CN104808179A - Cramer-rao bound based waveform optimizing method for MIMO radar in clutter background - Google Patents

Abstract

The invention provides a cramer-rao bound based waveform optimizing method for an MIMO radar in a clutter background, and belongs to the field of signal handling. The method comprises the steps of deducing CRB with parameters to be estimated according to a built signal model in the clutter environment; describing the optimal problem according to CRB based related rules. According to the method, a novel optimizing method based on diagonal loading is proposed to solve the obtained high non-linear optimization problem; the nonlinear optimization problem is converted into semi-define programming problem, so that the waveform covariance matrix can be efficiently optimized to achieve the minimized cramer-rao bound, and as a result, the system parameter estimation performances can be improved; compared with non-related waveform method, the method has the advantage that the parameter estimation performance for the MIMO radar system is obviously improved.

Description

Based on the MIMO radar waveform optimization method of Cramér-Rao lower bound under clutter environment

Technical field

The invention belongs to signal transacting field, further relate to the method that the MIMO radar waveform under the signal dependence clutter environment of waveform optimization technical field is optimized.

Background technology

In recent years, multiple-input and multiple-output (multiple-input multiple-output, MIMO) technology causes in communication and field of radar and pays close attention to widely and study, and waveform optimization is an important subject of MIMO radar.According to the difference needing the signal model optimized in waveform optimization problem, the optimization of waveform can be divided into two class methods for designing: (1) only designs transmitted waveform; (2) launch, receive waveform co-design.For the former, optimization aim is waveform covariance matrix or radar ambiguity function.For the latter, can combined optimization transmitted waveform and receive power, such as, maximum signal and interference plus noise are than the method for the detection perform to improve MIMO radar.

In the research process of waveform optimization method, many scholars achieve significant achievement.Such as, J.Li etc., based on the criterion about Cramér-Rao lower bound, optimize waveform to improve systematic parameter estimated performance.It should be noted that this document carries out under reception echo does not comprise the hypothesis depending on the clutter transmitted.But as everyone knows, in a lot of actual application, Received signal strength is inevitably subject to the pollution of clutter.Thus, from the viewpoint of parameter estimation, under being necessary very much research clutter environment, optimize waveform to improve the problem of parameter estimation performance.

Summary of the invention

The object of the invention is to the deficiency overcoming above-mentioned prior art, MIMO radar waveform optimization method based on Cramér-Rao lower bound under a kind of clutter environment is proposed, the method for the waveform optimization problem under clutter environment, by optimizing waveform covariance matrix (WCM) to minimize Cramér-Rao lower bound (CRB) thus to promote parameter estimation performance; The present invention is based on diagonal angle and load (DL) method, former nonlinearity problem is relaxed as convex optimization problem, thus ripe Semidefinite Programming (SDP) method can be utilized to carry out Efficient Solution.

Basic ideas of the present invention are: first set up waveform optimization model, then derive about the CRB of unknown parameter, finally relax nonlinearity problem into convex problem based on DL method, thus obtain Efficient Solution.Specific as follows:

Based on the MIMO radar waveform optimization method of Cramér-Rao lower bound under clutter environment, it comprises the steps:

The foundation of step one, MIMO radar system model

Based on the configuration of putting MIMO radar system altogether, if for transmitted waveform matrix, wherein represent the signal of i-th transmitter unit, M _tfor the number of MIMO radar system transmitter unit, L is transmitted waveform number of samples; Suppose that detectable signal is arrowband, and non-dispersive is propagated, then the signal that MIMO radar receives is expressed as:

$Y=\underset{k=1}{\overset{K}{\mathrm{\Σ}}}{\mathrm{\β}}_{k}a\left({\mathrm{\θ}}_k\right)v^T\left({\mathrm{\θ}}_k\right)S+\underset{i=1}{\overset{N_C}{\mathrm{\Σ}}}\mathrm{\ρ}\left({\mathrm{\θ}}_i\right)a_c\left({\mathrm{\θ}}_i\right)v_c^T\left({\mathrm{\θ}}_i\right)S+W---\left(1\right)$

Wherein for Received signal strength, M _rfor the number of MIMO radar system receiving element, for being proportional to the complex magnitude of target RCS (radar cross section), for target location parameter, K is the target numbers in considered range unit; Section 2 on the right of equation represents the clutter data received by receiver, ρ (θ _i) be θ _ithe reflection coefficient of place's clutter block, N _c(N _c>>M _tm _r) be clutter spatial sampling quantity, W represents interference noise, and it is independent of clutter, and suppose that the row of W are independent identically distributed round symmetric complex random vectors, average is 0, and covariance is unknown matrix B; A (θ _k) and v (θ _k) represent reception respectively, launch steering vector, be specifically expressed as:

$a\left({\mathrm{\θ}}_k\right)={[e^{j2\mathrm{\π}{f}_0{\mathrm{\τ}}_1},e^{j2\mathrm{\π}{f}_0{\mathrm{\τ}}_2},\·\·\·,e^{j2\mathrm{\π}{f}_0{\mathrm{\τ}}_{M_r}\left({\mathrm{\θ}}_k\right)}]}^T$

$v\left({\mathrm{\θ}}_k\right)={[e^{j2\mathrm{\π}{f}_0{\widetilde{\mathrm{\τ}}}_1\left({\mathrm{\θ}}_k\right)},e^{j2\mathrm{\π}{f}_0{\widetilde{\mathrm{\τ}}}_2\left({\mathrm{\θ}}_k\right)},\·\·\·,e^{j2\mathrm{\π}{f}_0{\widetilde{\mathrm{\τ}}}_{M_r}\left({\mathrm{\θ}}_k\right)}]}^T$

