A performance boundary characterization and optimization method for wireless covert communication with limited character input

Abstract

The present invention discloses a performance boundary characterization and optimization method for wireless covert communication with limited character input, comprising the following steps: (1) modeling the receiving end signal of wireless covert communication under limited character input, and using mutual information and KL divergence to describe the transmission performance and concealment of covert communication; (2) based on Taylor series expansion technology, accurately approximating the mutual information and KL divergence of covert communication, and obtaining closed-form solution expressions of mutual information and KL divergence; (3) accurately characterizing the performance boundary of wireless covert communication under limited character input; (4) optimizing the composition of the codebook, including the selection probability of constellation points, i.e., probabilistic constellation formation; the present invention gives for the first time the closed-form solution expressions of mutual information and KL divergence of wireless covert communication under limited character input, and separates the performance boundary characterization of covert communication under limited character input from the operation of complex integrals.

Description

Technical Field

[0001] The present invention relates to the field of communication technology, and in particular to a performance boundary characterization and optimization method for wireless covert communication with limited character input. Background Art

[0002] Wireless covert communication is a special communication method that mainly pursues concealment. It has important application prospects in complex confrontation environments and scenarios with extremely high security requirements. At present, the mainstream signal input for covert communication is Gaussian input. Gaussian input means that the input of the signal conforms to the Gaussian distribution (also known as the normal distribution). In digital communication systems, this usually means that the input of the signal is a continuous, smooth, infinite number of sample sequences whose amplitudes obey the Gaussian distribution. Gaussian input is often used in theoretical analysis and modeling because of the mathematical convenience of the Gaussian distribution.

[0003] It should be emphasized that in actual communication systems, due to the inherent characteristics of digital modulation technology, it is difficult to achieve continuous and unlimited signal inputs. Signal inputs are usually limited to a limited set of symbols (such as BPSK, QPSK, 16-QAM, 64QAM, etc.), and these symbols have a specific position or phase in the signal space. This signal input based on a limited set of symbols is also called finite alphabet inputs.

[0004] In addition, for Gaussian input signals and finite character input signals, existing theoretical simulations and engineering practices have shown that the two have significant analytical differences and obvious performance gaps in communication scenarios. More importantly, in view of the high security requirements of concealment constraints, the two have huge differences in statistical characteristics, performance analysis and boundary characterization. The existing analysis methods for Gaussian input cannot accurately characterize the performance boundary of covert communication under finite character input. Summary of the invention

[0005] Purpose of the invention: The purpose of the present invention is to provide a performance boundary characterization and optimization method for wireless covert communication with limited character input, so as to solve the problems existing in the background technology.

[0006] Technical solution: The performance boundary characterization and optimization method for wireless covert communication with limited character input described in the present invention includes the following steps:

[0007] (1) Model the receiving end signal of wireless covert communication under limited character input, and use mutual information and KL divergence to describe the transmission performance and concealment of covert communication;

[0008] (2) Based on the Taylor series expansion technology, the mutual information and KL divergence of covert communication are accurately approximated, and the complex integral operation is converted into a simple statistical summation operation to obtain the closed-form solution expressions of the mutual information and KL divergence;

[0009] (3) Based on the closed-form solution of mutual information and KL divergence, the performance boundary of wireless covert communication under limited character input is accurately characterized;

[0010] (4) Based on the obtained performance boundary of covert communication, the composition of the codebook is optimized, including the probability of selecting constellation points, i.e., probabilistic constellation forming.

[0011] Furthermore, step (1) includes the following steps:

[0012] (11) Modeling the codebook;

[0013] (12) Modeling the receiving end model;

[0014] (13) Modeling the detection end model;

[0015] (14) Convert the modeling problem in steps (11)-(13) into a mathematical problem.

[0016] Furthermore, in step (11), an input symbol is randomly selected from a discrete constellation set; the encoded message is an independent and identically distributed random codeword, the selection probability of the constellation point is defined, and the selected codebook is a probabilistic constellation shaping codebook.

[0017] Furthermore, in step (12), the channel transfer matrix and the distribution of received symbols are defined and described according to the input distribution and the constellation point selection probability; based on the additive white Gaussian noise channel, the mutual information is used to characterize the transmission rate.

[0018] Furthermore, in step (13), the channel transfer matrix and the distribution of received symbols are defined and described according to the input distribution and the probability of constellation point selection; based on the distribution of received symbols and the codebook composition, a binary hypothesis test is set; considering the highest degree of concealment, the KL divergence is used to characterize the concealment.

