CN106597441A - Multi-target ISAR imaging task-oriented MIMO radar waveform optimal design method - Google Patents

Abstract

The invention provides a multi-target ISAR imaging task-oriented MIMO radar waveform optimal design method comprising the following steps: S1, MIMO radar transmits a random-phase initial signal; S2, a waveform optimization model is designed and solved to get an optimal waveform; and S3, targets in multiple directions are imaged simultaneously by use of the designed optimal waveform. According to the multi-target ISAR imaging task-oriented MIMO radar waveform optimization design method proposed by the invention, the prior information such as the position, radar cross section (RCS) and speed of each target obtained by searching and tracking and the bandwidth limitation of imaging on radar transmitted waveform are taken as important constraint conditions of waveform optimization, and a multi-target imaging task-oriented waveform optimization model is designed on the basis and solved through a conjugate gradient method. The designed waveform can form a beam on each target direction, and the beams formed on different target directions match the target characteristics in the corresponding directions. Simulation results show that the designed waveform can image targets in multiple directions simultaneously, and has a good imaging effect.

Description

A kind of MIMO radar waveform Optimization Design towards multiple target ISAR imaging task

Technical field

The present invention relates to Signal and Information Processing technology, and in particular to a kind of MIMO for multiple target ISAR imaging task Radar waveform Optimization Design.

Background technology

MIMO (Multiple Input Multiple Output) radar is a kind of new of field of radar proposition in recent years Radar system, it has multiple transmittings and reception antenna.According to " distance " of each antenna distance, MIMO radar can be divided into statistical MIMO radar and coherent type MIMO radar.Coherent type MIMO radar has less antenna distance, and far field objects echo is for receipts It is all related to send out for antenna array, and its each antenna can launch unlike signal and obtain good waveform diversity gain.Therefore, MIMO radar can neatly design transmitted waveform according to practical application.

At present, MIMO radar waveform optimization design has become study hotspot, and achieves many achievements in research.To make radar In spatial domain, transmitting pattern is expected in synthesis to transmission signal, and Yang Xiao is superfine《Minimize the MIMO radar transmitting pattern optimization of secondary lobe Algorithm》(electronics and information journal, 2013,35 (12):2815-2822), Wang Xu etc. exists《A kind of MIMO thunders based on prior information Up to transmitting pattern method for designing》(electronics and information journal, 2013,35 (12):Propose respectively in 2802-2808) and accordingly do Method.

But, these algorithms are mostly the narrowband transmissions according to tracking, the needs design of detection Radar Task, not There is the demand for considering that imaging task is designed waveform optimization.Target imaging can provide important target characteristic for target recognition, Occupy an important position in Radar Task, and imaging task requires the broadband signal that radar emission matches with target characteristic.

For the MIMO radar waveform optimization design towards multiple target ISAR imaging task, the present invention will by search and with The modes such as track obtain the prior information of each target and are imaged the bandwidth restriction to radar emission waveform as the weight of waveform optimization Constraints is wanted, the waveform optimization model towards multi-target imaging task is set up on this basis, and by conjugate gradient algorithms Solved, have devised can be while realize the MIMO radar transmitted waveform being imaged to the target of different azimuth.

The content of the invention

It is an object of the invention to overcome above-mentioned weak point of the prior art, propose it is a kind of towards multiple target ISAR into As the MIMO radar waveform Optimization Design of task, comprise the following steps：

The first step：The random initial signal of MIMO radar transmitter, phase；

Radar emission assumes that the emission array of MIMO radar is the even linear array being made up of M omnidirectional antenna, array element distance For d, the signal of m-th array element transmitting can be expressed as

X in formula_mT () represents the baseband signal of m-th array element transmitting, f_cIt is signal carrier frequency, T_pIt is pulse width, then launches The signal that signal synthesizes at the θ directions of far field is

Therefore, in θ direction signals in frequency f_cThe power spectrum at+nB/N places can be written as

Wherein

Represent in frequency f_cThe steering vector at+nB/N places.

Second step：The model of waveform optimization is designed, and solution is carried out to model and obtain optimum waveform；

If the waveform of m-th antenna transmitting is expressed asWhereinRepresent x_mL the phase place of (), takes q=1； Define waveform phasing matrix be

Therefore, θ_kPower spectrum and θ of the direction signal at n-th frequency_kThe directional diagram in direction is represented by

P_n(θ_k, Φ)=| χ_n(θ_k,Φ)|²/N (7)

Wherein

Approach for directional diagram and approach the following cost function set up with regard to phasing matrix with power spectrum：

Wherein

τ_k(Φ)=[U (θ_k,Φ)/N-u(θ_k)/N] (10)

η_hn(Φ)=[P_n(θ_h,Φ)-p_hn] (11)

