CDMA multiuser communications - Semantic Scholar
Trellis-Coded DS/CDMA Multiuser Communications Huang Lee and Kwang-Cheng Chen Graduate Institute of Communication Engineering National Taiwan University, Taipei, Taiwan, R.O.C. [email protected] [email protected] Abstract— We investigate trellis coding with multiuser detection (MUD) applied in direct-sequence code-division multipleaccess (DS/CDMA) communications. A generalized model called multi-sequence model is applied to systematically design the transceivers. In multi-sequence model, each user can be assigned more than one spreading sequence and the coding, modulation, and spreading sequences are jointly considered based on the extended Ungerboeck’s set partitioning principle to maximize free distance. We derive the optimum detector under the maximum likelihood criterion. To measure the influence on the bit-errorrate due to coding and multiple-access interference, asymptotic multiuser coding gain (AMCG) is defined to provide lower and upper bounds for transceivers. The numerical experiments justify the precision of AMCG to evaluate system performance. Verified by numerical analysis, a transmitter designed by multi-sequence model can achieve significant coding gain over uncoded systems while applying optimum detector, and the performance of coded systems is severely degraded without MUD. Therefore, these facts not only demonstrate the value to jointly design coding, modulation, and spreading sequences in trellis-coded DS/CDMA communications but also emphasis the needs of MUD to optimize the system performance.
I. I NTRODUCTION Channel coding is normally employed in communications to improve performance. In particular, it was emphasized by Viterbi [1] [2] that channel coding increases bandwidth efficiency in direct-sequence code-division multiple-access (DS/CDMA) systems. In addition, trellis-coded modulation (TCM) [3] is an efficient modulation scheme over additive white Gaussian noise (AWGN) channels, and has been applied to many systems, where significant coding gain is obtained without sacrificing data rate or bandwidth efficiency. Some investigations have been done on coded DS/CDMA systems. Boudreau and Falconer [4] indicated that convolutionally coded DS/CDMA (CC-DS/CDMA) provides superior performance to trellis-coded DS/CDMA (TC-DS/CDMA). Woerner and Stark [5] introduced bi-orthogonal signal set (BSS) to increase normalized minimum Euclidean distance (free distance). Choe and Georghiades [6] introduced orthogonal plane sequence modulation (OPSM), which improves power and bandwidth efficiency. Nevertheless, aforementioned pioneer studies ignored the presence of multiple-access interference (MAI), which degrades the bit-error-rate (BER) performance. Fawer and Aazhang [7] applied multiuser detection (MUD) [8] to improve performance based on the model in [5]. Giallorenzi and Wilson [9] [10] derived optimum and sub-
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optimum MUD for CC-DS/CDMA communications. In [5] [6], by ignoring MAI, multiple spreading sequences are assigned to each user to improve performance. However, employing multiple spreading sequences also decreases the number of maximal allowable users with fixed spreading gain and may introduce larger MAI among users. On the other hand, employing TCM in systems can alleviate above problems at the price of decreased free distance. Thus, the design of optimal transmitters should consider the trade-off between TCM and the number of spreading sequences. To achieve above goal, we introduce a model that combines advantages of TCM and multiple spreading sequences and jointly design coding, modulation, and spreading sequences to maximize the free distance. In our model, multiple spreading sequences are assigned to each user to increase free distance. Moreover, the signal constellation can be expanded as TCM to balance the number of necessary spreading sequences. Because the design depends on the structure of encoders, the number of active users, and spreading sequences, the proposed model is very much desired to be generalized to adapt different conditions. In addition to transmitter design, we further propose the optimum detector that performs the functions of MUD and decoding jointly and is optimal in maximum likelihood (ML) sense, and demonstrate that the proposed detector is near-far resistant by both mathematical and numerical analyses. The asymptotic multiuser coding gain (AMCG) is defined as a performance measure to evaluate the proposed transceivers. AMCG can be regarded as generalized asymptotic multiuser efficiency (AME) [8] and used as a criterion for designing transceivers, since the ordering of AME of two schemes determines the ordering of corresponding BER for sufficiently high signal-to-noise ratio (SNR) regime [8]. The AMCG for CC-DS/CDMA is formulated in [9]. In this paper, we derived the AMCG and its upper and lower bounds of optimum detector for the generalized coded DS/CDMA model. ∗ T H Notation: Superscripts (·) , (·) , and (·) denote complex conjugate, transpose, and Hermitian adjoint operators, respectively. A circumflex over any variable represents an estimate of that variable, such as x for an estimate of x. The subspace spanned by columns of a matrix W is represented with angle brackets around the symbol for the matrix, such as W. Given a set Ω, |Ω| denotes the number of elements in Ω.
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Info. Source
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Mapping and Spreading K Channel gain K
Fig. 1.
Unlike conventional systems that the performance depends on the distance among symbols, in DS/CDMA the performance relies on the distance among transmitted sequences.
System block diagram of a coded DS/CDMA transceiver.
II. S YSTEM M ODEL The architecture of a coded DS/CDMA transceiver is illustrated in Fig. 1. Considering a synchronous DS/CDMA system with K users, we assume that (a) the encoders are identical with α input information bits, β output coded bits, and constraint length ν; (b) received amplitude and phase are perfectly estimated; (c) users send equal number of symbols. A. Multi-sequence Model of coded DS/CDMA systems We introduce multi-sequence model for transmission. For user k = 1, · · · , K, each user is assigned D orthonormal spreading sequences (OSS), and the sequences are represented T by Nc ×1 vectors sk,d √1N ck,d,1 · · · ck,d,Nc , d = 1, c 2 · · · , D where ck,d,n ∈ C are chip codes with |ck,d,n | = 1, and Nc > DK is the length of OSS. Moreover, each OSS is multiplied by a coefficient xk,d ∈ R or C. For user k, the transmitted sequence is given by D xk,d sk,d . (1) sk = d=1
In another point of view, the OSS of user k are used to compose a signal space Wk where Wk sk,1 · · · sk,D N ×D . Consequently, (1) is equivalent to c the procedure of linear combination, and any signal point sk for transmission in the signal space Wk can be arbitrarily designed by choosing different xk,1 , · · · , xk,D . Because the encoder outputs β bits each time, we need to choose 2β signal points that compose a signal constellation ΩW,k ∈ Wk with |ΩW,k | = 2β from Wk for transmission. Two important measures should be noted in the multisequence model while designing transmitters. One is the dimension of Wk , and the other is the distance among signal points. For user k who has D OSS, the dimension of Wk is D. However, because the coefficients xk,d can be complexvalued, we define the effective dimension (ED) of Wk as D if xk,d ∈ R for all d ED 2D otherwise. Consider the signal space composed of all user’s spreading sequences with length Nc . The maximum ED of the signal space is 2Nc . In DS/CDMA, multidimensional signal space can be constructed by multi-sequence model. Applying multidimensional constellation can increase free distance [11] [12] so that we can derive additional coding gain by increasing ED. The other measure is the distance among signal points 2
d2 (sk , sk ) sk − sk .
