CN104808190A - Steady waveform design method for improving worst parameter estimation performance of MIMO (multiple-input and multiple-output) radar - Google Patents

Abstract

A steady waveform design method for improving the worst parameter estimation performance of an MIMO radar belongs to the field of signal processing.

Description

Improve the sane waveform design method of MIMO radar the poorest parameter estimation performance

Technical field

The invention belongs to signal transacting field, further relate to the sane waveform design method of MIMO radar the poorest parameter estimation performance under the improvement clutter environment of Waveform Design technical field.

Background technology

In the last few years, the optimization of MIMO radar waveform was subject to the attention of increasing scholar and slip-stick artist.According to the object module used in waveform optimization problem, current waveform optimization method can be divided into following two classes: (1) is based on the waveform optimization of point target (point target); (2) based on the waveform optimization of Extended target (extended target).Based on the Waveform Design of point target, the object of optimization is waveform Correlation Matrix (WCM, waveform covariance matrix) or radar ambiguity function (radar ambiguityfunction).Waveform optimization method based on WCM only designs the spatial domain of transmitted waveform instead of the overall feature of transmitted waveform.Specifically, the people such as D.R.Fuhrmann and G.S.Antonio designs to realize specific energy spatial domain to WCM and distributes.And the people such as S.Peter have not only paid close attention to energy spatial domain and distribute, and have also contemplated that the spatial domain cross-correlation between different target, namely minimize spatial domain cross-correlation between different azimuth to improve the detection estimated performance of system.

Under the hypothesis that Received signal strength is not polluted by the clutter depending on transmitted waveform, the people such as J.Li propose the waveform optimization criterion of a few class based on CRB to optimize WCM thus the Parameter Estimation Precision of raising point target.Received signal strength is by under clutter pollution condition, and the people such as H.Y Wang consider target prior imformation and know under condition based on the MIMO radar waveform of CRB and the combined optimization problem of biased estimator.It should be noted that in these methods, solving of waveform optimization problem all needs parameter to know.But in Practical Project, these parameters must be obtained by estimation, thus inevitably there is evaluated error.Thus, optimizing the parameter estimation performance that obtains of waveform to evaluated error and uncertainty based on estimated parameter is than more sensitive.

Summary of the invention

The object of the invention under being to overcome clutter conditions traditional waveform optimization method to initial parameter evaluated error sensitive issue, propose a kind of sane waveform design method improving MIMO radar the poorest parameter estimation performance, the method comprises Parameter uncertainties convex set, based on the alternative manner solving-optimizing problem of DL technology, to alleviate parameter estimating error or the uncertain system sensitivity problem brought, thus improve the MIMO radar waveform optimize parameter estimate performance under worst case.

The basic ideas of the inventive method are: first MIMO radar Received signal strength model under structure clutter conditions, based on the CRB of this model inference parameter to be estimated, then set up the explicit sane waveform optimization model comprising parameter uncertainty, for solving this nonlinear optimal problem, propose a kind of iterative algorithm based on DL technology, each step of iteration can relax as Semidefinite Programming thus obtain Efficient Solution.After obtaining an optimum intermediate solution based on iterative algorithm, optimum waveform covariance matrix can reconstruct under least square meaning.

The present invention improves the sane waveform design method of MIMO radar the poorest parameter estimation performance, and it comprises the steps:

Step one, structure MIMO radar Received signal strength model

Suppose that MIMO radar Received signal strength is:

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

Wherein, for being proportional to the complex magnitude of target RCS (radar cross section), for target location parameter, K is target numbers, and ρ (θ) is for being in the reflection coefficient of θ position clutter block, and W represents interference noise, and often row are separate and with distribution circle symmetric complex random vector, have zero-mean, its covariance B the unknown, for the matrix that transmits, a (θ _k) and v (θ _k) represent reception respectively, launch steering vector, be specifically expressed as:

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

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

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

If rang ring is divided into N _c(N _c> > NML) individual resolution element, MIMO radar Received signal strength model is rewritten as

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

Wherein, represent clutter transport function, 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, a _c(θ _i) and v _c(θ _i) represent θ respectively _ithe reception of place's clutter block, transmitting steering vector; Vec (H _c) be the same multiple Gaussian random vector distributed, its average is zero, and covariance is $V=[v_1,v_2,...,v_{N_C}],v_i=v_c\left({\mathrm{\θ}}_i\right)\⊗a_c\left({\mathrm{\θ}}_i\right),i=\mathrm{1,2},...,N_C,\mathrm{\Ξ}=\mathrm{diag}\{{\mathrm{\σ}}_1^2,{\mathrm{\σ}}_2^2,...,{\mathrm{\σ}}_{N_C}^2\},$ ${\mathrm{\σ}}_i^2=E\left[\mathrm{\ρ}\left({\mathrm{\θ}}_i\right){\mathrm{\ρ}}^{*}\left({\mathrm{\θ}}_i\right)\right];$

Step 2, build sane waveform optimization model based on CRB

Consider unknown parameter θ=[θ ₁, θ ₂..., θ _k] ^t, cRB under condition, through deriving, this CRB can be expressed as follows:

$C=\frac{1}{2}{\left[\begin{array}{ccc}\mathrm{Re}\left(F_{11}\right)& \mathrm{Re}\left(F_{12}\right)& -\mathrm{Im}\left(F_{12}\right)\\ {\mathrm{Re}}^T\left(F_{12}\right)& \mathrm{Re}\left(F_{22}\right)& -\mathrm{Im}\left(F_{22}\right)\\ -{\mathrm{Im}}^T\left(F_{12}\right)& -{\mathrm{Im}}^T\left(F_{22}\right)& \mathrm{Re}\left(F_{22}\right)\end{array}\right]}^{-1}$

