SPACE–TIME CODES 287
At the end of the procedure, we obtain a orthonormal matrix Q, a triangular matrix L as
well as a set of permutation matrices P
µ
with 1 ≤ µ ≤ N
T
. It has to be mentioned that the
computational overheads compared with the conventional Gram–Schmidt procedure are
negligible.
At this point, we have no wish to conceal that the strategy does not always achieve the
optimum succession as is the case with the post-sorting algorithm. The algorithm especially
fails in situations where two column vectors q
µ
and q
ν
have large norms but point in similar
directions. Owing to their large norms, these vectors are among the latest columns to be
processed. When the smaller one has been chosen as the next vector to be processed, its
projection to the other vector is rather large on account of their similar directions. Hence,
the remaining orthogonal component may become small, leading to a very small diagonal
element in the upper left corner of L.
Fortunately, simulations demonstrate that those pathological events occur very rarely,
so that the SQLD performs nearly as well as the optimum post-sorting algorithm. Moreover,
it is possible to concatenate the sorted QL decomposition and the post-sorting algorithm.
This combination always ensures the best detection order. The supplemental costs are rather
small because only very few additional permutations are required owing to the presorting
of the SQLD. Hence, this algorithm is suited to perform the QL decomposition and to
provide a close-to-optimal order of detection.
5.5.6 Performance of Multilayer Detection Schemes
This section analyses the performance of the different detection schemes described above.
The comparison is drawn for a multiple-antenna system with N
T
= 4 transmit and N
R
= 4
receive antennas. The channel matrix H consists of independent identically distributed
complex channel coefficients h
µ,ν
whose real and imaginary parts are independent and
Gaussian distributed with zero mean and variance 1/2. Moreover, the channel is constant
during one coded frame and H is assumed to be perfectly known to the receiver, while the
transmitter has no channel knowledge at all. QPSK was chosen as a modulation scheme.
It has to be emphasised that all derived techniques work as well with other modulation
schemes. However, the computational costs increase exponentially with M in the case of
the optimal APP detector, while the complexity of the QL decomposition based approach
is independent of the modulation alphabet’s size.
Turbo Multilayer Detection
We start with the turbo detection approach from Subsection 5.5.2. For the simulations, we
used a simple half-rate convolutional code with memory m = 2 and generator polynomials
g
1
(D) = 1 +D
2
and g
2
(D) = 1 +D + D
2
. In all cases, a max-log MAP decoder was
deployed. Figure 5.52 shows the obtained results, where solid lines correspond to perfectly
interleaved channels, i.e. each transmitted vector x[k] experiences a different channel matrix
H[k]. Dashed lines indicate a block fading channel with only a single channel matrix
for the entire coded frame. In the left-hand diagram, the bit error rates of the max-log
MAP solution are depicted versus the signal-to-noise ratio E
b
/N
0
. We observe that the
performance increases from iteration to iteration. However, the largest gains are achieved
in the first iterations, while additional runs lead to only minor improvements. This coincides
with the observations made in the context of turbo decoding in Chapter 4. Moreover, the