A method for processing weak signal under alpha stable distribution noise

Abstract

A weak signal processing method under alpha stable distribution noise comprises the following steps: a) nonlinear amplitude limiting processing is performed on the alpha stable distribution noise to suppress the components with large amplitude in the time domain of the noise; b) Gaussian discrimination, parameter estimation and Gaussian distribution fitting are performed on the alpha stable distribution noise after the amplitude limiting processing, so that the alpha stable distribution noise is converted into approximate Gaussian noise; c) nonlinear bistable random resonance technology is used to perform pre-enhancement processing on the weak signal under the approximate Gaussian background noise; and d) traditional non-correlation receiving processing is performed on the output signal after the pre-enhancement processing, so that the receiving error rate performance of the weak signal under the alpha stable distribution noise is improved. The timing information of the signal is directly extracted from the signal after the pre-enhancement processing; the non-correlation receiving processing is performed on the signal after the pre-enhancement processing, so that the receiving error rate performance of the weak signal processing under the alpha stable distribution noise is improved, and the method has the advantages of small symbol synchronization overhead and low calculation complexity.

Description

Technical Field

[0001] The present invention belongs to the technical field of signal processing, and in particular relates to a weak signal processing method under Alpha stable distribution noise. Background Art

[0002] Traditional weak signal processing methods typically assume that background noise follows a Gaussian distribution. This allows them to exploit statistical properties such as the first-order moment (mean) and second-order moment (variance) of Gaussian noise. This not only greatly simplifies the mathematical analysis of signal processing, but also makes signal processing algorithms for Gaussian noise distributions easy to implement and computationally low in complexity. However, in many practical applications, background noise exhibits significant bursts and non-Gaussian characteristics. For example, atmospheric noise in low-frequency communications, reverberation noise in underwater acoustic communications, clutter in radar signal processing, and salt-and-pepper noise in image processing all exhibit strong non-Gaussian pulse characteristics. Traditional Gaussian noise models cannot accurately describe the distribution characteristics of non-Gaussian noise. Previous studies have shown that compared to non-Gaussian noise models such as the generalized Gaussian distribution and Gaussian mixture distribution, the alpha-stable distribution model satisfies the generalized central limit theorem and offers advantages such as good stability and low complexity. Theoretical research and measurement and analysis of real-world background noise data demonstrate that the alpha-stable distribution can more accurately model different types of impulse noise.

[0003] At present, alpha stable distribution has become the preferred choice for modeling non-Gaussian impulse noise. [1] Zhou Tao, Wang Jia. A review of alpha stable distribution [J]. Electroacoustic Technology, 2011, 35(03): 57-60. DOI: 10.16311 / j.audioe.2011.03.027; [2] Song Guoli, Guo Xinyi, Ma Li. Alpha stable distribution model in marine environmental noise [J]. Acta Acoustics, 2019, 44(02): 177-188. DOI: 10.15949 / j.cnki.0371-0025.2019.02.004.

[0004] When the characteristic exponent of alpha-stable noise is less than 2, there are no finite second-order moments or higher-order moments. When the characteristic exponent is less than 1, there is no finite first-order moment. Therefore, traditional linear signal processing methods based on first-order and second-order moments, as well as non-Gaussian signal processing methods based on higher-order moments, are no longer applicable to alpha-stable noise.

[0005] At present, the weak signal processing methods under Alpha stable distribution noise mainly include:

[0006] 1) Using nonlinear processing methods such as wave clipping and amplitude limiting. This method can suppress the components with larger amplitudes in the noise. However, since it does not fully consider the statistical distribution characteristics of the noise, its noise suppression ability is limited.

[0007] 2) Gaussian filtering and local optimal processing methods. These two methods mainly rely on the probability density function of noise. However, except for two special distribution types (Gaussian distribution and Cauchy distribution), the rest of the alpha stable distribution has no closed probability density function expression, which increases the difficulty and complexity of implementing Gaussian filtering and local optimal processing. Summary of the Invention

[0008] In order to overcome the shortcomings of the above-mentioned prior art, the purpose of the present invention is to provide a weak signal processing method under Alpha stable distribution noise, which performs nonlinear limiting, parameter estimation and Gaussian distribution fitting on the Alpha stable distribution noise, and uses stochastic resonance technology to pre-enhance the noise signal after limiting. The timing information of the signal can be directly extracted from the pre-enhanced signal, avoiding the additional overhead of code element synchronization in the traditional signal receiving method; the signal after pre-enhancement is then subjected to traditional non-correlated reception processing, which not only improves the receiving bit error rate performance of weak signal processing under Alpha stable distribution noise, but also has the advantages of low code element synchronization overhead and low computational complexity.

