CN104808190B - Improve the sane waveform design method of the worst parameter Estimation performance of MIMO radar - Google Patents

Improve the sane waveform design method of the worst parameter Estimation performance of MIMO radar Download PDF

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CN104808190B
CN104808190B CN201510164914.6A CN201510164914A CN104808190B CN 104808190 B CN104808190 B CN 104808190B CN 201510164914 A CN201510164914 A CN 201510164914A CN 104808190 B CN104808190 B CN 104808190B
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CN104808190A (en
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郭荣艳
李娜娜
贺娅莉
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Zhoukou Normal University
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S7/00—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00
    • G01S7/02—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S13/00
    • G01S7/42—Diversity systems specially adapted for radar

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  • Radar, Positioning & Navigation (AREA)
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  • Computer Networks & Wireless Communication (AREA)
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  • Radar Systems Or Details Thereof (AREA)
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Abstract

The sane waveform design method for improving the worst parameter Estimation performance of MIMO radar belongs to field of signal processing, and this method is：Initially set up MIMO radar receipt signal model under clutter scene, lower bound Cramér-Rao lower bound based on this model inference Parameter Estimation Precision to be estimated, evaluated error modeling to direction of arrival angle be present, initial parameter is then estimated that uncertain convex set is explicitly included into waveform optimization problem, establishes sane waveform optimization model；This optimization problem is solved using the alternative manner based on diagonal loading technique；The each step of iteration can all relax as Semidefinite Programming, it is hereby achieved that Efficient Solution, passes through proposed iterative algorithm to optimize waveform covariance matrix, and then improve the parameter Estimation performance under clutter environment under worst case afterwards.Irrelevant waveform and non-robust method are compared to, the present invention has preferable robustness, thus closer to engineer applied.

Description

Improve the sane waveform design method of the worst parameter Estimation performance of MIMO radar
Technical field
The invention belongs to field of signal processing, under the improvement clutter environment for further relating to Waveform Design technical field The sane waveform design method of the worst parameter Estimation performance of MIMO radar.
Background technology
In the last few years, MIMO radar waveform optimization is paid attention to by increasing scholar and engineer.It is excellent according to waveform The object module used in change problem, current waveform optimization method can be divided into two categories below：(1) it is based on point target (point Target waveform optimization)；(2) waveform optimization based on extension target (extended target).Waveform based on point target Design, the object of optimization is waveform Correlation Matrix (WCM, waveform covariance matrix) or radar ambiguity function (radar ambiguity function).Spatial domain rather than transmitting of the waveform optimization method based on WCM only to transmitted waveform The overall feature of waveform is designed.Specifically, D.R.Fuhrmann and G.S.Antonio et al. WCM is designed with Realize specific energy spatial domain distribution.And S.Peter et al. has not been concerned only with the distribution of energy spatial domain, and it have also contemplated that different mesh Spatial domain cross-correlation between mark, that is, the spatial domain cross-correlation between different azimuth is minimized to improve the detection of system estimation performance.
It is not relied under the hypothesis of the clutter pollution of transmitted waveform in reception signal, J.Li et al. proposes several classes and is based on CRB waveform optimization criterion improves the Parameter Estimation Precision of point target to optimize WCM.Reception signal is by clutter pollution condition Under, H.Y Wang et al. consider MIMO radar waveform and biased estimator based on CRB under the conditions of target prior information is known Combined optimization problem.It should be noted that the solution of waveform optimization problem is required for parameter to know in these methods.However, In Practical Project, these parameters must be obtained by estimation, thus evaluated error inevitably be present.Thus, joined based on estimation The parameter Estimation performance that number optimization waveform obtains is than more sensitive to evaluated error and uncertainty.
The content of the invention
Present invention aims to overcome that traditional waveform optimization method is to initial parameter evaluated error sensitivity under clutter conditions Problem, it is proposed that a kind of sane waveform design method for improving the worst parameter Estimation performance of MIMO radar, this method include parameter Uncertain convex set, the alternative manner solving-optimizing problem based on DL technologies, to mitigate parameter estimating error or not know what is brought System sensitivity problem, so as to improve the estimation performance of the MIMO radar waveform Optimal Parameters under worst case.
The basic ideas of the inventive method are：MIMO radar receipt signal model under clutter conditions is built first, based on this The CRB of model inference parameter to be estimated, the explicit sane waveform optimization model for including parameter uncertainty is then established, to solve This nonlinear optimal problem, proposes a kind of iterative algorithm based on DL technologies, and each step of iteration can relax as Semidefinite Programming Problem is so as to obtaining Efficient Solution.After obtaining an optimal intermediate solution based on iterative algorithm, optimum waveform covariance matrix can be most A young waiter in a wineshop or an inn multiplies to be reconstructed under meaning.
