MULTIUSER DETECTION IN CDMA SYSTEMS 241 matched filter bank cc first stage second stage y r 1 r N u M −1 1,1 M −1 1,1 M −1 1,1 M −1 N u ,N u M −1 N u ,N u M −1 N u ,N u ˜a (0) 1 ˜a (0) N u N u  u=2 M 1,u ˜a (0) u N u −1  u=1 M N u ,u ˜a (0) u ˜a (1) 1 ˜a (1) N u N u  u=2 M 1,u ˜a (1) u N u −1  u=1 M N u ,u ˜a (1) u ˜a (2) 1 ˜a (2) N u Figure 5.8 Structure of multistage detector for iterative parallel interference cancellation soft estimates ˆa (0) v=u = r v=u /M v,v . Subtracting them from r u leads to an improved estimate ˜a (1) u after the first iteration. The interference cancellation is simultaneously applied to all users and repeated with updated estimates ˜a (µ) u in subsequent iterations. In the µ-th iteration, the u-th symbol becomes ˜a (µ) u = M −1 u,u ·  r u − u−1  v=1 M u,v ˜a (µ−1) v − N u  v=u+1 M u,v ˜a (µ−1) v  . (5.44) The simultaneous application of (5.44) for all symbols a u ,1≤ u ≤ N u , is also called Jacobi algorithm and known as linear parallel interference cancellation (PIC). An implementation leads directly to a multistage detector depicted in Figure 5.8 (Honig and Tsatsanis 2000; Moshavi 1996). Several identical modules highlighted by the gray shaded areas are serially concatenated. Each module represents one iteration step so that we need m stages for m iterations. The choice of the matrix M determines the kind of detector that is approximated. For M = R, we approximate the decorrelator, and the coefficients M u,v = R u,v used in (5.44) equal the elements of the correlation matrix. The MMSE filter is approximated for M = R +σ 2 N /σ 2 A · I N u . Hence, the diagonal elements of M have to be replaced with M u,u = R u,u + N 0 /E s . Convergence Behavior of Decorrelator Approximation The convergence properties of this iterative algorithm depend on the eigenvalue distribution of M. Therefore, (5.44) is described using vector notations. The matrix A = diag(diag(R)) is diagonal and contains the diagonal elements of the correlation matrix R. The PIC approx- imating the decorrelator delivers ˜ a (0) ZF = A −1 · r ˜ a (1) ZF = A −1  r −  R −A  ˜ a (0) ZF  = A −1/2  I N u − A −1/2  R −A  A −1/2  A −1/2 r 242 MULTIUSER DETECTION IN CDMA SYSTEMS ˜ a (2) ZF = A −1  r −  R −A  ˜ a (1) ZF  = A −1/2  I N u − A −1/2  R − A  A −1/2 +  A −1/2  R −A  A −1/2  2  A −1/2 r . . . ˜ a (m) ZF = A −1/2 m  µ=0  A −1/2  A −R  A −1/2  µ A −1/2 r. (5.45) The output after the m-th iteration in (5.45) represents the m-th order Taylor series approx- imation of R −1 (M ¨ uller and Verdu 2001). Rewriting it with the normalized correlation matrix ¯ R = A −1/2 RA −1/2 yields ˜ a (m) ZF = A −1/2 m  µ=0  I N u − ¯ R  µ A −1/2 r. (5.46) This series only converges to the true inverse of R if the magnitudes of all eigenvalues of I N u − ¯ R are smaller than 1. This condition is equivalent to λ max ( ¯ R)<2. Since λ max tends asymptotically to (1 + √ β) 2 (M ¨ uller 1998), we obtain an approximation of the maximum load below which the Jacobi algorithm will converge. β max = N U,max N s <( √ 2 − 1) 2 ≈ 0.17. (5.47) Obviously, the Jacobi algorithm or, equivalently, the linear PIC converges toward the true decorrelator only for very low loads. Hence, this technique is not suited for highly loaded systems. Convergence Behavior of MMSE Approximation According to the last section, we have to replace the diagonal matrix A with the matrix D = A + σ 2 N /σ 2 A I N u to approximate the MMSE detector. With this substitution, we obtain the following estimates after different iterations. ˜ a (0) MMSE = D −1 · r ˜ a (1) MMSE = D −1  r −  R −A  ˜ a (0) MMSE  = D −1/2  I N u − D −1/2  R −A  D −1/2  D −1/2 r ˜ a (2) MMSE = D −1  r −  R −A  ˜ a (1) MMSE  = D −1/2  I N u − D −1/2  R −A  D −1/2 +  D −1/2  R − A  D −1/2  2  D −1/2 r . . . ˜ a (m) MMSE = D −1/2 m  µ=0  D −1/2  A −R  D −1/2  µ D −1/2 r. (5.48) MULTIUSER DETECTION IN CDMA SYSTEMS 243 To determine the convergence properties concerning the MMSE filter, (5.48) can be trans- formed into the form of (5.46) ˜ a (m) MMSE = D −1/2 m  µ=0  I N u − D −1/2  R + σ 2 N σ 2 A I N