[0009] In order to achieve the above object, the technical solution adopted by the present invention is:

[0010] A weak signal processing method under alpha stable distribution noise includes the following steps:

[0011] a) Perform nonlinear limiting processing on the alpha stable distribution noise to preliminarily suppress the components in the noise time domain whose amplitude is greater than the limiting amplitude;

[0012] b) performing Gaussian discrimination, parameter estimation, and Gaussian distribution fitting on the clipped Alpha stable distribution noise to convert the Alpha stable distribution noise into approximate Gaussian noise;

[0013] c) Using nonlinear bistable stochastic resonance technology to pre-enhance weak signals in near-Gaussian noise; the concept of "weak" in weak signals refers to the signal amplitude being very small relative to the noise, that is, the useful transmission signal is submerged in the strong background noise;

[0014] d) The output signal after pre-enhancement processing is further subjected to non-correlated reception processing to improve the reception bit error rate performance of weak signals under alpha stable distribution noise.

[0015] The step a) is specifically as follows:

[0016] Perform nonlinear limiting processing on the Alpha stable distribution noise w(t), set the limiting amplitude K, and the limiting function expression is:

[0017]

[0018] Wherein, x is the input signal of the limiting function, Lim(·) is the limiting function, Lim(x) is the limiting signal obtained after the input signal x is limited by the limiting function Lim(·), and the noise after limiting processing is recorded as Lim_w(t)=Lim(w(t)).

[0019] The step b) is specifically as follows:

[0020] The Gaussianity of the noise Lim_w(t) after clipping is judged. On this basis, the maximum likelihood estimation method is used to estimate the parameters of the approximately Gaussian distribution noise Lim_w(t) after clipping, and the mean and standard deviation estimates of the noise Lim_w(t) are obtained.

[0021] Finally, the Gaussian distribution obtained by cumulative empirical distribution function and parameter estimation of noise Lim_w(t) The cumulative distribution function (CDF) of the statistic is used to fit the distribution, which further verifies the correctness of Gaussian discrimination and the accuracy of parameter estimation. They represent the mean estimate and standard deviation estimate of the Gaussian distribution after parameter estimation.

[0022] Furthermore, a quantile plot (Quantile-Quantile plot, abbreviated as QQ plot) is used for discrimination; when using the QQ plot to discriminate the Gaussianity of sample data, if the scattered points on the QQ plot are approximately near a straight line, it means that the sample data approximately obeys a Gaussian distribution, and the slope of the straight line is the sample standard deviation, and the intercept is the sample mean.

[0023] The step c) is specifically as follows:

[0024] The baseband signal s(t) transmitted with a bipolar baseband signal is expressed as

[0025]

[0026] Where A is the signal amplitude, s k is the symbol, g(·) is the unit gate function with a duration of T, T is the symbol interval, and the parameter N represents the number of transmitted symbols; the mixed signal of the received noise and the transmitted signal after the limiting process is expressed as

[0027] r(t)=s(t)+Lim_w(t) (5)

[0028] The dynamic equation for processing the received mixed signal r(t) using a nonlinear bistable stochastic resonance system is usually expressed by the Langevin equation;

[0029]

[0030] Where a and b are the parameters of the bistable system, Lim_w(t) represents the noise x(t) after nonlinear limiting of the alpha-stable distributed noise w(t), and the output signal of the bistable system is obtained by the fourth-order Runge-Kutta method.

[0031] Furthermore, the bistable stochastic resonance system is realized by adding noise and adjusting the parameters of the bistable system;

[0032] Among them, when the background noise has exceeded the noise level required to generate cooperative resonance, the system parameters are adjusted;

[0033] The parameter adjustment implementation method of the bistable stochastic resonance system includes the following steps:

[0034] Firstly, based on the adiabatic approximate stochastic resonance theory, a bistable system model with normalized system parameters can be obtained, and the ordered signal, random noise and bistable system satisfy the cooperative resonance matching relationship;

[0035] Secondly, the Langevin equation expression (6) of the bistable system is transformed into τ = at, And compared with the bistable system model under normalized system parameters, the analytical expression of the bistable system parameters is obtained:

[0036]

[0037] Where T0 represents the signal symbol interval in the normalized bistable system model, represents the average power (or variance) of the Gaussian noise in the normalized bistable system model, T represents the symbol interval of the signal under the non-adiabatic approximation condition, Represents the average power estimate of the approximate Gaussian noise of the clipped noise obtained after the Alpha stable noise is clipped under the non-adiabatic approximation condition. Finally, according to the expression (7) of the bistable system parameters a, b, combined with the signal code element interval T and the estimated value of the average power of the clipping noise Lim_w(t), the values ​​of the bistable system parameters a, b are adjusted to ensure the occurrence of stochastic resonance in actual application scenarios.

[0038] The step d) is specifically as follows:

[0039] The mixed signal r(t)=s(t)+Lim_w(t) of the received noise and the transmission signal is subjected to stochastic resonance pre-enhancement processing to obtain the output signal x(t), and then the signal x(t) is subjected to non-correlated reception processing.

[0040] Further, performing stochastic resonance processing on the received signal after the limiting processing, and then performing non-correlated receiving processing;

[0041] In the numerical simulation, the received signal r(t)=s(t)+Lim_w(t) (step c) (the mixed signal of the received noise and the transmitted signal after the amplitude limiting process represented in equation (5)) is first subjected to the bistable stochastic resonance process in step c) to obtain the output signal x(t) of the bistable stochastic resonance system; the signal x(t) enhanced by the stochastic resonance is used to determine the timing information of the code element, obtain the start and end time of the transmitted code element, and determine the sampling decision time t of the code element. k = kT; the signal x(t) after the bistable stochastic resonance pre-enhancement process is sampled at the decision time t k =kT directly performs sampling judgment:

[0042]

[0043] The judgment result x k and the transmission code element symbol s k =+1 or s k = -1 (the expression is shown in formula (4) in step c)) and compare to count the number of error symbols N e4 The received bit error rate of the related receiving process is expressed as the ratio of the number of error symbols to the total number of transmitted symbols, P e4 =N e4 / N.

[0044] Beneficial effects of the present invention:

[0045] The present invention performs simple nonlinear clipping on alpha-stable distributed noise to initially suppress high-amplitude components in the noise time domain. Gaussianity determination, parameter estimation, and Gaussian distribution fitting are then performed on the clipped alpha-stable distributed noise, transforming the weak signal processing problem under alpha-stable distributed noise into a weak signal processing problem under approximately Gaussian background noise.

[0046] On this basis, with the help of the unique counter-intuitive characteristic of linear stochastic resonance technology of using harmful noise to enhance useful signals, pre-enhancement processing of weak signals under approximately Gaussian background noise is achieved. The timing information of the signal after pre-enhancement processing can be directly extracted, avoiding the additional overhead of code element synchronization in traditional signal reception methods.

[0047] Finally, the pre-enhanced signal is subjected to non-correlated reception processing. Compared with traditional correlated reception processing, this avoids the overhead of symbol synchronization and reduces the computational complexity caused by correlation operations. Compared with traditional non-correlated reception processing, the enhanced processing of weak signals by stochastic resonance technology increases the accuracy of the symbol sampling decision moment, thereby improving the receiving bit error rate performance of weak signal processing under alpha stable distribution noise. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] Figure 1 It is a schematic diagram of the weak signal processing flow under Alpha stable distribution noise.

[0049] Figure 2 It is a schematic diagram of the probability density function of Alpha stable distribution noise.

[0050] Figure 3 Figure 1. Schematic diagram of the Quantile-Quantile representation of alpha-stable noise and clipped noise. (a) is a time-domain simulation of alpha-stable noise w(t), (b) is a time-domain simulation of clipped noise Lim_w(t), (c) is a QQ-distributed simulation of alpha-stable noise w(t), and (d) is a QQ-distributed simulation of clipped noise Lim_w(t).

[0051] Figure 4 It is a schematic diagram of the cumulative empirical distribution curve of the noise after the Alpha stable distribution noise is limited (limited noise) and the noise cumulative distribution curve after the limited noise is Gaussian fitted.

[0052] Figure 5 This is a schematic diagram of the time domain processing of nonlinear stochastic resonance of weak signals under alpha stable distribution noise.

[0053] Among them, (a) is the time domain simulation diagram of the transmitted baseband signal s(t), (b) is the time domain simulation diagram of the noise Lim_w(t) after limiting processing, (c) is the time domain simulation diagram of the received mixed signal r(t)=s(t)+Lim_w(t) after limiting processing, and (d) is the time domain simulation diagram of the output signal x(t) obtained after the received mixed signal is pre-enhanced by random resonance.

[0054] Figure 6 This is a schematic diagram of the MATLAB numerical simulation comparison of the receiving bit error rate performance of different signal processing methods under Alpha stable distribution noise. DETAILED DESCRIPTION

[0055] The present invention will be further described in detail below with reference to the accompanying drawings.

[0056] The specific embodiment of the present invention is as follows: Figure 1 shown.

[0057] Reference Figure 1 The weak signal processing process under the Alpha stable distribution noise in the present invention includes the following steps:

[0058] a) The skew parameter (also called symmetry parameter) of the Alpha stable distribution is used to describe the degree of distortion of the Alpha stable distribution. If the skew parameter is 0, the Alpha stable distribution is called a symmetric Alpha stable distribution. The location parameter characterizes the offset of the probability density function of the Alpha stable distribution on the X-axis. Generally, the symmetric Alpha stable distribution with skew parameter and location parameter of 0 is the most widely used. Its characteristic function ψ α The expression of (t) can be simplified as

[0059] ψ α (t)=exp(-γ|t| α ) (1)

[0060] Among them, α∈(0,2] is the characteristic index, which generally takes a value between 1 and 2. The characteristic index describes the impact degree of the stable distribution and can also measure the thickness of the distribution function's tail. The smaller the characteristic index value, the stronger the impulse and the thicker the corresponding distribution tail. On the contrary, as the characteristic index value increases, the impact degree of the distribution decreases and the distribution tail becomes thinner. γ is the dispersion coefficient, which is a measure of the degree of dispersion of the sample relative to the mean, similar to the physical meaning of the variance in the Gaussian distribution. exp(·) is an exponential function with the natural constant e as the base. Except for the special Gaussian distribution (α=2) and Cauchy distribution (α=1), the remaining Alpha stable distribution probability density functions (PDF) do not have closed expressions and usually require numerical methods to perform inverse Fourier transform on the characteristic function to obtain them. The Alpha stable distribution probability density function f obtained by numerical methods α (x) can be expressed as

[0061]

[0062] The most notable characteristic of the Alpha stable distribution is the burst pulse characteristic with a "thick tail" (the probability of a large-value component appearing is high). The probability density function of the Alpha stable distribution noise is as follows: Figure 2 As shown. Take the Alpha stable distribution noise w(t) with characteristic index α = 1.8 and dispersion coefficient γ = 1 as an example;

[0063] First, a simple nonlinear limiting process is performed on the Alpha stable distribution noise w(t), and the limiting amplitude K is set to K=5 (the setting principle is to eliminate the components with larger amplitudes as much as possible while not affecting the useful signal). The limiting function expression is:

[0064]

[0065] Among them, x is the input signal of the limiting function, Lim(·) is the limiting function, and Lim(x) is the limiting signal obtained after the input signal x is limited by the limiting function Lim(·). Therefore, the limiting noise obtained after the Alpha stable distribution noise w(t) is limited is recorded as Lim_w(t) = Lim(w(t)). b) Perform Gaussianity judgment on the noise Lim_w(t) after limiting. In order to intuitively reflect the judgment results, the quantile graph (Quantile-Quantile graph, referred to as QQ graph) is used for judgment. The QQ graph is a scatter plot with the quantile of the standard normal distribution as the horizontal axis and the sample value as the vertical axis. When using the QQ graph to judge the Gaussianity of the sample data, if the scattered points on the QQ graph are approximately near a straight line, it means that the sample data approximately obeys the Gaussian distribution, and the slope of the straight line is the sample standard deviation, and the intercept is the sample mean. By Figure 3 The QQ distribution diagram corresponding to the Alpha stable distribution noise w(t) (such as Figure 3 (c)) and the corresponding QQ distribution diagram of the noise Lim_w(t) after amplitude limiting (as shown in Figure 3 As shown in (d), it can be seen that the noise distribution Lim_w(t) after limiting processing is closer to the Gaussian distribution.

[0066] On this basis, the maximum likelihood estimation method is used to estimate the parameters of the approximately Gaussian distribution noise Lim_w(t) after the clipping process, and the mean estimate of the noise Lim_w(t) is obtained. and standard deviation estimates

[0067] Finally, the Gaussian distribution obtained by cumulative empirical distribution function and parameter estimation of noise Lim_w(t) The cumulative distribution function (CDF) of the distribution is used for distribution fitting, which further verifies the correctness of Gaussian discrimination and the accuracy of parameter estimation. The fitting results are shown in Figure 4 As shown in the figure. The cumulative empirical distribution curve of the alpha stable distribution noise after clipping (clipping noise) describes the true cumulative empirical distribution of the clipping noise; the cumulative distribution curve of the clipping noise after parameter estimation and Gaussian fitting (the approximate Gaussian noise obtained after Gaussian fitting of the clipping noise) describes the cumulative distribution of the approximate Gaussian noise obtained by Gaussian fitting. By comparing the deviations of the two curves, the accuracy of the Gaussian fit can be intuitively observed. If the deviation between the cumulative distribution curve of the approximate Gaussian noise obtained after Gaussian fitting of the clipping noise is not large and the true cumulative empirical distribution of the clipping noise is small, it indicates that the Gaussian fit is relatively accurate. If the deviation between the two is large, it means that the approximate Gaussian noise after Gaussian fitting deviates from the true clipping noise distribution, indicating that the Gaussian fit is inaccurate.

[0068] The classical stochastic resonance theory assumes that the background noise obeys a Gaussian distribution. Therefore, the accuracy of the Gaussian fitting of the noise after the Alpha stable distribution noise is limited (limited noise) will directly determine the collaborative resonance effect and signal processing performance of the stochastic resonance signal processing.

[0069] Nonlinear stochastic resonance signal processing technology realizes the cross-integration of stochastic dynamics and information science, and shows unique advantages in weak signal processing. However, the classical theory of bistable stochastic resonance usually assumes that the background noise follows a Gaussian distribution.

[0070] from Figure 4 It can be seen that the deviation between the cumulative empirical distribution curve of the alpha stable distribution noise after limiting processing (limited noise) and the cumulative distribution curve of the limited noise after Gaussian fitting is small, which lays the foundation for further using stochastic resonance technology to enhance signal processing and improve signal processing performance.

[0071] c) In order to simplify the theoretical analysis of stochastic resonance signal processing, the embodiment assumes that the transmitted signal s(t) is a bipolar baseband signal, and the expression is:

[0072]

[0073] Where A is the signal amplitude, and its value is A=1. k is the code element symbol, and its value is s k =+1 or s k = -1. g(·) is a unit gate function with a duration of T (symbol interval), and the parameter N represents the number of transmitted symbols. Since the nonlinear limiting function has a limiting threshold value of K = 5, the limiting function will not affect the signal. Therefore, the mixed signal of the received noise and the transmitted signal after the limiting process is expressed as

[0074] r(t)=s(t)+Lim_w(t) (5)

[0075] The dynamic equation for processing the received mixed signal r(t) using a nonlinear bistable stochastic resonance system is usually expressed by the Langevin equation.

[0076]

[0077] Where a and b are the parameters of the bistable system, s(t) represents the transmitted baseband signal, Lim_w(t) represents the noise after nonlinear limiting of the alpha-stable distributed noise w(t), and x(t) is the output signal of the bistable system. As can be seen from Equation (6), the Langevin equation contains the random noise term Lim_w(t), which falls into the category of stochastic differential equations and is difficult to obtain an analytical solution. Therefore, the fourth-order Runge-Kutta method is usually used for numerical solution.

[0078] Bistable stochastic resonance systems are typically implemented using two methods: adding noise and adjusting the bistable system parameters. When the background noise level exceeds the level required for cooperative resonance, adding noise only further degrades signal processing performance, making the noise-adding method ineffective. In contrast, adjusting the system parameters is more flexible and feasible.

[0079] Furthermore, the parameter adjustment implementation method of the bistable stochastic resonance system includes the following steps:

[0080] First, according to the classical adiabatic approximation (T>>1s,σ 2 <1) Stochastic resonance theory can obtain the bistable system model under normalized system parameters, that is, under the normalized bistable system parameter conditions (a0=1, b0=1), when the average power of Gaussian noise is When the signal code element interval is T0 = 100s, the ordered signal, random noise and bistable system satisfy the cooperative resonance matching relationship. Existing studies have confirmed through theoretical analysis and numerical simulation that cooperative resonance can occur at this time.

[0081] Secondly, the Langevin equation expression (6) of the bistable system is transformed into τ = at, By comparing with the bistable system model under normalized system parameters, the analytical expression of the bistable system parameters can be obtained:

[0082]

[0083] Where a and b represent the parameters of the bistable system, T0 represents the signal symbol interval in the normalized bistable system model, and the value is T0 = 100s. Represents the average power (or variance) of the Gaussian noise in the normalized bistable system model, and takes the value T represents the symbol interval of the signal under the non-adiabatic approximation condition. In the embodiment, the value T is 0.01s. Represents the average power estimate of the approximate Gaussian noise obtained by limiting the Alpha stable noise under non-adiabatic approximation conditions. In the embodiment, the estimated value is

[0084] Finally, according to the expression (7) of the bistable system parameters a and b, combined with the signal symbol interval T (T = 0.01s in the embodiment) and the estimated value of the average power of the limiting noise Lim_w(t), Adjust the values ​​of the bistable system parameters a and b to a=10000, b=2.039×10 11, which can ensure that in practical application scenarios (not limited to the assumption of adiabatic approximation conditions, that is, T<<1s,δ 2 >1) The occurrence of stochastic resonance phenomenon.

[0085] Figure 5 Given the Alpha stable distribution noise (Alpha stable distribution noise w(t) is as follows Figure 3 Time domain MATLAB numerical simulation of the weak signal (shown in (a)) after nonlinear limiting processing and nonlinear bistable stochastic resonance enhancement processing.

[0086] Figure 5 (a) shows the transmitted bipolar baseband signal s(t), the signal code element interval is T = 0.01s, the number of code elements is N = 20, and the time domain signal range of the simulation analysis is 0 to 0.2s;

[0087] Figure 5 (b) shows the time domain noise signal Lim_w(t) obtained after the Alpha stable distribution noise w(t) is processed by nonlinear limiting. The specific limiting processing method refers to step b);

[0088] Figure 5 (c) shows the mixed signal r(t)=s(t)+Lim_w(t) of the transmission signal s(t) and the noise signal Lim_w(t) after the limiting process;

[0089] Figure 5 The received mixed signal r(t)=s(t)+Lim_w(t) shown in (d) is processed by bistable stochastic resonance enhancement, and the output signal x(t) of the nonlinear bistable stochastic resonance system is obtained. The specific implementation method refers to the bistable stochastic resonance system parameter adjustment implementation step in step c).

[0090] d) From step b) and the corresponding signal processing results, it can be seen that the Alpha stable distribution noise can be fitted into an approximate Gaussian distribution noise form after simple nonlinear limiting processing. At this time, the signal processing methods that can be used are mainly: 1) directly performing correlated reception processing on the received signal r(t) = s(t) + Lim_w(t); 2) directly performing non-correlated reception processing on the received signal r(t) = s(t) + Lim_w(t); 3) performing random resonance pre-enhancement processing on the received signal r(t) = s(t) + Lim_w(t) to obtain the output signal x(t), and then performing correlated reception processing on the signal x(t); 4) performing random resonance pre-enhancement processing on the received signal r(t) = s(t) + Lim_w(t) to obtain the output signal x(t), and then performing non-correlated reception processing on the signal x(t).

