# Symbol error rate analysis of relay-based wireless systems.

**ABSTRACT** In this paper, the symbol error rate (SER) performance of a relay-based amplify-and-forward (AF) system is analyzed over fading channels . The relay power-gain is optimized with the objective of maximizing the received signal-to-noise-ratio (SNR) at the destination, given that the fading statistics of the links are known at the relay node. The Gaussian finite mixture is utilized to mathematically formulate, in a simple and unified way, the statistics of the received SNR at optimal relay power-gain. These statistics include the probability density function (pdf) and the moment generating function (MGF). Using this technique, the SER for coherent and differentially coherent modulations are derived. Monte Carlo simulation results are presented to validate the derived expressions.

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**ABSTRACT:**In this paper, a novel approximation for the Gaussian Q-function in the form of Q(√x) is presented. The proposed approximation is compared with other known closed-form approximations of the Q-function in terms of accuracy and applicability. This approximation can be effciently used to simplify intractable problems which do not have explicit solutions. The approximation of the generalized probability density function (PDF) using Gaussian fnite mixture components is utilized with the proposed approximation to derive a generalized closed-form expression for the Bit Error Rate (BER) for coherent modulation techniques. The derived expression is evaluated over Nakagami-m and Weibull fading channels. Monte Carlo simulation of the average BER expression is provided over Nakagami-m and Weibull fading channels in order to validate the derived expression. The derived average SER expression can be used by engineers to evaluate the BER performance for any coherent modulation technique and over any fading channel given that the fading channel statistics are modeled using Gaussian fnite mixture.GCC Conference and Exhibition (GCC), 2013 7th IEEE; 01/2013 - SourceAvailable from: Rawan Alkurd
##### Conference Paper: Error Rate Performance Analysis of Cooperative SCR in VANETs over Generalized Fading Channels

Wireless Communications and Networking Conference (WCNC), 2014 IEEE; 04/2013

Page 1

Symbol Error Rate Analysis of Relay-based

Wireless Systems

Ibrahim Y. Abualhaol

Khalifa University, Sharjah, United Arab Emirates

Email: ibrahimee@ieee.org

Abstract—In this paper, the symbol error rate (SER) per-

formance of a relay-based amplify-and-forward (AF) system is

analyzed over fading channels . The relay power-gain is optimized

with the objective of maximizing the received signal-to-noise-ratio

(SNR) at the destination, given that the fading statistics of the

links are known at the relay node. The Gaussian finite mixture is

utilized to mathematically formulate, in a simple and unified way,

the statistics of the received SNR at optimal relay power-gain.

These statistics include the probability density function (pdf) and

the moment generating function (MGF). Using this technique,

the SER for coherent and differentially coherent modulations

are derived. Monte Carlo simulation results are presented to

validate the derived expressions.

I. INTRODUCTION

Cooperative relay-based wireless systems exploit the broad-

cast nature of the wireless medium and allow nodes to jointly

transmit information through relaying to improve the trans-

mission capacity and the performance. The first formulation

of a relaying problem appeared in the information theory

community in [1] and were served as the motivating cause

of the concurrent development of the ALOHA system at the

University of Hawaii. The traditional relay channel model is

comprised of three nodes (Fig. 1): a source (S) that transmits

information, a destination (D) that receives information, and

a relay (R) that both receives and transmits information to

enhance the communication between the source and the des-

tination. This model integrated with the understanding of the

benefits of MIMO (multiple-input-multiple-output) systems in

wireless channels makes the community realize that multiple

relays can emulate the strategies designed for MIMO sys-

tems and offer significant network performance enhancements

in terms of various metrics, including increased capacity,

improved reliability, and minimized symbol-error probability

(SEP). Therefore, the interest in distributed systems (i.e.,

virtual MIMO) has inspired the community to analyze the

statistics of the cooperative relay-based systems over fading

channels.

The authors in [2] analyzed the performance of two-hop

relay-based system over Rayleigh fading channels. In this

work the authors derived closed-form expressions for the pdf

of the received signal-to-noise-ratio (SNR) at the destination

without taking into consideration the direct path. A more

general cooperative models based on parallel relays have been

examined in [3] and [4]. The authors in [3] considered the

Destination

[D]

Source

[S]

Relay

[R]

?

