CN113702961A - MIMO-STAP radar transmit-receive space-time resource configuration method in clutter scene - Google Patents

MIMO-STAP radar transmit-receive space-time resource configuration method in clutter scene Download PDF

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CN113702961A
CN113702961A CN202110801357.XA CN202110801357A CN113702961A CN 113702961 A CN113702961 A CN 113702961A CN 202110801357 A CN202110801357 A CN 202110801357A CN 113702961 A CN113702961 A CN 113702961A
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clutter
array element
vector
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王洪雁
周贺
袁海
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Zhejiang Sci Tech University ZSTU
Zhejiang University of Science and Technology ZUST
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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
    • G01S13/00—Systems using the reflection or reradiation of radio waves, e.g. radar systems; Analogous systems using reflection or reradiation of waves whose nature or wavelength is irrelevant or unspecified
    • G01S13/02—Systems using reflection of radio waves, e.g. primary radar systems; Analogous systems
    • 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
    • G01S7/28—Details of pulse systems
    • G01S7/285—Receivers
    • 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
    • G01S7/41—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S13/00 using analysis of echo signal for target characterisation; Target signature; Target cross-section
    • G01S7/415—Identification of targets based on measurements of movement associated with the target

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Abstract

The invention provides a method for configuring space-time resources for receiving and transmitting an MIMO-STAP radar in a clutter scene. Firstly, constructing an MIMO-STAP radar signal model under a joint sparse optimization structure; secondly, taking a maximized output signal-to-noise-and-noise ratio (SCNR) as a performance index, combining sparse design of a transmitting waveform, a receiving weight, a transmitting-receiving array element and a pulse interval, solving the receiving weight based on a Minimum Variance Distortionless Response (MVDR) criterion, performing antenna pulse selection by Successive Convex Approximation (SCA), and optimizing the transmitting waveform by adopting a semi-definite programming (SDP) and randomization method; finally, an alternating iteration method is provided to solve the complex nonlinear optimization problem. Under the condition of not influencing the target detection performance, the algorithm provided by the invention can better select a small number of transmitting and receiving array elements and pulses so as to reduce the hardware cost and energy consumption of the system.

Description

MIMO-STAP radar transmit-receive space-time resource configuration method in clutter scene
Technical Field
The invention belongs to the field of array processing, and particularly relates to a MIMO-STAP radar transmit-receive space-time resource allocation method in a clutter scene.
Background
In recent years, a Multiple-input Multiple-output (MIMO) radar has a waveform diversity characteristic, so that the spatial degree of freedom is significantly higher than that of a conventional phased array radar, and thus the MIMO radar has excellent target detection and parameter estimation performances, and is further widely concerned by engineering and scientific researchers. According to the array element spacing distribution, the MIMO radar can be divided into a centralized type and a distributed type, wherein the centralized MIMO radar has smaller array element spacing, and the waveform diversity can expand the array virtual aperture so as to improve the target parameter estimation performance; the distributed MIMO radar transmitting-receiving array element interval is large, and the waveform diversity can inhibit the target space flicker characteristic, so that the target detection performance is improved.
Space Time Adaptive Processing (STAP) techniques proposed by Brennan and Reed are known to be effective methods of suppressing clutter to improve radar performance. The STAP method is based on the space domain freedom degree obtained by the array and the time domain freedom degree obtained by the pulse train, and the space-time two-dimensional joint processing is carried out on the target and the clutter to improve the detection and estimation performance of the system. Based on the advantage that the spatial domain degree of freedom is obviously improved due to the expansion of the virtual aperture, Bliss and Forsythe introduce STAP into an MIMO radar system for the first time to enhance the clutter suppression capability of the system and further improve the detection estimation performance of the MIMO radar.
As can be seen from the above, the performance of the MIMO radar depends on the array configuration and the transmission waveform to a large extent, that is: array aperture, pulse spacing, and baseband signal. It should be noted that due to the limited space and power resources of the radar platform, the high software and hardware costs and energy consumption caused by the array aperture and the waveform diversity become bottlenecks that limit the application of MIMO radar engineering. In particular, for airborne MIMO radar, the narrow platform space limits the number of array elements so that the array configuration must be optimized to improve spatial processing performance while reducing hardware costs; secondly, the limited platform power consumption restricts the pulse number, so that the pulse configuration needs to be optimized to improve the time domain processing capability and reduce the energy consumption; finally, the waveform diversity performance depends on the baseband signal, which must be designed to optimally configure the system aperture and power distribution.
For the problem of optimizing array configuration to improve airspace degree of freedom and simultaneously reduce hardware cost, s.joshi proposes a convex optimization array element selection method, the proposed method relaxes the array element selection problem into a convex optimization model, and then uses a convex optimization tool to solve the model to optimize array element configuration, however, the method has higher computational complexity, based on which, h.godrich proposes a greedy-based heuristic algorithm for selecting array elements, and the proposed method can obviously reduce computational complexity but lacks rigorous mathematical proof; in addition, the h.h.kim designs sparse array elements using Compressed Sensing (CS) based MIMO-STAP model reduces computational complexity, but hardware cost is not reduced, because the use of dense sampling matrix in CS limits the number of measurements, while all antennas are needed. The above method only considers the array configuration and does not involve transmit pulse selection. Aiming at the problem, E.Tohid uses a Cramer-Rao Lower Bound (CRLB) as a performance index of array element pulse joint selection, and provides two array element pulse selection algorithms based on convex optimization and sub-model optimization. Under the condition of single interference, the X.Wang provides an iterative min-max algorithm to select the optimal antenna pulse pair based on the space spectrum phase relation number; for the case of the interference number limitation, x.wang expands single interference into multiple interference and maximizes an output Signal-to-Noise Ratio (SCNR) by selecting an optimal subset of antenna pulse pairs, and simultaneously analyzes the influence of a space-time structure on STAP performance in detail based on a spatial spectrum correlation coefficient. In addition, the purpose of reducing the hardware cost is also achieved by optimizing the STAP weight value by W.Shi and then designing sparse antenna pulse to maximize the output SCNR. The method only relates to array pulse configuration, baseband signal design is not considered, however, the baseband signal structure and the correlation greatly determine the virtual array aperture configuration and the power spatial distribution of the MIMO radar system, thereby affecting the system target detection and estimation performance. B.Tang jointly optimizes and expands the waveform and the receiving filter into MIMO radar space-time adaptive processing under the condition that the target prior information is known; for the case that the target direction and the doppler frequency are unknown, a.aubry proposes a design method for optimizing the output signal-to-interference-and-noise ratio transmission waveform and the doppler filter under the worst case. Based on a B.Tang MIMO-STAP model, J.Li jointly designs the transmitting waveform and the receiving filter of the airborne MIMO radar under the constant modulus and energy constraint, and provides a new sequence optimization method to maximize the output SCNR. In order to obtain good waveform characteristics, similarity constraint is added on the basis of J.Li by Shi and S, a new algorithm for jointly designing the transmitting waveform and the receiving filter of the MIMO-STAP radar is provided, and the target detection performance is also improved. It should be noted that none of the above mentioned consider the MIMO-STAP transceiving space-time resource configuration at the same time, and thus the system performance improvement is limited.
