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Transformation matrix algorithm for reducing the computational complexity of multiuser receivers for DS-CDMA wireless systems

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Multiuser detection is an important technology in wireless CDMA systems for improving both data rate as well as user capacity. However, the computational complexity of multiuser detection prevents the widespread use of this technique. Most of the CDMA systems today and in the near future will continue to use the conventional matched filter with its comparatively low user capacity and a slow data rate. However, if we could lower the computational complexity of multiuser detectors, most of the CDMA systems would be likely to take advantage of this technique in order to gain increased system capacity and a better data rate. In this paper, a novel approach for reducing the computational complexity of multiuser receivers is proposed. It utilizes the transformation matrix technique to improve the performance of multiuser detectors. We show that the mathematical computations of the implementation complexity can turn in overall less complex system that has strong impact on the system's signal to noise ratio (SNR) and the bit error rate (BER). The performance measure adopted in this paper is the achievable bit rate for a fixed probability of error (10<sup>-7</sup>) and consistent values of SNR.
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Abstract- Multiuser detection is an important technology in
wireless CDMA systems for improving both data rate as well as
user capacity. However, the computational complexity of
multiuser detection prevents the widespread use of this
technique. Most of the CDMA systems today and in the near
future will continue to use the conventional matched filter with
its comparatively low user capacity and a slow data rate.
However, if we could lower the computational complexity of
multiuser detectors, most of the CDMA systems would be likely
to take advantage of this technique in order to gain increased
system capacity and a better data rate. In this paper, a novel
approach for reducing the computational complexity of
multiuser receivers is proposed. It utilizes the transformation
matrix technique to improve the performance of multiuser
detectors. We show that the mathematical computations of the
implementation complexity can turn in overall less complex
system that has strong impact on the system’s signal to noise
ratio (SNR) and the bit error rate (BER). The performance
measure adopted in this paper is the achievable bit rate for a
fixed probability of error (10
-7
) and consistent values of SNR.
I. INTRODUCTION
Multiuser direct-sequence code division multiple access
(DS-CDMA) has received wide attention in the field of
wireless communications [1, 2]. With the emergence of
multiple access techniques, there has been an increase in the
interest in performing simultaneous estimation and detection
over all users [4]. Multiple access interference (MAI) can be
prevented by selecting mutually orthogonal signature
waveforms for all the active users. However, it is not possible
to ensure perfect orthogonality among received signature
waveforms in a mobile environment, and thus MAI arises.
Multiuser detection is a technique to improve the capacity
and coverage in a CDMA system. Being a critical component
of this technique, the maximum likelihood (ML) multiuser
receiver has received extensive study. However, the
computational complexity of this receiver prevents the
widespread use of this technique. Due to the high
computational complexity, most of the CDMA systems today
and in the near future will continue to use the conventional
matched filter with comparatively low user capacity and a
slow data rate.
While many multiuser detectors could achieve optimal
performance, their relatively higher complexity prevents
CDMA systems from adapting this technology for signal
detection. However, if we could lower the computational
complexity of multiuser detectors, most CDMA systems
would likely take advantage of this technique in order to gain
increased system capacity and a better data rate. In this paper,
a novel approach for reducing the asymptotic computational
complexity of multiuser receivers is proposed that utilizes the
transformation matrix technique to improve the performance
of multiuser detectors. By using the proposed algorithm, the
computational complexity of multiuser detectors can be
reduced by several orders of magnitude. This is done by
realizing that much of the processing performed is
unnecessary. Since most of the decisions are correct, we can
reduce the number of computations by using the
transformation matrices only on those coordinates that are
most likely to lead to an incorrect decision. By doing this, we
can greatly reduce the processing that was required to make a
decision about the correct region or the coordinate. Thus, this
reduction in the computational complexity will likely give us
a considerable improvement in the performance of multiuser
receivers. The performance measure adopted in this paper is
the achievable bit rate for a fixed probability of error (10
-7
)
and consistent values of SNR.
The rest of this paper is organized as follows: Section 2
describes the research that has already been done in this area.
