Weak target signal-level localization method based on passive radar network collaboration without reference channel

Abstract

The present invention discloses a weak target signal-level positioning method for a networked, collaborative passive radar system without a reference channel. The method first initializes system parameters and establishes a target echo model for the passive radar without a reference channel that is contaminated by noise and direct wave interference. The target echo model of distributed multiple stations is then combined with unknown signals from non-cooperative external radiation sources to establish a complete data set, and its conditional likelihood function is derived. Finally, based on the expectation-maximization principle, the expected step and maximum step of the conditional likelihood function are iteratively executed to continuously update the target position estimate and suppress direct wave interference until the convergence conditions of the expectation-maximization algorithm are met to obtain a maximum likelihood estimate of the target position, thereby achieving joint direct wave suppression and target position estimate update. The method of the present invention solves the problems of low positioning accuracy, susceptibility to direct wave interference, unknown signals from non-cooperative external radiation sources, and strong dependence on reference channels in current passive radar systems for weak target detection.

Description

Technical Field

[0001] The present invention belongs to the technical field of passive radar target positioning, and in particular relates to a weak target signal-level positioning method for passive radar networking collaboration without a reference channel. Background Art

[0002] Passive radars detect target status by detecting reflected echoes from external emitters. Compared to active radars, which require dedicated emitters, passive radars offer advantages such as low electromagnetic pollution, low power consumption, and strong stealth. With the widespread deployment of digital video / audio broadcast base stations, cellular communication base stations, and communication / navigation satellites, the application of passive radar-based detection technology in civilian applications is increasing. To address the high computational complexity and limited detection accuracy associated with unknown signals from non-cooperative external emitters, passive radars are currently classified into two categories based on whether or not they are equipped with a reference channel. The first category is dual-channel passive radars. In addition to a monitoring channel for observing target echoes, a reference channel is also equipped, pointing toward the external emitter, to receive direct waves from the external emitter as a reference signal. Dual-channel passive radars primarily exploit the correlation between target echoes and the reference signal to obtain target status information. Therefore, this approach relies heavily on high-quality reference channel signals. In contrast, passive radars without a reference channel typically only have a single monitoring channel and use the correlation between target echoes from the same external emitter observed by different radars to detect the target. Reference-less passive radars avoid the stringent requirements of deploying a high-quality reference channel, reducing system hardware cost and complexity. However, the separate deployment of the passive radar's receiver and external radiation source inevitably results in the received target echo being superimposed on a direct wave—a signal emitted by the external radiation source and reaching the passive radar receiver directly. Because the direct wave is not reflected or attenuated by the target, it contains no target information and is typically much stronger than the target echo. This is also known as direct wave interference. Previous studies have shown that direct wave interference can reduce target detection probability and target parameter estimation accuracy.

[0003] Passive radar network collaboration involves combining multiple widely distributed passive radars and external radiation sources to form a distributed passive collaborative detection network, enabling multi-layered, multi-angle, and three-dimensional detection of targets. Compared to a single radar system, widely distributed passive radar network collaboration accumulates more samples to improve signal gain. Multi-angle observations suppress the glint of the target's radar cross-section, thereby achieving spatial diversity gain. Therefore, passive radar network collaboration offers unique advantages in anti-interference, counter-stealth, and low-altitude target detection.

[0004] Target localization is a core task in cooperative detection using passive radar networks. Methods can generally be categorized into two types. The first is a two-step localization method, also known as data-level localization. First, each passive radar independently estimates data parameters related to the target's state, such as the echo arrival time and arrival time difference, based on the received target echo. These data parameters are then combined to estimate the target's position. This method has the advantages of requiring minimal transmission bandwidth and low computational complexity. However, errors in the data parameter estimation generated in the first step can accumulate in the second step, affecting the final positioning accuracy, particularly in environments with small sample sizes or low signal-to-noise ratios. Furthermore, because each passive radar independently processes the target echo, effective signal-to-noise ratio accumulation for weak target echoes is difficult, limiting the detection capability. With the increase in communication bandwidth and computing power, direct localization, also known as signal-level localization, has gradually attracted widespread attention. This method bypasses the data parameter estimation step by directly maximizing the joint likelihood function constructed from all passive radar echoes to obtain the maximum likelihood estimate of the target's position. This theoretically achieves higher positioning accuracy and robustness, and is particularly effective in detecting weak targets.

[0005] While research has made some progress in signal-level target localization using cooperative passive radar networks, most approaches have ignored direct-arrival interference, which is stronger than the target echo, or assumed that direct-arrival interference is completely suppressed. Furthermore, some studies rely on high-quality reference channels, increasing system deployment cost and complexity. Consequently, there is currently a lack of effective solutions for high-precision signal-level target localization using cooperative passive radar networks without reference channels in direct-arrival interference environments with unknown signals from non-cooperative external emitters.

[0006] The document "Direct Target Localization for Distributed Passive Radars With Direct-Path Interference Suppression," IEEE Transactions on Signal Processing, 2024, vol. 72, pp. 3611-3625, designs a direct target localization algorithm by combining the reference channel and surveillance channel observations of a distributed passive radar, while also suppressing direct wave interference. However, this work is highly dependent on reference channel observations and is not suitable for signal-level target localization of passive radars without a reference channel. CN108519586A discloses a distributed passive radar system and its target localization method. The method first uses a reference signal to extract the echo arrival time difference, and then uses the echo arrival time difference estimated by multiple passive radars to achieve target localization. This work is essentially a two-step positioning method, with limited positioning performance in environments with a small sample size and low signal-to-noise ratio. In particular, it ignores the direct wave interference superimposed on the target echo, making it unable to operate in environments with strong direct wave interference. CN117872342A discloses a distributed passive radar signal-level robust positioning method with direct wave suppression. First, the reference channel observation signal is used to estimate the waveform of the external radiation source's emission signal. This waveform is then substituted into the direct target position positioning algorithm established by the monitoring channel observation signal, thereby achieving low-complexity signal-level target positioning for a distributed dual-channel passive radar. This method is suitable for target positioning problems in dual-channel distributed passive radars equipped with an additional reference channel. It is not applicable to scenarios such as direct wave suppression and signal-level positioning of radars without a reference channel.

