STUDY ON ADVANCED MODULATION FORMATS IN ...Study on Advanced Modulation Formats in Coherent Optical...

STUDY ON ADVANCED MODULATION FORMATS IN

COHERENT OPTICAL COMMUNICATION SYSTEMS

ZHANG HONGYU

NATIONAL UNIVERSITY OF SINGAPORE

2012

STUDY ON ADVANCED MODULATION FORMATS IN

COHERENT OPTICAL COMMUNICATION SYSTEMS

ZHANG HONGYU

(B.Sc., Dalian University of Technology, China)

A THESIS SUBMITTED

FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2012

Study on Advanced Modulation Formats in Coherent Optical Communication Systems

- I -

Acknowledgement

First and Foremost, I would like to express my deepest gratitude to my supervisors Dr. Yu

Changyuan and Prof. Kam Pooi-Yuen for their invaluable guidance and kind support

throughout my Ph.D study. From them, I learnt not only knowledge, research skills and

experience, but also the right attitudes and wisdom towards research and life.

I would like to thank fellow researchers Dr. Chen Jian, Zhang Shaoliang, Zhang

Banghong, Cao Shengjiao, Xu Zhuoran, Wu Mingwei, Da Bin, Gao Zhi, and many others for

their help in my research.

I would like to thank lab officer Mr. Goh Thiam Pheng for providing a perfect working

environment for me.

I would like to thank my closest friends for their emotional support.

Last but not least, I would like to express my sincere appreciation to my parents and other

family members for their love and encouragement.


- II -

Contents

Acknowledgement……………………………………………………I

Contents……………………………………………………………..II

Abstract……………………………………………………………...V

List of Figures……………………………………………………..VII

List of Tables………………………………………………………XII

List of Abbreviations…………………………………………….XIII

1 Introduction………………………………………………………1

1.1 Background……………………………………………………..1

1.2 Literature Review…………..…………………………………...5

1.2.1 Coherent Detection Systems………………………………...5

1.2.2 Polarization Demultiplexing………………………………...9

1.2.3 Frequency Offset Estimation……………………………….10

1.2.4 Phase Estimation…………………………………………...11

1.2.4.1 Viterbi &Viterbi Mth-power Phase Estimation……………...11

1.2.4.2 Decision-Aided Maximum Likelihood Phase Estimation…...13

1.3 Objectives and Contribution of the Study……………………..18

1.4 Organization of the Thesis…………………………………….19

2 Performance Analysis of 4-Point Modulation Formats………21

2.1 BER Derivation of BPSK……………………………………..22


- III -

2.1.1 BER of BPSK in AWGN Channel…………………………22

2.1.2 BER of BPSK with a Phase Estimation Error……………...26

2.1.3 Approximate BER of BPSK………………………………..28

2.2 Performance Analysis of QPSK……………………………….31

2.2.1 Conditional SER and BER of QPSK………………………31

2.2.2 Approximate SER and BER of QPSK……………………..35

2.3 Performance Analysis of 4-PAM……………………………...36

2.3.1 Conditional SER and BER of 4-PAM……………………...36

2.3.2 Approximate SER and BER of 4-PAM……………………40

2.4 Performance Analysis of (1, 3)………………………………..41

2.4.1 Conditional SER and BER of (1, 3)………………………..41

2.4.2 Approximate SER and BER of (1, 3)………………………45

2.5 Results and Discussions……………………………………….45

2.6 Cross-over SNR between QPSK and (1, 3)…………………...50

2.7 Conclusions……………………………………………………54

3 Performance Analysis of 8-Point Modulation Formats………55

3.1 Performance Analysis of 8-PSK………………………………56

3.2 Performance Analysis of Rectangular 8-QAM………………..57

3.2.1 Conditional BER of Rectangular 8-QAM………………..58

3.2.2 Approximate BER of Rectangular 8-QAM………………66

3.3 Performance Analysis of 8-point Star QAM…………………..68

3.3.1 Analysis of 8-star QAM………………………………….68

3.3.1.1 BER of 8-star QAM………………………………………….69

3.3.1.2 Ring Ratio Optimization……………………………………..73

3.3.2 Analysis of Rotated 8-star QAM…………………………75


- IV -

3.3.2.1 BER of Rotated 8-star QAM…………………………………75

3.3.2.2 Ring Ratio Optimization……………………………………..80

3.4 Performance Analysis of Triangular 8-QAM…………………81

3.5 Results and Discussions……………………………………….85

3.6 Conclusions……………………………………………………95

4 Performance Analysis of Higher-order Modulation Formats..97

4.1 Analysis of 16-Star QAM……………………………………..98

4.1.1 Introduction of 16-star QAM………………………………98

4.1.2 Craig’s Method……………………………………………..99

4.1.3 Conditional BER of 16-star QAM………………………..101

4.1.4 Results and Discussions…………………………………..104

4.2 Analysis of 64-QAM…………………………………………108

4.2.1 Conditional BER of 64-QAM…………………………….108

4.2.2 Results and Discussions…………………………………..110

4.3 Conclusions…………………………………………………..113

5 Experiments……………………………………………………114

5.1 Experimental Setup…………………………………………..114

5.2 Results and Discussions……………………………………...118

5.3 Conclusions…………………………………………………..127

6 Conclusions and Future Work………………………………..128

6.1 Conclusions…………………………………………………..128

6.2 Future Work………………………………………………….130

Appendix…………………………………………………………..132

References…………………………………………………………134

Publication List……………………………………………………148


- V -

Abstract

Recently, advanced modulation formats with digital coherent detection have attracted

extensive attention in coherent optical communications, since they can achieve a high spectral

efficiency (SE). One major issue in coherent optical communication systems is to recover

carrier phase, which is perturbed by phase noise generated from the laser linewidths of both

the transmitter and local oscillator (LO). Digital-signal-processing (DSP) based phase

estimation (PE) algorithms, such as Mth power and Decision-aided (DA) maximum likelihood

(ML), are preferred at the receiver for carrier phase recovery. The PE algorithm has a residual

phase estimation error which degrades the system performance.

This thesis systemically studies the performance of different advanced modulation

formats (4-point, 8-point, and higher-order) in the presence of laser phase noise and additive

white Gaussian noise (AWGN), and experimentally verifies our analysis and simulations in

the coherent optical B2B systems.

First of all, the conditional bit-error rates (BER) of different modulation formats are

derived in the presence of a random phase estimation error and AWGN. Through a series of

approximations, simple and accurate approximate BERs are obtained, which allow quick

estimations of the BER performance and laser linewidth (LLW) tolerance.

In addition, these approximate BERs can also lead to some initial results as follows:


- VI -

(1) The cross-over signal-to-noise ratio (SNR) algorithm between QPSK and (1, 3), which

allows a quick and accurate estimation of the cross-over SNR point between QPSK and (1, 3)

under different phase estimation error variance;

(2) The ring ratio optimization algorithms of 8-star QAM and rotated 8-star QAM,

respectively. These algorithms can be used to quickly find out the optimum ring ratio under

different conditions, such as SNR/bit, LLW, and symbol rate;

(3) The reason why 8-star QAM has more phase noise tolerance than rotated 8-star QAM

is discussed based on the approximate BERs.

We also evaluate the BER performance and LLW tolerance for higher-order modulation

formats, such as 16-star QAM and 64-QAM. The optimum ring ratio and ring ratio fluctuation

penalty is numerically studied for 16-star QAM. Moreover, coherent optical B2B experiments

are conducted in this thesis to verify our analysis and simulations.

This study illustrates the procedure in detail how to carry out the analysis for advanced

modulation formats, and how to optimize the performance for two-dimensional constellations.

More importantly, the results in the thesis provide algorithms and comments which can be used

for the electrical engineers to choose the best modulation formats and optimum parameters of

each constellation under different conditions.


- VII -

List of Figures

1.1 Block diagram of a coherent homodyne receiver………………………………………...5

1.2 The structure of a butterfly FIR filter…………………………………………………….9

1.3 Mth-power processing circuit diagram………………………………………………….11

1.4 The block diagram of the DA ML PE method…………………………………………..14

1.5 Simulated BER performance of 20-Gb/s QPSK (10-Gsymbol/s) with Mth-power, DE

DA ML, and PA DA ML (linewidth per laser 100 kHz )……………………….. 17

1.6 Simulated BER performance of 30-Gb/s 8-PSK (10-Gsymbol/s) with Mth-power, DE

DA ML, and PA DA ML (linewidth per laser 100 kHz )……………………….. 17

2.1 Constellation map and decision boundary of BPSK…………………………………….22

2.2 Constellation map and decision boundary of BPSK with a phase estimation error ...27

2.3 Comparison between exact numerical BER, MC simulation, and approximate BER when

2 2 21 10 rad ..……………………………………………………………………... 31

2.4 Constellation map, decision boundaries and Gray code mapping of QPSK with a phase

estimation error ……………………………………………………………………….32

2.5 Constellation map, decision boundaries and Gray code mapping of 4-PAM with a phase

estimation error ……………………………………………………………………….37


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2.6 Illustration of MSB and LSB……………………………………………………………39

2.7 Constellation map, decision boundaries and bit mapping of (1, 3) with a phase estimation

error …………………………………………………………………………………...42

2.8 The BERs from analysis and MC simulations in perfectly coherent case………………46

2.9 The BERs from analysis and MC simulations when 2 2 210 .rad ……………...47

2.10 Comparison of BER approximations with exact results and MC simulations when

2 3 210 .rad ………………………………………………………………………..47

2.11 Comparison of SER approximations with exact results and MC simulations when

2 3 210 .rad ………………………………………………………………………..48

2.12 The SNR penalty @BER=10-4 as a function of phase estimation error variance for

exact and approximate BER expressions…………………………………………………49

2.13 The cross-over SNR algorithm (2.49) between QPSK and (1, 3) and its

approximation (2.50) as a function of phase estimation error variance…………………..53

3.1 Constellation map, decision boundaries and Gray code mapping of 8-PSK with a phase

error . …………………………………………………………………………………56

3.2 Constellation map, decision boundaries and Gray code bit mapping of rectangular

8-QAM……………………………………………………………………………………58

3.3 Illustration of how the triangular region CDE is formed………………………………..61

3.4 Plot of the ignored term 4, |triP A …………………………………………………65

3.5 The comparison of exact BER expression (3.13 and 3.20) and approximation (3.22) for

rectangular 8-QAM……………………………………………………………………….67


- IX -

3.6 Constellation map, decision boundaries and Gray code mapping of 8-star QAM with a

phase error . ……………………………………………………………………………69

3.7 Constellation maps and decision boundaries of rotated 8-star QAM with a phase

estimation error . ……………………………………………………………………..75

3.8 Constellation maps and decision boundaries of triangular 8-QAM with a phase

estimation error . ……………………………………………………………………..82

3.9 The BERs from analysis and MC simulations in perfectly coherent case………………86

3.10 The BERs from analysis and MC simulations when 2 3 28 10 .rad ……….87

3.11 Comparison between numerical BERs, MC simulations, and approximate BERs

when 51 10sT with optimum memory length………………………………….87

3.12 The SNR/bit penalty as a function of linewidth per laser symbol duration product

( sT ) when BER=10-4 from numerical integration and approximation in different

8-point modulation formats……………………………………………………………….89

3.13 Optimum ring ratio as a function of SNR/symbol phase estimation error variance

product for 8-star QAM…………………………………………………………………..91

3.14 BER as a function of ring ratio from numerical integration given 15 dBb

and 2 3 21 10 rad for 8-star QAM and rotated 8-star QAM……………………...92

3.15 SNR penalties due to ring ratio shift for 8-star QAM and rotated 8-star QAM…….92

4.1 Constellation map and decision boundaries of 16-star QAM with a phase estimation

error ………………………………………………………………………………….98

4.2 One signal point and one of its decision regions………………………………………..99


- X -

4.3 The BERs of 16-star QAM from analysis and MC simulations before and after

optimization when 2 3 21 10 rad …………………………………………………..104

4.4 BER as a function of ring ratio with different 2 ................................................105

4.5 Optimum ring ratio versus different linewidth per laser symbol duration product........106

4.6 SNR penalty due to ring ratio fluctuation.......................................................................106

4.7 The SNR penalty as a function of linewidth per laser symbol duration product………107

4.8 Constellation map, decision boundaries, and Gray code mapping of 64-QAM with a

phase error …………………………………………………………………………108

4.9 Details in the decision region of signal S0……………………………………………..109

4.10 The BERs of 64-QAM and 16-PSK from analysis and MC simulations. (a)

2 41 10 rad2, (b) 2 46 10

rad2……………………………………………….110

4.11 The SNR penalty as a function of linewidth per laser symbol duration product

( sT ) when BER= 410 with optimum memory length……………………………..111

5.1 Optical transmitter for NRZ and RZ QPSK……………………………………………115

5.2 Experimental setups for optical RZ 8-point star QAMs (8-star QAM and rotated 8-star

QAM).…………………………………………………………………………………...116

5.3 Experimental setup for optical RZ 8-PSK ………………….........................................117

5.4 BER performance comparison between experimental and theoretical BERs (2.23b) for

RZ QPSK………………………………………………………………………………..118

5.5 Optimum ring ratios as a function of OSNR at LLW per laser of 100 kHz…………...120

5.6 Eye diagrams of signals at points a and b in figure 5.2 through direct detection. (a) RZ

QPSK (b) RZ 8-star QAM………………………………………………………………121


- XI -

5.7 BER performance comparison of experimental RZ 8-star QAM signal and theoretical

formula (3.34) under different values of ring ratio at LLW=100 kHz per laser………...121

5.8 BER performance comparison of experimental RZ rotated 8-star QAM signal and

theoretical formula (3.47) under different values of ring ratio at LLW=100 kHz per

laser ……………………………………………………………………………………..122

5.9 Constellation maps of (a) ring ratio=2.4 at OSNR=17 dB, (b) ring ratio=2.6 at OSNR=18

dB, and (c) ring ratio=2.0 at OSNR=19 dB……………………………………………..123

5.10 BER performance comparison of experimental RZ 8-star QAM signal and theoretical

analysis under different LLWs…………………………………………………………..124

5.11 Recovered constellation map of 8-star QAM ………………………………….….125

5.12 BER performance comparison of experimental RZ rotated 8-star QAM signal and

theoretical analysis under different LLWs………………………………………………125

5.13 Recovered constellation map of rotated 8-star QAM……………………………...126

5.14 BER performance comparison between experimental and theoretical BER (3.3) for

8-PSK……………………………………………………………………………………126

6.1 Constellation map and circular decision boundaries of (1, 3)…………………………131


- XII -

List of Tables

3.1 The LLW tolerance comparison between 8-point modulation formats ………………...89

4.1 Parameter details of 16-star QAM with a phase estimation error……………………...101

4.2 Parameter details of 16-star QAM in perfectly coherent case…………………………103

4.3 Optimum ring ratio in different range of sT ...........................................................105


- XIII -

List of Abbreviations

ADC Analog-to-Digital Converter

AM Amplitude Modulator

ASK Amplitude-Shift Keying

AWG Arbitrary Waveform Generator

AWGN Additive White Gaussian Noise

B2B Back-to-Back

BER Bit-Error Rate

CD Chromatic Dispersion

CMA Constant Modulus Algorithm

DA ML Decision-Aided Maximum Likelihood

DD Differential Decoding

DE Differential Encoding

DFB Distributed Feedback

DPSK Differential Phase-Shift Keying

DSP Digital Signal Processing

ECL External Cavity Laser

EDFA Erbium-Doped Fiber Amplifier


- XIV -

FEC Forward Error Control

FIR Finite Impulse Response

FOE Frequency Offset Estimation

IF Intermediate Frequency

IM/DD Intensity Modulation/Direct Detection

i.i.d independent identically distributed

IIR Infinite Impulse Response

IQ In-phase and Quadrature-phase

ISI Inter-symbol Interference

LD Laser Diode

LLW Laser Linewidth

LO Local Oscillator

LPF Low-Pass Filter

LSB Least Significant Bit

MB Middle Bit

MC Monte Carlo

MSB Most Significant Bit

MZM Mach-Zehnder Modulators

NRZ Non Return-to-Zero

OSNR Optical Signal-to-Noise Ratio

PA Pilot-Assisted

PAM Pulse Amplitude Modulation

PDF Probability Density Function


- XV -

PE Phase Estimation

PLL Phase-Locked Loop

PM Phase Modulator

PMD Polarization Mode Dispersion

PPG Pulse Pattern Generator

PRBS Pseudo-Random Bit Sequences

PSK Phase-shift Keying

QAM Quadrature Amplitude Modulation

RP Reference Phasor

RZ Return-to-Zero

SE Spectral Efficiency

SER Symbol-Error Rate

SNR Signal-to-Noise Ratio

SPM Self-Phase Modulation

V&V Viterbi & Viterbi

WDM Wavelength Division Multiplexing


–1–

Chapter 1

Introduction

1.1 Background

Optical communication systems refer to communication systems that use carriers of high

frequencies (~ 193.1 THz) and employ optical fibers for information transmission. Typically,

depending on the detection methods, optical communication systems can be classified into

two categories: direct detection and coherent detection. Intensity modulation with direct

detection (IM/DD) (Note that the acronym DD represents direct detection only if IM is in

front, otherwise, DD represents differential decoding) has been mainly used by researchers in

the early days of optical communications as it is a simple and cost effective scheme. However,

theoretically binary single-polarized IM/DD systems can only achieve a spectral efficiency

(SE) of 1 bit/s/Hz/polarization at most [1]. Besides, frequency and phase information cannot

be preserved when we use direct detection. As the increasing demand of high SE [2],

researchers began to employ some alternative modulation schemes, which transmit

information by modulating the phase of the optical carrier. Well-known modulation schemes


–2–

are such as phase-shift keying (PSK) (which only modulate the phase of the optical carrier),

and quadrature amplitude modulation (QAM) (which simultaneously modulate both the

amplitude and phase of the optical carrier). By using this kind of modulation formats, SE can

be raised up to more than 1 bit/s/Hz/polarization [3].

Therefore, during the 1980s, coherent optical transmission techniques attracted extensive

attention. There are two motivations behind using coherent optical systems. First, compared

with IM/DD systems, the shot noise limited receiver sensitivity of coherent communication

techniques can be improved by up to 20 dB, thus greatly extending the unrepeated

transmission distance [1, 4]. Second, spectral efficiency of coherent systems can also be

improved by the use of advanced modulation formats. During those years, experimental

optical phase-lock loop (PLL) was used to lock the phase of the local oscillator (LO) laser to

that of the incoming signal. This scheme is referred to as homodyne detection. Homodyne

detection based on an optical PLL can directly demodulate the incoming optical signal to

baseband stage, but it was difficult and unstable to implement and operate at optical domain

[5, 6]. Hence, researchers turned to use heterodyne detection in order to simplify the design of

the receiver [7]. The heterodyne receivers firstly down-convert the incoming optical signal to

an intermediate frequency (IF) which is in microwave region, and then an electrical PLL is

employed to lock the phase of the intermediate frequency signal [8, 9]. But the IF must be

much higher than the signal bit rate. Due to the large laser linewidth (LLW) and PLL

feedback delay, most early works only focused on some simple modulation formats, such as

binary phase-shift keying (BPSK) and differential phase-shift keying (DPSK) [10, 11].

However, the R&D activities of coherent systems were interrupted for almost twenty

years owing to the invention of erbium-doped fiber amplifiers (EDFA) and


–3–

wavelength-division multiplexing (WDM) techniques in the 1990s. Hence, the R&D interest

switched back to IM/DD systems from the year 1990 to around 2002. EDFA can be used to

increase the transmission distance [12], and WDM techniques can be employed to improve

the system capacity [13].

Since around 2002, coherent optical communication technologies, especially homodyne

detection systems, have been revived, due to the advent of high-speed analog-to-digital

converters (ADCs) [14-16]. Coherent detection schemes with the aid of the high-speed ADCs

are advantageous in that they provide the possibility to fully retrieve the amplitude and phase

information of an optical signal, and also to post-compensate for the chromatic dispersion

(CD), polarization-mode dispersion (PMD) and nonlinearity by using digital signal processing

(DSP) algorithms [17-24]. Because of many advantages of employing high-speed ADCs,

nowadays, the motivations in coherent communication systems are not only to improve the

receiver sensitivity, but also to improve the SE with the aid of high-order modulation formats

[25, 26]. Compared to the binary single-polarized IM/DD systems, whose SE is limited to 1

bit/s/Hz/polarization, advanced modulation formats with each symbol encoding m bits can

achieve an SE up to m bits/s/Hz/polarization. Moreover, coherent optical systems using

advanced modulation formats have higher tolerance to CD and PMD because they can reduce

symbol rate while keeping the same bit rate. The most commonly used multi-level modulation

formats in coherent communication systems are QPSK ( m = 2), 8-PSK, 8-QAM ( m = 3),

16-QAM ( m = 4) and 64-QAM ( m = 6). However, few studies have focused on analyzing the

performance of different advanced modulation formats in systems which include not only

AWGN noise but also linear phase noises.


