the university of calgary a logarithmic amplifier and hilbert transformer for optical single

THE UNIVERSITY OF CALGARY

A Logarithmic Amplifier and Hilbert Transformer

for Optical Single Sideband

by

Christopher Daniel Holdenried

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF ELECTRICAL AND COMPUTER

ENGINEERING

CALGARY, ALBERTA

February, 2005

c© Christopher Daniel Holdenried 2005

THE UNIVERSITY OF CALGARY

FACULTY OF GRADUATE STUDIES

The undersigned certify that they have read, and recommend to the Faculty of Graduate

Studies for acceptance, a thesis entitled “A Logarithmic Amplifier and Hilbert

Transformer for Optical Single Sideband” submitted by Christopher Daniel Holdenried in

partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY.

__________________________________________ Supervisor, Dr. James W. Haslett Department of Electrical and Computer Engineering __________________________________________ Dr. John G. McRory Department of Electrical and Computer Engineering __________________________________________ Dr. Robert J. Davies Department of Electrical and Computer Engineering __________________________________________ Dr. Brent Maundy Department of Electrical and Computer Engineering __________________________________________ Dr. Harvey Yarranton Department of Chemical and Petroleum Engineering __________________________________________ External Examiner, Dr. Calvin Plett Department of Electronics, Carleton University

___________________________ Date

ii

Abstract

Chromatic dispersion is the pulse spreading that occurs during transmission

through optical fiber and is due to the non-constant delay of fiber with wavelength.

Gigabit optical communication systems require some method of dispersion compen-

sation. Optical single sideband (OSSB) is commonly used to transmit narrow-band

signals in order to avoid power fading due to dispersion. However, in the absence

of special optical filters, a broadband Hilbert transformer and logarithmic ampli-

fier are required in order to generate OSSB for baseband gigabit data signals. This

thesis describes the development of unique gigabit logarithmic amplifier and Hilbert

transformer integrated circuits.

The Cherry-Hooper amplifier with emitter follower feedback is introduced as a

gigabit amplifier building block. This circuit is ultimately used to design a broadband

logarithmic amplifier for OSSB. The log amplifier architecture is developed using a

novel design procedure, with proof of a logarithmic response. A Hilbert transformer

integrated circuit is developed based on non-integrated Hilbert transformer designs

by previous researchers. It uses Q-enhanced on-chip LC transmission lines. The log

amplifier and Hilbert transformer designs were fabricated as integrated circuits, and

their performance is verified through measurements of the circuits.

Simulation results of an OSSB system are described and show that the above

mentioned circuits enable an OSSB system with immunity to dispersive power fading.

Actual OSSB transmitters were assembled and measured OSSB optical spectra are

presented for 5 and 10 Gb/s broadband signals and a 1.9 GHz narrow-band signal.

iii

Acknowledgements

I thank Jim Haslett for his superb guidance and sound judgement. I would not

have applied for and won certain awards without his encouragement. Thank you for

helping with designs and the long and productive hours spent writing and editing

articles. He was often able to see core mathematical ideas when I could not. I am

also grateful for his generosity which was demonstrated, for example, by allowing

me to attend conferences very early in my studies. This allowed me to see close up

what was expected of me and how to obtain it.

Thank you to John McRory for teaching me everything I know about microwave

circuits and for help designing the logarithmic amplifier. I thank him for negotiating

access to the NT35 technology at Nortel so that we could fabricate the first loga-

rithmic amplifier. I also gratefully acknowledge the financial support of TRLabs,

including the perks, made possible by John McRory, Roger Pederson, and George

Squires.

Thank you to Bob Davies for guidance with all of the optical communications

aspects of this thesis, and for the idea of this thesis. Thank you to the great minds

who are part of the TRLabs and ATIPS teams for many useful discussions and for

providing a challenging environment. Thank you to Dave Clegg and Chris Haugen

for assistance with experiments and equipment. Thank you to Bogdan Georgescu

for your hard work developing the coupled inductor Q-enhancement principles which

helped me to design the integrated Hilbert transformer. Thank you to Michael Lynch

for many useful discussions, work related and otherwise.

Thank you to A.J. Bergsma and Douglas Beards for their support and design

iv

ideas when designing the first logarithmic amplifier. Thank you to A.J. for teaching

me about IC layout and for working late some nights to finish the IC layout of the

first logarithmic amplifier. Thank you to Nortel for financial support and for allowing

me to work with A.J. and Doug.