Section 3 presents the ML algorithm and the proposed
algorithm, with section 3.1 covering the ML algorithm and
its corresponding computational complexity and section 3.2
covering the proposed transformation matrix algorithm. The
mathematical derivations for generating consistent values of
SNR and the standard formulas for BER are presented in
sections 3.3 and 3.4, respectively. The mathematical and
Transformation Matrix Algorithm for Reducing
the Computational Complexity of Multiuser
Receivers for DS-CDMA Wireless Systems
Syed S. Rizvi and Khaled M. Elleithy
Computer Science and Engineering Department
University of Bridgeport
Bridgeport, CT 06601
{srizvi, elleithy}@bridgeport.edu
Aasia Riasat
Department of Computer Science
Old Dominion University
Norfolk, VA 23529
ariasat@cs.odu.edu
1-4244-0697-8/07/.00 ©2007 IEEE.
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simulation results of SNR and BER performance are provided
in section 4. Finally, section 5 concludes the paper.
II. R
ELATED WORK
Multiuser receivers can be categorized in the following two
forms: optimal ML sequence estimation (MLSE) receivers
and suboptimal linear and nonlinear receivers. It has been
shown in [9] that DS-CDMA is not fundamentally MAI
limited and can be near-far resistant. In order to mitigate the
problem of MAI, Verdu [9] proposed and analyzed the
optimum multiuser detector for asynchronous Gaussian
multiple access channels. The ML receiver searches all the
possible demodulated bits in order to find the decision region
that maximizes the correlation metric given by [3]. The
practical application of this mechanism is limited by the
complexity of the receiver [10]. This optimal detector
outperforms the conventional detector, but unfortunately its
complexity grows exponentially with a complexity of O (2)
K
,
where K is the number of active users.
Much research has been done to reduce this receiver’s
computational complexity. Recently, Ottosson and Agrell [8]
proposed a ML receiver that uses the neighboring decent
(ND) algorithm. They implemented an iterative approach
using the ND algorithm to locate the region where the actual
observations belong. In order to reduce the computational
complexity of optimum receivers, the iterative approach using
the ND algorithm performs MAI cancellation linearly. The
linearity of the iterative approach increases noise components
at the receiving end. Due to the enhancement in the noise
components, the SNR and BER of the ND algorithm is more
affected by the MAI.
Several tree-search detection receivers have been proposed
in the literature [13, 14], in order to reduce the complexity of
the original ML detection scheme proposed by Verdu.
Specifically, [13] 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 were used by [14] in order to reduce the
complexity incurred by the optimum multiuser detectors. In
addition, an optimal MMSE receiver requires the inversion of
a large matrix. This computation takes a relatively long time
and makes the detection process slow and expensive [10, 11].
Xie, Rushforth, Short and Moon [15] 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. MLSE receivers
give optimum performance but at a cost of increased receiver
computational complexity.
In this paper, we employ a new approach using a
transformation matrix algorithm that 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
transformation matrices only on those coordinates which are
most likely to lead to an incorrect decision. By doing this, we
can greatly reduce the unnecessary processing involved in
making decisions about the correct region or coordinates. Our
mathematical results show that the proposed approach
successfully reduces the computational complexity of the
optimal ML receiver. The complexity of the proposed
algorithm is not polynomial with respect to the number of
users, but it still gives a comparatively reduced complexity
and provides much better performance in terms of SNR and
the BER than other well known multiuser detector algorithms
such as ML and ND.
III. T
HE COMPUTATIONAL COMPLEXITY OF MULTIUSER RECEIVERS
We consider a synchronous DS-CDMA system as a linear
time invariant (LTI) channel. In an LTI channel, the
probability of variations in the interference parameters, such
as the timing of all users is extremely low. The proposed
algorithm utilizes the complex properties of the existing
inverse matrix algorithms to construct the transformation
matrices and to determine the location of the transformation
points that may occur in any coordinates of the constellation
diagram. 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 ∈±
A. The Computational Complexity O
f The ML Algorithm
In order to mitigate the problem of MAI, Verdu [3]
proposed and analyzed the optimum multiuser detector for
Gaussian multiple access channels. When a 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 is
multiplied with user-1’s unique signature waveform,
()
1
Ct.