θ_kK-th discrete azimuth angle is represented, K represents discretization azimuth sum, u (θ_k) represent and expect transmitting pattern,Represent θ_hExpectation power spectrum on direction at n-th frequency,Represent target direction,Represent not Directional diagram at common-azimuth approaches weight,Represent θ_hPower spectrum on direction at different frequent points approaches weight；Generation The 1st of valency function represents that directional diagram is approached, and the 2nd represents that target direction power spectrum is approached；Therefore, Optimized model is

The determination method of target direction signal transmission power is as follows in the Optimized model：

The information such as flight speed, position and RCS according to the target for getting and distance by radar equation

Wherein (S/N)_oRepresent the output signal-to-noise ratio of radar receiver, P_tThe signal power of radar emission is represented, G represents thunder Up to the gain of antenna, λ represents wavelength, and σ represents Target scatter section area, and k is Boltzmann constant, T₀For normal room temperature, B_nTo connect Receipts machine noise bandwidth, F_nFor noise coefficient, R represents the distance of target and radar；In the certain condition of radar emission general power P Under, it is that the reception signal for ensureing H target direction can be detected, make the receiver output signal-to-noise ratio of H target direction equal For a certain determination value more than detection threshold, then the expectation transmission power of h-th target direction is

Wherein, R_hThe distance of target and radar for h-th target direction.

Target direction band method for determining width is as follows in the Optimized model：

It has been generally acknowledged that distance, speed and target direction of motion embody the Threat of target, air threat priority is defined For

Wherein, υ_hFor the target flight speed of h-th target direction,For the target flight course of h-th target direction；

The target that Threat is high and size is little needs higher range resolution ratio ρ_r；First, by by H target direction Target resolution be set to fixed valueApproximate size S of each target can be obtained_h；When range resolution ratio ρ_rIt is less than Or equal to target size 1/50 when, it is possible to obtain preferable imaging effect, thus by needed for the target of h-th target direction away from High Resolution ρ_rhIt is defined as

Therefore, the band of h-th target direction is a width of

Wherein c is the light velocity；In order to distinguish the echo-signal in different target direction, the transmission signal of each target direction should be distributed In orthogonal frequency band.

Target direction expects that the determination method of power spectrum is as follows in the Optimized model：

In the case of known radar transmission signal total bandwidth B and total frequency points N, frequency number shared by h-th target direction For N_h=round (B_hN/B) andTherefore, on h-th target direction, it would be desirable to transmission power P_thIt is uniformly distributed In N_hOn individual frequency, you can obtain the expectation power spectrum on h-th target direction.

Expect that the determination method of directional diagram is as follows in the Optimized model：

Desired orientation figure is made up of in the target direction the main lobe of traditional wave beam, and designed directional diagram on each target direction General power should with target direction expect power spectrum it is consistent.

The method for solving of the Optimized model is as follows：

Formula (12) is phasing matrix Φ unconstrained optimization problems, due to nonconvex property, it is impossible to ensure its optimal solution.But cost letter Number Q (Φ) is quadravalence trigonometric polynomial, and the locally optimal solution of quadravalence trigonometric polynomial object function is the 1/2 near of globally optimal solution Seemingly；Solution is optimized to waveform using the conjugate gradient method based on waveform bit matrix First-order Gradient information, and to cost function First-order Gradient expression formula derived, its idiographic flow is as follows：

Step 1：Given initial point Φ₀(with mode is randomly generated), and precision ε ＞ 0；

Step 2：IfThen optimization terminates and minimum point is Φ₀, otherwise execution step 3；

Step 3：The direction of search is setAnd make iterationses k=0；

Step 4：Step size mu is sought with one-dimensional linear search method_k(determine that step-length has interval and using yellow using advance and retreat method herein Golden split-run determines step-length) so that

And update phasing matrix Φ_k+1=Φ_k+μ_kD^(k)；

Step 5：Gradient of the calculation cost function with regard to phasing matrixCalculate

And the direction of search is set is

Step 6：Judge end conditionWhether set up, optimize if setting up and terminate, otherwise make k=k+1, Execution step 4；

Wherein, gradients of the cost function Q (Φ) with regard to phasing matrix Φ in algorithmFor

Wherein

For Hadamard products.

3rd step：The target in multiple orientation is imaged simultaneously using designed optimum waveform；

Understand that transmission signal is by formula (2)

WhereinPhasing matrix Φ is obtained by solving-optimizing model, then radar is received Echo-signal be

Wherein, σ_iRepresent the scattering coefficient of scattering point, t_lThe slow time is represented, t is full-time, R_hi(t_l) represent in h-th mesh Mark direction, i-th scattering point of target is in t_lThe distance at moment.

In each target direction, by the echo-signal for receiving in distance to after carrying out matched filtering process, in orientation To making Fourier transformation (FFT) to it, you can obtain the imaging results to respective direction target.

