I-a Related Work
For uniform linear arrays (ULAs), an adjacent antenna separation of no larger than half of the operating wavelength is used to avoid the introduction of grating lobes [1, 2]. This can become prohibitive in terms of the cost associated with the number of antennas required. Instead, sparse arrays become a desirable alternative due to the fact that the nonuniform nature of their adjacent antenna separations avoids grating lobes even when the mean adjacent antenna separation is greater than half the operating wavelength .
However, the sidelobe behaviour of sparse arrays is unpredictable. This means that optimisation of the antenna locations is required in order to achieve a desired beam response. Such optimisation can be achieved by stochastic optimisation methods such as genetic algorithms (GAs)[4, 5, 6], and simulated annealing (SA)[7, 8]. Difference sets and almost difference sets have also been successfully used in the design of sparse arrays, [9, 10], and merged with GAs to help give an improved performance, [11, 12]. The disadvantage of GAs, and similar design methods, is the potentially long computation time and the possibility of convergence to a non-optimal solution.
More recently, the area of compressive sensing (CS) has been explored , and CS-based methods have been proposed in the design of traditional sparse arrays [14, 15, 16, 17, 18, 19]. CS theory says that when certain conditions are met it is possible to recover some signals from fewer measurements than used by traditional methods . It is possible to use CS to design sparse sensor arrays by obtaining a close approximation of a desired beam response using as few array elements as possible.
Further work has also shown that it is possible to improve the sparseness of a solution by considering a reweighted norm minimisation problem [17, 20, 21, 22]. The aim of these methods is to bring the minimisation of the norm of the weight coefficients closer to that of the minimisation of the norm. To do this an iterative method is required to solve a series of reweighted minimisation problems, where locations with small weight coefficients are more heavily penalised than locations with large weight coefficients.
Alternatively, the problem can be converted into a probabilistic framework (termed Bayesian compressive sensing (BCS)) 
, with some suggested advantages to BCS as compared to traditional CS based implementations. However, an important point of interest is that the problem can be solved by the relevance vector machine (RVM) optimisation framework, which is efficient to use as also supported by the comparisons shown in the design examples section of this paper. Additionally, using BCS can remove the need to fine tune the error limits or sparsity associated with the implementations of CS above . Such approaches have been applied in the design of sparse arrays with real valued and complex valued weight coefficients [26, 27, 28], where the multi-task BCS scheme , is applied in the case of complex valued weight coefficients.
The methods discussed above have been implemented assuming the arrays consist of isotropic array elements. As a result, the polarisation of a signal is not taken into account when considering the performance of an array. Instead arrays based on vector sensors, [19, 30], provide a desirable alternative as they allow the measurement of both the horizontal and vertical components of the received waveform. For example, the vector sensors used could be crossed dipoles (two orthogonally orientated dipoles) [19, 31, 32, 33], or tripoles (three orthogonally orientated dipoles) [34, 35].
When tripoles are used it is possible to measure the full electromagnetic (EM) field at a given point 
. These arrays have been applied in the area of direction and polarisation estimation. Due to the close proximity of the three orthogonal dipoles that make up each tripole there can be issues with mutual coupling when implemented in practice. As a result, the concept of spatially stretched tripoles (SST) has been developed and used in the area of direction of arrival (DOA) estimation . An SST is a tripole where the three orthogonal dipoles are spread over a given geometry, leading to reduced mutual coupling effects.
In this work for the first time the problem of designing sparse SST arrays (SSSTAs) is addressed. Unlike for the design of traditional sparse arrays there are now two optimisation problems to solve, i.e. finding the optimal locations and orientations for the dipoles. It is proposed to use CS and BCS based design methods that go beyond the state of the art in order to solve these problems.
