Optimizing performance of multiple access multi carrier multilevel frequency shift keying systems

OPTIMIZING PERFORMANCE OF
MULTIPLE ACCESS MULTI-CARRIER
MULTILEVEL FREQUENCY SHIFT KEYING SYSTEMS

TAY HAN SIONG

NATIONAL UNIVERSITY OF SINGAPORE
2005


OPTIMIZING PERFORMANCE OF
MULTIPLE ACCESS MULTI-CARRIER
MULTILEVEL FREQUENCY SHIFT KEYING SYSTEMS

TAY HAN SIONG
(B.Eng.(Hons.), NUS)

A THESIS SUBMITTED
FOR THE DEGREE OF MASTER OF ENGINEERING
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
NATIONAL UNIVERSITY OF SINGAPORE
2005


Acknowledgement

First of all, I’ll like to thank my supervisors, Dr Chai Chin Choy and Professor
Tjhung Tjeng Thiang, for all their generous advice and unwavering patience. This
thesis will not be possible without their continuous support.

I’ll also like to thank Mr Ng Khai Sheng and Mr Thomas Sushil. Their invaluable

encouragement and friendship have put me through many rough times. This tenure
has certainly been more rewarding due to the two of you.

Also to the Institute of Infocomm Research, for giving me the opportunity to conduct
such exciting research.

Finally to my Mum and Dad for their endless support and understanding.

i


Table of Contents

Acknowledgement

i

Table of Contents

ii

Summary

v

List of Figures

viii

List of Tables

x

Chapter 1 Introduction

1

1.1

Literature Review

3

1.2

Thesis Overview

6

Chapter 2 Multi-Carrier MFSK System Model

8

2.1

Transmitter and Receiver

8

2.2

Decoder

10

2.3

System Capacity and Normalized Throughput

11

Chapter 3 Optimal Diversity Order of Multiple Access
Multi-Carrier MFSK Systems

13

3.1

Introduction

13

3.2

Derivation of Symbol Error Rate and Optimization of Diversity Order

14

3.2.1

Derivation of Symbol Error Rate

14

3.2.2

Optimization of Diversity Order

16

3.3

Numerical Results and Comparison

17

ii


3.4

Optimal Diversity Order for Maximum User Capacity Subject to Symbol
Error Probability Constraint

3.5

3.6

20

Throughput Maximization Subject to Symbol Error Probability Constraint
and Constant Number of Users

22

Conclusion

24

Chapter 4 Diversity Control in Multiple Access Multi-Carrier
MFSK Systems

25

4.1

Introduction

25

4.2

Symbol Error Probability, System Capacity and Throughput

26

4.3

Optimal Diversity Order for Multi-Carrier MFSK System

29

4.3.1

Optimal Diversity Order for Maximizing Individual Throughput

29

4.3.2

Optimal Diversity Order for Maximizing Total System Throughput 31

4.4

Adaptation of Diversity Order

33

4.5

Explanation Using Game Theory

35

4.6

Conclusion

38

Chapter 5 Balanced Incomplete Block Design to Improve
Performance of Multi-Carrier MFSK Systems

40

5.1

Introduction

40

5.2

Balanced Incomplete Block Design for Multi-Carrier MFSK

41

5.3

Analysis and Derivations

43

5.3.1

Derivation of User Capacity and Bandwidth Efficiency

43

5.3.2

Derivation of Error Probability

44

5.4

Effect and Selection of Various BIB Design Parameters

47

5.4.1

47

Optimal Diversity Order for Minimum Error Rate

iii


5.4.2

Optimal Modulation Level for Maximum User Capacity at
Constant Bandwidth Efficiency