Wherein,

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

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

${\left[F_{22}\right]}_{\mathrm{ij}}=h_i^H\left[{(I+(R_S\⊗B^{-1})R_{H_c})}^{-1}(R_S\⊗B^{-1})\right]h_j$

In formula, R _s=SS ^h, $R_{H_c}=E\left[\mathrm{vec}\left(H_c\right){\mathrm{vec}}^H\left(H_c\right)\right]$ For positive semidefinite Hermitian matrix, $h_k=v\left({\mathrm{\θ}}_k\right)\⊗a\left({\mathrm{\θ}}_k\right),$ ${\stackrel{\·}{h}}_k=\frac{\∂(v\left({\mathrm{\θ}}_k\right)\⊗a\left({\mathrm{\θ}}_k\right))}{\∂{\mathrm{\θ}}_k},k=\mathrm{1,2},...,K;$

Only consider direction of arrival angle, namely θ evaluated error is on the impact of system performance, can be as follows to a kth destination channel matrix modeling:

${\tilde{h}}_k=h_k+{\mathrm{\δ}}_k$

Wherein, h _kbe respectively a kth destination channel matrix that is actual and hypothesis, δ _kfor error, belong to following convex set:

And, wherein, be respectively h _kreal and suppose vector reciprocal, for error, belong to following convex set:

Based on foregoing, improving the sane waveform optimization problem of worst condition parameter estimation performance under clutter conditions can be expressed as: about under the constraint of WCM, based on Parameter uncertainties convex set optimize WCM to minimize the CRB under worst case; Under Trace-opt criterion, optimization problem can be described as:

$\begin{array}{ccc}\underset{R_S}{\min }& \underset{{\left\{{\mathrm{\δ}}_k\right\}}_{k=1}^K,{\left\{{\stackrel{\·}{\mathrm{\δ}}}_k\right\}}_{k=1}^K}{\max }& \mathrm{tr}\left(C\right)\end{array}$

tr(R _S)＝LP

Wherein, P represents total emissive power; In formula, the 3rd constraint establishment is because each transmitter unit emissive power can not be less than zero;

Solving of step 3, sane waveform internal layer optimization problem

Internal layer optimization problem solve based on following lemma 1:

Lemma 1. supposes that A is the positive semidefinite hermitian matrix of a M × M, then the inequality below is set up: time and if only if A is diagonal matrix, equation is set up; According to lemma 1, internal layer optimization problem can relax and be:

$\begin{array}{cc}\underset{{\left\{{\mathrm{\δ}}_k\right\}}_{k=1}^K,{\left\{{\stackrel{\·}{\mathrm{\δ}}}_k\right\}}_{k=1}^K}{\max }& \underset{k=1}{\overset{K}{\mathrm{\Σ}}}\frac{1}{{\left[2\mathrm{Re}\left(F\right)\right]}_{\mathrm{kk}}}\end{array}$

Based on CRB, above formula can be rewritten as:

$\begin{array}{cc}\underset{{\left\{{\mathrm{\δ}}_k\right\}}_{k=1}^K,{\left\{{\stackrel{\·}{\mathrm{\δ}}}_k\right\}}_{k=1}^K}{\max }& \underset{k=1}{\overset{K}{\mathrm{\Σ}}}\frac{1}{{\mathrm{\β}}_k^{*}{\stackrel{\·}{\tilde{h}}}_k^H\left[{(I+(R_S\⊗B^{-1})R_{H_c})}^{-1}(R_S\⊗B^{-1})\right]{\stackrel{\·}{\tilde{h}}}_{\text{k}}{\mathrm{\β}}_k+{\tilde{h}}_k^H\left[{(I+(R_S\⊗B^{-1})R_{H_c})}^{-1}(R_S\⊗B^{-1})\right]{\tilde{h}}_k}\end{array}$

Real part operational character Re{} is got in deletion, is because in above formula, each and item are real numbers;

From above formula, and in formula, the denominator of kth item only depends on δ _kwith two, the problem equivalent therefore in above formula in, under corresponding constraint, maximize and each in formula, can be expressed as:

$\begin{array}{cc}\underset{{\left\{{\mathrm{\δ}}_k\right\}}_{k=1}^K,{\left\{{\stackrel{\·}{\mathrm{\δ}}}_k\right\}}_{k=1}^K}{\max }& \frac{1}{{\mathrm{\β}}_k^{*}{\stackrel{\·}{\tilde{h}}}_k^H\left[{(I+(R_S\⊗B^{-1})R_{H_c})}^{-1}(R_S\⊗B^{-1})\right]{\stackrel{\·}{\tilde{h}}}_{\text{k}}{\mathrm{\β}}_k+{\tilde{h}}_k^H\left[{(I+(R_S\⊗B^{-1})R_{H_c})}^{-1}(R_S\⊗B^{-1})\right]{\tilde{h}}_k}\end{array}$

$\begin{array}{ccc}s.t.& {\left|\right|{\stackrel{\·}{\mathrm{\δ}}}_k\left|\right|}_F\≤{\mathrm{\σ}}_k,{\left|\right|{\mathrm{\δ}}_k\left|\right|}_F\≤{\mathrm{\ζ}}_k& k=\mathrm{1,2},...,K\end{array}$

For solving above formula, to R _sapplication diagonal angle loading technique, that is:

Wherein, ε < < λ _max(R _s) be load factor, λ _max() representing matrix eigenvalue of maximum, selects ε=λ _max(R _s)/1000; Use respectively replace the R in sane optimization problem _s, can obtain with respectively for and δ _kconvex;