As a result, it is now necessary to formulate the problem to include the fact that there are three potential dipoles at each point on the sampling grid and the signal model now includes polarisation information (requiring alterations to the CS and BCS formulations). It is possible to avoid co-located dipoles by viewing them as a special case of the minimum adjacent dipole separation not meeting a physical size constraint . However, if the methods in  are directly applied in this case, then although there will be a minimum spacing between antenna locations, there can still be multiple dipoles at each location. Therefore it is necessary to consider co-located dipoles as breaking the size constraint. Here, the design of SSSTAs utilising the size constraint is implemented in two ways: i) An iterative minimum distance sampling method (IMDSM) with CS and BCS; ii) an altered iterative reweighted minimisation scheme (AIRMS). When integrating the CS/BCS based method with the IMDSM it is also important to account for the response due to the previously fixed dipoles when deciding what the reference response in the current iteration is.
The remainder of this paper is structured as follows: Section II gives details of the proposed design methods, including the array model being used (II-A), a review of CS and BCS (II-B and II-C) and the proposed IMDSM and reweighted design methods for SSSTAs (II-D and II-E). In Section III design examples are presented to verify the effectiveness of the proposed methods and conclusions are drawn in Section IV.
Ii Proposed Design Methods
Ii-a Array Model
Figure 1 shows an example of a linear SSSTA. possible dipole locations are spread along the y-axis with an adjacent separation of . For each possible dipole location there are three potential orientation directions, one parallel to each axis. Also shown is a signal with its direction of arrival (DOA) defined by the angles and , with and [34, 35]. A plane-wave signal model is assumed, i.e. the signal impinges upon the array from the far field.
The spatial steering vector of the array is given by
where is the auxiliary polarization angle and is the polarization phase difference.
Now the array can be split into three sub-arrays, one parallel to each axis. With , the steering vector of each sub-array is given by:
The response of the array is given by
where is the complex weight coefficient for the dipole located at the point and orientated parallel to the -axis and denotes the Hermitian transpose. Note that for an SSSTA if , then , as there can be only one dipole present. Similarly
where is the contribution of the dipole located at the point to the overall steering vector parallel to the -axis.
Ii-B Compressive Sensing for SSSTA Design
Suppose is the desired beam response as a function of and . Then the problem is to match the designed response to this desired response for the full range of and values of interest while finding the optimised dipole locations and orientations.
First, consider Figure 1 as being a grid of potential dipole locations. Here is a large number and sparseness is then introduced by selecting the weight coefficients to give as few active dipoles as possible, or in other words as few non-zero valued weight coefficients as possible, while still giving a designed response close to the desired one. Note, a large means it is more likely that the optimal locations will appear on the grid thus allowing for a better performance. However, the tradeoff is that if is too large the efficiency of the algorithm deteriorates.
This problem is formulated as
where is the norm of the weight coefficients , 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 places a limit on the allowed difference between the desired and the designed responses. Minimising the norm has the effect of minimising the number of dipoles used, while the constraint ensures a reasonable approximation of the ideal reference response is achieved. If the size of is increased, more error can be introduced into the final response, which would be expected to allow a sparser solution to be achieved. Note, indicates the norm.
In detail, and S are respectively given by
where is the number of points sampled at each dimension of the desired beam response. In this work is the ideal response, i.e. a value of one for the mainlobe and zeros for the other entries. Note, has to be large enough to ensure all angular and polarisation points of interest are considered.
and for contains the real and imaginary components of the complex weight coefficient given by the entry in w. Here, the variable has been introduced and requires minimising. By keeping less than this value the effect is to minimise the norm of all of the absolute weight coefficients.
Now decompose to , , to reformulate (10). Note, the upper limit on the sum is as there are potential dipole orientations at each location.
In vector form, where and . Then (10) can be rewritten as
Note, a value of , means the second constraint in (12) ensures that the real and imaginary parts of the weight coefficient contained in will both be equal to zero. This allows the desired sparsity to be introduced.
where is the real component and is the imaginary component. Then, the final formulation is as follows
Note, the values for are included with the weight coefficients in . This is so that it is not necessary to predefine their values, instead the algorithm finds them at the same time as the optimised weight coefficients. As a result, it is necessary for the vector to select the values for minimisation and the zeros are introduced into to ensure the same values do not contribute to the error between the ideal reference response and the achieved response in the first constraint in (16). Finally, as the weight coefficients have been split into real and imaginary parts, the response given by the product will contain the real and imaginary parts of the achieved response separately. This means the reference pattern has to be split in a similar manner giving (14).