Thank you to the Canadian Microelectronics Corporation for paying for the fab-

rication of several ICs that are part of this thesis, for providing access to world class

design software, and for donating test equipment. Without this support I would

never have been able to obtain the results that I did.

I gratefully acknowledge the financial support of NSERC, Alberta iCORE, and

the IEEE. Without this support, my studies would have been cut short. I value the

many friends that I have made through these organizations.

Thank you to Leila Southwood, Pauline Cummings, Simon Arsenault, and Ella

Gee for their administrative support which makes this work possible. Thanks also

to Jonathan Eskritt, Paul Horbal, and Josh Nakaska for keeping the ATIPS system

and web site going when they weren’t busy with their own research. Behind every

strong researcher, there are even stronger administrators.

v

For Regina and Siddhartha.

Thank you for love and support.

vi

Table of Contents

Approval Page ii

Abstract iii

Acknowledgements iv

Dedication vi

Table of Contents vii

List of Tables x

List of Figures xi

List Of Symbols and Abbreviations xvi

1 Introduction 11.1 Research Objective and Scope . . . . . . . . . . . . . . . . . . . . . . 41.2 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Background 62.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Chromatic Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Types of Dispersion . . . . . . . . . . . . . . . . . . . . . . . . 72.2.2 Mathematical Definition of Dispersion . . . . . . . . . . . . . 8

2.3 Methods to Compensate for Dispersion . . . . . . . . . . . . . . . . . 92.3.1 Optical Techniques . . . . . . . . . . . . . . . . . . . . . . . . 102.3.2 Post Detection Compensation . . . . . . . . . . . . . . . . . . 12

2.4 Compatible Optical Single Sideband . . . . . . . . . . . . . . . . . . 142.4.1 Complex Envelope Representation of Bandpass Signals . . . . 142.4.2 COSSB Modulation . . . . . . . . . . . . . . . . . . . . . . . . 172.4.3 COSSB Implementation: The Ideal Minimum Phase Modula-

tor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4.4 Dispersion Effects on Double and Single Sideband Signals . . 212.4.5 Minimum Phase Dispersion Compensation . . . . . . . . . . . 262.4.6 Previous Experiments Using COSSB . . . . . . . . . . . . . . 272.4.7 The Mach-Zehnder Modulator . . . . . . . . . . . . . . . . . . 29

vii

2.4.8 Competing Technologies: Solitons, Coherent Detection Sys-tems, and Duobinary Transmission . . . . . . . . . . . . . . . 31

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 Analysis and Design of HBT Cherry-Hooper Amplifiers with Emit-ter Follower Feedback for Optical Communications 36

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2 Large Signal Performance . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.1 HBT β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3 Small Signal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3.1 Design Example . . . . . . . . . . . . . . . . . . . . . . . . . 443.4 Amplifier Noise Performance . . . . . . . . . . . . . . . . . . . . . . 473.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . 493.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4 A Novel Parallel Summation Logarithmic Amplifier 554.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2 Distinction and Comparison of Logarithmic Amplifiers . . . . . . . . 56

4.2.1 The Series Linear-Limit Logarithmic Amplifier . . . . . . . . 574.2.2 Parallel Summation Logarithmic Amplifiers . . . . . . . . . . 59

4.3 Design Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.3.1 Logarithmic Slope and Intercept . . . . . . . . . . . . . . . . . 684.3.2 The Delay Matched Progressive Compression Amplifier . . . . 69

4.4 Implementation 1: A DC-4 GHz Si BJT Logarithmic Amplifier . . . 704.4.1 Design of Implementation 1 . . . . . . . . . . . . . . . . . . . 704.4.2 Measurements of Implementation 1 . . . . . . . . . . . . . . . 73

4.5 Implementation 2: A DC-6 GHz SiGe HBT Logarithmic Amplifier . 804.5.1 Design of Implementation 2 . . . . . . . . . . . . . . . . . . . 804.5.2 Measurements of Implementation 2 . . . . . . . . . . . . . . . 87

4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5 A 10 Gb/s Hilbert Transformer with Q-Enhanced LC TransmissionLines 92

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.2 HT Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.3 Design of LC Transmission Lines . . . . . . . . . . . . . . . . . . . . 96

5.3.1 Q-Enhanced LC Transmission Lines . . . . . . . . . . . . . . . 975.4 Circuit Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

viii

6 Simulations of COSSB System Implementations 1166.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166.2 Performance of the Logarithmic Amplifier . . . . . . . . . . . . . . . 1166.3 Performance of the HT . . . . . . . . . . . . . . . . . . . . . . . . . 1216.4 Combined Performance of Logarithmic Amplifier and HT Circuits . . 125

