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IJCSNS International Journal of Computer Science and Network Security, VOL.9 No.6, June 2009

1

A Low

A LowA Low

A Low-

--

-Complexity Algorithm for Improving Computing

Complexity Algorithm for Improving Computing Complexity Algorithm for Improving Computing

Complexity Algorithm for Improving Computing

Power of Multiuser Receivers for Next Generation Wireless

Power of Multiuser Receivers for Next Generation Wireless Power of Multiuser Receivers for Next Generation Wireless

Power of Multiuser Receivers for Next Generation Wireless

Systems

SystemsSystems

Systems

Syed S. Rizvi

†

††

†

, Aasia Riasat

††

††††

††

, and Khaled M. Elleithy

†††

††††††

†††

srizvi@fuee.bridgeport.edu aasia.riasat@iobm.edu.pk elleithy@bridgeport.edu

†

††

†

Computer Science and Engineering Department, University of Bridgeport, Bridgeport, CT, 06604 USA

††

††††

††

Computer Science Department, Institute of Business Management, Karachi, 78100, Pakistan

†††

††††††

†††

Computer Science and Engineering Department, University of Bridgeport, Bridgeport, CT, 06604 USA

Summary

A new transformation matrix (TM) algorithm for reducing the

computational complexity of multiuser receivers for DS-CDMA

wireless system is presented. Next generation multiuser wireless

receivers will need to use low computational complexity

algorithm in order to perform both fast signal detection and error

estimation. Several multiuser signal detection algorithms have

proposed for next generation wireless receivers, which are

designed to give good performance in terms of signal to noise

ratio (SNR) and bit error rate (BER), are discarded for a direct

implementation as they have high computational complexity. In

this paper, we propose a new low-complexity TM algorithm that

can be used to perform fast signal detection for multiuser

wireless receives. This reduction in computational complexity

would likely to give us a considerable improvement in the

performance of multiuser wireless receivers such as high

computing power and low error rate. In addition, we also present

a formal mathematical proof for computational complexities that

verifies the low-complexity of the proposed algorithm

Key words:

Computational complexity, DS-CDMA systems, Multiuser

communications, Wireless receivers

1. Introduction

Code division multiple access (CDMA) has been widely

used and accepted for wireless access in terrestrial and

satellite applications. CDMA cellular systems use state of

the art digital communication techniques and build on

some of the most sophisticated aspects of modern

statistical communication theory. CDMA technique has

significant advantages over the analog and conventional

time-division-multiple access (TDMA) system. CDMA is

a multiple access (MA) technique that uses spread

spectrum modulation where each user has its own unique

chip sequence. This technique enables multiple users to

access a common channel simultaneously.

Multiuser direct-sequence code division multiple access

(DS-CDMA) has received wide attention in the field of

wireless communications [4, 8]. In CDMA communication

systems, several users are active on the same fringe of the

spectrum at the same time. Therefore, the received signal

results from the sum of all the contributions from the

active users [2]. Conventional spread spectrum

mechanisms applied in DS-CDMA are severely limited in

performance by MAI [3, 4], leading to both system

capacity limitations and strict power control requirements.

The traditional way to deal with such a situation would be

to process the received signal through parallel devices.

Verdu’s [1] proposed and analyzed the optimum

multiuser detector and the maximum likelihood sequence

detector, which, unfortunately, is too complex for practical

implementation, since its complexity grows exponentially

as the function of the number of users. Although the

performance of multiuser detector is optimum, it is not a

very practical system because the number of required

computations increases as 2

k

, where k is the number of

users to be detected. Multiuser detectors suffer from their

relatively higher computational complexity that prevents

CDMA systems to adapt this technology for signal

detection. However, if we could lower the complexity of

multiuser detectors, most of the CDMA systems would

likely to get advantage of this technique in terms of

increased system capacity and a better data rate.

In this paper, we employ a new approach of TM

technique that observes the coordinates of the constellation

diagram to determine the location of the transformation

points (TPs). Since most of the decisions are correct, we

can reduce the number of required computations by using

transformation matrixes only on those coordinates which

are most likely to lead to an incorrect decision. By doing

this, we can greatly reduce the unnecessary processing

involves in making decisions about the correct region or

the coordinate. Our mathematical results show that the

proposed approach successfully reduces the computational

complexity of the optimal ML receiver.

The rest of this paper is organized as follows. Section 2

describes the state of the art research that has already been

done in this area. Section 3 presents both the original ML

IJCSNS International Journal of Computer Science and Network Security, VOL.8 No.11, November 2008

2

algorithm and the proposed TM algorithm along with a

comprehensive discussion of their computational

complexities. The numerical and simulation results are

presented in Section 4. Finally, we conclude the paper in

Section 5.