Disclosure of Invention
Aiming at the problems of overhigh hardware cost and overlarge energy consumption of an MIMO radar system, the invention provides a MIMO-STAP radar transmitting and receiving space-time resource allocation method in a clutter scene, namely a radar transmitting waveform, a receiving weight, a transmitting and receiving array element and pulse interval combined optimization method for improving the MIMO-STAP detection performance. Firstly, constructing an MIMO-STAP radar signal model under a joint sparse optimization structure; secondly, taking a maximized output signal-to-noise-and-noise ratio (SCNR) as a performance index, combining sparse design of a transmitting waveform, a receiving weight, a transmitting-receiving array element and a pulse interval, solving the receiving weight based on a Minimum Variance Distortionless Response (MVDR) criterion, performing antenna pulse selection by Successive Convex Approximation (SCA)), and optimizing the transmitting waveform by adopting a semi-positive programming (SDP) and randomization method; finally, an alternating iteration method is provided to solve the complex nonlinear optimization problem.
In order to achieve the above object, the present invention provides a method for configuring space-time resources for receiving and transmitting a MIMO-STAP radar in a clutter scene, which specifically includes:
establishing an MIMO-STAP model;
obtaining a constant modulus constraint and array element pulse selection model;
establishing a joint sparse optimization problem with constant modulus constraint and array element pulse selection based on a maximized output SCNR criterion to obtain a complex nonlinear optimization problem about an optimization variable;
and solving the complex nonlinear optimization problem by adopting an alternating iteration mode to obtain an optimal transmitting waveform, a receiving weight, a selection vector and a corresponding output SCNR.
Further, establishing an MIMO-STAP model specifically comprises:
the speed and the height of the loader are respectively set as v and HThe linear radial velocity of the mark loader is v0Target azimuth angle theta, elevation angle
Figure BDA0003164874180000021
Cone angle phi satisfying geometric relationship
Figure BDA0003164874180000022
The receiving and transmitting array is a short-distance parallel uniform linear array, the number of array elements is M and N respectively, the spacing between the array elements is d ═ lambda/2, and lambda is the radar working wavelength; baseband transmit signal
Figure BDA0003164874180000023
Wherein
Figure BDA0003164874180000024
Sampling a baseband signal of an M-th transmitting array element, wherein M is 1,2, …, M and L are sample numbers; the pulse number in the Coherent Processing Interval (CPI) is K, and the pulse repetition period is T;
considering the point target model, the target signal received by the MIMO radar in the kth pulse is expressed as:
Figure BDA0003164874180000025
wherein epsilontIs the target echo amplitude, fdt＝2(vcosφ+v0) T/lambda is the target Doppler frequency, j is an imaginary unit; a istTransmitting a steering vector for the target, btReceive steering vectors for the target, i.e.:
at＝[1,exp(j2πfst)，…，exp(j2π(M-1)fst)]T,bt＝[1,exp(j2πfst)，…，exp(j2π(N-1)fst)]T (2)
wherein f isstD cos phi/lambda is normalized spatial frequency; properties based on the Kronecker product as follows:
Figure BDA0003164874180000031
coherent processing interval CPI entryThe target received signal is represented as:
Figure BDA0003164874180000032
wherein the content of the first and second substances,
Figure BDA0003164874180000033
for the target space-time steering vector(s),
Figure BDA0003164874180000034
is the product of Kronecker; dt＝[1,exp(j2πfdt)，…，exp(j2π(K-1)fdt)]TA target Doppler steering vector; i isKRepresenting an identity matrix of dimension K.
Further, clutter modeling is N considering a clutter modelcAnd (3) superposing the clutter blocks, and expressing the clutter in the coherent processing interval CPI as:
Figure BDA0003164874180000035
wherein epsilonc,iRepresenting the amplitude, u, of the echo of the ith clutter blockc,iSpace-time steering vector for the ith clutter block, namely:
Figure BDA0003164874180000036
wherein, ac,i，bc,i，dc,iRespectively for the ith clutter block transmitting guide vector, receiving guide vector and Doppler guide vector, namely:
ac,i＝[1,exp(j2πfsc,i)，…，exp(j2π(M-1)fsc,i)]T，bc,i＝[1,exp(j2πfsc,i)，…，exp(j2π(N-1)fsc,i)]T (6)
dc,i＝[1,exp(j2πfdc,i)，…，exp(j2π(K-1)fdc,i)]T (7)
wherein f issc,i＝d cosφiLambda and fdc,i＝2(v cosφi+v0) T/lambda is normalized space frequency and Doppler frequency of the ith clutter block respectively.
Further, consider the noise model, noise ynModeled as mean 0, covariance matrix σ2Gaussian white noise of I, radar echo signal within the coherent processing interval CPI is represented as follows:
y＝yt+yc+yn (8)
the maximum detection probability under the Gaussian noise condition is equivalent to the maximum output SCNR, and the output SCNR of the MIMO-STAP model is expressed as follows according to the output SCNR definition:
Figure BDA0003164874180000037
wherein w is a reception weight vector, Rc+nCovariance matrix of noise added to clutter, wHIs the conjugate transpose of w.
Further, acquiring constant modulus constraints specifically includes:
the constant modulus signal being implemented on the basis of phase encoding, i.e.
Figure BDA0003164874180000038
p＝1,2,…,ML，θpIs the phase of s (p);
based on the above, a constant modulus signal design model for improving the target detection performance is obtained as follows:
Figure BDA0003164874180000041
further, obtaining an array element pulse selection model specifically includes:
define matrix A ∈ {0,1}M×KSelecting a matrix for transmitting the array element pulse, wherein A m,k1 denotes that the mth array element transmits the kth pulse, am,kIf 0, no emission is carried out; defining vector z e {0,1}NSelecting a vector for a received array element, wherein z n1 indicates that the nth array element is selected, znIf 0, not selecting;
based on this, the reconstructed target and clutter space-time steering vectors are as follows:
Figure BDA0003164874180000042
Figure BDA0003164874180000043
further, the joint sparse optimization problem is represented as:
Figure BDA0003164874180000044
wherein the content of the first and second substances,
Figure BDA0003164874180000045
is the target echo power; 1 is a vector of all 1 s,
Figure BDA0003164874180000046
indicating that the number of pulses of the selected transmitting array elements is not more than the total number of pulses of the transmitting array elements;
Figure BDA0003164874180000047
indicating that the number of selected receive array elements does not exceed the total number of receive array elements.
