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

For small multipath delays, a typical spread system has R mm (t) ≃ 0, t ≠ 0, therefore
Φ mm ≃ ∆D H L c
∆D Lc ⊙ R mm (0). We partition R mm (0) as follows
⎡
⎤
R (1,1) · · · R (1,Lc)
R mm (0) =
.
⎢ . .. . ⎥
⎣
⎦ where R (i,j) = [s i 1, . . . , s i L t
] T [s j 1 , . . . , sj L t
]. (2.54)
R (Lc,1) · · · R (Lc,L c)
R (i,j) ≃ 0, i ≠ j, that is, well designed spreading codes have small cross-correlations
for multiple lags. This implies det(Φ mm ) ≈ ∏ L c
j=1 det(∆DH ∆D ⊙ R (j,j) ). Furthermore,
if R (i,i) ≃ R (j,j) , i ≠ j, then we can further approximate det(Φ mm ) ≈ [det(∆D H ∆D ⊙
R (1,1) )] Lc . Note that det(∆D H ∆D ⊙ R (1,1) ) corresponds to the coding gain for a single
path system. Hence we see that good flat fading channel codes tend to give large coding
gain even for multipath channels.
For a spread system, based on the consideration of random sequences, we can use the
following spreading code correlation model[AM00]:
(s l 1
i1
) T (s l 2
i2
) =
where ρ c is a common cross-correlation value.
⎧
1 if τ l 1
i 1
= τ l 2
i 2
and i 1 = i 2 ,
⎪⎨ ρ c if τ l 1
i 1
= τ l 2
i 2
and i 1 ≠ i 2 ,
ρ c × Lu−|τ l 1
i1 −τ l 2
i2 |
L u
if 0 < |τ l 1
i 1
− τ l 2
i 2
| < L u ,
⎪⎩ 0 else.
(2.55)
To examine the effects of multipath on well-designed single path codes, we consider
a specific code: the 2 × 2 rate 1 BPSK Class 1 code, which is discussed in depth in
Section 2.5. If ρ c = 0.3, L u = 32, for L c = 1 with delay profile [0; 0], and for L c = 2 with
delay profile [0 1; 0 1], we compare the distance spectra (the enumeration of the coding
gains of all codeword pairs) in Table 2.1. As we can see, the coding gain for L c = 2
is roughly the square of the coding gain for L c = 1, which agrees with our approximate
analysis above. For the L c = 2 case, we have searched for optimal codes with delay profile
[0, τ1 2; τ 2 1, τ 2 2], where 0 ≤ τ i l ≤ 30, the optimal codes all turn out to be the same as that
for the flat fading channel. An exhaustive search for L c = 3 and τ max ≤ 5 shows that the
flat fading channel code is still optimal. Figure 2.4 shows the union bounds for 2 × 2 rate
25