Content uploaded by Khaled Elleithy
Author content
All content in this area was uploaded by Khaled Elleithy
Content may be subject to copyright.
Content uploaded by Khaled Elleithy
Author content
All content in this area was uploaded by Khaled Elleithy
Content may be subject to copyright.
Content uploaded by Khaled Elleithy
Author content
All content in this area was uploaded by Khaled Elleithy
Content may be subject to copyright.
Abstract- A closed-form expression to determine an average
error rate for synchronous DS-CDMA multi-user detector is
derived based on the transformation matrix (TM) algorithm
proposed in [1]. The derived expression for average bit error
rate (BER) can be used to produce higher and consistent values
of signal-to-noise ratio (SNR) that consequently results good
quality signal. In addition, the closed-form expression can be
used to quantify the multiple access interference (MAI) for a
desirable BER performance by which unwanted signals or
interference can be suppressed relative to the desired signal at
the receiving end. The derived closed-form expression for BER is
not only shown to substantially improve the performance of the
multiuser detectors by means of higher values of SNR but also
has much lower MAI. The performance measure adopted in this
paper is the achievable bit rate for a fixed probability of bit error
(10
-7
) and the consistent values of SNR.
Index Terms—Bit error rate, multiple access interference,
multi-user detection, signal to noise ratio.
I. INTRODUCTION
Commercial applications that based on DS-CDMA systems
have adapted spread spectrum technology and they can extract
parallel benefits such as MAI immunity, low transmit power
density and multiple simultaneous transmissions [7]. In
addition, a spread system requires the same transmitter power
as an un-spread system on the additive white Gaussian noise
channel (AWGN) when MAI is absent. However, we consider
an AWGN channel where the MAI can severally affect the
BER performance. In this paper, we focus on the derivation of
a closed-form expression using the TM algorithm [1]. We
further expand the derived expression for BER to quantifying
the MAI for synchronous DS-CDMA multi-user systems. In
addition, our simulation results suggest that the closed-form
expression for average BER based on the TM algorithm can
be used to produce consistent values of SNR that provide a
strong immunity against MAI.
With the emergence of multiple access techniques, there has
been an increase in the interest in performing simultaneous
estimation and detection over all users [5]. MAI can be
prevented by selecting mutually orthogonal signature
waveforms for all the active users [6]. However, it is not
possible to ensure perfect orthogonality among received
signature waveforms in a mobile environment, and thus MAI
arises. In order to quantify an average MAI in a multiuser DS-
CDMA system, this paper provides a closed-form expression
for an average rate. The derived closed-form expression
demonstrates that how consistent values of SNR could be
used to achieve good quality signal by which unwanted
signals or interference can be suppressed relative to the
desired signal at the receiving end.
The rest of this paper is organized as follows: Section II
presents a brief discussion on the multi-user receivers. Section
III presents the derivation of the closed-form expression for
average BER in a synchronous DS-CDMA system. The
simulation results are provided in Section IV. Finally, Section
V concludes the paper.
II. SYNCHRONOUS MULTIUSER DETECTORS FOR DS-
CDMA COMMUNICATION SYSTEMS
Multi-user receivers can be categorized in the following
two forms: optimal ML sequence estimation (MLSE)
receivers and suboptimal linear and nonlinear receivers. It has
been shown in [5] that DS-CDMA is not fundamentally MAI
limited and can be near-far resistant. Optimal multiuser
detection consists of a matched filter followed by a ML
sequence detector implemented via a dynamic parallel
programming algorithm. In order to mitigate the problem of
MAI, Verdu [2] proposed and analyzed the optimum
multiuser detector for asynchronous Gaussian multiple access
channels. The optimum ML receiver searches all the possible
demodulated bits in order to find the decision region that
maximizes the correlation metric given by [3]. The practical
application of this mechanism is limited by the complexity of
the receiver [1]. This optimal detector outperforms the
conventional detector, but unfortunately its complexity grows
exponentially with a complexity of O (2)
K
, where K is the
number of active users. Consequently, Verdu’s work has
inspired researchers to search for suboptimal multiusers
Syed S. Rizvi and Khaled M. Elleithy
Computer Science and Engineering Department
University of Bridgeport
Bridgeport, CT 06601
{srizvi, elleithy}@bridgeport.edu
Aasia Riasat
Department of Computer Science
Institute of Business Management
Karachi, Pakistan 78100
Aasia.riasat@iobm.edu.pk
A Closed-Form Expression for BER to
Quantify MAI for Synchronous DS-CDMA
Multi-user Detector
978-1-4244-1870-1/08/$25.00 (c) 2008 IEEE
Authorized licensed use limited to: University of Bridgeport. Downloaded on February 24,2010 at 13:31:14 EST from IEEE Xplore. Restrictions apply.
