CODE DIVISION MULTIPLE ACCESS 203 user u are obtained by r u [l] = C T u [l] ·H H u [l] ·H u [l] ·C u [l] ·a[l] + ˜ n[l]. (4.58) Although this approach maximizes the SNR and perfectly exploits diversity, it does not consider MUI, which dramatically limits the system performance (Dekorsy 2000; Kaiser 1998). The diagonal matrix H H u [l] ·H u [l] between C T u [l]andC u [l] in (4.58) destroys the orthogonality of the spreading codes because the chips of the spreading codes are weighted with different magnitudes. The performance degradation is the same as in single-carrier CDMA systems. Orthogonal restoring combining (ORC) The influence of MUI can be easily overcome in OFDM-CDMA systems. Restoring the orthogonality is possible by perfectly equalizing the channel also known as ZF solu- tion (Fazel and Kaiser 2003). In OFDM-based systems, this is easily implemented by dividing each symbol in y[l] with the corresponding channel coefficient. With H −1 u [l] = diag  H −1 u [l, 0] ···H −1 u [l, N c − 1]  , we obtain r u [l] = E ORC u [l] ·y[l] = C T u [l] ·H −1 u [l] ·  H u [l] ·C[l] ·a[l] +n[l]  = C T u [l] ·C[l] ·a[l] +C T u [l] ·H −1 u [l] ·n[l]. (4.59) If the partial spreading codes c u [l, µ] of different users are mutually orthogonal, C T u [l] · C[l] = [0 N b ×(u−1)N b I N b 0 N b ×(N u −u)N b ] holds. Hence, the multiplication with C T u [l] sup- presses all users except user u and (4.59) becomes r u [l] = a u [l] +C T u [l] ·H −1 u [l] ·n[l]. (4.60) We see that the desired symbols a u [l] have been perfectly extracted, and only the modi- fied background noise disturbs a decision. However, this same background noise is often significantly amplified by dividing through small channel coefficients leading to high error probabilities, especially at low SNRs. This effect is well-known from ZF equalization (Kammeyer 2004) and linear multiuser detection (Moshavi 1996). A comparison with the linear ZF detector in Subsection 5.2.1 on page 234 shows the following equivalence. For a fully loaded system with N s = N u , C[l] is an orthogonal N u × N u matrix. Neglecting time indices, the ZF criterion (4.59) delivers with S = HC E =  S H S  −1 S H = C −1 H −1 H −H C −H C H H H = C T H −1 . (4.61) Obviously, (4.61) coincides with E ORC u [l] in (4.59). For the downlink, OFDM-CDMA allows a very efficient implementation of the ZF multiuser detector. Equal gain combining (EGC) Two approaches exist that try to find a compromise between interference suppression and noise amplification. In the first, instead of dividing through a channel coefficient, we could just correct the phase shift and keep the amplitude constant. Hence, all chips experience 204 CODE DIVISION MULTIPLE ACCESS the same ‘gain’ resulting in r u [l] = E EGC u [l] ·y[l] = C T u [l] ·     H ∗ u [l,0] |H u [l,0]| . . . H ∗ u [l,N c −1] |H u [l,N c −1]|     ( H u [l] ·C[l] ·a[l] +n[l] ) = C T u [l] ·     |H u [l,0]| 2 |H u [l,0]| . . . |H u [l,N c −1]| 2 |H u [l,N c −1]|     · C[l] ·a[l] + ˜ n[l]. (4.62) From (4.62) we see that the equalizer coefficients have unit magnitudes so that the noise is not amplified. The second impact is that amplitude variations of the channel transfer function are not emphasized by the equalizer so that the originally perfect correlation properties of the spreading codes become not so bad after equalization as for MRC. Minimum mean squared error (MMSE) A second possibility to avoid an amplification of the background noise is to use the MMSE solution. Starting with the MMSE criterion E u = argmin W E    Wy[l] −a u [l]   2  = argmin W E    W  H u [l]C[l]a[l] +n[l]  − a u [l]   2  , (4.63) for user u, a solution is obtained by setting the derivation with respect to E H u to zero and solving the equation system. This yields E MMSE u [l] = C T u [l] ·H H u [l] ·  H u [l] · H H u [l] + σ 2 N σ 2 A · I N c  −1 . (4.64) Since H u [l] is a diagonal matrix, the application of (4.64) results in r u [l] = E MMSE u [l] ·y[l] = C T u [l]      |H u [l,0]| 2 |H u [l,0]| 2 +σ 2 N /σ 2 A . . . |H u [l,N c −1]| 2 |H u [l,N c −1]| 2 +σ 2 N /σ 2 A      C[l]a[l] + ˜ n[l]. (4.65) Obviously, we have to add the ratio between noise power σ 2 N and signal power σ 2 A to the squared magnitudes in the denominators. This avoids the noise amplification at the subcarriers with deep fades. For infinite high SNR, σ 2 N /σ 2 A → 0 holds and the MMSE equalization equals the ORC scheme. CODE DIVISION MULTIPLE ACCESS 205 Similar to the ORC solution, we can compare (4.64) with the linear MMSE multiuser detector on page 238. For a fully loaded system with N s = N u and orthogonal spreading codes, C[l] is an orthogonal N u × N u matrix and C[] T C[] = I N u holds. The MMSE criterion in (5.37) delivers with S = HC E =  S H S + σ 2 N σ 2 A I N u  −1 S H =  C T H H HC + σ 2 N σ 2 A I N u  −1 C T H H = C −1  H H H + σ 2 N σ 2 A I N u  −1 C −T C T H H = C T  H H H + σ 2 N σ 2 A I N u  −1 H H . (4.66) Since the channel matrices are diagonal, (4.66) and (4.64) are identical. Hence, OFDM- CDMA allows a very efficient implementation of the MMSE multiuser detector for the downlink without matrix inversion. To evaluate the performances of the described equalization techniques, we consider the synchronous downlink of an OFDM-CDMA system with BPSK modulation. Scrambled Walsh codes with a spreading factor N s = 16 are employed. The choice of N c = 16 sub- carriers results in a mapping of one information bit onto one OFDM symbol. Moreover, a 4-path Rayleigh fading channel is used requiring a guard interval of length L t − 1 = 3 samples. The E b /N 0 loss due to the insertion of the cyclic prefix has not been considered because it is identical for all equalization schemes. As explained earlier, the frequency selectivity of the channel destroys the Walsh codes’ orthogonality and MUI disturbs the transmission. For a load of β = 1/2, we see from 0 5 10 15 20 10 −4 10 −3 10 −2 10 −1 10 0 0 5 10 15 20 10 −4 10 −3 10 −2 10 −1 10 0 E b /N 0 in dB →E b /N 0 in dB → BER → BER → a) N u = 8,β = 1/2 b) N u = 16,β = 1 N u = 1N u = 1 MRCMRC ORCORC EGCEGC MMSEMMSE Figure 4.25 Error rate performance of OFDM-CDMA system with G p = 16 and different equalization techniques for a 4-path Rayleigh fading channel a) N u = 8 active users, b) N u = 16 active users 206 CODE DIVISION MULTIPLE ACCESS 4 8 12 16 10 −4 10 −3 10 −2 10 −1 10 0 4 8 12 16 10 −4 10 −3 10 −2 10 −1 10 0 BER → BER → a) E b /N 0 = 8dB b) E b /N 0 = 12 dB N u = 1 N u →N u → MRC ORC EGC MMSE Figure 4.26 Error rate performance of OFDM-CDMA system with G p = 16 and different equalization techniques for a 4-path Rayleigh fading channel a) E b /N 0 = 8dB,b)E b /N 0 = 12 dB Figure 4.25a that the MMSE approach performs best over the whole range of SNRs. EGC comes very close to the MMSE solution at low and medium SNRs, but loses up to 3 dB for high SNRs. MRC performs much worse except in the low SNR regime where the background noise dominates the system reliability. In this area, ORC represents the worst approach; it can outperform MRC only for SNRs larger than 14 dB due to the noise amplification. None of the equalizing schemes can reach the single-user bound (SUB) that represents the achievable error rate in the absence of interference. Figure 4.25b depicts the results for N u = 16, that is, a fully loaded system with β = 1. The advantage of the MMSE solution becomes larger. Especially, EGC loses a lot and is even outperformed by ORC at high SNRs. The higher the load, the better is the performance of ORC compared to EGC and MRC because interference becomes the dominating penalty. As will be shown in Section 5.2, linear multiuser detection schemes are not able to reach the SUB for high load. The discussed effects are confirmed in Figure 4.26 where the bit error rate is depicted versus the number of users. First, we recognize that ORC is independent of the load β