272 
NEURAL NET RECEIVERS IN 
MULTIPLE-ACCESS COMMUNICATIONS 
Bernd-Peter Paris, Geoffrey Orsak, Mahesh Varanasi, Behnaam Aazhang 
Department of Electrical and Computer Engineering 
Rice University 
Houston, TX 77251-1892 
ABSTRACT 
The application of neural networks to the demodulation of 
spread-spectrum signals in a multiple-access environment is 
considered. This study is motivated in large part by the fact 
that, in a multiuser system, the conventional (matched fil- 
ter) receiver suffers severe performance degradation as the 
relative powers of the interfering signals become large (the 
"near-far" problem). Furthermore, the optimum receiver, 
which alleviates the near-far problem, is too complex to be 
of practical use. Receivers based on multi-layer percepttons 
are considered as a simple and robust alternative to the opti- 
mum solution. The optimum receiver is used to benchmark 
the performance of the neural net receiver; in particular, it is 
proven to be instrumental in identifying the decision regions 
of the neural networks. The back-propagation algorithm and 
a modified version of it are used to train the neural net. An 
importance sampling technique is introduced to reduce the 
number of simulations necessary to evaluate the performance 
of neural nets. In all examples considered the proposed neu- 
ral net receiver significantly outperforms the conventional 
receiver. 
INTRODUCTION 
In this paper we consider the problem of demodulating signals in a code-division 
multiple-access (CDMA) Gaussian channel. Multiple accessing in code domain is 
achieved by spreading the spectrum of the transmitted signals using preassigned 
code waveforms. The conventional method of demodulating a spread-spectrum sig- 
nal in a multiuser environment employs one filter matched to the desired signal. 
Since the conventional receiver ignores the presence of interfering signals it is reli- 
able only when there are few simultaneous transmissions. Furthermore, when the 
relative received power of the interfering signals become large (the "near-far" prob- 
lem), severe performance degradation of the system is observed even in situations 
with relatively low bandwidth efficiencies (defined as the ratio of the number of 
channel subscribers to the spread of the bandwidth) [Aazhang 87]. For this reason 
there has been an interest in designing optimum receivers for multi-user communica- 
tion systems [Verdu 86, Lupas 89, Poor 88]. The resulting optimum demodulators, 
Neural Net Receivers in Multiple-Access Communications 273 
however, have a variable decoding delay with computational and storage complexity 
that depend exponentially on the number of active users. Unfortunately, this com- 
putational intensity is unacceptable in many applications. There is hence a need 
for near optimum receivers that are robust to near-far effects with a reasonable 
computational complexity to ensure their practical implementation. 
In this study, we introduce a class of neural net receivers that are based on mul- 
tilayer percepttons trained via the back-propagation algorithm. Neural net receivers 
are very attractive alternatives to the optimum and conventional receivers due to 
their highly parallel structures. As we will observe, the performance of the neural 
net receivers closely track that of the optimum receiver in all examples considered. 
SYSTEM DESCRIPTION 
In the multiple-access network of interest, transmitters are assumed to share a radio 
band in a combination of the time and code domain. One way of multiple accessing 
in the code domain is spread spectrum, which is a signaling scheme that uses a much 
wider bandwidth than necessary for a given data rate. Let us assume that in a given 
time interval there are K active transmitters in the network. In a simple setting, 
the k th active user, in a symbol interval, transmits a signal from a binary signal 
set derived from the set of code waveforms assigned to the corresponding user. The 
signal is time limited to the interval [0, T], where T is the symbol duration. 
