Joint detection and estimation method for noise enhancement nonlinear system under Neyman-Pearson framework

Abstract

The invention discloses a noise enhancement nonlinear system joint detection and estimation method under a Neyman-Pearson framework, and belongs to the field of signal processing. Firstly, independent additive noise is added to a nonlinear system input signal, and noise-corrected nonlinear system output is obtained after the nonlinear system input signal passes through the nonlinear system. And secondly, under the Neyman-Pearson criterion, performing binary hypothesis judgment based on the output, and estimating unknown parameters in the signal of which the judgment result is hypothesis H1. And under the condition of ensuring that the detection probability and the false alarm probability meet certain constraints, constructing a noise enhancement nonlinear system joint detection and estimation model which minimizes the conditional estimation risk. The additive noise is the optimal solution of the model and is random distribution formed by convex combination of not more than three constant vectors with a certain weight. According to the method, noise enhancement and nonlinear system joint detection and estimation under the Neyman-Pearson criterion are combined, and the estimation performance is further improved under the condition that the detection performance is not reduced.

Description

TECHNICAL FIELD

[0001] The present application belongs to the field of signal processing, and particularly relates to the problem of hypothesis testing and estimation of binary signal under the Neyman-Pearson criterion for nonlinear systems, noise enhancement. BACKGROUND

[0002] Noise is ubiquitous, especially in complex systems. In classical signal processing, noise is regarded as an unwanted signal, and the more noise in the system, the smaller the channel capacity, thus the detection performance and estimation accuracy are both reduced. However, the influence of noise on the system is not all negative. Under certain conditions, noise can have a positive enhancing effect on the signal through a nonlinear system, which is called noise enhancement phenomenon. Existing researches have shown that adding noise to the input of some nonlinear systems or adjusting the background noise level can significantly improve the detection and / or estimation performance of the system. Existing researches mostly study noise enhancement and parameter estimation, signal detection independently, but in actual application scenarios, signal detection and parameter estimation are interdependent, separate processing cannot fully utilize prior information, and will increase unnecessary calculations. Joint detection and estimation is to put the detection problem and the estimation problem into the same framework, and then estimate the unknown parameters in the signal under the condition of detecting the signal, which can well overcome this shortcoming. The present application extends the noise enhancement joint detection and estimation framework to more general nonlinear systems. For the case where the prior probability is unknown, additive noise independent of the input signal of the nonlinear system is added to the input signal of the nonlinear system, and then the output signal of the noise-modified nonlinear system is used to determine which of the hypotheses H0 and H1 is true under the Neyman-Pearson criterion. When the result of the determination is that the hypothesis H1 is true, the unknown parameters in the signal are estimated. Under the condition that the detection probability and the false alarm probability meet certain constraints, the conditional Bayesian estimation risk is minimized as much as possible. SUMMARY

[0003] The purpose of the present application is to solve the problem of joint signal detection and estimation for nonlinear systems, combined with the principle of noise enhancement, a noise enhancement nonlinear system joint detection and estimation method under the Neyman-Pearson framework is proposed. Specifically, by adding additive noise independent of the input signal of the nonlinear system to the input signal of the nonlinear system, then using the output signal of the noise-added nonlinear system to determine which of the original hypothesis and the alternative hypothesis is true under the Neyman-Pearson criterion, and then estimating the unknown parameters in the signal whose result of the determination is H1. Under the premise of ensuring that the detection performance does not decrease, the conditional Bayesian estimation risk is minimized.

[0004] The present application specifically includes the following steps:

[0005] ① Add additive noise n to the input signal x of the nonlinear system;

[0006] ② After passing through the nonlinear system, the noise-corrected nonlinear system output z = T(x + n) is obtained; where T(·) represents the transfer function of the nonlinear system;

[0007] ③ Under the Neyman-Pearson criterion, the output z of the nonlinear system is used to determine which hypothesis H0 or H1 is true; the corresponding decision function under the Neyman-Pearson criterion is denoted as...