2. Related Work

Multiuser receivers can be categorized in the following

two forms: optimal maximum likelihood sequence

estimation (MLSE) receivers and suboptimal linear and

nonlinear receivers. Suboptimal multiuser detection

algorithms can be further classified into linear and

interference cancellation type algorithms. The figurative

representation of the research work that has been done so

far in this area is shown in Fig. 1. Optimal multiuser

wireless receiver consists of a matched filter followed by a

maximum likelihood sequence detector implemented via a

dynamic programming algorithm. In order to mitigate the

problem of MAI, Verdu [6] proposed and analyzed the

optimum multiuser detector for asynchronous Gaussian

multiple access channels. The optimum detector searches

all the possible demodulated bits in order to find the

decision region that maximizes the correlation metric

given by [1]. The practical application of this mechanism

is limited by the complexity of the receiver [7]. This

optimum detector outperforms the conventional detector,

but unfortunately its complexity grows exponentially in

the order of O (2)

K

, where K is the number of active users.

Much research has been done to reduce this receiver’s

computational complexity. Recently, Ottosson and Agrell

[5] proposed a new ML receiver that uses the neighbor

descent (ND) algorithm. They implemented an iterative

approach using the ND algorithm to locate the region

where the actual observations belong. In order to reduce

the computational complexity of optimum receivers, the

iterative approach uses the ND algorithm that performs

MAI cancellation linearly. The linearity of their iterative

approach increases noise components at the receiving end.

Due to the enhancement in the noise components, the SNR

and BER of ND algorithm is more affected by the MAI.

Several tree-search detection receivers have been

proposed in the literature [10, 11], in order to reduce the

computational complexity of the original ML detection

scheme proposed by Verdu. Specifically, [10] investigated

a tree-search detection algorithm, where a recursive,

additive metric was developed in order to reduce the

search complexity. Reduced tree-search algorithms, such

as the well known M-algorithms and T-algorithms [12],

were used by [11] in order to reduce the complexity

incurred by the optimum multiuser detectors.

In order to make an optimal wireless receiver that gives

minimum mean square error (MMSE) performance, we

need to provide some knowledge of interference such as

phase, frequency, delays, and amplitude for all users. In

addition, an optimal MMSE receiver requires the inversion

of a large matrix. This computation takes relatively long

time and makes the detection process slow and expensive

[7, 8]. On the other hand, an adaptive MMSE receiver

greatly reduces the entire computation process and gives

an acceptable performance. Xie, Rushforth, Short and

Moon [13] proposed an approximate MLSE solution

known as the pre-survivor processing (PSP) type

algorithm, which combined a tree search algorithm for

data detection with the aid of the recursive least square

(RLS) adaptive algorithm used for channel amplitude and

phase estimation. The PSP algorithm was first proposed by

Seshadri [14] for blind estimation in single user

ISI-contaminated channels.

3. Proposed Low-Complexity TM Algorithm

We consider a synchronous DS-CDMA system as a

linear time invariant (LTI) channel. In a LTI channel, the

probability of variations in the interference parameters,

such as the timing of all users, amplitude variation, phase

shift, and frequency shift, is extremely low. This property

makes it possible to reduce the overall computational

complexity at the receiving end. Our TM technique utilizes

the complex properties of the existing inverse matrix

algorithms to construct the transformation matrices and to

determine the location of the TPs that may occur in any

coordinate of the constellation diagram. The individual

TPs can be used to determine the average computational

complexity.

The system may consist of K users. User k can

transmit a signal at any given time with the power of W

k

.

With the binary phase shift keying (BPSK) modulation

technique, the transmitted bits belong to either +1 or -1,

(i.e.,

{ 1}

k

b

∈ ±

). The cross correlation can be reduced by

neglecting the variable delay spreads, since these delays

are relatively small as compared to the symbol

transmission time. In order to detect signals from any user,

the demodulated output of the low pass filter is multiplied

by a unique signature waveform assigned by a pseudo

Multiuser Receivers

Optimal MLSE

Receivers

Non-Linear

Suboptimal

Receivers

Linear

Fig.1.Multiuser optimal and suboptimal wireless receivers

IJCSNS International Journal of Computer Science and Network Security, VOL.8 No.11, November 2008

3

random number generator. It should be noted that we

extract the signal using the match filter followed by a

Viterbi algorithm [15].

3.1 Original Optimum Multiuser Receiver

The optimum multiuser receiver exists and permits to

relax the constraints of choosing the spreading sequences

with good correlation properties at a cost of increased

receiver complexity. Fig. 2 shows the block diagram of an

optimum receiver that uses a bank of matched filters and a

maximum likelihood Viterbi decision algorithm [15] for

signal detection. It should be noted in Fig. 2 that the

proposed TM algorithm is implemented in conjunction

with the Viterbi decision algorithm [15] with the feedback

mechanism. In order to detect signal from any user, the

demodulated output of the low pass filter is multiplied by a

unique signature waveform assigned by a pseudo random

number generator.

