CN109150304A - A kind of calculation method of free space light intensity channel up to capacity - Google Patents
A kind of calculation method of free space light intensity channel up to capacity Download PDFInfo
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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
Technical Field
The invention relates to the field of communication, in particular to a method for calculating the reachable capacity of a free space light intensity channel.
Background
Free Space Optics (FSO) has recently attracted considerable research attention in academia and industry due to its wide unlicensed spectrum, low electromagnetic interference, high security and high data rates (references: [1], [2], [3], [4 ]). Unlike conventional radio frequency communications, FSO communications employ an intensity modulation and direct detection (IM/DD) scheme. To meet eye safety and practical lighting considerations, the emitted signal limit should be controlled by meeting peak and average optical power. Under this setup, (references: [5], [6]) have demonstrated that the capacity-achieving distribution channel of FSO is discrete over a limited set of points, and thus the gaussian distribution of classical channel capacity that achieves RF cannot be applied to FSO channels. So far it is not known how to efficiently search for the achievable capacity dispersion of the channel for the FSO channel, but the current approach is only an exhaustive search at each signal-to-noise ratio (SNR) point.
To circumvent inefficient computations, upper and lower bounds for FSO channel capacity have been derived. In references [7], [8], [9], [10], the capacity is subject to the power constraint by the average optical limit. Based on the sphere packing method, the authors derive an upper and lower bound in reference [7], with a gap between the two bounds of approximately 0.5 bits per transmission. In reference [8], the lower bound on a series of discrete non-uniform distributions is proposed by maximizing the source entropy, and the upper bound is also demonstrated by sphere packing. The two boundaries in reference [8] asymptotically describe the capacity of the FSO channel at low signal-to-noise ratios. The upper bound developed in reference [9] improves the results in references [7], [8] by using a new approximation method to obtain the intrinsic volume simplex. The authors have derived upper and lower limits in reference [10] and the gap between the boundaries tends to zero average optical power towards infinity. Based on a new recursive approach, an upper bound is proposed in reference [11], which further improves the sphere filling upper bound in reference [10 ].
Further, references [10], [11], [12], [13] investigated the capacity limit optical power limits at peak and mean values. For a fixed ratio of average power to peak power, the gap goes to zero between the upper and lower limits in reference [10] when the SNR becomes infinite. Further, by using a truncated Gaussian distribution, it is shown that the lower bound derived in reference [11] approaches the spherical filling upper bound (reference [10]) at high signal-to-noise ratio. The achievable rate with discrete non-uniform input distribution is derived in reference [12] by maximizing the source, tight at high signal-to-noise ratios. The use of the entropy weight inequality and the lagrangian function method was developed in reference [13] under the constraints of peak optical power, average optical power and electrical power, a closed form lower bound (called the ABG lower bound).
In summary, the existing paper focuses on the lower or upper limit of the FSO channel capacity, and numerical results demonstrate the compactness of the channel boundaries. However, existing work does not directly give how to achieve FSO channel capacity with discrete input distribution. Therefore, there is no theoretical method to efficiently obtain the channel capacity of the FSO communication system.
Reference documents: [1] T.Komine and M.Nakagawa, "Fundamental analysis for visible-light communication systems using led lights," IEEETranss. Consum. Electron, vol.50, No.1, pp.100-107, Feb.2004.
[2]H.Elgala,R.Mesleh,and H.Haasn,“Indoor optical wirelesscommunication: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-andvarianceconstrained scalar gaussian channels,”Inf.Contr.,vol.18,no.3,pp.203–219,Feb.1971.
[6]S.H.T.Chan and F.Kschischang,“Capacity-achieving probabilitymeasure for conditionally gaussian channels withbounded inputs,”IEEETrans.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 capacityfor 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-packingperspective,”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 withclosed-form for siso channel and broadcast channel in visible lightcommunication networks,”J.Lightwave Technol.,vol.35,no.14,pp.2778–2787,Jul.2017.
Disclosure of Invention
The invention aims to develop an effective method for finding the optimal distribution capable of achieving the FSO channel capacity under the constraint of peak value and average optical power, and particularly provides a method for calculating the reachable capacity of a free space light intensity channel, which comprises the following steps:
step 1, setting a channel model;
step 2, solving the capacity of the channel model;
and 3, solving the optimal solution of the capacity of the channel model.
