Tài liệu Adaptive WCDMA (P14) docx

14 MMSE multiuser detectors 14.1 MINIMUM MEAN-SQUARE ERROR (MMSE) LINEAR MULTIUSER DETECTION If the amplitude of the user’s k signal in equation (13.7) is A k , then the vector of matched filter outputs y in equation (13.10) can be represented as y = RAb + n (14.1) where A is a diagonal matrix with elements A k A = diag||A k || (14.2) If the multiuser detector transfer function is denoted as M, then the minimum mean-square error (MMSE) detector is defined as min M ∈ R K×K E  ||b − My|| 2  (14.3) One can show that the MMSE linear detector outputs the following decisions [1–3]: ˆ b k = sgn  1 A k ([R + σ 2 A −2 ] −1 y) k  = sgn(([R + σ 2 A −2 ] −1 y) k ) (14.4) The block diagram of a linear MMSE detector is shown in Figure 14.1. Therefore, the MMSE linear detector replaces the transformation R −1 of the decorre- lating detector by [R + σ 2 A −2 ] −1 (14.5) where σ 2 A −2 = diag  σ 2 A 2 1 , , σ 2 A 2 K  (14.6) Adaptive WCDMA: Theory And Practice. Savo G. Glisic Copyright ¶ 2003 John Wiley & Sons, Ltd. ISBN: 0-470-84825-1 492 MMSE MULTIUSER DETECTORS Sync K Sync 2 Sync 1 Matched filter User 2 Matched filter User K y ( t ) Matched filter User 1 [R + σ 2 A −2 ] −1 , b 1 [ i ] ˆ b 2 [ i ] ˆ b K [ i ] ˆ Figure 14.1 MMSE linear detector for a synchronous channel. As an illustration for the two users case we have [R + σ 2 A −2 ] −1 =  1 + σ 2 A 2 1  1 + σ 2 A 2 2  − ρ 2  −1      1 + σ 2 A 2 2 −ρ −ρ 1 + σ 2 A 2 1      (14.7) and the detector is shown in Figure 14.2. y ( t ) + + − − y 1 y 2 1+ σ 2 A 2 2 1+ σ 2 A 1 2 S 2 ( t ) T ∫ 0 T ∫ 0 T ∫ 0 s 1 ( t ) r b 1 ˆ b 2 ˆ Figure 14.2 MMSE linear receiver for two synchronous users. MINIMUM MEAN-SQUARE ERROR (MMSE) LINEAR MULTIUSER DETECTION 493 Single-user matched filter Gaussian approximation Single-user matched filter exact MMSE exact & approx. Signal-to-noise ratio (dB) 10 12 16 18 22 10 −5.5 10 −5 10 −4.5 10 −4 10 −3.5 10 −3 10 −2.5 Probability of error 10 −2 14 20 Figure 14.3 Bit-error-rate with eight equal-power users and identical cross-correlations ρ kl = 0.1. −5 010 a b c d e −10 5 Near−far ratio A 2 / A 1 (dB) 10 −3 10 −2 10 −1 Bit error rate Figure 14.4 Bit-error-rate with two users and cross-correlation ρ = 0.8: a – single-user matched filter, b – decorrelator, c – MMSE, d – minimum (upper bound), e – minimum (lower bound). 494 MMSE MULTIUSER DETECTORS In the asynchronous case, similar to the solution in Section 13.3 of Chapter 13, the MMSE linear detector is a K-input, K-output, linear, time-invariant filter with trans- fer function [R T [1]z + R[0] + σ 2 A −2 + R[1]z −1 ] −1 (14.8) Performance results are illustrated in Figures 14.3. and 14.4. As expected, in Figure 14.3, the MMSE detector demonstrates better performance than the conventional detector deno- ted as a single-user matched filter receiver (MFR). In Figure 14.4 bit error rate (BER) is presented versus the near–far ratio for different detectors. One can see that MMSE shows better performance than decorrelator. In the figure signal-to-noise ratio (SNR) of the desired user is equal to 10 dB. 14.2 SYSTEM MODEL IN MULTIPATH FADING CHANNEL In this section the channel impulse response and the received signal will be presented as c k (t) = L