CN105319545A - MIMO radar waveform design method for improving STAP detection performance - Google Patents

Abstract

The invention belongs to the signal processing field, and relates to an MIMO-OFDM (Orthogonal Frequency Division Multiplexing) radar waveform design method for improving STAP detection performance. Under the constraint of a constant modulus, and based on an output SINR maximization criterion, the invention introduces the optimization problem of maximizing detection performance through designing emission waveforms. In order to solve a complex nonlinear optimization problem, the invention converts the nonlinear optimization problem into a semi-definite programming problem for obtaining an efficient solution based on a DL method, so as to maximize an output SINR, and thereby maximize system detection performance. The simulation result shows that compared with non-relevant emission waveforms, the emission waveforms obtained by the method can substantially increase an output SINR, thereby improving system detection performance. The method can obtain obviously improved detection performance compared with utilizing non-relevant waveforms.

Description

Improve the MIMO-OFDM radar waveform method for designing of STAP detection perform

Technical field

The invention belongs to signal transacting field, relate to a kind of method improving the MIMO-OFDM radar waveform design of STAP detection perform.The method can improve the letter miscellaneous noise ratio of output terminal, by setting up MIMO-OFDM-STAP model, under permanent modular constraint, maximize criterion based on output letter miscellaneous noise ratio (SINR), derived by design transmitted waveform to maximize the optimization problem of detection perform.

Background technology

By the impact of MIMO communication explosion, and radar is break through the demand self limited new theory and new technology, and MIMO radar concept is arisen at the historic moment.Compared with can only sending the phased-array radar of relevant waveform, MIMO radar can utilize multiple transmitter unit to launch almost random waveform.Based on array antenna spacing, MIMO radar system can be divided into following two classes: (1) splits radar, and (2) put radar altogether.The former adopts and splits Transmit-Receive Unit far away and launch desired signal, simultaneously object observing from different perspectives, thus space diversity can be utilized to overcome the hydraulic performance decline because target glint causes.On the contrary, the very near transmitter unit of the latter's service range to increase the virtual aperture of receiving array, thus makes its performance be better than phased-array radar, such as, and parameter identification, flexibly transmit beam direction G-Design.

OFDM (Orthogonalfrequencydivisionmultiplexing, OFDM) signal receives increasing concern as a kind of broadband low probability of intercept radar waveform.OFDM radar utilizes that multiple orthogonal subcarrier is parallel to be detected, thus effectively can resist the frequency selective fading that multipath transmisstion causes, and improves the noiseproof feature of system.OFDM and MIMO combine with technique is got up, the advantage of MIMO and OFDM can be given full play to, thus the detection perform to target can be significantly improved.

Space-time adaptive process (STAP) grew up from eighties of last century the early 1990s, for the technology processed airborne radar (airborneradar) data.STAP technology all has a wide range of applications in military and civilian, and such as, geology is monitored, early warning, Ground moving targets detection (GMTI), moving target detect (MTI), region investigation etc.For traditional phased-array radar, STAP fundamental research is quite ripe.Many for improving STAP complicacy and constringent algorithm was suggested already.These algorithms just can be applied to MIMO radar past slightly amendment.D.W.Bliss and K.W.Forsythe proposes the concept of MIMO-STAP.Because MIMO-STAP is the new ideas just proposed recent years, relevant document is also fewer.The people such as C.Y.Chen propose a kind of method of estimation clutter subspace newly, and the method make use of the geometrical feature of problem and the special block diagonal arrangement of interference covariance matrix, thus significantly can reduce computation complexity compared with universe adaptive approach.Under the scene of general transmitted waveform, WangG. etc. have extensively studied the clutter order of MIMO-STAP and the relation of transmitted waveform, and provide the criterion determining clutter order.

Although B.Friedlander and C.Y.Chen improves the detection perform of MIMO radar by optimizing transmitted waveform, but this thought is not applied in MIMO-OFDM-STAP.And although WangG. has carried out comparatively deep research to MIMO-STAP, but be also only limitted to the data processing of receiving end.For this problem, the present invention considers to improve MIMO-STAP detection perform by waveform optimization.

Summary of the invention

The object of the invention is to the deficiency overcoming above-mentioned prior art, propose a kind of waveform optimization method based on diagonal angle loading technique, this optimization problem is solved by positive semidefinite relaxing techniques, this nonlinear optimal problem is converted into the Semidefinite Programming that can obtain Efficient Solution, thus maximize output SINR, and then maximize systems axiol-ogy performance.Realizing basic ideas of the present invention is first set up waveform optimization model, then based on diagonal angle loading technique, this Optimized model is converted into Semidefinite Programming, finally introduces convex optimized algorithm and solves.

Technical scheme of the present invention is: the MIMO-OFDM radar waveform method for designing improving STAP detection perform, the steps include:

One, MIMO-OFDM-STAP system modelling

(1) MIMO-OFDM-STAP Received signal strength describes

MIMO-OFDM radar n-th receives array element, and the reception data in l PRI can be expressed as:

$\begin{array}{c}{x}_{n,l}=\underset{m=0}{\overset{M-1}{\Σ}}{\mathrm{\ρ}}_t{s}_m{e}^{j\[(2\mathrm{\π}/\mathrm{\λ})\left({\mathrm{sin\θ}}_t\right(d_{r}n+d_{t}m+2vt)+2v_{t}t)+2{\mathrm{\πf}}_m\]}+\\ {\∫}_0^{2\mathrm{\π}}\underset{m=0}{\overset{M-1}{\Σ}}{\mathrm{\ρ}}_{\left(\mathrm{\θ}\right)}{s}_m{e}^{j\[(2\mathrm{\π}/\mathrm{\λ})\left(\sin \mathrm{\θ}\right(d_{r}n+d_{t}m+2vt\left)\right)++2{\mathrm{\πf}}_m\]}d\mathrm{\θ}+z_{n,l}\end{array}$

In formula, be m the discrete form launching the complex baseband signal that array element is launched in each PRI, a _mfor corresponding signal amplitude, k is waveform sampling number, and f _m=f ₀+ m Δ f, f ₀for signal carrier frequency, Δ f is frequency interval, meets T Δ f=1; ρ _tbe respectively the complex magnitude of considered rang ring internal object with ρ (θ) and be positioned at the clutter refection coefficient of θ; V, v _trepresent the translational speed of Texas tower and target respectively, λ is waveform centre wavelength; In addition, represent the interference that the n-th reception array element receives in l PRI and noise.

