Method, system, and computer program product for optimizing the capacity of a communication channel using a Dirichlet process

Abstract

The present invention relates to a method for optimizing the capacity of a communication channel in a communication system, the communication system comprising at least a transmitter (10), a receiver (11), the communication channel (12) between the transmitter and the receiver, and a channel conditional probability distribution estimator (13). The transmitter (10) transmits a message conveyed by a signal associated with the transmission probability of an input signal probability distribution estimated according to the estimator (13). The transmitter (10) takes at least the message as an input and outputs a signal to be transmitted on the channel (12). The channel takes the transmitted signal as an input and outputs the received signal, and the received signal is processed at the receiver (11) to decode the transmitted message. The probability distribution denoted as p Y|X (y|x) is related to the output Y corresponding to the received signal given the input X corresponding to the transmitted signal X, and thus represents the channel conditional probability distribution of the probability of outputting the received signal Y given the transmitted signal X. By using a functional basis of the probability distribution, the probability distribution is estimated as an approximation of the channel conditional probability distribution p Y|X (y|x), and thus the channel conditional probability distribution is approximated for each possible signal transmitted by the transmitter by means of a mixture model having a mixture probability distribution from the functional basis. Then, the probability distribution for each possible transmitted signal is estimated at least from the output signal received at the receiver and by using collapsed Gibbs sampling depending on the Dirichlet process.

Description

Technical Field

[0001] The present disclosure relates to the optimization of communication channel capacity, and particularly to and is dedicated to the case of non-trivial channels where the optimal input distribution cannot be obtained theoretically. Background Art

[0002] The basic characteristics of a memoryless communication channel can be represented by the so-called "conditional probability distribution" of the output Y given the input X, denoted as p Y|X (y|x). Some examples of well-known communication channels are given in the following list:

[0003] · Additive white Gaussian noise channel, where Y = X + N and N is Gaussian distributed. The additive white Gaussian noise channel models wired communication subject to perturbations caused by thermal noise at the receiver.

[0004] · Fading channel Y = a·X + N, where a follows a fading distribution (such as Rayleigh distribution). The fading channel models transmission over a narrowband wireless channel in a radio propagation environment involving rich scattering.

[0005] · More complex channels can involve non-linear effects. This is the case, for example, of optical channels where, when too much transmit power is added in wavelength division multiplexing (so-called "WDM transmission"), the Kerr effect cannot be ignored and the channel capacity is reduced, which is driven by the non-linear Schrödinger equation.

[0006] Once the conditional probability distribution p Y|X (y|x) is accurately known, the following terms can be relied upon to optimize the communication system:

[0007] · Design of the input signal to maximize the mutual information between the input and output of the channel.

[0008] · Design of the optimal receiver, which typically relies on the processing of the likelihood probability p Y|X (y|x).

[0009] Thereafter, it is assumed that the channel conditional probability distribution p Y|X (y|x) has low fluctuations to enable learning and tracking.

[0010] Typically, a wireless communication system can be modeled by a linear system, and its parameters are estimated by transmitting training signals. However, when the number of parameters to be tracked is high, a large amount of pilot overhead is required and the system throughput is reduced. Moreover, when the channel is more complex, it is difficult or impossible to define the model to be tracked. The future of wireless communication systems is to continue to increase the carrier frequency in the goal of utilizing the available frequency bands. Terahertz communication (above 300 GHz) is a keyword for the envisioned new generation of telecommunications (6G). However, until today's knowledge, terahertz communication requires new materials and new radio devices in the new field between electronics and photonics. The strong nonlinear and stochastic characteristics of these new channels cannot be ignored in order to obtain the best benefits of these new frequency bands. The concern is to learn such a communication channel rather than relying on inaccurate models.

[0011] To optimize the transmission strategy, the optimized input distribution must be known in advance at the transmitter. The received signal is used to estimate the channel conditional probability distribution, which in turn is used to optimize the optimal input distribution that is ultimately provided to the transmitter. A compact and accurate representation of the channel conditional probability distribution is used so that:

[0012] ο When feeding back the channel conditional probability distribution from the receiver to the transmitter, the overhead on the feedback link is limited,

[0013] ο The computational complexity of calculating the optimal input distribution is limited. Summary of the Invention

[0014] However, the problem lies in jointly learning the channel conditional probability distribution and optimizing the transmission method accordingly. One of the main challenges is to learn the channel conditional probability distribution when the channel model is unknown. To solve the broadest class of channels, an accurate description of the channel is required, but this often leads to high complexity at the receiver. Therefore, it is better to find a compact and accurate representation of the channel conditional probability distribution.

[0015] The present disclosure aims to improve this situation.

[0016] To this end, a method for optimizing the capacity of a communication channel in a communication system is proposed, the communication system at least including a transmitter ( Figure 1 reference numeral 10 therein), a receiver (11), the communication channel (12) between the transmitter and the receiver, and a channel conditional probability distribution estimator (13).

[0017] The transmitter (10) transmits a message conveyed by a signal associated with the transmission probability of an input signal probability distribution estimated according to the estimator (13).

[0018] The transmitter (10) takes at least a message as input and outputs a signal to be transmitted over the channel (12), and the channel takes the transmitted signal as input and outputs the received signal, which is processed at the receiver (11) to decode the transmitted message.

[0019] Denoted as p Y|X (y|x) The probability distribution is related to the output Y corresponding to the received signal given the input X corresponding to the transmitted signal X, and thus represents the channel conditional probability distribution of the probability of outputting the received signal Y given the transmitted signal X.

[0020] More particularly, by using a functional basis of the probability distribution, the probability distribution is estimated as the channel conditional probability distribution p Y|X (y|x) approximation. Thus, through a mixture model with a mixture probability distribution from the said functional basis, the channel conditional probability distribution is approximated for each possible signal transmitted by the transmitter. The probability distribution for each possible transmitted signal is estimated at least from the output signal received at the receiver and by using collapsed Gibbs sampling depending on the Dirichlet process, as described in detail in the embodiments presented below.

[0021] Then, the implementation of the present disclosure enables an accurate representation of the channel conditional probability distribution and thus enhances the robustness of decoding.

[0022] The wording that the transmitter (10) "at least" takes a message as input refers to the fact that the transmitter can also take the estimated input signal probability distribution as input, and more particularly, in an alternative embodiment implementing the optimizer (14), takes the optimized input signal probability distribution as input. Similarly, the probability distribution of each possible transmitted signal can be estimated based on the output signal received at the receiver and also due to the optimized input signal probability distribution.

[0023] In an embodiment that seems advantageous in terms of the calculations to be performed, the aforementioned functional basis is selected from the exponential function family. Alternatively, other possible embodiments can use known numerical integration methods.