[0007] In summary, the existing distributed passive radar related target detection or positioning algorithms are not fully applicable to the direct wave suppression and weak target signal level target positioning problems of passive radar without reference channel. Summary of the Invention

[0008] To solve the above technical problems, the present invention provides a weak target signal-level positioning method for passive radar networking without reference channels, which can achieve low-complexity and high-precision robust positioning of weak target signal-level by passive radar without reference channels in a direct wave interference environment.

[0009] The technical solution adopted by the present invention is: a weak target signal-level positioning method using passive radar networking without a reference channel, the specific steps of which are as follows:

[0010] S1. Initialize system parameters;

[0011] Initialize the number N and position coordinates of external radiation sources Initialize the number M and position coordinates of passive radar receivers Target position coordinates p = [x, y] T .

[0012] Among them, the superscript (·) T Indicates transpose.

[0013] For a passive dual-base pair consisting of the nth external radiation source and the mth passive radar receiver, the signal amplitudes of the direct wave interference and the target echo in the received signal are expressed as α m,n and β m,n ; Set the number of passive radar sampling points to K and the sampling period to T s ; The noise of the superposition of the echo received by each passive radar receiver obeys zero mean and variance is are complex circularly symmetric white Gaussian distributions and are independent of each other.

[0014] S2. Establish a passive radar echo model without a reference channel contaminated by direct waves and noise, and construct a joint observation model for passive radar network collaboration to obtain spatial diversity gain;

[0015] S3. Based on the maximum likelihood principle, a signal-level likelihood function for cooperative observation of passive radar networks is constructed to enhance the detection capability of weak targets, thereby forming a non-convex and nonlinear high-dimensional target position estimation problem;

[0016] S4. Treat the unknown signals of non-cooperative external radiation sources as unobservable hidden variables, and conduct collaborative observations with passive radar networks to construct a complete data set and derive its conditional likelihood function.

[0017] S5. Design a weak target signal-level iterative estimation algorithm for direct wave interference suppression based on the expectation maximization principle. Each iteration first calculates the conditional expectation of the likelihood function with respect to the unknown external radiation source signal, and then maximizes the expectation based on a joint optimization mechanism of sequential and Newton iterations to achieve joint direct wave suppression and target position estimation update.

[0018] Furthermore, the step S2 is specifically as follows:

[0019] The target echo received by the mth passive radar receiver from the nth external radiation source is contaminated by the direct wave and noise The expression is as follows:

[0020]

[0021] Among them, the superscript (·) d and(·) e represent the direct path and target echo path respectively, α m,n and β m,n represent the complex amplitudes of the direct wave and the target echo respectively; represents the baseband transmission signal of the nth external radiation source at time t; represents the direct path propagation delay from the nth external radiation source to the mth passive radar receiver; represents the propagation delay of the echo path from the nth external radiation source through the target to the mth passive radar receiver, and c0 represents the propagation speed of electromagnetic waves; The variance is A complex circularly symmetric white Gaussian random variable.

[0022] Assuming that the noises of different passive radar receivers are independent of each other and white in space and time, the expression is as follows:

[0023]

[0024] Among them, the symbol Indicates time, symbol (·) * and δ(·) denote the statistical expectation, conjugate, and impulse functions, respectively.

[0025] set up and Respectively represent the time domain signal in formula (1) and The K-point discrete sampling vector, the sampling period is T s .

[0026] Among them, the superscript (·) d / e Indicates the direct path or target echo path.

[0027] Then calculate the frequency domain expression of the discrete vector to extract the delay of the direct wave interference and the target echo, and set represents the K-point discrete Fourier transform of the received signal, T represents the discrete Fourier transform matrix, and its (a, b)th element is expressed as The frequency domain expression of the discrete sampling vector of formula (1) is as follows:

[0028]

[0029] in, and The symbol diag{·} represents a diagonal matrix whose diagonal elements are vectors or matrices.

[0030] The observation model of MN passive dual-base pairs composed of M passive radars without reference channels and N external radiation sources is combined. The joint observation r of the passive radar network without reference channels is expressed as follows:

[0031]

[0032] Furthermore, the step S3 is specifically as follows:

[0033] Based on the Gaussian characteristics of the joint observation signal, the logarithmic form of the likelihood function of the joint observation r in formula (4) is as follows:

[0034]

[0035] in,

[0036]

[0037] and,

[0038]

[0039] Among them, the symbol represents the Kronecker product, the symbol det(·) represents the determinant of the matrix, (·) Η represents the conjugate transpose, (·) -1 Represents the inverse operation, matrix I K and I M Represents K-dimensional and M-dimensional identity matrices, matrices and Represent the noise w m,n and signal s n The covariance matrix of .

[0040] Maximum likelihood estimation of target location It is obtained by maximizing the log-likelihood function lnp(r) in formula (5), which is expressed as follows:

[0041]

[0042] Among them, the unknown parameter vector θ=[p T ,α T ,β T ] T , α=[α 1,1 ,…,α M,N ] T , β=[β 1,1 ,…,β M,N ] T .

