CN105974391B - The non-steady waveform design method of MIMO radar under the conditions of knowing target priori - Google Patents

Abstract

The invention belongs to field of signal processing, it is related to the non-design for knowing the steady waveform optimization of MIMO radar under the conditions of target priori.Its implementation is, initially set up MIMO radar receipt signal model under clutter scene, it need to estimate parameter Cramér-Rao lower bound (CRB) based on this model inference, initial parameter is then estimated that uncertain convex set is explicitly included into waveform optimization problem, establishes steady waveform optimization model；The nonlinear problem extremely complex about optimized variable, traditional optimization solution is difficult to be utilized, the present invention proposes that one kind, to solve this problem, can convert this challenge to Semidefinite Programming (SDP) problem, to obtain Efficient Solution based on Hadamard inequality methods.Compared with uncorrelated transmitted waveform and non-robust method, the present invention can significantly improve it is worst under the conditions of parameter Estimation performance, there is preferable robustness, thus closer to engineer application.

Description

MIMO radar robust waveform design method under condition of uncertain target priori knowledge

Technical Field

The invention belongs to the field of signal processing, and relates to a design method for MIMO radar robust waveform optimization under the condition of uncertain target priori knowledge.

Background

In recent years, a multiple-input multiple-output (MIMO) radar system has attracted extensive attention and research in the field of radar signal processing at home and abroad, and waveform optimization is an important research subject of MIMO radars. According to the different signal models to be optimized in the waveform optimization problem, the waveform optimization method can be divided into the following two categories: (1) optimizing a transmission waveform; (2) and jointly optimizing the transmitting waveform and the receiving weight. For the former, designers improve system performance by optimizing the waveform covariance matrix or the radar blur function. For the latter, the overall performance of the MIMO radar is improved by jointly optimizing the transmitting and receiving ends.

In the course of the study of the waveform optimization method, previous scholars have achieved some achievements. Li et al studied the problem of cramer-perot boundary (CRB) based waveform design under a point target model to improve parameter estimation performance. In a clutter environment, H.Y Wang et al consider the joint optimization problem of the waveform of the CRB-based MIMO radar and the biased estimator under the condition of the definite target prior information. It should be noted that the waveform optimization problem in these methods requires the parameters to be known. However, in actual engineering, these parameters must be estimated, and thus, estimation errors inevitably exist. Aiming at the problem that the parameter estimation performance of the system under the worst condition is improved through waveform design under the scene that the prior knowledge of the target of interest is unknown, the invention provides a robust waveform optimization method of the MIMO radar, which is used for the robust waveform design of the MIMO radar under the condition that the prior knowledge of the target is unknown and obviously improves the parameter estimation performance under the worst condition.

Disclosure of Invention

In order to improve the MIMO radar waveform optimization parameter estimation performance under the worst condition, the invention provides a method for designing the stable waveform of the MIMO radar under the condition of uncertain target prior knowledge, explicitly including a parameter estimation error model into the waveform design problem, and obtaining the mathematical expression of the stable waveform optimization under the constraints of transmitting waveform power and an uncertain convex set of initial parameters based on the trace criterion of constraining CRB under the minimum worst condition. In order to solve the obtained complex nonlinear optimization problem, firstly, the objective function is decomposed into a plurality of independent sub-problems by utilizing a Hadamard inequality. And then the inner-layer optimization problem, i.e., the maximization term, translates it into a minimization problem. The outer optimization problem, i.e., the minimization term, is then transformed, in conjunction with the inner optimization problem, into a semi-definite programming (SDP) problem, resulting in an efficient solution using a sophisticated optimization toolkit such as CVX. The technical scheme of the invention is as follows:

1. building robust waveform optimization problem model

1) Constructing MIMO radar signal model

Suppose a MIMO radar receives a signal:

wherein,in order to transmit the matrix of waveforms,a discrete baseband representation of the signal transmitted for the ith transmit unit, and a number of samples L,andthe RCS-proportional complex amplitude and target position parameters for the K targets of interest, respectively, are estimated based on the received data. W is a noise and interference matrix, each column of the noise and interference matrix can be modeled into independent and identically distributed circularly symmetric complex Gaussian random vectors, the mean value of the vectors is 0, and the covariance matrix is an unknown matrix Q. H_k＝a(θ_k)v^T(θ_k) Denotes the k-th target channel matrix, a (θ)_k) And v (θ)_k) Respectively indicate being located at theta_kReceive and transmit steering vectors of a targetIt can be expressed as:

in the formula (f)₀Representing the carrier frequency, τ_m(θ_k),m＝1,2,…M_rFor electromagnetic waves from being at theta_kTo the m-th receiving array element, andn＝1,2,…M_tthe transmission time from the nth transmit array element to the target.