–4–

Generally, with the aids of ADCs and DSP, it is preferred to use homodyne detection

(phase-diversity coherent receivers) rather than heterodyne detection to down-convert the

optical signals near the baseband [27]. This scheme allows for a free-running LO laser,

therefore, optical PLL is not needed to lock the phase of LO laser to the carrier phase. Some

literatures also refer to this scheme as digital coherent receiver [26]. The structure and theory

of digital coherent receiver are mainly discussed in [28-30]. After the digital coherent receiver,

ADCs are used to sample the electrical signals. Then several DSP algorithms can be

employed in electrical domain to recover the original signals instead of using bulky and costly

optical components [31-35]. The DSP algorithms can be used to compensate linear and

nonlinear distortions. Firstly, CD and PMD are needed to be compensated by using finite

impulse response (FIR) or infinite impulse response (IIR) filters [36-40]. After the CD and

PMD compensation, inter-symbol interference (ISI) between two polarization states has to be

removed with the aid of a butterfly-like FIR filter with the blind constant modulus algorithm

(CMA) [41-45]. Before the signals are fed into the symbol detector, frequency offset

estimation (FOE) and phase estimation (PE) are also needed to be carried out to recover the

carrier phase. The conventional FOE based on Mth-power operations is studied in [46-48].

Decision feedback method is also used in FOE to enlarge the estimation range [49, 50]. A

feed-forward FOE based on Gardner timing recovery algorithm [51] is investigated in [52]

which can be implemented in a real-time receiver. Moreover, the conventional phase

estimation method is Viterbi & Viterbi (V&V) Mth-power method [53]. A modified

Mth-power PE method can also be applied to QAM systems, but suffers from system

performance degradation [54-56]. However, V&V Mth-power method requires phase

unwrapping to deal with phase ambiguity, and also requires many nonlinear computations [18,


–5–

57]. Recently, researchers start to have interests on a computationally efficient decision-aided

(DA) maximum likelihood (ML) PE method [58, 59]. Adaptive version of this DA ML PE

method is also studied in [60, 61] which allows adaptively selection of the memory length.

In the following section, we will review in details the coherent detection systems and also

the DSP algorithms in the coherent receiver.

1.2 Literature Review

The first part of this section reviews the concept and implementation of coherent detection

systems. Several commonly-used DSP algorithms in the receiver are reviewed in the other

parts.

1.2.1 Coherent Detection Systems

Coherent optical communication systems started to attract extensive attention in the 1980s.

Since that time, the homodyne receiver is mainly used in coherent optical systems. Next we

will focus on discussing the concept and implementation of a homodyne receiver. The block

diagram of a single-polarization homodyne receiver is shown in figure 1.1.

90

PD1

PD2

PD3

PD4

LO

Figure 1.1 Block diagram of a coherent homodyne receiver.


–6–

Basically, in any coherent detection systems, a local oscillator is employed as a phase

reference to beat with the received optical signal for the purpose of demodulation. Normally,

an optical phase-diversity homodyne receiver, which composes of four 3-dB couplers and one

90 phase shifter, is used to introduce a 90 hybrid. In such a receiver (see figure 1.1), the

received optical signal mixes with four states of LO reference signals in the 90 hybrid in

order to create the in-phase and quadrature-phase (IQ) components of the signal [28]. Then

the four different light signals are delivered into two pairs of balanced photodetectors. Note

that this coherent homodyne receiver can be applied to any advanced modulation formats

[26].

The transmitted optical signal model can be assumed as

exp .s s c s sE t P j t j t j t (1.1)

Here, sP is the power of the received signal, ( )s t is the phase modulation of

corresponding signal, c is the angular carrier frequency of the signal, and s t is the

phase noise from the transmitter laser. Similarly, the electromagnetic field of the LO output

can be written as

exp ,LO LO LO LOE t P j t j t (1.2)

where LOP is the power of the LO, LO is the angular carrier frequency of the LO output,

and LO t is the phase noise from the LO laser.

The four inputs to the photodetectors can be represented as

1exp 1

2 2s LOE E j n

, with n=1, 2, 3, and 4. Here, n denotes the order of the


–7–

photodetectors as shown in figure 1.1. After the two balanced photodetectors, the IQ

components of the photocurrents can be expressed as

1cos ( ) ,s LO c LO s s LO shotI t R P P j t j t j t t n t (1.3a)

2sin ( ) ,s LO c LO s s LO shotQ t R P P j t j t j t t n t (1.3b)

where R represents the responsitivity of the photodetector, 1shotn t and 2shotn t are

the shot noise from the photodetector [1]. Note that 1shotn t and 2shotn t are

independent. Here, in equation (1.3), when LO frequency coincides with the signal carrier

frequency ( c LO ), the receiver is a homodyne receiver; on the other hand, when

c LO , the receiver is a heterodyne receiver. Homodyne receiver requires much less

bandwidth of the electrical low-pass filter (LPF) than the heterodyne receiver does [62], thus

makes it commonly used in coherent optical communication systems. Therefore, the received

base-band signal model in coherent homodyne receiver can be obtained by combining the IQ

components (1.3) together, and is given by

exp ( ) .s LO s s LO shot

r t I t j Q t

R P P j t j t t n t

(1.4)

It can be observed from equation (1.4) that the transmitted signals are rotated by a phase

noise s LOt t t from both the transmitter and LO lasers. Hence,

s LOt t t is also called laser phase noise. After sampling the signals by ADCs,

the sampled waveform can be expressed as [6]

exp ( ) 2 .

s s

s

r k I kT j Q kT

A j k jk f j k n k

(1.5)


–8–

Here, k represents the kth sample over the time interval , ( 1)s skT k T ( 1s sT R is the

symbol duration, and sR is the symbol rate), ( )r k is the received signal, s LOA R P P is

the amplitude, k is the laser phase noise, and n k is the shot noise. Note that the

frequency offset f between transmitter and LO lasers is considered here.

The laser phase noise stems from the LLW of both the transmitter and LO. It can be

modeled as a random walk process (Wiener process), which is described as [6]

( ) ( ),k

m

k m

(1.6)

where ( )m are independent identically distributed (i.i.d) Gaussian random variables with

distribution 20, pN . And the laser phase noise variance 2p in one symbol period sT

can be expressed as [63]

2 2 (2 ) .p sT (1.7)

Here, 2 means the sum of both LLWs of the transmitter and LO. It can be observed from

(1.6) that laser phase noise at current moment k is an accumulation of all previous phase

noise from the beginning until k . Therefore, the laser phase noise keeps rotating the phase of

the received signals, thus leading to performance degradation in coherent detection. Note that

laser phase noise is also named as linear phase noise. Besides linear phase noise, nonlinear

phase noise exists in long-haul transmission systems due to the self-phase modulation (SPM)

effect in the optical fiber [64-67]. Note that transmission is not taken into account in our

analysis and experiments. Furthermore, the shot noise n k can be assumed as an additive

white Gaussian noise (AWGN) with mean 0 and variance 20 2n N , where 0N is the


–9–

spectral height of the AWGN [68].

Now, we consider equation (1.5). In order to retrieve the phase modulation information

s k , we have to remove the frequency offset term f and phase noise term k . In

next sections, we will review how to do polarization demultiplexing, frequency offset

estimation, and phase estimation by using DSP algorithms.

1.2.2 Polarization Demultiplexing

Before the sampled received signals are sent to do frequency offset compensation and carrier

phase recovery, they need to remove the ISI between two polarization states. Note that CD is

assumed to be already compensated at this stage. In a polarization-diversity receiver, an

4M -tap butterfly FIR filter is used, and the structure of the butterfly FIR filter is shown in

figure 1.2 [41].

xxW

yxW

xyW

yyW

inX

inY

outX

outY

Figure 1.2 The structure of a butterfly FIR filter.

The 2M-dimensional input vector is X ,Yin inIn k k . Here, X and Y represent the two

polarization states. The output of the filter is described as

, where , ,Tout x x xx yxX k W In W W k W k (1.8a)


–10–

, where , .Tout y y xy yyY k W In W W k W k (1.8b)

Here, xxW k , yxW k , xyW k , and yyW k are 1M tap weight vectors of the filter,

T denotes matrix transpose, and k represents the value at time skT . Note that outX k

and outY k are both scalars.

Normally, the blind constant modulus algorithm (CMA) is used to adjust the values of the

weight vectors [42]. The weight vectors are adjusted by using

1 X ,xx xx x out inW k W k X k k (1.9a)

1 Y ,yx yx x out inW k W k X k k (1.9b)

1 X ,xy xy y out inW k W k Y k k (1.9c)

1 Y .yy yy y out inW k W k Y k k (1.9d)

Here, is the step size, 21x outX k , and 2

1y outY k .

We should notice that for coherent optical binary PSK (BPSK) systems, the tap weight

adaptation is different [69, 70]. The tap coefficients are adaptively adjusted as [71]

2 2 1 2 * *

*2 2 1 2 *

1 X

1 ,

X 1

j f k kout in

xx xxj f k k

out in

e b b X k k

W k W k

e b bX k k

(1.10)

where exp 2 1b j f k k , and * denotes complex conjugation. The other

tap coefficients adaptation terms are similar to (1.10).

1.2.3 Frequency Offset Estimation


–11–

After dealing with the cross-talk between two polarizations, we need to compensate frequency

offset and recover carrier phase. Consider FOE first. Now, (1.5) is the signal model.

Because of laser fabrication and heating issues, carrier frequencies between transmitter

and LO lasers are always mismatched [49]. A widely-used FOE method is implemented in

frequency domain. The estimated frequency offset can be written as [46]

1

1 2 0

1 1ˆ arg max exp 2 .N

LM

Nf n

f r n n j fM L

(1.11)

Here, in equation (1.11), M represents the number of different phases, and signals r n are

raised to the Mth power in order to remove the phase modulation s . L is the number of

symbols used to estimate the frequency offset, and Nf are the normalized frequency offsets.

Then we should use (1.11) to compensate the frequency offset.

1.2.4 Phase Estimation

After frequency offset compensation, the carrier phase which is distorted by laser phase noise

needs to be recovered. In this section, we will review the conventional V&V Mth-power PE

method as well as DA ML PE method.

1.2.4.1 Viterbi & Viterbi Mth-power Phase Estimation

k N

k N

M

r k

M

ˆ k

s k k

Figure 1.3 Mth-power processing circuit diagram.


–12–

After FOE, equation (1.5) now becomes like

exp ( ) .sr k A j k j k n k (1.12)

In order to retrieve the phase modulation information s k , phase noise k needs to be

estimated and removed. A well-known procedure is V&V Mth-power PE method, which

raises the received signals to the Mth-power for carrier phase recovery [53]. The structure of

Mth-power PE is shown in figure 1.3. At the lower branch, we take Mth-power of the received

signals r k , and (1.12) can be written as

exp ( ) exp .M M Msr k A jM k jM k A jM k (1.13)

This is the scenario for M-PSK signals. Note that additive noise n k is ignored in (1.13)

based on the principle of PEs. It can be observed that after Mth-power, phase modulation

s k is removed from r k . Then we average r k over (2N+1) samples from

si k N T to sk N T in order to estimate the carrier phase at time skT . The

estimated carrier phase at skT in Mth-power PE is given by

1ˆ arctan .s

s

k N TM

i k N T

k r iM

(1.14)

Subtracting the estimated phase reference ˆ k from arg r k , we can obtain the phase

modulation s k .

It should be noted that owing to the nature of arctan operation, the estimated phase

reference is always within the range between M and exhibit a phase ambiguity of

2 M . Differential encoding (DE) and differential decoding (DD) can be used to avoid


–13–

phase ambiguity. A phase jump of 2 M occurs only when the trajectory of the phase

noise exceeds the range between M [57]. Hence, in order to eliminate the phase jump,

the estimated phase reference ˆ k has to be compared with the previous one ˆ 1k .

This process is called phase unwrapping, and the procedure of phase unwrapping for M-PSK

is described as [18]

ˆ ˆ ˆ if 1

2ˆ ˆ ˆ ˆ if 1

2ˆ ˆ ˆ if 1

k k kM M

k k k kM M

k k kM M

(1.15)

The M-power PE algorithm can also be extended to QAM systems. However, the

performance will be degraded due to the fact that only a subgroup of symbols is used for

phase estimation [56].

1.2.4.2 Decision-Aided Maximum Likelihood Phase Estimation

Based on the review from previous section, it can be observed that V&V M-power PE

algorithm has some limitations, i.e., it requires phase unwrapping to deal with phase

ambiguity, and also requires many nonlinear computations (such as M and arctan

operations). To solve these issues, a computationally linear and efficient DA ML PE

algorithm is proposed in [58].

The DA ML PE scheme forms a reference phasor (RP) as [59]

1

1 *ˆ( ) ( ) ( ) ( ),k

l k L

V k U k r l m l

(1.16)


–14–

where ( )r l is the l th received signal, *ˆ ( )m l is the receiver’s decision on the l th received

symbol, superscript * denotes complex conjugation, L is the memory length (number of past

symbols from sk L T to 1 sk T used to estimate the carrier phase at current time point

skT ), and 1 2ˆ( ) | ( ) |

k

l k LU k m l

is a normalization factor. The received signal ( )r k is

rotated by *( )V k before being fed into the symbol detector. The structure of DA ML PE

method is summarized in figure 1.4. Here, in the figure, iC represents the signal

constellation points, and Re x denotes the real part of x.

11 *ˆ( ) ( ) ( ) ( )

k

l k L

V k U k r l m l

( )V k

Symbol DetectorˆSet ( ) ifdm k C

* *

2* *

max Re , for -ary PSK

max Re 2 , for -QAM

ii

d

i ii

r k V k C MC

r k V k C C M

ˆ ( )m k

( )r k RP EstimatorV k

Figure 1.4 The block diagram of the DA ML PE method.

Since any PE algorithm is imperfect, there will be a residual phase estimation error

which degrades the system performance. The residual phase estimation error k is

given by [72]

ˆ ,k k k (1.17)

where k and ˆ k are phase noise and estimated phase reference at time point k,

respectively. Here, in DA ML PE, ˆ arg ( )k V k . It is assumed in [59] that for DA ML PE,


–15–

the distribution of the phase error k is Gaussian 2(0, ).N The phase error variance

2 is given in terms of the laser phase noise variance 2

p (1.7), signal-to-noise ratio

(SNR) per symbol ,s and memory length L as [59]

2

2 2 2'

2 3 1 1,

6 2p nL L

L L

(1.18)

where 2' 1 .n s The accuracy of this expression is verified by using Monte Carlo (MC)

simulations in [59]. It is suggested that when additive noise is dominant, longer memory

length L should be used; on the other hand, when phase noise is dominant, shorter L should be

employed. The optimum L is given as

2 2'0.25 1 24 0.75 ,opt n pL

(1.19)

where x represents the largest integer less than or equal to .x The phase estimation error

variance 2 in Mth-power PE is also discussed for QPSK and can be expressed, as

described in [72], as

22

2 2 2''

1 4.51.

6 2n

p nL

L L

(1.20)

The feedback property of DA ML PE would cause error propagation if there are decision

feedback errors [73]. To avoid this problem, either differential encoding (DE) [74] or periodic

pilots [73] can be used. DE leads to approximate doubling of the BER compared with non-DE

case [68]. On the other hand, pilots take up a bit extra power depending on how many pilots

are used. Usually, DE and differential decoding (DD) are used in experiments to prevent error

propagation because of their simplicity [24].


–16–

MC simulations are conducted between Mth-power, DE DA ML, and pilot-assisted (PA)

DA ML for QPSK and 8-PSK, and the results are depicted in figure 1.5 and 1.6, respectively.

With forward error control (FEC) coding employed in optical communication systems,

41 10 can be set as a reasonable reference level to compare the bit-error rate (BER)

performance [75]. Note that BER= 41 10 is also used as the reference level for comparison

in the following chapters. For 20-Gb/s QPSK with 100 kHz , the memory length for

both Mth-power and DE DA ML and pilots for PA DA ML are all selected to be 5L , and

data length for PA DA ML is 1000. From figure 1.5, we can see that DE DA ML has

approximate doubling the BER of PA DA ML due to the use of DE. And DA ML can achieve

comparable performance with Mth-power. In addition, for 30-Gb/s 8-PSK with

100 kHz , the memory length for both Mth-power and DE DA ML are selected as 10.

For PA DA ML case, data length is still 1000, and two pilot lengths are selected as 10 and 23

for comparison. As can be seen in figure 1.6, the results are similar to that of QPSK. It should

be noted that decision-error propagation cannot be effectively overcome and causes BER

jump at low SNR region (BER > 210 ) where decision errors frequently occur at this region.

This is due to the fact that only a small number of pilots is used, e.g. 10L . This problem

can be solved by selecting a larger number of pilots, such as 23L in the MC simulations.

Also note that the error propagation is not of big concern when BER is less than 310 at

which region decision-feedback errors are infrequent [76].


–17–

7 7.5 8 8.5 9 9.5 1010

-5

10-4

10-3

10-2

SNR/bit [dB]

BE

R

DE DAMLPA DAMLMth Power

L=5

Figure 1.5 Simulated BER performance of 20-Gb/s QPSK (10-Gsymbol/s) with Mth-power,

DE DA ML, and PA DA ML (linewidth per laser 100 kHz ).

7 8 9 10 11 12 1310

-5

10-4

10-3

10-2

10-1

100

SNR/bit [dB]

BE

R

DE DA MLPA DA ML(pilots =10)Mth powerPA DA ML(pilots =23)

Solid line: L=10Dashed line: L=23

Figure 1.6 Simulated BER performance of 30-Gb/s 8-PSK (10-Gsymbol/s) with Mth-power,

DE DA ML, and PA DA ML (linewidth per laser 100 kHz ).


–18–

1.3 Objectives and Contribution of the Study

Recently, advanced modulation formats with coherent detection have attracted extensive

attention, since they can greatly improve the SE of optical systems. One major issue in a

coherent optical communication system is to recover carrier phase, which is perturbed by

phase noise generated from the LLWs of both the transmitter and LO. Since the PE is

imperfect, the residual phase estimation error degrades the system performance. The BER

performance of various modulation formats with phase estimation error has been considered

in [10, 72, 77-80]. Moreover, experiments using multi-level modulation formats and coherent

detection are also conducted. Both return-to-zero (RZ) QPSK and RZ 8-PSK signals have

been demonstrated in 100-Gb/s transmission systems [81, 82]. 8- and 16-ary QAM signals are

also generated in [83, 84].

The main objective of this study is to analytically and experimentally investigate and

compare the performance of different advanced modulation formats in the coherent optical

back-to-back (B2B) systems (in the presence of laser phase noise and AWGN). From

previous experience, it is important to optimize the signal constellations in the

two-dimensional space [85, 86]. In the presence of phase noise, our aim is to enlarge the

angular distance between adjacent symbols in order to give a robust protection over the phase

estimation error. Therefore, the main contributions of this research are summarized as below:

Conditional and approximate BER expressions of different 4-ary, 8-ary, and

higher-order modulation formats are analytically derived in the presence of a

random phase estimation error and AWGN. Based on the conditional and

approximate BER expressions, BER performance, phase error variance tolerance


–19–

and LLW tolerance are compared between different modulation formats.

Based on the approximate BERs, the cross-over SNR algorithm between QPSK

and (1, 3) is derived. This algorithm can be used to approximate the cross-over

SNR points between QPSK and (1, 3) under different LLW, so that it allows quick

decision for engineers to choose a preferable 4-ary constellation in coherent

optical communication systems.

Based on the approximate BERs, the ring ratio optimization algorithms are derived

to optimize the placement of points in 8-star QAM and rotated 8-star QAM,

respectively. Moreover, our analysis finds out the reason that 8-star QAM has

larger LLW tolerance than rotated 8-star QAM.

Some analyses are done in higher-order modulation formats (16-ary and 64-ary).

Experiments are conducted to verify our analysis and simulation results.

The approach in the study illustrates a procedure in detail how to carry out the analysis

and optimize the performance for two-dimensional advanced modulation formats. The results

of this research may have significant impact on theoretically and experimentally

understanding the principles of different advanced modulation formats. Also, this research

may have some practical uses in that it can give engineers some advices in choosing

parameter values and modulation formats under different conditions.

1.4 Organization of the Thesis

The rest of this thesis is organized as follows.

In Chapter 2, we analytically derive the conditional and approximate symbol-error rate

(SER) and BER expressions for three 4-ary modulation formats, i.e., QPSK, 4 pulse


–20–

amplitude modulation (PAM), and (1, 3), respectively. Cross-over SNR algorithm between

QPSK and (1, 3) is obtained based on the approximate BERs.

In Chapter 3, five 8-ary modulation formats, such as 8-PSK, rectangular 8-QAM, 8-star

QAM, rotated 8-star QAM, and triangular 8-QAM, are investigated in the presence of a phase

estimation error and AWGN. The conditional and approximate BERs are given for the above

8-ary modulation formats. Ring ratio optimization algorithms are derived for 8-star QAM and

rotated 8-star QAM to optimize their BER performance. In addition, the reason that 8-star

QAM tolerates larger LLW than rotated 8-star QAM is discussed.

In Chapter 4, higher-order modulation formats (16-star QAM and 64-QAM) is

analytically studied. The conditional BER of 16-star QAM is obtained by using Craig’s

method. The BER performance of 64-QAM is also investigated and compared with that of

16-PSK.