[0008] ④ If the decision result is that hypothesis H0 is true, then the parameter value is known as θ0 and no estimation is required; if the decision result is that hypothesis H1 is true, then the unknown parameter θ is estimated.

[0009] Furthermore, assuming that under H0, the probability density function of the input signal x of the nonlinear system is p0(x), (N is an integer greater than or equal to 1); Assuming under H1, the input signal x of the nonlinear system contains an unknown parameter θ, with a corresponding conditional probability density function p1(x|θ); the unknown parameter θ∈Λ follows a probability function p θ The distribution of (θ); the additive noise follows a probability density function p n The distribution of (n),

[0010] The decision function under the Neyman-Pearson criterion Represented as:

[0011]

[0012] Where η represents the decision threshold, and f0(z) represents the output of the nonlinear system. (M is an integer greater than or equal to 1) The probability density function under hypothesis H0:

[0013]

[0014] Where δ(·) represents the Dirac function; f1(z) represents the probability density function of the nonlinear system output z under the assumption H1:

[0015]

[0016] Furthermore, the additive noise n and the decision threshold η are solutions to the following optimization problem:

[0017]

[0018] in, and These represent the false alarm probability and the detection probability, respectively. Indicates an estimate; This indicates that the conditional Bayesian estimation risk is used; α0 and β0 represent the upper limit of the false alarm probability and the lower limit of the detection probability, respectively.

[0019] Furthermore, the expression for the conditional Bayesian estimation of risk is as follows:

[0020]

[0021] in, This represents the probability that the decision result is H1 if the hypothesis H1 is true. This represents the loss function.

[0022] Furthermore, the additive noise n is a random variable that follows a random distribution consisting of a convex combination of at most three constant vectors with certain weights, and its corresponding probability density function is:

[0023]

[0024] Where, n i (i = 1, 2, 3) are the weighting coefficients, 0 ≤ k i ≤1 and k1+k2+k3=1.

[0025] Furthermore, the constant vector n i and corresponding weight k i (i = 1, 2, 3) and the decision threshold η are determined by the following constrained optimization problem:

[0026]

[0027] in f1(z|n)=∫ Λ p θ (θ)f1(z|θ,n)dθ,

[0028] Furthermore, the constant vector n i and corresponding weight k i (i = 1, 2, 3) and the decision threshold η are obtained by a global optimization algorithm; the global optimization method includes, but is not limited to, particle swarm optimization, ant colony optimization or genetic algorithm.

[0029] This invention combines noise enhancement with the joint detection and estimation problem under the Neyman-Pearson criterion. By adding noise to the nonlinear input signal, the output signal of the noise-corrected nonlinear system is used to make a decision under the Neyman-Pearson criterion. Then, when the decision result is that hypothesis H1 is true, the unknown parameters in the signal are estimated. This achieves the goal of further reducing the risk of conditional estimation while ensuring that the detection probability and false alarm probability meet certain constraints.

[0030] The present application mainly adopts the method of simulation experiment for verification, and all steps and conclusions are verified correct on MATLAB R2018b. BRIEF DESCRIPTION OF DRAWINGS

[0031] Figure 1 is the workflow diagram of the present application.

[0032] Figure 2 is the noise-free and noise-corrected conditional Bayesian estimation risk value corresponding to different σ θ values in the simulation of the present application.

[0033] Figure 3 is the noise-free and noise-corrected detection probability value corresponding to different σ θ values in the simulation of the present application.

[0034] Figure 4 is the noise-free and noise-corrected false alarm probability value corresponding to different σ θ values in the simulation of the present application. DETAILED DESCRIPTION

[0035] The present application will be further described below in conjunction with examples, but should not be understood as limiting the above-mentioned subject matter of the present application to the following examples. Various substitutions and modifications can be made according to ordinary technical knowledge and conventional means in the art without departing from the above-mentioned technical idea of the present application, and all should be included in the protection scope of the present application.

