Disclosure of Invention
In order to overcome the problems in the prior art, the invention provides a Wald-based method for detecting a distance expansion target under an interference and clutter background.
The technical scheme for solving the technical problems is as follows:
the invention provides a Wald-based detection method for a distance expansion target under interference and clutter background, which comprises the following steps:
step 1, obtaining data to be detected from KP distance units to be detected as main data, obtaining KS observation data without target signals from a non-target distance unit adjacent to a detected unit as auxiliary data, carrying out subspace modeling on target signals and interference signals, modeling clutter into complex Gaussian distribution with zero mean value and unknown positive fixed clutter covariance matrix, and establishing a binary hypothesis test model;
Step 2, transforming the constructed binary hypothesis testing model by utilizing the oblique symmetry of the clutter covariance matrix, performing dimension reduction treatment on the transformed binary hypothesis testing model to obtain a new binary hypothesis testing model, and then constructing a joint probability density function of main data and auxiliary data;
Step 3, carrying out maximum likelihood estimation on unknown parameters in the joint probability density function, and constructing detection statistic etaRPD-2Wald under the Wald test of a two-step method;
And 4, setting a detection threshold TG according to the preset false alarm probability, comparing the detection statistic etaRPD-2Wald with the detection threshold TG, if etaRPD-2Wald≥TG, judging that the current KP distance units to be detected have distance expansion targets, otherwise, judging that the current KP distance units to be detected have no distance expansion targets if etaRPD-2Wald<TG.
Further, in the step 1, subspace modeling is performed on the target signal and the interference signal, clutter modeling is performed as complex gaussian distribution with zero mean and unknown positive definite covariance matrix, and a binary hypothesis testing model is established, which specifically includes:
where H0 denotes the assumption that there is no target signal, H1 denotes the assumption that there is a target signal, rk denotes the signal received by the kth distance unit,Representing clutter, p representing the coordinate vector of an unknown target signal in the subspace, qk representing the coordinate vector of interference in the subspace,Representing a multi-rank subspace matrix,Representing a multi-rank subspace matrix, alphak represents the value of the kth distance unit coordinate vector, k e ΩP≡{1,...,KP represents the number of primary data, and k e ΩS≡{KP+1,...,KP+KS represents the number of secondary data.
Further, in the step 2, the constructed binary hypothesis testing model is transformed by using the oblique symmetry of the clutter covariance matrix, and then the dimension reduction processing is performed on the transformed binary hypothesis testing model, so as to obtain a new binary hypothesis testing model, wherein the new binary hypothesis testing model is as follows:
Wherein,
In the above formula, H0 represents the assumption that there is no target signal, H1 represents the assumption that there is a target signal, and xk represents the position of the target signalThe signal received by the kth distance unit after matrix transformation, yk, is shown in the pairThe k distance unit after matrix transformation receives signals, J represents an interference subspace, qk represents coordinate vectors of interference in the subspace, nk represents vectors of Gaussian distribution, p represents coordinate vectors of unknown target signals in the subspace, and H represents a signal subspace; Representing the signal subspace in which the corresponding matrix transformation is performed; Representing the interference subspace subjected to corresponding matrix transformation; Representing signals received by a kth distance unit after matrix transformation of rk; Representing the signal received by the kth distance unit after matrix transformation of rk, rk representing the signal received by the kth distance unit, k ε ΩP≡{1,...,KP representing the number of primary data, k ε ΩS≡{KP+1,...,KP+KS representing the number of auxiliary data, (. Cndot.)H representing the conjugate transpose; representing a set of real matrices in the dimension r x1,Representing a set of complex matrices in dimension N x 1, where r e N, n=na×Nt, represents that Na antenna arrays sense KP distance units, each collecting Nt samples from each unit.