The step 1 comprises the following steps: setting a typical IM/DD (intensity modulation and direct detection) FSO (Free-space optical) channel which comprises an LED or a laser diode LD as a transmitter and a single-photon detector PD as a receiver;
the peak optical power and the average optical power of the input signal X are both constrained such that X is 0 ≦ X ≦ A, andwherein A is the amplitude of the signal,is the mean of the signals, 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 a mean of zero and a variance of σ2。
Since the information is embedded in the intensity of the optical signal, the transmitted signal X should be real, non-negative. In addition, due to eye safety standards and practical lighting requirements, both the peak optical power and the average optical power of signal X should be constrained such that X is 0 ≦ X ≦ A, and
the step 2 comprises the following steps:
for the channel model set in step 1, its capacity is defined as givenMaximum mutual information C with all possible inputs distributed in input-output channelsFSO:
Wherein I (X; Y) is mutual information, H (Y) is entropy of Y, H (Y | X) is joint entropy, P (X) represents distribution of X, f (Y | X) represents entropy of Y, f (Y | X) represents distribution of X, f (Y | X) represents entropy of Y, fY(Y) denotes the Y probability density function (pdf), obviously fY(y) is a function of P (X).
The step 3 comprises the following steps:
step 3-1, setting the input signal X as a discrete random variable with K nonnegative real numbers { Xk}1≤k≤KAnd satisfies the following conditions:
wherein Pr { X ═ Xk}=pkDenotes X ═ XkProbability value of time correspondence is pk,xkIs the k point, pkIs xkThe corresponding probability of the occurrence of the event,is a positive integer;
step 3-2: the noise Z follows a Gaussian distribution, fYThe pdf of (y) is converted to the following form:
solving the capacity of the channel model is equivalently written as the following optimization problem:
due to the variables K, { pk}1≤k≤KAnd { xk}1≤k≤KThe optimization problem is a hybrid discrete non-convex problem. Further, the objective function (5a) is non-integrable, and there is no analytical expression for the objective function (5 a). 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, defined as follows:
wherein phi (p) is the objective function and gamma is the constraint set, 1KIs a K × 1 vector, where all elements are equal to 1, then the problem (5a), i.e., 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 for the variable p; this makes it necessary to reduce the design variables by a fixed variable x.
Step 3-4, fixing x by using equal spacing:
from [0, A]Selecting K values { x) at equal intervals in the rangek}1≤k≤KNamely:
setting proposition 1: set K*Andis the optimal solution to the problem (7), gamma denotesBetween any two points in the middleA minimum distance of, i.e. Representing two arbitrary points, for a given epsilon0Is greater than 0, whenThere is a sequence xl}1≤l≤KSatisfies the following conditions:
is k*A set of values that can be taken;
and (3) proving that: without loss of generality, assumeIs an ascending order.Wherein K is 2*Let gamma denote the smallest dkI.e. by
For a given accuracy epsilon0> 0, construct a sequence { xk}1≤k≤KOne of which is large enough (typically 20)Satisfies the following formula
|xk-xk+1|≤ε0(10)
In the formula xkIs defined in (8).
Thus, for any pointWhereinPoint of presence xlSatisfies the following conditions:
under proposition 1, the optimal solutionIs straightforward. Must contain some x given arbitrary precisionk}1≤k≤K. Since K is greater than K, so in { x }k}1≤k≤KThere may be many redundant points in. However, this redundancy does not affect the objective function to achieve the maximum value, since the effect of the redundancy point can be reduced by optimizing the pdf of p. Thus, for a given K, one can be determined to be in { x }k}1≤k≤KIs approximately optimalAs shown in proposition 1.
And 3-5, solving the problem by adopting a gradient projection method (7).
The steps 3-5 comprise:
step 3-5-1, letThe gradient, which represents the objective function, equation (7a), is given by:
step 3-5-2, however, regardless of the objective function φ (p) or the gradientNone of which parse the expression. To solve this problem, a numerical integration method is used to separately pair phi (p) and phi (p)And (3) carrying out approximation: since 0. ltoreq. X. ltoreq.A and Z follows a Gaussian distribution, [ - τ1,A+τ1]And [ - τ [ - ]2,A+τ2]Respectively denotes phi (p) andintegration interval of τ1、τ2The small interval representing the integral is to approximate two very small values (which may take numbers between 0 and 1, e.g., 0.4, 0.5), where τ1> 0 and τ2> 0, letAndrespectively representing the approximate values of the objective function phi (p)To approximate values of (a):
let p be0Representing a feasible initiation point, pnDenote the nth iteration feasible point, where n is 1,2Gradient projection iteration pnAnd pn+1Given by:
formula (III) αn∈(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 vectorSo that it is connected withThe distance between them is minimal, and the projection (15b) constitutes the following optimization problem:
pn+1≥0 (17d)
the problem (17) is a convex quadratic programming problem and can be solved efficiently by using an off-the-shelf convex optimization solver, such as CVX.