6.4.1 Performance at a Mach-Zehnder Modulation Depth of 0.25 . 1256.4.2 Performance at a Mach-Zehnder Modulation Depth of 0.20 . 133

6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7 Measurements of COSSB Transmitters 1407.1 10 Gb/s COSSB Experiment Using the HT . . . . . . . . . . . . . . 1407.2 COSSB Experiments Using the HT and the Logarithmic Amplifier . . 145

7.2.1 Experiment Using a 1.9 GHz Sinusoid . . . . . . . . . . . . . 1457.2.2 Experiment Using Filtered 5 Gb/s Data . . . . . . . . . . . . 147

7.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

8 Conclusions 1558.0.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Bibliography 160

A Amplifier DC Transfer Characteristic 167

B Analysis of the Emitter Follower Load 169

C Derivation of Equation (3.11) 171

D Example Calculation of an Amplifier Noise Contribution 172

E Widlar Biasing 174

F Design of the Logarithmic Amplifier Test Fixture 178

G Description of Equipment Used for COSSB Experiments 182

ix

List of Tables

3.1 Differential output noise of the CHEF amplifier at 1 GHz. . . . . . . 49

4.1 Comparison of high frequency true log amplifiers. . . . . . . . . . . . 90

5.1 Noise figure of HT die C with Q-enhancement turned on. . . . . . . . 109

G.1 List of major equipment used in COSSB experiments. . . . . . . . . . 183G.2 Power characteristic of the Sumitomo intensity modulator. . . . . . . 190

x

List of Figures

1.1 Long-haul optical system architectures. . . . . . . . . . . . . . . . . . 2

2.1 Pulse spreading due to chromatic dispersion. . . . . . . . . . . . . . . 72.2 A typical loss and dispersion profile for single mode fiber. . . . . . . . 102.3 Narrow-band bandpass signal. . . . . . . . . . . . . . . . . . . . . . . 152.4 Ideal minimum phase modulator. . . . . . . . . . . . . . . . . . . . . 212.5 Electrical signals at points throughout the COSSB system. . . . . . . 222.6 Filtered 10 Gb/s DSB and SSB signals plotted against frequency nor-

malized to the carrier. . . . . . . . . . . . . . . . . . . . . . . . . . . 232.7 Eye diagram of detected SSB signal. . . . . . . . . . . . . . . . . . . 232.8 Detected 10 Gb/s DSB signal after 200 km of dispersive fiber. . . . . 252.9 Detected 10 Gb/s SSB signal after 200 km of dispersive fiber. . . . . 262.10 Ideal minimum phase dispersion compensator. . . . . . . . . . . . . . 282.11 Dual arm Mach-Zehnder modulator. . . . . . . . . . . . . . . . . . . . 302.12 Transfer function of a Mach-Zehnder with Vπ=1. . . . . . . . . . . . . 31

3.1 Cherry-Hooper amplifier with emitter follower feedback. . . . . . . . . 373.2 Plot of AC β. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.3 CHEF amplifier (a) differential mode half circuit and (b) small signal

equivalent including source and load impedances. . . . . . . . . . . . 413.4 Magnitude and group delay responses for a second order system. . . . 443.5 Eye diagrams for a 10 Gb/s signal filtered with second order systems

having (a) Q = 1/√

3 and (b) Q = 1.0. . . . . . . . . . . . . . . . . . 443.6 Plot of Q factor for different values of R1,R2, and R′

f . . . . . . . . . 463.7 Plot of 3 dB bandwidth for different values of R1,R2, and R′

f . . . . . 463.8 Plot of low frequency gain for different values of R1,R2, and R′

f . . . . 463.9 Schematic diagram of the CHEF amplifier test circuit. . . . . . . . . 503.10 CHEF amplifier IC microphotograph. . . . . . . . . . . . . . . . . . . 513.11 Comparison of theoretical gain based on equations (3.9) and (C.2),

simulated gain, and measured gain. . . . . . . . . . . . . . . . . . . . 523.12 Comparison of theoretical group delay based on equations (3.9) and (C.2)

, as well as simulated and measured group delay. . . . . . . . . . . . . 523.13 Measured eye diagrams at 10 Gb/s: (a) Through measurement at

20 mVpp and single-ended CHEF amplifier output for differential in-put signals of amplitude (b) 7 mVpp, (c) 20 mVpp, and (d) 400 mVpp. 54

