CN108987941B - One-dimensional subarray division method based on compressed sensing - Google Patents
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Abstract
The invention discloses a one-dimensional subarray division method based on compressed sensing, which comprises the following steps: step S01) dividing the given N-element linear array into M sub-arrays, adding excitation only at the ports of the sub-arrays, and determining the corresponding array element excitation vector Wele(ii) a Step S02) according to the array element excitation vector WeleConstructing a sparse vector S by compressibility, and establishing a sparse basis matrix T to enable WeleTS; step S03) takes the minimum subarray number as the optimization target, takes the main lobe width and the side lobe level requirement of the radiation directional diagram as the constraint conditions, and excites the vector W according to the array elementseleEstablishing a convex optimization model for sparse vector recovery by using the sparse basis matrix T and the sparse vector S; step S04) solving the convex optimization model established in the step S03), and calculating a sparse vector S, thereby determining the number of subarrays, the weight of the subarrays and the size of the subarrays; and completing the division of the N-element linear array. The method has high calculation efficiency, is simple and feasible, and reduces the manufacturing cost and the cost of the whole system.
Description
Technical Field
The invention relates to the field of wireless communication, in particular to a one-dimensional subarray division method based on compressed sensing.
Background
The Subarray division technique is widely used in the design of large Array Antennas because it can simplify the feed network of the Array Antenna and reduce the cost and complexity of the whole Antenna system (reference [1 ]: z.y.xiong, z.h.xu, s.w.chen and s.p.xiao, "surveillance Partition in Array Antenna Based on the algorithm X," IEEE Antennas Wireless performance. The subarray division means that the number of subarrays, the weight of the subarrays and the size of the subarrays are determined at the same time, and solving the multivariable non-convex nonlinear optimization problem is still a great challenge today.
Published studies show that genetic algorithms (reference [2 ]: r.l.haupt, "Optimized weighting of elementary matrices of elementary sizes," IEEE trans.antennas propag., vol.55, No.4, pp.1207-1210, apr.2007) and differential evolutionary algorithms (reference [3 ]: s.k.goudos, k.a.gotsis, k.siakara, e.e.vadais, and j.n.sahelos, "a multi-object approach based on elementary linear analysis on parametric differential evolution volume," IEEE trans.antennas propag., vol.61, No.6, pp.3042-3052, 2013) can be applied to both of these methods, although the relative evolutionary frequencies of these two methods are required to be calculated, and the relative subvaluation of the two sub-arrays is required.
In order to improve the overall efficiency of the subarray, a class of subarray division methods based on excitation matching criteria, such as the adjacent subarray division method (reference [4 ]: l.manica, p.rocca, m.benedetti, and a.massa, "Design of subraw linear and planar array anti-linear with SLL controlled base on excitation matching algorithm," IEEE trans.antennas pro pag., vol.57, No.6, pp.1684-1691, jun.2009), clustering subarray partitioning methods (reference [5 ]: z.y.xiong, z.h.xu, s.w.Chenand s.p.xiao, "Cluster Analysis for the Synthesis of partitioned monopulses extensions," IEEE trans.extensions, vol.62, No.4, pp.1738-1749, ap.2014) and regularized sparse solving methods (reference [6 ]: g.oliveri, m.salucci, and a.massa, "synthetic matrix controlled linear array Analysis of a sparse-segmented solvent," IEEE trans.extensions, vol.64, No.10, pp.4277-4287, etc.) are proposed as a set of prior information to solving the problem.
Disclosure of Invention
The invention aims to overcome the defects in the prior subarray division technology and provides a one-dimensional subarray division method based on compressed sensing, which utilizes the compressibility of an array element excitation vector to express the array element excitation vector as a sparse vector under a sparse basis, so that the root of the sparse vector is obtainedAccording to the compressed sensing theory, the multivariable non-convex optimization problem of simultaneously optimizing the number of sub-arrays, the weight of the sub-arrays and the size of the sub-arrays involved in sub-array division is converted into a convex optimization problem of sparse vector recovery, and iterative weighting l is utilized1The norm minimization algorithm is solved efficiently. The algorithm realizes the integral optimization of multivariable, and has high calculation efficiency and simple and easy operation. Meanwhile, the algorithm can realize the division of the minimum subarray number, thereby greatly simplifying the feed network of the array antenna and reducing the manufacturing cost and the cost of the whole system.
