Channel model substitution method for enabling parameter mapping by using moment convergence criterion

Abstract

The invention discloses a channel model substitution method for enabling parameter mapping by using a moment convergence criterion, which comprises the following specific steps of: setting a substitution channel model of an original channel model to obey a probability density function consisting of K unknown parameters; defining a general form of a moment mean square error between the original channel model and the alternative channel model by taking parameter mapping as a condition; the L-order moment is converged by minimizing the moment mean square error, a corresponding parameter with the minimum moment mean square error is obtained, and therefore a parameterized mapping relation is established; the replacement channel model is a probability density function obeying to be composed of parameters with the minimum moment mean square error. According to the method, the problem that a moment matching equation system is too complex and cannot solve and / or has no feasible solution can be solved, the application range is wide, and the problem that KLD minimization is only suitable for few conditions can be solved.

Description

Technical Field

[0001] The present invention relates to channel modeling and optimization, and particularly to a method for substituting a channel model enabling parameter mapping by using a moment convergence criterion, belonging to the technical field of wireless communication. Background Art

[0002] The performance analysis and optimization of a communication system largely depend on an accurate and tractable channel model. An accurate channel model is crucial because it quantifies the impact of channel attenuation on signal reception and processing, and equally important is to provide good mathematical tractability for closed-form and insightful analysis. However, since it is difficult to balance the relationship between modeling accuracy and mathematical tractability well, a channel model substitution (CMS) technique is proposed as a solution measure. This technique uses a tractable and simpler substitution model to replace a mathematically intractable channel model, and at the same time adopts a suitable parameter mapping method between the original model and the substitution model to ensure the substitution accuracy.

[0003] Therefore, in order to ensure the substitution accuracy, it is necessary to find a suitable parameter mapping method and determine the values of the distribution parameters of the substitution model according to the given parameter values of the original model. So far, there are mainly two main parameter mapping methods: moment matching and Kullback-Leibler divergence (KLD) minimization. Moment matching is the most widely used method, which mainly uses solving a system of multivariate nonlinear equations to establish the channel parameter mapping relationship. However, this method may lead to a highly complex system of equations, especially when dealing with a hybrid substitution model or many constrained substitution parameters need to be mapped. For example, when the proposed CMS moment matching system of equations is transcendental, the existence and uniqueness of its solution cannot be guaranteed.

[0004] At the same time, KLD minimization is only applicable to a few special cases. According to research, KLD minimization is only used for the theoretically optimal parameter mapping of CMS for gamma substitution lognormal, exponential substitution Pareto, and Rayleigh substitution K-distribution. It can be found from the above that neither of these two dominant parameter mapping methods fully exploits the potential of CMS. Summary of the Invention

[0005] Aiming at the deficiencies of the existing technology that when the moment matching equation system is too complex to be solved and / or has no feasible solution, and at the same time KLD minimization is only applicable to a few cases, the object of the present invention is to propose a method for substituting a channel model enabling parameter mapping by using a moment convergence criterion. The present invention does not need to solve a complex system of equations and has a wide application range, which not only solves the problem that the moment matching equation system is too complex to be solved and / or has no feasible solution, but also solves the limitation that KLD minimization is only applicable to a few cases.

[0006] The technical solution of the present invention is implemented as follows:

[0007] A method for replacing a channel model enabling parameter mapping using the moment convergence criterion is as follows

[0008] 1) Let represent the random channel fading envelope, which follows a probability density function f 1 , a 2 ,..., a N composed of N known parameters, and f G (g) is called the original channel model; the alternative channel model of the original channel model f G (g) is denoted as G which follows a probability density function composed of K unknown parameters b = [b , b 1 ,..., b 2 ,..., b K .

[0009] 2) Conditional on the parameter mapping b, define the general form of the mean square error of moments MMSE between the original channel model f G (g) and the alternative channel model as:

[0010]

[0011] where L represents the preset number of moments to converge, m l is the l-th order moment of the original channel model; is the l-th order moment of the alternative channel model; ρ 1 ≥ ρ 2 ≥... ρ L represents the importance of moments of different orders in the mean square error of moments, and satisfies the normalization property and is specified as a hyperparameter;

[0012] 3) Converge the L-th order moment by minimizing the mean square error of moments, and thus establish the following parametric mapping relationship:

[0013]

[0014] 4) The alternative channel model is the probability density function that follows a composition of K parameters b MC = [b MC:1 , b MC:2 ,..., b MC:K .

[0015] Among them, both the original channel model and the alternative channel model have L-th order moments.

