. In order to give an accurate solution, the simulation result of the present invention can achieve the same effect as exhaustive, and an imprecise gradient descent method is proposed to solve the mixed continuous discrete optimization problem. Theoretically, it shows that the The obtained optimal solution converges to the optimal discrete distribution, which can realize the FSO channel capacity. .

Description of drawings

The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments, and the advantages of the above or other aspects of the present invention will become clearer.

Figure 1a shows the relationship between the number of points and the achievable rate under different SNR conditions under the condition of φ=2.

Figure 1b shows the relationship between the number of points and the achievable rate under different SNR conditions under the condition of φ=3.

Figure 1c shows the relationship between the number of points and the achievable rate under different SNR conditions under the condition of φ=4.

Fig. 2 is a graph showing the variation of discrete maximum entropy and the optimal channel capacity proposed by the present invention with SNR under the condition of φ=4 and different number of points K.

FIG. 3 is a graph showing the channel capacity obtained by the exhaustive method and the optimal channel capacity proposed by the present invention with SNR at φ=4, discrete maximum entropy.

Detailed ways

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

The present invention provides a method for calculating the reachable capacity of a free-space optical intensity channel, comprising the following steps:

Step 1, set the channel model;

Step 2, solve the capacity of the channel model;

Step 3: Obtain the optimal solution of the capacity of the channel model.

Step 1 includes: setting a typical IM/DD (intensity modulation and directDetection) FSO (Free-space optical, free-space optical intensity) channel, which contains an LED or a laser diode LD as a transmitter, A single photon detector PD acts as a receiver;

Both the peak optical power and the average optical power of the input signal X are constrained such that 0≤X≤A, and Among them, A is the amplitude of the signal, is the mean value of the signal, μ is the electrical power; on the FSO channel, the received signal Y is given by:

Y=X+Z (1)

where Z is an independent Gaussian noise with zero mean and variance σ ² .

Since the information is embedded in the intensity of the optical signal, the transmitted signal X should be real non-negative. Furthermore, due to eye safety standards and practical lighting requirements, both the peak optical power and the average optical power of the signal X should be constrained such that 0≤X≤A, and

Step 2 includes:

For the channel model set in step 1, its capacity is defined as the maximum mutual information C ^FSO distributed over the input and output channels given all possible inputs:

where I(X; Y) is the mutual information, H(Y) is the entropy of Y, H(Y|X) is the joint entropy, P(X) is the distribution of X, and f _Y (y) is the probability density function of Y ( pdf, probability density function), obviously, f _Y (y) is a function of P(X).

Step 3 includes:

Step 3-1, set the input signal X to be a discrete random variable with K non-negative real numbers {x _k } _1≤k≤K , satisfying:

In the formula, Pr{X=x _k }=p _k indicates that the corresponding probability value is p _k when X=x _k , x _k is the kth point, and p _k is the corresponding probability of x _k , is a positive integer;

Step 3-2: The noise Z follows a Gaussian distribution, and the pdf of f _Y (y) is transformed into the following form:

Solving the capacity of the channel model is equivalently written as the following optimization problem:

Due to the variables K, {p _k } _1≤k≤K and {x _k } _1≤k≤K , the above optimization problem is a mixed discrete non-convex problem. Furthermore, the objective function (5a) is not integrable, and there is no analytical expression for the objective function (5a). Therefore, the above optimization problem is difficult to solve. The present invention will develop an efficient method to search for the optimal input distribution.

Step 3-3, define as follows:

where φ(p) is the objective function, Υ is the constraint set, where 1 _K is a K×1 vector, in which all elements are equal to 1, then problem (5a), the optimization problem in step 3-2, is equivalently rewritten as The following question (7):

s.t.p∈Υ (7b)

There are three key variables in the rephrased problem, namely K, p and x; when K and x are fixed, the problem is a convex problem of variable p; this makes it necessary to reduce the design variables by fixing the variable x.

Steps 3-4, use equal spacing to fix x:

Select K values {x _k } _1≤k≤K from the range [0,A] at medium intervals, namely:

Set Proposition 1: Set K ^* and is the optimal solution of problem (7), γ represents The minimum distance between any two points in represents any two points, for a given ε ₀ > 0, when There exists a sequence {x _l } _1≤l≤K satisfying:

is the set of values that k ^* can take;

Proof: Without loss of generality, suppose is an ascending order. where k=2,...,K ^* , let γ denote the smallest _dk , namely

For a given precision ε ₀ > 0, construct a sequence {x _k } _1≤k≤K , one of which is large enough (usually 20) satisfy the following formula

|x _k -x _k+1 |≤ε ₀ (10)

where x _k is defined in (8).

