performance of m-qam over generalized mobile fading...

DEPARTMENT OF ELECTRICAL ENGINEERING

COLLEGE OF ENGINEERING KING SAUD UNIVERSITY

PERFORMANCE OF M-QAM OVER GENERALIZED

MOBILE FADING CHANNELS USING MRC DIVERSITY

BY

IBRAHIM AL-SHAHRANI

A DISSERTATION

SUBMITTED TO THE DEANSHIP OF GRADUATE STUDIES IN PARTIAL

FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE

IN

ELECTRICAL ENGINEERING

SUPERVISED BY

PROF. ADEL AHMAD ALI

Feb 2007

II

We, the undersigned, have read the thesis and examined the candidate and do recommend the

thesis for acceptance.

Prof. Adel Ahmed Ali ----------------------------- (Supervisor)

III

ABSTRACT

Demands for faster data rates on wireless and cellular communication systems, such as

HSDPA and WiMax, have led to much current interest in the use of M-ary Quadrature

Amplitude Modulation (M-QAM) signaling formats due to its high spectral efficiency. In

wireless channel, it is well known that the fading phenomenon, which inherently exists in

most radio links, constitutes one of the boundary conditions of radio communications design.

A widely recognized practice for combating fading in digital communications over such a

time-varying channel is to use space diversity techniques. And the optimum liner diversity

combining technique is the Maximal-Ratio-Combining (MRC).

However, exact performance analyses of square coherent M-QAM in fading environments

have been reported recently in the literature, especially for MRC diversity systems. But all

the previous works were in integral form or in terms of special functions such as

hypergeometric function. In this thesis, new analytical, simple expressions for the exact

average symbol error rates (ASER) for M-QAM transmitted over slow, flat, identically

independently distributed (i.i.d) fading channels using MRC. Three types of fading channels

are considered: Rayleigh, Ricean, and Nakagami-m. These simple and efficient ASER

formulas make it possible for the first time to study, analyze and discuss the parameters of

various constellations of square M-QAM, diversity order and fading parameters precisely and

easily.

The obtained expressions are in the form of sum of exponentials where the number of terms

can be determined according to the required accuracy. Theoretically to get the exact solution,

the series must be infinite. But because the series converge rapidly, 10 terms are enough to

get an error less than 0.005 dB in the worst case which is at low order of modulation index

(M), diversity, and low fading parameter (K for Rician or m for Nakagami-m). The minimum

number of terms to approximate the ASER is two terms, so single term is also investigated

which is available in the literatures. The error for different number of terms are investigated

and tabulated.

IV

CONTENTS

Abstract III

Acknowledgements VI

1. Introduction 1

1.1 Background 1

1.2 Multipath Fading Environment 2

1.2.1 Rayleigh Fading 3

1.2.2 Ricean Fading 4

1.2.3 Nakagami-m Fading 5

1.3 Principles of Diversity Combining 6

1.3.1 Maximal Ratio Combining 7

1.3.2 Equal Gain Combining 9

1.3.3 Selection Combining 10

1.4 Literature Review 11

1.4.1 Ricean Fading channel 11

1.4.2 Nakagami-m Fading channel 13

2. Performance of M-QAM transmitted over AWGN Channel 15

2.1 Introduction 15

2.2 Exact representation of error function Q(x) in terms of a series of exponentials 17

2.3 Exact representation of error function squared Q2(x) in terms of a series of 22

exponentials

2.4 M-QAM Performance over AWGN channel using the exponential form 23

2.5 Concluding Remarks 25

3. Performance of M-QAM transmitted over Ricean Channel with MRC 33

3.1 Introduction 33

3.2 Performance of M-QAM over single Ricean channel 33

3.3 M-QAM Performance over Ricean channel with MRC Diversity 38

3.3.1 Approximation of M-QAM Performance over Ricean channel with MRC 43

Diversity

V

3.3.2 Analysis of Approximation 45


4. Performance of M-QAM transmitted over Nakagami-m Channel with MRC 63

4.1 Introduction 63

4.2 Performance of M-QAM over Nakagami channel with MRC Diversity 63

4.3 Results and Analysis of M-QAM over Nakagami channel with MRC Diversity 65

4.4 Approximation of M-QAM over Nakagami channel with MRC Diversity 69

4.4.1 Analysis of Approximation 71


5. Conclusions and Future Work 84

5.1 Conclusion 84

5.2 Future Work 85

References 86 Appendices 89 Appendix I Appendix II

VI

ACKNOWLEDGEMENTS

All praises be to the Almighty ALLAH, whose uniqueness, oneness, and wholeness is

unchangeable, and Who gave me the confidence, courage, and patience during the course of

this work. All respects are for the prophet MUHAMMAD (P.B.U.H) who enabled us to

recognize our creator.

During the course of my graduate studies, I have benefited tremendously from my

interactions with many extraordinary individuals. Foremost, I thank my advisor Prof. Adel A.

Ali for his guidance and support. His unique blend of vision, technical knowledge and

generosity will be an inspiring role model for my future career.

I thank all faculty members of the Electrical Engineering department for being so nice and

cooperative to me, especially Prof. Abdulaziz Al-Ruwais and Prof. Abdulrahman Al-Jabri.

I would like to thank the staff of STC and Huawei Technologies for their co-operation and

support in completing this research.

Last, but not the least, I would like to thank my wife and daughter and my parents.

1

CHAPTER 1

INTRODUCTION 1.1 Background Wireless communications has been one of the fastest growing segments in the

telecommunications industry. In 2005 with 140 million wireless telephone subscribers in the

Middle East and Africa and more than 2 billion around the world, the future looks even

brighter with potential for growth. The current mobile network is a combination of 2G and

3G cellular systems. The various advantages of 3G over 2G such as higher speed as well as

increased system capacity have been major motivations to move to 3G. In addition to

wireless voice service, markets in wireless broadband services, cordless phones, and direct-

to-home satellite broadcasting have been expanding.

The trends described above are sufficient to sustain a strong demand for wireless digital

communication systems in the future. The major driving force for wireless in the 21st

century, however, will likely lie in the increasingly popular Internet. The 90s have witnessed

the emergence of the Internet as a source for information access, a way to communicate

(email, on-line chat rooms) and an opportunity for business. The impact of the Internet on the

traditional telephone industry is far-reaching. For some time, it has been apparent that data

and telecommunication technologies are converging. Today operator companies not only

provide voice but also data services. Future communication networks, using both wired and

wireless interconnection, will seamlessly support a variety of services, including voice, video

and data. Such huge information flow can only be realized with high speed transmission.

Since present phone lines support data transmission at fixed points, high-speed wireless

mobile networks are being developed as a possible choice for transmission from a central

office to the end user.

People want high transmission speeds and considerable effort has been made to raise the

transmission speed of networks. The increases in speed of mobile systems during the past

twenty five years have come about in the main with a corresponding increase in the number

of signals in a two-dimensional (2-D) modulation format, where M-QAM is the best

candidate today. Knowledge of the error performance of M-QAM signal set in additive white

2

Gaussian noise (AWGN) and in fading channel is very useful to a particular system design,

such as reduced implementation complexity. In the latter case, it is then desirable to know the

error performance sacrificed in exchange for other gains. Evaluation of the exact symbol

error rate (SER) of M-QAM already available in the literatures but not in easy formulas such

as hypergeometric functions.

In this thesis, we represent the Q(x) function or the error function in terms of a summation of

exponential functions to make the analytical solution doable and simple. Then this new form

is used to find the ASER of M-QAM over different fading channels and also when the MRC

diversity technique is used. The objective is to provide new and simple analytical tools and

results that can be used by system designers and students to determine and compare the

performances of M-QAM coherent modulation schemes in fading channel when MRC is

used. This is essential to the development of high speed wireless data systems. Of the same

importance, or maybe more important, is the fact that this method can be used for a wide

range of fading conditions and environments , and are not limited to a specific fading model,

diversity technique, or modulation scheme.

1.2. Multipath Fading Environment

The mobile radio propagation environment places fundamental limitations on the

performances of wireless radio systems. There are roughly three independent phenomena that

together create a hostile transmission environment: path loss variation with distance, short

term (fast) multipath fading and slow log-normal shadowing. The underlying physical

principles behind these three phenomenon are different. Path loss is due to the decay of

electro-magnetic wave intensity in the atmosphere. Multipath fading is caused by multipath

propagation, while slow shadowing is due to the topographical variations along the

transmission path. In this thesis, we will focus on multipath fading.

In wireless radio systems, several signals with different amplitudes, phases and delays

corresponding to different transmission paths arrive at the receiver. The different signal

components add at the receiver constructively or destructively to give the resultant signal.

This multipath fading results in rapid variations typically as much as 30 to 40 dB in the

envelope of the received signal over a distance corresponding to a fraction of a wavelength.

The velocity of the mobile station, v, and the carrier frequency, fc, determine the fading rate

or fading bandwidth, fD, that is, fD = (v/c) fc, where c is the speed of light. Hence, faster

motion leads to more rapid fading. Multipath also causes time dispersion, because the

3

multiple replicas of the transmitted signal propagate over different transmission paths and

reach the receiver antenna with different time delays. This is called frequency selective

fading. On the other hand, the fading is said to be non selective or flat if time delays in

distinct paths are not large enough to result in resolvable replicas of the transmitted signal at

the receiver antenna. From the frequency domain perspective, all frequency components of

the transmitted signal undergo the same attenuation and phase shift through the channel in a

frequency-flat fading channel. This thesis is concerned with frequency-flat multipath fading.

In a frequency-flat fading environment, the received signal is simply the transmitted signal

multiplied by a complex-valued random process which introduces a fading envelope and a

random phase to the transmitted signal. A number of different models have been proposed in

the literature to describe the statistical behavior of the fading envelope of the received signal

[14]-[16]. Well known models are the Rayleigh, Ricean and Nakagami distributions. Next we

discuss and review the theoretical origins and the characteristics of the three models.

1.2.1 Rayleigh Fading

In Rayleigh fading, the composite received signal consists of a large number of plane waves

resulting from scattering at surface elements [15]. Using a central limit theorem, the received

complex low-pass signal g(t) = α(t) exp(jφ(t)) = gI(t) + jgQ(t) can be modeled as a complex

Gaussian random process. In the absence of a line-of-sight (LOS) or specular component,

gI(t) and gQ(t) have zero mean. At any time t, gI and gQ are Gaussian random variables (RVs)

with

(1.1)

where E[x] denotes the expected value of x and Var[x], the variance of x. Therefore, the

envelope α of the received signal has a Rayleigh distribution given by

(1.2)

The Rayleigh fading mode1 agrees well with macrocellular field measurements over the

frequency range from 50 to 11,200 MHz at distances of a few tens of wavelengths or greater

where the mean signal is sensibly constant [17]. It usually applies to scenarios where there is

4

no LOS path between the transmitter and receiver antennas. The phase φ of the received

signal is uniformly distributed from 0 to 2π at any time t, and the amplitude and phase are

statistically independent. Define a new variable γs = α2 /N0 proportional to the squared

envelope α2, where No denotes the one-sided power spectral density of Gaussian noise.

Variable γs, denotes received signal-to-noise ratio per symbol. The probability density

function (PDF) of γs, is given by

(1.3)

where, Λ= σ2 /N0 is the average signal-to-noise ratio per symbol and u(x) is the unit step function. 1.2.2 Ricean Fading If there is a LOS or specular component between the transmitter and receiver, gI(t) and gQ(t)

have non-zero mean m1 and m2 and the envelope α is a Ricean RV with PDF given by

(1.4)

where µ2 = m12 + m2

2 is the non-centrality parameter, and I0(x) is the zero-order modified

Bessel function of the first End. Ricean fading is often observed in microcellular and satellite

applications where a LOS path exists [18], [19]. The Rice K factor is the ratio of the power in

the specular and scattered components, i.e., K = µ2/2σ2. For K = 0, the channel exhibits

Rayleigh fading, and for K = ∞, the channel has no fading. It was reported in [20] that a

typical value of K for practical microcellular channels is about K = 7 dB. Rice factor K = 12

dB was reported for a smaller number of cases. Values of Rice factor in outdoor and indoor

systems usually range from 0 to 25 [20]. The PDF of SNR variable γs, in a Ricean fading

channel can be expressed in terms of the Rice K factor as

(1.5)

where, Λ = (µ2 + 2σ2) /N0 is the average SNR in the Ricean fading.

5

The phase is no longer uniformly distributed but has a preferred value owing to the presence

of a dominant component. The PDF of the phase is given in [21, Eqn. (5.62)].

1.2.3 Nakagami-m Fading

The Nakagami-m distribution was developed by Nakagami in the early 1940's to characterize

rapid fading in long distance High-Frequency (HF) channels [14]. It was shown to sometimes

have greater flexibility and accuracy in matching some experimental data than either the

Rayleigh, Ricean, or log-normal distributions [22]-[24]. The Nakagami-m distribution

describes the received envelope amplitude by

(1.6)

where, Ω = E[α2] is the average power of Nakagami distributed α, Γ(x) is the gamma

function, and the parameter m is defined as the ratio of moments, called the fading parameter,

(1.7)

The Nakagami-m distribution is a generalized distribution that can be used to model different

fading environments by changing the value of m. Rayleigh fading is obtained

for m = 1, and a one-sided Gaussian RV is described by m = 0.5. The non-fading case

corresponds to m = ∞. For values of m in the range 1/2 ≤ m ≤ 1, (1.6) models fading

conditions more severe than Rayleigh fading. For values of m > 1, (1.6) models less severe

fading than Rayleigh. Furthermore, the Ricean distribution and log-normal distribution can

sometimes be closely approximated by Nakagami fading under certain conditions [14], [22],

[23]. A summary of Nakagami- m distribution properties can be found in a review paper by

Nakagami [14].

Using a transformation of random variables, the SNR γs, in Nakagami fading has PDF,

(1.8)

where, Λ = Ω /N0 is the average SNR.

6

1.3. Principles of Diversity Combining

In order to improve the reliability of transmissions on wireless radio channels, some measures

have to be employed to reduce the seventy of multipath fading. Diversity techniques have

been known to be effective in combating the extreme and rapid signal variations associated

with the wireless radio transmission path. Basically, the diversity method requires that a

number of transmission paths be available, al1 carrying the same message but having

independent fading statistics. The mean signal strengths of the paths should also be

approximately the same. Diversity can be achieved by methods that can be placed into seven

categories [25]. In this section, we are concerned with space diversity where the distance

between the receiving antennas is made large enough to ensure independent fading. Usually a

spatial separation of about a half-wavelength will suffice (typically less than 30cm for

frequencies above 500 MHz) [17]. Excellent references on the topic of diversity systems are

[17], [25]-[29].

Diversity combining refers to the method by which the signals from the diversity branches

are combined. There are several ways of categorizing diversity combining methods.

Predetection combining refers to diversity combining that takes place before detection, while

postdetection combining takes place after detection. For diversity strategies incorporating

signal summing, summing after detection can be either equal or inferior to summing before

detection because a nonlinear effect is often experienced in a detection process [26]. With

ideal coherent detection, there is no performance difference between predetection and

postdetection combining [25]. In any case, diversity combining methods include maximal

ratio combining (MRC), equal gain combining (EGC) and selection combining (SC).

The received signal from the l th diversity branch is represented by zl = glsi + nl = yl + nl (1.9)

where, gl = αl exp(jφl) is a complex channel gain, si is the transmitted signal, yl = glsi is the

faded signal, and nl is additive Gaussian noise.

7

1.3.1 Maximal Ratio Combining

In this method, the individual branches must be first co-phased and weighted proportionately

to their channel gain and then summed. This is equivalent to weighting each branch by the

complex conjugate of its channel gain, i.e., cl = gl* = αl exp(–jφl), where cl is the weighting

coefficient of the l th branch. It is well known that MRC results in a maximum likelihood

(ML) receiver [25] and gives the best possible performance among the diversity combining

techniques. Fig. 1.1 shows general block diagrams of a coherent predetection and

postdetection L-branch maximal ratio combiner.

The combined noiseless signal is

(1.10) The noise powers Pn in al1 branches are assumed to be equal. Likewise, the total noise power

is the sum of the noise powers in each branch, weighted by the branch gain factors,

(1.11) The total SNR is

(1.12)

the sum of the branch SNRs.

(a)

8

(b)

Fig.1.1. Block diagram of coherent maximal ratio combining, (a) predetection (b)

postdetection

When all the diversity branches provide the same average power and they are uncorrelated,

the PDF of the total SNR with MRC in Rayleigh fading is given by [17]

(1.13)

where, Λ is the average signal-to-noise ratio per branch.

In Ricean fading with MRC [30]

(1.14)

where, )2/( 22

1σµ==∑

=

L

llT KK , lK )2/( 22 σµl= , 2µ = ∑

=

L

ll

1

2µ , Λ = (µ2/ L + 2σ2) /(2Pn) is the

mean SNR per branch, and IL-1 (x) is the (L –1)th-order modified Bessel function of the first

kind.