Further, in the step 2, a joint probability density function of the main data and the auxiliary data is constructed, wherein H0 assumes that the joint probability density function f0(RP; Q) is:
h1 assumes that the joint probability density function is:
In the above formula, H0 denotes an assumption that there is no target signal, H1 denotes an assumption that there is a target signal, KP denotes the number of distance units of data to be detected, n=na×Nt denotes that Na antenna arrays sense KP distance units, each antenna collects Nt samples from each unit, RP is a main data matrix, p denotes a coordinate vector of an unknown target signal in a subspace, α denotes a coordinate vector, Q denotes an interference coordinate covariance matrix, and M denotes a clutter covariance matrix; for a full column rank matrix function of p, H represents a signal subspace, J represents an interference subspace, and q is E N; A full row rank matrix function of q, tr (·) represents the trace of the matrix, (·)H represents the conjugate transpose,Representing a set of N x (q + 1) dimensional complex matrices,Represents a set of (q+1) x KP -dimensional complex matrices.
Further, the step 3 specifically includes:
Step 3-1, the unknown parameters in the joint probability density function comprise a clutter covariance matrix, a target coordinate and an interference coordinate covariance matrix, and maximum likelihood estimation is carried out on the clutter covariance matrix, the target coordinate and the interference coordinate covariance matrix in the joint probability density function;
And 3-2, substituting the maximum likelihood estimator of the unknown component in the target coordinates, the converted interference coordinate covariance estimator and the estimator of the clutter covariance matrix back into a joint probability density function, and constructing a detection statistic etaRPD-2Wald under the two-step Wald test.
Further, in the step 3-2, a detection statistic ηRPD-2Wald under the two-step Wald test is constructed:
where tr represents the trace of the matrix and (-)H represents the conjugate transpose;
master data matrix
Projection matrix
Guide vector
Auxiliary data estimation covariance matrix
PK isA principal vector of the matrix;
Wherein H represents a signal subspace,Representing the signal subspace after matrix transformation of H,Projection matrix representing projection to full column rank matrix, projection matrix
Compared with the prior art, the invention has the following technical effects:
(1) The invention combines a dimension reduction method and clutter covariance matrix oblique symmetry to improve the performance of a distance expansion target direction detector under the condition of auxiliary data deficiency under the condition of subspace interference and Gaussian clutter background, assumes that an interference subspace is independent of a signal subspace and two subspace coordinates are unknown, models clutter components into complex Gaussian vectors with zero mean value, solves the maximum likelihood estimation of unknown clutter covariance matrix, interference coordinate vectors and signal coordinate vectors by utilizing the dimension reduction subspace method and an oblique symmetry transformation matrix structure, thereby establishing an RPD-2Wald detector for target detection;
(2) The method aligns possible useful signals to an unknown direction, restricts the possible useful signals to belong to an observable given subspace, solves the self-adaptive detection problem under the condition that guide vectors are not matched, combines the dimension reduction method with clutter covariance matrix oblique symmetry to improve the performance of a distance expansion target direction detector under the condition of auxiliary data deficiency under the condition that subspace interference is increased by the clutter background, and improves the detection performance of the detector while guaranteeing the characteristic of constant false alarm rate based on Wlad detection strategies, thereby having wide potential popularization and application values.
Detailed Description
In order to further describe the technical means and effects adopted by the present invention to achieve the preset purpose, the following detailed description is given below of the specific implementation, structure, features and effects of the technical solution according to the present invention with reference to the accompanying drawings and preferred embodiments. The particular features, structures, or characteristics of one or more embodiments may be combined in any suitable manner. Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs.
