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Nghiên cứu kỹ thuật điều chế chỉ số lặp lại cho các hệ thống OFDM

MINISTRY OF EDUCATION & TRAINING

MINISTRY OF NATIONAL DEFENSE

MILITARY TECHNICAL ACADEMY

LE THI THANH HUYEN

REPEATED INDEX MODULATION
FOR OFDM SYSTEMS

A Thesis for the Degree of Doctor of Philosophy

HA NOI - 2020


MINISTRY OF EDUCATION & TRAINING

MINISTRY OF NATIONAL DEFENSE

MILITARY TECHNICAL ACADEMY


LE THI THANH HUYEN

REPEATED INDEX MODULATION
FOR OFDM SYSTEMS

A Thesis for the Degree of Doctor of Philosophy

Specialization: Electronic Engineering
Specialization code: 9 52 02 03

SUPERVISOR
Prof. TRAN XUAN NAM

HA NOI - 2020


ASSURANCE

I hereby declare that this thesis was carried out by myself under the
guidance of my supervisor. The presented results and data in the thesis are reliable and have not been published anywhere in the form of
books, monographs or articles. The references in the thesis are cited in
accordance with the university’s regulations.
Hanoi, May 17th, 2019
Author

Le Thi Thanh Huyen


ACKNOWLEDGEMENTS

It is a pleasure to take this opportunity to send my very great appreciation to those who made this thesis possible with their supports.
First, I would like to express my deep gratitude to my supervisor,
Prof. Tran Xuan Nam, for his guidance, encouragement and meaningful
critiques during my researching process. This thesis would not have been
completed without him.
My special thanks are sent to my lecturers in Faculty of Radio - Electronics, especially my lecturers and colleagues in Department of Communications who share a variety of difficulties for me to have more time
to concentrate on researching. I also would like to sincerely thank my
research group for sharing their knowledge and valuable assistance.
Finally, my gratitude is for my family members who support my studies with strong encouragement and sympathy. Especially, my deepest


love is for my mother and two little sons who always are my endless
inspiration and motivation for me to overcome all obstacles.
Author

Le Thi Thanh Huyen


TABLE OF CONTENTS

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iv

List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

List of tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Chapter 1. RESEARCH BACKGROUND . . . . . . . . . . . . . . .

8

1.1. Basic principle of IM-OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.1.1. IM-OFDM model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.1.2. Sub-carrier mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

1.1.3. IM-OFDM signal detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

1.1.4. Advantages and disadvantages of IM-OFDM . . . . . . . . . . . .

16

1.2. Related works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

1.3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

Chapter 2. REPEATED INDEX MODULATION FOR OFDM
WITH DIVERSITY RECEPTION . . . . . . . . . . . . . . . . . . . . . .

24

2.1. RIM-OFDM with diversity reception model . . . . . . . . . . . . . . . .

24

2.2. Performance analysis of RIM-OFDM-MRC/SC under perfect CSI
28
2.2.1. Performance analysis for RIM-OFDM-MRC . . . . . . . . . . . .
i

29


2.2.2. Performance analysis for RIM-OFDM-SC . . . . . . . . . . . . . . .

34

2.3. Performance analysis of RIM-OFDM-MRC/SC under imperfect
CSI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

2.3.1. Performance analysis for RIM-OFDM-MRC . . . . . . . . . . . .

35

2.3.2. Performance analysis for RIM-OFDM-SC . . . . . . . . . . . . . . .

40

2.4. Performance evaluation and discussion . . . . . . . . . . . . . . . . . . . . .

41

2.4.1. Performance evaluation under perfect CSI . . . . . . . . . . . . . .

41

2.4.2. SEP performance evaluation under imperfect CSI condition .
48
2.4.3. Comparison of the computational complexity . . . . . . . . . . .

49

2.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

Chapter 3. REPEATED INDEX MODULATION FOR OFDM
WITH COORDINATE INTERLEAVING . . . . . . . . . . . . . . .

51

3.1. RIM-OFDM-CI system model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

3.2. Performance analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

3.2.1. Symbol error probability derivation . . . . . . . . . . . . . . . . . . . . .

56

3.2.2. Asymptotic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

3.2.3. Optimization of rotation angle . . . . . . . . . . . . . . . . . . . . . . . . . .

60

3.3. Low-complexity detectors for RIM-OFDM-CI. . . . . . . . . . . . . . .

62

3.3.1. Low-complexity ML detector . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

3.3.2. LLR detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

3.3.3. GD detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

3.4. Complexity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

3.5. Performance evaluations and discussion. . . . . . . . . . . . . . . . . . . . .

69

ii


3.6. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

CONCLUSIONS AND FUTURE WORK . . . . . . . . . . . . . . .