since the whole interference is suppressed. Different SNRs just lead to a vertical shift of the curve (cf. Figs 4.26a and b). Moreover, ORC outperforms MRC and EGC for high loads and SNRs. MMSE equalization shows the best performance except for very low loads. In that region, EGC and, especially, MRC show a better performance because the interference power is low and optimizing the SNR ensures the best performance. Figure 4.27 points out another interesting aspect that holds for single-carrier CDMA systems also. Since the frequency selectivity destroys the orthogonality of spreading codes, there exists a rivalry between diversity and MUI. The trade-off depends on the kind of equalization that is applied. For the MMSE equalizer, the diversity gain dominates and the error rate performance is improved for growing L t . On the contrary, the MUI conceals the diversity effect for EGC and performance degrades for increasing L t . CODE DIVISION MULTIPLE ACCESS 207 0 5 10 15 20 10 −4 10 −3 10 −2 10 −1 10 0 E b /N 0 in dB → BER → L t = 2 L t = 3 L t = 4 L t = 8 EGC MMSE Figure 4.27 Error rate performance of OFDM-CDMA system with G p = 16 and N u = 16 for EGC and MMSE equalization and L t -path Rayleigh fading channels Quasi-Synchronous Uplink Transmission With respect to the uplink, an equalization is not as easy because each user is affected by an individual channel. For simplicity, we assume a coarse synchronization ensuring that the maximum delay κ between two users is limited to the length N g of the guard interval minus the maximum channel delay κ max . κ ≤ N g − κ max (4.67) In this case, a block-oriented processing is possible and a single FFT block can transform the OFDM symbols of all users simultaneously into the frequency domain. Hence, the signature matrix S[l] becomes S[l] =  s 1 [l] ··· s N u [l]  (4.68) with s u [l] = diag  H u [l, 0] ···H u [l, N c − 1]  · c u [l]. The signature of a user is obtained by multiplying the coefficients of the channel transfer function element-wise with the chips of the spreading code. The data vector a[l] is defined as described in (4.54). The simple MF provides the sufficient statistics, that is, we do not lose any information and an optimum overall processing is still possible. Hence, despreading with MRC has to be applied, resulting in r[l] = E MRC [l] ·y[l] = S H [l] ·y[l] = S H [l] ·S[l] ·a[l] +S H [l] ·n[l]. (4.69) Owing to the nondiagonal structure of S H [l] · S[l], MUI degrades the system performance. This is confirmed by the results shown in Figure 4.28. With growing β, error floors occur so that a reliable uncoded transmission is not possible for loads larger than 0.5. The larger β, the smaller is the influence of the background noise as depicted in Figure 4.28. Concluding, we can state that OFDM represents a pretty good technique for synchronous downlink transmissions while the discussed benefits cannot be exploited in the uplink. Here, 208 CODE DIVISION MULTIPLE ACCESS 0 4 8 12 16 20 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 0 0.5 1 1.5 2 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 E b /N 0 in dB → BER → BER → β = 1/16 β = 0.5 β = 1 β = 1.5 β = 2 SUB 0dB 5dB 10 dB 15 dB 20 dB a) b) β → Figure 4.28 Error rate performance of OFDM-CDMA uplink with G p = 16, BPSK, and 4-path Rayleigh fading channels each signal experiences its own channel so that a common equalization is not possible. Moreover, different carrier frequency offsets between transmitter and receiver pairs destroy even the orthogonality between subcarriers of the same user and cause ICI. Therefore, more sophisticated detection algorithms as presented in Chapter 5 are required. 