Analysis of Bit Error Probability of 
Direct-Sequence CDMA Multiuser 
Demodulators 
Toshiyuki Tanaka 
Department of Electronics and Information Engineering 
Tokyo Metropolitan University 
Hachioji, Tokyo 192-0397, Japan 
tanaka @ ee i. metro-u. ac.jp 
Abstract 
We analyze the bit error probability of multiuser demodulators for direct- 
sequence binary phase-shift-keying (DS/BPSK) CDMA channel with ad- 
ditive gaussian noise. The problem of multiuser demodulation is cast 
into the finite-temperature decoding problem, and replica analysis is ap- 
plied to evaluate the performance of the resulting MPM (Marginal Pos- 
terior Mode) demodulators, which include the optimal demodulator and 
the MAP demodulator as special cases. An approximate implementa- 
tion of demodulators is proposed using analog-valued Hopfield model 
as a naive mean-field approximation to the MPM demodulators, and its 
performance is also evaluated by the replica analysis. Results of the per- 
formance evaluation shows effectiveness of the optimal demodulator and 
the mean-field demodulator compared with the conventional one, espe- 
cially in the cases of small information bit rate and low noise level. 
1 Introduction 
The CDMA (Code-Division-Multiple-Access) technique [1] is important as a fundamental 
technology of digital communications systems, such as cellular phones. The important ap- 
plications include realization of spread-spectrum multipoint-to-point communications sys- 
tems, in which multiple users share the same communication channel. In the multipoint-to- 
point system, each user modulates his/her own information bit sequence using a spreading 
code sequence before transmitting it, and the receiver uses the same spreading code se- 
quence for demodulation to obtain the original information bit sequence. Different users 
use different spreading code sequences so that the demodulation procedure randomizes 
and thus suppresses multiple access interference effects of transmitted signal sequences 
sent from different users. 
The direct-sequence binary phase-shift-keying (DS/BPSK) [1] is the basic method among 
various methods realizing CDMA, and a lot of studies have been done on it. Use of 
Hopfield-type recurrent neural network has been proposed as an implementation of a mul- 
tiuser demodulator [2]. In this paper, we analyze the bit error probability of the neural 
multiuser demodulator applied to demodulation of DS/BPSK CDMA channel. 
Spreading Code Sequences 
2 
N 
Information Bits 
Gaussian Noise 
Received Signal 
) , 
Figure 1: DS/BPSK CDMA model 
2 DS/BPSK CDMA system 
We assume that a single Gaussian channel is shared by N users, each of which wishes 
to transmit his/her own information bit sequence. We also take a simplifying assumption, 
that all the users are completely synchronized with each other, with respect not only to the 
chip timing but also to the information bit timing. We focus on any of the time intervals 
corresponding to the duration of one information bit. Let i  {-1, 1} be the information 
bit to be transmitted by user i (i = 1, ..., N) during the time interval, and P be the number 
of the spreading code chips (clocks) per information bit. For simplicity, the spreading code 
sequences for the users are assumed to be random bit sequences {r/; t = 1, . .., P}, where 
r/'s are independent and identically distributed (i.i.d.) binary random variables following 
Prob[q = 4-1] = 1/2. 
User i modulates the information bit i by the spreading code sequence and transmits the 
modulated sequence {i r/; t = 1, ..., P} (with carrier modulation, in actual situations). 
Assuming that power control [3] is done perfectly so that every transmitted sequences ar- 
rive at the receiver with the same intensity, the received signal sequence (after baseband 
demodulation) is {yt; t = 1, ..., P}, with 
N 
i=1 
(1) 
where pt  N(0, %2) is i.i.d. gaussian noise. This system is illustrated in Fig. 1. 
At the receiver side, one has to estimate the information bits {i } based on the knowledge of 
the received signal {yt} and the spreading code sequences {r/} for the users. The demodu- 
lator refers to the system which does this task. Accuracy of the estimation depends on what 
demodulator one uses. Some demodulators are introduced in Sect. 3, and analytical results 
for their performance is derived in Sect. 4. 
3 Demodulators 
3.1 Conventional demodulator 
The conventional demodulator (CD) [ 1-3] estimates the information bit i using the spread- 
ing code sequence {r/; t = 1, . .., P} for the user i, by 
CD) = sgn(hi), hi = 
P 
N7 tt 
rli Y  
t=l 
(2) 
We can rewrite hi as 
P N P 
P 1 1 tvt. 
hi: + + .i 
t=l ki t=l 
(3) 
The second and third terms of the right-hand side represent the effects of multiple ac- 
cess interference and noise, respectively. CD would give the correct information bit in the 
single-user (N = 1), and no noise (v t -- 0) case, but estimation may contain some errors 
in the multiple-user and/or noisy cases. 