If the clutter echo in target range unit can be modeled as the superposition of some independent clutter blocks, and Received signal strength carries out down-converted, above formula can be expressed as again:

$X_l={\mathrm{\ρ}}_t{e}^{j2{\mathrm{\πf}}_{D}l}{\mathrm{ab}}^{T}S+\underset{i=0}{\overset{N_C-1}{\Σ}}{\mathrm{\ρ}}_i{e}^{j2{\mathrm{\π\βf}}_{s,i}l}{a}_i{b_i}^{T}S+Z_l$

Wherein, N _c(N _c>>NML) clutter ring number of samples is represented, S=[s ₁, s ₂..., s _m] ^trepresent the signal matrix in each PRI.

$b={\[1,e^{j2{\mathrm{\π\γf}}_s},...,e^{j2\mathrm{\π}(M-1){\mathrm{\γf}}_s}\]}^T$ With $b_i={\[1,e^{j2{\mathrm{\π\γf}}_{s,i}},...,e^{j2\mathrm{\π}(M-1){\mathrm{\γf}}_{s,i}}\]}^T$ Represent target respectively and be positioned at θ _ithe transmitting steering vector of clutter.f _s＝d _Rsinθ _t/λ,f _D＝2(vsinθ _t+v _t)T/λ,f _si＝d _Rsinθ _i/λ,γ＝d _T/d _R,β＝2vT/d _R。 $a={\[1,e^{j2{\mathrm{\πf}}_s},...,e^{j2\mathrm{\π}(N-1)f_s}\]}^T$ With $a_i={\[1,e^{j2{\mathrm{\πf}}_{s,i}},...,e^{j2\mathrm{\π}(N-1)f_{s,i}})}^T$ Represent target respectively and be positioned at θ _ithe reception steering vector of clutter, can suppose Z _lrow are independent identically distributed round symmetric complex random vectors, its average is 0, and covariance matrix is unknown matrix

(2) snap statement during sky in rang ring interested

For obtaining the statistic for target detection, we utilize S ^h(SS ^h) ^-1/2as matched filter, again then the output of corresponding vector quantization coupling can be expressed as follows:

${\tilde{x}}_l={\mathrm{\ρ}}_t{e}^{j2{\mathrm{\πf}}_{D}l}(\mathrm{\Φ}\⊗I_N)(b\⊗a)+\underset{i=0}{\overset{N_C-1}{\Σ}}{\mathrm{\ρ}}_i{e}^{j2{\mathrm{\π\βf}}_{s,i}l}(\mathrm{\Φ}\⊗I_N)(b_i\⊗a_i)+vec\left({\tilde{Z}}_l\right)$

Wherein, Φ=SS ^h(SS ^h) ^-1/2=diag{|a ₁|| a ₂| ... | a _m|, diag{} represents diagonal matrix, ${\tilde{Z}}_l=Z_l{S}^H{\left({\mathrm{SS}}^H\right)}^{-1/2}.$

By above formula we can obtain in interested rang ring total empty time snap be:

$\begin{array}{c}{X}_C={\mathrm{\ρ}}_t{U}_D\⊗\left(\right(\mathrm{\Φ}\⊗I_N\left)\right(b\⊗a\left)\right)+\\ \underset{i=0}{\overset{N_C-1}{\Σ}}{\mathrm{\ρ}}_t{U}_{D,i}\⊗\left(\right(\mathrm{\Φ}\⊗I_N\left)\right(b_i\⊗a_i\left)\right)+I_L\⊗vec\left({\tilde{Z}}_l\right)\\ ={\mathrm{\ρ}}_t\left(I_L\⊗\mathrm{\Φ}\⊗I_N\right)\left(U_D\⊗b\⊗a\right)+\\ \left(I_L\⊗\mathrm{\Φ}\⊗I_N\right)\underset{i=0}{\overset{N_C-1}{\Σ}}{\mathrm{\ρ}}_i\left(U_{D,i}\⊗b_i\⊗a_i\right)+I_L\⊗vec\left({\tilde{Z}}_l\right)\end{array}$

Wherein, $u_D={[1,e^{{j2\mathrm{\πf}}_D},...,e^{j2\mathrm{\π}(L-1)f_D}]}^T$ With $u_{D,i}={[1,e^{{j2\mathrm{\πf}}_{D,i}},...,e^{j2\mathrm{\π}(L-1)f_{D,i}}]}^T$ Represent target respectively and be positioned at θ _idoppler's steering vector of clutter.I _nrepresent the unit matrix of N × N.

Two, SINR statement is exported

(1) based on MVDR criterion, optimum SINR statement

Based on the undistorted criterion of minimum variance (MVDR), can obtain optimum output SINR can be expressed as:

$SINR={{\mathrm{\ρ}}_t}^2{\[(I_L\⊗\mathrm{\Φ}\⊗I_N)(U_D\⊗b\⊗a)\]}^H{R}_{i+n}^{-1}\[(I_L\⊗\mathrm{\Φ}\⊗I_N)(U_D\⊗b\⊗a)\]$

In formula,

$\begin{array}{c}{R}_{i+n}=E\[\left(\right(I_L\⊗\mathrm{\Φ}\⊗I_N\left)\underset{i=0}{\overset{N_C-1}{\Σ}}{\mathrm{\ρ}}_i\right(U_{D,i}\⊗b_i\⊗a_i)+I_L\⊗vec({\tilde{Z}}_l\left)\right)\\ {\left(\right(I_L\⊗\mathrm{\Φ}\⊗I_N\left)\underset{i=0}{\overset{N_C-1}{\Σ}}{\mathrm{\ρ}}_i\right(U_{D,i}\⊗b_i\⊗a_i)+I_L\⊗vec({\tilde{Z}}_l\left)\right)}^H\]\end{array}$

(2) clutter Gaussian distribution, and with the uncorrelated condition of interference under simplify the SINR of output

Suppose ρ _iindependent same distribution, and obedience average is 0, variance is gaussian distribution, under clutter and the incoherent hypothesis of interference plus noise item,

Export SINR and can be reduced to following expression:

$SINR=|{\mathrm{\ρ}}_t{|}^2{v_t}^H{(I+\tilde{A}{R}_C)}^{-1}\tilde{A}{v}_t$

Wherein, $V={\[v_1,v_2,...,v_{N_C}\]}^H,v_i=U_{D,i}\⊗b_i\⊗a_i,\mathrm{\Ξ}=diag({{\mathrm{\σ}}_1}^2,{{\mathrm{\σ}}_2}^2,...,{{\mathrm{\σ}}_{N_C}}^2),A=I_L\⊗\mathrm{\Φ}\⊗I_N,$

Three, under permanent modular constraint, waveform optimization problem is stated

It should be noted that, in engineering, state of saturation is operated in play its maximum efficiency in order to radar transmitter can be made, simultaneously for avoiding amplifier nonlinearity to make transmitted waveform distortion, usually require that radar emission waveform has constant modulus property, for this OFDM-MIMO radar, this requirement also must meet.

Can be obtained by above analysis, under total emission power and permanent modular constraint, the waveform optimization problem maximizing MIMO-OFDM-STAP detection probability can be expressed as:

$\begin{array}{cc}\underset{a_m}{max}& |{\mathrm{\ρ}}_t{|}^2{v_t}^H{(I+\tilde{A}{R}_C)}^{-1}\tilde{A}{v}_t\end{array}$

s.t.|a _m|＝C _m

$\underset{m=1}{\overset{M}{\Σ}}{a}_m^2=P$

||a _m|| ²≥0

In formula, C _mbe the amplitude of m transmitted waveform, second is constrained to total emission power constraint, and the 3rd constraint shows that the emissive power of each transmitting array element is more than or equal to zero, P and represents total emission power.

In order to can Efficient Solution above formula optimization problem, first we simplify this problem.Notice, matrix in comprise second and third constraint comprises this equally.Because first is constrained to the permanent modular constraint transmitted, thus can relax as a square constraint.Thus, this optimization problem can be expressed as again:

$\begin{array}{cc}\underset{a_m^2}{max}& |{\mathrm{\ρ}}_t{|}^2{v_t}^H{(I+\tilde{A}{R}_C)}^{-1}\tilde{A}{v}_t\end{array}$

$\begin{array}{cc}s.t.& a_m^2=D_m\end{array}$

$\underset{m=1}{\overset{M}{\Σ}}{a}_m^2=P$

||a _m|| ²≥0

Four, based on the waveform optimization problem solving of DL method

(1) based on the positive definite R of DL method _c

This optimization problem contains permanent modular constraint, is not obviously thus convex problem, is easily absorbed in locally optimal solution when solving globally optimal solution.Meanwhile, due to we cannot determine character.Objective function in optimization problem is about optimized variable a _mthe nonlinear function of more complicated.Thus, for this optimization problem, convex optimization method can not be utilized separate.And if utilize other numerical methods, such as gradient method, just may produce the problem of convergence.For solving this nonlinear optimal problem, we are to R _cutilize the diagonal angle loading method being usually applied to robust ada-ptive beamformer, can obtain

ε << λ in formula _max(R _c) i.e. so-called load factor (loadingfactor), λ _maxthe eigenvalue of maximum of () representing matrix.

(2) based on positive definite R _csimplify and export SINR

Will substitute into and export SINR expression formula, and utilize topology, can obtain

$\begin{array}{c}SINR=|{\mathrm{\&rho;}}_t{|}^2{v}_t^H(I-{\left(I+\tilde{A}{\tilde{R}}_C\right)}^{-1}){\tilde{R}}_C^{-1}{v}_t\\ =|{\mathrm{\&rho;}}_t{|}^2{v}_t^H{\tilde{R}}_C^{-1}{v}_t-|{\mathrm{\&rho;}}_t{|}^2{v}_t^H{({\tilde{R}}_C+{\tilde{R}}_C\tilde{A}{\tilde{R}}_C)}^{-1}{v}_t\end{array}$

(3) based on convex Optimization Solution waveform optimization problem

Based on above-mentioned discussion, waveform optimization problem can be expressed equivalently as following SDP problem:

$\begin{array}{cc}\underset{a_m^2,t}{min}& t\end{array}$

$a_m^2=D_m$

$\underset{m=1}{\overset{M}{\&Sigma;}}{a}_m^2=P$

||a _m|| ²≥0

In formula, t is auxiliary optimized variable.The optimum solution of above formula effectively obtains by CVX Optimization Toolbox.

The present invention compared with prior art has the following advantages:

The first, the detection perform of the present invention by utilizing transmitting terminal degree of freedom to improve MIMO-OFDM-STAP.By design transmitted waveform, the output SINR of system is improved, thus improves systems axiol-ogy performance.

The second, the present invention is based on diagonal angle loading method, the nonlinear waveform optimization problem of complexity is converted into semi definite programming problem, thus the Optimization Toolbox of comparative maturity can be utilized to solve.