[0016] Compared with the prior art, the present invention has the following beneficial effects:

[0017] 1. The present invention formulates an optimization problem to minimize the moment mean square error (MMSE) between the original channel model and its substitute. The parameter mapping method based on the moment convergence criterion of the present invention can solve the problem that the moment matching equation system is too complex to solve and / or has no feasible solution. This method is particularly useful when the moment matching equation system is too complex to solve and / or has no feasible solution.

[0018] The proposed parameter mapping method based on moment convergence relies on minimizing a linear combination of simple quadratic forms, solving the problem of solving non-linear multivariable equation systems and dealing with the KLD objective function given in integral form. This linear combination form and its additive property make the constructed optimization problem particularly suitable for parameter mapping problems involving a large number of unknown parameters and adopting a mixed distribution model. By transforming complex non-linear problems into linear combination optimization, this method not only maintains the simplicity of mathematical expressions but also significantly improves computational efficiency, providing an efficient and robust solution framework for mixed distribution modeling in high-dimensional parameter spaces.

[0019] 2. The parameter mapping method based on the moment convergence criterion has a wide range of applications and can solve the problem that KLD minimization is only applicable to a few cases. The parameter mapping method based on moment convergence proposed by the present invention only needs to meet one condition, that is, the original channel model and the selected alternative model both have L-order moments. This condition is much looser than the conditions of moment matching and KLD minimization and can be satisfied in most actual CMS applications, thus ensuring the general applicability of the proposed method. BRIEF DESCRIPTION OF THE DRAWINGS

[0020] Figure 1 - Flowchart of the channel model substitution based on the moment convergence criterion of the present invention.

[0021] Figure 2 - Schematic diagram of the comparison of evaluation indicators between the present invention and the existing model substitution method in two model substitution embodiments. Among them, Figure 2 (left) shows the relationship between KLD and ISE generated by three parameter mapping criteria of moment convergence, moment matching, and KLD minimization respectively and different μ and σ in the gamma substitution lognormal CMS; Figure 2 (right) shows the comparison of KLD and ISE based on the moment convergence criterion with those of moment matching in the inverse Gaussian substitution lognormal CMS.

[0022] Figure 3- The alternative probability density functions (left figure) and cumulative distribution functions (right figure) of the gamma distribution, inverse Gaussian (IG) distribution, and mixture gamma (MG) distribution obtained by the parameter mapping method based on moment convergence in the present invention are compared with the original lognormal shadow model in the schematic diagram. Detailed implementation manners

[0023] The implementation manners and principles of the present invention are further described in detail below with reference to the accompanying drawings.

[0024] In the field of wireless communication, the random channel fading envelope is a mathematical model that describes the random variation of the amplitude of a signal when it propagates in a wireless channel.

[0025] The random effects that cause the time-varying channel fading envelope can be characterized by a channel envelope model. Such a model is usually a continuous random distribution and has a support set of non-negative real numbers and a set of distribution parameters. Let represent the random channel fading envelope, which follows a probability density function (PDF) f 1 , a 2 ,..., a N consisting of N known parameters. G (g). f G (g) is called the original channel model and is also referred to as the reference distribution or target distribution in the CMS.

[0026] In practical applications, although the original channel model can obtain a good fitting result with the empirical observations, it may lead to difficulties in analysis or excessive complexity in simulation. Therefore, the goal of the CMS is to find another simpler random distribution model, whose PDF consists of K unknown parameters b = [b 1 , b 2 ,..., b K to serve as an alternative to the original channel model. is called the alternative channel model. To maintain the matching degree between the original channel model and the alternative channel model, the quantitative relationship between a = [a 1 , a 2 ,..., a N and b = [b 1 , b 2 ,..., b K is denoted as which is called parameter mapping. Note: K is not necessarily equal to N.

[0027] One of the most commonly used channel parameter mapping methods is to make the first K moments of the original channel model and its alternative channel model equal, resulting in the moment matching criterion. Therefore, the channel parameter mapping relationship can be established by solving a system of equations consisting of K equations, where b = [b 1 ,b 2 ,...,b K :

[0028]

[0029] where gives the expectation of the random variable. The K alternative parameters generated by solving such a moment matching system of equations can be written as b MM = [b MM:1 ,b MM:2 ,...,b MM:K . For the sake of simplicity in the following analysis and discussion, we denote the K-th moment as

[0030] Considering that most high-order moments are given in the form of a combination of polynomials and special functions, the moment matching system of equations given in Equation (1) is usually non-linear. Therefore, when K is large, the problem of whether there is a feasible solution arises. Even if there is, it is unlikely that the analytical solution of Equation (1) can be solved into an explicit form. Therefore, the moment matching method is very suitable for the CMS problem with K ≤ 3, because when K ≤ 3, sufficient mathematical tractability can be maintained.

