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

Similarly the key matrix Φ is defined as,
Φ = (D α − D β ) H R(D α − D β ) = ∆D H R∆D ∈ C LcLt×LcLt . (2.65)
Again, if nL u ≤ τ max < (n + 1)L u , and each transmit antenna uses the same fixed
spreading code at each symbol time, the correlation matrix R has the same structure as
R defined in (2.32).
For the quasi-static fading channel
⎡ ⎤
∆D(1)
∆D(2)
Φ =
⎢ . ⎥
⎣ ⎦
∆D(N c )
H
⎡
⎤
R(0) R(1) . . . R(n + 1) 0 ⎡ ⎤
R(1) T
.
R(0) .. . .. ∆D(1)
R(n + 1)
. 0 .. . .. . .. ∆D(2)
0
.
⎢
⎣
R(n + 1) T . .. . .. . .. ⎢ . ⎥
R(1) ⎥ ⎣ ⎦
⎦ ∆D(N c )
0 R(n + 1) T 0 R(1) T R(0)
(2.66)
Note that the difference between the uplink Φ and the downlink Φ resides only in the
form of ∆D(t) (defined in Equation (2.14)) and the ∆D(t) (defined in Equation (2.59)).
Note that R(0) and R(t) can be partitioned into K × K block matrices,
⎡
⎤
R (1,1) (0) · · · R (1,K) (0)
R(0) =
.
⎢ . .. . ⎥
⎣
⎦
R (K,1) (0) · · · R (K,K) (0)
thus ¯Φ can be simplified as
⎡
R(t) = ⎢
⎣
⎤
n (t) · · · R n
(1,K) (t)
.
. .. . ⎥
⎦ , (2.67)
n (t) · · · R (K,K) (t)
R (1,1)
R (K,1)
n
Φ =
where
K∑
K∑
i=1 j=1
(∆D (i)
L c
) H ∆D (j)
L c
⊙ R (i,j) (0)+
n∑
t=1
(∆D (i)
L c
) H Q n+1 ∆D (j)
L c
∆D (k)
(
⊙
R (i,j)
n
(t) + (R (i,j)
n (t)) T ) ∈ C LcLt×LcLt , (2.68)
L c
{ }} {
L c
= [ ∆D (k) , . . . , ∆D (k) ] ∈ C Nc×LcLt , (2.69)
with D (k) defined in Equation (2.2) and Q defined in Equation (2.37).
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