Furthermore, the complex nonlinear optimization problem is solved by adopting an alternating iteration mode, which specifically comprises the following steps:
firstly fixing a transmitting array element pulse selection matrix A and a receiving array element selection vector z, solving an optimal receiving weight vector w, and representing the sub-problem again as follows for solving the sub-problem:
Figure BDA0003164874180000048
wherein C is a constant; based on the minimum variance distortionless response criterion, solving the optimization problem to obtain the following weight vector:
Figure BDA0003164874180000049
will woThe optimization problem for a, z is re-expressed as follows, substituting equation (13):
Figure RE-GDA00032758039200000410
under the condition of uncorrelated clutter noise, the covariance matrix of clutter noise is expressed as:
Figure RE-GDA0003275803920000051
let εc,iIndependently and uniformly distributed, and satisfies that the mean value is 0 and the variance is sigmac,iThe above equation is restated as follows:
Figure BDA0003164874180000052
wherein the content of the first and second substances,
Figure BDA0003164874180000053
substituting the clutter noise covariance matrix into the objective function in the problem (16) to obtain:
Figure BDA0003164874180000054
based on the inverse lemma of the matrix,
Figure BDA0003164874180000055
the rewrite is as follows:
Figure BDA0003164874180000056
noise waves are considerably stronger than white noise, i.e.
Figure BDA0003164874180000057
Then
Figure BDA0003164874180000058
Wherein the content of the first and second substances,
Figure BDA0003164874180000059
delta is the power ratio of the noise clutter,
Figure BDA00031648741800000510
let D be [ VUc,Vut]，
Figure BDA00031648741800000511
Then based on the determinant property of the blocking matrix, the following results are obtained:
Figure BDA00031648741800000512
thereby, the following were obtained:
Figure BDA00031648741800000513
transforming the objective function to a logarithmic form, the optimization problem is rewritten as:
Figure BDA0003164874180000061
from the problem (24), it can be known that the target function is the difference between two concave functions, then the global optimal solution is located at the extreme point, but the constraint is binary constraint and has no convexity; to solve this problem, let
Figure BDA0003164874180000062
Comprising a polyhedron
Figure BDA0003164874180000063
And (4) relaxing the binary constraint to a frame constraint with convexity, and solving by using a convex optimization method.
Then fixing the transmitting array element pulse selection matrix A and the receiving weight vector w, solving a receiving array element selection vector z, and based on the problem, expressing the optimization problem of the receiving array element selection vector z again as follows:
Figure BDA0003164874180000064
wherein when R is fixedsWhen, VHV is regarded as a constant; the concave function maximum problem is equivalent to the convex function minimum problem, based on which the above formula is rewritten as follows:
Figure BDA0003164874180000065
which respectively approximate concave functions based on first-order Taylor decomposition iteration
Figure BDA0003164874180000066
To obtain a convex approximation, in particular, the (j +1) th iteration:
Figure BDA0003164874180000067
Figure BDA0003164874180000068
wherein the content of the first and second substances,
Figure BDA0003164874180000069
are respectively f (z)1、f(z)2At z(j)The gradient of (a) of (b) is,
Figure BDA00031648741800000610
Figure BDA00031648741800000611
thus, the above optimization problem is equivalent to:
Figure BDA00031648741800000612
according to the formula, the objective function is a convex problem about the variable z to be optimized, so that efficient solution can be realized based on a CVX convex optimization toolkit;
and fixing a receiving array element selection vector z and a receiving weight vector w, solving a transmitting array element pulse selection matrix A, and expressing the optimization problem about the variable A again as follows:
Figure BDA0003164874180000071
this problem is solved using the SCA method as described above, wherein,
Figure BDA0003164874180000072
Figure BDA0003164874180000073
thus, the above optimization problem is equivalent to:
Figure BDA0003164874180000074
the SCA is a local heuristic algorithm, and the obtained optimal solution depends on initial array element pulse selection, so a plurality of initial points should be randomly generated, and the variable value for obtaining the maximum SCNR is selected as the optimal solution.
As a further step, the pulse selection matrix A of the fixed transmitting array element, the selection vector z of the receiving array element and the receiving weight vectorQuantity w, solving for RsThe method specifically comprises the following steps:
Figure BDA0003164874180000075
order to
Figure BDA0003164874180000076
Wherein the content of the first and second substances,
Figure BDA0003164874180000077
then the process of the first step is carried out,
Figure BDA0003164874180000078
in the same way, the method for preparing the composite material,
Figure BDA0003164874180000079
in order to ensure that the water-soluble organic acid,
Figure BDA00031648741800000710
wherein woptSelecting an optimal receiving weight vector for the array element; then by the formula
Figure BDA00031648741800000711
Obtaining:
Figure BDA00031648741800000713
thereby obtaining the result that,
Figure BDA00031648741800000712
then
Figure BDA0003164874180000081
In the same way, the method for preparing the composite material,
Figure BDA0003164874180000082
based on tr (ab) ═ tr (ba), the output signal-to-noise ratio is expressed as:
Figure BDA0003164874180000083
thus, design RsThe optimization problem to maximize the output SCNR is represented as follows:
Figure BDA0003164874180000084
the constraint is relaxed to a convex constraint on the optimization variables, whereby the above problem is re-expressed as follows:
Figure BDA0003164874180000085
wherein the content of the first and second substances,
Figure BDA0003164874180000086
in order to solve the pseudo-convex problem (41), a semi-positive definite programming SDP mode is adopted for solving, and the method specifically comprises the following steps:
Figure BDA0003164874180000087
wherein H ═ tRs(ii) a The problem is efficiently solved by utilizing a CVX convex optimization toolkit; is provided (H)＊,t＊) To the optimal solution of the problem (43), then
Figure BDA00031648741800000812
Thereby obtaining an optimal solution of the pseudo-concave problem; the obtained result is a correlation matrix about the waveform S, and for solving the S, the following randomization method is adopted: generating a set of independent identically distributed complex gaussiansVector ηkWherein
Figure BDA0003164874180000088
k is 1, …, and Q is the randomization number; covariance matrix
Figure BDA0003164874180000089
Wherein the content of the first and second substances,
Figure BDA00031648741800000810
optimum waveform S＊The vectorized form is represented as follows:
Figure BDA00031648741800000811
wherein s isk(i)＝g*μ(ηk(i))，μ(ηk(i))＝exp(jarg(ηk(i)))，i＝1,…,ML。
Compared with the prior art, the invention has the following advantages: the designed waveform has good constant-modulus waveform characteristics and the beam pattern has low side lobes because the method optimizes the receiving weight, and can concentrate the power in the direction of the target and inhibit the echoes in other directions. The transmitting waveform, the receiving weight, the transmitting and receiving array elements and the pulse interval are designed in a combined sparse mode to output the SCNR to the maximum extent, and a small number of the transmitting and receiving array elements and the pulses can be selected well under the condition that the target detection performance is not influenced, so that the hardware cost and the energy consumption of a system are reduced.
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In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the drawings needed to be used in the description of the embodiments or the prior art will be briefly described below, it is obvious that the drawings in the following description are only some embodiments described in the present application, and other drawings can be obtained by those skilled in the art without creative efforts.
