332 CDMA We note that the same formula can easily be obtained in the discrete matched filter model v = RCs + m with v = G † r. In the noise-free channel, x = Cs holds and we simply obtain x by applying R −1 ,thatis, x = R −1 v, which is equivalent to Equation (5.43). Thus, the decorrelating receiver just extends the method of simple matrix inversion to the noisy channel. An estimate for the transmit signal vector s can be obtained by channel inversion as ˆ s = C −1 x = C −1 R −1 v. For user number k, we get the estimate ˆs k = c −1 k  R −1 v  k , (5.44) where the index k indicates the kth element of the vector. Thus, we need only to know the signature waveforms of all the other users. It is not necessary to know their fading amplitudes. The discrete transmission model including the decorrelation receiver is given by x = Cs + ˜ m with ˜ m = R −1 m. The autocorrelation matrix of the noise vector ˜ m = (m 1 , ,m K ) T is given by E  ˜ m ˜ m †  = N 0 R −1 . For user number k, we thus have the transmission model x k = c k s k +˜m k with noise variance given by E  | ˜m k | 2  = N 0  R −1  kk , where ( R ) kk denotes the kth diagonal element of the matrix R −1 . For BPSK transmission, the BER performance of user k for fixed channel amplitudes is then given by P (k) b = 1 2 erfc   | c k | 2  R −1  kk E b N 0  . We now evaluate this BER performance in an alternative way that gives some insight into the geometrical structure. We recall that the receiver calculates the coordinates of the relevant part r ♦ of the receive vector r with respect to the signature base vectors g k . Without losing generality, we consider user number 1. An error occurs if the first coordinate of the transmit vector corresponding to the base vector g 1 has a positive sign, but the first coordinate of the receive vector has a negative sign (or vice versa). We can visualize the CDMA 333 Transmit vector threshold Decision g 1   g 2 Figure 5.35 Distance to the decision threshold for the decorrelator for c 1 = c 2 = 1. geometrical situation as follows. The K − 1-dimensional hyperplane spanned by g 2 , ,g K divides the K-dimensional hyperplane given by the vector space V = span(g 1 , ,g K ) into two half planes. An error occurs if the (relevant) receive vector r ♦ lies in the other half plane of V than the transmit vector. For K = 2andc 1 = c 2 = 1, the situation is depicted in Figure 5.35 We may thus regard the K −1-dimensional hyperplane span(g 2 , ,g K ) as a decision threshold. Let  be the distance between the transmit vector and this threshold. The bit error probability for user 1 is then given by P b = Q   σ  . We note that the coordinates of all users are independent and uniquely defined, and the decision about the transmit symbol s 1 of user 1 does not depend on the values of the other transmit symbols s 2 , ,s K . For the performance analysis, we may thus set s 2 =···= s K = 0. This means that the distance of the transmit vector to the threshold is the same as the distance of c 1 g 1 to the threshold (see Figure 5.35). It is evident from the figure and the Pythagorean theorem that  2 = | c 1 | 2  E b  1 −   Pg 1   2  , where the vector Pg 1 is the orthogonal projection of the base vector g 1 onto the threshold. For two dimensions, we know from elementary vector geometry that Pg 1 = (g † 2 g 1 )g 2 = ρ 12 g 2 . 334 CDMA In that case, the projector P is given by the matrix P = g 2 g † 2 . For the general case of more dimensions, we calculate  by using the orthogonality prin- ciple. The formalism is similar to the derivation of the decorrelating receiver. We define G 1 = [g 2 , ,g K ]. Let k be the vector of coordinates that minimizes the distance between g 1 and the threshold, that is, k = arg min u  g 1 − G 1 u  2 . From the orthogonality condition G † 1 ( g 1 − G 1 k ) = 0, we obtain k = (G † 1 G 1 ) −1 G † 1 g 1 . We thus have  2 = | c 1 | 2  E b  g 1 − G 1 k  2 = | c 1 | 2  E b    g 1 − G 1 (G † 1 G 1 ) −1 G † 1 g 1    2 . We note that (G † 1 G 1 ) −1 G † 1 = G − 1 is the pseudoinverse of G 1 and P = G 1 G − 1 is the orthogonal projector onto the