Symbol: the lower case and upper case letters of bold type represent vectors and matrices, respectively. The transposed and Frobenius norm, rank, trace of the matrix and Kronecker product are expressed as (.)TAnd | | · |,and the elements of X are rounded to the nearest integer.
Has the advantages that: the invention solves the problem of free space optical communication channel capacity, solves the original problem, the previous methods are approximate, and can not achieve better effect (compared with exhaustion), the exhaustion method is time-consuming and can give accurate solution only with accurate precision. .
Drawings
The foregoing and other advantages of the invention will become more apparent from the following detailed description of the invention when taken in conjunction with the accompanying drawings.
Fig. 1a shows the relationship between the number of points and the achievable rate under different SNR conditions under the condition of phi 2.
Fig. 1b shows the relationship between the number of points and the achievable rate under different SNR conditions under the condition of phi being 3.
Fig. 1c shows the relationship between the number of points and the achievable rate under different SNR conditions under the condition of phi 4.
Fig. 2 is a graph of variation of discrete maximum entropy and the optimal channel capacity proposed by the present invention with SNR at different points K, where Φ is 4.
Fig. 3 is a graph of the variation of the channel capacity from the exhaustive method and the optimum channel capacity proposed by the present invention with SNR at 4, discrete maximum entropy.
Detailed Description
The invention is further explained below with reference to the drawings and the embodiments.
The invention provides a method for calculating the reachable capacity of a free space light intensity channel, which comprises the following steps:
step 1, setting a channel model;
step 2, solving the capacity of the channel model;
and 3, solving the optimal solution of the capacity of the channel model.
The step 1 comprises the following steps: setting a typical IM/DD (intensity modulation and direct detection) FSO (Free-space optical) channel which comprises an LED or a laser diode LD as a transmitter and a single-photon detector PD as a receiver;
the peak optical power and the average optical power of the input signal X are both constrained such that X is 0 ≦ X ≦ A, andwherein A is the amplitude of the signal,is the mean of the signals, 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 a mean of zero and a variance of σ2。
Since the information is embedded in the intensity of the optical signal, the transmitted signal X should be real, non-negative. In addition, due to eye safety standards and practical lighting requirements, both the peak optical power and the average optical power of signal X should be constrained such that X is 0 ≦ X ≦ A, and
the step 2 comprises the following steps:
for the channel model set in step 1, its capacity is defined as the maximum mutual information C in the input-output channel of all possible inputs givenFSO:
Wherein I (X; Y) is mutual information, H (Y) is entropy of Y, H (Y | X) is joint entropy, P (X) represents distribution of X, f (Y | X) represents entropy of Y, f (Y | X) represents distribution of X, f (Y | X) represents entropy of Y, fY(Y) denotes the Y probability density function (pdf), obviously fY(y) is a function of P (X).
The step 3 comprises the following steps:
step 3-1, setting the input signal X as a discrete random variable with K nonnegative real numbers { Xk}1≤k≤KAnd satisfies the following conditions:
wherein Pr { X ═ Xk}=pkDenotes X ═ XkProbability value of time correspondence is pk,xkIs the k point, pkIs xkThe corresponding probability of the occurrence of the event,is a positive integer;
step 3-2: the noise Z follows a Gaussian distribution, fYThe pdf of (y) is converted to the following form:
solving the capacity of the channel model is equivalently written as the following optimization problem:
due to the variables K, { pk}1≤k≤KAnd { xk}1≤k≤KThe optimization problem is a hybrid discrete non-convex problem. Further, the objective function (5a) is non-integrable, and there is no analytical expression for the objective function (5 a). 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, defined as follows:
wherein phi (p) is the objective function and gamma is the constraint set, and 1KIs a K × 1 vector, where all elements are equal to 1, then the problem (5a), i.e., 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 for the variable p; this makes it necessary to reduce the design variables by a fixed variable x.