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In [13], we showed that DMC-M PSK [13], TC-DS/CDMA [4], BSS [5] and OPSM [6] schemes are special cases of multi-sequence model. Here, we show that the DMC2PSK scheme constructed by defining xk,d ∈ {M -PSK} and −1 D = β (log2 M ) has the same distance property as CCDS/CDMA. Consider a encoder with β output coded bits. Each bit is modulated by a spreading sequence with length shortened to Nc /β, so the symbol duration is divided into β slots. The signal constellation of CC-DS/CDMA then can be regarded as a β-dimension hypercube constellation. Since we can construct a β-dimension hypercube constellation by βMC-2PSK, the signal constellations of βMC-2PSK and CCDS/CDMA are both hypercube, though their dimension is expanded by different approaches. Both schemes therefore have equal distance property. Compared with TC-DS/CDMA, hypercube signal constellations usually achieve larger free distance under the same bandwidth efficiency. This explains the results in [4]. Multi-sequence model provides the spreading sequences geometrical meaning and combines the modulation and spreading technique. It provides two approaches to expand signal constellations: increasing D and applying higher level modulation. The price of high-level modulation is transmission power, however, it is a considerable method once Nc is limited. On the other hand, when Nc is large, more OSS can be assigned to each user to increase ED. C. Coded DS/CDMA Transmitter At time index i, user k’s source generates α information bits α×1 which is denoted by an α × 1 vector bk [i] ∈ {+1, −1} . T T T We define the vectorbk bk [−I] · · · bk [I] ; the matrix B b1 · · · bK α(2I+1)×K ; and the set ΩB contains all possible matrices B. The user k’s ith convolutionally coded symbol uk [i] generated by passing α bits through the encoder is given by (3) uk [i] f (bk [i] , ψk [i]) ∈ 1, · · · , 2β and ψk [i + 1] g (bk [i] , ψk [i]). Function f describes that uk [i] depends on the corresponding bits bk [i] and encoder state ψk [i]. Function g describes state evolution. By mapping uk [i] into signal constellation ΩW,k that consists 2β distinct spreading sequences, we get user k’s ith transmitted waveform sk (t; uk [i])
D
xk,d [i] sk,d (t) m (uk [i]) ∈ ΩW,k (4)
d=1
Nc where sk,d (t) √1N n=1 ck,d,n Π(t − nTc ); m(·) is the c mapping function; xk,d [i] are coefficients transmitted at time i and depend on uk [i]; Π(·) is a rectangular function of unit amplitude on the interval [0, Tc ); and Tc is chip duration.
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The mapping (4) between coded symbols and transmitted sequences is a key to combine the coding and spreading technique. For each user, since the performance is dominated by free distance, while applying Viterbi algorithm, coding and mapping should be jointly designed such that the distinct paths through the trellis are separated by larger distance defined in (2). The Ungerboeck’s set partitioning principle is generalized in the multi-sequence model to maximize free distance. To apply Ungerboeck’s principle, we design the signal constellation ΩW,k that is geometrically uniform [14], and use two-way geometrically uniform partitioning over ΩW,k . If partitions admit binary isometric labelings, coded symbols can be one-to-one mapped to transmitted signals by Ungerboeck’s principle [14] such that the distance among distinct paths through the trellis are maximized. Proposition 1 in Appendix shows that the signal constellation of DMC-M PSK is geometrically uniform and geometrically uniform partitions admit binary isometric labelings. III. O PTIMUM M ULTIUSER D ETECTOR Assuming an AWGN channel, the received signal that is the sum of all K users’ signals is K I r(t) ak sk (t − iTs ; uk [i]) + nw (t) (5) k=1
i=−I
where |ak | and ak are the received amplitude and phase of user k; Ts = Nc Tc is the symbol duration; and nw (t) is circularly symmetric AWGN with power spectral density σ 2 /2 = N0 /4 for real and imaginary parts. After chip-matched filtering, which preserves sufficient statistic, the ML detector maximizes the function I
H H H 2y [i] Ax [i] − x [i] A RAx [i] Λ(B) = Re i=−I
where S s1,1 · · · s1,D · · · sK,D N ×KD ; x [i] c T x1,1 [i] · · · x1,D [i] · · · xK,D [i] is a KD × 1 vecT tor; y [i] y1,1 [i] · · · y1,D [i] · · · yK,D [i] is a
∞ ∗ KD × 1 vector; yk,d [i] r (t)s (t − iT )dt; k,d s −∞ H
R S S is the cross-correlation matrix; and A diag {a1 , · · · , a2 , · · · , aK } is a KD × KD diagonal matrix. The optimum MUD on information symbols is applied by maximizing Λ(B). The desired information is estimated by arg max Λ(B). B B∈ΩB
(6)
The maximization can be realized by Viterbi algorithm, and the metric of Viterbi algorithm is given by M [i − 1] + λ (B), i = −I, · · · , I M [i] (7) 0, i < −I λ (B) Re 2y H [i] Ax [i] − xH [i] AH RAx [i] . (8) The trellis at receiver is the combination of all users’ trellis structures, and is extremely complicated with totally 2Kν states while each state has 2Kα emanating and remerging branches. The complexity of optimum detector grows exponentially with the number of active users K.