[0024] In an embodiment, the probability distribution estimation is based on an approximation of the channel conditional probability distribution, which approximation depends on a basis that decomposes into the probability distribution function p(y|x; θ), where θ is a set of parameters θ j and the distribution function is an exponential family, and the parameters are the mean and variance, such that the approximation of p Y|X (y|x) by the basis of the probability distribution function p(y|x; θ) is given by is given, where N, and the set {θ j}, {w j} are the parameters to be estimated.

[0025] In this embodiment, the probability distribution function g(y|x; θ) comes from the exponential distribution family, and the prior on θ can be conjugate to the corresponding exponential distribution with parameter λ, where λ = {λ1, λ2} represents the hyperparameters of the conjugate prior on the parameter λ,

[0026] and then the following steps can be performed:

[0027] - N clusters are constructed for the parameter sets {θ j}, {w j}, and each cluster is used to group symbols together, which is achieved by associating an integer index from 1 to N with each symbol group.

[0028] - The cluster index is assigned to the i-th sample denoted as c i such that c -i becomes the cluster assignment index for all other clusters except the i-th cluster, and the posterior distribution p(c i = k|x 1:n , y n:n , c -i , λ) is calculated, where x 1:n represents the symbol of the transmitted signal, and y 1:n represents the input symbol of the received signal.

[0029] - And sampling is performed from the posterior distribution in order to update the parameters {θ j , {w j}, whereby the parameters N, {θ j , {w j} are estimated to provide an approximation of the channel conditional probability distribution.

[0030] In this embodiment, the updated parameters {θ j , {w j} can be sent to the transmitter and / or receiver to calculate the conditional probability distribution estimate. Then, this updated parameter can be used to enhance the signaling obtained at the receiver and / or transmitter. Then, in this embodiment, the transmitter can improve the accuracy of its knowledge of the conditional probability distribution, thereby more efficiently optimizing its transmission strategy. On the receiver side, the improved accuracy of the conditional probability distribution knowledge can be used to improve its decoding performance.

[0031] In an embodiment, the posterior distribution p(c i = k|x1:n , y 1:n , c -i , λ) can be given by the following formula:

[0032] p(c i = k|x 1:n , y 1:n , c -i , λ) = p(c i = k|c -u )p(y i |x 1:u-1 , x i+1:n , y 1:i-1 , y i+1:n , c -i , c i = k, λ), where:

[0033] - p(c i = k|c -i ) is calculated according to the Polya urn scheme,

[0034] - and p(y i |x 1:i-1 , x i+1:n , y 1:i-1 , y i+1:n , c -i , c i = k, λ) = ∫p(y i |x i , θ)p(θ|x 1:i-1 , x i+1:n , y 1:i-1 , y i+1:n , c -i , c i = k, λ)dθ.

[0035] In this embodiment, it is also possible to check whether a new cluster is to be created to determine the best value of the parameter N that provides the optimal approximation of the channel conditional probability distribution.

[0036] In this embodiment, a random selection can be made from the probability of verifying the following formula to create a new cluster:

[0037] p(y i |x i , P0) = ∫p(y i |x i , θ)dP0, where P0 is the conjugate of p(y|x; θ).

[0038] In the first embodiment and in the second embodiment, the message sent is transmitted by signals belonging to a finite signal set, and this signal set is represented by a corresponding set of possible symbols x. In the first embodiment, the symbols x sent are known at the receiver in the "training mode", while in the second embodiment, they are unknown.

[0039] Thus, in the first embodiment, the method can be applied only to a predefined value of the input X, which is known, and the associated output Y is considered only when the input X is equal to the predefined value.

[0040] In the third embodiment and in the fifth embodiment, the message sent is transmitted by signals that do not necessarily belong to a finite signal set. However, the third embodiment corresponds to the case where the input X is known, while the fourth embodiment corresponds to the case where the input X is unknown and the signals sent do not belong to a known finite signal set.

[0041] Thus, the second and fourth embodiments where the input X is unknown involve a tracking mode. In the tracking mode, the input signal probability distribution p X (x) is considered for estimating the channel conditional probability distribution. The value of the symbol x sent can be inferred using Bayesian methods such that each observed value y contributes to updating p Y|X (y|x) according to the current estimate of the conditional probability distribution using the probability weights corresponding to y being associated with x.

[0042] It should be noted that, in the aforementioned second embodiment, the input signal probability distribution p X (x) simply becomes a scalar (probability value), and p Y|X (y|x) becomes p Y|X (y|x = ω m ), where the weight corresponds to the probability of y being associated with x = ω m .

[0043] It should also be noted that, in the first and second embodiments, before constructing N clusters for the parameter sets {θ j}, {w j}, the output Y can be selected based on the set of possible inputs X. Moreover, the input X can typically be known in the first embodiment, which results in a significant reduction in complexity when implementing the first embodiment.

[0044] Otherwise, when the input X is unknown and even does not belong to a finite set of known symbols (or signals), typically in the fourth embodiment, a model for the non - parametric joint density can be defined as:

[0045] where: (21)

[0046] The parameters {θ j}, {w j}, and {ψ j} represent conditional density parameters, and j is a clustering index. The parameters (θ, ψ) are jointly obtained from the base measure of the Dirichlet process "DP" such that (θ, ψ) ∼ DP(αP 0θ ×P 0ψ ), where α is a scaling parameter.

[0047] According to the options related to the optimization calculation proposed above, the signal set is represented by the corresponding symbol set, and the parameters of the mixture model are optimized to define the optimized symbol positions and probabilities to be used at the transmitter in order to optimize the capacity of the approximate channel. Then, the optimized symbol positions and probabilities are provided to the receiver and / or transmitter.

[0048] Then, the input signal distribution optimizer ( Figure 1 label 14 therein) depends on the estimation of the channel probability distribution and is performed as explained above. When the function basis selected for channel estimation is the exponential family, a closed form expression can be derived and the algorithm converges to the optimal solution. Then, the optimized p X (x) delivered by the optimizer 14 can be distributed to the estimator 13 (and the transmitter 10 and / or the receiver 11).

[0049] The present disclosure also aims at a system that at least includes a transmitter (10), a receiver (11), a communication channel (12) between the transmitter and the receiver, and a channel conditional probability distribution estimator (13) for implementing the above method. In this system, the estimator:

[0050] · Obtains information related to the signal (x; p X (x)) used at the transmitter;

[0051] · Receives the signal y transmitted through the channel and characterized by the channel conditional probability distribution p Y|X (y|x);

[0052] · Estimates the channel conditional probability distribution from the received signal, which is approximated by a distribution model dependent on the function basis;

[0053] · Sends the estimated channel conditional probability distribution to the receiver and / or transmitter.

[0054] The system may also include an input signal probability distribution optimizer (14) configured to perform a method for achieving optimization. Here, the input signal probability distribution optimizer (14):

[0055] · obtains the estimated channel conditional probability distribution p Y|X (y|x)

[0056] · optimizes the input distribution p X (x)

[0057] · sends the optimized input signal distribution to the transmitter (10), to the receiver (11), and / or to the channel conditional probability distribution estimator (13).

[0058] The present disclosure also aims at a computer program including instructions which, when executed by a processor, implement the method as proposed above. The instructions of the program may be distributed on the receiver, the transmitter, the estimator, and optionally to the optimizer. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] When reading the following description of the embodiments given as examples, more details and advantages of the present invention will be understood, and these details and advantages will become apparent from the associated drawings, in which:

[0060] Figure 1 shows an overview of a general system according to an example implementation.

[0061] Figure 2 shows the steps implemented by the conditional probability distribution estimator 13 as an example of an embodiment Figure 1 illustrated by

[0062] Figure 3 shows the steps implemented by the input signal distribution optimizer 14 as an example of an alternative embodiment Figure 1 illustrated by

[0063] Figure 4 shows the steps implemented by the transmitter 10 as an example of an embodiment Figure 1 illustrated by

[0064] Figure 5 shows the steps implemented by the receiver 11 as an example of an embodiment Figure 1 illustrated by

[0065] Figure 6 illustrates estimating the channel probability distribution from y and x at the receiver (in the absence of the optimizer 14).

[0066] Figure 7Shows an example of the implementation of the Dirichlet process and Gibbs sampling for x belonging to {+1, -1, …, -3}.

[0067] Figure 8 Shows a system involving an input optimizer 14 according to an alternative embodiment of the present invention.

[0068] Figure 9 Shows the implementation of estimating the channel probability distribution by optimizing from y and the input signal distribution p X (x) at the receiver.

[0069] Figure 10 Shows according to Figure 9 An example of the implementation of a modified Dirichlet process and Gibbs sampling for x belonging to {+1, -1, …, -3} according to an embodiment.

[0070] Figure 11 Shows the details of the steps that can be implemented to estimate the channel probability distribution and thus can represent an example of an algorithm of a possible computer program according to the present disclosure. Detailed description of the specific implementation

[0071] The background of the present disclosure is described below, where, in addition to the estimation of the channel conditional probability distribution, the optimization of the channel conditional probability distribution is also performed according to alternative embodiments. Then, these two steps (estimation and optimization) can depend on the knowledge of the input signal or its probability distribution, or simply do not depend on any knowledge of the input signal (then perform the so-called blind "tracking" phase).

[0072] Therefore, the input signal probability distribution needs to be shared with the transmitter to optimize the channel capacity. The input signal probability distribution needs to be shared with the receiver to optimize the performance of the receiver. The input signal probability distribution needs to be shared with the channel conditional probability distribution estimation to improve the performance of the channel conditional probability distribution estimation.

[0073] By improving the input signal probability distribution based on channel knowledge, the channel capacity is improved, which further allows for the improvement of the channel conditional probability distribution estimation, etc. Therefore, an iterative method for jointly optimizing the channel conditional probability distribution estimation and the input signal probability distribution is preferred. Another advantage of this method is that it allows tracking the changes in the input signal probability distribution when its change speed is not too high.

[0074] The channel conditional probability distribution estimator can approximate the channel conditional probability distribution p Y|X (y|x) by using a function basis of the probability distribution, and the function basis of this probability distribution is preferably the exponential function family. In fact, specific calculations shown below are more easily manipulated using such exponential functions.

[0075] As a general approach, a transmitter uses a finite set of symbols over a communication channel characterized by a conditional probability distribution. For each possible symbol sent by the transmitter, the conditional probability distribution can be approximated by a mixture model with a mixture probability distribution from an exponential family. From at least the output symbol, and using folded Gibbs sampling relying on a Dirichlet process, the conditional probability distribution for each possible transmitted symbol in the finite set of symbols is estimated. The result is in the form of a mixture model, and the number of components is directly known from the observations. The exponential family makes it easier to implement Gibbs sampling.

[0076] The conditional probability distribution estimate is preferably computed at the receiver.

[0077] In a first embodiment, referred to as "training mode", the symbol x is considered to be known at the receiver.

[0078] In the second embodiment, referred to as "tracking mode", the symbol x is unknown at the receiver. However, the input signal probability distribution p is considered X (x) is known and is taken into account during the conditional probability distribution estimation step.

[0079] The two embodiments may be interleaved in time depending on the situation where pilot symbols are sent, either when the data has been correctly decoded and can be used as a pilot signal for channel estimation and approximation, or when the data has not been correctly decoded.

[0080] In addition, the parameters of the mixed model estimated by the conditional probability distribution estimator can be provided to an input distribution optimizer (according to the aforementioned optional embodiment), which defines the optimized symbol positions and probabilities to be used at the transmitter for optimizing the capacity of the approximated channel. The optimized symbol positions and probabilities are then provided to the transmitter and the receiver.

[0081] Reference Figure 1 , a system according to a possible embodiment comprises: a transmitter 10, a receiver 11, a transmission channel 12, a channel condition probability distribution estimator 13 and an optional input signal probability distribution optimizer 14. The transmitter 10 transmits a message conveyed by a signal belonging to a finite set of signals, each signal being associated with a transmission probability according to an (estimated, or optionally, optimized) input signal probability distribution. The transmitter 10 takes as input a message and an optionally estimated (or optionally, optimized) input signal probability distribution, and outputs a signal to be transmitted on the channel 12. The channel 12 takes as input the transmitted signal and outputs a received signal, which is processed at the receiver 11 in order to decode the transmitted message.

[0082] The channel conditional probability distribution that calculates the probability of outputting a given signal for a fixed input. The probability distribution can generally be defined over discrete or continuous input and / or output alphabets. Preferably, a continuous output alphabet is considered, and in this case the probability distribution is referred to as a "probability density function".

[0083] The channel conditional probability distribution estimator 13 takes as input the received signal and the input signal or its estimated (or optionally, optimized) probability distribution, and outputs the channel conditional probability distribution. The channel conditional probability distribution estimator 13 is preferably located in the receiver 11, the transmitter 10, or an external computing device.

[0084] Then, the input signal probability distribution optimizer 14 can take the conditional probability distribution estimate as input and output the optimized input signal probability distribution to the transmitter 10 and the receiver 11. Then, in this embodiment, the conditional probability distribution estimate can be used to calculate the optimized input signal probability distribution at the input signal probability distribution optimizer 14. Moreover, it is shown below that when the conditional probability distribution estimate is approximated by a mixture of exponential distributions, the optimization can be made more efficient.

[0085] The receiver 11 takes as input the received signal, the optimized input signal probability distribution, and the estimated channel conditional probability distribution, and performs an estimation of the message conveyed in the received signal.

[0086] In this example of the embodiment, the conditional probability distribution estimator 13 preferably implements the following steps as Figure 2 illustrated:

[0087] · S21: Obtain information related to the transmitted symbol x, such as the symbol x itself or the input signal distribution p X (x) used at the transmitter 10.