Step 3-4, fixing x by using equal spacing:
from [0, A]Selecting K values { x) at equal intervals in the rangek}1≤k≤KNamely:
setting proposition 1: set K*Andis the optimal solution to the problem (7), gamma denotesThe minimum distance between any two points in the space, i.e. Representing two arbitrary points, for a given epsilon0Is greater than 0, whenThere is a sequence xl}1≤l≤KSatisfies the following conditions:
is k*A set of values that can be taken;
and (3) proving that: without loss of generality, assumeIs an ascending order.Wherein K is 2*Let gamma denote the smallest dkI.e. by
For a given accuracy epsilon0> 0, construct a sequence { xk}1≤k≤KOne of which is large enough (typically 20)Satisfies the following formula
|xk-xk+1|≤ε0(10)
In the formula xkIs defined in (8).
Thus, for any pointWhereinPoint of presence xlSatisfies the following conditions:
under proposition 1, the optimal solutionIs straightforward. Must contain some x given arbitrary precisionk}1≤k≤K. Since K is greater than K*Thus, in { xk}1≤k≤KThere may be many redundant points in. However, such redundancy does not affect the maximization of the objective functionValue, since the effect of the redundancy point can be reduced by optimizing the pdf of p. Thus, for a given K, one can be determined to be in { x }k}1≤k≤KIs approximately optimalAs shown in proposition 1.
And 3-5, solving the problem by adopting a gradient projection method (7).
The steps 3-5 comprise:
step 3-5-1, letThe gradient, which represents the objective function, equation (7a), is given by:
step 3-5-2, however, regardless of the objective function φ (p) or the gradientNone of which parse the expression. To solve this problem, a numerical integration method is used to separately pair phi (p) and phi (p)And (3) carrying out approximation: since 0. ltoreq. X. ltoreq.A and Z follows a Gaussian distribution, [ - τ1,A+τ1]And [ - τ [ - ]2,A+τ2]Respectively denotes phi (p) andintegration interval of τ1、τ2The small interval representing the integral is to approximate two very small values, where τ1> 0 and τ2> 0, letAndrespectively representing the approximate values of the objective function phi (p)To approximate values of (a):
let p be0Representing a feasible initiation point, pnDenote the nth iteration feasible point, where n is 1,2Gradient projection iteration pnAnd pn+1Given by:
formula (III) αn∈(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 e y, so thatGet it andthe distance between them is minimal, and the projection (15b) constitutes the following optimization problem:
pn+1≥0 (17d)
the problem (17) is a convex quadratic programming problem and can be solved efficiently by using an off-the-shelf convex optimization solver, such as CVX.
In step 3-5-2, a straight-line backtracking line is adopted to select a proper step length in step (15a) to achieve degressive action, and the method specifically comprises the following steps:
step 3-5-2-1, initialization: k is selected to be more than or equal to 2 and lambdaK-1Not more than 0, set up c2,c3An iteration stop parameter;
step 3-5-2-2, setting n to 0, selecting a feasible initiation point p0∈Y;
Step 3-5-2-3, n ═ n +1, and then calculationAnd
step 3-5-2-4, calculate step αn;
Step 3-5-2-5, calculating
Step 3-5-2-6, if | | | pn-pn-1||≤c2Then stop, thenOtherwise, turning to the step 3-5-2-3;
step 3-5-2-7, if | λK-λK-1|≤c3Then stop and then output popt=pn,KoptK, otherwise K +1, and then to step 3-5-2-2, wherein K isoptRepresenting discrete points xkOptimum number of { lambda }, ofK-1Is an initial value of the objective function, poptIs the optimal probability value that satisfies the condition.
Due to the optimal discrete distribution K*,x*,p*Is a discrete random variable unique in a finite number of values, and the optimal number K can be obtained by a simple one-dimensional searchoptK +1 is used for the next iteration, where K is initialized to not less than 2. In summary, the proposed imprecise gradient descent method is listed in the above method.