hyperplane span(g 2 , ,g K ). An orthogonal projector P is a matrix with the following properties: it is Hermitian, that is, P = P † and has the property P 2 = P. Since  g 1  2 = 1, we readily get the expression  2 = | c 1 | 2  E b  1 − g † 1 Pg 1  or, equivalently,  2 = | c 1 | 2  E b  1 −   Pg 1   2  , which is just the Pythagorean Theorem. The bit error rate is then given by P b = 1 2 erfc   | c 1 | 2  1 −   Pg 1   2  E b N 0  . (5.45) CDMA 335 As a secondary result, we have thus seen that 1  R −1  11 =  1 −   Pg 1   2  . The quantity   Pg 1   2 = (G † 1 g 1 ) † (G † 1 G 1 ) −1 (G † 1 g 1 ) has the following structure. The matrix R 1 = G † 1 G 1 is just the autocorrelation matrix R with the first row and the first column deleted. The vector v 1 = G † 1 g 1 , is just the column of the K −1 detector outputs of g 1 for all other detectors g 2 , ,g K . From Equation (5.45), we see that the performance compared to ideal single-user BPSK is degraded by the channel attenuation |c 1 | 2 and the geometrical factor  1 −   Pg 1   2  .For Rayleigh fading, we simply average over the fading amplitude c 1 as shown in Subsec- tion 2.4.3 to get P b = 1 2    1 −       1 −   Pg 1   2  E b N 0 1 +  1 −   Pg 1   2  E b N 0    . (5.46) We now evaluate the expression (5.45) with   Pg 1   2 = | ρ 12 | 2 for the two-user performance in the AWGN channel and compare it with the SUMF receiver. We set the fading amplitude c 1 = 1. Figure 5.36 shows the bit error curves for both receivers for equal fading amplitudes c 1 = c 2 = 1 and different values of the correlation coefficient ρ 12 . We see that for these relatively high correlation coefficients, the decorrelator performs several decibels better than the SUMF receiver. The difference becomes smaller for small correlations, but the decorrelator is always better. In fact, one can show that for real-valued coefficients c 1 , c 2 and ρ 12 , the decorrelator is better than the SUMF receiver (see Problem 3). For complex values of the fading amplitudes, it may occasionally happen that     c ∗ 1 c 2 ρ 12    in Equation (5.30) becomes smaller than | c 1 ρ 12 | 2 in Equation (5.45) because of favorable channel phases. Comparing the performance of the decorrelator for ρ 12 = 0.6 with MLSE curve for the same ρ 12 in Figure 5.30, we note that the loss is always less than 1 dB. Figure 5.37 shows the bit error curves for both receivers for a fixed correlation coefficient ρ 12 = 0.6and the different values of the fading amplitudes. For small interferer amplitudes |c 2 ||c 1 |, the performance curves of the SUMF receiver approach the limit of the ideal single-user (SU) BPSK curve. However, for this relatively high correlation coefficient, the degradation becomes severe if the interferer power is of the same order (or even higher) as that of the user under consideration. The curve for the decorrelator does not depend on the amplitude of the interferer. It has a fixed degradation (caused by the geometry) of approximately 2 dB compared to the ideal BPSK. Therefore, for very low values of the interferer amplitude, the SUMF receiver performs better. 336 CDMA 0 2 4 6 8 10 12 14 16 18 10 −4 10 −3 10 −2 10 −1 10 0 E b /N 0 [dB] P b SU BPSK 4/5 4/5 1/2 1/2 3/5 Figure 5.36 Bit error curves for the SUMF receiver (−o−) and the decorrelator (−+−) for equal fading amplitudes c 1 = c 2 = 1andρ 12 = 1/2, 3/5, 4/5. 0 2 4 6 8 10 12 14 16 18 10 −4 10 −3 10 −2 10 −1 10 0 E b /N 0 [dB] P b SU BPSK 1.5 1.0 0.5 0.1 Figure 5.37 Bit error curves for the SUMF receiver (−o−) and the decorrelator (−+−) for ρ 12 = 3/5 and fading amplitudes c 1 = 1andc 2 = 0.1, 0.5, 1, 1.5. CDMA 337 The MMSE receiver If an estimate for the SNR is available, a linear receiver that is better than the decorrelator can be obtained from the minimum mean square error (MMSE) condition. We consider the orthonormal base discrete channel model r = GCs + n We look for a linear estimate ˆ s = (ˆs 1 , ,ˆs K ) T for s = (s 1 , ,s K ) T that minimizes the mean square error E  | ˆs k − s k | 2  for all k. A linear estimator is given by ˆs i = K  k=1 b ik r k or, in