Transformation Matrix System for Reducing the Computational Complexity of

Wireless Multi-user Receivers for DS-CDMA Systems

Syed S. Rizvi, 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@cbm.edu.pk

Abstract

This paper presents a new transformation matrix

system for reducing the computational complexity of

multiuser receiver. The proposed system observes the

coordinates of the constellation diagram to determine

the location of the transformation points. Since most of

the decisions are correct, we can reduce the number of

required computations by using the transformation

matrixes only on those coordinates which are most

likely to lead to an incorrect decision. Our results

show that the proposed system successfully reduces the

computational complexity of an optimal multiuser

receiver. The complexity of the proposed system is not

polynomial in the number of users, but it still gives

comparatively reduced complexity.

1. Introduction

Recovery of transmitted signals at the receiver is

achieved by demodulation techniques [4]. A widely

used technique called the single user matched filter

regards the Code division multiple access (CDMA)

channel as a single user channel and considers the

detection problem of each user individually. In the

absence of user interference, the single user matched

filter is optimal in the sense of minimizing the bit error

rate. However, this is no longer true in CDMA systems

where MAI is present. To overcome these

disadvantages of a single user matched filter, multiuser

detectors have been developed [2, 3]. Multiuser direct-

sequence (DS) CDMA has received wide attention in

the field of wireless communications [1, 3]. Verdu’s

[2] 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. In

this paper, we work on the Verdu’s original algorithm

that has the main advantage of reaching minimum

mean square error (MMSE) performance. We

demonstrate that the proposed system can reduce the

asymptotic computational complexity of multiuser

receivers by using transformation matrix technique.

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. In case of synchronous

CDMA system, two main criteria are employed,

namely the zero-forcing (ZF) and the MMSE. Both

mechanisms can implement in two possible ways [1].

In the first option, both of them can be implemented to

deal simultaneously with ISI and MAI where as in the

second option, they deal only with ISI. The two well

known implementations of a non-linear multiuser

receiver are SIC and PIC. In interference cancellation,

MAI is first estimated and then subtracted from the

received signal [3]. On the other hand, linear

multiusers receivers apply a linear transformation to an

observation vector, which serves as soft decision for

the transmitted data. Recently, Ottosson and Agrell [5]

proposed a new ML receiver that uses the neighbor

descent (ND) algorithm. They implemented a linear

iterative approach using the ND algorithm to locate the

region (s). 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.

3. Proposed transformation matrix system

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: y(m) = Tb

+ n where T represents the transformation matrix,

{1}b

k

∈± and n represents the noise components. In

addition, if vectors are regarded as points in K-

Fifth International Conference on Information Technology: New Generations

978-0-7695-3099-4/08 $25.00 © 2008 IEEE

DOI 10.1109/ITNG.2008.123

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Fifth International Conference on Information Technology: New Generations

978-0-7695-3099-4/08 $25.00 © 2008 IEEE

DOI 10.1109/ITNG.2008.123

1252

Fifth International Conference on Information Technology: New Generations

978-0-7695-3099-4/08 $25.00 © 2008 IEEE

DOI 10.1109/ITNG.2008.123

1251

Fifth International Conference on Information Technology: New Generations

978-0-7695-3099-4/08 $25.00 © 2008 IEEE

DOI 10.1109/ITNG.2008.123

1251

Fifth International Conference on Information Technology: New Generations

978-0-7695-3099-4/08 $25.00 © 2008 IEEE

DOI 10.1109/ITNG.2008.123

1251

dimensional space, then the vectors constitute the

constellation diagram that has K total points. In order

to compute the collective computational complexity of

the optimum receiver, it is essential to determine the

complexity of each individual transformation point.

The computational complexity of each individual

transformation point represents by X

¬

of the

transformation point which is equal to the collective

complexity of Q

¬

, R

¬

, and S

¬

. We consider the original

vector with respect to each transmitted symbol.

ii ji ki

XQ

XQ

X R ij jj kj XR

XS

XS

ik jk kk

¬¬¬

¬

¬¬¬¬

=

¬

¬¬¬

(1)

The second matrix on the right hand side of (1)

represents b, where as the first matrix on the right hand

side of (1) represents the actual transformation matrix.

Therefore, the transformation matrix from the global

reference points to a particular local reference point

can now be derived from (1)

XQ

XQ

TXRXR

LG

XS

XS

=

¬

¬

¬

(2)

Equation (2) can also be written as:

/

ii ji ki

Tijjjkj

LG

ik jk kk

¬¬¬

¬¬¬

=

¬¬¬

(3)

In (3), the dot products of the unit vectors of the two

reference points are in fact the same as the unit vectors

of the inverse transformation matrix of (2). We need to

compute the locations of the actual transformation

points described in (2) and (3). Therefore, (2) and (3)

can be rewritten as:

11 12 13

21 22 23

31 32 33

,,

,,

,,

ittt

Jttt

kttt

¬

¬

=

=

¬

=

(4)

By substituting the values of

and ,,ij k

¬¬ ¬

from

(4) into (3) and replacing T

L/G

from (4) into (2):

()()()

()()()

()()()

11 12 13 11 12 13 11 12 13

11 12 13

=

/ 212223 212223 212223 212223

31 32 33

31 32 33 31 32 33 31 32 33

it it jt k jt it jt k kt it jt k

ttt

T ititjtk jtitjtkktitjtk t t t

LG

ttt

it it jt k jt it jt k kt it jt k

++ ++ ++

=++++++

++ ++ ++

(5)

Equation (5) corresponds to the following standard

equation used for computing the computational

complexity at the receiving end:

{

}

{}

b

1, 1b

k

∈− +

ℵ= Τ . Using (5), a simple matrix

addition of the received demodulated bits can be used

to approximate the number of most correlated

transformation points. This implies that the total

number of demodulated bits that need to search out by

the decision algorithm can not exceed by 5

k

– 4

k

. In

other words, the total number of most correlated pairs

is upper bounded by 5

k

– 4

k

. Figure 3 shows the

computational complexities for a network of 10 users.

As we can see that the proposed system for a small

network of 10 users requires fewer computations as

compare to the ML and the ND system.

4. Conclusion

We proposed a new transformation matrix system

that can be used to significantly reduce the asymptotic

computational complexity of multiuser receiver. We

also presented a mathematical model which verifies the

implementation of the transformation matrix technique.

Furthermore, the numerical results for the

computational complexity of the proposed system

demonstrated the success of the proposed system over

the ML and the ND algorithms.

5. References

[1] 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.

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

Press, 1988.

[3] P. Tan and K. Lars, “Multiuser Detection in CDMA: A

Comparison of Relaxations, Exact, and Heuristic Search

Methods,” July 09, 2003.

[4] 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.

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

Multiuser Receiver,” Electronics Letters, Vol. 31, Issue-

18, pp. 1544-1555, August 1995.

1

10

100

1000

10000

12345678 910

Us ers

Co m p l ex ity

ML

ND

Proposed

Fig.1 The asymptotic computational complexities

versus small number of users

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