When receiver wants to detect the signal from user-1, it

first demodulates the received signal to obtain the

base-band signal. The base-band signal multiplies with

user-1’s unique signature waveform,

(

)

1

C t

.The resulting

signal,

1

( )

r t

, is applied to the input of the matched filter.

The matched filter integrates the resulting signal

1

{ ( )}

r t

over each symbol period T, and the output is read into the

decoder at the end of each integration cycle. The outputs

of the matched filter and the Verdu’s algorithm can be

represented by

( )

k

y m

and

( )

k

b m

, respectively where m is

the sampling interval. We also assume that the first timing

offset τ

1

is almost zero and τ

2

< T. The same procedure

applies to other users. The outputs of the matched filter for

the first two users at the m

th

sampling interval can be

expressed as follows:

( ) ( ) ( )

{ }

( )

( )

{

}

1

1 1 1

1

m T

m T

y m r t C t dt

T

+

=

∫

(1)

( ) ( ) ( )

{ }

( )

( )

{

}

2 1

2 2 2 2

2

1

m T

m T

y m r t C t dt

T

τ

+ +

+

= −

∫

(2)

The received signal

(

)

1

r t

and

(

)

2

r t

can be

expressed as follows:

( )

( )

( ) ( )

{ }

0.5

1 1 1

1

M

b

i M

r t E b i C t iT

C

=−

∑

= − (3)

( )

(

)

( ) ( )

{ }

0.5

2 2 2 2

2

M

C b

i M

r t E b i C t iT

τ

=−

= − −

∑

(4)

where

1

C

E

and

2

C

E

represent the original bit energy of the

received signals with respect to their unique signature

waveforms.

The received signals

(

)

1

r t

and

(

)

2

r t

can be treated

as a single signal

(

)

r t

that will be distinguished by the

receiver with respect to its unique signature waveform.

Based on the above analysis, we can combine equation (3)

and (4).

( )

(

)

( ) ( )

{ }

( )

( ) ( )

{ }

0.5

1 1

1

0.5

2 2 2

2

M

C b

i M

M

C b

i M

r t E b i C t iT

E b i C t iT

τ

=−

=−

= −

∑

+ − −

∑

(5)

Substitute (5) as an individual equation into (1), we have

( )

( )

( ) ( )

{ }

{

}

( 1)

0.5

1 1

1

( )

1

1

1

( )

m T

M

C b

i M

m T

E b i C t iT

y m

T

C t dt

+

=−

−

∑

∫

=

(6)

Substitute (5) as an individual equation into (2), we have

( )

( )

( ) ( )

{ }

{

}

( )

( 1)

0.5

2 2 2

2

( )

2

2 2

1

m T

M

C b

i M

m T

E b i C t iT

y m

T

C t dt

τ

τ

+

=−

− − ×

∑

∫

=

−

(7)

After performing integration over the given interval, we

get the following results with the noise components as

well as the cross correlation of signature waveforms.

( )

(

)

( )

(

)

( )

( )

( )

( )

( ) ( )

0.5 0.5

1 1 2 1

1 2

0.5 0.5

2 0 2 1 1

2 2

= 1

1

C C

C C

y m E b m E b m

E b m E b m n m

ρ

ρ ρ

−

+ − +

+ + +

(8)

( ) ( )

( ) ( ) ( )

0.5 0.5

( 1)

2 2 1 1

2 1

0.5 0.5

1

1 0 1 1 2

1 1

y m E b m E b m

C C

E b m E b m n m

C C

ρ

ρ ρ

= + − +

+ + +

−

(9)

where coefficients b

1

(m) and b

2

(m)

represent MAI,

1/ 0 / 1

ρ

− +

are cross-correlations of signature waveforms,

and n

1

(m) and n

2

(m) represent the minimum noise

components. Since the channel is LTI, the probability of

unwanted noise is minimum.

These symbols can now be decoded using a maximum

likelihood Viterbi decision algorithm [15]. Viterbi

algorithm can be used to detect these signals in much the

same way as convolution codes. This algorithm makes

decision over a finite window of sampling instants rather

IJCSNS International Journal of Computer Science and Network Security, VOL.8 No.11, November 2008

4

than waiting for all the data to be received [4]. The above

derivation can be extended from two users to K number of

users. The number of operations performed in the Viterbi

algorithm is proportional to the number of decision states,

and the number of decision states is exponential with

respect to the total number of users. The asymptotic

computational complexity of this algorithm can be

approximated as: O(2)

K

.