The present invention improves the sane waveform design method of the worst parameter Estimation performance of MIMO radar, and it comprises the following steps：
Step 1: structure MIMO radar receipt signal model
Assuming that MIMO radar reception signal is：
Wherein,To be proportional to the complex magnitude of target RCS (radar cross section),For target location parameter, K For target numbers, ρ (θ) is the reflectance factor in θ opening position clutter blocks, and W represents interference noise, and each column is separate and same Distribution circle symmetric complex random vector, has zero-mean, and its covariance B is unknown,For transmission signal square Battle array, a (θk) and v (θk) represent to receive respectively, launch steering vector, it is embodied as：
In formula, f0For carrier frequency, τm(θk), m=1,2 ... MrWithFor transmission time, ac(θ) And vc(θ) represents θ respectivelykLocate reception and the transmitting steering vector of target；
If rang ring is divided into NC(NC＞ ＞ NML) individual resolution cell, MIMO radar receipt signal model is rewritten as
Wherein,Represent clutter transmission function, ρ (θi) it is θiLocate the reflectance factor of clutter block, NC(NC＞ ＞ MtMr) it is clutter spatial sampling quantity, ac(θi) and vc(θi) θ is represented respectivelyiLocate the reception of clutter block, transmitting is oriented to Vector；vec(Hc) it is that its average is zero, and covariance is with the multiple Gauss random vector of distribution
Step 2: sane waveform optimization model of the structure based on CRB
Consider unknown parameter θ=[θ1,θ2,…,θK]T、Under the conditions of CRB, by deriving, this CRB can state such as Under：
Wherein,
In formula, RS=SSH,For positive semidefinite Hermitian matrix,
Only consider direction of arrival angle, be i.e. influence of the θ evaluated errors to systematic function, k-th of destination channel matrix can be built Mould is as follows：
Wherein,hkRespectively actual and hypothesis k-th of destination channel matrix, δkForError, belong to as Lower convex set：
AndWherein,Respectively hkVector reciprocal that is real and assuming,For's Error, belong to following convex set：
Based on the above, the sane waveform optimization problem for improving worst condition parameter Estimation performance under clutter conditions can be with It is expressed as：Under the constraint on WCM, based on Parameter uncertainties convex setOptimize WCM to minimize under worst case CRB；Under Trace-opt criterions, optimization problem can be described as：
tr(RS)=LP
Wherein, P represents total transmission power；In formula the 3rd constraint establishment be due to each transmitter unit transmission power not It is likely less than zero；
Step 3: the solution of sane waveform internal layer optimization problem
The solution of internal layer optimization problem is based on following lemma 1：
The positive semidefinite hermitian matrix that A is a M × M is assumed in lemma 1., then following inequality is set up：Equation is set up when and if only if A is diagonal matrix；According to lemma 1, internal layer optimization problem can relax For：
Based on CRB, above formula can be rewritten as：
Deletion takes real part operator Re { }, is due to each in above formula and item is real number；
δ is only relied upon from the denominator of kth item in above formula, and formulakWithTwo, thus it is of equal value the problem of in above formula In, under corresponding constraint, maximize and formula in each single item, be represented by：
To solve above formula, to RSUsing diagonal loading technique, i.e.,：
Wherein, ε ＜ ＜ λmax(RS) it is load factor, λmax() representing matrix eigenvalue of maximum, select ε=λmax(RS)/ 1000；Use respectivelyReplace the R in sane optimization problemS, can obtain WithRespectively for And δkIt is convex；
Thus, above formula can be rewritten as：
Above formula can tear the minimization problem for being write as following two independence open：
s.t. ||δk||F≤ζk
Above-mentioned two minimization problem can be solved by following lemma 2：
Lemma 2, assume hermitian matrix'sThen and if only ifWhen, ZWherein, Δ C =A-BHC-1The Schur that B is C in Z is mended；
By quoting lemma 2, above-mentioned two minimization problem can be converted into following SDP problems 1：
Wherein, t is auxiliary variable；
By more than two formulas obtainWithBring into sane optimization problem, consider outside optimization problem；
Step 4: the solution of sane waveform outer layer optimization problem
Outside optimization problem is solved using following proposition
Proposition：Using matrix manipulation, the constraint in sane optimization problem can be equivalent to following LMI：
Wherein
Using lemma 2 and above-mentioned proposition is combined, outer layer optimization problem can be expressed as following SDP problems：
Wherein, X is an auxiliary variable；
After optimal E is obtained, under least square meaning, RSFollowing model construction can be passed through：
s.t. tr(RS)=LP
Using lemma 2 and above-mentioned proposition is combined, above formula can be equivalent to following SDP problems 2：
tr(RS)=LP
Step 5: sane waveform optimization problem is solved using alternative manner
Step 5.1, given waveform covariance matrix initial value；
Step 5.2, above-mentioned SDP problems 1 are solved to obtain optimal δk,
Step 5.3, SDP problems 2 are solved to obtain optimal E；
Step 5.4, return to step 5.2 iteration again, until CRB is no longer substantially reduced.
Step 6: being based on least square method, optimal waveform covariance matrix is reconstructed, R can be obtainedS。
The beneficial effects of the invention are as follows：This method not can be used for releiving traditional waveform optimization method to parameter estimating error and not Certainty sensitive issue.MIMO radar receipt signal model under clutter scene is initially set up, is characterized and treated based on this model inference Estimate lower bound-Cramér-Rao lower bound (CRB) of Parameter Estimation Precision, Parameter uncertainties convex set is explicitly then included into conventional wave In shape optimization problem；To solve this nonlinear optimal problem, the present invention proposes a kind of changing based on diagonal loading (DL) technology For method, each step in iteration can all be converted into Semidefinite Programming (SDP) problem, it is hereby achieved that Efficient Solution, to realize The sane waveform optimization of MIMO radar under worst case, and then enable the parameter Estimation performance under worst case to be lifted, it is and non- Waveform correlation is compared, and this method is obviously improved to the parameter Estimation performance under worst case.
Brief description of the drawings
Fig. 1 is the flow chart that the present invention realizes；
Fig. 2 is the flow chart of the iterative algorithm of the present invention；
Fig. 3 is optimal launching beam side of the present invention when initial angle has evaluated error and array signal to noise ratio is 10dB Xiang Tu；
Fig. 4 be in the case where initial angle has evaluated error situation that the present invention carries algorithm and irrelevant waveform obtains with The CRB of ANSR changes.
Fig. 5 is optimal launching beam side of the present invention when array calibration has evaluated error and array signal to noise ratio is 10dB Xiang Tu；
Fig. 6 is in the case where array calibration has evaluated error situation, and the present invention carries algorithm and obtained most with irrelevant waveform The CRB changed in the case of bad with ANSR.
Embodiment
The present invention is described in further detail below in conjunction with the accompanying drawings.
As shown in Figures 1 to 6, the present invention improves the sane waveform design method of the worst parameter Estimation performance of MIMO radar Implementation process is as follows：
1st, sane waveform optimization problem model is established
1) MIMO radar signal model is built
Assuming that MIMO radar reception signal is：
Wherein,To be proportional to the complex magnitude of target RCS (radar cross section),For target location parameter, two Person is required for estimating.K is target numbers, and ρ (θ) is the reflectance factor in θ opening position clutter blocks, and W represents interference noise, each column It is separate and with circle symmetric complex random vector is distributed, there is zero-mean, its covariance B is unknown,For transmission signal matrix.a(θk) and v (θk) represent to receive respectively, launch steering vector, it is embodied as：
In formula, f0For carrier frequency, τm(θk), m=1,2 ... MrWithFor transmission time, ac(θ) And vc(θ) represents θ respectivelykLocate reception and the transmitting steering vector of target.
If rang ring is divided into NC(NC＞ ＞ NML) individual resolution cell, receipt signal model can be rewritten as
Wherein,Represent clutter transmission function, ρ (θi) it is θiLocate the reflectance factor of clutter block, NC(NC＞ ＞ MtMr) it is clutter spatial sampling quantity, ac(θi) and vc(θi) θ is represented respectivelyiLocate the reception of clutter block, transmitting is oriented to Vector.vec(Hc) the multiple Gauss random vector with distribution is may be considered as, its average is zero, and covariance isRHcIt can also be further represented as：Wherein,
2) the sane waveform optimization model based on CRB is built
Consider unknown parameter θ=[θ1,θ2,…,θK]T、Under the conditions of CRB, by deriving, this CRB can state such as Under：
Wherein,
In formula, RS=SSH,For positive semidefinite Hermitian matrix,
It is apparent that CRB be on θ,Hc, W function, and these parameters need to be obtained by estimation, thus can not Evaluated error be present with avoiding.Thus, the systematic parameter obtained using the CRB optimization waveforms based on certain group estimates of parameters is estimated Performance may be less than the more rational estimates of parameters of another set.And then in engineer applied, it is necessary to consider systematic function To the tender subject of initial parameter evaluated error.
In the present invention, direction of arrival angle, i.e. influence of the θ evaluated errors to systematic function are only considered.Thus, can be to k-th The modeling of destination channel matrix is as follows：
Wherein,hkRespectively actual and hypothesis k-th of destination channel matrix, δkForError, belong to as Lower convex set：
AndWherein,Respectively hkVector reciprocal that is real and assuming,For's Error, belong to following convex set：
Based on discussed above, the sane waveform optimization problem for improving worst condition parameter Estimation performance under clutter conditions can be with It is expressed as：Under the constraint on WCM, based on Parameter uncertainties convex setOptimize WCM to minimize under worst case CRB.Under Trace-opt criterions, optimization problem can be described as：
tr(RS)=LP
Wherein, P represents total transmission power；In formula the 3rd constraint establishment be due to each transmitter unit transmission power not It is likely less than zero.
It it is one on R it is obvious that the object function of the mark of CRB matrixes, i.e. above formulaSAnd δk,K=1,2 ..., K's Extremely complex nonlinear function, thus be very difficult to solve using conventional methods such as convex optimizations.
2. the solution of sane waveform optimization problem
1) solution of internal layer optimization problem
As described above, the object function of optimization problem is extremely complex nonlinear function, it is difficult to utilizes traditional optimization Method solves.To solve this problem, internal layer optimization problem is considered first.The solution of internal layer optimization problem is based on following lemma 1：
The positive semidefinite hermitian matrix that A is a M × M is assumed in lemma 1., then following inequality is set up： Equation is set up when and if only if A is diagonal matrix.According to lemma 1, internal layer optimization problem, which can relax, is：
Based on CRB, above formula can be rewritten as：
Deletion takes real part operator Re { }, is due to each in above formula and item is real number.
δ is only relied upon from the denominator of kth item in above formula, and formulakWithTwo, thus it is of equal value the problem of in above formula In, under corresponding constraint, maximize and formula in each single item, be represented by：
It should be noted that due toUnderstandFor indefinite matrix, therefore on Formula is difficult to solve, to solve this problem, to RSUsing diagonal loading technique, i.e.,：
Wherein, ε ＜ ＜ λmax(RS) it is load factor, λmax() representing matrix eigenvalue of maximum, below in l-G simulation test, Select ε=λmax(RS)/1000.Use respectivelyReplace the R in sane optimization problemS, can obtainIt is obvious thatWithRespectively forAnd δkIt is convex.
Thus, above formula can be rewritten as：
Similar, above formula can be write as the minimization problem of following two independence：
s.t. ||δk||F≤ζk
Two above problem can be solved by following lemma 2：
Hermitian matrix is assumed in lemma 2.'sThen and if only ifWhen,Wherein, Δ C=A-BHC-1The Schur that B is C in Z is mended.
By quoting lemma 2, two above problem can clearly be converted into following SDP problems：
Wherein, t is auxiliary variable.
By more than two formulas obtainWithBring into sane optimization problem, consider outside optimization problem.
2) solution of outside optimization problem
The present invention is the outside optimization problem of solution using following proposition
Proposition：Using matrix manipulation, the constraint in sane optimization problem can be equivalent to following LMI：
Wherein
Using lemma 2 and above-mentioned proposition is combined, outer layer optimization problem can be expressed as following SDP problems：
Wherein, X is an auxiliary variable.
After optimal E is obtained, under least square meaning, RSFollowing model construction can be passed through：
s.t. tr(RS)=LP
Similar to discussed above, above formula can be equivalent to following SDP problems：
tr(RS)=LP
3) iterative algorithm
Given WCM initial values, δk,And RSOptimized by following steps：
1. solving internal layer SDP problems obtains optimal δk,
2. solving outer layer SDP problems obtains E；
Repeat step 1. 2., until CRB is no longer substantially reduced.Hereafter, model constructed by solution can obtain RS。
The effect of the present invention can be further illustrated by following emulation：
Simulated conditions：
MIMO radar is that 3 hairs 3 are received, and using two MIMO radar systems, its antenna configuration is respectively：MIMO radar (0.5, 0.5), MIMO radar (1.5,0.5), the array element spacing in the numeral expression transmitter and receiver in bracket is (with wavelength here For unit).Systematic sampling points are 256.The definition of array signal to noise ratio isSpan is that -10dB arrives 30dB. Wherein, P refers to total transmission power,For additive white Gaussian noise variance.Modeling clutter is discrete sampling, and its RCS is modeled as independence It is with the gaussian random variable vector of distribution, average zero, varianceAnd assume fixed in coherent processing inteval. Clutter signal to noise ratio is defined asEqual to 30dB.- 5°There are a strong jamming, signal to noise ratio 60dB in direction.Only θ= There is the point target that a reflectance factor is 1 at 20 °.In following emulation, it is assumed that two kinds of situations, one is only considering that initial angle is estimated Error be present in meter；The second is only consider correction error present in transmitting-receiving array.
Emulation content：
A：Uncertain situation be present in initial angle estimation
Assuming that the uncertainty of initial angular estimation is Δ θ=[- 3 °, 3 °], i.e.,WhereinFor θ estimation, By calculating, data are obtained：MIMO (0.5,0.5) is ζ=5.4382, σ=7.6593, MIMO (1.5,0.5) be ζ= 27.6329 σ=29.6754.
Fig. 3 is optimal launching beam directional diagram under the conditions of ASNR=10dB.It is observed that transmission signal beam pattern Peak value be located at around target location, it means that, under the convex uncertain worst case systematic parameter estimation performance can obtain Improve.Further, since sparse emission array, graing lobe situation occurs in MIMO radar (1.5,0.5), as shown in Fig. 3 (b).