The MIMO radar waveform Optimization Design towards multiple target ISAR imaging task for being carried, will by search and with The modes such as track obtain the prior informations such as the position of each target, Radar Cross Section (RCS), speed and imaging to radar emission The bandwidth of waveform limits the important restrictions condition as waveform optimization, establishes on this basis towards multi-target imaging task Waveform optimization model, and solved by conjugate gradient method.Designed waveform can be upwardly formed wave beam, and different mesh in each target side The wave beam that mark side is upwardly formed matches with the target property of its respective direction, can simultaneously realize that the target to different azimuth is carried out The MIMO radar transmitted waveform of imaging.

Description of the drawings

Fig. 1 illustrates MIMO radar emission array schematic diagram；

Fig. 2 illustrates target direction one-dimensional range profile；

Fig. 3 (a) illustrates transmitting pattern, and Fig. 3 (b) illustrates transmission signal power spectrum；

Fig. 4 illustrates aircraft scattering model figure, and Fig. 4 (a) illustrates -20 ° of direction aircraft scattering model figures, and Fig. 4 (b) illustrates 20 ° Direction aircraft scattering model figure；

Fig. 5 illustrates designed optimization waveform imaging results, and Fig. 5 (a) is shown with the p- 20 ° of direction mesh of designed waveform Target imaging results, Fig. 5 (b) is shown with imaging results of the designed waveform to 20 ° of direction targets；

Fig. 6 illustrates linear FM signal imaging results, and Fig. 6 (a) is shown with the p- 20 ° of direction targets of linear FM signal Imaging results, Fig. 6 (b) is shown with imaging results of the linear FM signal to 20 ° of direction targets；

Fig. 7 illustrates the flow chart of the present invention.

Specific embodiment

Below in conjunction with the accompanying drawings with the example of the present invention, the invention will be further described.

The first step：MIMO radar produces transmission signal；

Fig. 1 shows MIMO radar emission array schematic diagram, and the emission array that MIMO radar is assumed in the present invention is by M= The even linear array that 4 omnidirectional antennas are constituted, array element distance is d=0.5*c/ (f_c+ B/2), then the signal of m-th array element transmitting can To be expressed as

Xm (t) represents the baseband signal of m-th array element transmitting, the present invention number of winning the confidence carrier frequency f in formula_c=10GHz, T_pFor arteries and veins Width, bandwidth B=800MHz are rushed, then the signal that transmission signal synthesizes at the θ directions of far field is

In actual applications, pulse width T_pInterior, MIMO radar launches L subpulse and a width of T of subpulse_s=1/ B, i.e. T_p=LT_s, L=200 is taken herein.Therefore, l-th baseband signal of m-th array element transmitting is in a pulse widthL=1,2 ..., L, its spectrum expression formula is

Wherein N represents the points of discrete Fourier transform, and N=L is taken herein.Therefore, array emitter discrete baseband signal Frequency spectrum is Z (n)=[Z₁(n),Z₂(n),…,Z_M(n)]^T=Xf_n, wherein f_n=[1, e-^j2πn/N,…,e^{-j2π(L-1)n/N}]^TRepresent N Conversion vector in the DFT of point at n-th frequency, X=[x₁,x2,…,x_L] represent array emitter baseband signal and x_l=[x₁ (l),x₂(l),…,x_M(l)]^T。

Therefore, in θ direction signals in frequency f_cThe power spectrum at+nB/N places can be written as

Wherein

Represent in frequency f_cThe steering vector at+nB/N places.

Second step：The model of waveform optimization is designed, and solution is carried out to model and obtain optimum waveform；

In order to the work efficiency transmitted waveform for maximizing transmitter is needed with constant modulus property.Therefore, m-th antenna transmitting Waveform can be expressed asWhereinRepresent x_mL the phase place of (), takes q=1；Define the phasing matrix of waveform For

Therefore θ_kPower spectrum and θ of the direction signal at n-th frequency_kThe directional diagram in direction is represented by

P_n(θ_k, Φ)=| χ_n(θ_k,Φ)|²/N (8)

Wherein

Can approach for directional diagram and approach the following cost function set up with regard to phasing matrix with power spectrum：

Wherein

τ_k(Φ)=[U (θ_k,Φ)/N-u(θ_k)/N] (11)

η_hn(Φ)=[P_n(θ_h,Φ)-p_hn] (12)

θ_kK-th discrete azimuth angle is represented, K represents discretization azimuth sum, takes K=181, u (θ_k) represent and expect transmitting Directional diagram,Represent θ_hExpectation power spectrum on direction at n-th frequency,Target direction is represented, is taken herein Two target directions, i.e. θ₁=-20 °, θ₂=20 °,Represent that the directional diagram at different orientations approaches weight,Represent θ_hPower spectrum on direction at different frequent points approaches weight, the azimuthal weight in 4, directional diagram two ends {α_k}_{K=1 ... 4,178 ..., 181}It is set as 5, other are set as 1, except the weight beta of specified otherwise power spectrum_hnIt is set to 1.Cost function The 1st represent directional diagram approach, the 2nd represent target direction power spectrum approach.Therefore, Optimized model is

It is as follows for the determination method of target direction signal transmission power in Optimized model：

The information such as flight speed, position and RCS according to the target for getting and distance by radar equation