76

PUBLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

iii


LIST OF ABBREVIATIONS

Abbreviation

Definition

AWGN

Additive White Gaussian Noise

BEP

Bit Error Probability

BER

Bit Error Rate

CI

Coordinate Interleaving

CS

Compressed Sensing

CSI

Channel State Information

D2D

Device to Device

ESIM-OFDM

Enhanced Sub-carrier Index Modulation for Orthogonal Frequency Division Multiplexing

FBMC

Filter Bank Multi-Carrier

FFT

Fast Fourier Transform

GD

Greedy Detection

ICI

Inter-Channel Interference

IEP

Index Error Probability

IFFT

Inverse Fast Fourier Transform

IM

Index Modulation

IM-OFDM

Index Modulation for OFDM

iv


IM-OFDM-CI

Index Modulation for OFDM with Coordinate
Interleaving

IoT

Internet of Things

ISI

Inter-Symbol Interference

ITU

International Telecommunications Union

LowML

Low-complexity Maximum Likelihood

LLR

Log Likelihood Ratio

LUT

Look-up Table

M2M

Machine to Machine

Mbps

Megabit per second

MGF

Moment Generating Function

MIMO

Multiple Input Multiple Output

ML

Maximum Likelihood

MM-IM-OFDM

Multi-Mode IM-OFDM

MRC

Maximal Ratio Combining

NOMA

Non-Orthogonal Multiple Access

OFDM

Orthogonal Frequency Division Multiplexing

OFDM-GIM

OFDM with Generalized IM

OFDM-I/Q-IM

OFDM with In-phase and Quadrature Index
Modulation

OFDM-SS

OFDM Spread Spectrum

PAPR

Peak-to-Average Power Ratio

PEP

Pairwise Error Probability

PIEP

Pairwise Index Error Probability
v


PSK

Phase Shift Keying

QAM

Quadrature Amplitude Modulation

RIM-OFDM

Repeated Index Modulation for OFDM

RIM-OFDM-MRC

Repeated Index Modulation for OFDM with
Maximal Ratio Combining

RIM-OFDM-SC

Repeated Index Modulation for OFDM with Selection Combining

RIM-OFDM-CI

Repeated Index Modulation for OFDM with Coordinate Interleaving

SC

Selection Combining

SEP

Symbol Error Probability

SIMO

Single Input Multiple Output

S-IM-OFDM

Spread IM-OFDM

SNR

Signal to Noise Ratio

SM

Spatial Modulation

SS

Spread Spectrum

UWA

Underwater Acoustic

V2V

Vehicle to Vehicle

V2X

Vehicle to Everything

xG

x-th Generation

vi


LIST OF FIGURES

1.1

Block diagram of an IM-OFDM system. . . . . . . . . . . . 10

2.1

Structure of the RIM-OFDM-MRC/SC transceiver. . . . . . 25

2.2

The SEP comparison between RIM-OFDM-MRC and the
conventional IM-OFDM-MRC system when N = 4, K =
2, L = 2, M = {4, 8}. . . . . . . . . . . . . . . . . . . . . . . 42

2.3

The SEP performance of RIM-OFDM-SC in comparison
with IM-OFDM-SC for N = 4, K = 2, L = 2, M = {4, 8}. . 43

2.4

The relationship between the index error probability of
RIM-OFDM-MRC/SC and the modulation order M in
comparison with IM-OFDM-MRC/SC for N = 4, K = 2,
M = {2, 4, 8, 16}. . . . . . . . . . . . . . . . . . . . . . . . . 44

2.5

The impact of L on the SEP performance of RIM-OFDMMRC and RIM-OFDM-SC for M = 4, N = 4, K = 2 and
L = {1, 2, 4, 6}. . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.6

The SEP performance of RIM-OFDM-MRC under influence of K for M = {2, 4, 8, 16}, N = {5, 8}, K = {2, 3, 4, 5}.