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IV. S YSTEM A NALYSIS A. Error Probability [i] We derive Pk Pr bk [i] = bk [i] , which is the ith symbol error probability of user k. At the receiver, the optimum detector makes decisions among all possible sequences, and any sequence can be one-to-one mapped to an unique matrix X x [−I] · · · x [I] KD×(2I+1) . For particular user k at time index i, the set of errors is given [i] by Ek (X) {E| (X − E) ∈ X and ek [i] = 0} where X k [i], and is the set of all possible X, ek [i] xk [i] − x T xk [i] xk,1 [i] · · · xk,D [i] is a D × 1 vector. = An error event denoted by X → X|X means that X X − E is preferred to X by the receiver when X is transmitted. After some calculations with (5), the pairwise probability of a detection error is given by I H [i ] AH RAe [i ] e i =−I . Pr X → X|X = Q σ The error probability is derived by averaging over all possible sequences and errors, and is bounded by the union bound [i] Pk ≤ Pr [X] · Pr X → X|X . (9) X∈X E∈E [i] (X) k
B. Minimum Distance As σ approaching zero, the minimum distance among sequences dominates the asymptotic behavior of the performance. For particular user k, the set of errors is defined as Ek (X) {E| (X − E) ∈ X and ek [i] = 0 for some i}. Assume that I is large enough such that the minimum distance df ree,k is independent of i, and is given by I d2f ree,k min min eH [i ] AH RAe [i ] . (10) X∈X E∈Ek (X)
i =−I
C. Asymptotic Multiuser Coding Gain of Optimum Detector By (10), we define AMCG of user k as I H H min min i =−I e [i ] A RAe [i ] /P X∈X E∈Ek (X) ηk (11) 2 |ak | · d2min,k /P where P and P are the average energies spent to transmit with coded and uncoded systems, respectively, and dmin,k is the minimum distance of uncoded DS/CDMA systems. AMCG is a way to quantify the performance gain and loss of coded DS/CDMA in high SNR regime relative to an uncoded single user system. When there is only one user or the signature waveforms among users are orthogonal, AMCG is equivalent to asymptotic coding gain (ACG) of user k: I H min min i =−I e [i ] e [i ] /P X∈X E∈Ek (X) ≥ 1. (12) ACGk d2min,d /P While users do not employ coding, AMCG is equal to AME. The near-far resistance of coded DS/CDMA can be defined as minimizing (11) with respect to ak : η k inf ηk . ak >0; k =k
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0
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where (14) is achieved by εk = λM (·) denotes the largest eigenvalue of the matrix, and (·)(p,q) denotes the pth row and qth column entry of the matrix. Because R is positive definite, lower bound (15) is always larger than zero, so the optimum receiver is near-far resistant. V. S IMULATION R ESULTS To compare systems fairly, we have to maintain the bandwidth, coding rate, and coding complexity identical in a
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Fig. 2. BER versus SNR of optimum and conventional detectors in five-user systems with equal received energy. Nc = 32. (a) A 4-state rate 1/2 code with generator polynomials 7, 5 is employed. (b) A 4-state rate 2/3 code with generator polynomials 10, 07, 05 is employed. 0
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= ACGk · γk (13) T T where ek ak ek [−I] · · · ak eTk [I] is a D(2I + 1)×1 T vector; εk eT1 · · · eTk−1 eTk+1 · · · eTK is a (K − 1)D(2I + 1)×1 vector; Ξk is a (K − 1)D(2I + 1)×(K − 1)D(2I+1) matrix by striking out all rows and columns related ξk is a (K −1)D(2I +1)×D(2I +1) matrix that to user k in R; is comprised of columns related to user k with the elements and R diag{R, R, · · · , R} is a of user k removed in R; KD(2I + 1)×KD(2I + 1) matrix. The γk in (13) is the ratio that represents MAI degrading performance. If γk = 1, the high SNR performance suffers no degradation due to other active users. Otherwise, γk < 1 means that performance is degraded by MAI. Furthermore, the ACGk in (13) shows that we can employ coding to improve performance. Proposition 2 in Appendix proves that ηk ≤ ACGk . It implies that the optimum detector may remove MAI completely. Once the equality is achieved, further implements of spreading sequences yield negligible gains in BER. We derive the lower bound of η k (ηk ) and prove it always larger than zero (near-far resistant). By (13), we obtain H I ξkH ek H ek εk ξk Ξk εk η k ≥ inf min ACGk H ak >0X∈X ; E∈Ek (X) ek ek H H −1 ek ξk Ξk ξk ek ≥ inf min ACGk 1 − ak >0X∈X ; E∈Ek (X) eH k k e (14) ACGk ≥ (R−1 )(kD−D+1,kD−D+1) ··· (R−1 )(kD−D+1,kD)
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The AMCG, which is calculated by searching over all possible X, can be used to evaluate transceivers. In order to give some interpretations about AMCG and derive its upper and lower bounds, we rewrite (11) in another form: I H H min min i =−I e [i ] A RAe [i ] X∈X E∈Ek (X) ηk = ACGk I 2 H |ak | · min min i =−I e [i ] e [i ] X∈X E∈Ek (X) H I ξkH ek H min min ek εk ξk Ξk εk X∈X E∈Ek (X) = ACGk H ek min min ek εH k εk X∈X E∈Ek (X)
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Fig. 3. BER versus SNR of optimum detectors for user 1 in two-user systems with near-far effect. Nc = 16. (a) An 8-state rate 2/3 code with generator polynomials 15, 23, 02 is employed. (b) An 8-state rate 3/4 code with generator polynomials 40, 15, 23, 02 is employed.
simulation. All simulations are in multiple-access channels. The spreading sequences are composed of complex-valued Gold code combined Walsh code. The solid and dashed lines in figures represent coded and uncoded systems, respectively. We show that the BER performance is improved significantly by coding while applying the optimum detector. The simulation results of employing different encoders are shown in Fig. 2. For the optimum detector, the coding gain of the codes that are considered ranges from 3 to 4 dB over the corresponding uncoded systems. On the other hand, the performance is severely degraded while applying conventional detector. In addition, two transmitter schemes are compared in Fig. 2 (b), and it is observed that increasing ED indeed improves performance. We show that assigning more than one OSS to each user in multiple-access channels may induce large MAI, and AMCG can be used to evaluate performance. Different transmitter schemes with corresponding encoders are employed as shown in Fig. 3. When there is only one user in the systems, employing multiple OSS can increase ED and ACG. However, multiple OSS may induce large MAI and degrade the performance in multiple-access channels. The results are organized in Table I, and it can be observed that the AMCG is an accurate measure in high SNR regime. We show that the near-far problem is alleviated by the optimum detector as shown in Fig. 4. The SNR of user 1 is fixed in 4 dB. The results verify that the optimum detector is indeed most effective by BER and near-far resistance.