The resulting signal,
1
()rt
, is applied to the input of the
matched filter. The outputs of the matched filter and Verdu’s
algorithm can be represented by
()
k
y
m
and
()
k
bm
,
respectively where m is the sampling interval. The outputs of
the matched filter for the first two users at the m
th
sampling
interval can be expressed as follows:
(1)
11
1
()
1
() () ()
mT
mT
ym rtCtdt
T
+
½
°°
®¾=
°°
¯¿
³
(1)
2( 1)
222
2
2( )
1
() () ( )
mT
mT
ym rtCt dt
T
τ
++
+
½
°°
®¾=−
°°
¯¿
³
(2)
The received signal
()
1
rt and
()
2
rtcan be expressed as
follows:
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(
)
0.5
11
1
1
() () ( )
M
C
b
iM
rt E biCt iT
=−
=−
¦
(3)
(
)
0.5
2
222
2
() () ( )
M
C
b
iM
rt E biC t iT
τ
=−
=−
¦
(4)
where
1
C
E
and
2
C
E
represent the original bit energy of the
received signals. Substitute (3) and (4) as an individual
equation into (1) and (2), respectively, and we get
() ()
()
{}
0.5
(1)
1
()
1111
1
()
mT
M
ym E biCtiT Ctdt
Cb
T
iM
mT
½+
½
§·°° °°
=−
¦
³
®® ¾¾
¨¸
©¹
°°
°°
=−
¯¿
¯¿
(5)
() ()
()
{}
()
0.5
(1)
1
222222
2
()
mT
M
ym E biCtiT Ct dt
Cb
T
iM
mT
ττ
½+
½
§·°° °°
=−
¦
³
®® ¾¾
¨¸
©¹
°°
°°
=−
¯¿
¯¿
(6)
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 0.5
=1
11 2120
12 2
0.5
1
211
2
ym E bm E b m E b m
CC C
Ebm nm
C
ρ
ρ
ρ
§· § · § ·
¨¸ ¨ ¸ ¨ ¸
+−+
¨¸ ¨ ¸ ¨ ¸
©¹ © ¹ © ¹
§·
¨¸
+++
¨¸
©¹
(7)
() () ()
() ()
0.5 0.5 0.5
(1)
221110
21 1
0.5
1
112
1
ym E bm E bm E bm
CC C
Ebm nm
C
ρρ
ρ
§· §· §·
¨¸ ¨¸ ¨¸
=++
¨¸ ¨¸ ¨¸
©¹ ©¹ ©¹
§·
¨¸
+++
¨¸
©¹
(8)
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. These
symbols can now be decoded using a ML Viterbi decision
algorithm. This algorithm makes a decision over a finite
window of sampling instants rather than waiting for all the
data to be received [5]. 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
computational complexity of this algorithm can be
approximated as: Ɉ (2)
k
.
B. The Computational Complexity Of The Proposed Algorithm
According to original Verdu’s algorithm, the outputs of the
matched filter
1
()ym, and
2
()ym can be considered as a single
output
()
ym. 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:
()
bym
η
+ where Τ represents
the transformation matrix,
{1}
k
b ∈± and
η
represents the
noise components. In addition, if the vectors are regarded as
points in K-dimensional space, then the vectors constitute a
constellation diagram that has K total points. This
constellation diagram can be mathematically expressed as:
{
}
{
}
1, 1
bbX where
∈−+
ҏwhere
X
represents the
collective computational complexity of a multiuser receiver.
The preceding equation is fundamental to 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 Ǖ ) that can only intersect each other at the following
points:
X = {Tb}
b {-1, 1}
K
\ Ǖ . Fig. 1 shows the constellation
diagram that consists of three different vectors (lines) with
the original vector ‘
X
Ȃȱ that represents the collective
complexity of the receiver.
Q, R, and S represent vectors or transformation points
within the coverage area of a cellular network (Fig. 1). In
addition, Q
¬
, R
¬
, and S
¬
represent the computational
complexity of each individual transformation point. 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 is
represented by X
¬
of the transformation point which is equal
to the collective complexity of Q
¬
, R
¬
, and S
¬
. 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.