2.7

46

The SEP performance of RIM-OFDM-SC under influence
of K when M = {2, 4, 8, 16}, N = {5, 8}, K = {2, 3, 4, 5}. . . 46

2.8

Influence of modulation size on the SEP of RIM-OFDMMRC/SC for N = 5, K = 4, and M = {2, 4, 8, 16, 32}. . . . . 47
vii


2.9

The SEP performance of RIM-OFDM-MRC in comparison with IM-OFDM-MRC under imperfect CSI when N =
2

4, K = 2, M = {4, 8}, and

= {0.01, 0.05}. . . . . . . . . . 48

2.10 The SEP performance of RIM-OFDM-SC in comparison
with IM-OFDM-SC under imperfect CSI when N = 4,
K = 2, M = {4, 8}, and

2

= 0.01. . . . . . . . . . . . . . . . 49

3.1

Block diagram of a typical RIM-OFDM-CI sub-block. . . . . 52

3.2

Rotated signal constellation. . . . . . . . . . . . . . . . . . . 60

3.3

Computational complexity comparison of LLR, GD, ML
and lowML detectors when a) N = 8, M = 16, K =
{1, 2, . . . , 7} and b) N = 8, K = 4, M = {2, 4, 8, 16, 32, 64}. . 68

3.4

Index error performance comparison of RIM-OFDM-CI,
IM-OFDM, IM-OFDM-CI and ReMO systems at the spectral efficiency (SE) of 1 bit/s/Hz, M = {2, 4}, N = 4,
K = {2, 3}. . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.5

SEP performance comparison between RIM-OFDM-CI,
IM-OFDM and CI-IM-OFDM using ML detection at the
spectral efficiency of 1 bit/s/Hz when M = {2, 4}, N = 4,
K = {2, 3}. . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.6

BER comparison between the proposed scheme and the
benchmark ones when N = 4, K = {2, 3}, M = {2, 4}. . . . 72

3.7

BER comparison between the proposed and benchmark
schemes at SE of 1.25 bits/s/Hz when N = {4, 8}, K =
{2, 4}, M = {2, 4, 8}. . . . . . . . . . . . . . . . . . . . . . . 73
viii


3.8

SEP performance of RIM-OFDM-CI and benchmark systems using different detectors. . . . . . . . . . . . . . . . . . 74

ix


LIST OF TABLES

1.1

An example of look-up table when N = 4, K = 2, p1 = 2 . . 13

2.1

Complexity comparison between the proposed schemes
and the benchmark. . . . . . . . . . . . . . . . . . . . . . . . 50

3.1

Example of LUT for N = 4, K = 2, pI = 2. . . . . . . . . . . 54

3.2

Complexity comparison between ML, LowML, LLR and
GD dectectors. . . . . . . . . . . . . . . . . . . . . . . . . . 68

x


LIST OF SYMBOLS

Symbol

Meaning

a

A complex number

aR

Real part of a

aI

Imaginary part of a

|a|

Modulus of a

a

A vector

A

A matrix

AH

The Hermitian transpose of A

AT

The transpose of A

c

Number of possible combinations of active indices

f (.)

Probability density function

G

Number of sub-blocks

K

Number of active sub-carriers

N

Number of sub-carriers in each sub-block

NF

Number of sub-carriers in IM-OFDM system

L

Number of receive antennas

P (.)

The probability of an event

PI

Index symbol error probability

PM

M -ary modulated symbol error probability
xi


Ps

Symbol error probability

Q (.)

The tail probability of the standard Gaussian
distribution

γ¯

Average SNR at each sub-carrier

I

Set of possible active sub-carrier indices

M (.)

The moment generating function.

S

Complex signal constellation



Rotated complex signal constellation

α

Index of an active sub-carrier
Channel estimation error variance

Θ

Big-Theta notation

φ

Rotation angle of signal constellation

φopt

Optimal rotation angle of signal constellation

.

2
F

Frobenius norm of a matrix

diag(.)

Diagonal matrix

C (N, K)

Binomial coefficient, C (N, K) =

x

N!
K!(N −K)!

Rounding down to the closest integer

log2 (.)

The base 2 logarithm

E {.}

Expectation operation.

xii


INTRODUCTION

Motivation
Wireless communication has been considered to be the fastest developing field of the communication industry. Through more than 30 years
of research and development, various generations of wireless communications have been born. The achievable data rate of wireless systems
has increased to several thousands of times higher (the fourth generation - 4G) than that of the second generation (2G) wireless systems.
Particularly, the 4G wireless communication systems, supported by key
technologies such as multiple-input multiple-output (MIMO), orthogonal
frequency division multiplexing (OFDM), cooperative communications,
have already achieved the data rate of hundreds Mbps [1].
The MIMO technique exploits the diversity of multiple transmit antennas and multiple receive antennas to enhance channel capacity without either increasing the transmit power or requiring more bandwidth.
Meanwhile, OFDM is known as an efficient multi-carrier transmission
technique which has high resistance to the multi-path fading.