In Chapter 5, experiments for QPSK, 8-PSK, 8-star QAM, and rotated 8-star QAM are

conducted in coherent optical B2B systems. The experimental results confirm our analytical

and simulation results.

Finally, conclusions are presented and future work is recommended in Chapter 6.


–21–

Chapter 2

Performance Analysis of 4-Point Modulation

Formats

In Section 1.2, we reviewed the coherent detection systems and DSP algorithms, and we also

did some MC simulations to compare the BER performance. From this chapter on, we can use

analytical tools to systematically study the performance of different advanced modulation

formats instead of doing time and resource-consuming MC simulations.

In this chapter, we focus on 4-ary modulation formats. We know that, in the presence of

phase noise, large angular distance between adjacent symbols gives a robust protection over

the phase estimation error. Therefore, we propose a new modulation format (1, 3). It has an

angle of 120 between adjacent signal points, which is larger than that of QPSK, namely,

90 . The BER performance and phase estimation error tolerance of QPSK, 4-PAM and (1, 3)

in the presence of a phase estimation error and AWGN are investigated in this chapter. Simple

and accurate approximate SERs and BERs are also presented. The accuracy of the

approximate BERs is verified by numerical integration of the exact conditional BERs and MC


–22–

simulations. This approximation results allow quick estimation of the SER and BER

performance, and also lead to an initial result of the cross-over SNR algorithm of QPSK and

(1, 3). Note that the analysis assumes no error propagation in the non-DE format. This is

because the impact of decision-feedback errors is found to be negligible at BER levels less

than 10-3 which we are most interested in.

2.1 BER Derivation of BPSK

We now start from the simplest modulation format to introduce the analytical procedure. This

procedure can be used to analyze the BER and SER of other advanced modulation formats in

the following chapters. We know that the simplest modulation format in coherent optical

communication systems is binary phase-shift keying (BPSK). It transmits only one bit per

symbol (“0” or “1”) at each symbol duration. We note that BPSK corresponds to

one-dimensional signals. Such signaling scheme is also called binary antipodal signaling [68].

2.1.1 BER of BPSK in AWGN Channel

0S1A

1S

0A

0n

bEbE

1n

Figure 2.1 Constellation map and decision boundary of BPSK. S0,S1: transmitted signal points;

A0, A1: decision regions; Eb: energy per bit; n0, n1: two independent AWGN components.


–23–

Figure 2.1 shows the constellation map and decision boundary of BPSK signals. Here, in

this figure, letter “I” means the in-phase axis, and the dotted line is the decision boundary. In

the case of BPSK, at each time we only have one of the two binary bits “0” or “1” transmitted,

and iS ( 0,1i ) denote the transmitted signal points. The two signal points 0S and 1S have

180 phase offset, and therefore they are antipodal. The amplitudes for 0S and 1S are

bE and bE , respectively. Hence, they have equal fixed energy per bit bE . Since

BPSK only transmits one bit per symbol, energy per symbol should be equal to energy per bit

s bE E . iA ( 0,1i ) represent the decision regions for iS , respectively. 0n and 1n are

two independent AWGN components. Statistically, they are identically independent (i.i.d)

Gaussian random variables, with mean 0 1[ ] [ ] 0E n E n and variance

2 2 20 1 0[ ] [ ] 2nE n E n N , where 0N is the spectral height of the AWGN. In the BPSK

case shown above, we can see geometrically that it is only the AWGN component 0n that

can cause a decision error. If 0S is transmitted, for example, a decision error occurs when

0n makes the received signal on the other side of the decision boundary, i.e., 1A . The

orthogonal noise component 1n cannot cause a decision error because it is parallel to the

decision boundary. Here, the received signal model (1.12)

exp ( )sr k A j k j k n k

is used. We should note that the assumption is made that CD and frequency offset are both

fully compensated, and also phase recovery is perfectly done (the phase noise term k in


–24–

the signal model can be removed). Therefore, this is the perfectly coherent case. Only AWGN

noise n k is taken into account in the analysis.

We now evaluate the BER expression for the BPSK signals in the case in which the

signals are equally likely ( 0 11

2P P ). This is the usual case in practice. The BER of BPSK

( )BP e can be written as

0 0 1 1

0 1

( ) ( | ) ( | )

1 ( | ) ( | ) ,

2

B B B

B B

P e P e S P P e S P

P e S P e S

(2.1)

where 0( | )BP e S and 1( | )BP e S are the error probabilities when signals 0S and 1S are

sent, respectively. By symmetry, it can be shown that 0 1( | ) ( | ).B BP e S P e S Therefore,

0( ) ( | ).B BP e P e S (2.2)

Now, we consider 0( | )BP e S . Conditioned on 0S , an error occurs when 0 .bn E

Therefore, we have

2

0

0

0

22

0

( ) ( | )

=

1 = .

22

b

B B

b

xN

E

P e P e S

P n E

e dxN

(2.3)

The integral can be expressed in one of two ways, i.e., either a complementary error function

(erfc function) or a Gaussian Q-function. In terms of a complementary error function, we have


–25–

2

0

2

0

0

0

0

0

1( ) =

1 2 =

2

1 =

2

1 = .

2

b

b

x

NB

E

x

N

E

N

b

b

P e e dxN

xe d

N

Eerfc

N

erfc

(2.4)

where 22

( ) v

t

erfc t e dv

is known as the complementary error function, and b is the

signal-to-noise ratio (SNR) per bit. Another way of expressing this error probability is

through Gaussian Q-function. The Gaussian Q-function is defined as 21

21( ) .

2

v

t

Q t e dv

Thus, Eq. (2.3) can also be expressed as

2

0

2

0

0

0

1

2

2

02

0

1( ) =

1 =

2

2

2 = Q

= Q 2 .

b

b

x

NB

E

x

N

E

N

b

b

P e e dxN

xe d

N

E

N

(2.5)

It is easy to show the relation 2 2erfc t Q t from the definitions.


–26–

Equations (2.4) and (2.5) are two different ways to express the BER of BPSK. As

mentioned above, BPSK transmits one bit per symbol. Therefore, the symbol error rate (SER)

sP e of BPSK should be equal to the BER as follows

1 1,

2 2s B b sP e P e erfc erfc (2.6)

where s is the SNR per symbol, and it is equal to SNR per bit b in the case of BPSK.

The SER of advanced modulation formats which transmit 2log M -bits per symbol

( 2log M >1) will be different from the BER of those modulation formats. It should be noted

that error function (erfc function) will be used to express the error probability in the following

chapters.

2.1.2 BER of BPSK with a Phase Estimation Error

In coherent optical detection, DSP based PE algorithms, such as Mth power [72] and DA ML

[59], are preferred at the receiver for carrier phase recovery. Since the PE is imperfect, the PE

algorithm has a residual phase estimation error which degrades the system performance.

We now consider the case in which both a phase estimation error and AWGN noise are

taken into account.

In the presence of a random phase error , the BER ( )BP e of a modulation format

can be numerically calculated as

( ) ( | ) ( ) ,B BP e P e p d

(2.7)

where ( | )BP e represents the BER of different modulation formats conditioned on a fixed

phase error value , and ( )p is the probability density function (PDF) of the phase


–27–

error , which can be assumed as a Gaussian PDF with mean zero and variance 2

[87].

I0S

01

0S

1A1S

0A1n

0n

bEbE

Figure 2.2 Constellation map and decision boundary of BPSK with a phase estimation

error .

Figure 2.2 shows the constellation map of BPSK signals with a residual phase estimation

error . The hollow point 0S represents the point when the original transmitted signal

point 0S is rotated by the phase estimation error . Similar to figure 2.1, 0n and 1n are

two independent AWGN components. 0n can cause a decision error if 0 cosbn E .

1n cannot cause a decision error because it is parallel to the decision boundary. Thus, the

average BER conditioned on a phase error can be written as


–28–

2

0

2

0

0

0

0

0cos

0cos

( | ) ( | , )

= cos

1 =

1 2 =

2

1 = cos .

2

b

b

B B

b

x

N

E

x

N

E

N

b

P e P e S

P n E

e dxN

xe d

N

erfc

(2.8)

Also, we have

( | ) ( | )s BP e P e (2.9)

for BPSK. Furthermore, we should notice that the perfectly coherent BER expression of

BPSK (2.4) can be obtained from the conditional BER (2.8) by setting 0 in equation

(2.8).

With the conditional BER ( | )BP e of BPSK obtained, we consider the PDF of the

phase estimation error ( ).p The distribution of the phase error is assumed to be

Gaussian 2(0, ).N Therefore, the analytical BER of BPSK can be obtained by

numerically integrating equation (2.7) for the BER conditioned on a phase estimation error

value against the PDF of the phase error. The analytical exact BER result will be shown later

and compared with the approximate BER curve.

2.1.3 Approximate BER of BPSK


–29–

For analytical purposes, it is desirable to have a simple and accurate approximate BER with

respect to SNR per bit b and phase estimation error variance 2 , instead of doing

numerical integration of the conditional BER. This approximation can be used to quickly

estimate the BER performance and LLW tolerance of a system.

Putting the equation (2.8) into equation (2.7), we obtain

2

22

1 1( ) cos exp .

2 22B bP e erfc d

(2.10)

The integral (2.10) can be approximated to

22

22

22

22 2

2

22 2

2

1 1 1( ) exp cos exp

2 22

1 1 = exp cos

2 22

1 cos 21 1 = exp

2 2 22

1 1

2 2

B b

b

b

P e d

d

d

2 2

22

222 2

4exp 1

2 2 2 2

1 1 1 = exp exp .

2 22

b b

b b

d

d

(2.11)

Here, the approximation 21( ) exp( )erfc x x

and 2cos( ) 1 ( ) / 2 are used in

the derivation. By using 2exp( )x dx

, (2.11) can be further expressed as


–30–

222 2

2

2 22 2

2

2 2

2

12 2

1 1 1( ) exp exp

2 22

1 1 1 1 1= exp exp

2 1 2 222

1 1 1= exp

2 122

1= 1 2 exp

2

B b b

b b b

b

b

b

b

P e d

d

.b

(2.12)

Equation (2.12) is the final approximate BER of BPSK.

Approximate BER (2.12) is plotted in Figure 2.3 for phase estimation error variance value

2 2 21 10 rad , which corresponds to rms phase error 5.7 in degree. Exact numerical

result obtained by numerical integration of (2.7) and Monte Carlo (MC) simulation are also

presented to show that the approximate BER is accurate. Note that the spectral height of

AWGN 0N is assumed to be 1 in our numerical evaluations and simulations. As can be seen

in figure 2.3, the numerical BER agrees quite well with the MC simulation. Also, it is shown

that the approximation (2.12) provides a very accurate estimate of the actual exact BER of

BPSK. The benefit achieved by the approximation (2.12) is its lower computational

complexity compared with the exact integration (2.10). We can see that exact BER (2.10) is

an integral of an erfc function, which is equivalent to a double integral of an exponential

function. Little information regarding the BER performance can be extracted from (2.10).

However, after approximation, equation (2.12) clearly shows that the BER performance can

be predicted by using an exponential function with respect to SNR per bit and phase error

variance. Therefore, approximation (2.12) has more value for researchers and engineers.


–31–

So far, the analytical procedure of BER of BPSK has been introduced. Next, we will

follow the procedure to derive the BER and SER of 4-point constellations.

4 6 8 10 12 14 16

10-15

10-10

10-5

100

SNR/bit [dB]

BE

R

exact numerical BER approximate BERMC simulation

Figure 2.3 Comparison between exact numerical BER, MC simulation, and

approximate BER when 2 2 21 10 rad .

2.2 Performance Analysis of QPSK

Section 2.1 has introduced the procedure to analyze the BER of the simplest modulation

format in coherent optical communication systems, i.e., BPSK. From this section on, we will

analyze the performance of advanced modulation formats beginning with a famous

modulation format, i.e., quadrature phase-shift keying (QPSK).

2.2.1 Conditional SER and BER of QPSK

The constellation map with Gray coding and decision boundaries of QPSK is shown in figure

2.4. Compared with the constellation map of BPSK shown in figure 2.1, QPSK maps its


–32–

signal points onto a two-dimensional space. Hence, in figure 2.4, letters “I” and “Q” mean the

in-phase and quadrature axis, respectively. QPSK has 4M signal points on the

two-dimensional space, which encodes 2log 2M bits per symbol. QPSK has four

modulation phases with 90 phase offset between adjacent symbols. Signal amplitude is sE .

Here, 2logs bE M E is the symbol energy.

2S

2A

1A

1S

3A

3S0A

0S

0S

0n

1nsE

Figure 2.4 Constellation map, decision boundaries and Gray code mapping of QPSK with a

phase estimation error .

A. Conditional Symbol-Error Rate (SER)

Symbol-error rate (SER) is simply the probability of deciding on the wrong symbol.

Hence, coding method is not considered in the analysis. Due to the symmetry (assuming that

the symbol 0S is transmitted) and equal probability of transmitting each symbol


–33–

( 1 4iP S ), and also following the theorem on total probability, the conditional SER of

QPSK is given by

3

0

0

0

( | ) ( | , )

( | , )

1 ( | , ),

s s i ii

s

s

P e P e S P S S

P e S

P c S

(2.13)

where 0( | , )sP e S is the probability of the received signal point falling outside the

decision region 0A when the symbol 0S is sent, and 0( | , )sP c S is the probability of

the received signal point falling in the decision region 0A . Thus,

0

0 1

0 1

0

( | ) 1 ( | , )

1 sin 4 , sin 4

1 sin 4 sin 4

1 1 1 sin

2 4

s s

s s

s s

s

P e P c S

P n E n E

P n E P n E

Eerfc

N

0

1 1 sin

2 4

1 1 sin sin

2 4 2 4

1 sin sin

4 4

s

s s

s s

Eerfc

N

erfc erfc

erfc erfc

.4

(2.14)

Here, we use the relation 1P n t P n t due to the fact that n is an i.i.d Gaussian

distribution. And 2logs bM is the SNR per symbol. By setting 0 in equation

(2.14), we get the SER of QPSK in perfectly coherent case as follows

21( ) sin sin .

4 4 4s s sP e erfc erfc

(2.15)


–34–

B. Conditional Bit-Error Rate (BER)

Consider the analysis of BER. BER is defined as the expected number of bit errors made

per bit detected [88]. Coding method must be taken into account in the derivation of BER.

Here, Gray coding is used in the bit mapping as shown in figure 2.4. The purpose of using

Gray code is to minimize the number of bit differences between nearest neighbor signal

points.

Now, we refer to figure 2.4, and assume that the transmitted symbol is 0S (“00”)

without loss of generality. Following the definition of BER, the average BER conditioned on

a phase error can be written as

0

1 0 2 0 3 0

( | ) ( | , )

1 | , 2 | , | , ,

2

B BP e P e S

P A S P A S P A S

(2.16)

where 0( | , )iP A S is the probability of the received signal point falling in the decision

region iA when the symbol 0S is sent. Observe that 1 1 2U A A and 2 2 3U A A are

half-planes, and thus equation (2.16) can be represented as

1 0 2 0

0 1

1( | ) | , | ,

21 1

sin 4 sin 42 2

1 1 sin sin .

4 4 4 4

B

s s

s s

P e P U S P U S

P n E P n E

erfc erfc

(2.17)

We compare equations (2.14) and (2.17). The two erfc multiplication in equation (2.16) is

usually much smaller than the single erfc function. Therefore, we can see that

1

( | ) ( | ),2B sP e P e (2.18)


–35–

when Gray coding is used.

By setting 0 in equation (2.17), we get the BER of QPSK in perfectly coherent

case as follows

1 1

( ) sin .2 4 2B s bP e erfc erfc

(2.19)

It can be seen that the BER of QPSK is same as that of BPSK in perfectly coherent case,

whereas the SE of QPSK is double that of BPSK. Therefore, QPSK is a quite famous

modulation format and is widely used in coherent optical communication systems.

2.2.2 Approximate SER and BER of QPSK

Similar to the case in BPSK, approximate SER and BER of QPSK are also needed for

analysis rather than doing numerical integration. These approximate SER and BER can be

used to quickly estimate the performance and phase estimation error tolerance of a system.

We will briefly introduce the procedure again. Putting the 1st erfc term of equation (2.14) into

the integral (2.7), we obtain

2

1 22

1 1( ) sin exp .

2 4 22s st sP e erfc d

(2.20)

The approximation 21( ) exp( )erfc x x

and 2cos( ) 1 ( ) / 2 are used in the

derivation, so (2.20) can be further approximated to

22 2 21 22 2

2 2

1 1 1 1( ) exp exp

2 2 22 2

1 1 1 = exp ,

2 22

s st s s s

s s

P e d

(2.21)


–36–

since 2exp( )x dx

. The 3rd term (two erfc multiplication) of equation (2.14) can be

approximated to

3

2 2

1sin sin

4 4 4

1 1 1 exp sin exp cos

4 4 4

1 exp ,

4

s s srd

s s

s

P e erfc erfc

(2.22)

using 21( ) exp( )erfc x x

. Finally, the approximate SER and BER in QPSK can be

expressed respectively as

2 21 1 1 1( , ) exp exp ,

2 2 42s s s sP e QPSK

(2.23a)

2 21 1 1( , ) exp .

2 24B s sP e QPSK

(2.23b)

2.3 Performance Analysis of 4-PAM

2.3.1 Conditional SER and BER of 4-PAM


–37–

0S2S

0S

2A 1A

1S

3A

3S

0A

1S

1D2D

Figure 2.5 Constellation map, decision boundaries and Gray code mapping of 4-PAM with a

phase estimation error .

In this section, we consider a modulation format named 4-pulse amplitude modulation (PAM).

The constellation map of 4-PAM is shown in figure 2.5. D1, Q-axis, and D2 are the decision

boundaries. From this figure, we can see that 4-PAM is also a 4-level amplitude-shift keying

(ASK) signal. The mapping of information bits is Gray coding. Note that the energies are not

equal for all the symbols. The absolute value of amplitude of signal points 0S and 2S is d ,

and the absolute value of amplitude of signal points 1S and 3S is 3d . For fairness of

comparison, therefore, the average symbol energy sE should be same as the one in QPSK,

and can be expressed as

2 2

2(3 )5 .

2sd d

E d

(2.24)

Thus, the amplitude of signal point 0S is 5.sd E


–38–

A. Conditional SER

We consider the SER of 4-PAM. Due to the symmetry (assuming that the symbols 0S

and 1S are transmitted) and equal probability of transmitting each symbol, the average SER

conditioned on a phase error can be written as

0 1

0 1

1 1( | ) 2 ( | , ) 2 ( | , )

4 41 1

( | , ) ( | , ).2 2

s s s

s s

P e P e S P e S

P e S P e S

(2.25)

Supposing that 0S is transmitted, the conditional SER can be expressed as

0 0 0 1

0 0

( | , ) ( distance from to Q-axis) ( distance from to )

( cos ) ( 2 cos )

1 cos 1 2 cos .

2 2

sP e S P n S P n S D

P n d P n d d

d d derfc erfc

N N

(2.26)

Similarly, supposing that symbol 1S is sent, the conditional SER 1( | , )sP e S is given by

1 1 1

0

( | , ) ( distance from to ) ( 3 cos 2 )

1 3 cos 2 .

2

sP e S P n S D P n d d

d derfc

N

(2.27)

Substituting equations (2.26) and (2.27) into (2.25), we obtain the conditional SER of 4-PAM

in the presence of a random phase estimation error. The final expression is shown in the

Appendix equation (A.1). By setting 0 , we obtain the SER of 4-PAM in perfectly

coherent case as follows

0

3.

4sd

P e erfcN

(2.28)

B. Conditional BER


–39–

The conditional BER ( | )BP e of 4-PAM can be computed by respectively computing

the error probabilities for the most significant bit (MSB) and the least significant bit (LSB)

[89], as indicated in figure 2.6. Note that each input bit is equally likely to end up by being

any one of the two bits.

0S

0S

Figure 2.6 Illustration of MSB and LSB.

Due to the symmetry between the MSB and LSB, the average BER conditioned on a

phase error can be represented by [90]

1| | | .

2B MSB B LSB BP e P e P e (2.29)

Without loss of generality, symbols 0S and 1S are considered to be sent. Therefore, we

have

0 11

| | , | , ,2MSB B MSB B MSB BP e P e S P e S (2.30a)

0 11

| | , | , .2LSB B LSB B LSB BP e P e S P e S (2.30b)

It is easy to show that


–40–

00

1 cos| , ,

2MSB Bd

P e S erfcN

(2.31a)

10

1 3 cos| , ,

2MSB Bd

P e S erfcN

(2.31b)

00 0

1 2 cos 1 2 cos| , ,

2 2LSB Bd d d d

P e S erfc erfcN N

(2.31c)

10 0

1 3 cos 2 1 3 cos 2| , .

2 2LSB Bd d d d

P e S erfc erfcN N

(2.31d)

Combing equation (2.29) through equation (2.31d), the conditional BER of 4-PAM can be

obtained (see Appendix equation (B.1)). Also, BER of 4-PAM in perfectly coherent case can

be obtained by setting 0

0 0 0

3 1 3 1 5.

