In this section, an explanation why RAB techniques are important for handling steering vector uncertainties and models for steering vector mismatch are provided. Furthermore, the most recent developed RAB algorithms are introduced and discussed.
2.5.1
Steering Vector Mismatch
When adaptive beamforming algorithms are applied to practical problems, the signal-to- interference-plus-noise ratio (SINR) performance may degrade when the data sample size is small and the convergence rate may reduce as the desired signal is presented in the training data [7]. Most importantly, the SINR performance of adaptive beamformers can suffer significant degradation because the underlying assumptions on the environment, signal sources or sensor array are usually non-ideal. This leads to a mismatch in the steering vector. To overcome the problem of steering vector mismatch, RAB techniques become a popular research area and various RAB algorithms have been developed.
In practical applications, different categories of steering vector mismatch include look direction and signal pointing errors, imperfect calibration and distorted antenna shape, manifold mismodeling due to source wavefront distortions, near-far field problem, signal fading and local scattering [7, 10].
Desired signal look direction mismatch is the simplest case for either modeling or handling. In this mismatch model, there exists an error for the DoA of the desired source signal (in some cases we also consider the interference signals). The error can be ei- ther a constant degree deviation or described by its statistical properties (e.g. uniform distribution within a certain range [14, 15]).
Near-far field mismatch is essentially caused by the spatial signature of the desired signal, which is assumed to be located in the near field of the antenna array, so that neither the array geometry nor the distance between the geometry center of the array and the signal source is negligible. In the case of ULA, the source is assumed to be located
on the line drawn from this geometrical center point in the normal direction to the array aperture, which is determined by the signal wavelengthλ and the number of sensors M . The choice for the distance must be compatible with the array geometry parameters and also depends onM and λ [7].
In the local scattering mismatch case as shown in Fig. 2.4 [9], we consider the source signal distributed (or scattered) due to the multipath scattering effect caused by the pres- ence of local scatterers [15]. This problem can be divided into two categories in terms of the signal signatures, called the coherent local scattering and incoherent local scattering.
object base station array
θi 2δi
Mobile i
Figure 2.4:Local Scattering Effect of Detecting a Moving Object from a Base Station Array
In coherent local scattering [7], the source signal is assumed to have time-invariant signature and the corresponding steering vector is modeled as
a = p + L X k=1 ejϕkb(θ k), (2.35)
where p corresponds to the direct path while b(θk)(k = 1,· · · , L) corresponds to the
scattered paths. The anglesθk(k = 1,· · · , L) are randomly and independently drawn in
each simulation run from a uniform generator with mean10◦ and standard deviation 2◦.
The anglesϕk(k = 1,· · · , L) are independently and uniformly taken from the interval
[0, 2π] in each simulation run. Notice that θk andϕk change from trials while remaining
constant over snapshots [7].
signature and the corresponding steering vector is modeled by a(i) = s0(i)p + L X k=1 sk(i)b(θk), (2.36)
where sk(i)(k = 0,· · · , L) are i.i.d zero mean complex Gaussian random variables in-
dependently drawn from a random generator. The angles θk(k = 0,· · · , L) are drawn
independently in each simulation run from a uniform generator with fixed mean value and standard deviation. This time,sk(i) changes both from run to run and from snapshot
to snapshot [7, 15].
2.5.2
Existing RAB Algorithms
In this subsection, we focus on the recently reported RAB algorithms. In [10], RAB design principles based on MVDR criterion have been discussed and summarized. These principles basically include: the generalized sidelobe canceller, diagonal loading [8, 9], eigenspace projection [18], worst-case optimization [7, 19] and steering vector estimation with presumed prior knowledge [11, 12].