detectors, which can achieve near-optimal performance with
comparatively less computational complexity. Suboptimal
receivers also simultaneously detect all user signals. However,
instead of ML detection, they perform a set of linear
transformations on the outputs of a matched filter that
significantly enhances the noise component. Multiuser
detection is therefore essential for the efficient operation of a
DS-CDMA system.
Several new algorithms have been proposed in the literature
[1, 4], in order to reduce the complexity of the original ML
detection scheme proposed by Verdu. Recently, [4] proposed
a ML receiver that uses the neighboring decent (ND)
algorithm. They implemented an iterative approach using the
ND algorithm to locate the region where the actual
observations belong. The linearity of the iterative approach
increases noise components at the receiving end. Due to the
enhancement in the noise components, the SNR and BER of
the ND algorithm are more affected by the MAI. Specifically,
[1] proposed a TM algorithm that observes the coordinates of
the constellation diagram to determine the location of the
transformation points. Since most of the decisions are correct,
the TM algorithm can reduce the number of required
computations by using transformation matrices only on those
coordinates which are most likely to lead to an incorrect
decision. By doing this, TM algorithm can greatly reduce the
unnecessary processing involved in making decisions about
the correct region or coordinates
III. THE CLOSED-FORM EXPRESSION FOR BER
We consider an AWGN channel where the MAI can
severally affect the BER performance. Furthermore, we
consider that the variations in the network load for an AWGN
channel introduce the presence of variance that represents
MAI. In other words, we use the values of variance in the
closed-form expression for the BER that may vary with
respect to the number of user presents in a communication
system.
A. Closed-Form Expression
We modeled the cellular network as a linear time invariant
(LTI) synchronous DS-CDMA system in which users utilize
an AWGN multi-path channel. Due to the AWGN channel
and the linearity property, the different signal components do
not experience deep fades. In other words, if the signal
changes during the transition, the receiver receives the
following signal:
( ) ( ) ( )
j
t e s t t
θ
η
−
ℜ = Α + +
(1)
where
A
is an attenuation factor,
θ
is a phase shift,
(
)
s t
is the desired signal, and
(
)
t
η
is the additive Gaussian noise.
Due to LTI characteristics, the derived expression is
independent of the phase shift. Therefore, the receiver
receives the following signal:
(
)
(
)
(
)
t s t t A
η
ℜ = + +
(2)
Since the attenuation factor
A
is uncorrelated with
(
)
t
η
,
we can use the expression of SNR derived in [1] directly in
the BER formula. Consider (3) to determine the BER in an
AWGN channel for a system where the transmitted bits are
modulated using the BPSK modulation technique.