[0088] ο The input signal distribution p X (x) can be provided by the input signal distribution optimizer, or

[0089] ο The input signal distribution p X (x) can be arbitrarily fixed by the transmitter (as presented in the following embodiments);

[0090] · S22: Receive the signal y transmitted through the channel, characterized by the channel conditional probability distribution p Y|X (y|x).

[0091] ο The channel conditional probability distribution p Y|X (y|x) can be known, or

[0092] ο The channel conditional probability distribution p Y|X(y|x) is unknown;

[0093] ·S23: When it is unknown, estimate the channel conditional probability distribution from the received signal, and the channel conditional probability distribution is approximated by the exponential distribution model presented in detail below;

[0094] ·S24: Send the estimated channel conditional probability distribution to the input signal distribution optimizer 14 (as a preferred embodiment) and the receiver 11.

[0095] In an exemplary embodiment, the input signal distribution optimizer 14 may implement Figure 3 the steps illustrated, where:

[0096] ·S31: Obtain the estimated channel conditional probability distribution p Y|X (y|x)

[0097] ·S32: Optimize the input distribution p X (x)

[0098] ·S33: Send the optimized input signal distribution to the transmitter 10, the receiver 11, and the channel conditional probability distribution estimator 13.

[0099] The transmitter 10 preferably implements Figure 4 the steps illustrated, where:

[0100] ·S41: Obtain the message to be sent

[0101] ·S42: Obtain the optimized input signal distribution p X (x)

[0102] ·S43: Create a signal to be sent for transmitting the message according to the optimized input signal distribution p X (x). This can be accomplished by using techniques of the prior art, typically involving a distribution matcher and coding modulation, such as:

[0103] -LDPC (“Low-Density Parity-Check”), as disclosed, for example, in “Bandwidth efficient and rate-matched low-density parity-check coded modulation” by G. Bocherer, F. Steiner, and P. Schulte, IEEE Trans. Commun, December 2015, Vol. 63, No. 12, pp. 4651–4665,

[0104] - or a polar code, as disclosed, for example, in: "Polar coded probabilistic amplitude shaping for short packets" by T. Prinz, P. Yuan, G. Bocherer, F. Steiner, O. Iscan, R. Bohnke, W. Xu, Int. workshop. On Signal Processing Advances in Wireless Communications (SPAWC), July 2017, pages 1 - 5,

[0105] · S44: And transmit the signal on channel 12.

[0106] The receiver 11 can implement Figure 5 the steps illustrated, where:

[0107] · S51: Obtain the received signal y

[0108] · S52: Obtain the optimized input signal distribution p X (x)

[0109] · S53: Obtain the estimated channel conditional probability distribution p Y|X (y|x)

[0110] · S54: Estimate the transmitted message.

[0111] The purpose of the conditional probability distribution estimator 13 is to estimate the distribution p Y|X (y|x) based on an approximation of the channel probability distribution, where the approximation of the channel probability distribution depends on a basis decomposed into a probability distribution function g(y|x; θ), where θ is a set of parameters. The distribution function is an exponential family, and the parameters are basically the mean and variance in the scalar case, and more generally, the mean vector and covariance matrix in the multivariate case. For one example, the set of parameters θ can include the mean vector and covariance matrix of the so-called "multivariate normal distribution". In another example, it can include the shape parameter and the spread parameter of the Nakagami distribution.

[0112] Generally, the function g(y|x; θ) under consideration is written in the following form

[0113] g(y|x; θ) = h(y, θ) exp(x T y - a(x, θ)),

[0114] where x, h(y, θ) are functions of y and θ, and a(x, θ) is the moment generating function. (x and y are vectors in this general case).

[0115] The conjugate prior of the above exponential model has the following form:

[0116] g(θ|λ) = g′(θ)exp(λ1 T θ - λ2a(θ) - b(λ1, λ2)),

[0117] where λ = {λ1, λ2}, g′(θ) is a scalar function of θ, the parameter λ1 has the same dimension as θ, λ2 is a scalar, and b(λ1, λ2) is the moment generating function of this prior chosen such that the prior integral is one.

[0118] Therefore, the purpose of the conditional probability distribution estimator 13 is to find the best approximation of p Y|X (y|x) in terms of the aforementioned basis:

[0119]

[0120] where N, and the sets {θ j}, {w j} are parameters to be estimated. The weight w j is a scalar parameter.

[0121] The function p Y|X (y|x) is a bivariate with variables x and y typically spanning a continuous domain.

[0122] In the first case, assume that the symbol x belongs to a finite alphabet Ω = {ω1,…,ω M} of cardinality M, which is specifically fixed and known. In the first embodiment according to the "training mode", the symbol x is known at the receiver. In the second embodiment according to the "tracking mode", the symbol x is unknown at the receiver, but the input signal probability distribution p X (x) is known.

[0123] In the second case, the symbol x can belong to a finite set of signals, but the set is not specifically fixed nor known. In the third embodiment according to the "training mode", the symbol x is known at the receiver. In the fourth embodiment according to the "tracking mode", the symbol x is unknown.

[0124] In the first case (same as the first and second embodiments above), the conditional probability distribution is fully characterized by the following formula

[0125]

[0126] where p X (x = ω m ) is one except for ω mEverywhere outside is null, where the value is equal to the transmitted symbol ω m probability. Thus, for the M possible values of x, seek to approximate only the M functions p of y Y|X (y|x = ω m ).

[0127] This can be done in parallel by using the M estimators stated as follows.

[0128] Therefore, now consider the following approximation:

[0129]

[0130] Here, each transition depends on the transmitted symbol (M possibilities, where M is the cardinality of the known symbol set in the first case shared by the first embodiment and the second embodiment).

[0131] Solving this estimation problem of Equation (1) is usually difficult. In fact, the number N m of the best approximations provided is unknown, which makes it difficult to use deterministic methods such as the expectation maximization algorithm.

[0132] Multiple embodiments are possible for the conditional probability distribution estimation and differ according to the knowledge of x or the assumptions about its statistics.

[0133] The embodiments are described below specifically for the conditional probability distribution estimation performed by the estimator 13. As will be understood below, these embodiments do not necessarily require any optimizer 14 (which is optional as shown above).

[0134] In the first embodiment related to the "training mode", it is assumed that the symbol x transmitted by the transmitter is known at the conditional probability distribution estimator 13. These symbols are used as pilot symbols in order to learn the best approximation as Figure 6 shown.

[0135] In this first embodiment related to the training mode, and in the second embodiment related to the tracking mode described in detail below, the optimizer 14 is not required, and finally only the estimation of p Y|X (y|x) is required at the receiver 11.