FIG. 1 is a flow chart of a resource allocation method implementation of the present invention;
FIG. 2 is a graph of real and imaginary distribution of the resulting waveform;
FIG. 3 is a beam pattern of the proposed method;
fig. 4 is a graph of P16, Q4, output SCNR versus SNR or CNR;
fig. 5 is a graph of P10, Q4, output SCNR versus SNR or CNR;
fig. 6 is a graph of P-16, Q-6, output SCNR versus SNR or CNR;
FIG. 7 is a diagram illustrating the effect of the receiving/transmitting array elements and the number of pulses on the output SCNR;
fig. 8 is a graph of P10, Q3, output SCNR versus SNR or CNR;
FIG. 9 is a diagram of the optimal transmit-receive array element pulse sparsity and its position distribution;
fig. 10 is a graph of output SCNR versus number of iterations.
Detailed Description
The embodiments of the present invention are implemented on the premise of the technical solution of the present invention, and detailed embodiments and specific operation procedures are given, but the scope of the present invention is not limited to the following embodiments.
As shown in fig. 1, the embodiment provides a method for configuring space-time resources for MIMO-STAP radar transceiving in a clutter scene, and the specific implementation steps may include:
step 1: establishing an MIMO-STAP model;
specifically, the speed and the height of the carrier are respectively set as v and H, and the radial speed of a target carrier connecting line is set as v0Target azimuth angle theta, elevation angle
Figure BDA0003164874180000091
Cone angle phi satisfying geometric relationship
Figure BDA0003164874180000092
The receiving and transmitting array is a short-distance parallel uniform linear array, the number of array elements is M and N respectively, the spacing between the array elements is d ═ lambda/2, and lambda is the radar working wavelength; baseband transmit signal
Figure BDA0003164874180000093
Wherein
Figure BDA0003164874180000094
The baseband signal of the M (M is 1,2, …, M) th transmitting array element is sampled, and L is the number of samples. The pulse number in the Coherent Processing Interval (CPI) is K, and the pulse repetition period is T.
Considering the point target model, the k-th intra-pulse MIMO radar reception target signal can be represented as:
Figure BDA0003164874180000095
wherein epsilontIs the target echo amplitude, fdt＝2(vcosφ+v0) T/λ is the target Doppler frequency, at，btRespectively transmitting and receiving guide vectors for a target, namely:
at＝[1,exp(j2πfst)，…，exp(j2π(M-1)fst)]T,bt＝[1,exp(j2πfst)，…，exp(j2π(N-1)fst)]T (2)
wherein f isstDcos phi/lambda is the normalized spatial frequency. Properties based on the Kronecker product as follows:
Figure BDA0003164874180000101
the target received signal in CPI may be expressed as:
Figure BDA0003164874180000102
wherein the content of the first and second substances,
Figure BDA0003164874180000103
for the target space-time steering vector(s),
Figure BDA0003164874180000104
is the product of Kronecker. dt＝[1,exp(j2πfdt)，...，exp(j2π(K-1)fdt)]TDoppler to targetA steering vector. I isKRepresenting an identity matrix of dimension K.
Considering a clutter model, clutter can be modeled as NcThe clutter blocks are added, the clutter in the CPI can be represented as:
Figure BDA0003164874180000105
wherein epsilonc,iRepresenting the amplitude of the echo of the ith clutter block. u. ofc,iSpace-time steering vector for the ith clutter block, namely:
Figure BDA0003164874180000106
wherein, ac,i，bc,i，dc,iThe transmission and receiving guide vectors and the Doppler guide vector of the ith clutter block are respectively, namely:
ac,i＝[1,exp(j2πfsc,i)，...，exp(j2π(M-1)fsc,i)]T，bc,i＝[1,exp(j2πfsc,i)，...，exp(j2π(N-1)fsc,i)]T (6)
dc,i＝[1,exp(j2πfdc,i)，…，exp(j2π(K-1)fdc,i)]T (7)
wherein f issc,i＝d cosφiLambda and fdc,i＝2(v cosφi+v0) T/lambda is normalized space frequency and Doppler frequency of the ith clutter block respectively.
Taking into account the noise model, noise ynModelled as mean 0, covariance matrix sigma2White gaussian noise of I.
Based on the above, the radar echo signal in the CPI can be expressed as follows:
y＝yt+yc+yn (8)
as is known, maximizing the detection probability under Gaussian noise conditions is equivalent to maximizing the output SCNR, and based on this, the invention considers optimizing MIMO-STAP transceiving space-time resource configuration to maximize the output SCNR. According to the output SCNR definition, the MIMO-STAP output SCNR can be expressed as follows:
Figure BDA0003164874180000107
wherein w is a reception weight vector, Rc+nIs a clutter plus noise covariance matrix.
Step 2: obtaining a constant modulus constraint and array element pulse selection model;
(1) obtaining constant modulus constraints
In particular, to avoid nonlinear effects, radar rf amplifiers are typically operated in an oversaturated state, so that the resulting waveform has a constant modulus characteristic. In practice, the constant modulus signal is usually implemented based on phase encoding, i.e.
Figure BDA0003164874180000111
p＝1,2,…,ML，θpIs the phase of s (p). Based on the above, a constant modulus signal design model for improving the target detection performance can be obtained as follows:
Figure BDA0003164874180000112
(2) obtaining array element pulse selection model
Define matrix A ∈ {0,1}M×KSelecting a matrix for transmitting the array element pulse, wherein Am,k1 means that the mth array element transmits the kth pulse, and 0 does not transmit. Defining vector z e {0,1}NSelecting a vector for a received array element, wherein z n1 means that the nth array element is selected, and 0 is not selected.
Based on the above, the reconfigurable target and clutter space-time guide vector is as follows:
Figure BDA0003164874180000113
Figure BDA0003164874180000114
and step 3: establishing a joint sparse optimization problem with constant modulus constraint and array element pulse selection based on a maximized output SCNR criterion to obtain a complex nonlinear optimization problem about an optimization variable;
specifically, the joint optimization problem of the transmitted signal, the receiving weight and the array element pulse selection can be expressed as follows:
Figure BDA0003164874180000115
wherein the content of the first and second substances,
Figure BDA0003164874180000116
is the target echo power. 1 is a vector of all 1 s,
Figure BDA0003164874180000117
indicating that the number of pulses of the selected transmitting array element is not more than the total number of pulses of the transmitting array element.
Figure BDA0003164874180000118
Indicating that the number of selected receive array elements does not exceed the total number of receive array elements.
From equation (13), the numerator of the objective function is the square of the correlation function with respect to the optimization variables a, z, s, w, and the denominator is the nonlinear function with respect to these four variables, so that the objective function is a complex nonlinear function with respect to the optimization variables, and the constant modulus constraint with respect to the transmit waveform is a non-convex constraint, so that the above problem cannot be solved directly by the conventional convex optimization method.
