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

“Concatenating” the r j (t), t = 1, . . . , N c into a large vector,
⎡
⎤ ⎡ ⎤
G
r = √ 1 (1) ∅ ∅ D 1 (1)
σ
.
t ⎢ ∅ .. ∅ ⎥ ⎢ . ⎥
⎣
⎦ ⎣ ⎦ h
∅ ∅ G 1 (N c ) D 1 (N c )
⎡
⎤ ⎡ ⎤ ⎡ ⎤
G
+ √ 2 (1) ∅ ∅ D 2 (1) n j (1)
σ
.
t ⎢ ∅ .. ∅ ⎥ ⎢ . ⎥
⎣
⎦ ⎣ ⎦ h + ⎢ . ⎥
⎣ ⎦
∅ ∅ G 2 (N c ) D 2 (N c ) n j (N c )
(2.74)
= √ σ t G 1 D 1 h + √ σ t G 2 D 2 h + n ∈ C (NcLu+τmax)×1 (2.75)
The matched filter, which consists of user 1’s spreading codes only, outputs
y = √ σ t G T 1 G 1 D 1¯h +
√
σt G T 1 G 2 D 2 h + m ∈ C LcLt×1 , (2.76)
where
m = G T 1 n. (2.77)
Treating √ σ t G T 1 G 2D 2 h + m as Gaussian noise ˜m, we have
E[ ˜m] = 0, E[ ˜m ˜m H ] = σ t G T 1 G 2 G T 2 G 1 + G T 1 G 1 . (2.78)
matrix
where
Following our prior approach, we end up with the new correlated codeword difference
˜Φ = ∆D1 H (ŘT 1 Ř−1 2 Ř 1 )∆D 1 ∈ C LcLt×LcLt (2.79)
Ř 1 = G T 1 (t)G 1 (t) ∈ C LcLt×LcLt ,
Ř 2 = σ t G T 1 (t)G 2 (t)G T 2 (t)G 1 (t)+G T 1 (t)G 1 (t) ∈ C LcLt×LcLt
(2.80)
The matrix ˜Φ can be simplified and approximated to
˜Φ ≃ (∆D (1) ) H ∆D (1) ⊙ ˜R ∈ C LcLt×LcLt , (2.81)
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