In formula, f ₀for carrier frequency, τ _m(θ _k), m=1,2 ... M _rwith for the transmission time, a _c(θ _i) and v _c(θ _i) represent θ respectively _ithe reception of place's clutter block, transmitting steering vector;

Through simplifying, formula (1) is expressed as again:

$Y=\underset{k=1}{\overset{K}{\mathrm{\Σ}}}{\mathrm{\β}}_{k}a\left({\mathrm{\θ}}_k\right)v^T\left({\mathrm{\θ}}_k\right)S+H_{c}S+W$

Wherein, represent clutter transport function, vec (H _c) be the same multiple Gaussian random vector distributed, its average is zero, and covariance is be expressed as further wherein,

$v=[v_1,v_2,\·\·\·,v_{N_{;C}}],v_i=v_c\left({\mathrm{\θ}}_i\right)\⊗a_c\left({\mathrm{\θ}}_i\right),i=\mathrm{1,2},\·\·\·,N_C,\mathrm{\Ξ}=\mathrm{diag}\{{\mathrm{\σ}}_1^2,{\mathrm{\σ}}_2^2,\·\·\·,{\mathrm{\σ}}_{N_C}^2\},{\mathrm{\σ}}_i^2=E\left[\mathrm{\ρ}\left({\mathrm{\θ}}_i\right){\mathrm{\ρ}}^{*}\left({\mathrm{\θ}}_i\right)\right]$

The CRB of the MIMO radar system model that step 2, step one are set up derives

Consider parameter unknown parameter θ=[θ under known conditions ₁, θ ₂..., θ _k] ^tcRB (namely retraining CRB), through derivation, be expressed as follows:

C _CCRB＝U(U ^HFU) ^-1U ^H

Wherein, U meets G (x) U (x)=0, U ^h(x) U (x)=I; And β _r=[β _{r, 1}, β _{r, 2}..., β _r,K] ^t, β _r=Re (β), β _i=Im (β), supposes for non-singular matrix; U is arranged on g (x) tangent line lineoid x meeting equality constraint, and F is the Fei Sheer information matrix (FIM) about x; Suppose complex amplitude matrix β=diag (β ₁, β ₂..., β _k) can accurately be estimated as:

g _i(x)＝β _R,i-1＝0,i＝1,…,K

g _j(x)＝β _I,j-1＝0,j＝K,…,K

Then g=[0 can be expressed as _{2K × K}, I _{2K × 2K}], corresponding U=[I _{k × K}0 _{k × 2K}] ^h;

Through deriving, the F about x can be expressed as follows:

$F=2\left[\begin{array}{ccc}\mathrm{Re}\left(F_{11}\right)& \mathrm{Re}\left(F_{12}\right)& -\mathrm{Im}\left(F_{12}\right)\\ R{e}^T\left(F_{12}\right)& \mathrm{Re}\left(F_{22}\right)& -\mathrm{Im}\left(F_{22}\right)\\ -{\mathrm{Im}}^T\left(F_{12}\right)& -{\mathrm{Im}}^T\left(F_{22}\right)& \mathrm{Re}\left(F_{22}\right)\end{array}\right]$

Wherein, ${\left[F_{11}\right]}_{\mathrm{ij}}={\mathrm{\β}}_i^{*}{\mathrm{\β}}_j{\stackrel{.}{h}}_i^H\left[{(I+(R_S\⊗B^{-1})R_{H_c})}^{-1}(R_S\⊗B^{-1})\right]{\stackrel{.}{h}}_j;$

${\left[F_{12}\right]}_{\mathrm{ij}}={\mathrm{\β}}_i^{*}{\stackrel{.}{h}}_i^H\left[{(I+(R_S\⊗B^{-1})R_{H_c})}^{-1}(R_S\⊗B^{-1})\right]h_j;{\left[F_{22}\right]}_{\mathrm{ij}}=h_i^H\left[{(I+(R_S\⊗B^{-1})R_{H_c})}^{-1}(R_S\⊗B^{-1})\right]h_j;$

$h_k=v\left({\mathrm{\θ}}_k\right)\⊗a\left({\mathrm{\θ}}_k\right);{\stackrel{.}{h}}_k=\frac{\∂(v\left({\mathrm{\θ}}_k\right)\⊗a\left({\mathrm{\θ}}_k\right))}{\∂{\mathrm{\θ}}_k},$ k＝1,2,…,K；R _S＝S ^*S ^T；

Step 3, based on Trace-Opt criterion, MIMO radar system waveform to be optimized

Under total emission power constraint, optimization problem can be expressed as follows:

$\begin{array}{cc}\underset{R_S}{\min }& \mathrm{tr}\left(C_{\mathrm{CCRB}}\right)\\ s.t.& \mathrm{tr}\left(R_S\right)=\mathrm{LP}\\ & R_S\≥0\end{array}$

Wherein, P is total emission power;

DL method is used R _son, can obtain

${\tilde{R}}_S=R_S+\mathrm{\ϵI}>0$

Wherein ε << λ _max(R _s), λ _maxthe eigenvalue of maximum of () representing matrix, uses substitute the R in optimization problem _s; Then, based on following proposition, the non-linear constrain in optimization problem is converted to linear restriction;

Proposition: by the computing between matrix and conversion, the constraint in optimization problem can become α I≤E≤β I, wherein

$E={(I+({\tilde{R}}_S\&CircleTimes;B^{-1})R_{H_c})}^{-1}({\tilde{R}}_S\&CircleTimes;B^{-1}),\mathrm{\&alpha;}=\frac{\mathrm{\&epsiv;}}{{\mathrm{\&lambda;}}_{\max }\left(B\right)+\mathrm{\&epsiv;}{\mathrm{\&lambda;}}_{\max }\left(R_{H_c}\right)},\mathrm{\&beta;}=\frac{\mathrm{LP}+\mathrm{\&epsiv;}}{{\mathrm{\&lambda;}}_{\min }\left(B\right)+(\mathrm{LP}+\mathrm{\&epsiv;}){\mathrm{\&lambda;}}_{\min }\left(R_{H_c}\right)};$