Fig. 4 is the CRB that changes obtained by proposed algorithm and uncorrelated waveform with ASNR.It is obvious that CRB is with ASNR's Increase and reduce.Furthermore, it is possible to it was observed that, parameter Estimation performance is better than uncorrelated ripple under the worst case that institute's extracting method obtains Shape.Moreover, with ASNR increase, CRB obtained by institute's extracting method is progressive in uncorrelated waveform.In addition, MIMO thunders shown in Fig. 4 (b) CRB up to (1.5,0.5) is significantly lower than the CRB of MIMO radar (0.5,0.5) shown in Fig. 4 (a).
B：Correction error be present in transmitting-receiving array
In this case, either transmitting and receiving array be assumed to have correction error (amplitude of sensor and Phase error and site error).Each element of transmitting and receiving array steering vector is disturbed by a disturbance variable, should Disturbance variable is the Cyclic Symmetry multiple Gauss stochastic variable of zero-mean, and variance isAfter calculating, obtain MIMO (0.5, 0.5) ζ=13.4764, σ=14.5712, MIMO (1.5,0.5) ζ=29.8362, σ=32.6573.
Fig. 5 features the optimal launching beam directional diagram that ASNR=10dB is obtained.From Fig. 5, it can be deduced that similar in appearance to Fig. 3's Conclusion.With the CRB under the worst case of ASNR changes as shown in fig. 6, being obtained from Fig. 6 obtained by carried algorithm and uncorrelated waveform Conclusion be similar to Fig. 4.
In summary, the present invention is directed to waveform optimization method under clutter conditions and initial parameter evaluated error sensitivity is asked Topic, it is proposed that the sane waveform optimization method based on the convex uncertain collection of parameter, and proposed for this complex nonlinear optimization problem It is a kind of based on the iterative method diagonally loaded.To improve the parameter Estimation robustness of MIMO radar system under clutter conditions Can, error condition be present first against direction of arrival angle and be modeled in the present invention, and by this parameter estimating error convex set explicitly It is included into waveform optimization problem, to solve this nonlinear optimal problem, the present invention proposes a kind of based on the iteration diagonally loaded Method carries out alternative optimization to transmitted waveform and parameter estimating error, to obtain optimal transmitted waveform covariance matrix.Repeatedly It is Semidefinite Programming that each step in generation, which can all be based on diagonal loading relaxation, it is hereby achieved that Efficient Solution.Begged for based on more than By understanding, institute's extracting method of the present invention can be the sane performance for improving radar parameter by designing transmitted waveform in engineer applied and estimating Solid theory is provided with realizing foundation.

Claims (1)

1. improve the sane waveform design method of the worst parameter Estimation performance of MIMO radar, it is characterised in that this method is included such as Lower step：
Step 1: structure MIMO radar receipt signal model
Assuming that MIMO radar reception signal is：
<mrow> <mi>Y</mi> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>K</mi> </munderover> <msub> <mi>&amp;beta;</mi> <mi>k</mi> </msub> <mi>a</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <msup> <mi>v</mi> <mi>T</mi> </msup> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mi>S</mi> <mo>+</mo> <msubsup> <mo>&amp;Integral;</mo> <mrow> <mo>-</mo> <mi>&amp;pi;</mi> </mrow> <mi>&amp;pi;</mi> </msubsup> <mi>&amp;rho;</mi> <mrow> <mo>(</mo> <mi>&amp;theta;</mi> <mo>)</mo> </mrow> <msub> <mi>a</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mi>&amp;theta;</mi> <mo>)</mo> </mrow> <msubsup> <mi>v</mi> <mi>c</mi> <mi>T</mi> </msubsup> <mrow> <mo>(</mo> <mi>&amp;theta;</mi> <mo>)</mo> </mrow> <mi>S</mi> <mi>d</mi> <mi>&amp;theta;</mi> <mo>+</mo> <mi>W</mi> </mrow>
Wherein,To be proportional to target radar cross section RCS complex magnitude,For target location parameter, K is number of targets Mesh, ρ (θ) are the reflectance factor in θ opening position clutter blocks, and W represents interference noise, and each column is separate and with distribution circle pair Claiming multiple Gauss random vector, there is zero-mean, its covariance B is unknown,For transmission signal matrix, a (θk) and v(θk) represent to receive respectively, launch steering vector, it is embodied as：
<mrow> <mi>a</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>&amp;lsqb;</mo> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mn>2</mn> <msub> <mi>&amp;pi;f</mi> <mn>0</mn> </msub> <msub> <mi>&amp;tau;</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>,</mo> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mn>2</mn> <msub> <mi>&amp;pi;f</mi> <mn>0</mn> </msub> <msub> <mi>&amp;tau;</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>,</mo> <mo>...</mo> <mo>,</mo> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mn>2</mn> <msub> <mi>&amp;pi;f</mi> <mn>0</mn> </msub> <msub> <mi>&amp;tau;</mi> <msub> <mi>M</mi> <mi>r</mi> </msub> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>&amp;rsqb;</mo> </mrow> <mi>T</mi> </msup> </mrow>
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In formula, f0For carrier frequency, τm(θk), m=1,2 ... MrWithN=1,2 ... MtFor transmission time, ac(θ) and vc (θ) represents θ respectivelykLocate reception and the transmitting steering vector of target；
If rang ring is divided into NC(NC＞ ＞ MtMr) individual resolution cell, MIMO radar receipt signal model is rewritten as
<mrow> <mi>Y</mi> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>K</mi> </munderover> <msub> <mi>&amp;beta;</mi> <mi>k</mi> </msub> <mi>a</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <msup> <mi>v</mi> <mi>T</mi> </msup> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mi>S</mi> <mo>+</mo> <msub> <mi>H</mi> <mi>c</mi> </msub> <mi>S</mi> <mo>+</mo> <mi>W</mi> </mrow>
