266 SPACE–TIME CODES
where
X
= diag[λ
1
···λ
r
] is a diagonal matrix containing the different power levels.
The transmit filter V
X
comprises those columns of V
H
that correspond to the r largest
singular values of H. Equivalently, the receive filter U
X
contains those columns of U
H
that are associated with the used eigenmodes. Using V
X
and U
X
leads to
y = U
H
X
· r =
H
·
X
· s +
˜
n .
Since
H
and
X
are diagonal matrices, the data streams are perfectly separated into
parallel channels whose signal-to-noise ratios amount to
γ
i
= σ
2
H,i
· λ
i
·
σ
2
X
σ
2
N
= σ
2
H,i
· λ
i
·
E
s
N
0
.
They depend on the squared singular values σ
2
H,i
and the specific transmit power levels λ
i
.
For digital communications, the input alphabet is not Gaussian distributed as assumed in
Section 5.3 but consists of discrete symbols. Hence, finding the optimal bit and power allo-
cation is a combinatorial problem and cannot be found by gradient methods as discussed on
241. Instead, algorithms presented elsewhere (Fischer and Huber, 1996; Hughes-Hartogs,
1989; Krongold et al., 1999; Mutti and Dahlhaus, 2004) have to be used. Different opti-
misation strategies are possible. One target may be to maximise the throughput at a given
average error rate. Alternatively, we can minimise the error probability at a given total
throughput or minimise the transmit power for target error and data rates.
No Channel Knowledge at Transmitter
In Section 5.3 it was shown that the resource space can be used even in the absence of
channel knowledge at the transmitter. The loss compared with the optimal waterfilling
solution is rather small. However, the price to be paid is the application of advanced signal
processing tools at the receiver. A famous example is the Bell Labs Layered Space–Time
(BLAST) architecture (Foschini, 1996; Foschini and Gans, 1998; Foschini et al., 1999).
Since no channel knowledge is available at the transmitter, the best strategy is to transmit
independent equal power data streams called layers. At least two BLAST versions exist.
The first, termed diagonal BLAST, distributes the data streams onto the transmit antennas
according to a certain permutation pattern. This kind of interleaving ensures that the symbols
within each layer experience more different fading coefficients, leading to a higher diversity
gain during the decoding process.
In this section, we focus only on the second version, termed vertical BLAST, which is
shown in Figure 5.39. Hence, no interleaving between layers is performed and each data
stream is solely assigned to a single antenna, leading to x = s. The name stems from the
vertical arrangement of the layers. This means that channel coding is applied per layer.
Alternatively, it is also possible to distribute a single coded data stream onto the transmit
antennas. There are slight differences between the two approaches, especially concerning
the detection at the receiver. As we will soon see, per-layer encoding makes it possible to
include the decoder in an iterative turbo detection, while this is not directly possible for a
single code stream.
From the mathematical description in Equation (5.78) we see that a superposition
r
µ
=
N
T
ν=1
h
µ,ν
· x
ν
+ n
µ