8 4 8Bd d d

P e erfc erfc erfcN N N

(2.32)

2.3.2 Approximate SER and BER of 4-PAM

Following the procedure as shown in section 2.1.3 and 2.2.2, we obtain the approximate SER

and BER of 4-PAM, respectively, as follows

1 12 22 2

2 2

20 0

102 22

0

1 2 1 21

( ,4 ) exp ,8

31 2

s

d d

N N dP e PAM

Nd

N

(2.33a)


–41–

1 12 22 2

2 2

20 0

102 2

2

0

12 2

2

20

12 2

2

0

1 2 1 2

exp

31 2

1( ,4 )

169

1 29

exp

31 2

B

d d

N N d

Nd

NP e PAM

d

N d

Nd

N

12 22

2

0 0 0

.

15 251 2 exp

d d

N N

(2.33b)

Actually, 2

0

9exp

d

N

and 2

0

25exp

d

N

in equation (2.33b) are much smaller than

2

0

expd

N

. Hence, we also have the relation 1

( , 4 ) ( , 4 )2B sP e PAM P e PAM for

4-PAM when Gray coding is used.

2.4 Performance Analysis of (1, 3)

2.4.1 Conditional SER and BER of (1, 3)


–42–

0A

1A2A

3A

0S

2S1S

3S

1S

a

Figure 2.7 Constellation map, decision boundaries and bit mapping of (1, 3) with a phase

estimation error .

From previous experience, it is important to optimize the signal constellations in the

two-dimensional space. In the presence of phase noise, large angular distance between

adjacent symbols gives a robust protection over the phase estimation error . In this

section, we propose a new modulation format (1, 3), as depicted in figure 2.7. It is shown in

this figure that (1, 3) has one signal point 0S locates on the origin and three other symbols

1,S 2 ,S and 3S locate on a circle with 120 phase difference between each other. The

angular distance of (1, 3) is larger than that of QPSK, namely, 90 . Therefore, it should

tolerate larger phase estimation error as well as LLW than QPSK. Note that phase noise does

not affect the signal point 0S which locates on the origin, and also 0S does not consume


–43–

any energy. “a” represents the amplitude of signal points 1,S 2 ,S and 3S . Note that Gray

code cannot be used in the bit mapping due to the fact that each symbol has three adjacent

error-decision regions. Two errors will occur when the error decision is made between A0 and

A2, and A1 and A3. The average energy per symbol sE is 23

4sa

E . Hence, the amplitude

of 1S is 4

3sE

a .

A. Conditional SER

Due to the symmetry (assuming that the symbols 0S and 1S are sent) and equal

probability of transmitting each symbol, the average conditional SER of (1, 3) can be

represented as

0 11 1

( | ) 1 ( | , ) 3 ( | , ).4 4s s sP e P e S P e S (2.34)

Supposing that 0S is transmitted, the conditional SER can be expressed as

00

3( | , ) 3 .

2 2 2s

a aP e S P n erfc

N

(2.35)

Here, we introduce some approximations in deriving equation (2.35). Some remote regions

which have very small probabilities are dropped to simplify the analysis, yet the

approximation does not affect the SER performance.

Similarly, supposing 1S is sent, it is easy to show that

10

1 2 cos( | , ) cos .

2 2 2s

a a aP e S P n a erfc

N

(2.36)


–44–

Therefore, the SER of (1, 3) conditioned on a random phase error is obtained by

substituting (2.35) and (2.36) into (2.34), and finally given by

0 0

3 3 2 cos( | ) .

8 82 2s

a a aP e erfc erfc

N N

(2.37)

The perfectly coherent SER is expressed as

0

3( ) .

4 2s

aP e erfc

N

(2.38)

B. Conditional BER

Similarly, the average conditional BER of (1, 3) can be represented as

0 11 1

( | ) 1 ( | , ) 3 ( | , ).4 4B B BP e P e S P e S (2.39)


0 1 0 2 0 3 0

0

1( | , ) | , 2 | , | ,

2

1 4 ,

2 2 2

BP e S P A S P A S P A S

a aP n erfc

N

(2.40a)

1 0 1 2 1 3 1

0

1( | , ) | , | , 2 | ,

2

1 1 2 cos cos .

2 2 4 2

BP e S P A S P A S P A S

a a aP n a erfc

N

(2.40b)

Finally, we have

0 0

1 3 2 cos( | ) .

4 162 2B

a a aP e erfc erfc

N N

(2.41)

The perfectly coherent BER is expressed as


–45–

0

7( ) .

16 2B

aP e erfc

N

(2.42)

2.4.2 Approximate SER and BER of (1, 3)

Following the same procedure, the approximate SER and BER of (1, 3) can be expressed

respectively as

12 222

0 00

3 3 1( , (1,3)) 1 exp ,

8 8 42s

a a aP e erfc

N NN

(2.43a)

12 222

0 00

1 3 1( , (1,3)) 1 exp ,

4 16 42B

a a aP e erfc

N NN

(2.43b)

Here, we can see that the value of ( , (1,3))BP e is a little bit different from that of

( , (1,3)) 2sP e . This is due to the fact that Gray code cannot be used in the bit mapping of (1,

3). Two errors occurring between A0 and A2, and A1 and A3 makes ( , (1,3))BP e a bit larger than

( , (1,3)) 2sP e .

2.5 Results and Discussions

Firstly, we verify our exact results by using MC simulations. The exact numerical results can

be obtained by numerically integrating (2.7) for the BER conditioned on a phase estimation

error value against the PDF of the phase error. Figure 2.8 and 2.9 illustrate that the analytical

BER expressions are verified by using MC simulations for different values of phase error

variance 2 . It shows in figure 2.8 that QPSK performs 1.7-dB better than (1, 3), and

3.8-dB better than 4-PAM at BER=10-4 in perfectly coherent case. The reason can be


–46–

explained by comparing their minimum Euclidean distances dmin as follows:

min, 2 1.414 ,QPSK s sd min,(1, 3) 4 3 1.155 ,s sd and

min,4 4 5 0.894 .PAM s sd Larger minimum Euclidean distance in a modulation

format gives a larger tolerance to AWGN, thus leading to a better BER performance in

perfectly coherent case. As can be seen, the results in figure 2.8 are consistent with the

minimum Euclidean distance analysis. However, figure 2.9 shows that the proposed (1, 3)

outperforms QPSK in terms of BER performance when phase estimation error variance is

large. This indicates that (1, 3) can tolerate larger phase estimation error as well as larger

LLW than QPSK. Among all three, 4-PAM shows the worst performance. 4-PAM only starts

to perform better than QPSK at high SNR region. This is due to the fact that 4-PAM has a

larger angular distance than QPSK.

5 10 15 20

10-10

10-8

10-6

10-4

10-2

100

SNR/bit b=E

b/N

0 [dB]

BE

R

QPSK4-PAM(1, 3)MC Simulations

Figure 2.8 The BERs from analysis and MC simulations in perfectly coherent case.


–47–

5 10 15 20 2510

-25

10-20

10-15

10-10

10-5

100

SNR/bit b=E

b/N

0 [dB]

BE

R

QPSK4-PAM(1, 3)MC Simulations

Figure 2.9 The BERs from analysis and MC simulations when 2 2 210 .rad

5 10 15 20

10-20

10-15

10-10

10-5

100

SNR/bit b=E

b/N

0 [dB]

BE

R

QPSK exactQPSK approximate4-PAM exact4-PAM approximate(1, 3) exact(1, 3) approximateMC simulations

Figure 2.10 Comparison of BER approximations with exact results and MC simulations when

2 3 210 .rad


–48–

10 15 20 25

10-20

10-15

10-10

10-5

100

SNR/symbol s=Es/N

0 [dB]

SE

R

QPSK exactQPSK approximate4-PAM exact4-PAM approximate(1, 3) exact(1, 3) approximateMC simulations

Figure 2.11 Comparison of SER approximations with exact results and MC simulations when

2 3 210 .rad

Approximations (2.23b), (2.33b), and (2.43b) are plotted in figure 2.10. Exact results

obtained by numerical integration together with MC simulations are also presented to show

that the approximations are accurate. It indicates that the approximate BERs can be used to

estimate the BER performance rather than doing numerical integration and time-consuming

MC simulations. Figure 2.11 shows the accuracy of the SER approximations. Note that only

BER results will be presented in the following sections since SER results are similar to the

BER results. Moreover, the approximate BERs can also be used to predict the phase

estimation error tolerance of a system. We use exact and approximate BER expressions to

calculate the SNR penalty due to the increasing phase estimation error, and the results are

shown in figure 2.12. The SNR references are the SNR/bit ( b ) at BER=10-4 in the perfectly

coherent cases, which are 8.37 dB, 12.19 dB, and 10.08 dB, for QPSK, 4-PAM, and (1, 3),


–49–

respectively. As shown in figure 2.12, when we use exact numerical integration to calculate

the tolerance, the phase estimation error variance leading to a 1-dB SNR penalty at BER=10-4

are found to be 21.2 10 , 23.7 10 , and 25.3 10 for QPSK, 4-PAM and (1, 3),

respectively.

10-3

10-2

10-1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

2 [rad2]

SN

R p

en

alty

@B

ER

=1

0-4 [d

B]

QPSK exactQPSK approximate4-PAM exact4-PAM approximate(1, 3) exact(1, 3) approximate

Figure 2.12 The SNR penalty @BER=10-4 as a function of phase estimation error variance for

exact and approximate BER expressions.

Then we use the derived approximate BERs to predict the phase error tolerance, and compare

the results with the ones using exact BER expressions. By using the approximate BERs, the

estimated values of 2 leading to a 1-dB SNR/bit ( b ) penalty at BER=10-4 is 21.0 10 ,

23.3 10 , and 25.0 10 for QPSK, 4-PAM and (1, 3), respectively. The estimated values

of phase estimation error tolerance are all smaller than the ones using exact numerical BERs.

This is because the approximate BERs are upper bounds of the exact numerical BERs.


–50–

Therefore, they provide a more rigid LLW requirement than the exact BERs. Finally, figure

2.12 also shows that the phase estimation error tolerance of (1, 3) outperforms that of QPSK

and 4-PAM, which proves that angular distance between adjacent symbols is an important

factor in coherent communication systems due to the effect of phase noise. This makes (1, 3)

a promising 4-point modulation format in coherent optical communication systems.

2.6 Cross-over SNR between QPSK and (1, 3)

Finally, we compare the BER performance of QPSK and (1, 3). 4-PAM is not taken into

account due to its bad performance. It can be seen from figure 2.9 that the BER performance

of (1, 3) surpasses that of QPSK above a certain SNR per bit ( b ). Therefore, it is desirable to

investigate the cross-over SNR between QPSK and (1, 3) to allow a quick decision for

engineers to choose a preferable modulation format under different parameters, such as LLW

and symbol rate.

Based on BER approximation formulas (2.23b) and (2.43b), the cross-over SNR can be

calculated when we let (2.23b) equal (2.43b). We observe that equation (2.23b) is an

exponential function, and equation (2.43b) is an erfc function plus an exponential function.

Therefore, (2.43b) needs to be approximated by using the approximation

21( ) exp( )erfc x x

, and is expressed as

12 22

2

0 00

1

22

0 0

1 3 1( , (1,3)) 1 exp

4 16 42

41 1 3 1 exp ,

4 16 3 3

B

s s

a a aP e erfc

N NN

E E

N N

(2.44)


–51–

where 4 3sa E is also substituted in the approximation. We let

( , ) ( , (1,3)),B BP e QPSK P e and then we need to solve the following equation

122 2 2 41 3 4

exp 1 ,2 2 3 8 3 3s s s

s

(2.45)

where 0s sE N represents the average SNR per symbol. (2.45) is still a nonlinear

function, and is impossible to be solved. In order to simplify (2.45), the exponential term in

equation (2.45) can be further approximated by using the approximation 1 ,xe x and

therefore, we have

122 2 2 41 3 4

1 1 .2 2 3 8 3 3s s s

s

(2.46)

The term 2 4

3s

in (2.46) can also be dropped since it is much less than 1, and we arrive

at the result

22 2 14 0,

3 4b b (2.47)

where 0 2b b sE N is the average SNR per bit. By solving (2.47), we get the solution

2

2

1 1 9.

12b

(2.48)

We should notice that SNR per bit ( b ) in (2.48) is a ratio. It can be converted into dB unit,

and is given by

2

10 2

1 1 9(in dB) 10log .

12b

(2.49)


–52–

When the phase error variance 2 is much smaller than 1, the term 29 in the square

root can be ignored. Therefore, (2.49) can be further approximated to

10 2

1(in dB) 10log .

6b

(2.50)

From equation (2.49), it can be seen that the cross-over SNR between QPSK and (1, 3) is

a function of phase estimation error variance 2 . Furthermore, after approximation,

equation (2.50) shows that the cross-over SNR behaves as the reciprocal of the phase error

variance. It is understandable that at small phase error region, the cross-over SNR is high;

whereas at large phase error region, the cross-over SNR is low. This indicates that the

performance of (1, 3) surpasses that of QPSK at low SNR when phase error is large, which is

consistent with the results shown in figure 2.9. The value of a phase estimation error variance

2 can be computed from given LLW and symbol rate values based on different PE

algorithms. Hence, the cross-over SNR algorithm (2.49) and its approximation (2.50) can be

applied to all PE algorithms. Also, we should note that all the approximate SERs and BERs

can be applied to any PE algorithms given the PDF of the phase estimation error in that PE

algorithm. For DA ML PE, the PDF of the phase estimation error is given in (1.18).

Equations (2.49) and (2.50) are plotted in figure 2.13. It shows in figure 2.13 that the

cross-over SNR per bit is 12.2 dB when 2 2 21 10 .rad Also, we see from figure 2.9

that given the same condition the cross-over SNR per bit is 12.8 dB from exact numerical

integration. Both equation (2.49) and its approximation (2.50) give the same results when

2 2 22 10 .rad When 2 2 22 10 ,rad

the approximation (2.50) gives a slightly


–53–

different result compared with (2.49). This is due to the fact that at large phase error variance

region, the value of 2 starts to affect the value in the square root in equation (2.49).

10-4

10-3

10-2

0

5

10

15

20

25

30

35

Phase error variance 2 [rad2]

Cro

ss-o

ver

SN

R/b

it [d

B]

equation (2.49)approximation (2.50)

Figure 2.13 The cross-over SNR algorithm (2.49) between QPSK and (1, 3) and its

approximation (2.50) as a function of phase estimation error variance.

Although numerical integration can be used to find out the cross-over SNR, it is time

consuming. (2.49) can give us a quick estimation of the cross-over SNR only if we substitute

a specific value of 2 . Therefore, cross-over SNR algorithm (2.49) can be used to

approximate the cross-over SNR points between QPSK and (1, 3) under different LLW and

symbol rate, so that it allows a quick decision for engineers to choose a preferable 4-ary

constellation in coherent optical communication systems.


–54–

2.7 Conclusions

In this chapter, the procedure of how to analyze the BER and SER performance for a

modulation format was illustrated in details. Based on this procedure, the conditional SER

and BER expressions of QPSK, 4-PAM and (1, 3) in the presence of a phase estimation error

and AWGN were derived. Approximate SERs and BERs were also presented analytically

through a series of approximations. Note that only BER results were plotted and compared in

the results and discussion section. The exact BER results can be obtained by numerically

integrating equation (2.7) for the BER conditioned on a phase error value against the PDF of

the phase error. The accuracy of the approximate BERs was verified via numerical integration

and MC simulations. Note that the approximate BERs can be used in any PE algorithms given

the PDF of the phase estimation error. It was found that the approximations can quickly

estimate the BER performance under different phase estimation errors, and also can predict

the lower bound on the phase estimation error requirement.

Furthermore, the cross-over SNR algorithm between QPSK and (1, 3) was theoretically

derived based on the approximate BERs. It gives a quick and accurate estimation of the

cross-over SNR points between QPSK and (1, 3) under different LLW and symbol rate rather

than doing extensive numerical integrations.


–55–

Chapter 3

Performance Analysis of 8-Point Modulation

Formats

In this chapter, we analyze the performance of five 8-point modulation formats. They are

8-PSK, rectangular 8-QAM, 8-star QAM, rotated 8-star QAM, and triangular 8-QAM. We

evaluate the BER performance of the 8-point modulation formats in the presence of a phase

estimation error and AWGN. We also obtain simple and accurate approximate BERs for each

proposed 8-point modulation format, which allow quick estimations of the BER performance

and LLW tolerance. Moreover, the ring ratio optimization algorithms are derived based on

our approximate BER expressions to optimize the placement of points in both 8-star QAM

and rotated 8-star QAM, thus giving the minimum BER. These ring ratio optimization

algorithms can be used to find out the optimum ring ratio under different linewidth per laser,

symbol rate, and SNR. Finally, LLW tolerance of 8-point constellations are also investigated

and compared to each other. Our work shows that the LLW tolerance in a coherent optical

communication system with 8-star QAM outperforms that with the other 8-ary modulation


–56–

formats. The reason of 8-star QAM has more LLW tolerance than rotated 8-star QAM is also

discussed. It can be explained from the approximate BER expressions and also from the

constellation maps.

3.1 Performance Analysis of 8-PSK

2A1A

5A

0AsE

3A

4A

6A

7A

Figure 3.1 Constellation map, decision boundaries and Gray code mapping of 8-PSK with a

phase error .

Start from a common 8-point modulation format, i.e., 8-PSK, as shown in figure 3.1. We can

see that 8-PSK is a one-ring constellation. All 8 signal points are located on one circle.

Therefore, space is wasted, and the angular distance between adjacent symbols is too small,

i.e., 8 (22.5 in degree). The BER expression conditioned on a phase error ( | )Bp e

for 8-PSK has already been derived and listed in [79], as follows:


–57–

1 1( | ) ( | ) sin sin

3 8 3 8

1 1 cos 1 sin

3 8 2 8

B B s s

s s

P e P e erfc erfc

erfc erfc

1 1 cos 1 sin ,

3 8 2 8s serfc erfc

(3.1)

where 2(log )s bM for a constellation encoding 2log M bits per symbol.

Note that for 8-PSK, the integration range is from 0 to instead of from to ,

since both ( | )BP e and ( | )BP e are used in the derivation of 8-PSK BER

expression. ( | )BP e and ( | )BP e are symmetry in the constellation. Also note that

only BER is taken into account in the following parts, since BER is more considered in

practice. The perfectly coherent BER of 8-PSK can be obtained by setting 0 in (3.1),

and is given by

1 1 1( ) sin cos 1 sin .

3 8 3 8 2 8B s s sP e erfc erfc erfc

(3.2)

Approximate BER of 8-PSK can also be derived by using the previous procedure, and is

written as

1 2 22 2

2

1 1 2( ,8 ) 1 2 exp 1

2 26 4 4 2

1 exp .

6

sB s s

s

s

P e PSK

(3.3)

where 1 11

2 2erfc t erfc t is used in the derivation.

3.2 Performance Analysis of Rectangular 8-QAM


–58–

3.2.1 Conditional BER of Rectangular 8-QAM

0A

1A

7A

6A

3A

2A

4A

5A

inr

outr

Figure 3.2 Constellation map, decision boundaries and Gray code bit mapping of rectangular

8-QAM.

Constellation map of rectangular 8-QAM is given in figure 3.2. Here, in this figure, the dotted

lines are the decision boundaries for this constellation. iA denote the decision regions. The

red dashed lines divide the polygonal regions (such as 0A , 2A , 4A , and 6A ) into two parts:

a semi-infinite rectangular part ,m rectA (such as region “BCEF”), and a triangular part

,m triA (such as region “CDE”, and 0, 2,4,6.m ) . This is for the convenience of further

derivation in the following part. Gray coding is used in the bit mapping. We see from figure


–59–

3.2 that rectangular 8-QAM is duo-ring constellation. Therefore, the inner ring radius is

defined to be ,inR and the outer ring radius is .outR

Before calculating the conditional BER, we should give a clear definition regarding inR

and .outR The ratio between outR and inR is fixed at 2 for rectangular 8-QAM, i.e.,

2out inR R . Since the mean of energies of the inner ring and outer ring should be equal to

the average symbol energy s , therefore,

2 2 2 2

22 3.

2 2 2in out in in

s inR R R R

R (3.4)

Note that for simplicity, notation s represents both the average SNR per symbol and

average symbol energy in the following parts due to the fact that spectral height of AWGN

0N is assumed to be 1 in our analysis and simulations. Based on (3.4), we obtain that

2

3in sR and 4

.3out sR (3.5)

Similar method of deriving the BER of 8-PSK will be used to calculate the

( | ) ( | )B BP e P e of rectangular 8-QAM. Due to the symmetry and equal probability

of transmitting each symbol (1 8 ), the conditional BER ( | ) ( | )B BP e P e of the

rectangular 8-QAM is equal to

4( | ) ( | ) [ ( | ,000) ( | ,000)]

84

[ ( | ,001) ( | ,001)].8

B b B B B

B B

P e P e P e P e

P e P e

(3.6)

In the beginning, supposing that “000” is transmitted, the BER conditioned on “000” is given

by


–60–

1 2 3

4 5 6

7

( | ,000) 2 ( | ,000) ( | ,000)1

( | ,000) 2 ( | ,000) 3 ( | ,000) 2 ( | ,000) ,3

( | ,000)B

P A P A P A

P e P A P A P A

P A

(3.7)

where ( | ,000)iP A denotes the probability of the received signal vector falling into the

regions iA when the transmitted signal is “000”. Note that the decision regions have a mirror

symmetry with respect to the phase error, i.e.,

1 7| ,000 | ,000 ,P A P A (3.8a)

2 6| ,000 | ,000 ,P A P A (3.8b)

3 5| ,000 | ,000 ,P A P A (3.8c)

4 4| ,000 | ,000 ,P A P A (3.8d)

producing

000 1 2 3

2, 6, 4 0,

2( | ,000) ( | ) [ ( | ) ( | ) ( | )

3 ( | )],

B b

tri tri tri

P e P e P U P U P U

P A A A A

(3.9)

where 1 2 3 4 5 6 0,triU A A A A A A , 2 1 2, 3rectU A A A , and

3 5 6, 7rectU A A A are all half -planes, and (3.8c) is used to obtain equation (3.9).