()
()
()
ijK
ijK
ijK
X Q Xi XQ XR XS
XQii XRji XSki
X R Xj XQ XR XS j
XQij XRjj XSkj
X S Xk XQ XR XS k
XQik XRjk XSkk
i
¬¬¬
== ++
¬¬¬
=++
¬¬ ¬
== ++
¬¬¬
=++
¬¬ ¬
==++
¬¬¬
=++
The following equation can be derived from the above system:
ii ji ki
XQ
XQ
X R ij jj kj XR
XS
XS
ik jk kk
§·
§·
¨¸
§·
¨¸
¨¸
¨¸
¨¸
¨¸
¨¸
¨¸
©¹
¨¸
¨¸
©¹
©¹
¬¬¬
¬
¬¬¬¬
=
¬
¬¬¬
(9)
Equation (9) represents the following: QRS with the unit
vectors
, , and ij k, and
,, and XQXR XS
¬¬ ¬
with
the inverse of the unit vectors
and ,,ij k
¬¬ ¬
. The second
matrix on the right hand side of (9) represents b, where as the
first matrix on the right hand side of (9) represents the actual
transformation matrix. Therefore, the transformation matrix
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from the global reference points to a particular local reference
point can now be derived from (9):
XQ
XQ
XR T XR
LG
XS
XS
§·
§·
¨¸
¨¸
¨¸
¨¸
¨¸
©¹
©¹
¬
¬
=
¬
(10)
Equation (10) can also be written as:
/
ii ji ki
Tijjjkj
LG
ik jk kk
¬¬¬
§·
¨¸
¨¸
¬¬¬
=
¨¸
¨¸
¬¬¬
¨¸
©¹
(11)
We need to compute the locations of the actual transformation
points described in (10) and (11). Let the unit vectors for the
local reference point be:
11 12 13
21 22 23
31 32 33
,,
,,
,,
ititjtk
jtitjtk
k t it jt k
¬
ªº
=
¬¼
¬
ªº
=
¬¼
¬
ªº
=
¬¼
(12)
Since,
()
iijk i
¬¬
++ = , where
()
1ijk++ =. The same is
true for the rest of the unit vectors. Therefore, (12) can be
rewritten as:
11 12 13
21 22 23
31 32 33
,,
,,
,,
ittt
Jttt
kttt
¬
¬
ªº
=
¬¼
ªº
=
¬¼
¬
ªº
=
¬¼
(13)
By substituting the values of
and
,,ij k
¬¬ ¬
from (13) into
(11), we obtain
()()()
()()()
()()()
11 12 13 11 12 13 11 12 13
11 12 13
=
/
21 22 23 21 22 23 21 22 23 21 22 23
31 32 33
31 32 33 31 32 33 31 32 33
it it jt k jtit jt k kt it jt k
ttt
T it it jt k jt it jt k kt it jt k t t t
LG
ttt
it it jt k jt it jt k kt it jt k
++ ++ ++
=++ ++ ++
++ ++ ++
§·
§·
¨¸
¨¸
¨¸
¨¸
¨¸
¨¸
¨¸
¨¸
¨¸
©¹
©¹
(14)
Substituting T
L/G
from (14) into (10), yields
11 12 13
21 22 23
31 32 33
ttt
XQ
XQ
XR t t t XR
XS
XS
tt t
¬ §·
§·
¨¸
¨¸
§·
¬
¨¸
¨¸
¨¸
=
¨¸
¨¸
¨¸
¬
©¹
¨¸
¨¸
©¹
©¹
(15)
Equation (15) corresponds to the following standard
equation used for computing the computational complexity at
the receiving end:
{}
{}
b b 1, 1
k
ℵ= Τ + . Using (15), a
simple matrix addition of the received demodulated bits can
be used to approximate the number of most correlated
transformation points. The entire procedure for computing
the number of demodulated bits that need to be searched out
by the decision algorithm can be used to approximate the
number of most correlated signals for any given set of
transformation points. This is because we need to check
whether or not the transformation points are closest to either
(+1, +1) or (-1, -1). The decision regions or the coordinates
where the transformation points lie for (+1, +1) and (-1, -1)
are simply the corresponding transformation matrices that
store the patterns of their occurrences. If the transformation
points do not exist in the region of either (+1, +1) or (-1, -1),
then it is just a matter of checking whether the transformation
points are closest to (+1, -1) or to (-1, +1). The minimum
search performed by the decision algorithm is conducted if
the transformation points 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. This implies that the total number of demodulated bits
that need to be searched out by the decision algorithm can not
exceed by 5
K
-4
K
. Thus, the total number of most correlated
pairs has an upper bound of 5
K
-4
K
.
Since most of the decisions are correct, we can reduce the
number of computations by using the transformation matrices
only on those coordinates that are most likely to lead to an
incorrect decision. Therefore, this greatly reduces the
unnecessary processing required to make a decision about the
correct region. Thus, the number of received demodulated
bits that need to be searched out can be approximated as 5
K
-
4
K
.
