Space-Time Block Codes for Wireless Systems - The ...

If we assume that the channel coefficients are independent for different transmit antennae,
a nontrivial Σ h will be a diagonal matrix with full rank, and the code design
will be independent of channel statistics. To simplify the analysis, in the sequel we will
assume this and focus on ¯Φ only.
2.2.3 The Correlated Codeword Sequence Difference Matrix Φ
We assume each transmit antenna uses the same fixed spreading code during the M
blocks
s i (1) = s i (2) = · · · = s i (MN c ) (2.31)
G 1 = · · · = G M and R mm exhibits a banded block Toeplitz structure. For the quasi-static
fading channels, Φ mm can be put in a more compact form.
If nL u ≤ τ max < (n + 1)L u , then R mm = G T mG m ∈ R NcKLcLt×NcKLcLt has the form
⎡
⎤
R mm (0) R mm (1) . . . R mm (n + 1) 0
R mm (1) T
.
R(0) .. . .. Rmm (n + 1)
R mm =
. .
.. . .. . .. .
, (2.32)
⎢
⎣
R mm (n + 1) T . .. . .. . .. Rmm (1) ⎥
⎦
0 R mm (n + 1) T . . . R mm (1) T R mm (0)
where R mm (0) captures the “on-time” correlation of the spreading codes,
R mm (0) = G(t) T G(t) ∈ R KLcLt×KLcLt , (2.33)
and R mm (t), 1 ≤ t ≤ n captures the “off-time” correlation,
⎡ ⎤
R mm (t) = ⎣ G(t) ⎦
0
⎤
⎣ 0 ⎦ ∈ R LcKLt×LcKLt (2.34)
G(t)
T ⎡
where, G(t) ∈ R (Lu+τmax)×KLcLt is defined in Equation (2.13), 0 ∈ R nLu×KLcLt is an
all-zero matrix whose size depends on n.
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