5.4.3

48

Selection of BIB Design Parameters for Maximum Error

Performance at Constant Bandwidth Efficiency

50

5.5

Performance Comparison with Conventional Multi-Carrier MFSK Systems 53

5.6

Conclusion

56

Chapter 6 Extension to Frequency-Hopping Multi-Carrier MFSK
Systems

58

6.1

Introduction

58

6.2

Frequency-Hopping Multi-Carrier MFSK System Model

59

6.3

Types of Random Frequency-Time Code and Comparison

62

6.3.1

Types of random Frequency-Time codes

63

6.3.2

Performance of Codes

67

6.3.3

Implementation Considerations of Codes

73

6.4

Derivations and Optimization of Frequency Diversity Order

74

6.4.1

Derivation of Symbol Error Probability

74

6.4.2

Derivation of System Bandwidth and Normalized Throughput

77

6.4.3

Optimization of Frequency Diversity Level

78

6.5

Effects of System Parameters

79

6.6

Conclusion

80

Chapter 7 Conclusion

82

References

85

iv


Summary

The Multi-carrier Multilevel Frequency Shift Keying (MC-MFSK) system is a form
of multi-tone MFSK systems, and it transmits on multiple frequency carriers
simultaneously. The number of frequency-carriers used is termed the diversity order.

We derive a new analytical solution for the optimal diversity order of the multipleaccess MC-MFSK system for achieving maximum throughput. The new formula
relates the optimal frequency diversity order to the modulation level and number of
users. We present numerically searched results for the optimal diversity order of MCMFSK systems in both Rayleigh and Rician fading channels based on previously
published works. We highlight that our formula gives very close results for optimal
diversity order compared to the numerically-searched ones at SNR above 40dB. We
also derive the optimal parameters for systems with several constraints such as error
probability limit and restricted number of users.

For the first time, we also derive the steady state solution of the MC-MSFK system
when control of the diversity orders is distributed to the users. We formulate the
diversity control problem for two scenarios: 1) non-cooperative system users, where
every user’s objective is to maximize its own throughput, 2) cooperative system users,

where every user’s objective is to maximize overall system throughput. For each
scenario, we present a steady state solution for the optimal diversity order. Using the
concept of game theory, the solution in the first scenario corresponds to a Nash

v


equilibrium point but is Pareto inefficient, while the solution in the second scenario
gives the desired Pareto efficient point.

Next we propose a method to select frequency-carriers in MC-MFSK systems to
improve error performance. The method uses a combinatorial construction called
Balanced Incomplete Block (BIB) Design to form selections of multiple frequencycarriers. With BIB design, any two selections will only coincide in at most one
frequency-carrier. The selections are uniquely assigned to each symbol of every user,
thus reducing the interference between the users in symbol transmission. We also
present a selection process for optimal BIB design parameters. The performance of
MC-MFSK systems using BIB design is compared to conventional MC-MFSK
systems in Rayleigh channels. Our results show significant improvement for the
proposed system for low number of user, while the performance is worse for larger
number of users. Given a suitable user number, the method can be employed in MCMFSK systems with the benefit of better error performance.

We also extend the MC-MSFK system to the Frequency-Hopping Multi-carrier
(FHMC)-MFSK system by introducing additional frequency-hopping. We present an
analysis for the frequency-time encoding techniques that provide maximum error
performance. We show that the optimal frequency diversity order has the same
relationship as the conventional MC-MFSK system, and is unaffected by the timediversity. Hence the frequency-hopping, which improves error rate exponentially, can
be used to achieve better error performance for the conventional MC-MFSK system.

vi

Thus we show the versatility of the MC-MFSK system, along with its maximum
capability in several practical conditions. We conclude that the MC-MFSK is a strong
candidate for future spread-spectrum communication systems, which required high
data rate and spectral efficiency.

vii


List of Figures

Fig. 2.1

MC-MFSK Transmitter

9

Fig. 2.2

MC-MFSK Receiver

Fig. 3.1

Symbol error rate Pe versus Diversity order L for analytical and

10

simulation Pe in non-fading AWGN channel with high SNR
channel at M = 256
Fig. 3.2

16

Numerically searched optimal diversity order Lopt versus SNR for
{M=1024, K=20}, {M=512, K=10}

Fig. 3.3

19

BER versus Diversity order L for fading channels for various M,
K, and SNR

20

Fig. 3.4

User capacity Kmax versus SER limit P0 at M = 256, 512, 1024

22

Fig. 4.1

Adaptation of Diversity order for User 1

34

Fig. 4.2

Adaptation of Diversity order for User 2

34

Fig. 4.3

Pe,1 and Pe,2 versus L1 for two user system, M = 256, L2 = 40&118

37

Fig. 5.1

BIB-MC-MFSK Transmitter

42

Fig. 5.2

Analytical BER versus number of users K for M=256, N=256 and
various diversity order L