[0136] The training mode is enabled at the transmitter 10 through given time / frequency resources known at the receiver 11, which also knows the transmitted message or the equivalent x. Therefore, referring to Figure 6 , the symbol x is specifically provided to the estimator 13. Generally, the knowledge of x can also be obtained at the receiver after correct decoding (data - assisted training). This can be checked by, for example, a cyclic redundancy check for the correctness of the decoded message.

[0137] When x is completely known at the receiver, solving Equation (2) can be achieved by using the method described based on Gibbs sampling, as disclosed, for example, in

[0138] NEAL00: Neal, Radford M. "Markov chain sampling methods for Dirichlet process mixture models." Journal of Computational and Graphical Statistics 9.2 (2000): 249 - 265.

[0139] It provides a computationally efficient and high - performance solution, especially when the probability distribution function g(y|x = ω m ; θ) belongs to the exponential family. Gibbs sampling is a randomized algorithm, i.e., it uses the generated random values to perform an approximation of statistical inference, rather than relying on deterministic methods. Usually, this involves an iterative method. Using sampling allows reducing the complexity by manipulating a finite number of elements instead of continuous functions. Using Gibbs sampling especially allows efficiently manipulating complex multivariate functions representing multivariate probability distributions used in statistical inference methods. Thus, the main principle for performing the estimation in (2) is to sample from the posterior distribution of the parameters {θ m,j}, {w m,j} with respect to the known observed sample y (i.e., obtaining a representative sample). In particular, the Dirichlet process can be used as the prior probability distribution in an infinite mixture model. It is a distribution over the possible parameters of the multinomial distribution and has the favorable property of being the conjugate prior of the multinomial distribution. This property helps simplify some steps of Gibbs sampling. Also, it is known that the Dirichlet process leads to few dominant components. Thus, by finding these N m dominant components, the approximation is compact and accurate.

[0140] Some modifications to the teachings of NEAL00 used in this paper are needed. Assume that x belongs to a finite alphabet known at the receiver. Associate each observed value y i with a given symbol x i among the M possible values of x, and feed one of the M parallel estimators according to the value of x i (the estimator is then one of the three estimators described in [NEAL00]). Finally, obtain M functions p Y|X (y|x = ω m ) from the M estimators. In particular, the parameters N m , {θ m,j}, {w m,j} are the outputs of the respective estimators and allow characterizing the M functions p Y|X (y|x = ωm )。

[0141] Figure 7 This is a visual example of an implementation where x belongs to {+1, -1, …, -3} here.

[0142] Details of an example of an implementation are given below with respect to the training mode, as Figure 11 shown. In a first implementation, when the received signal y is associated with the transmitted symbol x (such as x = ω m ), only the received signal y can be selected.

[0143] In step S10, the conditional probability distribution estimation is based on an approximation of the channel probability distribution, and the approximation of the channel probability distribution depends on the basis decomposed into the probability distribution function g(y|θ m,j ), where θ m,j is a set of parameters. The distribution function is an exponential family, and the parameters are basically the mean and variance in the scalar case, and more generally, the mean vector and covariance matrix in the multivariate case.

[0144] In step S11, the conditional probability distribution estimator can find the best approximation of p Y|X (y|x = ω m ) in the foregoing basis:

[0145]

[0146] where N m , and the set {θ m,j}, {w m,j} are the parameters to be estimated in step S12.

[0147] Solving this estimation problem is usually difficult. In fact, the number N m providing the best approximation is unknown, which makes it difficult to use deterministic methods such as the expectation-maximization algorithm.

[0148] However, the use of clustering described below with reference to Figure 11 steps S121 to S124 and more generally the Gibbs sampling-based method described in NEAL00 provide computationally efficient and high-performance solutions, especially when the probability distribution function g(y|x = ω m ; θ m,j ) belongs to the exponential family. Gibbs sampling is a randomized algorithm: it uses the generated random values to perform an approximation of statistical inference instead of relying on deterministic methods. Generally, this involves an iterative method.

[0149] Using sampling allows reducing complexity by manipulating a finite number of elements instead of a continuous function. Using Gibbs sampling in particular allows efficiently managing complex multivariate functions representing multivariate probability distributions used in statistical inference methods. Thus, the main principle for performing the estimation in (2) is that, at the input of the m-th estimator, samples are drawn (i.e., representative samples are obtained) from the posterior distribution of the parameters {θ m,j},{w m,j} with respect to the known observed samples y, i.e., after selecting the observed symbol y associated with the transmitted symbol x = ω m known to the receiver, i.e., for the time slots in which the transmitted symbol x = ω m is known, the observed symbol feeds the m-th estimator.

[0150] In particular, the Dirichlet process can be used as a prior probability distribution in an infinite mixture model. It is a distribution over the possible parameters of a multinomial distribution and has the property of being a conjugate prior of the multinomial (which helps simplify some steps of Gibbs sampling, as explained in [NEAL00]). Moreover, it is known that the Dirichlet process results in few dominant components. Thus, by finding these N m dominant components, the approximation is compact and accurate.

[0151] To determine these components in a tractable way, Monte Carlo methods (and more specifically, Markov chain Monte Carlo) can be applied, thus allowing samples to be drawn from a probability distribution. More specifically, Gibbs sampling provides an implementation of the relevant algorithm classes of the present disclosure. Gibbs sampling methods typically involve numerical integration steps, which are computationally demanding. Fortunately, when the probability distribution function g(y|x; θ) is a conjugate prior distribution (which is the case for the exponential family), the integration can be performed in a closed-form expression with significantly reduced computational complexity. Since o Y|X (y|x = ω m ) is a weighted sum of g(y|x; θ) functions, its conjugate prior is also known and can be computed efficiently. At the output of Gibbs sampling, the values N m , as well as the sets {θ m,j},{w m,j} are estimated, as shown in the total step S12 of Figure 11 .

[0152] In the current case, the collapsed Gibbs sampling of [NEAL00] is used in the specific implementation shown in Figure 11 , and is applied as follows in the specific case of the m-th estimator:

[0153] · In step S10, it is considered that the base g(y|θ) is from the exponential distribution family.

[0154] This allows considering the prior on θ to be conjugate to an exponential distribution with parameter λ. This is a mathematical trick that allows for a closed - form expression for the following steps.

[0155] λ = {λ1, λ2} represents the hyperparameters of the conjugate prior for the parameter.

[0156] · In step S11, perform the selection of the observed symbol {y m} associated with the transmitted symbol x = ω n}, where y n is the n - th symbol among the selected observed symbols. The expression x 1:n = ω m represents the fact that all transmitted symbols considered here are the same and equal to ω m .

[0157] Then, this overall step S12 can be decomposed into sub - steps S121 to S124 as follows:

[0158] · In step S121, consider N m,j clusters of the parameter sets {θ m,j}, {w m}:

[0159] ο Use the clusters to group the received symbols together. The clustering is achieved by associating an integer index from 1 to N m with each symbol group, as described below. This clustering assignment method can be justified by the fact that each received symbol is more likely to be associated with one of the N patterns that make up p Y|X (y|x = ω m ) in (2).