And 4, step 4: solving the complex nonlinear optimization problem by adopting an alternating iteration mode to obtain an optimal transmitting waveform, a receiving weight, a selection vector and a corresponding output SCNR:
aiming at the problems, the invention provides an alternate iteration method based on SDP, SCA and randomization, the method decomposes the problem into sub-problems which can be alternately solved by iteration, alternately solves a transmitting array element pulse selection matrix A, and receives an array element selection vector z and wavesForm correlation matrix Rs＝s*sTAnd receiving the weight vector w to maximize the output SCNR and based on a corresponding solution method to obtain an efficient solution.
4.1 fixation of RsSolving A, z and w, which is specifically divided into the following conditions;
fixing A and z, and solving the optimal receiving weight vector w. To solve this sub-problem, it can be re-expressed as follows:
Figure BDA0003164874180000121
where C is a constant. Based on the minimum variance distortionless response criterion, solving the optimization problem can obtain the following weight vector:
Figure BDA0003164874180000122
will woInstead of equation (13), the optimization problem for a, z can be re-expressed as:
Figure RE-GDA00032758039200000410
under the condition of clutter noise uncorrelated, the clutter noise covariance matrix can be expressed as:
Figure RE-GDA0003275803920000051
let εc,iIndependently and uniformly distributed, and satisfies that the mean value is 0 and the variance is sigmac,iThe above formula can be restated as follows:
Figure BDA0003164874180000125
wherein the content of the first and second substances,
Figure BDA0003164874180000126
substituting the clutter noise covariance matrix into the objective function in the problem (16) yields:
Figure BDA0003164874180000127
based on the inverse lemma of the matrix,
Figure BDA0003164874180000128
the following can be rewritten:
Figure BDA0003164874180000129
noise waves are considerably stronger than white noise, i.e.
Figure BDA00031648741800001210
Then
Figure BDA00031648741800001211
Wherein the content of the first and second substances,
Figure BDA0003164874180000131
delta can be viewed as the noise clutter power ratio,
Figure BDA0003164874180000132
let D be [ VUc,Vut]，
Figure BDA0003164874180000133
Then based on the determinant property of the blocking matrix, it is possible to obtain:
Figure BDA0003164874180000134
thus, it is possible to obtain:
Figure BDA0003164874180000135
transforming the objective function to a logarithmic form, the optimization problem can be rewritten as:
Figure BDA0003164874180000136
from the problem (24), the objective function is the difference between two concave functions, and the global optimal solution is located at the extreme point, but the constraint is binary constraint and has no convexity. To solve this problem, it is able to provide
Figure BDA0003164874180000137
Comprising a polyhedron
Figure BDA0003164874180000138
The binary constraint can be relaxed to a convex frame constraint, and the convex frame constraint can be solved by a convex optimization method.
Then fix A, w solve for z. Based on the above problem, the optimization problem with respect to variable z can be re-expressed as:
Figure BDA0003164874180000139
wherein when R is fixedsWhen, VHV may be considered a constant. As is well known, the concave function maximum problem is equivalent to the convex function minimum problem, based on which the above formula can be rewritten as follows:
Figure BDA00031648741800001310
the above equation can be regarded as two convex function Difference (DC) problems, and compared with the solution methods of constrained nonlinear programming problems such as multiplier method and feasible direction method, the non-convex problem is approximately converted into a convex problem in each iteration of continuous convex approximation (SCA) and has better convergence, so the SCA is adopted to solve the problems. Which respectively approximate concave functions based on first-order Taylor decomposition iteration
Figure BDA00031648741800001311
log|Du HDu+δI0To obtain a convex approximation, in particular, the (j +1) th iteration:
Figure BDA0003164874180000141
Figure BDA0003164874180000142
wherein the content of the first and second substances,
Figure BDA0003164874180000143
are respectively f (z)1、f(z)2At z(j)The gradient of (a) of (b) is,
Figure BDA0003164874180000144
Figure BDA0003164874180000145
thus, the above optimization problem can be equivalent to:
Figure BDA0003164874180000146
from the above equation, the objective function is a convex problem with respect to the variable z to be optimized, and thus can be efficiently solved based on the CVX convex optimization toolkit.
And fixing z and w to solve A. The optimization problem for variable a can be re-expressed as:
Figure BDA0003164874180000147
similarly, this problem can be solved using the SCA method as described above, wherein,
Figure BDA0003164874180000148
Figure BDA0003164874180000149
thus, the above optimization problem can be equivalent to:
Figure BDA00031648741800001410
as mentioned above, SCA is a local heuristic algorithm, and the obtained optimal solution depends on the initial array element pulse selection, therefore, several initial points should be randomly generated, and the variable value that can obtain the maximum SCNR is selected as the optimal solution.
4.2 solving for R by fixing A, z, wsThe method specifically comprises the following steps:
as can be seen from the above, the present invention,
Figure BDA00031648741800001411
order to
Figure BDA00031648741800001412
Wherein the content of the first and second substances,
Figure BDA00031648741800001413
then the process of the first step is carried out,
Figure BDA0003164874180000151
in the same way, the method for preparing the composite material,
Figure BDA0003164874180000152
in order to ensure that the water-soluble organic acid,
Figure BDA0003164874180000153
wopt＝vec(W)，
Figure BDA0003164874180000154
wherein woptAnd selecting the optimal receiving weight vector for the array element. Then by the formula
Figure BDA0003164874180000155
The following can be obtained:
Figure BDA0003164874180000156
thus, the method can obtain the product,
Figure BDA0003164874180000157
then
Figure BDA0003164874180000158
In the same way, the method for preparing the composite material,
Figure BDA0003164874180000159
based on tr (ab) ═ tr (ba), the output signal-to-noise ratio can be expressed as:
Figure BDA00031648741800001510
thus, design RsThe optimization problem to maximize the output SCNR can be expressed as follows:
Figure BDA00031648741800001511
as mentioned above, the objective function in the above formula is a quasi-convex problem about the optimization variable, and the constraint condition is a nonlinear constraint about the optimization variable, so that the traditional convex optimization method cannot be directly adopted for solving. To solve this problem, the constraint condition can be relaxed as a convex constraint on the optimization variables, and thus the above problem can be re-expressed as follows:
Figure BDA00031648741800001512
wherein the content of the first and second substances,
Figure BDA00031648741800001513
in order to solve the quasi-convex problem (41), a bisection method can be generally adopted, but a series of convex optimization problems need to be solved, and the method is troublesome. The above-described quasi-convex problem is thus solved by the following SDP:
Figure BDA0003164874180000161
wherein H ═ tRs. The above problem can be solved efficiently using the CVX convex optimization toolkit. Is provided (H)＊,t＊) To the optimal solution of the problem (43), then
Figure BDA0003164874180000167
Thereby obtaining the optimal solution of the simulated concave problem. The result is a correlation matrix for waveform S instead of S, and to solve for S, the following randomization method can be used: generating a set of independent identically distributed complex Gaussian vectors etakWherein
Figure BDA0003164874180000162
k is 1, …, and Q is the number of randomizations. Covariance matrix
Figure BDA0003164874180000163
Wherein the content of the first and second substances,
Figure BDA0003164874180000164
optimum waveform S＊The vectorized form can be expressed as follows:
Figure BDA0003164874180000165
wherein s isk(i)＝g*μ(ηk(i))，μ(ηk(i))＝exp(jarg(ηk(i)))，i＝1,…,ML。
The effects of the present invention can be further illustrated by the following simulations:
simulation conditions are as follows: the speed v of the airborne platform is 100m/s, the flying height H is 9000m, and the target speed v is010m/s, target azimuth angle theta 20 DEG and pitch angle
Figure BDA0003164874180000166
The cone angle φ is 62 °, Signal to Noise Ratio (SNR) SNR ∈ [10dB,50dB ] is]. The number of transmitting and receiving array elements M is 4, the interval of array elements is d lambda/2, the wavelength of carrier wave is 0.2016M, and the carrier frequency is fc1.488GHz, the signal bandwidth B16 MHz, the transmission pulse K4 in CPI, and the coding length L32. The target Radar Cross Section (RCS) is unchanged. The number Nc of Clutter uniform samples is 1000, the sampling points are independently and equally distributed, and the Noise to Noise Ratio (CNR) CNR belongs to [10dB,50dB ]]。
Simulation content:
simulation 1: the resulting real and imaginary distributions of the waveforms and beam patterns are shown in fig. 2 and 3.