The computational complexity of any multiuser receiver
can be quantified by its time complexity per bit [12]. The
collective computational complexity of the proposed
algorithm is achieved after performing the transformation
matrix sum. This implies that both quantities T and b from
Fig.1. A constellation diagram consisting of three different parameters
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our fundamental equation can be computed together and the
generation of all the values of the demodulated received bits b
can be done through the sum of the actual transformation
matrix T that approximately takes Ɉ (5/4)
k
operations with an
asymptotic constant. Using the Newton approximation
method given in MATLAB, we can directly come to an
approximation of Ɉ (5/4)
k
. The computational complexity of
the proposed algorithm 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 Ɉ (5/4)
k
, and therefore the time
complexity per bit is Ɉ (5/4)
k
.
C. C
omputing SNR Based On The Reduced Complexity
Consider the following points: (a)
is a computational
complexity that belongs to a certain coverage area, (b) SNR
(we represent SNR by
γ
) is uniformly distributed among all
the active users’ signals with respect to the computational
complexity, and (c) a certain cellular coverage area has K
users. Since SNR is uniformly distributed among all the
users’ signals at the receiving end, each user experiences an
average of
K
γ
SNR. In order to achieve maximum positive
values of SNR for most of the values of K, we propose that the
inverse of the computational complexity should equal the
difference between the inverse-normalization factor and the
product of inverse-normalization factor and SNR with respect
to the collective computational complexity of the system. This
hypothesis leads us to the following equation:
11
1
1
K
CC
C
γγ
−−
ªº
=− =
«»
ℵℵ
¬¼
(16)
where
C
in (16) represents the normalization factor, K
is the inverse of the computational complexity, and
γ
represents the SNR with respect to the collective
computational complexity. The main objective of (16) is to
ensure that we get maximum positive values of SNR for most
of the values of K. Using the complexity and the user-domain,
we can make an argument that the inverse of an average SNR
should be at least greater than zero. This argument
guarantees that the system does not work with a non-positive
value of SNR. In other words, the inverse of the average SNR
should equal to the difference of the normalization factor and
the inverse of the average computational complexity. The
above equation can be written as:
CK
γ
=ℵ (17)
Equation (17) can also be used to compute the values of
SNR in an ideal situation only if MAI does not affect the
received signals by K-1 users. However, in a practical DS-
CDMA system, this assumption can not be made. Therefore,
we should consider that the variations in the network load for
an AWGN channel introduces the presence of variance (we
represent variance by
2
σ
) that represents MAI. The selection
of variance is entirely dependent on the network load. In
order to compute the values of SNR in decibels (dB), we need
to change the linear quantity into decibels (dB) by
multiplying it with the base-10 logarithmic function as well
as with the variance. Since MAI is a multiplicative property
of SNR, the resultant approximation of SNR in dB is always a
product of the base-10 logarithmic function and the possible
variance with respect to the number of users.
()
2
10
10 log CK
γσ
=ℵ (18)
We use the precomputed values of variance, given in [6], in
our simulation that represents MAI for a range of users.
Furthermore, the normalization factor represents a varying
quantity that can be used to approximate the different values
of SNR with respect to the difference between average
computational complexity and average SNR. It should be
noted that (18) only gives approximate values that can be
closed to the actual values of SNR depending on both the
variance and the normalization factor.
D. C
omputing BER Based On The Reduced Complexity
We modeled the cellular network as a LTI synchronous
DS-CDMA system in which users utilize an AWGN
multipath channel. Due to the AWGN channel and the
linearity property, the different signal components do not
experience deep fades. In other words, if the signal changes
during the transition, the receiver receives the following
signal:
() () ()
j
te st t
θ
η
ℜ=Α + +
where
A is an attenuation factor,
θ
is a phase shift,
()
s
t is
the desired signal, and
()
t
η
is the additive Gaussian noise.
Due to LTI characteristics, the proposed algorithm is
independent of the phase shift, which permits us to ignore it
by simply setting the value of
θ
to zero. Therefore, the
receiver receives the following signal:
() () ()
tst tA
η
ℜ= + +
Since the attenuation factor
A is uncorrelated with
()
t
η
,
we can use the value of SNR directly in the BER formula.