Thus, above formula can be rewritten as:

$\begin{array}{cc}\underset{{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_k,{\mathrm{\&delta;}}_k}{\min }& {\mathrm{\&beta;}}_k^{*}{\stackrel{\&CenterDot;}{\tilde{h}}}_k^H\left[{(I+({\tilde{R}}_S\&CircleTimes;B^{-1})R_{H_c})}^{-1}({\tilde{R}}_S\&CircleTimes;B^{-1})\right]{\stackrel{\&CenterDot;}{\tilde{h}}}_k{\mathrm{\&beta;}}_k+{\tilde{h}}_k^H\left[{(I+({\tilde{R}}_S\&CircleTimes;B^{-1})R_{H_c})}^{-1}({\tilde{R}}_S\&CircleTimes;B^{-1})\right]{\tilde{h}}_k\end{array}$

$\begin{array}{cc}s.t.& {\left|\right|{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_k\left|\right|}_F\end{array}\&le;{\mathrm{\&sigma;}}_k,{\left|\right|{\mathrm{\&delta;}}_k\left|\right|}_F\&le;{\mathrm{\&zeta;}}_k$

Above formula can be torn open and be write as two independently minimization problems below:

$\begin{array}{cc}\underset{{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_k}{\min }& {\mathrm{\&beta;}}_k^{*}{\stackrel{\&CenterDot;}{\tilde{h}}}_k^H\left[{(I+({\tilde{R}}_S\&CircleTimes;B^{-1})R_{H_c})}^{-1}({\tilde{R}}_S\&CircleTimes;B^{-1})\right]{\stackrel{\&CenterDot;}{\tilde{h}}}_k{\mathrm{\&beta;}}_k\end{array}$

$\begin{array}{cc}s.t.& {\left|\right|{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_k\left|\right|}_F\end{array}\&le;{\mathrm{\&sigma;}}_k$

$\begin{array}{cc}\underset{{\mathrm{\&delta;}}_k}{\min }& {\tilde{h}}_k^H\left[{(I+({\tilde{R}}_S\&CircleTimes;B^{-1})R_{H_c})}^{-1}({\tilde{R}}_S\&CircleTimes;B^{-1})\right]{\tilde{h}}_k\end{array}$

s.t. ||δ _k|| _F≤ζ _k

Above-mentioned two minimization problems can be solved by lemma 2 below:

Lemma 2, suppose hermitian matrix $Z=\left[\begin{array}{cc}A& B^H\\ B& C\end{array}\right]$ 's then and if only if time, Z wherein, Δ C=A-B ^hc ^-1b is that the Schur of C in Z mends;

Lemma 2 by reference, above-mentioned two minimization problems can be converted into following SDP problem 1:

$\begin{array}{cc}\underset{t,{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_k}{\min }& t\end{array}$

$\begin{array}{cc}\underset{t,{\mathrm{\&delta;}}_k}{\min }& t\end{array}$

Wherein, t is auxiliary variable;

By what obtain from above two formulas with bring in sane optimization problem, consider outside optimization problem;

Solving of step 4, the sane outer optimization problem of waveform

Following proposition is utilized to solve outside optimization problem

Proposition: utilize matrix manipulation, the constraint in sane optimization problem can be equivalent to following LMI:

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

Use lemma 2 in conjunction with above-mentioned proposition, outer optimization problem can be expressed as following SDP problem:

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

Wherein, X is an auxiliary variable;

After obtaining optimum E, under least square meaning, R _sby following model construction:

$R_S\text{=arg}\underset{R_S}{\min }{\left|\right|{(E^{-1}-R_{H_c})}^{-1}-{\tilde{R}}_S\&CircleTimes;B^{-1}\left|\right|}_F$

s.t. tr(R _S)＝LP

Use lemma 2 in conjunction with above-mentioned proposition, above formula can be equivalent to following SDP problem 2:

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

tr(R _S)＝LP

Step 5, employing alternative manner solve sane waveform optimization problem

Step 5.1, given waveform covariance matrix initial value;

Step 5.2, solve above-mentioned SDP problem 1 to obtain optimum δ _k,

Step 5.3, solve SDP problem 2 to obtain optimum E;

Step 5.4, return step 5.2 iteration again, until CRB no longer significantly reduces.

Step 6, based on least square method, the waveform covariance matrix that reconstruct is optimum, can obtain R _s.

The invention has the beneficial effects as follows: the method can be used for releiving traditional waveform optimization method to parameter estimating error and uncertain sensitive issue.First set up MIMO radar Received signal strength model under clutter scene, characterize the lower bound-Cramér-Rao lower bound (CRB) of Parameter Estimation Precision to be estimated based on this model inference, then Parameter uncertainties convex set explicitly is comprised in traditional waveform optimization problem; For solving this nonlinear optimal problem, the present invention proposes a kind of alternative manner loading (DL) technology based on diagonal angle, each step in iteration all can be converted into Semidefinite Programming (SDP) problem, thus can Efficient Solution be obtained, to realize the sane waveform optimization of MIMO radar under worst case, and then the parameter estimation performance under worst case is promoted, compared with irrelevant waveform, the method has obvious lifting to the parameter estimation performance under worst case.

Accompanying drawing explanation

Fig. 1 is the process flow diagram that the present invention realizes;

Fig. 2 is the process flow diagram of iterative algorithm of the present invention;

Fig. 3 is the optimum transmit beam direction figure of the present invention when initial angle exists evaluated error and array signal to noise ratio (S/N ratio) is 10dB;

Fig. 4 under there is evaluated error situation in initial angle, the present invention carry the CRB changed with ANSR that algorithm and irrelevant waveform obtain.

