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Content uploaded by Khaled Elleithy

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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

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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

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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)

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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

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.

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Wireless Telecommunication Symposium (WTS 2007), Pomona,

California, April 26-28 2007.

[2] S. Verdu, Multiuser Detection. Cambridge University Press, 1988.

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Available at:

http://fibbla.ce.chalmers.se/CCN/publications/04/TanRas04TWC.pdf

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