11 1

| cos ,2 2 inP U erfc R

(3.10a)

21 1

| sin ,2 2 inP U erfc R

(3.10b)

21 1

| sin .2 2 inP U erfc R

(3.10c)


–61–

Let us see the remaining terms 2, 6, 4 0,tri tri triA A A A . Here, the minus term 0,triA is

due to the excess use of one 0,triA in order to produce a half-plane region 1U . We can make

it like this way 2, 6, 4 0, 0,2tri tri tri triA A A A A to simplify the derivation because

2, 6, 4 0,tri tri triA A A A is a semi-infinite rectangular region. It can be shown that

1

2, 6, 4 0,

2

1 1sin sin ,

2 2( | )

1cos

2

1 1 1 1= sin sin cos .

8 2 2 2

in in

tri tri tri

in

in in in

R n R

P A A A A P

n R

erfc R erfc R erfc R

(3.11)

Then, we see how to computer the probability 0,( | )triP A .

1n 2n

0,triA

Figure 3.3 Illustration of how the triangular region CDE is formed.


–62–

The probability 0,( | )triP A for the triangular region CDE in figure 3.2 can be calculated

by viewing the triangular region CDE as being formed in figure 3.3, i.e., as the difference

between two rectangular regions. Thus,

0, Re Re

1

2

1

1|

4

sin 2 sin4 4

cos 2 cos4 41

=4 1 3

cos cos2 2

tri big ct small ct

in in

in in

in in

P A P P

P R n R

P R n R

P R n R

21 1

sin sin2 2

sin sin 2)4 4

cos41

=64

in in

in in

in

P R n R

erfc R erfc R

erfc R

cos 2

4

1 3cos cos

2 2

1 1sin sin

2 2

in

in in

in in

erfc R

erfc R erfc R

erfc R erfc R

. (3.12)

Finally, we have


–63–

1 1 1 1cos sin

2 2 2 2

1 1 1 1sin sin

2 2 8 2

1 1sin cos

2 2

2( | ,000) ( | ,000)

3

in in

in in

in in

B B

erfc R erfc R

erfc R erfc R

erfc R erfc R

P e P e

sin sin 2)4 4

cos cos 24 41

32 1 3cos cos

2 2

in in

in in

in in

erfc R erfc R

erfc R erfc R

erfc R erfc R

.

1 1sin sin

2 2in inerfc R erfc R

(3.13)

Next, supposing that “001” is transmitted, the conditional BER is given by

0 2 3

4 5 6

7

( | ,001) ( | ,001) 2 ( | ,001)1

( | ,001) 3 ( | ,001) 2 ( | ,001) ( | ,001) .3

2 ( | ,001)B

P A P A P A

P e P A P A P A

P A

(3.14)

It is observed that

7 3| ,001 | ,001 ,P A P A (3.15)

producing

4 5

4 3 2, 4,

( | ,001) ( | ,001)2( | ,001) ( | ,001) .

( 2 | ,001)3B Brect tri

P U P UP e P e

P A A A A

(3.16)

where 4 4 2, 0 5 6 7triU A A A A A A and 5 3 4, 5rectU A A A are both half-planes,

and (3.15) is used to obtain equation (3.16). Hence, the probability of 4U and 5U is given

by

41 1

( | ) 2 sin ,2 4 2inP U erfc R

(3.17a)


–64–

51 1

( | ) 2 sin .2 4 2inP U erfc R

(3.17b)

Now, let us look at the remaining term 4 3 2, 4,2 rect triA A A A . This term actually can be

divided into three parts. 3 2,rectA A and 3 4,rectA A are two quadrants. 4,2 triA is an

arbitrary triangle. It is easy to show that

3 2,

1 1( | ) 2 cos

4 4 2

1 2 sin ,

2 4

rect in

in

P A A erfc R

erfc R

(3.18a)

3 4,

1 1( | ) 2 cos

4 4 2

1 2 sin .

2 4

rect in

in

P A A erfc R

erfc R

(3.18b)

The calculation of probability of 4, |triP A is shown as below

4,

2 12 cos 2 cos sin2 22 2

0 00 02 cos 2 sin

22 cos 2 cos ,

2|

12 sin 2 cos sin

2

1 1= exp exp[ ]

in

in in

in in

tri

in in

R

R R

R x R

P A P

R y R x

x ydx dy

N NN N

22 cos

2

0 02 cos

12 cos sin

2 sin1 2( ) ( ) ,

2

in

in

in

R x

Rin

in

R

R xR

erfc erfc f x dxN N

(3.19)

where 2

00

1( ) exp .

xf x

NN


–65–

From the constellation map, we already see that region 4,triA is far away from the signal

point “001”, so it should be unlikely for the signal point to fall into this area. Also, from

analysis (3.19), figure 3.4 shows that the BER descends dramatically fast when SNR per bit

relatively slowly increases. Even when SNR per bit equals to 7 dB, the probability of this

point falling into this area is only 1010 . Normally, BER should be around 210 at SNR/bit

of 7dB. In this case, this term 4, |triP A can be ignored.

6 8 10 12 14 16 18 20 2210

-300

10-250

10-200

10-150

10-100

10-50

100

SNR/bit [dB]

BE

R

Figure 3.4 Plot of the ignored term 4, |triP A .

Finally, we obtain


–66–

1 12 sin

2 4 2

1 1+ 2 sin

2 4 2

1 1+ 2 cos

4 4 22( | ,001) ( | ,001)

3 1 2 sin

2 4

1+

4

in

in

in

B B

in

erfc R

erfc R

erfc R

P e P e

erfc R

erf

.

12 cos

4 2

1 2 sin

2 4

in

in

c R

erfc R

(3.20)

Combining equations (3.13) and (3.20) together, we obtain the final conditional BER of

rectangular 8-QAM in the presence of phase noise. Note that the integration range is 0,

rather than , due to the fact that a pair of ( | )BP e and ( | )BP e are used in

the derivation. Moreover, by setting 0 , the perfectly coherent BER expression can be

written as

2

4 4

3 1 1 1 1

2 2 8 2 21( )

6 1 2 1

32 2 2

1 1 1 3

2 2 2 21

6 1 1 1

4 2 2

in in in

B

in in

in in

in

erfc R erfc R erfc R

P e

erfc R erfc R

erfc R erfc R

erfc R erfc R

.1 3 3

4 2 2in in inerfc R erfc R

(3.21)

3.2.2 Approximate BER of Rectangular 8-QAM


–67–

It can be seen from equations (3.13) and (3.20) that conditional BER of rectangular 8-QAM

contains not only some terms of single-erfc function, but also some terms of erfc

multiplication. The procedure of doing approximation for single-erfc function is shown in the

previous chapter. The erfc multiplication in (3.13) and (3.20) can be dropped for simplicity to

obtain the approximation, since the value of several erfc functions multiplication is usually

very small compared with the value of a single-erfc function. Therefore, the conditional BER

expression of rectangular 8-QAM can be approximated as

1 1 1 1( | ) ( | ) cos sin

6 2 6 2

1 1 1 1 sin + 2 sin

6 2 6 4 2

B B in in

in in

P e P e erfc R erfc R

erfc R erfc R

1 1 + 2 sin .

6 4 2 inerfc R

(3.22)

6 8 10 12 14 16 18 20

10-15

10-10

10-5

100

SNR/bit [dB]

BE

R

exact numerical integration from (3.13) and (3.20)after drop some unimportant terms (3.22)

Figure 3.5 The comparison of exact BER expression (3.13 and 3.20) and approximation (3.22)

for rectangular 8-QAM.


–68–

The accuracy of the approximation (3.22) is examined in figure 3.5. It is shown that after

dropping some unimportant terms, approximations (3.22) can be used to well approximate the

BER expression (3.13) and (3.20). It can be observed from (3.22) that only some single-erfc

functions are left in the expression. Therefore, the approximate BER of rectangular 8-QAM

can be expressed as

1 22 2 2

1 2 4 22 2 2

2 2

12 2 2

1 1( , rect 8-QAM) 1 exp

12 4

1 1 1 2 exp

6 4 2 4

1 1 5 + 1 2 exp 2

12 4

inB in

in inin

in

in

RP e R

R RR

R

R

2 4 22

2 2

2 4 212 2 22

2 2

2 2

2 2 2

2 21 1 5 + 1 2 exp 2 .

12 4 2 2 2

inin

in

inin in

in

RR

R

RR R

R

(3.23)

Note that the details of derivation are left to the reader for lack of space.

3.3 Performance Analysis of 8-point Star QAM

Rectangular 8-QAM is a duo-ring constellation. The ring ratio of rectangular 8-QAM is fixed

at 2 . In order to optimize the 8-point signal constellations in the two-dimensional space,

8-point star QAM is proposed. 8-point star QAM can be classified into two categories: 8-star

QAM and rotated 8-star QAM. In this section, we will theoretically investigate the

performance of 8-star QAM and rotated 8-star QAM.

3.3.1 Analysis of 8-star QAM


–69–

3.3.1.1 BER of 8-star QAM

0A

1A

2A

3A4A

5A

6A

7A

inR

outR

Figure 3.6 Constellation map, decision boundaries and Gray code mapping of 8-star QAM

with a phase error .

The constellation map with Gray coding and decision boundaries is shown in figure 3.6. It can

be seen from figure 3.6 that the angular distance between adjacent symbols in 8-star QAM is

2, which is larger than the other possible 8-ary constellations and 16-QAM, it makes

8-star QAM more noise-tolerant. Moreover, 8-star QAM is also a duo-ring modulation

formats (it can also be named as duo-ring QPSK), and the ring ratio of 8-star QAM can be

tuned to find the optimum location of the 8 signal points.

Before calculating the conditional BER, clear definitions of the inner ring radius inR and

outer ring radius outR , as shown in figure 3.6, should be given. The ring ratio is defined


–70–

as .out inR R inR can be written as ,in sR where is the coefficient of the

inner ring. outR can be expressed as .out in sR R Since the mean of

energies of the inner ring and outer ring should be equal to the average symbol energy ,s

therefore,

2 2 2 2 2

.2 2

in out s ss

R R

(3.24)

So we have the relation

2 2( 1)

1, >1 and 0< <1.2

(3.25)

Due to the symmetry (assuming that the symbols “000” and “100” are transmitted) and

equal probability of transmitting each symbol (1 8), the average BER conditioned on a phase

error can be written as

1 1

( | ) ( | ,000) ( | ,100).2 2B B BP e P e P e (3.26)

Supposing that an inner ring symbol “000” is sent, the conditional BER ( | ,000)BP e is

given by

1 2 3

4 5 6

7

( | ,000) 2 ( | ,000) ( | ,000)1

( | ,000) ( | ,000) 2 ( | ,000) 3 ( | ,000) .3

2 ( | ,000)B

P A P A P A

P e P A P A P A

P A

(3.27)

Observe that 1 1 5 2 6U A A A A and 2 2 6 3 7U A A A A are half-planes, and thus

equation (3.27) can be represented as

1 2 4 5 6 71

( | ,000) ( | ,000) ( | ,000) ( | ,000) .3BP e P U P U P A A A A (3.28)


–71–


11

( | ,000) sin sin ,4 2 4s sP U P n erfc

(3.29a)

21

( | ,000) cos cos ,4 2 4s sP U P n erfc

(3.29b)

4 5 6 7 1 2 3 0

1

2

( | ,000) 1 ( | ,000)

2cos 1 2cos 1 , 2 2=1

(2sin 1) ( 2sin 1)2 2

1 1=1 2cos 1 1 2cos 1

2 2 2 2

s s

s s

s s

P A A A A P A A A A

nP

n

erfc erfc

1 1(2sin 1) 1 ( 2sin 1) .

2 2 2 2s serfc erfc

(3.29c)

Similarly, supposing that signal “100” is transmitted, the conditional BER is given by

3 4 1 2 3 01

( | ,100) ( | ,100) ( | ,100) ( | ,100) ,3BP e P U P U P A A A A (3.30)

where 3 1 5 2 6U A A A A and 4 2 6 3 7U A A A A are half-planes. We get

31

( | ,100) sin ,2 4sP U erfc

(3.31a)

41

( | ,100) cos ,2 4sP U erfc

(3.31b)

1 2 3 0( | ,100)

1 1= (2 cos 1) 1 (2 cos 1)

2 2 2 2

1 1(2 sin 1) 1 ( 2 sin 1) .

2 2 2 2

s s

s s

P A A A A

erfc erfc

erfc erfc

(3.31c)


–72–

Combining equation (3.26) through equation (3.31c), the conditional BER in the presence

of a random phase estimation error in 8-star QAM can be obtained. The final expression

is presented in the Appendix equation (C.1). The perfectly coherent BER expression is given

by

1 ( 1)1

2 4 2

1 1 ( 3) ( 1)( ) 1

6 2 2 2

1 ( 1) 1

2 2

2

1 1 ( 1) +

6 4 2

ss

B s s

s

s

erfc erfc

P e erfc erfc

erfc

erfc

erfc

1 (3 1)

1 .2 2

( 1) 1 ( 1) 1

2 2 2

s s

s s

erfc

erfc erfc

(3.32)

It can be observed that in order to get the approximate BER, we need to simplify the four

erfc multiplication in equations (3.29c) and (3.31c) to a single erfc. Since

11 2cos 1

2 2serfc

and 1

(2 cos 1)2 2serfc

are the most

dominant terms in equations (3.29c) and (3.31c), respectively, therefore, the final conditional

BER expression can be approximated as


–73–

sin cos4 41

( | )12

2cos 12

sin cos4 41

12

(2 cos2

s s

B

s

s s

s

erfc erfc

P e

erfc

erfc erfc

erfc

.

1)

(3.33)

It can be seen that only some single erfc terms are left in equation (3.33). Finally, the

approximate BER in 8-star QAM can be expressed as

2 2 2 2 2 2 2 2

1 12 2 2 22 2

22

1 1exp 1 exp 1

2 2

1 1 1( ,8-star QAM) 1 1 1 1 .

12 2

1 exp

4

s s s s

B s s

s

P e

(3.34)

3.3.1.2 Ring Ratio Optimization

Generally, varying the radii of the two rings in 8-star QAM would severely affect its

performance. Therefore, we need to determine the optimum ring ratio. Based on equation

(3.34), an optimum ring ratio opt can be obtained to give the minimum BER ( ),BP e i.e.,

0BP e at opt .


–74–

From equation (3.34), the term 2 2 2 2 21exp 1

2 s s

can be dropped for

simplicity because it is much smaller than 2 2 21exp 1

2 s s

since the terms in the

parentheses are both negative and 2 1. And also the terms 2 21 s and

2 21 s can be dropped since they are much less than 1. The dominant terms are

left as

222 2 2 11 1 1

exp 1 exp .12 2 4s s s

(3.35)

We substitute the relation 22 1 into equation (3.35), use the approximation

1 ,xe x differentiate it by and make it equal to 0, drop 2 in the 22 term

since 2 1, and arrive at the result

2 3 22 2 2 1 0.s (3.36)

By solving the cubic equation (3.36), we get the solution

2

21,opt

opt

(3.37a)

3 32

1 1 3 1 3,

2 26 2opt

s

i iA B A B

(3.37b)

2 3 22 2

2 2 2,

848 864 ss s

A

(3.37c)


–75–

3

22 22

1 1.

6 72s s

B A

(3.37d)

From equation (3.37a) to (3.37d), it can be seen that optimum ring ratio opt is a

function of SNR/symbol s and phase error variance 2 . Note that the physical meaning

of opt is the optimum ratio of the inner ring radius to the average signal amplitude

,in optopt

s

R

.

3.3.2 Analysis of Rotated 8-star QAM

3.3.2.1 BER of Rotated 8-star QAM

0A

4A

1A5A

2A

6A

3A

7A

inR

outR

1L2L

3L

0S

0S

4S4S

Figure 3.7 Constellation maps and decision boundaries of rotated 8-star QAM with a phase

estimation error .


–76–

The constellation map and decision boundaries of rotated 8-star QAM is shown in figure 3.7.

Compared with points in the outer ring, the points in the inner ring of rotated 8-star QAM has

a 45 phase offset. This is the key difference between rotated 8-star QAM and 8-star QAM.

Here, we divide each decision region iA into two half parts for the convenience of

derivation. “+” and “-” in the figure are used to represent the two different half parts. Gray

coding is not used in the bit mapping due to the fact that each symbol has four adjacent

error-decision regions after 45 phase offset. Two errors will occur when the error decision is

made between A4 and A1, A5 and A3, A7 and A2, and A6 and A0. Same as 8-star QAM, inR is

defined as ,in sR and outR can be expressed as ,out in sR R

where is the ring ratio between outer and inner ring. Relation (3.25) can also be used for

rotated 8-star QAM.

Assume, without loss of generality, that the symbols corresponding to “000” and “100”

are transmitted. Then the average BER conditioned on a phase error can be derived by

1 1( | ) ( | , 000) ( | ,100).

2 2B B BP e P e P e (3.38)

Now, supposing that an inner ring symbol “000” is sent, the conditional BER is written as

1 2 3

4 5 6

7

( | ,000) ( | ,000) 2 ( | ,000)1

( | ,000) ( | ,000) 2 ( | ,000) 2 ( | ,000) .3

3 ( | ,000)B

P A P A P A

P e P A P A P A

P A

(3.39)

Observe that 1 4 1 5 3 7U A A A A A and 2 5 3 7 2 6U A A A A A are half-planes,

and thus (3.39) can be represented as

1 2

4 5 6 6 7 7

( | ,000) ( | ,000)1( | ,000) .

3 ( | ,000)B

P U P UP e

P A A A A A A

(3.40)


–77–


11

( | ,000) sin sin ,4 2 4s sP U P n erfc

(3.41a)

21

( | ,000) cos cos .4 2 4s sP U P n erfc

(3.41b)

Then we look at how to derive the remaining term in (3.40). Regions 5 ,A 7 ,A and 7A are

too remote for symbol “000”, therefore, the probability of falling into these three regions can

be ignored. The received signal will have a large probability of falling into the regions 4 ,A

6 ,A and 6 .A Since the shape of regions 4 ,A 6 ,A and 6A are arbitrary, it is difficult to

calculate the probability. Therefore, we need to extend the arbitrary regions to half-planes for

the convenience of analysis. For example, if we want to calculate the probability

6( | ,000)P A , we can extend the decision boundary line L1 to infinity to create a

half-plane. Yet 6( | ,000)P A is still the dominant probability in this half-plane. Thus,

6( | ,000)P A can be expressed as

6 0 1

2 2

2 2

| ,000 distance from to

2 2 2sin

2 4 4 2

1 2 2 2sin

2 2 4 4 2

s

P A P n S L

P n

erfc

.s

(3.41c)

In the following derivation, we use


–78–

2 22 2

2 4 4M

(3.41d)

for a notation. Term sM in (3.41c) represents the distance from signal point S0 to the

line L1. Similar to the derivation of 6( | ,000)P A , the other terms can be obtained as

61

| ,000 2sin ,2 2 sP A erfc M

(3.41e)

4 0 2| ,000 distance from to

2sin2

1 2sin .

2 2

s

s

P A P n S L

P n M

erfc M

(3.41f)

Similarly, supposing that signal “100” is transmitted, the conditional BER is given by

3 4

0 1 1 2 3 3

( | ,100) ( | ,100)1( | ,100) ,

3 ( | ,100)B

P U P UP e

P A A A A A A

(3.42)

where 3 1 5 3 7 2U A A A A A and 4 0 6 2 7 3U A A A A A are half-planes. It is

easy to get

31

| ,100 sin ,2 4sP U erfc

(3.43a)

41

| ,100 cos ,2 4sP U erfc

(3.43b)

0 4 21

| ,100 distance from to sin ,2 2 sP A P n S L erfc M

(3.43c)


–79–

1 1 4 3| ,100 | ,100 distance from to

1 sin .

2 2 s

P A P A P n S L

erfc M

(3.43d)

Combining equation (3.38) through (3.43d), the conditional BER in the presence of a

random phase estimation error in rotated 8-star QAM can be obtained. The final

expression is presented in the Appendix equation (D.1). By setting 0 , the perfectly

coherent BER expression of rotated 8-star QAM is given by

1( ) 3 .

6 2 2s s

B sP e erfc erfc erfc M

(3.44)

Next, we will derive the approximate BER of rotated 8-star QAM. It can be seen that only

some single-erfc terms are left in the final conditional BER expression. Putting the erfc term

in (3.41c) into (2.7) for example, we obtain

2

(3.41 ) 22

1 1( ) 2sin exp .