?

?

Fig. 1.Relay-based wireless communication system.

outage probability analysis of a relay-based network over

Nakagami-m fading channels. A closed-form outage probabil-

ity expression for the case of 푚 = 1 (i.e., Rayleigh fading) was

derived with the assumption of identically independent fadings

on the direct and relay links. The work in [3] did not include

the analysis of other possible fading models nor did it provide

closed-form expressions for the case of (푚 ∕= 1), which will be

one of the concerns in this paper. An important performance

metric is the symbol error rate (SER). The derivation of exact

SER for the same cooperative relay-based network in [3]

was addressed in [4]. The analysis was performed using the

moment generating function (MGF) approach over Rayleigh

fading channels (MPSK modulation scheme was considered).

However, the authors ignored the effect of noise at the relay in

their derivations, which will be another concern in this paper.

Another work reported in [5] focused on the derivation of

closed-form expression of the bit error rate (BER) for the

detect-and-forward relay-based system for differential BPSK

over Nakagami-m fading channels. This work involved diffi-

cult integrations that are not simplified for nonidentical fading

on the direct and the relay links and/or different fading models,

which are also concerns in this paper.

In this paper, a three nodes relay-based cooperative system

is considered as given in Fig. 1 where the source node

is communicating with the destination node directly and

indirectly through the relay node. In this model, the relay

power-gain is optimized, following similar formulation as

given in [6] and [7], to maximize the received SNR at

the destination, taking into consideration the effect of relay

noise. Then, the statistics of the received SNR in relay-

based nonregenerative cooperative system are formulated over

2011 IEEE 22nd International Symposium on Personal, Indoor and Mobile Radio Communications

978-1-4577-1348-4/11/$26.00 ©2011 IEEE1894

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nonidentical fading channels including Rayleigh, Weibull, and

Nakagami-m. After that, the average SEP is investigated for

coherent and differentially coherent modulations. The problem

formulation and the analysis of the model in this paper resulted

in complicated expressions that requires numerical integrations

for the targeted performance metric. Therefor, we revert to an

approximation technique to arrive at approximate expressions

based on the finite mixture decomposition [8] and [9].

The remainder of this paper is organized as follows. The

system and channel model are introduced in section II. The

received SNR statistics of the relay-based system are presented

in section III. The average SEP analysis is given in Section IV.

Numerical results are presented and validated by simulations

in Section V. Finally, the chapter is concluded in Section VI.

II. SYSTEM AND CHANNEL MODEL

Consider a relay-based wireless communication system as

in Fig. 1. Assume that terminal S is transmitting a signal 푠(푡),

which has an average power normalized to one (i.e., E[푠2(푡)] =

1), then the received complex baseband signal at R is

푟푅(푡) =˜푏푠(푡) + 푛푏(푡),

(1)

where˜푏 is the fading amplitude of the channel between

terminals S and R (here, assume that the fading phase can

be compensated), and 푛푏(푡) is an additive white gaussian

noise (AWGN) signal with one-sided power spectral density

(PSD) given as 푁푏. At terminal R, the received signal is

multiplied by the gain G of the nonregenerative (i.e., amplify

and forward) relay which then retransmits the signal using a

different frequency band to terminal D. The received complex

baseband signal at terminal D from terminal R can now be

written as

푟퐷,1(푡) = G˜ 푐(˜푏푠(푡) + 푛푏(푡)) + 푛푐(푡),

(2)

where ˜ 푐 is the fading amplitude of the channel between

terminals R and D, and 푛푐(푡) is an AWGN signal with one-

sided PSD 푁푐. On the other hand, the received complex

baseband signal at terminal D from terminal S can be written

as

푟퐷,2(푡) = ˜ 푎푠(푡) + 푛푎(푡),

(3)

where ˜ 푎 is the fading amplitude of the channel between

terminals S and D, 푛푎(푡) is an AWGN signal with one-

sided PSD 푁푎. Using equal gain combining (i.e., the diversity

branches are co-phased) for the received signals at terminal D

from terminal S and R (here the separation can be achieved

using frequency diversity) enables us to write the overall

received signal as

푟퐷(푡) = (G˜ 푐˜푏 + ˜ 푎)푠(푡) + G˜ 푐푛푏(푡) + 푛푐(푡) + 푛푎(푡).