[0031] It can be clearly seen from Equation (1) that the parameter mapping based on moment matching essentially determines the parameters through the local matching of finite-order statistical moments, which is a simplification and approximation of practical experience; another strictly parameter mapping method with mathematical proof is the method based on KLD minimization. By minimizing the KLD between the original channel model and the alternative channel model, the channel parameter mapping relationship can also be established:

[0032]

[0033] where represents the KLD between f G and under the premise that the parameter mapping relationship is b.

[0034] In CMS, the KLD quantifies when the original channel model f G (g) is replaced by using the parameter mapping b The error loss generated when approximating. Although the KLD-based CMS is optimal at the theoretical level, it has relatively high requirements for performing subsequent analysis and optimization in practical applications. This is because for most channel models, due to the complexity of the integration operation and the logarithmic function form of the ratio of probability density functions (PDFs) in the integrand, it is difficult to obtain a closed-form analytical expression for the Kullback-Leibler divergence (KLD).

[0035] The parameter mapping of moment convergence proposed by the present invention

[0036] As can be seen from the above two CMS parameter mapping methods, there are obvious drawbacks whether it is to solve complex equation systems or optimize implicit / integral forms of statistical distances. This prompts us to propose a new parameter mapping criterion, the core of which is to optimize a mathematically more tractable objective function, which ideally should have a closed-form analytical expression. The process of the present invention is as Figure 1 shown. Specifically, conditional on the parameter mapping b, the present invention defines the general form of the mean square error of moments (MMSE) between the original channel model and the alternative channel model as:

[0037]

[0038] where L represents the preset number of moments to be converged, m l is the l-th moment of the original channel model; is the l-th moment of the alternative channel model; ρ 1 ≥ρ 2 ≥...ρ L indicates the importance of moments of different orders in the MMSE and satisfies the normalization property and is specified as a hyperparameter (Hyperparameters are parameters that need to be preset before training a machine learning model and are not directly learned from the training data. They control the structure of the model, the key characteristics of the training process, and the optimization strategy, and have an important impact on the model performance).

[0039] In this method, we propose to converge the L-th moment by minimizing the MMSE defined in formula (3), thereby establishing the following parameterized mapping relationship:

[0040]

[0041] That is, it is called the moment convergence criterion. According to formula (3), taking the derivative of with respect to gives

[0042]

[0043] Comparing Equation (4) with Equations (1) and (2), it can be seen that the proposed parameter mapping method based on moment convergence depends on minimizing a linear combination of quadratic forms in a simple form, solving the problem of solving non-linear multivariate equations and dealing with the KLD objective function given in integral form. This form of linear combination and its additive property make the constructed optimization problem particularly suitable for parameter mapping problems with a large number of unknown parameters and using a mixture distribution model. By transforming the complex non-linear problem into a linear combination optimization, this method not only maintains the simplicity of mathematical expression but also significantly improves the computational efficiency, providing an efficient and robust solution framework for mixture distribution modeling in high-dimensional parameter spaces.

[0044] Another advantage of Equation (4) compared to Equation (2) is the symmetry of. Since is asymmetric, when the input order of the distribution changes, this may lead to inconsistent results, that is However, always has symmetry, manifested as

[0045]

[0046] In short, the proposed parameter mapping method based on moment convergence only needs to satisfy one condition: the original channel model and the selected alternative model both have L-order moments. This prerequisite is much looser than the conditions of moment matching and KLD minimization and can be satisfied in most practical CMS applications, thus ensuring the general applicability of the proposed parameter mapping method based on moment convergence.

[0047] The following selects three examples to illustrate the effectiveness and efficiency of the proposed parameter mapping method based on moment convergence. It should be noted that the proposed parameter mapping method based on moment convergence is not limited to the following examples, and its prerequisite for use is that the original channel model and the selected alternative model both have L-order moments.

[0048] Example 1. Replacement of the log-normal channel model with a gamma alternative

[0049] Assume that the characteristic parameters of a log-normal shadow model are: the location parameter μ and the scale parameter σ. Then the parameter mapping problem of the CMS with a gamma alternative to the log-normal can be expressed as where θ > 0, v > 0 are the shape and scale parameters of the gamma distribution alternative model. The closed forms of the k-order moments of the two functions are m k = exp(kμ + k 2 σ 2 / 2) and where Γ(·) represents the gamma function.

[0050] By equations (3) and (5), let L = K = 2 to obtain the MMSE and its first-order partial derivatives with respect to θ and v as

[0051]

[0052]

[0053] After the second derivative test, it can be found that regardless of the values of ρ 1 and ρ 2 , at θ MC =(exp(σ 2 ) - 1) -1 and v MC = exp(μ + σ 2 / 2), the minimum MMSE is equal to 0.

[0054] To verify the effect of the parameter mapping method based on moment convergence for completing the gamma substitution lognormal model compared with the moment matching and KLD minimization criteria, the present invention uses KLD and the integrated square error (ISE) as evaluation metrics. Among them The relationship with different μ and σ is as shown in Figure 2 (left). It can be clearly seen from the results shown in this figure that the KLD and ISE of the moment convergence and moment matching methods are the same, which proves that the moment convergence criterion proposed by the present invention can achieve the global optimum of parameter mapping in the sense of moments. When σ is small and / or μ is large, the accuracy of the proposed moment convergence criterion is also comparable to the best accuracy in the sense of KLD.

[0055] Example 2, inverse Gaussian substitution for lognormal

[0056] In addition to gamma substitution, the inverse Gaussian (IG) distribution is another commonly used and more accurate substitution for the lognormal shadow model. Similar to the previous example, this application will use the parameter mapping method based on moment convergence to represent the parameter mapping between the inverse Gaussian and the lognormal as where ζ > 0, λ > 0 are the mean and scale parameters of the IG substitution model. The k-th moment of the known IG substitution is where K (·) (·) represents the modified Bessel function of the second kind.

[0057] By equations (3) and (5), let L = K = 2 to obtain the MMSE and its first-order partial derivatives with respect to ζ and λ as

[0058]

[0059]

[0060] Through the second derivative test, the global minimum is at ζ MC= exp(μ + σ 2 / 2) and λ MC = (1 / 2)exp(μ)csch(σ 2 / 2), where csch(·) represents the hyperbolic cosecant function. Figure 2 (right) plots the KLD and ISE based on the moment convergence criterion in the inverse Gaussian substitution lognormal CMS, and compares them with the KLD and ISE of moment matching.

[0061] Example 3, Mixed Gamma Substitution Lognormal

[0062] For the previous two examples, we can use the second derivative to verify the optimality of the stationary points where the derived derivative is zero. However, it should be noted that when solving the optimization problem shown in Equation (4), finding the points where the first derivative is zero is not a necessary condition for finding the global or local minimum. In practical CMS applications, as long as the gradient of the moment mean square error (MMSE) can be expressed in an explicit form, various numerical methods (such as the interior point method IPM and sequential quadratic programming SQP) can be used to approximate the optimal solution of the parameter mapping problem. In addition, since L is not necessarily equal to K, it needs to be flexibly configured according to the accuracy requirements. Except for the case where the number of parameters matches (N = K), the effectiveness of the parameter mapping method based on the moment convergence criterion in the case of hyperparameter mapping (N < K) is also shown below. In addition, such problems cannot be solved by the parameter mapping method based on moment matching, which once again highlights the advantages of the parameter mapping method based on the moment convergence criterion.

[0063] To observe and analyze the attenuation effects of different scattering clusters on the propagation path, a mixed gamma distribution can be used as an alternative model to approximate the channel fading characteristics. In existing research, it is known that when using the moment matching method to determine the parameter mapping relationship for the mixed gamma distribution, it is necessary to solve very complex non - linear equations, resulting in high computational complexity. To address this problem, the parameter mapping method based on the moment convergence criterion proposed in this application is used below.

[0064] Consider a general mixed gamma substitution model, whose probability density function is composed of the PDFs of M weighted independent gamma distributions. Therefore, it is necessary to determine K = 3M unknown parameters. However, due to the normalization of the probability density function, these parameters are also subject to certain constraints. We can use the parameter mapping method based on moment convergence to achieve the parameter mapping from mixed gamma to lognormal CMS, expressed as where is the vector of weight, shape, and scale parameters, which needs to satisfy Due to the basic properties of the gamma distribution, we can obtain the k - th moment as:

[0065]

[0066] Therefore, from Equation (3), the corresponding MMSE can be obtained as follows:

[0067]

[0068] Thus, we can use the Lagrange multiplier to construct the Lagrangian function and solve the constrained parameter mapping problem of the hybrid gamma-to-log-normal CMS by the sequential quadratic programming (SQP) method:

[0069]

[0070] where η is the Lagrange multiplier. From Equation (11), the gradient of the closed-form Lagrangian function can be derived as