4.3 Low-Rate Channel Coding in CDMA Systems The previous sections illustrated that MUI dramatically degrades the system performance. Using OFDM-CDMA in a downlink transmission allows an appropriate equalization that suppresses the interference efficiently. However, this is not possible in an asynchronous uplink transmission. One possibility is the application of multiuser detection techniques that exploit the interference’s structure and are discussed in Chapter 5. Alternatively, we can interpret the interference as additional AWGN. This assumption is approximately fulfilled for a large number of users according to the central limit theorem. It is well-known that noise can be combated best by strong error-correcting codes. One important feature of CDMA systems is the inherent spectral spreading, already depicted in Figures 4.1 and 4.2. As shown in Figure 4.29, this spreading can also be described from Figure 4.24 as simply repeating each symbol a[] N s times and subsequent scrambling with a user-specific sequence c[, k] (Dekorsy 2000; Dekorsy et al. 2003; Frenger et al. 1998a; K ¨ uhn et al. 2000a,b; Viterbi 1990). Scrambling means that the repeated data stream is symbol-wise multiplied with the user-specific sequence without spectral spreading. There- fore, an ‘uncoded’ CDMA system with DS spreading can also be interpreted as a system with a scrambled repetition code of low rate 1/N s . The block matched filter in Figure 4.29 may describe the OFDM equalizers discussed in Subsection 4.2.2 (Figure 4.24) or a Rake receiver as depicted in Figure 4.4 excluding the summation over N s chips after the multiplication with c[, k]. The summation itself CODE DIVISION MULTIPLE ACCESS 209 repetition encoder repetition decoder matched filter a[]  k k c[, k] c[, k] N s h[k, κ] MUI+n[k] y[k] ˆa[] super channel Figure 4.29 Illustration of direct-sequence spreading as repetition coding and scrambling is common to OFDM-CDMA and single-carrier CDMA systems and is carried out by the repetition decoder. If the repetition is counted among the channel-coding parts of a communication system, only scrambling remains a CDMA-specific task and the system part between channel encoder and decoder depicted in Figure 4.29 can be regarded as a user-specific time-discrete super channel. However, repetition codes are known to have very poor error-correcting capabilities regarding their very low code rate. Hence, the task is to replace them with more powerful low-rate FEC codes that perform well at very low SNRs. This book does not claim to present the best code suited to this problem. In fact, some important aspects concerning the code design are illuminated and the performances of four different coding schemes are compared. Specifically, we look at traditional convolutionally encoded systems in which the rates of convolutional and repetition code are exchanged, a code-spread system, and serial as well as parallel code concatenations. The performance evaluation was carried out for an OFDM-CDMA uplink with N c = 64 subcarriers and a 4-path Rayleigh fading channel with uniform power delay profile. 7 Suc- cessive channel impulse responses are statistically independent, that is, perfect interleaving in the time domain is assumed. For notational simplicity, we restrict the analysis on BPSK although a generalization to multilevel modulation schemes is straightforward. In the next four subsections, the error rate performance of each coding scheme is analyzed for the single-user case. In Subsection 4.3.5, all schemes are finally compared in multiuser scenarios. 4.3.1 Conventional Coding Scheme (CCS) The first approach abbreviated as CCS does not change the classical DS spreading and can be interpreted as a concatenation of convolutional code and repetition code. It is illustrated in Figure 4.30. The convolutional code is described by its constraint length L c and the code rate R cc c = 1/n. Subsequent repetition encoding with rate R rc c = 1/N s = n/G p ensures a constant processing gain G p = R −1 c = (R cc c · R rc c ) −1 . The influence of different convolu- tional codes is illuminated by choosing different combinations of R cc c and R rc c while their product