Consider (19) that can be used to determine the BER in an
AWGN channel for a system where the transmitted bits are
modulated using the BPSK modulation technique.
BER
11QSNR
ª
º
=
¬
¼
(19)
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It should be noted that the signals from all the users are
synchronized with each other and the power of each users
signal is equal to the energy per bit with respect to time.
Since the attenuation factor and the white noise are
uncorrelated, the SNR can be directly placed in (19) as
follows:
BER
()
2
1
2
110QSNR
σ
+
ªº
=
¬¼
(20)
where
()
Qx
is the Gaussian
Q
function [7]. For simplicity,
(20) can also be written as:
BER =
2
10 1 10QSNR SNR
σ
ªº
+
¬¼
(21)
The second term in (20) represents the SNR degradation
due to MAI. This term depends on the cross-correlation
between the spreading code as well as the number of users. In
other words, an increase in K causes an increase in the second
term of (20) which causes a decrease in the overall BER
performance. Furthermore, the possibility of variance in
network load can not be ignored while calculating the BER
performance, since our numerical calculations for BER are
based on SNR which itself contains a small random amount
of variance, as shown in (20).
IV. P
ERFORMANCE ANALYSIS OF THE PROPOSED ALGORITHM
This section analyzes the computational complexity, SNR,
and the BER performance of the proposed algorithm and
compares it to the ND and the ML algorithms. The system is
modeled as a synchronous DS-CDMA system in a Gaussian
channel.
A. C
omplexity Analysis
The order of growth of a function is an important criterion
for analyzing the complexity and efficiency of an algorithm.
It gives a simple characterization of the algorithm’s efficiency
and also allows us to compare the relative performance of
algorithms with given input sizes. Fig. 2 shows the
computational complexities for a network that consists of 100
users. As we can see the proposed algorithm for a network of
100 users requires fewer computations as compared to the ML
and the ND algorithms. In addition, the proposed algorithm
greatly reduces the unnecessary computations involved in
signal detection by storing the pattern of occurrence of the
demodulated bits in the transformation matrix and uses it
only on those decision regions which are most likely to lead
to an incorrect decision. Also, from the subsequent sections,
we see that these significant computational savings do not
come at the expense of performance. 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. As the number of users increases in the
system, the computational complexity differences among the
three approaches will be obvious.
B. P
erformance Analysis Of SNR
Three detection algorithms are investigated, which are the
original ML algorithm, ND algorithm, and the proposed
algorithm. The values of SNR are approximated using (18).
In our simulation, we use one (i.e., C = 1) as a normalization
factor that remains same for all investigated algorithms. The
choice of a small value of
2
σ
is entirely based on the load of
the coverage area (K) and it is selected through a random
process for a certain range of users. For a lightly loaded
network, we expect that the value of variance may vary from
0.6 to 0.9 and for a heavily-loaded network, the value of
variance may vary from 0.1 to 1.
2 4 6 8 10 12 14 16 18 20 22
6
8
10
12
14
16
18
U S E R S
S N R
ML
ND
Proposed
Fig.3. Approximate value of SNR (dB) versus number of users (K =22) with a
random amount of variance for a synchronous DS-CDMA system in a Gaussian
channel
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
Asym ptotic Complexity
ML
ND
Proposed
Fig.2. Computational complexities versus users
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Fig. 3 shows a network where 22 users are active within
the coverage area of a cellular network. For a small value of
K, the proposed algorithm offers approximately 6.5 dB of
SNR where as the ND and the ML algorithms give 5.8 and
5.5 dB respectively. A slight increase in the value of K forces
the proposed algorithm to give an acceptable value of the
SNR that can be used to achieve a satisfactory BER
performance for a voice communication network. For a small
network, the three algorithms behave almost exactly the
same. When the system becomes heavily loaded such as K =
42, the divergence rate for the proposed algorithm increases.
This can be seen in Fig. 4, where the proposed algorithm has
more rapid divergence with respect to the number of users
when compared with the ND and the ML algorithms. The
divergence for SNR is an essential element in achieving a
minimum amount of BER. In other words, the divergence in
SNR is directly proportional to the convergence in BER
performance. In addition, it can be clearly observed from Fig.
4 that the linear increase in SNR for the proposed algorithm
is smoother and more uniform than the ND and the ML
algorithms.