[0036] Example 1:

[0037] The present embodiment discloses a noise-enhanced nonlinear system joint detection and estimation method under the Neyman-Pearson framework, comprising the following steps:

[0038] ① Add additive noise n to the input signal x of the nonlinear system;

[0039] ② After passing through the nonlinear system, obtain the noise-corrected nonlinear system output z = T(x + n); T(·) represents the transfer function of the nonlinear system;

[0040] ③ Under the Neyman-Pearson criterion, use the nonlinear system output z to determine which of the hypotheses H0 and H1 is true; the decision function under the Neyman-Pearson criterion is denoted as

[0041] ④ If the decision result is that the hypothesis H0 is true, the parameter value is known as θ0, and no estimation is needed; if the decision result is that the hypothesis H1 is true, the unknown parameter θ is estimated.

[0042] Example 2:

[0043] The embodiment discloses a noise-enhanced nonlinear system joint detection and estimation method under a Neyman-Pearson framework, and comprises the following steps.

[0044] ① inputting a signal into a nonlinear system adding an independent additive noise thereto N is an integer greater than or equal to 1. Under the assumption H0, the probability density function of x is p0(x). Under the assumption H1, x contains an unknown parameter θ, and the corresponding conditional probability density function is p1(x|θ), and the unknown parameter θ is subject to a distribution with a probability function p θ (θ). n The additive noise n is subject to a distribution with a probability density function p

[0045] ② obtaining a noise-modified nonlinear system output after the nonlinear system M is an integer greater than or equal to 1, and T(·) represents a transfer function of the nonlinear system. The probability density function of z under the assumption H0 is:

[0046]

[0047] wherein δ(·) represents a Dirac function, The probability density function of z under the assumption H1 is:

[0048]

[0049] wherein f1(z|n) = ∫ Λ p θ (θ)f1(z|θ,n)dθ,

[0050] ③ under the Neyman-Pearson criterion, the nonlinear system output z is used to determine which of the assumptions H0 and H1 is true, and the corresponding decision function under the Neyman-Pearson criterion is represented as:

[0051]

[0052] wherein η represents a decision threshold. Further, when the additive noise with the probability density function p n (n) is added to x, the decision function is selected as The obtained detection probability and false alarm probability may be respectively represented as

[0053]

[0054] wherein and ​This means selecting while adding a constant vector n as additive noise to x. The detection probability and false alarm probability are obtained as the decision function.

[0055] ④ If the decision result is that hypothesis H0 is true, then the parameter value is known as θ0, and no estimation is needed; if the decision result is that hypothesis H1 is true, then the unknown parameter θ is estimated, and the estimator is denoted as θ0. Under the Neyman-Pearson criterion, due to the assumption H i The prior probability of (i = 0, 1) is unknown, making it difficult to directly calculate the overall Bayesian risk estimate. To overcome this difficulty and simplify the problem, a conditional Bayesian risk estimate is adopted based on the assumption that the decision result of H1 is also H1. To evaluate and estimate performance, The calculation is as follows:

[0056]

[0057] in This represents the probability that the decision result is H1 if the hypothesis H1 is true. The loss function can be represented by mean squared error, absolute error, or uniform cost function, etc.

[0058] Furthermore, to minimize the risk of conditional Bayesian estimation without compromising detection performance, a joint detection and estimation model for noise-enhanced nonlinear systems under the Neyman-Pearson framework is constructed:

[0059]

[0060] Where α0 and β0 represent the upper limit of the false alarm probability and the lower limit of the detection probability, respectively. The additive noise n is the solution to the optimization problem in (14).

[0061] The additive noise n satisfying (14) is a random variable that follows a random distribution consisting of at most three constant vectors combined with certain weights in a convex combination, with the corresponding probability density function being

[0062]

[0063] Where, k i (i = 1, 2, 3) are the weighting coefficients, k i ∈[0,1] and k1+k2+k3=1. The proof is as follows:

[0064] Assuming the detector in the nonlinear joint detection and estimation system is fixed at φ1(·), then φ1(·) and T(·) together can be considered as a suboptimal detector. The results in (11)-(13) can be further analyzed. If replaced with φ1(·), the detection probability will be... False alarm probability And the risk of conditional Bayesian estimation Can be expressed as the expectation of a certain auxiliary function with respect to the additive noise distribution, as follows:

[0065]

[0066] Where

[0067] Further, fix the detector in (14) as φ1(·), a new noise-enhanced nonlinear joint detection and estimation model is obtained as follows:

[0068] At this time, the model in (19) is a convex optimization problem, and the corresponding optimal additive noise is a random distribution composed of no more than three constant vectors combined with certain weights. That is, when the detector is fixed as φ1(·), the optimal additive noise that minimizes the risk of conditional Bayesian estimation under the constraints of And can be expressed as a random distribution composed of no more than three constant vectors combined with certain weights.

[0069] Assuming that φ1 is the optimal additive noise modified detector in model (14) (i.e. ), the convex optimization problem in (19) still holds. At this time, the optimal additive noise in the optimization problem in (19) is also a random distribution composed of no more than three constant vectors. That is, the change of the detector does not affect the form of the optimal additive noise. Therefore, the optimal additive noise probability density function corresponding to the optimization problem in (14) is also as shown in (15).

[0070] Further, substitute in (15) into (14), and the non-convex constraint optimization problem in (14) can be simplified as follows:

[0071]

[0072] The constant vectors n i and the corresponding weights k i (i = 1, 2, 3) and the decision threshold η can be obtained by a global optimization algorithm. The global optimization method includes but is not limited to particle swarm optimization, ant colony optimization or genetic algorithm. The effect of the present application can be further illustrated by the following simulation experiment:

[0073] In this simulation experiment, the binary hypothesis testing problem is as follows:

[0074]

[0075] where x is the input signal of the nonlinear system, θ is the unknown parameter, and its probability density function is v is the zero-mean asymmetric Gaussian mixture background noise, and its probability density function is Here 0≤t≤1, and the probability density functions of x under H0 and H1 are denoted as p0(x) = p v (x) and p1(x) = ∫ Λ p v (x-θ)p θ (θ)dθ. When a constant vector n is added to the input signal x of the nonlinear system, the noise-modified output z of the amplitude-limiting system can be expressed as:

[0076]

[0077] where A is the amplitude-limiting threshold. The probability density functions of z under the hypotheses H i (i = 0, 1) are denoted as:

[0078]

[0079] Under the uniform cost allocation (UCA), C 10 = C 01 = 1, C 00 = C 11 = 0. Take A = 0.8, u θ = 0.5, μ = 0.5, σ = 0.1, σ θ = 0.2, and t = 0.6 as an example. When no noise is added to the input x of the nonlinear system, the detection probability, false alarm probability, and conditional Bayesian estimation risk value are 0.8318, 0.0596, and 0.3757, respectively. When random noise (which is a random distribution composed of constant vectors n1 = 1.201, n2 = 0.2711, and n3 = -0.416 with weights k1 = 0.0001, k2 = 0.9789, and k3 = 0.021) is added to the input signal of the nonlinear system, the corresponding detection probability, false alarm probability, and conditional Bayesian estimation risk value are 0.8319, 0.0592, and 0.3240, respectively. That is, by adding noise to the system, the detection performance is almost unchanged while the conditional Bayesian estimation risk value is reduced by more than 13.7%.

[0080] Keeping A = 0.8, u θ = 0.5, μ = 0.5, σ = 0.1, and t = 0.6 unchanged, the values of σ θ are gradually increased from 0 to 1.1, and the calculated noise-free and noise-modified conditional Bayesian estimation risk values, detection probability values, and false alarm probability values are shown in Figure 2 , Figure 3 and Figure 4 , respectively.

[0081] To further illustrate the results of Figure 2 , Figure 3 and Figure 4 , Table 1 gives the original decision threshold, conditional estimation risk r θ , detection probability P θ and false alarm probability P NP for different σ d values with A = 0.8, u fa = 0.5, μ = 0.5, σ = 0.1 and t = 0.6; Table 2 gives the optimal additive noise components k1 / k2 / n1 / n2 / n3, noise enhanced decision threshold, conditional estimation risk r θ , noise corrected detection probability P NP and false alarm probability P d for different σ fa values.

[0082] Table 1 Original decision threshold, conditional Bayesian estimation risk, detection probability and false alarm probability

[0083]

[0084] Table 2 Noise enhanced decision threshold, conditional Bayesian estimation risk, detection probability and false alarm probability

[0085]

Claims

1. A joint detection and estimation method for noise-enhanced nonlinear systems within the Neyman-Pearson framework, characterized in that: Includes the following steps: ① Add additive noise n to the input signal x of the nonlinear system; ② After passing through the nonlinear system, the noise-corrected nonlinear system output z = T(x + n) is obtained; where T(·) represents the transfer function of the nonlinear system; ③ Under the Neyman-Pearson criterion, the output z of the nonlinear system is used to determine which hypothesis H0 or H1 is true; the corresponding decision function under the Neyman-Pearson criterion is denoted as... ④ If the decision result is that hypothesis H0 is true, then the parameter value is known as θ0 and no estimation is required; if the decision result is that hypothesis H1 is true, then the unknown parameter θ is estimated.

2. The joint detection and estimation method for noise-enhanced nonlinear systems under the Neyman-Pearson framework according to claim 1, characterized in that: Under assumption H0, the probability density function of the input signal x of the nonlinear system is p0(x). (N is an integer greater than or equal to 1); Assuming under H1, the input signal x of the nonlinear system contains an unknown parameter θ, with a corresponding conditional probability density function p1(x|θ); the unknown parameter θ∈Λ follows a probability function p θ The distribution of (θ); the additive noise follows a probability density function p n The distribution of (n), The decision function under the Neyman-Pearson criterion Represented as: Where η represents the decision threshold, and f0(z) represents the output of the nonlinear system. (M is an integer greater than or equal to 1) The probability density function under hypothesis H0: Where δ(·) represents the Dirac function; f1(z) represents the probability density function of the nonlinear system output z under the assumption H1:

3. The joint detection and estimation method for noise-enhanced nonlinear systems under the Neyman-Pearson framework according to claim 2, characterized in that: The additive noise n and the decision threshold η are solutions to the following optimization problem: in, and These represent the false alarm probability and the detection probability, respectively. Indicates an estimate; This indicates that the conditional Bayesian estimation risk is used; α0 and β0 represent the upper limit of the false alarm probability and the lower limit of the detection probability, respectively.

4. The method for joint detection and estimation of noise-enhanced nonlinear systems under the Neyman-Pearson framework according to claim 3, characterized in that: The expression for the conditional Bayesian estimation of risk is as follows: in, This represents the probability that the decision result is H1 if the hypothesis H1 is true. This represents the loss function.

5. The method for joint detection and estimation of a noise-enhanced nonlinear system under the Neyman-Pearson framework according to claim 3, characterized in that: The additive noise n is a random variable that follows a random distribution consisting of a convex combination of at most three constant vectors with certain weights, and its corresponding probability density function is: Where, k i (i = 1, 2, 3) are the weighting coefficients, 0 ≤ k i ≤1 and k1+k2+k3=1.

6. The method for joint detection and estimation of a noise-enhanced nonlinear system under the Neyman-Pearson framework according to claim 5, characterized in that: The constant vector n i and corresponding weight k i (i = 1, 2, 3) and the decision threshold η are determined by the following constrained optimization problem: among them f1(z|n)=∫ Λ p θ (θ)f1(z|θ,n)dθ, 7. The method for joint detection and estimation of a noise-enhanced nonlinear system under the Neyman-Pearson framework according to claim 6, characterized in that: The constant vector n i and corresponding weight k i (i = 1, 2, 3) and the decision threshold η are obtained by a global optimization algorithm; the global optimization method includes, but is not limited to, particle swarm optimization, ant colony optimization or genetic algorithm.

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