Referring to fig. 1, in one embodiment of the present invention, there is provided a Wald-based method for detecting a range-extended target in an interference-plus-clutter background, comprising the steps of:
step 1, obtaining data to be detected from KP distance units to be detected as main data, obtaining KS observation data without target signals from a non-target distance unit adjacent to a detected unit as auxiliary data, carrying out subspace modeling on target signals and interference signals, modeling clutter into complex Gaussian distribution with zero mean value and unknown positive fixed clutter covariance matrix, and establishing a binary hypothesis test model;
Step 2, transforming the constructed binary hypothesis testing model by utilizing the oblique symmetry of the clutter covariance matrix, then performing dimension reduction treatment on the transformed binary hypothesis testing model to obtain a new binary hypothesis testing model, and then constructing a joint probability density function of the main data and the auxiliary data;
Step 3, carrying out maximum likelihood estimation on unknown parameters in the joint probability density function, and constructing detection statistic etaRPD-2Wald under the Wald test of a two-step method;
And 4, setting a detection threshold TG according to the preset false alarm probability, comparing the detection statistic etaRPD-2Wald with the detection threshold TG, if etaRPD-2Wald≥TG, judging that the current KP distance units to be detected have distance expansion targets, otherwise, judging that the current KP distance units to be detected have no distance expansion targets if etaRPD-2Wald<TG.
The following detailed development of each step is performed:
step 1, obtaining to-be-detected data from KP to-be-detected distance units as main data, obtaining KS observation data without target signals from non-target distance units adjacent to a detected unit as auxiliary data, assuming that an interference subspace is linearly independent of a signal subspace and two subspace coordinates are unknown, carrying out subspace modeling on a target signal and the interference signal, modeling clutter into complex Gaussian distribution with zero mean value and unknown positive fixed clutter covariance matrix, and establishing a binary hypothesis test model.
As an example, this step 1 may comprise the steps of:
Assuming that an array of Na antennas senses KP range units, each antenna collects Nt samples from each unit, the signal received by the kth range unit K ε ΩP≡{1,...,KP is denoted by rk, which is an N-dimensional complex vector, i.eRepresents an N x 1-dimensional vector received from the kth distance unit, where n=na×Nt t. Assuming that the disturbance is the sum of the colored clutter and the interference, the clutter is represented byAnd the interference signal ik are represented by two N x 1-dimensional vectors. Let H0 be assumed to contain only disturbances in rk, let H1 be assumed to contain disturbances in rk and useful target echoes
Based on the subspace representation, sk is modeled as sk＝αk s, hereRepresenting the value of the kth range bin coordinate vector,S belongs to a multi-rank subspace matrixSimilarly, the interference signal ik belongs to a multi-rank subspaceAnd q+r is less than or equal to N. As such,K is e.OMEGAP, whereRepresenting the coordinate vectors of the unknown target signal and the interference in the subspace, respectively. It is assumed here thatAnd (3) withIs known andThe full order matrix of the composition is also known, while αk, p and qk are unknown. ClutterModeled as zero mean and with an unknown positive definite covariance matrixIs a Gaussian vector of (1), i.eAt the same time, assume that a set of adjacent auxiliary data is adopted and is recorded asWhere K ε ΩS≡{KP+1,...,KP+KS, where KS +.gtN, and the auxiliary data contains only clutter components, the same statistics as the clutter components in the main data. Finally, assume that the clutter is independently co-distributed.
According to the above assumption, the problem of detecting the distance-extended target in the subspace interference plus unknown clutter environment can be expressed as a binary hypothesis test for judging whether the target exists in the data to be detected:
Where H0 denotes a hypothesis without a target signal, and H1 denotes a hypothesis with a target signal.
And 2, transforming the constructed binary hypothesis testing model by utilizing the oblique symmetry of the clutter covariance matrix, then performing dimension reduction processing on the transformed binary hypothesis testing model to obtain a new binary hypothesis testing model, and then constructing a joint probability density function of the main data and the auxiliary data.
As an example, this step 2 may include the steps of:
transforming the constructed model by using the oblique symmetry of the clutter covariance matrix, then performing dimension reduction processing on the transformed model to obtain a new binary hypothesis test model, and then constructing a joint probability density function of main data and auxiliary data, wherein the joint probability density function is specifically realized by using the oblique symmetry characteristic of radar detection signals, so that the method can be known in practiceAndHas an oblique symmetrical structure, and is expressed as follows:
In the above formula, D is a permutation matrix with an opposite angle of 1 and the rest positions of 0, and is shown in formula (3) (. Cndot.)* represents conjugation;
the next study included the oblique symmetry into the design program of the radar target direction detector,
Defining a unitary matrix T:
Wherein, IN2 represents an N/2-dimensional identity matrix, I(N-1)/2 represents an (N-1)/2-dimensional identity matrix, DN2 represents an N/2-dimensional permutation matrix, D(N-1)/2 represents an (N-1)/2-dimensional permutation matrix, the auxiliary diagonal element is 1, and the other elements are 0;i represent imaginary units.
