Least Squares Support Vector Machines for Channel Prediction in the MIMO System - Developing Concepts in Applied Intelligence - page 40

have been used for a nonlinear channel estimation for multiple-input multiple-
output systems. The SVM technique for a robust channel estimation in the
OFDM data structure was proposed by M.J. Fernandez-Getino Garcia et al. in
the paper [2]. However, none of these solutions can be used easily in on-line
adaptive algorithms.
In this paper, we propose a LS-SVM solution for channel prediction. There
are two novelties in our proposal. We use the LS-SVM for channel prediction in
a MIMO system in which inequality constraints are replaced by equality con-
straints and a quadratic error criterion is used.
This paper is organized as follows. Section 2 presents the system description
and the problem formulation. Section 3 introduces a recursive least squares al-
gorithm that can be implemented on-line. In section 4, the proposed algorithm
is then evaluated by means of computer simulations. Section 5 presents some
final conclusions.

2 The System Description and Problem Formulation

In this section, the MIMO received model for an un-coded and a beam-forming
system is introduced.
In the first case, we assume a MIMO wireless flat fading communication sys-
tem with N r receive antennas and N t transmit antennas modelled by

y = Hx + n (1)

where 1 transmitted symbol vector

with each x i belonging to constelation C with symbol energy E s ,and

y is the N r × 1 received vector, x is the N t ×

n is the

while noise vector of size N r × 1with n i ∼CN (0 ,N 0 ). The channel state matrix

H

gives a complex channel gain between the m -th receiver and the
n -th transmit antenna.
In the second case, the received symbols are expressed in two scenarios, when
the transmitter and the receiver have full channel information (CSI) and when
they have the prediction matrix. Then, the channel matrix H = U · D · V H
is a singular value decomposition (SVD) where

{h mn }

=

are unitary matrices
corresponding to the i -th non-zero singular value σ H ( i ), ( σ H (1)

U and V

≤ ...≤ σ H ( M ))

and M = rank ( H ). Assuming that x = v 1 · x wecanobtainfromEq.(1)the

received symbols u 1 as follows:

u 1 y = σ H (1) x + u 1 n

(2)

1

Assuming u = u n we can obtain

2 = N r

E | n | · N 0 (3)

Thus, the channel within a MIMO system is time-varying and can be expressed
in a matrix notation as

H + E . Therefore, the received symbols are as

H =

follows: