CN109683141B - Multi-input multi-output radar emission waveform design method based on Bayesian framework - Google Patents

Multi-input multi-output radar emission waveform design method based on Bayesian framework Download PDF

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CN109683141B
CN109683141B CN201910036014.1A CN201910036014A CN109683141B CN 109683141 B CN109683141 B CN 109683141B CN 201910036014 A CN201910036014 A CN 201910036014A CN 109683141 B CN109683141 B CN 109683141B
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戴奉周
张博
张玥玥
张宇
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Xidian University
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
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Abstract

The invention discloses a Bayesian framework-based multi-input multi-output radar emission waveform design method, which mainly solves the problem of inaccurate target detection in a related clutter environment in the prior art. The implementation scheme is as follows: 1) representing the possible existence of the target in the related clutter environment as a binary hypothesis testing problem; 2) establishing a Bayesian framework-based likelihood ratio test according to a binary hypothesis test; 3) and carrying out iterative solution on the likelihood ratio test until the maximum value of the likelihood ratio test is solved, wherein the transmitting signal corresponding to the maximum value is the designed transmitting signal waveform. The invention improves the inhibition capability to the clutter, reduces the calculation complexity, and can be used for target detection and tracking in the related clutter environment.

Description

Multi-input multi-output radar emission waveform design method based on Bayesian framework
Technical Field
The invention belongs to the technical field of radars, and further relates to a method for designing a multi-input multi-output radar transmitting waveform, which can be used for target detection and tracking in a related clutter environment.
Background
In recent years, mimo radar has received increasing attention from researchers and engineers. Different from the traditional phased array radar transmitting antenna which can only transmit coherent signals, the multi-input multi-output radar transmitting antenna can transmit any signals, and the target detection, tracking and identification capabilities of the radar can be obviously improved. With the research on the mimo radar, many systems of waveforms have been designed, but some conventional methods have certain problems, such as: the problem of high adaptive side lobe of a space domain synthesized signal exists in the orthogonal frequency division linear frequency modulation waveform; the orthogonality properties of the orthogonal polyphase coded waveforms are limited by the code length and are sensitive to doppler shifts.
P Stoica and J Li, in its published article "Waveform synthesis for diversity-based transmit antenna design" (IEEE Trans on Signal Processing,2008,56(6):2693-2598), propose a cyclic algorithm design Waveform, which keeps the amplitude of the transmitted Waveform always small fluctuation and satisfies the condition of low peak-to-average power ratio, but this method needs complicated iteration process and multiple times of iteration.
A minimum output mean square error estimation method is proposed in a paper 'MIMO radial wave Design in the Presence of the router' (IEEE Transactions on Aerospace & Electronic Systems 47.2(2011):770-781) published by Naghibi, T and F.Behnia, and can accurately detect a target in an uncorrelated Clutter environment, but the method is not suitable for the correlated Clutter environment.
Disclosure of Invention
The invention aims to provide a multi-input multi-output radar emission waveform design method based on a Bayesian framework aiming at the defects in the prior art so as to improve the detection capability of a radar on a target in a related clutter environment and reduce the calculation complexity.
In order to achieve the purpose, the technical scheme of the invention comprises the following steps:
(1) the possible existence of the target in the relevant clutter environment is expressed as a binary hypothesis test, namely:
Figure BDA0001945942130000021
wherein H0Indicating the presence of clutter signals only in the echo signal, H1Indicating the condition of target signal and clutter signal in the echo signal, x indicating the echo signal received by radar, xtRepresenting the target echo signal, xcRepresenting a clutter echo signal;
(2) establishing a Bayesian framework-based likelihood ratio test according to a binary hypothesis test:
(2a) calculating a clutter covariance matrix R;
(2b) and (3) constructing a probability density function f (R) based on a Bayesian framework by taking the inverse Wishart distribution as the prior distribution of the clutter covariance matrix R:
Figure BDA0001945942130000022
wherein v represents the degree of freedom, L represents the number of samples of the transmit pulse, NRIndicating the number of receiving antennas, | · non-wovenvA ν -th operation representing determinant,
Figure BDA0001945942130000023
v + N representing determinantROperation to the power of L, etr (-) represents the trace of the calculationBottom exponential operation, R-1The inverse of the covariance matrix R is represented,
Figure BDA0001945942130000024
expressing a gamma function, sigma expressing a diagonal loading coefficient, I expressing an identity matrix, and sigma expressing a concentration matrix;
(2c) establishing a Bayesian framework-based likelihood ratio test according to the probability density function in (2 b):
Figure BDA0001945942130000025
wherein, f (x | H)1) Is shown in H1Probability density function of x under the assumption, f (x | H)0) Is shown in H0A probability density function of x under the assumption condition;
f(x|H1) The formula of (1) is as follows:
Figure BDA0001945942130000026
wherein, f (x; R | H)1) Is shown in H1The joint probability density function of x and R, ^ denotes an indeterminate integral operation, under the assumed conditions;
f(x|H0) The formula of (1) is as follows:
Figure BDA0001945942130000031
wherein, f (x; R | H)0) Is shown in H0A joint probability density function of x and R under the assumed condition;
substituting the above two probability density functions into a Bayesian framework-based likelihood ratio test L (x), and updating L (x) to:
Figure BDA0001945942130000032
wherein x istRepresenting an objectWave signal, (.)HRepresenting the operation of solving conjugate transposition;
(3) solving a likelihood ratio test, the corresponding transmitted signal S being the maximum likelihood ratio testtI.e. the designed transmit signal waveform.