BER 1 1
Q SNR
=
(3)
Since the attenuation factor and the white noise are
uncorrelated, the SNR can be directly placed in (4) as follows:
BER
( )
2
1
2
1 10Q SNR
σ
−
+
=
(4)
For simplicity, (4) can also be written as:
BER =
2
10 1 10
Q SNR SNR
σ
+
(5)
The second term in (4) represents the SNR degradation due
to MAI. This term depends on the cross-correlation between
the spreading code as well as the number of users. In other
words, an increase in K causes an increase in the second term
of (4) which causes a decrease in the overall BER
performance. According to our initial assumption, our cellular
network is modeled as a LTI synchronous DS-CDMA system
for AWGN multipath channel. Taking this into account, if k
th
signal changes during the transition, (3) can be modified as:
k k k
j
e s N
θ
η
−
ℜ = Α + +
(6)
In (6), the first, second, and third term represent the MAI
component, the desired signal component, and the noise
component, respectively. Our model for cellular network is
not affected by a phase shift and frequency shift.. Therefore,
this simplifies (6) as:
k k k
s N A
η
ℜ = + +
(7)
where N is usefully interpreted as noise. The second term
in (7) is a zero mean Gaussian random variable with variance.
The third term of (7) is a MAI component that can be defined
as:
( )
2
cos
K
k k k
k
A A U
φ
=
=
∑
(8)
where
k
A
represents the envelop of a complex Gaussian
process with unit variance in each quadrature component
and
k
U
represents a non-faded amplitude of the
th
k
signal. In
(8),
k
φ
is a uniform random variable that represents the phase
Authorized licensed use limited to: University of Bridgeport. Downloaded on February 24,2010 at 13:31:14 EST from IEEE Xplore. Restrictions apply.
difference of the
th
k
user. Since we assume LTI channel, the
possibility of phase variation for K users can be neglected.
Even though, the right hand side of equation (8) is
independent and represents a Gaussian distribution process
over a range of 0 to 2
π
, the left hand side is not a pure
Gaussian function.
( )
2
2
10
1 10
k k
K
k k k
k
SNR
s SNR Q
SNR
A U
σ
η
=
ℜ = +
+
+
∑
(9)
The first two terms of (9) can be used to approximate the
error rate and the MAI for a user k. It should be noted that the
second term gives an average variance of the MAI over all
possible operating conditions that can be used to compute the
required SNR for a desirable BER performance. We use (9) in
conducting the simulation result and performing the
experimental verification by giving the non-folded amplitude
of the
th
k
signal and computing the corresponding MAI as a
Gaussian random variable with zero mean and conditional
variance that represents by
2
σ
. Similarly, the first terms of
(9) can be used to approximate the suppression of MAI for a
desirable BER performance. Since there is no closed-form
mathematical relationship exists between the first two terms of
(9), we believe this is the optimal approximation of the MAI
though the closed-form expression of average BER.
IV. PERFORMANCE ANALYSIS OF THE CLOSED-FORM
EXPRESSION FOR BER
Although, we have made some strict assumptions about the
perfect power control, the derived expression results
approximately 4 to 7 dB performance improvement over the
ND algorithm. In (9), we present a closed-form expression to
quantify MAI for the characteristic function involving both
the Q function and the SNR. Since the Q function is widely
available in many scientific software tools such as Matlab,
components of (9) can be programmed for direct evaluation.
A. Multiple Access Interference (MAI)
It should be noted that the MAI has a Gaussian-like shape
that decays exponentially with respect to the non-faded
amplitude of the
th
k
signal as shown in Fig. 1. As one can
observe in Fig. 1 that the convergence of MAI is exponential
due to the divergence in the SNR which is extremely a
desirable property for a cellular network. Thus the results of
Fig. 1 shows that the second term of (9) are extremely well
behaved in the sense that they are smooth, strictly non
negative, and decay exponentially due to the higher values of
SNR. In harmony with our expectations, as the number of
users, K, increased, the average SNR of the system decrease
linearly. This slight decrease in SNR causes a decrease in the
rate at which MAI diverges with respect to the SNR value as
shown in Fig. 2.
B. Consistent Signal-to-Noise Ratio (SNR)
Fig. 3 shows one of the possible cases of a heavily-loaded
network where 102 active users transmit BPSK modulated
signals. For a small value of K, the proposed TM algorithm
achieves approximately 6.5 dB of SNR where as the ND and
the ML algorithms give 5.8 and 5.5 dB, respectively. This
implies that a slight increase in the value of K forces the TM
algorithm to give an acceptable value of SNR that can be used
to achieve a satisfactory BER performance at least for a voice
communication network.
