# Design of Fixed Beamformers Based on Vector-Sensor Arrays

Authors: Matthew Hawes; Wei Liu
Journal: International Journal of Antennas and Propagation
 (2015)

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## Abstract

Vector-sensor arrays such as those composed of crossed dipole pairs are used as they can account for a signal’s polarisation in addition to the usual direction of arrival information, hence allowing expanded capacity of the system. The problem of designing fixed beamformers based on such an array, with a quaternionic signal model, is considered in this paper. Firstly, we consider the problem of designing the weight coefficients for a fixed set of vector-sensor locations. This can be achieved by minimising the sidelobe levels while keeping a unitary response for the main lobe. The second problem is then how to find a sparse set of sensor locations which can be efficiently used to implement a fixed beamformer. We propose solving this problem by converting the traditionall1 norm minimisation associated with compressive sensing into a modified l1 norm minimisation which simultaneously minimises all four parts of the quaternionic weight coefficients. Further improvements can be made in terms of sparsity by converting the problem into a series of iteratively solved reweighted minimisations, as well as being able to enforce a minimum spacing between active sensor locations. Design examples are provided to verify the effectiveness of the proposed design methods.


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## Body

## 1. Introduction

Traditionally fixed beamformers have been designed assuming the arrays consist of isotropic array elements [1, 2]. As a result the polarisation of a signal is not taken into account when considering the performance of an array. Instead, a vector-sensor array can be considered, allowing measurements of both the horizontal and vertical components of the received waveforms [3–14].In the past ten years, quaternion-valued signal processing has attracted more and more attention with application areas involving three- or four-dimensional signals [15], and a quaternionic signal model has been introduced into the field of vector-sensor arrays for both adaptive beamforming and direction of arrival (DOA) estimation, [6, 7, 9–11, 13, 16, 17]. A quaternionic formulation provides a compact and convenient representation of three- and four-dimensional signals, although there will not be improvement in performance in theory as equivalent formulations can be established in corresponding domains, that is, two real-valued numbers versus one complex-valued number, or four real-valued numbers (using a “long vector”) versus one quaternion-valued number. However, to the best of our knowledge the area of fixed beamformer design for vector-sensor arrays using such a signal model has not been considered yet (in particular, it would be the first time to address the problem of designing sparse vector-sensor arrays).If such a beamformer is to be implemented using a uniform linear array (ULA), it is well known that the adjacent sensor separation can be no larger than half the operating wavelength, in order to avoid grating lobes. This can be problematic when considering arrays with a large aperture size, due to the cost associated with the number of sensors required. As a result, sparse arrays become a desirable alternative [18], which allow separations to be greater than half a wavelength, while still avoiding grating lobes due to the randomness of sensor locations. However, the tradeoff in using sparse arrays is the unpredictable sidelobe behaviour and it is often necessary to optimise sensor locations in order to achieve a desired performance, (e.g., minimising the peak sidelobe level).Therefore, the second problem to consider when designing a fixed beamformer is how to find a sparse set of sensor locations that can be efficiently used to implement a desired fixed beamformer. Some nonlinear optimisation methods such as genetic algorithms (GAs) [19–23] and simulated annealing (SA) [24] have been regularly used to achieve the required location optimisation for sensor arrays. The disadvantage of these methods is the potentially long computation time and the possibility of convergence to a nonoptimal solution.More recently, the area of compressive sensing (CS) has been explored [25], and CS-based methods have been proposed in the design of sparse arrays [26–32] through l1 norm minimisation. It is also known that the sparsity of the final array can be improved by converting the problem into a series of iteratively reweighted minimisations [31–35]. This is achieved by the addition of a reweighting term which penalises small weight coefficients more heavily, meaning they are less likely to be repeated in the next iteration. In this paper we propose extending these schemes so that they can be used in designing a sparse set of vector-sensor locations with a quaternionic signal model. This is achieved by reformulating the problem into a modified l1 norm minimisation similar to what is used for complex-valued minimisation schemes [36] and it can readily be solved using existing convex optimisation toolboxes [37, 38].A third problem considered in this work is how to enforce a minimum spacing between active locations so that the vector-sensors with a nonzero physical size can fit into the resultant locations in practice. This is an extension of the work in [31], where a size constraint is employed for the design of traditional sparse scalar-sensor arrays.The rest of this paper is structured as follows: Section2 gives details of the proposed design method, including some basics about quaternions (Section 2.1), the array model being used (Section 2.2), the method to find the weight coefficients for a fixed set of locations (Section 2.3), the quaternionic CS-based (Section 2.4) and quaternionic reweighted minimisation based (Section 2.5) design methods, and methods of enforcing the size constraint (Section 2.6). Design examples are given in Section 3, with conclusions drawn in Section 4.


## 2. Proposed Design Method

### 2.1. Quaternions

A quaternion is a hypercomplex number defined as follows [39]: (1)q=Rq+iIq+jJq+kKq,where R(q) is the real part of the quaternion and I(q), J(q), and K(q) are the three imaginary components. Similarly for vectors and matrices of quaternions we have (2)v=R(v)+iI(v)+jJ(v)+kK(v),M=R(M)+iI(M)+jJ(M)+kK(M).The conjugate and modulus of a quaternion are given by(3)q∗=R(q)-iI(q)-jJ(q)-kK(q),q=R2(q)+I2(q)+J2(q)+K2(q).The imaginary units i, j, and k satisfy the following: (4)ii=jj=kk=ijk=-1.Finally{·}⊲ denotes the conjugate transpose of quaternionic vectors and matrices.It is worth noting that in many scenarios quaternions prove useful as they allow the easy representation of problems with four-dimensional data. However, care has to be taken when formulating a problem using quaternions as they are noncommutative.