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TABLE I K 1 1 1 2 2 1 1 1 2 2
Signal set 4-PSK 8-PSK 2D with 4-PSK 8-PSK 2D with 4-PSK Octonary AM-PM 16-QAM 4D with 4-PSK 16-QAM 4D with 4-PSK
D 1 1 2 1 2 1 1 4 1 4
ED 2 2 4 2 4 2 2 8 2 8
is to use low level modulation 4-PSK first. Further, when employing lower rate codes, we can increase D, apply high level modulation, or combine both methods. It relies on which strategy can provide larger AMCG. The general design in various channels of all parameters is subject to further study.
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AMCG Numerical Uncoded (baseline) 3.60 3.60 3.5 4.77 4.77 4.5 3.60 3.60 3.5 4.77 2.73 2.9 Uncoded (baseline) 3.98 3.98 3.3 6.99 6.99 6.7 3.98 3.22 3.2 6.99 3.42 3.4
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Fig. 4. BER versus near-far strength for user 1 in five-user systems. Nc = 32. A 4-state rate 1/2 code with generator polynomials 7, 5 is employed.
X∈X E∈Ek (X)
VI. C ONCLUSION We investigated coded DS/CDMA systems based on the multi-sequence model. In our model, multiple OSS are assigned to each user to increase ACG. However, large number of OSS may also induce more MAI and degrade performance in multiple-access channels. Expanding the signal constellation as TCM is needed to compensate the drawback of employing multiple OSS. By adjusting the number of OSS and the modulation, we can take advantages of both TCM and multiple OSS. By AMCG, we use multi-sequence model to design the transmitters such that the coding, modulation, and spreading sequences are jointly considered. In addition, since the multisequence model is a generalized model of coded DS/CDMA transmitters, the optimum detector and measure AMCG based on that model can be applied to many transmitter schemes. The optimum detector was formulated. Combining channel decoding with MUD, the receiver combats MAI and brings the coding into full play. Simulation demonstrated that significant coding gain is achieved by optimum detector with simple codes under multiple-access channels. Besides, the optimum detector has significant performance gain over the conventional detector and has capability to alleviate the near-far problem. AMCG is defined to quantify the performance loss and gain due to MAI and coding, and its upper and lower bounds were derived. The design, by the multi-sequence model, of signal points in multidimensional signal space depends on encoders, spreading sequences, and the number of users. Analysis verified that increasing ED can improve performance. However, increasing D may induce large MAI and degrade performance. Hence, when designing transmitters, a considerable method
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A PPENDIX Proposition 1: The signal constellation ΩW,k of the DMCM PSK scheme is geometrically uniform, and the geometrically uniform partitions admit binary isometric labelings. Proof: ΩW,k is geometrically uniform if there is a transitive symmetry group Γ (ΩW,k ) on ΩW,k . It was shown in [14] that M -PSK is geometrically uniform set with transitive symmetry group RM that is a set of rotations by multiples of 2π/M . Since ΩW,k is D Cartesian product of M-PSK, we D can get transitive symmetry group Γ (ΩW,k )=(RM ) on ΩW,k . So, ΩW,k is geometrically uniform. The partition ΩW,k /ΩW,k mD admits an isometric labeling by a group (Z2 ) because mD Γ (ΩW,k ) is isomorphic to (Z2 ) where Z2 is binary group, m=log2 M ; and ΩW,k is the subset. Proposition 2: For any user k, AMCG ηk of a coded DS/CDMA system is always smaller than or equal to ACGk that stands for ACG of the corresponding single user system. Proof: "Since εH k εk is# non-negative, the solutions of eH satisfy ek = 0 and εk = 0. e + εH min min k k k εk
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Hence, γk ≤ 1, and the upper bound of AMCG of any user k is given by ηk ≤ ACGk . R EFERENCES
[1] A. J. Viterbi, “Spread spectrum communications - myths and realities,” IEEE Commun. Mag., vol. 17, no. 5, pp. 11–18, May 1979. [2] ——, “When not to spread spectrum - a sequel,” IEEE Commun. Mag., vol. 23, no. 4, pp. 12–17, Apr. 1985. [3] G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE Trans. Inform. Theory, vol. IT-28, no. 1, pp. 55–67, Jan. 1982. [4] G. D. Boudreau, D. D. Falconer, and S. A. Mahmoud, “A comparison of trellis coded versus convolutionally coded spread-spectrum multipleaccess system,” IEEE J. Select. Areas Commun., vol. 8, no. 4, pp. 628– 640, May 1990. [5] B. D. Woerner and W. E. Stark, “Trellis-coded direct-sequence spreadspectrum communications,” IEEE Trans. Commun., vol. 42, no. 12, pp. 3161–3170, Dec. 1994. [6] S. Choe and C. N. Georghiades, “On the performance of a novel quasi-synchronous trellis-coded CDMA system,” IEEE Trans. Commun., vol. 50, no. 12, pp. 1984–1993, Dec. 2002. [7] U. Fawer and B. Aazhang, “Multiuser receivers for coded-division multiple-access systems with trellis-based modulation,” IEEE J. Select. Areas Commun., vol. 14, no. 8, pp. 1602–1609, Oct. 1996. [8] S. Verdu, Multiuser Detection, 1st ed. Cambridge Univ. Press, 1998. [9] T. R. Giallorenzi and S. G. Wilson, “Multiuser ML sequence estimator for convolutionally coded asynchronous DS-CDMA systems,” IEEE Trans. Commun., vol. 44, no. 8, pp. 997–1008, Aug. 1996. [10] ——, “Suboptimum multiuser receivers for convolutionally coded asynchronous DS-CDMA systems,” IEEE Trans. Commun., vol. 44, no. 9, pp. 1183–1196, Sept. 1996. [11] E. Biglieri, D. Divsalar, P. J. Mclane, and M. K. Simon, Introduction to Trellis-Coded Modulation with Applications, 1st ed. Macmillan, 1991. [12] L.-F. Wei, “Trellis-coded modulation with multidimensional constellations,” IEEE Trans. Inform. Theory, vol. IT-33, pp. 483–501, July 1987. [13] H. Lee and K.-C. Chen, “Near-optimium trellis-coded DS/CDMA multiuser communications,” in Proc. IEEE Vehicular Technology Conf., Orlando, FL, USA, Oct. 2003. [14] J. G. D. Forney, “Geometrically uniform codes,” IEEE Trans. Inform. Theory, vol. 37, no. 5, pp. 1241–1260, Sept. 1991.
