Optimal SISO Multiuser Detector
For the synchronous CDMA system (6.27), it is easily seen that a sufficient statistic for demodulating the ith code bits of the k users is given by the K-vector y[i] = [y1[i] ... yK[i]T, whose kth component is the output of a filter matched to sk(.) in the ith code-bit interval:
This sufficient statistic vector y[i] can be written as [520]
where R denotes the normalized cross-correlation matrix of the signal set s1, ..., sK:
; b[i] =[b1[i]
...bk[i]]T; and n[i] ~ N(0, s2 R)
is a Gaussian noise vector, independent of b[i].
In what follows, for notational simplicity, we drop the symbol index i whenever possible. Denote
From (6.32), the extrinsic information l1(bk) delivered by the SISO multiuser detector [cf. (6.29)] is then given by
where we have used the notation 
for bj 
{+1, –1}. The summations in the numerator
(respectively, denominator) in (6.34) are
over all the 2K-1 possible vectors b in 
(respectively, 
). We
have
Note that the first term in (6.35) is independent of b and therefore will be canceled in (6.34). The third term in (6.35) can be written as
where (6.36) follows
from the fact that bj {+1, –1}.
The first term in (6.37) is also
independent of b and will be canceled in
(6.34). In (6.34) the a priori
probabilities of the code bits can be expressed in terms of their LLRs l2(bj[i]), as follows.
Since
after some manipulations, we have for bj {+1, –1},
where (6.38) follows from a derivation similar to that of (6.37). Substituting (6.35), (6.37), and (6.39) into (6.34), we obtain
It is seen from (6.40)
that the extrinsic information l1(bk[i]) at the output of the SISO multiuser detector
consists of two parts; the first term contains the channel value of the desired
user yk[i]}, and the
second term is the information extracted from the other users' channel values
{yj[i]}j
k as well as their prior information {l2(b2[i])}jk.