Fig. 5 is the optimum transmit beam direction figure of the present invention when array calibration exists evaluated error and array signal to noise ratio (S/N ratio) is 10dB;

Fig. 6 under there is evaluated error situation in array calibration, the present invention put forward the worst case that algorithm and irrelevant waveform obtain under the CRB that changes with ANSR.

Embodiment

Below in conjunction with accompanying drawing, the present invention is described in further detail.

As shown in Figures 1 to 6, to improve the implementation procedure of the sane waveform design method of MIMO radar the poorest parameter estimation performance as follows in the present invention:

1, sane waveform optimization problem model is set up

1) MIMO radar signal model is built

Suppose that MIMO radar Received signal strength is:

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

Wherein, for being proportional to the complex magnitude of target RCS (radar cross section), for target location parameter, both need to estimate.K is target numbers, and ρ (θ) is for being in the reflection coefficient of θ position clutter block, and W represents interference noise, and often row are separate and with distribution circle symmetric complex random vector, have zero-mean, its covariance B the unknown, for the matrix that transmits.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 for the transmission time, a _c(θ) and v _c(θ) θ is represented respectively _kthe reception of place's target and transmitting steering vector.

If rang ring is divided into N _c(N _c> > NML) individual resolution element, Received signal strength model can be rewritten as

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

Wherein, represent clutter transport function, 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, a _c(θ _i) and v _c(θ _i) represent θ respectively _ithe reception of place's clutter block, transmitting steering vector.Vec (H _c) may be thought of as the same multiple Gaussian random vector distributed, its average is zero, and covariance is $R_{H_c}=E\left[\mathrm{vec}\left(H_c\right){\mathrm{vec}}^H\left(H_c\right)\right].$ R _hccan also be expressed as further: wherein, $V=[v_1,v_2,...,v_{N_C}],v_i=v_c\left({\mathrm{\&theta;}}_i\right)\&CircleTimes;a_c\left({\mathrm{\&theta;}}_i\right),i=\mathrm{1,2},...,N_C,\mathrm{\&Xi;}=\mathrm{diag}\{{\mathrm{\&sigma;}}_1^2,{\mathrm{\&sigma;}}_2^2,...,{\mathrm{\&sigma;}}_{N_C}^2\},{\mathrm{\&sigma;}}_i^2=E\left[\mathrm{\&rho;}\left({\mathrm{\&theta;}}_i\right){\mathrm{\&rho;}}^{*}\left({\mathrm{\&theta;}}_i\right)\right]$

2) the sane waveform optimization model based on CRB is built

Consider unknown parameter θ=[θ ₁, θ ₂..., θ _k] ^t, cRB under condition, through deriving, this CRB can be expressed as follows:

$C=\frac{1}{2}{\left[\begin{array}{ccc}\mathrm{Re}\left(F_{11}\right)& \mathrm{Re}\left(F_{12}\right)& -\mathrm{Im}\left(F_{12}\right)\\ {\mathrm{Re}}^T\left(F_{12}\right)& \mathrm{Re}\left(F_{22}\right)& -\mathrm{Im}\left(F_{22}\right)\\ -{\mathrm{Im}}^T\left(F_{12}\right)& -{\mathrm{Im}}^T\left(F_{22}\right)& \mathrm{Re}\left(F_{22}\right)\end{array}\right]}^{-1}$

Wherein,

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

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

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

In formula, R _s=SS ^h, $R_{H_c}=E\left[\mathrm{vec}\left(H_c\right){\mathrm{vec}}^H\left(H_c\right)\right]$ For positive semidefinite Hermitian matrix, $h_k=v\left({\mathrm{\&theta;}}_k\right)\&CircleTimes;a\left({\mathrm{\&theta;}}_k\right),$ ${\stackrel{\&CenterDot;}{h}}_k=\frac{\&PartialD;(v\left({\mathrm{\&theta;}}_k\right)\&CircleTimes;a\left({\mathrm{\&theta;}}_k\right))}{\&PartialD;{\mathrm{\&theta;}}_k},k=\mathrm{1,2},...,K.$

Significantly, CRB be about θ, h _c, W function, and these parameters need by estimate obtain, thus inevitably there is evaluated error.Thus, the systematic parameter estimated performance utilizing the CRB optimization waveform based on certain group estimates of parameters to obtain is worse than other one group of more rational estimates of parameters possibly.And then, in engineer applied, the tender subject of system performance to initial parameter evaluated error must be considered.

In the present invention, only consider direction of arrival angle, namely θ evaluated error is on the impact of system performance.Thus, can be as follows to a kth destination channel matrix modeling:

${\tilde{h}}_k=h_k+{\mathrm{\&delta;}}_k$

Wherein, h _kbe respectively a kth destination channel matrix that is actual and hypothesis, δ _kfor error, belong to following convex set:

And, wherein, be respectively h _kreal and suppose vector reciprocal, for error, belong to following convex set:

Based on above-mentioned discussion, improving the sane waveform optimization problem of worst condition parameter estimation performance under clutter conditions can be expressed as: about under the constraint of WCM, based on Parameter uncertainties convex set optimize WCM to minimize the CRB under worst case.Under Trace-opt criterion, optimization problem can be described as:

$\begin{array}{ccc}\underset{R_S}{\min }& \underset{{\left\{{\mathrm{\&delta;}}_k\right\}}_{k=1}^K,{\left\{{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_k\right\}}_{k=1}^K}{\max }& \mathrm{tr}\left(C\right)\end{array}$

tr(R _S)＝LP

Wherein, P represents total emissive power; In formula, the 3rd constraint establishment is because each transmitter unit emissive power can not be less than zero.