Fig. 5.3

Analytical BER versus number of users K for η = 1

48

32

, L = 4 and

various M
Fig. 5.4

50

BER versus number of users K for BIB-MC-MFSK and
conventional MC-MFSK systems in Rayleigh Channel and with

Fig. 6.1

Bit SNR = 40 dB

54

Frequency-Time matrix representation

60
viii


Fig. 6.2

FHMC-MFSK system decoding process

62

Fig. 6.3

Type I code

64

Fig. 6.4

Type II code

64

Fig. 6.5

Type III code

65

Fig. 6.6

Type IVa code

66

Fig. 6.7

Type IVb code

67

Fig. 6.8

Auto-correlation of Frequency-Time codes with M=256, L=10,
H=5

Fig. 6.9

Analytical and simulation SER for M=256, H=1, SNR= 40dB in
Rayleigh channels

Fig. 6.10

72

76

Analytical and simulation SER for M=256, H=2, SNR= 40dB in
Rayleigh channels

77

ix


List of Tables

Table 5.1

Examples of existent BIB designs

52

Table 6.1

Frequency-Time Code Types

63

Table 6.2

Distribution, mean and variance of auto-correlation for all code
types

71

x


Chapter 1
Introduction

In recent years, much research interest has been focused on multiple-access spread
spectrum systems. This is due to the need for a new generation of communication
systems, capable of delivering high data rate at wide bandwidth to mobile users. The
system must also be spectrally efficient.

One of such candidates is the Multi-carrier Multilevel Frequency Shift Keying (MCMFSK) system, which is proposed recently in [1,2] by Sinha as a candidate for future
high-speed spread spectrum communication systems. The performance of this system
is further analyzed for the Rician channel in [3] by Yu. It is a form of multi-tone
MFSK system, and MC-MFSK systems transmit on multiple frequency carriers
simultaneously. The system allows multiple-user access with its users sharing the
same frequency and time space. These multiple users are differentiated by the unique
permutations of frequency carriers, which each user uses to transmit its symbol. This
system has several desirable properties such as frequency diversity and immunity to

near-far effect. It also allows for an OFDM based multi-carrier implementation.

It is shown in [2] that the MC-MFSK system is able to achieve better performance
than Goodman’s frequency-hopping MFSK system. This motivated us to study the
MC-MFSK system in greater depth. We discovered that the MC-MFSK system has
the potential of delivering better performance, but so far, no research has been carried
out to optimize its performance. In this thesis, our objective is to exploit the maximum
1


capability of the MC-MFSK system. Based on our results, the MC-MFSK system is
presented as a strong candidate for future spread-spectrum communication systems.

In MC-MFSK systems, the number of frequency-carriers used per symbol
transmission is termed the diversity order L. Along with the modulation level M, they
are the main parameters in the MC-MFSK system. Throughout this thesis, we
optimize the system with respect to these two parameters. In addition, we derive
optimal parameters for the system under several system constraints such as error
probability limit and fixed number of users.

We also optimize the MC-MFSK system when control of the diversity orders is
distributed to the users. In this case, the diversity control problem is formulated for
two scenarios: 1) non-cooperative system users, where every user’s objective is to
maximize its own throughput without any regard to other users. 2) cooperative system
users, where every user’s objective is to maximize the overall system throughput.

Next we propose a novel method of selecting the multiple sub-channels used by all
users for symbol transmission. The selection of sub-channels improves the error
performance of the multiple-access MC-MFSK system by reducing the degree of
interference between the users. This method uses a combinatorial construction called

Balanced Incomplete Block (BIB) design to form a collection of sub-channels
selections, where any two selections will coincide in at most one sub-channel. These
selections of sub-channels are uniquely assigned to each symbol of every user. Thus
on symbol transmission, the effect of multiple-access interference is reduced.

2


Lastly, we extend the conventional MC-MFSK system to the Frequency-Hopping
MC-MFSK system by introducing additional frequency-hopping to every user. In this
system, each symbol transmission will span over several time hops and a different
permutation of sub-channels will be used per hop. We analyze the error performance
of this frequency-hopping MC-MFSK system, and optimize the system throughput
with respect to the diversity order.

