Intradyne Signal Recovery and Equalization for Optical Satellite Links

Intradyne Signal Recovery and Equalization for Optical Satellite Links

PHILIP CONROY

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF ELECTRICAL ENGINEERING

and Equalization for Optical Satellite Links

PHILIP CONROY

Master of Science in Electrical Engineering Date: September 11, 2017

Supervisor: Nickolay Ivchenko (KTH), Janis Surof (DLR) Examiner: Tomas Karlsson

School of Electrical Engineering

Performed at the DLR Institute of Communication and Navigation

Abstract

An optical communication system employing intradyne reception with

offline digital signal processing for a geostationary satellite communi-

cation scenario is presented. The digital signal processing is improved

via the implementation and comparison of several timing recovery al-

gorithms, and the inclusion of an equalization stage. The system is

tested over a 10.45 km link through the atmosphere, in which 40 Gbit/s

transmission using binary phase shift keying in the optical C-band is

demonstrated. Results show that the system’s performance in the field

closely matches back-to-back measurements. Simultaneous channel

measurements show that turbulence in the atmospheric channel rep-

resents a worst-case scenario for a geostationary satellite uplink.

Sammanfattning

Häri presenteras ett optiskt kommunikationssystem för en satellit i

geostationär bana som använder intradyn mottagning med nedkopp-

lad digital signalbehandling. Flertalet algoritmer för timing recovery

implementeras och jämförs för att tillsammans med ett utjämnare för-

bättra den digitala signalbehandlingen. Systemet testas genom att sän-

da en 40 Gbit/s sändning med binära fasskiften i det optiska C-bandet

på en 10,45 km lång länk genom atmosfären. Fältmätningarna stäm-

mer väl överens med systemets resultat i back-to-back mätningar. Si-

multana mätningar på kanalen visar att turbulens i atmosfärskanalen

ger ett värsta scenario för upplänkning till en geostationär satellit.

First and foremost I would like to thank Janis Surof from DLR, super- visor of this thesis. Your constant help and guidance were invaluable and made this project possible. I couldn’t have asked for better super- visor. I would also like to thank Drs. Juraj Poliak and Ramon Mata Calvo for lending their expertise, and for giving me the opportunity to work on such an exciting project. Thank you to all my other colleagues at DLR who made this such a great workplace, and to the other stu- dents for showing me around Munich. I will miss the beer gardens.

On the KTH side I want to thank Drs. Nickolay Ivchenko and Tomas Karlsson for helping supervise and facilitate the project, as well as for teaching me some of my favourite courses in my degree. I would like to give a big thank you to Karl Bolmgren for helping me translate the abstract into Swedish. Thanks to all the wonderful friends I’ve met over the past two years in Sweden. My experience at KTH has been an overwhelmingly positive one, and that is due in no small part to all the great people I’ve been privileged to meet.

Finally, I would like to say thank you to my family and friends back in Canada, and especially to Julia. I would never have made it this far without your constant love and support.

Philip Oberpfaffenhofen, August 30, 2017

v

Introduction 1

1 Ground-to-GEO Communication 3

1.1 Free-Space Optical Channel . . . . 4

1.2 Coherent Communications . . . 10

1.2.1 PSK Modulation . . . 10

1.2.2 Intradyne Receiver . . . 11

1.2.3 SNR and Theoretical Performance Limits . . . 14

1.3 Digital Signal Processing . . . 17

1.3.1 Polarization Demultiplexing . . . 18

1.3.2 Recentering . . . 18

1.3.3 Timing Recovery and Downsampling . . . 19

1.3.4 Imbalance Compensation . . . 19

1.3.5 Frequency Offset Compensation . . . 19

1.3.6 Carrier Phase Recovery . . . 20

1.3.7 Equalization . . . 21

1.3.8 Demodulation . . . 21

2 Timing Recovery 23 2.1 Feedback-Based Methods . . . 25

2.1.1 Gardner . . . 26

2.1.2 Mueller and Müller . . . 28

2.2 Feedforward-Based Methods . . . 30

2.2.1 Oerder and Meyr (O&M) . . . 30

2.2.2 Lee . . . 32

2.3 Maximum Offset . . . 34

2.4 Frame Length Extrapolation . . . 35

2.5 Results and Discussion . . . 37

2.5.1 Discussion . . . 39

vi

3 Equalization and Matched Filtering 42

3.1 Pulse Shaping and Matched Filtering . . . 42

3.1.1 Theory of the Matched Filter . . . 43

3.1.2 Raised-Cosine Pulse . . . 44

3.2 Equalization . . . 47

3.2.1 Finite Impulse Response Filter . . . 48

3.2.2 Equalizer Structure . . . 48

3.2.3 Adaptive Equalization . . . 51

3.3 Results and Discussion . . . 55

3.3.1 Receivers . . . 55

3.3.2 Signal Statistics . . . 55

3.3.3 Sensitivity Results . . . 56

4 Outdoor Transmission 60 4.1 Experimental Setup . . . 61

4.2 Turbulence Conditions . . . 64

4.2.1 Scintillation Measurement . . . 66

4.2.2 Log-Normal Distribution . . . 66

4.2.3 Gamma-Gamma Distribution . . . 68

4.2.4 Distribution Fit . . . 69

4.2.5 Fading Time Scale . . . 70

4.3 Transmission Results and Discussion . . . 72

4.3.1 40 GBaud Results . . . 72

4.3.2 Preliminary 56 GBaud Results . . . 73

Conclusion 75 A Optical IQ Modulator 82 A.1 Mach-Zehnder Modulator . . . 83

B Intradyne Receiver 85 B.1 90 Degree Hybrid . . . 86

B.2 Balanced Photodetector . . . 86

B.3 Analog to Digital Converter . . . 88

C Lin CMA 89

1.1 GEO Feeder Link Scenario . . . . 3

1.2 Gaussian Beam . . . . 4

1.3 Power Scintillation . . . . 8

1.4 Optical IQ Modulator . . . 10

1.5 PSK Constellation Diagrams . . . 11

1.6 Intradyne Receiver . . . 12

1.7 Raw Intradyne Signal . . . 13

1.8 DSP Structure . . . 17

1.9 Polarization Demultiplexing . . . 18

1.10 Constellation Diagram Evolution . . . 22

2.1 Timing Error . . . 24

2.2 Feedback Timing Recovery . . . 26

2.3 Gardner Timing Error Detector . . . 27

2.4 Gardner S-Curve . . . 28

2.5 Feedforward Timing Recovery . . . 30

2.6 Lee vs. O&M Output . . . 33

2.7 Maximum Timing Offset . . . 35

2.8 Frame Length Extrapolation . . . 36

2.9 Timing Recovery 10GBaud BER Curves . . . 37

2.10 Timing Recovery 20GBaud BER Curves . . . 38

2.11 Timing Recovery 30GBaud BER Curves . . . 38

2.12 Timing Recovery 40GBaud BER Curves . . . 39

3.1 Matched Filter . . . 43

3.2 Raised Cosine Pulse . . . 45

3.3 Raised Cosine Spectrum . . . 45

3.4 Sinc Pulse Train . . . 46

3.5 Equalized Signal in Time Domain . . . 47

3.6 FIR Filter Block Diagram . . . 48

viii

3.7 Decision Feedback Equalizer . . . 49

3.8 Equalizer Histograms . . . 55

3.9 25GRX 20 GBaud Sensitivity . . . 56

3.10 25GRX 30 GBaud Sensitivity . . . 57

3.11 40GRX 30 GBaud Sensitivity . . . 57

3.12 40GRX 40 GBaud Sensitivity . . . 58

4.1 Outdoor transmission setup . . . 61

4.2 Optical Transmitter . . . 62

4.3 Transmitter Viewfinder . . . 62

4.4 Optical Receiver . . . 63

4.5 Transmission Hight Profile . . . 64

4.6 Turbulence profile for THRUST testbed . . . 65

4.7 Theoretical Ground-to-GEO Scintillation . . . 65

4.8 Measured σ

. . . 66

4.9 Lognormal Histograms . . . 67

4.10 Gamma-Gamma Histograms . . . 69

4.11 Power Scintillation . . . 70

4.12 Fading Power Spectrum . . . 71

4.13 40 GBaud Outdor Sensitivity . . . 72

4.14 56 GBaud Field Test . . . 73

4.15 56 GBaud Timing Error . . . 74

A.1 Optical IQ Modulator . . . 82

A.2 Mach-Zehnder Modulator . . . 83

B.1 Intradyne Receiver . . . 85

B.2 90 Degree Hybrid . . . 86

B.3 Balanced Photodetector . . . 87

2.1 Timing Recovery Algorithm Summary . . . 25

2.2 Maximum BPG Frequency Offset . . . 35

2.3 Timing Recovery Decision Matrix . . . 41

3.1 Example Pre- and Post- Equalization Statistics . . . 56

4.1 Comparison of Measured and Estimated σ

Values . . . 69

x

ADC Analog-to-Digital Converter AVN Absolute Value Nonlinear AWGN Additive White Gaussian Noise

BER Bit Error Rate

BPD Balanced Photodetector BPG Bit Pattern Generator BPSK Binary Phase Shift Keying CMA Constant Modulus Algorithm CPR Carrier Phase Recovery

CWL Continuous-Wave Laser

DAC Digital-to-Analog Converter

DD Decision-Directed

DFE Decision Feedback Equalizer

DLR Deutsches Zentrum für Luft- und Raumfahrt DSP Digital Signal Processing

DWDM Dense Wavelength-Division Multiplexing

FC Fiber Coupling

FIR Finite Impulse Response

xi

FO Frequency Offset

FOC Frequency Offset Compensation FPGA Field-Programmable Gate Array FSM Fast Steering Mirror

FSO Free-Space Optical

GEO Geostationary Earth Orbit ISI Intersymbol Interference LFE Linear Feedforward Equalizer

LPF Low-Pass Filter

LO Local Oscillator

LOGN Logarithmic Nonlinear

LoL Loss-of-Lock

MCMA Modified Constant Modulus Algorithm MHV Modified Huffnagel-Valley

MZM Mach-Zehnder Modulator

M&M Mueller and Müller

NCO Numerically Controlled Oscillator

NDA Non-Data-Aided

OGS Optical Ground Station

OOK On-Off Keying

OPLL Optical Phase-Locked Loop O&M Oerder and Meyr

PBS Polarization Beam Splitter

PD Photodetector

PLL Phase-Locked Loop

PSK Phase Shift Keying

QPSK Quadrature Phase Shift Keying

RF Radio Frequency

RX Receiver

SNR Signal-to-Noise Ratio TED Timing Error Detector TEM Transverse Electromagnetic

THRUST Terabit Throughput Satellite System Technology TIA Transimpedance Amplifier

TX Transmitter

VCO Voltage-Controlled Oscillator

VOA Variable Optical Attenuator

Optical technologies have revolutionized modern communications.

Fiber optics can now achieve extraordinarily high data rates in the hundreds of Gbit/s [1] per modulated wave, and up to tens of Tbit/s [2] per light source. On the other hand, ground-satellite com- munications have traditionally been limited to radio-frequency (RF) technologies, which limits the available bandwidth a given ground station can provide. The growing demand for high-speed internet in more sparsely populated regions of the Globe necessitates the devel- opment of satellite links due to the uneconomical nature of construct- ing fiber connections for rural areas.

The European Commission Digital Single Market Broadband Strategy seeks to provide high-speed internet access for all its citizens by the year 2020, and satellite broadband (22 Mbit/s) has already been used to achieve previous targets in 2013 [3]. The increasing amount of satel- lite broadband users requires the establishment of high-speed feeder- links to satellites which are capable of meeting this demand.

The Terabit Throughput Satellite System Technology (THRUST) Project [4] aims to establish fast and reliable satellite uplinks at sev- eral ground stations throughout Europe by means of free-space opti- cal communications. A single optical ground station (OGS), employ- ing dense wavelength-division multiplexing (DWDM) in combination with high modulation rates, can provide all the required bandwidth for a connection on the order of Tbit/s, whereas typical RF technolo- gies would require dozens of gateways to achieve the same [5]. In October 2016, the THRUST project set a world record in free-space op- tical communications by demonstrating a transmission speed of 1.72 Tbit/s using intensity modulation and DWDM [6].

1

The focus of this thesis is on coherent communications. The objec-

tive is to improve the signal processing of the coherent receiver in or-

der to achieve faster and more reliable performance, and to verify this

performance in the field. The thesis describes the work carried out

at the DLR Institute of Communications and Navigation in Oberpfaf-

fenhofen, Germany, and is organized into four chapters. Chapter 1

gives an overview of the ground-to-GEO communication scenario. It

describes the atmospheric channel effects and the coherent communi-

cation system. Chapter 2 describes the work done on improving the

timing recovery subsystem. Chapter 3 describes the implementation

of an equalizer which performs distortion compensation and carrier

phase recovery. Chapter 4 describes the outdoor tests of the system

and scintillation measurements made in the THRUST GEO feeder-link

testbed. The system performance in the field at varying levels of opti-

cal turbulence is compared to back-to-back measurements made in the

laboratory.

Ground-to-GEO Communication

Optical uplinks in the C-band (1530 - 1565 nm) will provide data to satellites placed in geostationary Earth orbit (GEO), which then sepa- rate the data and transmit back to users on Earth via RF link, as shown in Figure 1.1. Current designs use the Ka-band, but user links employ- ing the Q- or W-bands are proposed [7].

The free-space optics problem of intermittent blockages by fog or cloud cover can be overcome by placing various ground stations throughout Europe in locations with anti-correlated weather patterns to achieve the required availability [4, 8]. Thus, 12 optical ground stations are proposed around the Mediterranean and Black Seas, the Canary Is- lands, and the British Isles.

Figure 1.1: GEO feeder link scenario.

3

1.1 Free-Space Optical Channel

The effects of the free-space optical uplink channel are briefly described in this section. The optical signal in free space is modeled as a Gaus- sian beam propagating in the transverse electromagnetic 0,0 (TEM00) mode [9], as shown in Figure 1.2. This allows for an estimation of the atmospheric impacts on the beam, which travels approximately 35000 km from the OGS to the satellite in GEO. The first 20 km of this path are in the troposphere, which contains the weather and turbu- lence that affects the beam. The remainder of the path is thin enough to be modeled as a vacuum [10]. The effects of the optical channel can be broken down into four relevant phenomena: geometrical power loss, atmospheric extinction, intensity scintillation, and the atmospheric phase piston. The electric field of a Gaussian beam propagating in the z direction in cylindrical coordinates is [11]

E(r, φ, z) = E

w

w(z) exp



−j[kz − η(z)] − r

 1

w

(z) + jk 2R(z)



(1.1) where w(z) is the beam spot size, k is the wavenumber, R(z) is the wavefront radius of curvature, and η(z) is the phase factor, given by

η(z) = tan

 λz πw

 (1.2)

where λ is the wavelength.

Figure 1.2: Projection of a Gaussian beam in the xz plane.

Geometrical Loss and Atmospheric Extinction

Geometrical attenuation and atmospheric extinction are the main ef- fects which cause static power loss as the beam travels through the atmosphere. These losses are taken as constant values, since the atmo- spheric composition can be viewed as static for our purposes. Inten- sity scintillations due to atmospheric turbulence can then be viewed as a perturbation on this constant power loss. The beam wavelength of 1550.15 nm is chosen such that it does not correspond with any at- mospheric absorption lines in the C-band.

The beam divergence angle, θ, shown in Figure 1.2, is responsible for the geometric attenuation.

θ = tan

λ

πw

(1.3)

The attenuation due to beam divergence (in decibels) having traveled a distance L is [10]

α

(dB) = 10 log  πθ

L

4D



(1.4) where D is the receiver telescope aperture size. The constant losses due to atmospheric extinction are denoted as α

. Constant losses due to coupling the beam out of the transmitter, and into the receiver are called α

and α

respectively. The total static power loss along the channel is

P

(dB) = α

+ α

+ α

+ α

(1.5) This calculation is important for determining the system link budget, and is not an objective of this thesis.

Intensity Scintillation

Intensity Scintillation is caused by pointing errors and index of refrac-

tion turbulence in the troposphere. Techniques to ensure correct point-

ing of the beam are beyond the scope of this thesis, and only the tur-

bulence is considered. The index of refraction turbulence causes parts

of the beam to refract and slightly change direction through the at-

mosphere. This creates areas of destructive (fading) and constructive

(surging) interference along the phase front. Additionally, larger struc- tures can slightly shift the entire beam, leading to beam wander. This also causes losses in received power as the beam is no longer centered on the receiver aperture.

A detailed examination of index of refraction turbulence is given in [12] and is based on the work of Kolmogorov, who originally stud- ied turbulent motion in fluids with statistical methods. The index of refraction n is sensitive to small fluctuations in temperature, and is modeled as a constant mean value with an added perturbation. At a given point in space R and time t, the index of refraction is

n(R, t) = n

+ n

(R, t) (1.6) where n

= hn(R, t)i ≈ 1 is the mean index of refraction, and n

(R, t) is the random perturbation. Bounded by an outer scale L

and an inner scale l

, if the turbulence is statistically homogeneous and isotropic, the covariance of n between two points is

B

(R) = hn

(R

)n

(R

+ R)i (1.7) which only depends upon the scalar distance between the two points R = |R

−R

|. In this case, a structure function D

(R) of the turbulence exhibits the asymptotic behavior

D

(R) = 2[B

(0) − B

(R)] =

( C

l

R

, 0 ≤ R  l

C

R

, l

 R  L

(1.8) C

is called the (refractive index) turbulence structure parameter, and is a measure of the amount of energy in the turbulence. C

values in the range of 10

m

indicate weak turbulence, and values in the range of 10

m

indicate strong turbulence. It is advantageous to work with the structure function D

(R) as opposed to the random process n

(R) directly, because while n

(R) is not a stationary process, it has stationary increments, and thus D

(R) behaves like a stationary process.

A commonly used turbulence profile is the Hufnagel-Valley model.

However, this model is only valid at sea level, and the OGS sites are

all located at higher altitudes. Therefore a modified profile must be

used, called the modified Hufnagel-Valley (MHV) [9]:

C

(h) = 0.00594 v 27

(10

h)

e

+ 2.7 · 10

e

+ A

e

(1.9)

This equation provides the turbulence structure parameter C

for a given height h above sea level. H

is the height of the OGS above sea level, and v is the effective cross-wind velocity. A

is a reference sea- level turbulence value. This model describes stationary layers with different levels of turbulence which depend on the crosswind.

A measure of the beam coherence is given by the Fried parameter r

, which is a function of the wavenumber k, zenith angle ζ, and the tur- bulence structure parameter C

:

r

= h

1.45k

sec(ζ) Z

HOGS

C

(h)dh i

(1.10) The received beam spot contains speckles due to amplitude scintilla- tion along the phase front. These distortions have a structure size of r

, which can be estimated by analyzing the intensity speckle patterns at the receiver, as described in [13].

Beam wander is also dependent upon the Fried parameter. The stan- dard deviation of the beam wander angle is

phα

i = 0.73  λ 2w

 2w

r



(1.11) These effects combine to produce the scintillations in received power at the receiver, pictured in Figure 1.3. The strength of the combined fading and surging effects can be characterized by the power scintilla- tion index, σ

:

σ

= hP

i − hP

i

hP

i

(1.12)

The measurement of σ

is described further in Chapter 4. The in-

tensity scintillation must be taken into account when determining the

link budget, as the actual received power can vary drastically. For the

receiver, it is important to use systems which are robust to scintilla-

tion. For instance, a phase-locked loop (PLL) must be able to quickly

reestablish a lock after a deep fade.

Figure 1.3: Plot of received power vs. time of an optical beam trans- mitted in a GEO feeder-link testbed.

Phase Piston

The Taylor hypothesis states that the turbulent layers of the atmo- sphere are frozen, and the whole structure which produces the phase screen moves laterally at the wind velocity, V (capitalized here for clar- ity) [14]. The phase-spatial power spectrum of a given layer of atmo- sphere for a plane wave in Kolmogorov turbulence is

F

(ν) = 0.033(2π)

k

C

∆hν

(1.13) where ν = pν

+ ν

is the magnitude of the spatial frequency vector.

The thickness of the atmospheric layer is ∆h. This spatial frequency spectrum can be related to a one-dimensional temporal spectrum by applying the Taylor hypothesis and numerically integrating, yielding the result

L

(f ) ≈ 0.055(2π)

k

C

∆hV

f

1 + 37.5D

(f /V )

(1.14)

where D is the aperture size and f is the frequency of the phase piston

spectrum. The phase piston is a slow process; for typical values of C

and V , L

(f > 100Hz) < 10

.

Final Assumptions

When considering the communications channel, the following assump- tions are made:

• The communications system is static and has limited bandwidth.

The optical channel introduces no bandwidth limitation.

• Intensity scintillation in the optical channel is flat across the sig- nal spectrum, and thus does not introduce any distortion, only changes in amplitude.

• The time scale of the modulation frequency is much shorter than

those of the atmospheric disturbances, ie. T

 T

, T

. The

amplitude and phase scintillation caused by the atmosphere at

any given time can therefore be viewed as a constant offset.

1.2 Coherent Communications

1.2.1 PSK Modulation

Coherent communications are a promising candidate for use in the op- tical feeder link. While intensity modulation with direct detection (ie.

on-off keying [OOK]) provides a straightforward implementation, it is sensitive to fading, and does not provide the possibility of using higher-order modulation schemes like a coherent system does.

IQ modulation is a common way to produce coherent modulation schemes. Two waveforms, I(t) and Q(t) are amplitude modulated onto carrier waves in quadrature (ie. the carrier waves differ by 90

in phase). This produces two orthogonal functions. By adding the two quadrature signals, it is possible to produce phase- and amplitude- modulated signals at the output. This allows the modulator to reach any point in the complex plane. Our system uses phase shift keying (PSK) modulation, in which the amplitude of the wave remains con- stant, and all information is encoded in the phase. The transmitted signal E

(t) is:

E

(t) = I(t) exp j(2πf

t) + jQ(t) exp j(2πf

t) 

= A exp j(2πf

t + φ(t))  (1.15) where f

is the carrier frequency, A is the amplitude of the modulated wave, and φ(t) is the modulated data. A basic block diagram demon- strating the concept is shown in Figure 1.4. The amplitude modulation is done by a Mach-Zehnder Modulator (MZM). This device uses ma- terials which cause optical waves propagating through the crystal

Figure 1.4: Block diagram of an optical IQ modulator

structure to self-interfere, and controlling this effect can produce am- plitude or phase modulation. The modulator is described in greater detail in Appendix A. To produce a BPSK signal, I(t) is varied be- tween 1 and -1, and Q(t) is held constant at 0. Thus all the energy is contained within the in-phase branch of the signal. To produce a QPSK signal, both I(t) and Q(t) are varied between 1 and -1, and the energy is evenly distributed between the in-phase and quadrature branches of the signal. This information can be represented by a constellation diagram, as shown in Figure 1.5.

(a) BPSK constellation diagram (b) QPSK constellation diagram

Figure 1.5: Constellation diagrams for (a) BPSK and (b) QPSK. E

(t) has constant magnitude, but changes in phase with the given trans- mitted symbol.

1.2.2 Intradyne Receiver

A coherent receiver can detect the amplitude, phase, and polarization of a signal. In general, in order to use a coherent modulation scheme, an exact replica of the carrier wave must be produced at the receiver to demodulate the received signal. One common solution is to use an optical PLL (OPLL) to steer a local oscillator (LO) to match the carrier frequency and phase [15]. This approach is known as homodyne de- tection.

Another solution, called intradyne detection (or digital homodyne),

uses a free-running LO to demodulate the received signal without the

need for an OPLL. The residual frequency offset (FO) and phase off-

set on the signal is subsequently compensated for by digital signal

Figure 1.6: Block diagram of the intradyne receiver. Blue connections denote optical connections, red electrical, and green digital.

processing (DSP). This is advantageous at optical frequencies because OPLL design becomes increasingly cumbersome as the modulation or- der is increased. This system is also more robust in the presence of fading, since there is no restabilization time necessary after signal loss.

The intradyne receiver is a proven technology in the fiber optics in- dustry, and allows the system complexity to be shifted from hardware over to software. The complete receiver is pictured in Figure 1.6. The received optical beam is first collimated and then coupled into an op- tical fiber using a fast steering mirror (FSM). This process is beyond the scope of the thesis, and here only the optical signal in the fiber is considered. A description of the optical bench is given in [5]. The FSM coupling efficiency varies over time due to environmental factors, which adds to the fluctuations in received power. The received optical signal is

E

= A

exp j(2πf

t + φ

(t)) 

(1.16) A polarization beam splitter (PBS) is used to split the beam into two orthogonal polarizations, X and Y. Next, the optical signal in the fiber is mixed with the LO, which is a continuous-wave laser, linearly polar- ized at 45

relative to the received beam. The LO is passed through a PBS, so that 50% of its energy becomes X-polarized, and the other 50%

is Y-polarized. The LO field is

E

= A

exp j(2πf

t + φ

) 

(1.17)

The remaining description of the receiver is continued in Appendix B.

The final digital signal which is input to the DSP, after sampling by the analog-to-digital converter (ADC), is

r

(k) = gA

A

exp 2πf

kT /N + φ

(k) − φ

+ φ

(k) + n(k) r

(k) = gA

A

exp 2πf

kT /N + φ

(k) − φ

+ φ

(k) + n(k)

(1.18) where g is a slowly-varying gain, f

= f

−f

is the beat frequency be- tween the received carrier and the free-running LO, φ

(k) is the com- bined phase noise of the lasers and the optical channel, and n(k) is additive noise. The noise terms are discussed further in Section 1.2.3.

The effect of the FO is clearly seen in BPSK signals, where it causes the energy in the received signal to oscillate between its in-phase and quadrature branches. From this point onwards, all further treatment of the signal is done in the digital domain. An example digitized in- tradyne signal is shown in Figure 1.7.

Figure 1.7: Discrete time domain image of a raw intradyne BPSK sig-

nal.

1.2.3 SNR and Theoretical Performance Limits

Shot Noise Limit

For a coherent system, there are two significant noise sources which enter the photocurrent: thermal noise and shot noise [16]. The genera- tion of the photocurrent is described in Appendix B. The mean square noise current is given by the sum of the two noise variances:

hi

i = σ

= σ

+ σ

(1.19) The first term is thermal noise which is caused by the random motion of electrons. This is modeled as a stationary Gaussian random process, which has a constant power spectral density. Integrating the power spectral density over the bandwidth of the receiver and taking into account amplifier noise yields the result:

σ

= 4k

T

R

F

∆f (1.20)

where k

is Boltzmann’s constant, T is the temperature in Kelvin, R

is the load resistance, F

is the amplifier noise figure, and the ∆f is the signal bandwidth.

The second term is shot noise, which is caused by the fact that the pho- tocurrent production is a quantum mechanical process with inherent randomness. This can be modeled as a stationary process with Pois- son statistics, which are often approximated with Gaussian statistics.

The shot noise power spectral density is also flat, and the following variance can be obtained:

σ

= 2qi

∆f (1.21)

where q is the electron charge, i

is the total diode current, and ∆f is the same bandwidth as is used for the thermal noise determination.

The diode photocurrent is given by

i

= rP + i

(1.22)

where r is the diode responsivity, P is the optical power incident on

the diode, and i

is the leakage current which flows through the

diode when no light is shining on the diode. Now the signal-to-noise ratio (SNR) can be determined:

SNR = hi

i

σ

= 2r

P

P

2q∆f (rP

+ i

) + σ

(1.23) If the LO power is great enough such that

P

 σ

2qr∆f (1.24)

two approximations can be made:

1. The optical power on each photodiode is constant and much greater than the dark current.

2. The shot noise dominates the thermal noise.

The SNR can then be approximated as SNR ≈ rP

q∆f (1.25)

This result is significant because it shows that by raising the LO power, the shot noise can be made to dominate and make the thermal noise insignificant. Indeed, this is one of the main advantages of coherent re- ceivers over direct detection receivers [16]. A theoretical performance limit in terms of the bit error rate (BER) of the detection system can be estimated using the complimentary error function [17]:

BER = 1

2 erfc(Q) (1.26)

For the case of a coherent PSK receiver, Q = √

SNR. Thus the best possible BER performance in the shot noise limit is

BER = 1 2 erfc 

s rP

q∆f

 (1.27)

Phase Noise

The phase noise φ

is a combination of the noise in the carrier and LO lasers and the effect of the atmospheric phase piston:

φ

(t) = φ

(t) + φ

(t) + φ

(t) (1.28)

where φ

(t) and φ

(t) are the phase noise of the carrier and LO lasers used. In comparison to φ

(t) and φ

(t), φ

(t) is slowly- varying and adds what can be considered to be a constant offset.

For a digital communication system, the phase noise of the carrier and LO lasers are described as a discrete-time random walk process [18]:

φ

(k) + φ

(k) = φ

(k) = φ

(k − 1) + ∆φ

(k) (1.29) The quantity ∆φ

(k) is the step size of the walk and is a zero-mean Gaussian random variable, which has the variance

σ

= 2π∆f

T (1.30)

where ∆f

is the sum of the carrier and LO laser linewidths, and T

is the symbol period.

1.3 Digital Signal Processing

Digital Signal Processing (DSP) is now a standard method of process- ing high-speed digital signals. The received signal is sent through a series of stages in which software algorithms correct various aspects of the signal. This section describes the stages which make up the DSP structure. The system block diagram is pictured in Figure 1.8. The evo- lution of the constellation diagram for a given BPSK signal is shown in Figure 1.10. The highlighted timing recovery and equalization DSP blocks are the subject of work in this thesis. A description of the initial development of the DSP structure is available in [10, 19].

Figure 1.8: DSP structure for intradyne PSK reception.

1.3.1 Polarization Demultiplexing

The state of polarization (SOP) of the received optical signal is un- known because some of the system components are non-polarization- maintaining. This effect is illustrated in Figure 1.9. The algorithm therefore seeks to determine the SOP of the incoming signal and sep- arate it into its original X and Y components, by means of a butterfly filter. This method can be modified to accommodate either single- or dual-polarization transmission. For single-polarization transmission, the algorithm restores all the signal energy back into one polarization:

r

(k) 0



= h

h

h

h

 r

(k) r

(k)



(1.31) To estimate the coefficients h

, a variant of the constant modulus algo- rithm (CMA) is used, as described by [17]. A description of the CMA is given in Chapter 3.

Figure 1.9: SOP of a dual-polarized signal before and after polarization demultiplexing.

1.3.2 Recentering

The recentering block computes a sliding window average with length L

of the signal and subtracts this result from the signal in order to re-

move any DC offset which may be present, similar to the effect of a

DC blocking capacitor.

1.3.3 Timing Recovery and Downsampling

The phase of the received symbols must be detected, so that when the signal is downsampled to the Baud rate, the samples are taken at the center of the received symbols. See Chapter 2 for a complete description of the timing recovery stage.

1.3.4 Imbalance Compensation

Gain imbalances in the transmitter or receiver can distort the balance of power between the I and Q branches of the signal, which distorts the constellation diagram from a circle (in the presence of FO) into an ellipse. The following filter is used to compensate this imbalance:

 r

(k) r

(k)



= g

g

g

0

  r

(k) r

(k)



(1.32) The coefficients g

are determined using a Gram-Schmidt orthogonal- ization procedure, as described in [20].

1.3.5 Frequency Offset Compensation

As mentioned in Section 1.2.2, the signal carries an FO due to the fre- quency mismatch between the free running LO laser and the carrier wave. Due to mechanical vibrations and changes in temperature, the laser frequencies do not remain constant [21]. The thermal drift is slow enough that it can be treated as a static offset, and the quicker, smaller variations due to mechanical vibration are treated as a dynamic change centered about the static offset [10]. The signal constellation therefore rotates at the beat frequency f

= f

− f

.

The basic frequency offset compensation (FOC) procedure is as fol-

lows: first, a coarse frequency offset is estimated and applied to a large

section of the signal. Then a fine compensation is applied to a small

section of the signal. Both the coarse and fine stages calculate the offset

in the same general way, but vary in the length of the observation win-

dow used, and how much of the signal is used in the calculation. First,

the data in the signal is removed by raising it to the M -th power, where

M is the modulation order. Because of this, a factor of 1/M must be

applied to obtain the correct result. Next, the Fast Fourier Transform

(FFT) of the signal is taken, and a peak at the (dynamic) beat frequency is obtained. The estimated beat frequency is therefore:

f ˆ

= 1 M max

f

|X(f )| (1.33)

where X(f ) is the Fourier Transform of the received signal raised to the power M . Additionally, the average carrier phase of the window can be estimated:

φ = ˆ 1

M arg[X( ˆ f

)] (1.34)

The phase correction can shift the received constellation into the cor- rect place, but does not compensate for any quickly-varying phase noise. The corrections can then be applied to the signal:

r

(k) = r(k)e

(1.35)

1.3.6 Carrier Phase Recovery

After the FOC stage, the constellation is no longer spinning and the remaining phase error can be determined. The carrier phase error is caused by several factors: the laser phase noise of both the carrier and the LO, the residual phase offset due to the FO, and by phase scintilla- tion (phase piston) caused by the atmosphere.

A feedforward Viterbi-Viterbi Window method is used for carrier phase recovery (CPR) [22]. The same as with the FOC stage, the signal is first raised to the the power M , where M is the modulation order. This prevents the signal data from affecting the phase estimation. Then, the average phase offset of a frame with length L

is determined:

φ = ˆ 1 M

1 L

LC

X

k=1

arg r(k)



(1.36)

The phase correction can then be applied to the signal:

r

(k) = r(k)e

(1.37)

There is no significant change observable after the CPR stage in Figure

1.10 because there is very little phase noise in the signal due to the

narrow linewidth lasers used.

1.3.7 Equalization

Distortions introduced by the communications system can severely degrade the signal quality. Equalization is a technique which can be used to restore the signal. See Chapter 3 for a complete description of the equalization stage.

1.3.8 Demodulation

A hard decision boundary is applied to the processed signal in order to retrieve a bit or bits. For BPSK, this amounts to checking the sign of the current sample, and applying the mapping:

( sgn [r(k)] > 0 → 1

sgn [r(k)] < 0 → 0 (1.38) where r(k) is the processed signal. There are two decision boundaries for QPSK, and the mapping follows similar logic:



 

 



 

 



sgn [r

(k)] > 0, sgn [r

(k)] > 0 → {1, 1}

sgn [r

(k)] > 0, sgn [r

(k)] < 0 → {1, 0}

sgn [r

(k)] < 0, sgn [r

(k)] > 0 → {0, 1}

sgn [r

(k)] < 0, sgn [r

(k)] < 0 → {0, 0}

(1.39)

where the subscripts I and Q refer to the in-phase and quadrature

branches of the signal, respectively.

Figure 1.10: The evolution of the constellation diagram of a 40GBaud

BPSK signal as it is treated by the various DSP stages. The signal has

power -30 dBm.

Timing Recovery

In a real communication system, the oscillator producing the signal is not perfectly synchronized with the oscillator driving the sampler.

This degrades the signal by producing variations in symbol timing.

Ideally, both oscillators should be operating at exactly the frequency corresponding to the Baud rate (or a known multiple of the Baud rate in the case of oversampling), however, varying environmental con- ditions and noise prevent the oscillators from behaving ideally. This produces a timing jitter as well as a slowly-varying timing drift in the signal. Together, these effects produce the symbol timing error (or off- set), shown in Figure 2.1. The result is sampling being done at subop- timal points in the signal, and a loss of received symbol power. The goal of the timing recovery stage, also referred to as clock recovery or synchronization, is to determine and compensate for this symbol tim- ing offset. The offset is described as a slowly-varying phase term in the received signal. Thus the received baseband signal can be written as:

r(t) = X

m

a

g[t − mT − τ (t)] + n(t) (2.1) where a

is the mth complex PSK symbol with |a

| = 1, T is the sym- bol duration, n(t) is additive Gaussian noise, and τ (t) is the symbol timing error which we are trying to compensate for, either by driving τ to zero, or by estimating its value and interpolating the signal. g(t) is the symbol pulse shape. The pulse shape describes how the symbol is modulated onto the carrier wave, and is described further in Sec- tion 3.1. The timing recovery stage is blind to the carrier phase, so the intradyne FO and phase noise can be omitted from equation 2.1.

23

Figure 2.1: Plot of timing error for a typical 10 GBaud (8x oversam- pled) signal with respect to time. The error wraps around when the TED selects a new sample which is closer to the center of the symbol.

Two basic classes of timing error detectors (TED) are investigated. The first class of timing error detectors are feedback-loop based detectors.

They use the baseband signal to produce a timing error signal, which

is used as the control signal to adjust the sampling done at the ADC by

means of a VCO or NCO in a feedback loop. The other class is a set of

feedforward estimators, which use a fixed sampling rate set by a free-

running oscillator. A section of the received baseband signal is used to

directly estimate the timing error. This result can then be fed to an in-

terpolator to recover the full amplitude of the symbol. With the excep-

tion of the Mueller-Müller detector, which is a decision-directed (DD)

system, all of the TEDs which are investigated are known as non-data-

aided (NDA) algorithms, which refers to the fact that there is no infor-

mation encoded in the signal itself which is used to perform the timing

recovery. The different timing recovery algorithms which are investi-

gated are summarized in table 2.1. The parameter L is the length of

the frame in symbols. The currently used system has an ADC with a sampling rate which is not controllable, and thus the focus is placed on feedforward TEDs. Additionally, modifications are made to the Gard- ner algorithm so that it can run in a feedforward configuration.

Table 2.1: Timing Recovery Algorithm Summary

Algorithm Type Oversampling Ratio Computational Burden [23]

Gardner NDA Feedback 2 2L Multiplications + PLL Mueller & Müller Decision-Directed 1 2L Multiplications + PLL + symbol decision

Oerder & Meyr NDA Feedforward 4 8L Multiplications AVN NDA Feedforward 4 8L Multiplications + L√

· operations LOGN NDA Feedforward 4 12L Multiplications + L log (·) operations

Lee NDA Feedforward 2 16L Multiplications

2.1 Feedback-Based Methods

The timing recovery problem has traditionally been solved by control- ling the sampling rate at the receiver. In general, the timing drift is slowly-varying compared to the Baud rate, so a PLL can lock onto the correct sampling intervals. The feedback-based algorithms presented here produce a timing error signal at the TED, the output of which is passed through a low-pass filter (LPF) and then fed to an oscillator that controls the sampling rate. The LPF filters out the higher frequency timing jitter and leaves the slower drift. As with any PLL, the loop filter determines how much noise is filtered out vs. how responsive the overall system is. A block diagram of the feedback configuration is shown in Figure 2.2.

It is desirable for as much of the receiver to be implemented digitally as possible. In such an architecture, the entire PLL structure, ie. the TED, loop filter, and NCO are implemented in the digital domain, which ul- timately supplies a control signal to the ADC. The various algorithms considered correspond to different implementations of the TED, which produces a control signal for the NCO.

One drawback of using a PLL is the possibility of loss-of-lock (LoL)

during fading. LoL occurs when a PLL loses connection to the input

signal (ie. during periods of strong fading). The system must then

resynchronize once the fade has passed. One possible workaround

would be the implementation of a Kalmann filter which would save

the previous timing information in memory and extrapolate the tim- ing drift during the fade. However, such a system could be quite ex- pensive from a computational standpoint.

Figure 2.2: Block diagram of a feedback timing recovery system.

2.1.1 Gardner

The Gardner TED compares the amplitudes of the current and previ- ous symbols, the difference of which is then multiplied by the sample at the midpoint in between. It requires a minimum oversampling of 2 samples per symbol, and the oversampling rate must be an integer multiple of 2 in order to function [24]. It is summarized by the formula:

e(k) = r

( [k − 1/2]T

2 )r

( kT

2 ) − r

( [k − 1]T 2 )  + r

( [k − 1/2]T

2 )r

( kT

2 ) − r

( [k − 1]T

2 ), k = 1, 2, 3...

(2.2)

where the subscripts I and Q refer to the in-phase and quadrature components of the signal. Using complex numbers, this can also be written as:

e(k) = Re

n r( kT

2 ) − r( [k − 1]T

2 )r

( [k − 1/2]T

2 )

o

(2.3)

As shown in Figure 2.3, during instants of perfect sampling, ie. when

the sampling point r(

/

) coincides with the center of the symbol r(mT ),

the difference in amplitude between the current symbol r(

/

) and the

previous symbol r(

/

) is minimized, and the midpoint r(

/

)

should lie on the zero-crossing. Thus, the error signal e(k) goes to

zero. At instants of imperfect sampling, there is a mismatch between

the amplitudes of the current and previous symbols, and the midpoint

(a) Early: e(k) > 0

(b) On Time: e(k) = 0

(c) Late: e(k) < 0

Figure 2.3: Gardner error detector in cases of (a) early sampling, (b) on time sampling, and (c) late sampling.

no longer lies on a zero-crossing. Thus, a non-zero error signal is pro- duced. In general, the relationship between the mean error signal and the true timing error is given by the following equation:

τ = 1

2π arcsin(−Ahe(k)i) (2.4)

where A is a gain factor which is a function of the received pulse shape, oversampling rate, and frame length over which the averaging takes place.

Using the technique of calculating the mean Gardner error signal he(k)i

for a given frame of samples and applying the correct gain, it is indeed

possible to adapt this algorithm to a feedforward structure for an in-

teger oversampling rate that is a multiple of 2. Care must be taken to

ensure that the samples being used for the calculation lie on the in-

ner part of the detector S-curve, the green area shown in Figure 2.4.

Figure 2.4: The S-curve shows how the detector output relates to the true timing error.

Otherwise, the system will report a low error when in fact the true timing error is greater than a quarter symbol. In the worst case, the sampling will be off by a half of a symbol, and the error signal will go to zero. This can be verified by inspecting equation 2.4. This problem can be overcome by comparing the relative amplitudes of the samples available to the detector and choosing the most appropriate point to track. In a typical feedback architecture, the correct sampling point is found passively. The zero-crossings of the detector output character- istic at the half-symbol intervals are unstable equilibrium points, and the sampling point will naturally be drawn toward the stable equilib- rium point which corresponds to the correct sampling point [25]. The Gardner algorithm is computationally light and simple to implement in a real-time system. However, it is quite sensitive to noise and fad- ing, which can compromise the performance of the system. See Section 2.5 for more discussion.

2.1.2 Mueller and Müller

The algorithm proposed by Mueller and Müller (M&M) has the major advantage of being able to work with sampling at the Baud rate [26].

The algorithm works by comparing the magnitudes of the current and

previous received signals. If the current symbol has a greater mag-

nitude than the previous symbol, then it can be inferred that the sam- pling is occurring faster than the Baud rate. If the current symbol has a lower magnitude than the previous symbol, then it can be inferred that the sampling is occurring at a slower rate than the Baud rate. Thus, the algorithm only detects changes in the sampling time relative to the previous sample. The algorithm correctly modifies the sign of the symbols when comparing them by multiplying the received symbols with the decided symbols.

e(k) = ˆ a

g[kT ] − ˆ a

g[(k − 1)T ] (2.5)

where ˆ a is the decided symbol. The fact that the system requires a

stream of decided symbols in order to work in real time puts it at odds

with the current feedforward, frame-based DSP architecture. The tim-

ing recovery (and downsampling) stage was placed near the beginning

of the DSP chain in order to reduce the computational overhead in the

rest of the system. The algorithm can therefore not be implemented

without significant changes to the rest of the DSP. Despite this, the

Mueller and Müller algorithm remains an attractive alternative due to

its low bandwidth requirements, and may be considered for use in the

future if a sample-based system is implemented, and a controllable

ADC is available.

2.2 Feedforward-Based Methods

An alternative to implementing a feedback-based timing recovery sys- tem is to directly estimate the timing offset section-by-section of the received signal. The benefits of using such a design are that there is no initial acquisition period where the system needs to adjust to the cor- rect sampling frequency, and more importantly, there are no problems due to loss-of-lock.

The feedforward timing error detector directly estimates the timing error from the signal. With this information, the signal can be down- sampled to the Baud rate at the correct point by interpolating the sym- bol at that point. This is shown in Figure 2.5. Because the timing error is slowly-varying, it can be assumed to remain constant for a section of signal with length L. The timing error can then be calculated for that section. Determining L is a trade-off between the accuracy and responsiveness of the system. Using large values of L provides the detector with more information to make an estimate, but inherently assumes that the timing offset will be constant for a longer period of time. This is studied in Section 2.3.

Figure 2.5: Block diagram of a feedforward timing recovery system.

2.2.1 Oerder and Meyr (O&M)

The most well-known and computationally inexpensive feedforward algorithm is the one proposed by Oerder and Meyr (O&M) [27]. The received signal is first squared, and then the Fourier spectral compo- nent of the resultant signal is computed at 1/T . The symbol phase, or in other words, the timing offset, is the phase of the Fourier spectral component. The algorithm can be summarized by the formula:

ˆ τ

T = −1 2π arg

LN −1

X

k=0

r  kT N

   

2

e

!

(2.6)

where N is the number of samples per symbol, L is the number of symbols per frame, and ˆ τ is the resulting estimated symbol timing er- ror. The quantity

can therefore be interpreted as the timing error in fractional symbols.

The reason for squaring the incoming signal is to introduce some mem- oryless nonlinearity f (·) such that E{f [r(

)]} 6= 0 . The signal r(

) can be viewed as a cyclostationary process with zero mean (ie.

E{r(

)} = 0) [25]. For a given cyclostationary process x(t) with Fourier transform X(2πf ),

E{X} =

LN −1

X

k=0

E

 x

 t − kT

N − τ



e

= LN

T X(1/T )e

(2.7)

From equation 2.7 it can be seen that the timing information τ can be extracted from the argument of the signal’s Fourier coefficient at 1/T , but only if E{x(t)} 6= 0.

Oerder and Meyr were the first to propose this idea, however several other variations on the same theme of introducing a different non- linearity f to the signal have since been developed. The Absolute Value Nonlinear (AVN) algorithm is a variation of the O&M algorithm, whereby the introduced nonlinearity f is changed from a squaring op- eration to an absolute value [23]. This results in the formula:

ˆ τ

T = −1 2π arg

LN −1

X

k=0

r  kT N

   

e

!

(2.8)

Another variant was proposed by Morelli, D’Andrea, and Mengali [28], called the Logarithmic Nonlinear (LOGN) algorithm. It introduces a nonlinearity f = log(1 + |r(t)|

C), where C is a constant given by the approximate SNR. The timing estimate is:

ˆ τ

T = −1 2π arg

LN −1

X

k=0

log

"

1 +    

r  kT N

   

2

C

#

e

!

(2.9)

2.2.2 Lee

A major drawback of the O&M, AVN, and the LOGN algorithms is the fact that they require an oversampling ratio greater than 2. In the literature the required oversampling is commonly reported as 4 [23, 25]. This can significantly limit the final rate at which the system can receive data. The algorithm proposed by Lee is a modification of the Oerder and Meyr algorithm (Section 2.2.1) which allows the system to work with an oversampling rate of N = 2 samples per symbol [29].

The modification to equation 2.6 is:

ˆ τ

T = −1 2π arg

LN

X

k=1

r  kT N

   

2

e

+ Re

 r  kT

N



r

 (k − 1)T N



e

! (2.10)

When N = 2, the term e

reduces to (−1)

. Thus it can be seen that the first term in the summation in equation 2.10, which is the stan- dard Oerder and Meyr algorithm, will only produce real numbers. The output of the TED is therefore constrained to the values {-0.5, 0, 0.5}, essentially limiting the resolution of the TED to 1/2 of a symbol. The second term in equation 2.10 is introduced to solve this problem. At N = 2 , the term e

becomes j · (−1)

. This allows for a com- plex argument in equation 2.10 and allows the detector to operate at 2 samples per symbol. This is pictured in Figure 2.6.

To better show how the additional term allows an estimate of τ at 2x oversampling (N = 2), equation 2.10 can be rewritten by expanding the two terms in the summation:

ˆ τ

T = −1

2π arg A + jB

(2.11) Defining T

≡ T /N , A can be rewritten as:

A =

2L

X

k=1

|r(kT

)|

e

=

L

X

k=1

|r(2kT

)|

− |r([2k + 1]T

)|

(2.12)

Figure 2.6: Comparison of the timing error estimates of the Lee and Oerder & Meyr algorithms at 2x oversampling.

The second term jB can be rewritten as:

jB =

2L

X

k=1

Rer(kT

)r

([k − 1]T

) e

= j

L

X

k=1

Re nh

r([2k − 1]T

) − r([2k + 1]T

) i

r

(2kT

) o

(2.13)

Equation 2.13 is strongly reminiscent of the Gardner timing error de- tector (2.3). It has been shown in [24] and [29] that A and B have the characteristic:

A ∝ cos(−2πτ /T ) (2.14)

B ∝ sin(−2πτ /T ) (2.15)

For max|A| ≈ max|B|, the estimator is approximately unbiased, and ˆ τ can be found:

ˆ τ

T = −1

2π arg  cos(−2πτ /T ) + j sin(−2πτ /T ) (2.16) Some authors [30] have identified a small bias in the Lee algorithm.

If g(t) has a raised-cosine characteristic, this bias becomes significant

at large values of roll-off factor, β. The following correction has been proposed:

ˆ τ

T = −1

2π arg γA + jB

(2.17)

where γ is given by:

γ = 2 sin(πβ/2)

πβ(1 − β

/4) (2.18)

2.3 Maximum Offset

The timing error in the signal originates from an offset between the sig- nal generator and the sampler. Therefore it is a natural question to ask what offset the system can tolerate. All timing recovery systems work under the assumption that τ (t) is slowly-varying and approximately constant over a given frame of symbols. This is a valid assumption as long as this offset does not rise too drastically.

It was found that the system will tolerate an increasing offset with no

BER penalty up to a certain point, after which the system will fail, as

shown in Figure 2.7. The maximum offset was found for two frame

lengths in the laboratory by manually adjusting the timing drift and

recording the BER. The maximum frame length is linearly related to

the oscillator offset: L

∝ ∆f

. The results are shown in Figure 2.7

summarized in table 2.2.

(a) Frame Length: 2800 symbols

(b) Frame Length: 500 symbols

Figure 2.7: BER vs. Oscillator offset for frame lengths: (a) 2800 and (b) 500 symbols. Signal: 10GBaud BPSK, power: -30dBm.

Table 2.2: Maximum BPG Frequency Offset Frame Length Baud Rate Maximum Offset

2800 10 GBaud 1.56 MHz

500 10 GBaud 8.65 MHz

2.4 Frame Length Extrapolation

Apart from accuracy and oversampling, another consideration is the

computational burden a given TED imposes on the system. For the

class of feedforward timing estimators, it was found that a significant amount of computation could be saved by computing the timing off- set for part of a given frame, but applying that estimation to the en- tire frame. This is possible because the timing drift is slow enough to remain constant over the frame, but significantly fewer symbols are required to calculate an accurate estimate. For instance, for a given frame with a length of 2800 symbols, it was found that the timing off- set could be accurately calculated using only 500 symbols (0.18·L), but that the result could still be applied to all 2800 symbols in the frame with a minimal BER penalty. The error increases drastically when less than 200 (0.07·L) symbols are used in the calculation. Results of this investigation are shown in Figure 2.8. An optimum point is at about 700 symbols (0.25·L), where the error due to extrapolation is not yet significant, but a great amount of computation can be saved.

Figure 2.8: BER results for various frame lengths. A constant extrapo-

lation length of 2800 symbols is used.

2.5 Results and Discussion

The timing recovery algorithms are tested offline, using back-to-back fiber transmission recorded in the lab. A 40 GHz bandwidth receiver, and an 80 Gsample/s ADC are used. Details about the experimental setup are in Chapter 4.

Figure 2.9: Receiver sensitivity curves comparing various timing re-

covery algorithms for 8 samples per symbol (10 GBaud BPSK).

Figure 2.10: Receiver sensitivity curves comparing various timing re- covery algorithms for 4 samples per symbol (20 GBaud BPSK).

Figure 2.11: Receiver sensitivity curves comparing various timing re-

covery algorithms for 8/3 samples per symbol (30 GBaud BPSK).