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I. I NTRODUCTION Channel coding is normally employed in communications to improve performance. In particular, it was emphasized by Viterbi [1] [2] that channel coding increases bandwidth efficiency in direct-sequence code-division multiple-access (DS/CDMA) systems. In addition, trellis-coded modulation (TCM) [3] is an efficient modulation scheme over additive white Gaussian noise (AWGN) channels, and has been applied to many systems, where significant coding gain is obtained without sacrificing data rate or bandwidth efficiency. Some investigations have been done on coded DS/CDMA systems. Boudreau and Falconer [4] indicated that convolutionally coded DS/CDMA (CC-DS/CDMA) provides superior performance to trellis-coded DS/CDMA (TC-DS/CDMA). Woerner and Stark [5] introduced bi-orthogonal signal set (BSS) to increase normalized minimum Euclidean distance (free distance). Choe and Georghiades [6] introduced orthogonal plane sequence modulation (OPSM), which improves power and bandwidth efficiency. Nevertheless, aforementioned pioneer studies ignored the presence of multiple-access interference (MAI), which degrades the bit-error-rate (BER) performance. Fawer and Aazhang [7] applied multiuser detection (MUD) [8] to improve performance based on the model in [5]. Giallorenzi and Wilson [9] [10] derived optimum and sub-
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optimum MUD for CC-DS/CDMA communications. In [5] [6], by ignoring MAI, multiple spreading sequences are assigned to each user to improve performance. However, employing multiple spreading sequences also decreases the number of maximal allowable users with fixed spreading gain and may introduce larger MAI among users. On the other hand, employing TCM in systems can alleviate above problems at the price of decreased free distance. Thus, the design of optimal transmitters should consider the trade-off between TCM and the number of spreading sequences. To achieve above goal, we introduce a model that combines advantages of TCM and multiple spreading sequences and jointly design coding, modulation, and spreading sequences to maximize the free distance. In our model, multiple spreading sequences are assigned to each user to increase free distance. Moreover, the signal constellation can be expanded as TCM to balance the number of necessary spreading sequences. Because the design depends on the structure of encoders, the number of active users, and spreading sequences, the proposed model is very much desired to be generalized to adapt different conditions. In addition to transmitter design, we further propose the optimum detector that performs the functions of MUD and decoding jointly and is optimal in maximum likelihood (ML) sense, and demonstrate that the proposed detector is near-far resistant by both mathematical and numerical analyses. The asymptotic multiuser coding gain (AMCG) is defined as a performance measure to evaluate the proposed transceivers. AMCG can be regarded as generalized asymptotic multiuser efficiency (AME) [8] and used as a criterion for designing transceivers, since the ordering of AME of two schemes determines the ordering of corresponding BER for sufficiently high signal-to-noise ratio (SNR) regime [8]. The AMCG for CC-DS/CDMA is formulated in [9]. In this paper, we derived the AMCG and its upper and lower bounds of optimum detector for the generalized coded DS/CDMA model. ∗ T H Notation: Superscripts (·) , (·) , and (·) denote complex conjugate, transpose, and Hermitian adjoint operators, respectively. A circumflex over any variable represents an estimate of that variable, such as x for an estimate of x. The subspace spanned by columns of a matrix W is represented with angle brackets around the symbol for the matrix, such as W. Given a set Ω, |Ω| denotes the number of elements in Ω.
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Info. Source
Encoder 1
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. . Info. Source
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B. Applications of Multi-sequence Model
Mapping and Spreading K Channel gain K
Fig. 1.
Unlike conventional systems that the performance depends on the distance among symbols, in DS/CDMA the performance relies on the distance among transmitted sequences.
System block diagram of a coded DS/CDMA transceiver.
II. S YSTEM M ODEL The architecture of a coded DS/CDMA transceiver is illustrated in Fig. 1. Considering a synchronous DS/CDMA system with K users, we assume that (a) the encoders are identical with α input information bits, β output coded bits, and constraint length ν; (b) received amplitude and phase are perfectly estimated; (c) users send equal number of symbols. A. Multi-sequence Model of coded DS/CDMA systems We introduce multi-sequence model for transmission. For user k = 1, · · · , K, each user is assigned D orthonormal spreading sequences (OSS), and the sequences are represented T by Nc ×1 vectors sk,d √1N ck,d,1 · · · ck,d,Nc , d = 1, c 2 · · · , D where ck,d,n ∈ C are chip codes with |ck,d,n | = 1, and Nc > DK is the length of OSS. Moreover, each OSS is multiplied by a coefficient xk,d ∈ R or C. For user k, the transmitted sequence is given by D xk,d sk,d . (1) sk = d=1
In another point of view, the OSS of user k are used to compose a signal space Wk where Wk sk,1 · · · sk,D N ×D . Consequently, (1) is equivalent to c the procedure of linear combination, and any signal point sk for transmission in the signal space Wk can be arbitrarily designed by choosing different xk,1 , · · · , xk,D . Because the encoder outputs β bits each time, we need to choose 2β signal points that compose a signal constellation ΩW,k ∈ Wk with |ΩW,k | = 2β from Wk for transmission. Two important measures should be noted in the multisequence model while designing transmitters. One is the dimension of Wk , and the other is the distance among signal points. For user k who has D OSS, the dimension of Wk is D. However, because the coefficients xk,d can be complexvalued, we define the effective dimension (ED) of Wk as D if xk,d ∈ R for all d ED 2D otherwise. Consider the signal space composed of all user’s spreading sequences with length Nc . The maximum ED of the signal space is 2Nc . In DS/CDMA, multidimensional signal space can be constructed by multi-sequence model. Applying multidimensional constellation can increase free distance [11] [12] so that we can derive additional coding gain by increasing ED. The other measure is the distance among signal points 2
d2 (sk , sk ) sk − sk .