k  l=1 c (n) k, δ(t − τ k, ) (14.9) r(t) = N b −1  n=0 K  k=1 L  l=1 A k b (n) k c (n) k,l s k (t − nT − τ k,l ) + n(t) (14.10) The received signal is time-discretized, by antialias filtering and sampling r(t) at the rate 1/T s = S/T c = SG/T ,whereS is the number of samples per chip and G = T/T C is the processing gain. The received discrete-time signal over a data block of N b symbols is r = SCAb + n ∈ C SGN b (14.11) where r = [r T (0) , ,r T (N b −1) ] T ∈ C SGN b (14.12) is the input sample vector with r T (n) ={r[T s (nSG + 1)], ,r[T s (n + 1)SG]}∈C SG (14.13) SYSTEM MODEL IN MULTIPATH FADING CHANNEL 495 S = [S (0) , S (1) , ,S (N b −1) ] ∈ R SGN b ×KLN b =              S (0) (0) 0 ··· 0 . . . S (1) (0) . . . . . . S (0) (D) . . . . . . 0 0S (1) (D) . . . S (N b −1) (0) . . . . . . . . . . . . 0 ··· 0S (N b −1) (D)              (14.14) is the sampled spreading sequence matrix, D = (T + T m )/T . In a single-path channel, D = 1 due to the asynchronity of users. In multipath channels, D ≥ 2 due to the multi- path spread. The code matrix is defined with several components (S (n) (0), ,S (n) (D)) for each symbol interval to simplify the presentation of the cross-correlation matrix com- ponents. T m is the maximum delay spread, S (n) = [s (n) 1,1 , ,s (n) 1,L , ,s (n) K,L ] ∈ R SGN b ×KL (14.15) where s (n) k,l =                    0 T SGN b ×1 n = 0 τ k,l = 0 [[s k [T s (SG − τ k,l + 1)], ,s k (T s SG)] T , 0 T (SGN b −τ k,l )×1 ] T n = 0 τ k,l > 0 [0 T [(n−1)SG+τ k,l ]×1 , s T k , 0 T [SG(N b −n)−τ k,l ]×1 ] T 0 <n<N b − 1 (0 [SG(N b −1)+τ k,l ]×1 , {s k (T s ), ,s k [T s (SG − τ k,l )]}) T n = N b − 1 (14.16) where τ k,l is the time-discretized delay in sample intervals and s k = [s k (T s ), ,s k (T s SG)] T ∈ R SG (14.17) is the sampled signature sequence of the kth user. By analogy with equation (13.59) C = diag  C (0) , ,C (N b −1)  ∈ C KLN b ×KN b (14.18) is the channel coefficient matrix with C (n) = diag  c (n) 1 , ,c (n) K  ∈ C KL×K (14.19) and c (n) k = [c (n) k,1 , ,c (n) k,L ] T ∈ C L (14.20) 496 MMSE MULTIUSER DETECTORS Equation (14.2) now becomes A = diag[A (0) , ,A (N b −1) ] ∈ R KN b ×KN b (14.21) the matrix of total received average amplitudes with A (n) = diag[A 1 , ,A K ] ∈ R K×K (14.22) Bit vector from equation (13.56) becomes b = [b T (0) , ,b T (N b −1) ] T ∈ℵ KN b (14.23) with the modulation symbol alphabet ℵ [with binary phase shift keying (BPSK) ℵ= {−1, 1}]and b (n) = [b (n) 1 , ,b (n) K ] ∈ℵ K (14.24) and n ∈ C SGN b is the channel noise vector. It is assumed that the data bits are independent identically distributed random variables independent from the channel coefficients and the noise process. The cross-correlation matrix equation (13.70) for the spreading sequences can be formed as R = S T S ∈ R KLN b ×KLN b =              R (0,0) ··· R (0,D) 0 KL ··· 0 KL . . . . . . . . . . . . . . . R (D,0) . . . . . . . . . 0 KL 0 KL . . . . . . . . . R (N b −D,N b −1) . . . . . . . . . . . . . . . 