### 2.2. Quaternionic Array Model

Figure1 shows the array structure being considered. There are M potentially active crossed dipole pairs located along the y-axis, uniformly spaced a distance d apart. At each location one of the dipoles is parallel to the x-axis and the other to the y-axis. Also shown is a signal with its direction of arrival (DOA) defined by the angles θ and ϕ. Without loss of generality, we assume the signals impinge upon the array from the y-z plane; that is, ϕ=π/2 or ϕ=-π/2. The angle θ is limited to 0≤θ≤π/2. A plane-wave signal model is considered; that is, the signal impinges upon the array from the far field.Figure 1
Array model being considered with crossed dipole elements.The spatial steering vector of the array is given by(5)ssθ,ϕ=1,e-j2πdsinθsinϕ/λ,WW…,e-j2π(M-1)dsinθsinϕ/λT,where λ is the wavelength of the signal of interest and {·}T denotes the transpose operation.For crossed dipoles the spatial-polarization coherent vector contains information about a signals polarisation and is given by [3–5, 11] (6)sp(θ,ϕ,γ,η)=-cos⁡γ,cos⁡θsinγejηforϕ=π2cos⁡γ,-cos⁡θsinγejηforϕ=-π2,where γ∈[0,π/2] is the auxiliary polarization angle and η∈[-π,π) is the polarization phase difference.Now the array structure can be split into two subarrays, that is, one parallel to thex-axis and one to y-axis. The steering vector of each of these subarrays is complex-valued and given by(7)sx(θ,ϕ,γ,η)=-cos⁡γssθ,ϕforϕ=π2cos⁡γssθ,ϕforϕ=-π2,sy(θ,ϕ,γ,η)=cos⁡θsinγejηss(θ,ϕ)forϕ=π2-cos⁡θsinγejηss(θ,ϕ)forϕ=-π2.These are then combined to give an overall quaternionic steering vector as follows:(8)sθ,ϕ,γ,η=sxθ,ϕ,γ,η+isyθ,ϕ,γ,η,=Rsxθ,ϕ,γ,η+jJsxθ,ϕ,γ,η+iRsyθ,ϕ,γ,η+ijJsyθ,ϕ,γ,η,=Rsxθ,ϕ,γ,η+iRsyθ,ϕ,γ,η+jJsxθ,ϕ,γ,η+kJsyθ,ϕ,γ,η.The response of the array is given by(9)Pθ,ϕ,γ,η=w⊲sθ,ϕ,γ,η,where w is the quaternionic weight coefficient vector defined as (10)w=w1w2⋯wMT,and wm is a quaternionic value for m=1,2,…,M.


### 2.3. Weight Vector Design for a Fixed Set of Locations

The first problem we consider here is that of designing the weight coefficients for a given array geometry, where the sensor locations could be that of a ULA or a known sparse structure or other layouts.In order to achieve a desirable response (or reference response)Pr(θ,ϕ,γ,η), we can minimize the error between the designed and desired responses subject to a set of linear constraints to meet any specific requirements of the design, which can be formulated as follows:(11)min⁡w⁡WWWIpr-w⊲S2subjecttow⊲SC=f,where ·2 denotes the l2 norm,(12)pr=Prθ1,ϕ1,γ1,η1,…,PrθL,ϕL,γL,ηL,S=sθ1,ϕ1,γ1,η1,…,sθL,ϕL,γL,ηL.L is the number of points sampled in the desired beam response, SC is the constraint matrix, and f is the corresponding response vector.In this work, we use the ideal response forPr(θ,ϕ,γ,η), that is, a value of one for the main lobe and zeros for the other entries. Here we consider a special case of the above formulation, where we only minimize the design error over the sidelobe region and the designed response is constrained to be unity at the main lobe region. In this case, we have(13)min⁡w⁡WWWipr-w⊲S2subjecttow⊲SML=1,where pr and S will only cover the sidelobe region, SML is constructed in a similar manner to S but only considers the main lobe of interest, and 1 is a vector with a value of unity for all its entries.To convert the problem into a form that can easily be solved, the quaternionic values have to be split into real and imaginary parts that can be considered separately. This gives the problem in the following form:(14)min⁡w~⁡WWWIp^r-w~TS~2subjecttoRw~TS~ML=1WWWWWIIw~TS~ML=0WWWWWIJw~TS~ML=0WWWWWIKw~TS~ML=0,where the following definitions are used:(15)w~=R(w1),-I(w1),-J(w1),-K(w1),…,WWRwM,-IwM,-JwM,-KwMTp^r=Rpr,Ipr,Jpr,Kpr,S~=R(s1)I(s1)J(s1)K(s1)-I(s1)R(s1)-K(s1)J(s1)-J(s1)K(s1)R(s1)-I(s1)-K(s1)-J(s1)I(s1)R(s1)⋮⋮⋮⋮R(sM)I(sM)J(sM)K(sM)-I(sM)R(sM)-K(sM)J(sM)-J(sM)K(sM)R(sM)-I(sM)-K(sM)-J(sM)I(sM)R(sM),where sm is the mth row vector of the matrix S and S~ML is constructed in a similar way to S~ but only using the main lobe.As mentioned this design method can be used with any known array geometry. However, in many cases, the array geometry is not known in advance and a location optimisation process is needed to find the set of array sensor locations, such as in the sparse array design problem, which will be dealt with in the next subsection.


