Multicode Sparse-Sequence CDMA: Approach to

Wireless Pers Commun DOI 10.1007/s11277-012-0859-0

Multicode Sparse-Sequence CDMA: Approach to Optimum Performance by Linearly Complex WSLAS Detectors Yi Sun · Jizhong Xiao

© Springer Science+Business Media New York 2012

Abstract This paper investigates the performance-complexity tradeoff of the wide-sense likelihood ascent search (WSLAS) detectors in large multicode sparse-sequence CDMA. It is illustrated that when each sequence has sparsely only 16 nonzero chips, in a channel load up to 1.05 bits/s/Hz and a broad SNR region, the linearly complex WSLAS detectors can achieve the benchmark optimum BER while the complexity is significantly reduced from 0.5 times bit number to a constant less than 30 additions per bit by the sequence sparsity. The evaluation result of multiuse efficiency also shows that the sparse sequences of 16 nonzero chips can already provide a sufficient degree of freedom. Keywords

Sparsity · CDMA · Maximum likelihood · Large system · Multicode

1 Introduction It is well known [1,2] that the NP-hard optimum global maximum likelihood (GML) detector achieves the single-user bit error rate (BER) in the high SNR regime in the limit of large random spreading (LRS) CDMA. The belief propagation (BP) algorithm can be applied to approach the GML detection in LRS-CDMA with Gaussian approximation [3,4]. Compared with conventional dense spreading CDMA, the sparsely spread CDMA employs the sparse spreading sequences consisting of only few nonzero chips and therefore can substantially reduce computational complexity. Meanwhile, statistical analysis for the large sparsely spread (LSS) CDMA [4–7] shows that when a spreading sequence has only L = 16 nonzero chips, the BP algorithm can still achieve near optimum performance with a much lower complexity by sequence sparsity. Nevertheless, the BP is exponentially complex in computation. The

Y. Sun (B) · J. Xiao Department of Electrical Engineering, The City College of City University of New York, New York, NY 10031, USA e-mail: [email protected] J. Xiao e-mail: [email protected]

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tradeoff between the computational complexity and bandwidth efficiency at different channel load is studied in [8]. Spatial coupling is recently applied to improve performance of multiuse detector in sparsely spread CDMA [9]. In this paper, we investigate and report the performance-complexity tradeoff of the WSLAS detectors in the LSS CDMA via simulations. It has been shown [10,11] that with a per-bit complexity less than 0.5 times the number of bits per transmission (i.e. a linear complexity), the wide-sense sequential likelihood ascent search (WSLAS) detectors are proved to achieve the single-bit performance in the high SNR regime in the LRS CDMA with channel load less than 1/2 − 1/(4ln2) bits/s/Hz. In simulations, the WSLAS detectors approach the benchmark GML BER in all SNR with the channel load up to 1.05 bit/s/Hz. In this paper, it is shown that the BER using spars sequences with only L = 16 nonzero chips is already near the benchmark GML BER [1,2] using conventional dense sequences, confirming the result in [5], while the sequence sparsity significantly reduces the WSLAS complexity. Moreover, evaluation of multiuse efficiency demonstrates that the sparse sequences of 16 nonzero chips can already provide a sufficient degree of freedom [12] for multiuse detection.

2 LSS-CDMA and WSLAS Detectors 2.1 LSS-CDMA Consider a K -user bit-synchronous Gaussian CDMA channel that the bit period is Tb and the chip period is Tc . The bandwidth is approximately equal to W = 1/Tc and the processing gain is N = Tb /Tc . When K and N are finite and fixed, a quasi-large sparse sequence CDMA system can be obtained by the multicode technique [10] as follows. User k multiplexes and simultaneously transmits Bk bits bk j , j = 1, . . . , Bk , which are extended by a factor of √ B to occupy √ B bit periods of BTb seconds and spread by the sparse sequences sk j ∈ {−1/ L, 0, 1/ L} B N , each having BN chips, L nonzero chips that are randomly distributed, and unit length. The signal amplitude for bk j is Ak j . During an extended bit period BTb , the chip MF outputs a vector r=

BK K  

sk j Ak j bk j + m

(1)

k=1 j=1

where m∼ N (0, σ 2 I B N ) is a white Gaussian noise vector. The total number of bits is equal K to U = k=1 Bk . The L nonzero chips in sk j are randomly located and equiprobably take on √ ±1/ L’s. A more general distribution of the nonzero chips can be considered [5], though. After spreading, the bit energy is spread only over the L nonzero chips. In general, L is much smaller than BN and thus the spreading is sparse. When L = B N , the spreading becomes the ordinary dense random spreading. The ratios βk = Bk /B of the bit multiplexing factors Bk and the temporal spreading factor B for k = 1, . . . , K are fixed and then the channel load is α=

K K 1  1  Bk = βk W BTb N k=1

(2)

k=1

bits/s/Hz. Since Bk and B can be arbitrarily large, an LSS CDMA system can be obtained by increasing Bk and B. The signal model of Eq. (1) can be written in the matrix form r = SAb + m with S, A properly written. The MF bank ST outputs a sufficient statistic y = ST r = RAb + n where R = ST S and n = ST m ∼ N (0, σ 2 R). Let H = ARA.

