3.5.1 Numerical Analysis
In this section, a closed-form expression for the average bit error rate (BER) over AWGN channel is presented. Channel estimation accompanying with the pilot-channel aided SIC employing three types of cancellation-ordering method in asynchronous systems (but assumed chip synchronous, i.e., τ ≠0 and τ′=0 where τ and τ′ are defined in Appendix C is analyzed.
Channel Estimation
For sake of simplicity, the long scramble sequences are viewed as random sequences where {cs} are modeled as i.i.d. random variables. Thus, code correlations are assumed to be zero-mean complex-valued Gaussian random variables where
) 2 /(
1 )]) ( (Re[
)]) (
(Re[ (;) , Var (;) , SF
Var λknJ τkJ = μknJ τkJ = when chip synchronous assumption is made, and Var(Re[λk(n;J)(τk,J)])=Var(Re[μk(;nJ)(τk,J)])=1/(3SF) when chip asynchronous assumption is
made [90]. Iαˆ(Jn) is also a zero-mean complex-valued Gaussian random variable. It can be shown that the mean-squared error (MSE) of the channel estimation is given by
⎟⎟⎠
Without loss of generality, the time dependency is negligible in the following discussions.
Thus, from (3-12), the decision statistics at the u-th stage are given as follows.
] z SIC I: Ordering Based on Average Power
In this type of SIC, user data is detected and removed according to the descending average power. According to the central limit theorem, YˆSIC<u>|b<u> are assumed as Gaussian-distributed random variables with mean P<u>b<u> and variance
( ) ( )
since the decision statistics are the sum of many variables. This assumption is commonly made in the case of successive cancellation [54]. As shown in the next section, it provides good approximation to the AWGN-dominant systems. The variation of signal power is neglected for simplicity, and the interference components are assumed to be uncorrelated with b<u>. According to ˆ ( );
; τ
ϕ′SICk<u> , λ;k;<u>(τ) and ψˆ′SIC;k;<u>(τ) in (3-17),(3-19) and (3-16),
respectively, it is shown that code correlations are considered in (3-23). Also,
( )
⎟⎟ where the first and the third terms come from channel estimation errors. Strictly speaking, the BER analysis must be analyzed with the method of robust statistics as far as error detection is concerned [66]. Nevertheless, for simplicity, the error probability for the u-th canceled can be approximated as⎟⎠
z SIC II: Ordering Based on RAKE Outputs after G-bit Cancellation of One User
This SIC finds the next detected user after each cancellation of the currently detected user,
i.e., decision statistics of undetected users at each stage are used as ordering bases. For SIC II and SIC III, the BER analysis of the cases with G>1 is very complicated, and thus only the case of G=1 is performed. Fortunately, as shown in the next section, when G increases, the BER of the three types of SIC are comparable in the AWGN channel. The error probability for the u-th canceled user is given as follows.
dx
π is the probability distribution
function (pdf) of Gaussian random variables YˆSIC<u>|b<u> with mean P<u> and variance random variables with zero mean and variance equal to
[ ( ) ] ( )
⎥ Equation (31) denotes that the decision variables of the u-th detected user falling on x1,x2, …, and xu−1 at the first, second, …, and the (u-1)-th stage, respectively, are smaller than
those of users <1>,<2>, …,<u-1> which are decided to be detected at the first, second, …, and (u-1)-th stage, respectively. When it comes to the u-th stage, the decision variables fall on xu since interferences from previously detected u-1 users are canceled, and there are K-u users whose decision variables are smaller than that of user <u> where decision error occurs when xu≥0. To simplify the calculation, we assume that xk+1≈xk since g<u>,<k>(xk+1−xk) is small when it is compared to other random variables in (31). Then the error probability is approximated as
∑ ∑ ∑ ∑ ∑
z SIC III: Ordering Based on RAKE Outputs at First Stage in Each G-bit Interval
In this type of SIC, user data is detected and canceled according to descending signal strength at the correlator outputs of the first stage, i.e., decision statistics of all users at the first stage are used as the ordering basis. Thus, the error probability for the u-th canceled user is modified to
∑ ∑ ∑ ∑ ∑ ∑ ∑
the first stage was decided to be canceled in the u-th stage since there are u-1 users having larger value of decision variables and there are K-u users with smaller value of decision variables than user <u>. When it comes to the u-th stage, the value of decision variables becomes x2 since interferences from previously detected u-1 users are canceled, and the
decision error occurs when x2≥0 in the case b<u>=−1. The decision errors of previously detected users are ignored for simplicity in both SIC II and SIC III. Then, the average BER for a system with K users is p Kk prTIIIK k K
K
r,TIII( ) =
∑
=1 ,( ),< >/ .From the above analyses, channel estimation errors are shown to lead to nonlinear influence on the decision statistics as well as system performance. Analytical methods used in SIC II and SIC III can be viewed as the modification of the order statistics [21] where the ordering basis of SIC II changes after each cancellation.
3.5.2 Computational Complexity
Ordering in SIC I only takes the transmitted power and long-term channel gain into consideration. In SIC II and SIC III, they fully take advantages of instant received signal strength, i.e., ordering is based on the compromise between reliability and channel estimates (weighting of MAI to others) where SIC III is the simplified version of SIC II. In Table 3-1, the reordering frequency, throughput, latency, and computational complexity of all SICs are summarized. The processes in Part I of all SICs can be pipelined since they do not need feedback information from Part II.