Based on this proposition, optimization problem can be changed to SDP problem by following lemma 1;

Hermitian matrix is supposed in lemma 1. $Z=\left[\begin{array}{cc}A& B^H\\ B& C\end{array}\right]$ C > 0, then during and if only if Δ C>=0, Z>=0, wherein, Δ C=A-B ^hc ^-1b is that the Schur of C in Z mends;

Based on lemma 1, above-mentioned optimization problem can be expressed as following SDP problem:

$\begin{array}{cc}\underset{X,E}{\min }& \mathrm{tr}\left(X\right)\\ s.t.& \mathrm{\&alpha;I}\&le;E\&le;\mathrm{\&beta;I}\\ & \left[\begin{array}{cc}X& U\\ U^H& U^H\mathrm{FU}\end{array}\right]\&GreaterEqual;0\end{array}$

Wherein, X is auxiliary optimized variable;

Step 4, based on Det-Opt criterion, MIMO radar system waveform to be optimized

From step 3, the determinant minimizing CRB is equivalent to maximize U ^hthe determinant of FU, therefore, the optimization problem based on Det-Opt criterion can be expressed as follows SDP problem:

$\begin{array}{cc}\underset{E}{\min }& -\log \det \left(U^H\mathrm{FU}\right)\\ s.t.& \mathrm{\&alpha;I}\&le;E\&le;\mathrm{\&beta;I}\end{array}$

Step 5, based on least square fitting R _s

Two class optimization problems in solution procedure three and step 4 can obtain optimum E, then, can obtain demarcate as follows:

$\begin{array}{c}\mathrm{tr}\left(\mathrm{\&mu;}{\tilde{R}}_{\mathrm{SB}}\right)=\mathrm{tr}\left(({\tilde{R}}_S\&CircleTimes;B^{-1})\right)\\ =(\mathrm{LP}+M_t\mathrm{\&epsiv;})\mathrm{tr}\left(B^{-1}\right)\end{array}$

Wherein, μ is a scalar, and it meets equality constraint; Diagonal loading coefficient ε=LP/1000;

Given by least square method matching R _s, it can be expressed as follows:

$\begin{array}{c}{R}_S=\arg \underset{R_S}{\min }\left|\right|{\tilde{R}}_{\mathrm{SB}}-({\tilde{R}}_S\&CircleTimes;B^{-1})\left|\right|\\ s.t.\mathrm{tr}\left(R_S\right)=\mathrm{LP}\\ R_S\&GreaterEqual;0\end{array}$

Above formula can equivalent statements as follows:

$\begin{array}{c}\underset{R_S,t}{\min }t\\ s.t.\left|\right|{\tilde{R}}_{\mathrm{SB}}-({\tilde{R}}_S\&CircleTimes;B^{-1})\left|\right|\&le;t\\ \mathrm{tr}\left(R_S\right)=\mathrm{LP}\\ R_S\&GreaterEqual;0\end{array}$

Based on lemma 1, above formula also can be converted into following SDP problem:

$\begin{array}{c}\underset{R_S,t}{\min }t\\ s.t.\left[\begin{array}{cc}t& {\mathrm{vec}}^H({\tilde{R}}_{\mathrm{SB}}-({\tilde{R}}_S\&CircleTimes;B^{-1}))\\ \mathrm{vec}({\tilde{R}}_{\mathrm{SB}}-({\tilde{R}}_S\&CircleTimes;B^{-1}))& I\end{array}\right]\\ \mathrm{tr}\left(R_S\right)=\mathrm{LP}\\ R_S\&GreaterEqual;0\end{array}\&GreaterEqual;0.$

Beneficial effect of the present invention is as follows:

The first, for the problem improving MIMO radar system parameter estimation performance under clutter scene, the present invention adopts the waveform design method loaded based on diagonal angle to improve MIMO radar system parameter estimation performance.

The second, based on diagonal angle loading method, nonlinear waveform optimization problem can be converted into semi definite programming problem, thus the Optimization Toolbox of comparative maturity can be utilized to obtain Efficient Solution.

Accompanying drawing explanation

Fig. 1 is the process flow diagram based on the MIMO radar waveform optimization method of Cramér-Rao lower bound under clutter environment of the present invention;

Fig. 2 is the geometric graph of MIMO radar system signal model in the present invention;

Optimum transmit beam direction figure when Fig. 3 is ASNR=50dB, the CNR=10dB under Trace-opt, Det-opt criterion;

Fig. 4 is that the funtcional relationship of two angle on targets to ASNR or CNR under Trace-opt, Det-opt criterion represents, and the corresponding situation of uncorrelated waveform;

When Fig. 5 is ASNR=-10dB, CNR=50dB, CRB is as the function representation of angle or clutter evaluated error.

Embodiment

Below in conjunction with drawings and Examples, the present invention is described in further detail.