The steps 3-5-2-4 comprise:
step 3-5-2-4-1, selection Is an initial step length, rho is a step length reduction factor, c is a parameter, and a number between 0 and 1 is generally taken;
step 3-5-2-4-2, repeat until satisfying:
wherein For the value of the objective function for the next iteration,for the value of this iteration of the process,is a projection;
3-5-2-4-3,and ← represents assignment, namely, the step size of the iteration is multiplied by the step size reduction factor to be assigned to the step size of the next time;
3-5-2-4-4, ending repetition;
step 3-5-2-4-5, whenThen the process is terminated;
optimality of the imprecise gradient descent method: it is worth noting that if τ1Is sufficiently large, [ - τ [ -T ]1,A+τ1]Close to [ - ∞, ∞ ] with an arbitrarily small gap]. In addition, the following theorem indicates approximationsPhi (p) can be approached with an arbitrarily small error.
Theorem 1: for a given accuracy ε1> 0, there is a sufficiently large parameter to satisfy τ1>σ
Then, there are:
wherein phi (p) andare given in (6d) and (14a), respectively, and further,
the following was demonstrated:
εtotalrepresentsAnd phi (p), is given by
Wherein,
furthermore,. epsilontotalThe upper limit of the absolute value is given by:
the upper bound of (A) is:
due to (y-x)k)2≥y2The inequality (25b) holds for y ≦ - τ1≤0。
In the following, it will be shown that with τ1The right two terms of (25c) become zero. Specifically, the integral of the equationGiven by:
in addition, the formulaThe integral of (d) is given by:
substituting (26b) and (27b) into (25c) yields:
wherein,
in addition, due toComprises the following steps:
the formulaThe upper limit is:
due to (y-x)k)2≥(y-A)2The inequality (30b) holds, for y ≧ A + τ1。
In the following, it will be shown that with τ1The right two terms of (25c) become zero.
Integral of the equationGiven by:
furthermore, the integral of this termGiven by:
by substituting (31b) and (32b) into (30c), we obtained:
also, becauseIn time, there are:
by combining (22a), (29) and (34) with
When tau is1When the pressure is larger than or equal to sigma,andboth of which are non-negative monotonically decreasing functions,
then the
Since erfc (x) is a non-negative monotonically decreasing function andis provided with
Thus, it is possible to provideFor any given precision ε1> 0, there is a large parameter τ1σ satisfies:
then, there are:
also, for a sufficiently large τ2,Can be approached by an arbitrarily small difference
Theorem 2: for a given accuracy ε2> 0, presence of tau2σ is more than or equal to the following condition:
then, there are:
andare given in (13) and (14b), respectively.
The following was demonstrated:
is provided withTo representAndthe vector difference between, is given by:
in addition to this, the present invention is,the elements of (c) can be written as:
wherein,
in addition to this, the present invention is,upper limit of absolute value of element:
thenAndthe upper bound of the norm difference is:
in the following, it will be shown that with τ2The right term of (42) will become zero. In particular, the termGiven by:
then, integration of the termGiven by:
furthermore, the integral of this equationGiven by:
finally, the integral of this equationGiven by:
by substituting (44b), (45b) and (46) into (43c), we obtain:
also, the termThe upper limit of (2):
then, integration of the termGiven by:
furthermore, the integral of this equationGiven by:
finally, the integral of this equationGiven by:
by substituting (49b), (50b), and (51) into (48c), we obtain:
by combining (47) and (52), there are:
also, when τ is2When the pressure is larger than or equal to sigma,andboth are non-negative monotonically decreasing functions, and then
Since erfc (x) is a non-negative monotonically decreasing function andis provided with
Thus, for any given precision ε2> 0, there is a large parameter τ2σ satisfies:
then, there are:
therefore, according to the inaccurate gradient descent method, the optimal discrete input distribution can be efficiently calculated. Furthermore, the sequence of probability distributions { p } obtained by Algorithm 2nThe convergence to the optimal distribution, which is justified by the following theorem:
theorem 3 (Convergence analysis) for a given K, { pnConverges on the optimal solution of the problem (12), and accordingly, { phi (p) }n) The corresponding converges to the optimal value of the problem (12).
The following was demonstrated:
suppose { pnConverge to a non-stationary pointThis needs to be proven:
the condition (56a) holds, and the inequality holds, as demonstrated by propositions 2.3.1 in references [14] D.P.Bertsekekekekekekekekekekes and D.P.Berksekes, nonlinear programming, Athena Scientific,1999,
wherein,
then, the user can use the device to perform the operation,comprises the following steps:
since (57) the inequality (58c) is also true; inequality (58d) is due toWhen in useThen, the equation holds, where | κ | ═ ien||。
According to theorem 2, | | enThe value of | can be arbitrarily small. Thus, there is a parameter κ that satisfiesIn addition, due toIs non-stationary, condition (56b) holds.