remains constant. The employed convolutional codes are summarized in Table 4.2. They have been found by a nested code search (Frenger et al. 1998b) and represent codes with maximum free distance and minimum number of sequences with weight d f 7 Similar results can be obtained for single-carrier CDMA systems. The differences concern only the path crosstalk of the Rake receiver and the E b /N 0 -loss due to the cyclic prefix for OFDM-CDMA. 210 CODE DIVISION MULTIPLE ACCESS convolutional code repetition code d[i] ˜ b[l] b[k] a[k] c[l, k] x[k]  BPSK CCS R cc c = 1 n R rc c = 1 N s = n G p Figure 4.30 Conventional coding scheme (CCS) consisting of outer convolutional code, interleaver, and inner repetition code Table 4.2 Parameters of coding schemes for OFDM- CDMA system with processing gain G p = R −1 c = 64 L c R cc c generators R rc c d f CCS 2 7 1/2 133 8 , 171 8 1/32 10 CCS 4 7 1/4 117 8 , 127 8 , 155 8 , 171 8 1/16 20 CCS 8 7 1/8 117 8 , 127 8 , 155 8 , 171 8 1/8 40 135 8 , 173 8 , 135 8 , 145 8 CSS 7 1/64 (Frenger et al. 1998b) 1 320 We see from Figure 4.31 that the performance can be improved by decreasing the code rate. The largest gains are obtained by changing from R cc c = 1/2toR cc c = 1/4 while a fur- ther reduction of R cc c leads only to minor improvements. The reason is that convolutional codes of very low rate incorporate a repetition of parity bits as well. The contribution of repeated bits becomes larger for decreasing constraint lengths and code rates. Therefore, no large gains can be expected for extremely low-rate convolutional codes. This is confirmed by the free distances summarized in Table 4.2, which grow in the same way as R c is reduced. 4.3.2 Code-Spread Scheme (CSS) Reducing R cc c to the minimum value of R c = 1/G p results in a single very low-rate con- volutional code and the repetition code is discarded. The corresponding structure of the transmitter is depicted in Figure 4.32. The convolutional encoder already performs the entire spreading so that the coded sequence is directly scrambled with the user-specific sequence. Many ideas of the so-called code-spreading are encapsulated in Viterbi (1990). In (Frenger et al. 1998b) an enormous number of low-rate convolutional codes found by computer search are listed. These codes have a maximum free distance d f and a minimum number of sequences with weight d f . However, the obtained codes also include a kind of unequal repetition code, that is, different bits of a code word are repeated unequally (Frenger et al. 1998b). Therefore, the performance of CSS is comparable to that of CCSs, as the results in Figure 4.31 show. CODE DIVISION MULTIPLE ACCESS 211 0 2 4 6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 E b /N 0 in dB → BER → CCS 2 CCS 4 CCS 8 CSS Figure 4.31 Performance of single-user OFDM-CDMA system with N c = 64 subcarriers, 4-path Rayleigh fading and different convolutional codes from Table 4.2 convolutional code d[i] b[k] a[k] c[l, k] x[k]  BPSK R cc c = 1 G p CSS Figure 4.32 CSS consisting of single low-rate convolutional code 4.3.3 Serially Concatenated Coding Scheme (SCCS) Instead of reducing the code rate of the convolutional code, we know from Section 3.6 that parallel and serial concatenations of very simple component codes lead to extremely powerful codes. Hence, the inner repetition code should be at least partly replaced by a stronger code. With regard to the serial concatenation, we know from Section 3.6 that the inner code should be a recursive convolutional code in order to exploit the benefits of large interleavers (Benedetto et al. 1996). In the following part, two different concatenated coding schemes are considered: a serial concatenation of two convolutional codes serial concatenated convolutional code (SCCC) and a serial concatenation of an outer convolutional code, and an inner Walsh code (SCCW) (Dekorsy et al. 1999a,b). The latter scheme is used in the uplink of IS95 (Gilhousen et al. 1991; Salmasi and Gilhousen 1991) where Walsh codes are employed as an orthogonal modulation scheme allowing a simple noncoherent demodulation. Although Walsh codes are not recursive convolutional codes, they offer the advantage of a small code rate (large spreading) and low computational decoding costs even for soft-output decoding (see Fast Hadamard Transform in Subsection 3.4.5). 