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In [13], we showed that DMC-M PSK [13], TC-DS/CDMA [4], BSS [5] and OPSM [6] schemes are special cases of multi-sequence model. Here, we show that the DMC2PSK scheme constructed by defining xk,d ∈ {M -PSK} and −1 D = β (log2 M ) has the same distance property as CCDS/CDMA. Consider a encoder with β output coded bits. Each bit is modulated by a spreading sequence with length shortened to Nc /β, so the symbol duration is divided into β slots. The signal constellation of CC-DS/CDMA then can be regarded as a β-dimension hypercube constellation. Since we can construct a β-dimension hypercube constellation by βMC-2PSK, the signal constellations of βMC-2PSK and CCDS/CDMA are both hypercube, though their dimension is expanded by different approaches. Both schemes therefore have equal distance property. Compared with TC-DS/CDMA, hypercube signal constellations usually achieve larger free distance under the same bandwidth efficiency. This explains the results in [4]. Multi-sequence model provides the spreading sequences geometrical meaning and combines the modulation and spreading technique. It provides two approaches to expand signal constellations: increasing D and applying higher level modulation. The price of high-level modulation is transmission power, however, it is a considerable method once Nc is limited. On the other hand, when Nc is large, more OSS can be assigned to each user to increase ED. C. Coded DS/CDMA Transmitter At time index i, user k’s source generates α information bits α×1 which is denoted by an α × 1 vector bk [i] ∈ {+1, −1} . T T T We define the vectorbk bk [−I] · · · bk [I] ; the matrix B b1 · · · bK α(2I+1)×K ; and the set ΩB contains all possible matrices B. The user k’s ith convolutionally coded symbol uk [i] generated by passing α bits through the encoder is given by (3) uk [i] f (bk [i] , ψk [i]) ∈ 1, · · · , 2β and ψk [i + 1] g (bk [i] , ψk [i]). Function f describes that uk [i] depends on the corresponding bits bk [i] and encoder state ψk [i]. Function g describes state evolution. By mapping uk [i] into signal constellation ΩW,k that consists 2β distinct spreading sequences, we get user k’s ith transmitted waveform sk (t; uk [i])
D
xk,d [i] sk,d (t) m (uk [i]) ∈ ΩW,k (4)
d=1
Nc where sk,d (t) √1N n=1 ck,d,n Π(t − nTc ); m(·) is the c mapping function; xk,d [i] are coefficients transmitted at time i and depend on uk [i]; Π(·) is a rectangular function of unit amplitude on the interval [0, Tc ); and Tc is chip duration.
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The mapping (4) between coded symbols and transmitted sequences is a key to combine the coding and spreading technique. For each user, since the performance is dominated by free distance, while applying Viterbi algorithm, coding and mapping should be jointly designed such that the distinct paths through the trellis are separated by larger distance defined in (2). The Ungerboeck’s set partitioning principle is generalized in the multi-sequence model to maximize free distance. To apply Ungerboeck’s principle, we design the signal constellation ΩW,k that is geometrically uniform [14], and use two-way geometrically uniform partitioning over ΩW,k . If partitions admit binary isometric labelings, coded symbols can be one-to-one mapped to transmitted signals by Ungerboeck’s principle [14] such that the distance among distinct paths through the trellis are maximized. Proposition 1 in Appendix shows that the signal constellation of DMC-M PSK is geometrically uniform and geometrically uniform partitions admit binary isometric labelings. III. O PTIMUM M ULTIUSER D ETECTOR Assuming an AWGN channel, the received signal that is the sum of all K users’ signals is K I r(t) ak sk (t − iTs ; uk [i]) + nw (t) (5) k=1
i=−I
where |ak | and ak are the received amplitude and phase of user k; Ts = Nc Tc is the symbol duration; and nw (t) is circularly symmetric AWGN with power spectral density σ 2 /2 = N0 /4 for real and imaginary parts. After chip-matched filtering, which preserves sufficient statistic, the ML detector maximizes the function I
H H H 2y [i] Ax [i] − x [i] A RAx [i] Λ(B) = Re i=−I
where S s1,1 · · · s1,D · · · sK,D N ×KD ; x [i] c T x1,1 [i] · · · x1,D [i] · · · xK,D [i] is a KD × 1 vecT tor; y [i] y1,1 [i] · · · y1,D [i] · · · yK,D [i] is a
∞ ∗ KD × 1 vector; yk,d [i] r (t)s (t − iT )dt; k,d s −∞ H
R S S is the cross-correlation matrix; and A diag {a1 , · · · , a2 , · · · , aK } is a KD × KD diagonal matrix. The optimum MUD on information symbols is applied by maximizing Λ(B). The desired information is estimated by arg max Λ(B). B B∈ΩB
(6)
The maximization can be realized by Viterbi algorithm, and the metric of Viterbi algorithm is given by M [i − 1] + λ (B), i = −I, · · · , I M [i] (7) 0, i < −I λ (B) Re 2y H [i] Ax [i] − xH [i] AH RAx [i] . (8) The trellis at receiver is the combination of all users’ trellis structures, and is extremely complicated with totally 2Kν states while each state has 2Kα emanating and remerging branches. The complexity of optimum detector grows exponentially with the number of active users K.