[0043] Furthermore, the step S4 is specifically as follows:

[0044] Unknown external radiation source signal is considered to be an unobservable hidden vector. Then the observation r in equation (4) is considered to be an incomplete data set, and the expression of the complete data set y is defined as follows:

[0045] y=[r T ,s T ] T (9)

[0046] According to Bayesian theory, the conditional log-likelihood function expression of the complete data set y is as follows:

[0047]

[0048] Among them, the function p(r,s,θ) represents the joint likelihood function of r, s and θ; the function p(s,θ) represents the joint likelihood function of s and θ; and the function p(θ) represents the likelihood function of θ.

[0049] The functions lnp(r|s,θ) and lnp(s|θ) represent the conditional log-likelihood functions of the networked collaborative joint observation r and the non-cooperative external radiation source signal s, respectively, and are expressed as follows:

[0050]

[0051] in, represents the covariance matrix of the external radiation source signal s, μ represents the conditional mean of the observation r, and the specific expression is as follows:

[0052]

[0053] Combining Equations (11) and (12), the conditional log-likelihood function expression of the complete data set y in Equation (10) is as follows:

[0054] lnp(y|θ)=f1(s)+f2(s,θ) (14)

[0055] in,

[0056]

[0057] Furthermore, the step S5 is specifically as follows:

[0058] S51, performing the expectation step, i.e. finding the conditional expectation of the likelihood function of the complete data set;

[0059] The conditional expectation of the likelihood function lnp(y|θ) of the complete data set y in formula (14) is The expression is as follows:

[0060]

[0061] Among them, the superscript (·) (l) represents the lth iteration, Denotes the parameter estimate for the lth iteration. Discarding the term f1(s) that is unrelated to the estimated parameter θ and calculating the conditional expectation of f2(s,θ), we get the following expression:

[0062]

[0063] in,

[0064]

[0065] The symbol Tr{·} represents the trace of the matrix; the vector and matrix Respectively represent s n The minimum mean squared error estimate and conditional covariance matrix of .

[0066] Due to s n and r n obey a joint Gaussian distribution with mean zero, then and The closed-form expression is as follows:

[0067]

[0068] in,

[0069]

[0070] and,

[0071]

[0072] S52, executing the maximum step, that is, maximizing the conditional expectation of the likelihood function of the complete data set obtained in step S51 based on the joint optimization mechanism of sequential and Newton iteration;

[0073] By maximizing the conditional expectation of formula (16) The estimated parameter vector for the (l+1)th iteration The expression is as follows:

[0074]

[0075] Then maximize the function in (25) That is, the target position p and the signal amplitudes α and β are jointly optimized. Using the sequential estimation strategy, Transformed into two low-dimensional sub-estimation problems, the expressions are as follows:

[0076]

[0077] in, and They represent the estimation of the signal amplitude given by the lth iteration respectively.

[0078] The optimization problems of equations (26) and (27) are solved as follows:

[0079] A. Solve the optimization problem (26), i.e., Newton iterative estimation of the target position;

[0080] The solution of the optimization problem (26) is solved by the Newton iteration method, and the expression is as follows:

[0081]

[0082] Here, i represents the index of the Newton iteration. and Represents functions respectively About estimated location The first and second derivatives of are expressed as follows:

[0083]

[0084] in,

[0085]

[0086] In formula (30), the path delay Derivatives of the target position coordinates x and y and The expressions are as follows:

[0087]

[0088] In formula (30), the cost function Q1 is related to the path delay The derivative of The expression is as follows:

[0089]

[0090] in,

[0091]

[0092] and,

[0093]

[0094] The Newton iteration process of formula (28) continues until the difference between two consecutive estimates meets the convergence criterion:

[0095] Among them, ε Newton Indicates a smaller value, set according to the actual scenario and positioning accuracy.

[0096] B. Solve the optimization problem (27), i.e., estimate the signal amplitude;

[0097] Let function about and The conjugate partial derivatives of are equal to zero, and the expressions are as follows:

[0098]

[0099] By solving equation (35), the solution of the optimization problem (27) is as follows:

[0100]

[0101] Based on steps A and B, the updated estimated parameters for the l+1th iteration of problem (25) have all been obtained, namely, and in formula (36) and It will be used as input for the l+2th iteration.

[0102] S53, repeating steps S51-S52 until the expectation maximization iteration convergence condition is met, thereby achieving joint direct wave suppression and target position estimation update;

[0103] The maximization iterative convergence condition is

[0104] Among them, ε EM Set according to the actual scenario and task accuracy requirements.

[0105] Furthermore, in step S5, the weak target signal-level iterative estimation algorithm for direct wave interference suppression performs an initial estimation on unknown parameters, specifically as follows:

[0106] In problem (26), the unknown parameter and is initialized uniformly randomly on each trial, and then a coarse 2D grid search is performed on the target position to find the initial estimate of the target position that maximizes the problem (26)

[0107] After the iterative estimation algorithm converges, that is, after the maximum iterative convergence condition is reached, the simplified expression of the target position estimation problem in equation (26) is as follows:

[0108]

[0109] Among them, Indicates that from the observation r m,n Subtract the estimated direct wave interference from item Indicates the estimated target echo.

[0110] The beneficial effects of the present invention are as follows: the method of the present invention first initializes system parameters and establishes a target echo model of a passive radar without a reference channel that is contaminated by noise and direct wave interference, then combines the target echo models of distributed multiple stations and unknown signals of non-cooperative external radiation sources to establish a complete data set, and derives its conditional likelihood function, and finally, based on the expectation maximization principle, it iteratively executes the expected step and maximum step of the conditional likelihood function to continuously update the target position estimation and suppress direct wave interference until the convergence condition of the expectation maximization algorithm is met to obtain the maximum likelihood estimation of the target position, thereby realizing joint direct wave suppression and target position estimation update. The method of the present invention can be applied to the field of passive radar target positioning. A non-cooperative external radiation source with unknown signals is used as a signal source. A passive radar without a reference channel is used to perform weak target signal-level positioning of a weak target in a networked collaborative manner with direct wave interference suppression. The method is particularly suitable for realizing low-complexity robust weak target signal-level positioning of a weak target in a direct wave interference environment by using a non-cooperative external radiation source with unknown signals. At the same time, the method gets rid of the dependence on the reference channel, reduces the system deployment cost and complexity, and solves the problems of low positioning accuracy, susceptibility to direct wave interference, unknown signals of the non-cooperative external radiation source, and strong dependence on the reference channel in weak target detection of current passive radar systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0111] Figure 1 The present invention provides a flow chart of a weak target signal level positioning method for passive radar networking without a reference channel.