Accompanying drawing explanation

Fig. 1 is MIMO-OFDM-STAP model;

Fig. 2 is the process flow diagram that the present invention realizes;

Fig. 3 is that of the present invention output believes the Changing Pattern figure of miscellaneous noise ratio along with the signal to noise ratio (S/N ratio) of array;

Fig. 4 is that of the present invention output believes the Changing Pattern figure of miscellaneous noise ratio along with miscellaneous noise ratio;

Fig. 5 is the figure of the output signal-to-noise ratio of the present invention when initial angle exists evaluated error and array signal to noise ratio (S/N ratio) is 30dB;

Effect of the present invention further illustrates by following emulation:

Embodiment

Below in conjunction with accompanying drawing 2, performing step of the present invention is described in further detail:

One, MIMO-OFDM-STAP system modelling

(1) MIMO-OFDM-STAP Received signal strength describes

MIMO-OFDM radar n-th receives array element, and the reception data in l PRI can be expressed as:

$\begin{array}{c}{x}_{n,l}=\underset{m=0}{\overset{M-1}{\&Sigma;}}{\mathrm{\&rho;}}_t{s}_m{e}^{j\&lsqb;(2\mathrm{\&pi;}/\mathrm{\&lambda;})\left({\mathrm{sin\&theta;}}_t\right(d_{r}n+d_{t}m+2vt)+2v_{t}t)+2{\mathrm{\&pi;f}}_m\&rsqb;}+\\ {\&Integral;}_0^{2\mathrm{\&pi;}}\underset{m=0}{\overset{M-1}{\&Sigma;}}{\mathrm{\&rho;}}_{\left(\mathrm{\&theta;}\right)}{s}_m{e}^{j\&lsqb;(2\mathrm{\&pi;}/\mathrm{\&lambda;})\left(\sin \mathrm{\&theta;}\right(d_{r}n+d_{t}m+2vt\left)\right)++2{\mathrm{\&pi;f}}_m\&rsqb;}d\mathrm{\&theta;}+z_{n,l}\end{array}$

In formula, be m the discrete form launching the complex baseband signal that array element is launched in each PRI, K is waveform sampling number, and a _mfor corresponding signal amplitude, f _m=f ₀+ m Δ f, f ₀for signal carrier frequency, Δ f is frequency interval, meets T Δ f=1; ρ _tbe respectively the complex magnitude of considered rang ring internal object with ρ (θ) and be positioned at the clutter refection coefficient of θ; V, v _trepresent the translational speed of Texas tower and target respectively, λ is waveform centre wavelength; In addition, represent the interference that the n-th reception array element receives in l PRI and noise.

If the clutter echo in target range unit can be modeled as the superposition of some independent clutter blocks, and Received signal strength carries out down-converted, above formula can be expressed as again:

$X_l={\mathrm{\&rho;}}_t{e}^{j2{\mathrm{\&pi;f}}_{D}l}{\mathrm{ab}}^{T}S+\underset{i=0}{\overset{N_C-1}{\&Sigma;}}{\mathrm{\&rho;}}_i{e}^{j2{\mathrm{\&pi;\&beta;f}}_{s,i}l}{a}_i{b_i}^{T}S+Z_l$

Wherein, N _c(N _c>>NML) clutter ring number of samples is represented, S=[s ₁, s ₂..., s _m] ^trepresent the signal matrix in each PRI.

$b={\&lsqb;1,e^{j2{\mathrm{\&pi;\&gamma;f}}_s},...,e^{j2\mathrm{\&pi;}(M-1){\mathrm{\&gamma;f}}_s}\&rsqb;}^T$ With $b_i={\&lsqb;1,e^{j2{\mathrm{\&pi;\&gamma;f}}_{s,i}},...,e^{j2\mathrm{\&pi;}(M-1){\mathrm{\&gamma;f}}_{s,i}}\&rsqb;}^T$ Be respectively target and be positioned at θ _ithe transmitting steering vector of clutter.f _s＝d _Rsinθ _t/λ,f _D＝2(vsinθ _t+v _t)T/λ,f _si＝d _Rsinθ _i/λ,γ＝d _T/d _R,β＝2vT/d _R。

$a={\&lsqb;1,e^{j2{\mathrm{\&pi;\&gamma;f}}_s},...,e^{j2\mathrm{\&pi;}(N-1)f_s}\&rsqb;}^T$ With $a_i={\&lsqb;1,e^{j2{\mathrm{\&pi;f}}_{s,i}},...,e^{j2\mathrm{\&pi;}(N-1)f_{s,i}})}^T$ Represent target respectively and be positioned at θ _ithe reception steering vector of clutter, can suppose Z _lrow are independent identically distributed round symmetric complex random vectors, its average is 0, and covariance matrix is unknown matrix

(2) snap statement during sky in rang ring interested

For obtaining the statistic for target detection, we utilize S ^h(SS ^h) ^-1/2as matched filter, again then the output of corresponding vector quantization coupling can be expressed as follows:

${\tilde{x}}_l={\mathrm{\&rho;}}_t{e}^{j2{\mathrm{\&pi;f}}_{D}l}(\mathrm{\&Phi;}\&CircleTimes;I_N)(b\&CircleTimes;a)+\underset{i=0}{\overset{N_C-1}{\&Sigma;}}{\mathrm{\&rho;}}_i{e}^{j2{\mathrm{\&pi;\&beta;f}}_{s,i}l}(\mathrm{\&Phi;}\&CircleTimes;I_N)(b_i\&CircleTimes;a_i)+vec\left({\tilde{Z}}_l\right)$

Wherein, Φ=SS ^h(SS ^h) ^-1/2=diag{|a ₁|| a ₂| ... | a _m|, diag{} represents diagonal matrix, ${\tilde{Z}}_l=Z_l{S}^H{\left({\mathrm{SS}}^H\right)}^{-1/2}.$

By above formula we can obtain in interested rang ring total empty time snap be:

$\begin{array}{c}{X}_C={\mathrm{\&rho;}}_t{U}_D\&CircleTimes;\left(\right(\mathrm{\&Phi;}\&CircleTimes;I_N\left)\right(b\&CircleTimes;a\left)\right)+\\ \underset{i=0}{\overset{N_C-1}{\&Sigma;}}{\mathrm{\&rho;}}_t{U}_{D,i}\&CircleTimes;\left(\right(\mathrm{\&Phi;}\&CircleTimes;I_N\left)\right(b_i\&CircleTimes;a_i\left)\right)+I_L\&CircleTimes;vec\left({\tilde{Z}}_l\right)\\ ={\mathrm{\&rho;}}_t\left(I_L\&CircleTimes;\mathrm{\&Phi;}\&CircleTimes;I_N\right)\left(U_D\&CircleTimes;b\&CircleTimes;a\right)+\\ \left(I_L\&CircleTimes;\mathrm{\&Phi;}\&CircleTimes;I_N\right)\underset{i=0}{\overset{N_C-1}{\&Sigma;}}{\mathrm{\&rho;}}_i\left(U_{D,i}\&CircleTimes;b_i\&CircleTimes;a_i\right)+I_L\&CircleTimes;vec\left({\tilde{Z}}_l\right)\end{array}$

Wherein, $u_D={\&lsqb;1,e^{j2{\mathrm{\&pi;f}}_D},...,e^{j2\mathrm{\&pi;}(L-1)f_D}\&rsqb;}^T$ With $u_{D,i}={\&lsqb;1,e^{j2{\mathrm{\&pi;f}}_{D,i}},...,e^{j2\mathrm{\&pi;}(L-1)f_{D,i}})}^T$ Represent target respectively and be positioned at θ _idoppler's steering vector of clutter.I _nrepresent the unit matrix of N × N.

Two, SINR statement is exported

(1) based on MVDR criterion, optimum SINR statement

Based on the undistorted criterion of minimum variance (MVDR), can obtain optimum output SINR can be expressed as:

$SINR={{\mathrm{\&rho;}}_t}^2{\&lsqb;(I_L\&CircleTimes;\mathrm{\&Phi;}\&CircleTimes;I_N)(U_D\&CircleTimes;b\&CircleTimes;a)\&rsqb;}^H{R}_{i+n}^{-1}\&lsqb;(I_L\&CircleTimes;\mathrm{\&Phi;}\&CircleTimes;I_N)(U_D\&CircleTimes;b\&CircleTimes;a)\&rsqb;$

In formula,

$\begin{array}{c}{R}_{i+n}=E\&lsqb;\left(\right(I_L\&CircleTimes;\mathrm{\&Phi;}\&CircleTimes;I_N\left)\underset{i=0}{\overset{N_C-1}{\&Sigma;}}{\mathrm{\&rho;}}_i\right(U_{D,i}\&CircleTimes;b_i\&CircleTimes;a_i)+I_L\&CircleTimes;vec({\tilde{Z}}_l\left)\right)\\ {\left(\right(I_L\&CircleTimes;\mathrm{\&Phi;}\&CircleTimes;I_N\left)\underset{i=0}{\overset{N_C-1}{\&Sigma;}}{\mathrm{\&rho;}}_i\right(U_{D,i}\&CircleTimes;b_i\&CircleTimes;a_i)+I_L\&CircleTimes;vec({\tilde{Z}}_l\left)\right)}^H\&rsqb;\end{array}$

(2) clutter Gaussian distribution, and with the uncorrelated condition of interference under simplify the SINR of output

Suppose ρ _iindependent same distribution, and obedience average is 0, variance is gaussian distribution, under clutter and the incoherent hypothesis of interference plus noise item,

Export SINR and can be reduced to following expression:

$SINR=|{\mathrm{\&rho;}}_t{|}^2{v_t}^H{(I+\tilde{A}{R}_C)}^{-1}\tilde{A}{v}_t$

Wherein,

$v_i=U_{D,i}\&CircleTimes;b_i\&CircleTimes;a_i,\mathrm{\&Xi;}=\mathrm{diag}({{\mathrm{\&sigma;}}_1}^2,{{\mathrm{\&sigma;}}_2}^2,...,{{\mathrm{\&sigma;}}_{N_C}}^2).$

Three, under permanent modular constraint, waveform optimization problem is stated

It should be noted that, in engineering, state of saturation is operated in play its maximum efficiency in order to radar transmitter can be made, simultaneously for avoiding amplifier nonlinearity to make transmitted waveform distortion, usually require that radar emission waveform has constant modulus property, for this OFDM-MIMO radar, this requirement also must meet.

Can be obtained by above analysis, under total emission power and permanent modular constraint, the waveform optimization problem maximizing MIMO-OFDM-STAP detection probability can be expressed as:

$\begin{array}{cc}\underset{a_m}{max}& |{\mathrm{\&rho;}}_t{|}^2{v_t}^H{(I+\tilde{A}{R}_C)}^{-1}\tilde{A}{v}_t\end{array}$

s.t.|a _m|＝C _m

$\underset{m=1}{\overset{M}{\&Sigma;}}{a}_m^2=P---\left(1\right)$

||a _m|| ²≥0

In formula, C _mbe the amplitude of m transmitted waveform, second is constrained to total emission power constraint, and the 3rd constraint shows that the emissive power of each transmitting array element is more than or equal to zero, P and represents total emission power.

In order to can Efficient Solution above formula optimization problem, first we simplify this problem.Notice, matrix in comprise second and third constraint comprises this equally.Because first is constrained to the permanent modular constraint transmitted, thus can relax as a square constraint.Thus, this optimization problem can be expressed as again:

$\begin{array}{cc}\underset{a_m^2}{max}& |{\mathrm{\&rho;}}_t{|}^2{v_t}^H{(I+\tilde{A}{R}_C)}^{-1}\tilde{A}{v}_t\end{array}$

$\begin{array}{cc}s.t.& a_m^2=D_m\end{array}$

$\underset{m=1}{\overset{M}{\&Sigma;}}{a}_m^2=P$

||a _m|| ²≥0

Four, based on the waveform optimization problem solving of DL method

(1) based on the positive definite R of DL method _c

This optimization problem contains permanent modular constraint, is not obviously thus convex problem, is easily absorbed in locally optimal solution when solving globally optimal solution.Meanwhile, due to we cannot determine character.Objective function in optimization problem is about optimized variable a _mthe nonlinear function of more complicated.Thus, for this optimization problem, convex optimization method can not be utilized separate.And if utilize other numerical methods, such as gradient method, just may produce the problem of convergence.For solving this nonlinear optimal problem, we are to R _cutilize the diagonal angle loading method being usually applied to robust ada-ptive beamformer, can obtain

${\tilde{R}}_C=R_C+\mathrm{\&epsiv;}I>0$

ε << λ in formula _max(R _c) i.e. so-called load factor (loadingfactor), λ _maxthe eigenvalue of maximum of () representing matrix.