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IV. S YSTEM A NALYSIS A. Error Probability [i] We derive Pk Pr bk [i] = bk [i] , which is the ith symbol error probability of user k. At the receiver, the optimum detector makes decisions among all possible sequences, and any sequence can be one-to-one mapped to an unique matrix X x [−I] · · · x [I] KD×(2I+1) . For particular user k at time index i, the set of errors is given [i] by Ek (X) {E| (X − E) ∈ X and ek [i] = 0} where X k [i], and is the set of all possible X, ek [i] xk [i] − x T xk [i] xk,1 [i] · · · xk,D [i] is a D × 1 vector. = An error event denoted by X → X|X means that X X − E is preferred to X by the receiver when X is transmitted. After some calculations with (5), the pairwise probability of a detection error is given by I H [i ] AH RAe [i ] e i =−I . Pr X → X|X = Q σ The error probability is derived by averaging over all possible sequences and errors, and is bounded by the union bound [i] Pk ≤ Pr [X] · Pr X → X|X . (9) X∈X E∈E [i] (X) k
B. Minimum Distance As σ approaching zero, the minimum distance among sequences dominates the asymptotic behavior of the performance. For particular user k, the set of errors is defined as Ek (X) {E| (X − E) ∈ X and ek [i] = 0 for some i}. Assume that I is large enough such that the minimum distance df ree,k is independent of i, and is given by I d2f ree,k min min eH [i ] AH RAe [i ] . (10) X∈X E∈Ek (X)
i =−I
C. Asymptotic Multiuser Coding Gain of Optimum Detector By (10), we define AMCG of user k as I H H min min i =−I e [i ] A RAe [i ] /P X∈X E∈Ek (X) ηk (11) 2 |ak | · d2min,k /P where P and P are the average energies spent to transmit with coded and uncoded systems, respectively, and dmin,k is the minimum distance of uncoded DS/CDMA systems. AMCG is a way to quantify the performance gain and loss of coded DS/CDMA in high SNR regime relative to an uncoded single user system. When there is only one user or the signature waveforms among users are orthogonal, AMCG is equivalent to asymptotic coding gain (ACG) of user k: I H min min i =−I e [i ] e [i ] /P X∈X E∈Ek (X) ≥ 1. (12) ACGk d2min,d /P While users do not employ coding, AMCG is equal to AME. The near-far resistance of coded DS/CDMA can be defined as minimizing (11) with respect to ak : η k inf ηk . ak >0; k =k
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0
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···
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where (14) is achieved by εk = λM (·) denotes the largest eigenvalue of the matrix, and (·)(p,q) denotes the pth row and qth column entry of the matrix. Because R is positive definite, lower bound (15) is always larger than zero, so the optimum receiver is near-far resistant. V. S IMULATION R ESULTS To compare systems fairly, we have to maintain the bandwidth, coding rate, and coding complexity identical in a
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Fig. 2. BER versus SNR of optimum and conventional detectors in five-user systems with equal received energy. Nc = 32. (a) A 4-state rate 1/2 code with generator polynomials 7, 5 is employed. (b) A 4-state rate 2/3 code with generator polynomials 10, 07, 05 is employed. 0
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= ACGk · γk (13) T T where ek ak ek [−I] · · · ak eTk [I] is a D(2I + 1)×1 T vector; εk eT1 · · · eTk−1 eTk+1 · · · eTK is a (K − 1)D(2I + 1)×1 vector; Ξk is a (K − 1)D(2I + 1)×(K − 1)D(2I+1) matrix by striking out all rows and columns related ξk is a (K −1)D(2I +1)×D(2I +1) matrix that to user k in R; is comprised of columns related to user k with the elements and R diag{R, R, · · · , R} is a of user k removed in R; KD(2I + 1)×KD(2I + 1) matrix. The γk in (13) is the ratio that represents MAI degrading performance. If γk = 1, the high SNR performance suffers no degradation due to other active users. Otherwise, γk < 1 means that performance is degraded by MAI. Furthermore, the ACGk in (13) shows that we can employ coding to improve performance. Proposition 2 in Appendix proves that ηk ≤ ACGk . It implies that the optimum detector may remove MAI completely. Once the equality is achieved, further implements of spreading sequences yield negligible gains in BER. We derive the lower bound of η k (ηk ) and prove it always larger than zero (near-far resistant). By (13), we obtain H I ξkH ek H ek εk ξk Ξk εk η k ≥ inf min ACGk H ak >0X∈X ; E∈Ek (X) ek ek H H −1 ek ξk Ξk ξk ek ≥ inf min ACGk 1 − ak >0X∈X ; E∈Ek (X) eH k k e (14) ACGk ≥ (R−1 )(kD−D+1,kD−D+1) ··· (R−1 )(kD−D+1,kD)
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The AMCG, which is calculated by searching over all possible X, can be used to evaluate transceivers. In order to give some interpretations about AMCG and derive its upper and lower bounds, we rewrite (11) in another form: I H H min min i =−I e [i ] A RAe [i ] X∈X E∈Ek (X) ηk = ACGk I 2 H |ak | · min min i =−I e [i ] e [i ] X∈X E∈Ek (X) H I ξkH ek H min min ek εk ξk Ξk εk X∈X E∈Ek (X) = ACGk H ek min min ek εH k εk X∈X E∈Ek (X)
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Fig. 3. BER versus SNR of optimum detectors for user 1 in two-user systems with near-far effect. Nc = 16. (a) An 8-state rate 2/3 code with generator polynomials 15, 23, 02 is employed. (b) An 8-state rate 3/4 code with generator polynomials 40, 15, 23, 02 is employed.
simulation. All simulations are in multiple-access channels. The spreading sequences are composed of complex-valued Gold code combined Walsh code. The solid and dashed lines in figures represent coded and uncoded systems, respectively. We show that the BER performance is improved significantly by coding while applying the optimum detector. The simulation results of employing different encoders are shown in Fig. 2. For the optimum detector, the coding gain of the codes that are considered ranges from 3 to 4 dB over the corresponding uncoded systems. On the other hand, the performance is severely degraded while applying conventional detector. In addition, two transmitter schemes are compared in Fig. 2 (b), and it is observed that increasing ED indeed improves performance. We show that assigning more than one OSS to each user in multiple-access channels may induce large MAI, and AMCG can be used to evaluate performance. Different transmitter schemes with corresponding encoders are employed as shown in Fig. 3. When there is only one user in the systems, employing multiple OSS can increase ED and ACG. However, multiple OSS may induce large MAI and degrade the performance in multiple-access channels. The results are organized in Table I, and it can be observed that the AMCG is an accurate measure in high SNR regime. We show that the near-far problem is alleviated by the optimum detector as shown in Fig. 4. The SNR of user 1 is fixed in 4 dB. The results verify that the optimum detector is indeed most effective by BER and near-far resistance.
