SPACE–TIME CODES 285
are summarised in Figure 5.50. The QL decomposition presented in Equation (5.118) now
delivers a unitary (N
R
+ N
T
) × N
T
matrix Q and a lower triangular N
T
× N
T
matrix L.
Certainly, Q
and L are not identical with Q and L, e.g. Q has N
T
more rows than the
original matrix Q. It can be split into the submatrices Q
1
and Q
2
such that Q
1
has the
same size as Q for the ZF approach.
Similarly to the V-BLAST algorithm, the order of detection has to be determined
with respect to the error covariance matrix given in Equation (5.119). Please note that
underlined vectors and matrices are associated with the extended system model. Since the
only difference between ZF and MMSE is the use of H
instead of H, it is not surprising
that the row norm of the inverse extended triangular matrix L
determines the optimum
sorting. However, in contrast to the ZF case, L
need not be explicitly inverted. Looking at
Equation (5.118), we recognise that the lower part of the equation delivers the relation given
in Equation (5.120). Hence, L
−1
is gained as a byproduct of the initial QL decomposition
and the optimum post-sorting algorithm exploits the row norms of Q
2
. This compensates
for the higher computational costs due to QL decomposing a larger matrix H
.
Since the MMSE approach represents a compromise between matched and zero-forcing
filters, residual interference remains in its outputs. This effect will now be considered in
more detail. Using the extended channel matrix requires modification of the received vector
r as well. An appropriate way is to append N
T
zeros. The detection starts by multiplying r
with Q
H
, yielding the result in Equation (5.121a). In fact, only a multiplication with Q
H
1
,is
performed, having the same complexity as filtering with Q
H
for the zero-forcing solution.
However, in contrast to Q and Q
, Q
1
does not contain orthogonal columns because it
consists of only the first N
R
rows of Q. Hence, the noise term Q
H
1
n is coloured, i.e. its
samples are correlated and a symbol-by-symbol detection as considered here is suboptimum.
Furthermore, the product Q
1
Q does not equal the identity matrix any more. In order
to illuminate the consequence, we will take a deeper look at the product of the extended
matrices Q
and H. Inserting their specific structures given in Equation (5.118) results in
Q
H
H = Q
H
1
H + Q
H
2
·
σ
N
σ
X
I
N
T
!
= L ⇔ Q
H
1
H = L −
σ
N
σ
X
· Q
H
2
(5.122)
Replacing the term Q
H
1
· H in Equation (5.121a) delivers Equation (5.121b). We observe the
desired term L
x, the coloured noise contribution Q
H
1
n and a third term in the middle also
depending on x. It is this term that represents the residual interference after filtering with
Q
H
1
. If the noise power σ
2
N
becomes small, the problem of noise amplification becomes
less severe and the filter can concentrate on the interference suppression. For σ
2
N
→ 0,
the MMSE solution tends to the zero-forcing solution and no interference remains in the
system.
Sorted QL Decomposition
We saw from the previous discussion that the order of detection is crucial in successive
interference cancellation schemes. However, reordering the layers by the explained post-
sorting algorithm requires rather large computational costs owing to several permutations
and Householder reflections. They can be omitted by realising that the QL decomposition
is performed column by column of H. Therefore, it should be possible to change the order