(2) based on positive definite R _csimplify and export SINR

Will substitute into and export SINR expression formula, and utilize topology, can obtain

$\begin{array}{c}SINR=|{\mathrm{\&rho;}}_t{|}^2{v}_t^H(I-{\left(I+\tilde{A}{\tilde{R}}_C\right)}^{-1}){\tilde{R}}_C^{-1}{v}_t\\ =|{\mathrm{\&rho;}}_t{|}^2{v}_t^H{\tilde{R}}_C^{-1}{v}_t-|{\mathrm{\&rho;}}_t{|}^2{v}_t^H{({\tilde{R}}_C+{\tilde{R}}_C\tilde{A}{\tilde{R}}_C)}^{-1}{v}_t\end{array}$

(3) based on convex Optimization Solution waveform optimization problem

Based on above-mentioned discussion, waveform optimization problem can be expressed equivalently as following SDP problem:

$\begin{array}{cc}\underset{a_m^2,t}{min}& t\end{array}$

$a_m^2=D_m$

$\underset{m=1}{\overset{M}{\&Sigma;}}{a}_m^2=P$

||a _m|| ²≥0

In formula, t is auxiliary optimized variable.The optimum solution of above formula effectively obtains by CVX Optimization Toolbox.

Effect of the present invention further illustrates by following emulation:

Simulated conditions: MIMO radar is 42 and receives, and reception array element distance is half-wavelength, and launching array element distance is 2 times of wavelength, and umber of pulse is 3, adopt two MIMO radar to detect target, be respectively A (0.5,0.5), B (1.5,0.5), array signal to noise ratio (S/N ratio) is defined as wherein, P refers to total emissive power, the variance of the white hot noise attached by finger, the signal to noise ratio (S/N ratio) of array is from 10 to 50 decibels of changes, and clutter block number is 10000, and miscellaneous noise ratio is from 10 to 50 decibels of changes, and interference noise ratio is 60 decibels, and sampling number is 256.This document assumes that have target in the direction of 4 °, the modeling of clutter uses discrete point, and its RCS is modeled as independent identically distributed gaussian random variable vector, and average is zero, and variance is and hypothesis is fixed on coherent processing inteval.In emulation, algorithm in this paper and uncorrelated waveform are contrasted, the improvement situation of signal to noise ratio (S/N ratio) can be seen.

Emulation content:

Emulation A: initial angle estimates the situation that there is not error

Can be obtained by Fig. 3, output signal-to-noise ratio increases along with the increase of array signal to noise ratio (S/N ratio), radar B is compared with radar A, the amplitude that radar B increases is larger, also can see simultaneously, the signal to noise ratio (S/N ratio) that institute's extracting method obtains is higher than the signal to noise ratio (S/N ratio) obtained under uncorrelated waveform, and clearly under institute's extracting method, the ability of detections of radar target is stronger.

Can be obtained by Fig. 4, output signal-to-noise ratio reduces along with the increase of array signal to noise ratio (S/N ratio), radar B is compared with radar A, the amplitude that radar B reduces is larger, also can see simultaneously, the signal to noise ratio (S/N ratio) that institute's extracting method obtains is higher than the signal to noise ratio (S/N ratio) obtained under uncorrelated waveform, and clearly under institute's extracting method, the ability of detections of radar target is stronger.

Emulation B: initial angle estimates the situation that there is error

As can be seen from Figure 5, when initial angle exists error, output signal-to-noise ratio is very responsive to angle change, along with angle changes a lot.The Waveform Design illustrating herein is not a sane waveform design method.

In sum, the present invention, by optimizing transmitted waveform, improves the detection perform of MIMO-OFDM-STAP.Under permanent modular constraint, based on diagonal angle loading technique, the nonlinear waveform optimization problem of this complexity is relaxed as SDP problem, thus convex optimization method can be utilized to obtain Efficient Solution.Thus, institute of the present invention extracting method can provide solid theory for improving systems axiol-ogy performance by design transmitted waveform in engineer applied and realize foundation.

Claims (1)

1. improve the MIMO-OFDM radar waveform method for designing of STAP detection perform, comprise the steps:

One, MIMO-OFDM-STAP system modelling

(1) MIMO-OFDM-STAP Received signal strength describes

MIMO-OFDM radar n-th receives array element, and the reception data in l PRI can be expressed as:

$\begin{array}{c}{x}_{n,l}=\underset{m=0}{\overset{M-1}{\&Sigma;}}{\mathrm{\&rho;}}_t{s}_m{e}^{j\&lsqb;\left(2\mathrm{\&pi;}/\mathrm{\&lambda;}\right)\left({\mathrm{sin\&theta;}}_t\left(d_{r}n+d_{t}m+2vt\right)+2v_{t}t\right)+2{\mathrm{\&pi;f}}_m\&rsqb;}+\\ {\&Integral;}_0^{2\mathrm{\&pi;}}\underset{m=0}{\overset{M-1}{\&Sigma;}}{\mathrm{\&rho;}}_{\left(\mathrm{\&theta;}\right)}{s}_m{e}^{j\&lsqb;\left(2\mathrm{\&pi;}/\mathrm{\&lambda;}\right)\left(\sin \mathrm{\&theta;}\left(d_{r}n+d_{t}m+2vt\right)\right)++2{\mathrm{\&pi;f}}_m\&rsqb;}d\mathrm{\&theta;}+z_{n,l}\end{array}$

