Noise enhancement nonlinear system joint detection and estimation method under Bayesian framework

Abstract

The invention discloses a noise enhancement nonlinear system joint detection and estimation method under a Bayesian 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 signal passes through a nonlinear system. And secondly, under the Bayesian criterion, judging which hypothesis in the binary hypotheses is established by utilizing the output of a nonlinear system of noise correction, and estimating unknown parameters in the signal of which the judgment result is H1. On the premise that the detection performance is not reduced, a noise enhancement nonlinear system joint detection and estimation model which minimizes the estimation risk is constructed. The additive noise is the optimal solution of the model and is random distribution formed by not more than two constant vectors. According to the method, noise enhancement and nonlinear system joint detection and estimation under the Bayesian framework are combined, and the Bayesian estimation risk is further reduced under the condition that the Bayesian detection cost is not increased.

Description

Technical Field

[0001] This invention belongs to the field of signal processing, specifically relating to nonlinear systems, noise enhancement, and the hypothesis testing and estimation of binary signals under Bayesian criteria. Background Technology

[0002] Noise is ubiquitous, especially in complex systems. Generally, noise is considered a source of interference with system performance, and mainstream research has focused on noise suppression. However, the impact of noise on systems is not always negative. Under certain conditions, noise can positively enhance signals through nonlinear systems; this phenomenon is known as noise enhancement. Existing research shows that adding noise to the input of some nonlinear systems or adjusting the background noise level can significantly improve the system's detection and / or estimation performance. Current research often studies noise enhancement independently from parameter estimation and signal detection. However, in practical applications, signal detection and parameter estimation are interdependent; separate processing makes it difficult to fully utilize prior information and increases unnecessary computation. Joint detection and estimation, on the other hand, incorporates the detection and estimation problems into the same framework, estimating unknown parameters in the signal only after detection, effectively overcoming this shortcoming. This invention extends the noise enhancement joint detection and estimation framework to more general nonlinear systems. For cases where the prior probability is known, additive noise is added to the input signal of the nonlinear system, and the output signal of the nonlinear system with noise correction is used to make a decision under the Bayesian criterion. The decision result is to estimate the unknown parameters when hypothesis H1 is true, and further minimize the risk of Bayesian estimation without reducing the detection performance. Summary of the Invention

[0003] The purpose of this invention is to address the problem of joint signal detection and estimation for nonlinear systems. Combining the principle of noise enhancement, a Bayesian framework-based method for joint detection and estimation of noise-enhanced nonlinear systems is proposed. Specifically, additive noise is added to the input signal of the nonlinear system. Then, under the Bayesian criterion, the output signal of the noisy nonlinear system is used to determine which of the null and alternative hypotheses is true. The unknown parameters in the signal with the decision result H1 are then estimated. This minimizes the risk of Bayesian estimation without degrading detection performance.

[0004] The present invention 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 Bayesian 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 Bayesian 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, 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] Furthermore, the decision function under the Bayesian criterion Represented as:

[0011]

[0012] in, Let c denote the decision threshold, P(H0) and P(H1) represent the prior probabilities of hypothesis H0 and hypothesis H1, respectively. i,j (i,j=0,1) indicates that when H is assumed j If true, the judgment is H. i The cost function; 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 is the solution to the following optimization problem:

[0017]

[0018] in Indicates the risk of Bayesian detection. Indicates when H is assumed jIf true, the judgment is H. i The probability of; Indicates an estimate; J0 represents the risk estimated by Bayesian estimation; J0 represents the upper limit of the risk detected by Bayesian estimation.

[0019] Furthermore, the additive noise n is a random variable that follows a random distribution consisting of a convex combination of at most two constant vectors with certain weights, and its corresponding probability density function is:

[0020]

[0021] Where, k i (i = 1, 2) are the weighting coefficients, 0 ≤ k i ≤1 and k1+k2=1.

[0022] Furthermore, the constant vector n i and corresponding weight k i (i = 1, 2) is determined by the following constrained optimization problem:

[0023]

[0024] in

[0025] This represents the loss function.

[0026] Furthermore, the constant vector n i and corresponding weight k i (i = 1, 2) is 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.

[0027] This invention combines noise enhancement with the joint detection and estimation problem under the Bayesian 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 Bayesian 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 estimation risk while ensuring detection performance.

[0028] This invention is mainly verified by simulation experiments, and all steps and conclusions have been verified to be correct on MATLAB R2018b. Attached Figure Description

[0029] Figure 1 This is a flowchart of the process of the present invention.

[0030] Figure 2 These are the Bayesian estimated risk values ​​for different σ values ​​in the simulation of this invention, both without noise and with noise correction.

[0031] Figure 3 These are the Bayesian detection risk values ​​without noise and with noise correction corresponding to different σ values ​​in the simulation of this invention. Detailed Implementation

[0032] The present invention will be further described below with reference to embodiments, but it should not be construed that the scope of the present invention is limited to the following embodiments. Various substitutions and modifications made based on ordinary technical knowledge and common practices in the art without departing from the above-described technical concept of the present invention should be included within the scope of protection of the present invention.

[0033] Example 1:

[0034] This embodiment discloses a joint detection and estimation method for noise-enhanced nonlinear systems under a Bayesian framework, including the following steps:

[0035] ① Add additive noise n to the input signal x of the nonlinear system;

[0036] 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;

[0037] ③ Under the Bayesian 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 Bayesian criterion is denoted as...

[0038] 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.

[0039] Example 2:

[0040] This embodiment discloses a joint detection and estimation method for noise-enhanced nonlinear systems under a Bayesian framework, including the following steps:

[0041] ① Input signal to nonlinear system Add independent additive noise N is an integer greater than or equal to 1. Under hypothesis H0, the probability density function of x is p0(x). Under hypothesis H1, x contains an unknown parameter θ, with a corresponding conditional probability density function p1(x|θ), and the unknown parameter θ∈Λ follows a probability function p θ The distribution of (θ). The additive noise n follows a probability density function p. n The distribution of (n).

[0042] ② After passing through the nonlinear system, obtain the noise-corrected nonlinear system output. M is an integer greater than or equal to 1, and T(·) denotes the transfer function of the nonlinear system. The probability density function of z under the assumption H0 is:

[0043]

[0044] Where δ(·) represents the Dirac function, The probability density function of z under hypothesis H1 is:

[0045]

[0046] Where, f1(z|n)=∫ Λ p θ (θ)f1(z|θ,n)dθ,

[0047] ③ Under the Bayesian criterion, the output z of the nonlinear system is used to determine which hypothesis H0 or H1 is true. This corresponds to the decision function under the Bayesian criterion. Represented as:

[0048]

[0049] in, Let c denote the decision threshold, P(H0) and P(H1) represent the prior probabilities of hypothesis H0 and hypothesis H1, respectively. i,j (i,j=0,1) indicates that when H is assumed j If true, the judgment is H. i The cost function. In this case, the corresponding Bayesian detection risk. Calculated as:

[0050]

[0051] in Indicates when H is assumed j If true, the judgment is H. i The probability of; These represent the addition of probability density functions p to the input of a nonlinear system. n (n) simultaneously select When constructing the decision function, the corresponding false alarm probability and detection probability are calculated.

[0052] 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. The overall Bayesian estimated risk is:

[0053]

[0054] in, Indicates when H j If true, the judgment is H. i The risk is estimated conditionally for (i,j=0,1), and there is

[0055]

[0056] in, The loss function can be represented by mean squared error, absolute error, or uniform cost function, etc. Combining equations (10) and (11), the overall Bayesian estimate of risk is:

[0057]

[0058] in

[0059]

[0060] Furthermore, to minimize the Bayesian estimation risk without increasing the risk of Bayesian detection, a joint detection and estimation model for noise-enhanced nonlinear systems under a Bayesian framework is constructed:

[0061]

[0062] Where J0 represents the upper limit of the Bayesian detection risk. The additive noise n is the solution to the optimization problem in (20).

[0063] The additive noise n satisfying (20) is a random variable that follows a random distribution consisting of at most two constant vectors combined with certain weights in a convex combination, with the corresponding probability density function being

[0064]

[0065] Where k i (i = 1, 2) are the weighting coefficients, 0 ≤ k i ≤1 and k1+k2=1. The proof is as follows:

[0066] 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 (13) and (16) can be further analyzed. If replaced with φ1(·), then the Bayesian detection risk J(φ1,p) n ) and Bayesian risk estimation It can be expressed as the expectation of a specific auxiliary function with respect to the additive noise distribution, as follows:

[0067]

[0068] in F0(n)dn,

[0069]

[0070] in

[0071]

[0072] in p θ (θ)dθdz,

[0073] Furthermore, by fixing the detector in (20) to φ1(·), a novel noise-enhanced nonlinear joint detection and estimation model is obtained as follows:

[0074]

[0075] Where C0 = J0 - P(H0)c 00 -P(H1)c 01 At this point, the model in (27) is a convex optimization problem. Through derivation, the corresponding optimal additive noise is a random distribution consisting of no more than two constant vectors combined with certain weights in a convex configuration. That is, in J(φ1,p... n Under the constraint that J0 = J(φ1,0), the optimal additive noise that minimizes the risk of Bayesian estimation is a random distribution consisting of a convex combination of no more than two constant vectors with certain weights.

[0076] Assume φ1 is the detector with optimal additive noise correction in model (20) (i.e. If the convex optimization problem in (27) still holds, then the optimal additive noise in the optimization problem in (27) is also a random distribution consisting of no more than two constant vectors. That is to say, the change of the detector will not affect the form of the optimal additive noise. Therefore, the optimal additive noise probability density function corresponding to the optimization problem in (20) is also as shown in (21).