[0160] ο The clustering assignment index of the i - th observed symbol is c i .

[0161] ο Let c -i be the clustering assignment indices of all other clusters except the i - th cluster.

[0162] · In step S122, under the assumption that the transmitted symbol is ω m , calculate the posterior distribution p(c i = k|y 1:n , c -i , λ, x 1:n = ω m ), where y 1:n represents the input symbol of the received signal y. It can be shown that

[0163] ο p(ci = k|y 1:n , c -i , λ, x 1:n = ω m ) = p(c i = k|c -i )p(y i |y 1:i-1 , y i+1:n , c -i , c i = k, λ, x 1:n = ω m )

[0164] οp(c i = k|c -i ) For example, it is calculated due to the Polya urn model

[0165] οp(y i |y 1:i-1 , y i+1:n , c -i , c i = k, λ, x 1:n = ω m ) = ∫p(y i |x i = ω m , θ)p(θ|y 1:i-1 , y i+1:n , c -i , c i = k, λ, x 1:n = ω m )dθ, which is computable because p(y i |x i = ω m , θ) follows an approximation where g(y|θ) is from the exponential distribution family, which involves a known conjugate prior

[0166] · In step S123, sampling is performed starting from the posterior distribution to update the parameters {θ m,j}, {w m,j}.

[0167] · It is also preferable to check whether a new cluster must be created. This helps to determine the optimal value N that provides the best approximation of the channel conditional probability distribution m . This is done in step S124 by first starting a new cluster, which is defined as

[0168] p(y i |x i = ω m , P0) = ∫p(y i |xi = ω m , θ) dP0

[0169] where P0 is the conjugate of p(y|x = ω m ; θ m,j )(with parameter λ). Then, a random selection is made from this probability to create a new cluster. Thus, a random selection is made from the probability that validates this probability to create a new cluster.

[0170] At the output of this process, the parameters N m , {θ m,j , {w m,j} are estimated, and an approximation of the channel conditional probability distribution p Y|X (y|x = ω m ) from (2) is provided. This process is repeated or executed in parallel for M estimators.

[0171] The second embodiment is given below, which implements a "tracking mode" using statistical knowledge. This mode is used efficiently after the first embodiment. In this embodiment, the transmitted symbol x is not known, but as shown below Figure 9 it is still possible to use the knowledge of p X (x) to evaluate (2), and then provide it to estimator 13 (as well as receiver 11 and transmitter 10). In fact, there is no step of selecting from the observed symbols. However, M estimators are still calculated, and any observed symbol is considered as an input. The purpose of Bayesian inference is to weight the observed symbols according to the confidence that the observed symbols correspond to the transmission of the symbol x = ω m associated with the m-th estimator.

[0172] The main difference from the training mode is that the transmitted symbol x is unknown. However, the value of x can be inferred using Bayesian methods, i.e., each observed value y contributes to updating p m (y|x = ω Y|X ) with a weight corresponding to the probability associated with y being related to ω m according to the current estimate of the conditional probability distribution.

[0173] Details of the second embodiment for implementing the aforementioned tracking mode are given below.

[0174] Clustering techniques will be used. If (θ i ) belongs to the j-th cluster, let the cluster assignment label be s i = j. Let ρ n = [s1,..., s n T ​A vector representing the cluster assignment labels. Obtained using predictions in a Dirichlet process (DP) according to a covariate-dependent urn model:

[0175]

[0176] For each estimator, in order to update the conditional probability distribution p Y|X (y|x = ω m ), one can compute

[0177]

[0178] where ζ 0,y (y|x n+1 ) = ∫p(y|x n+1 , θ)dP0,

[0179] Then, the clusters can be updated similarly to the first embodiment. For example, details of the prediction principle in a Dirichlet process (DP) according to a covariate-dependent urn model are given in the following literature: "Improving prediction from Dirichlet process mixtures via enrichment", Sara K. Wade, David B. Dunson, Sonia Petrone, Lorenzo Trippa, Journal of Machine Learning Research (Journal of Machine Learning Research, November 15, 2013)

[0180] In the third embodiment, the symbol x is not assumed to belong to a finite alphabet. Therefore, for all possible x values, the bivariate conditional probability p Y|X (y|x) must be learned.

[0181] In step S10, the conditional probability distribution estimation is based on an approximation of the channel probability distribution, which approximation depends on a basis decomposed into a probability distribution function g(y|x, θ j ), where θ j is a set of parameters. The distribution function is an exponential family, and the parameters are basically the mean and variance in the scalar case, and more generally, the mean vector and covariance matrix in the multivariate case. The conditional probability distribution estimator can find the best approximation of p Y|X (y|x) in terms of the aforementioned basis:

[0182]

[0183] where N, and the sets {θ j}, {wj} are the parameters to be estimated.

[0184] This solution relies on the same algorithm as in the first embodiment, with the main differences as follows: there is only one estimator instead of M estimators. The transmitter will send the symbol x that is known at the receiver. The performance will be improved by using these transmitted symbols from a pseudo-random generator that will generate x values over the domain of interest (e.g., following a Gaussian distribution). By knowing the initial parameters of the pseudo-random generator, both the transmitter and the receiver can know the x symbols.

[0185] The estimator uses the folded Gibbs sampling method as in the first embodiment. In the current case, again, the folded Gibbs sampling of NEAL00 is used in the specific embodiment shown, and it is applied in the specific case for the estimator as: Figure 11 In step S10, consider that the base g(y|x,θ) is from the exponential distribution family.

[0186] · This allows considering that the prior about θ is conjugate to the exponential distribution with parameter λ. This is a mathematical trick that allows having a closed-form expression for the following steps.

[0187] ο λ = {λ1,λ2} represents the hyperparameters of the conjugate prior about the parameters.

[0188] ολ = {λ1,λ2} represents the hyperparameters of the conjugate prior about the parameters.

[0189] · Here, step S11 now corresponds to the selection of the observed symbols {y n} associated with the transmitted symbols {x n}.

[0190] This overall step S12 can be decomposed again into:

[0191] · In step S121, consider N clusters of the parameter sets {θ j}, {w j}.

[0192] ο The clusters are used to group the received symbols together. The clustering is achieved by associating an integer index from 1 to N with each symbol group, as described below. This clustering assignment method can be justified by the fact that each received symbol is more likely to be associated with one of the N modes that make up p Y|X (y|x) in (2).

[0193] ο The clustering assignment index of the i-th observed symbol is c i .

[0194] ο Let c -i be the clustering assignment indices of all other clusters except the i-th cluster.