3.2 MAP demodulator 
The accuracy of the estimation would be significantly improved if the demodulator knows 
the spreading code sequences for all N users and makes full use of them by simultane- 
ously estimating the information bits for all the users (the multiuser demodulator). This 
is the case, for example, for a base station receiving signals from many users. A common 
approach to the multiuser demodulation is to use the MAP decoding, which estimates the 
information bits {si = i } by maximizing the posterior probability P({i }l {yt}). We call 
this kind of multiuser demodulator the MAP demodulator 1 . 
When we assume uniform prior for the information bits, the posterior probability is explic- 
itly given by 
p(sl{yt}) = Z - exp(-fisH(s)), (4) 
where 
H(s) ---- lsrWs -hrs, (5) 
fis -- N/% 2, s ---- (si), h -- (hi), and W -- (wij) is the sample covariance of the spreading 
code sequences, 
P 
1 
wij =   r/t.r/- (6) 
t J' 
t=l 
The problem of MAP demodulation thus reduces to the following minimization problem: 
(mP) = arg min H(s). (7) 
se{-1, 1} ;v 
3.3 MPM demodulator 
Although the MAP demodulator is sometimes referred to as "optimal," actually it is not so 
in terms of the common measure of performance, i.e., the bit error probability Pb, which is 
1The MAP demodulator refers to the same one as what is frequently called the "maximum- 
likelihood (ML) demodulator" in the literature. 
N 
related to the overlap M ---- (l/N) i=1 ii between the original information bits {i } and 
their estimates {i } as 
1-M 
Pb -- (8) 
2 
The 'MPM (Marginal Posterior Mode [4]) demodulator,' with the inverse temperature fi, is 
defined as follows: 
/(MPM) = sgn((si)?), (9) 
where (.)? refers to the average with respect to the distribution 
p?(s) = Z(fi) - exp(-H(s)). (10) 
Then, we can show that the MPM demodulator with  = s is the optimal one minimizing 
the bit error probability Pb. It is a direct consequence of general argument on optimal 
decoders [5]. Note that the MAP demodulator corresponds to the MPM demodulator in the 
 --> +oo limit (the zero-temperature demodulator). 
4 Analysis 
4.1 Conventional demodulator 
In the cases where we can assume that N and P are both large while ot -- PIN = O(1), 
evaluation of the overlap M, and therefore the bit error probability Pb, for those demodu- 
lators are possible. For CD, simple application of the central limit theorem yields 
where 
is the error function. 
M = erf 2(1+ 1/fis) ' 
2 fox 
erf(x) =  e-t2dt 
(12) 
4.2 MPM demodulator 
For the MPM demodulator with inverse temperature , we have used the replica analysis 
to evaluate the bit error probability Pb. Assuming that N and P are both large while ot = 
PIN = O (1), and that the macroscopic properties of the demodulator are self-averaging 
with respect to the randomness of the information bits, of the spreading codes, and of the 
noise, we evaluate the quenched average of the free energy ((log Z)) in the thermodynamic 
limit N --> oo, where ((.)) denotes averaging over the information bits and the noise. 
Evaluation of the overlap M (within replica-symmetric (RS) ansatz) requires solving 
saddle-point problem for scalar variables {m, q, E, F}. The saddle-point equations are 
E z 
f Dz tanh(x/z + E), 
1 q- fi(1 - q)' 
q = f Dz tanh2(x/z + E) 
[ 
F = tfi2 1 - 2m + q + 
[1 q- fl(1 - q)12 
(13) 
where Dz = (1/x/-)e-Z2/2dz is the gaussian measure. The overlap M is then given by 
M = f Dz sgn(x/z q- E), (14) 
from which Pb is evaluated via (8). This is the first main result of this paper. 
4.3 MAP demodulator: Zero-temperature limit 
Taking the zero-temperature limit/5 --> +cx) of the result for the MPM demodulator yields 
the result for the MAP demodulator. Assuming that q --> 1 as/5 --> +cx), while/5(1 - q) 
remains finite in this limit, the saddle-point equations reduce to 
M = m = erf 2(2- 2m + 1//5s) ' 
It is found numerically, however, that the assumption q --> 1 is not valid for small or, so 
that we have to solve the original saddle-point equations in such cases. 
4.4 Optimal demodulator: The case 
Letting /5 = /5s in the result for the MPM demodulator gives the optimal demodulator 
minimizing the bit error probability. In this case, it can be shown that m = q and E = F 
hold for the solutions of the saddle-point equations (13). 