u  D −1/2  µ D −1/2 r. (5.49) Now, the same argumentation as for the decorrelator can be applied and the condition for convergence becomes (Grant and Schlegel 2001) max u=1 N u A 2 u,u λ u + σ 2 N /σ 2 A A 2 u,u + σ 2 N /σ 2 A < 2 ⇒ β< min u=1 N u       2 + σ 2 N A 2 u,u σ 2 A − 1   2 . The first difference compared to the decorrelator is that the maximum load β depends on the SNR σ 2 N /σ 2 A = N 0 /E s . This term increases the convergence area a little bit. However, for high SNR σ 2 N /σ 2 A becomes small and both decorrelator and MMSE filter are approached only for low loads. This behavior is illustrated in Figure 5.9a showing the results for the first five iterations and a load β = 0.5. Only for very low SNR (large σ 2 N /σ 2 A ) the iterative approximation reaches the true MMSE filter. For higher SNRs, β = 0.5 is beyond the convergence region and the PIC performs even worse than the matched filter. Figure 5.9b shows the results for E b /N 0 = 10 dB versus β. Again, it is confirmed that convergence can be ensured only for low load. 5.2.4 Linear Successive Interference Cancellation (SIC) The poor convergence properties of the linear PIC can be substantially improved. Imagine that the interference cancellation described in (5.44) is carried out successively for different 0 5 10 15 20 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 0 0.5 1 1.5 2 10 −3 10 −2 10 −1 10 0 E b /N 0 in dB → BER → BER → β → a) β = 0.5 b) E b /N 0 = 10 dB 1 iteration 2 iterations 3 iterations 4 iterations 5 iterations Figure 5.9 Performance of linear PIC approximating the MMSE filter (upper dashed line: matched filter; lower dashed line: true MMSE filter) 244 MULTIUSER DETECTION IN CDMA SYSTEMS users starting with u = 1 and ending with u = N u . Considering the µ-th iteration for user u, only estimates ˜a (µ−1) v=u of the previous iteration µ − 1 are used. However, updated estimates ˜a (µ) v<u of the µ-th iteration are already available for users 1 ≤ v<u. Replacing all old estimates ˜a (µ−1) v<u in (5.44) with their updated versions ˜a (µ) v<u of the current iteration results in the Gauss-Seidel algorithm ˜a (µ) u = M −1 u,u ·  r u − u−1  v=1 M u,v ˜a (µ) v − N u  v=u+1 M u,v ˜a (µ−1) v  . (5.50) Besides improved convergence properties another advantage is the in-place implementation, that is, updated estimates can directly overwrite old values because they are not used any longer, thereby saving valuable memory. The analysis of the convergence behavior is not as easy as for the PIC. In Golub and van Loan (1996) it is shown that the algorithm always converges for Hermitian positive definite matrices M. Fortunately, in the context of our CDMA system M represents the correlation matrix R or R + σ 2 N /σ 2 A I N u . Hence, M can be assumed to be Hermitian and positive definite so that the Gauss-Seidel algorithm always converges. Figure 5.10a confirms the promised convergence properties. Considering a half-loaded system, five iterations suffice to approach the true MMSE filter. At low SNRs, the perfor- mance of the MMSE filter is reached with even less iterations. Figure 5.10b shows that with increasing load more iterations are needed. For loads above β = 1, the first iteration can perform even worse than the matched filter. However, successive iterations substantially improve the performance. Comparing the computational costs of a direct matrix inversion with the iterative approx- imations in terms of number of multiplications, we see from (5.50) that N u multiplications per iteration and user are needed. For m iterations, this leads to mN 2 u multiplications 0 5 10 15 20 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 0 0.5 1 1.5 2 10 −3 10 −2 10 −1 10 0 E b /N 0 in dB → BER → BER → β → a) β = 0.5 b) E b /N 0 = 10 dB 1 iteration 2 iterations 3 iterations 4 iterations 5 iterations Figure 5.10 Performance of linear SIC approximating the MMSE filter (upper dashed line: matched filter, lower dashed line: true MMSE filter) MULTIUSER DETECTION IN CDMA SYSTEMS 245 compared to a complexity of O(N 3 u ) for the direct