[0195] · In step S122, calculate the posterior distribution p(c i = k|y 1:n , x 1:n , c -i , λ), where y 1:n represents the input symbol of the received signal y. It can be shown that

[0196] ο(c i = k|y 1:n , x 1:n , c -i , λ) = p(c i = k|c -i )p(y i |y 1:i-1 , y i+1:n , x 1:n , c -i , c i = k, λ)

[0197] οp(c i = k|c -i ) is calculated, for example, due to the Polya urn model

[0198] οp(y i |y 1:i-1 , y i+1:n , x 1:n , c -i , c i = k, λ) = ∫p(y i |x i , θ)p(θ|y 1:i-1 , y i+1:n , x 1:n , c -i , c i = k, λ)dθ, which is computable because p(y i |x i , θ) follows an approximation and g(y|x, θ) is from the exponential distribution family, which involves known conjugate priors.

[0199] · In step S123, perform sampling starting from the posterior distribution in order to update the parameters {θ m}, {w m}.

[0200] · It is also preferable to check whether a new cluster must be created. This helps to determine the optimal value N m that provides the best approximation of the channel conditional probability distribution. This is done in step S124 by first starting a new cluster, which is defined as

[0201] p(y i |x, P0) = ∫ p(y i |x, θ) dP0,

[0202] where P0 is the conjugate of p(y|x; θ j )(with parameter λ), which is computable because p(y|x; θ j ) is a weighted sum of g(y|x, θ j ) functions belonging to the exponential family. Then, a random selection is made from this probability to create a new cluster. Thus, a random selection is made from the probability that validates this probability to create a new cluster.

[0203] Details of the use of the Polya urn model can be obtained from the following literature:

[0204] “Ferguson distributions via Polya urn schemes”, D. Blackwell and J. B. MacQueen, The Annals of Statistics, 1973, Vol. 1, No. 2, pp. 353–355.

[0205] In a fourth embodiment involving a tracking pattern related to the aforementioned second case, attention is paid to the fact that the input does not belong to a finite alphabet and the transmitted symbol is unknown. This pattern is used efficiently after the third embodiment. The model for the non-parametric joint density is defined as:

[0206]

[0207] The parameters {θ j}, {w j}, and {ψ j} represent conditional density parameters, which can be the mean and covariance, magnitude, and corresponding parameters of the input distribution in the case of Gaussian density, respectively. Generally, the parameters (θ, ψ) are jointly obtained from the base measure of the Dirichlet process “DP”, i.e., (θ, ψ) ~ DP(αP 0θ ×P 0ψ ), where α is a scaling parameter. The corresponding conditional probability density can be written in the following non-parametric form

[0208]

[0209] It should be noted that in the prediction stage, considering the fact that the placement of the particles is fixed and optimized according to the training: the denominator in (22) and p(x|ψ j ) act as scaling factors.

[0210] Clustering techniques will be used. If (θi , ψ i If it belongs to the j-th cluster, then set the cluster assignment label as s i = j. Let ρ n = [s1, …, s n T denote the vector of cluster assignment labels. Using the prediction in the DP according to the covariate-related urn model, obtain:

[0211]

[0212] where

[0213]

[0214] where n j is the number of topic indices in the j-th cluster.

[0215] Further assume that in the tracking phase of channel probability density estimation, the particle set is fixed. This is a practical assumption because the positions of particles in the constellation are fixed in most practical applications. Here, the notation x n+1 ∈ x 1:n is used to emphasize sending a new signal from the set of fixed constellation points used during training. The estimated density p(y|y 1:n , x 1:n , x n+1 ∈ x 1:n ) for the newly received signal y is obtained as

[0216] p(y|y 1:n , x 1:n , x n+1 ∈ x 1:n ) =

[0217] ∑ ρn ∑ sn+1 p(y|y 1:n , x 1:n , x n+1 ∈ x 1:n , ρ n , s n+1 )

[0218] × p(s n+1 |y 1:n , x 1:n , x n+1 ∈ x 1:n , ρ n )

[0219] × p(ρ n |y 1:n , x 1:n , x n+1 ∈ x​1:n ),(27)

[0220] where, if (θ i , ψ i ) belongs to the j-th cluster, then ρ n = [s1, …, s n T represents the cluster assignment label vector with s i = j. Using (24), (27) can be simplified to

[0221]

[0222] where h is the number of groups in the partition ρ n and

[0223]

[0224] where represents the input set and output set of the j-th cluster. Thus, the estimated conditional probability density is obtained according to Equation (28).

[0225] After any one of the above embodiments, an estimate of the conditional density probability function p T|X (y|x) is obtained. In the first two embodiments, it is obtained for a fixed constellation of the transmitted symbol x, while for the third and fourth embodiments, it is referred to as a bivariate function of y and x, and in particular for any value of x.

[0226] This knowledge of p Y|X (y|x) is used in the receiver to calculate the likelihood of each received symbol y assuming the transmitted symbol x. In maximum likelihood decoding of the transmitted symbol, this likelihood is required by selecting the symbol x that maximizes the likelihood of any received symbol x. This likelihood is also a component for calculating the log-likelihood ratio provided at the input of a soft-input decoder, such as in: "Soft-input soft-output lattice sphere decoder for linear channels" by J. Boutros, N. Gresset, L. Brunel, and M. Fossorier, GLOBECOM'03. IEEE Global Telecommunications Conference (IEEE Catalog Number 03CH37489), San Francisco, California, 2003, Volume 3, pp. 1583 - 1587.

[0227] Therefore, the receiver is interested in knowing the estimate of p Y|X (y|x).

[0228] ​The present invention can be advantageously used to provide a compact representation of p j (y|x) through parameters {w j} and {θ Y|X} provided to the receiver by, for example, a channel conditional probability distribution estimator.

[0229] In another option, knowledge of p Y|X (y|x) can be advantageously used at the transmitter to optimize the input distribution. In fact, based on knowledge of a given distribution p X (x) (which is discrete with fixed x positions and varying probabilities in most applications), the capacity of the channel characterized by p X (x) with input p Y|X (y|x) can be evaluated. Thus, the input distribution can be optimized.

[0230] In a first case of such optimization, the input distribution that provides the highest capacity among a predefined set of input distributions is selected. Such a set is obtained, for example, by sampling Gaussian input distributions with different variance values. For example, sampling is performed on the x positions following a QAM (e.g., 256QAM) constellation. In a second case of such optimization, the set is provided by a plurality of known constellations, such as QPSK, 16-QAM, 64-QAM, 256QAM, 8-PSK, 32-PSK, etc. In a third case of such optimization, the positions of the symbols x and their associated probabilities are randomly selected, and the best-found random constellation is selected after each capacity calculation.

[0231] In a last case of such optimization, it is advantageous to select a function g(y|x,θ j ) from the exponential family. In fact, p Y|X (y|x) can be calculated in closed form for a fixed value of x as a linear combination of such functions, and thus the derivative function p Y|X (y|x) can be calculated. Therefore, gradient descent methods can be used to optimize the capacity with respect to the input distribution p X (x).