9

In Nakagami-m fading with MRC [31],

(1.15)

where, mT = ∑=

L

llm

1 and ΛT = ∑

=

ΛL

ll

1, Λl is the mean SNR of the lth branch, and the ratio

lm /Λl is the same for all diversity branches. 1.3.2 Equal Gain Combining

Maximal ratio combining requires complete knowledge of channel branch gains. Equal gain

combining is similar to MRC because the diversity branches are co-phased, but simpler than

MRC as the gains are set equal to a constant value of unity. The block diagram for EGC is

the same as Fig.1.1 except for the weighting coefficients cl = exp(–jφl). That is, the channel

estimator in EGC only needs to estimate the channel phase but not the channel amplitude.

The performance of EGC is not as good as optimal MRC but is comparable to MRC. In

practice, coherent postdetection EGC is useful for modulation schemes having equal energy

symbols, such as M-ary phase shift keying (MPSK) because only channel phase information

is required. For signals of unequal energy, complete channel knowledge is required for

coherent detection and therefore postdetection MRC is usually used. Predetection EGC,

however, still has merits to be used with unequal energy signals because apart from the L

channel phase estimators, predetection EGC only requires one AGC to estimate channel

amplitude after the matched filter while predetection MRC would require L AGC's to obtain

L channel amplitude knowledge.

The combined noiseless signal is given by

(1.16) and the SNR of the combiner output is

(1.17)

To find the probability distribution function of an EGC output y which is a sum of Rayleigh

RV's is a difficult task. We will adopt Beaulieu's infinite series result for the distribution of

the combiner output signal in Rayleigh, Ricean and Nakagami-m fading [32]-[34].

10

1.3.3 Selection Combining Selection diversity is generally the simplest method of dl. Its performance suffers some loss

compared to MRC and EGC. Ideal selection combining chooses the branch giving the highest

SNR at any instant. Fig. 1.2 illustrates the principle of selection combining. In practice, the

branch with the largest (S +N) is usually selected, since it is difficult to measure SNR. For

radio links using continuous transmission, e.g., frequency division multiple access (FDMA)

systems, SC is not very practical, since it requires continuous monitoring of al1 the diversity

branches. If such monitoring is performed, it is probably better to use maximal ratio

combining since the implementation is marginally more complex and the performance is

better [25]. In time division multiple access (TDMA) systems, however, a form of SC can be

implemented where the diversity branch is selected prior to the transmission of a TDMA

burst. The selected branch is then used for the transmission of the entire burst [25]. In the

following analysis, we assume ideal continuous branch selection. As far as the statistics of the

output signal are concerned, it is immaterial where the selection is done. The antenna signals

could be sampled, for example, and the best one sent to the receiver.

Fig.1.2. Block diagram of predetection selection combining.

Assuming all L diversity branches are independent and identically distributed (iid), the PDF

of the signal amplitude at the output of a selection combiner in Ricean fading is given by [33]

(1.18)

11

where, Ω = E[α2] is the total signal power in each Ricean channel, and Q(a,b) =

dxaxIxeb

xa

∫∞ +−

)(02

22

is the Marcum-Q function.

The selection combining Nakagami-m distributed γs is derived for iid diversity branches as

(1.19)

where, γ(α,x) = dttex

t∫ −−

0

1α is the incomplete gamma function [35, 8.35] and Λ = E[γs].

For selection combining and Rayleigh fading, the PDF of γs, assuming iid diversity branches

is given by [17]

(1.20)

1.4 Literature Review In this section we will discuss the previous work done on the performance of square MQAM

of fading channels using Maximal Ratio Combining (MRC) reception. Both Ricean and

Nakagami-m channels will be analyzed and Rayleigh as a special case of either Ricean or

Nakagami-m channel.

1.4.1 Ricean Fading Channel Recently, Zhang et al [7] presented a closed form series expression for the error probability

of MRC multichannel reception of MQAM coherent systems over a flat Ricean fading

channels. The derived expression was

(1.21)

where,

and,

12

Seo et al [8] derived the exact SER expression of the coherent square M-QAM scheme with

an MRC diversity reception in Ricean fading channels. Their results were in terms of

confluent hypergeometric and hypergeometric functions.

Ps(e) = X1 – X2 (1.22)

Where,

Where,

denotes the confluent hypergeometric

function.

where,

2F1(a, b; c; z) is the hypergeometric function and A1 = 1–1/ M , A2 = 1.5 log2 (M)/(M–1).

Kl is the Ricean factor of the lth diversity channel.

is the average SNR per bit at the combiner output, is

the average SNR per bit on the lth diversity channel.

Manjeet et al [9] came up with a closed form expression for SER of MQAM in Ricean fading

environment under MRC diversity reception. The derived expression is in terms of a single

finite integral with an integrand composed of elementary functions.

(1.23)

13

where, g = 3/[2(M – 1)], B = 1–1/ M , sgkh γθθ += 2sin)( and k = K + L.

1.4.1 Nakagami-m Fading Channel Falujah et al [10] proposed a method for computing the exact average symbol error

probability (SEP) of the square MQAM with MRC diversity over independent Nakagami-m

fading channels with arbitrary fading index m. They proposed a closed form expression for

the average SEP over L independent diversity channels in terms of Gaussian hypergeometric

functions.

2122)( HHqqePM −+−= (1.24)

where,

(1.25)

q = 1–1/ M , p = 1.5 log2 (M)/(M–1) and 2F1(w, x; y; z) is the Gaussian hypergeometric

function.

(1.26)

The same authors [11] proposed another solution in terms of Gaussian hypergeometric

function and Appell's hypergeometric function. The total average SEP of MQAM is

2122)( HHqqePM −+−=

Where,

given that for Re(b) > 0, and |k| < |h|; (Re(.) denotes the real value). H2 is given by

14

(1.27)

given that F2 (d; a1, a2; b1, b2; x, y) is the Appell's hypergeometric function.

Manjeet et al [12] presented the symbol error rate (SER) performance of coherent square

MQAM with l th order diversity in frequency non-selective Nakagami-m fading environment

corrupted by additive white Gaussian Noise (AWGN). They derived an expression for SER in

terms of a single finite integral with an integrand composed of elementary function.

(1.28)

Annamalai et al [13] derived an exact integral expression for calculating the SER of MQAM

in conjunction with L-fold antenna diversity on Nakagami-m fading channel with MRC

reception. The solution was derived using Parseval's theorem. It was given in terms of Gauss

hypergeometric series.

21)( IIPES −=ε (1.29)

where,

(1.30)

(1.31)

where, )/( λλ +=∧

pp .

15

CHAPTER 2 PERFORMANCE OF M-QAM TRANSMITTED OVER AWGN CHANNEL 2.1. Introduction

M-ary Quadrature Amplitude Modulation (M-QAM) is the most spectrum efficient

modulation technique. The transmitted signal varies in both the carrier phase and carrier

amplitude between known constellation points in the (I, Q) plane. The information sequence

mj is separated into 2 branches Im, Qm. The second is modulated by a quadrature carrier

of the first. The signal waveforms can be expressed as [5]:

sm(t) = Im g(t) cos (2πfct) - Qm g(t) sin(2πfct,)

where, g(t) is the pulse shape signal, Im and Qm are the information-bearing signal amplitudes

corresponding to M = 2k possible k-bit blocks of symbols. Im and Qm take the discrete values

Im = Qm = (2m-1-M)d m=1,2, … M.

The signal waveforms have two terms, each one is M -PAM signal. For square M-QAM

constellation the k-bit per sample is even integer (i.e. M = 4, 16, 64 …).

Symbol Error Rate (SER) for square constellation M-QAM can be obtained from the SER of

M -PAM, where the SER for symmetric M-PAM is given by

( ) ⎟⎟⎠⎞

⎜⎜⎝

⎛

−⎟⎠

⎞⎜⎝

⎛−=

16112)( 2

0 MNE

QM

ESER sPAM (2.1)

Since QAM is composed of quadrature combination of 2 M −PAM, each with half the total

power and since a correct QAM decision is made only when a correct symbol decision is

made independently on each of these modulations, the probability of correct symbol decision

Ps(c) for QAM can be expressed as [5,2]:

16

[ ]

[ ]22

)(11)(

)()(

ESERESER

cPcP

PAMMQAM

PAMMsQAMs

−−=

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

−⎟⎟⎠

⎞⎜⎜⎝

⎛−−⎟

⎟⎠

⎞⎜⎜⎝

⎛

−⎟⎟⎠

⎞⎜⎜⎝

⎛−=

)1(3114

)1(3114)(

0

22

0 MNE

QMMN

EQ

MESER ss

QAM

(2.2)

or

( ) ( )( )γγ bQabQaESER QAM24)( −=

(2.3)

where, ⎟⎠

⎞⎜⎝

⎛ −=M

a 11 ,)1(

3−

=M

b and0N

Es=γ

As can be seen from equation (2.3) we have two terms; the first one with Q(x) and the second

has Q2(x). For simplicity equation (2.3) can be approximated by the first term only so that the

Q2(x) is ignored since Q2(x) << Q(x) [5]. Fig.2.1 shows equation (2.3) with and without

Q2(x).

12 12.5 13 13.5 14 14.5 15 15.5 1610-2

10-1SER of 16-MQAM on AWGNC for (3) with & without Q2(x)

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r

(3) with squer Q(x)(3) without squer Q(x)

Fig.2.1. M-QAM over AWGN channel with and without Q2(x).

17

From Fig.2.1, we can see that at 10-2

, the difference is only 0.0025 dB, so the squared term in

equation (2.3) can be omitted without major loss in the accuracy. Then, equation (2.3) can be

approximated as [5]

⎟⎟⎠

⎞⎜⎜⎝

⎛

−⎟⎟⎠

⎞⎜⎜⎝

⎛−≈

13114)(,, M

QM

EP AWGNQAMsγ (2.4)

where, γ = Es/No.

An upper-bound on BER, which is good to within 1 dB for M > 4 is given by [6]:

( ), , 2( ) 0.2exp 1.5 /( 1) logs QAM AWGNP E M Mγ< − − (2.5)

Equation (2.5) doesn’t have the error function. It has important advantage; it consists of only

elementary functions, exponential and the log functions in particular. However, the

approximation error is in the order of 1 dB. We will call this approximation Goldsmith in the

rest of this thesis. In the next section we will consider an expansion of the error function in

terms of exponential function.

2.2 Exact representation of Error Function Q(x) in terms of a series of exponentials In communication theory problems we often face Q(x), where

∫∞

−=x

u duexQ 2/2

21)(π

(2.6)

Few years ago, the following form appeared [1]

∫ ⎟⎟

⎠

⎞⎜⎜⎝

⎛ −=

2/

02

2

sin2exp1Q(x)

π

θπx

(2.7)

This new form has the following advantages [2]:

1) The integration limits are independent of the argument of the function.

2) The integrand has a Gaussian form with respect to the function.

18

3) It has finite integration limits [0, π/2].

Both equations (2.6) and (2.7) can not be evaluated in closed form. While it is not possible to

get a solution for equation (2.7) in a simple closed form expression not involving infinite

sums, so a numerical approximation needs to be implemented. Equation (2.7) can be written

in the following improved exponential form by applying numerical integration using right

rectangular rule [1]:

∫∑−

⎟⎟⎠

⎞⎜⎜⎝

⎛ −≤

=

i

i

dxxQi

N

i

θ

θ

θθπ

1

2

2

1 sin2exp1)(

( )∑=

−≤N

iii xbaxQ

1

2exp)(

where, πθθ 1−−

= iiia ,

iib

θ2sin21

= andN

ii 2

πθ = .

or [3] ∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −≤

N

i iNR

xN

xQ1

2

2

, sin2exp

21)(

θ (2.8)

where N

ii 2

πθ =

By increasing N, the bound tends to the exact value [1]. For example:

N=1, ⎟⎟⎠

⎞⎜⎜⎝

⎛ −≤

2exp

21)(

2

1xxQR (2.9)

N=2, ( ) ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −+−≤

2expexp

41)(

22

2xxxQR (2.10)

N=3, ( ) ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −+⎟⎟

⎠

⎞⎜⎜⎝

⎛ −+−≤

2exp

32exp2exp

61)(

222

3xxxxQR (2.11)

N=6, ( ) ( ) ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −+⎟⎟⎠

⎞⎜⎜⎝

⎛

+−

+⎟⎟⎠

⎞⎜⎜⎝

⎛ −+−+−+⎟⎟

⎠

⎞⎜⎜⎝

⎛

−−

≤2

exp32

2exp32expexp2exp

322exp

121)(

22222

2

6xxxxxxxQR

(2.12)

Fig.2.2 shows the plot of equations (2.7), (2.9), (2.10), (2.11), and (2.12) respectively. For the

case when N=1, we have the Chernoff-Rubin bound [1].

19

0 2 4 6 8 10 12 1410-6

10-5

10-4

10-3

10-2

10-1

x(dB)= 20 log (x)

log

Q(x

)

ExactN=1N=2N=3N=6

Fig. 2.2: Q(x) Exact given by equation (2.7) Vs. approximations given by (2.9), (2.10),

(2.11), & (2.12).

If we apply trapezoidal rule to equation (2.7), then we have [4]:

∑−

=

−

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+=

1

12

22/

, sin2exp

21

4)(

2 N

i i

x

NTx

NNexQ

θ (2.13)

where, N

ii 2

πθ =

Substituting in equation (2.13) for different values of N

N=1, 2/1,

2

41)( x

T exQ −≤ (2.14)

N=2, 22

41

81)( 2/

2,xx

T eexQ −− +≤ (2.15)

N=3, ( )3/222/3,

222

61

121)( xxx

T eeexQ −−− ++≤ (2.16)

N=4, ( )2222 )22/(2)22/(22/4, 8

1161)( xxxx

T eeeexQ −+−−−− +++≤ (2.17)

20

N=6, ( )3/22)32/(2)32/(22/6,

222222

121

241)( xxxxxx

T eeeeeexQ −−−+−−−− +++++≤ (2.18)

For the case when N=2 for an arbitrary pointθ , we have

( )θπθθ

222 sin2/2/2, 4

124

1),( xxT eexgQ −− +⎟

⎠⎞

⎜⎝⎛ −=≈ (2.19)

For optimumθ , 3/πθ =opt [1], equation (2.19) is given by

3/22/2,

22

41

121 xx

optT eeQ −− +≈ (2.20)

For high order of N (N ≥ 3) the optimum θ is almost like the equispaced one. Fig. 2.3 shows

Q(x) exact and three approximations for N=1, 2, 2opt. Fig.2.4 is zoomed version of Fig. 2.3.

0 2 4 6 8 10 12 1410-6

10-5

10-4

10-3

10-2

10-1

x(dB)= 20 log (x)

log

Q(x

)

ExactQT1QT2

QT2op

Fig.2.3. Exact Vs. approximations of equations (2.14), (2.15) and (2.20)

21

0 1 2 3 4 5 6 7 8 9 1010-3

10-2

10-1

x(dB)= 20 log (x)

log

Q(x

)ExactQT1QT2

QT2op

Fig 2.4: Exact Vs. approximations of equations (2.14), (2.15) and (2.20)

From Fig 2.4, we can see that equation (2.20) gives a tight upper bound with maximum

difference at x = 5dB or 0.5dB at 10-1.4. Equation (2.15) gives good approximation but it is

under the exact for 3.5 < x < 9 dB and the difference is maximum at x = 14dB. Fig. 2.5 shows

that for N ≥ 3 the trapezoidal overlaps the exact.

0 2 4 6 8 10 12 1410-6

10-5

10-4

10-3

10-2

10-1

x(dB)= 20 log (x)

log

Q(x

)

ExactQT3QT4

QT6

Fig.2.5. Exact Vs. approximations of equations (2.16), (2.17) and (2.18)

22

Table 2.1 and 2.2 provides a summary of the differences between the exact and the

approximation using rectangular rule and Trapezoidal rule in dB for different number of

terms respectively.

Table 2.1: Rectangular rule

Difference in dB using Rectangular Rule Log Q(x) N=1 N=2 N=3 N=6

10-1 2.922 1.587 1.132 0.6375 10-2 1.601 0.8045 0.6135 0.3415 10-3 1.1445 0.634 0.4325 0.25 10-4 0.905 0.536 0.351 0.2035 10-5 0.7545 0.467 0.3065 0.174 10-6 0.6505 0.4145 0.277 0.1535

Table 2.2: Trapezoidal Rule

Difference in dB using Trapezoidal Rule Log Q(x) N=1 N=2 N=2opt N=3 N=4 N=6

10-1 0.476 0.118 0.73 -0.052 -0.004 0.0005 10-2 0.754 -0.079 0.2775 0.0125 -0.0025 0 10-3 0.6305 0.062 0.0995 -0.0085 0.001 0 10-4 0.536 0.134 0.0415 -0.005 0 0 10-5 0.467 0.159 0.0315 0.012 -0.0015 0 10-6 0.415 0.1655 0.0375 0.0295 -0.0005 0

Comparing tables 2.1 and 2.2, it is clear that trapezoidal rule converges faster hence it is

better to be used in evaluating Q(x) numerically. For N > 6, equation (2.13) represents Q(x)

nearly exactly using sum of exponentials.