matrix inversion. Hence, as long as the number of iterations is smaller than N u , we save computational costs. Besides parallel and SIC strategies, there exist further iterative approaches like the conjugate gradient method and a general polynomial series expansion of the inverse (M ¨ uller 1998). These approaches are not pursued here. All linear techniques described so far do not reach the SUB, that is, interference remains in the system after filtering. The information theoretic analysis in Section 4.3 showed that the optimum detector performs much better than linear techniques. Therefore, we have to look for nonlinear approaches that come closer to the optimum solution. These techniques exploit the finite signal alphabet to improve the MUD. 5.3 Nonlinear Iterative Multiuser Detection A major drawback of the previously introduced linear detectors is not exploiting the discrete nature of the transmit signals. This shortcoming can be easily overcome by introducing nonlinear devices into the multistage structure to exploit the discrete alphabets. This means that the signals ˜a (µ) v=u in (5.44) or (5.50) are passed through a suited nonlinear device before they are used for interference cancellation. For simplicity, we restrict the analysis to a normalized BPSK, that is, we transmit x =±1. An extension to quaternary phase shift keying (QPSK) that treats real and imaginary parts separately is straightforward while schemes with more levels need more sophisticated methods. 5.3.1 Nonlinear Devices The simplest nonlinearity is naturally a hard decision, that is, determining the sign of a signal Q HD (y) = sgn(y). (5.51) If the tentative decision is correct, the interference can be cancelled perfectly. However, if the decision is wrong, which may be very likely in the early stages of the detection process, especially for large β, interference is not reduced but in fact increased and the situation becomes worse. Therefore, more sophisticated functions taking into account the reliability of the signals should be preferred. A selection analysed by K ¨ uhn et al. (2002) is depicted in Figure 5.11. To keep the influence of wrong decisions as small as possible, it is advantageous not to decide on unreliable small samples but to keep them small. Obviously, interference is generally not perfectly cancelled by these approaches, but the error made by wrong decision is remarkably reduced. The simplest form that follows this strategy is the clipper or limiter. It has a linear shape for |y|≤1 and outputs ±1 for larger inputs |y| > 1 Q clip (y) =      −1fory<−1 y for |y|≤1 +1fory>+1. (5.52) Hence, the clipper exploits the fact that the transmitted signals cannot be larger than 1. 3 Interference is totally cancelled if the signal has the correct sign and a magnitude larger 3 For notational simplicity, we assume the normalization to E s /T s = 1. 246 MULTIUSER DETECTION IN CDMA SYSTEMS 11 11 1 1 11 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 NL 1 NL 2 α αα α HD clipper tanh Figure 5.11 Examples for nonlinear devices than 1. For small values, the reliability is low and the interference can only be partly reduced. In case of a wrong sign, the degradation is not as large as for the hard decision. A smooth version of the clipper is obtained with the tanh-function avoiding sharp edges. We know from Section 3.4 on page 110 that the expectation of a bit is obtained from its log-likelihood ratio L by tanh(L/2). However, the LLR can be determined only if the signal to interference plus noise ratio (SINR) is perfectly known. This represents a big difficulty because we do not know the exact interference level in each iteration. Therefore, we introduce a parameter α according to Q tanh (y) = tanh(αy) (5.53) that depends on the SNR as well as the effective interference and has to be optimized with respect to a minimum error rate. Figure 5.12 compares the tanh-function for different α with the hard decision and the clipper. For small α, the tanh is very smooth and its output is pretended to be unreliable even for large