Clearly, CRB matrix trace, the i.e. objective function of above formula, be one about R _sand δ _k, k=1,2 ..., the very complicated nonlinear function of K, thus utilizes the classic methods such as such as convex optimization to be very difficult to solve.

2. the solving of sane waveform optimization problem

1) the solving of internal layer optimization problem

As mentioned above, the objective function of optimization problem is very complicated nonlinear function, is difficult to utilize traditional Optimization Method.For solving this problem, first consider internal layer optimization problem.Internal layer optimization problem solve based on following lemma 1:

Lemma 1. supposes that A is the positive semidefinite hermitian matrix of a M × M, then the inequality below is set up: time and if only if A is diagonal matrix, equation is set up.According to lemma 1, internal layer optimization problem can relax and be:

$\begin{array}{cc}\underset{{\left\{{\mathrm{\&delta;}}_k\right\}}_{k=1}^K,{\left\{{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_k\right\}}_{k=1}^K}{\max }& \underset{k=1}{\overset{K}{\mathrm{\&Sigma;}}}\frac{1}{{\left[2\mathrm{Re}\left(F\right)\right]}_{\mathrm{kk}}}\end{array}$

Based on CRB, above formula can be rewritten as:

$\begin{array}{cc}\underset{{\left\{{\mathrm{\&delta;}}_k\right\}}_{k=1}^K,{\left\{{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_k\right\}}_{k=1}^K}{\max }& \underset{k=1}{\overset{K}{\mathrm{\&Sigma;}}}\frac{1}{{\mathrm{\&beta;}}_k^{*}{\stackrel{\&CenterDot;}{\tilde{h}}}_k^H\left[{(I+(R_S\&CircleTimes;B^{-1})R_{H_c})}^{-1}(R_S\&CircleTimes;B^{-1})\right]{\stackrel{\&CenterDot;}{\tilde{h}}}_{\text{k}}{\mathrm{\&beta;}}_k+{\tilde{h}}_k^H\left[{(I+(R_S\&CircleTimes;B^{-1})R_{H_c})}^{-1}(R_S\&CircleTimes;B^{-1})\right]{\tilde{h}}_k}\end{array}$

Real part operational character Re{} is got in deletion, is because in above formula, each and item are real numbers.

From above formula, and in formula, the denominator of kth item only depends on δ _kwith two, the problem equivalent therefore in above formula in, under corresponding constraint, maximize and each in formula, can be expressed as:

$\begin{array}{cc}\underset{{\left\{{\mathrm{\&delta;}}_k\right\}}_{k=1}^K,{\left\{{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_k\right\}}_{k=1}^K}{\max }& \frac{1}{{\mathrm{\&beta;}}_k^{*}{\stackrel{\&CenterDot;}{\tilde{h}}}_k^H\left[{(I+(R_S\&CircleTimes;B^{-1})R_{H_c})}^{-1}(R_S\&CircleTimes;B^{-1})\right]{\stackrel{\&CenterDot;}{\tilde{h}}}_{\text{k}}{\mathrm{\&beta;}}_k+{\tilde{h}}_k^H\left[{(I+(R_S\&CircleTimes;B^{-1})R_{H_c})}^{-1}(R_S\&CircleTimes;B^{-1})\right]{\tilde{h}}_k}\end{array}$

$\begin{array}{ccc}s.t.& {\left|\right|{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_k\left|\right|}_F\&le;{\mathrm{\&sigma;}}_k,{\left|\right|{\mathrm{\&delta;}}_k\left|\right|}_F\&le;{\mathrm{\&zeta;}}_k& k=\mathrm{1,2},...,K\end{array}$

It should be noted that due to known for indefinite matrix, therefore above formula is difficult to solve, for solving this problem, to R _sapplication diagonal angle loading technique, that is:

Wherein, ε < < λ _max(R _s) be load factor, λ _max() representing matrix eigenvalue of maximum, below in l-G simulation test, select ε=λ _max(R _s)/1000.Use respectively replace the R in sane optimization problem _s, can obtain clearly, with respectively for and δ _kconvex.

Thus, above formula can be rewritten as:

$\begin{array}{cc}\underset{{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_k,{\mathrm{\&delta;}}_k}{\min }& {\mathrm{\&beta;}}_k^{*}{\stackrel{\&CenterDot;}{\tilde{h}}}_k^H\left[{(I+({\tilde{R}}_S\&CircleTimes;B^{-1})R_{H_c})}^{-1}({\tilde{R}}_S\&CircleTimes;B^{-1})\right]{\stackrel{\&CenterDot;}{\tilde{h}}}_k{\mathrm{\&beta;}}_k+{\tilde{h}}_k^H\left[{(I+({\tilde{R}}_S\&CircleTimes;B^{-1})R_{H_c})}^{-1}({\tilde{R}}_S\&CircleTimes;B^{-1})\right]{\tilde{h}}_k\end{array}$

$\begin{array}{cc}s.t.& {\left|\right|{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_k\left|\right|}_F\end{array}\&le;{\mathrm{\&sigma;}}_k,{\left|\right|{\mathrm{\&delta;}}_k\left|\right|}_F\&le;{\mathrm{\&zeta;}}_k$

Similar, above formula can be write as two independently minimization problems below:

$\begin{array}{cc}\underset{{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_k}{\min }& {\mathrm{\&beta;}}_k^{*}{\stackrel{\&CenterDot;}{\tilde{h}}}_k^H\left[{(I+({\tilde{R}}_S\&CircleTimes;B^{-1})R_{H_c})}^{-1}({\tilde{R}}_S\&CircleTimes;B^{-1})\right]{\stackrel{\&CenterDot;}{\tilde{h}}}_k{\mathrm{\&beta;}}_k\end{array}$

$\begin{array}{cc}s.t.& {\left|\right|{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_k\left|\right|}_F\end{array}\&le;{\mathrm{\&sigma;}}_k$

$\begin{array}{cc}\underset{{\mathrm{\&delta;}}_k}{\min }& {\tilde{h}}_k^H\left[{(I+({\tilde{R}}_S\&CircleTimes;B^{-1})R_{H_c})}^{-1}({\tilde{R}}_S\&CircleTimes;B^{-1})\right]{\tilde{h}}_k\end{array}$

s.t. ||δ _k|| _F≤ζ _k

Above two problems can be solved by lemma 2 below:

Hermitian matrix is supposed in lemma 2. $Z=\left[\begin{array}{cc}A& B^H\\ B& C\end{array}\right]$ 's then and if only if time, wherein, Δ C=A-B ^hc ^-1b is that the Schur of C in Z mends.

Lemma 2 by reference, above two problems can clearly be converted into following SDP problem:

$\begin{array}{cc}\underset{t,{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_k}{\min }& t\end{array}$

$\begin{array}{cc}\underset{t,{\mathrm{\&delta;}}_k}{\min }& t\end{array}$

Wherein, t is auxiliary variable.

By what obtain from above two formulas with bring in sane optimization problem, consider outside optimization problem.

2) the solving of outside optimization problem

The present invention utilizes following proposition for solving outside optimization problem

Proposition: utilize matrix manipulation, the constraint in sane optimization problem can be equivalent to following LMI:

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

Use lemma 2 and in conjunction with above-mentioned proposition, outer optimization problem can be expressed as following SDP problem:

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

Wherein, X is an auxiliary variable.

After obtaining optimum E, under least square meaning, R _sby following model construction:

$R_S\text{=arg}\underset{R_S}{\min }{\left|\right|{(E^{-1}-R_{H_c})}^{-1}-{\tilde{R}}_S\&CircleTimes;B^{-1}\left|\right|}_F$

s.t. tr(R _S)＝LP

Be similar to above-mentioned discussion, above formula can be equivalent to following SDP problem:

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

tr(R _S)＝LP

3) iterative algorithm

Given WCM initial value, δ _k, and R _sbe optimized by following steps:

1. solve internal layer SDP problem and obtain optimum δ _k,

2. solve outer SDP problem and obtain E;

Repeat step 1. 2., until CRB no longer significantly reduces.After this, solve constructed model R _s.

Effect of the present invention further illustrates by following emulation:

Simulated conditions:

MIMO radar is 33 and receives, utilize two MIMO radar system, its antenna configuration is respectively: MIMO radar (0.5,0.5), MIMO radar (1.5,0.5) the numeral transmitter, here in bracket and the array element distance (in units of wavelength) in receiver.It is 256 that systematic sampling is counted.Array signal to noise ratio (S/N ratio) is defined as span is-10dB to 30dB.Wherein, P refers to total emissive power, for additive white Gaussian noise variance.Modeling clutter is discrete sampling, and its RCS is modeled as independent identically distributed gaussian random variable vector, and average is zero, and variance is and suppose coherent processing inteval internal fixtion.Clutter signal to noise ratio (S/N ratio) is defined as equal 30dB.-5 ^°there is a strong jamming in direction, and signal to noise ratio (S/N ratio) is 60dB.Only in θ=20, ° place has a reflection coefficient to be the point target of 1.In following emulation, suppose two kinds of situations, the first only considers that initial angle estimates to there is error; It two is the correction errors only considering to exist in transmitting-receiving array.

Emulation content:

A: initial angle estimates to there is uncertain situation

Suppose that the uncertainty that initial angle is estimated is Δ θ=[-3 °, 3 °], namely wherein for the estimation of θ, through calculating, obtain data: MIMO (0.5,0.5) is ζ=5.4382, σ=7.6593, MIMO (1.5,0.5) is ζ=27.6329, σ=29.6754.

Fig. 3 is optimum transmit beam direction figure under ASNR=10dB condition.Can observe, the peak value of the beam pattern that transmits is positioned at around target location, this means, under this convex uncertain worst case, systematic parameter estimated performance can improve.In addition, due to sparse emission array, MIMO radar (1.5,0.5) there will be graing lobe situation, as shown in Fig. 3 (b).

The CRB of Fig. 4 for being changed with ASNR by proposed algorithm and uncorrelated waveform gained.Clearly, CRB reduces with the increase of ASNR.In addition, can observe, under the worst case that institute's extracting method obtains, parameter estimation performance is better than uncorrelated waveform.And along with the increase of ASNR, institute extracting method gained CRB is progressive in uncorrelated waveform.In addition, the CRB of MIMO radar (1.5,0.5) shown in Fig. 4 (b) is starkly lower than the CRB of MIMO radar (0.5,0.5) shown in Fig. 4 (a).

B: transmitting-receiving array exists the situation of correction error

In this case, be no matter that transmitting and receiving array is assumed that to have correction error (amplitude of sensor and phase error and site error).Each element of transmitting and receiving array steering vector disturb by the disturbance variable, this disturbance variable is the multiple Gaussian random variable of Cyclic Symmetry of zero-mean, and variance is after calculating, obtain ζ=13.4764 of MIMO (0.5,0.5), σ=14.5712, ζ=29.8362 of MIMO (1.5,0.5), σ=32.6573.