However, unlike the norm, the norm does not penalise all non-zero valued coefficients equally. Instead, larger coefficients are penalised more heavily. To further improve the sparseness of the array and get a better approximation of the norm minimisation, large reweighting terms can be applied to the smaller weight coefficients so that they are penalised more heavily [17, 18, 20, 21, 22].
When applied to the above modified norm minimisation problem we get the following
where now and Here is the current iteration, holds the current estimate of the weight coefficients, contains the weight coefficients, from the previous iteration, for the dipole and is a small value roughly equal to the minimum desired weight coefficient. The iterative algorithm would then follow the steps below:
Set and find an initial estimate of the weight coefficients by solving (16).
, and find the reweighting terms .
Repeat steps 2 to 3 until i.e. until the number of active locations has remained the same for three iterations. Here define .
The addition of the reweighting term, which is calculated using coefficients from the previous iteration, means all non-zero valued coefficients are penalised in a more uniform manner.
It is worth noting that as it stands the solutions to (16) and (17) do not strictly give an SSSTA in the result. This is because currently there is no way of guaranteeing there can only be a single dipole at a given location. In other words the proposed methods are in effect finding a sparse weight coefficient vector without considering the locations of the associated dipoles. The methods detailed in Section II-D and Section II-E can both be used to overcome this issue and ensure that there are no co-located dipoles, guaranteeing an SSSTA.
Ii-C Bayesian Compressive Sensing for SSSTA Design
When considering BCS for sparse array design, [26, 27, 28, 37], there are two formulations of BCS that can be used. Firstly there is a single task (ST) BCS formulation  which can be implemented using a RVM [24, 38]. Alternatively multi task (MT) BCS, , can be used when there are multiple CS measurements and the statistical relationships between them can be exploited. This could include measurements at multiple time instances, or in the case of sparse array design if multiple or complex weight coefficients have to be minimised. As a result MT-BCS is well suited to the problem being addressed and is formulated in what follows. However, the ST-BCS based design methodology for SSSTA design is provided in the appendix for the interested reader.
Firstly, consider matching the real and imaginary parts of the achieved array response to that of the ideal reference response:
where , and
are zero mean Gaussian error vectors, with a variance of, , , , , and . The problem now is to find the solutions to solve
It is known that for the likelihood function and the priors and , the following applies
This allows the problem to be written as
The prior is the same as to model the relationship between the real and imaginary parts of the weight coefficients, while still enforcing sparsity. It is given by and found as follows:
is the multi-task shared hyperpriors,
, given by a Gamma distribution, andis a shared Gamma hierarchial prior, where
which after integrating over and simplifying gives:
Equation (23) considers points as there are three potential dipoles at each location.
and from Bayes’ theorem
From (23), the fact that a Gamma hierarchial prior is placed on and the fact that can be modelled as a Gaussian likelihood, then
where and are parameters associated with the MT-BCS process chosen to encourage sparsity. In (II-C) the mean and covariance are given by:
respectively, where . Note, this gives a Student’s t-distribution for .
with a mode given by
As the mode of a student-t distribution is equal to its mean the resulting weight coefficients are given by 
The final optimal weight coefficient vector is then given by
Note, that as for the CS formulation discussed in the previous subsection the MT-BCS scheme detailed here is unable to guarantee an SSSTA as an outcome. This is because it is in effect finding a sparse weight coefficient vector without considering where the associated dipoles are located. As a result, it is possible that there could be multiple dipoles present at the optimised locations (optimised locations refers to the locations with one or more non-zero valued weight coefficients). This means the desired reduction in mutual coupling effects when implemented in practice will not be achieved. Instead to ensure an SSSTA the methods discussed in the following subsections should be considered.
Ii-D Iterative Minimum Distance Sampling Method for SSSTAs
In the above two formulations, there is no way to ensure that an SSSTA is achieved. This is due to the fact that only the weight coefficients associated with a given dipole are minimised, rather than considering if there are any co-located dipoles.