### 2.4. CS-Based Design of Sparse Vector-Sensor Arrays

As before, supposePr(θ,ϕ,γ,η) is a reference pattern which we wish to achieve. First, consider Figure 1 as being a grid of potentially active crossed dipole locations. In this instance, (M-1)d is the aperture of the array and M is a large number. Sparseness is then introduced by selecting the weight coefficients to give as few active crossed dipoles as possible, while still giving a designed response that is close to the desired one.This problem is formulated as(16)min⁡⁡WWWIw0subjecttopr-w⊲S2≤α,where w0 is the number of nonzero weight coefficients in w, pr is the vector holding the desired beam response at the sampled angular and polarisation points of interest, S is the matrix composed of the corresponding steering vectors, and α places a limit on the allowed difference between the desired and the designed responses.In practice, (16) is approximated by a minimisation of the l1 norm of the weight coefficients [25]; that is,(17)min⁡⁡WWWIw1subjecttopr-w⊲S2≤α.This formulation is effective for the design of narrowband sparse scalar-sensor arrays. When considering quaternionic coefficients the problem has to be reformulated to ensure the real and three imaginary parts of the quaternion are simultaneously minimised. This is achieved by following a scheme similar to that used when considering thel1 minimisation of complex data [36]. When all four parts of a quaternionic coefficient are equal to zero, the crossed dipoles can be considered inactive/not present, and as a result sparsity is introduced.First we rewrite (17) as(18)min⁡⁡WWWIt∈R+subjecttopr-w⊲S2≤α,w1≤t,where(19)w1=∑m=1M‍RwmIwmJwmKwm2.Now we decomposet to t=∑m=1Mtm‍, tm∈R+. In vector form, we have(20)t=[1,…,1]t1⋮tM=1Tt.Then (18) can be rewritten as(21)min⁡t⁡WWWI1Ttsubjecttopr-w⊲S2≤αWWWWIIIRwmIwmJwmKwm2≤tm,m=1,…,M.Now define(22)w^=t1,Rw1,-Iw1,-Jw1,-Kw1,…,WWRwM,-IwM,-JwM,-KwMTc^=1,0,0,0,0,…,1,0,0,0,0T,S^=0000R(s1)I(s1)J(s1)K(s1)-I(s1)R(s1)-K(s1)J(s1)-J(s1)K(s1)R(s1)-I(s1)-K(s1)-J(s1)I(s1)R(s1)⋮⋮⋮⋮0000R(sM)I(sM)J(sM)K(sM)-I(sM)R(sM)-K(sM)J(sM)-J(sM)K(sM)R(sM)-I(sM)-K(sM)-J(sM)I(sM)R(sM).Finally we arrive at the final formulation for the sparse vector-sensor array design problem(23)min⁡w^⁡WWWIc^Tw^subjecttop^r-w^TS^2≤αWWWWIIIRwmIwmJwmKwm2≤tm,m=1,…,M.


### 2.5. Iteratively Solved Reweighted Quaternionic Minimisations

For the design of sparse arrays consisting of isotropic array elements with real-valued coefficients, reweightedl1 minimisations are used in order to get a closer approximation to the l0 minimisation [31, 34, 35]. This is done by solving a series of reweighted l1 minimisations, where the reweighting term penalises smaller weight coefficients more heavily.Following the idea, we introduce a reweighting parameter to each quaternionic coefficient. This leads to (23) being altered to(24)min⁡w^⁡WWWIc^Tw^subjecttop^r-w^TS^2≤αWWWWWIamiRwmIwmJwmKwm2≤tmi,WWWWWIIWm=0,…,M-1,where we now have(25)w^=t1i,R(w1i),-I(w1i),-J(w1i),-K(w1i),…,WWR(wMi),-I(wMi),-J(wMi),-K(wMi)T,c^=a0i,0J,a1i,0J,…,0JT,ami=RwmIwmJwmKwm2+ϵ-1with ϵ being set to slightly below the minimum implemented absolute coefficient value for a given location.The problem is iteratively solved as follows.(1)
Seti=1 and obtain an initial estimate of the weight coefficients by solving (23).(2)
Consideri=i+1; find the reweighting terms ami for all m and solve (24).(3)
Repeat step 2 until the number of active sensor locations has remained constant for three or more iterations of the algorithm.Note that it is the addition of the reweighting term that improves the sparsity of the solution. If a small nonzero valued combined weight coefficient is found in the previous iteration, it results in a large reweighting term in the current iteration. As a result the nonzero value is unlikely to be repeated, therefore improving the sparsity of the solution. Conversely, a large nonzero valued coefficient will give a small reweighting term. As a result the large nonzero value is likely to be repeated.