0 KL ··· 0 KL ··· R (N b −1,N b −1)              (14.25) where equation (13.20) now becomes R (n,n−j) = D−j  i=0 S T (n) (i)S (n−j) (i + j),j ∈{0, ,D} (14.26) and R (n−j,n) = R T (n,n−j) . The elements of the correlation matrix can be written as R (n,n  ) =     R (n,n  ) 1,1 ··· R (n,n  ) 1,K . . . . . . . . . R (n,n  ) K,1 R (n,n  ) K,K     ∈ R KL×KL (14.27) MMSE DETECTOR STRUCTURES 497 and R (n,n  ) k,k  =     R (n,n  ) k1,k  1 ··· R (n,n  ) k1,k  L . . . . . . . . . R (n,n  ) kL,k  1 ··· R (n,n  ) kL,k  L     ∈ R L×L (14.28) where equation (13.71) now becomes R (n,n  ) kl,k  l  = SG−1+τ k,l  j=τ k,l s k [T s (j − τ k,l )]s k  {T s [j − τ k  l  + (n  − n)SG]}=s T (n) k,l s (n  ) k  ,l  (14.29) and represents the correlation between users k and k  , lth and l  th paths, between their nth and n  th symbol intervals. 14.3 MMSE DETECTOR STRUCTURES One of the conclusions in Chapter 13 was that noise enhancement in linear Multi-user detection (MUD) causes system performance degradation for large product KL.Inthis section we consider the possibility of reducing the site of the matrix to be inverted by using multipath combining prior to MUD. The structure is called the postcombining detector and the basic block diagram of the receiver is shown in Figure 14.5 [4]. The starting point in the derivation of the receiver structure is the cost function E{|b − ˆ b| 2 } Matched filter 1, 1 Matched filter 1, L Matched filter K , L 1/ T s K × K Multiuser detection Matched filter K , 1 Multipath combining Multipath combining r ( n ) Figure 14.5 Postcombining interference suppression receiver. 498 MMSE MULTIUSER DETECTORS where ˆ b = L H [post] r (14.30) The detector linear transform matrix is given as L [post] = SCA(AC H RCA + σ 2 I) −1 ∈ C SGN b ×KN b (14.31) This result is obtained by minimizing the cost function, and derivation details may be found in any standard textbook on signal processing. Here, R = S T S is the signature sequence cross-correlation matrix defined by equation (14.25). The output of the post- combining LMMSE receiver is y [post] = (AC H RCA + σ 2 I) −1 (SCA) H r ∈ C K (14.32) where (SCA) H r is the multipath [maximum ratio (MR)] combined matched filter bank output. For nonfading additive white Gaussian noise (AWGN), L [post] = S(R + σ 2 (A H A) −1 ) −1 (14.33) The postcombining LMMSE receiver in fading channels depends on the channel com- plex coefficients of all users and paths. If the channel is changing rapidly, the optimal LMMSE receiver changes continuously. The adaptive versions of the LMMSE receivers have increasing convergence problems as the fading rate increases. The dependence on the fading channel state can be removed by applying a precombining interference suppression type of receiver. The receiver block diagram in this case is shown in Figure 14.6 [4]. The transfer function of the detector is obtained by minimizing each element of the cost function E{|h − ˆ h| 2 } (14.34) 1/ T s Multipath combining Multipath combining KL × KL Multiuser detection r ( n ) MF 1, L MF K ,1 MF K , L MF 1,1 Figure 14.6 Precombining interference suppression receiver. MMSE DETECTOR STRUCTURES 499 where h = CAb (14.35) and ˆ h = L T [pre] r is the estimate (14.36) The solution of this minimization is [4] L [pre] = S(R + σ 2 R −1 h ) −1 ∈ R SGN b ×KLN b (14.37) R h = diag  A 2 1 R c 1 , ,A 2 K R c k  ∈ R KLN b ×KLN b (14.38) R c k = diag  E  |c k,1 | 2  , ,E  |c k,L | 2  ∈ R L×L (14.39) y [pre] = (R + σ 2 R −1 h ) −1 S T r ∈ C KL (14.40) The