Wherein,Represent clutter transmission function, ρ (θi) it is θiLocate the reflectance factor of clutter block, ac(θi) And vc(θi) θ is represented respectivelyiLocate reception, the transmitting steering vector of clutter block；vec(Hc) it is the same multiple Gauss random vector being distributed, Its average is zero, and covariance is For positioned at θiIt is miscellaneous The power of ripple block；E [] is to ask expectation computing to accord with；
Step 2: sane waveform optimization model of the structure based on CRB
Consider unknown parameter θ=[θ1,θ2,…,θK]T、Under the conditions of CRB, by deriving, this CRB can be expressed as follows：
<mrow> <mi>C</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>Re</mi> <mrow> <mo>(</mo> <msub> <mi>F</mi> <mn>11</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>Re</mi> <mrow> <mo>(</mo> <msub> <mi>F</mi> <mn>12</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <mi>Im</mi> <mrow> <mo>(</mo> <msub> <mi>F</mi> <mn>12</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mi>Re</mi> <mi>T</mi> </msup> <mrow> <mo>(</mo> <msub> <mi>F</mi> <mn>12</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>Re</mi> <mrow> <mo>(</mo> <msub> <mi>F</mi> <mn>22</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <mi>Im</mi> <mrow> <mo>(</mo> <msub> <mi>F</mi> <mn>22</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msup> <mi>Im</mi> <mi>T</mi> </msup> <mrow> <mo>(</mo> <msub> <mi>F</mi> <mn>12</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msup> <mi>Im</mi> <mi>T</mi> </msup> <mrow> <mo>(</mo> <msub> <mi>F</mi> <mn>22</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>Re</mi> <mrow> <mo>(</mo> <msub> <mi>F</mi> <mn>22</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow>
Wherein,
<mrow> <msub> <mrow> <mo>&amp;lsqb;</mo> <msub> <mi>F</mi> <mn>11</mn> </msub> <mo>&amp;rsqb;</mo> </mrow> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>&amp;beta;</mi> <mi>i</mi> <mo>*</mo> </msubsup> <msub> <mi>&amp;beta;</mi> <mi>j</mi> </msub> <msubsup> <mover> <mi>h</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>i</mi> <mi>H</mi> </msubsup> <mo>&amp;lsqb;</mo> <msup> <mrow> <mo>(</mo> <mi>I</mi> <mo>+</mo> <mo>(</mo> <mrow> <msub> <mi>R</mi> <mi>S</mi> </msub> <mo>&amp;CircleTimes;</mo> <msup> <mi>B</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> <msub> <mi>R</mi> <msub> <mi>H</mi> <mi>c</mi> </msub> </msub> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>S</mi> </msub> <mo>&amp;CircleTimes;</mo> <msup> <mi>B</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mover> <mi>h</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>j</mi> </msub> </mrow>
<mrow> <msub> <mrow> <mo>&amp;lsqb;</mo> <msub> <mi>F</mi> <mn>12</mn> </msub> <mo>&amp;rsqb;</mo> </mrow> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>&amp;beta;</mi> <mi>i</mi> <mo>*</mo> </msubsup> <msubsup> <mover> <mi>h</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>i</mi> <mi>H</mi> </msubsup> <mo>&amp;lsqb;</mo> <msup> <mrow> <mo>(</mo> <mi>I</mi> <mo>+</mo> <mo>(</mo> <mrow> <msub> <mi>R</mi> <mi>S</mi> </msub> <mo>&amp;CircleTimes;</mo> <msup> <mi>B</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> <msub> <mi>R</mi> <msub> <mi>H</mi> <mi>c</mi> </msub> </msub> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>S</mi> </msub> <mo>&amp;CircleTimes;</mo> <msup> <mi>B</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mi>h</mi> <mi>j</mi> </msub> </mrow>
<mrow> <msub> <mrow> <mo>&amp;lsqb;</mo> <msub> <mi>F</mi> <mn>22</mn> </msub> <mo>&amp;rsqb;</mo> </mrow> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>h</mi> <mi>i</mi> <mi>H</mi> </msubsup> <mo>&amp;lsqb;</mo> <msup> <mrow> <mo>(</mo> <mi>I</mi> <mo>+</mo> <mo>(</mo> <mrow> <msub> <mi>R</mi> <mi>S</mi> </msub> <mo>&amp;CircleTimes;</mo> <msup> <mi>B</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> <msub> <mi>R</mi> <msub> <mi>H</mi> <mi>c</mi> </msub> </msub> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>S</mi> </msub> <mo>&amp;CircleTimes;</mo> <msup> <mi>B</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mi>h</mi> <mi>j</mi> </msub> </mrow>
In formula, RS=SSH,For positive semidefinite Hermitian matrix, E [] is to ask expectation computing to accord with；
Only consider direction of arrival angle, i.e. influence of the θ evaluated errors to systematic function can be to k-th of destination channel matrix modeling such as Under：
<mrow> <msub> <mover> <mi>h</mi> <mo>~</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <msub> <mi>h</mi> <mi>k</mi> </msub> <mo>+</mo> <msub> <mi>&amp;delta;</mi> <mi>k</mi> </msub> </mrow>
Wherein,hkRespectively actual and hypothesis k-th of destination channel matrix, δkForError, belong to following convex Collection：
AndWherein, Respectively hkVector reciprocal that is real and assuming,ForError, category In following convex set：
Wherein, ζk,σkRespectively error deltak,The mould upper bound, | | | |FFor general Frobenius norm operators；
Based on the above, improving the sane waveform optimization problem of worst condition parameter Estimation performance under clutter conditions can state For：Under the constraint on WCM, based on Parameter uncertainties convex setOptimize WCM to minimize the CRB under worst case； Under Trace-opt criterions, optimization problem can be described as：
<mrow> <munder> <mi>min</mi> <msub> <mi>R</mi> <mi>S</mi> </msub> </munder> <munder> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mrow> <msubsup> <mrow> <mo>{</mo> <msub> <mi>&amp;delta;</mi> <mi>k</mi> </msub> <mo>}</mo> </mrow> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>K</mi> </msubsup> <mo>,</mo> <msubsup> <mrow> <mo>{</mo> <msub> <mover> <mi>&amp;delta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>k</mi> </msub> <mo>}</mo> </mrow> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>K</mi> </msubsup> </mrow> </munder> <mi>t</mi> <mi>r</mi> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow>
tr(RS)=LP
RS≥0
Wherein, P represents total transmission power；The 3rd constraint establishment is due to that each transmitter unit transmission power is impossible in formula Less than zero；L is transmission signal matrixIn each transmitted waveform dimension, namely waveform sampling number；
Step 3: the solution of sane waveform internal layer optimization problem
The solution of internal layer optimization problem is based on following lemma 1：