3.2 Proposed Transformation Matrix (TM) Algorithm

According to original Verdu’s algorithm, the outputs of

the matched filter

(

)

1

y m

, and

(

)

2

y m

can be

Z

Carrier

Recovery

Carrier

Recovery

C

1

(t)

r

2

(t)

X

X

y

1

(m)

y

2

(m)

X

Band Pass

Filter

Low Pass

Filter

X

Matched Filter

∫ [.] dt

Matched Filter

∫ [.] dt

C

2

(t)

r

1

(t)

r

x

(t)

Viterbi

Decision

Algorithm

Low Pass

Filter

Feedback

Path

Extend the fundamental

equation for K transformation

points

Set up the equation for K points with

respect to their individual computational

complexity

Create transformation matrix

that consists of K

transformation points

Transformation Matrix Technique

Perform transformation

matrix sum

Approximation of an average

asymptotic computational

complexity

Fig.2 Implementation of proposed transformation matrix (TM) algorithm with the optimum multiuser receiver

IJCSNS International Journal of Computer Science and Network Security, VOL.8 No.11, November 2008

5

considered as a single output

(

)

y m

.In order to minimize

the noise components and to maximize the received

demodulated bits, we can transform the output of the

matched filter, and this transformation can be expressed as

follows:

(

)

by m

η

= Τ +

where

Τ

represents the

transformation matrix,

{ 1}

b

k

∈ ±

and

η

represents the

noise components. In addition, if vectors are regarded as

points in K-dimensional space, then the vectors constitute

the constellation diagram that has K total points. This

constellation diagram can be mathematically expressed as:

{

}

{

}

1, 1

b where b

ℵ = Τ ∈ − +

. We use this equation as

a fundamental equation of the proposed algorithm.

According to the detection rule, the constellation diagram

can be partitioned into 2

K

lines (where the total possible

lines in the constellation diagram can be represented as ſ )

that can only intersect each other at the following points:

ℵ

= {Tb}

b ∈{-1, 1}

K

¥ ſ

Fig. 3 shows the constellation diagram that consists of

three different vectors (lines) with the original vector ‘Χ’

that represents the collective complexity of the receiver. Q,

R, and S represent vectors or TPs within the coverage area

of a cellular network as shown in Fig. 3. In addition, Q

¬

,

R

¬

, and S

¬

represent the computational complexity of each

individual TP. In order to compute the collective

computational complexity of the optimum wireless

receiver, it is essential to determine the complexity of each

individual TP. The computational complexity of each

individual TP represents by X

¬

of the TP which is equal to

the collective complexity of Q

¬

, R

¬

, and S

¬

. A TM defines

how to map points from one coordinate space into another.

A transformation does not change the original vector,

instead it alters the components. In order to derive the

value of the original vector X, we need to perform the

following derivations. We consider the original vector with

respect to each transmitted symbol or bit. The following

system can be derived from the above equations:

i i ji k i

X Q

X Q

X R ij jj k j X R

X S

X S

i k jk k k

¬ ¬ ¬

¬

¬ ¬ ¬ ¬

=

¬

¬ ¬ ¬

(10)

Equation (10) represents the following: QRS with the

unit vectors

, , and

i j k

;

, , a n d

X Q X R X S

¬ ¬ ¬

with the inverse of the

unit vectors

and

, ,

i j k

¬ ¬ ¬

. The second matrix on the

right hand side of (10) represents b, where as the first

matrix on the right hand side of (10) represents the actual

TM. Therefore, the TM from the global reference points

(which could be Q, R, or S) to a particular local reference

point can now be derived from (10):

/

X Q

X Q

X R T X R

L G

X S

X S

¬

¬

=

¬

(11)

Equation (11) can also be written as:

/

ii ji ki

T ij j j k j

L G

ik jk k k

¬ ¬ ¬

¬ ¬ ¬

=

¬ ¬ ¬

(12)

In equation (12), the dot products of the unit vectors of

the two reference points are in fact the same as the unit

vectors of the inverse TM of (11). We need to compute the

locations of the actual TPs described in equations (11) and

(12). Let the unit vectors for the local reference point be:

( ), ( ) , ( )

1 1 1 2 1 3

( ), ( ), ( )

2 1 2 2 2 3

( ) , ( ) , ( )

3 1 3 2 3 3

i t i t j t k

j t i t j t k

k t i t j t k

¬

=

¬

=

¬

=

(13)

Since,

(

)

i i j k i

¬ ¬

+ + =

, where

(

)

1

i j k

+ + =

. The

same is true for the rest of the unit vectors (i.e.,

i i

¬ ¬

=

).

Therefore, (13) can be rewritten as:

1 1 1 2 1 3

2 1 2 2 2 3

3 1 3 2 3 3

, ,

, ,

, ,

i t t t

j t t t

k t t t

¬

=

¬

=

¬

=

(14)

By substituting the values of

and

, ,

i j k

¬ ¬ ¬

from (14)

into (12), we obtain

(

)

(

)

(

)

( ) ( ) ( )

( ) ( ) ( )

11 12 13 11 12 13 11 12 13

/ 21 22 23 21 22 23 21 22 23

31 32 33 31 32 33 31 32 33

11 12 13

/ 21 22 23

31 32 33

i t i t j t k j t i t j t k k t i t j t k

T i t i t j t k j t i t j t k k t i t j t k

L G

i t i t j t k j t i t j t k k t i t j t k

t t t

T t t t

L G

t t t

+ + + + + +

= + + + + + +

+ + + + + +

=

(15)

Substituting T

L/G

from (15) into (11), yields

1 1 1 2 1 3

2 1 2 2 2 3

3 1 3 2 3 3

t t t

X Q

X Q

X R t t t X R

X S

X S

t t t

¬

¬

=

¬

(16)

IJCSNS International Journal of Computer Science and Network Security, VOL.8 No.11, November 2008

6

Equation (16) corresponds to the following standard

equation used for computing the computational complexity

at the receiving end:

{

}

{ }

b 1, 1

b

k

∈ − +ℵ = Τ If the target

of one transformation

(

)

:

U Q R

→

is the same as the

source of other transformation

(

)

:

T R S

→

, then we can

combine two or more transformations and form the

following composition:

( ) ( )

(

)

: ,

TU Q S TU Q T U Q

→ =

.

This composition can be used to derive the collective

computational complexity at the receiving end using (16).

Since the channel is assumed to be LTI, the TPs may occur

in any coordinate of the constellation diagram. The

positive and negative coordinates of the constellation

diagram do not make any difference for a LTI propagation

channel. In addition, the TPs should lie within the

specified range of the system. Since we assumed that the

transmitted signals are modulated using BPSK which can

at most use 1 bit out of 2 bits (that is,

{ 1}

b

k

∈ ±

),

consider the following set of TPs to approximate the

number of demodulated received bits that need to search

out by decision algorithm:

( ) ( )

( ) ( )

1 1 1 1 1 1

1 1 1 1 0 1 1 1 1 1 1 1

1 1 1 1 1 1

1 1 1 1 1 1

1 -1 1 1 1 1 1 -1 1 1 1 1

1 1 1 1 1 1

K

K

−

ℵ = + −

−

− − − − −

+ − − + − − −

− − − − −

Using (16), a simple matrix addition of the received

demodulated bits can be used to approximate the number

of most correlated TPs. The set of TPs correspond the

actual location with in the TM as shown in (16). The entire

procedure for computing the number of demodulated bits

that need to search out by the decision algorithm can be

used to approximate the number of most correlated signals

for any given set of TPs. This is because, we need to check

weather or not the TPs are closest to either (+1, +1) or (-1,

-1). The decision regions or the coordinates where the TPs

lie for (+1, +1) and (-1, -1) are simply the corresponding

transformation matrixes that store the patterns of their

occurrences. If the TPs do not exist in the region

(coordinate) of either (+1, +1) or (-1, -1), then it just a

matter of checking weather the TPs are closest to (+1, -1)

or to (-1, +1). In other words, the second matrix on the

right hand side of (16) requires a comprehensive search of

at most 5

K

demodulated bits that indirectly correspond to

one or more users. The minimum search performed by the

decision algorithm is conducted if the TPs exist within the

incorrect region. Since the minimum search saves

computation by one degree, the decision algorithm has to

search at least 4

k

demodulated bits. The average number of

computations required by a system on any given set

always exists between the maximum and the minimum

number of computations performed in each operational

cycle [9]. This implies that the total number of

demodulated bits that need to search out by the decision

algorithm can not exceed by5

K

-4

K

. In other words, the

total numbers of most correlated pairs are upper-bounded

by5

K

-4

K

.

Since most of the decisions are correct, we can reduce

the number of computations by using the transformation

matrixes only on those coordinates that are most likely

lead to an incorrect decision. In other words, TM does not

process those coordinates which are most likely lead to a

correct decision. By doing this, we greatly reduce the

X

S

Q

R

¬

R

S

¬

Q

¬

X

S

Q

R

¬

R

S

¬

Q

¬

Fig. 3 A constellation diagram consisting of three different vectors

IJCSNS International Journal of Computer Science and Network Security, VOL.8 No.11, November 2008

7

unnecessary processing that requires to make a decision

about the correct region or the coordinate. Thus, the

number of received demodulated bits that need to search

out can be approximated as: 5

K

-4

K

. The total number of

pair in the upper-bound describes the computational

complexity at the receiving end.

The computational complexity of any multiuser receiver

can be quantified by its time complexity per bit [9]. The

collective computational complexity of the proposed

algorithm is achieved after performing the TM sum using

the complex properties of the existing inverse matrix

algorithms. In other words, the computational complexity

can be computed by determining the number of operations

required by the receiver to detect and demodulate the

transmitted information divided by the total number of

demodulated bits. Therefore, both quantities T and b from

our fundamental equation can be computed together and

the generation for all the values of demodulated received

bits b can be done through the sum of the actual T that

approximately takes O (5/4)

K

operations with an

asymptotic constant. We determine the collective

complexity of optimum multiuser receiver by performing

the TM sum.

After selecting the BPSK modulated bits (

{ 1}

k

b

∈ ±

)

and the TPs that may occur in any coordinate of the

constellation diagram, the collective asymptotic

computational complexity of the optimal ML receiver can

be approximated after performing the TM sum. The

resultant approximation has no concern with the decision

algorithm, since the approximate result can only be used to

analyze the number of operations performed by the

receiver. The computational complexity of the proposed

algorithm for multiuser detection is not polynomial in the

number of users, instead the number of operations required

to maximize the demodulation of the transmitted bits and

to choose an optimal value of b is O (5/4)

K

, and therefore

the time complexity per bit is O (5/4)

K

. Even though, the

computational complexity of the proposed algorithm is not

polynomial in terms of the total number of users, but it still

gives significantly reduced computational complexity.

3.2 Proofs for Computational Complexity

This section provides the formal mathematical proof of

the above discussion that proves the efficiency of the

proposed algorithm with given input sizes. We provide a

mathematical proof for both the upper bound and the lower

bound of the proposed algorithm over the ND and the ML

algorithms.

Proof (1):

(

)

f x

is upper bound of

(

)

1

g x

and

(

)

2

g x

For the sake of this proof, we consider each algorithm

represents by the growth of a function as follows: Let

( )

(

)

5

4

K

f x = for the proposed algorithm,

( ) ( )

1

2

K

g x = for the ML algorithm, and

( )

(

)

2

3

2

K

g x = for the ND algorithm. Equation (17)

shows that the proposed algorithm

(

)

f x

is in the lower

bound of both

(

)

1

g x

and

(

)

2

g x

. Therefore, the values

of the function

(

)

f x

, with different input sizes, always

exist as a lower limit of both

(

)

1

g x

and

(

)

2

g x

. In

order to prove this hypothesis mathematically, we need to

consider the following equations:

(

)

(

)

(

)

(

)

(

)

(

)

1 2

and

f x g x f x g x

= =

Ο Ο

Ο ΟΟ Ο

Ο Ο

(17)

( )

(

)

( )

( )

( )

( )

( )

( )

1 1

2 2

5

,

4

5

4

K

K

f x c g x

f x c g x

= <

= <

Solving for

( )

g x

, we get the following two equations:

( )

(

)

( )

1

5

2.0

4

K

K

f x c= <

(18)

( )

(

)

(

)

2

5 3

4 2

K K

f x c= < (19)

Solving for

(

)

1

g x

, we can write an argument using

(18), such as:

(

)

f x

is said to be

(

)

(

)

1 1

c g x

×

Ο

ΟΟ

Ο

, if

and only if there exists a constant

1

c

and the threshold

o

n

such that:

(

)

(

)

1

f x c g x

< whenever

o

x n

>

.

(

)

(

)

(

)

1

f x c g x

= ×

Ο

ΟΟ

Ο

Thus this is proved using (18). It should be noted that the

n

0

is the threshold value at which both functions

approximately approaches each other. Solving for

(

)

2

g x

,

we can write a similar argument using (19), such as:

(

)

f x

is said to be

(

)

(

)

2 2

c g x

×

Ο

ΟΟ

Ο

,

If and only if there exists a constant

2

c

and the threshold

o

n

such that:

IJCSNS International Journal of Computer Science and Network Security, VOL.8 No.11, November 2008

8

(

)

(

)

2

f x c g x

< Whenever

o

x n

>

.

(

)

(

)

(

)

2

f x c g x

= ×

Ο

ΟΟ

Ο

Thus this is proved using (19).

Proof (2):

(

)

f x

is lower bound of

(

)

1

g x

and

(

)

2

g x

In order to analyze the lower bound, we provide a proof

in the reverse order to define a lower bound for the

function

(

)

f x

. Equation (20) demonstrates that both

functions

(

)

1

g x

and

(

)

2

g x

is the upper bounds for

the function

(

)

f x

. The corresponding values of

(

)

1

g x

and

(

)

2

g x

with different input sizes always lie as a

maximum upper limit of

(

)

f x

, and hence both functions

(

)

1

g x

and

(

)

2

g x

always yield a greater complexity.

In order to prove this hypothesis mathematically, we need

to consider the following equations:

(

)

(

)

(

)

1

g x f x

= Ω and

(

)

(

)

(

)

2

g x f x

= Ω (20)

( ) ( ) ( )

(

)

( )

( )

( )

( )

1 1

2 1

2.0

3

2

K

K

g x c f x

g x c f x

= >

= >

Solving for

(

)

f x

, we get the following two equations:

( ) ( ) ( )

1 1

2.0 5 4

K K

g x c= > (21)

( ) ( ) ( )

2 2

3 2 5 4

K K

g x c= > (22)

Solving for

(

)

1

g x

, we can make the following argument

using (21), such as

(

)

1

g x

is said to be

(

)

(

)

1

c f x

Ω ×

If and only if there exists a constant

1

c

and the threshold

o

n

such that:

(

)

(

)

1 1

g x c f x

> whenever

o

x n

>

.

(

)

(

)

(

)

1 1

g x c f x

= Ω ×

Thus this is proved using (21). Solving for

(

)

2

g x

, we can

claim a similar argument using (22), such as

(

)

2

g x

is said to be

(

)

(

)

2

c f x

Ω × ,

If and only if there exists a constant

2

c

and the threshold

o

n

such that:

(

)

(

)

2 2

g x c f x

>

(

)

(

)

(

)

2 2

g x c f x

= Ω ×

Thus this is proved using (22). As we have proved here

(referring (17) and (20)) that:

(

)

(

)

(

)

(

)

(

)

(

)

and

f x c g x g x c f x

= × = Ω ×

Ο

ΟΟ

Ο

4. Performance Analysis and Experimental

Verifications

The order of growth of a function is an important

criterion for proving the complexity and efficiency of an

algorithm. It gives simple characterization of the

algorithm’s efficiency and also allows us to compare the

relative performance of algorithms with given input sizes.

In this section, we present a comparative analysis of the

asymptotic computational complexity of the proposed

algorithm over the ML and the ND algorithms. The original

asymptotic computational complexity of the ML optimal

receiver is (2)

k

[1]. Another research paper [5] has reduced

the complexity from (2)

k

to (3/2)

k

. This paper [5], also

known as ND algorithm, has reduced the computational

complexity after considering a synchronous DS-CDMA

system.

According to our numerical results, we successfully

reduced the computational complexity at an acceptable

BER after considering the DS-CDMA synchronous LTI

system. The numerical results show the asymptotic

computational complexities with respect to the number of

users as shown in Fig. 4 and 5 for 10 and 100 users,

0 1 2 3 4 5 6 7 8 9 10

10

0

10

1

10

2

10

3

10

4

U s e r s

As ymptotic Complexity

ML

ND

Proposed

Fig. 4. The

asymptotic computational complexities versus small number of

users

IJCSNS International Journal of Computer Science and Network Security, VOL.8 No.11, November 2008

9

respectively. As the number of users increases in the

system, the computational complexity differences among

the three approaches will be obvious.

Fig. 4 shows the computational complexities for a

network that consists of 10 users. As we can see that the

proposed algorithm for a small network of 10 users requires

fewer computations as compare to the ML and the ND

algorithms. In addition, the proposed algorithm greatly

reduces the unnecessary computations involve in signal

detection by storing the pattern of occurrence of the

demodulated bits in the TM and uses it only on those

coordinates or decision regions which are most likely lead

to an incorrect decision. The computational complexity for

a network that consists of 100 users is shown in Fig. 5. It

should be noted that the computational complexity curve

for the proposed algorithm is growing in a linear order

rather than in an exponential order. The computational

linearity of the proposed algorithm comes by employing the

TM technique that avoids considering all the decision

variables and thus provides much better performance over

the ND and the ML algorithms. In other words, this can be

considered as an extension of the former results that

demonstrates the consistency in the linear growth for the

required computations of the proposed algorithm. As we

increase the number of users in the system, more

transformation matrixes will be used to determine that

which coordinate(s) or decision region(s) within the

constellation diagram is most likely to produce errors.

5. Conclusion

In this paper, a novel approach for reducing the

computational complexity of multiuser receivers was

proposed that utilizes the TM technique to improve the

performance of multiuser receiver. In addition to the

low-complexity algorithm, we provided a complete

implementation of the proposed algorithm with the support

of a well driven mathematical model. In order to prove the

low-complexity and the correctness of the proposed

algorithm, we provided the formal mathematical proofs for

both the upper and the lower bounds of the proposed

complexity. The mathematical proofs for both bounds

demonstrated that the computational complexity of the TM

algorithm with any input size always be less than the ML

and the ND algorithms. The reduction in computational

complexity increases the computing power of a multiuser

receiver. Consequently, the increase in computing power

would likely to result fast signal detection and error

estimation which do not come at the expense of

performance. For the future work, it will be interesting to

implement the proposed approach for asynchronous

systems with non-linear time variant properties of the

channel.

References

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

[2] P. Tan, K. Lars, and Teng Lim, “Constrained Maximum-Likelihood

Detection in CDMA” IEEE Transaction on Communications, VOL.

49, No. 1, pp. 142 – 153, January 2001.

[3] S. Moshavi, “Multiuser Detection for DS-CDMA communications,”

IEEE Communications Magazine, Vol. 34, No. 10, pp. 124–36,

October. 1996.

[4] M. Sarraf and R. Karim, W-CDMA and CDMA 2000 for 3G Mobile

Networks. McGraw Hill Telecom Professional, 2002.

[5] T. Ottosson and E. Agrell, “An ML optimal CDMA Multiuser

Receiver,” Department of Information Theory, Chalmers University

of Technology, Sweden.

[6] 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.