### 2.6. Enforcing the Physical Size Constraint

In above formulations, the solutions do not take the size of the vector-sensors into account. As a result we could end up with an array that could not be implemented in practice due to the vector-sensors not fitting in their deigned locations. Therefore, a minimum spacing of the vector-sensors’ physical size should be applied to the optimisation.This can be achieved using the methods recently proposed in [31], where the postprocessing method and iterative minimum distance sampling method can be directly applied to our sparse vector-sensor array design. These two methods involve some degree of merging locations that are too close together. This is a form of steering vector error that may degrade the performance of the array. As a result, a constraint is required on the minimisation to limit the effects of this error, given by limiting εw~2 to a small value in the design, where ε is the limit on the expected norm-bounded steering vector error, as normally employed in the design of robust beamformers. To further improve the performance of the array, the weight vector for the final sensor location can be redesigned using the method proposed in Section 2.3.A third method to enforce the size constraint is to alter the reweighting scheme so that all locations not meeting the size constraint are heavily penalised in order to avoid replication in the next iteration. The modified reweighting terms are given by(26)δmi=wmi-1+ϵ-1m=1wmi-1+ϵ-1m>1andconstraintmetϵ-1otherwise.The iterative procedure detailed in Section2.5 is now also repeated until a solution complying with the size constraint is met (rather than looking at how many iterations the number of nonzero valued coefficients has remained constant for).


## 2.1. Quaternions

A quaternion is a hypercomplex number defined as follows [39]: (1)q=Rq+iIq+jJq+kKq,where R(q) is the real part of the quaternion and I(q), J(q), and K(q) are the three imaginary components. Similarly for vectors and matrices of quaternions we have (2)v=R(v)+iI(v)+jJ(v)+kK(v),M=R(M)+iI(M)+jJ(M)+kK(M).The conjugate and modulus of a quaternion are given by(3)q∗=R(q)-iI(q)-jJ(q)-kK(q),q=R2(q)+I2(q)+J2(q)+K2(q).The imaginary units i, j, and k satisfy the following: (4)ii=jj=kk=ijk=-1.Finally{·}⊲ denotes the conjugate transpose of quaternionic vectors and matrices.It is worth noting that in many scenarios quaternions prove useful as they allow the easy representation of problems with four-dimensional data. However, care has to be taken when formulating a problem using quaternions as they are noncommutative.


## 2.2. Quaternionic Array Model

Figure1 shows the array structure being considered. There are M potentially active crossed dipole pairs located along the y-axis, uniformly spaced a distance d apart. At each location one of the dipoles is parallel to the x-axis and the other to the y-axis. Also shown is a signal with its direction of arrival (DOA) defined by the angles θ and ϕ. Without loss of generality, we assume the signals impinge upon the array from the y-z plane; that is, ϕ=π/2 or ϕ=-π/2. The angle θ is limited to 0≤θ≤π/2. A plane-wave signal model is considered; that is, the signal impinges upon the array from the far field.Figure 1
Array model being considered with crossed dipole elements.The spatial steering vector of the array is given by(5)ssθ,ϕ=1,e-j2πdsinθsinϕ/λ,WW…,e-j2π(M-1)dsinθsinϕ/λT,where λ is the wavelength of the signal of interest and {·}T denotes the transpose operation.For crossed dipoles the spatial-polarization coherent vector contains information about a signals polarisation and is given by [3–5, 11] (6)sp(θ,ϕ,γ,η)=-cos⁡γ,cos⁡θsinγejηforϕ=π2cos⁡γ,-cos⁡θsinγejηforϕ=-π2,where γ∈[0,π/2] is the auxiliary polarization angle and η∈[-π,π) is the polarization phase difference.Now the array structure can be split into two subarrays, that is, one parallel to thex-axis and one to y-axis. The steering vector of each of these subarrays is complex-valued and given by(7)sx(θ,ϕ,γ,η)=-cos⁡γssθ,ϕforϕ=π2cos⁡γssθ,ϕforϕ=-π2,sy(θ,ϕ,γ,η)=cos⁡θsinγejηss(θ,ϕ)forϕ=π2-cos⁡θsinγejηss(θ,ϕ)forϕ=-π2.These are then combined to give an overall quaternionic steering vector as follows:(8)sθ,ϕ,γ,η=sxθ,ϕ,γ,η+isyθ,ϕ,γ,η,=Rsxθ,ϕ,γ,η+jJsxθ,ϕ,γ,η+iRsyθ,ϕ,γ,η+ijJsyθ,ϕ,γ,η,=Rsxθ,ϕ,γ,η+iRsyθ,ϕ,γ,η+jJsxθ,ϕ,γ,η+kJsyθ,ϕ,γ,η.The response of the array is given by(9)Pθ,ϕ,γ,η=w⊲sθ,ϕ,γ,η,where w is the quaternionic weight coefficient vector defined as (10)w=w1w2⋯wMT,and wm is a quaternionic value for m=1,2,…,M.


