US is a matrix representation of the signal subspace where the columns consist of the signal
eigenvectors. UN is a matrix representation of the noise subspace where the columns consist of the
noise eigenvectors.
The subspaces are related by the following identity.
(6.39)
Also the subspaces possess the following property: the signal eigenvectors are orthogonal to the noise
subspace, which can be written as
(6.40)
We will make use of this property when we discuss the Eigen Canceller in the following chapters. The
Eigen Canceller weights become equal to [8]
(6.41)
With this information the ESD theorem allows us to write the matrix in the following manner (M is
the size of the matrix).
(6.42) RXX aM
i1
li # vi # v*i
wEC [I US # U*
S ] # rxd
U*
S # UN 0
I US # U*
S UN # U*
N
RXX US # DS # U*
S UN # DN # U*
N
RXX U # D # U*
RXX E5x # x*6
r^
Xd(t)
1
K
# aK1
j0
x(t j) # d*(t j)
R^
XX(t)
1
K
# aK1
j0
x(t j) # x*(t j)
wopt R1
XX # rXd
RECEIVER DIGITAL SIGNAL PROCESSING 303
where is the ith eigenvalue of the covariance matrix and is the ith eigenvector of RXX associated
with the ith eigenvalue.
The proposed solution is to then calculate the eigenvalues and eigenvectors of the matrix so the
matrix inversion can be simply written as follows:
(6.
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