[0112] Figure 2 The figure is a schematic diagram of a two-dimensional simulation scenario of collaborative target positioning by passive radar networking without a reference channel in an embodiment of the present invention.

[0113] Figure 3 This is a likelihood plane diagram of the weak target signal level positioning method for passive radar networking without reference channels in an embodiment of the present invention.

[0114] Figure 4 2 is a schematic diagram showing a comparison of positioning errors between the method of the present invention and other target positioning algorithms under different direct wave interference intensities under weak target echo intensity in an embodiment of the present invention.

[0115] Figure 5 3 is a schematic diagram showing a comparison of positioning errors between the method of the present invention and other target positioning algorithms under different direct wave interference intensities under strong target echo intensity in an embodiment of the present invention.

[0116] Figure 6 2 is a schematic diagram comparing positioning errors of the method of the present invention and other target positioning algorithms under different target echo intensities in an embodiment of the present invention. DETAILED DESCRIPTION

[0117] In order to facilitate the description of the present invention, the following terms are first defined:

[0118] 1. Passive radar: It obtains the target status by receiving the reflected echo from the external radiation source to the target without the need for a dedicated transmitter.

[0119] 2. Passive radar network collaboration: Passive radars and external radiation sources are widely distributed to form a passive collaborative detection network, which observes targets from multiple perspectives, suppresses target radar cross-sectional area flicker, and obtains spatial diversity gain.

[0120] 3. External radiation sources: Various signal emission sources that are widely present in space, including but not limited to commercial radiation sources such as broadcast / communication base stations, navigation / communication satellites, and military signal emission sources such as active radar / jamming.

[0121] 4. Non-cooperative external radiation source: The signal waveform and other working parameters of the external radiation source are unknown.

[0122] 5. Direct wave interference: The signal received by the radar that comes directly from an external radiation source and is superimposed on the target echo.

[0123] 6. Cramer-Rao lower bound: the lower bound of the variance of any unbiased estimator.

[0124] The present invention is mainly verified by simulation experiments, and all steps and conclusions are verified to be correct on Matlab R2020b. The method of the present invention is further described below with reference to the accompanying drawings and examples.

[0125] like Figure 1 As shown in FIG, a flow chart of a weak target signal-level positioning method for passive radar networking without a reference channel of the present invention is shown, and the specific steps are as follows:

[0126] S1. Initialize system parameters;

[0127] Initialize the number N and position coordinates of external radiation sources Initialize the number M and position coordinates of passive radar receivers Target position coordinates p = [x, y] T .

[0128] Among them, the superscript (·) T Indicates transpose.

[0129] For a passive dual-base pair consisting of the nth external radiation source and the mth passive radar receiver, the signal amplitudes of the direct wave interference and the target echo in the received signal are expressed as α m,n and β m,n ; Set the number of passive radar sampling points to K and the sampling period to T s ; The noise of the superposition of the echo received by each passive radar receiver obeys zero mean and variance is are complex circularly symmetric white Gaussian distributions and are independent of each other.

[0130] S2. Establish a passive radar echo model without a reference channel contaminated by direct waves and noise, and construct a joint observation model for passive radar network collaboration to obtain spatial diversity gain;

[0131] S3. Based on the maximum likelihood principle, a signal-level likelihood function for cooperative observation of passive radar networks is constructed to enhance the detection capability of weak targets, thereby forming a non-convex and nonlinear high-dimensional target position estimation problem;

[0132] S4. Treat the unknown signals of non-cooperative external radiation sources as unobservable hidden variables, and conduct collaborative observations with passive radar networks to construct a complete data set and derive its conditional likelihood function.

[0133] S5. Design a weak target signal-level iterative estimation algorithm for direct wave interference suppression based on the expectation maximization principle. Each iteration first calculates the conditional expectation of the likelihood function with respect to the unknown external radiation source signal, and then maximizes the expectation based on a joint optimization mechanism of sequential and Newton iterations to achieve joint direct wave suppression and target position estimation update.

[0134] In this embodiment, step S2 is specifically as follows:

[0135] The target echo received by the mth passive radar receiver from the nth external radiation source is contaminated by the direct wave and noise The expression is as follows:

[0136]

[0137] Among them, the superscript (·) d and(·) e represent the direct path and target echo path respectively, α m,n and β m,n represent the complex amplitudes of the direct wave and the target echo respectively; represents the baseband transmission signal of the nth external radiation source at time t; represents the direct path propagation delay from the nth external radiation source to the mth passive radar receiver; represents the propagation delay of the echo path from the nth external radiation source through the target to the mth passive radar receiver, and c0 represents the propagation speed of electromagnetic waves; The variance is A complex circularly symmetric white Gaussian random variable.

[0138] Assuming that the noises of different passive radar receivers are independent of each other and white in space and time, the expression is as follows:

[0139]

[0140] Among them, the symbol t represents time, and the symbol (·) * and δ(·) denote the statistical expectation, conjugate, and impulse functions, respectively.

[0141] set up and Respectively represent the time domain signal in formula (1) and The K-point discrete sampling vector, the sampling period is T s .

[0142] Among them, the superscript (·) d / e Indicates the direct path or target echo path.