Therefore, for any point in There exists a point x _l that satisfies:

Under Proposition 1, the optimal solution is direct. must contain some {x _k } _1≤k≤K at a given arbitrary precision. Since K is greater than K ^* , there may be many redundant points in {x _k } _1≤k≤K . However, this redundancy does not affect the objective function to achieve the maximum value, because the influence of redundant points can be reduced by optimizing the pdf of p. Therefore, for a given K, a subset of {x _k } _1≤k≤K can be determined to be approximately optimal As Proposition 1 shows.

Steps 3-5, use gradient projection to solve problem (7).

Steps 3-5 include:

Step 3-5-1, let represents the gradient of the objective function, Eq. (7a), given by:

Step 3-5-2, however, whether it is the objective function φ(p) or the gradient Neither has an analytic expression. In order to solve this problem, the numerical integration method is used for φ(p) and Approximation: Since 0≤X≤A, and Z follows a Gaussian distribution, [-τ ₁ ,A+τ ₁ ] and [-τ ₂ ,A+τ ₂ ] denote φ(p) and The integral interval of , τ ₁ and τ ₂ represent the small interval of the integral, which are two very small values taken for approximation, where τ ₁ >0 and τ ₂ >0, let and respectively represent the approximate value of the objective function φ(p) and An approximation of , namely:

Let p ₀ denote a feasible initial point and p _n the nth iteration feasible point, where n = 1, 2, ..., the inexact gradient The gradient projection iterations _pn and _pn+1 are given by:

where α _n ∈(0,1] is the step size of the nth iteration,

In the formula, According to the projection definition (16), the projection operation (15b) is to find a vector pn+1∈Υ such that it is the same as The distance between them is the smallest, and the projection (15b) constitutes the following optimization problem:

p _n+1 ≥ 0 (17d)

Problem (17) is a convex quadratic programming problem and can be solved efficiently by using off-the-shelf convex optimization solvers such as CVX.

In step 3-5-2, a straight backtracking line is used to select an appropriate step size in (15a) to achieve decrement, which specifically includes:

Step 3-5-2-1, initialization: select K≥2, λ _K-1 ≤0, set c ₂ , c ₃ as iteration stop parameters;

Step 3-5-2-2, set n=0, select a feasible initial point p ₀ ∈ Y;

Step 3-5-2-3, n=n+1, then calculate and

Step 3-5-2-4, calculation step size α _n ;

Step 3-5-2-5, calculation

Step 3-5-2-6, stop if ||p _n -p _n-1 ||≤c ₂ , then Otherwise, go to step 3-5-2-3;

Step 3-5-2-7, if |λ _K -λ _K-1 |≤c ₃ , stop, then output p ^opt = _pn , K ^opt =K, otherwise K=K+1, then turn to step 3 -5-2-2, where K ^opt represents the optimal number of discrete points {x _k }, λ _K-1 is the initial value of the objective function, and p ^opt is the optimal probability value that satisfies the conditions.

Since the optimal discrete distribution {K ^* ,x ^* ,p ^* } is the only discrete random variable in a finite number of values, the optimal number K ^opt can be obtained by a simple one-dimensional search, K=K+1 for the next time Iterate, where K is initialized not less than 2. In summary, the proposed imprecise gradient descent method is listed in the above methods.

Steps 3-5-2-4 include:

Step 3-5-2-4-1, select is the initial step size, ρ is the step size reduction factor, and c is a parameter, generally a number between 0 and 1;

Steps 3-5-2-4-2, repeat until:

in is the objective function value for the next iteration, is the value of this iteration, for projection;

Step 3-5-2-4-3, ← means assignment, that is, multiply the step size of this iteration by the step size reduction factor and assign it to the next step size;

Step 3-5-2-4-4, end the repetition;

Steps 3-5-2-4-5, when terminated;

Optimality of Inexact Gradient Descent: It is worth pointing out that if the value of τ ₁ is large enough, [-τ ₁ ,A+τ ₁ ] approaches [-∞,∞] by an arbitrarily small margin. Furthermore, the following theorem shows that the approximation φ(p) can be approximated with arbitrarily small errors.