[0019] Furthermore, step (2) includes the following steps:

[0020] (21) The mutual information and KL divergence are approximated by Taylor series, and a preliminary expansion is performed on the transmission amplitude gain to obtain the second-order expansion of the mutual information and the second-order expansion of the KL divergence;

[0021] (22) Analyze the obtained mutual information and KL divergence expansion to obtain the necessary conditions for the optimal input distribution;

[0022] (23) Based on the necessary conditions for the optimal input distribution, the expansion of the mutual information is modified to obtain the closed-form expression of the mutual information in the optimal case;

[0023] (24) The KL divergence is further expanded to the fourth order, and the closed-form expression of the KL divergence in the optimal case is obtained.

[0024] Furthermore, step (4) includes the following steps:

[0025] (41) Transform the mathematical problem in step (14) into a standard convex problem;

[0026] (42) Use the interior point method or the primal-dual interior point method to solve convex problems.

[0027] Beneficial effects: Compared with the prior art, the present invention has the following significant advantages: for the first time, the closed-form solution expressions of mutual information and KL divergence of wireless covert communication under limited character input are given, and the boundary characterization of covert communication performance under limited character input is separated from the operation of complex integrals. Simple statistics can be used to accurately characterize the covert communication performance under limited character input, and a new and convenient method is given for the performance evaluation of covert communication systems under limited character input. In addition, a covert communication optimization method under limited character input is given based on theoretical boundaries, which can achieve stronger concealment and higher transmission rate. BRIEF DESCRIPTION OF THE DRAWINGS

[0028] Figure 1 is a flow chart of the present invention;

[0029] Figure 2 This is a diagram for verifying the accuracy of the Taylor series approximation method of the present invention;

[0030] Figure 3 It is a performance comparison chart of the optimization method of the present invention and the original method. DETAILED DESCRIPTION

[0031] The technical solution of the present invention is further described below in conjunction with the accompanying drawings.

[0032] like Figure 1 As shown, an embodiment of the present invention provides a performance boundary characterization and optimization method for wireless covert communication with limited character input, comprising the following steps:

[0033] (1) Modeling wireless covert communication under limited character input

[0034] The sender's message is randomly encoded. Encoded into a codeword , where the codeword It is generated randomly using independent and identical distribution (iid), and the formula is as follows:

[0035] (1)

[0036] Different from the traditional covert communication optimization, the present invention assumes that the input data does not use the traditional Gaussian codebook. It assumes that the input symbol is from a known discrete constellation set (e.g., -PSK, -QAM). In addition, each discrete constellation set There are N discrete constellation points ,in, , i is the number of the constellation point. And, in the discrete constellation set middle, The probability of choosing is , for each symbol Use its x component and quantity To characterize, .

[0037] The present invention considers optimizing the input distribution, that is, optimizing the selection probability of the constellation points. is not equally distributed and is defined as:

[0038] (2)

[0039] In the above formula The physical meaning of the constellation point The probability of selecting (i.e., the codeword in the codebook) The probability of occurrence of The receiver then uses his knowledge to decode the hidden message. The transmission rate is determined by the discrete input and the channel output at the receiver It is measured by the mutual information between them, which is specifically expressed as:

[0040] (3)

[0041] The receiver has prior knowledge of the construction of the codebook, including all potential discrete constellation sets and input distributions. The symbols he receives are contaminated by additive white Gaussian noise, which can be further given by and The joint distribution of is as follows:

[0042] (4)

[0043] According to the parallel characteristics of the sub-channels, the channel probability transfer matrix It can be expressed as:

[0044] (5)

[0045] in,

[0046] (6)

[0047] in, represents the variance of the complex Gaussian noise of the receiver on the x-axis and y-axis, represents the amplitude coefficient of the channel from the sender to the receiver, Indicates the amplitude gain coefficient of the sender. For constellation points The x-axis component of For constellation points The y-axis component of .

[0048] In order to ensure the highest level of concealment, the present invention takes the worst case as the target scenario for covert communication, that is, the detector knows the composition of the codebook, including the specific location of each constellation point and the transmission probability of each constellation point. In order to determine whether covert communication is in progress, the covert communication detector performs a Specifically, the null hypothesis ( ) indicates that there is no communication, where each sample is an independent and identically distributed complex Gaussian random variable, and its distribution conforms to ; Alternative hypothesis ( ) indicates that communication is in progress. The goal of the covert communication detector is to distinguish between these two hypotheses:

[0049]

[0050] (7)

[0051] set up (Correspondingly, ) indicates that when (Correspondingly, ) is true. The false alarm probability is Reject if True The probability of It means that the probability of missed detection is Accept if false The probability of Indicates. Assuming the distribution and is known to the covert communication detector, and the covert communication detector can perform optimal statistical hypothesis testing ,in yes and The covert communication detector can perform blind detection when the sum of the error probabilities is equal to 1, that is, Therefore, just make sure The value of is very small, so small that it can be ignored, and covert communication can be achieved. Correspondingly, this constraint can be converted into a covert constraint based on KL divergence, that is,

[0052] (8)

[0053] in, It is a very small positive number in mathematical notation, representing "approaching zero" in mathematical terms.