As shown in Figure 1, the MIMO radar waveform optimization method implementation procedure based on Cramér-Rao lower bound under clutter environment of the present invention is as follows:

The proposition of 1.MIMO radar system model and CRB derive

1) proposition of signal model

As shown in Figure 2, this model is based on putting MIMO radar configuration altogether, and transmitter and receiver are all enough close to some target making them share equal angular for MIMO radar system geometric figure to be optimized.Wherein, M _tand M _rbe respectively the transmitting of this MIMO radar system, receiving element number.If for transmitted waveform matrix, wherein represent the signal of i-th transmitter unit, L is transmitted waveform number of samples.Suppose that detectable signal is arrowband, and non-dispersive is propagated, then the signal that MIMO radar receives can be expressed as:

$Y=\underset{k=1}{\overset{K}{\mathrm{\&Sigma;}}}{\mathrm{\&beta;}}_{k}a\left({\mathrm{\&theta;}}_k\right)v^T\left({\mathrm{\&theta;}}_k\right)S+\underset{i=1}{\overset{N_C}{\mathrm{\&Sigma;}}}\mathrm{\&rho;}\left({\mathrm{\&theta;}}_i\right)a_c\left({\mathrm{\&theta;}}_i\right)v_c^T\left({\mathrm{\&theta;}}_i\right)S+W---\left(1\right)$

Wherein for Received signal strength, for being proportional to the complex magnitude of target RCS, for target location parameter, both need to estimate, K is the target numbers in considered range unit.Section 2 on the right of equation represents the clutter data received by receiver, ρ (θ _i) be θ _ithe reflection coefficient of place's clutter block, N _c(N _c>>M _tm _r) be clutter spatial sampling quantity, W represents interference noise, and it is independent of clutter.Suppose that the row of W are independent identically distributed round symmetric complex random vectors, average is 0, and covariance is unknown matrix B.A (θ _k) and v (θ _k) represent reception respectively, launch steering vector, be specifically expressed as:

$a\left({\mathrm{\&theta;}}_k\right)={[e^{j2\mathrm{\&pi;}{f}_0{\mathrm{\&tau;}}_1},e^{j2\mathrm{\&pi;}{f}_0{\mathrm{\&tau;}}_2},\&CenterDot;\&CenterDot;\&CenterDot;,e^{j2\mathrm{\&pi;}{f}_0{\mathrm{\&tau;}}_{M_r}\left({\mathrm{\&theta;}}_k\right)}]}^T$

$v\left({\mathrm{\&theta;}}_k\right)={[e^{j2\mathrm{\&pi;}{f}_0{\widetilde{\mathrm{\&tau;}}}_1\left({\mathrm{\&theta;}}_k\right)},e^{j2\mathrm{\&pi;}{f}_0{\widetilde{\mathrm{\&tau;}}}_2\left({\mathrm{\&theta;}}_k\right)},\&CenterDot;\&CenterDot;\&CenterDot;,e^{j2\mathrm{\&pi;}{f}_0{\widetilde{\mathrm{\&tau;}}}_{M_r}\left({\mathrm{\&theta;}}_k\right)}]}^T$

In formula, f ₀for carrier frequency, τ _m(θ _k), m=1,2 ... M _rwith for the transmission time, a _c(θ _i) and v _c(θ _i) represent θ respectively _ithe reception of place's clutter block, transmitting steering vector.

In order to simply, formula (1) can be expressed as again:

$Y=\underset{k=1}{\overset{K}{\mathrm{\&Sigma;}}}{\mathrm{\&beta;}}_{k}a\left({\mathrm{\&theta;}}_k\right)v^T\left({\mathrm{\&theta;}}_k\right)S+H_{c}S+W$

Wherein, represent clutter transport function, vec (H _c) may be thought of as the same multiple Gaussian random vector distributed, its average is zero, and covariance is can also be expressed as further: wherein, $v=[v_1,v_2,\&CenterDot;\&CenterDot;\&CenterDot;,v_{N_{;C}}],v_i=v_c\left({\mathrm{\&theta;}}_i\right)\&CircleTimes;a_c\left({\mathrm{\&theta;}}_i\right),i=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,N_C,$ $\mathrm{\&Xi;}=\mathrm{diag}\{{\mathrm{\&sigma;}}_1^2,{\mathrm{\&sigma;}}_2^2,\&CenterDot;\&CenterDot;\&CenterDot;,{\mathrm{\&sigma;}}_{N_C}^2\},{\mathrm{\&sigma;}}_i^2=E\left[\mathrm{\&rho;}\left({\mathrm{\&theta;}}_i\right){\mathrm{\&rho;}}^{*}\left({\mathrm{\&theta;}}_i\right)\right].$

2) CRB derives

Consider parameter unknown parameter θ=[θ under known conditions ₁, θ ₂..., θ _k] ^tcRB (namely retraining CRB), through deriving, can be expressed as follows:

C _CCRB＝U(U ^HFU) ^-1U ^H

Wherein, U meets G (x) U (x)=0, U ^h(x) U (x)=I.And β _r=[β _{r, 1}, β _{r, 2}..., β _r,K] ^t, β _r=Re (β), β _i=Im (β), supposes for non-singular matrix.U is arranged on g (x) tangent line lineoid x meeting equality constraint, and F is the Fei Sheer information matrix about x.In the present invention, suppose complex amplitude matrix β=diag (β ₁, β ₂..., β _k) can accurately be estimated as:

g _i(x)＝β _R,i-1＝0,i＝1,…,K

g _j(x)＝β _I,j-1＝0,j＝K,…,K

Then g=[0 can be expressed as _{2K × K}, I _{2K × 2K}], corresponding U=[I _{k × K}0 _{k × 2K}] ^h.