inputs. On the contrary, α = 1 comes close to the clipper in the nearly linear area around the origin and large α>1 approach the hard decision. Next, two further nonlinear functions are proposed. The first one (NL 1) has a linear shape around the origin and hops to ±1 for values larger than a certain threshold α Q NL1 (y) =      −1fory<−α y for |y|≤α +1fory>α. (5.54) The difference compared to the clipper is that this nonlinearity starts to totally remove the interference for values smaller than 1. The parameter α has to be optimized according to the load and the SNR. The second function (NL 2) avoids any cancellation for unreliable MULTIUSER DETECTION IN CDMA SYSTEMS 247 −4 −3 −2 −1 0 1 2 3 4 −1 −0.5 0 0.5 1 x → q(x) → α = 0.5 α = 1 α = 2 α = 4 HD clipper Figure 5.12 Comparison of tanh for different α with hard decision and clipper values and allows interference reduction only above a threshold α Q NL2 (y) =                −1fory<−1 y for − 1 ≤ y ≤−α 0for|y| <α y for α ≤ y ≤+1 +1fory>+1. (5.55) Obviously, it reduces to a simple clipper for α = 0. Finally, we will look at coded CDMA systems. If the computational costs do not represent a restriction, the channel decoder can be used as a nonlinear device (Hagenauer 1996a). Since it exploits the redundancy of the code, it can increase the reliability of the estimates remarkably. Again, we have to distinguish between hard-output and soft-output decoding. For convolutional codes presented in Chapter 3, hard-output decoding can be performed by the Viterbi algorithm while soft-output decoding can be carried out by the BCJR or Max-Log-MAP algorithms. 5.3.2 Uncoded Nonlinear Interference Cancellation Uncoded Parallel Interference Cancellation First, we have to optimize the parameter α for the nonlinear functions NL 1, NL 2, and tanh. We start our analysis with the PIC whose structure for the linear case in Figure 5.8 has to be extended. Figure 5.13 shows the µ-th stage of the resulting multistage receiver. Prior to the interference cancellation, the interference reduced signals ˜r (µ−1) v of the previous iteration are scaled with coefficients M −1 v,v . The application of the nonlinear function now yields estimates for all signals ˜a (µ−1) v = Q  M −1 v,v ·˜r (µ−1) v  . (5.56) 248 MULTIUSER DETECTION IN CDMA SYSTEMS cc r 1 r N u ˜r (µ−1) 1 ˜r (µ−1) N u ˜r (µ) 1 ˜r (µ) N u M −1 1,1 M −1 1,1 M −1 N u ,N u M −1 N u ,N u N u  v=2 M 1,v ˜a (µ−1) v N u −1  v=1 M N u ,v ˜a (µ−1) v ˜a (µ−1) 1 ˜a (µ−1) N u ˜a (µ) 1 ˜a (µ) N u µ-th stage Q(·) Q(·) Q(·) Q(·) Figure 5.13 µ-th stage of a multistage detector for nonlinear parallel interference cancel- lation For user u, all estimates ˜a (µ−1) v=u are first weighted with the correlation coefficients M u,v=u , then summed up and finally subtracted from the matched filter output r u ˜r (µ) u = r u −  v=u M u,v ·˜a (µ−1) v . (5.57) After this cancellation step is performed for all users, the procedure is repeated. If the iterative scheme converges and the global optimum is reached, the interference is can- celled more and more until the single-user performance is obtained. However, the iterative algorithm may get stuck in a local optimum. Figure 5.14 shows the performance of the nonlinearities NL 1 and NL 2 versus the design parameter α. Looking at NL 2, we observe that α NL 2 opt = 0 is always the best choice regardless of the number of iterations. Hence, NL 2 reduces to a simple clipper. With regard to NL 1, the optimum α depends on the iteration. In the first stage, the minimum BER is also delivered by a clipper obtained with α NL 1 opt = 1. For the fifth stage, 0.3 ≤ α NL 1 opt ≤ 0.4 is the best choice. Moreover, the comparison of NL 2 with NL 1 shows that NL 1 is at least as good as NL 2 and generally outperforms NL 2 (Zha and Blostein 2003). The same analysis has been performed for the tanh-function. From Figure 5.15 we recognize that 1 ≤ α ≤ 2 is an appropriate choice for a large variety of loads. With growing β, the optimum α becomes smaller and approaches 1 for β = 1.25. This indicates that the SINR is small