Wherein (S/N)_oRepresent the output signal-to-noise ratio of radar receiver, P_tThe signal power of radar emission is represented, P is taken herein_t =8000W, G represent the gain of radar antenna, and λ represents wavelength, and σ represents Target scatter section area, and dissipating for two targets is taken herein Penetrate coefficient and be respectively σ₁=3, σ₂=2.K is Boltzmann constant, T₀For normal room temperature, B_nFor receiver noise bandwidth, F_nTo make an uproar Sonic system number, R represents the distance of target and radar.It is to ensure H=2 target side under conditions of radar emission general power P is certain To reception signal can be detected, make H target direction receiver output signal-to-noise ratio be it is a certain more than detection threshold Determination value, then the expectation transmission power of h-th target direction be

Wherein, R_hThe distance of target and radar for h-th target direction, the distance that two targets are taken herein is respectively R₁ =14700m, R₂=14000m.The expectation radiant power that can be calculated target direction by above-mentioned parameter is respectively P_t1=3581W, P_t2=4419W.

For target direction band method for determining width is as follows in Optimized model：

It has been generally acknowledged that distance, speed and target direction of motion embody the Threat of target.By target prestige in the present invention Stress is defined as

Wherein, υ_hFor the target flight speed of h-th target direction, the speed that both direction target is taken herein is respectively υ₁ =300m/s, υ₂=400m/s,For the target flight course of h-th target direction, the speed of both direction target is taken herein RespectivelyBy above-mentioned parameter it can be calculated that the Threat of both direction target is respectively V₁=2, V₂=2.7.

The target that Threat is high and size is little needs higher range resolution ratio ρ_r.First, by by H target direction Target resolution be set to fixed valueApproximate size S of each target can be obtained_h, emulate according to circumferential edge The approximate size that both direction target can be obtained is S₁=36m, S₂=60m.By experience, when range resolution ratio ρ_rLess than or wait When the 1/50 of target size, it is possible to obtain preferable imaging effect.Thus by the target required separation distance of h-th target direction point Resolution ρ_rhIt is defined as

Therefore, the band of h-th target direction is a width of

Wherein c is the light velocity.In order to distinguish the echo-signal in different target direction, the transmission signal of each target direction should be distributed In orthogonal frequency band.By above-mentioned parameter, it is ρ that can calculate the range resolution ratio needed for target direction imaging_r1=0.36, ρ_r2 =0.44 with a width of B₁=416MHz, B₂=340MHz.

For target direction expects that the determination method of power spectrum is as follows in Optimized model：

In the case of known radar transmission signal total bandwidth B and total frequency points N, frequency number shared by h-th target direction For N_h=round (B_hN/B) andTherefore, on h-th target direction, it would be desirable to transmission power P_thIt is uniformly distributed In N_hOn individual frequency, you can obtain the expectation power spectrum on h-th target direction, two target sides can be calculated to obtain by above-mentioned analysis 104 and 85 are respectively to the shared frequency number of expectation.

It is as follows for the determination method of expectation directional diagram in Optimized model：

Desired orientation figure is made up of in the target direction the main lobe of traditional wave beam, and designed directional diagram on each target direction General power should with target direction expect power spectrum it is consistent.

It is as follows for the method for solving of Optimized model：

Formula (13) is phasing matrix Φ unconstrained optimization problems, due to nonconvex property, it is impossible to ensure its optimal solution, but cost letter Number Q (Φ) is quadravalence trigonometric polynomial, and the locally optimal solution of quadravalence trigonometric polynomial object function is the 1/2 near of globally optimal solution Seemingly.Further, since the dimension of phasing matrix is larger, the computation complexity of the Hess matrixes of cost function is higher, therefore this The bright conjugate gradient method using based on waveform bit matrix First-order Gradient information is optimized solution to waveform, and to cost function First-order Gradient expression formula is derived.Its idiographic flow is as follows：

Step 1：Given initial point Φ₀(with mode is randomly generated), and precision ε ＞ 0；

Step 2：IfThen optimization terminates and minimum point is Φ₀, otherwise execution step 3；

Step 3：The direction of search is setAnd make iterationses k=0；

Step 4：Step size mu is sought with one-dimensional linear search method_k(determine that step-length has interval and using yellow using advance and retreat method herein Golden split-run determines step-length) so that

And update phasing matrix Φ_k+1=Φ_k+μ_kD^(k)；

Step 5：Gradient of the calculation cost function with regard to phasing matrixCalculate

And the direction of search is set is

Step 6：Judge end conditionWhether set up, optimize if setting up and terminate, otherwise make k=k+1, Execution step 4；

Wherein, gradients of the cost function Q (Φ) with regard to phasing matrix Φ in algorithmFor

Wherein

For Hadamard products.

3rd step：The target in multiple orientation is imaged simultaneously using designed optimum waveform；

Understand that transmission signal is by formula (2)

WhereinPhasing matrix Φ is obtained by solving-optimizing model, then radar is received Echo-signal be

Wherein, σ_iRepresent the scattering coefficient of scattering point, t_lThe slow time is represented, t is full-time, R_hi(t_l) represent in h-th mesh Mark direction, i-th scattering point of target is in t_lThe distance at moment.

In the present invention, it is assumed that motion compensation has been completed.Due to the spectrum orthogonal in target direction transmission signal, this is utilized Condition can separate the echo-signal in different target direction in echo.In each target direction, the echo-signal for receiving is existed Distance makees Fourier transformation (FFT) to it, you can obtain to respective direction mesh to carrying out after matched filtering process in orientation Target imaging results.

Example：Towards the MIMO radar waveform method for designing of multiple target ISAR imaging

Emulation experiment：If a MIMO radar system includes M=4 transmitting array element, array element distance d=0.5*c/ (f_c+B/ 2), signal carrier frequency f_c=10GHz, radar emission signal total bandwidth signal bandwidth B=800MHz, subpulse number L=200, total frequency Points N=L=200, discretization azimuth sum K=181, in order to suppress the secondary lobe in directional diagram far field, 4, directional diagram two ends side Weight { the α of parallactic angle_k}_{K=1 ... 4,178 ..., 181}It is set as 5, other are set as 1, except the weight beta of specified otherwise power spectrum_hnIt is set to 1, the traditional wave beam for generating desired orientation figure is weighed using 30dB Chebyshevs, and termination threshold value is ε=10^-3, radar emission general power It is set to P=8000W.The information got further according to search lighting, tracking etc. is as shown in table 1.

The target component of table 1

First, the target resolution in -20 ° of directions and 20 ° of directions is set to into 2, i.e.,Can obtain such as Fig. 2 institutes The one-dimensional range profile to target for showing.From Fig. 2 (a), the approximate size of -20 ° of direction targets is S₁=36m.By Fig. 2 (b) Understand, 20^°The approximate size of direction target is S₂=60m.According to formula (16), formula (17), formula (18) and table 2 can be further It is ρ to extrapolate the range resolution ratio needed for target direction imaging_r1=0.36, ρ_r2=0.44 with a width of B₁=416MHz, B₂= 340MHz, thus can obtain two target directions and expect that shared frequency number is respectively 104 and 85.In two target directions, -20 ° of sides To target range and target scattering coefficient it is big than the value in 20 ° of directions, comprehensive the two parameters, according to formula (15) and table 2 The expectation power that two target direction transmission signals can be derived is respectively -1.82dB and 0dB.

Such as Fig. 3 (a) is shown in solid for the directional diagram of designed waveform, it can be seen that designed waveform is in -20 ° of directions and 20 ° Direction defines two wave beams, and approaches desired orientation figure.Fig. 3 (b) gives the power spectrum of two target direction transmission signals, It can be seen that frequency band non-overlapping copies residing for target direction transmission signal, and designed power spectrum with expect that power spectrum has and preferably force Nearly effect.Fig. 4 is target direction aircraft scattering model, and Fig. 5 and Fig. 6 is respectively target direction and is formed with designed optimization ripple The result of picture and the result being imaged with linear FM signal.The imaging results of contrast Fig. 5 and Fig. 6, can calculate in -20 ° of directions, scheme The Y-PSNR of 5 (a) and Fig. 6 (a) is 35.31, is 34.72 in the Y-PSNR of 20 ° of directions, Fig. 5 (b) and Fig. 6 (b), Thus the effectiveness of provable this paper institutes extracting method.

The inventive method utilizes the waveform diversity advantage of MIMO radar, it is proposed that one kind is towards multiple target ISAR imaging task MIMO radar waveform Optimization Design, the method will by search for and track etc. mode obtain the position of each target, radar The prior informations such as scattering resonance state (RCS), speed and imaging are limited as waveform optimization the bandwidth of radar emission waveform Important restrictions condition, establishes on this basis the waveform optimization model towards multi-target imaging task, and by conjugate gradient Method is solved.Designed waveform can be upwardly formed wave beam in each target side, and the wave beam that different target side is upwardly formed is corresponding to its The target property in direction matches.Simulation result shows, designed waveform the target of multiple directions can be imaged simultaneously simultaneously and With preferable imaging effect.

Claims (9)

1. a kind of MIMO radar waveform Optimization Design towards multiple target ISAR imaging task, comprises the following steps：

The first step：The random initial signal of MIMO radar transmitter, phase；

Second step：The model of waveform optimization is designed, and solution is carried out to model and obtain optimum waveform；

3rd step：The target in multiple orientation is imaged simultaneously using designed optimum waveform.

2. a kind of MIMO radar waveform optimization design side towards multiple target ISAR imaging task according to claim 1 Method, the wherein first step are specially：

Radar emission assumes that the emission array of MIMO radar is the even linear array being made up of M omnidirectional antenna, and array element distance is d, The signal of m-th array element transmitting can be expressed as

$s_m\left(t\right)=x_m\left(t\right)e^{j2{\mathrm{\πf}}_{c}t},0\≤t\≤T_p---\left(1\right)$

Xm (t) represents the baseband signal of m-th array element transmitting, f in formula_cIt is signal carrier frequency, T_pPulse width, then transmission signal The signal of synthesis is at the θ directions of far field

$\begin{array}{c}s(\mathrm{\θ},t)=\underset{m=0}{\overset{M-1}{\Σ}}{S}_m(t+\frac{mdsin\mathrm{\θ}}{c})\\ =\underset{m=0}{\overset{M-1}{\Σ}}{x}_m(t+\frac{md\sin \mathrm{\θ}}{c})e^{j2{\mathrm{\πf}}_c(t+\frac{md\sin \mathrm{\θ}}{c})},0\≤t\≤T_p\end{array}---\left(2\right)$

Therefore, in θ direction signals in frequency f_cThe power spectrum at+nB/N places can be written as

$\begin{array}{c}{P}_n\left(\mathrm{\θ}\right)=|a_n^T\left(\mathrm{\θ}\right)Z\left(n\right){|}^2/N\\ =a_n^T\left(\mathrm{\θ}\right){\mathrm{Xf}}_n{f}_n^H{X}^H{a}_n^{*}\left(\mathrm{\θ}\right)/N\end{array}---\left(3\right)$

Wherein

$\begin{array}{c}{a}_n\left(\mathrm{\θ}\right)={\[1,e^{j2\mathrm{\π}(f_c+nB/N)\frac{d\sin \mathrm{\θ}}{c}},\mathrm{...},e^{j2\mathrm{\π}(f_c+nB/N)\frac{(M-1)d\sin \mathrm{\θ}}{c}}\]}^T,\\ n=-N/2,-N/2+1,\mathrm{...},-N/2-1\end{array}---\left(4\right)$

Represent in frequency f_cThe steering vector at+nB/N places.

3. a kind of MIMO radar waveform optimization design side towards multiple target ISAR imaging task according to claim 1 Method, wherein second step are specially：

If the waveform of m-th antenna transmitting is expressed asWhereinRepresent x_mL the phase place of (), takes q=1；Definition The phasing matrix of waveform is

Therefore, θ_kPower spectrum and θ of the direction signal at n-th frequency_kThe directional diagram in direction is represented by

P_n(θ_k, Φ)=| χ_n(θ_k,Φ)|²/N (7)

$U({\mathrm{\θ}}_k,\mathrm{\Φ})=\underset{n=-N/2}{\overset{N/2-1}{\Σ}}|{\mathrm{\χ}}_n({\mathrm{\θ}}_k,\mathrm{\Φ}){|}^2/N---\left(8\right)$

Wherein

Approach for directional diagram and approach the following cost function set up with regard to phasing matrix with power spectrum：

$Q\left(\mathrm{\Φ}\right)=\underset{k=1}{\overset{K}{\Σ}}{\mathrm{\α}}_k{\[{\mathrm{\τ}}_k\left(\mathrm{\Φ}\right)\]}^2+\frac{1}{N}\underset{h=1}{\overset{H}{\Σ}}\underset{n=-N/2}{\overset{N/2-1}{\Σ}}{\mathrm{\β}}_{hn}{\[{\mathrm{\η}}_{hn}\left(\mathrm{\Φ}\right)\]}^2---\left(9\right)$

Wherein

τ_k(Φ)=[U (θ_k,Φ)/N-u(θ_k)/N] (10)

η_hn(Φ)=[P_n(θ_h,Φ)-p_hn] (11)

θ_kK-th discrete azimuth angle is represented, K represents discretization azimuth sum, u (θ_k) represent and expect transmitting pattern,Represent θ_hExpectation power spectrum on direction at n-th frequency,Represent target direction,Represent different Directional diagram at azimuth approaches weight,Represent θ_hPower spectrum on direction at different frequent points approaches weight；Cost The 1st of function represents that directional diagram is approached, and the 2nd represents that target direction power spectrum is approached；Therefore, Optimized model is

$\underset{\mathrm{\Φ}}{\min }Q\left(\mathrm{\Φ}\right)---\left(12\right)$ $.$

4. a kind of MIMO radar waveform optimization design side towards multiple target ISAR imaging task according to claim 2 Method, the determination method of target direction signal transmission power is as follows in the Optimized model：

The information such as flight speed, position and RCS according to the target for getting and distance by radar equation

${\left(\frac{S}{N}\right)}_o=\frac{P_t{G}^2{\mathrm{\λ}}^2\mathrm{\σ}}{{\left(4\mathrm{\π}\right)}^3{\mathrm{kT}}_0{B}_n{F}_n{R}^4}---\left(13\right)$

Wherein (S/N)_oRepresent the output signal-to-noise ratio of radar receiver, P_tThe signal power of radar emission is represented, G represents radar day The gain of line, λ represents wavelength, and σ represents Target scatter section area, and k is Boltzmann constant, T₀For normal room temperature, B_nFor receiver Noise bandwidth, F_nFor noise coefficient, R represents the distance of target and radar；Under conditions of radar emission general power P is certain, it is Ensure H target direction reception signal can be detected, make H target direction receiver output signal-to-noise ratio be it is a certain More than the determination value of detection threshold, then the expectation transmission power of h-th target direction is

$P_{th}=\frac{P}{\underset{h=1}{\overset{H}{\Σ}}\frac{{\mathrm{\σ}}_1{R_h}^4}{{\mathrm{\σ}}_h{R_1}^4}}\·\frac{{\mathrm{\σ}}_1{R_h}^4}{{\mathrm{\σ}}_h{R_1}^4}---\left(14\right)$

Wherein, R_hThe distance of target and radar for h-th target direction.

5. a kind of MIMO radar waveform optimization design side towards multiple target ISAR imaging task according to claim 2 Method, target direction band method for determining width is as follows in the Optimized model：

It has been generally acknowledged that distance, speed and target direction of motion embody the Threat of target, air threat priority is defined as

Wherein, υ_hFor the target flight speed of h-th target direction,For the target flight course of h-th target direction；

The target that Threat is high and size is little needs higher range resolution ratio ρ_r；First, by by the mesh of H target direction Mark resolution is set to fixed valueApproximate size S of each target can be obtained_h；When range resolution ratio ρ_rLess than or wait When the 1/50 of target size, it is possible to obtain preferable imaging effect, thus by the target required separation distance of h-th target direction point Resolution ρ_rhIt is defined as

${\mathrm{\ρ}}_{rh}=\left\{\begin{array}{cc}\frac{1}{50}{S}_h& r_h\≤1\\ \frac{1}{50V_h}{S}_h& V_h>1\end{array}\right.---\left(16\right)$

Therefore, the band of h-th target direction is a width of

$B_h=\frac{c}{2{\mathrm{\ρ}}_{rh}}---\left(17\right)$

Wherein c is the light velocity；In order to distinguish the echo-signal in different target direction, the transmission signal of each target direction should be distributed in just In the frequency band of friendship.

6. a kind of MIMO radar waveform optimization design side towards multiple target ISAR imaging task according to claim 2 Method, target direction expects that the determination method of power spectrum is as follows in the Optimized model：

In the case of known radar transmission signal total bandwidth B and total frequency points N, frequency number shared by h-th target direction is N_h= round(B_hN/B) andTherefore, on h-th target direction, it would be desirable to transmission power P_thIt is evenly distributed on N_hIt is individual On frequency, you can obtain the expectation power spectrum on h-th target direction.

7. a kind of MIMO radar waveform optimization design side towards multiple target ISAR imaging task according to claim 2 Method, expects that the determination method of directional diagram is as follows in the Optimized model：

Desired orientation figure is made up of in the target direction the main lobe of traditional wave beam, and on each target direction designed directional diagram it is total Power should expect that power spectrum is consistent with target direction.

8. a kind of MIMO radar waveform optimization design side towards multiple target ISAR imaging task according to claim 2 Method, the method for solving of the Optimized model is as follows：

Formula (12) is phasing matrix Φ unconstrained optimization problems, due to nonconvex property, it is impossible to ensure its optimal solution, but cost function Q (Φ) it is quadravalence trigonometric polynomial, the locally optimal solution of quadravalence trigonometric polynomial object function is the 1/2 approximate of globally optimal solution； Solution is optimized to waveform using the conjugate gradient method based on waveform bit matrix First-order Gradient information, and to the one of cost function Rank pressure gradient expression formula is derived, and its idiographic flow is as follows：

Step 1：Given initial point Φ₀(with mode is randomly generated), and precision ε ＞ 0；

Step 2：IfThen optimization terminates and minimum point is Φ₀, otherwise execution step 3；

Step 3：The direction of search is setAnd make iterationses k=0；

Step 4：Step size mu is sought with one-dimensional linear search method_k(determine that step-length has interval and using gold point using advance and retreat method herein The method of cutting determines step-length) so that

$Q({\mathrm{\Φ}}_k+{\mathrm{\μ}}_k{D}^{\left(k\right)})=\underset{\mathrm{\μ}\≥0}{\min }Q({\mathrm{\Φ}}_k+{\mathrm{\μ}}_k{D}^{\left(k\right)})$

And update phasing matrix Φ_k+1=Φ_k+μ_kD^(k)；

Step 5：Gradient of the calculation cost function with regard to phasing matrixCalculate

${\mathrm{\γ}}_{k+1}=\frac{Tr\[{(\▿Q({\mathrm{\Φ}}_{k+1}\left)\right)}^T(\▿Q({\mathrm{\Φ}}_{k+1})-\▿Q\left({\mathrm{\Φ}}_k\right))\]}{Tr\[{(\▿Q({\mathrm{\Φ}}_k\left)\right)}^T\▿Q\left({\mathrm{\Φ}}_k\right)\]}$