Through deriving, the F about x can be expressed as follows:

$F=2\left[\begin{array}{ccc}\mathrm{Re}\left(F_{11}\right)& \mathrm{Re}\left(F_{12}\right)& -\mathrm{Im}\left(F_{12}\right)\\ R{e}^T\left(F_{12}\right)& \mathrm{Re}\left(F_{22}\right)& -\mathrm{Im}\left(F_{22}\right)\\ -{\mathrm{Im}}^T\left(F_{12}\right)& -{\mathrm{Im}}^T\left(F_{22}\right)& \mathrm{Re}\left(F_{22}\right)\end{array}\right]$

Wherein, ${\left[F_{11}\right]}_{\mathrm{ij}}={\mathrm{\&beta;}}_i^{*}{\mathrm{\&beta;}}_j{\stackrel{.}{h}}_i^H\left[{(I+(R_S\&CircleTimes;B^{-1})R_{H_c})}^{-1}(R_S\&CircleTimes;B^{-1})\right]{\stackrel{.}{h}}_j;$

${\left[F_{12}\right]}_{\mathrm{ij}}={\mathrm{\&beta;}}_i^{*}{\stackrel{.}{h}}_i^H\left[{(I+(R_S\&CircleTimes;B^{-1})R_{H_c})}^{-1}(R_S\&CircleTimes;B^{-1})\right]h_j;{\left[F_{22}\right]}_{\mathrm{ij}}=h_i^H\left[{(I+(R_S\&CircleTimes;B^{-1})R_{H_c})}^{-1}(R_S\&CircleTimes;B^{-1})\right]h_j;$

$h_k=v\left({\mathrm{\&theta;}}_k\right)\&CircleTimes;a\left({\mathrm{\&theta;}}_k\right);{\stackrel{.}{h}}_k=\frac{\&PartialD;(v\left({\mathrm{\&theta;}}_k\right)\&CircleTimes;a\left({\mathrm{\&theta;}}_k\right))}{\&PartialD;{\mathrm{\&theta;}}_k},$ k＝1,2,…,K；R _S＝S ^*S ^T。

2.MIMO radar waveform is optimized

1) based on the waveform optimization of Trace-Opt criterion

Under total emission power constraint, optimization problem can be expressed as follows:

$\begin{array}{cc}\underset{R_S}{\min }& \mathrm{tr}\left(C_{\mathrm{CCRB}}\right)\\ s.t.& \mathrm{tr}\left(R_S\right)=\mathrm{LP}\\ & R_s\&GreaterEqual;0\end{array}$

Wherein P is total emission power.

Can find out, C _cCRBbe one about the inverse nonlinear function of F, and F is about R _snonlinear function.Therefore, this problem is comparatively complicated nonlinear optimal problem, is difficult to utilize traditional Optimization Method.For solving this problem, the DL method considered normal in robust ada-ptive beamformer is used R by the present invention _son, can obtain

${\tilde{R}}_S=R_S+\mathrm{\&epsiv;I}>0$

Wherein ε << λ _max(R _s), λ _maxthe eigenvalue of maximum of () representing matrix, uses substitute the R in optimization problem _s.Then, based on following proposition, the non-linear constrain in optimization problem can be converted to linear restriction.

Proposition: by the computing between matrix and conversion, the constraint in optimization problem can become α I≤E≤β I, wherein

$E={(I+({\tilde{R}}_S\&CircleTimes;B^{-1})R_{H_c})}^{-1}({\tilde{R}}_S\&CircleTimes;B^{-1}),\mathrm{\&alpha;}=\frac{\mathrm{\&epsiv;}}{{\mathrm{\&lambda;}}_{\max }\left(B\right)+\mathrm{\&epsiv;}{\mathrm{\&lambda;}}_{\max }\left(R_{H_c}\right)},\mathrm{\&beta;}=\frac{\mathrm{LP}+\mathrm{\&epsiv;}}{{\mathrm{\&lambda;}}_{\min }\left(B\right)+(\mathrm{LP}+\mathrm{\&epsiv;}){\mathrm{\&lambda;}}_{\min }\left(R_{H_c}\right)};$

Based on this proposition, optimization problem can be changed to Semidefinite Programming by following lemma 1.

Hermitian matrix is supposed in lemma 1. $Z=\left[\begin{array}{cc}A& B^H\\ B& C\end{array}\right]$ C > 0, then during and if only if Δ C>=0, Z>=0, wherein, Δ C=A-B ^hc ^-1b is that the Schur of C in Z mends.

Based on lemma 1, above-mentioned optimization problem can be expressed as following SDP problem:

$\begin{array}{cc}\underset{X,E}{\min }& \mathrm{tr}\left(X\right)\\ s.t.& \mathrm{\&alpha;I}\&le;E\&le;\mathrm{\&beta;I}\\ & \left[\begin{array}{cc}X& U\\ U^H& U^H\mathrm{FU}\end{array}\right]\&GreaterEqual;0\end{array}$

Wherein, X is auxiliary optimized variable.

2) based on the waveform optimization of Det-Opt criterion

From the above-mentioned waveform optimization based on Trace-Opt criterion, the determinant minimizing CRB is equivalent to maximize U ^hthe determinant of FU.Thus, the optimization problem based on Det-Opt criterion can be expressed as follows SDP problem:

$\begin{array}{cc}\underset{E}{\min }& -\log \det \left(U^H\mathrm{FU}\right)\\ s.t.& \mathrm{\&alpha;I}\&le;E\&le;\mathrm{\&beta;I}\end{array}$

3) matching R under least square meaning _s

Solve above two class optimization problems and can obtain optimum E, then, can obtain demarcate as follows:

$\begin{array}{c}\mathrm{tr}\left(\mathrm{\&mu;}{\tilde{R}}_{\mathrm{SB}}\right)=\mathrm{tr}\left(({\tilde{R}}_S\&CircleTimes;B^{-1})\right)\\ =(\mathrm{LP}+M_t\mathrm{\&epsiv;})\mathrm{tr}\left(B^{-1}\right)\end{array}$

Wherein, μ is a scalar, and it meets equality constraint.