(4)

Following similar formulation as [6], the maximum SNR is

achieved when G = 2

˜푏

˜ 푎 ˜ 푐and can be given as

˜ 훾푚푎푥=E[푠2(푡)]

푁표

[1

2˜ 푎2+˜푏2] =1

2˜ 훾푎+ ˜ 훾푏,

(5)

where ˜ 훾푎 = ˜ 푎2/푁표 and ˜ 훾푏 =˜푏2/푁표 are the received SNR

over S-D link and S-R link, respectively. The pdf of 훾푚푎푥,

given that the two links in (5) are statistically independent,

can be found by referring to [10] as

∫∞

where 훾표 = E[푠2(푡)]/푁표. The moment generating function

(MGF) of ˜ 훾푚푎푥can be written as

푓˜ 훾푚푎푥(훾) =

0

푓˜ 푎2

(

훾 − (2

훾표)휂

)

푓˜푏2

(휂

훾표

)

푑휂,

(6)

푀˜ 훾푚푎푥(푠) = 푀˜ 훾푎

(푠

2

)

푀˜ 훾푏(푠),

(7)

where the resultant MGF depends on the fading statistics over

both links 푎 and 푏. Three fading scenarios are considered

here; Rayleigh, Nakagami-푚, and Weibull fading. If identical

independent fading is assumed in the two links (i.e., 푎 and 푏),

then the MGF of 훾푚푎푥for Rayleigh and Nakagami-푚 as [11,

Table 2.2, pp. 21]

푀˜ 훾푚푎푥(푠) =

(

1 −푠훾표

1 −푠훾표

2

)−1

)−푚(

(1 − 푠훾표)−1,

1 −푠훾표

푚

(8)

푀˜ 훾푚푎푥(푠) =

(

2푚

)−푚

,

(9)

where 푚 is the Nakagami-푚 fading parameter. The MGF of

훾푚푎푥 for Weibull is given in (10) on the top of next page

where G

fading parameter.

These MGF expressions can be used to evaluate an im-

portant performance metrics of cooperative system operating

over such fading channels. In this paper, an approach based

on Gaussian finite mixture approximation will be followed to

unify the SER analysis over various types of fading channels.

In this approach, the fading pdfs are represented in terms of

a weighted sum of Gaussian pdfs with suitable parameters.

These weights and parameters represent a signature of the

fading. This kind of decomposition inherits two important

advantages. The first advantage is the simplicity of the analysis

for various identical or nonidentical fading models for the

targeted performance metric. Secondly, this approach unifies

the analysis over various fading scenarios [6].

훽/2,1

1,훽/2(.) is the Meijer’s G-function and 훽 is the Weibull

III. SNR STATISTICS OF THE RELAY-BASED SYSTEM

The pdfs of ˜ 푎2and˜푏2can be expressed mathematically

as the weighted sum of Gaussian pdfs using the expectation

maximization algorithm as follows:

푓˜ 푎2(푎2) ≃

푁푎

∑

푖=1

푤푎,푖

√2휋휎푎,푖

exp

(

−((푎2)2− 휇푎,푖)2

2휎2

푎,푖

)

,

(11)

푓˜푏2(푏2) ≃

푁푏

∑

푗=1

푤푏,푗

√2휋휎푏,푗

exp

(

−((푏2)2− 휇푏,푗)2

2휎2

푏,푗

)

,

(12)

where 푁푎 and 푁푏 are the number of Gaussian components

which are required to approximate the pdfs of ˜ 푎2and˜푏2,

respectively. As it will be shown by numerical examples,

1895

Page 3

푀˜ 훾푚푎푥(푠) =훽

2(2휋)

2−훽

2

⎛

⎝Γ

(

1 +4

훽

)

훾표

⎞

⎠

훽

2 (

−√2푠

훽

)훽

×

⎡

⎢

⎢

⎣G

훽/2,1

1,훽/2

⎛

⎜

⎜

⎝

⎛

⎝Γ

(

1 +4

훽

)

훾표

⎞

⎠

−훽

4 (−2푠

훽

)훽/2

1

1,1 + 2/훽,...,1 + (훽 − 2)/훽

⎞

⎟

⎟

⎠

⎤

⎥

⎥

⎦

2

,

(10)

this kind of approximation coincides with the Monte Carlo

simulation to an acceptable degree of accuracy (∣error∣<1%).