According to reference [14]]Bertsekas and D.P.Bertsekas, NonlinerProgramming, Athena Scientific,1999 proposition 2.3.1, known as, { pnEvery limit point of the is stable. Furthermore, the stable point is the global optimum point because the problem (12) is convex with a fixed K.
Theorem 3 ensures that the solution obtained by the method provided by the invention is the reachable capacity distribution, i.e. the optimal input dispersion distribution. In the examples, numerical simulations were used to verify the theoretical results of the present invention.
Examples
The numerical results are used to illustrate the performance of the proposed inaccurate gradient descent method and the source entropy maximization method, which approximates the channel capacity by maximizing the source entropy, which is set as the reference method. In addition, exhaustive search methods are compared, with increasing computational complexity of points, defining parameters which are
Fig. 1a, 1b and 1c show the relationship between the number of points and the achievable rate under different SNR conditions at 2, 3 and 4, respectively. In fig. 1a, a method where the achievable rate by an inexact gradient algorithm is higher than the maximum source entropy can be observed. Furthermore, as K increases, the achievable rate of the inexact gradient method also increases, however, the maximum source entropy method is initially incremented and then decremented. This is because the objective function h (X) of the maximum source entropy, instead of the mutual information I (X; Y). As the SNR increases, the achievable rates for the inaccurate gradient projection method and the maximum source entropy method increase, but the difference between the two decreases. The same result is shown in FIG. 1b and FIG. 1c, comparing FIG. 1a, FIG. 1b and FIG. 1c, and it can be seen that the achievable rates for both methods increase with increasing φ, but the difference decreases.
Fig. 2 plots the relationship between different points K and achievable rate at 4. Fig. 2 shows that the achievable rate of the inexact gradient method is higher than the maximum source entropy. As the SNR increases, a larger K may achieve optimal channel capacity.
In fig. 3, the number of points K that are optimal to achieve the achievable rate at different SNRs is given at 4, and the exhaustive method is also compared. In fig. 3, the achievable rate of inaccurate gradients is higher than the maximum source entropy, especially under high signal-to-noise conditions. Furthermore, the achievable rate of inaccurate gradients is the same as exhaustive methods, which may also prove the optimality of inaccurate gradients.
Finally, the calculation times for the three methods are shown in the table. Table 1 shows the CPU time for the different points to complete the three methods under the conditions of phi 4 and SNR 0 dB. All simulations used MATLAB (R2016b), with a 3.4GHz CPU and 16GB RAM. In addition, the maximize source entropy method utilizes 1stOpt software to solve the 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 maximize source entropy method varies little. This is because the non-linear equations are computed by the 1st Opt software, which is a key step in the method of maximizing the source entropy. Table 2 is the optimal channel capacity parameter.
Table 1 calculation of time comparison (phi 4, SNR 0dB)
TABLE 2 optimal channel capacity parameter
Parameter(s) | Value taking |
Noise power σ2 | 1 |
Phi, i.e. PAR | [2,3,4] |
Peak optical power | sqrt(NOISE_VARIANCE)*10.^(SNR/10) |
Peak value | SOURCE_VARIANCE*PAR |
The invention provides an inaccurate gradient descent method to solve the problem of mixed continuous discrete optimization, theoretically shows that the obtained optimal solution converges on the optimal discrete distribution, and can realize the FSO channel capacity. .
The present invention provides a method for calculating the achievable capacity of a free-space optical intensity channel, and a plurality of methods and approaches for implementing the technical solution, and the above description is only a preferred embodiment of the present invention, it should be noted that, for those skilled in the art, a plurality of modifications and embellishments can be made without departing from the principle of the present invention, and these modifications and embellishments should also be regarded as the protection scope of the present invention. All the components not specified in the present embodiment can be realized by the prior art.
Claims (7)
1. A method for calculating the reachable capacity of an FSO channel is characterized by comprising the following steps:
step 1, setting a channel model;
step 2, solving the capacity of the channel model;
and 3, solving the optimal solution of the capacity of the channel model.
2. The method of 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 and a single photon detector PD as a receiver;
the peak optical power and the average optical power of the input signal X are both constrained such that X is 0 ≦ X ≦ A, andwherein A is the amplitude of the signal,is the mean of the signals, 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 a mean of zero and a variance of σ2。
3. The method of claim 2, wherein step 2 comprises:
for the channel model set in step 1, its capacity is defined as the maximum mutual information C in the input-output channel of all possible inputs givenFSO:
Wherein I (X; Y) is mutual information, H (Y) is entropy of Y, H (Y | X) is joint entropy, P (X) represents distribution of X, f (Y | X) represents entropy of Y, f (Y | X) represents distribution of X, f (Y | X) represents entropy of Y, fY(Y) represents the Y probability density function pdf, fY(y) is a function of P (X).
4. The method of claim 3, wherein step 3 comprises:
step 3-1, setting the input signal X as a discrete random variable with K nonnegative real numbers { Xk}1≤k≤KAnd satisfies the following conditions:
wherein Pr { X ═ Xk}=pkDenotes X ═ XkProbability value of time correspondence is pk,xkIs the k point, pkIs xkThe corresponding probability of the occurrence of the event,is a positive integer;
step 3-2: the noise Z follows a Gaussian distribution, fYThe pdf of (y) is converted to the following form:
solving the capacity of the channel model is equivalently written as the following optimization problem:
step 3-3, defined as follows:
wherein phi (p) is the objective function and gamma is the constraint set, 1KIs a K × 1 vector, where all elements are equal to 1, then the problem (5a), i.e., 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 for the variable p;
step 3-4, fixing x by using equal spacing:
from [0, A]Selecting K values { x) at equal intervals in the rangek}1≤k≤KNamely:
setting proposition 1: set K*Andis questions ofOptimal solution to problem (7), gammaThe minimum distance between any two points in the space, i.e. Representing two arbitrary points, for a given epsilon0Is greater than 0, whenThere is a sequence xl}1≤l≤KSatisfies the following conditions:
is k*A set of values that can be taken;
and 3-5, solving the problem by adopting a gradient projection method (7).
5. The method of claim 4, wherein steps 3-5 comprise:
step 3-5-1, letThe gradient, which represents the objective function, equation (7a), is given by:
step 3-5-2, respectively aligning phi (p) and phi (p) by adopting a numerical integration methodAnd (3) carrying out approximation: since 0. ltoreq. X. ltoreq.A and Z follows a Gaussian distribution, [ - τ1,A+τ1]And [ - τ [ - ]2,A+τ2]Respectively denotes phi (p) andof where τ is1> 0 and τ2>0,τ1、τ2The small interval between the integrals is expressed in order to approximate two very small valuesAndrespectively representing the approximate values of the objective function phi (p)To approximate values of (a):
let p be0Representing a feasible initiation point, pnDenote the nth iteration feasible point, where n is 1,2Gradient projection iteration pnAnd pn+1Given by:
formula (III) αn∈(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+1E γ, such that it andthe distance between them is minimal, and the projection (15b) constitutes the following optimization problem:
pn+1≥0 (17d)。
6. the method according to claim 5, wherein in step 3-5-2, a straight-line backtracking line is used to select a suitable step size in (15a) to achieve the decrement, specifically comprising:
step 3-5-2-1, initialization: k is selected to be more than or equal to 2 and lambdaK-1Not more than 0, set up c2,c3An iteration stop parameter;
step 3-5-2-2, setting n to 0, selecting a feasible initiation point p0∈Y;
Step 3-5-2-3, n ═ n +1, and then calculationAnd
step 3-5-2-4, calculate step αn;
Step 3-5-2-5, calculating
Step 3-5-2-6, if | | | pn-pn-1||≤c2Then stop, thenOtherwise, turning to the step 3-5-2-3;
step 3-5-2-7, if | λK-λK-1|≤c3Then stop and then output popt=pn,KoptK, otherwise K +1, and then to step 3-5-2-2, wherein K isoptRepresenting discrete points xkOptimum number of { lambda }, ofK-1Is an initial value of the objective function, poptIs the optimal probability value that satisfies the condition.
7. The method of claim 6, wherein steps 3-5-2-4 comprise:
step 3-5-2-4-1, selectionρ,c∈(0,1),Is an initial step length, rho is a step length reduction factor, c is a parameter, and a number between 0 and 1 is generally taken;
step 3-5-2-4-2, repeat until satisfying:
wherein For the value of the objective function for the next iteration,for the value of this iteration of the process,is a projection;
3-5-2-4-3,← representing assignment;
3-5-2-4-4, ending repetition;
step 3-5-2-4-5, whenAnd then terminates.
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