2) Construction of a robust waveform optimization model based on constrained CRB

Now considerThe unknown parameter theta is [ theta ] under the known condition₁,θ₂,L,θ_K]^TThe CRB of (a), the so-called constrained CRB, can be expressed as:

in the formula,

C₁₁＝(2Re(F₁₁))^-1

and

in the formula, R_S＝SS^H，k＝1,2,…,K。

Assuming that the derivative of the target channel matrix is uncertain but belongs to a certain convex support, it can be modeled as:

in the formula,andrespectively representing the true and assumed derivatives of the kth target signal matrix,is composed ofWhich belongs to the following set of lower tension branches:

in order not to produce a trivial solution, it is assumed herein

Based on the modeling, the robust waveform design problem of improving the parameter estimation performance under the worst condition can be briefly described as follows: optimizing WCM to minimize convex set under power constraints with respect to WCMCRB under the above worst conditions. Based on trace criteria, this optimization problem can be described as follows:

tr(R_S)＝LP

wherein P is the total transmit power. The third constraint is due to the fact that the transmission power of any array element is greater than or equal to 0.

Obviously, the objective function in the above optimization problem is with respect to R_SAndk is a complex nonlinear function of 1,2, …, K. Thus, it is relatively difficult to solve using conventional methods such as convex optimization.

2. Solution of robust waveform optimization problem

According to the Hadamard inequality, the maximization problem in the above optimization problem can be relaxed as follows:

from the above equation, the k-th addition term depends only onThus, the equation is equivalent to maximizing each term added under the corresponding constraint, i.e., this problemCan be expressed as K independent problems as follows:

note that the parameters to be optimized in the above equation are complex numbers. To express this problem as a convex problem, we can translate this problem into the following problem with real numbers:

in the formula,

due to Q_RIs a positive definite matrix, andto relate toThen the above optimization problem can be equated to:

in the formula,t is an auxiliary variable.

Based on the Schur's theorem, two constraint conditions in the above optimization problem are rewritten, which are:

wherein η is not less than 0.

Based on the above, the optimization problem in step 1 is finally converted into the following SDP problem:

η≥0

to obtain the channel matrix derivative in the worst case, given the optimal T case, one can obtain

Similarly, the above formula can be rewritten as

This problem is equivalent to the SDP problem:

will obtainAnd R_SThe optimal worst-case CRB is obtained by substituting it into the objective function in the following optimization problem:

tr(R_S)＝LP

for the SDP problem described above, an efficient solution can be obtained using many mature algorithms at the present stage. It should be noted, however, that the proposed method only results in the WCM and not the final transmit waveform.

The invention has the beneficial effects that: the method can be used for reducing the problem that the waveform optimization method is sensitive to parameter estimation errors and uncertainty. An initial parameter error model is explicitly included in a waveform optimization problem and a robust Waveform Covariance (WCM) design problem is obtained based on a Cramer-Roman boundary (CRB), so that sensitivity is integrally reduced, parameter estimation accuracy under worst conditions (worst-case) is finally improved, meanwhile, in order to solve the obtained complex nonlinear optimization problem, a Hadamard Inequality is firstly utilized to decompose an objective function into a plurality of mutually independent sub-problems, then an inner layer optimization problem can be converted into a minimization problem, and finally an outer layer optimization problem and an inner layer optimization problem can be jointly converted into a semi-definite programming (SDP) problem, so that the robust optimization problem can be efficiently solved. Simulation results show that compared with uncorrelated transmitted waveforms and non-robust methods, the method can significantly improve the parameter estimation performance under the worst condition, has better robustness, and is closer to engineering application.