[0077] Furthermore, in (21) Substituting into (20), the non-convex constraint optimization problem in (20) can be simplified as follows:

[0078]

[0079] in,

[0080] The constant vector n i and corresponding weight k i(i = 1, 2) 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.

[0081] The effects of this invention can be further illustrated by the following simulation experiments:

[0082] In this simulation experiment, the binary hypothesis testing problem is as follows:

[0083]

[0084] Where x is the input signal of the nonlinear system, θ is an unknown parameter, and its probability density function is: v represents zero-mean asymmetric Gaussian mixture background noise, corresponding to the probability density function. Here, 0 ≤ t ≤ 1, then the probability density functions of x under H0 and H1 are expressed 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-corrected output z of the limiting system can be expressed as:

[0085]

[0086] Where A is the amplitude limiting threshold. z assumes H i The probability density function for (i = 0, 1) is expressed as:

[0087]

[0088] Under Uniform Cost Allocation (UCA), C 10 =C 01 =1,C 00 =C 11 =0. For ease of understanding, the decision function is rewritten as follows:

[0089]

[0090] Where Γ1 represents the observation space of hypothesis H1.

[0091] Take A=1,u θ =0.1, μ=0.5, σ=0.51, σ θTaking t=0.5 and t=0.5 as examples, without adding noise to the input x of the nonlinear system, the Bayesian detection risk and Bayesian estimation risk are 0.3460 and 0.5435, respectively. When random noise is added to the input signal of the nonlinear system (the random noise is a random distribution composed of constant vectors n1=-0.8143 and n2=-0.8641 with weights k1=0.2441 and k2=0.2559), and the observation space Γ1 is adjusted from (-∞,-1.018]∪[0.5556,+∞) to (-∞,-1.1001]∪[-0.2956,+∞), the corresponding Bayesian detection risk and Bayesian estimation risk are 0.3458 and 0.2542, respectively. That is to say, by adding noise to the system and adjusting the detector, the Bayesian detection risk remains almost unchanged while the Bayesian estimation risk is reduced by more than 53%.

[0092] Keep A=1, u θ =0.45, μ=0.5, σ θ With t=0.5 and t=0.5 unchanged, the value of σ is gradually increased from 0 to 2.3. The calculated Bayesian estimated risk values ​​and Bayesian detection risk values ​​for the un-noiseed and noise-corrected versions are as follows: Figure 2 , Figure 3 As shown.

[0093] To further explain Figure 2 and Figure 3 The results are shown in Table 1 for A=1, u θ =0.1, μ=0.5, σ θ For σ = 0.5 and t = 0.5, the original Bayesian detection risk value J and the Bayesian estimated risk value R corresponding to different σ values ​​are: B The observation space Γ1 corresponding to the decision hypothesis H1 holds; Table 2 gives the noise components λ / n1 / n2 of the optimal additive noise, the noise-corrected Bayesian detection risk J, and the Bayesian estimation risk R for different σ values. B And the observation space Γ1 corresponding to when hypothesis H1 holds.

[0094] Table 1. Original Judgment Scheme, Bayesian Estimation Risk, and Detection Risk

[0095]

[0096] Table 2. Noise Enhancement Decision Scheme, Bayesian Estimation Risk, and Detection Risk

[0097]

Claims

1. A joint detection and estimation method for noise-enhanced nonlinear systems within a Bayesian 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 Bayesian 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 Bayesian 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 method for joint detection and estimation of noise-enhanced nonlinear systems under a Bayesian 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), Decision function under Bayesian criterion Represented as: in, Let c denote the decision threshold, P(H0) and P(H1) represent the prior probabilities of hypothesis H0 and hypothesis H1, respectively. i,j (i,j=0,1) indicates that when H is assumed j If true, the judgment is H. i The cost function; 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 method for joint detection and estimation of a noise-enhanced nonlinear system under a Bayesian framework according to claim 2, characterized in that: The additive noise n is the solution to the following optimization problem: in, Indicates the risk of Bayesian detection. Indicates when H is assumed j If true, the judgment is H. i The probability of; Indicates an estimate; J0 represents the risk estimated by Bayesian estimation; J0 represents the upper limit of the risk detected by Bayesian estimation.

4. The method for joint detection and estimation of a noise-enhanced nonlinear system under a Bayesian 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 two constant vectors with certain weights, and its corresponding probability density function is: Where, k i (i = 1, 2) are the weighting coefficients, 0 ≤ k i ≤1 and k1+k2=1.

5. The method for joint detection and estimation of a noise-enhanced nonlinear system under a Bayesian framework according to claim 4, characterized in that: The constant vector n i and corresponding weight k i (i = 1, 2) is determined by the following constrained optimization problem: in This represents the loss function.

6. The method for joint detection and estimation of a noise-enhanced nonlinear system under a Bayesian framework according to claim 5, characterized in that: The constant vector n i and corresponding weight k i (i = 1, 2) is 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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