Given by least square method matching R _s, it can be expressed as follows:

$\begin{array}{c}{R}_S=\arg \underset{R_S}{\min }\left|\right|{\tilde{R}}_{\mathrm{SB}}-({\tilde{R}}_S\&CircleTimes;B^{-1})\left|\right|\\ s.t.\mathrm{tr}\left(R_S\right)=\mathrm{LP}\\ R_S\&GreaterEqual;0\end{array}$

Above formula can equivalent statements as follows:

$\begin{array}{c}\underset{R_S,t}{\min }t\\ s.t.\left|\right|{\tilde{R}}_{\mathrm{SB}}-({\tilde{R}}_S\&CircleTimes;B^{-1})\left|\right|\&le;t\\ \mathrm{tr}\left(R_S\right)=\mathrm{LP}\\ R_S\&GreaterEqual;0\end{array}$

Based on lemma 1, similarly, above formula also can be converted into following SDP problem:

$\begin{array}{c}\underset{R_S,t}{\min }t\\ s.t.\left[\begin{array}{cc}t& {\mathrm{vec}}^H({\tilde{R}}_{\mathrm{SB}}-({\tilde{R}}_S\&CircleTimes;B^{-1}))\\ \mathrm{vec}({\tilde{R}}_{\mathrm{SB}}-({\tilde{R}}_S\&CircleTimes;B^{-1}))& I\end{array}\right]\\ \mathrm{tr}\left(R_S\right)=\mathrm{LP}\\ R_S\&GreaterEqual;0\end{array}\&GreaterEqual;0$

In emulation experiment below, select best diagonal loading coefficient ε=LP/1000.

Effect of the present invention further illustrates by following emulation:

Simulated conditions:

Conveniently, all parameters used by emulation have all been listed in table 1, refer to additivity white thermal noise variance, Clutter modeling uses discrete point model, and its RCS is modeled as independent identically distributed gaussian random variable vector, and average is zero, and variance is and suppose in the Coherent processing time constant.Jammer is modeled into point source, and it is launched and the incoherent White Gaussian signal of MIMO radar signal.

Table 1

In order to check the validity of the method, the present invention considers emphatically following two kinds of situations:

The first has the CRB of clearly known initial parameter; It two is the impacts on CRB of initial parameter evaluated error.

Emulation content:

A: without the CRB of initial parameter evaluated error

Fig. 3 shows under Trace-opt or Det-opt criterion, and the best during ASNR=50dB and CNR=10dB sends beam pattern.Can find out, there is a power recess significantly in interference radiating way.In addition, can find out and differ greatly between two target gained power, this is because the present invention only considers the CRB that minimizes total CRB instead of minimize each parameter.

Fig. 4 illustrates under Trace-opt or Det-opt criterion, and two angles are to the funtcional relationship of ASNR or CNR, and the corresponding situation of uncorrelated waveform.Can also see, the CRB obtained by method proposed by the invention or uncorrelated waveform reduces with the increase of ASNR, and increases along with the reduction of CNR.But the no matter size of ASNR or CNR, the CRB under Trace-opt or Det-opt criterion is more much lower than the situation of uncorrelated waveform.In addition, Trace-opt criterion can obtain the CRB lower than Det-opt criterion.The CRB of MIMO radar (2.5,0.5) is lower than the CRB of MIMO radar (0.5,0.5).

B: initial parameter evaluated error is on the impact of CRB

Fig. 5 shows under ASNR=-10dB, CNR=50dB condition, and CRB is with the situation of change of angle or clutter evaluated error.Can see, CRB is obvious with the evaluated error fluctuation of angle or clutter, and this shows that method of the present invention is very responsive to error.

In sum, under the present invention is directed to clutter scene, improve the problem of MIMO radar system parameter estimation performance, propose to optimize transmitted waveform to improve the method for systematic parameter estimated performance based on diagonal angle loading technique.The present invention is MIMO radar Received signal strength model under modeling clutter scene first, and based on this model inference unknown parameter Cramér-Rao lower bound, then based on Trace-Opt and Det-Opt criteria construction waveform optimization problem.Due to the complex nonlinear problem that this problem is about optimized variable, be difficult to utilize traditional Optimization Method.For this problem, the present invention proposes a kind of method based on diagonal angle loading technique to relax this nonlinear problem into Semidefinite Programming, thus can obtain Efficient Solution.Known compared with uncorrelated transmitted waveform, the present invention can significantly improve systematic parameter estimated performance.Known based on above discussion, institute of the present invention extracting method can be in engineer applied and is provided solid theory by design transmitted waveform raising radar parameter estimated performance and realized foundation.

Claims (1)

1. under clutter environment based on the MIMO radar waveform optimization method of Cramér-Rao lower bound, it is characterized in that, the method comprises the steps:

The foundation of step one, MIMO radar system model

Based on the configuration of putting MIMO radar system altogether, if for transmitted waveform matrix, wherein i=1,2 ..., M _trepresent the signal of i-th transmitter unit, M _tfor the number of MIMO radar system transmitter unit, L is transmitted waveform number of samples; Suppose that detectable signal is arrowband, and non-dispersive is propagated, then the signal that MIMO radar receives is expressed as:

$Y=\underset{k=1}{\overset{K}{\mathrm{\&Sigma;}}}{\mathrm{\&beta;}}_{k}a\left({\mathrm{\&theta;}}_k\right)v^T\left({\mathrm{\&theta;}}_k\right)S+\underset{i=1}{\overset{N_C}{\mathrm{\&Sigma;}}}\mathrm{\&rho;}\left({\mathrm{\&theta;}}_i\right)a_c\left({\mathrm{\&theta;}}_i\right)v_c^T\left({\mathrm{\&theta;}}_i\right)S+W---\left(1\right)$