In order to achieve the above object, the present invention provides a compressed sensing-based one-dimensional subarray division method, which includes:
step S01) dividing the given N-element linear array into M sub-arrays, adding excitation only at the ports of the sub-arrays, and determining the corresponding array element excitation vector Wele；
Step S02) according to the array element excitation vector WeleConstructing a sparse vector S by compressibility, and establishing a sparse basis matrix T to enable Wele＝TS；
Step S03) takes the minimum subarray number as the optimization target, takes the main lobe width and the side lobe level requirement of the radiation directional diagram as the constraint conditions, and excites the vector W according to the array elementseleEstablishing a convex optimization model for sparse vector recovery by using the sparse basis matrix T and the sparse vector S;
step S04) solving the convex optimization model established in the step S03), and calculating a sparse vector S, thereby determining the number of subarrays, the weight of the subarrays and the size of the subarrays; and completing the division of the N-element linear array.
As a modification of the above method, the step S01) is specifically:
dividing a given N-element linear array into M sub-arrays, and adding excitation only at sub-array ports to obtain a sub-array weighting vector Wsub＝[a1,a2,…,aM]TWherein a ismRepresenting the weight of the mth subarray port, M is 1, …, M, then the array element excitation vector Wele＝[w1,w2,…,wN]T，wnRepresenting the nth array elementN is 1, …, N, WeleIs particularly shown as
Wherein N ismIs the number of array elements contained in the mth sub-array, and is used for representing the size of the corresponding sub-array.
As a modification of the above method, in step S01), the number M of sub-arrays is much smaller than the number N of array elements, i.e., M < < N.
As a modification of the above method, step S02) specifically includes:
step S02-1) array element excitation vector WeleDifferencing adjacent elements to obtain a vector S':
step S02-2) exciting the first array element1Concatenated with the vector S' to form a sparse vector S, i.e.
The sparse vector S is a vector with the length of N and comprises M nonzero elements;
step S02-3) according to the array element excitation vector WeleWith the numerical characteristics of the vector S, constructing a basis matrix T, i.e.
And has:
Wele＝TS (5)。
as a modification of the above method, the step S03) specifically includes:
step S03-1), establishing a linear array directional diagram:
F(θ)＝A(θ)Wele(6)
where A (θ) is the steering vector:
wherein the wave number k is 2 pi/lambda, where lambda is the operating wavelength of the antenna, dnIs the position of the nth array element, where N is 1, … N, and θ is the observation angle relative to the broadside direction;
step S03-2) when the array structure is symmetric, the steering vector a (θ) is equivalent to:
A(θ)＝[2cos(kd1sin(θ),2cos(kd2sin(θ)),…,2cos(kdNsin(θ))](8)
F(θ)＝A(θ)TS＝ψ(θ)S (9)
step S03-3) according to the given main lobe width FNBW and the upper limit requirement UB of the side lobe level, and in combination with the compressive sensing theory, establishing a convex optimization model for minimizing the number of sub-arrays:
wherein l1,l2Respectively representing 1 norm and 2 norm, thetamaxIs the main beam direction; the sideline region is denoted as the side lobe region, determined by the main lobe width FNBW.
As a modification of the above method, the step S04) specifically includes:
step S04-1) with iterative weighting l1The norm minimization algorithm solves the convex optimization model established in the step S03), and thus a sparse vector S is calculated;
the ith iteration of the convex optimization model is represented as:
wherein, X(i)Is a diagonal matrix, defines an initial X(0)Is a unit matrix, after each iteration is finished, the element value on the X diagonal line is updated according to the calculated value of the sparse vector S, and the updated result is used as the initial value of the next iteration, so that the ith iteration isIn the course of a second iteration, X(i)The value of the nth element on the diagonal is:
typically ξ is 0.0001 for 5 iterations;
step S04-2) brings the sparse vector S into WeleCalculating array element excitation vector W as TSeleExtracting the weight a of the subarraymM is 1, …, M, and the number of elements N included in each sub-arraymI.e. the sub-array size.