Drawings

FIG. 1 is a flow chart of the present invention;

fig. 2 is an optimal transmission beam pattern obtained under the condition that ASNR is 10 dB;

FIG. 3 is a worst case CRBs as a function of ASNR for the uncorrelated waveforms of the invention;

FIG. 4 is a graph of the variation of the resulting worst-case CRB mean with ASNR for the present invention compared to uncorrelated waveforms and non-robust methods

Detailed Description

The invention is described in further detail below with reference to fig. 1 to 4 and the examples: the specific implementation steps of the MIMO radar steady waveform design under the condition of the uncertain target priori knowledge are as follows:

1. building robust waveform optimization problem model

1) Constructing MIMO radar signal model

Suppose a MIMO radar receives a signal:

wherein,in order to transmit the matrix of waveforms,a discrete baseband representation of the signal is transmitted for the ith transmit unit and the number of samples is L.Andthe RCS-proportional complex amplitude and target position parameters for the K targets of interest, respectively, are estimated based on the received data. W is a noise plus interference matrix, each column of which can be modeled as an independent identityThe mean value of the distributed circularly symmetric complex Gaussian random vectors is 0, and the covariance matrix is an unknown matrix Q. H_k＝a(θ_k)v^T(θ_k) Denotes the k-th target channel matrix, a (θ)_k) And v (θ)_k) Respectively indicate being located at theta_kThe receive and transmit steering vectors of the target of (1), may be expressed as:

in the formula (f)₀Is the carrier frequency, τ_m(θ_k),m＝1,2,…M_rFor electromagnetic waves from being at theta_kTo the m-th receiving array element, andthe transmission time from the nth transmit array element to the target.

2) CRB derivation

Consider thatThe unknown parameter theta is [ theta ] under the known condition₁,θ₂,L,θ_K]^TThe constrained CRB of (a) may be represented as:

C_CCRB＝U(U^HFU)^-1U^H

in the formula, U satisfies

G(x)U(x)＝0,U^H(x)U(x)＝I

F is aboutFisher Information Matrix (FIM).

it is assumed herein that the amplitude matrix β ═ diag (β)₁，β₂，L，β_K) In the known manner, it is known that,having the form G ═ 0_2K×K,I_2K×2K]The corresponding null space U may be represented as U ═ I_K×K0_K×2K]^HThe FIM for x is calculated as follows:

then

Therefore, the temperature of the molten metal is controlled,

F(θ,θ)＝2Re(F₁₁)，

F^T(β_R,θ)＝F(θ,β_R)＝2Re(F₁₂)

F^T(β_I,θ)＝F(θ,β_I)＝-2Im(F₁₂)

F(β_R,β_R)＝F(β_I,β_I)＝2Re(F₂₂)

F(β_R,β_I)＝F^T(β_I,β_R)＝-2Im(F₂₂)

in the formula,

based on the above discussion, FIM F for x can be expressed as follows

Substituted into the above formulas to obtain

3) Construction of a robust waveform optimization model based on CRB

Assuming that the derivative of the target channel matrix is uncertain but belongs to a certain convex support, it can be modeled as:

error of the measurementBelonging to the following convex support sets:

based on the above discussion, the robust waveform design problem that improves the performance of parameter estimation under worst-case conditions can be briefly described as follows:

tr(R_S)＝LP

2. solution of robust waveform optimization problem

Assuming that the derivative of the target channel matrix is uncertain but belongs to a certain convex support, it can be modeled as:

in the formula,andrespectively representing the true and assumed derivatives of the kth target signal matrix,is composed ofWhich belongs to the following set of lower tension branches:

in order not to produce a trivial solution, it is assumed herein

Based on the modeling, the robust waveform design problem of improving the parameter estimation performance under the worst condition can be briefly described as follows: optimizing WCM to minimize convex set under power constraints with respect to WCMCRB under the above worst conditions. Based on trace criteria, this optimization problem can be described as follows:

tr(R_S)＝LP

wherein P is the total transmit power. The third constraint is due to the fact that the transmission power of any array element is greater than or equal to 0.

Obviously, the objective function in the above optimization problem is with respect to R_SAndis used to generate a complex non-linear function of (2). Thus, it is relatively difficult to solve using conventional methods such as convex optimization.

3. According to the Hadamard inequality, the maximization problem in the above optimization problem can be relaxed as follows:

from the above equation, the k-th addition term depends only onThus, the equation is equivalent to maximizing each term added under the corresponding constraint, i.e., this problem can be expressed as K mutually independent problems as follows:

note that the parameters to be optimized in the above equation are complex numbers. To express this problem as a convex problem, we can translate this problem into the following problem with real numbers:

in the formula,

due to Q_RIs a positive definite matrix, andto relate toThen the above optimization problem can be equated to:

in the formula,t is an auxiliary variable.