In this paper we will concentrate on symbol-synchronous CDMA systems. Syn- 
chronous systems find applications in time slotted channels with the central (base) 
station transmitting to remote (mobile) terminals and also in relays between cen- 
tral stations. The synchronous problem will also be construed as providing us with 
a manageable setting to better understand the issues in the more difficult asyn- 
chronous situation. In a synchronous CDMA system, the users maintain time syn- 
chronism so that the relative time delays associated with all users are assumed to be 
zero. To illustrate the potentials of the proposed multiuser detector, we present the 
application to binary PSK direct-sequence signals in coherent systems. Therefore, 
the signal at a given receiver is the superposition of the K transmitted signals in 
additive channel noise (see [Aazhang 87, Lupas 89] and references within) 
P K 
i--1 k--1 
where P is the packet length, A is the signal amplitude, we is the carrier frequency, 
0 is the phase angle. The symbol b? )  {-1, +1} denotes the bit that the k th user is 
transmitting in the ita time interval. In this model, nt is the additive channel noise 
which is assumed to be a white Gaussian random process. The time-limited code 
waveform, denoted. by a (t), is derived from the spreading sequence assigned to the 
k t user. That is, a(t) = Y7  a)p(t- jT) where p(t) is the unit rectangular 
pulse of duration T and N is the length of the spreading sequence. One code 
period a_  = [a?),a?),...,a)_l] is used for spreading the signal per symbol so 
274 Paris, Orsak, Varanasi and Aazhang 
that T = NTe. In this system, spectrum efficiency is measured as the ratio of the 
number of channel users to the spread factor, KIN. 
In the next two sections, we first consider optimum synchronous demodulation of 
the multiuser spread-spectrum signal. Then, we introduce the application of neural 
networks to the multiuser detection problem. 
OPTIMUM RECEIVER 
Multiuser detection is an active research area with the objective of developing strate- 
gies for demodulation of information sent by several transmitters sharing a channel 
[Verdu 86, Poor 88, Varanasi 89, Lupas 89]. In these situations with two or more 
users of a multiple-access Gaussian channel, one filter matched to the desired signal 
is no longer optimum since the decision statistics are effected by the other signals 
(e.g., the statistics are disturbed by cross-correlations with the interfering signals). 
Employing conventional matched filters, because of its structural simplicity, may 
still be justified if the system is operating at a low bandwidth efficiency. However, 
as the number of users in the system with fixed bandwidth grows or as the rel- 
ative received powers of the interfering signals become large, severe performance 
degradation of the conventional matched filter is observed [Aazhang 87]. For direct- 
sequence spread-spectrum systems, optimum receivers obtained by Verdu and Poor 
require an extremely high degree of software complexity and storage, which may be 
unacceptable for most multiple-access systems [Verdu 86, Lupas 89]. Despite imple- 
mentation problems, studies on optimum demodulation illustrate that the effects of 
interfering signals in a CDMA system, in principle, can be neutralized. 
A complete study of the suboptimum neural net receiver requires a review 
of the maximum likelihood sequence detection formulation. Assuming that all 
possible information sequences are independent and equally likely, and defining 
b_ (i) = [b? ) b? ) b?]', it is easy to see that an cptimum decision on b (i) is a 
, ,'', m 
one-shot decision in that it requires the observation of the received signal only in 
the i th time interval. Without loss of generality, we will therefore focus our attention 
on i = 0 and drop the time superscript and consider the demodulation of the vector 
of bits b_ with the observation of the received signal in the interval [0, T]. 
In a K-user Gaussian channel, the most likely information vector is chosen as 
that which maximizes the log of the likelihood function (see [Lupas 89]) 
b_.op  = arg max 2 (t)r(t)dt - [ bS(t)]2dt 
where S(t) = Aa(t) cos(wet + 0) is the modulating signal of the k th user. The 
optimum decision can also be written as 
b_.op  = arg max { 2y'b - bHb}, 
b_{_ i,.i_i}K -- -- 
(3) 
where H is the K x K matrix of signal cross-correlations such that the (k, l) ta 
element is h,l =< S(t), Sl(t) >. The vector of sufficient statistics y consists of the 
Neural Net Receivers in Multiple-Access Communications 275 
outputs of a bank of K filters each matched to one of the signals 
T 
y = r(t)S(t)dt, for k = 1,2,...,K. (4) 
The maximization in (3) has been shown to be NP-complete [Lupas 89], i.e., no 
algorithm is known that can solve the maximization problem in polynomial time in 
K. This computational intensity is unacceptable in many applications. In the next 
section, we consider a suboptimum receiver that employs artificial neural networks 
for finding a solution to a maximization problem similar to (3). 
NEURAL NETWORK 
Until now the application of neural networks to multiple-access communications has 
not drawn much attention. In this study we employ neural networks for classifying 
different signals in synchronous additive Gaussian channels. We assume that the 
information bits of the first of the K signals is of interest, therefore, the phase 
angle of the desired signal is assumed to be zero (i.e., 01 = 0). Two configurations 
with multi-layer perceptrons and sigmoid nonlinearity are considered for multiuser 
detection of direct-sequence spread-spectrum signals. 
One structure is depicted in Figure 1.b where a layered network of percep- 
ttons processes the sufficient statistics (4) of the multi-user Gaussian channel. In 
this structure the first layer of the net (referred to as the hidden layer) processes 
[y, Y2,..., YK]. The output layer may only have one node since there is only one 
signal that is being demodulated. This feed-forward structure is then trained using 
the back-propagation algorithm [Rumelhart 86]. 
In an alternate configuration, the continuous-time received signal is converted to 
an N-dimensional vector by sampling the output of the front-end filter at the chip 
rate T ' as illustrated in Figure 1.a. The input vector to the net can be written so 
that the demodulation of the first signal is viewed as a classification problem: 
+ Aa_O) + _ + I_ or - Aa_ O) + _ + I_, (5) 
where a_O) is the spreading code vector of the first user, _ is a length-N vector 
of filtered Gaussian noise samples and/_.= '.=2bkA cos(0k)a  is the multiple- 
access interference vector with A = A:Tc/2, k = 1,2,..., K. The layered neural 
net is then trained to process the input vector for demodulation of the first user's 
information bits via the back-propagation algorithm. For this configuration we 
consider two training methods, first the multi-layer receiver is trained, via the back- 
propagation algorithm, to classify the parity of the desired signal (referred to as the 
"trained" example) [Lippmann 87]. In another attempt (referred to as the "preset" 
example), the input layer of the net is preset as Gaussian classifiers and the other 
layers are trained using the back propagation algorithm [Gschwendtner 88]. 
Since we are interested in understanding the internal representation of knowledge 
by the weights of the net, a signal space method is developed to illustrate decision 
regions. In a K-user system where the spreading sequences are not orthogonal, the 
276 Paris, Orsak, Varanasi and Aazhang 
signals can be represented by orthonormal bases using the Gram-Schmidt procedure. 
The optimum decision regions in the signal space for the demodulation of bl are 
known [Poor 88] and can be directly compared to ones for the neural net. Figure 2 
illustrates decision regions for the optimum receiver and for "preset" and "trained" 
neural net receivers. In this example, two users are sharing a channel with N = 3, 
signal to noise ratio of user I (SNRi) equal to 8dB and relative energies of the 
two user, E2/E = 6dB. As it is seen in this figure the decision region of the 
"preset" example is almost identical to the optimum boundary, however, the decision 
boundary for the "trained" example is quite conservative. Such comparisons are 
instrumental not only in identifying the pattern by which decisions are made by 
the neural networks but also in understanding the characteristics of the training 
algorithms. 
PERFORMANCE ANALYSIS 
In this paper, we motivate the application of neural nets to single-user detection 
in multiuser channels by comparing the performance of the receivers in Figure I to 
that of the conventional and the optimum [Poor 88]. Since exact analysis of the bit 
error probabilities for the neural net receivers are analytically intractable, we con- 
sider Monte Carlo simulations. This method can produce very accurate estimates 
of bit-error probability if the number of simulations is sufficiently large to ensure 
occurrence of several erroneous decisions. The fact that these multiuser receivers 
operate with near optimum error rates puts a tremendous computational burden on 
the computer system. The new variance reduction scheme, developed by Orsak and 
Aazhang in [Orsak 89], first shifts the simulated channel noise to bias the simula- 
tions and then scales the error rate to obtain an unbiased estimate with a reduced 
variance. This importance sampling technique, which proved to be extremely effec- 
tive in single-user detection [Orsak 89], is applied to the analysis of the multiuser 
systems. 
As discussed in [Orsak 89], the fundamental issue is to generate more errors by 
biasing the simulations in cases where the error rate is very small. This strategy 
is better described by the two-user Gaussian example in Figure 2. In this example 
the simulation is carried out by generating zero-mean Gaussian noise vectors _, 
random phase 2 and random values of the interfering bit b. Considering b = 1 
(corresponding to signals +a + a2 or +a - a2 which are marked by "+" in Figure 
2) error occurs if the statistics fall on the left side of the decision boundary. It can 
be shown that the most efficient biasing scheme corresponds to a shift of the mean 
of the Gaussian noise and the multiple-access interference such that the mean of the 
statistics are placed on the decision boundary (the shifted signals are marked by 
,,rq,, in Figure 2). Since this strategy generates much more errors than the standard 
Monte Carlo, errors are weighted to obtain an unbiased estimate of the error rate. 
The importance sampling technique substantially reduces the number of simulation 
trials compared to standard Monte Carlo for a given accuracy. In Figure 3 the gain 
which is defined as the ratio of the number of trials required for a fixed variance 
using Monte Carlo to that using the importance sampling method, is plotted versus 
Neural Net Receivers in Multiple-Access Communications 277 
the bit-error probability. In this example, the spreading sequence length, N is equal 
3 and relative energies of the two user, E2/E1 = 6dB. The gain in this example 
of severe near-far problem is inversely proportional to the error rate. Furthermore, 
results from extensive analysis indicated that the proposed importance sampling 
technique is well suited for problems in multi-user communications and less than 
100 trials is sufficient for an accurate error probability estimate. 
NUMERICAL RESULTS 
The performance of the conventional, optimum [Poor 88] and the neural net re- 
ceivers are compared via Monte Carlo simulations employing the importance sam- 
pling method. Except for a difference in length of training periods, the two configu- 
rations in Figure I result in similar average bit-error probabilities. Results presented 
here correspond to the neural net receiver in Figure 1.a. 
A two-user Gaussian channel is considered with severe near-far problem where 
E2/E -- 6dB and spreading sequence length N - 3. In Figure 4, the average 
bit-error probabilities of the four receivers (conventional, optimum, neural nets for 
the "trained" and "preset" examples) are plotted versus the signal to noise ratio of 
the first user (SNR). It is clear from this figure that the two neural net receivers 
outperform the matched filter receiver over the range of SNR. Figure 5 depicts 
these average error probabilities versus the relative energies of the two users (i.e., 
E2/Ei ) for a fixed SNR - 8dB and N - 3. As expected the conventional receiver 
becomes multiple-access limited as E2 increases, however, the performance of the 
neural net receivers closely track that of the optimum receiver for all values of E2. 
We also considered a three-user Gaussian example with a high bandwidth effi- 
ciency and severe near-far problem where spreading sequence length N - 3 and first 
and third users have equal energy and second user has four times more energy (i.e., 
E2/E -- 6dB ). The average error probabilities of the four receivers versus SNRx 
are depicted in Figure 6. The neural net receivers maintained their near optimum 
performance even in this three user example with a spread factor of 3 corresponding 
to a bandwidth efficiency of 1. 
CONCLUSIONS 
In this paper, we consider the problem of demodulating a signal in a multiple- 
access Gaussian channel. The error probability of different neural net receivers were 
compared with the conventional and optimum receivers in a symbol-synchronous 
system. As expected the performance of the conventional receiver (matched filter) is 
very sensitive to the strength of the interfering users. However, the error probability 
of the neural net receiver is independent of the strength of the other users and is 
at least one order of magnitude better than the conventional receiver. Except for a 
difference in the length of training periods, the two configurations in Figure i result 
in similar average bit-error probabilities. However, the training strategies, "preset" 
and "trained", resulted in slightly different error rates and decision regions. 