0 0.05 0.1 0.15 0.2 0.25
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
MAI for a range of variance versus non-faded amplitude
non-faded amplitude (U
k
)
MAI for a range of variance
MAI for a range of variance
Fig.2 MAI for a range of
2
σ
with K=10, and SNR=12 dB, N is computed
using the first term of (9)
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
MAI for a range of variance versus non-faded amplitude
non-faded amplitude (U
k
)
MAI for a range of variance
MAI for a range of variance
Fig.1 MAI for a range of
2
σ
with K=5, and SNR=14 dB, N is computed
using the first term of (9)
Authorized licensed use limited to: University of Bridgeport. Downloaded on February 24,2010 at 13:31:14 EST from IEEE Xplore. Restrictions apply.
Fig. 3 shows that the linear increase in SNR is consistent
not only for a lightly-loaded network (i.e., the value of K is
less than 50) but also for a heavily-loaded network (i.e., when
K is greater than 100). However, this can also be noticed from
Fig. 3 that as the number of users increase in the system, the
differences between the SNR values for the proposed
algorithm and the other two ML and the ND algorithms
become wider. From Fig. 3, the TM algorithm gives
approximately 36 dB for K = 72 which is more than what we
expect to achieve for an optimal BER performance. In
addition to that, the random amount of variance is more
affected on the SNR values in a heavily-loaded case than in a
lightly-loaded case.
C. Bit Error Rate (BER) Performance Analysis
The standard performance criterion in digital
communications is the probability of BER. Some voice-band
modem applications permit error rates no greater than 10
-5
,
whereas other applications such as digitized voice tolerate
error rates as high as 10
-2
to 10
-3
. Simulation results show that
the closed-form expression provides better approximation of
BER for the proposed TM algorithm than the ML and the ND
algorithms for all values of SNR. Fig. 4 and 5 show a plot of
three BER versus SNR curves. These curves were plotted
using (9) in an AWGN channel.
The results for a lightly-loaded network (Fig. 4)
demonstrate that an optimal BER performance can be
achieved for a reasonable range of SNR. It should be noted
that the BER performance of the TM algorithm is always
better than the ML and the ND algorithms as shown in Fig. 4.
For the first few values of SNR, the ND algorithm almost
approaches the ML algorithm whereas the TM algorithm still
maintains a reasonable performance difference. It can be seen
in Fig. 4 that the TM algorithm achieves less than 10
-2
BER
for SNR = 8 dB which is quite closed to the required
reasonable BER performance for a voice communication
system. For small values of SNR, the BER for these three
algorithms is almost equal, but as we increase the value of
SNR, typically more than 10 dB, we can observe the
difference in the BER performance.
Fig. 5 shows the plot of BER versus SNR curves for
synchronous DS-CDMA system for large values of K. We
can see that as the number of users increases in the system, the
BER advantage of the proposed TM algorithm over the ND
and the ML algorithms decreases as shown in Fig. 5. The
BER performance analysis for a heavily-loaded case
corresponds to the fact that as the BER increases, the range of
the precomputed desire values of an average SNR decreases
0 1 2 3 4 5 6 7 8
10
-3
10
-2
10
-1
10
0
SNR (dB)
BE R
ML
ND
Proposed
Fig.4. BER versus SNR (0<dB<9) curves for a lightly loaded network
2 12 22 32 42 52 62 72 82 92 102
5
10
15
20
25
30
35
40
45
50
U S E R S
S N R
ML
ND
Proposed
Fig.3 Approximate value of SNR (dB) versus number of users (K =102,
heavily-loaded network) with a random amount of variance for a synchronous
system in an AWGN channel.