1.1

Literature Review

The MC-MFSK system is first introduced by Sinha in [1]. The MC-MFSK system is a
multiple-access system based on OFDM implementation [1]. By making use of
advances in OFDM technologies, the system can be easily implemented with the
IFFT/FFT operations, which eliminate the need for banks of oscillators [4]. The MCMFSK system has some advantages over both FH-MFSK and conventional DirectSequence (DS)-CDMA systems as follows. Firstly, compared to FH-MFSK system,
the MC-MFSK system is more robust against the effect of large delay spreads as it
has a lower signaling rate on individual sub-channels. Secondly, the DS-CDMA
system is highly susceptible to the near-far effect [5] while the MC-MSFK system is
immune to this effect. Thirdly, the MC-MFSK system achieves frequency diversity.
Due to these advantages, Sinha et al. propose the MC-MFSK system as a strong
candidate for future high-speed wireless system [1].

In [2], Sinha presents a derivation for the error performance of the MC-MFSK
system. The main assumptions made in this evaluation are: 1) the system is under a
Rayleigh fading channel, and 2) all its users have the same diversity order. The
derived upper-bound for the symbol error rate of the system can be found in [2].

3


Using this expression, Sinha proves that the MC-MFSK system achieves a higher user
capacity than conventional FH-MFSK systems. The expression is also shown to be an
effective upper-bound for the error probability, with the bound been tighter for higher
SNR. However, the proposed expression is mathematically complicated. The results
in [2] show that there exists an optimal diversity order, which maximizes the MCMFSK system performance. However, no further attempt has been made to evaluate
this optimal diversity order analytically.

Yu et al. [3] evaluate the error performance of MC-MFSK systems for Rician fading
channels. For a Rician channel, the authors use a novel approach of combining the
line-of-sight (LOS) carriers of the multiple signals into a single LOS carrier, and
combining the multipath components of multiple signals with other Gaussian noises
to form a single Gaussian process. Hence, the probability density function (pdf) for
the output of envelope detectors in MC-MFSK systems is first derived. This pdf is
then used to evaluate of the false alarm and deletion probability of the tones in MCMFSK systems. The false alarm refers to erroneous detection of a tone when none is
actually sent, while deletion refers to failure to detect a tone when it is actually sent.
Applying these probabilities using Sinha’s analysis in [2], they derive the upperbound for the error probability of the system. The difference between Sinha and Yu’s
analysis is in the expression for the false alarm and deletion probabilities. Therefore
the error performance expression in [3] for the Rician channel is also as
mathematically complicated as its counterpart in [2] for the Rayleigh channel. Similar
to Sinha’s analysis in [2], an optimal diversity order is also observed in [3]. Again no
effort has been taken to optimize the performance of the MC-MFSK system.

4


In [6], Atkin et al. propose to use a combinatorial construction, called Balance
Incomplete Block (BIB) design, for selection of frequency carriers in multi-tone
MFSK modulation. Multi-tone (MT)-MFSK systems are an extension of basic MFSK
systems, where the MT-MFSK system utilizes a permutation of frequency carriers for
signaling instead of one carrier in basic MFSK systems. The authors in [6] use the
BIB design to form the permutations of the frequency carriers. The interesting
attribute of the BIB design is that the permutations will overlap on at most λ carriers,
where λ is a user-defined parameter in BIB design. The authors show that the system
using BIB design achieves a better performance than other MT-MFSK systems using
designs such as Hadamard matrices. As for more details on the BIB design, we advise
readers to refer to the works in [7] and [8].

The MC-MFSK system is evolved from the multiple-access Frequency Hopping
(FH)-MFSK system proposed by Goodman et al. in [9]. In this thesis, the term FHMFSK system refers to Goodman’s system in [9]. The FH-MFSK system is different
from the conventional FH spread spectrum system [10-12] as follows. In FH-MFSK
systems, there is no segregation of bandwidth into sub-bands and the entire bandwidth
is made up of M orthogonal sub-channels, where M is also the modulation level of the
system. As for conventional FH spread spectrum systems, the entire system
bandwidth B is segregated into multiple sub-bands each of M orthogonal subchannels. To transmit a symbol m in the FH-MFSK system, the frequency-hopping
sequence is generated by cyclic-shifting the user’s hop-address by the value m. Since
the pioneering work of [9], the FH-MFSK system has been studied by several
researchers and it has been shown to offer a higher capacity than its conventional
counterpart [13,14].