Compared with the prior art, the invention has the following advantages:
first, because the waveform designed by the invention is based on the bayesian framework, the computational complexity is reduced compared with the existing waveform designed based on the gaussian distribution framework.
Secondly, compared with the existing orthogonal linear frequency modulation waveform and orthogonal multiphase coding waveform, the emission signal waveform based on the Bayesian framework designed by the invention improves the clutter suppression capability, thereby improving the target detection accuracy.
Drawings
FIG. 1 is a general flow chart of an implementation of the present invention;
FIG. 2 is a graph of cost function versus iteration number for use in the present invention;
FIG. 3 is a comparison of the beam direction of the waveform designed by the present invention and the existing random R Gaussian-based waveform, determining R Gaussian-based waveform;
Detailed Description
The embodiments and effects of the present invention will be described in further detail with reference to the accompanying drawings
Referring to fig. 1, the implementation steps of the invention are as follows:
step 1, representing the possible existence condition of the target in the related clutter environment as a binary hypothesis test.
(1a) Target echo signal xtSum clutter echo signal xcRespectively, as follows:
Figure BDA0001945942130000033
Figure BDA0001945942130000034
where α represents the amplitude of the target echo signal, (. alpha)TIndicating a transpose operation, S a transmit signal,
Figure BDA0001945942130000041
representing a kronecker operation, NRWhich indicates the number of the receiving antennas,
Figure BDA0001945942130000042
represents NRDimension Unit matrix, vec (-) denotes vectorization operation, θtRepresenting the angle of arrival, b (θ), of the target echo signalt) Denotes the angle of arrival as θtA (θ) of the received array steering vectort) Denotes the angle of arrival as θtThe transmit array steering vector of (a);
Ncrepresenting the number, x, of clutter units distributed within the range ring of the targetc,kExpressing the kth clutter unit signal, and satisfying the following formula:
Figure BDA0001945942130000043
wherein, deltac,kRepresenting the amplitude, theta, of the k-th cluttered element signalc,kRepresents the angle of arrival, b (θ), of the k-th clutter unit signalc,k) Denotes the angle of arrival as θc,kA (θ) of the received array steering vectorc,k) Denotes the angle of arrival as θc,kThe transmit array steering vector of (a);
(1b) establishing a binary hypothesis test:
Figure BDA0001945942130000044
wherein H0Indicating the presence of clutter signals only in the echo signal, H1The method is characterized in that the method represents the situation that a target signal and a clutter signal exist in an echo signal, and x represents the echo signal received by a radar.
And 2, establishing a Bayesian framework-based likelihood ratio test according to the binary hypothesis test.