212 CODE DIVISION MULTIPLE ACCESS conv. code rep. code inner code d[i] b 1 [l] b 2 [l  ] b[k] a[k] c[l, k] x[k]  BPSK R cc c = 1 n R inner c R rc c = 1 N s SCCS Figure 4.33 Structure of serially concatenated coding scheme (SCCS)   −1 L( ˆ b 2 [l  ]) L e ( ˆ b 1 [l]) L a ( ˆ b 1 [l]) ˆ d[i] rep. dec. dec. inner convolutional decoder Figure 4.34 Decoder structure of serially concatenated coding scheme (SCCS) Figure 4.33 shows the structure of the SCCS. The outer convolutional encoder is fol- lowed by an interleaver and an inner code that can be chosen as described above. The final repetition code may be necessary to ensure a constant processing gain. Since we are not interested in interleaver design for concatenated codes, we simply use random interleavers as described in Chapter 3 and vary only the length L π . The corresponding decoder structure is shown in Figure 4.34. First, the received sig- nal is equalized in the frequency domain according to the MRC principle including the descrambling. 8 Next, an integrate-and-dump filter decodes the repetition code and delivers the log-likelihood ratios (LLRs) L( ˆ b 2 [l  ]). Now, the iterative decoding process starts with the inner soft-in soft-out decoder. The extrinsic part L e ( ˆ b 1 [l]) of its output is deinterleaved and fed to the outer soft-output convolutional decoder. Again, extrinsic information is extracted and fed back as a priori information L a ( ˆ b 1 [l]) to the inner decoder. This iterative turbo processing is carried out several times until convergence is obtained (cf. Section 3.6). Owing to the high number of parameters, we fix the code rate of the outer convolutional code to R cc c = 1/2. Hence, introducing the inner code affects only the repetition code whose code rate R rc c increases in the same way as R inner c decreases (see. Table 4.3). Although theoretical analysis tells us that the minimum distance of the outer code should be as large as possible (see page 138), the iterative decoding process benefits from a stronger inner code. This is confirmed by simulation results showing that lower rates of the outer convolutional code, for example, R cc c = 1/6, coming along with higher rates of the inner codes, for example, R rc c = 1, lead to a significant performance loss. The interleaver  between the outer convolutional and the inner encoder is a randomly chosen interleaver of length N = 600 or N = 6000. 9 8 In single-carrier CDMA, this corresponds to the Rake receiver of Figure 4.4 excluding the summation over N s chips after the multiplication with c[, k]. 9 The shorter interleaver may be suited for full duplex speech transmission, while the longer one is restricted to data transmission with weaker delay constraints. [...]... inner code rep code g1 = 78 , g2 = 58 wh Walsh, Rc = 4/16 rc Rc = 1/8 SCCW 2 g1 = 78 , g2 = 58 rc Rc = 1/3 SCCW 3 g1 = 78 , g2 = 58 wh Walsh, Rc = 6/64 wh Walsh, Rc = 8/256 wh Walsh, Rc = 6/64 wh Walsh, Rc = 6/64 r g1 = 78 , g2 = 58 SCCW 1 SCCW 4 g1 = 238 , g2 = 558 SCCW 5 g1 = 1338 , g2 = 171 8 SCCC 1 g1 = 78 , g2 = 58 SCCC 2 g1 = 78 , g2 = 58 g1 = 238 , g2 = 278 = 358 , g4 = 378 r g3 rc Rc = 1/3 rc... systems, cu [ ] is independent of the time index while it varies from symbol to symbol in long code systems The obtained sequence xu Wireless Communications over MIMO Channels Volker K¨ hn u  2006 John Wiley & Sons, Ltd 228 MULTIUSER DETECTION IN CDMA SYSTEMS is transmitted over the individual channel with impulse response hu [ ] At the receiver, that is, a common base station, all transmitted signals... 