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Content uploaded by Khaled Elleithy
Author content
All content in this area was uploaded by Khaled Elleithy
Content may be subject to copyright.
Content uploaded by Khaled Elleithy
Author content
All content in this area was uploaded by Khaled Elleithy
Content may be subject to copyright.
Abstract—Wireless Multiuser receivers suffer from their
relatively higher computational complexity that prevents
widespread use of this technique. In addition, one of the main
characteristics of multi-channel communications that can
severely degrade the performance is the inconsistent and low
values of SNR that result in high BER and poor channel
capacity. It has been shown that the computational complexity
of a multiuser receiver can be reduced by using the
transformation matrix (TM) algorithm [4]. In this paper, we
provide quantification of SNR based on the computational
complexity of TM algorithm. We show that the reduction of
complexity results high and consistent values of SNR that can
consequently be used to achieve a desirable BER performance.
In addition, our simulation results also suggest that the high
and consistent values of SNR can be achieved for a desirable
BER performance. The performance measure adopted in this
paper is the consistent values of SNR.
Keywords—Computational complexity, DS-CDMA,
wireless multiuser receivers, signal to noise ratio
I. INTRODUCTION
From the design standpoint, for a given modulation and
the coding scheme there is a one to one correspondence
between the bit error rate (BER) and the signal-to-noise ratio
(SNR). From the user standpoint, SNR is not the favorite
criterion for the performance evaluation of digital
communication links, because the user measures the quality
of a system by the number of errors in the received bits and
prefers to avoid the technical detail of modulation or coding.
However, using received SNR rather than BER will allow us
to relate our performance criteria to the required transmitted
power, which is very important for battery-operated wireless
operations. Using SNR rather than BER has two advantages.
First, SNR is the criterion used for accessing both digital and
analog modulation techniques. Second, SNR is directly
related to the transmitted power, which is an important
design parameter.
A significant amount of efforts have been made in order
to achieve high values of SNR [3, 5]. However, none of
these methods relate the complexity of multiuser receivers
for achieving high SNR values. On the other hand, the TM
algorithm is a low complexity, but synchronous transmission
technique that is able to reduce the number of computations
performs by a multiuser receiver for signal detection. The
TM algorithm therefore provides fast multiuser signal
detection which can be further used to achieve high SNR
values. The contribution of this research work is the
quantification of SNR using the TM algorithm proposed by
Rizvi [4]. At high SNR values, the error rate for multi
channel can be reduced as well the capacity of the channel
can be well approximated.
Verdu [1] proposed the optimum multiuser detector for
asynchronous systems. The complexity of multiuser receiver
grows exponentially in an order of O (2)
K
, where K is the
number of active users. Recently, [2] proposed a ML
receiver that uses the neighboring decent (ND) algorithm
with an iterative approach to locate the regions. The linearity
of the iterative approach increases noise components at the
receiving end. The TM algorithm [4] observes the
coordinates of the constellation diagram to determine the
location of the transformation points. Since most of the
decisions are correct, the TM algorithm can reduce the
number of computations by using the transformation
matrices only on those coordinates which are most likely to
lead to an incorrect decision.
II. THE PROPOSED QUANTIFICATION OF SNR
In this section, we derive an expression to provide
quantification of SNR for the signals received at the DS-
CDMA multiuser receiver. The reduced complexity of the
TM algorithm provides faster detection rate. The faster
Analyzing SNR Performance of a Low-
Complexity Wireless Multiuser Receiver for
DS-CDMA 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
Institute of Business Management
Karachi, Pakistan 78100
aasia.riasat@iobm.edu.pk
detection rate results high and consistent values of SNR.
Once we determine the values of SNR, we can relate them to
the BER performance and the channel capacity
approximation for a wireless multiuser receiver. Also, MAI
causes the SNR degradation resulting in a degraded SNR
performance for a particular value of E
b
/N
o
. We present that
due to the reduced complexity, the SNR performance of the
TM algorithm would remain consistent in terms of the
desired values even for a large value of K. This consistency
in SNR performance yields an optimal BER performance.
A. System Model and Key Assumptions
Our fundamental assumption is that the system is linear
time invariant (LTI) which leads us to the fact that the
transmitted signals experience no deep fades. Due to the
linearity and time invariant properties of the system, we can
ignore the phase shift, and deep fades. In other words, the
overall SNR of the received signals has a slow convergence
rate compared to the convergence rate of the BER.