2 2 22B c sP e erfc M d

(3.45)

The approximations 21( ) exp( )erfc x x

, 2cos( ) 1 ( ) / 2 , and sin

are used in the derivation, so (3.45) can be further approximated to

22 22 2

(3.41 ) 2 22 2

22 22

2 2 2

2 2

21( ) exp

1 22

21 exp

2 1 2

1 1 = 1 2

2

sB c s

s

ss

s

s

MP e M

Md

22 212 22

2 2

2exp ,

1 2

ss

s

MM

(3.46)


–80–

since 2exp( ) .x dx

The details of derivation of the other terms are left to the reader.

Finally, the approximate BER in rotated 8-star QAM can be expressed as

2 2 2 2 2 2 2 2

22 212 2 2 22

2 2

21 222 2 2 2 2

1 1exp 1 exp 1

2 2

21 1 3( , rotated 8-star QAM) 1 2 exp

12 2 1 2

3 11 exp

2 2

s s s s

sB s s

s

ss s

MP e M

MM

2

2 2 2

.

2 s

(3.47)

3.3.2.2 Ring Ratio Optimization

We need to determine the optimum ring ratio of rotated 8-star QAM in this section. Based on

(3.47), an optimum ring ratio opt can be obtained to give the minimum BER ( ).BP e From

approximation (3.47), the term 2 2 2 2 21exp 1

2 s s

can be dropped for simplicity

because it is much smaller than 2 2 21exp 1

2 s s

since the terms in the parentheses

are both negative and 2 1. And also the terms 2 2s and 2 2 2

s can be

approximated to zero. The dominant terms are left as

22 2 2 2 2 2 2

22 22 2

1 3exp 1 exp 2

2 21 1

( ) .12 3

exp2 2

s s s s

Bs

s

M M

P e MM

(3.48)


–81–

We substitute (3.41d) and the relation 22 1 into (3.48), use the approximation

1 ,xe x differentiate it by and make it equal to 0, drop 2 in the 22 term

since 2 1, and arrive at the result

2 2 3 2 2 2 255 3 3 9 3(9 1) 2 2 0.

16 8 4 4 4s s s s s s

(3.49)

We see that (3.49) is a cubic function with a normal form written as 3 2 0ax bx cx d .

Here, we can set 2 255,

16 sa 23 3(9 1) ,

8 4s sb

292 ,

4 s sc

and

232 .

4 sd By solving equation (3.49), we get the solution as follows

2

21,opt

opt

(3.50a)

3 31 3 1 3,

3 2 2optb i i

A B A Ba

(3.50b)

33 2

23 2 2

, .2 327 6 9

b bc d c bA B A

a aa a a

(3.50c)

3.4 Performance Analysis of Triangular 8-QAM

Finally, we will derive the conditional and approximate BERs of triangular 8-QAM to

complete the study of 8-point modulation formats. Constellation map with decision

boundaries of triangular 8-QAM is shown in figure 3.8. Triangular 8-QAM is a constellation

with 6 equilateral triangles inside, such as signal points “100”, “000”, and “001” form one

equilateral triangle. We assume that the length of one side of the equilateral triangles is d.

Furthermore, triangular 8-QAM is a tri-ring modulation format. Signal points “000” and


–82–

“011” are on one ring, “001” and “010” are on the second ring, and points “100”, “101”,

“111”, and “110” are on the third ring. And the amplitudes can be written as 1

2d ,

3

2d , and

0A

1A

2A

3A

4A5A

6A

7A

Figure 3.8 Constellation maps and decision boundaries of triangular 8-QAM with a phase

estimation error .

7

2d for signal points “000”, “001”, and “100”, respectively. Therefore, the average SNR

per symbol is expressed as

2 22

2

3 72 2 4

2 2 2 9.

8 8s

d d d

d

(3.51)

Here, we have 8

.9 sd Note that Gray coding cannot be used in the bit mapping.


–83–

Assume, without loss of generality, that symbols corresponding to “000”, “001”, and

“100” are sent. Thus, we have

1 1 1( | ) ( | , 000) ( | ,001) ( | ,100).

4 4 2B B B BP e P e P e P e (3.52)

Supposing that symbol “000” is transmitted, the conditional BER ( | ,000)BP e is given

by

sin cos 3 sin cos2 2 6 2 2 2 6 21

( | ,000) .6

sin cos2 2 6 2

B

d derfc d erfc d

P ed

erfc d

(3.53a)

Similarly, we obtain

3 sin cos 3 sin cos2 2 3 2 2 2 3 21

( | ,001) ,6

3 sin sin 2 3 sin sin2 2 2 2 2 2 2 2

B

d derfc d erfc d

P ed d

erfc d erfc d

(3.53b)

7 sin cos2 2 2 21

( | ,100) .6 3

2 7 sin sin arctan2 2 2 2

B

derfc d

P ed

erfc d

(3.53c)

Note that the details of derivation are left to the reader because they are similar to that of

rotated 8-star QAM. We see from equations (3.53a) through (3.53c) that a lot of sine-cosine

product terms appear in the equations. Therefore, (3.53a) through (3.53c) can be simplified by

using the product-to-sum formula, and the results are given by


–84–

3sin 3 sin

4 2 6 4 2 61( | )

24 3sin

4 2 6

3 5 3sin sin

4 2 3 4 2 31 +

24

B

d d d derfc erfc

P ed d

erfc

d d d derfc erfc

3 3sin 2 sin

2 2 2 2

1 7 7cos

2 21

.12 1 7 cos arctan 3 2 7 3

2 cos arctan2 2 2

d d d derfc erfc

d derfc

d derfc

(3.54)

where sin cos sin sin 2 and

sin sin cos cos 2 are used in the simplification.

Since (3.54) only contains some single-erfc functions, approximate BER of triangular

8-QAM can also be obtained as follows


–85–

12 2 2 222

2 2

12 2 2 222

2 2

2 2

31 exp 1

8 4 81 1( | )

245 3

4 1 exp 18 4 5 8

31

8

1 1

24

B

d dd

dP e

d dd

d

d

12 222

2 2

12 2 2 222

2 2

12 2 2 222

2 2

3exp 1

4 8 3

9 31 exp 1

8 4 9 8

3 33 1 exp 1

2 4 3 2

dd

d

d dd

d

d dd

d

12 2 2271 1 1

1 exp 2 .6 2 2 4

d dS

(3.55a)

where

1 22

2 120

0

1 12 exp ,

2 4 4

AdS A

A

(3.55b)

2 20 2

37 7cos arctan

21 7 3 3cos 2arctan cos arctan ,

4 2 4 22A d d

(3.55c)

2 21

37 7 cos arctan

27 3 3sin 2arctan sin arctan .

4 2 2 2A d d

(3.55d)



–86–

Figure 3.9 and 3.10 illustrate that the analytical BER expressions are verified by using MC

simulations in perfectly coherent case and in the case of 2 38 10 , respectively. We

can see from figure 3.9 that triangular 8-QAM performs the best in perfectly coherent case,

and rotated 8-star QAM performs the second; whereas 8-PSK and 8-star QAM have nearly

the same performance, and both of them performs the worst. The reason can be explained by

comparing their minimum Euclidean distances dmin as follows: min,8

0.943 ,9tri s sd

min, 2 0.64 0.905 ,rotated s sd min,2

0.816 ,3rect s sd

min,8 2 sin 0.765 ,8PSK s sd and min,8 2 0.54 0.764 .star s sd We can

see that the results in figure 3.9 are consistent with the minimum Euclidean distance analysis.

6 8 10 12 14 16 18 20

10-15

10-10

10-5

100

SNR/bit b=E

b/N

0 [dB]

BE

R

8-PSKrectangular 8-QAM8-star QAMrotated 8-star QAMtriangular 8-QAMMC simulations

Figure 3.9 The BERs from analysis and MC simulations in perfectly coherent case.


–87–

5 10 15 20 2510

-15

10-10

10-5

100

SNR/bit b=Eb/N

0 [dB]

BE

R


Figure 3.10 The BERs from analysis and MC simulations when 2 3 28 10 .rad

6 8 10 12 14 16 18

10-10

10-8

10-6

10-4

10-2

100

SNR/bit b=Eb/N

0 [dB]

BE

R


Solid line: numerical exact BERsDashed line: Approximate BERs

Ts = 110 -5

Figure 3.11 Comparison between numerical BERs, MC simulations, and approximate BERs

when 51 10sT with optimum memory length.


–88–

On the other hand, when 2 increases to 38 10 , it is shown in figure 3.10 that triangular

8-QAM, rectangular 8-QAM, and 8-PSK start to have an error floor at high SNR region. Only

8-star QAM and rotated 8-star QAM can perform well at large phase error variance. Moreover,

8-star QAM also starts to outperform rotated 8-star QAM in terms of BER when phase

estimation error variance is large, as depicted in figure 3.10. This indicates that 8-star QAM

should have the largest LLW tolerance among all the 8-point modulation formats.

Furthermore, the approximate BERs of all above five 8-point modulation formats are

plotted in figure 3.11 when 51 10sT , which corresponds to linewidth per laser

of 100 kHz in a 10-Gsymbol/s system. Note that the PDF of phase estimation error in DA ML

PE (1.18) is used from this chapter on. The phase error variance 2 can be calculated

given linewidth per laser symbol duration product sT , SNR/symbol s , and memory

length L. Here, optimum memory length (1.19) is employed to plot the figure. Exact

numerical integration results and MC simulations are also presented to show that the

approximations are accurate. Additionally, it is also found that the derived approximate BERs

can be used to quickly estimate the LLW tolerance of the 8-point modulation formats in DA

ML PE. The LLW tolerance is examined and plotted in figure 3.12 based on our numerical

integration and approximate results. The SNR references are the SNR/bit ( b ) at BER=10-4 in

the perfectly coherent case, which are 11.7 dB, 11.1 dB, 11.7 dB, 10.6 dB, and 10.3 dB for

8-PSK, rectangular 8-QAM, 8-star QAM, rotated 8-star QAM, and triangular 8-QAM,

respectively. The values of sT leading to a 1-dB SNR/bit ( b ) penalty at BER=10-4

from exact numerical BERs and approximated BERs can be summarized in table 3.1.


–89–

10-6

10-5

10-4

10-3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Linewidth per laser / symbol rate (TS)

SN

R p

en

alty

@ B

ER

=1

0 -4 [d

B]

8-PSKrectangular 8-QAM8-star QAMrotated 8-star QAMtriangular 8-QAM

Solid line: From analysisDashed line: From approximation

Figure 3.12 The SNR/bit penalty as a function of linewidth per laser symbol duration product

( sT ) when BER=10-4 from numerical integration and approximation in different 8-point

modulation formats.

TABLE 3.1: The LLW tolerance comparison between 8-point modulation formats

sT 8-PSK Rectangular

8-QAM

8-star QAM Rotated

8-star QAM

Triangular

8-QAM

From exact

numerical

BERs

53.6 10 57.1 10 46.5 10 42.2 10 41.3 10

From

approximate

BERs

52.7 10 55.5 10 43 10 41.4 10 59 10


–90–

It can be seen from figure 3.12 and table 3.1 that 8-star QAM indeed has the largest LLW

tolerance among 8-point modulation formats. Moreover, figure 3.12 and table 3.1 show that

the estimated values of LLW tolerance are all smaller than the ones using exact numerical

BERs. This is because the approximate BERs are upper bounds of the exact numerical BERs.

Therefore, they provide a more rigid LLW requirement than the exact numerical BERs.

Next, we will show the results for the ring ratio optimization algorithms (3.37) and (3.50).

From equation (3.37a) to (3.37d) and equation (3.50a) to (3.50c), it can be seen that optimum

ring ratio opt is a function of SNR/symbol s and phase error variance 2 . Especially,

for the ring ratio optimization algorithm of 8-star QAM (3.37a) to (3.37d), opt is a function

of 2s (also can be written as 2

0

sE

N ). The dependence of opt on 2

0

sE

N can be

interpreted as that the ring ratio is optimized to a point where the impact between the phase

error and AWGN gets balanced at that point. For example, when phase error variance 2

dominates, errors mostly occur among the 4 points in the inner ring. Therefore, ring ratio

needs to be small to enlarge the distance between adjacent points in the inner ring. Yet when

AWGN N0 dominates, ring ratio needs to be relatively large to increase the distance between

adjacent points between inner and outer ring.

The term opt as a function of 2s for 8-star QAM is plotted in figure 3.13. We

also scan the numerical BERs of (2.7) as a function of the ring ratio given a fixed average

15 dBb and 2 3 21 10 rad in the cases of 8-star QAM and rotated 8-star QAM, and

the results are shown in figure 3.14.


–91–

10-4

10-3

10-2

10-1

2

2.1

2.2

2.3

2.4

2.5

2.6

s2

op

t

optimum ring ratio for 8-star QAM

Figure 3.13 Optimum ring ratio as a function of SNR/symbol phase estimation error variance

product for 8-star QAM.

It is shown that given the above condition the optimum ring ratio from exact numerical

integration is 2.4 and 2.0 for 8-star QAM and rotated 8-star QAM, respectively. Then we

substitute the same parameters into the ring ratio optimization algorithms, and obtain the

optimum ring ratio values of 2.45 and 2.0, for 8-star QAM and rotated 8-star QAM,

respectively. The ring ratio shift for 8-star QAM is only 0.05, which corresponds to an SNR

per bit penalty of 0.015 dB at BER=10-4. Therefore, the ring ratio optimization algorithms can

be used to optimize the ring ratio in 8-star QAM and rotated 8-star QAM. In real experiments,

it is difficult to maintain a stable ring ratio. Hence, we also examine the SNR penalty due to

the ring ratio shift under this condition as depicted in figure 3.15. This figure illustrates that

both 8-star QAM and rotated 8-star QAM can tolerate a large ring ratio fluctuation around

their corresponding optimum ring ratio values. Although numerical integration can be used to

determine the optimum ring ratio, it is time consuming.


–92–

2 4 6 8 1010

-10

10-8

10-6

10-4

10-2

100

BE

R

8-star QAM, opt

=2.4

rotated 8-star QAM, opt

=2.0

From numerical integration:

b=15dB, 2 =110-3 rad2

Figure 3.14 BER as a function of ring ratio from numerical integration given 15 dBb

and 2 3 21 10 rad for 8-star QAM and rotated 8-star QAM.

1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

SN

R p

en

alty

@B

ER

=1

0-4 [d

B]

8-star QAMrotated 8-star QAM

Figure 3.15 SNR penalties due to ring ratio shift for 8-star QAM and rotated 8-star QAM.


–93–

As can be seen in figure 3.14, a large range of values of the ring ratio has to be scanned to

locate the optimum values. The ring ratio optimization algorithms give us an explicit way to

see that the optimum ring ratio is not a constant, but depend on two parameters, i.e.,

SNR/symbol and phase estimation error variance. Especially for 8-star QAM, the ring ratio

optimization algorithm depends on the product of SNR/symbol and phase estimation error

variance. They also allow engineers to make a quick estimation of the optimum ring ratio for

8-star QAM and rotated 8-star QAM under different parameters.

Finally, although BER performance of rotated 8-star QAM is better than that of 8-star

QAM at a small phase error variance (as shown in figure 3.9), the performance of 8-star QAM

will surpass that of rotated 8-star QAM at a large phase error variance (as can be seen in

figure 3.10). Therefore, next we will discuss the reason why the LLW tolerance of 8-star

QAM is larger than that of rotated 8-star QAM, as illustrated in figure 3.12 and table 3.1. To

analyze the reason, we need to compare the approximate BER dominant terms (3.35) and

(3.48) of 8-star QAM and rotated 8-star QAM.

At a small phase error variance 2 , (3.35) and (3.48) can be further approximated to

222 11 1 1

| 8-star exp exp ,12 2 4B s sP e

(3.56)

and

2 2 21 1 1| rotated 8-star exp 3exp ,

12 2B s sP e M

(3.57)

respectively. We substitute

,8 2.4opt star , ,8 0.54opt star , (3.58a)

, 2.0opt rotated , , 0.64opt rotated (3.58b)


–94–

into (3.56) and (3.57), respectively. Obviously, we get

2 21 1exp 0.54 exp 0.64

2 2s s in each first exponent term in (3.56) and (3.57),

and 2 2exp 0.54 0.49 3exp 0.64 0.54s s

in the second exponent terms.

This proves that | 8-star | rotated 8-starB BP e P e when the phase error variance is

small.

When the phase error variance becomes larger, we look at (3.35) and (3.48). 2 does

not affect the 2nd exponent term in (3.35), yet the values of the 2nd and 3rd exponents in (3.48)

become larger due to the increasing positive values of the terms 22 22 s M and

22 2 2s M in the exponents as 2 becomes larger. This leads to worse BER

performance of rotated 8-star QAM at a large phase estimation error variance, thus causing

the smaller LLW tolerance of rotated 8-star QAM compared with 8-star QAM.

Furthermore, we will physically explain the reason from the constellation maps and

decision boundaries of 8-star QAM and rotated 8-star QAM. From figure 3.6 (8-star QAM

constellation), the 2nd exponent term in (3.35) relates to the probability that phase-error

rotated “000” falls into the error-decision regions 4 5 6 7A A A A (outside the square

region) and also the probability that phase-error rotated “100” falls into the error-decision

regions 0 1 2 3A A A A (the square region). It can be seen that phase-error rotated “100”

will get a bit closer to the straight-line decision boundary, and phase-error rotated “000” will

get even further away from the decision boundary at a large phase error variance. This cancels

out the impact of the phase error which can be seen in the 2nd exponent term in (3.35), thus


–95–

leading to a larger phase noise tolerance for 8-star QAM. However, from figure 3.7 (rotated

8-star QAM constellation), the 2nd exponent term in (3.48) relates to the probability that 0S

falls into the error-decision regions 4 ,A 6 ,A and 6 ,A and the 3rd exponent term relates to

the probability that 4S falls into the error-decision regions 1 ,A 1,A and 0 .A We can see

that the rotated signal point 0S will get much closer to the decision boundaries L2 and L1,

and 4S will get much closer to the decision boundaries L3 and L2 at a large phase error

variance. This is due to the fact that decision boundaries L1, L2, and L3 are tilted which makes

the phase-error rotated signal points 0S and 4S easier to get close to. And also this is the

reason why rotated 8-star QAM tolerates smaller phase estimation error variance. Therefore, a

large-phase-error rotated signal point in rotated 8-star QAM has more probability of falling

into an error-decision region. This can physically explain why BER performance of rotated

8-star QAM is worse than 8-star QAM at a large phase error variance.

3.6 Conclusions

In this chapter, the conditional and approximate BER expressions of five 8-point modulation

formats (8-PSK, rectangular 8-QAM, 8-star QAM, rotated 8-star QAM, and triangular

8-QAM) were derived in the presence of a phase estimation error and AWGN. The

approximate BERs can be employed in any PE algorithms given the PDF of the phase

estimation error. Here, DA ML PE was used in the analysis as an illustration. It was found

that these approximations can quickly estimate the BER performance under different

parameters, such as SNR/bit, LLW and symbol rate, and also can predict the lower bound on


–96–

the LLW requirement. The accuracy of the approximations was confirmed via numerical

integrations and MC simulations.

To optimize the BER performance in the duo-ring modulation formats (such as 8-star

QAM and rotated 8-star QAM), the approximate BERs was minimized with respect to the

ring ratio. This led to two new ring ratio optimization algorithms. It was found that the

optimum ring ratios are not constants, but depend on two parameters, namely, SNR/symbol

and phase error variance. These algorithms can provide us a quick estimation of the optimum

ring ratio for different parameters, instead of using time-consuming numerical integration and

scanning over a large range of values of ring ratio. Therefore, our work showed how to place

the signal points to optimize the performance in 8-star QAM and rotated 8-star QAM, and can

also be applied for other duo-ring modulation formats.

Finally, it was found that rotated 8-star QAM has better BER performance than 8-star

QAM at a small LLW, yet 8-star QAM outperforms rotated 8-star QAM in terms of BER at a

large LLW. The reason was also discussed analytically and physically in the results and

discussions part. Thus, 8-star QAM can be considered as a promising modulation format in

coherent optical communication systems due to its high phase noise tolerance.


–97–

Chapter 4

Performance Analysis of Higher-order

Modulation Formats

In this chapter, we start to analyze higher-order modulation formats ( 16M ). Since the

theoretical analysis of higher-order modulation formats ( 16M ) is complicated, we only

focus on two promising modulation formats, i.e., 16-star QAM (Type I [91]) and 64-QAM.

The conditional BER expression of 16-star QAM is analytically obtained based on the Craig’s

method in the presence of a random phase error and AWGN. Based on this new BER

expression, the corresponding optimum ring ratio of 16-star QAM under different LLWs is

presented. LLW and ring ratio fluctuation tolerances of 16-star QAM are also investigated. It

is shown that 16-star QAM has larger LLW tolerance than 16-PSK and 16-QAM (Type III

[91]).

Furthermore, we know that the SE of 64-QAM is 1.5 times that of 16-PSK. Therefore, in

this chapter, the exact conditional BER and LLW tolerance of 64-QAM are analytically


–98–

examined. Our results show that the LLW tolerance of DA ML PE in 64-QAM and 16-PSK

are nearly the same, whereas 64-QAM has a higher SE than 16-PSK.