The invention has the advantages that:
1. the method takes the minimum subarray number as an optimization target, and realizes the simultaneous optimization of the subarray number, the subarray weight and the subarray size;
2. the method utilizes the compressible characteristic of the array element excitation vector to establish the sparse representation of the array element excitation vector under a sparse basis matrix, thereby converting the multi-parameter non-convex subarray optimization problem into a convex optimization problem of sparse vector recovery, and effectively solving the complex optimization problem by adopting a convex optimization algorithm;
3. the method has high calculation efficiency, is simple and easy to implement, can quickly synthesize the optimal subarray meeting the requirements of the main lobe width and the side lobe level, simplifies the feed network of the array antenna, and reduces the manufacturing cost and the cost of the whole system.
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FIG. 1 is a flow chart of a compressed sensing-based one-dimensional subarray division method of the present invention;
FIG. 2 is a schematic diagram of one-dimensional subarray division according to the present invention;
FIG. 3 is a diagram of the subarray distribution of the present invention for 200 element uniform linear subarray division;
fig. 4 is a far field pattern of the subarray implementation shown in fig. 3.
Detailed Description
The invention will now be further described with reference to the accompanying drawings.
As shown in fig. 1, a compressed sensing-based one-dimensional subarray division method of the present invention includes the following steps:
step S01: determining an array element excitation vector;
dividing a given N-element linear array into M sub-arrays, adding excitation only at the ports of the sub-arrays, and determining corresponding array element excitation vectors Wele. As shown in FIG. 1, an N-element linear array is divided into M sub-arrays, and excitation is added only to the ports of the sub-arrays, and the excitation weight of the M-th sub-array port is amAnd array elements in the same subarray are excited identically.
Step S01 specifically includes the following processes:
dividing a given N-element linear array into M sub-arrays, and adding excitation only at the sub-array ports, i.e. sub-array weighting vectors Wsub＝[a1,a2,…,aM]TWherein a ismThe weight of the mth subarray port is represented (M is 1, …, M), and then the corresponding array element excitation vector Wele＝[w1,w2,…,wN]T(wnThe excitation value representing the nth array element, N being 1, …, N) may be specifically expressed as
Wherein N ismIs the number of array elements contained in the mth sub-array, and is used for representing the size of the corresponding sub-array. It should be noted that the number of subarrays M is much smaller than the number of array elements N, i.e. M<<N。
Step S02: establishing sparse representation of array excitation vectors;
according to array element excitation vector WeleThe compressibility of the array element is controlled, a sparse vector S is constructed, a sparse basis matrix T is established, and the vector S is an array element excitation vector WeleSparse representation under a sparse basis T;
step S02 specifically includes the following steps:
exciting vector W of array elementeleThe difference between adjacent elements can obtain a vector S', which is specifically expressed as
Exciting the first array element with a value w1Concatenated with vector S' to form vector S, i.e.
It can be seen that the vector S includes only M non-zero elements, and is a sparse vector, and the number of non-zero elements is the same as the number of sub-arrays. According to array element excitation vector WeleThe numerical characteristics of the vector S form a basis matrix T, which is expressed in detail as
That is, T is a lower triangular matrix, and elements above the main diagonal are all 0, and elements below the main diagonal and the main diagonal are all 1. And is provided with
Wele＝TS (5)
That is to say that the vector S is WeleSparse representation under the sparse basis matrix T.