## 2.3. Weight Vector Design for a Fixed Set of Locations

The first problem we consider here is that of designing the weight coefficients for a given array geometry, where the sensor locations could be that of a ULA or a known sparse structure or other layouts.In order to achieve a desirable response (or reference response)Pr(θ,ϕ,γ,η), we can minimize the error between the designed and desired responses subject to a set of linear constraints to meet any specific requirements of the design, which can be formulated as follows:(11)min⁡w⁡WWWIpr-w⊲S2subjecttow⊲SC=f,where ·2 denotes the l2 norm,(12)pr=Prθ1,ϕ1,γ1,η1,…,PrθL,ϕL,γL,ηL,S=sθ1,ϕ1,γ1,η1,…,sθL,ϕL,γL,ηL.L is the number of points sampled in the desired beam response, SC is the constraint matrix, and f is the corresponding response vector.In this work, we use the ideal response forPr(θ,ϕ,γ,η), that is, a value of one for the main lobe and zeros for the other entries. Here we consider a special case of the above formulation, where we only minimize the design error over the sidelobe region and the designed response is constrained to be unity at the main lobe region. In this case, we have(13)min⁡w⁡WWWipr-w⊲S2subjecttow⊲SML=1,where pr and S will only cover the sidelobe region, SML is constructed in a similar manner to S but only considers the main lobe of interest, and 1 is a vector with a value of unity for all its entries.To convert the problem into a form that can easily be solved, the quaternionic values have to be split into real and imaginary parts that can be considered separately. This gives the problem in the following form:(14)min⁡w~⁡WWWIp^r-w~TS~2subjecttoRw~TS~ML=1WWWWWIIw~TS~ML=0WWWWWIJw~TS~ML=0WWWWWIKw~TS~ML=0,where the following definitions are used:(15)w~=R(w1),-I(w1),-J(w1),-K(w1),…,WWRwM,-IwM,-JwM,-KwMTp^r=Rpr,Ipr,Jpr,Kpr,S~=R(s1)I(s1)J(s1)K(s1)-I(s1)R(s1)-K(s1)J(s1)-J(s1)K(s1)R(s1)-I(s1)-K(s1)-J(s1)I(s1)R(s1)⋮⋮⋮⋮R(sM)I(sM)J(sM)K(sM)-I(sM)R(sM)-K(sM)J(sM)-J(sM)K(sM)R(sM)-I(sM)-K(sM)-J(sM)I(sM)R(sM),where sm is the mth row vector of the matrix S and S~ML is constructed in a similar way to S~ but only using the main lobe.As mentioned this design method can be used with any known array geometry. However, in many cases, the array geometry is not known in advance and a location optimisation process is needed to find the set of array sensor locations, such as in the sparse array design problem, which will be dealt with in the next subsection.


## 2.4. CS-Based Design of Sparse Vector-Sensor Arrays

As before, supposePr(θ,ϕ,γ,η) is a reference pattern which we wish to achieve. First, consider Figure 1 as being a grid of potentially active crossed dipole locations. In this instance, (M-1)d is the aperture of the array and M is a large number. Sparseness is then introduced by selecting the weight coefficients to give as few active crossed dipoles as possible, while still giving a designed response that is close to the desired one.This problem is formulated as(16)min⁡⁡WWWIw0subjecttopr-w⊲S2≤α,where w0 is the number of nonzero weight coefficients in w, pr is the vector holding the desired beam response at the sampled angular and polarisation points of interest, S is the matrix composed of the corresponding steering vectors, and α places a limit on the allowed difference between the desired and the designed responses.In practice, (16) is approximated by a minimisation of the l1 norm of the weight coefficients [25]; that is,(17)min⁡⁡WWWIw1subjecttopr-w⊲S2≤α.This formulation is effective for the design of narrowband sparse scalar-sensor arrays. When considering quaternionic coefficients the problem has to be reformulated to ensure the real and three imaginary parts of the quaternion are simultaneously minimised. This is achieved by following a scheme similar to that used when considering thel1 minimisation of complex data [36]. When all four parts of a quaternionic coefficient are equal to zero, the crossed dipoles can be considered inactive/not present, and as a result sparsity is introduced.First we rewrite (17) as(18)min⁡⁡WWWIt∈R+subjecttopr-w⊲S2≤α,w1≤t,where(19)w1=∑m=1M‍RwmIwmJwmKwm2.Now we decomposet to t=∑m=1Mtm‍, tm∈R+. In vector form, we have(20)t=[1,…,1]t1⋮tM=1Tt.Then (18) can be rewritten as(21)min⁡t⁡WWWI1Ttsubjecttopr-w⊲S2≤αWWWWIIIRwmIwmJwmKwm2≤tm,m=1,…,M.Now define(22)w^=t1,Rw1,-Iw1,-Jw1,-Kw1,…,WWRwM,-IwM,-JwM,-KwMTc^=1,0,0,0,0,…,1,0,0,0,0T,S^=0000R(s1)I(s1)J(s1)K(s1)-I(s1)R(s1)-K(s1)J(s1)-J(s1)K(s1)R(s1)-I(s1)-K(s1)-J(s1)I(s1)R(s1)⋮⋮⋮⋮0000R(sM)I(sM)J(sM)K(sM)-I(sM)R(sM)-K(sM)J(sM)-J(sM)K(sM)R(sM)-I(sM)-K(sM)-J(sM)I(sM)R(sM).Finally we arrive at the final formulation for the sparse vector-sensor array design problem(23)min⁡w^⁡WWWIc^Tw^subjecttop^r-w^TS^2≤αWWWWIIIRwmIwmJwmKwm2≤tm,m=1,…,M.


## 2.5. Iteratively Solved Reweighted Quaternionic Minimisations

For the design of sparse arrays consisting of isotropic array elements with real-valued coefficients, reweightedl1 minimisations are used in order to get a closer approximation to the l0 minimisation [31, 34, 35]. This is done by solving a series of reweighted l1 minimisations, where the reweighting term penalises smaller weight coefficients more heavily.Following the idea, we introduce a reweighting parameter to each quaternionic coefficient. This leads to (23) being altered to(24)min⁡w^⁡WWWIc^Tw^subjecttop^r-w^TS^2≤αWWWWWIamiRwmIwmJwmKwm2≤tmi,WWWWWIIWm=0,…,M-1,where we now have(25)w^=t1i,R(w1i),-I(w1i),-J(w1i),-K(w1i),…,WWR(wMi),-I(wMi),-J(wMi),-K(wMi)T,c^=a0i,0J,a1i,0J,…,0JT,ami=RwmIwmJwmKwm2+ϵ-1with ϵ being set to slightly below the minimum implemented absolute coefficient value for a given location.The problem is iteratively solved as follows.(1)
Seti=1 and obtain an initial estimate of the weight coefficients by solving (23).(2)
Consideri=i+1; find the reweighting terms ami for all m and solve (24).(3)
Repeat step 2 until the number of active sensor locations has remained constant for three or more iterations of the algorithm.Note that it is the addition of the reweighting term that improves the sparsity of the solution. If a small nonzero valued combined weight coefficient is found in the previous iteration, it results in a large reweighting term in the current iteration. As a result the nonzero value is unlikely to be repeated, therefore improving the sparsity of the solution. Conversely, a large nonzero valued coefficient will give a small reweighting term. As a result the large nonzero value is likely to be repeated.


