Blind and Semiblind Channel Estimation - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 270

8.3.7 Finite Alphabet

The finite alphabet (FA) property of digital modulation schemes may be ex-
ploited in blind equalization. In the FA structure the source signal is chosen
from a finite set such as

±

1 for BPSK (binary phase shift keying) or a set of
phase shifts for DQPSK signal. Methods using FA property fit the received
data to the unknown channel taps and project the estimated symbols onto
an FA. Typically, these methods estimate the channel and symbols in an al-
ternating manner using the well-known least-squares method, or advanced
versions of EM-algorithm such as SAGE. The FA property has also been used
in conjunction with other signal properties such as cyclostationarity.

8.3.8 Special Matrix Structures

Multiple-output SIMO and MIMO models can lead to special matrix struc-
tures, for example, Block-Hankel or Block-Toeplitz matrices for channel coef-
ficients or symbols. Blind equalization algorithms can be developed to exploit
or enforce such structures. In addition, the full column rank property of chan-
nel matrix is a useful property. The column or row space of the data matrix
may be used to find the column space of the channel matrix or the row space
of the matrix where the symbols are stacked, respectively.
Subspace methods exploit the fact that the low-rank model is applicable,
and one can perform the decomposition to signal and noise subspaces based
on the pattern of the eigenvalues of the data correlation matrix, or singular
values of the data matrix. Signal subspace is spanned by eigenvectors cor-
responding to dominant eigenvalues of the correlation matrix, or singular
vectors corresponding to dominant singular values of the data matrix. The
same subspace is spanned by the columns of the channel matrix

N . 35 , 39 Well-
known signal or noise subspace estimation methods such as MUSIC may be
applied to determine the channel coefficients. This approach is explained in
more detail in the context of blind MIMO channel identification algorithms.

H

8.4 Receivers for MIMO Systems

8.4.1 I-MIMO and FIR-MIMO

Over the years, several models 40 of the transmission process have been used.
We start from the basic linear model that relates the received signals and the
transmitted ones by

y k = Hs k + w k , (8.14)

where H is the channel matrix associated with m transmitter, n receiver MIMO
system, s is a column vector of m transmitted signals, y is a column vector