Step S03: establishing a convex optimization model for sparse vector recovery;
according to the compressed sensing theory, the minimum subarray number is taken as an optimization target, the main lobe width and side lobe level requirements of a radiation pattern are taken as constraint conditions, and the array element excitation vector W obtained in the step S02 is combinedeleEstablishing a convex optimization model for sparse signal recovery by using a sparse basis matrix T and a sparse vector S;
the specific process of step S03 is as follows:
the linear array direction diagram is shown as:
F(θ)＝A(θ)Wele(6)
wherein the steering vector a (θ) is:
wherein the wave number k is 2 pi/lambda, where lambda is the antennaOperating wavelength of dnIs the position of the nth array element, where N is 1, … N, and θ is the observation angle relative to the broadside direction;
if the array structure is symmetric, the steering vector can be equivalent to:
A(θ)＝[2cos(kd1sin(θ),2cos(kd2sin(θ)),…,2cos(kdNsin(θ))](8)
substituting equation (5) into equation (6) yields a directional pattern represented by a sparse vector S, i.e.
F(θ)＝A(θ)TS＝ψ(θ)S (9)
Then, according to a given main lobe width FNBW and a side lobe level upper limit requirement UB, a convex optimization model for minimizing the number of sub-arrays is established by combining a compressive sensing theory, and the expression of the convex optimization model is as follows:
wherein, thetamaxIs the main beam direction; the sideline region is represented by a side lobe region which can be determined by the main lobe width FNBW.
Step S04: outputting the number of subarrays, the weight of the subarrays and the size of the subarrays;
weighting by iteration l1The norm minimization algorithm solves the convex optimization model established in step S03, and calculates the sparse vector S, thereby determining the number of subarrays, the weight of the subarrays, and the size of the subarrays.
The specific process of step S04 is as follows:
to enhance the sparsity of the solution, here an iterative weighting/is used1The norm minimization algorithm is used for solving the convex optimization model established in the step S03 in a solver CVX, and the ith iteration is expressed as
Wherein l1,l2Respectively representing 1 norm and 2 norm, X is a diagonal matrix, defining an initial X(0)Is a unit array, and is based on the sparse vector after each iteration is finishedUpdating the element value on the diagonal line of X by the calculated value of S, and taking the updated result as the initial value of the next iteration, so that X is used in the ith iteration process(i)The value of the nth element on the diagonal is
Usually, ξ value in the iterative process is 0.0001, the iterative times is 5, from this, the sparse vector S is calculated, its non-zero element number is the sub-array number M, the sparse vector S is substituted into the equation (5) to calculate the array element excitation vector WeleThereby extracting the weight a of the subarraym(M-1, …, M), and the number of elements N included in each sub-arraymI.e. the sub-array size.
The one-dimensional subarray division method based on compressed sensing provided by the invention can be further verified and explained by the following specific simulation examples.
Simulation example:
in the example, even interval linear arrays with array elements of 200 and adjacent element intervals of 0.5 lambda are subjected to sub-array division so as to realize the far field radiation characteristic that the main beam direction is 0 degree, the main lobe width is 1.6 degrees and the side lobe level is less than-30 dB. It has been found that the above-mentioned desired radiation characteristics can be achieved by applying chebyshev excitation weighting to the 200 array elements, and it should be noted that even if such excitation weighting has a symmetric distribution characteristic, it is somewhat convenient to design the feeding network, but since it is necessary to add an excitation port to each array element, the corresponding feeding network is relatively complex and heavy. By adopting the one-dimensional subarray division method based on compressive sensing provided by the invention for design, simulation results show that the number of subarrays required by the method is only 11, and corresponding subarray weights and subarray sizes are shown in fig. 3, so that the subarray distribution has symmetry and is beneficial to implementation of a feed network. The numerical parameters of each sub-array are listed in table 1. The synthesized far-field pattern is shown in fig. 4, and by contrast, the synthesized subarrays of the method of the present invention achieve the desired radiation characteristics. Compared with array element-level excitation weighting, the subarray division method provided by the invention can realize the expected radiation pattern only by adding excitation at 11 subarray ports, and greatly simplifies the feed network, thereby reducing the system cost and application value, and having very obvious engineering significance and application value.
TABLE 1
Finally, it should be noted that the above embodiments are only used for illustrating the technical solutions of the present invention and are not limited. Although the present invention has been described in detail with reference to the embodiments, it will be understood by those skilled in the art that various changes may be made and equivalents may be substituted without departing from the spirit and scope of the invention as defined in the appended claims.
Claims (3)
1. A one-dimensional subarray division method based on compressed sensing, the method comprising:
step S01) dividing the given N-element linear array into M sub-arrays, adding excitation only at the ports of the sub-arrays, and determining the corresponding array element excitation vector Wele；
Step S02) according to the array element excitation vector WeleConstructing a sparse vector S by compressibility, and establishing a sparse basis matrix T to enable Wele＝TS；
Step S03) takes the minimum subarray number as the optimization target, takes the main lobe width and the side lobe level requirement of the radiation directional diagram as the constraint conditions, and excites the vector W according to the array elementseleEstablishing a convex optimization model for sparse vector recovery by using the sparse basis matrix T and the sparse vector S;
step S04) solving the convex optimization model established in the step S03), and calculating a sparse vector S, thereby determining the number of subarrays, the weight of the subarrays and the size of the subarrays; completing the division of the N-element linear array;
the step S01) is specifically:
dividing a given N-element linear array into M sub-arrays, and adding excitation only at sub-array ports to obtain the sub-arraysWeighting vector Wsub＝[a1,a2,…,aM]TWherein a ismRepresenting the weight of the mth subarray port, M is 1, …, M, then the array element excitation vector Wele＝[w1,w2,…,wN]T，wnDenotes the excitation value of the nth array element, N is 1, …, N, WeleIs particularly shown as
Wherein N ismThe array element number contained in the mth subarray is used for representing the size of the corresponding subarray;
in step S01), the number M of subarrays is much smaller than the number N of array elements, i.e., M < < N;
step S02) specifically includes:
step S02-1) array element excitation vector WeleDifferencing adjacent elements to obtain a vector S':
step S02-2) exciting the first array element1Concatenated with the vector S' to form a sparse vector S, i.e.
The sparse vector S is a vector with the length of N and comprises M nonzero elements;
step S02-3) according to the array element excitation vector WeleWith the numerical characteristics of the vector S, constructing a basis matrix T, i.e.
And has:
Wele＝TS (5)。
2. the compressed sensing-based one-dimensional subarray division method according to claim 1, wherein the step S03) specifically comprises:
step S03-1), establishing a linear array directional diagram:
F(θ)＝A(θ)Wele(6)
where A (θ) is the steering vector:
wherein the wave number k is 2 pi/lambda, where lambda is the operating wavelength of the antenna, dnIs the position of the nth array element, where N is 1, … N, and θ is the observation angle relative to the broadside direction;
step S03-2) when the array structure is symmetric, the steering vector a (θ) is equivalent to:
A(θ)＝[2cos(kd1sin(θ),2cos(kd2sin(θ)),…,2cos(kdNsin(θ))](8)
F(θ)＝A(θ)TS＝ψ(θ)S (9)
step S03-3) according to the given main lobe width FNBW and the upper limit requirement UB of the side lobe level, and in combination with the compressive sensing theory, establishing a convex optimization model for minimizing the number of sub-arrays:
wherein l1,l2Respectively representing 1 norm and 2 norm, thetamaxIs the main beam direction; the sideline region is denoted as the side lobe region, determined by the main lobe width FNBW.
3. The compressed sensing-based one-dimensional subarray division method according to claim 2, wherein the step S04) specifically comprises:
step S04-1) with iterative weighting l1The norm minimization algorithm solves the convex optimization model established in the step S03), and thus a sparse vector S is calculated;
the ith iteration of the convex optimization model is represented as:
wherein, X(i)Is a diagonal matrix, defines an initial X(0)Is a unit matrix, after each iteration is finished, the element value on the diagonal line of X is updated according to the calculated value of the sparse vector S, and the updated result is used as the initial value of the next iteration, so that in the ith iteration process, X is used as the initial value of the next iteration(i)The value of the nth element on the diagonal is:
ξ is 0.0001, i is 5;
step S04-2) brings the sparse vector S into WeleCalculating array element excitation vector W as TSeleExtracting the weight a of the subarraymM is 1, …, M, and the number of elements N included in each sub-arraymI.e. the sub-array size.
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