2.3 Exact representation of Error Function Squared Q2(x) in terms of a series of

exponentials

Another function which needs to be evaluated in communication theory especially when we

deal with QAM is Q2(x) which is defined as [2] follows:

∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

4/

02

22

sin2exp1(x)Q

π

θπx

23

Since Q(x) and Q2(x) are both monotonically increasing functions, we can apply trapezoidal

rule on Q2(x) as we did for Q(x) to get

∑−

=

−

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+=

1

12

22

sin2exp

41

8)(

2 N

i i

x xNN

exQθ

(2.21)

where, N

ii 4

πθ =

In appendix I it is shown that the best form to approximate Q2(x) using the minimum number

of exponential is

( )22 exp81)( xxQ −≈ (2.22)

2.4. M-QAM Performance over AWGN Channel using the Exponential Form

Using the exponential form of Q(x) and Q2(x) equation (2.3) can be written as:

2 2

/ 2 1 1/ 2sin / 2sin

1 1

1( ) 44 2 8 4

i i

b bm nb b

QAMi i

e ae aSER E a e em m n n

γ γγ α γ β

− −− −− −

= =

⎧ ⎫= + − −⎨ ⎬

⎩ ⎭∑ ∑

(2.23)

m

ii 2

πα = n

ii 4

πβ =

where, ∑−

=

−

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+=

1

12

22/

sin2exp

21

4)(

2 m

i i

x xmm

exQθ

, with m

ii 2

πθ =

and ∑−

=

−

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+=

1

12

22

sin2exp

41

8)(

2 n

i i

x xnn

exQθ

, with n

ii 4

πθ =

Let m = 2n then equation (2.23) will be

2 2/ 2 2 1 1

/ 2sin / 2sin

1 1

1( ) 48 4 8 4

i i

b bn nb b

QAMi i

e ae aSER E a e en n n n

γ γγ θ γ θ

− −− −− −

= =

⎧ ⎫= + − −⎨ ⎬

⎩ ⎭∑ ∑

24

and n

ii 4

πθ =

2 2/ 2 2 1 1

/ 2sin / 2sin

1 1( )

2 2i i

b bn nb b

QAMi i

a e aeSER E e a en

γ γγ θ γ θ

− −− −− −

= =

⎧ ⎫= + − −⎨ ⎬

⎩ ⎭∑ ∑

2 2/ 2 1 2 1 1

/ 2sin / 2sin / 2sin

1 1( )

2 2i i i

b bn n nb b b

QAMi i n i

a e aeSER E e e a en

γ γγ θ γ θ γ θ

− −− − −− − −

= = =

⎧ ⎫= + + − −⎨ ⎬

⎩ ⎭∑ ∑ ∑

( ) 2 2/ 2 1 2 1

/ 2sin / 2sin

1( ) 1

2 2i i

b b n nb b

QAMi i n

a e aeSER E a e en

γ γγ θ γ θ

− − − −− −

= =

⎧ ⎫= − + − +⎨ ⎬

⎩ ⎭∑ ∑ (2.24)

where, ⎟⎠

⎞⎜⎝

⎛ −=M

a 11 ,)1(

3−

=M

b ,0N

Es=γ and n

ii 4

πθ =

The probability of symbol error of M-QAM with square constellation over AWGN channel in

exponential form is given by equation (2.24). Next equation (2.24) will be examined for

different values of n by comparing equation (2.24) with the exact value computed using the

built-in MATLAB function (erfc).

n =1, ( )/ 2( ) (2 )2

b bQAM

aSER E e a eγ γ− −≈ + − (2.25)

n = 2, ( )/ 2

2 /(2 2 ) 2 /(2 2 )(2 )( ) 12 2 2

b bb b

QAMa e a eSER E a e e

γ γγ γ

− −− − − +⎛ ⎞−

≈ + + − +⎜ ⎟⎝ ⎠

(2.26)

n = 3, ( )( )/ 2

2 /(2 3) 2 2 /3 2 /(2 3)(2 )( ) 12 2 2

b bb b b b

QAMa e a eSER E a e e e e

γ γγ γ γ γ

− −− − − − − +⎛ ⎞−

≈ + + − + + +⎜ ⎟⎝ ⎠

(2.27)

Fig.2.6 shows the exact and approximate SER given by equation (2.25) for M= 4, 16 and 64

respectively. We can see that even for n =1 we have excellent approximation with maximum

difference of 0.3 dB for M = 64 at SER = 10-1, 0.27dB and 0.2dB for M=16 and M=4,

respectively. If we plot equation (2.26) with n = 2 we can see that both exact and approximate

25

overlap as shown in fig.2.8. The maximum difference between equations (2.26) and (2.3) for

all M is at SER= 10-1, which are 0.04dB, 0.05dB and 0.06dB for M=4, 16 and 64

respectively. For n = 3, the largest difference is 0.025dB at M = 64 and SER = 10-1. For n =

6, the largest difference is 0.006dB at M = 64 and SER = 10-1. Making n=16 will lead to

difference less than 0.0008 dB for all M and for 0 < SNR< 30 dB.

Also the approximation proposed by Chiani [1] can be used to express the system

performance by using equations (2.20) and (2.22) in (2.3) to get what we will call Chiani

approximation in the rest of the thesis. Then, the SER of square M-QAM over AWGN

channel using Chiani approximation is

/ 2

2 / 3

3 2

b bb

AWGNe aeSER a e

γ γγ

− −−⎛ ⎞

≈ + −⎜ ⎟⎝ ⎠

(2.28)

2.5 Concluding Remarks

The essence of the above results is summarized below:

1. The error function can be represented by an infinite series of exponential functions

using numerical integration.

2. Numerical integration using trapezoidal rule provide much better results as compared

to rectangular rule.

3. Error rate for M-QAM over Gaussian channels can be represented as a sum of N

exponential functions.

4. The above error rate analysis can be obtained with any required accuracy by simply

increasing N.

26

0 5 10 15 20 25 3010-6

10-5

10-4

10-3

10-2

10-1

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

rExactApp

M=4M=4

M=16

M=64

Fig.2.6. M-QAM on AWGN for both exact and approximate when n =1.

4 5 6 7 8 9 10 11 12 13 1410

-6

10-5

10-4

10-3

10-2

10-1

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r

ExactApp,n=1

Fig.2.7. 4-QAM on AWGN for both exact and approximate when n =1.

27

4 5 6 7 8 9 10 11 12 13 1410-6

10-5

10-4

10-3

10-2

10-1

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

rExactApp

Fig.2.8. 4-QAM on AWGN for both exact and approximate when n = 2.

4 5 6 7 8 9 10 11 12 13 1410

-6

10-5

10-4

10-3

10-2

10-1

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r

ExactApp

Fig.2.9. 4-QAM on AWGN both exact & approximate when n = 3.

28

12 13 14 15 16 17 18 19 20 2110-6

10-5

10-4

10-3

10-2

10-1

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

rExactApp


12 13 14 15 16 17 18 19 20 2110

-6

10-5

10-4

10-3

10-2

10-1

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r

ExactApp


29

12 13 14 15 16 17 18 19 20 2110-6

10-5

10-4

10-3

10-2

10-1

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

rExactApp


19 20 21 22 23 24 25 26 27 2810

-6

10-5

10-4

10-3

10-2

10-1

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r

ExactApp


30

19 20 21 22 23 24 25 26 27 2810-6

10-5

10-4

10-3

10-2

10-1

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

rExactApp


19 20 21 22 23 24 25 26 27 2810

-6

10-5

10-4

10-3

10-2

10-1

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r

ExactApp


31

19 19.5 20 20.5 21 21.5 22 22.5 23 23.5 2410

-3

10-2

10-1

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

rExactApp

Fig.2.16. Zoomed Version of 64-QAM on AWGN for both exact & approximate when n = 3.

19 19.5 20 20.5 21 21.5 2210

-2

10-1

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r

ExactApp

Fig.2.17. Zoomed version of 64-QAM on AWGN for both exact & approximate when n = 3.

32

0 5 10 15 20 25 3010-6

10-5

10-4

10-3

10-2

10-1

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

rExactApp

Fig.2.18. M-QAM on AWGN for both exact and approximate when n = 2

33

CHAPTER 3

PERFORMANCE OF M-QAM TRANSMITTED OVER RICEAN CHANNEL WITH MRC 3.1 Introduction

In this chapter, the average symbol error rate (ASER) performance of coherent square M-

QAM with L-th order diversity in frequency nonselective slowly Ricean fading environment

for single channel and multiple reception using MRC are presented. Ricean distribution is

considered here, because this distribution is not only the best fit for digital signals received in

line-of-sight (LOS) communication links [5], but also because there is a close agreement

between the Ricean distribution and variety of propagation paths spanning nearly all

frequency bands [2]. Moreover, like Nakagami distribution, Ricean distribution is also a two

parameter distribution, that provides more flexibility and accuracy in matching the observed

signal statistics. M-QAM considered here is the widely used modulation technique due to its

high spectral efficiency.

We assume that there are L diversity channels carrying the same transmitted signal. Each

channel is modeled as frequency non-selective slowly Ricean fading channel corrupted by

additive white Gaussian noise (AWGN) process. The fading processes among the L diversity

channels are assumed to be mutually statistically independent. The noise processes in the L

diversity channels are assumed to be mutually statistically independent, with identical

distribution (iid). For a M-QAM demodulator [5] which makes its decision based on the

output of a linear filter operating on an undistorted symbol waveform, it is well known that

the probability of symbol error caused by AWGN depends only on the instantaneous SNR (γ)

associated with each symbol.

3.2 Performance of M-QAM over Single Ricean Channel

From chapter 2, we have seen that the Symbol Error Rate (SER) for square constellation M-

QAM over AWGN channel is given by [5]

34

⎟⎟⎠

⎞⎜⎜⎝

⎛

−⎟⎠

⎞⎜⎝

⎛−−⎟

⎟⎠

⎞⎜⎜⎝

⎛

−⎟⎠

⎞⎜⎝

⎛−=

)1(3114

)1(3114

0

22

0 MNEQ

MMNEQ

MSER ss

QAM (3.1)

or ( ) ( )[ ]γγ bQabQaESERQAM24)( −= (3.2)

where, ⎟⎠

⎞⎜⎝

⎛ −=M

a 11 , )1(

3−

=M

b , 0N

Es=γ

Writing equation (3.1) in the exponential form we have,

( )⎭⎬⎫

⎩⎨⎧

+−+−= ∑ ∑−

=

−

=

−−−− 1

1

12//

2/

122

)(n

i

n

ni

SbSbbb

QAMii eeaaee

naESER γγ

γγ

(3.3)

where, ⎟⎠

⎞⎜⎝

⎛ −=M

a 11 , )1(

3−

=M

b , 0N

Es=γ , n

ii 4

πθ = 22sini iS θ=

To find the Average Symbol Error Rate (ASER) of QAM system over a single Ricean

channel we need first to determine the pdf of the instantaneous SNR, γ . In fading γ becomes

a random variable and for Ricean fading environment, the probability density function (pdf)

of instantaneous SNR γ at the output of channel is given as [2]

⎟⎟⎠

⎞⎜⎜⎝

⎛ +⎥⎦

⎤⎢⎣

⎡ +−−+=

γγ

γγ

γγγ

)1(20)1(exp1)( KKIKKeKp (3.4)

where, γ is the expected value ofγ , K is the Ricean parameter and 0I (.) is the zeroth-order

modified Bessel function of the first kind. Equation (3.4) can be written in a compact form as

[ ] ( )γγγγ rKIrKrep 20exp)( −−= (3.5)

where, γ

Kr +=

1

35

Once the statistics of the instantaneous SNR γ are determined as a function of the Ricean

parameter K and the average instantaneous SNRγ , the Average Symbol Error Rate of Ricean

channel (ASERR) as a function of K and γ of the system can be calculated by averaging the

conditional probability of SER over the pdf ofγ . i.e.,

γγγ dpEPASER S∫∞

=0

)()./( (3.6)

where, )/( γEPS is the conditional probability of symbol error which is the SER over AWGN

channel given by equation (3.3). Substituting equations (3.3) & (3.5) in equation (3.6) to

derive ASERR we have

( )⎪⎩

⎪⎨⎧ −×−−

= ∫∞

0

202

2/γγ

γγdrKI

reben

KareASERR ( )∫∞ −×−

−0

202γγ

γγdrKI

rebae

( )∑∫−

=

∞

××−+1

1 0

20/

)1(n

i

dIreiSbea γβγγγ

( )⎪⎭

⎪⎬⎫

××+ ∑∫−

=

∞12

0

20/n

ni

dIreiSbe γβγγγ

(3.7)

Using the following relationship [35]

( ) ( )∫∞

⎟⎠⎞

⎜⎝⎛=−

0exp120exp

ccdxxIcx ββ

Equation (3.7) can be written as

( )rbrb

Kra

rbrb

Kr

n

KareASERR +

⎟⎠⎞

⎜⎝⎛

+−

⎪⎪⎩

⎪⎪⎨

⎧

+

⎟⎠⎞

⎜⎝⎛

+−=

2

exp

22

2exp

( )∑−

= +

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−+

1

1

exp1n

i i

i

ii

rSbrSb

KrSSa

⎪⎪⎭

⎪⎪⎬

⎫

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+

+∑−

=

1

1

expn

i i

i

ii

rSbrSb

KrSS

36

( )rbrb

Kba

rbrb

Kb

narASERR +

⎟⎠⎞

⎜⎝⎛

+−

−

⎪⎪⎩

⎪⎪⎨

⎧

+

⎟⎠⎞

⎜⎝⎛

+−

=2

exp

22

exp

( )

⎪⎪⎭

⎪⎪⎬

⎫

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

++

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

−+ ∑∑

−

=

−

=

121

1

expexp1n

ni i

iin

i i

ii

rSbrSb

KbS

rSbrSb

KbSa (3.8)

( ) ( )( )

⎪⎪⎩

⎪⎪⎨

⎧

++

⎟⎟⎠

⎞⎜⎜⎝

⎛++

−+

=Kb

KbKb

nKaASERR 12

12exp

1γγ

γ

( )

( )( )KbKb

Kba

++

⎟⎟⎠

⎞⎜⎜⎝

⎛++

−

−12

1exp

γγ

γ

( ) ( )

( )( )

( )⎪⎪⎭

⎪⎪⎬

⎫

++

⎟⎟⎠

⎞⎜⎜⎝

⎛++

−

+++

⎟⎟⎠

⎞⎜⎜⎝

⎛++

−−

+ ∑∑−

=

−

=

121

1 11

exp

11

exp1n

ni i

iin

i i

ii

SKbSKb

KbS

SKbSKb

KbSa

γγ

γ

γγ

γ

(3.9)

Equation (3.9) represents the ASER of square QAM over single Ricean fading channel as a

function of K (Ricean parameter) and the average SNRγ . If we let K= ∞ in equation (3.9) we

will get equation (3.3) which is the SER of M-QAM over AWGN channel. By substituting

K=0 in equation (3.9) which represents Rayleigh fading channel, we get

⎩⎨⎧

+=

21γbn

aASERRay ( )12 +−

γba ( )

⎭⎬⎫

++

+−

+ ∑∑−

=

−

=

121

1

1 n

ni i

in

i i

i

SbS

SbSa

γγ

⎩⎨⎧

+=

21γbn

aASERRay ( )12 +−

γba

⎭⎬⎫

+−

++ ∑∑

−

=

−

=

1

1

12

1

n

i i

in

i i

i

SbaS

SbS

γγ (3.10)

Fig.3.1 shows 16-QAM over Ricean fading channel for different values of K that are 0, 2, 6,

and 12. These values usually used in the literatures because they close to the values observed

in the experimental work. From fig.3.1 we can see that small change in K leads to big change

in the system performance. Both figs.3.2 and 3.3 shows the performance of square M-QAM

37

over single Ricean channel. Where fig.3.2 is for both K= 0 and 2. And fig.3.3 for K=6 and

12.

0 5 10 15 20 25 30 35 4010-6

10-5

10-4

10-3

10-2

10-1

100

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r

K=0K=2K=5K=12AWGN

16-QAM

Fig.3.1. Exact 16-QAM over single Ricean with K= 0, 2, 6 & 12.

0 5 10 15 20 25 30 35 40

10-4

10-3

10-2

10-1

100

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r

K=0,RayleighK=2

+ 4 -QAM* 16-QAMo 64-QAM

Fig.3.2. Exact M-QAM over single Ricean with K= 0 & 2.

