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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.