The multi-layer perceptton was very successful in the classification problem in the 
presence of interfering signals. In all the examples that were considered, two layers 
278 Paris, Orsak, Varanasi and Aazhang 
of perceptrons proved to be sufficient to closely approximate the decision boundary 
of the optimum receiver. We anticipate that this application of neural networks 
will shed more light on the potentials of neural nets in digital communications. The 
issues facing the project were quite general in nature and are reported in many neural 
network studies. However, we were able to address these issues in multiple-access 
communications since the disturbances are structured and the optimum receiver 
(which is NP-hard) is well understood. 
References 
[Aazhang 87] 
[Gschwendtner 88] 
[Lippmann 87] 
[Lupas 89] 
[Orsak 89] 
[Poor 88] 
[Rumelhart 86] 
[Varanasi 89] 
[Verdu 86] 
B. Aazhang and H. V. Poor. Performance of DS/SSMA Com- 
munications in Impulsive Channels-Part I: Linear Correlation 
Receivers. IEEE Trans. Commun., COM-35(11):1179-1188, 
November 1987. 
A. B. Gschwendtner. DARPA Neural Network Study. AFCEA 
International Press, 1988. 
R. P. Lippmann and B. Gold. Neural-Net Classifiers Useful for 
Speech Recognition. In IEEE First Conference on Neural Net- 
works, pages 417-425, San Diego, CA, June 21-24, 1987. 
R. Lupas and S. Verdu. Linear Multiuser Detectors for Syn- 
chronous Code-Division Multiple-Access Channels. IEEE Trans. 
Info. Theory, IT-34, 1989. 
G. Orsak and B. Aazhang. On the Theory of Importance Sam- 
pling Applied to the Analysis of Detection Systems. IEEE 
Trans. Commun., COM-37, April, 1989. 
H. V. Poor and S. Verdu. Single-User Detectors for Multiuser 
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M. K. Varanasi and B. Aazhang. Multistage Detection in 
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Neural Net Receivers in Multiple-Access Communications 279 
Sampler 
(n+l)T 
r(t) 
cos(COcO 
r(t) 
4 
m 
-2- 
-4 
-3 
(a) 
Figure 1. Two Neural Net Receiver Structures. 
o 
Matched Filter  
Neural Net (preset) 
' 
[ ,,di Neural Net ( trained ) 
-2 -1 0 1 2 3 
(b) 
Figure 2. Decision Boundaries of the Various Receivers. 
10 12 
1010 
10 8 
10 6 
10 4 
10 2 
10 0 
10 -13 
Opt. Receiver 
Net (preset) 
Net (trained) 
Matched Filte/ 
10 -11 10 -9 10 -7 10 -5 10 -3 10 -1 
Prob. of Error 
Figure 3. Importance Sampling Gain versus Error Rate for 2-user Example. 
280 Paris, Orsok, Varanasi and Aazhang 
10 -1 
10'2 
lO 
lO 
.1o 
10' 
10 
lO -1 
2 4 6 sSN R 1o 12 14 16 
in dB 
Figure 4. Prob. of Error as a Function of the SNR (E2/E1 = 4). 
10 -1 
Matched Filter 
Neural Net (trained) 
Neural Net (preset) 
Opt. Receiver 
10-4 
0 1 2 3 4 
E2/E1 
Figure 5. Influence of MA-Interference (SNR = 8dB). 
10 -1 . 
10 -2 
10 -3 
10-4 . 
-5 
10 
10 -6 . 
-7 . 
IO 
lO -8 . 
10 -9 . 
10 -10. 
10 -11. 
I0 -12. 
10 -13 
Filter 
Neural Net (trained) 
Neural Net (preset) 
Opt. Receiver 
 ' I ' ' I '  I ' ' I ' ' I '  I ' ' 
2 4 6 8 I0 12 14 
SNR in dB 
Figure 6. Error Curves for the 3-User Example. 
16 