38

0 5 10 15 20 25 30 35 4010-6

10-5

10-4

10-3

10-2

10-1

100

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

rK=6K=12

+ 4 -QAM* 16-QAMo 64-QAM

Fig.3.3. Exact M-QAM over single Ricean with K= 6 & 12.

3.3 M-QAM Performance over Ricean Channel with MRC Diversity.

In the previous section, the performance of square M-QAM over single Ricean channel was

obtained. In this section, the system performance will be considered when the Maximal Ratio

Combiner (MRC) is used. The MRC diversity system has L antennas. With an assumption

that the received signal from each antenna is subjected to slow and flat Ricean channel and

these channels are independent and identical distributed (iid). Then the probability density

function (pdf) of γ after MRC diversity combining, is given by [7]-[9].

( )⎟⎟⎠

⎞⎜⎜⎝

⎛ +−⎥

⎦

⎤⎢⎣

⎡ +−−−

⎟⎟⎠

⎞⎜⎜⎝

⎛ ++=

γγ

γγ

γγ

γγγ

)(21)(exp2

1

)( KLKLIKLKe

L

KKLKLp

where, L is the number of branches, γ is the expected value of γ (SNR), K is the Ricean

parameter and 1−LI (.) is the modified Bessel function of L-1 order.

39

Let γ

KLr += &

21−

=Lv

[ ] ( )γγγγγ rKvIrKev

Krrp v 22exp)( −−⎟⎠⎞

⎜⎝⎛= (3.11)

The ASER as functions of K, L, and γ of the system can be calculated by averaging the

conditional probability of error over the pdf ofγ . This could be done by substituting

equations (3.3) & (3.11) in equation (3.6)

( )⎪⎩

⎪⎨⎧ −−

⎟⎠⎞

⎜⎝⎛−

= ∫∞

0, 222

2/γγ

γγγ drKvIberevv

Kr

n

KareASER MRCRic

( )∫∞ −−

−0

222γγ

γγγ drKvIbereva

( ) ( ) γγγ γγ drKvIeean

i

Sbrv i 2211

1 0

/∑∫−

=

∞−−−+ ( )

⎭⎬⎫

+ ∑∫−

=

∞−− γγγ γγ drKvIee

n

ni

Sbrv i 2212

0

/

( )( )

⎪⎩

⎪⎨⎧ +−

⎟⎠⎞

⎜⎝⎛−

= ∫∞

0, 222

2/2γγ

γγ drKvIrbevv

Kr

n

KareASER MRCRic

( ) ( )∫

∞ +−

−0

222γγγ γ

drKvIea rbv

( ) ( ) ( ) γγγ γ drKvIean

i

SrSbv ii 2211

1 0

/∑∫−

=

∞+−−+ ( ) ( )

⎭⎬⎫

+ ∑∫−

=

∞+− γγγ γ drKvIe

n

ni

SrSbv ii 2212

0

/

(3.12)

Using the following relationship [35]

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛∫∞

= +−−

αββαβα

2exp

0)2(2

2)12( vvxv dxxvIex

40

( )⎥⎦⎤

⎢⎣⎡

+−

⎪⎩

⎪⎨⎧ +

⎟⎠⎞

⎜⎝⎛

+⎟⎠⎞

⎜⎝⎛=

rbKbv

rb

v

Kr

n

vrKarASER MRCRic 2exp

12

22

21

,

⎥⎦⎤

⎢⎣⎡

+−

⎟⎠⎞

⎜⎝⎛

+−

+

rbKb

rba v

exp12

12

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

+−

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−+ ∑−

=

+

irSbKb

rSbSa

n

i

v

i

i exp11

1

12

⎪⎭

⎪⎬⎫

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+−

⎟⎟⎠

⎞⎜⎜⎝

⎛+

+ ∑−

=

+

irSbKb

rSbSn

ni

v

i

i exp12 12

(3.13)

⎥⎦⎤

⎢⎣⎡

+−

⎪⎩

⎪⎨⎧

⎟⎠⎞

⎜⎝⎛

+=

rbKbL

rbr

naASER MRCRic 2

exp2

221

, ⎥⎦⎤

⎢⎣⎡

+−

⎟⎠⎞

⎜⎝⎛

+−

rbKb

rbra L

exp2

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

+−

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−+ ∑−

= irSbKb

rSbrSa

n

i

L

i

i exp11

1

⎪⎭

⎪⎬⎫

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+−

⎟⎟⎠

⎞⎜⎜⎝

⎛+

+ ∑−

= irSbKb

rSbrSn

ni

L

i

i exp12

(3.14)

where, ⎟⎠

⎞⎜⎝

⎛ −=M

a 11 , )1(

3−

=M

b , 0N

Es=γ , n

ii 4

πθ = ,

γKLr +

= , Si=2sin2 (θi) , γ = mean(γ )

Equation (3.14) is the ASER of QAM using MRC of L branches over Ricean channels that is

iid. Then it is varied in γ (average signal-noise ratio), L (number of MRC branches), Ricean

parameter, ∑=

=L

jjKK

1, and M (the modulation index).

For L=1, equation (3.14) is reduced to equation (3.9). If K= ∞ then equation (3.14) will be

equation (3.3) where the MRC doesn't improve the system. When K=0 then equation (3.14)

represents a Rayleigh flat slowly fading channel and equation (3.14) is reduced to

LL

MRCRay LbLa

LbL

naASER ⎟⎟

⎠

⎞⎜⎜⎝

⎛+

−⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=γγ 22

221

, ( )∑−

=⎟⎟⎠

⎞⎜⎜⎝

⎛+

−+1

1

1n

i

L

i

i

LSbLSa

γ

⎪⎭

⎪⎬⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛+

+ ∑−

=

12n

ni

L

i

i

LSbLS

γ

(3.15)

Fig.3.4 shows the performance of Rayleigh channel using MRC for different values of L (1,

2, 3, 4, 8, 16) for 16-QAM and fig.3.5 shows 16-QAM over Ricean channel with K=8. In

41

fig.3.5 we can see 16-QAM with diversity over both Rayleigh and Ricean with K=2 and in

fig.3.7 the same system but the Ricean parameter K is 6 and 12. Form figs.3.4 – 3.7 we can

see that the improvement in the system performance decreases when the K increase and also

when the order of diversity increase (the number of used branches).

15 20 25 30 3510-6

10-5

10-4

10-3

10-2

10-1

SNR per Symbol (dB)

AS

ER

L=1L=2L=3L=4L=8L=16

Fig.3.4. Exact 16-QAM over Rayleigh for different L.

42

15 20 25 30 3510-6

10-5

10-4

10-3

10-2

10-1

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r

L=1L=2L=3L=4L=8L=16

Fig.3.5. Exact16-QAM with MRC of different L over Ricean with K = 8.

0 5 10 15 20 25 30 35 4010-6

10-5

10-4

10-3

10-2

10-1

100

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r

L1L=2L=4L=8

o K=0+ K=2* AWGN

16-QAM

Fig.3.6. Exact 16-QAM over Ricean with K=0, 2 and MRC with L = 1,2,4 and 8.

43

15 20 25 3010-6

10-5

10-4

10-3

10-2

10-1

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r

L1L=2L=4L=8

o K=6+ K=12* AWGN

16-QAM

Fig.3.7. Exact 16-QAM over Ricean with K=6, 12 and MRC with L = 1,2,4 and 8.

3.3.1 Approximation of M-QAM Performance over Ricean Channel with MRC

Diversity.

In the previous sections we discussed the exact system performance where we get equation

(3.14). This equation is good because it is a simple closed form but we need to make n very

large to get the exact solution. In this section the error between the exact and different

approximation formulas will be analyzed for square M-QAM over single Ricean channel and

when MRC reception is used. These approximated formulas are based on equation (3.14),

Goldsmith, and Chiani. Where Goldsmith, and Chiani were introduced in chapter 2.

The first approximation we will call it " n = 1 " which can be obtained by letting n=1 in

equation (3.14) we get

,2 exp

2 2 2

L

Ric MRCa r KbASER

b r b r−⎛ ⎞ ⎡ ⎤≈ ⎜ ⎟ ⎢ ⎥+ +⎝ ⎠ ⎣ ⎦

2

exp2

La r Kbab r b r

⎛ ⎞ −⎛ ⎞ ⎡ ⎤+ −⎜ ⎟⎜ ⎟ ⎢ ⎥+ +⎝ ⎠ ⎣ ⎦⎝ ⎠

(3.16)

44

where, ⎟⎠

⎞⎜⎝

⎛ −=M

a 11 , )1(

3−

=M

b , 0N

Es=γ , γ

KLr += , γ = mean(γ )

For single Ricean channel we let L=1 in equation (3.16) to have the approximate system

performance as

,2 exp

2 2 2Ric MRCa r KbASER

b r b r−⎛ ⎞ ⎡ ⎤≈ ⎜ ⎟ ⎢ ⎥+ +⎝ ⎠ ⎣ ⎦

2

exp2

a r Kbab r b r

⎛ ⎞ −⎛ ⎞ ⎡ ⎤+ −⎜ ⎟⎜ ⎟ ⎢ ⎥+ +⎝ ⎠ ⎣ ⎦⎝ ⎠

(3.17)

As a special case for single Rayleigh channel where K=0 and L=1 (3.16) reduced to

22 2Raya rASER

b r⎛ ⎞≈ ⎜ ⎟+⎝ ⎠

2

2a ra

b r⎛ ⎞⎛ ⎞+ −⎜ ⎟⎜ ⎟+⎝ ⎠⎝ ⎠

(3.18)

And for Rayleigh channel with MRC diversity the system performance can be approximated

when n=1 as

,2

2 2

L

Ray MRCa rASER

b r⎛ ⎞≈ ⎜ ⎟+⎝ ⎠

2

2

La rab r

⎛ ⎞⎛ ⎞+ −⎜ ⎟⎜ ⎟+⎝ ⎠⎝ ⎠ (3.19)

Of course by increasing n we will get better approximation as we will see later.

The second approximation that we consider is Goldsmith approximation. From chapter 2, we

have symbol error rate of M-QAM over AWGN channel is

AWGN 21.5SER 0.2exp log

1M

Mγ−⎡ ⎤≈ ⎢ ⎥−⎣ ⎦

(3.20)

By substituting equations (3.20) and (3.11) in equation (3.6) and using the relation [35]

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛∫∞

= +−−

αββαβα

2exp

0)2(2

2)12( vvxv dxxvIex

We will get the second approximation of M-QAM over Ricean with MRC as

( )( )( )( ) ( )( )Ri-MRC 2

1 1.5ASER 0.2 log exp1 1.5 1 1.5

LM L K KM

M L K M L Kγ

γ γ

⎛ ⎞ ⎡ ⎤− + −≈ ⎜ ⎟ ⎢ ⎥⎜ ⎟− + + − + +⎢ ⎥⎝ ⎠ ⎣ ⎦

(3.21)

The last approximate expression is based on Chiani approximation. By going though the

same steps in the previous expressions but with SER over AWGN channel as

45

/ 2

2 / 3

3 2

b bb

AWGNe aeSER a e

γ γγ

− −−⎛ ⎞

≈ + −⎜ ⎟⎝ ⎠

Then approximate system performance can be expressed as

,1 2 exp3 2 2

L

Ric MRCr KbASER a

b r b r⎧ −⎪ ⎛ ⎞ ⎡ ⎤≈ ⎨ ⎜ ⎟ ⎢ ⎥+ +⎝ ⎠ ⎣ ⎦⎪⎩

3 2exp2 3 2 3

Lr Kbb r b r

−⎛ ⎞ ⎡ ⎤+⎜ ⎟ ⎢ ⎥+ +⎝ ⎠ ⎣ ⎦

exp2

La r Kbb r b r

⎫− ⎪⎛ ⎞ ⎡ ⎤− ⎬⎜ ⎟ ⎢ ⎥+ +⎝ ⎠ ⎣ ⎦⎪⎭

(3.22)

3.3.2 Analysis of Approximation

In fig. 3.8, we can see 16-QAM over single Rayleigh for both exact and approximate with

n=1 we find that the difference is 0.2 dB at ASER= 10-2. And in fig.3.9, we see zoom in of

fig. 3.8 but with Goldsmith and Chiani approximations where the differences are 0.2 and 0.6

dB for Chiani and Goldsmith respectively. These results are listed in more detailed in table

3.1.The figures 3.10 to 3.21 show different combination of K (Ricean parameter) and L

(diversity order) for exact and the approximations that presented in section 3.4.1. The tables

3.1 to 3.10 summarized the error difference between the exact and the approximation when

n= 1,2,3,4, and 10 also the Goldsmith and Chiani approximation for different M, L and K.

Where equations (3.21) and (3.22) are used for the approximation of Goldsmith and Chiani

respectively. And equation (3.14) was used in the approximation for n= 1,2,3,4, and 10 where

equation (3.16) is special case of (3.14).



1. The system performance can be expressed by a simple finite series of elementary

functions that is the exponential.

2. The error decays rapidly with increasing values of n.

3. At fixed n, the error increases when the modulation index is increased.

4. At fixed n, K and L, the error decreases when SNR is increased.

46

15 20 25 30 3510-3

10-2

10-1

SNR per Symbol (dB)

AS

ER

n=1Exact

Fig.3.8. ASER of 16-QAM on Rayleigh channel for n=1 and Exact.

27.6 27.8 28 28.2 28.4 28.6 28.8 29 29.2

10-2

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r

16-QAM, Single Rayleigh

Exactn=1GoldsmithChiani

Fig.3.9. Exact and Approx. of 16-QAM over Single Rayleigh (K=0).

47

24 24.2 24.4 24.6 24.8 25 25.2 25.4 25.6

10-2

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r

16-QAM, Single Rician (K=2)


Fig.3.10 Exact and Approx. of 16-QAM over Single Ricean (K=2).

15 20 25 3010-4

10-3

10-2

10-1

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r

K=6,n=1K=12,n=1K=6,ExactK=12,Exact

Fig.3.11. ASER of 16-QAM on Ricean channel of K=6 &12 for n =1 and Exact.

48

24 24.2 24.4 24.6 24.8 25 25.2

10-3

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r

16-QAM, Single Rician( K=6)


Fig.3.12 Exact and Approx. of 16-QAM over Single Ricean (K= 6).

23.1 23.2 23.3 23.4 23.5 23.6 23.7 23.8 23.9 24

10-4

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r

16-QAM, Single Rician (K=12)


Fig.3.13 Exact and Approx. of 16-QAM over Single Ricean (K=12).

49

15 20 25 3010-4

10-3

10-2

10-1

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

rAppr,n=1ExactL=1

L=2

L=3L=8

Fig.3.14. 16-QAM over Rayleigh for different L and for both exact and approximate, n = 1.

31 31.2 31.4 31.6 31.8 32 32.2 32.4 32.6 32.8

10-4

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r

16-QAM, MRC-L=2 over Rician with K=0


Fig.3.15 Exact and Approx. of 16-QAM over Rayleigh with MRC of L = 2.

50

27.2 27.3 27.4 27.5 27.6 27.7 27.8 27.9 28 28.1

10-5

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r

16-QAM, MRC-L=4 over Rician with K=0 (Rayleigh)



23.3 23.4 23.5 23.6 23.7 23.8 23.9 24

10-5

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r

16-QAM, MRC-L=8 over Rician with K=0 (Rayleigh)



51

20 21 22 23 24 25 26 27 28 29 30

10-4

10-3

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

rExactExactExactExactn=1n=1n=1n=1

L=1

L=8

L=4

L=2

Fig.3.18. 16-QAM over Ricean (K=6) for different L and for both Exact and Approx., n =1.

30.1 30.2 30.3 30.4 30.5 30.6 30.7 30.8 30.9 31

10-5

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r



Fig.3.19 Exact and Approx. of 16-QAM over Ricean (k = 6) with MRC of L = 2.

52

25.3 25.4 25.5 25.6 25.7 25.8 25.9 26 26.1 26.2

10-5

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r




22.8 22.9 23 23.1 23.2 23.3 23.4 23.5

10-5

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r




53

Table 3.1: Difference in dB between exact and approximate for M-QAM over single Rayleigh fading channel.

ASER for M-QAM over Single Rayleigh(K=0) 1.E-01 1.E-02 1.E-03 1.E-04 1.E-05 1.E-06

M n Difference (dB) 4 Goldsmith 0.9839 0.5891 0.5588 0.5558 4 Chiani 0.1796 0.2256 0.2285 0.2288 4 1 0.1879 0.1685 0.1665 0.1663 4 2 0.0449 0.0402 0.0397 0.0397 4 3 0.0198 0.0177 0.0175 0.0175 4 4 0.0111 0.01 0.0098 0.0098 4 10 0.0018 0.0016 0.0016 0.0015 16 Goldsmith 0.9194 0.9171 0.9166 16 Chiani 0.1188 0.1554 0.1582 16 1 0.2922 0.2682 0.2658 16 2 0.0693 0.0634 0.0629 16 3 0.0305 0.0279 0.0277 16 4 0.0171 0.0157 0.0155 16 10 0.0027 0.0025 0.0025 64 Goldsmith 2.2647 2.1368 64 Chiani 0.0809 0.1169 64 1 0.35 0.3231 64 2 0.0825 0.076 64 3 0.0363 0.0334 64 4 0.0204 0.0187 64 10 0.0033 0.0029

54

Table 3.2: Difference in dB between exact and approximate for 4-QAM over single Ricean fading channel (K = 2,6,&12).