C. P
erformance Analysis Of BER
The standard performance criterion in digital
communications is the probability of BER. Some voice-band
modem applications permit error rates no greater than 10
-5
,
whereas other applications such as digitized voice tolerate
error rates as high as 10
-2
to 10
-3
. Simulation results show
that the proposed algorithm performs better than the ML and
the ND algorithms for all values of BER. Fig. 5 and Fig. 6
show a plot of three BER versus SNR curves. These curves
were plotted using (21) in an AWGN channel for a small
range of users. The simulation results of the proposed
algorithm for a lightly-loaded network demonstrate that an
optimal BER performance can be achieved for a reasonable
range of SNR. It should be noted that the BER performance
of the proposed algorithm is always better than the ML and
the ND algorithms as shown in Fig. 5. For the first few values
of SNR, the ND algorithm almost approaches the ML
algorithm whereas the proposed algorithm still maintains a
reasonable performance difference. It can be seen in Fig. 5
that the proposed algorithm achieves less than 10
-2
BER for
SNR = 8 dB which is quite closed to the required reasonable
BER performance for a voice communication system. For
small values of SNR, the BER for these three algorithms is
almost equal, but as we increase the value of SNR, typically
more than 10 dB, one can clearly observe the difference in the
BER performance.
The BER curve in Fig. 6 is calculated using (21). The
former result demonstrates a slight improvement over the
BER performance shown in Fig. 5 for all SNR values above 9
2 6 10 14 18 22 26 30 34 38 42
6
8
10
12
14
16
18
20
22
24
26
U S E R S
S N R
ML
ND
Proposed
Fig.4. Approximate value of SNR (dB) versus number of users (K =42) with a
random amount of variance for a synchronous DS-CDMA system in a Gaussian
channel
0 2 4 6 8 10 12 14
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
SNR (dB)
BER
ML
ND
Proposed
Fig.6. BER versus SNR (0<dB<14) curves
0 1 2 3 4 5 6 7 8
10
-3
10
-2
10
-1
10
0
SNR (dB)
BER
ML
ND
Proposed
Fig.5. BER versus SNR (0<dB<9) curves
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dB. Even for small values of SNR, the proposed algorithm
gives better performance than the ML and the ND algorithms.
As the value of SNR increases, the BER performance of the
proposed algorithm over the ND and the ML algorithms
becomes more and more substantial because the probability of
having more divergent values of SNR increases. It can also be
noticed in Fig. 6 that the proposed algorithm achieves less
that 10
-3
BER for SNR = 10 dB which is more than what we
desire for a voice communication system. Therefore, based
on the BER performance results, the proposed algorithm
achieves better performance in terms of BER than the other
well-known multiuser receiver algorithms while at the same
time the proposed algorithm simplifies the design and greatly
reduces the computational complexity of multiuser receivers.
Furthermore, the proposed algorithm achieves 10
-7
BER
performance for SNR = 14 dB as shown in Fig. 6. This is
more than an acceptable value of BER for data
communication such as FTP and is quite close to the desired
value of BER (typically 10
-8
BER is required) for high fidelity
digital audio systems. In addition, as the number of users
increases in the system, we get consistent values of SNR
which further maintains the overall BER performance of
multiuser receiver.
V. C
ONCLUSION
In this paper, a novel approach for reducing the
computational complexity of multiuser receivers is proposed,
which utilizes the transformation matrix technique to
improve the performance of multiuser detectors. We present a
mathematical model which verifies the implementation of the
transformation matrix technique. The numerical results for
the computational complexity of the proposed algorithm
demonstrate the success of the proposed algorithm over the
MD and the ND algorithms. Furthermore, we present a new
mathematical model for computing the values of SNR. The
main advantage of the proposed mathematical model for SNR
is that it guarantees that the receiver does not process the
signals that have non-positive values of SNR. In order to
show the consistency and the correctness of the proposed
approach, we present comprehensive simulation results for
computing SNR with different ranges of users. The
simulation results for SNR demonstrate the consistency of the
desired values required to achieve an optimal BER
performance. In addition, we present BER results not only
for a lightly-loaded network but also for a heavily-loaded
network. The BER simulation results of the proposed
algorithm suggest that the proposed algorithm achieves better
BER performance for all values of SNR than the other well-
known multiuser detection algorithms.
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