[7] G. Woodward and B. Vucetic, “Adaptive Detection for DS-CDMA,”

Proceedings of the IEEE, Vol. 86, No. 7, pp. 1413 -1434, July 1998.

[8] C. Piero, Multiuser Detection in CDMA Mobile Terminals. Artech

House, Inc., 2002.

[9] Thomas H. Cormen, Charles E. Leiserson, and Ronald L. Rivest,

Introduction to Algorithms. Eastern Economy Edition, MIT press,

Cambridge, MA, USA, 1990.

[10] L. K. Rasmussen, T. J. Lim, and T. M. Aulin, “Breadth-first

Maximum Likelihood Detection in Multiuser CDMA,” IEEE

Transactions on Communications, Vol. 45, pp. 1176-1178, October

1997.

[11] L. Wei, L. K. Rasmussen, and R. Wyrwas, “Near Optimum

Tree-Search detection Schemes for Bit-Synchronous Multiuser

CDMA Systems over Gaussian and Two-Path Rayleigh-Fading

Channels,” IEEE Transactions on Communications, Vol. 45, pp.

691-700, June 1997.

[12] J. B. Anderson and S. Mohan, “Sequential Coding Algorithms: A

Survey and Code Analysis,” IEEE Transactions on Communications,

Vol. 32, pp. 169-176, February 1984.

[13] Z. Xie, C. K. Rushforth, R. T. Short, and T. K. Moon, “Joint Signal

detection and Parameter Estimations,” IEEE Transactions on

Communications, Vol. 41, pp. 1208-1216, August 1993.

[14] N. Seshadri, “Joint Data and Channel Estimation using Blind Trellis

Search Techniques,” IEEE Transactions on Communications, Vol.

42, pp. 1000-1011, Feb/Mar/Apr 1994.

[15] A. J. Viterbi, “The Orthogonal-Random Wave form Dichotomy for

Digital Mobile Personal Communications,” IEEE Personal

Communications Magazine, Vol. 1, Issue. 1, pp. 18–24, February

1994.

0 10 20 30 40 50 60 70 80 90 100

10

0

10

5

10

10

10

15

10

20

10

25

10

30

10

35

U s e r s

Asymptotic Complexity

ML

ND

Proposed

Fig. 5. The asymptotic computational complexities versus intermediate

number of users

IJCSNS International Journal of Computer Science and Network Security, VOL.8 No.11, November 2008

10

Syed S. Rizvi is a Ph.D. student of

Computer Science and Engineering

at University of Bridgeport. He

received a B.S. in Computer

Engineering from Sir Syed

University of Engineering and

Technology and an M.S. in

Computer Engineering from Old

Dominion University in 2001 and

2005, respectively. In the past, he

has done research on

bioinformatics projects where he investigated the use of Linux

based cluster search engines for finding the desired proteins in

input and outputs sequences from multiple databases. For last

three year, his research focused primarily on the modeling and

simulation of wide range parallel/distributed systems and the web

based training applications. Syed Rizvi is the author of 68

scholarly publications in various areas. His current research

focuses on the design, implementation and comparisons of

algorithms in the areas of multiuser communications, multipath

signals detection, multi-access interference estimation,

computational complexity and combinatorial optimization of

multiuser receivers, peer-to-peer networking, network security,

and reconfigurable coprocessor and FPGA based architectures.

Aasia Riasat is an Associate

Professor of Computer Science at

Collage of Business Management

(CBM) since May 2006. She

received an M.S.C. in Computer

Science from the University of

Sindh, and an M.S in Computer

Science from Old Dominion

University in 2005. For last one

year, she is working as one of the

active members of the wireless and mobile communications

(WMC) lab research group of University of Bridgeport,

Bridgeport CT. In WMC research group, she is mainly

responsible for simulation design for all the research work. Aasia

Riasat is the author or co-author of more than 40 scholarly

publications in various areas. Her research interests include

modeling and simulation, web-based visualization, virtual reality,

data compression, and algorithms optimization.

Khaled Elleithy received the B.Sc.

degree in computer science and

automatic control from Alexandria

University in 1983, the MS Degree

in computer networks from the same

university in 1986, and the MS and

Ph.D. degrees in computer science

from The Center for Advanced

Computer Studies at the University

of Louisiana at Lafayette in 1988

and 1990, respectively. From 1983 to 1986, he was with the

Computer Science Department, Alexandria University, Egypt, as

a lecturer. From September 1990 to May 1995 he worked as an

assistant professor at the Department of Computer Engineering,

King Fahd University of Petroleum and Minerals, Dhahran,

Saudi Arabia. From May 1995 to December 2000, he has worked

as an Associate Professor in the same department. In January

2000, Dr. Elleithy has joined the Department of Computer

Science and Engineering in University of Bridgeport as an

associate professor. Dr. Elleithy published more than seventy

research papers in international journals and conferences. He has

research interests are in the areas of computer networks, network

security, mobile communications, and formal approaches for

design and verification.