ASER for 4-QAM over Single Ricean 1.E-01 1.E-02 1.E-03 1.E-04 1.E-05 1.E-06

K n Difference (dB) 2 Goldsmith 0.5633 0.4213 0.533 0.5531 2 Chiani 0.287 0.245 0.2309 0.229 2 1 0.1719 0.1581 0.1648 0.1662 2 2 0.0428 0.0376 0.0393 0.0396 2 3 0.0188 0.0166 0.0173 0.0175 2 4 0.0106 0.0093 0.0097 0.0098 2 10 0.0017 0.0014 0.0015 0.0016 6 Goldsmith 0.1069 0.227 0.1214 0.2235 0.4957 6 Chiani 0.4266 0.307 0.2786 0.2578 0.2342 6 1 0.1675 0.1143 0.1061 0.1407 0.1623 6 2 0.0428 0.0247 0.0255 0.0337 0.0387 6 3 0.0186 0.0109 0.0113 0.0148 0.0171 6 4 0.0104 0.0061 0.0064 0.0083 0.0096 6 10 0.0017 0.001 0.001 0.0013 0.0016

12 Goldsmith 0.0374 0.3892 0.4108 0.3952 0.3421 0.1688 12 Chiani 0.4619 0.2733 0.2048 0.1958 0.2213 0.2591 12 1 0.1821 0.0973 0.0298 0.0152 0.0368 0.0862 12 2 0.0422 0.0173 0.0108 0.0105 0.0142 0.0226 12 3 0.0185 0.0077 0.0049 0.0048 0.0063 0.01 12 4 0.0104 0.0043 0.0028 0.0027 0.0036 0.0056 12 10 0.0017 0.0007 0.0005 0.0004 0.0005 0.0009

55



K n Difference (dB) 2 Goldsmith 0.9409 0.9147 0.916 2 Chiani 0.2114 0.1748 0.1606 2 1 0.2709 0.2537 0.2637 2 2 0.0635 0.0599 0.0624 2 3 0.0278 0.0263 0.0275 2 4 0.0156 0.0148 0.0154 2 10 0.0024 0.0023 0.0025 6 Goldsmith 0.9782 0.8834 0.8819 0.9067 6 Chiani 0.3418 0.2621 0.2341 0.1933 6 1 0.2613 0.1618 0.1706 0.2307 6 2 0.058 0.0367 0.0409 0.0547 6 3 0.0252 0.0162 0.018 0.0241 6 4 0.0141 0.009 0.0101 0.0135 6 10 0.0022 0.0014 0.0016 0.0022


56



K n Difference (dB) 2 Goldsmith 2.0191 2.0549 2 Chiani 0.1727 0.137 2 1 0.3225 0.3063 2 2 0.0746 0.072 2 3 0.0327 0.0317 2 4 0.0183 0.0178 2 10 0.0029 0.0028 6 Goldsmith 1.6639 1.441 1.5996 6 Chiani 0.3061 0.2407 0.2096 0.1581 6 1 0.302 0.1875 0.2061 0.2801 6 2 0.0657 0.0431 0.0492 0.0661 6 3 0.0285 0.0189 0.0217 0.0291 6 4 0.0159 0.0106 0.0121 0.0163 6 10 0.0025 0.0017 0.0019 0.0026


57

Table 3.5: Difference in dB between exact and approximate for 4-QAM over Ricean fading channel (K= 0 and 2) with MRC of L branches.

ASER for 4-QAM over Ricean With MRC 1.E-01 1.E-02 1.E-03 1.E-04 1.E-05 1.E-06

K L n Difference (dB) 0 2 Goldsmith 0.308 0.1344 0.1997 0.2166 0.2215 0.2231 0 2 Chiani 0.392 0.3438 0.3224 0.3156 0.3134 0.3128 0 2 1 0.1527 0.1277 0.1127 0.1076 0.1061 0.1055 0 2 2 0.0435 0.03 0.0265 0.0255 0.0252 0.025 0 2 3 0.0191 0.0132 0.0117 0.0112 0.0111 0.011 0 2 4 0.0107 0.0074 0.0066 0.0063 0.0062 0.0062 0 2 10 0.0017 0.0012 0.0011 0.001 0.001 0.001 0 4 Goldsmith 0.0431 0.3533 0.4033 0.4151 0.4188 0.4201 0 4 Chiani 0.4521 0.3021 0.2337 0.2024 0.1868 0.1787 0 4 1 0.1695 0.1119 0.0553 0.0256 0.0102 0.0019 0 4 2 0.0438 0.0207 0.0134 0.0105 0.0092 0.0084 0 4 3 0.0187 0.0092 0.006 0.0047 0.0042 0.0038 0 4 4 0.0105 0.0051 0.0034 0.0027 0.0024 0.0022 0 4 10 0.0017 0.0008 0.0006 0.0004 0.0004 0.0003 0 8 Goldsmith 0.0711 0.4218 0.4429 0.4347 0.424 0.4149 0 8 Chiani 0.4705 0.2556 0.1566 0.1103 0.0868 0.0739 0 8 1 0.1876 0.0923 0.0065 0.0591 0.0874 0.1036 0 8 2 0.0418 0.015 0.0059 0.0028 0.0017 0.0013 0 8 3 0.0186 0.0066 0.0028 0.0015 0.0009 0.0007 0 8 4 0.0104 0.0037 0.0016 0.0009 0.0005 0.0004 0 8 10 0.0016 0.0006 0.0003 0.0001 0.0001 0.0001 2 2 Goldsmith 0.2127 0.1791 0.2182 0.2232 0.2238 0.2238 2 2 Chiani 0.4081 0.3346 0.3157 0.3128 0.3125 0.3125 2 2 1 0.1577 0.1227 0.1081 0.1056 0.1053 0.1053 2 2 2 0.0432 0.0281 0.0255 0.0251 0.025 0.025 2 2 3 0.0188 0.0124 0.0112 0.0111 0.011 0.011 2 2 4 0.0105 0.0069 0.0063 0.0062 0.0062 0.0062 2 2 10 0.0017 0.0011 0.001 0.001 0.001 0.001 2 4 Goldsmith 0.0236 0.3615 0.4066 0.4165 0.4193 0.4203 2 4 Chiani 0.4542 0.2966 0.2282 0.1984 0.1842 0.1772 2 4 1 0.1721 0.1091 0.0508 0.0219 0.0078 0.0004 2 4 2 0.0434 0.02 0.0129 0.0101 0.009 0.0084 2 4 3 0.0186 0.0088 0.0058 0.0046 0.0041 0.0038 2 4 4 0.0104 0.0049 0.0033 0.0026 0.0024 0.0022 2 4 10 0.0016 0.0008 0.0005 0.0004 0.0004 0.0003 2 8 Goldsmith 0.0745 0.4231 0.4432 0.4344 0.4235 0.4144 2 8 Chiani 0.4707 0.254 0.155 0.109 0.0858 0.0732 2 8 1 0.1884 0.0914 0.0082 0.0606 0.0885 0.1045 2 8 2 0.0417 0.0148 0.0057 0.0028 0.0017 0.0013 2 8 3 0.0187 0.0065 0.0027 0.0015 0.0009 0.0007 2 8 4 0.0104 0.0037 0.0016 0.0008 0.0006 0.0004 2 8 10 0.0017 0.0006 0.0002 0.0001 0.0001 0.0001

58

Table 3.6: Difference in dB between exact and approximate for 4-QAM over Ricean fading channel (K = 6 and 12) with MRC of L branches.

ASER for 4-QAM over Ricean with MRC 1.E-01 1.E-02 1.E-03 1.E-04 1.E-05 1.E-06

K L n Difference (dB) 6 2 Goldsmith 0.0384 0.3272 0.3474 0.3203 0.2791 0.2475 6 2 Chiani 0.4466 0.3008 0.2594 0.266 0.2864 0.3016 6 2 1 0.1716 0.108 0.0695 0.0707 0.0855 0.0971 6 2 2 0.0429 0.0213 0.0172 0.0184 0.0213 0.0235 6 2 3 0.0185 0.0094 0.0077 0.0082 0.0094 0.0104 6 2 4 0.0104 0.0053 0.0043 0.0046 0.0053 0.0059 6 2 10 0.0016 0.0008 0.0007 0.0007 0.0009 0.001 6 4 Goldsmith 0.0314 0.3927 0.4238 0.4255 0.424 0.4228 6 4 Chiani 0.4626 0.2758 0.1986 0.1695 0.1602 0.1593 6 4 1 0.1805 0.0998 0.0271 0.0045 0.0154 0.0173 6 4 2 0.0425 0.0174 0.0099 0.0076 0.007 0.0069 6 4 3 0.0185 0.0077 0.0045 0.0035 0.0033 0.0032 6 4 4 0.0103 0.0043 0.0025 0.002 0.0019 0.0018 6 4 10 0.0016 0.0007 0.0004 0.0003 0.0003 0.0003 6 8 Goldsmith 0.0886 0.4291 0.4443 0.4327 0.4203 0.4105 6 8 Chiani 0.4723 0.2471 0.146 0.1007 0.0791 0.068 6 8 1 0.1913 0.0874 0.0167 0.0694 0.0963 0.1109 6 8 2 0.0413 0.014 0.005 0.0023 0.0015 0.0011 6 8 3 0.0186 0.0061 0.0024 0.0013 0.0008 0.0006 6 8 4 0.0103 0.0035 0.0014 0.0007 0.0005 0.0003 6 8 10 0.0016 0.0005 0.0002 0.0001 0.0001 0.0001

12 2 Goldsmith 0.0547 0.4041 0.427 0.4219 0.4116 0.3925 12 2 Chiani 0.4656 0.2654 0.1859 0.1626 0.1684 0.194 12 2 1 0.1848 0.0945 0.0157 0.0127 0.0114 0.0093 12 2 2 0.042 0.0162 0.0088 0.0072 0.008 0.0106 12 2 3 0.0185 0.0072 0.004 0.0034 0.0037 0.0048 12 2 4 0.0103 0.004 0.0023 0.0019 0.0021 0.0027 12 2 10 0.0016 0.0007 0.0004 0.0004 0.0004 0.0005 12 4 Goldsmith 0.0792 0.4215 0.4397 0.4324 0.4253 0.4217 12 4 Chiani 0.4702 0.2529 0.1596 0.1221 0.1084 0.1068 12 4 1 0.1894 0.0894 0.0053 0.0491 0.0665 0.0703 12 4 2 0.0415 0.0147 0.0063 0.0038 0.0032 0.0032 12 4 3 0.0186 0.0065 0.003 0.0019 0.0016 0.0015 12 4 4 0.0103 0.0036 0.0017 0.0011 0.0009 0.0009 12 4 10 0.0016 0.0006 0.0003 0.0002 0.0002 0.0001 12 8 Goldsmith 0.1072 0.4376 0.4458 0.429 0.4129 0.4007 12 8 Chiani 0.4745 0.2369 0.1314 0.086 0.0659 0.0567 12 8 1 0.1954 0.0817 0.0306 0.0853 0.1117 0.125 12 8 2 0.0407 0.0129 0.0037 0.0015 0.0009 0.0008 12 8 3 0.0186 0.0056 0.0019 0.0009 0.0005 0.0004 12 8 4 0.0103 0.0032 0.0011 0.0005 0.0003 0.0003 12 8 10 0.0016 0.0005 0.0002 0.0001 0 0.0001

59



K L n Difference (dB) 0 2 Goldsmith 0.9815 0.92 0.9005 0.8944 0.8925 0 2 Chiani 0.3136 0.2909 0.2786 0.2745 0.2731 0 2 1 0.2626 0.1954 0.1733 0.1664 0.1642 0 2 2 0.0626 0.0449 0.0403 0.039 0.0386 0 2 3 0.0274 0.0197 0.0177 0.0171 0.0169 0 2 4 0.0153 0.011 0.0099 0.0096 0.0095 0 2 10 0.0024 0.0018 0.0015 0.0015 0.0015 0 4 Goldsmith 0.987 0.8618 0.8027 0.7728 0.757 0.7484 0 4 Chiani 0.3662 0.2625 0.2111 0.1865 0.1741 0.1676 0 4 1 0.2658 0.1368 0.0705 0.0386 0.0224 0.0138 0 4 2 0.0582 0.0288 0.0192 0.0154 0.0136 0.0126 0 4 3 0.0252 0.0127 0.0086 0.007 0.0062 0.0058 0 4 4 0.0141 0.0071 0.0048 0.004 0.0035 0.0033 0 4 10 0.0022 0.0011 0.0007 0.0007 0.0006 0.0005 0 8 Goldsmith 0.984 0.8163 0.7234 0.667 0.6305 0.6057 0 8 Chiani 0.3803 0.2201 0.1408 0.1022 0.0822 0.0711 0 8 1 0.2722 0.0907 0.0117 0.0622 0.0891 0.1045 0 8 2 0.0548 0.0189 0.0077 0.004 0.0025 0.0018 0 8 3 0.024 0.0085 0.0038 0.0021 0.0013 0.0009 0 8 4 0.0133 0.0048 0.0022 0.0012 0.0008 0.0005 0 8 10 0.0021 0.0008 0.0004 0.0002 0.0001 0.0001 2 2 Goldsmith 0.9801 0.9091 0.894 0.8919 0.8916 2 2 Chiani 0.3268 0.2854 0.2744 0.2728 0.2726 2 2 1 0.2607 0.1837 0.1662 0.1635 0.1633 2 2 2 0.0606 0.042 0.0389 0.0384 0.0384 2 2 3 0.0264 0.0184 0.0171 0.0169 0.0169 2 2 4 0.0148 0.0103 0.0096 0.0094 0.0095 2 2 10 0.0024 0.0016 0.0015 0.0015 0.0015 2 4 Goldsmith 0.9859 0.8565 0.7974 0.7688 0.7543 0.7467 2 4 Chiani 0.3676 0.2578 0.2065 0.1833 0.172 0.1663 2 4 1 0.266 0.1315 0.0648 0.0344 0.0197 0.0122 2 4 2 0.0575 0.0278 0.0184 0.0149 0.0133 0.0125 2 4 3 0.0249 0.0123 0.0083 0.0068 0.0061 0.0057 2 4 4 0.0139 0.0069 0.0046 0.0038 0.0034 0.0033 2 4 10 0.0022 0.0011 0.0007 0.0006 0.0005 0.0005 2 8 Goldsmith 0.9838 0.815 0.7216 0.6651 0.6287 0.6043 2 8 Chiani 0.3805 0.2187 0.1392 0.101 0.0812 0.0704 2 8 1 0.2724 0.0891 0.0135 0.0637 0.0902 0.1053 2 8 2 0.0546 0.0185 0.0075 0.0039 0.0025 0.0018 2 8 3 0.0239 0.0083 0.0037 0.002 0.0013 0.0009 2 8 4 0.0133 0.0046 0.0022 0.0012 0.0008 0.0006 2 8 10 0.0021 0.0007 0.0004 0.0002 0.0001 0.0001

60



K L n Difference (dB) 6 2 Goldsmith 0.9828 0.8649 0.834 0.8449 0.8668 0.8816 6 2 Chiani 0.36 0.2612 0.2337 0.2405 0.2556 0.2657 6 2 1 0.264 0.1387 0.1019 0.1124 0.1357 0.1522 6 2 2 0.0573 0.0304 0.0261 0.0288 0.0333 0.0364 6 2 3 0.0249 0.0134 0.0116 0.0128 0.0147 0.016 6 2 4 0.0139 0.0075 0.0065 0.0072 0.0082 0.009 6 2 10 0.0022 0.0012 0.001 0.0012 0.0013 0.0015 6 4 Goldsmith 0.9841 0.8363 0.7677 0.7379 0.7276 0.7267 6 4 Chiani 0.3738 0.2394 0.1805 0.1581 0.1513 0.1512 6 4 1 0.2685 0.1108 0.0338 0.0035 0.0061 0.007 6 4 2 0.0559 0.0234 0.014 0.0112 0.0105 0.0105 6 4 3 0.0243 0.0104 0.0064 0.0052 0.0049 0.0048 6 4 4 0.0135 0.0059 0.0036 0.003 0.0028 0.0027 6 4 10 0.0021 0.001 0.0006 0.0005 0.0005 0.0004 6 8 Goldsmith 0.9829 0.8086 0.7119 0.6538 0.6171 0.5932 6 8 Chiani 0.3813 0.2121 0.131 0.0935 0.0751 0.0656 6 8 1 0.2733 0.0821 0.0234 0.073 0.0981 0.1117 6 8 2 0.0542 0.0173 0.0064 0.0032 0.0021 0.0016 6 8 3 0.0237 0.0078 0.0032 0.0017 0.0011 0.0008 6 8 4 0.0132 0.0044 0.0019 0.001 0.0007 0.0005 6 8 10 0.0021 0.0007 0.0003 0.0002 0.0001 0.0001