The positive semidefinite hermitian matrix that A is a M × M is assumed in lemma 1., then following inequality is set up：When and Equation is set up only when A is diagonal matrix；According to lemma 1, internal layer optimization problem, which can relax, is：
<mrow> <munder> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mrow> <msubsup> <mrow> <mo>{</mo> <msub> <mi>&amp;delta;</mi> <mi>k</mi> </msub> <mo>}</mo> </mrow> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>K</mi> </msubsup> <mo>,</mo> <msubsup> <mrow> <mo>{</mo> <msub> <mover> <mi>&amp;delta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>k</mi> </msub> <mo>}</mo> </mrow> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>K</mi> </msubsup> </mrow> </munder> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>K</mi> </munderover> <mfrac> <mn>1</mn> <msub> <mrow> <mo>&amp;lsqb;</mo> <mn>2</mn> <mi>Re</mi> <mrow> <mo>(</mo> <mi>F</mi> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> </mrow> <mrow> <mi>k</mi> <mi>k</mi> </mrow> </msub> </mfrac> </mrow>
Wherein, F is the F matrix during above-mentioned CRB is represented, subscript kk is then k-th of diagonal element for taking matrix 2Re (F)；
Based on CRB, above formula can be rewritten as：
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <munder> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mrow> <msubsup> <mrow> <mo>{</mo> <msub> <mi>&amp;delta;</mi> <mi>k</mi> </msub> <mo>}</mo> </mrow> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>K</mi> </msubsup> <mo>,</mo> <msubsup> <mrow> <mo>{</mo> <msub> <mover> <mi>&amp;delta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>k</mi> </msub> <mo>}</mo> </mrow> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>K</mi> </msubsup> </mrow> </munder> </mtd> <mtd> <mrow> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>K</mi> </munderover> <mfrac> <mn>1</mn> <mrow> <msubsup> <mi>&amp;beta;</mi> <mi>k</mi> <mo>*</mo> </msubsup> <msubsup> <mover> <mover> <mi>h</mi> <mo>~</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> <mi>k</mi> <mi>H</mi> </msubsup> <mo>&amp;lsqb;</mo> <msup> <mrow> <mo>(</mo> <mi>I</mi> <mo>+</mo> <mo>(</mo> <mrow> <msub> <mi>R</mi> <mi>S</mi> </msub> <mo>&amp;CircleTimes;</mo> <msup> <mi>B</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> <msub> <mi>R</mi> <msub> <mi>H</mi> <mi>c</mi> </msub> </msub> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>S</mi> </msub> <mo>&amp;CircleTimes;</mo> <msup> <mi>B</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mover> <mover> <mi>h</mi> <mo>~</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> <mi>k</mi> </msub> <msub> <mi>&amp;beta;</mi> <mi>k</mi> </msub> <mo>+</mo> <msubsup> <mover> <mi>h</mi> <mo>~</mo> </mover> <mi>k</mi> <mi>H</mi> </msubsup> <mo>&amp;lsqb;</mo> <msup> <mrow> <mo>(</mo> <mi>I</mi> <mo>+</mo> <mo>(</mo> <mrow> <msub> <mi>R</mi> <mi>S</mi> </msub> <mo>&amp;CircleTimes;</mo> <msup> <mi>B</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> <msub> <mi>R</mi> <msub> <mi>H</mi> <mi>c</mi> </msub> </msub> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>S</mi> </msub> <mo>&amp;CircleTimes;</mo> <msup> <mi>B</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mover> <mi>h</mi> <mo>~</mo> </mover> <mi>k</mi> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mfenced>
Deletion takes real part operator Re { }, is due to each in above formula and item is real number；
δ is only relied upon from the denominator of kth item in above formula, and formulakWithTwo, therefore be equivalent to the problem of in above formula, in phase Under the constraint answered, each single item in maximization and formula, it is represented by：
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <munder> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mrow> <msubsup> <mrow> <mo>{</mo> <msub> <mi>&amp;delta;</mi> <mi>k</mi> </msub> <mo>}</mo> </mrow> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>K</mi> </msubsup> <mo>,</mo> <msubsup> <mrow> <mo>{</mo> <msub> <mover> <mi>&amp;delta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>k</mi> </msub> <mo>}</mo> </mrow> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>K</mi> </msubsup> </mrow> </munder> </mtd> <mtd> <mfrac> <mn>1</mn> <mrow> <msubsup> <mi>&amp;beta;</mi> <mi>k</mi> <mo>*</mo> </msubsup> <msubsup> <mover> <mover> <mi>h</mi> <mo>~</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> <mi>k</mi> <mi>H</mi> </msubsup> <mo>&amp;lsqb;</mo> <msup> <mrow> <mo>(</mo> <mi>I</mi> <mo>+</mo> <mo>(</mo> <mrow> <msub> <mi>R</mi> <mi>S</mi> </msub> <mo>&amp;CircleTimes;</mo> <msup> <mi>B</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> <msub> <mi>R</mi> <msub> <mi>H</mi> <mi>c</mi> </msub> </msub> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>S</mi> </msub> <mo>&amp;CircleTimes;</mo> <msup> <mi>B</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mover> <mover> <mi>h</mi> <mo>~</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> <mi>k</mi> </msub> <msub> <mi>&amp;beta;</mi> <mi>k</mi> </msub> <mo>+</mo> <msubsup> <mover> <mi>h</mi> <mo>~</mo> </mover> <mi>k</mi> <mi>H</mi> </msubsup> <mo>&amp;lsqb;</mo> <msup> <mrow> <mo>(</mo> <mi>I</mi> <mo>+</mo> <mo>(</mo> <mrow> <msub> <mi>R</mi> <mi>S</mi> </msub> <mo>&amp;CircleTimes;</mo> <msup> <mi>B</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> <msub> <mi>R</mi> <msub> <mi>H</mi> <mi>c</mi> </msub> </msub> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>S</mi> </msub> <mo>&amp;CircleTimes;</mo> <msup> <mi>B</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mover> <mi>h</mi> <mo>~</mo> </mover> <mi>k</mi> </msub> </mrow> </mfrac> </mtd> </mtr> </mtable> </mfenced>
To solve above formula, to RSUsing diagonal loading technique, i.e.,：
<mrow> <msub> <mover> <mi>R</mi> <mo>~</mo> </mover> <mi>S</mi> </msub> <mo>=</mo> <msub> <mi>R</mi> <mi>S</mi> </msub> <mo>+</mo> <mi>&amp;epsiv;</mi> <mi>I</mi> <mo>&gt;</mo> <mn>0</mn> </mrow>
Wherein, ε ＜ ＜ λmax(RS) it is load factor, λmax() representing matrix eigenvalue of maximum, select ε=λmax(RS)/1000； Use respectivelyReplace the R in sane optimization problemS, can obtain WithRespectively for And δkIt is convex；
Thus, above formula can be rewritten as：