The robust Capon beamformer (RCB) as discussed in [8] utilizes a diagonal loading method in which the loading factor is calculated based on a presumed uncertainty set for the SoI. It firstly started from an estimate of the desired signal power which is given by
˜
σ2 = 1
aHR−1a, (2.37)
wherea is the mismatched steering vector and R is the data covariance matrix. Further- more, the RCB approach leads to the optimization problem given by
min a a HR−1 a subject to (a− ¯a)HC−1 (a− ¯a) 6 1, (2.38)
where botha and C¯ −1are given. In the algorithm steps, an eigendecomposition technique
and Newton’s method are required to deliver an estimate of the loading factor λ, which further helps with the estimation of the desired signal power σ˜2. This method has a
Several of the most well-known online optimization programming based RAB ap- proaches include the worst-case optimization method [7], the sequential quadratic pro- gramme (SQP) method [12] and the method in [11], all of which aim to solve online optimization programmes (i.e. second order cone programme (SOCP) and semi-definite programme (SDP)) with presumed prior knowledge so that to obtain an estimate for the desired signal steering vector. The worst-case optimization method and [11] use the same uncertainty constraint for the steering vector mismatch as in the RCB method. How- ever, the SQP method uses a presumed steering vector that belongs to an uncertainty set A , {p + e, kek 6 } where p is the presumed steering vector, e is the mismatch and is a known constant to restrict the uncertainty range. The presumed steering vectorp is then iteratively updated by adding the orthogonal part of the errore and by enforcing that the updated version ofp orthogonal to a subspace matrix P⊥
p, which is also orthogonal to the
actual steering vectorp + e. This process can be expressed as an optimization programme described by min e (p + e) HRˆ−1 (p + e) subject to pHe = 0, P⊥p(p + e) = 0 (p + e)HC(p + e) 6 p¯ HCp,¯ (2.39)
where ˆR is the SCM and ¯C =R
¯ θ
p(θ)pH(θ)dθ, where ¯θ is the complement of θ, which is
the angular sector in which the desired signal is assumed to be located,p(θ) is the steering vector associated with a particular directionθ, [11, 12, 14]. However, because of the very high computational cost (at least O(M3.5)) for the online optimization programmes, the methods of [7,11,12] lack computation efficiency. Additionally, the direct implementation of SCM in both the optimization objective function and computation for the weight may reduce the accuracy and final SINR performance.
Some recent design approaches have considered combining different design principles together to improve RAB performance. In the algorithms of [14, 15], the data covariance matrix and the desired signal steering vector are separately and sequentially estimated. In both of these algorithms, the steering vector is estimated using the SQP method. However, the data covariance matrix in [14] is estimated by a linear shrinkage model expressed as
˜
R = ˆβ ˆR + ˆαI, (2.40)
where ˆR is the SCM, ˆβ and ˆα are positive shrinkage parameters which are derived by min- imizing the mean squared error (MSE)MSE( ˜R) = E[k ˜R− Rk2], where R is the actual
covariance variance rather than the SCM [14]. Essentially this shrinkage method belongs to the class of diagonal loading approaches with the loading factorα/ ˆˆ β computable and adaptable. The algorithm in [15] approaches the covariance matrix estimation in a totally different way, which directly estimates the interference-plus-noise covariance(INC) ma- trix ˜Ri+n based on a INC matrix reconstruction method. It employs the Capon spatial
spectrum estimator
ˆ
P (θ) = 1
dH(θ) ˆR−1d(θ), (2.41)
whereθ can be any possible angle, d(θ) is the steering vector associated with angle θ and ˆ
R is the SCM. Furthermore, it is used for the INC matrix reconstruction as ˜ Ri+n = Z ¯ θ ˆ P (θ)d(θ)dH(θ)d(θ). (2.42)
The outstanding performance of [15] can be extremely close to the optimum SINR. How- ever, it has high potential computational cost when the number of sample points taken within the angular sector ¯θ is large.
The common point of all the above algorithms introduced is the difficulty of estimating the steering vector in a computationally efficient way. Efforts have been made to avoid high complexity especially with online optimization programmes and an attractive algo- rithm named low-complexity mismatch estimation (LOCME) has been developed in [13]. LOCME aims to estimate the steering vector mismatch with a cost ofO(M3) and does not
require any optimization programme or additional information from the steering vector. It describes the estimation of the array steering vector as the projection onto a prede- fined subspace of the correlation between the beamforming output signal and the array observation vector as [13]
ˆ
a =√M Pd
kPdk, (2.43)
whereP is the eigensubspace projection matrix which can be obtained if the angular range in which the steering vector is located is known andd is cross-correlation vector between the array observation datax and the beamformer output y, which is computed directly by d = E[xy].