0 2 4 6 8 10 12 14 16 18 20 22 24 26
10
-5
10
-4
10
-3
10
-2
10
-1
SNR (dB)
BE R
ML
ND
Proposed
Fig.5. BER versus SNR (0<dB<26) curves for a heavily-loaded network
Authorized licensed use limited to: University of Bridgeport. Downloaded on February 24,2010 at 13:31:14 EST from IEEE Xplore. Restrictions apply.
and hence the BER performance of the proposed TM
algorithm slightly deteriorates.
According to the simulation results for a heavily-loaded
case, the proposed TM algorithm maintains around 10
-4
BER
performance as SNR goes to infinity. On the other hand, the
ND algorithm maintains no less than 0.5 X 10
-3
BER
performance whereas the ML algorithm maintains
approximately 10
-3
BER performance as SNR reaches to
infinity.
It was expected that as the number of users, K, in the system
increased, the BER performance would also suffer due to the
increase in MAI. Recall that the second term of (9) is directly
proportional to the number of users, K. The increase in the
second term of (9) causes an increase in MAI that degrades
the values of SNR. The degradation in SNR performance
increases the BER. However, it should be noted that the
performance differences for SNR = 0 to 10 dB are small
compared to the performance differences for SNR = 10 to 26
dB. This was evident in simulation results using MATLAB, as
shown in Fig. 5.
Furthermore, for a heavily-loaded network, we increase the
number of users in the system that would implicitly increase
the value of the variance, which is used in the calculation of
the BER. Consequently, the BER performance for the TM
algorithm in a heavily-loaded network is slightly affected by
increasing the value of K. But due to the reduced
computational complexity of the proposed TM algorithm, we
comparatively achieve better BER performance over the ND
and the ML algorithms.
V. CONCLUSION
In this paper, we presented a closed-form expression of
average error rate in synchronous multi-user DS-CDMA
systems. The principle advantage of the derived closed-form
expression is that it can be used to correctly quantify the MAI
for a desirable BER performance. In order to show the
consistency and the correctness of the closed-form expression,
we presented the simulation results for both lightly and
heavily loaded networks to compute the average BER for
different ranges of SNR. The simulation results of the MAI
demonstrated that the unwanted signals or interference can be
effectively reduced relative to the desired signal at the
receiving end. The simulation results of BER suggested that
the derived closed-form expression provides better BER
performance of the TM algorithm for all values of SNR.
REFERENCES
[1] Syed S. Rizvi, K. M. Elleithy, and Aasia Riasat, “Transformation
Matrix Algorithm for Reducing the Computational Complexity of
Multiuser Receivers for DS-CDMA Wireless Systems,” accepted in
Wireless Telecommunication Symposium (WTS 2007), Pomona,
California, April 26-28 2007.
[2] S. Verdu, Multiuser Detection. Cambridge University Press, 1988.
[3] S. Verdu, “Minimum probability of Error for Asynchronous Gaussian
Multiple access Channels,” IEEE Transaction on Information Theory,
Vol. IT-32, Issue-1, pp. 85–96, January 1986.
[4] T. Ottosson and E. Agrell, “ML optimal CDMA Multiuser Receiver,”
Electronics Letters, Vol. 31, Issue-18, pp. 1544-1555, August 1995.
[5] P. Tan and K. Lars, “Multiuser Detection in CDMA: A Comparison of
Relaxations, Exact, and Heuristic Search Methods,” July 09, 2003.
Available at:
http://fibbla.ce.chalmers.se/CCN/publications/04/TanRas04TWC.pdf
[6] Z. Lei, and T. Lim, “Simplified Polynomial Expansion Linear
Detection for DS-CDMA systems,” IEEE Electronic Letters, Vol. 34,
No. 16, pp. 1561-1563, August 1998.
[7] R.L. Peterson, R.E. Ziemer, and D.E. Borth, Introduction to Spread
Spectrum Communications, Englewood Cliffs, N.J., Prentice Hall,
1995.
Authorized licensed use limited to: University of Bridgeport. Downloaded on February 24,2010 at 13:31:14 EST from IEEE Xplore. Restrictions apply.