Using random variable transformation and by assuming 훾표=

E[푠2(푡)]/푁표, the pdf of ˜ 훾푎and ˜ 훾푏can be obtained from (11)

and (12) as

푓˜ 훾푎(훾푎) ≃

푁푎

∑

푁푏

∑

푖=1

푤푎,푖

√2휋휎푎,푖

훾표

exp

(

(

−(훾푎/훾표− 휇푎,푖)2

2휎2

푎,푖

)

)

, (13)

푓˜ 훾푏(훾푏) ≃

푖=1

푤푏,푖

√2휋휎푏,푖

훾표

exp

−(훾푏/훾표− 휇푏,푖)2

2휎2

푏,푖

. (14)

Our objective is to find the pdf and the MGF of ˜ 훾푚푎푥.

A. The MGF of ˜ 훾푚푎푥

The MGF of a random variable ˜ 푥 with pdf 푓˜ 푥(푥), where

푥 > 0, is defined as

∫∞

From (11) and (12), and by using (15), it can be shown after

some straightforward manipulation that

[

[

푀˜ 푥(s) = E[exp(s푥)] =

0

exp(s푥)푓˜ 푥(푥)푑푥.

(15)

푀˜ 훾푎(s) =

푁푎

∑

푁푏

∑

푖=1

푤푎,푖exp(훾표휇푎,푖)s +(훾2

표휎2

푎,푖)s2

2

]

]

,

(16)

푀˜ 훾푏(s) =

푖=1

푤푏,푖exp(훾표휇푏,푖)s +(훾2

표휎2

푏,푖)s2

2

.

(17)

From (7), and by assuming ˜ 푎 and˜푏 to be independent random

variables, which is an acceptable assumption because the two

links are independents, the MGF of ˜ 훾푚푎푥, after scaling ˜ 훾푎by

1/2, can be given as follows:

푀˜ 훾푚푎푥(s) =

푁푎

∑

푖=1

푁푏

∑

푗=1

푤푖푗,푒푞exp

[

(휇푖푗,푒푞)s +(휎2

푖푗,푒푞)s2

2

]

(18)

,

where,

푤푖푗,푒푞= 푤푖,푎푤푗,푏

휇푖푗,푒푞= 훾표

[휇푎,푖

(휎푎,푖

2

+ 휇푏,푗

]

푏,푗

휎2

푖푗,푒푞= 훾2

표

[

2

)2+ 휎2

]

.

(19)

B. The pdf of ˜ 훾푚푎푥

By noticing that (18) represents a double summation of

MGF of Gaussian random variables with parameters 휇푖푗,푒푞

and 휎푖푗,푒푞, it can be proved that the pdf of ˜ 훾푚푎푥is given by

푓˜ 훾푚푎푥(훾) =

푁푎

∑

푖=1

푁푏

∑

푗=1

푤푖푗,푒푞

√2휋휎푖푗,푒푞

exp

[

−(훾 − 휇푖푗,푒푞)2

2휎2

푖푗,푒푞

]

. (20)

The derived expression of 푓˜ 훾푚푎푥(훾) contains (푁푎+푁푏) means,

(푁푎+ 푁푏) variances, and (푁푎+ 푁푏) weighting coefficients.

IV. SYMBOL ERROR PROBABILITY

The MGF approach is very useful to derive the average

SEP of a communication system over fading channels. In this

section, we discuss how the MGF in (18) can be used to

simplify the derivation of analytical expressions of the average

SEP over generalized fading channels that can be represented

as finite weighted sum of Gaussian pdfs. The author in [12]

summarizes general SEP expressions over AWGN channel for

coherent modulations as

푃푠(훾) ≃ 훼 푄(√휌 훾),

(21)

where 훼, 휌 depend on the type of modulation as given in Table

I, and 푄(푥) is the Q-function defined as

∫∞

. In particular, the nearest neighbor approximation has the

form as given in (21), where 훼 is the number of nearest

neighbors to a constellation at the minimum distance, and 휌

is a constant that relates the minimum distance to the average

symbol energy. Table I summarizes the SEP (i.e., 푃푠(훾)) for

PSK, QAM, and FSK modulations. Moreover, as an example

of differentially coherent modulation, 푃푠(훾) of DPSK is also

given in Table I. To get an approximation of the bit error

rate (BER) we use the same expression but we replace 훼 by

훼

log2(푀), where 푀 is the modulation order.