5

1.2

Thesis Overview

In Chapter 2, we describe the MC-MFSK system model. The decoding process of the
system is explained. We also present derivations of the MC-MFSK system capacity
and system throughput.

In Chapter 3, we derive a mathematically simpler expression for the error probability
of the MC-MFSK system, by assuming that the channel is non-fading and of high
SNR. We make use of this expression to derive a new analytical solution for the
optimal diversity order, which maximizes throughput and minimizes error probability.
By comparing it with the numerical results from previous works in [2,3], we verify
that the optimal diversity order is valid for fading channels at SNR above 40 dB. By
using our error probability expression again, we maximize the throughput of the MCMFSK system under the constraints of an error probability limit and constant number
of users. We also maximize the user capacity for the MC-MFSK system constrained
by an error probability limit.

For Chapter 4, we study the diversity control problem in the MC-MFSK system when
control of the diversity orders is distributed to each user. We formulate the objective
functions for two different scenarios: 1) system users are non-cooperative and each
user’s objective is to maximize its own throughput; 2) system users are cooperative
and their objective is to maximize the total system throughput. We then derive a new
steady state expression for the solution of optimal diversity in each case. The
solutions are then explained using game theory.

6


Next, in Chapter 5, we propose a method of sub-channels selection based on a

combinatorial construction called Balanced Incomplete Block (BIB) design in MCMFSK systems. The method improves the MC-MFSK error performance by limiting
the overlapping of selected sub-channels between any two users to at most a single
sub-channel. We will also introduce the properties of the BIB design and describe its
deployment into MC-MFSK systems for our method of sub-channels selection. We
derive the error probability and user capacity of the system, and use these derivations
to analyze the effect of various BIB design parameters. Based on our analysis on the
parameters, we also propose a method in selecting a suitable parameter pair for BIB
design, which will maximize the error performance of the system. We will simulate
and compare the performances of both MC-MFSK systems using our proposed
method and conventional MC-MFSK systems.

In Chapter 6, we extend the MC-MFSK system to the Frequency-Hopping Multicarrier (FHMC)-MFSK system. This is achieved by introducing additional frequencyhopping to the MC-MFSK system. Frequency-time code is needed for the system to
select the permutation of frequency sub-channels at different time-hops. Thus we
examine all practical forms of the frequency-time code, the distribution of their
correlations, as well as their implementations. Using the same approach as in Chapter
3, we derive the error probability, bandwidth and optimal frequency diversity order of
the FHMC-MFSK system. Based on these derivations, we also study the effect of
system parameters such as time diversity, frequency diversity and modulation level,
on the system measures like error probability and bandwidth. Finally we conclude the
thesis in Chapter 7.

7


Chapter 2
Multi-Carrier MFSK System Model

The MC-MFSK system is adapted from a multiple-access Frequency-Hopping MFSK
(FH-MFSK) system proposed by Goodman in [9]. The MC-MFSK system uses
address code to generate a permutation of frequency-carriers for each symbol. A tone

is sent simultaneously on each of the selected carriers for a given symbol duration.
The system is also viewed as a special case of the Multi-tone Frequency-Hopping
MFSK system [15,16] when the time diversity equals to unity.

2.1

Transmitter and Receiver

In the MC-MFSK system, the total bandwidth is divided into M sub-channels, each
with an orthogonal carrier frequency like the MFSK system in [9]. M=2k is also the
modulation level of the data, where k is the number of bits per symbol.

The block diagrams of the transmitter and receiver are shown on Figures 2.1 and 2.2
respectively. All system users are assumed to have the same diversity order L, and
each one of them is assigned a unique address code, represented by a binary vector a
of length and Hamming weight equal M and L, respectively. The operators ⊕ and
represent the modulation and demodulation process, and Si represents a cyclic shift
operation by i position.

8


At the transmitter, a transmit vector is formed by cyclically shifting the user address
code by the symbol value m. Each entry in the transmit vector represents a subchannel. The presence of frequency tones at the set of sub-channels selected to
transmit the symbol is indicated by a “1” on the corresponding entries. For unique
mapping of each transmit vector to a user-specific symbol, every address code has to
be a-periodic; and must not be cyclic shifted versions of one another.