4.1 Series linear-limit logarithmic amplifier. . . . . . . . . . . . . . . . . . 574.2 Linear-limit logarithmic amplifier response. . . . . . . . . . . . . . . . 58

xi

4.3 High gain limiter and unity gain buffer in parallel. . . . . . . . . . . . 594.4 Progressive compression, parallel summation logarithmic amplifier. . . 604.5 Parallel amplification, parallel summation logarithmic amplifier. . . . 614.6 Parallel summation logarithmic amplifier transfer function. . . . . . . 624.7 An example of a three stage delay matched progressive compression

log amplifier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.8 Parallel summation logarithmic amplifier implementation. . . . . . . . 714.9 Amplifier used as a gain or delay cell. . . . . . . . . . . . . . . . . . . 724.10 Summing/limiting amplifier. . . . . . . . . . . . . . . . . . . . . . . . 724.11 Input matching circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . 724.12 Microphotograph of the Si logarithmic amplifier integrated circuit. . . 734.13 Measured return loss and gain. . . . . . . . . . . . . . . . . . . . . . . 744.14 Measured group delay response. . . . . . . . . . . . . . . . . . . . . . 754.15 Measured logarithmic responses, peak voltages shown. . . . . . . . . 774.16 Logarithmic error for separate and broadband line fits. . . . . . . . . 784.17 Measured and ideal logarithmic amplifier output spectrum for a 1.8 GHz

input tone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.18 Real time oscilloscope plot of single ended output voltage. . . . . . . 794.19 Simulated log amplifier differential responses to sinusoidal inputs with

and without capacitive delay tuning. . . . . . . . . . . . . . . . . . . 814.20 Logarithmic amplifier block diagram. . . . . . . . . . . . . . . . . . . 824.21 Schematic diagram of the input impedance match circuit and first

high gain stage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.22 Schematic diagram of the DC offset error reduction circuit. . . . . . . 844.23 Schematic diagram of the output summation circuit. . . . . . . . . . . 844.24 Simulated DC transfer characteristic of logarithmic amplifier over one

hundred Monte Carlo iterations. . . . . . . . . . . . . . . . . . . . . . 864.25 Simulated logarithmic response of SiGe logarithmic amplifier at 4 GHz

for three different temperatures. . . . . . . . . . . . . . . . . . . . . . 864.26 Microphotograph of the SiGe logarithmic amplifier integrated circuit. 874.27 Measured single ended logarithmic response from 100 MHz to 6 GHz. 884.28 Measured real time logarithmic amplifier single ended output waveforms. 89

5.1 Response of filter with an infinite number of taps. . . . . . . . . . . . 935.2 Tapped delay implementation of an HT. . . . . . . . . . . . . . . . . 935.3 Magnitude responses of four tap HTs for three different values of Υ. . 955.4 Spectrum of a COSSB signal generated with a four tap HT. . . . . . 955.5 Schematic diagram of the LC transmission line used in the HT. . . . 975.6 Layout and loss of passive LC transmission line. . . . . . . . . . . . . 985.7 Transformer based Q-enhanced floating inductor. . . . . . . . . . . . 99

xii

5.8 Delay line with emitter follower tap buffers. . . . . . . . . . . . . . . 1005.9 Efficient Q-enhancement circuit using both signal currents. . . . . . . 1015.10 A floating inductor which is Q-enhanced using a simple cross coupled

pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.11 HT summing amplifier. . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.12 Integrated HT microphotograph. . . . . . . . . . . . . . . . . . . . . 1045.13 Supply independent current source. . . . . . . . . . . . . . . . . . . . 1045.14 Measured S11 and S22 of three dice with Q-enhancement on. . . . . . 1065.15 Plot of (a) S21 and (b) group delay simulated with resistive and ca-

pacitive layout parasitics. . . . . . . . . . . . . . . . . . . . . . . . . . 1075.16 Plot of (a) measured S21 and normalized theoretical S21 and (b) group

delay for three die with Q-enhancement on and off. . . . . . . . . . . 1085.17 Measured phase of three die with a phase shift corresponding to 120 ps

of delay subtracted, and theoretical four tap HT phase response witha phase shift corresponding to 90 ps of delay subtracted. . . . . . . . 108

5.18 Responses of four tap HTs to the repeated binary pattern 01001000. . 1115.19 Output of transmission line to a sequence of pulses. . . . . . . . . . . 1125.20 Responses of four tap HTs to the repeated binary pattern 10. . . . . 1135.21 Responses of four tap HTs to the repeated binary pattern 1000. . . . 1145.22 Responses of four tap HTs to the repeated binary pattern 0111. . . . 115

