Transformation Matrix System for Reducing the Computational Complexity of Wireless Multi-User Receivers for DS-CDMA Systems

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.

Full text

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}bk∈± 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
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
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 TL/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 jtit jt k ktit jt k ttt
T ititjtk jtitjtk ktitjtk t t t
LG
ttt
itit jt k jt it jt k kt it j t 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 5k – 4k. 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
12345 678910
Users
Com plex ity
ML
ND
Proposed
Fig.1 The asymptotic computational complexities
versus small number of users
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