## 2.6. Enforcing the Physical Size Constraint

In above formulations, the solutions do not take the size of the vector-sensors into account. As a result we could end up with an array that could not be implemented in practice due to the vector-sensors not fitting in their deigned locations. Therefore, a minimum spacing of the vector-sensors’ physical size should be applied to the optimisation.This can be achieved using the methods recently proposed in [31], where the postprocessing method and iterative minimum distance sampling method can be directly applied to our sparse vector-sensor array design. These two methods involve some degree of merging locations that are too close together. This is a form of steering vector error that may degrade the performance of the array. As a result, a constraint is required on the minimisation to limit the effects of this error, given by limiting εw~2 to a small value in the design, where ε is the limit on the expected norm-bounded steering vector error, as normally employed in the design of robust beamformers. To further improve the performance of the array, the weight vector for the final sensor location can be redesigned using the method proposed in Section 2.3.A third method to enforce the size constraint is to alter the reweighting scheme so that all locations not meeting the size constraint are heavily penalised in order to avoid replication in the next iteration. The modified reweighting terms are given by(26)δmi=wmi-1+ϵ-1m=1wmi-1+ϵ-1m>1andconstraintmetϵ-1otherwise.The iterative procedure detailed in Section2.5 is now also repeated until a solution complying with the size constraint is met (rather than looking at how many iterations the number of nonzero valued coefficients has remained constant for).


## 3. Design Examples

In this section design examples will be presented in order to verify the effectiveness of the proposed design methods. This will include one design example based on a ULA and one example to illustrate how the methods can be used to design a sparse vector-sensor array and how the reweighted method can improve the sparsity of the solution. Finally, an example will be given to illustrate the performance of arrays designed while enforcing a size constraint.For all of the figures that follow, positive values ofθ indicate the value range θ∈[0∘,90∘] for ϕ=90∘, while negative values of θ∈[-90∘,0∘] indicate an equivalent range of θ∈[0∘,90∘] with ϕ=-90∘. The main lobe is designed to be at the single point defined by θ=0∘ and ϕ=90∘, with the sidelobes given by θ=[10∘,90∘] for ϕ=90∘ and ϕ=-90∘.


### 3.1. ULA Based Design Example

First we consider a ULA consisting of ten crossed dipoles with an adjacent separation of0.5λ. Both here and in all that follows, λ is the wavelength of the signal of interest. The polarisations are given by (γ,η)=(30∘,25∘). By solving (14) to obtain the weight coefficients we end up with the response shown in Figure 2. It can be clearly seen that the main lobe is in the correct location and sufficient sidelobe attenuation has been achieved; that is, the minimisation has successfully designed a fixed beamformer.Figure 2
Beam response for the design example based on a ULA.


### 3.2. Sparse Location Optimisation Design Examples

Now we will consider using the nonreweighted and reweighted minimisations to design sets of sparse locations. In this instance the maximum aperture of the array is15λ and the polarisation of the signal of interest is defined by (γ,η)=(10∘,-10∘).Firstly, (23) is solved with a value α=0.45. However, this resulted in 44 active locations over an aperture of 13.39λ, giving a mean adjacent sensor separation of 0.31λ. As a result, a ULA of equivalent length could be implemented using a smaller number of crossed dipoles. Therefore, there is no saving in cost and so forth.By increasing the value ofα to 0.94, the designed array instead has 9 active locations over an aperture of 5.13λ, giving a mean adjacent separation of 0.64λ. Therefore, less sensors are required than an equivalent length ULA. On the other hand, as the value of α has been significantly increased there will be more error between the desired and designed responses than the case with α=0.45. The responses for both cases are shown in Figure 3. This clearly shows the degradation in the performance of the array response. The main lobe width has been widened and the sidelobe levels raised. For completeness the locations found for the value of α=0.94 are shown in Table 1.Table 1
Locations found from solving (23) with a value of α=0.94.
n
d
n
/
λ
n
d
n
/
λ
n
d
n
/
λ
1
4.93
4
6.75
7
8.46
2
5.79
5
7.50
8
9.21
3
6.59
6
8.31
9
10.07Figure 3
Beam response for the locations found by solving (23).As an alternative, we can iteratively solve the series of reweighted minimisations given by (24) while using all the original parameters. So there is no increase in the value of α and we still use α=0.45, which suggests there should not be degradation in the performance of the array’s response. This results in 13 active locations spread over an aperture of 10.37λ as detailed in Table 2. Here the mean adjacent separation is 0.86λ, meaning the array can be implemented with fewer crossed dipoles than a ULA of equivalent length. Figure 4 shows the resulting response and it is clear that an acceptable performance has been achieved.Table 2
Locations found from solving (24) with a value of α=0.45.
n
d
n
/
λ
n
d
n
/
λ
n
d
n
/
λ
1
2.32
6
6.64
10
10.07
2
3.22
7
7.50
11
10.92
3
4.08
8
8.36
12
11.78
4
4.93
9
9.21
13
12.68
5
5.79Figure 4
Beam response for the locations found by solving (24).