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <munder> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mrow> <msub> <mover> <mi>&amp;delta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>k</mi> </msub> <mo>,</mo> <msub> <mi>&amp;delta;</mi> <mi>k</mi> </msub> </mrow> </munder> </mtd> <mtd> <mrow> <msubsup> <mi>&amp;beta;</mi> <mi>k</mi> <mo>*</mo> </msubsup> <msubsup> <mover> <mover> <mi>h</mi> <mo>~</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> <mi>k</mi> <mi>H</mi> </msubsup> <mo>&amp;lsqb;</mo> <msup> <mrow> <mo>(</mo> <mi>I</mi> <mo>+</mo> <mo>(</mo> <mrow> <msub> <mover> <mi>R</mi> <mo>~</mo> </mover> <mi>S</mi> </msub> <mo>&amp;CircleTimes;</mo> <msup> <mi>B</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> <msub> <mi>R</mi> <msub> <mi>H</mi> <mi>c</mi> </msub> </msub> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mover> <mi>R</mi> <mo>~</mo> </mover> <mi>S</mi> </msub> <mo>&amp;CircleTimes;</mo> <msup> <mi>B</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mover> <mover> <mi>h</mi> <mo>~</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> <mi>k</mi> </msub> <msub> <mi>&amp;beta;</mi> <mi>k</mi> </msub> <mo>+</mo> <msubsup> <mover> <mi>h</mi> <mo>~</mo> </mover> <mi>k</mi> <mi>H</mi> </msubsup> <mo>&amp;lsqb;</mo> <msup> <mrow> <mo>(</mo> <mi>I</mi> <mo>+</mo> <mo>(</mo> <mrow> <msub> <mover> <mi>R</mi> <mo>~</mo> </mover> <mi>S</mi> </msub> <mo>&amp;CircleTimes;</mo> <msup> <mi>B</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> <msub> <mi>R</mi> <msub> <mi>H</mi> <mi>c</mi> </msub> </msub> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mover> <mi>R</mi> <mo>~</mo> </mover> <mi>S</mi> </msub> <mo>&amp;CircleTimes;</mo> <msup> <mi>B</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mover> <mi>h</mi> <mo>~</mo> </mover> <mi>k</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced>
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>s</mi> <mo>.</mo> <mi>t</mi> <mo>.</mo> </mrow> </mtd> <mtd> <mrow> <mo>|</mo> <mo>|</mo> <msub> <mover> <mi>&amp;delta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>k</mi> </msub> <mo>|</mo> <msub> <mo>|</mo> <mi>F</mi> </msub> <mo>&amp;le;</mo> <msub> <mi>&amp;sigma;</mi> <mi>k</mi> </msub> <mo>,</mo> <mo>|</mo> <mo>|</mo> <msub> <mi>&amp;delta;</mi> <mi>k</mi> </msub> <mo>|</mo> <msub> <mo>|</mo> <mi>F</mi> </msub> <mo>&amp;le;</mo> <msub> <mi>&amp;zeta;</mi> <mi>k</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced>
Above formula can tear the minimization problem for being write as following two independence open：
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <munder> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <msub> <mover> <mi>&amp;delta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>k</mi> </msub> </munder> </mtd> <mtd> <mrow> <msubsup> <mi>&amp;beta;</mi> <mi>k</mi> <mo>*</mo> </msubsup> <msubsup> <mover> <mover> <mi>h</mi> <mo>~</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> <mi>k</mi> <mi>H</mi> </msubsup> <mo>&amp;lsqb;</mo> <msup> <mrow> <mo>(</mo> <mi>I</mi> <mo>+</mo> <mo>(</mo> <mrow> <msub> <mover> <mi>R</mi> <mo>~</mo> </mover> <mi>S</mi> </msub> <mo>&amp;CircleTimes;</mo> <msup> <mi>B</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> <msub> <mi>R</mi> <msub> <mi>H</mi> <mi>c</mi> </msub> </msub> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mover> <mi>R</mi> <mo>~</mo> </mover> <mi>S</mi> </msub> <mo>&amp;CircleTimes;</mo> <msup> <mi>B</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mover> <mover> <mi>h</mi> <mo>~</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> <mi>k</mi> </msub> <msub> <mi>&amp;beta;</mi> <mi>k</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced>
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>s</mi> <mo>.</mo> <mi>t</mi> <mo>.</mo> </mrow> </mtd> <mtd> <mrow> <mo>|</mo> <mo>|</mo> <msub> <mover> <mi>&amp;delta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>k</mi> </msub> <mo>|</mo> <msub> <mo>|</mo> <mi>F</mi> </msub> <mo>&amp;le;</mo> <msub> <mi>&amp;sigma;</mi> <mi>k</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced>
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <munder> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <msub> <mi>&amp;delta;</mi> <mi>k</mi> </msub> </munder> </mtd> <mtd> <mrow> <msubsup> <mover> <mi>h</mi> <mo>~</mo> </mover> <mi>k</mi> <mi>H</mi> </msubsup> <mo>&amp;lsqb;</mo> <msup> <mrow> <mo>(</mo> <mi>I</mi> <mo>+</mo> <mo>(</mo> <mrow> <msub> <mover> <mi>R</mi> <mo>~</mo> </mover> <mi>S</mi> </msub> <mo>&amp;CircleTimes;</mo> <msup> <mi>B</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> <msub> <mi>R</mi> <msub> <mi>H</mi> <mi>c</mi> </msub> </msub> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mover> <mi>R</mi> <mo>~</mo> </mover> <mi>S</mi> </msub> <mo>&amp;CircleTimes;</mo> <msup> <mi>B</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mover> <mi>h</mi> <mo>~</mo> </mover> <mi>k</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced>
s.t.||δk||F≤ζk
Above-mentioned two minimization problem can be solved by following lemma 2：
Lemma 2, assume hermitian matrix'sDuring then and if only if Δ C >=0, Z >=0, wherein, Δ C=A- BHC-1The Schur that B is C in Z is mended；
By quoting lemma 2, above-mentioned two minimization problem can be converted into following SDP problems 1：
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <munder> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mrow> <mi>t</mi> <mo>,</mo> <msub> <mover> <mi>&amp;delta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>k</mi> </msub> </mrow> </munder> </mtd> <mtd> <mi>t</mi> </mtd> </mtr> </mtable> </mfenced>
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>s</mi> <mo>.</mo> <mi>t</mi> <mo>.</mo> </mrow> </mtd> <mtd> <mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>&amp;sigma;</mi> <mi>k</mi> </msub> </mtd> <mtd> <msubsup> <mover> <mi>&amp;delta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>k</mi> <mi>H</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>&amp;delta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>k</mi> </msub> </mtd> <mtd> <mi>I</mi> </mtd> </mtr> </mtable> </mfenced> <mo>&amp;GreaterEqual;</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mfenced>