Theorem 1: For a given precision ε ₁ > 0, there is a parameter large enough to satisfy τ ₁ >σ

Then there is:

where φ(p) and are given in (6d) and (14a), respectively, in addition,

The proof is as follows:

ε _total represents and φ(p), given by

in,

Furthermore, the upper bound for the absolute value of ε _total is given by:

The upper bound is:

Since (yx _k ) ² ≥ y ² , inequality (25b) holds for y≤-τ ₁ ≤0.

In the following, it will be shown that as τ ₁ increases, the right term of (25c) becomes zero. Specifically, the integral of this formula is given by:

Furthermore, this formula The integral of is given by:

Substituting (26b) and (27b) into (25c), we get:

in,

Furthermore, since Have:

the formula The upper limit is:

Since (yx _k ) ² ≥(yA) ² , inequality (30b) holds for y≥A+τ ₁ .

In the following, it will be shown that as τ ₁ increases, the right term of (25c) becomes zero.

integral of the formula is given by:

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

By substituting (31b) and (32b) into (30c), we get:

Likewise, because , there are:

By combining (22a), (29) and (34), we have

When τ ₁ ≥σ, and Both are non-negative monotonically decreasing functions,

Then

Because erfc(x) is a non-negative monotonically decreasing function and Have

Therefore, for any given precision ε ₁ > 0, there exists a large parameter τ ₁ >σ such that:

Then there is:

Likewise, for a sufficiently large τ ₂ , can be approached by arbitrarily small gaps

Theorem 2: For a given precision ε ₂ > 0, there exists τ ₂ ≥σ satisfying:

Then, there is:

and are given in (13) and (14b), respectively.

The proof is as follows:

Assume express and The vector difference between , given by:

also, The elements can be written as:

in,

also, The upper limit of the absolute value of an element:

So and The upper bound of the norm difference is:

In the following, it will be shown that as τ ₂ increases, the right-hand term of (42) will become zero. Specifically, the term is given by:

Then, the integral of the term is given by:

Furthermore, the integral of this formula is given by:

Finally, the integral of this formula is given by:

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

Moreover, the term The upper limit of:

Then, the integral of the term is given by:

Furthermore, the integral of this formula is given by:

Finally, the integral of this formula is given by:

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

By combining (47) and (52), we have:

Similarly, when τ ₂ ≥σ, and Both are non-negative monotonically decreasing functions, then

Since erfc(x) is a non-negative monotonically decreasing function and Have

Therefore, for any given precision ε ₂ > 0, there exists a large parameter τ ₂ >σ such that:

Then there is:

Therefore, according to the inexact gradient descent method, the optimal discrete input distribution can be computed efficiently. Moreover, the sequence {p _n } of probability distributions obtained by Algorithm 2 converges to the optimal distribution, which is proved by the following theorem:

Theorem 3: (Convergence Analysis) For a given K, {p _n } converges to the optimal solution of problem (12), correspondingly, {φ(p _n )} converges to the optimal solution of problem (12) correspondingly. figure of merit.

The proof is as follows:

Suppose {p _n } converges to a non-stationary point Need to prove this:

According to the proof of Proposition 2.3.1 in reference [14] D.P. Bertsekas and D.P. Bertsekas., Nonlinear Programming., Athena Scientific, 1999, condition (56a) holds, and the following inequality holds,

in,

Then, Have:

Since (57) holds, inequality (58c) also holds; inequality (58d) holds because when , the equation holds, where |κ|=|| _en ||.

According to Theorem 2, the value of ||e _n || can be arbitrarily small. Therefore, there exists a parameter κ that satisfies Furthermore, since is non-stationary, and condition (56b) holds.

According to Proposition 2.3.1 in Reference [14] DPBertsekas and DPBertsekas., Nonlinear Programming., Athena Scientific, 1999, every limit point of {p _n } is stable. Furthermore, since problem (12) is convex with fixed K, the stable point is the global optimum.

Theorem 3 guarantees that the solution obtained by the method proposed in the present invention is the reachable capacity distribution, that is, the optimal input discrete distribution. In the examples, numerical simulations were performed to verify the theoretical results of the present invention.