The

OFDM system offers a variety of advantages such as inter-symbol interference (ISI) resistance, easy implementation by inverse fast Fourier
transform/fast Fourier transform (IFFT/FFT). It can also provide higher
spectral efficiency over the single carrier system since its orthogonal sub1


carriers overlap in the frequency domain.
Due to vast developments of smart terminals, new applications with
high-density usage, fast and continuous mobility such as cloud services,
machine-to-machine (M2M) communications, autonomous cars, smart
home, smart health care, Internet of Things (IoT), etc, the 5G system has promoted challenging researches in the wireless communication
community [2]. It is expected that ubiquitous communications between
anybody, anything at anytime with high data rate and transmission reliability, low latency are soon available [3]. Although there are several
5G trial systems installed worldwide, so far there have not been any
official standards released yet. The International Telecommunications
Union (ITU) has set 2020 as the deadline for the IMT-2020 standards.
According to a recent report of the ITU [3], 5G can provide data rate
significantly higher, about tens to hundreds of times faster than that of
4G. For latency issue, the response time to a request of 5G can reduce
to be about 1 millisecond compared to that around 120 milliseconds and
between roughly 15-60 milliseconds of 3G and 4G, respectively [3].
In order to achieve the above significant improvement, the 5G system
continues employing OFDM as one of the primary modulation technologies [2]. Meanwhile, based on OFDM, index modulation for OFDM
(IM-OFDM) has been proposed and emerged as a promising multicarrier transmission technique. IM-OFDM utilizes the indices of active
sub-carriers of OFDM systems to convey additional information bits.
There are several advantages over the conventional OFDM proved for
IM-OFDM such as the improved transmission reliability, energy effi2


ciency and the flexible trade-off between the error performance and the
spectral efficiency [4], [5]. However, in order to be accepted for possible
inclusion in the 5G standards and have a full understanding about the
IM-OFDM capability, more studies should be carried out.
Inspired by the motivation of OFDM in the framework of 5G and the
application potentials of IM-OFDM to the future commercial standards,
the present thesis has adopted IM-OFDM as the research theme for its
study with the title “Repeated index modulation for OFDM systems”.
Within the scope of the research topic, the thesis aims to conduct a
thorough study on the IM-OFDM system, and make its contributions to
enhance performances of this attractive system.
Research Objectives
Motivated by the application potentials of IM-OFDM and the fact
that its limitations, such as high computational complexity and limited
transmission reliability, which may prevent it from possible implementation, this research aims at proposing enhanced IM-OFDM systems to
tackle these problems. Moreover, a mathematical framework for the
performance analysis is also developed to evaluate the performance of
the proposed systems under various channel conditions. The specific
objectives of the thesis research can be summarized as follows:
• Upon studying the related IM-OFDM systems in the literature, efficient signal processing techniques such as repetition code and coordinate interleaving are proposed to employ in the considered systems.
• Efficient signal detectors for the IM-OFDM system, which can bal3


ance the error performance with computational complexity, are studied and proposed for the considered systems.
• Developing mathematical frameworks for performance analysis of
the proposed systems, which can give an insight into the system
behavior under the impacts of the system parameters.
Research areas
• Wireless communication systems under the impact of different fading conditions.
• Multi-carrier transmission using OFDM and index modulation.
• Detection theory and complexity analysis.
Research method
In this thesis, both the theoretical analysis and the Monte-Carlo simulation are used to evaluate the performance of the considered systems.
• The analytical methods are used for calculating the computational
complexity of the detection algorithms and to derive the closedform expressions for symbol error and bit error probabilities of the
proposed systems.
• The Monte-Carlo simulation is applied to validate the analytical
results and to make comparison between the performance of the
proposed systems and that of the benchmarks.
Thesis contribution
The major contributions of the thesis can be summarized as follows:
4


• Contributions to IM-OFDM with diversity reception
– Based on the concept of IM-OFDM with diversity reception [6],
an enhanced IM-OFDM system with spatial diversity using the
maximal ratio combination and selection combination (abv. as
RIM-OFDM-MRC and RIM-OFDM-SC, respectively) is proposed
to improve the error performance over the conventional IM-OFDM
system with diversity reception.
– The closed-form expressions for the index error probability (IEP)
and symbol error probability (SEP) of RIM-OFDM-MRC and
RIM-OFDM-SC under both perfect and imperfect channel state
information (CSI) conditions are derived to analyze the error
performance and the impacts of the system parameters on the
transmission reliability. Simulation results are also provided to
validate the theoretical analysis.
• Contributions to IM-OFDM with coordinate interleaving
– Based on the idea of IM-OFDM with coordinate interleaving
(IM-OFDM-CI) [7], an enhanced scheme of IM-OFDM, referred
to as repeated IM-OFDM-CI (RIM-OFDM-CI) is proposed to
improve the transmission reliability and flexibility of the conventional IM-OFDM-CI system. The closed-form expressions for
symbol and bit error probabilities of the proposed system are
also derived.
– Three low-complexity detectors for RIM-OFDM-CI, which can
significantly reduce the computational complexity while still achiev5