4.1 Analysis of 16-star QAM

4.1.1 Introduction of 16-star QAM

inR outR

Figure 4.1 Constellation map and decision boundaries of 16-star QAM with a phase

estimation error .

Figure 4.1 shows the constellation map and decision boundaries of a 16-star QAM. Same as

8-star QAM, the inner ring radius inR is defined as ,in sR and outer ring radius

outR can be expressed as ,out in sR R where is the ring ratio between


–99–

outer and inner ring. Relation (3.25) 2 2( 1) 2 1, >1 and 0< <1 can also be

applied to 16-star QAM. Note that here 4s b because 4 bits transmit as a symbol.

It can be observed that the decision regions are polygons. Therefore, we will introduce

how to use Craig’s method [92] to calculate the exact SER in polar coordinates.

4.1.2 Craig’s Method

Figure 4.2 One signal point and one of its decision regions.

In [92], Craig proposed a simple method of calculating exact SER using polar coordinates.

This approach is more convenient for calculating the SER when the decision regions are

polygonal. Figure 4.2 is an illustrative figure showing how to calculate the probability of one

signal point falling into one of its corresponding error regions (an arbitrary-shaped region as

the shaded area shown in the figure). Here, kx is the distance from the signal point to one of

the corners of its decision region, k is the angle of the corner, and k is the spreading

angle between the signal point and its decision boundary, as can be seen in figure 4.2.


–100–

For the assumed AWGN having independent in-phase and quadrature components, the

bivariate Gaussian density function can be expressed in rectangular coordinates as

2 2

, 2 2

1( , ) exp ,

2 2X Y

n n

x yf x y

(4.1)

where 20 2n N is the variance of the AWGN. Since cosx r and siny r , so the

bivariate Gaussian density function can be converted into polar coordinates’ form as

2

2 2( , ) exp .

2 2n n

r rp r

(4.2)

Therefore, considering the case as shown in figure 4.2, the probability of this signal point

falling into the shaded area of error can be calculated by

2

2 20 0

2 2

2 20 0

( , ) exp2 2

1 1 exp exp .

2 22 2

k k

k k

kn nD D

Dn n

r rP d p r dr d dr

r Dd d

(4.3)

Here, each subscript k represents a different decision region as shown in figure 4.2. D is

the distance from the signal point to a point A on the boundary, and can be obtained by

using the sine theorem

sin

.sin( )

k k

k

xD

(4.4)

Substituting (4.4) into (4.3), it gives that

2 2

2 20

sin1exp[ ] .

2 2 sin ( )

kk k

kn k

xP d

(4.5)


–101–

Therefore, in the case of 16-star QAM as shown in figure 4.1, the exact SER conditioned on a

phase error can be written as:

2 28

2 21 0

sin1( | ) exp .

2 2 sin

kk k

sk n k

xP e d

(4.6)

There are two advantages by using Craig’s SER expression. Firstly, it simplifies the

conditional SER expression from a sum of double integrals to a sum of single integrals.

Additionally, the signal integral is easier to be numerically evaluated due to its finite

integration limits and exponential integrand.

4.1.3 Conditional BER of 16-star QAM

In [80], the conditional BERs of 16-PSK and 16-QAM have been derived, and the

corresponding phase error tolerances have been studied. 16-star QAM has an angle of 45

between two adjacent signal points, which is larger than those of 16-PSK and 16-QAM,

22.5 and 31 , respectively. Therefore, in this section, we will analytically derive the

approximate conditional BER of 16-star QAM by using the previously discussed Craig’s

method.

According to equation (4.6), we need to calculate three parameters kx , k , and k in 8

different regions as indicated in figure 4.1, and these parameters are given in TABLE 4.1.

TABLE 4.1: Parameter details of 16-star QAM with a phase estimation error

k

kx k k

1 ( 1)

2cosin

inR

R

2

1 ( 2 1)( 1) 2sintan

( 1) 2cos


–102–

2 inR 8

1 ( 2 1)( 1) 2sintan

( 1) 2cos

3 ( 1)

2cosin

inR

R

2

1 ( 2 1)( 1) 2 sintan

2 cos ( 1)

4 4x 4 4

5 ( 1)

2cosin

inR

R

2

1 ( 2 1)( 1) 2sintan

( 1) 2cos

6 inR 8

1 ( 2 1)( 1) 2sintan

( 1) 2cos

7 ( 1)

2cosin

inR

R

2

1 ( 2 1)( 1) 2 sintan

2 cos ( 1)

8 8x 8 8

Here, we have

2 2

42 1 1

( 1) sin cos ,2 2inx R

14

( 2 1)( 1) 2 sintan ,

2 cos ( 1) 8

14

7 ( 2 1)( 1) 2 sintan ,

8 2 cos ( 1)

2 2

82 1 1

( 1) sin cos ,2 2inx R

18

( 2 1)( 1) 2 sintan ,

2 cos ( 1) 8


–103–

18

7 ( 2 1)( 1) 2 sintan .

8 2 cos ( 1)

In perfectly coherent case (by setting 0 ), region 2 (1, 3, and 4) will be symmetric to

the region 6 (5, 7, and 8) as indicated in figure 4.1. And the detailed parameters are given in

TABLE 4.2.

TABLE 4.2: Parameter details of 16-star QAM in perfectly coherent case

k kx k k

1 ( 1)

2inR

2

1 ( 2 1)( 1)

tan( 1)

2 inR

8

1 ( 2 1)( 1)

tan( 1)

3 ( 1)

2inR

2

1 ( 2 1)( 1)

tan( 1)

4 4x 4 4

Here, we have

2 2 24 ( 2 1) ( 1) ( 1) ,

2inR

x

14

( 2 1)( 1)tan ,

( 1) 8

14

7 ( 2 1)( 1)tan .

8 ( 1)

Since it is more meaningful to evaluate a system in terms of its BER, the approximate

conditional BER when the Gray code is considered can be calculated as [68]


–104–

1

( | ) ( | ).4B sP e P e (4.7)

4.1.4 Results and Discussions

10 15 20 2510

-20

10-15

10-10

10-5

100

SNR/bit b [dB]

BE

R

= 2.6

Marker +: MC simulationsLines: Analytical BER

= 1.8

Figure 4.3 The BERs of 16-star QAM from analysis and MC simulations before and after

optimization when 2 3 21 10 rad .

The approximate BER of 16-star QAM is verified using MC simulations, as shown in figure

4.3. Note that the analytical BERs are a bit larger than the ones obtained from MC simulations

due to the approximation. It can be seen that the SNR penalty as a result of the approximation

(4.7) is less than 0.4 dB at a BER of 410 .

Generally, varying the radii of the two rings in 16-star QAM would severely affect its

performance, as depicted in figure 4.3. To find the optimum ring ratio, we scan the analytical

BER as a function of the ring ratio given a fixed average SNR per bit 15 dB, and the

results are shown in figure 4.4. As can be seen from figure 4.4, optimum ring ratio, which

corresponds to the minimum BER, should be different under different LLW.


–105–

0 2 4 6 8 1010

-5

10-4

10-3

10-2

10-1

100

BE

R

2 =110-4,

opt=1.8

2 =410-3,

opt=1.7

Figure 4.4 BER as a function of ring ratio with different 2 .

It is difficult to derive the ring ratio optimization algorithm for 16-star QAM. Hence, we

can only numerically investigate the optimum ring ratio. When the SNR per bit is fixed at 15

dB, and the derived BER expression is also used, the corresponding optimum ring ratio is

obtained as a step function of different range of linewidth per laser symbol duration product

( sT ) as shown in figure 4.5 as well as in Table 4.3. Since resolution of 0.1 is used to

calculate the optimum ring ratio, step-like characteristics is shown in figure 4.5. If the

resolution goes to infinitesimal, the curve in figure 4.5 will be continuous. Also, an increase

of LLW leads to a decrease of optimum ring ratio when SNR is fixed (explanation on p. 90).

TABLE 4.3: Optimum ring ratio in different range of sT

Range of sT Corresponding optimum ring ratio

0 ~ 52.8 10 1.8

52.9 10 ~ 43 10 1.7

> 43 10 1.6


–106–

As can be read from figure 4.5, for 51 10sT and in a 40-Gbit/s system, which

corresponds to a linewidth per laser ( ) of 100 kHz, then opt has to be chosen as 1.8.

10-7

10-6

10-5

10-4

10-3

1.55

1.6

1.65

1.7

1.75

1.8

1.85

Linewidth per laser / symbol rate (Ts)

Op

timu

m r

ing

ra

tio

opt

Figure 4.5 Optimum ring ratio versus different linewidth per laser symbol duration product.

1.6 1.7 1.8 1.9 2 2.1 2.20

0.2

0.4

0.6

0.8

1

1.2

1.4

SN

R P

en

alty

@B

ER

=1

0-4 [d

B]

opt=1.8T

s=2.710-6

Figure 4.6 SNR penalty due to ring ratio fluctuation.


–107–

By using the optimization results obtained here, we see that 16-star QAM with the optimized

ring ratio ( 1.8opt ) can achieve about 2.5-dB improvement over the non-optimal one at

BER=10-4, as shown in figure 4.3.

However, in real experiments, ring ratio may fluctuate around optimum value. Ring ratio

fluctuation will lead to SNR penalty, which causes a degradation of the system performance.

SNR penalty at a BER of 10-4 due to ring ratio fluctuation is depicted in figure 4.6. As

sT is chosen to be 62.7 10 , opt should be 1.8. As shown in the figure, the range of

ring ratio leading to less than 1-dB SNR penalty is from 1.6 to 2.17. This illustrates that

16-star QAM has a medium tolerance to ring ratio fluctuation. We should note that the ring

ratio fluctuation tolerance of 16-star QAM is smaller than that of 8-star QAM and rotated

8-star QAM.

10-8

10-7

10-6

10-5

10-4

0

0.2

0.4

0.6

0.8

1


SN

R p

en

alty

@ B

ER

=1

0 -4 [d

B]

16-PSK16-QAM16-star QAM

Figure 4.7 The SNR penalty as a function of linewidth per laser symbol duration product.


–108–

Finally, we see from figure 4.7 that the linewidth per laser symbol duration products

leading to a 1-dB SNR penalty at BER= 410 are found to be 68.3 10 , 54.5 10 , and

58.5 10 for 16-PSK, 16-QAM and 16-star QAM, respectively, corresponding to 83 kHz,

450 kHz and 850 kHz linewidth for each laser at a 40-Gb/s system. It shows that 16-star

QAM has more LLW tolerance than 16-PSK and 16-QAM.

4.2 Analysis of 64-QAM

4.2.1 Conditional BER of 64-QAM

0S

Figure 4.8 Constellation map, decision boundaries, and Gray code mapping of 64-QAM with

a phase error .

The constellation map with Gray coding and decision boundaries of 64-QAM is shown in

figure 4.8. Here, the conditional BER ( | )BP e of 64-QAM can be computed by


–109–

computing the error probabilities for the most significant bit (MSB), the middle bit (MB), and

the least significant bit (LSB), as indicated in figure 4.9. Note that each input bit is equally

likely to end up by being any one of the six bits. The average symbol energy is given by

242 ,s d therefore, 42.sd

110110

0S

0 d

d

MSB

Middle bit

LSB

I

Q

Figure 4.9 Details in the decision region of signal S0.

Due to the symmetry between the MSB, MB, and LSB, the average BER conditioned on a

phase error can be represented by

1| | | | .

6B MSB B MB B LSB BP e P e P e P e (4.8)

Note that there are two groups of MSB, MB, and LSB as shown in figure 4.9. Without loss of

generality, symbols in the first quadrant are considered to be sent. Therefore, we have

Contains 16 symbols in the first quadrant

| ,110110 | ,110111 ......1| .

16 ...... | ,100100

MSB B MSB BMSB B

MSB B

P e P eP e

P e

(4.9)

This is same for |MB BP e and |LSB BP e .


–110–

Here, we only take symbol “110110” as an example. It is easy to show that

1| ,110110 2 cos ,

2 4MSB BP e erfc d

(4.10a)

1| ,110110 4 2 cos

2 4

1 4 2 cos ,

2 4

MB BP e erfc d d

erfc d d

(4.10b)

1| ,110110 2 2 cos

2 4

1 1 6 2 cos 2 2 cos

2 4 2 4

1 6 2 cos .

2 4

LSB BP e erfc d d

erfc d d erfc d d

erfc d d

(4.10c)

The detailed derivation of the other terms is left to the reader for lack of space. The perfectly

coherent BER expression is given by setting 0 .

4.2.2 Results and Discussions

5 10 15 20 2510

-10

10-8

10-6

10-4

10-2

100

SNR/bit [dB]

BE

R

2 =110-4 rad2

Solid blue line: 64-QAMDashed red line: 16-PSKMarker + : MC simulations

5 10 15 20 2510

-10

10-8

10-6

10-4

10-2

100

SNR/bit [dB]

BE

R

2 =610-4 rad2

Solid blue line: 64-QAMDashed red line: 16-PSKMarker + : MC simulations

Figure 4.10 The BERs of 64-QAM and 16-PSK from analysis and MC simulations.

(a) 2 41 10 rad2, (b) 2 46 10

rad2.


–111–

Figure 4.10 illustrates that the derived BER expression is justified by using MC

simulations. Moreover, the BER performance of 64-QAM is compared with that of 16-PSK. It

can be seen that the BER performance of 64-QAM is only slightly worse than that of 16-PSK.

At the BER level of 410 , the SNR penalty is only 0.3 dB when 2 41 10 rad2, and is 0.5

dB when the phase error variance increases to 46 10 rad2.

10-7

10-6

10-5

0

0.2

0.4

0.6

0.8

1


SN

R p

en

alty

@ B

ER

=1

0 -4 [d

B]

64-QAM16-PSK

Figure 4.11 The SNR penalty as a function of linewidth per laser symbol duration product

( sT ) when BER= 410 with optimum memory length.

Finally, the LLW tolerance of DA ML PE in the two modulation formats is also examined,

as shown in figure 4.11. The values of sT leading to a 1-dB penalty are 68.3 10 and

69.6 10 for 16-PSK and 64-QAM, respectively. This is due to the fact that the average

angular distance between two adjacent signal points in 64-QAM is larger than that in 16-PSK.


–112–

We should note that same sT leads to different phase estimation error variance 2 for

64-QAM and 16-PSK because average SNR per symbol s is different for the two modulation

formats.

So far, we have done all the analytical work in this thesis. The analytical approaches in this

thesis are of significance in that they illustrate procedures in detail how to do the analysis and

optimize the performance of an advanced modulation format in the two-dimensional space.

Specifically, some important points are listed as follows. Firstly, the procedure of analyzing

QPSK and 8-PSK can be applied to all other higher-order PSK modulation formats. However,

the performance analysis of higher-order PSKs is not that important due to the fact that they

have increasingly small angular distance between adjacent symbols, which makes them less

noise tolerant. It can be seen that the performance of 16-PSK is similar to that of 64-QAM.

Hence, advanced QAM modulation formats are more worth being investigated. Moreover, the

BER analysis and ring ratio optimization procedures of star QAMs (8-star QAM, rotated 8-star

QAM, and 16-star QAM) can be extended to other multi-ring QAM formats. The BER analysis

approach of 64-QAM can also be extended to other higher-order ( 22 -ary, 1, 2,3...n n ) square

QAM systems, such as 256-QAM and 1024-QAM. Finally, some novel modulation formats,

such as (1, 3) and triangular 8-QAM, are also proposed. The analysis of them is difficult to do,

yet still can be done based on our general analytical procedure through some approximations. It

is shown that by better designing the constellation, (1, 3) can tolerate larger phase estimation

error than QPSK because it has larger angular distance between adjacent symbols. In the next

chapter, we will carry out experiments to verify our analysis and show the off-line processed

results.


–113–

4.3 Conclusions

The 16-star QAM performance has been optimized in terms of ring ratio under different LLW

based on our analytically derived BER. It was found that 16-star QAM can tolerate moderate

ring ratio fluctuation. More importantly, the LLW tolerance of 16-star QAM outperformed

those of 16-PSK and 16-QAM.

Moreover, based on our analytically derived BER and the DA ML PE algorithm, it was

found that the SE of 64-QAM outperforms that of 16-PSK while keeping a larger LLW

tolerance. It suggests that 16-star QAM and 64-QAM are two promising modulation formats

in coherent optical communication systems.


–114–

Chapter 5

Experiments

In this chapter, we conduct coherent optical B2B experiments to verify our analysis.

Experiments are carried out for 10-Gsymbol/s QPSK, 8-PSK, 8-star QAM, and rotated 8-star

QAM systems, respectively. After off-line DSP, our experimental results verify the

theoretically derived approximate BERs of the above four modulation formats. Moreover, the

ring ratio optimization algorithms of 8-star QAM and rotated 8-star QAM are also confirmed

by our experiments, respectively. Two PE schemes (Mth-power and DA ML) are used in the

off-line DSP for comparison. DA ML PE shows comparable performance with V&V

Mth-power PE but with much less computational complexity.

5.1 Experimental Setup

Figure 5.1 shows the transmitter for optical non return-to-zero (NRZ) and return-to-zero (RZ)

QPSK signals. Here, Data1 and Data2 are 10-Gb/s, length 211-1, pseudo-random bit sequences

(PRBS) of binary signals generated by a pulse pattern generator (PPG). An integrated optical

I/Q modulator is used to generate an NRZ QPSK signal. The integrated I/Q modulator has


–115–

two Mach-Zehnder Modulators (MZM) and one phase modulator (PM) inside [93]. The two

MZMs, operating at push-pull mode, modulate the drive signals Data1 and Data2 onto the

lightwave from the laser source. The PM introduces a 90 phase shift to the signals in either

one of the two branches. This can put the two signals in quadrature to each other. The output

of the I/Q modulator can be expressed as

1 2cos cos .2 2out inV t V t

E t E t jV V

(5.1)

Here, inE t is the incoming light (carrier) from the laser source, outE t is the modulated

light, 1V t and 2V t are the voltages of the drive signals Data1 and Data2, and V (null

point) is the half-wave voltage, which represents the required voltage switching the output

light intensity of the MZM from its maximum to its minimum value [94].

Delay

Figure 5.1 Optical transmitter for NRZ and RZ QPSK. PPG: pulse pattern generator; NRZ:

non return-to-zero; MZM: Mach-zehnder modulators; RZ: return-to-zero.

Before Data1 and Data2 are fed into the I/Q modulator, a component “Delay” is used in either


–116–

one of the data streams. We can tune the “Delay” in order to synchronize the two data streams.

Finally, 20-Gbit/s (10-Gsymbol/s) QPSK signal is generated, and the four points of QPSK are

in the four quadrants, respectively. Note that the QPSK signal is NRZ at this point. The NRZ

QPSK signal is converted to RZ format using a MZM driven by a 10-GHz clock signal. The

duty cycle of the RZ signals can be changed by varying the bias of the MZM, the amplitude,

and the frequency of the sinusoidal clock signal [95]. The output of the MZM is a 20-Gb/s RZ

QPSK signal. Note that in figure 5.1, the solid pointers represent that the outputs are electrical,

and the dotted pointers mean the outputs are optical.

The principle of the I/Q modulator indicates that it can generate M2-QAM when the drive

signals 1V t and 2V t are M-level. For example, 16-QAM signal can be generated by

feeding two 4-level signals into the I/Q modulator. The electrical M-level signals can be

obtained by using either an arbitrary waveform generator (AWG) or a serial of MZMs.

Usually, the quality of the electrical M-level signals generated from AWG is better than that

generated from a serial of MZMs, yet with higher cost.

Figure 5.2 Experimental setups for optical RZ 8-point star QAMs (8-star QAM and rotated

8-star QAM). LD: laser diode; LO: local oscillator; DSP: digital signal processing.


–117–

Next, we will generate optical RZ 8-point star QAM signals based on the RZ QPSK

signals. Figure 5.2 shows our experimental setup for optical RZ 8-point star QAMs. Data1, 2,

and 3 are 10-Gb/s, length 211-1, PRBS of binary signals generated by a PPG. The output of

the I/Q modulator is a 20-Gb/s NRZ QPSK signal. The QPSK signal is converted to RZ

format using a MZM driven by a 10-GHz clock signal. An amplitude modulator (AM) is used

after the RZ QPSK signal to generate a duo-ring QPSK, which is also named as 8-point star

QAMs as shown in figure 5.2. A normal AM is used to generate the 8-star QAM constellation

as shown in figure 3.6. On the other hand, a balanced single drive MZM is used as an AM

after the RZ QPSK signal to generate rotated 8-star QAM signals as shown in figure 3.7. A

10-Gb/s PRBS binary signal as Data3 is fed into the AM. We tune the bias of the AM to

choose different values of the ring ratio of 8-point star QAMs. The transmitter laser diode

(LD) we use in the experiment is either an external cavity laser (ECL) or a distributed

feedback (DFB) laser, which has 100 kHz or 5 MHz LLW, respectively. Now, a 30-Gb/s

optical RZ 8-point star QAM transmitter is built up. At the receiver side, the local oscillator

(LO) laser uses the same kind of laser as the transmitter LD does.

Figure 5.3 Experimental setup for optical RZ 8-PSK


–118–

Coherent detection and off-line DSP (as illustrated in Section 1.2 of Chapter 1) are used in

our scheme to calculate the BER and to draw the constellation map. Note that differential

decoding (DD) is employed to detect the phase changes rather than the absolute values of the

phase. The PE methods used in our off-line DSP processing are V&V Mth-power and DA

ML. We compare the BER performances between these two PE methods in the next section.

Figure 5.3 shows our experimental scheme for optical RZ 8-PSK. The difference between

figure 5.3 and figure 5.2 is that we change the AM to a phase modulator (PM). Note that

although RZ format is employed in our experiments, NRZ can also be used for experimental

verification. The comparison between RZ and NRZ formats may be the future work.


8 9 10 11 12 13

10-4

10-3

10-2

10-1

OSNR [dB]

BE

R

TheorectialExperimental MthPowerExperimental DA ML

Figure 5.4 BER performance comparison between experimental and theoretical BERs (2.23b)

for RZ QPSK.

Firstly, we conduct an experiment for QPSK. The transmitter and LO lasers used in the

experiment are both ECL lasers with 100 kHz LLW. The measured B2B BER performance of


–119–

QPSK is shown in figure 5.4. Theoretical ideal curve plotted from equation (2.23b) for is also

presented as a reference. We should note that the optical signal-to-noise ratio (OSNR) in the

experiment is different from the SNR/symbol s in our algorithm. The relation between s

and OSNR is given by [3]

OSNR ,2

ss

ref

R

B (5.2)

where sR is the symbol rate (10-Gsymbol/s in our experiment), and refB is the reference

optical bandwidth, which is commonly chosen to be 12.5 GHz.

Compared with theoretical ideal curve, the experimental results require approximately

1.5-dB higher OSNR to achieve a BER level of 10-4. This is due to the fact that several

implementation degradation issues, such as the bandwidth limitation of real devices, IQ

imbalance, and some other noise, are not taken into account in our analysis. Our derivation

only considers the laser phase noise and AWGN, thus giving the theoretical limit of the BER.

Moreover, we did not assume DD in our analysis. In the experiment, DD is used, which leads

to approximate doubling of the BER. This is because if one symbol is received incorrectly,

the output decision on this symbol and the next symbol will be both incorrect. This

approximate doubling of the BER accounts for around 0.8-dB OSNR penalty between the

theoretical and experimental results in figure 5.4. Therefore, the actual power penalty from

implementation itself is only 0.7 dB.

Furthermore, two PE methods are used to recover carrier phase of the received signal.

BER performance comparison between V&V Mth-power PE and DA ML PE is also plotted in

figure 5.4. In our off-line DSP, memory length L is selected to be 15 due to the small LLW of

ECLs. As can be seen in figure 5.4, DA ML can achieve comparable performance with


–120–

Mth-power. Moreover, DA ML is more computationally efficient than Mth-power, and does

not require phase unwrapping.

Next, we conduct experiments for 8-star QAM and rotated 8-star QAM. Before we

conduct the experiments, we need to find the optimum ring ratios for both 8-star QAM and

rotated 8-star QAM at total LLW of 200 kHz (transmitter and LO laser each with 100 kHz

LLW) by using our ring ratio optimization algorithms (3.37a) to (3.37d) and (3.50a) to

(3.50c). The optimum ring ratios opt are plotted as a function of different values of OSNR

in figure 5.5. From figure 5.5, we see that the optimum ring ratio opt of 8-star QAM varies

from 2.47 to 2.35 when OSNR changes from 11 dB to 19 dB. We choose ,8 2.4opt star

for 8-star QAM in our experiment for convenience. By using the approximate BER formula

(3.34), we find that the small ring ratio fluctuation around the chosen ,8opt star leads to

only a 0.01-dB OSNR penalty at BER=10-4, which can be ignored.

12 14 16 181.8

1.9

2

2.1

2.2

2.3

2.4

2.5

2.6

OSNR [dB]

Op

timu

m r

ing

ra

tio

opt

rotated 8-star QAM8-star QAM

Figure 5.5 Optimum ring ratios as a function of OSNR at LLW per laser of 100 kHz.


–121–

Similarly, the optimum ring ratio opt of rotated 8-star QAM is selected as

, 8 2.0opt rotated star for convenience at LLW=100 kHz per laser in our experiments.

Figure 5.6 Eye diagrams of signals at points a and b in figure 5.2 through direct detection. (a)

RZ QPSK (b) RZ 8-star QAM

Figure 5.6 shows the eye diagrams measured through direct detection at points a and b, which

are marked in figure 5.2. Figures 5.6(a) and (b) are the eye diagrams of RZ QPSK signal and

RZ 8-star QAM signal, respectively. Two levels can be clearly seen in figure 5.6(b).

11 12 13 14 15 16 17 18 19

10-4

10-3

10-2

OSNR [dB]

BE

R

Ring ratio=2.4 (optimum)Ring ratio=2.6Ring ratio=2.0

Theoretical

Experimental

Dashed line: DA ML PESolid line: Mth-power PE

8-star QAMLLW=100 kHz per laser

Figure 5.7 BER performance comparison of experimental RZ 8-star QAM signal and

theoretical formula (3.34) under different values of ring ratio at LLW=100 kHz per laser


–122–

11 12 13 14 15 16 17 18

10-4

10-3

10-2

OSNR [dB]

BE

R

Ring ratio=2 (optimum)Ring ratio=2.4Ring ratio=1.6

Theoretical

Experimental


rotated 8-star QAMLLW=100 kHz per laser

Figure 5.8 BER performance comparison of experimental RZ rotated 8-star QAM signal and

theoretical formula (3.47) under different values of ring ratio at LLW=100 kHz per laser

The measured B2B BER performance of 8-star QAM is shown in figure 5.7. Three values

of ring ratio are used in our experiment. Besides the optimum ring ratio of 2.4, another two

values of ring ratio, which are 2.6 and 2.0, are also measured for comparison. Theoretical

ideal curves plotted from (3.34) for three ring ratios are also presented as a reference.

Compared with theoretical ideal curves, the experimental results require approximately 2.3

dB, 2.9 dB and 3 dB higher OSNR to achieve a BER level of 10-4, for ring ratios equal to 2.4,

2.6 and 2.0, respectively. The reason of difference between experimental and theoretical

results is given previously. However, the trends of the curves from analysis and experiment

are the same. The experiment verifies our analysis that the BER performance of ring ratio 2.4

is better than those of ring ratio 2.6 and 2.0. Therefore, 2.4 is confirmed by experiment as the

optimum ring ratio for 8-star QAM when LLW per laser = 100 kHz. Moreover, it can be seen


–123–

from figure 5.8 that 2.0 is verified by experiment as the optimum ring ratio for rotated 8-star

QAM when LLW per laser = 100 kHz. Note that ring ratio drift occurs during the experiments

which adds some extra penalty to the performance, e.g. ring ratio=2.6 for 8-star QAM and

ring ratio=1.6 for rotated 8-star QAM.

In addition, BER performance comparison between V&V Mth-power PE and DA ML PE

is also presented, as plotted in figure 5.7 and figure 5.8. In our off-line DSP, memory length L

is selected to be 23 due to the small LLW of ECLs. As shown in the two figures, DA ML can

achieve comparable performance with Mth-power.

Figure 5.9 Constellation maps of (a) ring ratio=2.4 at OSNR=17 dB, (b) ring ratio=2.6 at

OSNR=18 dB, and (c) ring ratio=2.0 at OSNR=19 dB

The constellation maps of 8-star QAM under three ring ratios are depicted in figure 5.9.

Figure 5.9(b) shows the constellation map at ring ratio=2.6. We see that errors mostly occur

among the 4 points in the inner ring. On the other hand, errors mostly occur in adjacent points

between inner and outer rings in the case of ring ratio=2.0, as shown in figure 5.9(c). Note

that the results are consistent with the physical explanation on p. 90. Therefore, it is shown

that ring ratio 2.4 can optimize the constellation points of 8-star QAM and give the best BER

performance.


–124–

12 14 16 18 20

10-4

10-3

10-2

OSNR [dB]

BE

R

LLW=100 kHz (optimum ring ratio=2.4)LLW=5 MHz (optimum ring ratio=2.2)

Theoretical


8-star QAM

Experimental

Figure 5.10 BER performance comparison of experimental RZ 8-star QAM signal and

theoretical analysis under different LLWs.

Finally, we replace the transmitter and LO lasers with DFB lasers (each with LLW=5

MHz), and conduct the experiments. By using the ring ratio optimization algorithms, the

optimum ring ratios are selected to be 2.2 and 2.3 for 8-star QAM and rotated 8-star QAM,

respectively. Note that the memory length L is chosen to be 7 due to the large LLW of DFBs.

The BER results are shown in figure 5.10 and figure 5.12. As can be seen in the two figures,

8-star QAM has 1.9-dB OSNR penalty compared to rotated 8-star QAM when LLW=100 kHz.

However, 8-star QAM can still have a good BER performance at LLW=5 MHz, yet rotated

8-star QAM cannot. This proves the previous discussion that 8-star QAM can tolerate larger

LLW than rotated 8-star QAM as shown in section 3.5. Furthermore, the constellation map of

8-star QAM at LLW=5 MHz is depicted in figure 5.11. It is shown that at LLW= 5 MHz,


–125–

8-star QAM still can have a good recovered constellation. Yet rotated 8-star QAM cannot

clearly recover the received constellation as shown in figure 5.13. Also, we notice that rotated

8-star QAM may have more intensity fluctuation than 8-star QAM in the experiments.

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

Figure 5.11 Recovered constellation map of 8-star QAM.

8 10 12 14 16 18 20

10-4

10-3

10-2

OSNR [dB]

BE

R

LLW=100 kHz (optimum ring ratio =2.0 )LLW=5 MHz (optimum ring ratio =2.3 )

Dotted red line: TheoreticalDashed black line: DA ML PESolid blue line: Mth-power PE

rotated 8-star QAM

Figure 5.12 BER performance comparison of experimental RZ rotated 8-star QAM signal and

theoretical analysis under different LLWs.


–126–

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

Figure 5.13 Recovered constellation map of rotated 8-star QAM.

12 14 16 18 20 22

10-4

10-3

OSNR [dB]

BE

R

TheoreticalExperimental

Figure 5.14 BER performance comparison between experimental and theoretical BER

(3.3) for 8-PSK

At last, the experiment of a 10-Gsymbol/s 8-PSK system is conducted. Lasers used as a

transmitter and a LO are ECL lasers with LLW=100 kHz. The BER performance of 8-PSK is

shown in figure 5.14. Note that we only use Mth-power PE to recover the carrier phase. We

see that there are about 6-dB OSNR penalty between the experimental and theoretical results.


–127–

This is due to the phase chirp introduced by the PM [96]. Usually, 8-PSK is not a common

modulation format in coherent optical communication systems due to its small LLW

tolerance.

We should note that the experiments for the other modulation formats, such as (1, 3),

rectangular 8-QAM and triangular 8-QAM, etc., cannot be conducted due to the limitation of

the integrated I/Q modulator. These modulation formats can be optically generated with the

aid of an arbitrary waveform generator (AWG). Moreover, we also should note that the

cross-over SNR, optimum ring ratio, and phase noise tolerance comparison are all symbol rate

dependent due to the fact that laser phase noise variance 2 2 (2 )p sT is a function of

LLW and symbol duration.

5.3 Conclusions

We have experimentally generated optical RZ QPSK, 8-PSK, 8-star QAM, and rotated 8-star

QAM signals at a symbol rate of 10-Gsymbol/s, by using a parallel I/Q modulator followed

by an AM or a PM with binary drive signals. After off-line DSP, our experimental results

were compared with theoretical ones to show the accuracy of the analysis. The ring ratio

optimization algorithms of the two duo-ring modulation formats (8-star QAM and rotated

8-star QAM) were also demonstrated in our experiments. More importantly, from previous

discussions, 8-star QAM should perform better than rotated 8-star QAM in terms of BER

when the LLW is large. This phenomenon can also be observed in our experimental results.

Lastly, experimental data processed with DA ML PE presents comparable results with that

processed using the conventional V&V Mth-power PE.


–128–

Chapter 6

Conclusions and Future Work

6.1 Conclusions

The primary objective of this study was to systematically analyze the performance of different

advanced modulation formats in the presence of linear phase noise and AWGN. The specific

advanced modulation formats which have been studied in the thesis were 4-point, 8-point, and

higher-order signal constellations. Besides, coherent optical B2B experiments were also carried

out to verify our analytical work and simulations.

In this study, the exact conditional BER expressions and closed-form approximations of the

exact BER expressions of all the proposed advanced modulation formats have been derived in

the presence of linear phase noise and AWGN. This BER derivation approach gave an

illustration in detail on how to carry out the analysis for two-dimensional modulation formats.

Also, the approximate BERs can contribute to quick estimation and prediction of the BER

performance and LLW tolerance of a coherent optical system.

The approximate BERs can also be used to further study some advanced modulation

formats. First of all, in Chapter 2, the approximate BERs led to an initial result of the


–129–

cross-over SNR algorithm between QPSK and (1, 3). The cross-over SNR algorithm can be

employed to approximate the cross-over SNR points between QPSK and (1, 3) under different

LLW. Moreover, the ring ratio optimization algorithms of 8-star QAM and rotated 8-star

QAM were obtained in Chapter 3 based on the approximate BERs. These algorithms showed

that the optimum ring ratios are not constants, but as a function of SNR/symbol and phase

estimation error variance. These algorithms can also provide us a quick estimation of the

optimum ring ratio under different parameters, which can be used in real experiments. The

procedure of optimization can be extended to other duo-ring modulation formats to optimize

their performance. At last, the approximate BERs were used to find out the reason why 8-star

QAM has more phase noise tolerance than rotated 8-star QAM in Chapter 3.

Higher-order modulation formats, such as 16-star QAM and 64-QAM, were also

analytically investigated in Chapter 4 of this thesis. The optimum ring ratio under different

LLW of 16-star QAM was numerically examined. The results showed that 16-star QAM has

the largest LLW tolerance compared with 16-PSK and 16-QAM due to the fact that it has the

largest angular distance between adjacent symbols. Additionally, it showed that the LLW

tolerance of DA ML PE in 64-QAM is larger than that of 16-PSK, whereas 64-QAM has a

higher SE than 16-PSK.

These results above are of considerable importance since they provide equations and

comments which can be used for the electrical engineers to choose the best modulation formats

under different parameters, such as SNR/bit, LLW, symbol rate, ring ratio, and SE. Coherent

optical B2B experiments were conducted to verify our analysis in Chapter 5. Our experimental

results confirmed the theoretically derived approximate BERs and ring ratio optimization

algorithms. Also, DA ML PE showed comparable performance with V&V Mth-power PE in


–130–

our experiments.

6.2 Future Work

Several interesting directions are recommended for the future work:

One possible avenue is to continue the analysis in higher-order modulation formats

( 16M ). For 16-point modulation formats, rotated 16-star QAM and 4-ringed QPSK

may be worthy of investigation. However, it has to be pointed out that it is difficult to

analyze the higher-order modulation formats.

Nonlinear phase noise caused by fiber transmission can also be included in the

analysis. Notice that the performance analysis in the thesis is based on phase

estimation error and AWGN. Therefore, adding nonlinear phase noise in will not

affect any conclusions in the thesis. Future work can be done in analyzing the PDF,

especially the variance of the phase estimation error 2 in the case of including

both laser and nonlinear phase noise, and in renewing equations (1.18), (1.19) and

(1.20). Intensity noise may also be considered. Yet the intensity-noise problem is

much less severe when a balanced receiver is employed [1].

Another direct extension of this work is to carry out experiments for other advanced

modulation formats. Due to the constraint of our experimental setup, some advanced

modulation formats, such as 16-QAM, are difficult to generate. Serial MZMs

implementation is difficult to generate good multi-level signals. Good multi-level

electrical signals can be obtained using an AWG. We can further do some experiments

after an AWG is put into our setup.


–131–

Another interesting area for future work is to do performance analysis in circular

decision boundary. The decision boundary used in this thesis is a conventional one,

which is a bisector between adjacent symbols. The circular decision boundary for (1, 3)

is shown in figure 6.1. Compared with the conventional decision boundary as shown

in figure 2.7, the outer signal point, such as 1S , has more space to swing in the

presence of a phase error. This will lead to larger phase noise tolerance compared to

the conventional decision boundary, which also means better BER performance at

large phase estimation error region. This circular decision boundary can be applied to

other duo-ring modulation formats. It is worthy to analyze the performance and

optimum ring ratio in circular decision boundary, and to compare the results with that

in conventional decision boundary.

0A

1A2A

3A

0S

2S1S

3S

1S

Figure 6.1 Constellation map and circular decision boundaries of (1, 3).


–132–

Appendix

Conditional BER and SER Results in the

Presence of a Phase Error

A. Conditional SER in 4-PAM

0 0 0

1 cos 1 2 cos 1 3 cos 2( | ) .

4 4 4sd d d d d


(A.1)

B. Conditional BER in 4-PAM

0 0 0

0 0 0

1 cos 1 3 cos 1 2 cos|

8 8 8

1 2 cos 1 3 cos 2 1 3 cos 2 .

8 8 8

Bd d d d


d d d d d derfc erfc erfc

N N N

(B.1)


–133–

C. Conditional BER in 8-star QAM

1 1sin cos

2 4 2 4

1 1 1( | ) 1 2cos 1 1 2cos 1

6 2 2 2 2

1 1 (2sin 1) 1 ( 2sin 1)

2 2 2 2

s s

B s s

s s

erfc erfc

P e erfc erfc

erfc erfc

1 1sin cos

2 4 2 4

1 1 1 + (2 cos 1) 1 (2 cos 1)

6 2 2 2 2

1(2 sin 1)

2 2

s s

s s

s

erfc erfc

erfc erfc

erfc

.

11 ( 2 sin 1)

2 2serfc

(C.1)

D. Conditional BER in rotated 8-star QAM

1 1sin cos

2 4 2 41( | )

6 12sin 2sin

2 2 2

1 1sin

2 4 21

6

s s

B

s s

s

erfc erfc

P e

erfc M erfc M

erfc e

cos4

.1

sin sin2 2 2

s

s s

rfc

erfc M erfc M

(D.1)


–134–

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Publication List

Journal Papers

1. Hongyu Zhang, Pooi-Yuen Kam, and Changyuan Yu, “Performance analysis of coherent

optical 8-star QAM systems using decision-aided maximum likelihood phase estimation,”

Opt. Express, vol. 20, no. 8, pp.9302-9311, 2012.

2. Hongyu Zhang, Banghong Zhang, Changyuan Yu, and Pooi-Yuen Kam, “Experiments on

Coherent Optical 8-star QAM Systems using Decision-Aided Maximum Likelihood Phase

Estimation,” IEEE Photon. Technol. Lett, vol. 24, no. 23, pp. 2139-2142, 2012.

3. Hongyu Zhang, Banghong Zhang, Pooi-Yuen Kam, and Changyuan Yu, “Performance

Analysis and Experiments of Coherent Optical 8QAMs using Decision-Aided Carrier

Phase Estimation,” under preparation for J. Lightw. Technol.

4. Hongyu Zhang, Pooi-Yuen Kam, and Changyuan Yu, “Symbol-Error Rate Performance

of 4-Point Constellations in the Presence of Phase Estimation Error,” submit to IEEE

Photon. Technol. Lett.

5. Banghong Zhang, Hongyu Zhang, Changyuan Yu, Xiaofei Cheng, Yong Kee Yeo,

Pooi-Kuen Kam, Jing Yang, Yu Hsiang Wen, and Kai-Ming Feng, “An All-Optical


–149–

RZ-8QAM Signal Conversion Scheme Based on Four-Wave Mixing in Highly

Non-Linear Fiber,” accepted by IEEE Photon. Technol. Lett.

Conference Papers

1. Hongyu Zhang, Shaoliang Zhang, Pooi-Yuen Kam, Changyuan Yu, and Jian Chen,

“Optimized Phase Error Tolerance of 16-Star Quadrature Amplitude Modulation in

Coherent Optical Communication Systems,” OptoElectronics and Communications

Conference (OECC) 2010, paper 8P-15, Sapporo, Japan, July 2010.

2. Hongyu Zhang, Pooi-Yuen Kam, and Changyuan Yu, “Laser Linewidth Tolerance of

Coherent Optical 64-QAM and 16-PSK using Decision Aided Maximum Likelihood

Phase Estimation,” Conference on Lasers and Electro-Optics (CLEO) 2011, paper

JWA17, Baltimore, USA, May 2011.

3. Hongyu Zhang, Pooi-Yuen Kam, and Changyuan Yu, “Optimal Ring Ratio of 16-Star

Quadrature Amplitude Modulation in Coherent Optical Communication Systems,”

OptoElectronics and Communications Conference (OECC) 2011, paper 7P3_030,

Kaohsiung, Taiwan, July 2011.

4. Yi Yu, Hongyu Zhang, Changyuan Yu, “107Gb/s WDM DQPSK Systems with 50 GHz

Channel Spacing by Low-Crosstalk Demodulators,” OptoElectronics and

Communications Conference (OECC) 2010, paper 7P-23, Sapporo, Japan, July 2010.

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