Example

Numerical results are used to illustrate the performance of the proposed imprecise gradient descent method and source entropy maximization method, which approximates the channel capacity by maximizing the source entropy, which is set as the benchmark method. In addition, the exhaustive search method is also compared. As the number of points increases, the computational complexity is

Figure 1a, Figure 1b and Figure 1c show the relationship between the number of points and the achievable rate under different SNR conditions under the conditions of φ=2, φ=3 and φ=4, respectively. In Fig. 1a, it can be observed that the imprecise gradient algorithm obtains a higher achievable rate than the method with maximum source entropy. Furthermore, as K increases, the achievable rate of the imprecise gradient method also increases, however, the maximum source entropy method starts to increase and then decreases. This is because the objective function H(X) of maximum source entropy replaces the mutual information I(X; Y). As the SNR increases, the achievable rates of the imprecise gradient projection method and the maximum source entropy method increase, but the gap between them decreases. The same results are shown in Fig. 1b and Fig. 1c. Comparing Fig. 1a, Fig. 1b and Fig. 1c, it can be seen that with the increase of φ, the attainable rate of both methods increases, but the gap between them decreases.

Figure 2 plots the relationship between the number of points K and the achievable rate at φ=4. Figure 2 shows that the achievable rate of the imprecise gradient method is higher than that of the maximum source entropy. As the SNR increases, a larger K can achieve the optimal channel capacity.

In Fig. 3, under φ=4, under different SNRs, the optimal number of points K to reach the achievable rate is given, and the exhaustive method is also compared. In Figure 3, the achievable rate of the imprecise gradient is higher than that of the maximum source entropy, especially at high signal-to-noise ratio conditions. In addition, the reachability rate of the imprecise gradient is the same as the exhaustive method, which can also prove the optimality of the imprecise gradient.

Finally, the computation times of the three methods are shown in the table. Table 1 shows the CPU time of the three methods with different points under the condition of φ=4 and SNR=0dB. All simulations use MATLAB (R2016b) with 3.4GHz CPU and 16GB RAM. In addition, the maximizing source entropy method utilizes 1stOpt software to solve nonlinear equations. As shown in Table 1. As the number of K points increases, the CPU time consumed by the exhaustive method increases rapidly, while the CPU time of the proposed method increases slowly. Note that the CPU time of the maximizing source entropy method is almost unchanged. This is because the nonlinear equation is calculated by the 1stOpt software, which is a key step in the method of maximizing the source entropy. Table 2 is the optimal channel capacity parameters.

Table 1 Comparison of calculation time (φ=4, SNR=0dB)

Table 2 Optimal channel capacity parameters

parameter value Noise powerσ² 1 φ, or PAR [2,3,4] peak optical power sqrt(NOISE_VARIANCE)*10.^(SNR/10) peak SOURCE_VARIANCE*PAR

The invention proposes an imprecise gradient descent method to solve the mixed continuous discrete optimization problem, and theoretically shows that the obtained optimal solution converges to the optimal discrete distribution and can realize the FSO channel capacity. .

The present invention provides a method for calculating the reachable capacity of a free-space optical intensity channel. There are many specific methods and approaches for realizing the technical solution. The above are only the preferred embodiments of the present invention. For those skilled in the art, without departing from the principle of the present invention, several improvements and modifications can also be made, and these improvements and modifications should also be regarded as the protection scope of the present invention. All components not specified in this embodiment can be implemented by existing technologies.

Claims (7)