[0091] Theoretical comparison and analysis of the above four signal processing methods are as follows:

[0092] 1) Directly performing correlation receiving processing on the received signal after amplitude limiting processing

[0093] According to the classical linear signal processing theory, performing correlation receiving processing on the bipolar baseband signal under the approximate Gaussian distribution noise can obtain the approximate optimal signal receiving bit error rate performance. However, the premise is that when the received signal and the local sample signal achieve symbol synchronization, the sampling time of the symbol can be accurately determined, at this time, the correlation between the received signal and the sample signal is the best, and the correlation receiving processing can obtain the approximate optimal signal receiving performance. For the baseband signal in the embodiment, the pilot insertion method is usually used to achieve symbol synchronization, that is, a pilot signal is inserted at the zero point of the baseband signal spectrum at the sending end, and the pilot signal is extracted by using a narrowband filter at the receiving end, and then the symbol synchronization between the received signal and the sample signal is achieved after processing. Therefore, without considering the overhead of symbol synchronization and the complexity of correlation operation, directly performing correlation receiving processing on the mixed signal of the weak signal after nonlinear amplitude limiting processing under the Alpha stable distribution noise will obtain the approximate optimal receiving performance.

[0094] 2) Directly performing non-correlation receiving processing on the received signal after amplitude limiting processing

[0095] The mixed signal of the weak signal after nonlinear amplitude limiting processing under the Alpha stable distribution noise is processed by using non-correlation receiving processing. Since the statistical characteristics of the noise signal are not fully utilized, compared with the correlation receiving signal processing method, the receiving bit error rate performance is poor. However, the non-correlation receiving processing can avoid the complexity of the correlation operation between the transmission symbol and the sample symbol. In addition, the non-correlation receiving processing also needs to transmit the start and end time information of the symbol to accurately determine the sampling judgment time of the signal symbol.

[0096] 3) Performing random resonance processing on the received signal after amplitude limiting processing, and then performing correlation receiving processing

[0097] The noise signal after nonlinear amplitude limiting processing approximately obeys the Gaussian distribution, which meets the application condition of the classical bistable random resonance theory. By analyzing the cooperative resonance mechanism of the signal, the noise and the bistable system, and by means of random resonance, the counterintuitive characteristic of random noise is used to enhance the ordered signal, and the received weak signal is enhanced. Referring to the numerical simulation results of step c) Figure 5 It can be seen from the numerical simulation results of (c) that the signal enhanced directly by using random resonance can distinguish the starting position of the transmission symbol, without additional symbol synchronization overhead. Since the approximate Gaussian noise after processing by the random resonance system no longer obeys the Gaussian distribution but obeys the Lorentz distribution, at this time, the correlation receiving processing can obtain better receiving bit error rate performance, but the performance is slightly worse than that of the direct correlation receiving processing.

[0098] 4) Perform random resonance processing on the received signal after the limiting processing, and then perform non-correlated reception processing

[0099] By using the counter-intuitive property of noise enhancement signal by means of stochastic resonance, the received weak signal can be pre-enhanced. Figure 5 The numerical simulation results of (c) show that the signal enhanced by stochastic resonance can be directly used to determine the starting position of the transmitted codeword, without the need for additional codeword synchronization overhead. Non-correlated reception processing of the signal after stochastic resonance pre-enhancement can make more accurate sampling decisions compared to direct non-correlated reception processing of the received signal. Therefore, the bit error rate performance is better than direct non-correlated processing of the received signal, but slightly worse than direct correlation processing. Since non-correlated reception processing does not require correlation operations, the computational complexity is lower. Taking into account the signal reception bit error rate performance, codeword synchronization overhead, and computational complexity, it is expected that this method will have potential practical application value in weak signal processing under alpha-stable distributed noise.

[0100] In summary, the theoretical basis, signal reception performance, and advantages and disadvantages of the above four signal processing methods are shown in Table 1.

[0101] Table 1 Performance comparison of different signal processing methods

[0102]

[0103]

[0104] In order to further verify the theoretical analysis results of the above four signal processing methods, MATLAB numerical simulation is used for further verification.

[0105] First, the definition of generalized signal-to-noise ratio (GSNR) for weak signal processing under alpha stable distribution noise used in numerical simulation is given.

[0106]

[0107] Among them, the parameters Represents the average power of the signal. For the bipolar baseband signal in the embodiment (the expression is defined in formula (4)), the average power of the signal is A represents the amplitude of the signal, and the parameter γ represents the dispersion coefficient of the Alpha stable distribution noise (refer to the definition in formula (1)).

[0108] The numerical simulation focuses on the low signal-to-noise ratio (GSNR) situation from -20dB to -5dB, which represents the weak signal strength under Alpha stable distribution noise.

[0109] 1) Numerical simulation implementation of direct correlation reception processing

[0110] The timing information of the code element is obtained by inserting the pilot code element synchronization method to achieve accurate synchronization between the transmission code element and the sample code element. In the numerical simulation, the sample code element s0(t) = -1, s1(t) = 1, t∈[0,T] is defined. The received signal r(t) = s(t) + Lim_w(t) (the mixed signal of the received noise and the transmission signal after the limiting processing represented in step c) in equation (5) is correlated with the sample code element. The receiving processing rule is to perform correlation operation on the transmission code element s(t) and the sample code element s0(t) = -1, s1(t) = 1 at the sampling decision time T and compare them.