To solve this problem it is proposed to extend the idea of imposing a physical size constraint on the optimisation from . However, when directly applied these methods only ensure that there is a minimum distance between the optimised antenna locations. Therefore, in this instances they could not guarantee an SSSTA as there can potentially be three dipoles at each antenna location. As a result, it is necessary to also consider the fact that co-located dipoles at a given location can also be seen as breaking the minimum separation of a physical size constraint. In this work we use the idea of the IMDSM and AIRMS algorithms proposed in  to ensure an SSSTA is achieved as the final solution.
Note, that the iterative nature of the IMSDM based approaches means that the relationship between or and the algorithms performance becomes less predictable. Consider the fact that the value of used affects where the first dipole is located. This then defines the remaining aperture, which is again sampled using grid points. As a result the density of the sampling grid in the next iteration varies depending on where the previous dipole was placed and the value of , which in turn makes it difficult to predict how the performance will be effected by . The effects of can also be hard to predict for similar reasons.
Ii-D1 CS Based IMDSM
To begin with, the full aperture of the array is uniformly sampled and an estimate of the weight coefficients found using (16), with the first cluster of dipoles that are too close together being merged to give the first location as shown in Figure 2. At this point if there are multiple dipoles at the merged location the least significant are discarded to leave a single dipole present. The remainder of the aperture is then uniformly sampled, ensuring that the next dipole will be at least the distance of the size constraint away. This process is then repeated until there is no room for further dipoles.
It is worth noting that this method has involved the merger of dipole locations and has the potential for some dipoles to be discarded in order to avoid co-located dipoles. As a result the weight coefficients may no longer be optimal for the given dipole locations and orientations. However, the locations and orientations can be used to efficiently implement a fixed beamformer, by minimising the sidelobe levels while keeping a unitary response for the mainlobe location. This is detailed below in Section II-D3.
Ii-D2 MT-BCS Based IMDSM
In essence the same iterative procedure is followed in this instance. The initial set of weight coefficients used to find the first cluster is instead found using the MT-BCS procedure detailed in Section II-C. For subsequent iterations some changes have to be made to ensure that the method of solving the problem can account for the fact that some dipole locations and orientations have been fixed and will be contributing to the overall response.
As a result, consider the following
where and are found by subtracting the response due to the locations fixed in the previous iteration from the reference response in the previous iteration. Then from the remaining uniformly sampled aperture in the current iteration we construct and the resulting estimate of the weight coefficients are given by . Following the MT-BCS scheme detailed in Section II-C the solution is
This process is repeated, with the merging and discarding of dipoles. As a result it is again necessary to use the method for redesigning the weight coefficients detailed below.
Ii-D3 Fixed Beamformer Design for Given Dipole Locations and Orientations
After obtaining the dipole locations and orientations using the CS-IMDSM or BCS-IMDSM, it is necessary to re-design the coefficients of the array to provide a closer approximation to the desired responses. This is a classic fixed beamformer design problem and can be solved using the method described below, which is applicable to any arbitrary array geometry.
The redesign of the weight coefficients is achieved by minimising the sidelobe levels subject to a unitary response for the mainlobe direction. This can be formulated as
where and is a series of 1s and 0s to ensure only the correct dipole orientations are used, , only considers the mainlobe direction and denotes the Hadamard product.
Ii-E Altered Iterative Reweighted Minimisation Scheme for SSSTAs
To avoid the merging and discarding of dipoles as required for IMDSM, this work also proposes an AIRMS. Here the reweighting scheme in (17) is adapted to also penalise dipole locations that are too close together . This gives the following reweighting scheme
Now the iterative procedure is repeated until a solution that complies with the size constraint being enforced is obtained.
Unfortunately, this algorithm will not always guarantee a viable solution, due to the presence of in the calculation of reweighting terms. The inclusion of is required for numerical stability, but prevents a zero weight coefficient in the current iteration guaranteeing a zero weight coefficient in the next iteration. Based on the authors’ experience with different design parameters, if a solution is possible it will usually be achieved in less than iterations.
It is also hard to predict if a solution will be achieved, or the performance level achieved, based on the selection of . This is as the choice of greatly effects how likely we are to get a solution that meets the size constraint value. It may be expected that increasing should allow an improvement in the algorithms performance as it is more likely to get the optimal locations included on the sampling grid. This also makes it more likely that two or more dipoles will be located closer together than the size constraint making it harder to get a valid solution.