The average SEP (i.e., 푆퐸푃) over generalized fading chan-

nels can be derived as follows:

∫∞

=훼

휋

00

2sin2(휙)

=훼

휋

0

2sin2(휙)

푄(푥) =

푥

1

√2휋exp(−푢2

2)푑푢.

(22)

푆퐸푃 =

0

푃푠(훾)푓˜ 훾푚푎푥(훾)푑훾

∫휋/2

∫휋/2

Substituting (18) in (23), we get

∫∞

exp

(

−휌훼

)

푓˜ 훾푚푎푥(훾) 푑휙 푑훾

푀˜ 훾푚푎푥

(

−휌훼

)

푑휙.

(23)

푆퐸푃 ≃훼

휋

푁푎

∑

푖=1

푁푏

∑

푗=1

푤푖푗,푒푞

∫휋/2

0

exp

[−휇푖푗,푒푞휌

푖푗,푒푞휌2

8sin4(휙)

2sin2(휙)

]

+

휎2

푑휙.

(24)

1896

Page 4

For differential modulation, the average SEP of binary

differential phase shift keying (DPSK) can be simplified to

be

푆퐸푃 =1

2푀˜ 훾(푠)∣푠=1≃

푁푎

∑

푖=1

푁푏

∑

푗=1

푤푖푗,푒푞

2

exp[(휇푖푗,푒푞)

+

(휎2

푖푗,푒푞)

2

]

.

(25)

It is obvious from (24) and (25) that using Gaussian finite

mixture representation enabled us to find expressions for the

average SEP in simple forms.

TABLE I

SEP FOR VARIOUS MODULATION SCHEMES

Modulation

BFSK

BPSK

DPSK

QPSK,4-QAM

M-PAM

M-PSK

푃푠(훾)

= 푄(√훾 )

= 푄(√2훾 )

=1

2exp(−훾)

≃ 2푄(√훾

푀

푄

≃ 2푄(√2훾 sin(휋/푀))

푄

√

)

≃2(푀−1)

≃4(√푀−1)

≃ 4푄(

(√

(√

3훾

푀−1)

6훾

푀2−1

)

)

Rectangular M-QAM

√푀

3훾

푀−1

Non-rectangular M-QAM

V. SIMULATION AND NUMERICAL RESULTS

In this simulation, the cooperative relay-based wireless

system is assumed as shown in Fig. 1. The fading amplitudes

˜ 푎,˜푏, and ˜ 푐 are assumed to be independent but not necessarily

identical random variables. Three types of fading are chosen;

Rayleigh, Nakagami-m, and Weibull fading. For more infor-

mation about these types of fading, the reader can refer to [11].

The fading amplitude ˜ 푎 is assumed to be a Rayleigh random

variable with 퐸[˜ 푎2] = 1, where the S-D link is assumed to

be the worst link. The fading amplitude˜푏 is assumed to be a

Weibull random variable with 퐸[˜푏2] = 1 and fading parameter

(훽 = 4) which represents a more reliable link than the S-D

link. The fading amplitude ˜ 푐 is assumed to be a Nakagami-m

random variable with 퐸[˜ 푐2] = 1 and fading parameter (m = 4

), which represents a reliable R-D link. A finite mixture with

expectation maximization algorithm is used to approximate the

pdfs of ˜ 푎2,˜푏2, and ˜ 푐2using ten Gaussian components for each

with tolerance 휀 = 10−3[8]. The choice of ten components

is for demonstration of the analysis and can be increased if

more accuracy is required. The parameters and the weighting

coefficients estimation for 푓˜ 푎2(푎2), 푓˜푏2(푏2), and 푓˜ 푐2(푐2) are

given in tables II, III, and IV, respectively.

Comparison between the analytical results for the BER

of BPSK, BFSK, and DPSK, in (24), and the Monte Carlo

simulation (105simulation runs) are given in Fig. 2. In this

figure, the analytical results are very near to the simulation for

moderate values of 훾표. On the other hand, the approximation

deviate for large and small values of 훾표. This deviation is ex-

pected because of the use of approximated expression of SEP

TABLE II

PARAMETERS AND WEIGHTING COEFFICIENTS FOR 푓˜ 푎2(푎2)

i

푤푎,푖

0.1139

0.0087

0.6147

0.0437

0.1752

0.0115

0.0017

0.0198

0.0034

0.0074

휇푎,푖

0.2821

4.9530

0.9306

2.8815

0.5280

4.0052

6.9460

2.4941

0.2566

3.4896

휎2

0.0398

0.4716

0.4448

0.2984

0.1466

0.2828

1.3330

0.3010

0.0415

0.3592

푎,푖

1

2

3

4

5

6

7

8

9

10

TABLE III

PARAMETERS AND WEIGHTING COEFFICIENTS FOR 푓˜푏2(푏2)

i

1

2

3

4

5

6

7

8

9

10

푤푏,푖

0.0920

0.7100

0.0436

0.0844

0.0271

0.0176

0.0183

0.0024

0.0023

0.0022

휇푏,푖

0.6081

0.9287

0.5662

1.5697

1.9191

2.1649

1.3026

0.4655

2.5025

2.9282

휎2

0.0375

0.0982

0.0337

0.0500

0.0566

0.0819

0.0809

0.0264

0.0636

0.1070

푏,푖

in Table I. Furthermore, to investigate the approximation with

higher order modulations, a comparison between the analytical

results for the BER of QPSK, 8-QAM, 16-QAM, and 64-

QAM, in (24), and the Monte Carlo simulation (105simulation

runs) are given in Fig. 3. Here, it is also obvious that the

simulation coincide with analytical results to acceptable degree

of accuracy.

01234567

10

−3

10

−2

10

−1

γo [dB]

Average BER

DPSK SIM

DPSK ANA

BPSK SIM

BPSK ANA

BFSK SIM

BFSK ANA

Fig. 2.

BFSK, and DPSK , in (24), and the Monte Carlo simulation (105simulation

runs).

Comparison between the analytical results of the BER for BPSK,

1897

Page 5

TABLE IV

PARAMETERS AND WEIGHTING COEFFICIENTS FOR 푓˜ 푐2(푐2)

i

1

2

3

4

5

6

7

8

9

푤푐,푖

0.1863

0.1230

0.4457

0.1815

0.0172

0.0269

0.0104

0.0065

0.0015

0.0010

휇푐,푖

1.3307

0.4337

0.8707

0.5790

0.3380

1.6683

1.9818

2.1361

2.6075

0.2962

휎2

0.1051

0.0402

0.0995

0.0645

0.0267

0.0854

0.0584

0.0614

0.0672

0.0294

푐,푖

10

0510 1520

10

−3

10

−2

10

−1

γo [dB]

Average BER

QPSK SIM

QPSK ANA

8−QAM SIM

8−QAM ANA

16−QAM SIM

16−QAM ANA

64−QAM SIM

64−QAM ANA

Fig. 3.

8-QAM, 16-QAM, and 64-QAM, in (24), and the Monte Carlo simulation

(105simulation runs).

Comparison between the analytical results of the BER for QPSK,

VI. CONCLUSION

In this paper, the performance analysis in terms of SER

for cooperative relay-based wireless system is considered over

generalized fading channels. General analytical formulations

of SNR statistics, including pdf and MGF are extracted in

closed-forms. These expressions provided a tool to study the

SER performance of a relay-based system over generalized

fading channels. The SEP expressions for coherent and dif-

ferentially coherent modulations are derived. In the numerical

demonstrations, Weibull, Nakagami-m, and Rayleigh fading

channels are used as examples to show the effectiveness of

using finite mixture decomposition in simplifying the SER

perfomance analysis of relay-based systems.

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