Based on the Schur's theorem, two constraint conditions in the above optimization problem are rewritten, which are:

wherein η is not less than 0.

Based on the above, the optimization problem in step 1 is finally converted into the following SDP problem:

η≥0

to obtain the channel matrix derivative in the worst case, given the optimal T case, one can obtain

Similarly, the above formula can be rewritten as

This problem is equivalent to the SDP problem:

will obtainAnd R_SThe optimal worst-case CRB is obtained by fitting it to the objective function in the following optimization problem.

For the SDP problem described above, an efficient solution can be obtained using many mature algorithms at the present stage. It should be noted, however, that the proposed method only results in the WCM and not the final transmit waveform.

Simulation conditions are as follows:

the experiment was based on 3 patientsA 3-receive MIMO radar having two configurations: MIMO radar (0.5 ) and MIMO radar (1.5, 0.5), the parameter in parentheses is the interval (in wavelength) of adjacent array elements of the transmitting array and the receiving array respectively. The number of system samples is L256. Array signal-to-interference-and-noise ratio (ASNR) is defined asThe value range is [ -10,30dB []Wherein, P is the total transmitting power,representing the variance of additive white noise. There is interference at-5 ° in the scene, and the array interference ratio (JNR) is defined as the interference power vs. M_rProduct ofThe quotient of (d) is set to 60 dB. The target with unity amplitude is at 20 o.

Simulation content:

let us assume that the uncertainty of the initial angle estimate is Δ θ [ -3o,3o [ ]]I.e. byWhereinAnd (3) calculating to obtain data as the estimation of theta: it can be seen that σ is 7.5634 for MIMO radar (0.5 ) and 29.5438 for MIMO radar (1.5, 0.5).

Fig. 2 shows the best transmit beam pattern obtained by the proposed method when ASNR is 10 dB. As can be seen from fig. 2, this method places a peak near the target, which means that the worst parameter estimation performance of the MIMO radar on the convex uncertainty set can be improved. Furthermore, as can be taken from fig. 2(b), the MIMO radar (1.5, 0.5) can generate grating lobes due to the sparse transmit array.

To verify the improved performance of the parameter estimation under the worst condition, the CRBs obtained by the proposed method and the uncorrelated waveforms vary with the ASNR as shown in the figure3, the comparison is based on completenessThe CRB resulting from the non-robust method of knowledge is the reference. Clearly, the worst-case CRBs resulting from the proposed method and uncorrelated waveform increase with increasing ASNR, based on completenessThe CRB obtained by the non-robust method of knowledge is also the same. Secondly, regardless of the ASNR value, the proposed method is significantly better than the uncorrelated waveform in terms of the worst-case CRB, since optimizing the waveform concentrates the transmit energy on the uncorrelated waveformWhile the non-correlated waveforms are transmitted omni-directionally. In addition, it can be seen that the difference between the CRBs under the worst condition obtained by the proposed method and the non-robust method under the certain conditions is relatively small, which indicates that the proposed method can effectively improve the parameter estimation performance under the worst condition. As can be seen from a comparison between fig. 3 (a) and (b), the CRB of the MIMO radar (1.5, 0.5) is smaller than that of the MIMO radar (0.5 ), because the former has a larger dummy receiving array aperture than the latter.

FIG. 4 shows the variation of the CRB mean with ASNR (based on 100 Monte Carlo tests) under the worst conditions obtained by the proposed method. From fig. 4, it can be seen that the proposed method results in an optimized waveform with better worst-case CRB than a non-robust, non-correlated waveform, in other words, the proposed method relates toHas better steady performance.

In summary, aiming at the problem of waveform optimization of the robust MIMO radar, the invention explicitly includes the parameter uncertainty set into the waveform optimization problem, improves the system parameter estimation performance under the worst condition based on the constrained CRB, and provides a method based on the Hadamard inequality to solve the problem aiming at the obtained complex nonlinear problem, and the provided method can convert the complex problem into the SDP problem, thereby obtaining efficient solution. Numerical simulation shows that compared with a non-robust method and a non-correlated waveform, the method can significantly reduce the CRB under the worst condition, thereby improving the robust performance of system parameter estimation.