### 3.3. Design Examples with Size Constraint

Two examples will now be considered to show the effectiveness of two methods for enforcing the size constraint. For both, the size of the crossed dipoles being considered is assumed to be0.8λ.


#### 3.3.1. Postprocessing Design Example

For this example we consider an aperture of15λ split into 300 potential sensor locations. In this instance the polarisation of the signal of interest is given by (γ,η)=(0∘,0∘). The values of α=0.75, β=0.1, and ε=1 were used in the constraints placed on the minimisation.This resulted in 15 crossed dipoles spread over11.77λ as shown in Table 3. Here the minimum adjacent separation is 0.81λ indicating that the size constraint has been successfully enforced. The resulting response is shown in Figure 5 which shows a desirable response has been achieved.Table 3
Locations for the postprocessing design example.
n
d
n
/
λ
n
d
n
/
λ
n
d
n
/
λ
1
1.61
6
5.82
11
10.06
2
2.44
7
6.64
12
10.92
3
3.29
8
7.49
13
11.74
4
4.11
9
8.35
14
12.57
5
4.96
10
9.16
15
13.38Figure 5
Beam response for the postprocessing design example.In order to ensure an acceptable performance here we had to redesign the weight coefficients for the final merged locations. It is reasonable to expect the same from the iterative minimum distance sampling method due to the fact that some merger of location is still required. A similar performance would also be achieved and as a result an example of it is not given here. However, a reweighted design example with size constraint is considered below as an alternative that does not require a redesigning of weight coefficients due to no locations being merged.


#### 3.3.2. Reweighted Design Example

In this example we are now considering 600 possible locations over the same aperture. However, due to the improved sparsity for a given amount of error offered by the reweighted method we can now use a value ofα=0.3.This results in 14 crossed dipole locations shown in Table4. In this instance the aperture of the designed array is 11.29λ with a minimum adjacent separation of 0.85λ. Again, from Figure 6 we can see that an acceptable response has been achieved.Table 4
Locations for the reweighted design example.
n
d
n
/
λ
n
d
n
/
λ
n
d
n
/
λ
1
2.26
6
6.67
11
10.94
2
3.26
7
7.53
12
11.79
3
4.11
8
8.38
13
12.64
4
4.97
9
9.23
14
13.55
5
5.82
10
10.08Figure 6
Beam response for the reweighted design example.


## 3.1. ULA Based Design Example

First we consider a ULA consisting of ten crossed dipoles with an adjacent separation of0.5λ. Both here and in all that follows, λ is the wavelength of the signal of interest. The polarisations are given by (γ,η)=(30∘,25∘). By solving (14) to obtain the weight coefficients we end up with the response shown in Figure 2. It can be clearly seen that the main lobe is in the correct location and sufficient sidelobe attenuation has been achieved; that is, the minimisation has successfully designed a fixed beamformer.Figure 2
Beam response for the design example based on a ULA.


## 3.2. Sparse Location Optimisation Design Examples

Now we will consider using the nonreweighted and reweighted minimisations to design sets of sparse locations. In this instance the maximum aperture of the array is15λ and the polarisation of the signal of interest is defined by (γ,η)=(10∘,-10∘).Firstly, (23) is solved with a value α=0.45. However, this resulted in 44 active locations over an aperture of 13.39λ, giving a mean adjacent sensor separation of 0.31λ. As a result, a ULA of equivalent length could be implemented using a smaller number of crossed dipoles. Therefore, there is no saving in cost and so forth.By increasing the value ofα to 0.94, the designed array instead has 9 active locations over an aperture of 5.13λ, giving a mean adjacent separation of 0.64λ. Therefore, less sensors are required than an equivalent length ULA. On the other hand, as the value of α has been significantly increased there will be more error between the desired and designed responses than the case with α=0.45. The responses for both cases are shown in Figure 3. This clearly shows the degradation in the performance of the array response. The main lobe width has been widened and the sidelobe levels raised. For completeness the locations found for the value of α=0.94 are shown in Table 1.Table 1
Locations found from solving (23) with a value of α=0.94.
n
d
n
/
λ
n
d
n
/
λ
n
d
n
/
λ
1
4.93
4
6.75
7
8.46
2
5.79
5
7.50
8
9.21
3
6.59
6
8.31
9
10.07Figure 3
Beam response for the locations found by solving (23).As an alternative, we can iteratively solve the series of reweighted minimisations given by (24) while using all the original parameters. So there is no increase in the value of α and we still use α=0.45, which suggests there should not be degradation in the performance of the array’s response. This results in 13 active locations spread over an aperture of 10.37λ as detailed in Table 2. Here the mean adjacent separation is 0.86λ, meaning the array can be implemented with fewer crossed dipoles than a ULA of equivalent length. Figure 4 shows the resulting response and it is clear that an acceptable performance has been achieved.Table 2
Locations found from solving (24) with a value of α=0.45.
n
d
n
/
λ
n
d
n
/
λ
n
d
n
/
λ
1
2.32
6
6.64
10
10.07
2
3.22
7
7.50
11
10.92
3
4.08
8
8.36
12
11.78
4
4.93
9
9.21
13
12.68
5
5.79Figure 4
Beam response for the locations found by solving (24).