By using the above formula, multiplying formula (1) by T yields:
Wherein,
In the above-mentioned method, the step of,Representing signals received by a kth distance unit after matrix transformation of rk; Representing signals received by a kth distance unit after matrix transformation of rk; Representing the signal subspace in which the corresponding matrix transformation is performed; Representing the interference subspace subjected to corresponding matrix transformation; is a gaussian distributed vector.
Further using dimension reduction means to multiply formula (5) byThe method comprises the following steps:
Wherein,
In the above formula, xk is shown in the pairThe signal received by the kth distance unit after matrix transformation, yk, is shown in the pairThe signal received by the kth distance unit after matrix transformation, J represents the interference subspace,Representing the coordinate vector of the disturbance in the subspace, nk representing the vector of the gaussian distribution,Representing a coordinate vector of an unknown target signal in the subspace, and H represents the signal subspace; Representing signals received by a kth distance unit after matrix transformation of rk; Representing signals received by a kth distance unit after matrix transformation of rk; Representing the signal subspace in which the corresponding matrix transformation is performed; Representing the interference subspace with corresponding matrix transformation, nk is the vector covariance matrix of Gaussian distributionRepresenting a set of real matrices in the dimension r x1,Representing a set of complex matrices in the N x1 dimension.
For convenience of representation, a distance unit signal sensed by the radar antenna is optimally represented, and R= [ RP RS ] is set, whereinAs a matrix of primary data,For auxiliary data matrix, letInterference coordinate covariance matrixCoordinate vectorAnd the total detection distance unit number k=kP+KS.
Constructing a joint probability density function:
wherein H0 assumes that the joint probability density function is:
f0(R;M,Q)＝(π)-NK|M|K[exp{-tr[M-1T0]}]K, (9)
in the above, useSimplified expression in which the covariance matrix is aided
H1 assumes that the joint probability density function is:
f1(R;p,α,M,Q)＝(π)-NKMK[exp{-tr[M-1T1]}]K, (10)
In the above-mentioned method, the step of,
Constructing a two-step joint probability density function:
H0 assumes that the two-step joint probability density function is:
h1 assumes that the two-step joint probability density function is:
Here, theAs a full column rank matrix function of p,A full row rank matrix function of q.
And 3, carrying out maximum likelihood estimation on unknown parameters in the joint probability density function, and constructing detection statistic etaRPD-2Wald under the Wald test of a two-step method.
Step 1 indicates that the unknown parameters in the joint probability density function include a clutter covariance matrix M, a target coordinate αk p, and an interference coordinate covariance matrix Q, so that the maximum likelihood estimation is performed on the clutter covariance matrix M, the target coordinate αk p, and the interference coordinate covariance matrix Q in the joint probability density function.