It can be seen from fig. 2 that the values of the designed waveforms all fall on the unit circle, and have constant envelope characteristics. Fig. 3 shows the beam pattern obtained by the proposed method, which places a peak at θ ═ 20 °, indicating that the transmission power can be concentrated in the area where the target is located, so that the target detection performance can be effectively improved.
Simulation 2: under the condition that the number P of the transmission array element pulse pairs is 16 and the number Q of the receiving array elements is 4, a graph of the change of the output SCNR along with the SNR or the CNR is shown in fig. 4; it is the curve of SCNR output with SNR or CNR obtained by the proposed method, uncorrelated waveforms, waveform-only design algorithm, and waveform and receive weight joint design algorithm.
As can be seen from FIG. 4, the SCNR outputs obtained by the above methods all increase with the increase of SNR and decrease with the increase of CNR; secondly, compared with uncorrelated waveforms, the proposed method and the waveform-only design algorithm, the waveform and reception right joint design algorithm all have higher output SCNR; thirdly, the output SCNR obtained by the method is superior to a waveform design algorithm only, which can be attributed to that only the waveform design algorithm considers the design of the transmitted waveform and does not consider the optimization of the receiving weight; in addition, the difference between the output SCNR obtained by the proposed method and the waveform and reception weight joint design algorithm is small, because the initial waveform of the waveform and reception weight joint design algorithm is a Linear Frequency Modulation (LFM) signal, which indicates that the proposed method can significantly improve the output SCNR for any waveform.
Simulation 3: when the number of the transmitting array element pulse pairs is P equal to 10 and the number of the receiving array elements Q equal to 4, a graph of the change of SCNR with SNR or CNR is output, as shown in fig. 5, which is a change curve of the output SCNR with SNR or CNR obtained by the proposed method, enumeration method, iterative min-max algorithm, and greedy submode algorithm.
As can be seen from FIG. 5, the SCNR outputs obtained by the above methods all increase with the increase of SNR and decrease with the increase of CNR; secondly, the output SCNR obtained by the enumeration method is the highest in all the methods, and because the enumeration method is used for global search, a global optimal solution can be obtained, but the calculation amount is large, and the calculation complexity is high; thirdly, the method is very close to the output SCNR obtained by the enumeration method and has better effect than the iteration min-max algorithm and the greedy sub-mode algorithm, because the greedy sub-mode algorithm belongs to the greedy algorithm, although the calculated amount is lower, the local optimization is easily caused; the iterative min-max algorithm is not directly associated with the output SCNR.
And (4) simulation: when the number P of the transmission array element pulse pairs is 16 and the number Q of the reception array elements is 6, a graph of the change of SCNR with SNR or CNR is output, and the number of the reception array elements is increased to N10, as shown in fig. 6. The SCNR output curve along with SNR or CNR is obtained by the method, the enumeration method, the iterative min-max algorithm and the greedy submode algorithm. As can be seen from fig. 6, the method still has better performance in the aspect of receiving array element selection, similar to the effect of simulation 3.
And (5) simulation: fig. 7 is a graph showing the variation of output SCNR with the transmitting and receiving array elements and the number of pulses obtained by the proposed method, enumeration method, iterative min-max algorithm and greedy submode algorithm under the conditions that SNR is 20dB and CNR is 20 dB.
As can be seen from fig. 7, with the increase of the selection of the transmission array element pulse pair, the output SCNR obtained by the above method tends to increase, and tends to stabilize when the logarithm P of the transmission array element pulse is about 8, and the output SCNR tends to stabilize when the number Q of the reception array element is about 6.
And (6) simulation: under the conditions that the number of the transmitting array element pulse pairs is P-10, the number of the receiving array elements is Q-3 (N-4) and the constant-modulus waveform is jointly designed, a graph of the variation of the SCNR along with the SNR or the CNR and an optimal transmitting and receiving array element pulse sparse position distribution graph are output, as shown in fig. 8 and fig. 9. In order to verify the influence of the proposed method on the whole system, the variation curve of the output SCNR obtained by the proposed method, the enumeration method, the iterative min-max algorithm and the greedy submode algorithm along with the SNR or the CNR is obtained.
As can be seen from fig. 8, the performance of the proposed method is the best, mainly because the proposed method jointly designs the antenna pulse and the transmission waveform and uses an alternating iteration method to find the optimal value.
And (7) simulation: the number P of transmission array element pairs is 10, the number Q of reception array elements is 3(N is 4), and the output SCNR varies with the number of iterations in the case where SNR is 20dB and CNR is 20dB, as shown in fig. 10.
As can be seen from fig. 10, as the number of iterations increases, the fluctuation of the output SCNR obtained by the proposed method gradually decreases, and after the 5 th iteration, the fluctuation tends to be stable, indicating that the proposed method has better convergence.
Simulation results show that the designed waveform has good constant-modulus waveform characteristics and the beam pattern has low side lobes because the method optimizes the receiving weight to concentrate the power in the direction of the target and inhibit echoes in other directions. The transmitting waveform, the receiving weight, the transmitting and receiving array elements and the pulse interval are designed in a combined sparse mode to output the SCNR to the maximum extent, and a small number of the transmitting and receiving array elements and the pulses can be selected well under the condition that the target detection performance is not influenced, so that the hardware cost and the energy consumption of a system are reduced.
The foregoing descriptions of specific exemplary embodiments of the present invention have been presented for purposes of illustration and description. It is not intended to limit the invention to the precise form disclosed, and obviously many modifications and variations are possible in light of the above teaching. The exemplary embodiments were chosen and described in order to explain certain principles of the invention and its practical application to enable one skilled in the art to make and use various exemplary embodiments of the invention and various alternatives and modifications as are suited to the particular use contemplated. It is intended that the scope of the invention be defined by the claims and their equivalents.