## 3.3. Design Examples with Size Constraint

Two examples will now be considered to show the effectiveness of two methods for enforcing the size constraint. For both, the size of the crossed dipoles being considered is assumed to be0.8λ.


### 3.3.1. Postprocessing Design Example

For this example we consider an aperture of15λ split into 300 potential sensor locations. In this instance the polarisation of the signal of interest is given by (γ,η)=(0∘,0∘). The values of α=0.75, β=0.1, and ε=1 were used in the constraints placed on the minimisation.This resulted in 15 crossed dipoles spread over11.77λ as shown in Table 3. Here the minimum adjacent separation is 0.81λ indicating that the size constraint has been successfully enforced. The resulting response is shown in Figure 5 which shows a desirable response has been achieved.Table 3
Locations for the postprocessing design example.
n
d
n
/
λ
n
d
n
/
λ
n
d
n
/
λ
1
1.61
6
5.82
11
10.06
2
2.44
7
6.64
12
10.92
3
3.29
8
7.49
13
11.74
4
4.11
9
8.35
14
12.57
5
4.96
10
9.16
15
13.38Figure 5
Beam response for the postprocessing design example.In order to ensure an acceptable performance here we had to redesign the weight coefficients for the final merged locations. It is reasonable to expect the same from the iterative minimum distance sampling method due to the fact that some merger of location is still required. A similar performance would also be achieved and as a result an example of it is not given here. However, a reweighted design example with size constraint is considered below as an alternative that does not require a redesigning of weight coefficients due to no locations being merged.


### 3.3.2. Reweighted Design Example

In this example we are now considering 600 possible locations over the same aperture. However, due to the improved sparsity for a given amount of error offered by the reweighted method we can now use a value ofα=0.3.This results in 14 crossed dipole locations shown in Table4. In this instance the aperture of the designed array is 11.29λ with a minimum adjacent separation of 0.85λ. Again, from Figure 6 we can see that an acceptable response has been achieved.Table 4
Locations for the reweighted design example.
n
d
n
/
λ
n
d
n
/
λ
n
d
n
/
λ
1
2.26
6
6.67
11
10.94
2
3.26
7
7.53
12
11.79
3
4.11
8
8.38
13
12.64
4
4.97
9
9.23
14
13.55
5
5.82
10
10.08Figure 6
Beam response for the reweighted design example.


## 3.3.1. Postprocessing Design Example

For this example we consider an aperture of15λ split into 300 potential sensor locations. In this instance the polarisation of the signal of interest is given by (γ,η)=(0∘,0∘). The values of α=0.75, β=0.1, and ε=1 were used in the constraints placed on the minimisation.This resulted in 15 crossed dipoles spread over11.77λ as shown in Table 3. Here the minimum adjacent separation is 0.81λ indicating that the size constraint has been successfully enforced. The resulting response is shown in Figure 5 which shows a desirable response has been achieved.Table 3
Locations for the postprocessing design example.
n
d
n
/
λ
n
d
n
/
λ
n
d
n
/
λ
1
1.61
6
5.82
11
10.06
2
2.44
7
6.64
12
10.92
3
3.29
8
7.49
13
11.74
4
4.11
9
8.35
14
12.57
5
4.96
10
9.16
15
13.38Figure 5
Beam response for the postprocessing design example.In order to ensure an acceptable performance here we had to redesign the weight coefficients for the final merged locations. It is reasonable to expect the same from the iterative minimum distance sampling method due to the fact that some merger of location is still required. A similar performance would also be achieved and as a result an example of it is not given here. However, a reweighted design example with size constraint is considered below as an alternative that does not require a redesigning of weight coefficients due to no locations being merged.


## 3.3.2. Reweighted Design Example

In this example we are now considering 600 possible locations over the same aperture. However, due to the improved sparsity for a given amount of error offered by the reweighted method we can now use a value ofα=0.3.This results in 14 crossed dipole locations shown in Table4. In this instance the aperture of the designed array is 11.29λ with a minimum adjacent separation of 0.85λ. Again, from Figure 6 we can see that an acceptable response has been achieved.Table 4
Locations for the reweighted design example.
n
d
n
/
λ
n
d
n
/
λ
n
d
n
/
λ
1
2.26
6
6.67
11
10.94
2
3.26
7
7.53
12
11.79
3
4.11
8
8.38
13
12.64
4
4.97
9
9.23
14
13.55
5
5.82
10
10.08Figure 6
Beam response for the reweighted design example.


## 4. Conclusions

In this paper the problem of designing fixed beamformers based on vector-sensor arrays (with a quaternionic formulation for compact and convenient representation) has been considered. This problem can be split into two parts: first, designing the coefficients for a fixed set of vector-sensor locations and, second, finding a set of sparse locations that can be used to more efficiently implement a fixed beamformer.The first part of the problem can be solved by minimising the sidelobe levels in the response while keeping a unitary response at the main lobe. For the second part of the problem a reformulation of the traditionall1 norm minimisation into a modified l1 norm minimisataion has been proposed. This is required in order to convert the problem into a form that can be solved, while still ensuring that all four parts of the quaternionic weight coefficients are simultaneously minimised. It is also possible to further improve the performance of the proposed method by converting the problem into a series of iteratively reweighted minimisations, as well as enforcing a minimum spacing of the physical size of the sensors being considered. Design examples have been provided to verify the effectiveness of the proposed design methods.



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*Source: 181937-2015-04-09.xml*