Wherein for Received signal strength, M _rfor the number of MIMO radar system receiving element, for being proportional to the complex magnitude of target RCS, for target location parameter, K is the target numbers in considered range unit; Section 2 on the right of equation represents the clutter data received by receiver, ρ (θ _i) be θ _ithe reflection coefficient of place's clutter block, N _c(N _c> > M _tm _r) be clutter spatial sampling quantity, W represents interference noise, and it is independent of clutter, and suppose that the row of W are independent identically distributed round symmetric complex random vectors, average is 0, and covariance is unknown matrix B; A (θ _k) and v (θ _k) represent reception respectively, launch steering vector, be specifically expressed as:

$a\left({\mathrm{\&theta;}}_k\right)={[e^{j2\mathrm{\&pi;}{f}_0{\mathrm{\&tau;}}_1\left({\mathrm{\&theta;}}_k\right)},e^{j2\mathrm{\&pi;}{f}_0{\mathrm{\&tau;}}_2\left({\mathrm{\&theta;}}_k\right)},...,e^{j2\mathrm{\&pi;}{f}_0{\mathrm{\&tau;}}_{M_r}\left({\mathrm{\&theta;}}_k\right)}]}^T$

$v\left({\mathrm{\&theta;}}_k\right)={[e^{j2\mathrm{\&pi;}{f}_0{\widetilde{\mathrm{\&tau;}}}_1\left({\mathrm{\&theta;}}_k\right)},e^{j2\mathrm{\&pi;}{f}_0{\widetilde{\mathrm{\&tau;}}}_2\left({\mathrm{\&theta;}}_k\right)},...,e^{j2\mathrm{\&pi;}{f}_0{\widetilde{\mathrm{\&tau;}}}_{M_t}\left({\mathrm{\&theta;}}_k\right)}]}^T$

In formula, f ₀for carrier frequency, τ _m(θ _k), m=1,2 ... M _rwith n=1,2 ... M _tfor the transmission time, a _c(θ _i) and v _c(θ _i) represent θ respectively _ithe reception of place's clutter block, transmitting steering vector;

Through simplifying, formula (1) is expressed as again:

$Y=\underset{k=1}{\overset{K}{\mathrm{\&Sigma;}}}{\mathrm{\&beta;}}_{k}a\left({\mathrm{\&theta;}}_k\right)v^T\left({\mathrm{\&theta;}}_k\right)S+H_{c}S+W$

Wherein, represent clutter transport function, vec (H _c) be the same multiple Gaussian random vector distributed, its average is zero, and covariance is $R_{H_c}=E\left[\mathrm{vec}\left(H_c\right){\mathrm{vec}}^H\left(H_c\right)\right];$ be expressed as further $R_{H_c}=\mathrm{V\&Xi;}{V}^H\&GreaterEqual;0;$ Wherein, $V=[v_1,v_2,...,v_{N_C}],v_i=v_c\left({\mathrm{\&theta;}}_i\right)\&CircleTimes;a_c\left({\mathrm{\&theta;}}_i\right),$ i＝1,2,…,N _C， $\mathrm{\&Xi;}=\mathrm{diag}\{{\mathrm{\&sigma;}}_1^2,{\mathrm{\&sigma;}}_2^2,...,{\mathrm{\&sigma;}}_{N_C}^2\},{\mathrm{\&sigma;}}_i^2=E\left[\mathrm{\&rho;}\left({\mathrm{\&theta;}}_i\right){\mathrm{\&rho;}}^{*}\left({\mathrm{\&theta;}}_i\right)\right];$

The CRB of the MIMO radar system model that step 2, step one are set up derives

Consider parameter unknown parameter θ=[θ under known conditions ₁, θ ₂..., θ _k] ^tcRB (namely retraining CRB), through derivation, be expressed as follows:

C _CCRB＝U(U ^HFU) ^-1U ^H

Wherein, U meets G (x) U (x)=0, U ^h(x) U (x)=I; And β _r=[β _{r, 1}, β _{r, 2}..., β _r,K] ^t, β _r=Re (β), β _i=Im (β), supposes for non-singular matrix; U is arranged on g (x) tangent line lineoid x meeting equality constraint, and F is the Fei Sheer information matrix about x; Suppose complex amplitude matrix β=diag (β ₁, β ₂..., β _k) can accurately be estimated as:

g _i(x)＝β _R,i-1＝0,i＝1,…,K

g _j(x)＝β _I,j-1＝0,j＝K,…,K

Then g=[0 can be expressed as _{2K × K}, I _{2K × 2K}], corresponding U=[I _{k × K}0 _{k × 2K}] ^h;

Through deriving, the F about x can be expressed as follows:

$F=2\left[\begin{array}{ccc}\mathrm{Re}\left(F_{11}\right)& \mathrm{Re}\left(F_{12}\right)& -\mathrm{Im}\left(F_{12}\right)\\ {\mathrm{Re}}^T\left(F_{12}\right)& \mathrm{Re}\left(F_{22}\right)& -\mathrm{Im}\left(F_{22}\right)\\ -{\mathrm{Im}}^T\left(F_{12}\right)& -{\mathrm{Im}}^T\left(F_{22}\right)& \mathrm{Re}\left(F_{22}\right)\end{array}\right]$

Wherein, ${\left[F_{11}\right]}_{\mathrm{ij}}={\mathrm{\&beta;}}_i^{*}{\mathrm{\&beta;}}_j{\stackrel{\&CenterDot;}{h}}_i^H\left[{(I+(R_S\&CircleTimes;B^{-1})R_{H_c})}^{-1}(R_S\&CircleTimes;B^{-1})\right]{\stackrel{\&CenterDot;}{h}}_j;$