(2a) Calculating a clutter covariance matrix R;
Figure BDA0001945942130000045
wherein E (-) represents the desired operation;
substituting formula <2> into formula <5>, updating R to:
Figure BDA0001945942130000046
wherein, (.)HRepresents a conjugate transpose operation;
(2b) and (3) constructing a probability density function f (R) based on a Bayesian framework by taking the inverse Wishart distribution as the prior distribution of the clutter covariance matrix R:
Figure BDA0001945942130000047
wherein v represents the degree of freedom, L represents the number of samples of the transmit pulse, NRIndicating the number of receiving antennas, | · non-wovenvA ν -th operation representing determinant,
Figure BDA0001945942130000051
v + N representing determinantRThe power of L is operated, etr (DEG) represents exponential operation with the trace of the calculation as the base, R-1The inverse of the covariance matrix R is represented,
Figure BDA0001945942130000052
expressing a gamma function, sigma expressing a diagonal loading coefficient, I expressing an identity matrix, and sigma expressing a concentration matrix;
(2c) establishing a Bayesian framework-based likelihood ratio test according to the probability density function in (2 b):
Figure BDA0001945942130000053
wherein, f (x | H)1) Is shown in H1Probability density function of x under the assumption, f (x | H)0) Is shown in H0A probability density function of x under the assumption condition;
f(x|H1) The formula of (1) is as follows:
Figure BDA0001945942130000054
wherein, f (x; R | H)1) Is shown in H1The joint probability density function of x and R, ^ denotes an indeterminate integral operation, under the assumed conditions;
f(x|H0) The formula of (1) is as follows:
Figure BDA0001945942130000055
wherein, f (x; R | H)0) Is shown in H0A joint probability density function of x and R under the assumed condition;
substituting formula <9> and formula <10> into formula (8), and updating L (x) to:
Figure BDA0001945942130000056
wherein x istRepresenting a target echo signal, (.)HIndicating a conjugate transpose operation.
Step 3, solving likelihood ratio test, when likelihood ratio test reaches maximum, corresponding transmitting signal StI.e. the designed transmit signal waveform.
The prior art optimization algorithm for solving likelihood ratio test includes: gradient descent method, steepest ascent method, newton method and conjugate gradient method.
The example uses, but is not limited to, the steepest-rise method in the prior art to solve the likelihood ratio test, and the implementation steps are as follows:
(3a) the likelihood ratio test l (x) is taken as the base 10 logarithm, expressed as:
lnL(x)＝ln|(x-xt)(x-xt)H+(ν-NRL)Σ|-(ν+1)-ln|xxH+(ν-NRL)Σ|-(ν+1)， <12>
wherein |. non chlorine-(v+1)The power- (v +1) operation of the determinant is expressed, the concentration matrix is expressed by sigma, and the following formula is satisfied:
Figure BDA0001945942130000061
wherein, sigma represents a diagonal loading coefficient, and I represents an identity matrix;
(3b) according to lnL (x). varies.. lt | n | xxH+(ν-NRL)Σ|-ln|(x-xt)(x-xt)H+(ν-NRL) a relationship of Σ | is given by:
h(x)＝ln|xxH+(ν-NRL)Σ|-ln|(x-xt)(x-xt)H+(ν-NRL)Σ|， <14>
wherein, oc represents a proportional relationship, ln | represents an exponential operation taking the absolute value of · to base 10;
(3c) substituting formula <1> and formula <13> into formula <14> yields the following cost function:
Figure BDA0001945942130000062
wherein S ist＝STS denotes a transmission signal, STDenotes the transposition of S, St *Denotes StX represents the echo signal received by the radar, v represents the degree of freedom,
Figure BDA0001945942130000063
(3d) solving a cost function h (S)t,St *) Of the maximum value of (a), S corresponding to the maximum valuetNamely the designed transmitting signal waveform, the implementation steps are as follows:
(3d1) starting from m-1, giving the initial value of the transmitted signal
Figure BDA0001945942130000064
And an iteration step size mu;
(3d2) the iteration point of the (m-1) th iteration
Figure BDA0001945942130000065
Substituting the following iteration point formula:
Figure BDA0001945942130000066
wherein vec (-) denotes vectorizing · St (m)Represents the iteration point of the mth iteration (·)*A companion matrix is represented that represents the companion matrix,
Figure BDA0001945942130000067
and expressing the corresponding cost function gradient of the iteration point of the (m-1) th iteration, and satisfying the following formula:
Figure BDA0001945942130000071
wherein the content of the first and second substances,
Figure BDA0001945942130000072
denotes a partial derivative operation, h (S)t (m-1),(St (m-1))*) Denotes St (m-1)A corresponding cost function;
(3d3) substituting equation <16> into the following equation:
Figure BDA0001945942130000073
wherein S ist *(m)Representing argument of an iteration point of the mth iteration, exp representing exponential operation with natural number as base, j representing imaginary number, and arg (·) representing argument operation;
(3d4) let m be m +1, repeatedly perform (3d2) and (3d3) until m is m +1
Figure BDA0001945942130000074
Figure BDA0001945942130000075
Corresponding to
Figure BDA0001945942130000076
Is the maximum value of a cost function, wherein St (k)The iteration point of the k-th iteration is indicated,
Figure BDA0001945942130000077
denotes k → ∞ time
Figure BDA0001945942130000078
Is measured.
The effect of the invention is further explained by simulation experiments.
1. Conditions of the experiment
The hardware platform of the simulation experiment of the invention is as follows: multiple input multiple output radar, MATLAB R2017 a.
The radar is provided with four transmitting antennas and four receiving antennas. Assuming that the arrival angle of the point target is 0 ° and the arrival angle range of the clutter is θ e (-180 °,180 °), the amplitude of the clutter unit signal is expressed as:
Figure BDA0001945942130000079
wherein k is 1,2c，Nc＝361。
2. Analysis of experimental content and results
Experiment 1: setting the amplitude alpha of the point target signal as 100, the iteration step length mu as 10, the sampling number L as 16, and the degree of freedom v as 4NRThe relationship between the cost function and the iteration number of the invention is simulated by MATLAB R2017a software with L being 256, and the result is shown in FIG. 2, wherein the abscissa in FIG. 2 represents the iteration number and the ordinate represents the generationA cost function.
As seen from fig. 2: the likelihood ratio test solving of the embodiment only needs to iterate for 4 times, and the cost function can reach the maximum value, so that the likelihood ratio test solving of the invention has the advantages of less iteration times and simple calculation.
Experiment 2: MATLAB R2017a software is used for respectively simulating the waveform designed by the invention and the existing random R Gaussian distribution-based waveform and determining the R Gaussian distribution-based beam pattern, and the result is shown in figure 3, wherein the abscissa in figure 3 represents the arrival angle of a target, and the ordinate represents the beam direction.
As can be seen from fig. 3: the clutter side lobe height of the waveform designed by the invention is-27 db, the clutter side lobe height of the existing random R based on Gaussian distribution waveform and the determined R based on Gaussian distribution waveform is-23 db, and the clutter suppression capability of the transmitted signal waveform designed by the invention is good.

Claims (5)

1. A multi-input multi-output radar emission waveform design method based on a Bayesian framework is characterized by comprising the following steps:
(1) the possible existence of the target in the relevant clutter environment is expressed as a binary hypothesis test, namely:
Figure FDA0002887802220000011
wherein H0Indicating the presence of clutter signals only in the echo signal, H1Indicating the condition of target signal and clutter signal in the echo signal, x indicating the echo signal received by radar, xtRepresenting the target echo signal, xcRepresenting a clutter echo signal;
(2) establishing a Bayesian framework-based likelihood ratio test according to a binary hypothesis test:
(2a) calculating a clutter covariance matrix R;
(2b) and (3) constructing a probability density function f (R) based on a Bayesian framework by taking the inverse Wishart distribution as the prior distribution of the clutter covariance matrix R:
Figure FDA0002887802220000012
wherein v represents the degree of freedom, L represents the number of samples of the transmit pulse, NRIndicating the number of receiving antennas, | · non-wovenvA ν -th operation representing determinant,
Figure FDA0002887802220000013
v + N representing determinantRThe power of L is operated, etr (DEG) represents exponential operation with the trace of the calculation as the base, R-1The inverse of the covariance matrix R is represented,
Figure FDA0002887802220000014
represents a gamma function, Σ represents a density matrix;
(2c) establishing a Bayesian framework-based likelihood ratio test according to the probability density function in (2 b):
Figure FDA0002887802220000015
wherein, f (x | H)1) Is shown in H1Probability density function of x under the assumption, f (x | H)0) Is shown in H0A probability density function of x under the assumption condition;
f(x|H1) The formula of (1) is as follows:
Figure FDA0002887802220000016
wherein, f (x; R | H)1) Is shown in H1The joint probability density function of x and R, ^ denotes an indeterminate integral operation, under the assumed conditions;
f(x|H0) The formula of (1) is as follows:
Figure FDA0002887802220000021
wherein, f (x; R | H)0) Is shown in H0A joint probability density function of x and R under the assumed condition;
substituting the above two probability density functions into a Bayesian framework-based likelihood ratio test L (x), and updating L (x) to:
Figure FDA0002887802220000022
wherein x istRepresenting a target echo signal, (.)HRepresenting the operation of solving conjugate transposition;
(3) solving a likelihood ratio test, the corresponding transmitted signal S being the maximum likelihood ratio testtI.e. the designed transmit signal waveform.