Iii Design Examples
This section provides design examples to verify the effectiveness of the proposed methods. All examples are implemented on a computer with an Intel Xeon CPU E3-1271 (3.60GHz) and 16GB of RAM.
For all of the figures that follow positive values of indicate the value range for , while negative values of indicate an equivalent range of with .
Here a broadside design example and two off-broadside design examples are considered to illustrate the effectiveness of the proposed design methods, when designing linear SSSTAs. Although the AIRMS does not necessarily require the weight coefficients to be redesigned, they have been here in order to allow a fairer comparison between all three design methods considered. Unless otherwise stated, the examples consider the scenario of with a maximum possible aperture of 10. For the design examples using MT-BCS the values of and are set as suggested in , with the value of being found from the CS-IMDSM and AIRMS design examples. In this work the CS-IMDSM and AIRMS are implemented using cvx, a package for specifying and solving convex programs [39, 40].
Note, the selection of has been made to get close to the sampling density suggested in , while also accounting for the fact that the proposed methods have to consider three antennas at each grid point rather than a single antenna. As discussed for the proposed methods it is also hard to predict how changing will effect the performance of the algorithms (in the case of the AIRMS a solution is not even always guaranteed). Experience with different design examples suggest that for a aperture usually ensures a suitable solution will be achieved by at least one of the three proposed methods.
For the three examples the response from an equivalent ULA is also provided as a further comparison. To ensure optimised dipole locations and orientations for the ULAs, solve the minimisation in (38) with to allow the three dipole orientations at each location to be considered. Then a new is constructed in order to keep only the most significant dipole orientations at each location. The minimisation in (38) is then resolved to give the final optimised dipole orientations and locations.
Iii-a Broadside Example
For the broadside design example, the mainlobe is given by for , with the sidelobe regions defined by for and being sampled every . The polarisation information is given by and . For the CS-IMDSM and AIRMS examples the value of is used.
The responses for the CS-IMDSM, BCS-IMDSM and AIRMS design examples are shown in Figure 3. For all three of the proposed methods the correct mainlobe location has been achieved (whereas the ULA example gave a error), along with sufficient sidelobe attenuation. For completeness the resulting dipole locations are shown in Tables I, II and III, respectively, where it is clear the size constraint has been successfully enforced in all cases. Figures. 4, 5 and 6 illustrate the orientations of the dipoles for each of the three broadside examples and the ULA orientations are shown in Figure 7. Note, the dipole positions shown in the figures do not accurately reflect the true dipole locations. The true locations should instead be determined from the corresponding tables provided.
The following performance measures are summarised in Table IV: aperture length, mean adjacent dipole separation (), number of dipoles required (also given as a reduction as compared to an equivalent ULA), norm of the error between the desired and achieved responses (, where are the optimised weight coefficients for a given method), the amplitude of the peak sidelobe closest to the mainlobe, the computation time and the number of iterations required by each method.
Firstly, as expected, it can be seen that there are reasonably small error values, suggesting that a good match to the desired response has been achieved in each case. For two of the three proposed methods the error between the designed and desired response is less than that for the ULA. This suggests a better approximation of the ideal response has been achieved, despite requiring less dipoles (48 less for BCS-IMDSM and 57 less for AIRMS) and the introduction of sparsity. It can also be seen that by comparing the values of a comparable amount of sparseness has been introduced by each of the design methods, with the BCS-IMDSM performing slightly better (and also giving the lowest response error).
