Kernel Methods for Smart Antennas
Angel Navia-Vazquez, M. Martínez-Ramón, R. Jordan and C.G. Christodoulou
Intelligent Systems: Techniques and Applications
, Maastricht, The Netherlands
In many array processing applications in communications it is essential to reach a
reasonable performance level, particularly in those environments which change
rapidly. For example, in mobile communications, users continuously change in
number and allocation. Also, channels may vary due to moving objects. In order
to extend the capacity of the channels, beamforming is used to allow users to
reuse frequencies. In such situations, adaptive arrays are used for tracking issues.
Linear LMS or RLS are used because of their properties of fast convergence
and low computational cost. The limits on their performance are related to the
maximum available number K of array elements and the number N of data
available for training the array parameters. K is lower bounded by the number of
interferences that have to be cancelled. Also, K is upper bounded by many
practical reasons, but the one that concerns us here is its associated number of
parameters. If it is too large, then, a large N will be needed for a reasonable
convergence; with N small, the system may overfit ( Haykin, 1996).
These bounds can be extended with kernel algorithms. They consist of a
nonlinear transformation of the data to a higher (possibly infinite) dimensional
Hilbert space H provided with a dot product which may be expressed as a
nonlinear function of the input data (kernel) (Aizerman, Braverman et al., 1964).
Then, linear algorithms can be applied to H, which is nonlinear from the point of
view of the input data. Their main drawback is that overfitting may occur due to
the fact that the complexity of the system will increase the number of data needed
for training. Nevertheless, control of complexity can be applied by the use of
Support Vector Machines (SVMs) (Vapnik, 1995), thus giving nonlinear
algorithms which need the same number of data as their linear counterparts -see,
e. g., (Martínez-Ramón, Rojo-Álvarez et al., 2007).
The main drawback of standard SVMs is that they are not adaptive.
Nevertheless, there exist a set of techniques that use an iterative re-weighted least
squares for online training of the machine, plus techniques to online adapt the
subspace in which this procedure is applied, thus making possible an adaptive
version of these kernel machines (Navia-Vazquez, Pérez-Cruz et al., 2001).
We present here an adaptive SVM, formulated in its complex valued version
for antenna array processing. Examples, presented here, cover signal detection
and tracking in multiuser nonstationary environments.