two detectors are compared in Figure 14.7. The postcombining scheme performs better. BEP 0 5 10 15 20 25 30 Number of users 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 10 −9 10 −8 10 −7 10 −6 Precomb. LMMSE Postcomb. LMMSE 0 dB 5 dB 10 dB 15 dB Figure 14.7 Bit error probabilities as a function of the number of users for the postcombining and precombining LMMSE detectors in an asynchronous two-path fixed channel with different SNRs, and bit rate 16 kb s −1 , Gold code of length 31, td/T = 4.63 × 10 −3 , maximum delay spread 10 chips [5]. Reproduced from Latva-aho, M. (1998) Advanced Receivers for Wideband CDMA Systems. Ph.D. Thesis, University of Oulu, Oulu, by permission of IEEE. 500 MMSE MULTIUSER DETECTORS RAKE LMMSE-RAKE Two-path fading channel SNR = 20 dB 2 users, the other one 20 dB stronger 10 −4 10 −3 10 −2 10 −1 10 0 BEP 10 −5 4 8 16 322 Spreading factor ( G ) Figure 14.8 Bit error probabilities as a function of the near–far ratio for the conventional RAKE receiver and the precombining LMMSE (LMMSE-RAKE) receiver with a different spreading factor (G) in a two-path Rayleigh fading channel with maximum delay spreads of 2 µs for G = 4, and 7 µs for other spreading factors. The average signal-to-noise ratio is 20 dB, the data modulation is BPSK, the number of users is 2, the other user has 20-dB higher power. Data rates vary from 128 kb s −1 to 2.048 Mbit s −1 ; no channel coding is assumed [5]. Reproduced from Latva-aho, M. (1998) Advanced Receivers for Wideband CDMA Systems. Ph.D. Thesis, University of Oulu, Oulu, by permission of IEEE. The illustration of LMMSE-RAKE receiver performance in near–far environment is shown in Figure 14.8 [5]. Considerable improvement compared to conventional RAKE is evident. 14.4 SPATIAL PROCESSING When combined with multiple receiver antennas, the receiver structures may have one of the forms shown in Figure 14.9 [4, 6–8]. The channel impulse response for the kth user’s ith sensor can be now written as c k,i (t) = L k  l=1 c (n) k,l e j2πλ −1 e(φ k,l ),ε i  δ[t − (τ k,l,i )] (14.41) [...]... by permission of IEEE Table 14.1 The BERs of different blind adaptive receivers at an SNR of 20 dB in a two-path Rayleigh fading channel at vehicle speeds of 40 km h−1 The acronyms used are adaptive LMMSE-RAKE (LR), adaptive MOE (MOE), Griffiths’ algorithm (GRA), constant modulus algorithm with average channel tap powers (CMA2), constrained adaptive LMMSE-RAKE (C-LR), constrained constant modulus algorithm... in Figure 14.10, [9–17] (14.45) 504 MMSE MULTIUSER DETECTORS Channel estimator (n) ˆ C k, l * Adaptive FIR wkl(n) (n) yk, l (n) + d k, l − (n) LMS e k, l (n) r ˆ bk (n) Σ Channel estimator (n) ˆ C k, L * Adaptive FIR wkl(n) (n) yk, L (n) + d k, L − (n) LMS Figure 14.10 e k, L General block diagram of the adaptive LMMSE-RAKE receiver By using notation r(n) = [rT(n−D) , , rT(n) , , rT(n+D) ]T ∈... k,l k,l We decompose equation (14.53) into adaptive and fixed components as w(n) = sk,l + x(n) ∈ CMSG k,l k,l where x(n) is the adaptive filter component and k,l T T T sk,l = [0T (DSG+τk,l )×1 , sk , 0(DSG−τk,l )×1 ] (14.53) 506 MMSE MULTIUSER DETECTORS 1 Pilot MF 2N + 1 (n − N ) ˆ ck, l ˆ (n − N ) bk Σ ∗ To combiner 1/T r MF sk, l (n) (n − N ) yk, l N·T 1/T Adaptive FIR xk,l(n) + (n − N ) dk, l − (n... 0.03 0.02 0.01 0 m = 1/10 K = 10 A – constant modulus algorithm B – Griffiths’ algorithm C – blind adaptive MOE D – adaptive LMMSE-RAKE 100 200 300 400 D 500 600 700 Number of iterations (symbol intervals) Figure 14.13 Excess mean squared error as a function of the number of iterations for different blind adaptive receivers in a two-path fading channel with vehicle speeds of 40 km h−1 , the number of... 10 A – constant modulus algorithm B – Griffiths’ algorithm C – blind adaptive MOE D – adaptive LMMSE-RAKE Excess MSE 0.07 0.06 0.05 0.04 A 0.03 0.02 B 0.01 C 0 D 1000 2000 3000 4000 5000 6000 Number of iterations (symbol intervals) Figure 14.14 Excess mean squared error as a function of the number of iterations for different blind adaptive receivers in a two-path fading channel with vehicle speeds of... 0.03 0.02 0.01 0 m = 1/10 K = 20 A – constant modulus algorithm B – Griffiths’ algorithm C – blind adaptive MOE D – adaptive LMMSE-RAKE 100 200 300 400 500 Number of iterations (symbol intervals) 600 Figure 14.15 Excess mean squared error as a function of the number of iterations for different blind adaptive receivers in a two-path fading channel with vehicle speeds of 40 km h−1 , the number of active... = 1/100 K = 20 A – constant modulus algorithm B – Griffiths’ algorithm C – blind adaptive MOE D – adaptive LMMSE-RAKE 0.04 0.03 A 0.02 B C 0.01 0 D 1000 2000 3000 4000 5000 Number of iterations (symbol intervals) 6000 Figure 14.16 Excess mean squared error as a function of the number of iterations for different blind adaptive receivers in a two-path fading channel with vehicle speeds of 40 km h−1 ,... L[TMS] = SCA(ACH RCA + σ 2 I)−1 14.5 SINGLE-USER LMMSE RECEIVERS FOR FREQUENCY-SELECTIVE FADING CHANNELS 14.5.1 Adaptive precombining LMMSE receivers ˆ In this case, Mean-Square Error (MSE) criterion E{|h − h|2 } requires that the reference signal h = CAb is available in adaptive implementations For adaptive single-user receivers, the optimization criterion is presented for each path separately, that is,... 205–211 9 Rapajic, P B and Vucetic, B S (1995) Linear adaptive transmitter-receiver structures for asynchronous SCMA systems Eur Trans Telecommun., 6(1), 21–27 10 Miller, S L (1995) An adaptive direct-sequence code-division multiple-access receiver for multiuser interference rejection IEEE Trans Commun., 43, 1746–1755 11 Rapajic, P B and Vucetic, B S (1994) Adaptive receiver structures for asynchronous CDMA... (1996) Orthogonalization based adaptive interference suppression for direct-sequence code-division multiple-access systems IEEE Trans Commun., 44(9), 1082–1085 13 Miller, S L (1996) Training analysis of adaptive interference suppression for direct-sequence code-division multiple-access systems IEEE Trans Commun., 44(4), 488–495 14 Latva-aho, M and Juntti, M (1997) Modified adaptive LMMSE receiver for DS-CDMA . σ 2 A −2 ] −1 (14.5) where σ 2 A −2 = diag  σ 2 A 2 1 , , σ 2 A 2 K  (14.6) Adaptive WCDMA: Theory And Practice. Savo G. Glisic Copyright ¶ 2003 John Wiley. requires that the refer- ence signal h = CAb is available in adaptive implementations. For adaptive single-user receivers, the optimization criterion is