matrix notation, by ˆ s = Br with (B) ik = b ik . The method to find the matrix B is quite similar to the derivation of the Wiener estimator in Subsection 4.3.3. We apply the orthogonality theorem of probability theory that says that the MMSE condition is equivalent to the orthogonality condition E  ( ˆs i − s i ) r ∗ k  = 0. In matrix notation, this can be written as E  ( ˆ s − s ) r †  = 0. We insert the linear estimator for ˆ s and get BE  rr †  = E  sr †  . Using the fact that the signal and the noise are uncorrelated and that the symbols are independently modulated, the left hand side can be evaluated as E  rr †  = GCE  ss †  ( GC ) † + E  nn †  = E S GC ( GC ) † + N 0 I. For the right-hand side, we get E  sr †  = E S ( GC ) † . We thus have B  E S GC ( GC ) † + N 0 I  = E S ( GC ) † or B = ( GC ) †  GC ( GC ) † + N 0 E S I  −1 . We note that for a matrix A the relation A †  AA † + tI  −1 =  A † A + tI  −1 A † (5.47) 338 CDMA holds for any real number t (if the inverses mentioned above exist) (see Problem 4). We can thus write B =  ( GC ) † GC + N 0 E S I  −1 ( GC ) † . Using elementary matrix operations, we can write this as B = C −1  G † G + N 0 E S  CC †  −1  −1 G † . We thus have ˆ s = C −1  G † G + N 0 E S  CC †  −1  −1 G † r. (5.48) We note that in the limit N 0 → 0 this approaches the decorrelating receiver. For very noisy channels it approaches the SUMF receiver. The treatment is similar for the discrete matched filter model v = RCs + m. We look for a linear estimator ˆ s = Av. The orthogonality principle in matrix notation reads as E  ( ˆ s − s ) v †  = 0. We insert the linear estimator for ˆ s and get AE  vv †  = E  sv †  . The left-hand side can be evaluated as E  vv †  = RCE  ss †  ( RC ) † + E  mm †  = E S RCC † R † + N 0 R. For the right-hand side, we get E  sv †  We thus have A  E S RCC † R + N 0 R  = E S C † R, where we have used the fact that R is Hermitian. We assume that the matrices R and C are regular, and multiply from the right-hand side by R −1  C †  −1 C −1 and obtain A  E S R + N 0  CC †  −1  = E S C −1 and, finally, A = E S C −1  E S R + N 0  C † C  −1  −1 . The estimate for the transmit vector is then given by ˆ s = C −1  R + N 0 E S  CC †  −1  −1 v. (5.49) CDMA 339 5.3.4 Suboptimal nonlinear receiver structures In this subsection, we present the method of successive interference cancellation (SIC) (see e.g. (Frenger et al. 1999)). There are two variations of that theme. The serial SIC should be preferred if the different users have very different power levels as is the case in the absence of power control. We see that, in this case, the SUMF receiver may totally fail for the users with low signal level. The parallel SIC is the better choice if the different users have approximately the same power levels as it is the case in the presence of power control. For this method, it is necessary that the SUMF receiver gives correct results at least for a significant part of the users. Serial successive interference cancellation The idea is simple: if the reception is corrupted by strong interferers with known signature pulses and known fading amplitudes, one can combat the near–far problem by adjusting the decision threshold to the situation. We illustrate the idea for the case of two users with the same parameters as in Figure 5.27 and Figure 5.29. In those figures, the fading amplitudes are c 1 = 1andc 2 = 2, and the transmit symbol energy has been normalized to E S = 1. The signature vectors are given by g 1 =  1 0  , g 2 = 1 5  3 4  . The situation is depicted in Figure 5.38. In the beginning, we do not have any knowledge about the BPSK transmit symbols s 1 and s 2 . The receive vector is given by r = s 1 g 1 + 2s 2 g 2 + n, where n is the two-dimensional AWGN. Because the signal of user 2 is stronger than that of user 1, the SUMF receiver will work quite well for that user by utilizing Threshold 2 in Figure 5.38 to provide us with a reliable estimate ˆs 2 = sign ( g 2 · r ) of symbol s 2 .Ifs 2 has been correctly decided as ˆs 2 =+1, we may now decide whether s 00 =+g 1 + 2g 2 or s 10 =−g 1 + 2g 2 has been transmitted. These two cases corresponding to Region 1 and Region 2 in Figure 5.38 are separated by Threshold 1a which is a straight half line with the same distance between s 00 and s 10 . The half line ends at Threshold 2. For ˆs 2 =−1, we have to decide whether s 01 =+g 1 − 2g 2 or s 11 =−g 1 − 2g 2 340 CDMA Threshold 2 00 10 0111 c 1 g 1 c 2 g 2 Region 1 Region 2 Region 3 Region 4 Figure 5.38 Two-user BPSK SIC receiver for unequal fading amplitudes. corresponding to Region 3 and Region 4, respectively. These regions are separated by Threshold 1b. Formally, the decision on s 1 is given by ˆs 1 = sign ( g 1 · ( r − 2ˆs 2 g 2 )) . If s 2 has been correctly decided, this corresponds to the maximum likelihood decision for s 1 . However, the estimate ˆs 2 may be wrong. Comparing Figure 5.29 and Figure 5.38, we note that there are two stripe-like shaped regions where the decisions on s 2 differ. However, there are only two small triangular regions where this affects the decision ˆs 1 . Therefore, s 1 may be correctly decided even if the decision on s 2 is erroneous. This correct decision ˆs 1 now provides us with a better estimate for s 2 , which is given by ˆs 2 = sign ( g 2 · ( r − ˆs 1 g 1 )) . Thus, this successive interference cancellation also improves the decisions for the stronger signal. We now formalize the procedure and generalize to K users. For simplicity, we restrict ourselves to BPSK. The generalization on QPSK is straightforward. The receive signal is CDMA 341 then given by r = K  k=1 c k s k g k + n. Without losing generality, we assume that the signals are ordered according to their receive powers, that is, | c 1 | 2 ≤ | c 2 | 2 ≤ | c K | 2 . We start with the estimate for the strongest signal and get ˆs K = sign  c ∗ K g † K r  . We now make decisions for the other K − 1 symbols s K−1 ,s K−2 , ,s 1 according to the descending signal strength order as ˆs k = sign  c ∗ k g † k  r − K  i=k+1 c i ˆs i g i  . After this first iteration, estimates ˆs 1 , ,ˆs K for all K symbols are available. We may now start a second iteration. We replace the old estimate ˆs K with the new one ˆs (new) K = sign  c ∗ k g † k  r − K−1  i=1 c i ˆs i g i  , thereby using the information available about all other symbol decisions obtained in the first iteration. We now make decisions for the other K − 1 signals according to the descending signal strength order as ˆs (new) k = sign  c ∗ k g † k  r − k−1  i=1 c i ˆs i g i − K  i=k+1 c i ˆs (new) i g i  . Note that the estimates ˆs i ,i<k of the last iteration are used together with the already available new estimates ˆs (new) i ,i>k. When all estimates are calculated in descending order, we may start a third iteration, and so forth. The method described above is called serial SIC because the estimates are obtained serially in an order given by the signal strengths. It is especially useful if there are significant differences in the signal strengths. This is the case in the absence of power control. In case of an efficient power control, all signal powers are of the same order, and there is no reason for one of these signals to be privileged. In that case, we should apply the parallel SIC receiver. Parallel successive interference cancellation In a parallel SIC receiver, all iterations are performed in parallel. Again, we consider BPSK transmission and the discrete channel model r = K  k=1 c k s k g k + n. [...]... IEEE 802.11 In 199 7, the Institute of Electrical and Electronics Engineers (IEEE) released the standard IEEE 802.11 for wireless local area network (WLAN) applications Systems according to this standard offer a wireless access to existing local area computer networks via the socalled access points as well as direct wireless interconnections within a small group of computers Within the standard, three... Repeat reQuest (ARQ) scheme, not a forward error correction, is applied Further types of IEEE 802.11 – the types IEEE 802.11a and IEEE 802.11g mentioned in Subsection 4.6.3 – use OFDM to handle data rates of up to 54 Mbit/s More details on the