1. a calculation method of FSO channel reachable capacity, is characterized in that, comprises the steps: Step 1, set the channel model; Step 2, solve the capacity of the channel model; Step 3: Obtain the optimal solution of the capacity of the channel model. 2. The method according to claim 1, wherein step 1 comprises: setting a typical IM/DD FSO free space light intensity channel, which comprises an LED or a laser diode LD as a transmitter, a single photon detection The device PD acts as a receiver; Both the peak optical power and the average optical power of the input signal X are constrained such that 0≤X≤A, and Among them, A is the amplitude of the signal, is the mean value of the signal, μ is the electrical power; on the FSO channel, the received signal Y is given by: Y=X+Z (1) where Z is an independent Gaussian noise with zero mean and variance σ ² . 3. The method according to claim 2, wherein step 2 comprises: For the channel model set in step 1, its capacity is defined as the maximum mutual information C ^FSO distributed over the input and output channels given all possible inputs: where I(X; Y) is the mutual information, H(Y) is the entropy of Y, H(Y|X) is the joint entropy, P(X) is the distribution of X, and f _Y (y) is the probability density function pdf of Y , f _Y (y) is a function of P(X). 4. The method according to claim 3, wherein step 3 comprises: Step 3-1, set the input signal X to be a discrete random variable with K non-negative real numbers {x _k } _1≤k≤K , satisfying: In the formula, Pr{X=x _k }=p _k indicates that the corresponding probability value is p _k when X=x _k , x _k is the kth point, and p _k is the corresponding probability of x _k , is a positive integer; Step 3-2: The noise Z follows a Gaussian distribution, and the pdf of f _Y (y) is transformed into the following form: Solving the capacity of the channel model is equivalently written as the following optimization problem: Step 3-3, define as follows: where φ(p) is the objective function, Υ is the constraint set, 1 _K is the K×1 vector, where all elements are equal to 1, then the problem (5a), the optimization problem in step 3-2, is equivalently rewritten as the following problem (7): s.t.p∈Υ (7b) There are three key variables in the rewritten problem, namely K, p and x; when K and x are fixed, the problem is a convex problem of variable p; Steps 3-4, use equal spacing to fix x: Select K values {x _k } _1≤k≤K from the range [0,A] at medium intervals, namely: Set Proposition 1: Set K ^* and is the optimal solution of problem (7), γ represents The minimum distance between any two points in represents any two points, for a given ε ₀ > 0, when There exists a sequence {x _l } _1≤l≤K satisfying: is the set of values that k ^* can take; Steps 3-5, use gradient projection to solve problem (7). 5. The method according to claim 4, wherein steps 3-5 comprise: Step 3-5-1, let represents the gradient of the objective function, Eq. (7a), given by: Step 3-5-2, using numerical integration method to calculate φ(p) and Approximation: Since 0≤X≤A, and Z follows a Gaussian distribution, [-τ ₁ ,A+τ ₁ ] and [-τ ₂ ,A+τ ₂ ] denote φ(p) and , where τ ₁ >0 and τ ₂ >0, τ ₁ and τ ₂ represent the small interval of the integral, which are two very small values to be approximated, let and respectively represent the approximate value of the objective function φ(p) and An approximation of , namely: Let p ₀ denote a feasible initial point and p _n the nth iteration feasible point, where n = 1, 2, ..., the inexact gradient The gradient projection iterations _pn and _pn+1 are given by: where α _n ∈(0,1] is the step size of the nth iteration, In the formula, According to the projection definition (16), the projection operation (15b) is to find a vector p _n+1 ∈ Υ such that it is the same as The distance between them is the smallest, and the projection (15b) constitutes the following optimization problem: p _n+1 ≥ 0 (17d). 6. The method according to claim 5, characterized in that, in step 3-5-2, a straight backtracking line is used to select an appropriate step size in (15a) to achieve decreasing, specifically comprising: Step 3-5-2-1, initialization: select K≥2, λ _K-1 ≤0, set c ₂ , c ₃ as iteration stop parameters; Step 3-5-2-2, set n=0, select a feasible initial point p ₀ ∈ Y; Step 3-5-2-3, n=n+1, then calculate and Step 3-5-2-4, calculation step size α _n ; Step 3-5-2-5, calculation Step 3-5-2-6, stop if ||p _n -p _n-1 ||≤c ₂ , then Otherwise, go to step 3-5-2-3; Step 3-5-2-7, if |λ _K -λ _K-1 |≤c ₃ , stop, then output p ^opt = _pn , K ^opt =K, otherwise K=K+1, then turn to step 3 -5-2-2, where K ^opt represents the optimal number of discrete points {x _k }, λ _K-1 is the initial value of the objective function, and p ^opt is the optimal probability value that satisfies the conditions. 7. The method according to claim 6, wherein step 3-5-2-4 comprises: Step 3-5-2-4-1, select ρ,c∈(0,1), is the initial step size, ρ is the step size reduction factor, and c is a parameter, generally a number between 0 and 1; Steps 3-5-2-4-2, repeat until: in is the objective function value for the next iteration, is the value of this iteration, for projection; Step 3-5-2-4-3, ← means assignment; Step 3-5-2-4-4, end the repetition; Steps 3-5-2-4-5, when terminated when.