12 2 Goldsmith 0.983 0.8266 0.755 0.7307 0.7385 0.7693 12 2 Chiani 0.3759 0.2299 0.1695 0.1529 0.1605 0.1842 12 2 1 0.2698 0.1005 0.0204 0.0033 0.0042 0.0338 12 2 2 0.0551 0.0213 0.0124 0.0109 0.0125 0.0169 12 2 3 0.024 0.0095 0.0057 0.0051 0.0057 0.0076 12 2 4 0.0134 0.0053 0.0032 0.0029 0.0032 0.0043 12 2 10 0.0021 0.0008 0.0005 0.0005 0.0005 0.0007 12 4 Goldsmith 0.9828 0.8143 0.7266 0.6814 0.6614 0.6578 12 4 Chiani 0.3795 0.2179 0.1444 0.1144 0.1038 0.1035 12 4 1 0.2721 0.0878 0.0082 0.0489 0.064 0.0659 12 4 2 0.0544 0.0185 0.0084 0.0055 0.0048 0.0049 12 4 3 0.0238 0.0083 0.0041 0.0027 0.0024 0.0024 12 4 4 0.0132 0.0047 0.0024 0.0016 0.0014 0.0014 12 4 10 0.0021 0.0007 0.0004 0.0002 0.0003 0.0002 12 8 Goldsmith 0.9819 0.7993 0.6956 0.6324 0.5922 0.5663 12 8 Chiani 0.3827 0.2023 0.1172 0.0797 0.0628 0.0551 12 8 1 0.2748 0.0717 0.0395 0.0898 0.1139 0.1261 12 8 2 0.0534 0.0153 0.0047 0.002 0.0013 0.0011 12 8 3 0.0234 0.0069 0.0025 0.0011 0.0007 0.0005 12 8 4 0.013 0.0039 0.0015 0.0007 0.0004 0.0003 12 8 10 0.002 0.0006 0.0002 0.0001 0 0.0001

61



K L n Difference (dB) 0 2 Goldsmith 1.815 1.5504 1.4865 1.4677 0 2 Chiani 0.276 0.265 0.2562 0.2531 0 2 1 0.3152 0.2306 0.2051 0.1972 0 2 2 0.0722 0.0525 0.0474 0.046 0 2 3 0.0315 0.023 0.0208 0.0202 0 2 4 0.0176 0.0129 0.0116 0.0113 0 2 10 0.0028 0.0021 0.0018 0.0018 0 4 Goldsmith 1.6123 1.2598 1.1348 1.0771 1.0476 1.0319 0 4 Chiani 0.3312 0.2461 0.2011 0.1793 0.168 0.1621 0 4 1 0.3052 0.1493 0.0786 0.0454 0.0288 0.0199 0 4 2 0.0654 0.0328 0.0222 0.0178 0.0159 0.0148 0 4 3 0.0283 0.0145 0.01 0.0081 0.0073 0.0068 0 4 4 0.0158 0.0081 0.0056 0.0046 0.0041 0.0039 0 4 10 0.0026 0.0013 0.0009 0.0007 0.0007 0.0006 0 8 Goldsmith 1.5185 1.1197 0.9542 0.8633 0.8071 0.7702 0 8 Chiani 0.3467 0.2071 0.1348 0.0991 0.0803 0.0699 0 8 1 0.302 0.0908 0.013 0.0631 0.0895 0.1046 0 8 2 0.0607 0.0206 0.0085 0.0044 0.0029 0.0021 0 8 3 0.0263 0.0093 0.0042 0.0023 0.0015 0.0011 0 8 4 0.0146 0.0053 0.0024 0.0013 0.0009 0.0006 0 8 10 0.0023 0.0009 0.0004 0.0002 0.0001 0.0001 2 2 Goldsmith 1.7477 1.5033 1.4649 1.4598 2 2 Chiani 0.29 0.2614 0.2531 0.2518 2 2 1 0.3084 0.2156 0.1967 0.1941 2 2 2 0.0694 0.0491 0.0458 0.0454 2 2 3 0.0302 0.0215 0.0201 0.02 2 2 4 0.0169 0.0121 0.0113 0.0112 2 2 10 0.0027 0.0019 0.0018 0.0018 2 4 Goldsmith 1.597 1.245 1.1232 1.0691 1.0426 1.0289 2 4 Chiani 0.3329 0.242 0.197 0.1763 0.1662 0.161 2 4 1 0.3038 0.1427 0.0724 0.041 0.0259 0.0182 2 4 2 0.0645 0.0315 0.0212 0.0172 0.0154 0.0146 2 4 3 0.0279 0.0139 0.0095 0.0078 0.007 0.0067 2 4 4 0.0156 0.0078 0.0054 0.0044 0.004 0.0038 2 4 10 0.0025 0.0012 0.0008 0.0007 0.0006 0.0006 2 8 Goldsmith 1.5156 1.1162 0.9508 0.8602 0.8045 0.768 2 8 Chiani 0.3468 0.2057 0.1334 0.0979 0.0795 0.0692 2 8 1 0.3018 0.089 0.015 0.0647 0.0907 0.1055 2 8 2 0.0605 0.0203 0.0083 0.0043 0.0028 0.0021 2 8 3 0.0262 0.0092 0.0041 0.0022 0.0015 0.0011 2 8 4 0.0145 0.0052 0.0023 0.0013 0.0008 0.0006 2 8 10 0.0023 0.0009 0.0004 0.0002 0.0001 0.0001

62

Table 3.10: Difference in dB between exact and approximate for 64-QAM over Ricean fading channel (K= 6 and 12) with MRC of L branches.

ASER for 64-QAM over Ricean With MRC 1.E-01 1.E-02 1.E-03 1.E-04 1.E-05 1.E-06

K L n Difference (dB) 6 2 Goldsmith 1.6093 1.2991 1.2567 1.3102 1.3829 6 2 Chiani 0.3251 0.244 0.2207 0.2264 0.2386 6 2 1 0.3017 0.1548 0.1194 0.1345 0.1622 6 2 2 0.0643 0.035 0.0307 0.0342 0.0394 6 2 3 0.0279 0.0154 0.0136 0.0152 0.0174 6 2 4 0.0155 0.0087 0.0077 0.0085 0.0097 6 2 10 0.0024 0.0014 0.0012 0.0014 0.0015 6 4 Goldsmith 1.5518 1.1833 1.0556 1.0061 0.9913 0.9914 6 4 Chiani 0.3397 0.2252 0.173 0.153 0.1471 0.1469 6 4 1 0.3016 0.1168 0.0379 0.0079 0.0013 0.0017 6 4 2 0.0622 0.0261 0.016 0.0129 0.0122 0.0123 6 4 3 0.0269 0.0116 0.0073 0.006 0.0056 0.0057 6 4 4 0.015 0.0065 0.0041 0.0034 0.0032 0.0033 6 4 10 0.0024 0.001 0.0006 0.0005 0.0005 0.0005 6 8 Goldsmith 1.5037 1.1001 0.9321 0.8409 0.7859 0.751 6 8 Chiani 0.3478 0.1995 0.1254 0.0907 0.0735 0.0647 6 8 1 0.3012 0.0806 0.0253 0.0741 0.0985 0.1119 6 8 2 0.0599 0.0187 0.0072 0.0036 0.0024 0.0018 6 8 3 0.0259 0.0085 0.0036 0.0019 0.0013 0.0009 6 8 4 0.0144 0.0048 0.0021 0.0011 0.0008 0.0006 6 8 10 0.0023 0.0007 0.0004 0.0001 0.0001 0.0001

12 2 Goldsmith 1.5323 1.157 1.0313 1.0005 1.0311 1.1123 12 2 Chiani 0.3419 0.2163 0.1626 0.1483 0.1559 0.1778 12 2 1 0.3007 0.1044 0.0238 0.0019 0.0124 0.0466 12 2 2 0.0611 0.0238 0.0142 0.0128 0.0149 0.0202 12 2 3 0.0264 0.0106 0.0066 0.0059 0.0068 0.0091 12 2 4 0.0147 0.006 0.0037 0.0034 0.0038 0.0051 12 2 10 0.0023 0.001 0.0006 0.0006 0.0006 0.0008 12 4 Goldsmith 1.5117 1.1178 0.9649 0.8945 0.8665 0.8648 12 4 Chiani 0.3458 0.205 0.1386 0.1112 0.1017 0.1017 12 4 1 0.3009 0.0882 0.0085 0.0483 0.0625 0.0636 12 4 2 0.0602 0.0204 0.0095 0.0063 0.0056 0.0058 12 4 3 0.026 0.0092 0.0046 0.0031 0.0028 0.0028 12 4 4 0.0145 0.0052 0.0027 0.0018 0.0016 0.0017 12 4 10 0.0023 0.0009 0.0005 0.0003 0.0003 0.0003 12 8 Goldsmith 1.4879 1.076 0.901 0.8043 0.7459 0.7095 12 8 Chiani 0.3492 0.19 0.1121 0.0773 0.0616 0.0545 12 8 1 0.3008 0.0683 0.0423 0.0912 0.1146 0.1265 12 8 2 0.059 0.0165 0.0052 0.0024 0.0015 0.0011 12 8 3 0.0255 0.0076 0.0028 0.0013 0.0008 0.0005 12 8 4 0.0142 0.0043 0.0016 0.0008 0.0005 0.0003 12 8 10 0.0022 0.0007 0.0003 0.0001 0.0001 0

63

CHAPTER 4

PERFORMANCE OF M-QAM TRANSMITTED OVER NAKAGAMI-m CHANNEL WITH MRC 4.1 Introduction

In this chapter, the Average Symbol Error Rate (ASER) performance of a coherent square M-

QAM with L-th order diversity in frequency nonselective slowly Nakagami-m fading

environment using MRC reception is presented. Nakagami-m distribution is considered here,

because this distribution is not only the best fit for digital signals received in urban radio

multipath fading channels [5], but also because there is a close agreement between the

Nakagami-m distribution and variety of propagation paths spanning nearly all frequency

bands [2]. Moreover, like Rice distribution, Nakagami-m distribution is also a two parameter

distribution, that provides more flexibility and accuracy in matching the observed signal

statistics. M-QAM considered here is the widely used modulation technique due to its high

spectral efficiency.

We assume that there are L diversity channels carrying the same transmitted signal. Each

channel is modeled as frequency non-selective slowly Nakagami-m fading channel corrupted

by additive white Gaussian noise (AWGN) process. The fading processes among the L

diversity channels are assumed to be mutually statistically independent. The noise processes

in the L diversity channels are assumed to be mutually statistically independent, with

identical distribution (iid). For a M-QAM demodulator [5] which makes its decision based on

the output of a linear filter operating on an undistorted symbol waveform, it is well known

that the probability of symbol error caused by AWGN depends only on the instantaneous

SNR (γ) associated with each symbol.

4.2 Performance of M-QAM over Nakagami Channel with MRC diversity

In fading, γ becomes a random variable and for Nakagami-m fading environment, the

probability density function (pdf) of instantaneous SNR γ at the output of the kth channel is

given as [10]-[12],

64

( ) ( )( )

1 exp /k km m

k k kkk

k k

mmpm

γ γ γγ

γ

− −⎛ ⎞= ⎜ ⎟ Γ⎝ ⎠

where Г(.) is the gamma function, mk is the fading parameter of kth channel and kγ is the

average of γ.

For identical and independent channels (iid), the pdf of γ at the output of the MRC may be

written as [10]-[12],

( ) ( )( )

1 exp /mL mL mLm LpmL

γ γ γγ

γ

− −⎛ ⎞= ⎜ ⎟ Γ⎝ ⎠

(4.1)

where, m is the Nakagami fading parameter, which is assumed to be identical for all channels

(mk= m) and kγ is the average SNR associated with each symbol, which is related to γ as

1

L

kk

γ γ=

=∑

Once the statistics of the instantaneous SNR γ are determined as a function of the Nakagami

parameter m and the average instantaneous SNRγ , the Average Symbol Error Rate of

Nakagami channel (ASERNa) as a function of m and γ of the system can be calculated by

averaging the conditional probability of SER over the pdf ofγ . i.e.,

γγγ dpEPASER S∫∞

=0

)()./( (4.2)

where, )/( γEPS is the conditional probability of symbol error which is the SER over non-

fading channel corrupted by AWGN channel. From chapter 2, the SER of QAM which is

given in the exponential form and can be written as

( )⎭⎬⎫

⎩⎨⎧

+−+−= ∑ ∑−

=

−

=

−−−− 1

1

12//

2/

122

)(n

i

n

ni

SbSbbb

QAMii eeaaee

naESER γγ

γγ

(4.3)

We now derive the expression for average SER of coherent square MQAM with L-th order

diversity in frequency non-selective slowly Nakagami-m fading channel using MRC

reception by substituting equations (4.1) & (4.3) in equation (4.2) to get ASERMRC-Na to get

65

( ) ( )⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡+−+−⎟⎟

⎠

⎞⎜⎜⎝

⎛ −Γ⎟⎟

⎠

⎞⎜⎜⎝

⎛= ∫ ∑ ∑

∞ −

=

−

=

−−−−

−−

0

1

1

12//

2/1 1

22exp

n

i

n

ni

SbSbbb

mLmL

NaMRCii eeaaeedmL

mLnamLASER γγ

γγ

γγγγ

γ

(4.4)

Using the following relationship given in Appendix II

( ) ( ) kk ukdxuxX −∞

− Γ=−∫0

1 exp

( ) ( ) ( )

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛+−+⎟⎟

⎠

⎞⎜⎜⎝

⎛+−⎟⎟

⎠

⎞⎜⎜⎝

⎛+Γ⋅

Γ⎟⎟⎠

⎞⎜⎜⎝

⎛=

−−

=

−

=

−−−

− ∑∑mLn

ni i

n

i

mL

i

mLmLmL

NaMRC SbmL

SbmLabmLabmLmL

mLnamLASER

121

1

1222

1γγγγγ

On further simplification, we get

( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛+−+⎟

⎠⎞

⎜⎝⎛ +−⎟

⎠⎞

⎜⎝⎛ += ∑∑

−

=

−−−

=

−−

−

121

11111

221

21 n

ni

mL

i

mLn

i i

mLmL

NaMRC mLSb

mLSba

mLba

mLb

naASER γγγγ

(4.5)

4.3 Results and Analysis of M-QAM over Nakagami Channel with MRC diversity

Equation (4.5) represents the ASER of coherent square M-QAM over Nakagami-m fading

channels that are iid using MRC reception. Then it is varied in γ (average signal-noise ratio),

L (number of MRC branches), Nakagami parameter m, and M (the modulation index).

For L=1 equation (4.5) represent the ASER of M-QAM over single Nakagami-m channel,

which is given by

( )1 2 1

1

1 1 1 1 1 12 2 2

m mm m n n

Sin Nai i ni i

a b a b b bASER an m m mS mS

γ γ γ γ− −− − − −

−= =

⎧ ⎫⎛ ⎞ ⎛ ⎞⎪ ⎪⎛ ⎞ ⎛ ⎞= + − + + − + + +⎨ ⎬⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭

∑ ∑

(4.6)

If we let m = ∞ in equation (4.6) we will get equation (4.3) which is the SER of M-QAM over

AWGN channel. By substituting m =1 in equation (4.6) which represents Rayleigh fading

channel, we get

66

⎩⎨⎧

+=

21γbn

aASERRay ( )12 +−

γba ( )

⎭⎬⎫

++

+−

+ ∑∑−

=

−

=

121

1

1 n

ni i

in

i i

i

SbS

SbSa

γγ

⎩⎨⎧

+=

21γbn

aASERRay ( )12 +−

γba

⎭⎬⎫

+−

++ ∑∑

−

=

−

=

1

1

12

1

n

i i

in

i i

i

SbaS

SbS

γγ (4.7)

And the performance of Rayleigh fading channel when MRC is used can be obtained by

making m = 1 in equation (4.5) to get

( )1 2 1

1

1 1 1 1 1 12 2 2

L LL L n n

MRC Rayi i ni i

a b a b b bASER an L L LS LS

γ γ γ γ− −− − − −

−= =

⎧ ⎫⎛ ⎞ ⎛ ⎞⎪ ⎪⎛ ⎞ ⎛ ⎞= + − + + − + + +⎨ ⎬⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭

∑ ∑

(4.8)

Equation (4.8) can be simplified more to be identical to equation (3.15) which is

1 22 2 2

L L

MRC Raya L a LASERn b L b Lγ γ−

⎧ ⎛ ⎞ ⎛ ⎞⎪= −⎨ ⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠⎪⎩( )∑

−

=⎟⎟⎠

⎞⎜⎜⎝

⎛+

−+1

11

n

i

L

i

i

LSbLSa

γ

⎪⎭

⎪⎬⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛+

+ ∑−

=

12n

ni

L

i

i

LSbLS

γ (4.9)

Fig.4.1 shows 16-QAM over Nakagami-m fading channel for different values of m that are

0.5, 1, 2, and 5. These values are usually used in the literatures because they are close to the

values observed in the experimental work. From fig.4.1 we can see that small change in m

leads to big change in the system performance. For m = 0.5 that represent the worst case or

more severe than Rayleigh that has m=1. Both figs.4.2 and 4.3 show the performance of

square M-QAM over single Nakagami-m channel. Where figure 4.2 is for both m= 0.5 and 1.

