entary signal estimator (ESE) and linear MMSE (LMMSE) estimator. Without loss of generality, we consider sin- gle symbol processing. Let's assume that there are J users and K resources (spreading fac- tor). The received signal at resource k can be written as yk = ;=1 hkjXkj where hkj is the channel coefficient corresponding to resource k from user j, Xkj is the transmitted signal by user j on resource k, and n ~C(0,No). For linear spreading codes, each user is assigned a spreading code sequence. Let Ckj be the kth coefficient of the spreading code for user j. We assume that all users share the same modulation alphabet W = WM}. Then, = 1,2, ., K where Sj W is the transmitted symbol by user j. The received sig- nal can be written as y = Hs + n where s=[s]...sj]T, n=[n1...nj]T, and [hkilk, is a K X J matrix with entries hkj = hkjCkj. Let hj denote the jth column of matrix H. The multi-user detector estimates the LLRs for S based on y [23]. Matched filter and ESE, which is a generalization of the matched filter that can accommo- date soft interference cancellation, can be used as multi-user detectors. The ESE multi-user detector first compresses the received signals to scalar values for each UE by matched filter- ing. The output of the matched filter can be written as y = [ = H4y. In order to take advantage of soft information computed at the channel decoder, we can apply the ESE to y, which approximates signal and interference as Gaussian random variables as follows hswhere Ej is residual interference plus noise. The residual interference Ej is approximated as a Gaussian random variable which can be described by its mean and variance as follows: M(E))= 358 Chapter 3 where u(si) and o2(si) are the a priori mean and variance of the symbol transmitted from the ith user, which can be computed using a priori bit LLRs. The LMMSE estimation can be used to estimate Sj from . From the estimation, LLR for each bit can be derived from conventional marginalization. 16 It can be noted that the ESE multi-user detector without mat