B. Closed Form Expression for SNR
Consider the following assumptions for an AWGN
channel:
(a)
ℵ
represents the computational complexity that
belongs to a certain coverage area.
(b) SNR (we represent SNR by
γ
) is uniformly distributed
among all the active user’s signals with respect to
computational complexity.
(c) A certain cellular coverage area has K users.
Based on these above assumptions, we can give the
following hypothesis:
{
}
1 2 3
, , ,.................,
i K
ℵ ℵ ℵ ℵ ℵ
∈
(1)
where
1, 2, 3,
....................
K
ℵ ℵ ℵ ℵ
indicates the indicates
the computational complexity-domain and
{
}
1 2 3
, , ,................,
i K
h h h h h
∈
(2)
where
1 2 3
, , ,................,
K
h h h h
indicates the user-
domain.
Complexity-domain can be considered as a simple data
structure for storing the patterns of occurrences of all active
users. User-Domain is the number of active users present in
the certain coverage area of a cellular network. The
collective computational complexity can be expressed as:
1
1, 2,.....,
K
i
i
where i K
=
ℵ = ℵ =
∑
(3)
Since each user has
th
h
part of the computational
complexity such as:
1 1 2 2
, ,......,
K K
h h h
ℵ ℵ ℵ
∈ ∈ ∈
.
This implies that each active user in a certain area of a
cellular network has an average of
K
ℵ computational
complexity. Since SNR is uniformly distributed among all
the user’s signals at the receiving end, each user experiences
an average of
K
γ
SNR. Therefore, this argument leads us
to:
(
)
(
)
1 1 1
1K C C C
γ γ
− − −
ℵ = − ℵ = − ℵ
(4)
where C in (4) represents the normalization factor,
K
ℵ
is
the inverse of the computational complexity, and
γ
ℵ
represents the SNR with respect to average computational
complexity.
Equation (4) can be interpreted that the inverse of
computational complexity equals to the difference between
the inverse-normalization factor and the product of the
inverse-normalization factor and SNR with respect to the
collective computational complexity. The main objective of
(4) is to make sure that we should get maximum positive
values of SNR for most of the values of K.
C. Proof for
γ
ℵ
If the previous assumptions are valid for an AWGN
channel, the following approximation must be true for both
the complexity and the user domains:
approximation
K C K
γ
ℵ → + (5)
We present our hypothesis that the difference between the
average computational complexity and the average SNR
should equal to the normalization factor. The main objective
of (5) is to get maximum positive values of SNR for most of
the values of K. Equation (5) can also be written as:
(
)
(
)
K K C
γ
ℵ − =
(6)
Based on (6), we can write the following equation:
(
)
1
K
C
γ
= −
ℵ ℵ
(7)
Since the right hand side of (7) represents the inverse of
the average computational complexity with the
normalization factor, the number of required operations can
not be less than zero. It should be noted that the right hand
side of (7) always gives us a positive value of SNR for any
value of K which is greater than 10. Equation (7) can also be
rewritten as:
(
)
[
]
1
1K C
γ
−
ℵ = − ℵ
(8)
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. Recall (4):
(
)
(
)
1
K C CK
γ γ
−
ℵ = ℵ − ℵ = ℵ−
(9)
Equation (9) represents SNR by determining the
difference between the power of the transmitted signal from
the computational complexity-domain and the number of
users from the user-domain. Equation (9) 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 does not
exist. 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
Φ
) that
represents MAI.
The selection of variance is entirely dependent on the
network load. The variance is a linear function of the active
users (K) and it should increase as we increase the value of
K. In order to compute the values of SNR, we need to
change the linear quantity into decibels (dB) by multiplying
it to the base-10 logarithmic function as well as with the
variance. This leads us to the following expression for SNR:
(
)
2
10
10 log
CK
γ
= Φ ℵ− (10)
We use the values of variance in our simulation that
represents MAI with respect to K.
III. EXPERIMENTAL VERIFICATION AND SIMULATION
RESULTS
It has been shown that the SNR degradation depends on
the number of users, K, [4]. An increase in K would degrade
the performance because it would increase the cross
correlation between the received signals from all the users
(i.e., K-1 users). Mathematically, we can express this as:
K
∝
MAI
∝
high BER
∝
1/SNR. This shows that a slight
increase in K would degrade the SNR performance that
consequently increases the BER. However, a large increase
in value of K forces MAI to reach its peak value that limits
the divergence of SNR for the TM algorithm.