223 equation CMF,u = 2Cu Eb /N0 1 · log2 1 + 2 1 + 2βCu Eb /N0 1 · log2 (1 + x) 2 = (4 .75 ) Resolving x= 2Cu Eb /N0 1 + 2βCu Eb /N0 with respect to N0 /Eb and substituting Cu = 1 2 (4 .76 ) · log2 (1 + x) yields N0 = log2 (1 + x) · Eb 1 −β x (4 .77 ) that has to be numerically solved The value x can be inserted into (4 .75 ) to obtain the CMF,u or the spectral efficiency ηMF = β · CMF,u Zero-Forcing Receiver... ¯ E C(S) C = Nu Nu (4 .71 ) ¯ The division of C by the spreading factor Ns delivers the spectral efficiency η= ¯ C = β · Cu Ns (4 .72 ) 220 CODE DIVISION MULTIPLE ACCESS already defined in (4. 17) It describes the average number of information bits transmitted per chip and is measured in bits/chip While (4 .72 ) assumes a perfect coding scheme ensuring an error-free transmission, (4. 17) considers a practical... the large scale analysis (Nu → ∞, Ns → ∞, β constant) in Verdu and Shamai (1999) we obtain CZF,u = Es 1 · log2 1 + 2(1 − β) 2 N0 (4 .78 ) and consequently ηZF = Es β · log2 1 + 2(1 − β) 2 N0 (4 .79 ) Comparing (4 .78 ) with the spectral efficiency for orthogonal codes given in (4 .73 ) we observe that the only difference is an SNR loss depending on the load β Hence, the decorrelator totally removes the interference... the pseudo inverses that also include the nonsingular case S =V † † U =V −1 0 H 0 −1 SH S SH 0 H U = −1 0 SH SSH rank(S) = Nu rank(S) = Ns (5.27a) and R† = SH S † =V −2 0 0 −1 SH S 0 H V = 0 SH SSH −2 rank(S) = Nu S rank(S) = Ns (5.27b) Using (5.27a) and (5.27b), we obtain the general result WH = S† ZF Near-Far Scenarios In scenarios with near-far effects (cf page 183), the received signal can be... sequence associated with the hypothesis d Inserting ˜ and applying the (5.4) into (5.3), neglecting all terms independent of the hypothesis d, 2 natural logarithm, we obtain with N N = σN I ˆ dmld = argmax log exp − ˜ y − Sa(d) ˜ d ˜ 2 = argmin y − S · a(d) H ˜ y − Sa(d) 2 σN (5.5) ˜ d From (5.5), we recognize that the joint maximum likelihood detector searches for that ˜ hypothesis d which minimizes... the hypotheses a within the summations and, therefore, are identical in the numerator and denominator so that they can be cancelled Furthermore, the u-th contribution in the inner sums of numerator and denominator in (5.16) is constant due to the restrictions of the outer summations over a with au = +1 or au = −1 Hence, we can extract the common factor exp[La (au )] from the ratio and obtain ˆ (5.17a)... choice and for high SNR, the code with M = 256 (SCCW 3) shows the best asymptotical performance Moreover, increasing the interleaver length from Lπ = 600 to Lπ = 6000 leads to improvements of 0.5 dB for SCCW 1, 0 .7 dB for SCCW 2, and 1 dB for SCCW 3 Compared to a single convolutional code with Lc = 7, the SCCWs perform better for medium and high SNR, but not for extremely low SNR However, the low SNR... SSH + 2 σN 2 σA INs −1 (5.36) which can be shown to be equivalent to2 WH MMSE = S S+ H −1 2 σN · SH = R + I 2 Nu σA N0 IN Es u −1 · SH (5. 37) Analyzing the result in (5. 37) , we see that the MMSE detector starts like the decorrelator with a matched filter bank Moreover, WMMSE represents a compromise between matched 2 filter and decorrelator For σN → 0, that is, infinitely large SNR, the identity matrix . R rc c d f CCS 2 7 1/2 133 8 , 171 8 1/32 10 CCS 4 7 1/4 1 17 8 , 1 27 8 , 155 8 , 171 8 1/16 20 CCS 8 7 1/8 1 17 8 , 1 27 8 , 155 8 , 171 8 1/8 40 135 8 , 173 8 , 135 8 , 145 8 CSS 7 1/64 (Frenger. g 2 = 171 8 Walsh, R wh c = 6/64 R rc c = 1/3 SCCC 1 g 1 = 7 8 , g 2 = 5 8 g 1 = 7 8 , g r 2 = 5 8 R rc c = 1/16 SCCC 2 g 1 = 7 8 , g 2 = 5 8 g 1 = 23 8 ,g 2 = 27 8 g r 3 = 35 8 ,g 4 = 37 8 R rc c =. inner code rep. code SCCW 1 g 1 = 7 8 , g 2 = 5 8 Walsh, R wh c = 4/16 R rc c = 1/8 SCCW 2 g 1 = 7 8 , g 2 = 5 8 Walsh, R wh c = 6/64 R rc c = 1/3 SCCW 3 g 1 = 7 8 , g 2 = 5 8 Walsh, R wh c = 8/256