In some embodiments, this step may include the sub-steps of:
Step 3-1-1, carrying out maximum likelihood estimation on a clutter covariance matrix M, a target coordinate alphak p and an interference coordinate covariance matrix Q in the joint probability density function in detection statistics etaRPD-GLRT under GLRT test construction;
Solving the maximum likelihood estimation value of the unknown parameter Q and the clutter covariance matrix M at H0 according to the formula (9)Maximum likelihood estimation value of the sum clutter covariance matrix M at H0The partial derivative of the unknown parameters is calculated by the above formula and is set to be zero, and the following can be obtained:
Maximum likelihood estimate valueAnd maximum likelihood estimateSubstitution formula (9) yields the probability density function under the assumption of H0:
Similar to the H0 hypothesis, the estimated values of M and E in the H1 hypothesis can be obtained by maximum likelihood estimation, and then the estimated value of α can be obtained by substituting the estimated value into formula (10):
Now only the maximum likelihood estimate of p has to be found, i.e. the maximum value of equation (17) containing p:
formula (17) may be converted to formula (18):
Here the number of the elements is the number,
HS′＝ZHHS,For a unitary slice matrix, HS' represents the matrix obtained by matrix transforming HS, HS represents the matrix obtained by matrix transforming H,The projection matrix for HS' p can be reconstructed asU is a unit norm vector belonging to HSRepresenting the identity matrix of the cell,Representing the projection matrix projected onto the full column rank matrix, IN representing the identity matrix, and substituting it into equation (18) yields the following equation (20):
det(A)(1-uHZHS-1/2RPA-1RPHS-1/2Zu)(20)
The maximum likelihood estimate for p is now equivalent to solving for the maximum of the latter term of equation (20), resulting in equation (21)
Substituting equation (21) into equation (18) and then into equation (16) yields the probability density function assumed by H1, see equation (22).
Step 3-1-2, carrying out maximum likelihood estimation according to a clutter covariance matrix M, a target coordinate alphak p and an interference coordinate covariance matrix Q in the joint probability density function, and substituting the maximum likelihood estimation amount of an unknown component in the target coordinate, the converted interference coordinate covariance estimation amount and the estimation amount of the clutter covariance matrix back into the joint probability density function to construct detection statistic etaRPD-GLRT of the RPD-GLRT detector based on a GLRT test criterion;
under the GLRT test, the test statistic can be expressed as:
Wherein ηRPD-GLRT is a test statistic and γ is a detection threshold under GLRT test, and substituting formulas (22) and (15) into formula (23) can obtain:
In the above-mentioned method, the step of,Lambdamax represents the maximum eigenvalue of the matrix parameters.
Step 3-2-1, carrying out maximum likelihood estimation on unknown parameters in the joint probability density function in the detection statistic etaRPD-2GLRT under the GLRT test by constructing a two-step method;
The first step is to solve the maximum likelihood estimation value of the unknown parameter Q at H0 according to the equation (11) assuming that the clutter covariance matrix is knownThe partial derivative of the unknown parameters is calculated by the above formula and is set to be zero, and the following can be obtained:
JM＝M-1/2 J, maximum likelihood estimateSubstituting formula (11) yields a probability density function under the assumption of H0 as formula (26), where the projection matrix
Similar to the H0 hypothesis, the estimated value of E in the H1 hypothesis can be obtained through maximum likelihood estimation, further the estimated value of alpha can be obtained, and the estimated value of alpha can be substituted into the formula (12) to obtain the formula (27), wherein the projection matrix
Now, only the maximum likelihood estimation of p is needed, and the maximum likelihood estimation of p can be obtained as follows, similar to the maximum likelihood estimation of p in the step 3-1-1:
Substituting equation (28) into (27) yields the probability density function of the H1 hypothesis:
Step 3-2-2, constructing detection statistic etaRPD-2GLRT under the GLRT test of a two-step method according to maximum likelihood estimation of a target coordinate alphak p and an interference coordinate covariance matrix Q in the joint probability density function;
second, estimating covariance matrix using auxiliary dataAnd replacing the clutter covariance matrix M to obtain detection statistic etaRPD-2GLRT under the two-step GLRT test:
Gamma1 is the threshold of the detector. Lambdamax represents the maximum eigenvalue of the matrix parameters.
