Vectorsensor 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 vectorsensor 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 traditional
Traditionally fixed beamformers have been designed assuming the arrays consist of isotropic array elements [
In the past ten years, quaternionvalued signal processing has attracted more and more attention with application areas involving three or fourdimensional signals [
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 [
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) [
More recently, the area of compressive sensing (CS) has been explored [
A third problem considered in this work is how to enforce a minimum spacing between active locations so that the vectorsensors with a nonzero physical size can fit into the resultant locations in practice. This is an extension of the work in [
The rest of this paper is structured as follows: Section
A quaternion is a hypercomplex number defined as follows [
The conjugate and modulus of a quaternion are given by
Finally
It is worth noting that in many scenarios quaternions prove useful as they allow the easy representation of problems with fourdimensional data. However, care has to be taken when formulating a problem using quaternions as they are noncommutative.
Figure
Array model being considered with crossed dipole elements.
The spatial steering vector of the array is given by
For crossed dipoles the spatialpolarization coherent vector contains information about a signals polarisation and is given by [
Now the array structure can be split into two subarrays, that is, one parallel to the
The response of the array is given by
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)
In this work, we use the ideal response for
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:
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.
As before, suppose
This problem is formulated as
In practice, (
This formulation is effective for the design of narrowband sparse scalarsensor 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 the
First we rewrite (
Now we decompose
Now define
Finally we arrive at the final formulation for the sparse vectorsensor array design problem
For the design of sparse arrays consisting of isotropic array elements with realvalued coefficients, reweighted
Following the idea, we introduce a reweighting parameter to each quaternionic coefficient. This leads to (
The problem is iteratively solved as follows.
Set
Consider
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.
In above formulations, the solutions do not take the size of the vectorsensors into account. As a result we could end up with an array that could not be implemented in practice due to the vectorsensors not fitting in their deigned locations. Therefore, a minimum spacing of the vectorsensors’ physical size should be applied to the optimisation.
This can be achieved using the methods recently proposed in [
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
The iterative procedure detailed in Section
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 vectorsensor 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 of
First we consider a ULA consisting of ten crossed dipoles with an adjacent separation of
Beam response for the design example based on a ULA.
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 is
Firstly, (
By increasing the value of
Locations found from solving (







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Beam response for the locations found by solving (
As an alternative, we can iteratively solve the series of reweighted minimisations given by (
Locations found from solving (







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Beam response for the locations found by solving (
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 be
For this example we consider an aperture of
This resulted in 15 crossed dipoles spread over
Locations for the postprocessing design example.







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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.
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 of
This results in 14 crossed dipole locations shown in Table
Locations for the reweighted design example.







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Beam response for the reweighted design example.
In this paper the problem of designing fixed beamformers based on vectorsensor 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 vectorsensor 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 traditional
The authors declare that there is no conflict of interests regarding the publication of this paper.