[0143] Then calculate the frequency domain expression of the discrete vector to extract the delay of the direct wave interference and the target echo, and set represents the K-point discrete Fourier transform of the received signal, T represents the discrete Fourier transform matrix, and its (a, b)th element is expressed as The frequency domain expression of the discrete sampling vector of formula (1) is as follows:

[0144]

[0145] in, and The symbol diag{·} represents a diagonal matrix whose diagonal elements are vectors or matrices.

[0146] The observation model of MN passive dual-base pairs composed of M passive radars without reference channels and N external radiation sources is combined. The joint observation r of the passive radar network without reference channels is expressed as follows:

[0147]

[0148] In this embodiment, step S3 is specifically as follows:

[0149] Based on the Gaussian characteristics of the joint observation signal, the logarithmic form of the likelihood function of the joint observation r in formula (4) is as follows:

[0150]

[0151] in,

[0152]

[0153] and,

[0154]

[0155] A=diag{α1,…,αN},α n =[α 1,n ,…,α M,n ] T

[0156] B=diag{β1,…,β N},β n =[β 1,n ,…,β M,n ] T (7)

[0157] Among them, the symbol represents the Kronecker product, the symbol det(·) represents the determinant of the matrix, (·) Η represents the conjugate transpose, (·) -1 Represents the inverse operation, matrix I K and I M Represents K-dimensional and M-dimensional identity matrices, matrices and Represent the noise w m,n and signal s n The covariance matrix of .

[0158] Maximum likelihood estimation of target location It is obtained by maximizing the log-likelihood function lnp(r) in formula (5), which is expressed as follows:

[0159]

[0160] Among them, the unknown parameter vector θ=[p T ,α T ,β T ] T , α=[α 1,1 ,…,α M,N ] T , β=[β 1,1 ,…,β M,N ] T .

[0161] The optimization problem in Equation (8) is highly nonconvex and nonlinear, making it difficult to find an analytical solution. Furthermore, its computational complexity increases exponentially with the number of external emitters and passive radars, N and M, making the computational effort prohibitive. To reduce the computational complexity, an iterative estimation algorithm for weak target signal levels based on the expectation-maximization principle is employed.

[0162] In this embodiment, step S4 is specifically as follows:

[0163] Unknown external radiation source signal is considered to be an unobservable hidden vector. Then the observation r in equation (4) is considered to be an incomplete data set, and the expression of the complete data set y is defined as follows:

[0164] y=[r T ,s T ] T (9)

[0165] According to Bayesian theory, the conditional log-likelihood function expression of the complete data set y is as follows:

[0166]

[0167] Among them, the function p(r,s,θ) represents the joint likelihood function of r, s and θ; the function p(s,θ) represents the joint likelihood function of s and θ; and the function p(θ) represents the likelihood function of θ.

[0168] The functions lnp(r|s,θ) and lnp(s|θ) represent the conditional log-likelihood functions of the networked collaborative joint observation r and the non-cooperative external radiation source signal s, respectively, and are expressed as follows:

[0169]

[0170] in, represents the covariance matrix of the external radiation source signal s, μ represents the conditional mean of the observation r, and the specific expression is as follows:

[0171]

[0172] Combining Equations (11) and (12), the conditional log-likelihood function expression of the complete data set y in Equation (10) is as follows:

[0173] lnp(y|θ)=f1(s)+f2(s,θ) (14)

[0174] in,

[0175]

[0176] In this embodiment, step S5 is specifically as follows:

[0177] S51, performing the expectation step, i.e. finding the conditional expectation of the likelihood function of the complete data set;

[0178] The conditional expectation of the likelihood function lnp(y|θ) of the complete data set y in formula (14) is The expression is as follows:

[0179]

[0180] Among them, the superscript (·) (l)represents the lth iteration, Denotes the parameter estimate for the lth iteration. Discarding the term f1(s) that is unrelated to the estimated parameter θ and calculating the conditional expectation of f2(s,θ), we get the following expression:

[0181]

[0182] in,

[0183]

[0184] The symbol Tr{·} represents the trace of the matrix; the vector and matrix Respectively represent s n The minimum mean squared error estimate and conditional covariance matrix of .

[0185] Due to s n and r n obey a joint Gaussian distribution with mean zero, then and The closed-form expression is as follows:

[0186]

[0187] in,

[0188]

[0189] and,

[0190]

[0191] S52, executing the maximum step, that is, maximizing the conditional expectation of the likelihood function of the complete data set obtained in step S51 based on the joint optimization mechanism of sequential and Newton iteration;

[0192] By maximizing the conditional expectation of formula (16) The estimated parameter vector for the (l+1)th iteration The expression is as follows:

[0193]

[0194] Then maximize the function in (25) That is, the target position p and the signal amplitudes α and β are jointly optimized. This is also a high-dimensional optimization problem with high computational complexity. Here, a sequential estimation strategy is adopted. Transformed into two low-dimensional sub-estimation problems, the expressions are as follows:

[0195]

[0196] in, and They represent the estimation of the signal amplitude given by the lth iteration respectively.

[0197] The optimization problems of equations (26) and (27) are solved as follows:

[0198] A. Solve the optimization problem (26), i.e., Newton iterative estimation of the target position;

[0199] The solution of the optimization problem (26) is solved by the Newton iteration method, and the expression is as follows:

[0200]

[0201] Here, i represents the index of the Newton iteration. and Represents functions respectively About estimated location The first and second derivatives of are expressed as follows:

[0202]

[0203] in,

[0204]

[0205] In formula (30), the path delay Derivatives of the target position coordinates x and y and The expressions are as follows:

[0206]

[0207] In formula (30), the cost function Q1 is related to the path delay The derivative of The expression is as follows:

[0208]

[0209] in,

[0210]

[0211] and,

[0212]

[0213] The Newton iteration process of formula (28) continues until the difference between two consecutive estimates meets the convergence criterion:

[0214] Among them, ε Newton Indicates a smaller value, set according to the actual scenario and positioning accuracy.

