I Introduction
Electromagnetic (EM) vector sensor arrays can track the direction of arrival (DOA) of impinging signals as well as their polarization. A crosseddipole sensor array, firstly introduced in [1] for adaptive beamforming, works by processing the received signals with a long polarization vector. Based on such a model, the beamforming problem is studied in detail in terms of output signaltointerferenceplusnoise ratio (SINR) [2]. In [3, 4], further detailed analysis was performed showing that the output SINR is affected by DOA and polarization differences.
Since there are four components for each vector sensor output in a crosseddipole array, a quaternion model instead of long vectors has been adopted in the past for both adaptive beamforming and direction of arrival (DOA) estimation [5, 6, 7, 8, 9]. In [10], the wellknown Capon beamformer was extended to the quaternion domain and a quaternionvalued Capon (QCapon) beamformer was proposed with the corresponding optimum solution derived.
However, in most of the beamforming studies, the signal of interest (SOI) is still complexvalued, i.e. with only two components: inphase (I) and quadrature (Q). Since the output of a quaternionvalued beamformer is also quaternionvalued, only two components of the quaternion are used to recover the SOI, which leads to redundancy in both calculation and data storage. However, with the development of quaternionvalued wireless communications [11, 12, 13], it is very likely that in the future we will have quaternionvalued signals as the SOI, where two traditional complexvalued signals with different polarisations arrive at the array with the same DOA. In such a case, a full quaternionvalued array model is needed to compactly represent the fourcomponent desired signal and also make sure the four components of the quaternionvalued output of the beamformer are fully utilised. In this work, we develop such a model and propose a new quaternionvalued Capon beamformer, where both its input and output are quaternionvalued.
This paper is structured as follows. The full quaternionvalued array model is introduced in Section II and the proposed quaternionvalued Capon beamformer is developed in Section III. Simulation results are presented in IV, and conclusions are drawn in Section V.
Ii Quaternion model for array processing
A quaternion is constructed by four components [14, 15], with one real part and three imaginary parts, which is defined as , where are three different imaginary units and are realvalued. The multiplication principle among such units is
(1) 
and
(2) 
The conjugate of is .
A quaternion number can be conveniently denoted as a combination of two complex numbers , where the complex number and . We will use this form later to represent our quaternionvalued signal of interest.
Consider a uniform linear array with crosseddipole sensors, as shown in Fig. 1, where the adjacent vector sensor spacing equals half wavelength, and the two components of each crosseddipole are parallel to and axes, respectively. A quaternionvalued narrowband signal impinges upon the vector sensor array among other uncorrelated quaternionvalued interfering signals , with background noise . can be decomposed into
(3) 
where and are two complexvalued subsignals with the same DOA but different polarizations.
Assume that all signals are ellipsepolarized. The parameters, including DOA and polarization of the th signal are denoted by for the first subsignal and for the second subsignal. Each crosseddipole sensor receives signals both in the and subarrays.
For signal , the corresponding received signals at the and subarrays are respectively given by [16]:
(4) 
where represents the received part in the xsubarray, represents the part in the ysubarray, and and are the polarizations of the two complex subsignals in and directions, respectively, which are given by [17],
(5)  
Note that and are the steering vectors for the two subsignals, which are equal to each other since the two subsignals share the same DOA .
(6) 
A quaternion model can be constructed by combining the two parts as below:
(8)  
where can be considered as the composite quaternionvalued steering vector. Combining all source signals and the noise together, the result is given by:
(9) 
where is the quaternionvalued noise vector consisting of the two subarray noise vectors and .
Iii Full quaternion Capon beamformer
Iiia The Full QCapon Beamformer
To recover the SOI among interfering signals and noise, the basic idea is to keep a unity response to the SOI at the beamformer output and then reduce the power/variance of the output as much as possible
[18, 19]. The key to construct such a Capon beamformer in the quaternion domain is to design an appropriate constraint to make sure the quaternionvalued SOI can pass through the beamformer with the desired unity response.Again note that the quaternionvalued SOI can be expressed as a combination of two complex subsignals. To construct such a constraint, one choice is to make sure the first complex subsignal of the SOI pass through the beamformer and appear in the real and components of the beamformer output, while the second complex subsignal appear in the and components of the beamformer output. Then, with a quaternionvalued weight vector w, the constraint can be formulated as
(10) 
where is the Hermitian transpose (combination of the quaternionvalued conjugate and transpose operation), , and .
With this constraint, the beamformer output is given by
(11)  
Clearly, the quaternionvalued SOI has been preserved at the output with the desired unity response.
Now, the fullquaternion Capon (full QCapon) beamformer can be formulated as
(12) 
where
(13) 
Applying the Lagrange multiplier method, we have
(14) 
where is a quaternionvalued vector.
The minimum can be obtained by setting the gradient of (14) with respect to equal to a zero vector [20]. It is given by
(15) 
Considering all the constraints above, we obtain the optimum weight vector as follows
(16) 
A detailed derivation for the quaternionvalued optimum weight vector can be found at the Appendix.
In the next subsection, we give a brief analysis to show that by this optimum weight vector, the interference part at the beamformer output in (11) has been suppressed effectively.