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TABLE I K 1 1 1 2 2 1 1 1 2 2
Signal set 4-PSK 8-PSK 2D with 4-PSK 8-PSK 2D with 4-PSK Octonary AM-PM 16-QAM 4D with 4-PSK 16-QAM 4D with 4-PSK
D 1 1 2 1 2 1 1 4 1 4
ED 2 2 4 2 4 2 2 8 2 8
is to use low level modulation 4-PSK first. Further, when employing lower rate codes, we can increase D, apply high level modulation, or combine both methods. It relies on which strategy can provide larger AMCG. The general design in various channels of all parameters is subject to further study.
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X∈X E∈Ek (X)
VI. C ONCLUSION We investigated coded DS/CDMA systems based on the multi-sequence model. In our model, multiple OSS are assigned to each user to increase ACG. However, large number of OSS may also induce more MAI and degrade performance in multiple-access channels. Expanding the signal constellation as TCM is needed to compensate the drawback of employing multiple OSS. By adjusting the number of OSS and the modulation, we can take advantages of both TCM and multiple OSS. By AMCG, we use multi-sequence model to design the transmitters such that the coding, modulation, and spreading sequences are jointly considered. In addition, since the multisequence model is a generalized model of coded DS/CDMA transmitters, the optimum detector and measure AMCG based on that model can be applied to many transmitter schemes. The optimum detector was formulated. Combining channel decoding with MUD, the receiver combats MAI and brings the coding into full play. Simulation demonstrated that significant coding gain is achieved by optimum detector with simple codes under multiple-access channels. Besides, the optimum detector has significant performance gain over the conventional detector and has capability to alleviate the near-far problem. AMCG is defined to quantify the performance loss and gain due to MAI and coding, and its upper and lower bounds were derived. The design, by the multi-sequence model, of signal points in multidimensional signal space depends on encoders, spreading sequences, and the number of users. Analysis verified that increasing ED can improve performance. However, increasing D may induce large MAI and degrade performance. Hence, when designing transmitters, a considerable method
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A PPENDIX Proposition 1: The signal constellation ΩW,k of the DMCM PSK scheme is geometrically uniform, and the geometrically uniform partitions admit binary isometric labelings. Proof: ΩW,k is geometrically uniform if there is a transitive symmetry group Γ (ΩW,k ) on ΩW,k . It was shown in [14] that M -PSK is geometrically uniform set with transitive symmetry group RM that is a set of rotations by multiples of 2π/M . Since ΩW,k is D Cartesian product of M-PSK, we D can get transitive symmetry group Γ (ΩW,k )=(RM ) on ΩW,k . So, ΩW,k is geometrically uniform. The partition ΩW,k /ΩW,k mD admits an isometric labeling by a group (Z2 ) because mD Γ (ΩW,k ) is isomorphic to (Z2 ) where Z2 is binary group, m=log2 M ; and ΩW,k is the subset. Proposition 2: For any user k, AMCG ηk of a coded DS/CDMA system is always smaller than or equal to ACGk that stands for ACG of the corresponding single user system. Proof: "Since εH k εk is# non-negative, the solutions of eH satisfy ek = 0 and εk = 0. e + εH min min k k k εk
384
Hence, γk ≤ 1, and the upper bound of AMCG of any user k is given by ηk ≤ ACGk . R EFERENCES
[1] A. J. Viterbi, “Spread spectrum communications - myths and realities,” IEEE Commun. Mag., vol. 17, no. 5, pp. 11–18, May 1979. [2] ——, “When not to spread spectrum - a sequel,” IEEE Commun. Mag., vol. 23, no. 4, pp. 12–17, Apr. 1985. [3] G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE Trans. Inform. Theory, vol. IT-28, no. 1, pp. 55–67, Jan. 1982. [4] G. D. Boudreau, D. D. Falconer, and S. A. Mahmoud, “A comparison of trellis coded versus convolutionally coded spread-spectrum multipleaccess system,” IEEE J. Select. Areas Commun., vol. 8, no. 4, pp. 628– 640, May 1990. [5] B. D. Woerner and W. E. Stark, “Trellis-coded direct-sequence spreadspectrum communications,” IEEE Trans. Commun., vol. 42, no. 12, pp. 3161–3170, Dec. 1994. [6] S. Choe and C. N. Georghiades, “On the performance of a novel quasi-synchronous trellis-coded CDMA system,” IEEE Trans. Commun., vol. 50, no. 12, pp. 1984–1993, Dec. 2002. [7] U. Fawer and B. Aazhang, “Multiuser receivers for coded-division multiple-access systems with trellis-based modulation,” IEEE J. Select. Areas Commun., vol. 14, no. 8, pp. 1602–1609, Oct. 1996. [8] S. Verdu, Multiuser Detection, 1st ed. Cambridge Univ. Press, 1998. [9] T. R. Giallorenzi and S. G. Wilson, “Multiuser ML sequence estimator for convolutionally coded asynchronous DS-CDMA systems,” IEEE Trans. Commun., vol. 44, no. 8, pp. 997–1008, Aug. 1996. [10] ——, “Suboptimum multiuser receivers for convolutionally coded asynchronous DS-CDMA systems,” IEEE Trans. Commun., vol. 44, no. 9, pp. 1183–1196, Sept. 1996. [11] E. Biglieri, D. Divsalar, P. J. Mclane, and M. K. Simon, Introduction to Trellis-Coded Modulation with Applications, 1st ed. Macmillan, 1991. [12] L.-F. Wei, “Trellis-coded modulation with multidimensional constellations,” IEEE Trans. Inform. Theory, vol. IT-33, pp. 483–501, July 1987. [13] H. Lee and K.-C. Chen, “Near-optimium trellis-coded DS/CDMA multiuser communications,” in Proc. IEEE Vehicular Technology Conf., Orlando, FL, USA, Oct. 2003. [14] J. G. D. Forney, “Geometrically uniform codes,” IEEE Trans. Inform. Theory, vol. 37, no. 5, pp. 1241–1260, Sept. 1991.
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