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A Low-Complexity Algorithm for Improving Computing Power of Multiuser Receivers for Next Generation Wireless Systems

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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
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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 receivers
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
= 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:
= {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
+ = 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
×
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
× ,
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
= × = ×
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,
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[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.
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