for large β. However, the differences are rather small in this interval. Only very low values of α result in a severe degradation because no interference is cancelled for α = 0 leading to the matched filter performance. If α is chosen too large, the tanh function saturates for most inputs and the error rate performance equals that of a hard decision. Figure 5.16a now compares all proposed nonlinearities for a fully loaded OFDM-CDMA system with β = 1 and five iterations. The tanh-function with optimized α shows the best performance among all schemes. NL 1 and clipper come closest to the tanh. The hard decision already loses 2 dB compared to the tanh. Although the nonlinearities consider the finite nature of the signal alphabet and all nonlinearities clearly outperform the matched MULTIUSER DETECTION IN CDMA SYSTEMS 249 0 0.2 0.4 0.6 0.8 1 10 −3 10 −2 10 −1 10 0 0 0.2 0.4 0.6 0.8 1 10 −3 10 −2 10 −1 10 0 BER → BER → a) first iteration b) fifth iteration α →α → β = 0.5β = 0.5 β = 0.75β = 0.75 β = 1β = 1 β = 1.25β = 1.25 SUBSUB Figure 5.14 PIC optimization for NL 1 and NL 2 in an uncoded OFDM-CDMA system with a 4-path Rayleigh fading channel and E b /N 0 = 8 dB (solid lines: NL 1, dashed lines: NL 2) 0 0.4 0.8 1.2 1.6 2 10 −3 10 −2 10 −1 10 0 0 0.4 0.8 1.2 1.6 2 10 −3 10 −2 10 −1 10 0 BER → BER → a) first iteration b) fifth iteration α →α → β = 0.5β = 0.5 β = 0.75β = 0.75 β = 1β = 1 β = 1.25β = 1.25 SUBSUB Figure 5.15 PIC optimization for tanh in an uncoded OFDM-CDMA system with a 4-path Rayleigh fading channel and E b /N 0 = 8dB filter, we observe an error floor that the SUB cannot be reached. Figure 5.16b illustrates this loss versus β.Ataloadofβ = 1, the error rate is increased by one decade compared to the single-user case; for β = 1.5, only the tanh can achieve a slight improvement compared to a simple matched filter. Therefore, we can conclude that nonlinear devices taking into account the finite nature of the signal alphabet improve the convergence behavior of PIC. The SUB is approximately reached up to loads of β = 0.5. For higher loads, performance degrades dramatically until no benefit to the matched filter can be observed. 250 MULTIUSER DETECTION IN CDMA SYSTEMS 0 2 4 6 8 10 10 −3 10 −2 10 −1 10 0 0 0.5 1 1.5 10 −3 10 −2 10 −1 10 0 E b /N 0 in dB → BER → BER → a) β = 1 b) E b /N 0 = 8dB β → MF MF HD HD NL 1 NL 1 clip clip tanh tanh Figure 5.16 PIC performance comparison of different nonlinearities with optimized α in an uncoded OFDM-CDMA system with 4-path Rayleigh fading channel Uncoded Successive Interference Cancellation From linear interference cancellation techniques, we already know that SIC according to the Gauss-Seidel algorithm converges much better than the PIC. Consequently, we now analyze on the nonlinear SIC. Figure 5.17 illustrates the influence of the parameter α for NL 1 and NL 2 on the SIC performance. As already observed for PIC, α NL 2 opt = 0isthe best choice regardless of the load and the considered iteration, and reduces nonlinearity 0 0.2 0.4 0.6 0.8 1 10 −3 10 −2 10 −1 10 0 0 0.2 0.4 0.6 0.8 1 10 −3 10 −2 10 −1 10 0 BER → BER → α →α → β = 0.5β = 0.5 β = 0.75β = 0.75 β = 1β = 1 β = 1.25β = 1.25 a) first iteration b) fifth iteration SUBSUB Figure 5.17 SIC optimization for NL 1 and NL 2 in an uncoded OFDM-CDMA system with a 4-path Rayleigh fading channel and E b /N 0 = 8 dB (solid lines: NL 1, dashed lines: NL 2) [...]... brief overview of different strategies without claiming to be comprehensive Principally, two different categories can be distinguished The first objective is to improve the link reliability, that is, the ergodic error probability or the outage probability are reduced This can be accomplished by enhancing the instantaneous signal-to-noise ratio (SNR) (beamforming) Wireless Communications over MIMO Channels. .. subsequent detection steps because of error propagation The overall performance and the convergence speed can be improved by a linear suppression of the interference prior to the first detection stage The Bell Labs Layered Space-Time (BLAST) detection of the Bell Labs (Foschini 1996; Foschini and Gans 19 98; Golden et al 19 98; Wolniansky et al 19 98) pursues this approach for multiple antenna systems It... iteration 0 10 = 0.5 = 0.75 =1 = 1.25 β β β β −1 10 BER → 10 251 −2 −2 10 SUB 10 SUB −3 0 = 0.5 = 0.75 =1 = 1.25 −3 0.4 0 .8 1.2 α→ 1.6 2 10 0 0.4 0 .8 1.2 α→ 1.6 2 Figure 5. 