And the direction of search is set is

Step 6：Judge end conditionWhether set up, optimize if setting up and terminate, otherwise make k=k+1, perform Step 4；

Wherein, gradients of the cost function Q (Φ) with regard to phasing matrix Φ in algorithmFor

$\begin{array}{c}\▿Q\left(\mathrm{\Φ}\right)=2\underset{k=1}{\overset{K}{\Σ}}{\mathrm{\α}}_k{\mathrm{\τ}}_k\left(\mathrm{\Φ}\right)\frac{\∂{\mathrm{\τ}}_k\left(\mathrm{\Φ}\right)}{\∂\mathrm{\Φ}}\\ +\frac{2}{N}\underset{h=1}{\overset{H}{\Σ}}\underset{n=-N/2}{\overset{N/2-1}{\Σ}}{\mathrm{\β}}_{hn}\frac{\∂{\mathrm{\η}}_{hn}\left(\mathrm{\Φ}\right)}{\∂\mathrm{\Φ}}\end{array}---\left(18\right)$

Wherein

$\begin{array}{c}\frac{\∂{\mathrm{\τ}}_k\left(\mathrm{\Φ}\right)}{\∂\mathrm{\Φ}}=\frac{1}{N}\·\frac{\∂U({\mathrm{\θ}}_k,\mathrm{\Φ})}{\∂\mathrm{\Φ}}\\ =\frac{1}{N^2}\·\underset{n=-N/2}{\overset{N/2-1}{\Σ}}\frac{\∂|{\mathrm{\χ}}_n({\mathrm{\θ}}_k,\mathrm{\Φ}){|}^2}{\∂\mathrm{\Φ}}\\ =\frac{1}{N^2}\·\underset{n=N/2}{\overset{N/2-1}{\Σ}}\[\frac{\∂{\mathrm{\χ}}_n({\mathrm{\θ}}_k,\mathrm{\Φ})}{\∂\mathrm{\Φ}}\·{\mathrm{\χ}}_n^H({\mathrm{\θ}}_k,\mathrm{\Φ})\\ +{\mathrm{\χ}}_n({\mathrm{\θ}}_k,\mathrm{\Φ})\·\frac{\∂{\mathrm{\χ}}_n^H({\mathrm{\θ}}_k,\mathrm{\Φ})}{\∂\mathrm{\Φ}}\]\end{array}---\left(19\right)$

$\begin{array}{c}\frac{\∂{\mathrm{\η}}_{hn}\left(\mathrm{\Φ}\right)}{\∂\mathrm{\Φ}}=\frac{\∂P_n({\mathrm{\θ}}_h,\mathrm{\Φ})}{\∂\mathrm{\Φ}}\\ =\frac{1}{N}\·\frac{\∂|{\mathrm{\χ}}_n({\mathrm{\θ}}_h,\mathrm{\Φ}){|}^2}{\∂\mathrm{\Φ}}\\ =\frac{1}{N}\·\[\frac{\∂{\mathrm{\χ}}_n({\mathrm{\θ}}_h,\mathrm{\Φ})}{\∂\mathrm{\Φ}}\·{\mathrm{\χ}}_n^H({\mathrm{\θ}}_h,\mathrm{\Φ})\\ +{\mathrm{\χ}}_n({\mathrm{\θ}}_h,\mathrm{\Φ})\·\frac{\∂{\mathrm{\χ}}_n^H({\mathrm{\θ}}_h,\mathrm{\Φ})}{\∂\mathrm{\Φ}}\]\end{array}---\left(20\right)$

⊙ is Hadamard products.

9. a kind of MIMO radar waveform optimization design side towards multiple target ISAR imaging task according to claim 1 Method, wherein the 3rd step is specially：

Understand that transmission signal is by formula (2)

$s(\mathrm{\θ},t)=\underset{h=1}{\overset{H}{\Σ}}rect\left(\frac{t}{T_p}\right)s({\mathrm{\θ}}_h,t)---\left(22\right)$

WhereinPhasing matrix Φ is obtained by solving-optimizing model, then what radar was received returns Ripple signal is

$\begin{array}{c}{s}_r(\mathrm{\θ},t)=\underset{h=1}{\overset{H}{\Σ}}\underset{m=1}{\overset{M}{\Σ}}\underset{i}{\Σ}{\mathrm{\σ}}_{i}rect\left(\frac{t-2R_{hi}\left(t_l\right)/c}{T_p}\right)\\ \·s_m({\mathrm{\θ}}_h,t-2R_{hi}(t_l)/c)\end{array}---\left(23\right)$

Wherein, σ_iRepresent the scattering coefficient of scattering point, t_lThe slow time is represented, t is full-time, R_hi(t_l) represent in h-th target side To i-th scattering point of target is in t_lThe distance at moment；In each target direction, by the echo-signal for receiving distance to After carrying out matched filtering process, in orientation Fourier transformation (FFT) is made to it, you can obtain to respective direction target into As result.