${\left[F_{12}\right]}_{\mathrm{ij}}={\mathrm{\&beta;}}_i^{*}{\stackrel{\&CenterDot;}{h}}_i^H\left[{(I+(R_S\&CircleTimes;B^{-1})R_{H_c})}^{-1}(R_S\&CircleTimes;B^{-1})\right]h_j;{\left[F_{22}\right]}_{\mathrm{ij}}=h_i^H\left[{(I+(R_S\&CircleTimes;B^{-1})R_{H_c})}^{-1}(R_S\&CircleTimes;B^{-1})\right]h_j;$

$h_k=v\left({\mathrm{\&theta;}}_k\right)\&CircleTimes;a\left({\mathrm{\&theta;}}_k\right);{\stackrel{\&CenterDot;}{h}}_k=\frac{\&PartialD;(v\left({\mathrm{\&theta;}}_k\right)\&CircleTimes;a\left({\mathrm{\&theta;}}_k\right))}{\&PartialD;{\mathrm{\&theta;}}_k},$ k＝1,2,…,K；R _S＝S ^*S ^T；

Step 3, based on Trace-Opt criterion, MIMO radar system waveform to be optimized

Under total emission power constraint, optimization problem can be expressed as follows:

$\begin{array}{cc}\underset{R_S}{\min }& \mathrm{tr}\left(C_{\mathrm{CCRB}}\right)\end{array}$

s.t. tr(R _S)＝LP

R _S≥0

Wherein, P is total emission power;

DL method is used R _son, can obtain

${\tilde{R}}_S=R_S+\mathrm{\&epsiv;I}>0$

Wherein ε < < λ _max(R _s), λ _maxthe eigenvalue of maximum of () representing matrix, uses substitute the R in optimization problem _s; Then, based on following proposition, the non-linear constrain in optimization problem is converted to linear restriction;

Proposition: by the computing between matrix and conversion, the constraint in optimization problem can become α I≤E≤β I, wherein $E={(I+({\tilde{R}}_S\&CircleTimes;B^{-1})R_{H_c})}^{-1}({\tilde{R}}_S\&CircleTimes;B^{-1}),\mathrm{\&alpha;}=\frac{\mathrm{\&epsiv;}}{{\mathrm{\&lambda;}}_{\max }\left(B\right)+\mathrm{\&epsiv;}{\mathrm{\&lambda;}}_{\max }\left(R_{H_c}\right)},\mathrm{\&beta;}=\frac{\mathrm{LP}+\mathrm{\&epsiv;}}{{\mathrm{\&lambda;}}_{\min }\left(B\right)+(\mathrm{LP}+\mathrm{\&epsiv;}){\mathrm{\&lambda;}}_{\min }\left(R_{H_c}\right)};$

Based on this proposition, optimization problem can be changed to SDP problem by following lemma 1;

Hermitian matrix is supposed in lemma 1. $Z=\left[\begin{array}{cc}A& B^H\\ B& C\end{array}\right]$ C > 0, then during and if only if Δ C>=0, Z>=0, wherein, Δ C=A-B ^hc ^-1b is that the Schur of C in Z mends;

Based on lemma 1, above-mentioned optimization problem can be expressed as following SDP problem:

$\begin{array}{cc}\underset{X,E}{\min }& \mathrm{tr}\left(X\right)\end{array}$

s.t. αI≤E≤βI

$\left[\begin{array}{cc}X& U\\ U^H& U^H\mathrm{FU}\end{array}\right]\&GreaterEqual;0$

Wherein, X is auxiliary optimized variable;

Step 4, based on Det-Opt criterion, MIMO radar system waveform to be optimized

From step 3, the determinant minimizing CRB is equivalent to maximize U ^hthe determinant of FU, therefore, the optimization problem based on Det-Opt criterion can be expressed as follows SDP problem:

$\begin{array}{cc}\underset{E}{\min }& -\log \det \left(U^H\mathrm{FU}\right)\end{array}$

s.t. αI≤E≤βI

Step 5, based on least square fitting R _s

Two class optimization problems in solution procedure three and step 4 can obtain optimum E, then, can obtain demarcate as follows:

$\begin{array}{c}\mathrm{tr}\left(\mathrm{\&mu;}{\tilde{R}}_{\mathrm{SB}}\right)=\mathrm{tr}\left(({\tilde{R}}_S\&CircleTimes;B^{-1})\right)\\ =(\mathrm{LP}+M_t\mathrm{\&epsiv;})\mathrm{tr}\left(B^{-1}\right)\end{array}$

Wherein, μ is a scalar, and it meets equality constraint; Diagonal loading coefficient ε=LP/1000;

Given by least square method matching R _s, it can be expressed as follows:

$R_S=\arg \underset{R_S}{\min }\left|\right|{\tilde{R}}_{\mathrm{SB}}-({\tilde{R}}_S\&CircleTimes;B^{-1})\left|\right|$

s.t. tr(R _S)＝LP

R _S≥0

Above formula can equivalent statements as follows:

$\begin{array}{cc}\underset{R_S,t}{\min }& t\end{array}$

$\begin{array}{cc}s.t.& \left|\right|{\tilde{R}}_{\mathrm{SB}}-({\tilde{R}}_S\&CircleTimes;B^{-1})\left|\right|\&le;t\end{array}$

tr(R _S)＝LP

R _S≥0

Based on lemma 1, above formula also can be converted into following SDP problem:

$\begin{array}{cc}\underset{R_S,t}{\min }& t\end{array}$

$\begin{array}{cc}s.t.& \left[\begin{array}{cc}t& {\mathrm{vec}}^H({\tilde{R}}_{\mathrm{SB}}-({\tilde{R}}_S\&CircleTimes;B^{-1}))\\ \mathrm{vec}({\tilde{R}}_{\mathrm{SB}}-({\tilde{R}}_S\&CircleTimes;B^{-1}))& I\end{array}\right]\end{array}\&GreaterEqual;0.$

tr(R _S)＝LP

R _S≥0