ing near-optimal and optimal system error performance of the
ML detector, are proposed.
Thesis structure
The thesis is organized in three chapters as follows:
• Chapter 1: Research background
This chapter introduces the research background of IM-OFDM and
related studies. Particularly, it presents a comprehensive review on
the recent studies of IM-OFDM and outlines several challenging open
problems which motivate the contributions of the thesis in the subsequent chapters.
• Chapter 2: Repeated IM-OFDM with diversity reception
This chapter proposes an enhanced IM-OFDM system with diversity reception using maximal ratio combination (RIM-OFDM-MRC)
and selection combination (RIM-OFDM-SC). Performance analysis
is carried out to determine the diversity and coding gains of the proposed system under both perfect and imperfect CSI conditions. Performance comparisons between the proposed system and the related
benchmark ones are provided using numerical and simulation results.
• Chapter 3: Repeated IM-OFDM with coordinate interleaving
In this chapter, a repeated IM-OFDM with coordinate interleaving (RIM-OFDM-CI) is proposed. Three low-complexity detectors,
namely low-complexity ML (lowML), log-likelihood ratio (LLR), and
greedy detection (GD) are presented for the RIM-OFDM-CI system
6


to relax the detection complexity. An optimal rotation angle for
the M -QAM modulation constellation is determined to improve the
error performance of the system. Numerical and simulation results
are provided to evaluate the RIM-OFDM-CI system performance of
against benchmark systems.

7


Chapter 1
RESEARCH BACKGROUND

This chapter provides research background for the present thesis. The
first section introduces the basic principle of IM-OFDM and outlines
the advantages and disadvantages of IM-OFDM over the conventional
OFDM. The next section offers a literature review of the related studies
on IM-OFDM and identifies the research scope for the present thesis.
1.1. Basic principle of IM-OFDM
Index modulation for OFDM is an OFDM-based transmission technique which utilizes the sub-carrier index to convey more data bits in
addition to the M -ary modulation. The idea of IM-OFDM is similar
to that of spatial modulation (SM) in which additional data bits can
be transmitted by means of indexing separate channels in either spatial or frequency domain using a portion of bits. The concept of IMOFDM was first introduced in [8] and then developed in [9]. In literature, it was referred to as different names such as sub-carrier index
modulation OFDM (SIM-OFDM) [8], OFDM with index modulation
(OFDM-IM) [9], multi-carrier index keying OFDM (MCIK-OFDM) [10],
OFDM with sub-carrier index modulation (OFDM-SIM) [11], selecting sub-carrier modulation (SSCM) [12], etc. Despite the variations in
names, their basic principles are the same. Throughout this thesis, for
8


the sake of consistence and except otherwise stated explicitly, the term
“index modulation for OFDM (IM-OFDM)” will be used instead.
Similar to SM [13], the incoming data bits in IM-OFDM are divided
into two parts. The first part is used to select the indices of active
sub-carriers, while the second part is fed to an M -ary mapper as in the
conventional OFDM system. However, it differs from the conventional
OFDM that IM-OFDM only activates a subset of sub-carriers, leaving
the remaining sub-carriers to be zero padded. Since the information
bits are transferred not only by the M -ary modulated symbols but also
by the indices of the active sub-carriers, IM-OFDM can attain better
transmission reliability and higher energy efficiency than that of the
conventional OFDM system [9].
1.1.1. IM-OFDM model
The block diagram of a typical IM-OFDM system is illustrated in
Fig. 1.1. The system consists of NF sub-carriers which are separated into
G sub-blocks, each with N sub-carriers. At the transmitter, a sequence
of incoming m bits is first separated into G groups of p bits. For the
g-th sub-block, the incoming p bits are then split into two bit sequences.
The first p1 = log2 (C (N, K)) bits are fed to a corresponding index
mapper to select K out of N sub-carriers, where N = {2, 4, 8, 16, 32, 64},
1 ≤ K ≤ N and C (N, K) =

N!
K!(N −K)!

is the binomial coefficient. Note

that when all the available sub-carriers are activated, i.e. K = N , IMOFDM becomes the conventional OFDM system. The set of active subcarrier indices in the g-th sub-block is denoted by θg = {α1 , . . . , αK },
9


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