[0071]

[0072] Taking σ = 0.5 and using the hybrid gamma-to-log-normal channel model to replace the CMS as an example, the moment convergence results obtained by the sequential quadratic programming (SQP) algorithm in 10 3 repeated trials are shown in Table I below. At the same time, two cases are also considered: that is, the importance of the two is equal [ρ 1 , ρ 2 ,..., ρ L = [1 / L, 1 / L,..., 1 / L] and the importance of the two decreases continuously [ρ 1 , ρ 2 ,..., ρ L-1 , ρ L = [(1 / 2) 1 , (1 / 2) 2 ,...,(1 / 2) L-1 , (1 / 2) L-1 .

[0073] Table I

[0074]

[0075] From the parameter mapping results shown in Table I, it can be obtained that the sequential quadratic programming (SQP) algorithm can effectively find the optimal alternative parameter set that minimizes the MMSE; increasing the number of hybrid kernels (i.e., parameter M) does not necessarily lead to a decrease in the MMSE. Similarly, increasing the number of moments (i.e., L) involved in calculating the MMSE does not necessarily effectively reduce the MMSE, which reflects that when applying the parameter mapping method based on moment convergence, it is very important to pre-set an appropriate distribution substitution form (i.e., select a suitable M) and the objective function.

[0076] In addition, for the case of a large M, since By solving The corresponding moment matching equations cannot obtain a feasible solution. This phenomenon indicates the flexibility and universality demonstrated by the moment convergence criterion during the parameter mapping process when using a mixture distribution substitution model, once again proving its theoretical value.

[0077] To make the results more intuitive, this application Figure 3 shows the alternative probability density functions (PDFs) and cumulative distribution functions (CDFs) of the gamma distribution, inverse Gaussian (IG) distribution, and mixture gamma (MG) distribution obtained by the parameter mapping method based on moment convergence, and compares them with the original lognormal shadow model. For simplicity of analysis, equal weights are used, i.e., [ρ 1 , ρ 2 ,..., ρ L = [1 / L, 1 / L,..., 1 / L]. Figure 3 Intuitively verifies the effectiveness and efficiency of the parameter mapping method based on moment convergence.

[0078] The comparison of the moment convergence criterion proposed in this invention with the two common existing mapping methods, namely KLD minimization (also called relative entropy minimization) and moment matching, in terms of parameter mapping accuracy, mathematical rigor, and computational complexity is shown in Table II. It can be seen from Table II that moment matching has a relatively high accuracy, relatively low mathematical rigor, and relatively low computational complexity; while KLD minimization has a medium accuracy, relatively high mathematical rigor, and relatively high computational complexity; the accuracy of this invention is relatively high, the mathematical rigor is between moment matching and KLD minimization, and the computational complexity is also between moment matching and KLD minimization.

[0079] Table II

[0080]

[0081] Finally, it should be noted that the above examples of this invention are only illustrations for explaining this invention, and are not limitations on the implementation manners of this invention. Although the applicant has described this invention in detail with reference to the preferred embodiments, for those of ordinary skill in the art, other different forms of changes and modifications can be made based on the above description. It is impossible to enumerate all the implementation manners here. Any obvious changes or modifications derived from the technical solutions of this invention still fall within the protection scope of this invention.

Claims

1. A channel model substitution method using moment convergence criteria to enable parameter mapping, characterized in that: The steps are as follows, 1) Set represents the random channel fading envelope, which is governed by N known parameters a=[a1,a2,...,a N ] is composed of the probability density function f G (g), f G (g) is called the original channel model; the original channel model f G The alternative channel model of (g) is denoted as is subject to K unknown parameters b=[b1,b2,...,b K ]probability density function composed of; 2) Define the original channel model f based on the parameter mapping b G (g) and alternative channel models The general form of the mean square error MMSE is: Where L represents the preset number of moments that need to converge, m l is the l-th order moment of the original channel model; is the l-th order moment of the alternative channel model; ρ1≥ρ2≥...ρ L Indicates the importance of moments of different orders in the moment mean square error and satisfies the normalization property and are specified as hyperparameters; 3) By minimizing the moment mean square error to converge the L-order moment, the parameterized mapping relationship is established as follows: 4) Alternative channel models That is, it is subject to K parameters b MC =[b MC:1 ,b MC:2 ,...,b MC:K ] is the probability density function composed of .

2. The method for channel model substitution using moment convergence criterion to enable parameter mapping according to claim 1, characterized in that: Both the original channel model and the replacement channel model have L-order moments.

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