In formula, be m the discrete form launching the complex baseband signal that array element is launched in each PRI, a _mfor corresponding signal amplitude, k is waveform sampling number, and f ₀for signal carrier frequency, Δ f is frequency interval, meets T Δ f=1; ρ _tbe respectively the complex magnitude of considered rang ring internal object with ρ (θ) and be positioned at the clutter refection coefficient of θ; V, v _trepresent the translational speed of Texas tower and target respectively, λ is waveform centre wavelength; In addition, represent the interference that the n-th reception array element receives in l PRI and noise;

If the clutter echo in target range unit can be modeled as the superposition of some independent clutter blocks, and Received signal strength carries out down-converted, above formula can be expressed as again:

$X_l={\mathrm{\&rho;}}_t{e}^{j2{\mathrm{\&pi;f}}_{D}l}{\mathrm{ab}}^{T}S+\underset{i=0}{\overset{N_C-1}{\&Sigma;}}{\mathrm{\&rho;}}_i{e}^{j2{\mathrm{\&pi;\&beta;f}}_{s,i}l}{a}_i{b_i}^{T}S+Z_l$

Wherein, N _c(N _c> > NML) represent clutter ring number of samples, S=[s ₁, s ₂..., s _m] ^trepresent the signal matrix in each PRI. $b={\&lsqb;1,e^{j2{\mathrm{\&pi;\&gamma;f}}_s},...,e^{j2\mathrm{\&pi;}\left(M-1\right){\mathrm{\&gamma;f}}_s}\&rsqb;}^T$ With $b_i={\&lsqb;1,e^{j2{\mathrm{\&pi;\&gamma;f}}_{s,i}},...,e^{j2\mathrm{\&pi;}\left(M-1\right){\mathrm{\&gamma;f}}_{s,i}}\&rsqb;}^T$ Represent target respectively and be positioned at θ _ithe transmitting steering vector of clutter,

f _s＝d _Rsinθ _t/λ,f _D＝2(vsinθ _t+v _t)T/λ,f _si＝d _Rsinθ _i/λ,γ＝d _T/d _R,β＝2vT/d _R， $a={\&lsqb;1,e^{j2{\mathrm{\&pi;f}}_s},...,e^{j2\mathrm{\&pi;}\left(N-1\right)f_s}\&rsqb;}^T$ With $a_i={\&lsqb;1,e^{j2{\mathrm{\&pi;f}}_{s,i}},...,e^{j2\mathrm{\&pi;}\left(N-1\right)f_{s,i}}\&rsqb;}^T$

Represent target respectively and be positioned at θ _ithe reception steering vector of clutter, can suppose Z _lrow are independent identically distributed round symmetric complex random vectors, its average is 0, and covariance matrix is unknown matrix

(2) snap statement during sky in rang ring interested

For obtaining the statistic for target detection, utilize S ^h(SS ^h) ^-1/2as matched filter, again then the output of corresponding vector quantization coupling can be expressed as follows:

${\tilde{x}}_l={\mathrm{\&rho;}}_t{e}^{j2{\mathrm{\&pi;f}}_{D}l}(\mathrm{\&Phi;}\&CircleTimes;I_N)(b\&CircleTimes;a)+\underset{i=0}{\overset{N_C-1}{\&Sigma;}}{\mathrm{\&rho;}}_i{e}^{j2{\mathrm{\&pi;\&beta;f}}_{s,i}l}(\mathrm{\&Phi;}\&CircleTimes;I_N)(b_i\&CircleTimes;a_i)+vec\left({\tilde{Z}}_l\right)$

Wherein, Φ=SS ^h(SS ^h) ^-1/2=diag{|a ₁|| a ₂| ... | a _m|, diag{} represents diagonal matrix, ${\tilde{Z}}_l=Z_l{S}^H{\left({\mathrm{SS}}^H\right)}^{-1/2},$

Can be obtained fom the above equation in interested rang ring total empty time snap be:

$\begin{array}{c}{X}_C={\mathrm{\&rho;}}_t{U}_D\&CircleTimes;\left(\left(\mathrm{\&Phi;}\&CircleTimes;I_N\right)\left(b\&CircleTimes;a\right)\right)+\\ \underset{i=0}{\overset{N_C-1}{\&Sigma;}}{\mathrm{\&rho;}}_i{U}_{D,i}\&CircleTimes;\left(\left(\mathrm{\&Phi;}\&CircleTimes;I_N\right)\left(b_i\&CircleTimes;a_i\right)\right)+I_L\&CircleTimes;vec\left({\tilde{Z}}_l\right)\\ ={\mathrm{\&rho;}}_t\left(I_L\&CircleTimes;\mathrm{\&Phi;}\&CircleTimes;I_N\right)\left(U_D\&CircleTimes;b\&CircleTimes;a\right)+\\ \left(I_L\&CircleTimes;\mathrm{\&Phi;}\&CircleTimes;I_N\right)\underset{i=0}{\overset{N_C-1}{\&Sigma;}}{\mathrm{\&rho;}}_i\left(U_{D,i}\&CircleTimes;b_i\&CircleTimes;a_i\right)+I_L\&CircleTimes;vec\left({\tilde{Z}}_l\right)\end{array}$

Wherein, $u_D={\&lsqb;1,e^{j2{\mathrm{\&pi;f}}_D},...,e^{j2\mathrm{\&pi;}\left(L-1\right)f_D}\&rsqb;}^T$ With $u_{D,i}={\&lsqb;1,e^{j2{\mathrm{\&pi;f}}_{D,i}},...,e^{j2\mathrm{\&pi;}\left(L-1\right)f_{D,i}}\&rsqb;}^T$ Represent target respectively and be positioned at θ _idoppler's steering vector of clutter, I _nrepresent the unit matrix of N × N;

Two, SINR statement is exported

(1) based on MVDR criterion, optimum SINR statement

Based on the undistorted criterion of minimum variance (MVDR), can obtain optimum output SINR can be expressed as:

$SINR={{\mathrm{\&rho;}}_t}^2{\&lsqb;(I_L\&CircleTimes;\mathrm{\&Phi;}\&CircleTimes;I_N)(U_D\&CircleTimes;b\&CircleTimes;a)\&rsqb;}^H{R}_{i+n}^{-1}\&lsqb;(I_L\&CircleTimes;\mathrm{\&Phi;}\&CircleTimes;I_N)(U_D\&CircleTimes;b\&CircleTimes;a)\&rsqb;$

In formula,

$\begin{array}{c}{R}_{i+n}=E\&lsqb;\left(\left(I_L\&CircleTimes;\mathrm{\&Phi;}\&CircleTimes;I_N\right)\underset{i=0}{\overset{N_C-1}{\&Sigma;}}{\mathrm{\&rho;}}_i\left(U_{D,i}\&CircleTimes;b_i\&CircleTimes;a_i\right)+I_L\&CircleTimes;vec\left({\tilde{Z}}_l\right)\right)\\ {\left(\left(I_L\&CircleTimes;\mathrm{\&Phi;}\&CircleTimes;I_N\right)\underset{i=0}{\overset{N_C-1}{\&Sigma;}}{\mathrm{\&rho;}}_i\left(U_{D,i}\&CircleTimes;b_i\&CircleTimes;a_i\right)+I_L\&CircleTimes;vec\left({\tilde{Z}}_l\right)\right)}^H\&rsqb;\end{array}$

(2) clutter Gaussian distribution, and with the uncorrelated condition of interference under simplify the SINR of output

Suppose ρ _iindependent same distribution, and obedience average is 0, variance is gaussian distribution, under clutter and the incoherent hypothesis of interference plus noise item,

Export SINR and can be reduced to following expression:

$SINR=|{\mathrm{\&rho;}}_t{|}^2{v_t}^H{(I+\tilde{A}{R}_C)}^{-1}\tilde{A}{v}_t$

Wherein, $V={\&lsqb;v_1,v_2,...,v_{N_C}\&rsqb;}^H,v_i=U_{D,i}\&CircleTimes;b_i\&CircleTimes;a_i,\mathrm{\&Xi;}=diag({{\mathrm{\&sigma;}}_1}^2,{{\mathrm{\&sigma;}}_2}^2,...,{{\mathrm{\&sigma;}}_{N_C}}^2),A=I_L\&CircleTimes;\mathrm{\&Phi;}\&CircleTimes;I_N,$

Three, under permanent modular constraint, waveform optimization problem is stated

State of saturation is operated in play its maximum efficiency in order to radar transmitter can be made, simultaneously for avoiding amplifier nonlinearity to make transmitted waveform distortion, usually require that radar emission waveform has constant modulus property, for this OFDM-MIMO radar, this requirement also must meet;

Can be obtained by above analysis, under total emission power and permanent modular constraint, the waveform optimization problem maximizing MIMO-OFDM-STAP detection probability can be expressed as:

$\underset{a_m}{max}|{\mathrm{\&rho;}}_t{|}^2{v_t}^H{(I+\tilde{A}{R}_C)}^{-1}\tilde{A}{v}_t$

s.t.|a _m|＝C _m

$\underset{m=1}{\overset{M}{\&Sigma;}}{a}_m^2=P$

||a _m|| ²≥0

In formula, C _mbe the amplitude of m transmitted waveform, second is constrained to total emission power constraint, and the 3rd constraint shows that the emissive power of each transmitting array element is more than or equal to zero, P and represents total emission power;

In order to can Efficient Solution above formula optimization problem, first we simplify this problem.Notice, matrix in comprise second and third constraint comprises this equally, and because first is constrained to the permanent modular constraint transmitted, thus can relax as a square constraint, thus, this optimization problem can be expressed as again:

$\underset{a_m^2}{max}|{\mathrm{\&rho;}}_t{|}^2{v_t}^H{(I+\tilde{A}{R}_C)}^{-1}\tilde{A}{v}_t$

$\begin{array}{cc}s.t.& a_m^2=D_m\end{array}$

$\underset{m=1}{\overset{M}{\&Sigma;}}{a}_m^2=P$

||a _m|| ²≥0

Four, based on the waveform optimization problem solving of DL method

(1) based on the positive definite R of DL method _c

This optimization problem contains permanent modular constraint, is not obviously thus convex problem, is easily absorbed in locally optimal solution when solving globally optimal solution, meanwhile, due to cannot determine character, the objective function in optimization problem is about optimized variable a _mthe nonlinear function of more complicated, thus, for this optimization problem, convex optimization method can not be utilized separate, and if utilize other numerical methods, such as gradient method, just may produce the problem of convergence, for solving this nonlinear optimal problem, to R _cutilize the diagonal angle loading method being usually applied to robust ada-ptive beamformer, can obtain

ε < < λ in formula _max(R _c) i.e. so-called load factor-loadingfactor, λ _maxthe eigenvalue of maximum of () representing matrix;

(2) based on positive definite R _csimplify and export SINR

Will substitute into and export SINR expression formula, and utilize topology, can obtain

$\begin{array}{c}SINR=|{\mathrm{\&rho;}}_t{|}^2{v}_t^H\left(I-{\left(I+\tilde{A}{\tilde{R}}_C\right)}^{-1}\right){\tilde{R}}_C^{-1}{v}_t\\ =|{\mathrm{\&rho;}}_t{|}^2{v}_t^H{\tilde{R}}_C^{-1}{v}_t-|{\mathrm{\&rho;}}_t{|}^2{v}_t^H{\left({\tilde{R}}_C+{\tilde{R}}_C\tilde{A}{\tilde{R}}_C\right)}^{-1}{v}_t\end{array}$

(3) based on convex Optimization Solution waveform optimization problem

Based on above-mentioned discussion, waveform optimization problem can be expressed equivalently as following SDP problem:

$\underset{a_m^2,t}{min}t$

$a_m^2=D_m$

$\underset{m=1}{\overset{M}{\&Sigma;}}{a}_m^2=P$

||a _m|| ²≥0

In formula, t is auxiliary optimized variable, and the optimum solution of above formula effectively obtains by CVX Optimization Toolbox.