S ma

Fig. 2.1 MC-MFSK Transmitter

At the receiver, envelope detector is used on each sub-channel to make a hard
decision on the received tone. Note also that individual envelope detector cannot

9


differentiate between the tones sent amongst the users, and the same output is given
even when more than one user transmits on the same sub-channel. A binary received
signal vector v, is then formed at the spectrum analyzer. The vector v represents subchannels that are transmitted on by at least one user. This vector is then passed to the
decoder.

• S M −1a

• S 0a

Fig. 2.2 MC-MFSK Receiver

2.2

Decoder

The desired signal is decoded by comparing the correlations of v with all possible M
cyclic-shifted versions of the address vector a. The shift associated with the largest
10


correlation will be decoded as the desired symbol. Mathematically, the decoding rule
can be expressed as

mˆ = arg max {v ⋅ S i a } ,

(2.1)

i

where mˆ denotes the decoded symbol, and · denotes the dot product operator. In the
case where 2 or more shifts have the maximum correlation value, one of these
contending shifts is chosen randomly.

The decoding process can also be seen as the selection of a completely occupied row
from the M x L decision array. We show the M x L decision array of the MC-MFSK
system in Figure 2.2. Each row corresponds to one of the possible symbols. The
number of occupied entries in a row reflects the correlation of that symbol.

For a non-fading AWGN channel with high SNR, we assume that each envelop
detector makes its decision based on the received tone without error. Hence, the
desired symbol always has a complete row filled by its L transmitted tones, while
interference from other users and self-interference will scatter and occupy entries in
other rows. Decoding error occurs when the interfering tones fill up the row of any
erroneous symbol, and the erroneous symbol is selected in the random choice.

2.3

System Capacity and Normalized Throughput

We consider system capacity as the amount of useful information that can be
transmitted through the system of symbol error rate, Pe. The diversity order L and
modulation level M have significant effect on the error performance of the MC-MFSK

system, and hence on the system capacity and throughput. Thus we formulate the
11


system capacity and normalized throughput for the MC-MFSK system to show the
relationship with the system parameters. The MC-MFSK system capacity C is similar
to the capacity of the multiple-access FH-MFSK system, which is given in [17] as
C = Pe ln (Pe ) + (1 − Pe ) ln (1 − Pe ) + ln M − Pe ln (M − 1) ,

(2.2)

in nats per channel use.

We use normalized system throughput as our performance measure, which is defined
as

W =

KC
,
B Ts

(2.3)

where K, B and Ts denote respectively, the number of user, system bandwidth and
symbol duration. Since the MC-MFSK system has M sub-channels and frequency
separation of 1 is used to preserve the orthogonality of each sub-channel, the
Ts

bandwidth is therefore equal to B = M ×

W=

1
. We can then simplify (2.3) into
Ts

KC
.
M

(2.4)

In the next chapter, we will derive the optimal system parameters for MC-MFSK
systems that will maximize the throughput. We will consider the maximization of
throughput for systems subjected to error probability and user number constraints. We
also derive the optimal diversity which maximizes the user capacity of MC-MFSK
systems with error probability constraint.

12


Chapter 3
Optimal Diversity Order of Multiple Access Multi-Carrier
MFSK Systems

3.1

Introduction

In MC-MFSK systems, the number of frequency-carriers used per symbol
transmission is termed the frequency diversity order L. For a given total frequency
bandwidth, the diversity order is directly related to the amount of multiple-access
interference (MAI) experienced by all users. Hence a trade-off exists between the
diversity gain and MAI. Another parameter of interest is the modulation level M.
Similar to conventional MFSK systems, the value of M refers to the alphabet size, and
also the number of orthogonal sub-channels. We use these two parameters to optimize
the MC-MFSK system performance.

Previous analyses [1-3] have evaluated the analytical error probability for MC-MFSK
systems in Rayleigh and Rician channels. Computational results show that there exists
an optimal diversity order. However, an analytical evaluation of this optimal diversity
order has not been made, plausibly due to the complexity of the evaluations.

The objective of this chapter is to work out an analytical solution for the optimal
diversity order that maximizes throughput and minimizes error probability. We
approach the problem by re-evaluating the system for a non-fading AWGN channel
with high SNR, focusing only on the diversity gain and MAI trade-off. A simpler

13