[0054] The covert communication detector has all the knowledge about the codebook construction, including all potential discrete constellation sets and input distributions, and the symbols he receives are interfered by additive Gaussian white noise. Therefore, The distribution of can be expressed as:

[0055] (9)

[0056] and Similar, in The following distribution can be given:

[0057] (10)

[0058] in, yes The marginal distribution can be expressed as:

[0059] (11)

[0060] in, Similar to , and can be expressed as:

[0061] (12)

[0062] The position of the constellation set is fixed and only focuses on optimizing the amplitude gain And the input distribution probability of the constellation set Based on the above objectives, the problem of wireless covert communication under limited character input can be expressed as:

[0063]

[0064]

[0065]

[0066]

[0067] (13)

[0068] (2) Use Taylor series expansion to approximate mutual information and KL divergence

[0069] According to the basic theory of Taylor series expansion and "square root law", at the moment of amplitude gain , the mutual information of hidden information with code length n can be expressed as:

[0070] (14)

[0071] The KL divergence of hidden information can be expressed as:

[0072] (15)

[0073] First, the mutual information in formula (13) is calculated to obtain:

[0074] (16)

[0075] in,

[0076] (17)

[0077] (18)

[0078] in,

[0079] (19)

[0080] Therefore, we can get

[0081]

[0082] (20)

[0083] as well as

[0084] ;

[0085]

[0086] (twenty one)

[0087] Based on algebraic calculations, we can get:

[0088]

[0089] (twenty two)

[0090] and At that moment, we can get:

[0091]

[0092] twenty three)

[0093] Based on Taylor series expansion theory, when At this moment, we can get:

[0094] (twenty four)

[0095] Combining the above formula, we can finally get:

[0096] (25)

[0097] as well as

[0098] (26)

[0099] Combining formulas (16), (24)-(26), we can deduce:

[0100] (27)

[0101] in, The physical meaning is the expected value of the power of the transmitted character (constellation point). is the expected value of the amplitude component of the transmitted character (constellation point) on the x-axis, is the expected value of the amplitude component of the transmitted symbol (constellation point) on the y-axis.

[0102] By transforming the KL divergence in equation (13), the n-dimensional KL divergence can be expressed as:

[0103] (28)

[0104] You can get:

[0105] (29)

[0106] in,

[0107]

[0108] (30)

[0109] For ease of representation, define the following function:

[0110] (31)

[0111] in,

[0112] (32)

[0113] Therefore, we can get

[0114]

[0115] (33)

[0116] Function about Taking the derivative, we can get the first-order and second-order derivatives as:

[0117] (34)

[0118] Similarly, the third-order derivative is:

[0119] (35)

[0120] Similarly, the fourth-order derivative is:

[0121] (36)

[0122] then , we can get the following results:

[0123] (37)

[0124] Combining the above parameters, we can get:

[0125] (38)

[0126] (39)

[0127] (40)

[0128] (41)

[0129] Based on Taylor series expansion theory, the following formula can be obtained:

[0130] (42)

[0131] Furthermore, we can get:

[0132] (43)

[0133] as well as

[0134] (44)

[0135] in, , You can get:

[0136] (45)

[0137] When the conditions are met At this moment, we can get:

[0138] (46)

[0139] (47)

[0140] (48)

[0141] (49)

[0142] Combining formulas (45), (46)-(49), we can get its KL divergence:

[0143] (50)

[0144] At the same time, the transmission rate can be expressed as:

[0145] (51)

[0146] If you meet the conditions or , the KL divergence can be expressed as:

[0147] (52)

[0148] And, the transmission rate can be expressed as:

[0149] (53)

[0150] According to formula (50), when Can get , and for any input distribution , the transmission rate is If the necessary optimal conditions ( or ) The transmission rate is Much smaller than , so it can be determined that the optimal covert communication codebook will satisfy . In the subsequent section, optimization is performed under the conditions of .

[0151] (3) Using closed-form solutions to characterize the performance of covert communication

[0152] Based on the above formula derivation and analysis, the original problem in formula (13) can be converted into:

[0153] (54)

[0154] Known, , , , At this point, the original problem (13) has been converted into a closed-form solution. The obtained mutual information and KL divergence expressions have been greatly simplified compared to the complex integral forms of equations (16) and (29), and retain a high degree of accuracy.