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2.2 WSLAS Detectors In each step a LAS detector updates a set of pre-specified bits such that the new bit vector has higher likelihood. The computation is efficient by comparison of current likelihood gradient with an optimal threshold [11]. The WSLAS detectors that eventually update one bit a step can be presented in the following unified form. WSLAS detector: Specify L(n) ⊆ {1, . . . , U }, ∀n ≥ 0 such that for all n ≥ n  with some n  > 0, L(n) = {(n mod U ) + 1}. Given an initial vector b(0) ∈ {−1, 1}U and let g(0) = −Hb(0) + Ay. At step n the bits for k ∈ L(n) are updated by ⎧ ⎨ +1, if bk (n) = −1, and gk (n) > tk (n), bk (n + 1) = −1, if bk (n) = +1, and gk (n) < −tk (n), (3) ⎩ bk (n), otherwise,  where the kth threshold is tk (n) = j∈L(n) |Hk j |; the bits for k ∈ / L(n) remain unchanged bk (n+1) = bk (n). Then update  g(n + 1) = g(n) + 2 bi (n)Hi (4) i∈L p (n)

where L p (n) is the index set of bits flipped in Eq. (3). bˆ is the demodulated vector if b(n) = ˆ ∀n ≥ n ∗ ≥ n  U . 

b, Since the computational complexity is mainly incurred by updating the BN-dimensional g(n), the sequence sparsity in the LSS CDMA makes most elements of H zeros and therefore significantly reduces the overall complexity. 3 Simulation Results 3.1 BER In the simulation, for simplicity and without loss of generality we consider Bk = B for all users. Since all WSLAS detectors approach the same performance in the large systems [10], the performance of the SLAS detector (a particular WSLAS detector) that updates cyclically one bit per step is illustrated in figures. The initial bit vector is produced by the MF. All bits have the equal power and then the BER is averaged over all bits. For U = B K ≤ 128 bits, a set of sparse sequences is randomly selected in each transmission. For B K > 128, five sets of sparse sequences are randomly selected and fixed for all transmissions, and the BER’s are estimated and shown together with their averages. As a complexity measure, the number of additions per bit counted from the core operation in Eq. (4) is also estimated. In all simulations, only the multiplication BK is given and the results are applicable to any pair of integers B and K with their multiplication equal to the given BK. The benchmark optimum GML BER with the dense spreading, which is analytically obtained by the statistic mechanics approach [1,2], is also illustrated as comparison of performance. Figures 1 and 2 illustrate, respectively, BER and complexity versus bit number BK with α = 0.8 bits/s/Hz and SNR = 11 dB. As BK increases, BER’s for all L monotonically decrease. This justifies the proposal of constructing a quasi-LSS CDMA by multiplexing a large number of bits though the multicode technique [10]. As L increases, BER approaches the limit BER of dense spreading with L = B N , which approaches the benchmark optimum GML BER and the single-bit performance in the high SNR regime when B K ≥ 500. It is

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Fig. 2 Complexity versus bit number BK in the same channel of Fig. 1

interesting that when L = 16, BER is already very close to the optimum GML BER using dense spreading. The complexity (additions/bit) monotonically increases with increasing BK and L but is saturated with respect to BK for small L (≤16). Hence, using sparse sequences with L = 16 nonzero chips can approach the optimum GML BER while complexity is significantly reduced in comparison with the dense spreading. The same conclusion can be obtained from Figs. 3 and 4 which respectively illustrate BER and complexity versus L with α = 0.8 and SNR = 11 dB.

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BK = 128 BK = 256 BK = 512 BK = 1024 BK = 2048

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L (chips) Fig. 4 Complexity versus nonzero-chip number L in the same channel of Fig. 3

Figure 5 illustrates BER versus SNR with B K = 1, 024 and α = 0.8. As L increases, BER monotonically decreases. In particular, the BER with L = 16 is already indistinguishable from the optimum GML BER using the dense spreading and approaches the single-bit bound in the high SNR regime. As can be seen, the complexity of the SLAS detector is insensitive to SNR. In all the simulations, the WSLAS detectors approach almost the same performance of the SLAS detector since all of them are local maximum likelihood (LML) detectors. However,

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the former has a little lower complexity. All the simulation results show that the variation of BER’s over different sequence samples is large when the number of nonzero chips L is small (say L = 4, 6) but decreases as L increases. 3.2 Multiuser Efficiency All WSLAS detectors are LML detectors. It is obtained in [11] that the distance from SAb to the LML point region of an error signal SA(b − 2ε) with error vector ε is lower bounded by d(ε) =

εT (2H − A2 )ε  . εT Hε

(5)

For equal power system, if d(ε)/A < 1 is almost surely true, then the WSLAS detector achieves unit asymptotic multiuse efficiency (AME) and achieves the single-bit performance with probability one. Thus,  the probability that the single-bit performance is not achieved is measured by p(L) = ε∈E p(L , w) where E is the set of error vectors, w is the weight of error vector ε, and p(L , w) = Pr(d(ε)/A < 1)

(7)

is the probability that the multiuse efficiency for an error vector ε is smaller than one. By simple algebra, we can obtain that √ 1 (1 + 1 + 8w)). (8) 4 Simulations are carried out to evaluate p(L , w) and p(L) with B N = 1, 024 chips per bit and 220 = 1, 048, 576 trials. As illustrated in Table 1, p(L , w) decreases as L and w increase, respectively and p(L) monotonically decreases with L increasing. When L = 16, p(L) is already sufficiently small, implying that the sparse sequences can provide sufficient degree of freedom compared with the dense spreading. p(L , w) = Pr(||Sε||