4.5 Demodulator using naive mean-field approximation 
Since solving the MAP or MPM demodulation problem is in general NP complete, we have 
to consider approximate implementations of those demodulators which are sub-optimal. A 
straightforward choice is the mean-field approximation (MFA) demodulator, which uses 
the analog-valued Hopfield model as the naive mean-field approximation to the finite- 
temperature demodulation problem 2. The solution {mi } of the mean-field equations 
mi = tanhi/5(- Ewijmj + hi)] 
J 
(16) 
gives an approximation to { (si)? }, from which we have the mean-field approximation to 
the MPM estimates, as 
]vw^) = sgn(mi). (17) 
The macroscopic properties of the MFA demodulator can be derived by the replica analysis 
as well, along the line proposed by Bray et al. [6] We have derived the following saddle- 
point equations: 
f Dz f(z), 
l q- /5 ' 
;- x Dzzf(z), q = Dz[f(z)] 2 
F -- t/52 1 - 2m + q + 
[1 q-/5;]2 ' 
(18) 
where f(z) is the function defined by 
f (z) = tanh[x/z - E f (z) + E]. (19) 
f(z) is a single-valued function of z since E is positive. The overlap M is then calculated 
by 
34 = f Dzsgn(f(z)). (20) 
This is the second main result of this paper. 
2The proposal by Kechriofis and Manolakos [2] is to use the Hopfield model for an approximation 
to the MAP demodulation. The proposal in this paper goes beyond theirs in that the analog-valued 
Hopfield model is used to approximate not the MAP demodulator in the zero-temperature limit but 
the MPM demodulators directly, including the optimal one. 
1 
O.Ol 
o.oool 
10 -6 
10 -8 
10-1o 
0.1 
MFA ...... \  1 
;::'7;: ........... \...\.1 
1 10 100 
(a)/5, = 1 
1 
O.Ol 
o.oool 
10 -6 
10 -8 
lo-lO 
o. 
MFA ...... \ . 
1 10 100 
(b) = 20 
Figure 2: Bit error probability for various demodulators. 
4.6 AT instability 
The AT instability [7] refers to the bifurcation of a saddle-point solution without replica 
symmetry from the replica-symmetric one. In this paper we follow the usual convention 
and assume that the first such destabilization occurs in the so-called "replicon mode [8]." 
As the stability condition of the RS saddle-point solution for the MPM demodulator, we 
obtain 
ot- E 2 f Dz sech4(x/z q- E) = 0. (21) 
For the MFA demodulator, we have 
a - E 2 Dz 1 q- 1 --z) 2) : 0. 
The RS solution is stable as long as the left-hand side of (21) or (22) is positive. 
(22) 
5 Performance evaluation 
The s addle-p oint equations (13) and (18) c an be solved numeric ally to evaluate the bit error 
probability Pb of the MPM demodulator and its naive mean-field approximation, respec- 
tively. We have investigated four demodulators: the optimal one (/5 = /5s), MAP, MFA 
(with/5 = /5s, i.e., the naive mean-field approximation to the optimal one), and CD. The 
results are summarized in Fig. 2 (a) and (b) for two cases with/5s = 1 and 20, respectively. 
Increasing a corresponds to relatively lowering the information bit rate, so that Pb should 
become small as a gets larger, which is in consistent with the general trend observed in 
Fig. 2. The optimal demodulator shows consistently better performance than CD, as ex- 
pected. The MAP demodulator marks almost the same performance as the optimal one 
(indeed the result of the MAP demodulator is nearly the same as that of the optimal de- 
modulator in the case/5s = 1, so they are indistinguishable from each other in Fig. 2 (a)). 
We also found that the performance of the optimal, MAP, and MFA demodulators is signif- 
icantly improved in the large-a region when the variance %2 of the noise is small relative 
to N, the number of the users. For example, in order to achieve practical level of bit error 
probability, Pb  10 -5 say, in the/5s = 1 case the optimal and MAP demodulators allow 
information bit rate 2 times faster than CD does. On the other hand, in the/5s = 20 case 
they allow information bit rate as much as 20 times faster than CD, which demonstrates that 
significant process gain is achieved by the optimal and MAP demodulators in such cases. 
The MFA demodulator with fl = rs showed the performance competitive with the optimal 
one for the 15s = 1 case. Although the MFA demodulator fell behind the optimal and MAP 
demodulators in the performance for the rs = 20 case, it still had process gain which al- 
lows about 10 times faster information bit rate than CD does. Moreover, we observed, using 
(22), that the RS saddle-point solution for the MFA demodulator with fl = rs was stable 
with respect to replica symmetry breaking (RSB), and thus RS ansatz was indeed valid 
for the MFA solution. It suggests that the free energy landscape is rather simple for these 
cases, making it easier for the MFA demodulator to find a good solution. This argument 
provides an explanation as to why finite-temperature analog-valued Hopfield models, pro- 
posed heuristically by Kechriotis and Manolakos [2], exhibited better performance in their 
numerical experiments. We also found that the RS saddle-point solution for the optimal 
demodulator was stable with respect to RSB over the whole range investigated, whereas 
the solution for the MAP demodulator was found to be unstable. This observation suggests 
the possibility to construct efficient near-optimal demodulators using advanced mean-field 
approximations, such as the TAP approach [9, 10]. 
Acknowledgments 
This work is supported in part by Grant-in-Aid for Scientific Research from the Ministry 
of Education, Science, Sports and Culture, Japan. 
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