[0155] (4) Optimization of covert communication parameters

[0156] For a known constellation set, that is, a fixed constellation point position, the transmission amplitude and constellation point probability After optimization, formula (54) can be rewritten as:

[0157] (55)

[0158] in,

[0159] (56)

[0160] (57)

[0161] (58)

[0162] The original problem is transformed into a standard second-order cone program (SOCP) problem, which is a classic convex problem and can be solved using common methods including the interior point method and the primal-dual interior point method. The constellation point output probability and transmission power in the codebook are optimized multiple times until convergence is reached and the optimal parameters are output, including the optimal transmission amplitude of covert communication and the constellation point output probability in the codebook, providing theoretical guidance for the actual communication system of covert communication.

[0163] Figure 2 The proposed Taylor series approximation method is demonstrated to have an approximate accuracy of KL divergence and mutual information. Referring to the method of the reference formula, the present invention inversely solves the transmission amplitude in the KL divergence and then brings it into the mutual information, and then verifies the accuracy of the two indicators, mutual information and KL divergence, using only the hiddenness parameter. As shown in the figure, the present invention verifies the BPSK codebook and the QPSK codebook, and the approximate method used is highly accurate, which fully meets the accuracy requirements of the problem optimization.

[0164] Figure 3The transmission rate comparison between the optimization method of the present invention and the original method is shown. Considering that there are few optimizations for the limited character input codebook in covert communication, and we first gave the closed-form solution expressions of mutual information and KL divergence for covert communication under limited character input, we designed a comparative experiment. We compared the fixed transmission amplitude and the optimal constellation point probability, the optimal transmission amplitude and the fixed constellation point probability, and the optimal transmission amplitude and the optimal constellation point probability. The proposed method has good convergence and the covert transmission performance can reach the highest value.

Claims

1. A performance boundary characterization and optimization method for wireless covert communication with limited character input, characterized in that: The following steps are involved: (1) Modeling the receiving end signal of wireless covert communication under limited character input, and using mutual information and KL divergence to describe the transmission performance and concealment of covert communication; including the following steps: (11) Modeling the codebook; (12) Modeling the receiving end model; (13) Modeling the detection end model; (14) Convert the modeling problem in steps (11)-(13) into a mathematical problem; the formula is as follows: ； in, Represents discrete Channel output at input and receiver The mutual information between represents the input distribution of n observations of the covert communication detector when communication is ongoing is true; represents the input distribution of n observations of the covert communication detector when no communication is true; represents the amplitude gain coefficient of the sender; is the probability of selecting the constellation point; ; (2) Based on the Taylor series expansion technology, the mutual information and KL divergence of covert communication are accurately approximated, and the complex integral operation is converted into a simple statistical summation operation to obtain the closed-form solution expression of the mutual information and KL divergence; the following steps are included: (21) The mutual information and KL divergence are approximated by Taylor series, and a preliminary expansion is performed on the transmission amplitude gain to obtain the second-order expansion of the mutual information and the second-order expansion of the KL divergence; the KL divergence is expressed as: ； in, is the expected value of the amplitude component of the transmitted character, i.e. the constellation point on the x-axis, is the expected value of the amplitude component of the sent character on the y-axis; And, the transmission rate is expressed as: ； in, The expected value of the power of the transmitted character; represents the variance of the complex Gaussian noise of the receiver on the x-axis and y-axis; Represents the amplitude coefficient of the channel from the sender to the receiver; (22) We analyze the mutual information and KL divergence expansions obtained to obtain the necessary conditions for the optimal input distribution; specifically, the optimal covert communication codebook must satisfy the condition ; (23) Based on the necessary conditions for the optimal input distribution, the expansion of the mutual information is modified to obtain the closed-form expression of the mutual information in the optimal case; (24) The KL divergence is further expanded to the fourth order, and the closed-form expression of the KL divergence in the optimal case is obtained; specifically, it is as follows: ； in, , , , ; (3) Based on the closed-form solution of mutual information and KL divergence, the performance boundary of wireless covert communication under limited character input is accurately characterized; (4) Based on the obtained covert communication performance boundary, the codebook composition is optimized, including the probability of constellation point selection, i.e., probabilistic constellation formation; the details are as follows: The following steps are involved: (41) Convert the mathematical problem in step (14) into a standard convex problem; specifically, as follows: ； in, ； ； ； (42) Use the interior point method or the primal-dual interior point method to solve the convex problem; optimize the output probability of the constellation points and the transmission power in the codebook multiple times until convergence is achieved and the optimal parameters are output, including the optimal transmission amplitude for covert communication and the output probability of the constellation points in the codebook.

Images