Claims (9)

1. A method for configuring MIMO-STAP radar transmitting and receiving space-time resources in a clutter scene is characterized by specifically comprising the following steps:
establishing an MIMO-STAP model;
obtaining a constant modulus constraint and array element pulse selection model;
establishing a joint sparse optimization problem with constant modulus constraint and array element pulse selection based on a maximized output SCNR criterion to obtain a complex nonlinear optimization problem about an optimization variable;
and solving the complex nonlinear optimization problem by adopting an alternating iteration mode to obtain an optimal transmitting waveform, a receiving weight, a selection vector and a corresponding output SCNR.
2. The method for configuring space-time resources for transceiving of the MIMO-STAP radar in the clutter scene according to claim 1, wherein the MIMO-STAP model is established, specifically:
the speed and the height of the carrier are respectively set as v and H, and the radial speed of a target carrier connecting line is set as v0Target azimuth angle theta, elevation angle
Figure FDA0003164874170000011
Cone angle phi satisfying geometric relationship
Figure FDA0003164874170000012
The receiving and transmitting array is a short-distance parallel uniform linear array, the number of array elements is M and N respectively, the spacing between the array elements is d ═ lambda/2, and lambda is the radar working wavelength; baseband transmit signal
Figure FDA0003164874170000013
Wherein
Figure FDA0003164874170000014
Sampling a baseband signal of an M-th transmitting array element, wherein M is 1,2, …, M and L are sample numbers; the pulse number in the Coherent Processing Interval (CPI) is K, and the pulse repetition period is T;
considering the point target model, the target signal received by the MIMO radar in the kth pulse is expressed as:
Figure FDA0003164874170000015
wherein epsilontIs the target echo amplitude, fdt＝2(vcosφ+v0) T/lambda is the target Doppler frequency, j is an imaginary unit; a istTransmitting a steering vector for the target, btReceive steering vectors for the target, i.e.:
at＝[1,exp(j2πfst)，…，exp(j2π(M-1)fst)]T,bt＝[1,exp(j2πfst)，…，exp(j2π(N-1)fst)]T (2)
wherein f isstD cos phi/lambda is normalized spatial frequency; properties based on the Kronecker product as follows:
Figure FDA0003164874170000016
the target received signal within the coherent processing interval CPI is represented as:
Figure FDA0003164874170000017
wherein the content of the first and second substances,
Figure FDA0003164874170000018
for the target space-time steering vector(s),
Figure FDA0003164874170000019
is the product of Kronecker; dt＝[1,exp(j2πfdt)，…，exp(j2π(K-1)fdt)]TA target Doppler steering vector; i isKRepresenting an identity matrix of dimension K.
3. The method according to claim 2, wherein the clutter model is considered and the clutter is modeled as NcAnd (3) superposing the clutter blocks, and expressing the clutter in the coherent processing interval CPI as:
Figure FDA0003164874170000021
wherein epsilonc,iRepresenting the amplitude, u, of the echo of the ith clutter blockc,iSpace-time steering vector for the ith clutter block, namely:
Figure FDA0003164874170000022
wherein, ac,i，bc,i，dc,iRespectively for the ith clutter block transmitting guide vector, receiving guide vector and Doppler guide vector, namely:
ac,i＝[1,exp(j2πfsc,i)，…，exp(j2π(M-1)fsc,i)]T，bc,i＝[1,exp(j2πfsc,i)，…，exp(j2π(N-1)fsc,i)]T (6)
dc,i＝[1,exp(j2πfdc,i)，…，exp(j2π(K-1)fdc,i)]T (7)
wherein f issc,i＝dcosφiLambda and fdc,i＝2(vcosφi+v0) T/lambda is normalized space frequency and Doppler frequency of the ith clutter block respectively.
4. Root of herbaceous plantThe MIMO-STAP radar transmit-receive space-time resource allocation method in clutter scene according to claim 2 or 3, characterized in that the noise model is considered, and the noise y is considerednModeled as mean 0, covariance matrix σ2Gaussian white noise of I, radar echo signal within the coherent processing interval CPI is represented as follows:
y＝yt+yc+yn (8)
the maximum detection probability under the Gaussian noise condition is equivalent to the maximum output SCNR, and the output SCNR of the MIMO-STAP model is expressed as follows according to the output SCNR definition:
Figure FDA0003164874170000023
wherein w is a reception weight vector, Rc+nCovariance matrix of noise added to clutter, wHIs the conjugate transpose of w.
5. The method for configuring space-time resources for transceiving of the MIMO-STAP radar in the clutter scene according to claim 1, wherein a constant modulus constraint is obtained, specifically:
the constant modulus signal being implemented on the basis of phase encoding, i.e.
Figure FDA0003164874170000024
θpIs the phase of s (p); based on the above, a constant modulus signal design model for improving the target detection performance is obtained as follows:
Figure FDA0003164874170000025
6. the method for configuring space-time resources for transceiving of the MIMO-STAP radar in the clutter scene according to claim 1, wherein the obtaining of the array element pulse selection model specifically comprises:
define matrix A ∈ {0,1}M×KSelecting a matrix for transmitting the array element pulse, wherein Am,k1 denotes that the mth array element transmits the kth pulse, am,kIf 0, no emission is carried out; defining vector z e {0,1}NSelecting a vector for a received array element, wherein zn1 indicates that the nth array element is selected, znIf 0, not selecting;
based on this, the reconstructed target and clutter space-time steering vectors are as follows:
Figure FDA0003164874170000031
Figure FDA0003164874170000032
7. the method for configuring space-time resources for MIMO-STAP radar transceiving in a clutter scene according to claim 1, wherein the joint sparse optimization problem is expressed as:
Figure FDA0003164874170000033
wherein the content of the first and second substances,
Figure FDA0003164874170000034
is the target echo power; 1 is a vector of all 1 s,
Figure FDA0003164874170000035
indicating that the number of pulses of the selected transmitting array elements is not more than the total number of pulses of the transmitting array elements;
Figure FDA0003164874170000036
indicating that the number of selected receive array elements does not exceed the total number of receive array elements.