[0232] The present invention can be advantageously used to provide a compact representation of p j (y|x) through parameters {w j} and {θ Y|X} provided to the transmitter by, for example, a channel conditional probability distribution estimator.

Claims

1. A method for optimizing the capacity of a communication channel in a communication system, the communication system at least including a transmitter, a receiver, the communication channel between the transmitter and the receiver, and a channel conditional probability distribution estimator; The transmitter transmits a message conveyed by a signal associated with a transmission probability of an input signal probability distribution estimated according to the estimator, The transmitter takes at least the message as an input and outputs a signal to be transmitted on the channel, the channel takes the transmitted signal as an input and outputs a received signal, and the received signal is processed at the receiver to decode the transmitted message, is denoted as p Y|X The probability distribution of (y|x) is related to the output Y corresponding to the received signal given the input X corresponding to the transmitted signal X, and thus represents the channel conditional probability distribution of the probability of outputting the received signal Y given the transmitted signal X. Among them, By using a functional basis of probability distributions, estimating the probability distribution as an approximation of the channel conditional probability distribution p Y|X (y|x), whereby the channel conditional probability distribution is approximated for each possible signal transmitted by the transmitter by means of a mixture model having a mixture probability distribution from the functional basis, and the probability distribution for each possible transmitted signal is estimated at least from the output signal received at the receiver and using collapsed Gibbs sampling that depends on a Dirichlet process.

2. The method according to claim 1, wherein The function basis is a family of exponential functions.

3. The method according to claim 2, wherein, The probability distribution estimation is based on an approximation of the channel conditional probability distribution, which depends on a basis decomposed into a probability distribution function g(y|x; θ), where θ is a set of parameters θ j , the distribution function is an exponential family, and the parameters are the mean and variance, such that the approximation of p Y|X (y|x) by the basis of the probability distribution function p(y|x; θ) is given by , where N and the set {θ j}, {w j} are parameters to be estimated.

4. The method according to claim 3, wherein The probability distribution function g(y|x;θ) comes from the exponential distribution family, and the prior about θ is conjugate to the corresponding exponential distribution with parameter λ, where λ = {λ1,λ2} represents the hyperparameter of the conjugate prior about the parameter λ, And wherein: - N clusters are constructed for the parameter sets {θ j}, {w j}, and each cluster is used to group symbols together, and the clustering is achieved by associating an integer index from 1 to N with each symbol group. - The clustering index is assigned to the i-th sample denoted as c i such that c -i becomes the clustering assignment index for all other clusters except the i-th cluster, and the posterior distribution p(c i = k|x 1:n , y 1:n , c -i , λ) is calculated, where x 1:n represents the symbol of the transmitted signal, and y 1:n represents the input symbol of the received signal. - And perform sampling from the posterior distribution in order to update the parameters {θ j},{w j}, thereby estimating the parameters N, {θ j},{w j} to provide an approximation of the channel conditional probability distribution 5. The method according to claim 4, wherein, The updated parameters {θ j}, {w j} are sent to the transmitter and / or the receiver to calculate the conditional probability distribution estimate.

6. The method according to claim 4 or 5, wherein The posterior distribution: p(c i = k|x 1:n , y 1:n , c -i , λ) is given by the following formula: p(c i = k|x 1:n , y 1:n , c -i , λ) = p(c i = k|c -i )p(y i |x 1:i-1 , x i+1:n , y 1:i-1 , y i+1:n , c -i , c i = k, λ), where: -p(c i = k|c -i ) is calculated according to the Polya urn model, - And (y i |x 1:i-1 , x i+1:n , y 1:i-1 , y i+1:n , c -i , c i = k, λ) = ∫ p(y i |x i , θ) p(θ|x 1:i-1 , x i+1:n , y 1:i-1 , y i+1:n , c -i , c i = k, λ) dθ。 7. The method according to claim 6, wherein It also checks whether a new cluster is to be created to determine the best value of the parameter N that provides the optimal approximation of the channel conditional probability distribution.

8. The method according to claim 7, wherein, A random selection is made from the probability of verifying the following formula to create a new cluster: p(y i |x i ; P0) = ∫p(y i |x i ; θ)dP0, where P0 is the conjugate of p(y|x; θ).

9. The method according to any one of claims 1 to 5, wherein the method is only applied to a predefined value of the input X, and wherein, The input X is known, and only when the input X is equal to the predefined value, the associated output Y in the training mode is considered.

10. The method according to any one of claims 1 to 5, wherein In the tracking mode, consider using the input signal probability distribution p X (x) for the channel conditional probability distribution estimation, and use the Bayesian method to infer the value of the transmitted symbol x, such that each observed value y helps to update p Y|X (y|x) according to the current estimate of the conditional probability distribution, using the weight corresponding to the probability associated with y and x.

11. The method according to claim 10, wherein, The model for the non-parametric joint density is defined as: Among them, the parameters {θ j}, {w j}, and {ψ j} represent conditional density parameters, and j is the clustering index. The parameters (θ, ψ) are jointly obtained from the base measure of the Dirichlet process "DP" such that (θ, ψ) ~ DP(αP 0θ ×P 0ψ ), where α is the scaling parameter.

12. The method according to any one of claims 1 to 5, wherein, The set of transmitted signals is represented by a set of corresponding symbols, and the parameters of the hybrid model are optimized to define the optimized symbol positions and probabilities to be used at the transmitter so as to optimize the capacity of the approximated channel, and then the optimized symbol positions and probabilities are provided to the transmitter and / or the receiver.

13. A system, the system comprising at least a transmitter, a receiver, a communication channel between the transmitter and the receiver, and a channel conditional probability distribution estimator for implementing the method according to any one of claims 1 to 11, wherein, The estimator: · Obtain information related to the signal (x; p X (x)) transmitted and used at the transmitter; ● Receive the signal y sent through the channel and characterized by the channel conditional probability distribution p Y|X (y|x); ● Estimates the channel conditional probability distribution from the received signal, and the channel conditional probability distribution is approximated by a distribution model dependent on the basis of the function; ● Sends the estimated channel conditional probability distribution to the receiver and / or the transmitter.

14. The system according to claim 13, the system further including an input signal probability distribution optimizer for performing the following method: The set of transmitted signals is represented by a set of corresponding symbols, and the parameters of the hybrid model are optimized to define the optimized symbol positions and probabilities to be used at the transmitter so as to optimize the capacity of the approximated channel, and then the optimized symbol positions and probabilities are provided to the transmitter and / or the receiver, Among them, The input signal distribution optimizer: · Obtain the estimated channel conditional probability distribution p Y|X (y|x), · Optimize the input distribution p X (x), · Sends the optimized input signal distribution to the transmitter and / or the receiver, and to the channel conditional probability distribution estimator.

15. A computer program product including computer instructions, which when executed by a processor implement the method according to any one of claims 1 to 12.

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