Step 3-3, constructing detection statistic etaRPD-2Wlad under the Wald test of a two-step method;
under Wald test, the test statistic can be expressed as:
in the above formula, gamma3 represents a detection threshold, Θ1 represents a row vector constructed by total unknown parameters under the assumption of H1, expressed asWherein Θr1＝p,Θs1＝[αT,ν1,vecT(Q)]T; a maximum likelihood estimate denoted Θ1; Represented as a maximum likelihood estimate of Θr1 under the H1 hypothesis, Θr0 represents the value of Θr under the H0 hypothesis, whereOne submatrix denoted as I-1 (Θ), and I (Θ) is a Fisher information matrix about Θ, which can be expressed as:
Wherein the method comprises the steps ofThe likelihood function is averaged after the logarithm is derived,The likelihood function is averaged after the logarithm is derived,The likelihood function is averaged after the logarithm is derived,And taking the logarithm of the likelihood function, performing bias derivative, and taking the average value.
This can be achieved by:
Taking logarithm first and then respectively taking logarithmAndThe derivation can be performed as follows:
then, substitution of formulas (35) and (36) into formula (33) can give:
Because Θr0＝0r×1, substituting equation (37) into equation (32) yields a Wald detection of fixed p, α, and M:
ηwald＝αHαpHHHM-1Hp (38)
Wherein the maximum likelihood estimate of α is the first column number of the maximum likelihood estimate of equation E, and is obtained according to the block matrix inversion formula:
substituting equation (39) into equation (38) and then estimating the oblique symmetry covariance matrix using the assistance dataReplacing the clutter covariance matrix M to obtain detection statistic etaRPD-2Wlad under the Wald test of a two-step method:
Wherein,
Master data matrix
Projection matrix
Guide vector
Auxiliary data estimation covariance matrix
PK isA principal vector of the matrix;
Wherein H represents a signal subspace,Representing the signal subspace after matrix transformation of H,Projection matrix representing projection to full column rank matrix, projection matrix
And 4, setting a detection threshold TG according to the preset false alarm probability, comparing the detection statistic etaRPD-2Wald with the detection threshold TG, if etaRPD-2Wald≥TG, judging that the current KP distance units to be detected have distance expansion targets, otherwise, judging that the current KP distance units to be detected have no distance expansion targets if etaRPD-2Wald<TG.
The proposed RPD-GLRT detector, RPD-2GLRT detector and RPD-2Wald detector were demonstrated to assume CFAR characteristics for the clutter covariance matrix M at H0, specifically:
it is necessary to verify that ηRPD-GLRT、ηRPD-2GLRT and ηRPD-2Wlad are independent of the clutter covariance matrix M under the assumption of H0 that M has CFAR characteristics.
For the RPD-GLRT detector and the RPD-2GLRT detector, it is first noted that the column of matrix Z in equation (24) is tensed into the orthogonal complement space of matrix JS, so that under the assumption of H0 there is RP＝JQ+NP, then there is:
ZHS-1/2RP＝ZHS-1/2(JQ+NP)＝ZHS-1/2NP＝F (41)
Here, theSubstituting RP＝JQ+NP into formula (24) can yield:
Here, theUsing whitening transformation and QR decomposition and blocking matrix theory, equation (42) can be rewritten as:
Wherein,
Is a matrix, which is listed as independent and equidistributed complex normal vector, the mean value is zero, the covariance matrix is Ir, andN2B,Are independent random matrices.
It is now apparent that under the assumption of H0, the distribution of ηRPD-GLRT in equation (24) is independent of M, and that the statistic is composed of two independent random matrices N2B andThe maximum eigenvalue of the matrix is determined and the distribution of the two matrices is independent of M. Thus demonstrating the CFAR properties of the aforementioned derived RPD-GLRT detector.
Since the RPD-GLRT detector is only inferior to the RPD-2GLRT detector by the A matrix, the CFAR characteristics of the RPD-2GLRT detector can be demonstrated in the same way.
For the RPD-2Wald detector, its statistics can be written as:
Wherein, since pK isThe principal vector of the matrix can be obtained:
Multiplying (46) byThe method comprises the following steps:
substitution of formula (47) into formula (45) can result in:
the statistical properties for β have been verified in published papers to be independent of M under the H0 assumption. The latter half is then verified. Definition of the definitionIs a unitary matrix of slices.
