SPACE–TIME CODES 257
5.4 Orthogonal Space–Time Block Codes
In this chapter we pursue the goal of obtaining spatial diversity by deploying several anten-
nas at the transmitter but only a single antenna at the receiver. However, a generalisation to
multiple receive antennas is straightforward (K
¨
uhn, 2006). Furthermore, it is assumed that
the average energy E
s
per transmitted symbol is constant and, in particular, independent of
N
T
and the length of a space–time block. Since aiming for diversity is mostly beneficial if
the channel between one transmit and one receive antenna provides no diversity, we con-
sider frequency-non-selective channels. Moreover, the channel is assumed to be constant
during one space–time code word.
We saw from Section 5.1 that spatial receive diversity is simply achieved by maximum
ratio combining the received samples, resulting in a coherent (constructive) superposi-
tion, i.e. the squared magnitudes have been summed. Unfortunately, transmitting the same
symbol s[] from all N
T
transmit antennas generally leads to an incoherent superposition
r[] =
N
T
ν=1
h
ν
[] ·
s[]
√
N
T
+ n[] = s[] ·
1
√
N
T
·
N
T
ν=1
h
ν
[] + n[] = s[] ·
˜
h[] +n[]
at the receive antenna. The factor 1/
√
N
T
ensures that the average transmit power per
symbol is independent of N
T
. In the case of i.i.d. Rayleigh fading coefficients, the new
channel coefficient
˜
h[] has the same distribution as each single coefficient h
ν
[] and
nothing has been won. If the transmitter knows the channel coefficients, it can predistort
the symbols so that the superposition at the receiver becomes coherent. This strategy is
known as beamforming and is not considered in this book. Therefore, more sophisticated
signalling schemes are required in order to achieve a diversity gain.
Although orthogonal space –time block codes do not provide a coding gain, they have
the great advantage that decoding simply requires some linear combinations of the received
symbols. Moreover, they provide the full diversity degree achievable with a certain number
of transmit and receive antennas. In order to have an additional coding gain, they can be
easily combined with conventional channel coding concepts, as discussed in the previous
chapters.
5.4.1 Alamouti’s Scheme
Before we give a general description of space–time block codes, a famous but simple
example should illustrate the basic principle. We consider the approach introduced by
Alamouti (Alamouti, 1998) using two transmit antennas and a single receive antenna. The
original structure is depicted in Figure 5.33. As we will see, each symbol s[] is transmitted
twice. In order to keep the transmit power per symbol constant, each instance of the symbol
is normalised by the factor 1/
√
2.
Two consecutive symbols s[ − 1] and s[] are collected from the input sequence.
They are denoted as s
1
= s[ − 1] and s
2
= s[] respectively and are mapped onto the
N
T
= 2 transmit antennas as follows. At the first time instant, x
1
[] = s
1
/
√
2 is sent over
the first antenna and x
2
[] = s
2
/
√
2 over the second one. At the receiver, we obtain the
superposition
r[] = h
1
x
1
[] + h
2
x
2
[] + n[] =
1
√
2
·
h
1
s
1
+ h
2
s
2
+ n[] .