And fig.4.3 for m = 2 and 5.

Fig.4.4 shows the performance of Nakagami-m channel with m = 0.5 and 1 using MRC for

different values of L (1, 2, & 4) for 16-QAM and fig.4.5 shows 16-QAM over Nakagami

channel with m = 2 and 5 with the same order of diversity. Form figs.3.4 & 3.5 we can see

that the improvement in the system performance decreases when the m increase and also

when the order of diversity increase (the number of used branches).

67

0 5 10 15 20 25 30 35 4010-6

10-5

10-4

10-3

10-2

10-1

100

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r

m=0.5m=1m=2m=5AWGN

Fig.4.1.Exact 16-QAM over single Nakagami with m=0.5, 1, 2 and 5

0 5 10 15 20 25 30 35 4010

-4

10-3

10-2

10-1

100

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r

m=0.5m=1

o 4QAM+ 16QAM* 64QAM

Fig.4.2.Exact M-QAM over single Nakagami with m=0.5 & 1.

68

0 5 10 15 20 25 30 35 4010-6

10-5

10-4

10-3

10-2

10-1

100

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r

m=0.5m=1

o 4QAM+ 16QAM* 64QAM

Fig.4.3. Exact M-QAM over single Nakagami with m = 2 and 5

0 5 10 15 20 25 30 35 4010-6

10-5

10-4

10-3

10-2

10-1

100

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r

+ m = 0.5o m = 1* AWGN

m=0.5,L=1

m=0.5,L=2 &m=1, L=1m=0.5,L=4 &m=1, L=2

m=1,L=4

Fig.4.4. Exact 16-QAM over Nakagami with m=0.5, 1 and L=1, 2 & 4.

2 5

69

12 14 16 18 20 22 24 26 28 3010-6

10-5

10-4

10-3

10-2

10-1

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r

m=2,L=1m=2,L=2m=2,L=4m=5,L=1m=5,L=2m=5,L=4AWGN

Fig.4.5. Exact 16-QAM over Nakagami with m = 2, 5 and L=1, 2 & 4

4.4 Approximation of M-QAM over Nakagami-m Channel with MRC diversity

In the previous sections we discussed the exact system performance where we get equation

(4.5). This equation is good because it is a simple closed form but we need to make n very

large to get the exact solution. In this section, the error between the exact and different

approximation formulas will be analyzed for square M-QAM over single Nakagami-m

channel and when MRC reception is used. These approximated formulas are based on

equation (4.5), Goldsmith [6], and Chiani [1]. Where Goldsmith, and Chiani were introduced

in chapter 2.

The first approximation we call it " n = 1 " which can be obtained by making n=1 in equation

(4.5) we get

( )1 2 12 2

mL mL

MRC Naa b bASER a

mL mLγ γ− −

−

⎧ ⎫⎪ ⎪⎛ ⎞ ⎛ ⎞≈ + + − +⎨ ⎬⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭

(4.10)

where, ⎟⎠

⎞⎜⎝

⎛ −=M

a 11 , )1(

3−

=M

b , 0N

Es=γ , γ = mean(γ )

For single Nakagami-m channel we let L=1 in equation (4.10) to have the approximate

system performance as

70

( )1 2 12 2

m m

MRC Naa b bASER a

m mγ γ− −

−

⎧ ⎫⎪ ⎪⎛ ⎞ ⎛ ⎞≈ + + − +⎨ ⎬⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭

(4.11)

As a special case for single Rayleigh channel where m=1 and L=1 equation (4.10) reduced to

22 2Raya rASER

b r⎛ ⎞≈ ⎜ ⎟+⎝ ⎠

2

2a ra

b r⎛ ⎞⎛ ⎞+ −⎜ ⎟⎜ ⎟+⎝ ⎠⎝ ⎠

(4.12)

And for Rayleigh channel with MRC diversity the system performance can be approximated

when n=1 as

,2

2 2

L

Ray MRCa rASER

b r⎛ ⎞≈ ⎜ ⎟+⎝ ⎠

2

2

La rab r

⎛ ⎞⎛ ⎞+ −⎜ ⎟⎜ ⎟+⎝ ⎠⎝ ⎠ (4.13)

Of course by increasing n we will get better approximation as we will see later.

The second approximation that we consider is Goldsmith approximation. The pdf of γ at the

output of MRC of Nakagami channel is given by equation (4.1)

( ) ( )( )

1 exp /mL mL mLm LpmL

γ γ γγ

γ

− −⎛ ⎞= ⎜ ⎟ Γ⎝ ⎠

The ASER is given by equation (4.2)

γγγ dpEPASERA SNaMRCp ∫∞

− =0

)()./(

From chapter 2 we have symbol error rate of M-QAM over AWGN channel is

AWGN 21.5( / ) SER 0.2exp log

1SP E MM

γγ −⎡ ⎤= ≈ ⎢ ⎥−⎣ ⎦ (4.14)

Therefore,

( )( )

2 1

0

(0.2) log 1.5exp1

mLmL

MRC Na

MmL LmASER dmL M

γ γγ γγ γ

∞−

−

⎛ ⎞ ⎛ ⎞−≈ −⎜ ⎟ ⎜ ⎟Γ −⎝ ⎠ ⎝ ⎠

∫

( )( ) ( )2(0.2) log 1.5

1

mL mLMmL LmmLmL Mγ γ

−⎛ ⎞ ⎛ ⎞

≈ Γ +⎜ ⎟ ⎜ ⎟Γ −⎝ ⎠ ⎝ ⎠

71

( )( )21.5 0.2 log

1

mL mLmL Lm M

Mγ γ

−⎛ ⎞ ⎛ ⎞

≈ + ⋅⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠

( )21.50.2 1 log( 1)

mL

MRC NaASER MmL M

γ−

−

⎛ ⎞≈ +⎜ ⎟−⎝ ⎠

(4.15)

The last approximate expression is based on Chiani approximation. By going though the

same steps in the previous expressions but with SER over AWGN channel as

/ 2

2 / 3

3 2

b bb

AWGNe aeSER a e

γ γγ

− −−⎛ ⎞

≈ + −⎜ ⎟⎝ ⎠

(4.16)

Then approximate system performance can be expressed as

( )

2 / 3/ 21

0

1 exp3 1 4

bmL bmL b

MRC NamL Lm e e aASER a e d

mL

γγγγγ γ

γ γ

−∞ −− −

−

⎛ ⎞⎛ ⎞ ⎛ ⎞−≈ + −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟Γ⎝ ⎠ ⎝ ⎠⎝ ⎠

∫

( )

2 / 3/ 21

0

1 exp3 1 4

bmL bmL bmL Lm e e aa e d

mL

γγγγγ γ

γ γ

−∞ −− −

⎛ ⎞⎛ ⎞ ⎛ ⎞−≈ + −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟Γ⎝ ⎠ ⎝ ⎠⎝ ⎠

∫

1 21 1 13 2 3 4

mL mL mL

MRC Nab b a bASER amL mL mLγ γ γ− − −

−

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞≈ + + + − +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

(4.17)

4.4.1 Analysis of Approximation

Table 4.1 - 4.6 shows the difference in dB for 4-QAM, 16-QAM and 64-QAM for different

values of m and L respectively. The figs.4.6– 4.15 shows the same information in the tables

but graphically. From the tables and figures we can see that Goldsmith approximation is good

to be used if we accept an error of 1 dB for 16-QAM only since its formula has one term only

but for other modulation indexes like 4 or 64-QAM we have maximum error of 2.8 and 3.2

dB respectively at ASER= 10-1 and single Nakagami-m parameter m = 0.5. Also we can see

72

that the approximation of n =1 is better than Chiani approximation although n =1 consists of

2 terms only. And Chiani is better only at high m and SNR.



1. The system performance can be expressed by a simple finite series of elementary

functions that is the exponential.

2. The error decays rapidly with increasing values of n.

3. At fixed n, the error increases when the modulation index is increased.

4. At fixed n, m and L, the error decreases when SNR is increased.

17.2 17.4 17.6 17.8 18 18.2 18.4

10-1

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r

Exacy

n=1

Chiani

Goldsmith

m=1L=116QAM

Fig.4.6. Exact and Approx. of 16-QAM over Nakagami with m =1 and L =1

73

14.4 14.6 14.8 15 15.2 15.4 15.6

10-1

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r

Exacyn=1ChianiGoldsmith

m=2L=116QAM

Fig.4.7. Exact and Approx. of 16-QAM over Nakagami with m = 2 and L=1

13 13.2 13.4 13.6 13.8 14

10-1

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r


m=5L=116QAM

Fig.4.8. Exact and Approx. of 16-QAM over Nakagami with m = 5 and L =1

74

17.4 17.6 17.8 18 18.2 18.4

10-1

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r


m=0.5L=216QAM

Fig.4.9. Exact and Approx. of 16-QAM over Nakagami with m = 0.5 and L =2

31.4 31.6 31.8 32 32.2 32.4

10-4

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r


m=1L=216QAM

Fig.4.10. Exact and Approx. of 16-QAM over Nakagami with m =1 and L =2

75

21.4 21.5 21.6 21.7 21.8 21.9 22 22.1 22.2 22.3 22.4

10-3

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r


m=2L=216QAM

Fig.4.11. Exact and Approx. of 16-QAM over Nakagami with m = 2 and L = 2.

19.5 19.6 19.7 19.8 19.9 20 20.1 20.2

10-3

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r


m=2L=416QAM

Fig.4.12. Exact and Approx. of 16-QAM over Nakagami with m = 2 and L =4

76

20.7 20.8 20.9 21 21.1 21.2 21.3 21.4 21.5

10-3

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r


m=5L=116QAM

Fig.4.13. Exact and Approx. of 16-QAM over Nakagami with m = 5 and L=1

19 19.1 19.2 19.3 19.4 19.5 19.6 19.7

10-3

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r


m=5L=216QAM

Fig.4.14. Exact and Approx. of 16-QAM over Nakagami with m = 5 and L = 2

77

18.4 18.5 18.6 18.7 18.8 18.9 19

10-3

SNR per Symbol (dB)

Pro

babi

lity

of S

ymbo

l erro

r


m=5L=416QAM

Fig.4.15. Exact and Approx. of 16-QAM over Nakagami with m = 5 and L = 4

78

Table 4.1: Difference in dB for 4-QAM over Nakagami fading channel for m = 0.5, 1 and L = 1, 2 and 4.

ASER for 4-QAM 1.E-01 1.E-02 1.E-03 1.E-04 1.E-05 1.E-06

m L n Difference (dB) 0.5 1 Goldsmith -2.832 -2.6626 0.5 1 Chiani 0.18 0.1912 0.5 1 1 0.4769 0.4634 0.5 1 2 0.1158 0.1127 0.5 1 3 0.0512 0.0498 0.5 1 4 0.0287 0.028 0.5 1 10 0.0046 0.0045 0.5 2 Goldsmith -0.9839 -0.5891 -0.5588 -0.5558 0.5 2 Chiani 0.5375 0.5069 0.5035 0.5 2 1 0.1879 0.1685 0.1665 0.1663 0.5 2 2 0.0449 0.0402 0.0397 0.0397 0.5 2 3 0.0198 0.0177 0.0175 0.0175 0.5 2 4 0.0111 0.0099 0.0098 0.0098 0.5 2 10 0.0018 0.0016 0.0016 0.0016 0.5 4 Goldsmith -0.308 0.1344 0.1997 0.2166 0.2215 0.2231 0.5 4 Chiani 0.5923 0.4469 0.4088 0.3975 0.394 0.3929 0.5 4 1 0.1527 0.1276 0.1127 0.1077 0.106 0.1055 0.5 4 2 0.0436 0.03 0.0265 0.0255 0.0252 0.0251 0.5 4 3 0.0191 0.0132 0.0117 0.0112 0.0111 0.011 0.5 4 4 0.0107 0.0074 0.0065 0.0063 0.0062 0.0062 0.5 4 10 0.0017 0.0012 0.001 0.001 0.001 0.001 1 1 Goldsmith -0.9839 -0.5891 -0.5588 -0.5558 1 1 Chiani 0.5375 0.5069 0.5035 1 1 1 0.1879 0.1685 0.1665 0.1663 1 1 2 0.0449 0.0402 0.0397 0.0397 1 1 3 0.0198 0.0177 0.0175 0.0175 1 1 4 0.0111 0.0099 0.0098 0.0098 1 1 10 0.0018 0.0016 0.0016 0.0016 1 2 Goldsmith -0.308 0.1344 0.1997 0.2166 0.2215 0.2231 1 2 Chiani 0.5923 0.4469 0.4088 0.3975 0.394 0.3929 1 2 1 0.1527 0.1276 0.1127 0.1077 0.106 0.1055 1 2 2 0.0436 0.03 0.0265 0.0255 0.0252 0.0251 1 2 3 0.0191 0.0132 0.0117 0.0112 0.0111 0.011 1 2 4 0.0107 0.0074 0.0065 0.0063 0.0062 0.0062 1 2 10 0.0017 0.0012 0.001 0.001 0.001 0.001 1 4 Goldsmith -0.0431 0.3533 0.4033 0.4151 0.4188 0.4201 1 4 Chiani 0.5858 0.3435 0.258 0.2209 0.2028 0.1935 1 4 1 0.1694 0.1119 0.0553 0.0256 0.0102 0.002 1 4 2 0.0438 0.0207 0.0134 0.0105 0.0092 0.0085 1 4 3 0.0187 0.0092 0.006 0.0048 0.0042 0.0039 1 4 4 0.0105 0.0051 0.0034 0.0027 0.0024 0.0022 1 4 10 0.0017 0.0008 0.0005 0.0004 0.0004 0.0004

79

Table 4.2: Difference in dB for 4-QAM over Nakagami fading channel for m = 2, 5 and L = 1, 2 and 4.


m L n Difference (dB) 2 1 Goldsmith -0.308 0.1344 0.1997 0.2166 0.2215 0.2231 2 1 Chiani 0.5923 0.4469 0.4088 0.3975 0.394 0.3929 2 1 1 0.1527 0.1276 0.1127 0.1077 0.106 0.1055 2 1 2 0.0436 0.03 0.0265 0.0255 0.0252 0.0251 2 1 3 0.0191 0.0132 0.0117 0.0112 0.0111 0.011 2 1 4 0.0107 0.0074 0.0065 0.0063 0.0062 0.0062 2 1 10 0.0017 0.0012 0.001 0.001 0.001 0.001 2 2 Goldsmith -0.0431 0.3533 0.4033 0.4151 0.4188 0.4201 2 2 Chiani 0.5858 0.3435 0.258 0.2209 0.2028 0.1935 2 2 1 0.1694 0.1119 0.0553 0.0256 0.0102 0.002 2 2 2 0.0438 0.0207 0.0134 0.0105 0.0092 0.0085 2 2 3 0.0187 0.0092 0.006 0.0048 0.0042 0.0039 2 2 4 0.0105 0.0051 0.0034 0.0027 0.0024 0.0022 2 2 10 0.0017 0.0008 0.0005 0.0004 0.0004 0.0004 2 4 Goldsmith 0.0711 0.4218 0.4429 0.4347 0.424 0.4149 2 4 Chiani 0.5743 0.2758 0.1642 0.1142 0.0892 0.0756 2 4 1 0.1876 0.0922 -0.0065 -0.0591 -0.0873 -0.1036 2 4 2 0.0418 0.0149 0.0059 0.0029 0.0018 0.0013 2 4 3 0.0186 0.0066 0.0028 0.0015 0.001 0.0007 2 4 4 0.0104 0.0037 0.0016 0.0009 0.0006 0.0004 2 4 10 0.0016 0.0006 0.0003 0.0001 0.0001 0.0001 5 1 Goldsmith 0.004 0.3841 0.4244 0.43 0.4297 0.4284 5 1 Chiani 0.5818 0.3175 0.2212 0.1777 0.1557 0.1435 5 1 1 0.1759 0.1056 0.0339 -0.005 -0.0262 -0.0382 5 1 2 0.0432 0.0185 0.0104 0.0072 0.0058 0.005 5 1 3 0.0187 0.0082 0.0047 0.0034 0.0027 0.0024 5 1 4 0.0105 0.0046 0.0027 0.0019 0.0016 0.0014 5 1 10 0.0017 0.0007 0.0004 0.0003 0.0003 0.0002 5 2 Goldsmith 0.0925 0.4324 0.4457 0.4316 0.4162 0.4033 5 2 Chiani 0.5713 0.2613 0.1452 0.0945 0.0705 0.0583 5 2 1 0.1921 0.0864 -0.022 -0.0781 -0.1071 -0.1233 5 2 2 0.0412 0.0137 0.0044 0.0017 0.001 0.0008 5 2 3 0.0187 0.006 0.0022 0.001 0.0006 0.0003 5 2 4 0.0104 0.0034 0.0013 0.0006 0.0003 0.0002 5 2 10 0.0016 0.0005 0.0002 0.0001 0.0001 0 5 4 Goldsmith 0.1338 0.4501 0.4463 0.4187 0.3923 0.3703 5 4 Chiani 0.5648 0.2313 0.1082 0.06 0.0425 0.0379 5 4 1 0.2019 0.0723 -0.0555 -0.1145 -0.1409 -0.1531 5 4 2 0.0399 0.0112 0.0017 0.0004 0.0007 0.0006 5 4 3 0.0187 0.0048 0.0012 0.0003 0.0001 0 5 4 4 0.0103 0.0027 0.0007 0.0002 0.0001 0 5 4 10 0.0016 0.0004 0.0001 0 0 0