[0111]

[0112]

[0113] Set the total number of transmitted code elements to N = 10000, compare the judgment result with the transmitted code element s(t) = +1 or s(t) = -1 (the expression is shown in formula (4) in step c), and count the number of error code elements N e1 The received bit error rate of the related receiving process is expressed as the ratio of the number of error symbols to the total number of transmitted symbols, P e1 =N e1 / N.

[0114] 2) Numerical simulation implementation of direct non-correlated reception processing

[0115] The insertion pilot symbol synchronization method is used to obtain the symbol timing information, obtain the start and end time of the transmission symbol, and determine the sampling decision time t of the symbol. k = kT. The received signal r(t) = s(t) + Lim_w(t) (step c) is a mixed signal of the noise and the transmitted signal after the amplitude limiting process represented in equation (5) at the sampling decision time t k =kT directly performs sampling judgment

[0116]

[0117] The judgment result k and the transmission code element symbol s k =+1 or s k = -1 (the expression is shown in formula (4) in step c)) and compare to count the number of error symbols N e2 The received bit error rate of non-correlated reception processing is expressed as the ratio of the number of error symbols to the total number of transmitted symbols, P e2 =N e2 / N.

[0118] 3) Numerical simulation of stochastic resonance + correlation reception processing

[0119] In the numerical simulation, the sample code element s0(t)=-1, s1(t)=1, t∈[0,T] is defined, and the received signal r(t)=s(t)+Lim_w(t) (step c) The mixed signal of the received noise and the transmitted signal after the amplitude limiting process represented in equation (5) is first subjected to the bistable stochastic resonance process in step c) to obtain the output signal x(t) of the bistable stochastic resonance system. Refer to step c) Figure 5 The numerical simulation results of (c) show that the starting position of the transmission codeword can be directly determined by using the signal x(t) enhanced by stochastic resonance, without the need for additional codeword synchronization overhead. The receiving and processing rules for the correlation between the output signal of the bistable stochastic resonance system and the sample codeword are as follows: at the sampling decision time T, the transmission codeword s(t) and the sample codewords s0(t) = -1, s1(t) = 1 are correlated and compared:

[0120]

[0121]

[0122] Set the total number of transmitted code elements to N = 10000, compare the judgment result with the transmitted code element s(t) = +1 or s(t) = -1 (the expression is shown in formula (4) in step c), and count the number of error code elements N e3 The received bit error rate of the related receiving process is expressed as the ratio of the number of error symbols to the total number of transmitted symbols, P e3 =N e3 / N.

[0123] 4) Numerical simulation of stochastic resonance + non-correlated reception processing

[0124] In the numerical simulation, the received signal r(t)=s(t)+Lim_w(t) (step c) is represented by the formula (5) after the amplitude limiting process, and the mixed signal of the received noise and the transmitted signal) is first subjected to the bistable stochastic resonance process in step c) to obtain the output signal x(t) of the bistable stochastic resonance system. Figure 5 From the numerical simulation results of (c), we can see that we can directly use the signal x(t) enhanced by stochastic resonance to determine the timing information of the code element, obtain the start and end time of the transmitted code element, and determine the sampling decision time t of the code element. k =kT. The signal x(t) after the bistable stochastic resonance pre-enhancement process is sampled at the decision time t k =kT directly performs sampling judgment:

[0125]

[0126] The judgment result x k and the transmission code element symbol s k =+1 or s k = -1 (the expression is shown in formula (4) in step c)) and compare to count the number of error symbols N e4 The received bit error rate of the related receiving process is expressed as the ratio of the number of error symbols to the total number of transmitted symbols, P e4 =N e4 / N.

[0127] According to the above numerical simulation steps, Figure 6 MATLAB numerical simulation results are presented to compare the bit error rate performance of four signal processing methods for weak signal reception in the presence of alpha-stable distributed noise. The theoretical analysis results are consistent with the numerical simulation results, further validating the correctness of the theoretical analysis. Furthermore, since non-correlated processing does not require correlation operations, its computational complexity is low. Stochastic resonance pre-enhancement, on the other hand, directly determines the start and end positions of symbols using the resonance signal, obtaining symbol timing and synchronization information, and reducing the overhead of symbol synchronization. Compared with direct non-correlated reception processing of the received signal, the combination of stochastic resonance pre-enhancement and non-correlated reception processing enables more accurate sampling decisions at the sampling decision moment, resulting in better bit error rate performance than direct non-correlated reception processing. Considering the signal reception bit error performance, symbol synchronization overhead, and computational complexity, it is expected that the combination of stochastic resonance and non-correlated reception has potential practical application value in weak signal processing in the presence of alpha-stable distributed noise.

[0128] The above is only an example embodiment of the present invention, but the protection scope of the present invention is not limited thereto. Any changes or replacements made within the technical scope of the present invention fall within the protection scope of the present invention.