When considering the computation time it can be seen that there is a difference between the three methods. The AIRMS has given a shorter computation compared to the CS-IMDSM which is explained by the fact that it requires fewer iterations as dipoles are not placed individually. There is also a significant reduction in the computation time between the CS-IMDSM and BCS-IMDSM design examples. This would suggest that the BCS-IMDSM design method is the more computationally efficient IMDSM based design method. The authors’ experience with different design examples also suggests that this is consistently the case and that the difference increases with the problem size.
|closest sidelobe (dB)||-20.02||-31.47||-30.55||-26.83|
To illustrate the effects of the value of used, now consider the same design example again with the values and , along with the original value of . The performance measures for the three proposed methods are summarised in Tables V-VII.
|closest sidelobe (dB)||-14.18||-17.85||-20.02||-10.47|
|closest sidelobe (dB)||-22.41||-20.56||-31.47||-19.63|
|closest sidelobe (dB)||NA||-24.61||-30.55||-29.88|
As expected, increasing the value of has increased the computation for the three proposed design methods. This is because the design methods now consider a larger sampling grid for each iteration, which in turn means a longer computation time. However, the effect on the other performance measures used has proven to be harder to predict.
For each of the design methods varying can alter the aperture of the designed array and the dipoles required to implement it in practice. The mean adjacent dipole separation has remained reasonably constant and for the CS-IMSDM method the smallest separation has even occurred for the largest value of . However, for the design of traditional sparse arrays using CS-based methods, increasing the value of would lead to an expected increase in the mean adjacent dipole separation. This is because a denser grid will be able to give a closer approximating to the ideal locations and as a result uses less dipole in total. By looking at the error between the designed responses and the ideal response, along with the amplitudes of the closest sidelobes, it can be seen that the effect on the desirability of the designed response is similarly hard to predict in advance. The same is true when off-broadside examples are considered. So for the remainder of this broadside design example and the two off-broadside design examples that follow only the original value of is used.
Finally, now consider the effect of on the performance of the CS-IMDSM and AIRMS for the broadside design example. Two further values of will be considered, and , respectively. The performance of the two methods for these values is summarised in Table VIII. For traditional CS based problems it would be expected to see that increasing the value of would increase the amount of error allowed, thus allowing extra sparsity to be introduced. However, here we can see the iterative nature of the algorithms has made predicting the effects of difficult. As a result, in what follows a single value of that gives a solution for both methods will be used in the off-broadside examples to allow a fair comparison. Note, the reason why no results are shown for AIRMS with is that no solution was obtained in this case.
|closest sidelobe (dB)||-16.83||-21.22||-26.26|
Iii-B Off-Broadside Example 1
For the first off-broadside design example consider a mainlobe location of for , with the sidelobe regions defined as for and for , which are sampled every . The polarisation information is given by and . The value is used to place a limit on the allowed error in responses.
Figure 8 shows the resulting responses for the three design examples. The CS-IMDSM design example has the mainlobe at the correct location, while for the other two examples and the ULA comparison the mainlobe is located at . In all three cases sufficient sidelobe attenuation has also been achieved. Again, for completeness the resulting dipole locations and orientations are shown in Tables IX, X and XI and Figures 9, 10 and 11, respectively, where it is clear the size constraint has been successfully enforced in all three cases. The comparison ULA dipole orientations are shown in Figure 12. Note, the distances in the dipole orientation figures are again not intended to be accurate. Instead, the dipole location information should be taken from the tables provided.
|closest sidelobe (dB)||-13.84||-19.20||-24.15||-22.02|
Table XII compares the performance measures for the off-broadside design examples. The first thing to note is that the error in the responses has significantly been increased for all three cases. This is expected as we used a larger value of and can be predicted after having looked at the three designed beam responses. It can be seen that the BCS-IMDSM has given the most accurate estimate of the desired response (as compared to the CS-IMDSM and AIRMS), but this has come at the expense of a reduced adjacent dipole separation. Again, the BCS-IMDSM has been shown to be the most computationally efficient of the proposed SSSTA design methods. Finally, although the error values show a worse approximation of the ideal response has been achieved by the proposed methods, as compared to the comparison ULA, a reasonable approximation has still been achieved despite using less dipoles and the introduction of sparsity.
|closest sidelobe (dB)||-31.07||-16.48||-14.96||-14.38|
|closest sidelobe (dB)||-13.77||-14.46||-7.24||-4.80|
|closest sidelobe (dB)||-10.24||-12.23||-12.57||-14.66|
|closest sidelobe (dB)||-10.86||-7.32||-7.11||-15.58|
|closest sidelobe (dB)||-22.19||-17.00|