18 SIC optimization for tanh in an uncoded OFDM-CDMA system with a 4-path Rayleigh fading channel and Eb /N0 = 8 dB NL 2 to a simple clipper However, the influence of α on the error rate performance is much larger than for PIC For α... −1 −1 −2 10 −3 0 10 MF HD NL 1 clip tanh 2 4 6 8 Eb /N0 in dB → BER → BER → 10 10 b) Eb /N0 = 8 dB 0 10 MF HD NL 1 clip tanh −2 10 −3 10 10 0 0.5 1 1.5 β→ Figure 5.19 Performance comparison of different nonlinearities with optimized α for SIC in an uncoded OFDM-CDMA system with 4-path Rayleigh fading channel (five iterations) BER → 10 10 10 a) Eb /N0 = 8 dB 0 b) Eb /N0 = 20 dB 0 10 −1 −1 10 BER → 10... decomposition (SQLD) Moreover, the QL decomposition based on the MMSE criterion was also derived Compared to pure SIC without pre-filtering, an improved convergence, especially, for extremely high loads was observed 6 Multiple Antenna Systems Multiple antenna systems became popular roughly a decade ago as a result of the fundamental work of Alamouti (19 98) , Foschini (1996), Foschini and Gans (19 98) , K¨ hn and... (Holma and Toskala 2004) More sophisticated methods are under discussion for further evolutions (3GPP 2005a; Hanzo et al 2002b) This chapter gives a brief overview of different multiple-input multiple-output (MIMO) strategies for point-to-point communications without claiming to be comprehensive Multiuser scenarios are briefly discussed in Section 2.4 After the introduction, Section 6.2 addresses spatial... = 3 0 10 −1 −1 10 −2 BER → 10 −3 −4 −5 0 −2 10 −3 10 −4 10 BPSK QPSK 2 4 6 8 10 Eb /N0 in dB → −5 12 10 0 BPSK QPSK 2 4 6 8 Eb /N0 in dB → 10 Figure 5.27 Sorted SIC performance for half-rate coded OFDM-CDMA system with β = 2, a 4-path Rayleigh fading channel and different convolutional codes (bold line: single-user bound) 2 58 MULTIUSER DETECTION IN CDMA SYSTEMS always the best choice and that the coding... 1.5 that correlation among the subchannels reduces the diversity gain In the case of a strong line-of-sight component (Rice fading), diversity is also not an appropriate means because fading is not a severe problem If we can exploit other sources of diversity, for example, frequency diversity with the Rake receiver or time diversity due to coding over time-varying channels, we are probably already... different nonlinearities 0 10 BPSK QPSK SUB −1 NL 1 HD tanh −1 QPSK 10 BER → 10 −2 −2 10 BPSK 10 −3 0 −3 0.2 0.4 0.6 α→ 0 .8 1 10 0 5 10 15 Eb /N0 in dB → 20 Figure 5.31 Performance of ZF-SQLD-SIC for uncoded OFDM-CDMA system, 4-path Rayleigh fading channel, and β = 1; a) Eb /N0 = 8 dB b) optimized α user-specific error probabilities Figure 5.31a compares different nonlinearities Their performances are... BPSK and QPSK However, for both detection schemes a large loss compared to the SUB can be observed Moreover, it is demonstrated that even sub-optimum sorting for the SQLD-SIC improves the performance compared to unsorted QLD-SIC Applying the optimum PSA, yields an additional gain of more than 3 dB over the SQLD-SIC We now have a look at the variations of user-specific error rates From Subsection 5.2.1, . a 4-path Rayleigh fading channel and E b /N 0 = 8 dB (solid lines: NL 1, dashed lines: NL 2) 0 0.4 0 .8 1.2 1.6 2 10 −3 10 −2 10 −1 10 0 0 0.4 0 .8 1.2 1.6 2 10 −3 10 −2 10 −1 10 0 BER → BER → a). fading channel and E b /N 0 = 8 dB (solid lines: NL 1, dashed lines: NL 2) MULTIUSER DETECTION IN CDMA SYSTEMS 251 0 0.4 0 .8 1.2 1.6 2 10 −3 10 −2 10 −1 10 0 0 0.4 0 .8 1.2 1.6 2 10 −3 10 −2 10 −1 10 0 BER. (five iterations) 0 0.2 0.4 0.6 0 .8 1 10 −3 10 −2 10 −1 10 0 0 0.2 0.4 0.6 0 .8 1 10 −3 10 −2 10 −1 10 0 BER → BER → α →α → first iteration fifth iteration 10th iteration a) E b /N 0 = 8dB b) E b /N 0 = 20 dB Figure