IiiB Interference Suppression
Expanding the covariance matrix, we have
(17) 
where are the power of the two subsignals of SOI and denotes the covariance matrix of interferences plus noise. Using the ShermanMorrison formula, we then have
(18) 
where is a quaternion vector.
Applying left eigendecomposition for quaternion matrix [21, 22, 23],
(19) 
with , where denotes the noise power.
With sufficiently high interference to noise ratio (INR), the inverse of can be approximated by
(20) 
Then, we have
(21) 
where is a quaternionvalued constant. Clearly, is the right linear combination of , and .
For those interfering signals, their quaternion steering vectors belong to the space rightspanned by the related eigenvectors, i.e. . As a result,
(22) 
which shows that the beamformer has eliminated the interferences effectively.
IiiC Complexity Analysis
In this section, we make a comparison of the computation complexity between the QCapon beamformer in [10] and our proposed full QCapon beamformer. To deal with a quaternionvalued signal, the QCapon beamformer has to process the two complex subsignals separately to recover the desired signal completely, which means we need to apply the beamformer twice for a quaternionvalued SOI. However, for the full QCapon beamformer, the SOI is recovered directly by applying the beamformer once.
For the QCapon beamformer, the weight vector is calculated by , where is the steering vector for the complexvalued SOI. As an example, we use Gaussian elimination to calculate the matrix inversion and quaternionvalued multiplications are needed, equivalent to realvalued multiplications. Additionally, requires realvalued multiplications, while real multiplications are needed for . In total, real multiplications are needed. When processing a quaternionvalued signal, this number will be doubled and the total number of real multiplications becomes .
For the proposed full QCapon beamformer, in addition to calculating , real multiplications are required to calculate and real multiplications for . In total, the number of realvalued multiplications is , which is roughly half of that of the QCapon beamformer.
Iv Simulations Results
In our simulations, we consider pairs of crossdipoles with half wavelength spacing. All signals are assumed to arrive from the same plane of and all interferences have the same polarization parameter . For the SOI, the two subsignals are set to (90°, 1.5°, 90°, 45°) and (90°, 1.5°, 0°, 0°), with inferences coming from (90°, 30°, 60°, 80°), (90°, 70°, 60°, 30°), (90°, 20°, 60°, 70°), (90°, 50°, 60°, 50°), respectively. The background noise is zeromean quaternionvalued Gaussian. The power of SOI and all interfering signals are set equal and SNR (INR) is 20dB.
Fig. 2 shows the resultant 3D beam pattern by the proposed beamformer, where the interfering signals from ()=(30°, 80°), (70°, 30°), (20°, 70°) and (50°, 50°) have all been effectively suppressed.
In the following, the output SINR performance of the two Capon beamformers (full QCapon and QCapon) is studied with the DOA and polarization (90°, 1.5°, 90°, 45°) and (90°, 1.5°, 0°, 0°) for SOI and (90°, 30°, 60°, 80°), (90°, 70°, 60°, 30°), (90°, 20°, 60°, 70°), (90°, 50°, 60°, 50°) for interferences. Again, we have set SNR=INR=20dB. All results are obtained by averaging 1000 MonteCarlo trials.
Fig. 3 shows the output SINR performance versus SNR with snapshots, where the solidline is for the optimal beamformer with infinite number of snapshots. For most of the input SNR range, in particular the lower range, the proposed full QCapon beamformer has a better performance than the QCapon beamformer. For very high input SNR values, these two beamformers have a very similar performance.
Next, we investigate their performance in the presence of DOA and polarization errors. The output SINR with respect to the number of snapshots is shown in Fig. 4 in the presence of error for the SOI, where the real DOA and polarization parameters are (91°,2.5°,91°,46°) and (91°,2.5°,1°,1°). It can be seen that the full QCapon beamformer has achieved a much higher output SINR than the QCapon beamformer, and this gap increases with the increase of snapshot number. Fig. 5 shows a similar trend in the presence of a error. Overall, we can see that the proposed full QCapon beamformer is more robust against array pointing errors.
V Conclusions
In this paper, a full quaternion model has been developed for adaptive beamforming based on crosseddipole arrays, with a new full quaternion Capon beamformer proposed. Different from previous studies in quaternionvalued adaptive beamforming, we have considered a quaternionvalued desired signal, given the recent development in quaternionvalued wireless communications research. The proposed beamformer has a better performance and a much lower computational complexity than a previously proposed QCapon beamformer and is also shown to be more robust against array pointing errors, as demonstrated by computer simulations.
The gradient of a quaternion vector with respect to can be calculated as below:
(23) 
where , is the th quaternionvalued coefficient of the beamformer. Then,
(24) 
where
(25) 
Since
is realvalued, with the chain rule
[20], we have(26)  
Similarly,
Hence,
(27) 
where the subscript in the last item means taking the first entry of the vector.
Finally,
(28) 
The gradient of the quaternion vector with respect to can be calculated in the same way:
(29)  
Similarly,
(30)  
Thus, the gradient can be expressed as
(31) 
Finally,
(32) 
The gradient of can be calculated as follows.
(33) 
(34) 
Now we calculate the gradient of with respect to the four components of .
(35)  
The other three components are,
Hence,
(36) 
Finally,
(37) 
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