Note that the columns of matrix Z in equation (49) are expanded into a matrixThus, under the assumption of H0, there is RP＝JQ+NP, then:
Here, theSubstituting formula (50) into formula (49) can yield:
Using whitening transformation and QR decomposition and blocking matrix theory, equation (51) can be rewritten as:
Wherein,
Is a matrix, which is listed as independent and equidistributed complex normal vector, the mean value is zero, the covariance matrix is Ir, andN2B,Are independent random matrices.
It is now apparent that under the assumption of H0, the distribution of ηRPD-2Wlad in equation (40) is independent of M, and that the statistic is composed of two independent random matrices N2B andThe maximum eigenvalue of the matrix is determined and the distribution of the two matrices is independent of M. Thus demonstrating the CFAR properties of the previously derived RPD-2Wald detector.
(1) The method combines the dimension reduction method and the oblique symmetry of the clutter covariance matrix to improve the performance of the distance expansion target direction detector under the background of subspace interference and Gaussian clutter, and establishes a subspace-based distance expansion target signal model under the condition of interference and unknown clutter. Based on the GLRT strategy and Wlad inspection strategy, the direction detector of the distance expansion target under the interference and clutter background based on the GLRT inspection, the two-step GLRT inspection and the two-step Wald inspection is deduced, the constant false alarm rate characteristic is ensured, the detection performance of the detector is improved, and the method has wide potential popularization and application values.
The effect of the invention can be illustrated by the following simulation experiment:
CRAF characteristics and detection performance of RPD-GLRT were analyzed by monte carlo simulation. Let the false alarm probability (Probabilities of FALSE ALARM, PFA) be 10-4, the number of simulations to acquire the threshold be 100/PFA, and the number of simulations to acquire the detection probability (Probabilities of Detection, PD) be 5000. The signal-to-noise ratio SCR is defined as:
SCR=αHαpHHHM-1Hp; (54)
The dry-to-hybrid ICR is defined as:
ICR=tr(QHJHM-1JQ)。 (55)
Without loss of generality, the signal subspace H is selected as:
H=[t(0.18),t(0.21),...,t(0.18+(r-1)0.03)], (56)
the interfering subspace J is:
J=[t(-0.03),t(0),...,t(-0.03+(q-1)0.03)], (57)
when N is even, t (f) is:
When N is an odd number, t (f) is:
The element of the mth row and n column in the clutter covariance matrix is denoted by [ M ]m,n＝σ2ρm-n,σ2, and ρ represents the correlation coefficient, so that ρ=0.9. Without loss of generality, the interference coordinate vector qk is represented as a zero-mean covariance matrixIs a complex normal vector of independent co-distribution, whereIs interference power, p is represented as a complex normal vector with zero mean covariance matrix Ir, and moreover the phase of |alphak|＝α,αk is an independent random variable with the same distribution and obeys the uniform distribution of (0, 2 pi).
The CFAR characteristics of the RPD-GLRT detector, the RPD-2GLRT detector and the RPD-2Wald detector on the clutter power sigma2 and the clutter covariance matrix are verified, and the results are shown in FIG. 2 (a), FIG. 2 (b) and FIG. 2 (c). Fig. 2 (a), 2 (b) and 2 (c) show that the fixed first-order hysteresis coefficient gamma only changes the clutter power level sigma2, the false alarm probability under the same threshold is basically unchanged, the fixed clutter power level sigma2 only changes the first-order hysteresis coefficient gamma, and the false alarm probability under the same threshold is very similar, so that the detection performances of the RPD-GLRT detector, the RPD-2GLRT detector and the RPD-2Wald detector are basically not influenced by the correlation of an interference coordinate covariance matrix and clutter power, and the fact that the detector provided by the invention has constant false alarm rate characteristics on the clutter power level sigma2 and the clutter covariance matrix is verified, which is consistent with the theoretical analysis before.
In order to test the anti-interference performance of the detector, detection performance simulation experiments under different interference-to-noise ratios are set, wherein a first-order hysteresis coefficient gamma is set to be 0.9, and clutter power sigma2 is set to be 1. The results are shown in FIG. 3 (a), FIG. 3 (b) and FIG. 3 (c). It can be seen from FIGS. 3 (a), 3 (b) and 3 (c) that the detection performance of the RPD-GLRT detector, the RPD-2GLRT detector and the RPD-2Wald detector are almost the same under different ICRs, which indicates that the detectors have good anti-interference performance.
Next, the influence of the change in the amount of main data on the detection performance of the detector is analyzed, and the results are shown in fig. 4 (a), 4 (b) and 4 (c). As can be seen from fig. 4 (a), 4 (b) and 4 (c), the number of main data KP increases, the detection probability decreases under the same SCR, and the detection performance of the RPD-GLRT detector, the RPD-2GLRT detector and the RPD-2Wald detector gradually decreases, because it is possible that the distance occupied by the target is fixed and increasing KP results in introducing too many target-free but interference-containing units, thereby resulting in a decrease in detection performance.
The effect of the amount of auxiliary data on the detection performance of the RPD-GLRT detector, the RPD-2GLRT detector and the RPD-2Wald detector was further studied, and the results are shown in FIG. 5 (a), FIG. 5 (b) and FIG. 5 (c). It can be seen from fig. 5 (a), 5 (b) and 5 (c) that the detection performance of the RPD-GLRT detector, the RPD-2GLRT detector and the RPD-2Wald detector improves with the increase of the auxiliary data, because the auxiliary data increases, and the accuracy of the clutter covariance matrix of the corresponding estimation improves, thereby indirectly improving the detection probability.
The above-mentioned research on the detector is based on the premise of subspace matching, because the situation that signal subspace mismatch or interference subspace mismatch occurs inevitably in the process of target detection, taking signal mismatch as an example, the signal mismatch can be understood as that the actual subspace steering vector deviates from the nominal steering vector, so that the detection performance suffers from different degrees. The invention thus investigates the detection performance of the detector under mismatch conditions of the signal subspace and the interference subspace of the proposed detector.
The signal subspace mismatch and the interference subspace mismatch are then analyzed. The degree of mismatch of the nominal signal subspace H and the actual interference subspace H0 is expressed by cos2 theta:
Where cos2 θ=1 represents the case where the nominal signal subspace H exactly matches the actual signal subspace H0, and cos2 θ=0 represents the case where the nominal signal subspace H exactly mismatches the actual signal subspace H0. The specific detection performance loss conditions under the signal subspace mismatch condition are shown in fig. 6 (a), fig. 6 (b) and fig. 6 (c).
Fig. 7 (a), 7 (b) and 7 (c) show detection performance curves compared with different detectors under different assistance data. It can be seen from fig. 7 (a), 7 (b) and 7 (c) that the proposed RPD-GLRT detector, RPD-2GLRT detector and RPD-2Wald detector can perform target detection under the condition of shortage of auxiliary data and have better effect than the same type of PD-GLRT detector, PD-2GLRT detector and PD-2Wald detector because the detector is not active when the amount of auxiliary data is smaller than the size of N, and the limitations of the PD-GLRT detector, PD-2GLRT detector and PD-2Wald detector structure cause the detector to be not active when the amount of auxiliary data is smaller than the size of N/2 because the auxiliary data is difficult to be acquired in the actual environment, and the proposed RPD-GLRT detector, RPD-2GLRT detector and RPD-2Wald detector have wide popularization value when the auxiliary data amount is small and more accurate target detection is required.
The foregoing embodiments are merely for illustrating the technical solution of the present invention, but not for limiting the same, and although the present invention has been described in detail with reference to the foregoing embodiments, it should be understood by those skilled in the art that the technical solution described in the foregoing embodiments may be modified or substituted for some of the technical features thereof, and that these modifications or substitutions should not depart from the spirit and scope of the technical solution of the embodiments of the present invention and should be included in the protection scope of the present invention.