2. The method of claim 1, wherein the target echo signal x in (1)tSum clutter echo signal xcRespectively, as follows:
Figure FDA0002887802220000023
Figure FDA0002887802220000024
where α represents the amplitude of the target echo signal, (. alpha)TIndicating a transpose operation, S a transmit signal,
Figure FDA0002887802220000025
representing a kronecker operation, NRWhich indicates the number of the receiving antennas,
Figure FDA0002887802220000026
represents NRDimension Unit matrix, vec (-) denotes vectorization operation, θtRepresenting the angle of arrival, b (θ), of the target echo signalt) Denotes the angle of arrival as θtA (θ) of the received array steering vectort) Denotes the angle of arrival as θtThe transmit array steering vector of (a);
Ncrepresenting the number, x, of clutter units distributed within the range ring of the targetc,kExpressing the kth clutter unit signal, and satisfying the following formula:
Figure FDA0002887802220000031
wherein, deltac,kRepresenting the amplitude, theta, of the k-th cluttered element signalc,kRepresents the angle of arrival, b (θ), of the k-th clutter unit signalc,k) Denotes the angle of arrival as θc,kA (θ) of the received array steering vectorc,k) Denotes the angle of arrival as θc,kThe transmit array steering vector.
3. The method of claim 2, wherein the clutter covariance matrix R is calculated in (2a) by the following equation:
Figure FDA0002887802220000032
wherein E (-) represents the desired operation;
combining clutter echo signals xcSubstituting the formula, updating R as:
Figure FDA0002887802220000033
wherein, (.)HRepresenting a conjugate transpose operation.
4. The method of claim 2, wherein the solution likelihood ratio test in (3) is implemented as follows:
(3a) the likelihood ratio test l (x) is taken as the base 10 logarithm, expressed as:
lnL(x)＝ln|(x-xt)(x-xt)H+(ν-NRL)Σ|-(ν+1)-ln|xxH+(ν-NRL)Σ|-(ν+1)，
wherein |. non chlorine-(v+1)The power- (v +1) operation of the determinant is expressed, the concentration matrix is expressed by sigma, and the following formula is satisfied:
Figure FDA0002887802220000034
wherein, sigma represents a diagonal loading coefficient, and I represents an identity matrix;
(3b) according to lnL (x). varies.. lt | n | xxH+(ν-NRL)Σ|-ln|(x-xt)(x-xt)H+(ν-NRL) a relationship of Σ | is given by:
h(x)＝ln|xxH+(ν-NRL)Σ|-ln|(x-xt)(x-xt)H+(ν-NRL)Σ|，
wherein, oc represents a proportional relationship, ln | represents an exponential operation taking the absolute value of · to base 10;
(3c) target echo signal xtAnd substituting the concentration matrix sigma into the equation h (x) to obtain the following cost function:
Figure FDA0002887802220000041
wherein S ist＝STS denotes a transmission signal, STDenotes the transposition of S, St *Denotes StX represents the echo signal received by the radar, v represents the degree of freedom,
Figure FDA0002887802220000042
(3d) solving a cost function h (S)t,St *) Of the maximum value of (a), S corresponding to the maximum valuetIs the designed hairA signal waveform is emitted.
5. The method of claim 4, wherein the cost function h (S) is solved in (3d)t,St *) Is achieved as follows:
(4a) starting from m-1, giving the initial value of the transmitted signal
Figure FDA0002887802220000043
And an iteration step size mu;
(4b) the iteration point of the (m-1) th iteration
Figure FDA0002887802220000044
Substituting the following iteration point formula:
Figure FDA0002887802220000045
where μ denotes the iteration step, vec (-) denotes vectorizing · St (m)Represents the iteration point of the mth iteration (·)*A companion matrix is represented that represents the companion matrix,
Figure FDA0002887802220000046
and expressing the corresponding cost function gradient of the iteration point of the (m-1) th iteration, and satisfying the following formula:
Figure FDA0002887802220000047
wherein the content of the first and second substances,
Figure FDA0002887802220000048
denotes a partial derivative operation, h (S)t (m-1),(St (m-1))*) Denotes St (m-1)A corresponding cost function;
(4c) point of iteration
Figure FDA0002887802220000049
Substituting into the following argument formula:
Figure FDA00028878022200000410
wherein S ist *(m)Representing argument of an iteration point of the mth iteration, exp representing exponential operation with natural number as base, j representing imaginary number, and arg (·) representing argument operation;
(4d) repeating (4b) and (4c) until m is m +1
Figure FDA0002887802220000051
Figure FDA0002887802220000052
Corresponding to
Figure FDA0002887802220000053
Is the maximum value of a cost function, wherein St (k)The iteration point of the k-th iteration is indicated,
Figure FDA0002887802220000054
denotes k → ∞ time
Figure FDA0002887802220000055
Is measured.
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