Fig. 5 features the optimum transmit beam pattern that ASNR=10dB obtains.From Fig. 5, the conclusion similar in appearance to Fig. 3 can be drawn.Carry CRB under the worst case that algorithm and uncorrelated waveform gained change with ASNR as shown in Figure 6, the conclusion obtained from Fig. 6 is similar to Fig. 4.

In sum, under the present invention is directed to clutter conditions, waveform optimization method is to initial parameter evaluated error sensitive issue, propose the sane waveform optimization method based on the convex uncertain collection of parameter, and propose a kind of iterative method loaded based on diagonal angle for this complex nonlinear optimization problem.For the sane performance of parameter estimation of MIMO radar system under raising clutter conditions, first there is error condition for direction of arrival angle and carry out modeling in the present invention, and this parameter estimating error convex set explicitly is comprised in afferent echo shape optimization problem, for solving this nonlinear optimal problem, the present invention proposes a kind of alternative manner loaded based on diagonal angle and carries out alternative optimization to transmitted waveform and parameter estimating error, to obtain optimum transmitted waveform covariance matrix.Each step of iteration all can load based on diagonal angle and relax as Semidefinite Programming, thus can obtain Efficient Solution.Known based on discussing above, institute of the present invention extracting method be can be the sane performance estimated by design transmitted waveform raising radar parameter in engineer applied and provides solid theory and realize foundation.

Claims (1)

1. improve the sane waveform design method of MIMO radar the poorest parameter estimation performance, it is characterized in that, the method comprises the steps:

Step one, structure MIMO radar Received signal strength model

Suppose that MIMO radar Received signal strength is:

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

Wherein, for being proportional to the complex magnitude of target RCS (radar cross section), for target location parameter, K is target numbers, and ρ (θ) is for being in the reflection coefficient of θ position clutter block, and W represents interference noise, and often row are separate and with distribution circle symmetric complex random vector, have zero-mean, its covariance B the unknown, for the matrix that transmits, a (θ _k) and v (θ _k) represent reception respectively, launch steering vector, be specifically expressed as:

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

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

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) and v _c(θ) θ is represented respectively _kthe reception of place's target and transmitting steering vector;

If rang ring is divided into N _c(N _c>>NML) individual resolution element, MIMO radar Received signal strength model is rewritten as

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

Wherein, represent clutter transport function, 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, a _c(θ _i) and v _c(θ _i) represent θ respectively _ithe reception of place's clutter block, transmitting steering vector; Vec (H _c) be the same multiple Gaussian random vector distributed, its average is zero, and covariance is $v_i=v_c\left({\mathrm{\&theta;}}_i\right)\&CircleTimes;a_c\left({\mathrm{\&theta;}}_i\right),i=\mathrm{1,2},...,N_C,\mathrm{\&Xi;}=\mathrm{diag}\{{\mathrm{\&sigma;}}_1^2,{\mathrm{\&sigma;}}_2^2,...,{\mathrm{\&sigma;}}_{N_C}^2\},$ ${\mathrm{\&sigma;}}_i^2=E\left[\mathrm{\&rho;}\left({\mathrm{\&theta;}}_i\right){\mathrm{\&rho;}}^{*}\left({\mathrm{\&theta;}}_i\right)\right];$

Step 2, build sane waveform optimization model based on CRB

Consider unknown parameter θ=[θ ₁, θ ₂..., θ _k] ^t, cRB under condition, through deriving, this CRB can be expressed as follows:

$C=\frac{1}{2}{\left[\begin{array}{ccc}\mathrm{Re}\left(F_{11}\right)& \mathrm{Re}\left(F_{12}\right)& -\mathrm{Im}\left(F_{12}\right)\\ {\mathrm{Re}}^T\left(F_{12}\right)& \mathrm{Re}\left(F_{22}\right)& -\mathrm{Im}\left(F_{22}\right)\\ -{\mathrm{Im}}^T\left(F_{12}\right)& -I{m}^T\left(F_{22}\right)& \mathrm{Re}\left(F_{22}\right)\end{array}\right]}^{-1}$

Wherein,

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

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

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

In formula, R _s=SS ^h, for positive semidefinite Hermitian matrix, ${\stackrel{\&CenterDot;}{h}}_k=\frac{\&PartialD;(v\left({\mathrm{\&theta;}}_k\right)\&CircleTimes;a\left({\mathrm{\&theta;}}_k\right))}{{\&PartialD;\mathrm{\&theta;}}_k},k=\mathrm{1,2},...,K;$

Only consider direction of arrival angle, namely θ evaluated error is on the impact of system performance, can be as follows to a kth destination channel matrix modeling:

${\tilde{h}}_k=h_k+{\mathrm{\&delta;}}_k$

Wherein, be respectively a kth destination channel matrix that is actual and hypothesis, δ _kfor error, belong to following convex set:

And, wherein, be respectively h _kreal and suppose vector reciprocal, for error, belong to following convex set:

Based on foregoing, improving the sane waveform optimization problem of worst condition parameter estimation performance under clutter conditions can be expressed as: about under the constraint of WCM, based on Parameter uncertainties convex set , optimize WCM to minimize the CRB under worst case; Under Trace-opt criterion, optimization problem can be described as:

Wherein, P represents total emissive power; In formula, the 3rd constraint establishment is because each transmitter unit emissive power can not be less than zero;

Solving of step 3, sane waveform internal layer optimization problem

Internal layer optimization problem solve based on following lemma 1:

Lemma 1. supposes that A is the positive semidefinite hermitian matrix of a M × M, then the inequality below is set up: time and if only if A is diagonal matrix, equation is set up; According to lemma 1, internal layer optimization problem can relax and be:

Based on CRB, above formula can be rewritten as:

real part operational character Re{} is got in deletion, is because in above formula, each and item are real numbers;

From above formula, and in formula, the denominator of kth item only depends on δ _kwith two, the problem equivalent therefore in above formula in, under corresponding constraint, maximize and each in formula, can be expressed as:

$\begin{array}{c}\underset{{\left\{{\mathrm{\&delta;}}_k\right\}}_{k=1}^K,{\left\{{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_k\right\}}_{k=1}^K}{\max }\frac{1}{{\mathrm{\&beta;}}_k^{*}{\stackrel{\stackrel{*}{~}}{h}}_k^H\left[{(I+(R_S\&CircleTimes;B^{-1})R_{H_c})}^{-1}(R_S\&CircleTimes;B^{-1})\right]{\stackrel{\stackrel{\&CenterDot;}{~}}{h}}_k{\mathrm{\&beta;}}_k+{\tilde{h}}_k^H\left[{(I+(R_S\&CircleTimes;B^{-1})R_{H_c})}^{-1}(R_S\&CircleTimes;B^{-1})\right]{\tilde{h}}_k}\\ s.t.{\left|\right|{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_k\left|\right|}_F\&le;{\mathrm{\&sigma;}}_k,{\left|\right|{\mathrm{\&delta;}}_k\left|\right|}_F\&le;{\mathrm{\&zeta;}}_k,k=\mathrm{1,2},...,K\end{array}$ For solving above formula, to R _sapplication diagonal angle loading technique, that is:

Wherein, ε << λ _max(R _s) be load factor, λ _max() representing matrix eigenvalue of maximum, selects ε=λ _max(R _s)/1000; Use respectively replace the R in sane optimization problem _s, can obtain with respectively for and δ _kconvex;

Thus, above formula can be rewritten as:

$\begin{array}{c}\underset{{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_k,{\mathrm{\&delta;}}_k}{\min }{\mathrm{\&beta;}}_k^{*}{\stackrel{\stackrel{\&CenterDot;}{~}}{h}}_k^H\left[{(I+({\tilde{R}}_S\&CircleTimes;B^{-1})R_{H_c})}^{-1}({\tilde{R}}_S\&CircleTimes;B^{-1})\right]{\stackrel{\stackrel{\&CenterDot;}{~}}{h}}_k{\mathrm{\&beta;}}_k+{\tilde{h}}_k^H\left[{\left(I({\tilde{R}}_S\&CircleTimes;B^{-1})R_{H_c}\right)}^{-1}({\tilde{R}}_S\&CircleTimes;B^{-1})\right]{\tilde{h}}_k\\ s.t.{\left|\right|{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_k\left|\right|}_F\&le;{\mathrm{\&sigma;}}_k,{\left|\right|{\mathrm{\&delta;}}_k\left|\right|}_F\&le;{\mathrm{\&zeta;}}_k\end{array}$

Above formula can be torn open and be write as two independently minimization problems below:

$\begin{array}{c}\underset{{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_k}{\min }{\mathrm{\&beta;}}_k^{*}{\stackrel{\stackrel{\&CenterDot;}{~}}{h}}_k^H\left[{(I+({\tilde{R}}_S\&CircleTimes;B^{-1})R_{H_c})}^{-1}({\tilde{R}}_S\&CircleTimes;B^{-1})\right]{\stackrel{\stackrel{\&CenterDot;}{~}}{h}}_k{\mathrm{\&beta;}}_k\\ s.t.{\left|\right|{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_k\left|\right|}_F\&le;{\mathrm{\&sigma;}}_k\end{array}$

$\begin{array}{c}\underset{{\mathrm{\&delta;}}_k}{\min }{\tilde{h}}_k^H\left[{(I+({\tilde{R}}_S\&CircleTimes;B^{-1})R_{H_c})}^{-1}({\tilde{R}}_S\&CircleTimes;B^{-1})\right]{\tilde{h}}_k\\ s.t.{\left|\right|{\mathrm{\&delta;}}_k\left|\right|}_F\&le;{\mathrm{\&zeta;}}_k\end{array}$

Above-mentioned two minimization problems can be solved by lemma 2 below:

Lemma 2, suppose hermitian matrix $Z=\left[\begin{array}{cc}A& B^H\\ B& C\end{array}\right]$ 's , then and if only if time, , wherein, Δ C=A-B ^hc ^-1b is that the Schur of C in Z mends;

Lemma 2 by reference, above-mentioned two minimization problems can be converted into following SDP problem 1:

Wherein, t is auxiliary variable;

By what obtain from above two formulas with bring in sane optimization problem, consider outside optimization problem;

Solving of step 4, the sane outer optimization problem of waveform

Following proposition is utilized to solve outside optimization problem

Proposition: utilize matrix manipulation, the constraint in sane optimization problem can be equivalent to following LMI:

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

Use lemma 2 in conjunction with above-mentioned proposition, outer optimization problem can be expressed as following SDP problem:

Wherein, X is an auxiliary variable;

After obtaining optimum E, under least square meaning, R _sby following model construction:

Use lemma 2 in conjunction with above-mentioned proposition, above formula can be equivalent to following SDP problem 2:

Step 5, employing alternative manner solve sane waveform optimization problem

Step 5.1, given waveform covariance matrix initial value;

Step 5.2, solve above-mentioned SDP problem 1 to obtain optimum

Step 5.3, solve SDP problem 2 to obtain optimum E;

Step 5.4, return step 5.2 iteration again, until CRB no longer significantly reduces.

Step 6, based on least square method, the waveform covariance matrix that reconstruct is optimum, can obtain R _s.