Claims

1. A weak signal processing method under alpha stable distribution noise, characterized in that: The following steps are included: a) Perform nonlinear limiting processing on the alpha stable distribution noise to preliminarily suppress the components in the noise time domain whose amplitude is greater than the limiting amplitude; b) performing Gaussian discrimination, parameter estimation, and Gaussian distribution fitting on the clipped Alpha stable distribution noise to convert the Alpha stable distribution noise into approximate Gaussian noise; c) Using nonlinear bistable stochastic resonance technology to pre-enhance weak signals under approximately Gaussian noise; d) Performing non-correlated reception processing on the output signal after pre-enhancement processing to improve the reception bit error rate performance of weak signals under alpha stable distribution noise; The step a) is specifically as follows: Perform nonlinear limiting processing on the Alpha stable distribution noise w(t), set the limiting amplitude K, and the limiting function expression is: Wherein, x is the input signal of the limiting function, Lim(·) is the limiting function, Lim(x) is the limiting signal obtained after the input signal x is limited by the limiting function Lim(·), and the noise after the limiting process is recorded as Lim_w(t)=Lim(w(t)); The step b) is specifically as follows: The Gaussianity of the noise Lim_w(t) after the clipping process is judged, and the parameters of the approximately Gaussian distribution noise Lim_w(t) after the clipping process are estimated using the maximum likelihood estimation method to obtain the mean estimate and standard deviation estimate of the noise Lim_w(t); Finally, the Gaussian distribution obtained by cumulative empirical distribution function and parameter estimation of noise Lim_w(t) The cumulative distribution function (CDF) of They represent the mean estimate and standard deviation estimate of the Gaussian distribution obtained after parameter estimation; Quantile plots are used for discrimination. When the QQ plot is used to discriminate the Gaussianity of sample data, if the scattered points on the QQ plot are approximately near a straight line, it means that the sample data approximately obeys the Gaussian distribution, and the slope of the straight line is the sample standard deviation, and the intercept is the sample mean; The step c) is specifically as follows: The baseband signal s(t) transmitted with a bipolar baseband signal is expressed as Where A is the signal amplitude, s k is the symbol, g(·) is the unit gate function with a duration of T, T is the symbol interval, and the parameter N represents the number of transmitted symbols; the mixed signal of the received noise and the transmitted signal after the limiting process is expressed as r(t)=s(t)+Lim_w(t) (5) The dynamic equation for processing the received mixed signal r(t) using a nonlinear bistable stochastic resonance system is expressed by the Langevin equation; Where a and b are the parameters of the bistable system, Lim_w(t) represents the noise after the alpha-stable distributed noise w(t) is processed by nonlinear limiting, and x(t) is the output signal of the bistable system. The fourth-order Runge-Kutta method is used for numerical solution. The bistable stochastic resonance system is realized by adding noise and adjusting the parameters of the bistable system. Among them, when the background noise has exceeded the noise level required to generate cooperative resonance, the system parameters are adjusted; The parameter adjustment method of the bistable stochastic resonance system includes the following steps: Firstly, based on the adiabatic approximate stochastic resonance theory, a bistable system model with normalized system parameters is obtained, and the ordered signal, random noise and bistable system satisfy the cooperative resonance matching relationship. Secondly, the Langevin equation expression (6) of the bistable system is transformed into τ = at, And compared with the bistable system model under normalized system parameters, the analytical expression of the bistable system parameters is obtained: Where T0 represents the signal symbol interval in the normalized bistable system model, represents the average power of Gaussian noise in the normalized bistable system model, T represents the symbol interval of the signal under the non-adiabatic approximation condition, Represents the average power estimate of the approximate Gaussian noise of the clipped noise obtained after the Alpha stable noise is clipped under the non-adiabatic approximation condition. Finally, according to the expression (7) of the bistable system parameters a, b, combined with the signal code element interval T and the estimated value of the average power of the clipping noise Lim_w(t), the values ​​of the bistable system parameters a, b are adjusted to ensure the occurrence of stochastic resonance in actual application scenarios.

2. The method for processing weak signals under alpha stable distribution noise according to claim 1, characterized in that: The step d) is specifically as follows: The mixed signal r(t)=s(t)+Lim_w(t) of the received noise and the transmission signal is subjected to stochastic resonance pre-enhancement processing to obtain the output signal x(t), and then the signal x(t) is subjected to non-correlated reception processing.

3. The method for processing weak signals under alpha stable distribution noise according to claim 1, characterized in that: Performing stochastic resonance processing on the received signal after the limiting processing, and then performing non-correlated receiving processing; In the numerical simulation, the received signal r(t)=s(t)+Lim_w(t) is first processed by the bistable stochastic resonance in step c) to obtain the output signal x(t) of the bistable stochastic resonance system. The signal x(t) enhanced by the stochastic resonance is used to determine the timing information of the code element, obtain the start and end time of the transmission code element, and determine the sampling decision time t of the code element. k = kT; the signal x(t) after the bistable stochastic resonance pre-enhancement process is sampled at the decision time t k =kT directly performs sampling judgment: The judgment result x k and the transmission code element symbol s k =+1 or s k = -1 for comparison, and count the number of error symbols N e4 The received bit error rate of the related receiving process is expressed as the ratio of the number of error symbols to the total number of transmitted symbols, P e4 =N e4 / N.

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