<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mi>t</mi> </mtd> <mtd> <mrow> <msubsup> <mi>&amp;beta;</mi> <mi>k</mi> <mo>*</mo> </msubsup> <msubsup> <mover> <mover> <mi>h</mi> <mo>~</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> <mi>k</mi> <mi>H</mi> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;beta;</mi> <mi>k</mi> </msub> <msub> <mover> <mover> <mi>h</mi> <mo>~</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> <mi>k</mi> </msub> </mrow> </mtd> <mtd> <mrow> <msubsup> <mover> <mi>R</mi> <mo>~</mo> </mover> <mi>S</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>&amp;CircleTimes;</mo> <mi>B</mi> <mo>+</mo> <msub> <mi>R</mi> <msub> <mi>H</mi> <mi>c</mi> </msub> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>&amp;GreaterEqual;</mo> <mn>0</mn> </mrow>
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <munder> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mrow> <mi>t</mi> <mo>,</mo> <msub> <mi>&amp;delta;</mi> <mi>k</mi> </msub> </mrow> </munder> </mtd> <mtd> <mi>t</mi> </mtd> </mtr> </mtable> </mfenced>
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>s</mi> <mo>.</mo> <mi>t</mi> <mo>.</mo> </mrow> </mtd> <mtd> <mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>&amp;zeta;</mi> <mi>k</mi> </msub> </mtd> <mtd> <msubsup> <mi>&amp;delta;</mi> <mi>k</mi> <mi>H</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&amp;delta;</mi> <mi>k</mi> </msub> </mtd> <mtd> <mi>I</mi> </mtd> </mtr> </mtable> </mfenced> <mo>&amp;GreaterEqual;</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mfenced>
<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mi>t</mi> </mtd> <mtd> <msubsup> <mover> <mi>h</mi> <mo>~</mo> </mover> <mi>k</mi> <mi>H</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>h</mi> <mo>~</mo> </mover> <mi>k</mi> </msub> </mtd> <mtd> <mrow> <msubsup> <mover> <mi>R</mi> <mo>~</mo> </mover> <mi>S</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>&amp;CircleTimes;</mo> <mi>B</mi> <mo>+</mo> <msub> <mi>R</mi> <msub> <mi>H</mi> <mi>c</mi> </msub> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>&amp;GreaterEqual;</mo> <mn>0</mn> </mrow>
Wherein, t is auxiliary variable；
By more than two formulas obtainWithBring into sane optimization problem, consider outside optimization problem；
Step 4: the solution of sane waveform outer layer optimization problem
Outside optimization problem is solved using following proposition
Proposition：Using matrix manipulation, the constraint in sane optimization problem can be equivalent to following LMI：
WhereinRepresenting matrix, L is transmitted waveform hits, λmin() represents to take the minimal eigenvalue of matrix；
Using lemma 2 and above-mentioned proposition is combined, outer layer optimization problem can be expressed as following SDP problems：
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <munder> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mrow> <mi>X</mi> <mo>,</mo> <mi>E</mi> </mrow> </munder> </mtd> <mtd> <mrow> <mi>t</mi> <mi>r</mi> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced>
<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mi>X</mi> </mtd> <mtd> <mi>I</mi> </mtd> </mtr> <mtr> <mtd> <mi>I</mi> </mtd> <mtd> <mi>F</mi> </mtd> </mtr> </mtable> </mfenced> <mo>&amp;GreaterEqual;</mo> <mn>0</mn> </mrow>
Wherein, X is an auxiliary variable, and F is the F matrix in CRB statements；
After optimal E is obtained, under least square meaning, RSFollowing model construction can be passed through：
<mrow> <msub> <mi>R</mi> <mi>S</mi> </msub> <mo>=</mo> <mi>arg</mi> <munder> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <msub> <mi>R</mi> <mi>S</mi> </msub> </munder> <mo>|</mo> <mo>|</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>E</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>-</mo> <msub> <mi>R</mi> <msub> <mi>H</mi> <mi>c</mi> </msub> </msub> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>-</mo> <msub> <mover> <mi>R</mi> <mo>~</mo> </mover> <mi>S</mi> </msub> <mo>&amp;CircleTimes;</mo> <msup> <mi>B</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>|</mo> <msub> <mo>|</mo> <mi>F</mi> </msub> </mrow>
s.t.tr(RS)=LP
RS≥0
Using lemma 2 and above-mentioned proposition is combined, above formula can be equivalent to following SDP problems 2：
<mrow> <munder> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mrow> <msub> <mi>R</mi> <mi>s</mi> </msub> <mo>,</mo> <mi>t</mi> </mrow> </munder> <mi>t</mi> </mrow>
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>s</mi> <mo>.</mo> <mi>t</mi> <mo>.</mo> </mrow> </mtd> <mtd> <mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mi>t</mi> </mtd> <mtd> <mrow> <msup> <mi>vec</mi> <mi>H</mi> </msup> <mrow> <mo>(</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>E</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>-</mo> <msub> <mi>R</mi> <msub> <mi>H</mi> <mi>c</mi> </msub> </msub> </mrow> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>-</mo> <msub> <mover> <mi>R</mi> <mo>~</mo> </mover> <mi>S</mi> </msub> <mo>&amp;CircleTimes;</mo> <msup> <mi>B</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>v</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>E</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>-</mo> <msub> <mi>R</mi> <msub> <mi>H</mi> <mi>c</mi> </msub> </msub> </mrow> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>-</mo> <msub> <mover> <mi>R</mi> <mo>~</mo> </mover> <mi>S</mi> </msub> <mo>&amp;CircleTimes;</mo> <msup> <mi>B</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mi>I</mi> </mtd> </mtr> </mtable> </mfenced> <mo>&amp;GreaterEqual;</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mfenced>
tr(RS)=LP
RS≥0
Step 5: sane waveform optimization problem is solved using alternative manner
Step 5.1, given waveform covariance matrix initial value；
Step 5.2, above-mentioned SDP problems 1 are solved to obtain optimal δk,
Step 5.3, SDP problems 2 are solved to obtain optimal E；
Step 5.4, return to step 5.2 iteration again, until CRB is no longer substantially reduced；
Step 6: being based on least square method, optimal waveform covariance matrix is reconstructed, R can be obtainedS。
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