[0215] B. Solve the optimization problem (27), i.e., estimate the signal amplitude;

[0216] The optimization problem (27) has an analytical solution, which is because the function It is a quadratic function of the unknown parameters α and β, as shown in formula (17).

[0217] Let function about and The conjugate partial derivatives of are equal to zero, and the expressions are as follows:

[0218]

[0219] By solving equation (35), the solution of the optimization problem (27) is as follows:

[0220]

[0221] Based on steps A and B, the updated estimated parameters for the l+1th iteration of problem (25) have all been obtained, namely, and in formula (36) and It will be used as input for the l+2th iteration.

[0222] S53, repeating steps S51-S52 until the expectation maximization iteration convergence condition is met, thereby achieving joint direct wave suppression and target position estimation update;

[0223] The maximization iterative convergence condition is

[0224] Among them, ε EM Set according to the actual scenario and task accuracy requirements.

[0225] In this embodiment, in step S5, the weak target signal level iterative estimation algorithm for direct wave interference suppression performs an initial estimation on the unknown parameters, specifically as follows:

[0226] In problem (26), the unknown parameter and is initialized uniformly randomly on each trial, and then a coarse 2D grid search is performed on the target position to find the initial estimate of the target position that maximizes the problem (26)

[0227] After the signal-level iterative algorithm converges, that is, after the maximum iterative convergence condition is reached, the simplified expression of the target position estimation problem in equation (26) is as follows:

[0228]

[0229] Among them, Indicates that from the observation r m,n Subtract the estimated direct wave interference from item In essence, the target position estimation is based on the observation of the direct wave interference. Estimated target echo The correlation between them is used to continuously update the estimated target position and suppress direct wave interference during the iteration process.

[0230] Considering that the target echo in the passive radar observation without reference channel is contaminated by noise and direct wave interference, a high-dimensional joint direct target positioning algorithm involving the target position and other unknown parameters is first designed. However, due to the highly nonlinear nature of the algorithm, there is no closed-form solution. In this embodiment, based on the weak target signal-level iterative estimation algorithm with direct wave interference suppression based on the expectation maximization principle, a weak target signal-level positioning method for collaborative passive radar network without reference channel is designed. The weak target signal-level positioning method iteratively estimates the target position and other unknown parameters, significantly reducing the computational complexity. In addition, a sequential estimation strategy is adopted to transform the high-dimensional estimation problem into several low-dimensional estimation problems, further reducing the computational burden. The weak target signal-level positioning method continuously subtracts the estimated direct wave interference from the received signal throughout the iterative process, and then updates the target position estimate until convergence, achieving joint direct wave suppression and target position estimation. In addition, unlike the two-step method of data-level positioning, the weak target signal-level positioning method of the present invention directly combines the received signals of multiple stations to estimate the target position, which can effectively address the problem of limited estimation performance caused by weak weak target echoes. It also gets rid of the existing passive radar's dependence on reference channels, reduces hardware costs, avoids the strict requirements of deploying high-quality reference channels, and has a wider applicability. Based on this, the weak target signal-level positioning method of the present invention can use non-cooperative external radiation sources with unknown signals to achieve signal-level low-complexity robust target positioning in a direct wave interference environment without reference channels and passive radar network collaboration. It can be applied to fields such as target positioning of passive radars.

[0231] The target position estimation accuracy is evaluated by the root mean square error (RMSE), which is expressed as follows:

[0232]

[0233] Wherein, L represents the number of Monte Carlo experiments. In this embodiment, L=300; represents the target position estimate of the lth Monte Carlo trial.

[0234] In this embodiment, the direct wave interference to target echo intensity ratio INR and the target echo to noise intensity ratio SNR are defined as follows:

[0235]

[0236] The weak target signal-level positioning method designed in this embodiment is a signal-level target positioning method based on the expectation maximization algorithm, so it is called EMDL (expectation maximization-based direct localization). In order to prove the direct wave suppression performance and positioning performance of the positioning algorithm EMDL of the present invention, three positioning algorithms are given for comparison: the positioning algorithm EMDL-NDPI (expectation maximization-based direct localization with nodirect-path interference), that is, signal-level target positioning based on the expectation maximization algorithm but without considering direct wave interference; the positioning algorithm CCTSL (cross-correlation-based two-step localization), that is, two-step target positioning that first estimates the intermediate parameters and then estimates the target position. In addition, it also includes a theoretical lower bound of the estimation error that can be obtained by the positioning algorithm under the observation model described in the present invention, namely RCRLB (square root of the Cramer-Rao lower bound CRLB).

[0237] The simulation scenario of this embodiment is as follows Figure 2 As shown, it contains M = 4 passive radars, located at and N = 1 external radiation source, located at A position at p = [5,6] T Km of target to be located.

[0238] By traversing the 2D grid plane of the monitoring area, the likelihood planes of the positioning algorithms EMDL-NDPI and EMDL under different direct wave interference intensities are obtained, as shown in Figure 3 As shown (the maximum likelihood point corresponds to the minimum point in the figure). Figure 3 (a) and 3(b) are the likelihood function planes of the positioning algorithm EMDL-NDPI when INR = -10dB and INR = 15dB, respectively; Figure 3 (c) and 3(d) are the likelihood function planes of the positioning algorithm EMDL when INR = -10dB and INR = 15dB respectively. Figure 3(a) and 3(c) show that when INR = -10dB, the maximum likelihood points of the positioning algorithms EMDL-NDPI and EMDL both appear at the true target position. When the intensity of the direct wave interference increases to INR = 15dB, as shown in Figure 3 As shown in Figures (b) and 3(d), the maximum likelihood point of the positioning algorithm EMDL-NDPI is far away from the true target position, while the maximum likelihood point of the positioning algorithm EMDL is still at the true target position, indicating that the invented positioning algorithm can work normally in a strong direct wave interference environment and estimate the target position.

[0239] Under the target echo intensity SNR=5dB, change the direct wave interference intensity INR, repeat the steps S1-S5 of the method of the present invention, and obtain Figure 4 .from Figure 4 As can be seen, as the intensity of direct wave interference increases, the positioning algorithms EMDL-NDPI and CCTSL quickly deviate from RCRLB, and positioning fails. However, the positioning error RMSE of the positioning algorithm EMDL of the present invention gradually decreases and always converges to RCRLB until the INR exceeds the tolerance threshold of 10dB. This means that the positioning algorithm of the present invention has good tolerance for direct wave interference, and that appropriate direct wave interference can help improve target positioning performance.

[0240] Under the target echo intensity SNR=10dB, change the direct wave interference intensity INR, repeat the steps S1-S5 of the method of the present invention, and obtain Figure 5 .contrast Figure 4 and 5 It can be seen that as the target echo intensity increases, the direct wave interference tolerance threshold of the positioning algorithm EMDL of the method of the present invention increases from INR=10dB to INIR=15dB, and has a stronger direct wave interference tolerance capability.

[0241] Under the direct wave interference intensity INR=-5dB, change the target echo intensity SNR, repeat the steps S1-S5 of the present invention, and obtain Figure 6 .from Figure 6 It can be seen that with the increase of SNR, the positioning algorithm EMDL of the present invention can first converge to RCRLB, which is effective. However, the positioning algorithms EMDL-NDPI and CCTSL require a larger SNR to converge to RCRLB.

[0242] In summary, the method of the present invention can be applied to the field of passive radar target positioning. It uses a non-cooperative external radiation source with unknown signals as a signal source, and utilizes a passive radar without a reference channel to perform weak target signal-level positioning with direct wave interference suppression on weak targets in a networked collaborative manner. It is particularly suitable for using a non-cooperative external radiation source with unknown signals to achieve signal-level low-complexity robust weak target positioning in a direct wave interference environment in a passive radar networked collaborative manner. At the same time, it gets rid of the dependence on the reference channel, reduces the system deployment cost and complexity, and solves the problems of current passive radar systems in weak target detection, such as low positioning accuracy, susceptibility to direct wave interference, unknown signals of non-cooperative external radiation sources, and strong dependence on reference channels.

[0243] Those skilled in the art will appreciate that the embodiments described herein are intended to help readers understand the principles of the present invention, and it should be understood that the scope of protection of the present invention is not limited to such specific descriptions and embodiments. Those skilled in the art can make various other specific variations and combinations based on the technical teachings disclosed in the present invention without departing from the essence of the present invention, and such variations and combinations are still within the scope of protection of the present invention.