80

Table 4.3: Difference in dB for 16-QAM over Nakagami fading channel for

m = 0.5, 1 and L = 1, 2 and 4. ASER for 16-QAM 1.E-01 1.E-02 1.E-03 1.E-04 1.E-05 1.E-06

m L n Difference (dB) 0.5 1 Goldsmith 0.6098 0.5 1 Chiani 0.6204 0.5 1 1 0.4689 0.5 1 2 0.114 0.5 1 3 0.0504 0.5 1 4 0.0283 0.5 1 10 0.0045 0.5 2 Goldsmith 0.9194 0.9171 0.9166 0.5 2 Chiani 0.6403 0.5954 0.5912 0.5 2 1 0.2922 0.2682 0.2658 0.5 2 2 0.0693 0.0634 0.0629 0.5 2 3 0.0305 0.0279 0.0277 0.5 2 4 0.0171 0.0156 0.0155 0.5 2 10 0.0027 0.0025 0.0025 0.5 4 Goldsmith 0.9815 0.92 0.9005 0.8944 0.8925 0.5 4 Chiani 0.576 0.4437 0.4103 0.4004 0.3973 0.5 4 1 0.2626 0.1954 0.1733 0.1664 0.1642 0.5 4 2 0.0626 0.0449 0.0403 0.039 0.0386 0.5 4 3 0.0274 0.0197 0.0177 0.0171 0.017 0.5 4 4 0.0153 0.011 0.0099 0.0096 0.0095 0.5 4 10 0.0024 0.0018 0.0016 0.0015 0.0015 1 1 Goldsmith 0.9194 0.9171 0.9166 1 1 Chiani 0.6403 0.5954 0.5912 1 1 1 0.2922 0.2682 0.2658 1 1 2 0.0693 0.0634 0.0629 1 1 3 0.0305 0.0279 0.0277 1 1 4 0.0171 0.0156 0.0155 1 1 10 0.0027 0.0025 0.0025 1 2 Goldsmith 0.9815 0.92 0.9005 0.8944 0.8925 1 2 Chiani 0.576 0.4437 0.4103 0.4004 0.3973 1 2 1 0.2626 0.1954 0.1733 0.1664 0.1642 1 2 2 0.0626 0.0449 0.0403 0.039 0.0386 1 2 3 0.0274 0.0197 0.0177 0.0171 0.017 1 2 4 0.0153 0.011 0.0099 0.0096 0.0095 1 2 10 0.0024 0.0018 0.0016 0.0015 0.0015 1 4 Goldsmith 0.987 0.8618 0.8027 0.7728 0.757 0.7484 1 4 Chiani 0.5241 0.3187 0.246 0.2137 0.1979 0.1897 1 4 1 0.2658 0.1368 0.0705 0.0386 0.0224 0.0138 1 4 2 0.0582 0.0288 0.0193 0.0154 0.0136 0.0127 1 4 3 0.0252 0.0127 0.0086 0.007 0.0062 0.0058 1 4 4 0.0141 0.0071 0.0049 0.004 0.0035 0.0033 1 4 10 0.0022 0.0011 0.0008 0.0006 0.0006 0.0005

81



m L n Difference (dB) 2 1 Goldsmith 0.9815 0.92 0.9005 0.8944 0.8925 2 1 Chiani 0.576 0.4437 0.4103 0.4004 0.3973 2 1 1 0.2626 0.1954 0.1733 0.1664 0.1642 2 1 2 0.0626 0.0449 0.0403 0.039 0.0386 2 1 3 0.0274 0.0197 0.0177 0.0171 0.017 2 1 4 0.0153 0.011 0.0099 0.0096 0.0095 2 1 10 0.0024 0.0018 0.0016 0.0015 0.0015 2 2 Goldsmith 0.987 0.8618 0.8027 0.7728 0.757 0.7484 2 2 Chiani 0.5241 0.3187 0.246 0.2137 0.1979 0.1897 2 2 1 0.2658 0.1368 0.0705 0.0386 0.0224 0.0138 2 2 2 0.0582 0.0288 0.0193 0.0154 0.0136 0.0127 2 2 3 0.0252 0.0127 0.0086 0.007 0.0062 0.0058 2 2 4 0.0141 0.0071 0.0049 0.004 0.0035 0.0033 2 2 10 0.0022 0.0011 0.0008 0.0006 0.0006 0.0005 2 4 Goldsmith 0.984 0.8163 0.7234 0.667 0.6305 0.6057 2 4 Chiani 0.4935 0.2452 0.1507 0.1075 0.0855 0.0734 2 4 1 0.2722 0.0907 -0.0117 -0.0622 -0.089 -0.1044 2 4 2 0.0548 0.0188 0.0077 0.004 0.0025 0.0019 2 4 3 0.0239 0.0084 0.0037 0.0021 0.0013 0.001 2 4 4 0.0133 0.0047 0.0022 0.0012 0.0008 0.0006 2 4 10 0.0021 0.0008 0.0004 0.0002 0.0001 0.0001 5 1 Goldsmith 0.9862 0.8449 0.7735 0.7344 0.7118 0.6983 5 1 Chiani 0.5122 0.2901 0.2082 0.1705 0.1511 0.1402 5 1 1 0.268 0.1202 0.0401 0.0001 -0.0212 -0.0332 5 1 2 0.0569 0.025 0.0145 0.0104 0.0084 0.0074 5 1 3 0.0247 0.0111 0.0066 0.0048 0.004 0.0035 5 1 4 0.0138 0.0062 0.0038 0.0028 0.0023 0.002 5 1 10 0.0022 0.001 0.0006 0.0004 0.0004 0.0003 5 2 Goldsmith 0.983 0.8061 0.7051 0.642 0.5998 0.5704 5 2 Chiani 0.487 0.2299 0.1318 0.0882 0.0673 0.0566 5 2 1 0.2739 0.0794 -0.0302 -0.0828 -0.1098 -0.1248 5 2 2 0.054 0.0167 0.0056 0.0024 0.0014 0.001 5 2 3 0.0237 0.0075 0.0029 0.0014 0.0008 0.0005 5 2 4 0.0131 0.0042 0.0017 0.0008 0.0005 0.0003 5 2 10 0.0021 0.0007 0.0003 0.0001 0.0001 0.0001 5 4 Goldsmith 0.9806 0.7841 0.666 0.5886 0.5343 0.4945 5 4 Chiani 0.4735 0.1984 0.0954 0.0552 0.041 0.0378 5 4 1 0.2777 0.0545 -0.0679 -0.1204 -0.1436 -0.1543 5 4 2 0.0526 0.0122 0.002 0.0006 0.0008 0.0006 5 4 3 0.0231 0.0056 0.0014 0.0004 0.0001 0.0001 5 4 4 0.0128 0.0032 0.0009 0.0003 0.0001 0 5 4 10 0.002 0.0005 0.0001 0 0 0

82

Table 4.5: Difference in dB for 64-QAM over Nakagami fading channel for m = 0.5, 1 and L = 1, 2 and 4.


m L n Difference (dB) 0.5 1 Goldsmith 3.2381 0.5 1 Chiani 0.8592 0.5 1 1 0.4672 0.5 1 2 0.1136 0.5 1 3 0.0502 0.5 1 4 0.0282 0.5 1 10 0.0045 0.5 2 Goldsmith 2.2647 2.1368 0.5 2 Chiani 0.6923 0.6425 0.5 2 1 0.3499 0.3231 0.5 2 2 0.0825 0.076 0.5 2 3 0.0363 0.0334 0.5 2 4 0.0203 0.0187 0.5 2 10 0.0032 0.003 0.5 4 Goldsmith 1.815 1.5504 1.4865 1.4677 0.5 4 Chiani 0.5718 0.4435 0.4115 0.402 0.5 4 1 0.3152 0.2306 0.2051 0.1973 0.5 4 2 0.0723 0.0525 0.0475 0.046 0.5 4 3 0.0316 0.023 0.0208 0.0202 0.5 4 4 0.0176 0.0129 0.0117 0.0113 0.5 4 10 0.0028 0.002 0.0019 0.0018 1 1 Goldsmith 2.2647 2.1368 1 1 Chiani 0.6923 0.6425 1 1 1 0.3499 0.3231 1 1 2 0.0825 0.076 1 1 3 0.0363 0.0334 1 1 4 0.0203 0.0187 1 1 10 0.0032 0.003 1 2 Goldsmith 1.815 1.5504 1.4865 1.4677 1 2 Chiani 0.5718 0.4435 0.4115 0.402 1 2 1 0.3152 0.2306 0.2051 0.1973 1 2 2 0.0723 0.0525 0.0475 0.046 1 2 3 0.0316 0.023 0.0208 0.0202 1 2 4 0.0176 0.0129 0.0117 0.0113 1 2 10 0.0028 0.002 0.0019 0.0018 1 4 Goldsmith 1.6123 1.2598 1.1348 1.0771 1.0476 1.0319 1 4 Chiani 0.5022 0.3097 0.2413 0.2109 0.1957 0.1879 1 4 1 0.3052 0.1493 0.0786 0.0454 0.0287 0.0199 1 4 2 0.0653 0.0328 0.0222 0.0178 0.0158 0.0148 1 4 3 0.0283 0.0144 0.0099 0.0081 0.0072 0.0068 1 4 4 0.0158 0.0081 0.0056 0.0046 0.0041 0.0039 1 4 10 0.0025 0.0013 0.0009 0.0007 0.0007 0.0006

83



m L n Difference (dB) 2 1 Goldsmith 1.815 1.5504 1.4865 1.4677 2 1 Chiani 0.5718 0.4435 0.4115 0.402 2 1 1 0.3152 0.2306 0.2051 0.1973 2 1 2 0.0723 0.0525 0.0475 0.046 2 1 3 0.0316 0.023 0.0208 0.0202 2 1 4 0.0176 0.0129 0.0117 0.0113 2 1 10 0.0028 0.002 0.0019 0.0018 2 2 Goldsmith 1.6123 1.2598 1.1348 1.0771 1.0476 1.0319 2 2 Chiani 0.5022 0.3097 0.2413 0.2109 0.1957 0.1879 2 2 1 0.3052 0.1493 0.0786 0.0454 0.0287 0.0199 2 2 2 0.0653 0.0328 0.0222 0.0178 0.0158 0.0148 2 2 3 0.0283 0.0144 0.0099 0.0081 0.0072 0.0068 2 2 4 0.0158 0.0081 0.0056 0.0046 0.0041 0.0039 2 2 10 0.0025 0.0013 0.0009 0.0007 0.0007 0.0006 2 4 Goldsmith 1.5185 1.1197 0.9542 0.8633 0.8071 0.7702 2 4 Chiani 0.4652 0.2345 0.1459 0.105 0.0841 0.0726 2 4 1 0.302 0.0908 -0.0131 -0.0631 -0.0895 -0.1046 2 4 2 0.0608 0.0206 0.0085 0.0045 0.0029 0.0021 2 4 3 0.0263 0.0093 0.0042 0.0023 0.0015 0.0011 2 4 4 0.0146 0.0052 0.0024 0.0014 0.0009 0.0007 2 4 10 0.0023 0.0008 0.0004 0.0002 0.0002 0.0001 5 1 Goldsmith 1.5742 1.2033 1.0628 0.9928 0.954 0.9314 5 1 Chiani 0.4877 0.2803 0.2033 0.1677 0.1491 0.1388 5 1 1 0.3038 0.1277 0.0439 0.0031 -0.0184 -0.0305 5 1 2 0.0636 0.0281 0.0165 0.0119 0.0097 0.0086 5 1 3 0.0275 0.0124 0.0076 0.0056 0.0046 0.0041 5 1 4 0.0153 0.007 0.0043 0.0032 0.0026 0.0023 5 1 10 0.0024 0.0011 0.0007 0.0005 0.0004 0.0004 5 2 Goldsmith 1.5005 1.0923 0.918 0.8195 0.7568 0.7142 5 2 Chiani 0.4576 0.219 0.1271 0.086 0.0661 0.056 5 2 1 0.3015 0.0771 -0.0328 -0.0843 -0.1106 -0.1253 5 2 2 0.0598 0.018 0.0061 0.0027 0.0016 0.0011 5 2 3 0.0258 0.0082 0.0032 0.0015 0.0009 0.0006 5 2 4 0.0144 0.0046 0.0019 0.0009 0.0005 0.0003 5 2 10 0.0023 0.0007 0.0003 0.0002 0.0001 0.0001 5 4 Goldsmith 1.4649 1.0377 0.8456 0.7316 0.6556 0.6013 5 4 Chiani 0.4421 0.1871 0.0911 0.0536 0.0406 0.0377 5 4 1 0.3007 0.0478 -0.0721 -0.1224 -0.1446 -0.1548 5 4 2 0.0577 0.0127 0.0021 0.0008 0.0008 0.0006 5 4 3 0.0249 0.006 0.0015 0.0004 0.0002 0.0001 5 4 4 0.0138 0.0034 0.0009 0.0003 0.0001 0 5 4 10 0.0022 0.0006 0.0002 0.0001 0 0

84

CHAPTER 5

CONCLUSIONS AND FUTURE WORK 5.1 Conclusions

In this thesis, we obtained simple closed form expression to determine the performance of M-

QAM transmitted over slow, flat, identically independently distributed (i.i.d) fading channels

and using space diversity in terms of ASER. Three types of fading channels are considered:

Rayleigh Ricean, and Nakagami. These simple and efficient ASER formulas make it possible

for the first time to study, analyze and discuss the parameters of various constellations of

square M-QAM, diversity order and fading parameters precisely and easily. MRC was used

as combining technique to overcome the fading channel. The main advantage of these

expressions, that they show the relation between the square M-QAM modulation index (M)

and the diversity order (L) and the fading parameter (K for Ricean or m for Nakagami-m)

versus SNR in one simple formula.

We used an expression for the SER of M-QAM modulation over a Gaussian channel

(AWGN). This expression is then used to obtain the ASER of square M-QAM using MRC

over both Ricean and Nakagami fading channels. For Rayleigh, it was obtained as a special

case of both Ricean (K=0) and Nakagami-m (m=1). The obtained expressions are in the form

of sum of exponentials or power series where the number of terms can be determined

according to the required accuracy. Theoretically, to get the exact solution the series must be

infinite. But because the series converge rapidly, 10 terms are enough to get an error less

than 0.005 dB in the worst case which is at low order of modulation index (M), diversity, and

low fading parameter (K for Ricean or m for Nakagami-m). The minimum number of terms

to approximate the ASER is two terms, so single term is also investigated which is available

in the literatures. The error for different number of terms are investigated and tabulated. The

error was limited to one dB for n=1 which is 2 terms for all M, L, and the fading parameter

(K or m) but for single term which we call Goldsmith the difference in dB between the exact

and approximate is increased with M and decreased with L, and the fading parameter.

85

5.2 Future Work

Our work could be extended for different types of combining techniques. Also in this thesis

we have assumed that the fading is i.i.d but this scenario no longer exists in real life if the

diversity is used in the mobile terminals, so the work presented here can be extended to the

case of correlated fading. Another possible extension of our work would be to obtain exact

ASER of M-QAM over slow, flat, fading when the fading channels are non-identical.

86

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