Claims (1)

1. The MIMO radar robust waveform design method under the condition of the unknown target priori knowledge is characterized by comprising the following steps:

1) building robust waveform optimization problem model

(1) Constructing an MIMO radar signal model:

suppose a MIMO radar receives a signal:

wherein,in order to transmit the matrix of waveforms,a discrete baseband representation of the signal transmitted for the ith transmit unit, and a number of samples L,andthe complex amplitude and the target position parameters of the K interested targets which are proportional to the RCS are obtained through estimation based on received data, W is a noise and interference matrix, each row of the noise and interference matrix can be modeled into independent and identically distributed circularly symmetric complex Gaussian random vectors, the mean value of the vectors is 0, and the covariance matrix is an unknown matrix Q and H_k＝a(θ_k)v^T(θ_k) Denotes the k-th target channel matrix, a (θ)_k) And v (θ)_k) Respectively indicate being located at theta_kThe receive and transmit steering vectors of the target of (1), may be expressed as:

in the formula (f)₀Representing the carrier frequency, τ_m(θ_k),m＝1,2,…M_rFor electromagnetic waves from being at theta_kTo the m-th receiving array element, andthe transmission time from the nth transmitting array element to the target;

(2) CRB derivation

Consider thatThe unknown parameter theta is [ theta ] under the known condition₁,θ₂,L,θ_K]^TThe constrained CRB of (a) may be represented as:

C_CCRB＝U(U^HFU)^-1U^H

in the formula, U satisfies

G(x)U(x)＝0,U^H(x)U(x)＝I

F is aboutThe Fisher Information Matrix (FIM),

amplitude matrix β ═ diag (β)₁，β₂，L，β_K) In the known manner, it is known that,having the form G ═ 0_2K×K,I_2K×2K]The corresponding null space U may be represented as U ═ I_K×K0_K×2K]^HThe FIM for x is calculated as follows:

then

Wherein R is_S＝SS^H；

Therefore, the temperature of the molten metal is controlled,

F(θ,θ)＝2Re(F₁₁)，

F^T(β_R,θ)＝F(θ,β_R)＝2Re(F₁₂)

F^T(β_I,θ)＝F(θ,β_I)＝-2Im(F₁₂)

F(β_R,β_R)＝F(β_I,β_I)＝2Re(F₂₂)

F(β_R,β_I)＝F^T(β_I,β_R)＝-2Im(F₂₂)

in the formula,based on the above discussion, FIM F for x can be expressed as follows

Substituted into the above formulas to obtain

Wherein, C₁₁＝(2Re(F₁₁))^-1

(3) Construction of a robust waveform optimization model based on CRB

Assuming that the derivative of the target channel matrix is uncertain but belongs to a certain convex support, it can be modeled as:

error of the measurementBelonging to the following convex support sets:

based on the modeling, the robust waveform design problem of improving the parameter estimation performance under the worst condition can be briefly described as follows:

tr(R_S)＝LP

andrespectively representing true and assumed derivatives of the kth target signal matrix;

2) solution of robust waveform optimization problem

According to the Hadamard inequality, the maximization problem in the above optimization problem can be relaxed as follows:

from the above equation, the k-th addition term depends only onThus, the equation is equivalent to maximizing each term added under the corresponding constraint, i.e., this problem can be expressed as K mutually independent problems as follows:

note that the parameter to be optimized in the above formula is a complex number, and in order to express this problem as a convex problem, this problem is converted into the following problem with respect to real numbers:

in the formula,

Q_x＝Re(Q),Q_y＝Im(Q)，

due to Q_RIs a positive definite matrix, andto relate toThen the above optimization problem can be equated to:

in the formula,t is an auxiliary variable;

based on the Schur's theorem, two constraint conditions in the above optimization problem are rewritten, which are:

wherein η is more than or equal to 0;

based on the above, the optimization problem in step 1 is finally converted into the following SDP problem:

to obtain the channel matrix derivative in the worst case, given the optimal T case, one can obtain

Similarly, the above formula can be rewritten as

This problem is equivalent to the SDP problem:

will obtainAnd R_SThe optimal worst-case CRB is obtained by substituting it into the objective function in the following optimization problem: