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MEHRDAD MIRSHAFIEI

UWB PULSE SHAPING USING FIBER BRAGG GRATINGS

Mmoire prsent la Facult des tudes suprieures de l'Universit Laval

dans le cadre du programme de matrise en gnie lectrique pour l'obtention du grade de matre s science (M.Sc.)

DPARTEMENT DE GNIE LECTRIQUE ET DE GNIE INFORMATIQUE FACULT DES SCIENCES ET DE GNIE

UNIVERSIT LA VAL QUBEC

2009

Mehrdad Mirshafiei, 2009

Rsum

Dans ce mmoire, nous concevons et gnrons des impulsions ultra large bande (UWB)

qui exploitent efficacement le masque spectral de la "US Federal Communications

Commission" (FCC). Une impulsion efficace amliore le rapport signal bruit au rcepteur

en utilisant la majorit de la puissance disponible sous le masque spectral dfini par la FCC,

ce qui rduit la probabilit d'erreur. Pour trouver les formes d'onde efficaces, nous

combinons plusieurs impulsions de type monocycle Gaussien spares avec certains dlais.

Chaque monocycle Gaussien a une amplitude inconnue. Les amplitudes sont trouves par

un processus d'optimisation qui maximise la puissance de l'impulsion en respectant le

masque spectral de la FCC sur toute la largeur de bande alloue aux communications

UWB.

Les impulsions efficaces sont ralises par des filtres rseaux de Bragg (FBG) dans le

domaine optique. L'impulsion temporelle est crite dans le domaine frquentiel , et une fibre

inonomode fait la conversion frquence--temps. La forme d'onde est inscrite dans le

domaine frquentiel par un FBG. Un photo dtecteur balanc limine l' impulsion

rectangulaire non-dsire qui est superpose la forme d'onde dsire. Une excellente

concordance entre les designs et les mesures est accomplie.

Les formes d'ondes gnres sont propages entre des antennes large bande. La rponse

impulsionnelle non-idale des antennes dgrade l'impulsion dsire, ce qi rduit

l'efficacit. Nous mesurons la rponse impulsionnelle de l'antenne et l'utilisons dans le

processus d'optimisation pour concevoir une forme d'onde efficace adapte la rponse de

l'antenne. Comme avant, cette forme d'onde est gnre avec un FBG. Les rsultats

exprimentaux montrent une excellente concordance avec la thorie et une amlioration

significative de l'efficacit de puissance.

11

Resume

In this thesis, we design and generate ultra-wideband (UWB) pulses that efficiently

exploit the US Federal Communications Commission (FCC) spectral mask. An efficient

pulse results in higher signal to noise ratio at the receiver by utilizing most of the available

power under the FCC spectral mask, lowering the probability of error. To find efficient

UWB waveforms, we combine several Gaussian monocycle pulses separated by certain

time delays. Each Gaussian monocycle has an unknown amplitude weight. The weights are

found by an optimization process which maximizes the power of the pulse while respecting

the FCC spectral mask over the entire UWB bandwidth.

We implement the efficient pulses by fiber Bragg grating (FBG) filters in the optical

domaine The time domain pulse shape is written in the frequency domain, and a single

modefiber performs the frequency-to-time conversion. The waveform is inscribed in the

frequency domain by the pulse shaping FBG. A balanced photodetector removes an

unwanted rectangular pulse superimposed on the desired waveform, assuring compliance at

low frequency. Excellent match between the designed and measured pulses is achieved.

The generated waveforms are propagated from wideband antennas. The non-flat impulse

response of the antenna degrades the designed pulse, lowering its efficiency. We measure

the antenna impulse response and take it into account in the optimization process to design

an efficient pulse adapted to the antenna response. As before, this pulse is generated by its

proper pulse shaping FBG. Experimental results show great match with theory and

significant improvement in terms of power efficiency.

III

Acknowledgement

There are many people who have helped me throughout the course ofmy graduate study. l

would like to thank my advisor Professor Leslie A. Rusch, for her guidance, encouragement

and support throughout my graduate career. The opportunities for growth and the

excitement of working in our group are deeply appreciated; l am honored to have had the

chance to be part of it. l would also like to thank my co-advisor, Professor Sophie

LaRocheIle. l truly appreciate her invaluable discussions about the fiber Bragg grating

aspects of my project.

Many thanks to my colleagues: Dr. Mohammad Abtahi, Serge Doucet, Julien Magn, and

aIl the rest, who helped me in various forms to finalize my project. Greatest thanks to

Mohamm"ad without whom this project would have never started. He was always there with

motivating discussions; taught me experimental work, and shared with me his technical

experience. AlI those cumbersome experiments would have been far from completion

without his efforts. Thanks to Serge for teaching me how to write FBGs and to Julien for

the mode locked laser. l would like to thank the COPL technicians, particularly Patrick

LaRocheIle for his assistance in the labo

My deepest appreciation goes to my parents for their support, and kindness in aIl and

every stage of my life. They have always been there for me, provided me with the best

education and encouraged me to leam. FinaIly, many thanks to aIl my teachers over the

years, from the first grade of elementary school to the university.

IV

Table of Contents

Rsum . ........... ........................................................................................ i

Resume .. ................................................................................................... ii

Acknowledgement .................................................................................... . iii

List of Figures ........ ...................................................................... .... ........ vi

List of Tables . .......................................................................................... ix

List of Acronyms ........ .............................................................................. x

Chapter 1 Introduction .. .. .. ............................................. .. ................ : ........................ .. ... .. ... ... ... ... ... .. 1

1.1 . UWB Basics ............... ....................... ....... .. .. ........ ..... ................ ... ............................................. .... .. ... 2

1.2. UWB Compared to Other Wireless Technologies ...... .. .................................................. .. .......... ........ 5

1.3. Applications ........ ....................... .... ................ ... .... .... ... ....................... ... ..... ............. ....... .......... .... ...... 7

1.4. Structure of the Thesis .................................................. ........ .. ................... .............. .. ..... ..... ... ... ... .... .. 8

Chapter 2 Optimal UWB Waveforms ................................. .... ........ .. .. ... .... ........... ... ..... ...... ....... 10

2.1. Pulse Shaping Techniques ............................................................ .......... ... .... ..... ... ...... ........ ..... ..... ... Il

2.1 .1. Pulse shaping in the Electrical Domain ........................ ..... .. .. ................................................ .. ......... 12

2.1 .2. Pulse shaping in the Optical Domain .......................................... .. ............... ... .. ... .. ... ...... ... .. ..... .. ... ... 13

2.2. Optimization Process Based on Sampling ......................................................................................... 14

2.3 . Optimized Sum of Weighted Gaussian Monocycles .......... ...... .. .... ............. ................. ...... .. .......... .. 21

2.3.1. Optimization Procedure ...................................................... ........................ ' ............ .. .................... .... 22

2.3.2. Multiband UWB Pulse Design .. .......... ..... ... ........ .. ... ...... .................................... ........... ............ ....... 26

2.4. COl1clusions ............................................................... .. ........... ................................................... .. ..... 28

Chapter 3 Optical UWB Pulse Shaping Using FBGs ................................................ .. ......... 29

3.1. Optical Pulse Shaping Methods ................................................................................... ......... ............ 30

3.2. UWB Pulse Shaping Using FBGs .................................................................... .. .............................. 34

3.2.1. A Balanced Receiver Approach ......................... ..... .. .. .. ...... .. .. .. .. ..................................................... 34

3.2.2. FBG Design and Fabrication ..................... ................................................................ ....................... 35

v

3.2.3. Simulation Results ............................ ............ ............................................... .. .................................. . 41

3.2.4. Experimental Results ............................................ ..... ................................................. ...................... 43

3.3. ' Waveform Tuning Using a Band-pass Filter .................................................................................... 48

3.4. Conclusion ........................................................................................................................................ 52

Chapter 4 UWB Pulse Propagation and EIRP Optimization ........................................... 53

4.1. EIRP ......................................................................... ...... .. ... ..... ....... ........ ........... .............................. 54

4.2. Antenna Frequency Response .......................................................................................................... 56

4.2.1. UWB Antenna Characteristics .......................................................................................................... 56

4.2.2. Experimental Measurements .............. .............................................................................................. 58

4.3. EIRP Measurements for Various Waveforms .................................................................................. 62

4.3.1. Link Transfer Function ..................................................................................................................... 62

4.3.2. EIRP and Output Measurements ............................................ ' .......................................................... 63

4.3.3. Conclusion on EIRP Measurements ................................................................................................. 67

4.4. EIRP Optimization Using the Channel Frequency Response ........................................................... 67

4.4.1. Optimization Process ........................................................................................................................ 68

4.4.2. EIRP-Optimized Pulse Generation ................................................................................................... 69

4.5. Conclusion ............................. ... ......................................................................................................... 73

Summary and Future Research Direction ................................................................................ 74

Appendix A MA TLAB Programs ................................................................................................ 77

References .............................................................................................................................................. 81

VI

List of Figures

Figure 1.1 FCC spectral masks for indoor and outdoor communication applications ........... 3

Figure 1.2 UWB spectrum coexisting with other narrowband communication

systems [5] ............................................................................................................ 4

Figure 1.3 WiMedia landscape of UWB compared to Wireless local area networks

(WLAN) [5] .......................................................................................................... 5

Figure 2.1 Autocorrelation coefficients which result in an optimal response for

R(e Jw ) under the FCC spectral mask ................................................................. 17

Figure 2.2 Normalized Autocorrelation spectrum, R(e Jw ) ................................................... 18 Figure 2.3 Optimal pulse samples ........................................................................................ 18

Figure 2.4 Time response of the optimal pulse, L = 20 ........................................................ 19

Figure 2.5 Optimal spectral response, L = 20 ....................................................................... 19

Figure 2.6 Normalized spectral response for L = 100 (a) and the corresponding time

domain response (b) ............................................................................................ 20

Figure 2.7 The top row gives time domain waveforms and the bottom row power

spectral densities for (a,d) Gaussian pulse, (b,e) Gaussian monocycle, and

(c,f) Gaussian doublet ......................................................................................... 22

Figure 1.8 Optimal UWB pulse shapes for L=2, 3, 7, 14 and 30 (a), and the

corresponding spectra (b) ................................................................................... 26

Figure 2.9 Efficiency vs. L for the optimization method based on sampling (red) and

combining Gaussian monocycle pulses (dashed blue) ....................................... 26

Figure 2.10 Normalized spectra ofUWB sub-band pulses; (a) 3.5----5GHZ, (b)

6----7.5GHZ, and (c) 8.5----10GHZ ......................................................................... 28

Figure 3.1 Spatial shaping using the SLM [16] .................................................................... 30

Figure 3.2 Broadband RF waveform generator, (a) Experimental apparatus. (b)

Reflective geometry Fourier transform [21] ....................................................... 31

Figure 3.3 UWB pulse generation based on spectral shaping of a MLFL ........................... 32

Figure 3.4 AlI optical UWB pulse generation based on phase modulation and

frequency discrimination .................................................................................... 33

VIl

Figure 3.5 Concept of arbitrary pulse generation by spectral pulse shaping ........................ 33

Figure 3.6 Block diagram of the UWB waveform generator ............................................... 35

Figure 3.7 Interference pattern ofa phase mask ................................................................... 36

Figure 3.8 MLFL normalized power spectral density .......................................................... 37

Figure 3.9 Flattening filter (FBG1) design; (a) required normalized spectrum, (b)

apodization profile .............................................................................................. 38

Figure 3.1 0 (a) Flattening filter transmission response measured using an optical

vector analyzer (b) a detailed view of the filter response ................................... 39

Figure 3.11 Pulse shaping filter (FBG2) design; (a) time domain target pulse, (b)

filter transmission profile, (c) apodization profile .............................................. 40

Figure 3.12 (a) L = 14 pulse shaping filter transmission -response measured using an

optical vector analyzer (b) same measurement after averaging ......................... 41

Figure 3.13 Simulation results for L=14. (a) Transmittivity ofFBGs, (b) PSD at

upper and lower arms, and (c) simulated and designed output pulse ................. 42

Figure 3.14 Experimental results for L = 14. (a) PSD at upper and lower arms, (b)

measured UWB pulse, and (c) the spectrum ...................................................... 45

Figure 3.15 Experimental results for L = 7. (a) measured UWB pulse, and (b) the

spectrum. The enlarged part shows the sinusoidal variations due to

multiple reflections ............................................................................................. 46

Figure 3.16 Experimental results for L = 30. (a) measured UWB pulse, and (b) the

spectrum ................................................................. ! 46

Figure 3.17 Schematic diagram of the tunable UWB waveform generator ......................... 48

Figure 3.18 (a) Transmitti~ity of the pulse shaping FBG , tunable Filterl and Filter2,

(b) designed UWB waveform and filters' shapes ............................................... 49

Figure 3.19 Tuning filters transmission responses (a) the low pass filter (b) the high

pass filter (c) a band pass filter ........................................................................... 49

Figure 3.20 Generated and target waveforms and their spectrum: (a, b) Gaussian, (c,

d) monocycle, '( e, f) doublet and (g, h) FCC-compliant, power efficient

pulses .................................................................................................................. 51

Figure 4.1 (a) SkyCross (SMT-3T010M) UWB antenna, (b) azimuth radiation

pattern at 4.9 GHz ............................................................................................... 57

VIn

Figure 4.2 Antenna measurements, (a) antenna frequency response, (b) antenna link

delay, (c) antenna phase response, (d) antenna reflection response ................... 60

Figure 4.3 (a) Smoothed antennas frequency response, (b) normalized time response ....... 61

Figure 4.4 The wireless link, (a) setup block diagram, (b) PA frequency response, (c)

antenna frequency response, (d) LNA frequency response, (e) PA,

antennas and LNA frequency response .............................................................. 62

Figure 4.5Transmit pulses (1), spectrums (2), EIRPs (3), for Gaussian monocycle,

doublet and FCC-optimized pulses ..................................................................... 64

Figure 4.6 Received pulses (1) and spectrums (2), for Gaussian monocycle, doublet

and FCC-optimized pulses ....... .. ... .......................................... ... ........ .. ....... ... ..... 65

Figure 4.7 The FCC and the effective spectral masks .......................................................... 68

Figure 4.8 EIRP optimized pulse shaping, (a) time domain pulse shape, (b)

transmission response of the pulse shaping FBG, (c) apodization profile

of the FBG, (d) experimental insertion loss of the pulse shaping FBG

written in 3 sweeps ............................................................................................. 70

Figure 4.9 (a) the designed (dashed line) and the measured (solid line) pulse shapes,

(b) Measured PSD and the FT of the measured time domain pulse shape,

( c) The PSD of the designed and generated p.ulse and the corresponding

EIRPs ............................................................................................................. : .... 71

Figure 4.10 Waveform comparison (a) time domain measurements (b) PSDs

compared to the effective mask, (c) PSDs compared to the FCC mask ............. 72

- - ----- --------------

IX

List of Tables

Table 1.1 FCC EIRP limits for indoor and outdoor UWB Applications ................................ 3

Table 1.2 Categories of applications approved by FCC ..................... .................................... 7

Table 3.1 Cutoff Wavelengths of the Filters for Different UWB Waveforms ..................... 49

Table 4.1 Peak to peak voltage (Vpp), Average total power and PE for the Gaussian

monocycle, doublet and the FCC-optimized waveforms ..................................... 66

List of Acronyms

UWB FCC UMTS

' EIRP SNR PE LNA PAM OOK PPM HDTV WPAN FBG FIR RF LO BJT CMOS BER BBCS SLM PSD DFT SMF FT OIE LCM BPD DL ATT UV 1FT EDFA MLFL OSA FWHM Tx Rx PCB MLA VNA PA

Ultra-wideband US Federal Communications Commission Univers al Mobile Telecommunication System Equivalent Isotropically Radiated Power Signal to Noise Ratio Power Efficiency Low Noise Amplifier Pulse Amplitude Modulation On-Off Keying Pulse Position Modulation High-Definition Television Wireless Personal Area Network Fiber Bragg Grating Finite Impulse Response Radio Frequency Localoscillator Bipolar Junction Transistor Complementary Metal Oxide Semiconductor Bit Error Rate Broadband Coherent Source Spatial Light Modulator Power Spectral Density Discrete Fourier Transform Single Mode Fiber Fourier Transform Optical Electrical Conversion Liquid Crystal Modulator Balanced Photodetector Delay Line Attenuator Ultra-violet Inverse Fourier Transform Erbium Doped Fiber Amplifier Mode Locked Fiber Laser Optical Spectrum Analyzer Full Width HalfMaximum Transmi tter Receiver Printed Circuit Board Meander Line Antenna Vector N etwork Anal yzer Power Amplifier

x

Xl

Vpp Peak-to-Peak Voltage BR Bit Rate

Chapter i: introduction ,

Chapter 1

Introduction

Connectivity for "everybody and everything at any place and any time" is the vision of

wireless systems beyond the third generation. Short-range wireless technology will play a

key role in scenarios of ubiquitous communications over different types of links [1]. Novel

devices based on ultra-wideband (UWB) radio technology have the potential to provide

solutions for many of today's problems in the area of spectrum management and radio

system engineering.

2 Chapter 1: introduction

1.1. UWB Basics

UWB technology has existed since the 1980s [2]; it mainly has been used for radar

applications because of the wideband nature of the signal that results in very accurate

timing information. In the early days UWB was referred to as impulse radio, where an

extremely short pulse with no carrier was used instead of modulating a sinusoid to transmit

information. These sub-nanosecond pulses occupy several GHz of bandwidth and are

tr&nsmitted with very low duty cycles. In April 2002, after extensive commentary from

industry, the US Federal Communications Commission (FCC) issued its first report on

UWB technology, thereby providing regulations to support deployment of UWB radio

systems [3]. These regulations allowed the UWB radios to coexist with already allocated

narrowband radio frequency (RF) emissions.

The band allocated to UWB communications lS 7.5 GHz wide, by far the largest

allocation of bandwidth to any commercial terrestrial system. The FCC UWB rulings

allocated 1500 times the spectrum allocation of a single UMTS (universal mobile

telecommunication system) license [4]. However, the available power levels are very low.

If the entire 7.5 GHz band is optimally utilized, the maximum power available to a

transmitter is approximately 0.5 m W. This effectively relegates UWB to indoor, short-

range, communications for high data rates, or very low data rates for substantial link

distances. In principle, trading data rate for link distance can be as simple as increasing the

number of pulses used to carry 1 bit. The more pulses per bit, the lower the data rate, and

the greater the achievable transmission distance.

UWB devices are intentional radiators under FCC Part 15 Rules. The FCC report

introduced four different categories for allowed UWB applications, and set radiation masks

for them. For a radiator to be considered UWB the fractional bandwidth defined as

must be at least 0.2. In the formula above, fH and JL are the higher and lower -10 dB bandwidths, respectively.

Chapter i: introduction

Table 1.1 Fee EIRP limits for indoor and outdoor UWB Applications

Freq uency (GHz)

0.96-1.61

1.61-1.99

1.99-3.1

3.1-10.6

Above 10.6

.96 : 1.99 3.1 1.61

-63.3 1

-75,.1 1

Indoor EIRP (dBm)

-75.3

-53.3

-51.3

-41.3

-51.3

-41.3

..... lndoor Outdoor

frequency (GI'Iz)

Outdoor El RP (dBm)

-75.3

-63.3

-61.3

-41.3

-61.3

'.

-51.3

,61 ~ ....... _-

10.6

Figure 1.1 Fee spectral masks for indoor and outdoor communication applications

3

Also, according to the Fee UWB rulings the signal is recognized as UWB if the signal

bandwidth, i.e. , IH - I L' is 500MHz or more. The radiation limits set by the Fee are

presented in Table 1.1 for indoor and outdoor data communication applications. These

limitations are expressed in terms of equivalent isotropically radiated power (EIRP). EIRP

is the product of the transmit power from the antenna and the antenna gain in a given

direction relative to an isotropic antenna. Further discussion about EIRP will follow in

section 4.1. Figure 1.1 shows the EIRP limits imposed by the Fee spectral masks for the

indoor and outdoor communication systems. Other applications such as vehicular radar, are

4 Chapter 1: introduction

restricted by different masks. Allowed UWB emission levels are less than or equal to the

level allowed for unintentional radiators such as computers and other electronic devices

(-41.3 dBm/MHz). Thus the UWB transmitter can be treated like noise by other

communication systems.

The strict power limitations imposed by the FCC spectral mask necessitate spectral pulse

shaping: designing spectrally efficient pulses that eke out most of the power available under

the FCC mask. UWB system performance highly depends on the signal to noise (SNR)

ratio. Therefore choosing efficient pulses for UWB. communication systems is of critical

importance. In this thesis, we design and generate sorne efficient UWB waveforms which

show significant improvements in terms of transmit power over the widely adopted

Gaussian UWB waveform ~amily.

The power efficiency (PE) is the average power of a pulse normalized by the total

admissible power under the FCC spectral mask (----0.5 m W). The goal of this work is to

maximize the PE for the indoor UWB communication systems by generating efficient

pulses that exploit the FCC mask in the best way.

Emitted Signal Power

Bluetooth, Zigbee WLAN 802.11 b WPAN 802.15.4

3G Cellular

WiMax Indoor

4G Cellular Indoor

(Future)

UW8 Spectrum FCC "Part 15 limits"

2.4 2.7 3.1 3.4 3.8 4.2 4.8 5.5 Frequency (GHz)

10.6

Figure 1.2 UWB spectrum coexisting with other narrowband communication systems 15]

Chapter i : introduction

1.2. UWB Compared to Other Wireless Technologies

1000

UWB 480Mbps @ 2m Short 200Mbps @ 4m

Distance . Fast

Download... UWB 110Mbps @ 10m _------_ _ ""Room-range

High-definition Quality of service,

streaming

Bluetooth

ZigBee

802.11 a/b/g/n Data Networking

.11n promises 100Mbps @ 100m

UWB, low data-rates, location & tracking

10 Range (m) 100

Figure 1.3 WiMedia landscape of UWB compared to Wireless local area networks (WLAN) [5]

5

Figure 1.2 highlights the low power but wideband nature of the UWB compared to other

wireless networks. UWB coexists with other narrowband networks; the interference caused

by a UWB transmitter can be viewed as a wideband interferer, and it has the effect of

raising the noise floor of the narrowband receiver. A major benefit to UWB of low power

constraint is preserving battery life. Another benefit of low PSD is low probability of

detection which is a concem for both military and consumer applications. The weak UWB

pulses are inherently short range which makes the operation of multiple independent links

possible within the same house. The broadband property of the UWB signal makes it

resistant to interference because any interfering signal is likely to affect a small portion of

the desired signal spectrum.

UWB can provide very high speed but short distance communication links. Figure 1.3

. shows the WiMedia landscape ofUWB services compared to IEEE 802.11 networks [5]. It

can be seen that UWB is the fastest in close range while the IEEE 802.11 is more suitable

for distances more than 10 m. The spatial capacity, an indicator of data intensity in a

transmission medium, is over 106 bit/s/m2 for UWB, whereas just 1000 bit/s/m2 for IEEE

802.11 b [2]. No system is capable of reaching a spatial capacity as high as that of a UWB

6 Chapter 1: Introduction

system. A reason is the Shannon channel capacity theorem [6]. The upper ' bound on the

capacity of a channel grows linearly with the available bandwidth. Thus, the UWB systems

,occupying several GHz of bandwidth show great potential for the future high capacity

wireless networks.

Sorne important issues attributable to UWB are discussed here.

a) Antennas: Antennas have a filtering effect on the UWB pulse. Good impedance

matching over the entire UWB bandwidth is desired to reduce reflection losses from

the antennas. The impulse response of the antenna changes with angles in azimuth and

elevation. Therefore, the transmitted pulse is differently distorted at every angle.

b) Low noise amplifiers (LNAs): Design of amplifiers ~s another challenge for UWB

applications. Due to the low power and wideband nature of the UWB signal, very low

noise and wideband amplifiers are essential at the receiver side.

c) Modulation: For pulsed UWB systems, the widely used modulation schemes are pulse

amplitude modulation (P AM), on-off keying (OOK), and pulse position modulation

(PPM). The OOK scheme results in energy detection receivers of lower complexity,

whereas the PPM shows better error performance but lower bit rates.

d) Multipath: In the indoor environment the signal bounces off objects located between

the transmitter and receiver creating multipath reflections. If the delay spread of the

echoes is smaller than the pulse width, the echoes can combine destructively leading

to multipath fadi,ng. However, for an indoor UWB system with a range of 10 m, the

delay spread is typically several nanoseconds [7]; significantly more than a typical

UWB signal pulse width. This makes UWB resistant to multipath interference. To

maximize the received energy, one can use a RAKE receiver to combine the signaIs

coming over resolvable propagation paths. However, combining many multipath

components increases the complexity of the receiver.

e) Multiband: Multiband UWB provides a method where the FFC approved 7.5 GHz

UWB bandwidth is split into several smaller frequency bands, each having a

minimum of 500 MHZ bandwidth. The signaIs do not interfere with each other

7 Chapter 1: Introduction

Table 1.2 Categories of applications approved by FCC 18]

Class/ Application Frequency band of operation

Communications and measurement systems 3.1-10.6 GHz

Imaging: ground penetrating radar, wall ,

8 Chapter 1: introduction

b) Wireless Personal Area Networking (WP AN): A high speed wireless UWB link can

connect cell phones, laptops, cameras, Mp3 players. This technology provides mu ch

higher data rates than Bluetooth or 802.11. UWB a portable MP3 player could stream

audio to high-quality surround-sound speakers anywhere in the room.

c) Vehicle collision avoidance: UWB can provide enough resolution to distinguish cars,

people, and poles on or near the road. This information can be used to alert the driver

and prevent collisions. UWB radar has the resolution to sense road conditions (i.e. ,

potholes, bumps, and gravel vs. pavement) and provide information to dynamically

adjust suspension, braking, and other drive systems.

d) Radar: UWB can provide centimeter accuracy in ranging because of its high time

resolution. Improved object identification (greater resolution) is achieved because the

received signal carries the information not only about the target as a whole, but also

about its separate elements. The low power of the UWB signaIs reduces the

probability of detection by hostile interceptors [10].

e) Other applications of UWB are public safety systems including motion detection

applications, RF tag for personal and asset tracking, medical monitoring and so forth.

Our proposed optical UWB pulse generation technique targets the applications a) and b)

rnentioned above. These applications require high speed, high range wireless

communications and our approach in efficient pulse generation can help fulfill these needs.

1.4. Structure of the Thesis

The basic concepts of the emerging UWB communications were outlined in this chapter.

The power restrictions imposed by the FCC spectral mask emphasizes the importance of

using efficient pulses as the building blocks of a UWB communication link. In Chapter 2,

'we develop a linear optimization program -to find optimal, pulses to exploit the power

available under the FCC spectral mask. In Chapter 3, we propose a method to implement

the efficient UWB waveforms using Fiber Bragg gratings (FBGs). We demonstrate

experimental results that accurately respect the FCC mask while maximizing the permitted

-- ---- - ---------- - - ------,

9 Chapter 1: introduction

power. Later, we generate several UWB pulses with the same setup by introducing a band-

pass tunable filter. In Chapter 4, we discuss the effect of antennas on the transmission of

UWB pulses. The impulse response of the antenna is measured using a network analyzer.

This non-ideal response affects the received signal, degrading the power efficiency at the

receiver. Subsequently, the antenna impulse response is taken into account in the

optimization process to design a new pulse which maximizes the EIRP. The

implementation of this newly designed pulse shows exceptior11 power efficiency and great

match to theoretical calculations.

This work has been truly coIlaborative in aIl the steps. My individual contributions are

detailed in Chapters 2 (development and generation of optimal UWB waveforms) and

Chapter 4 (characterization ofUWB pulse through transmit and receive antennas, and EIRP

optimization). Chapter 3 reports experiments run with Dr. Abtahi. While Dr. Abtahi was

responsible for supervising FBG development, deterrnining the experimental layout and

overaIl measurement strategies, l was an active participant in the measurements. The FBGs

used in Chapter 3 were fabricated by J. Magn, however l also participated in fabrication

and ,am now trained in writing FBGs. l wrote aIl FBGs presented in Chapter 4.

10 Chapter 2: Optimal UWB Waveforms

Chapter 2

Optimal UWB Waveforms

In UWB systems the conventional analog waveform, representing a message symbol, is a

simple pulse that in general is directly radiated. These short pulses have typical widths in

the pico second range, and thus bandwidths of over 1 GHz. In the literature, the most

common of these pulses are Gaussian monocycle and doublet pulse shapes [11]. Although

traditionally employed for UWB systems, these shapes poorly exploit the permissible

power under the FCC mask. Performance of a UWB system is mainly decideq by the

received SNR. To achieve the highest possible SNR, we have to eke out aIl the permissible

power under the FCC spectral mask. Therefore choosing the most efficient pulse for UWB

communication systems is of critical importance.

Il Chapter 2: Optimal UWB Waveforms

Recently, Giannakis et al. [12] suggested a digital finite impulse response (FIR) filter

approach to synthesizing UWB pulses and proposed filter design techniques by which

optimal waveforms that closely match the spectral rna~k can be obtained efficiently. For

single pulse design, a convex formulation is developed for the design of the FIR filter

coefficients that maximizes the spectrum utilization in terms of both the bandwidth and the

power allowed by th spectral mask. Although we do not intend to irnplement optimal

pulses with an FIR filter approach, still we use this method to find optimal pulse

waveforms. The resuIting waveforms will be implernented not via FIR fiItering but by a

cornpletely different method in Chapter 3 using optics.

In 2.1.1 , we address sorne of the common RF rnethods to generate UWB pulses. We

discuss optical . pulse shaping methods in section 2.1.2. The shortcomings of these

techniques in the generation of highly efficient UWB pulses lead us to design new

waveforms in sections 2.2 and 2.3.

2.1. Pulse Shaping Techniques

Transrnitter architectures for pulse-based UWB signaIs in the FCC UWB band (3.1-10.6

GHz) can be grouped into two categories depending on how the pulse is generated. The

first category of transmitters generate a pulse at baseband and up-convert it to a center

frequency in the UWB band by mixing with a local oscillator (L). The second category

generates a pulse directly in the UWB band without frequency translation. A base band

impulse may excite a filter that shapes the pulse, or the pulse may be directly synthesized at

RF without requiring additional filtering. The up-conversion architecture generally offers

more diversity and control over the frequency spectrum, but at the cost of higher power,

since an L must operate at the pulse center frequency.

Electrical pulse shapers can be designed based on either of the two approaches, but optical

pulse shaping methods directly generate the desired baseband waveform. Sorne examples of

electrical and optical approaches of pulse shaping are discussed in this section.

12 Chapter 2: Optimal UWB Waveforms

2.1.1. Pulse shaping in the Electrical Domain

We take a brief look at the electrical pulse shaping methods, although we will focus on

optical methods in this work. As mentioned before, the pulses are either directly shaped in

the UWB bandwidth or are up-converted from baseband. As an example of up-conversion

of a wideband pulse, we consider generation of a multiband UWB pulse [13]. A near-

Gaussian pulse is shaped from a triangular input signal by exploiting the exponential

properties 0 a bipolar junction transistor (BJT). The pulse is up-converted to one of the

fourteen 528 MHz-wide channels in the 3.1-10.6 GHz UWB band. Pulse shaping is

integrated into the mixer performing up-conversion, fabricated in a 0.18 ",m SiGe

BiCMOS process. We will see that the Gaussian s4ape poorly exploits the FCC spectral

mask.

ln [14] , an overall design of pulse generator and transmit antenna is proposed. They

design a chip . to generate Gaussian monocycle pulses for use with pulse position

modulation (PPM). The impulse generatoris preceded by a programmable pulse-position

modulator. The impulse generator consists of a triangular pulse generator and a cascade of

complex first -ord~r systems made up of differential pairs each approximating a Gaussian

monocycle waveform. The complete pulse generator is fabricated in IBM 0.18-,um Bi-

CMOS IC technology. The minimum attaina~le pulse width is about 375 ps, with 330 ps

offset for pulse-position modulation.

ln another technique, a single-chip CMOS pulse generator with pulse shaping is proposed

that combines various delayed pulses to form a short pulse that is filtered to obtain the

UWB pulse [15]. A separate band-pass filter (3.1 to 5 GHz) is ,used to obtain an FCC-

compliant pulse with duraton of about 1.5 ns; the generated pulse is, however, not power

efficient.

The major advantages of electrical generation of UWB pulses are low-cost and possibility

of integration on a single chip. The major drawback is the imprecision of the generated

pulses leading to violation of the FCC mask. As a result, the pulse power should be

lowered, reducing the SNR and worsening the bit error rate (BER). Another disadvantage is

13 Chapter 2: Optimal UWB Waveforms

that the electrical methods normally do not cover aIl of the available bandwidth, which

degrades the spectral utilization.

2.1.2. Pulse shaping in the Optical Domain

Optical pulse generation techniques for UWB have been proposed based on optical

spectral shaping and frequency-to-time conversion. The general concept is to shape the

spectrum of a broadband coherent source (BBCS) to match the desired time-domain

waveform. The spectral shape is converted to a time domain shape (a pulse shape) by

passing through a dispersive medium such as dispersive fiber or a crnrped fiber Bragg

grating. In this section, we present a concise review of previous UWB optical pulse shaping

techniques. These methods, along with our proposed method ofUWB pulse generation will

be thoroughly covered in Chapter 3.

Consider first a free space optical implementation. The pulse shaping device in [16] is a

4-f grating and lens apparatus consisting of two free space bulky gratings, two large focal

length lenses to angularly disperse the frequency components, and a spatial light modulator

(SLM) to modulate the, amplitude of frequency components. Although this setup is tunable

and can generate various pulses, it suffers from high free-space losses and bulky packaging.

A more realistic method of implementation is using optical fibers. An all-fiber pulse

shaper was proposed in [17] in which two optical filters with complementary spectra are

placed in two arms of an interferometer to shape the power spectrum to a Gaussian

monocycle or doublet pulse. In general, optical pulse shaping methods are of higher

precision compared to their electrical counterparts. However, research has usually focused

on generation of Gaussian pulses which have a po or coverage of the FCC mask. In the next

section, we will discuss the design of more sophisticated but efficient pulses which can be

generated using low-cost FBGs.

]4 Chapter 2: Optimal UWB Waveforms

2.2. Optimization Process Based on Sampling

To best exploit the UWB bandwidth, the subject of stringent FFC emission regulations,

we seek an optimized time domain pulse shape which maximizes the transmit power

subject to the FCC spectral mask. The normalized spectral mask for indoor

communications is plotted in Figure 1.1

Let pet) be a UWB pulse that we sample at Fs = 28 GHz. As the FCC spectral mask is

nonflat up to 10.1 GHz, this sampling rate is sufficient up to 14 GHZ per the N yquist

sampling criterion [18]. The sampling theorem states that the UWB pulse can be related to

the samples,p[k] , by

00

p(t) = 2:p[k ]sinc [Ct - kTo)~. ] (2.1) k =o

where Ta is the sample spacing and Fs = liTa is the sampling rate. We can approximate pet)

as a truncated sum of L terms; the UWB pulses have very short time do main responses,

justifying keeping only L terms, as samples quickly approach zero. Of course, the larger

that Lis, the better the approximation.

Our optimization criteria is the maximization of the transmit power, which is calculated as

the integral of the power spectral density (PSD) of the signal. The PSD of p(t) is Ip ( ej @ )1 2 , where (j) is the angular frequency and P ( e}fj ) is the Fourier transform of pet). In other

words, the problem is to

(2.2)

subject to Ip (e }{ )1 fal.ling under the FCe mask over the entire frequency range. The parameters a and fJ set the bandwidth of interest. Meeting this condition for aIl

frequencies would require an infinite number of constraint equations. The infinite number

of inequalities can be reduced to a finite number by sampling the frequency range. The

15 Chapter 2: Optimal UWB Waveforms

number of frequency samples should be high enough to assure good precision and at the

same time offer reasonable computation time.

The FCC mask amplitude constraints are not convex in the optimization variables p[k] and

hence algorithms for solving it must deal with local optima. As proposed in [19] , we

examine autocorrelation coefficients instead of the pulse itself; this converts the problem to

a linear optimization problem. Autocorrelation of p[ k] is defined as:

L- k

r[k]= LP[i)P[i+k] k=-L, ... ,O, ... ,L (2.3) ;=0

Note that r [k] = r [-k]. Taking the discrete Fourier transform (DFT) ofr [k] , we find

R ( el'" ) = f r [ k ] e - i",k k=-oo

L

= r [0] + 2 L r [k] cos (km) (2.4) k=1

From (2.2) and (2.4), the optimization goal becomes:

(2.5)

subject to R ( ejOJ ) falling under the FCC spectral mask. We also require that R ( ejOJ ) be

strictly non-negative. This is a necessary and sufficient condition for existence of p[ k]

satisfying (2.3) by the Fejer-Riesz theorem [19]:

Suppose C denotes the complex numbers field. If a complex function Wez):

C ~ C satisfies

m

W(z)= L w(n)z-n and W(z)~O Vlzl=l, n=-m

then there exists fez): C -) C and y(O), ... , y(m) E C such that

Chapter 2: Optimal UWB Waveforms

m

y (z) = l y( n) z-n and w (z) =/ y (z) /2 'v' / z /= 1 . n=-m

Y(z) is unique if we further impose the condition that aU its roots be in the

unit circle 1 z I~ 1 .

16

Now we have a linear objective in terms of r[k] and the constraints are also sets of infinite

inequalities in r[k]. This type of optimization problems can be solved using a convex

optimization solver. We choose SeDuMi, a MA TLAB optimization toolbox developed at

McMaster University [20]. This toolbox can solve the following problem:

maxBTy (2.6)

such that c. - A. Y > 0 for i = 1, 2, ... , n. 1 1

Therefore, we just need to convert our optimization goal and its constraints to this format.

The A, Band C matrices are found to be:

2cosaJ1 2cos 2aJ1 2cosLaJ\

2cos0J2 2 cos 2OJ2 2cosL0J2

1 2 cos aJn 2cos20Jn 2 cos aJn A= -1 -2 cos OJ1 -2cos20J\ -2cosL0J1 -1 -2 cos OJ2 -2cos 2OJ2 -2cosL0J2

1 -2cosOJn -2 cos2aJn -2cosLOJn (2.7)

B=[,B-a 2(sin,B-sina) ... L(sin,B-sina)]T

Chapter 2: Optimal UWB Waveforms

0.5

0.4

~ 0.3 '"1::'

cJ al 0.2 ()

~ u 0.1 c 0

~ ~ 0

18 Chapter 2: Optimal UWB Waveforms

The optimal pulse samples, p[k] , are obtained by the spectral factorization method from

the autocorrelation function using the program explained in Appendix A; the pulse samples

are plotted in Figure 2.3.

0

-10 ,-- -\

~ -20 1

E \: :1 2 ~ 1 i -30 Il , CI) c 0 ~ -40 ~

cs g :5 -50 ct

-60

-70 0 2 4 6 8 10 12 14

Frequency (GHz)

Figure 2.2 Normalized Autocorrelation spectrum, R (ejw )

0.4

0.3

0.2

~ cr 0.1 ct Q)

a. 0 E

cu CI)

Q) CIl -0.1 "3 a..

-0 .2

-0.3

-0.4 0 5 10 15 20 25

Figure 2.3 Optimal pulse samples

19 Chapter 2: Optimal UWB Waveforms

0.8

0.6

0.4

CL 0.2 "0 Q.l

.~ 0 ij E Cs -0.2 z

-0.4

-0.6

-0.8

-1 -1 -0.5 o 0.5 1.5

t (ns)

Figure 2.5 Time response of the optimal pulse, L = 20.

Having the samples, p[k], it is straightforward to find the continuous time response pet)

using (2.1 ). Figure 2.5 shows pet) obtained from the p[k] coefficients in Figure 2.3. We can

see sorne slowly decaying variations on either side of the pulse, attributed to the tails of the

sinc function in (2.1). The power spectrum of the optimal pulse is illustrated in Figure 2.4.

The power efficiency

o .. _ .. ---r-~V:.: " -~'--'-----"-~\" : : 1

-10 1"

-20 CI en :\ a.. "0 .~ -30

\1\ ni E 0 z

-40

-50 1

\i

-60 0 2 4 6 8 10 12 14

f(GHz)

Figure 2.4 Optimal spectral response, L = 20.

20 Chapter 2: Optimal UWB Waveforms

(2.8)

is the average power of the pulse normalized by the total admissible power under SFCC( (J)) ,

the Fee mask. BW is the UWB bandwidth over which we integrate to find the power.

We note from Figure 2.4 that the optimized pulse completely satisfies the Fee spectral

mask and has 75% efficiency. By increasing the pulse length (or equivalently, L), the

spectrum of the pulse better fits the mask, and the spectral utilization factor or power

efficiency increases. The larger the number of taps, the higher the power efficiency but also

the greater the complexity. Because of this trade-off between efficiency and pulse length, a

reasonable value for the number of taps should be chosen.

To examine an extreme case, L is set to 100. Figure 2.6a shows the spectrum in this case:

an extremely tight fit to the mask and efficiency of 94%. The resulting time domain pulse is

plotted in Figure 2.6b. The pulse has become very lengthy with many small variations at

the tail that are impractical to implement. We must strive for a simple, spectrally-efficient

pulse that is easy to implement as discussed in the next chapter.

o (/)

CL

-10

-20

~ -30

1 z -40

-50

\

-60 L--.w...J......J_--'_--'-_----'-_--'-_ ----'-_---' o 6 8 10 12

1 (GHz)

(a)

0.8

0.6

0.4

f 0.2 ~ .~ 0 ii E ~ -0.2

-0.4

-0.6

-0.8

-1 L-----'----'--'------'----'-_-'------'-_-'-----'--_'-----' -1 -0.5 0.5 1.5

t (ns)

(b)

2.5 3.5

Figure 2.6 Normalized spectral response for L = 100 (a) and the corresponding time domain response (b).

21 Chapter 2: Optimal UWB Waveforms

FinaIly, we conclude from our results that although many samples provide more power

efficient pulses, a high number of samples is not easily implemented. This is mainly caused

by the slowly decaying tail of the sinc function. The techniques used in this section can be

applied to a more realistic implementation, i.e., with smaller L. In section 2.3, we take the

concept of optimizing a sum of weighted sinc functions, and instead apply the optimization

to a sum ofweighted Gaussian monocycles [12].

2.3. Optim,ized Sum of Weighted Gaussian Monocycles.

Gaussian waveforms, ploned in Figure 2.7, are a family of functions deriving from a

Gaussian pulse defined as,

(2.9)

where A is an amplitude scaling factor and Tg is a time scaling factor. The Gaussian

monocycle (2.10) is the first derivative of the Gaussian pulse, and the second derivate

results in the Gaussian doublet (2.11).

(2.10)

(2.11 )

At each derivative, one zero-crossing is added. The Gaussian has no zero-crossing, a

monocycle has one, and so on. Furthermore, at each additional derivative the fractional

bandwidth decreases, while the center frequency increases (Figure 2.7).

An important aspect of these Gaussian waveforms is their wideband spectrum. By

combining many multi-GHz pulses, we can carve the desired spectral shape. In the time

domain, Gaussian waveforms are smooth and weIl behaved functions, making them easier

to implement, as discussed in Chapter 4. In addition, a Gaussian pulse is easy to directly

Chapter 2: Optimal UWB W. avef orms

V "'0

.~ 0.. := E !: ~ ~ ~ O. 5 Co . ~ ~ ~

E o

Gaussian

200 t eps)

(a)

Z (~--~--~~----o 5 10 15 20

f(GHz) (d)

v 1 "'0 .~ 0.. E ~

"'0 0 V .~ ~ E

~-I -200

v "'0

.~ 0.. E ~

~ O. ~ . ~ ~ E 0 Z 0

0

Monocycle

( 100 200 t (p)

(b)

J 10 15 _0

f(GHz) (e)

v "'0

.~ 0.. E ~

"'0 V .~ ~-

E 0 Z

v "'0

.~ 0.. E ~

-0 0. :-v

. ~ ~ E 0 z

Doublet

te ps)

(c)

22

200

;) JO j 5 20

f(GHz) (f)

Figure 2.7 The top row gives time domain waveforms and the bottom row power spectral densities for (a,d) Gaussian pulse, (b,e) Gaussian monocycle, and (c,f) Gaussian doublet.

generate ln electronics. For these reasons we choose the Gaussian monocycle as the

building block of the optimization process for this chapter as suggested in [12].

2.3.1~ Optimization Procedure

The optimization procedure to find the weights of combined Gaussian monocycles is quite

similar to the method discussed in section 2.2 for the sinc. The desired pulse shape, pet), is

written as a summation ofweighted Gaussian monocycles,

L -I

pet) = Lw[k]gm(t -kTa) (2.12) k=o

where gm is the Gaussian monocycle, Ta is the pulse spaclng, {w [k] } ~ :~- I are real

coefficients to be determined by the optimization process. By increasing the number of

coefficients, L, we can obtain a better power efficiency, but the pulse duration will be

greater.

23 Chapter 2: Optimal UWB Waveforms

Pif) , the F ourir transform of pet), can be expressed as

1 P (f) 1 = Iw (ej2 ~.fTo ) IIGm (f) 1 (2.13)

where G m cr) is the Fourier transform of the Gaussian monocycle and W is the discrete

Fouriertransform ofvector w defined by w = [w [O] ,w [l] , . ,w [L -1] J.

The UWB pulse pet) should be designed with an optimization process to maximize" the

permitted power within the UWB frequency range,

2

max Ir IpU)1 dl. p(t) l' p

(2.14)

where Fp is the desired UWB bandwidth. Tbis maximization problem is subject to the PSD

restrictions imposed by the FCC spectral mask. The power spectrum, Ip (/)1 2 , should be under the FCC spectral mask over the desired frequency range. This optimization problem

is non-convex, requiring rigorous numerical methods. To transform this to a convex

optimization problem we again turn to the autocorrelation of w, defined as

r[k]==LiW[i]w[i+k] (2.15)

with vector representation by r == [r [0], r [1], ... , r [L -.-: 1 ]]T We define two auxiliary

vectors, v Cf, L) and Cf, L) by

- (1 L) == [1 j27rj To j27rf2To j27rf (L-I)To JT v, ,e ,e "e (2.16)

(2.17)

The following equalities are useful in simplifying the problem.

(2.18)

24 Chapter 2: Optimal UWB .Waveforms

(2.19)

(2.20)

. where H indicates Hermitian transpose of a matrix and we have used the property

r[ k] = r[ -k] in (2.19). From (2.19) and (2.20), we can find

(2.21)

From (2.13), (2.14), and (2.21) the optimization goal can be simplified as

(2.22)

= fT (f,L )rpm (f)1 2d! = B T .r Fp

where B = f(f,L)P(f)12

d!, 1:;P

The FCC-imposed limit can be expressed as f' (eJ27rj7o )12 p (f)I~ s S FCC (f), Therefore the optimization problem is simplified to

maxB T .r (2.23)

subject to T (f,L)f s S FCC (1)/ Pm (1)12 f E Fp

To ensure a valid autocorrelation vector we also require T (f, L ) r ~ 0 f E Fp

These constraints, forming a convex semi-infinite linear optimization problem, can be

made discrete to form a finite linear program. While this gives an approximate solution,

enough samples ensure acceptable precision of the solution. The problem can be solved

25 Chapter 2: Optimal UWB Waveforms

using a convex cone optimization toolbox such as SeDuMi optimization tool, as in section

2.2.

Transforming our optimization problem (2.23) to the SeDuMi format (2.6) results in the

following matrixes

i = 1 Bi =

2 fPm (f)12 COS (27rf(i -l)To)df i = 2, ... ,L Fp

rp

{C , } 2n = 0 1 i=n+1 (2.24)

j = 1

j = 2, ... ,L'

A =- A , ' { }i=2n)=L {}i=n)=L

Ij i =n + 1) = 1 IJ i = 1 J' = 1

, where n is the number of equally spaced frequency samples over the UWB bandwidth F p

After obtaining the optimal autocorrelation vector r using SeDuMi, we find the optimal

filter tap coefficients m by spectral factorization [19]. Once we know the optimal tap coefficients, finding the UWB pulse which optimally exploits and respects the Fee mask is

trivial via (2.12).

Figure 2.8 shows the power efficient UWB pulses generated by combining different

number of Gaussian monocycles with parameters Ta = 35.7 ps and Tg = 46.5 ps. We see

clearly that increasing L results in longer and more complicated waveforms which have

better power efficiency. For L = 2 the result is a Gaussian monocycle and L = 3 yields a

Gaussian doublet. Any pulse, can become compliant by reducing their power to respect the

Fee mask especially at the 1.6 GHz edge. This leads to a very poor power efficiency, as

for the Gaussian monocycle withjust 0.12% of efficiency.

Chapter 2: Optimal UWB Waveforms

o --

-10

co ~ -20

ID ~ o

a.. -30 "0 ID .~ ro E -40 o Z

-50

-60

_-- :,;\\

: 1- L=30 L=~~ .. o ........ . . o .. L=3 1 1

1 1 1 1 .--0- - ' 0_. L=2 1 1

L=2 SE=0.12 % L=3 SE=1 .38 % L=7 SE=47.5 % L=14 SE=67.0 % L=30 SE=75.1 % Norm~lized FCC Mask

- -- ------- ---- -

26

_70~o----~----~----~----~----~----~ ____ ~ o 0.2 0.4 0.6 0.8 1.2

Time (ns) (a)

o 2 4 6 8 Frequency (GHz)

(b)

10 12 14

Figure 2.8 Optimal UWB pulse shapes for L = 2, 3~ 7, 14 and 30 (a), and the correspo.nding spectra (b).

Figure 2.9 shows the spectral efficiency versus L for both optimization methods used, i.e.

the optimization based on sampling and optimization by combining Gaussian monocycle

pulses. Interpolating curves have been fitted to the calculated points. We can see that using

pulse samples will result in a better efficiency, especially for large values of L. However,

larg~ values of L lead to very complex pulses. In the reasonable range of L ~ 30 , the two

methods have similar efficiencies. In this region, the Gaussian pulse combination method is

preferable because it offers smoother and shorter pulses. This graph helps us to choose an

appropriate value of L for good efficiency.

2.3.2. Multiband UWB Pulse Design

AlI the waveforms we studied up to now correspond to traditional UWB technology based

on single-band systems that directly modulate data into a sequence of pulses which occupy

the available bandwidth from 3.1 to 10.6 GHz. In multiband UWB schemes, as proposed in

[21] , the UWB frequency band is divided into several sub-bands, each with a bandwidth of

at least 500 MHz in compliance with the FCC regulation~ [21]. By interleaving the

27 Chapter 2: Optimal UWB Waveforms

transmitted symbols across sub-bands, multiband UWB systems can still maintain the

average transmit power as if the large GHz bandwidth is used. The advantage is that the

information can be processed over much smaller bandwidth, thereby reducing overall

design complexity, as weIl as improving spectral flexibility and worldwide compliance.

Multiband UWB facilitates coexistence with legacy systems and worldwide deployment by

enabling sorne sub-bands to be tumed off in order to avoid interference and comply with

different regulatory requirements. In addition, multiband systems provide another

dimension for multiple access via frequency division. Different users can use different

pulses for multiple access, and frequency hopping can also be easily implemented by

switching among those [12].

We can easily generate sub-band UWB pulses by our MATLAB code by changing the

bandwidth Fp We choose L = 70, To = 32.5 ps, rg == 62 ps and divide the UWB bandwidth

in 3 regions of (3.5~5GHZ), (6~7.5GHZ), and (8.5-10GHZ). We have put 1GHz of guard

band between the sub-bands. The resulting spectra of UWB sub-band pulses are plotted in

Figure 2.10. The high suppression of the out of band frequencies is to avoid interference.

9 ~ ........ ~ ......... ~ ......... ~ ....... . . ~ ......... ~ ........ . ~ ......... : .......... ~ ..... ~

. . . . . . o' . . .. .:+: .

8 . . . . . ._.+.~.+-.. ~~~.~ . . . . . .

. . . . --; - Sa~pling ~ 7 ............... . . ..... ':' ....... ':' ....... ':' ....... ':' ... .-...., .. ~ . . Ga~ssian . ~

5 j ..... /j ........ . j ....... . . j ..... .... j .... ..... j ......... j ......... j ........ . j .... .... ~ : /: : : : : : : : : : 1. ~ : ~ ~ ~ ~ ~ ~

4 .. ..... ~ ........ : ........ ~ ......... : ......... : ..... .. .. ~ .' ....... : ........ ~ . ..... . . ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 : ........ : ......... ~ .. .. . . ... : ......... : . .. ... ... : . ..... . .. : ......... ~ ........ ~ . ....... ~ . . . . . . .

i 5 10 15 20 25 30 35 40 45 50

L

Figure 2.9 Efficiency vs. L for the optimization method based on sampling (red) and combining Gaussian monocycle pulses (dashed blue)

Chapter 2: Optimal UWB Waveforms

1

" ~ (\ .1( . 2(

.3( ~ .3( - -

-41.1\ 1\ Il ft Il 1\ n f1 11 11 f1 fi J -4{ll\lIl1ft fl l\ f1 Il .5( sc

-6C

1

.2(

. 3(

. 5(

-6C

28

\

6 8 10 12 14 .7o-----J....I-~U---.I.--L...L.-6 ----=-"--'-8 ~~10 -L--J.

1L-2 ~1 4 .7()---J-.....J....I-L-L-I--U-..I.-~6 ~U--8 -.1.-,;10 --l-.l..-'-U-

12 ---L-..I-J

14 Frequency (GHz) Frequency (GHz) Frequency (GHz)

(a) (b) (c)

Figure 2.10 Normalized spectra ofUWB sub-band pulses; (a) 3.S--SGHZ, (b) 6--7.SGHZ, and (c) 8.S-10GHZ.

2.4. Conclusions

In this chapter, we di scussed the importance of pulse design in the performance of UWB

systems. We developed a linear optimization pro gram to find pulses that maximize the PE

vis-a-vis the FCC spectral mask. A MATLAB pro gram finds the pulse samples that

maximize the PSD while respecting the FCC mask. ~ptimization based on combining

weighted Gaussian monocycles did a better job in producing power efficient and temporally

smooth waveforms. Although we do not intend to implement optimal pulses with an FIR

filter approach, we used this method to find the optimal waveforms. We next look at optical

implementation of sorne of the optimally designed waveforms in Chapter 3.

29 Chapter 3: Optical UWB Pulse Shaping Using FBGs

Chapter 3

Optical UWB Pulse Shaping

Using FBGs

Various electrical and optical pulse shaping architectures have been proposed to generate

UWB waveforms. In the past, most of the research had been focused on generating the

widely adopted Gaussian, monocycle and doublet pulses [11] and [13-17]. In this chapter,

after sorne literature review of optical pulse shaping methods in section 3.1, we propose a

new approach to generate UWB pulses that are FCC-compliant and maximize the transmit

power. This method is based on balanced photodetection of a spectrally shaped

30 Chapter 3: Optical UWB Pulse Shaping Using FEGs

femtosecond laser source. Balanced detection not only, to sorne extent, reduces different

noise components but also eliminates the undesired superimposed rectangular pulse

imprinted on the desired pulse during conversion to the time domaine A simple method to

tune the setup to generate various UWB waveforms will be discussed in section 3.3.

3.1. Optical Pulse Shaping Methods

As first stated in section 2.1.2, there are numerous optical pulse generator architectures;

most focus on the widely adopted Gaussian, monocycle and doublet pulses. The most

promising optical pulse generation techniques are based on optical spectral shaping and

frequency-to-time conversion using a dispersive medium.

As an example, in [16] , lalali et al. showed an RF-photonic arbitrary waveform generator.

As illustrated in Figure 3.1 , a wide band optical pulse is spectrally shaped by a spatial light

modulator (SLM) after being diffracted by a diffraction grating. The 128 pixel SLM is

controlled by a computer and has a maximum optical dynamic range of 30 dB in amplitude

modulation. The resulting shape in spectrum is mapped in time do main by frequency-to-

time conversion using a certain length of single mode fiber (SMF). The total amount of

dispersion determines the pulse duration. The time domain pulse is generated by

sc

Desiree Waveform \'oltage

~F~ ~ .... :",. : ... .. : .. ", : ... i. . ::! ... " .... i.;.!',.i;.i.. t:IDri~c' r' ''c ---+, ~. :;%;~:, .

~pmal lJgtit Mcdubtor (SLM)

Ue n er.a1l!d

Figure 3.1 Spatial shaping using the SLM [16].

T 0 Dis.-persWe Medi .. m

31 Chapter 3: Optical UWB Pulse Shaping Using FBGs

photodetection of the optical pulse. The maXImum bandwidth IS limited by the

photodetector and the repetition rate by the laser source.

A similar work is demonstrated in [22]. Broadband RF waveforms suitable for UWB

systems are photonically synthesized via open-Ioop reflection-mode dispersive Fourier

transform optical pulse shaping. This method relies on the ability to shape the optical power

spectrum in a Fourier transform (FT) pulse . shaper followed by frequency-to-time

conversion in a dispersive medium. A block diagram of the experimental apparatus is

illustrated in Figure 3.2a. Short pulses from a mode locked erbium doped fiber laser (100

fs , 30 nm bandwidth) are spectrally filtered in a reflective FT pulse shaper. This allows the

impression of an arbitrary filter function onto the optical spectrum. These shaped pulses are

then dispersed in 5.5 km of single mode fiber. After optical-to-electrical (OIE) conversion

of the time-domain optical waveform, the measured RF waveform exhibits the shape of the

filter function applied to the optical power spectrum. Figure 3.2b shows the reflective

geometry FT pulse shaper configuration. The dispersed frequency components are

amplitude modulated in parallel under voltage control by the combination of the 128-pixel

(a)

(b)

Fs pulse

~ drculator .--____ ---. ---+lC-:0 2 Reflective

Ta port3

1 'v FT Pu Ise Shaper

Incidentfrom port2 of ci rcutator

PBS ~~

Al=f LeM Len~~'l~""~ . ' / . , { \ ~ -----~ J7 mirror:~/ :.:. I l. 1. ---~ k

~~ . \ ' i -_. __ ._~~'" V Grating !of . . .' ._ ............... \ 'r r.' --. 1/. ., /1, \JU V

Y'l .. Wave plate \.. ................... ,( ................. .J \.. ................. y ................ ..l

f f

Figure 3.2 Broadband RF waveform generator, (a) Experimental apparatus. (b) Reflective geometry Fourier transform 1221.

Chapter 3: Optical UWB Pulse Shaping Using FBGs

'--w /fJ' .

Figure 3.3 UWB pulse generation based on spectral shaping of a MLFL.

32

liquid crystal modulator (LCM). After modulation, the frequency components are

recombined by the lens/ grating combination.

Although these arbitrary waveform generators offer tremendous flexibility, and can

generate the desired UWB pulse, they cannot be used in many applications due to their

large size and high opticalloss.

A research group in Ottawa has recently proposed sorne pulse shaping techniques based

on fiber optics. In [17], two optical filters with complementary spectra are placed in two

arms of an interferometer (Figure 3.3). They use a mode locked fiber laser as the source.

The spectrum of the ultra-short pulse from port 1 is shaped by a tunable optical filter; and

the spectrum of the pulse from port 2 is spectrally shaped by a fiber Bragg grating (FBG),

acting as a transmission filter with a center wavelength that can also be slightly tuned by

applying tension. By adjusting the spectral widths and the center wavelengths of the two

optical filters, Gaussian monocycle or doublet pulses can be generated. The generated

pulses, however, do not resemble the desired waveform, and the RF spectrums con~ain non-

FCC-compliant baseband spectral content below 1 GHz. In addition, the interferometric

structure of this pulse shaper leads to sensitivity to environmental changes such as

temperature or vibration.

In another approach [23], a femtosecond pulse laser is spectrum sliced to the required

pulse width. The optical pulse train is then injected into a nonlinear fiber, together with a

CW probe laser, to create cross-phase modulation (Figure 3.4). An FBG is used as a

frequency discriminator. By locating the probe laser at the linear or the quadrature slopes of

Chapter 3: Optica/ UWB Pu/se Shaping Using FBGs

r----------------------------: r ~~~~::;;:~:: l t P f\ I ! source i UL 1 L.. .. _ ... _ ... _ ... _ .............. _.l 1

1....-------, 1 l '------' 1

1

Rf T-\A ~

Circulator

- - - -- - - -- - - - - - -- - - --UWBPulse auiPt-TLD: Tunable Laser Diode PC: Polarizatjon Controller

OA: Optical Amplifier PD: Photodetector

UFBG: Uniform Fiber Bragg Grating NLF: Nonlinear Fiber P: Optical Power R: Reflectivity

a: Amplitude of electrical pulse C 0

Figure 3.4 Ali optical UWB pulse generation based on phase modulation and frequeilcy discrimination.

33

the FBG reflection spectrum, UWB monocycle or doublet pulses are generated. The two

laser sources used in this technique makes it complex and costly. More importantly,

additional electrical filtering is required to remove the non-compliant spectral content

below 1.6 GHz.

From this brief literature review, we understand that one of the principle methods of

generating UWB waveforms is to spectrally shape a broadband laser source. Frequency-to-

time conversion maps the spectral shape to time domain and finally a photodetector

converts the pulse from optics to RF. This basic concept is shown in Figure 3.5.

One of the drawbacks of the discussed methods is that instead of generating FCC-

compliant, efficient pulses, they are confined to generation of simple Gaussian waveforms.

Coherent BBS

Spectral Pulse Shaper

Dispersive Medium

Detector

Figure 3.5 Concept of arbitrary pulse generation by spectral pulse shaping.

34 Chapter 3: Optical UWB Pulse Shaping Using FBGs

In the next section, we propose a new approach to generate UWB pulses that is both FCC-

compliant and maximizes the transmitted power. Our technique is also of the form shown

in Figure 3.5, with an FBG for spectral pulse shaping.

3.2. UWB Pulse Shaping Using FBGs

In this section, we propose and experimentally demonstrate the use of FBGs for spectral

pulse shaping. The setup is discussed in section 3.2.1. Basic concepts in FBG design and

fabrication are presented in section 3.2.2. We will also see th.e characteristics and the

measured transmission responses of our gratings. We will show promising experimental

. pulse shaping results in section 3.2.4, where the FBGs achieve high precision target

matching in both time and frequency domains.

3.2.1. A Balanced Receiver Approach

We use the general concept as in Figure 3.5 to generate the optimally designed pulses of

section 2.3 [24, 25]. A mode-Iocked fiber laser (MLFL) with large full width half

maximum (FWHM) bandwidth is used as a coherent broadband source. The spectral pulse

shaper in our design is a fiber Bragg grating in transmission with a transfer function

proportional to the desired pulse. We use an appropriate length of SMF as the dispersive

medium to generate the total required dispersion for the frequency-to-time conversion.

The particular form of our embodiment is heavily influenced by the requirement to

remove the undesired superimposed rectangular pulses imprinted on the desired pulse

during conversion to the time domain. Recall that aIl pulses generated by optical pulse

shaping techniques using frequency-to-time conversion contain an unwanted additive

rectangular pulse superimposed on the desired pulse shape, leading to strong, unwanted

spectral components in low frequencies -----1 GHz) that cannot be removed by a dc-block.

We use a balanced photodetector (BPD) to completely remove unwanted low frequency

components, as seen in Figure 3.6.

Chapter 3: Optical UWB Pulse Shaping Using FBGs

FBG2

SMF ~ MLFL FBG1

t //--....",- )

Desire pulse plus

~ rectangular pulse

+

BPD

Rectangular pulse

35

Measuring Deviee

Figure 3.6 Block diagram of the UWB waveform generator.

The block diagram of our proposed technique is shown in Figure 3.6. FBG 1 is used to

flatten the mode-locked source spectrum over the desired bandwidth. The optical signal is

then divided into two arms. In the first arm, we use a second chirped grating, FBG2, with a

complex apodization profile optimized to imprint the desired pulse shape on the spectrum

of the source. In the second arm, the optical delay line (DL) and the variable attenuator

(A TT) are used to balance the amplitude and the delay of the two arms. We used an isolator

to prevent back and forth reflections between the two FBGs. The SMF may be placed

anywhere along the generator; placing it before spectral shaping avoids requiring SMF in

both arms of the BPD.

3.2.2. FBG Design and Fabrication

A fiber Bragg grating (FBG) is a longitudinal periodic perturbation of the glass refractive

index induced in the optical fiber core [26]. This index modulation forms a filter reflecting

certain wavelengths. To produce this perturbation, we expose an uncoated piece of

photo sensitive optical fiber to the interference pattern of an ultraviolet (UV) laser. The

refractive index increases in the regions where the light intensity is high, thus causing a

periodic modulation of the index of refraction in the core. The interference pattern is

normally obtained from a so-called phase mask.

Phase masks are surface relief gratings etched in fused silica. In most applications, a

phase mask essentially serves as a precision diffraction grating that divides an incident

36 Chapter 3: Optical UWB Pulse Shaping Using FBGs

Laser beam

+ 1 thorder

Figure 3.7 Interference pattern of a phase mask.

monochromatic beam into two outgoing beams (Figure 3.7). The incident radiation is

usually in the UV range. By generating two outgoing beams, a phase mask creates an

interference pattern in the region the beams overlap. Typically the, phase masks are

operated in the + 1/-1 configuration where the power is maximum in + 1 and -1 diffraction

orders. In this case, the laser beam is directed perpendicularly to the phase mask. The

period of the fringe pattern created by the interference of the + 1 and -1 beams is exactly

one half of the period of the phase mask, regardless of the wavelength of the incident

radiation. Exposing a photo sensitive optical fiber to the interference pattern results in an

FBG. The UV exposure increases the average refractive index, &Ide' and also introduces a

sinusoidal modulation, b.nae , with a period equal to the period of the fringe pattern. In

apodized gratings these index changes can vary along the length of the grating.

Perturbations in core refractive index ' can couple the incoming light in the fundamental

fiber mode to the reflected fundamental mode, or to the cladding depending on the grating

period. The Bragg wavelength, defined as the- wavelength of maximal reflection can be

obtained from [27]

(3.1)

37 Chapter 3: Optical UWB Pulse Shaping Using FBGs

where B is the Bragg wavelength, nef!' is the effective refractive index, and A is the fringe

period. Chirped FBGs can be obtained using chirped phase masks. In a chirped mask, the

period of the groove changes along the mask according to A ( z ) = Ao + Cmz , where Ao is

the initial period, z is the length and Cm is the chirp expressed in nmlcm. A chirped grating

enlarges the reflection bandwidth because, from (3.1), a range of Bragg wavelengths are

produced along the grating as the etching period changes.

The grating apodization is the slowly varying envelope of the grating profile. A uniform

grating has no envelope variation. In general , finding the apodization profile of a grating

operating in transmission is a relatively easy task, as long as the target spectral profile does

not vary too rapidly. The chirped grating's spectral response, TCA) , and the apodization

profile is typically related by [28]

(3.2)

This equation links the index modulation amplitude, lYlac () to the desired transmission

profile T(). In (3.2), Cm is the index modulation chirp, and r is the ratio of modal power that overlaps with the grating (r is the confinement factor if the grating is confined in the

fiber core). Once the apodization profile is obtained, we simulate the grating spectral

0.8 E 2 t5 ~ 0.6

Cf)

"0 Q)

~ 04 ro . E L-

o Z

0.2

o ~~ ____ ~ ____ ~~ ____ ~ ____ ~~ 1536 1540 1544 1548 1552

Wavelength (nm)

Figure 3.8 MLFL normalized power spectral density.

38 Chapter 3: Optical UWB Pulse Shaping Using FBGs

response using a standard transfer matrix method performed by IFO Gratings software

available from Optiwave Corporation. This grating spectral response is then compared to

the target response; slight modifications are made to the apodization profile to tune it to the

target response. After several iterations the apodization profile leading to the best spectral

response is achieved.

After looking at the basic concepts of FBGs, we next start to design the gratings we need,

as shown in the setup (Figure 3.6). The PSD of the passive mode locked fiber laser (MLFL)

is measured using an optical spectrum analyzer (OSA), as shown in Figure 3.8. This

spectrum can be approximated by

(3.3)

where (j) and ())o are the angular and the centre frequencies, respectively, and a is a

constant. Referring again to Figure 3.6, FBG 1 is used to flatten the source spectrum,

therefore, its ideal transfer function is

0.9

E 0.8 :::J Z. 0.7 U

~ 0.6 CI)

"'0 0.5 Q)

~ 0.4 cu E 0.3 o Z 0.2

0.1 1

1570 o~--~ __ ~~J __ ~\~~ __ ~ __ ~. 1510 1520 1530 1540 1550

Wavelength (nm) (a)

1560

( !1()) !1()))

()) E OJ -- ()) +-c 2' c 2 (3.4)

otherwise

\ l 0.9

Q) 0 .8

~ 0 0 .7

cl: 0.6

C 0

~ 0.5 N :0 0.4 0 0.. 0.3

0.2

0.1 ~ 0 0 10 12 14

Length (cm)

(b)

Figure 3.9 Flattening filter (FBGl) design; (a) required normalized spectrum, (b) apodization profile.

Chapter 3: Optical UWB Pulse Shaping Using FBGs

0

-10 m ~ c -20 0 en o~ E -30 en c ~ r -40 (9 en LL -50

-6q530 1535 1540 1545 1550 Wavelength (nm)

(a)

1555

-2 x: 1539 Y : -20563

m 3 ~ -c o

0(i5 -4 en E en ~ -5 r (9 -6 CD LL

-7

1539 1541 1543 1545 1547 Wavelength (nm)

(b)

39

Figure 3.10 (a) Flattening filter transmission response measured using an optical vector analyzer (b) a detailed view of the filter response.

where ~ OJ is the desired bandwidth around (j) c From (3.3), (3.4), and by choosing

E(1539, 1548) as the desired bandwidth for the flattening filter, we can obtain the FBG 1

transmission spectrum, H FBGI () , plotted in Figure 3.9a. The apodization profile, shown in

Figure 3.'9b, is obtained by a transformation of Figure 3.9a. The y axis which shows the

apodization profile is found from (3.2) by setting nef!' = 1.452, r = 0.84 and Cm = 2.5

nm/cm. The abscissa is obtained by mapping the wavelength range to the 14 cm available

mask length. We fabricate the FBG using a standard phase mask scanning technique with a

244 nm UV laser beam. For FBG1, we used a 14 cm mask with a chirp rate of2.5 nm/cm

and a H2-loaded photo sensitive specialty fiber (UVS-INT fiber from Corative). The grating

apodization was performed by phase-mask dithering during the UV scan. Next, the fiber is

annealed to stabilize the response for future use. The flattening filter (FBG 1) transmission

response is plotted in Figure 3.1 O.

The transfer function of FBG2 can be obtained by a time-to-frequency mapping of the

designed UWB pulse shape, p(t). In this case, the pulse duration, flT, is mapped to the LU

linewidth corresponding to !1OJ = 2nc!1 / ~ , the available bandwidth. The ratio ~T / ~ is

equal to the total required dispersion of the fiber (i.e., D x L f ) and determines the required

length of SMF for converting the pulse to the time domaine

40 Chapter 3: Optical UWB Pulse Shaping Using FBGs

0.9

0.8 ~ ~0. 8 t= t= 2 0.7 0

0.6 0- 0:0.6 i:

c c 0 'V; .2

0.4 . ~ 0 . 5 ~

~ .t::0.4 "0 c 8. ~

0.2 ~ 0.3

41 Chapter 3: Optical UWB Pulse Shaping Using FBGs

non-uniforrnities in fiber photosensitivity. Sorne errors occur during the FBG writing setup

alignment and include the UV laser bearn quality, the laser emission angle on the fiber, the

parallelisrn of the fiber and the phase rnask.

3.2.3. Simulation Results

In this section, we sirnulate the output response of the UWB pulse shaping setup shown in

Figure 3.6. The functionality of each component in the setup can be rnodeled by a transfer

function in order to simulate the generated pulse shape at the BPD output. Neglecting the

propagation delay of the pulse envelope, the transfer function of a lossless dispersive SMF

can be rnodeled with very good precision [30] by

(3.6)

where L f is the fiber length. Also, /32 = a2 /3(~)

aOJ IS known as the second order

dispersion parameter and J3 (OJ) is the mode propagation constant. D (ps/km.nm) is related

to the dispersion parameter through /32 = -~ D / 271C The center wavelength is

o = 21CC / OJo and C is the light speed. For typical SMF, the third order dispersion parameter

-3

-4 iD "0

~ -5 o "iii .~ -6 ri)

c ~ -7 ..... N

as -8 u..

-9

-10

1535 1540 1545 1550 1555

iD ~-3 c o "iii .~ -4 E ri) c ~ -5 N C> fe -6 "0 CIl

g-7 Ci> >

-- -- - --~ - -- --------- - ---.,

42 Chapter 3: Optical UWB Pulse Shaping Using FBGs

~0.8 e E ~ 0.6 ro .= ~ 0.4 u..

0.2

i\

- F8G2 ---- F8G1

O~~~-~~--'--~~

1538 1542 1546

Wavelength (nm) (a)

1550

0.5

o 0.4 CI) Cl..

~ 0.3 . ~ ro E 0.2 o Z

0.1

rf""A-~~HV'l

1 1

Cl> "0 .2 ~ 0.4 E "0

. Ot--~-' ro E

~ -0.4

- Simulation - --- Design

o "-"""----"_-"------'-_....&....-_____ -0.8 ~--'-_ _'______''---__"'_ _ _'____J 1538 1542 1546 1550 0 0.2 0.4 0.6 0.8

Wavelength (nm) Time (ns) (b) (c)

Figure 3.13 Simulation results for L=14. (a) Transmittivity of FBGs, (b) PSD at upper and lower arms, and (c) simulated and designed output pulse.

is very small and thus third order chromatic dispersion is negligible over short distances

and for narrow bandwidths. In the present case, we assume that the fiber group delay is

linear over the frequency band of interest.

The Fourier transform of the optical signaIs at the inputs to the balanced photodetector

can be expressed by

El (m) == al As (m) H SMl - (m) H FBG l (m) H FBG 2 (m) (3.7)

(3.8)

where al represents the total loss in the first arm, and a 2 can be adjusted by variable

attenuator to balance the power in the two arms. A variable delay r in the second arm

compensates for any delay between these two lines.

Finally, the detected signal at the output of the balanced photodetector is

(3.9)

where ei(t) == 1FT {~({)}, i == 1,2 ; 1FT stands for inverse Fourier transform. We supposed

that the BPD has a fiat transfer function over the signal bandwidth. In practice the BPD

transfer function is not fiat and decays near the cut-off frequency.

43 Chapter 3: Optical UWB Pulse Shaping Using FBGs

We now examine our pulse shaping strategy via simulation. The measured spectrum of

the broadband source is used to design the apodization profile of FBG 1, as explained in

section 3.2.2. The role of this filter is to carve out the desired wavelength band, and

compensate for the non-flat spectrum of the broadband source. The simulated transmittivity

of FBG 1 is given by the dotted line in Figure 3.13a. Kinks in this curve are the result of a

finite duration apodization profile. FBG2 was designed to realize the optimized pulse shape

when L = 14 taps are used in (2.12). The simulated transmittivity ofFBG2 is given by the

solid line in Figure 3.13a. The simulated spectra of the output of FBG 1 using the fitted

curve for the broadband source spectra as input, is given by the dashed line in Figure 3.13b.

We see ringing at the cutoff wavelengths. Subsequent filtering by FBG2 yields the spectra

given by the solid line in Figure 3.13b. We see the ringing of FBG 1 now imprinted on the

output of FBG2. We now add 5.46 km of SMF with 16.3 ns/kmInm d