For lightly-loaded network, (2< K<50) where as for
heavily-loaded network (2< K<50) as shown in Fig.1 and
Fig. 2. LTI synchronous DS-CDMA over an AWGN
channel with small variation in
2
Φ
are used. The choice of
a small value of variance is entirely based on the value of K
and it is selected through a random process. 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.
A. Lightly and Heavily Loaded Networks
Fig. 1 shows one of the possible cases of a lightly-loaded
network where 22 active users transmit BPSK modulated
signals. For a small value of K, the proposed TM algorithm
achieves approximately 6.5 dB of SNR where as the ND and
the ML algorithms give 5.8 and 5.5 dB, respectively.
This implies that a slight increase in the value of K forces
the TM algorithm to give an acceptable value of SNR that
can be used to achieve a satisfactory BER performance at
least for a voice communication network. This can be seen
in Fig. 2 that the TM algorithm has more rapid divergence
with respect to the number of users than the ND and the ML
algorithms. The divergence in SNR is directly proportional
to the convergence in BER performance.
In addition, it can be clearly observed in Fig. 2 that the
linear increase in SNR for the TM algorithm is more
uniform and smoother over the ND and the ML algorithms.
Fig. 3 shows that the linear increase in SNR is consistent not
only for a lightly-loaded network but also for a heavily-
loaded network.
However, this can also be noticed from Fig. 3 that as the
number of users increase in the system, the differences
between the SNR values for the proposed algorithm and the
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.1 Approximate values
of SNR (dB) versus number of users (K=22) with
a random amount of variance for a synchronous system in an AWGN
channel.
other two ML and the ND algorithms become wider. From
Fig. 3, the TM algorithm gives approximately 36 dB for K =
72 which is more than what we expect to achieve for an
optimal BER performance. In addition to that, the random
amount of variance is more affected on the SNR values in a
heavily-loaded case than in a lightly-loaded case.
IV. CONCLUSION
In this paper, we presented the quantification of SNR
based on the TM algorithm. We have shown that the
reduction in the computational complexity of a multiuser
receiver can be used to achieve high and consistent values of
SNR. The simulation results suggest that due to a low
complexity domain, the SNR performance of the TM
algorithm is more uniform and smoother over the other well
known algorithms. For the future work, it will be interesting
to implement the proposed approach for asynchronous
systems to achieve desirable BER performance and
approximate the capacity of a multi channel.
REFRENCES
[1] S. Verdu, Multiuser Detection. Cambridge University Press, 1988.
[2] T. Ottosson and E. Agrell, “ML optimal CDMA Multiuser Receiver,”
Electronics Letters, Vol. 31, Issue-18, pp. 1544-1555, August 1995.
[3] N. Jindal, “High SNR analysis of MIMO broadcast channels,”
Proceedings of international symposium on information theory,
Vol.4, no. 9, pp. 2310 – 2314, Sept. 2005
[4] Syed S. Rizvi, Khaled M. Elleithy, and Aasia Riasat, “Transformation
Matrix Algorithm for Reducing the Computational Complexity of
Multiuser Receivers for DS-CDMA Wireless Systems,” Wireless
Telecommunication Symposium (WTS 2007), Pomona, California,
April 26-28 2007.
[5] A. Lozano, M. Antonia, and S. Verdú, “High-SNR Power Offset in
Multiantenna Communication,” IEEE Trans on information theory,
Vol. 51, No, 12, pp. 4134- 4151, Dec 2005.
2 12 22 32 42 52 62 72 82 92 102
5
10
15
20
25
30
35
40
45
50
U S E R S
S N R
ML
ND
Proposed
Fig.3 Approximate value of SNR (dB) versus number of users (K =102,
heavily-loaded network) with a random amount of variance for a
synchronous system in an AWGN channel.
2 5 8 11 14 17 20 23 26 29 32
6
8
10
12
14
16
18
20
22
U S E R S
S N R
ML
ND
Proposed
Fig.2 Approximate value of SNR (dB) versus number of users (K =32)
with a
random amount of variance for a synchronous system in an AWGN channel.