8. The method for configuring space-time resources for transceiving of the MIMO-STAP radar in the clutter scene according to claim 1, wherein the complex nonlinear optimization problem is solved in an alternating iterative manner, specifically:
firstly fixing a transmitting array element pulse selection matrix A and a receiving array element selection vector z, solving an optimal receiving weight vector w, and representing the sub-problem again as follows for solving the sub-problem:
Figure RE-FDA0003275803910000037
wherein C is a constant; based on the minimum variance distortionless response criterion, solving the optimization problem to obtain the following weight vector:
Figure RE-FDA0003275803910000038
will woThe optimization problem for a, z is re-expressed as follows, substituting equation (13):
Figure RE-FDA0003275803910000039
under the condition of uncorrelated clutter noise, the covariance matrix of clutter noise is expressed as:
Figure RE-FDA00032758039100000310
let εc,iIndependently and uniformly distributed, and satisfies that the mean value is 0 and the variance is sigmac,iThe above equation is restated as follows:
Figure RE-FDA00032758039100000311
wherein the content of the first and second substances,
Figure RE-FDA00032758039100000312
covariance of clutter noiseThe matrix is substituted into the objective function in the problem (16) to obtain:
Figure RE-FDA0003275803910000041
based on the inverse lemma of the matrix,
Figure RE-FDA0003275803910000042
the rewrite is as follows:
Figure RE-FDA0003275803910000043
noise waves are considerably stronger than white noise, i.e.
Figure RE-FDA0003275803910000044
Then
Figure RE-FDA0003275803910000045
Wherein the content of the first and second substances,
Figure RE-FDA0003275803910000046
delta is the power ratio of the noise clutter,
Figure RE-FDA0003275803910000047
let D be [ VUc,Vut]，
Figure RE-FDA0003275803910000048
Then based on the determinant property of the blocking matrix, the following results are obtained:
Figure RE-FDA0003275803910000049
thereby, the following were obtained:
Figure RE-FDA00032758039100000410
transforming the objective function to a logarithmic form, the optimization problem is rewritten as:
Figure RE-FDA00032758039100000411
from the problem (24), it can be known that the target function is the difference between two concave functions, then the global optimal solution is located at the extreme point, but the constraint is binary constraint and has no convexity; to solve this problem, let
Figure RE-FDA00032758039100000412
Comprising a polyhedron
Figure RE-FDA00032758039100000413
The binary constraint is relaxed to a frame constraint with convexity, and a convex optimization method is used for solving;
then fixing the transmitting array element pulse selection matrix A and the receiving weight vector w, solving a receiving array element selection vector z, and based on the problem, expressing the optimization problem of the receiving array element selection vector z again as follows:
Figure RE-FDA0003275803910000051
wherein when R is fixedsWhen, VHV is regarded as a constant; the concave function maximum problem is equivalent to the convex function minimum problem, based on which the above formula is rewritten as follows:
Figure RE-FDA0003275803910000052
which respectively approximate concave functions based on first-order Taylor decomposition iteration
Figure RE-FDA0003275803910000053
log|Du HDu+δI0To obtain a convex approximation, in particular, the (j +1) th iteration:
Figure RE-FDA0003275803910000054
Figure RE-FDA0003275803910000055
wherein the content of the first and second substances,
Figure RE-FDA0003275803910000056
are respectively f (z)1、f(z)2At z(j)The gradient of (a) of (b) is,
Figure RE-FDA0003275803910000057
Figure RE-FDA0003275803910000058
thus, the above optimization problem is equivalent to:
Figure RE-FDA0003275803910000059
according to the formula, the objective function is a convex problem about the variable z to be optimized, so that efficient solution can be realized based on a CVX convex optimization toolkit;
and fixing a receiving array element selection vector z and a receiving weight vector w, solving a transmitting array element pulse selection matrix A, and expressing the optimization problem about the variable A again as follows:
Figure RE-FDA00032758039100000510
this problem is solved using the SCA method as described above, wherein,
Figure RE-FDA0003275803910000061
Figure RE-FDA0003275803910000062
thus, the above optimization problem is equivalent to:
Figure RE-FDA0003275803910000063
the SCA is a local heuristic algorithm, and the obtained optimal solution depends on initial array element pulse selection, so a plurality of initial points should be randomly generated, and the variable value for obtaining the maximum SCNR is selected as the optimal solution.
9. The method according to claim 8, wherein the method for configuring space-time resources for MIMO-STAP radar transmission and reception in clutter scene comprises fixing a transmission array element pulse selection matrix A, a reception array element selection vector z, a reception weight vector w, and solving RsThe method specifically comprises the following steps:
Figure FDA0003164874170000064
order to
Figure FDA0003164874170000065
Wherein the content of the first and second substances,
Figure FDA0003164874170000066
then the process of the first step is carried out,
Figure FDA0003164874170000067
in the same way, the method for preparing the composite material,
Figure FDA0003164874170000068
in order to ensure that the water-soluble organic acid,
Figure FDA0003164874170000069
wherein woptSelecting an optimal receiving weight vector for the array element; then by the formula
Figure FDA00031648741700000610
Obtaining:
Figure FDA00031648741700000611
thereby obtaining the result that,
Figure FDA00031648741700000612
then
Figure FDA00031648741700000613
In the same way, the method for preparing the composite material,
Figure FDA00031648741700000614
based on tr (ab) ═ tr (ba), the output signal-to-noise ratio is expressed as:
Figure FDA0003164874170000071
thus, design RsThe optimization problem to maximize the output SCNR is represented as follows:
Figure FDA0003164874170000072
the constraint is relaxed to a convex constraint on the optimization variables, whereby the above problem is re-expressed as follows:
Figure FDA0003164874170000073
wherein the content of the first and second substances,
Figure FDA0003164874170000074
in order to solve the pseudo-convex problem (41), a semi-positive definite programming SDP mode is adopted for solving, and the method specifically comprises the following steps:
Figure FDA0003164874170000075
wherein H ═ tRs(ii) a The problem is efficiently solved by utilizing a CVX convex optimization toolkit; is provided (H)＊,t＊) To the optimal solution of the problem (43), then
Figure FDA0003164874170000076
Thereby obtaining an optimal solution of the pseudo-concave problem; the obtained result is a correlation matrix about the waveform S, and for solving the S, the following randomization method is adopted: generating a set of independent identically distributed complex Gaussian vectors etakWherein
Figure FDA0003164874170000077
The randomization times; covariance matrix
Figure FDA0003164874170000078
Wherein the content of the first and second substances,
Figure FDA0003164874170000079
optimum waveform S＊The vectorized form is represented as follows:
Figure FDA00031648741700000710
wherein s isk(i)＝g*μ(ηk(i))，μ(ηk(i))＝exp(jarg(ηk(i)))，i＝1,…,ML。
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Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN114594425A (en) * 2022-03-14 2022-06-07 电子科技大学 Clutter interference resistant short-time pulse train waveform design method
CN114675238A (en) * 2022-02-24 2022-06-28 中国人民解放军国防科技大学 Radar communication integrated waveform direct optimization method and system

Cited By (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN114675238A (en) * 2022-02-24 2022-06-28 中国人民解放军国防科技大学 Radar communication integrated waveform direct optimization method and system
CN114675238B (en) * 2022-02-24 2023-11-03 中国人民解放军国防科技大学 Radar communication integrated waveform direct optimization method and system
CN114594425A (en) * 2022-03-14 2022-06-07 电子科技大学 Clutter interference resistant short-time pulse train waveform design method
CN114594425B (en) * 2022-03-14 2023-05-16 电子科技大学 Short-time pulse train waveform design method for resisting clutter interference

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