Claims

1. A method for weak target signal-level positioning using passive radar networking without a reference channel, the specific steps are as follows: S1. Initialize system parameters; Initialize the number N and position coordinates of external radiation sources Initialize the number M and position coordinates of passive radar receivers Target position coordinates p = [x, y] T ; in, Superscript (·) T represents transpose; For a passive dual-base pair consisting of the nth external radiation source and the mth passive radar receiver, the signal amplitudes of the direct wave interference and the target echo in the received signal are expressed as α m,n and β m,n ; Set the number of passive radar sampling points to K and the sampling period to T s ; The noise of the superposition of the echo received by each passive radar receiver obeys zero mean and variance is The complex circularly symmetric white Gaussian distribution of and are independent of each other; S2. Establish a passive radar echo model without a reference channel contaminated by direct waves and noise, and construct a joint observation model for passive radar network collaboration to obtain spatial diversity gain; S3. Based on the maximum likelihood principle, a signal-level likelihood function for weak targets in cooperative observation by passive radar networks is constructed to enhance the detection capability of weak targets, thereby forming a non-convex and nonlinear high-dimensional target position estimation problem. S4. Treat the unknown signals of non-cooperative external radiation sources as unobservable hidden variables, and conduct collaborative observations with passive radar networks to construct a complete data set and derive its conditional likelihood function. S5. Design an iterative estimation algorithm for weak target signal level with direct wave interference suppression based on the expectation maximization principle. Each iteration first calculates the conditional expectation of the likelihood function with respect to the unknown external radiation source signal. Then, this expectation is maximized based on a joint optimization mechanism of sequential and Newton iterations to achieve joint direct wave suppression and target position estimation update. The step S2 is specifically as follows: The target echo received by the mth passive radar receiver from the nth external radiation source is contaminated by the direct wave and noise The expression is as follows: Among them, the superscript (·) d and(·) e represent the direct path and target echo path respectively, α m,n and β m,n represent the complex amplitudes of the direct wave and the target echo respectively; represents the baseband transmission signal of the nth external radiation source at time t; represents the direct path propagation delay from the nth external radiation source to the mth passive radar receiver; represents the propagation delay of the echo path from the nth external radiation source through the target to the mth passive radar receiver, and c0 represents the propagation speed of electromagnetic waves; The variance is A complex circularly symmetric white Gaussian random variable; Assuming that the noises of different passive radar receivers are independent of each other and white in space and time, the expression is as follows: Among them, the symbol Indicates time, symbol (·) * and δ(·) denote the statistical expectation, conjugate, and impulse functions, respectively; set up and Respectively represent the time domain signal in formula (1) and The K-point discrete sampling vector, the sampling period is T s ; Among them, the superscript (·) d / e Indicates the direct path or target echo path; Then calculate the frequency domain expression of the discrete vector to extract the delay of the direct wave interference and the target echo, and set represents the K-point discrete Fourier transform of the received signal, T represents the discrete Fourier transform matrix, and its (a, b)th element is expressed as f Δ =1 / T s K; then the frequency domain expression of the discrete sampling vector of formula (1) is as follows: in, and The symbol diag{·} represents a diagonal matrix whose diagonal elements are vectors or matrices; The observation model of MN passive dual-base pairs composed of M passive radars without reference channels and N external radiation sources is combined. The joint observation r of the passive radar network without reference channels is expressed as follows: The step S3 is specifically as follows: Based on the Gaussian characteristics of the joint observation signal, the logarithmic form of the likelihood function of the joint observation r in formula (4) is as follows: in, and, Among them, the symbol represents the Kronecker product, the symbol det(·) represents the determinant of the matrix, (·) Η represents the conjugate transpose, (·) -1 Represents the inverse operation, matrix I K and I M Represents K-dimensional and M-dimensional identity matrices, matrices and Represent the noise w m,n and signal s n The covariance matrix of Maximum likelihood estimation of target location It is obtained by maximizing the log-likelihood function lnp(r) in formula (5), which is expressed as follows: where the unknown parameter vector θ = [p T , α T , β T T , α = [α 1,1 , …, α M,N T , β = [β 1,1 , …, β M,N T ;​​​ The step S4 is specifically as follows: Unknown external radiation source signal is considered to be an unobservable hidden vector; then the observation r in formula (4) is considered to be an incomplete data set, and the expression of the complete data set y is defined as follows: y=[r T ,s T ] T (9) According to Bayesian theory, the conditional log-likelihood function expression of the complete data set y is as follows: Among them, the function p(r,s,θ) represents the joint likelihood function of r, s and θ; the function p(s,θ) represents the joint likelihood function of s and θ; the function p(θ) represents the likelihood function of θ; The functions lnp(r|s,θ) and lnp(s|θ) represent the conditional log-likelihood functions of the networked collaborative joint observation r and the non-cooperative external radiation source signal s, respectively, and are expressed as follows: in, represents the covariance matrix of the external radiation source signal s, μ represents the conditional mean of the observation r, and the specific expression is as follows: Combining Equations (11) and (12), the conditional log-likelihood function expression of the complete data set y in Equation (10) is as follows: lnp(y|θ)=f1(s)+f2(s,θ) (14) in, The step S5 is specifically as follows: S51, performing the expectation step, i.e. finding the conditional expectation of the likelihood function of the complete data set; The conditional expectation of the likelihood function lnp(y|θ) of the complete data set y in formula (14) is The expression is as follows: Among them, the superscript (·) (l) represents the lth iteration, Represents the parameter estimate of the lth iteration; discard the term f1(s) that is not related to the estimated parameter θ, calculate the conditional expectation of f2(s,θ), and the expression is as follows: in, The symbol Tr{·} represents the trace of the matrix; the vector and matrix Respectively represent s n The minimum mean square error estimate and conditional covariance matrix of ; Due to s n and r n obey a joint Gaussian distribution with mean zero, then and The closed-form expression is as follows: in, and, S52, executing the maximum step, that is, maximizing the conditional expectation of the likelihood function of the complete data set obtained in step S51 based on the joint optimization mechanism of sequential and Newton iteration; By maximizing the conditional expectation of formula (16) The estimated parameter vector for the l+1th iteration The expression is as follows: Then maximize the function in (25) That is, the target position p and the signal amplitudes α and β are jointly optimized; a sequential estimation strategy is adopted to Transformed into two low-dimensional sub-estimation problems, the expressions are as follows: in, and They represent the estimation of the signal amplitude given by the lth iteration respectively; The optimization problems of equations (26) and (27) are solved as follows: A. Solve the optimization problem (26), i.e., Newton iterative estimation of the target position; The solution of the optimization problem (26) is solved by the Newton iteration method, and the expression is as follows: Where i represents the index of Newton iteration; and Represents functions respectively About estimated location The first and second derivatives of are expressed as follows: in, In formula (30), the path delay Derivatives of the target position coordinates x and y and The expressions are as follows: In formula (30), the cost function Q1 is related to the path delay The derivative of The expression is as follows: in, and, The Newton iteration process of formula (28) continues until the difference between two consecutive estimates meets the convergence criterion: Among them, ε Newton Indicates a smaller value, set according to the actual scenario and positioning accuracy; B. Solve the optimization problem (27), i.e., estimate the signal amplitude; Let function about and The conjugate partial derivatives of are equal to zero, and the expressions are as follows: By solving equation (35), the solution of the optimization problem (27) is as follows: Based on steps A and B, the updated estimated parameters for the l+1th iteration of problem (25) have all been obtained, namely, and in formula (36) and It will be used as input for the l+2th iteration; S53, repeating steps S51-S52 until the expectation maximization iteration convergence condition is met, thereby achieving joint direct wave suppression and target position estimation update; The maximization iterative convergence condition is Among them, ε EM Set according to the actual scenario and task accuracy requirements.

2. The method for weak target signal level positioning based on passive radar network collaboration without reference channel according to claim 1, characterized in that: In step S5, the weak target signal-level iterative estimation algorithm for direct wave interference suppression performs an initial estimation on unknown parameters, specifically as follows: In problem (26), the unknown parameter and is initialized uniformly randomly on each trial, and then a coarse 2D grid search is performed on the target position to find the initial estimate of the target position that maximizes the problem (26) After the iterative estimation algorithm converges, that is, after the maximum iterative convergence condition is reached, the simplified expression of the target position estimation problem in equation (26) is as follows: Among them, Indicates that from the observation r m,n Subtract the estimated direct wave interference from item Indicates the estimated target echo.

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