IEEE 802.11 standard can be found, for example, in (O’Hara and Petrick 199 9) CDMA 355 dk = 00 dk = 01 dk = 10 dk = 11 → → → → sk = 0 sk = 1 sk = 2 sk = 3 cs1 =... Overview of mobile communication systems The first releases of these standards were published between 198 9 and 199 4 Since then, several hundred networks according to these standards were established all over the world serving more than 1.3 billion subscribers (June 2004) as illustrated by Figure 5.477 During the same time, standards for cordless telecommunications as the Digital Enhanced Cordless Telecommunications... (DECT) or the Personal Handy-phone System (PHS) evolved The main common features of these systems of the second generation (2G) are • a digital transmission technique • an enhanced network capacity • speech and data services with data rates of about 10 kbit/s • security mechanisms • international roaming capabilities Between 199 5 and 199 8, the main focus for new releases of these standards was on creating... rates and a packet switched mode The increase of the data rate was accomplished by allocating several channels for one connection and by using 7 The data of the Figure have been extracted from the internet pages of the GSM Association (www.gsmworld.com) and of the CDMA Development Group (www.cdg.org) where the interested reader may find the recent statistics 360 CDMA CDMA WCDMA 3rd generation cdma2 000... as Time Division Synchronous CDMA (TD-SCDMA) While the WCDMA and TD -CDMA- based modes use a chip rate of 3.84 Mchip/s and a carrier separation of about 5 MHz, the chip rate for the TD-SCDMA mode is 1.28 Mchip/s at a carrier separation of about 1.6 MHz In the first phase, UMTS networks are operated mainly in the FDD mode UMTS networks are and will be established in Europe and other countries with GSM... frequency bands have been considered within the 3G standardization process One evolutionary path starts from the IS-136 and the GSM/EDGE technology to create the UWC-136 standard (UWC: Universal Wireless Communications) using TDMA as the multiple access scheme The starting point for the CDMA evolution path to 3G is cdmaOne (IS -95 ) using a chip rate of 1.2288 Mchip/s and a carrier spacing of 1.25 MHz... parallel cdmaOne carriers Further details on cdmaOne and cdma2 000 can be found in Subsections 5.5.6 and 5.5.7, respectively Standardization of cdma2 000 is performed in a second 3GPP group called 3GPP2 , which is formed by standardization bodies from the United States, Korea, China and Japan To complete the discussion on the IMT-2000 family, it should be mentioned that an evolution of the DECT standard... systems can be found, for example, in (Holma and Toskala 2001; Konh¨ user et al 2000; Steele et al a 2001) 5.5.4 Wideband CDMA Within this subsection, an overview of the physical layer of the Wideband CDMA system, which is specified as the UTRA FDD mode, is given More details can be found, for CDMA 363 example, in (Holma and Toskala 2001) or within the standardization documents quoted in the following... security and capacity problems, standardization bodies in Europe, America and Japan developed the following standards for public mobile communication networks: • Global System for Mobile Communications (GSM, Europe) • Digital American Mobile Phone System (DAMPS) or Interim Standard 136 (IS-136, USA) • Interim Standard 95 (IS -95 ), nowadays called cdmaOne (USA) • Personal Digital Cellular (PDC, Japan) CDMA . +n. This channel model applies to synchronous wideband MC -CDMA and asynchronous wide- band DS -CDMA as well. However, the interpretation of matrix H is different and both cases have to be discussed separately. The. level of one or more high-power level users. 5.4 Receiver Structures for MC -CDMA and Asynchronous Wideband CDMA Transmission 5.4.1 The RAKE receiver As already pointed out at the beginning of this. + N 0 E S  CC †  −1  −1 v. (5. 49) CDMA 3 39 5.3.4 Suboptimal nonlinear receiver structures In this subsection, we present the method of successive interference cancellation (SIC) (see e.g. (Frenger et al. 199 9)). There