6.1 Minimum phase COSSB transmitter. . . . . . . . . . . . . . . . . . . 1176.2 5 Gb/s COSSB signals obtained through transient simulation of vari-

ous HTs and of the logarithmic amplifier IC. . . . . . . . . . . . . . . 1206.3 Transmitted 10 Gb/s COSSB spectrum and eye diagram with tran-

sient simulation of HT IC without logarithmic amplifier. . . . . . . . 1226.4 Eye diagrams of COSSB system using only the HT and using only

self-homodyning post detection equalization. . . . . . . . . . . . . . . 1236.5 Eye diagrams of COSSB system using only the HT and using only

minimum phase post detection equalization. . . . . . . . . . . . . . . 1246.6 Scaled optical signal envelope and its logarithm for a modulation

depth of 0.25. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.7 Spectra and eye diagram of 5 Gb/s COSSB signals filtered at 2.75 GHz

for a modulation depth of 0.25. . . . . . . . . . . . . . . . . . . . . . 1286.8 Eye diagram of 5 Gb/s signal recovered from a DSB system after

400 km of uncompensated dispersive fiber for a modulation depth of0.25. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

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6.9 Eye diagram of 5 Gb/s signal recovered from COSSB system with HTIC, without logarithmic amplifier, and using only self-homodyningpost detection equalization after 400 km. The modulation depth is0.25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.10 Eye diagrams of 5 Gb/s signals recovered from COSSB system withHT IC and with logarithmic amplifier and using only self-homodyningpost detection equalization. The modulation depth is 0.25. . . . . . . 130

6.11 Eye diagrams of 5 Gb/s signals recovered from COSSB system withHT IC and with logarithmic amplifier and using only minimum phasepost detection equalization. The modulation depth is 0.25. . . . . . . 131

6.12 Spectra recovered from OSSB systems after 400 km with and withoutlogarithmic amplifier. . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.13 Scaled optical signal envelope and its logarithm for a modulationdepth of 0.20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.14 Spectra and eye diagram of 5 Gb/s COSSB signals filtered at 2.75 GHzfor a modulation depth of 0.20. . . . . . . . . . . . . . . . . . . . . . 134

6.15 Eye diagram of 5 Gb/s signal recovered from a DSB system after400 km of uncompensated dispersive fiber for a modulation depth of0.20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.16 Eye diagrams of 5 Gb/s signals recovered from COSSB system withHT IC and without logarithmic amplifier and using only self-homodyningpost detection equalization. The modulation depth is 0.20. . . . . . . 136

6.17 Eye diagrams of 5 Gb/s signals recovered from COSSB system withHT IC and without the logarithmic amplifier and using only minimumphase post detection equalization. The modulation depth is 0.20. . . 137

6.18 Eye diagrams of 5 Gb/s signals recovered from COSSB system withHT IC and with logarithmic amplifier and using only self-homodyningpost detection equalization. The modulation depth is 0.20. . . . . . . 138

6.19 Eye diagrams of 5 Gb/s signals recovered from COSSB system withHT IC and with logarithmic amplifier and using only minimum phasepost detection equalization. The modulation depth is 0.20. . . . . . . 139

7.1 COSSB 10 Gb/s measurement system. . . . . . . . . . . . . . . . . . 1427.2 Spectrum and eye diagram of 10 Gb/s COSSB signal for 16 dBm of

intensity modulation power. . . . . . . . . . . . . . . . . . . . . . . . 1447.3 Spectrum and eye diagram of 10 Gb/s COSSB signal for 20 dBm of

intensity modulation power. . . . . . . . . . . . . . . . . . . . . . . . 1457.4 Spectrum of 1.9 GHz COSSB signal for 23 dBm of intensity modula-

tion power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1477.5 COSSB measurement system including the logarithmic amplifier. . . . 148

xiv

7.6 Logarithmic amplifier waveforms. . . . . . . . . . . . . . . . . . . . . 1507.7 Spectrum of 5 Gb/s COSSB signal for 17 dBm of intensity modulation

power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1517.8 Spectrum of 5 Gb/s COSSB signal and eye diagrams of 5 Gb/s COSSB

and DSB signals for 20 dBm of intensity modulation power. . . . . . 153

B.1 Schematic diagram of (a) emitter follower output buffers and differen-tial pair load and (b) high frequency small signal circuit of one emitterfollower and a differential mode half circuit of the differential pair. . 170

D.1 Half of the amplifier small signal circuit including dominant noisesources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

E.1 Differential pair. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175E.2 Differential pair with Widlar current biasing. . . . . . . . . . . . . . . 176

F.1 Logarithmic amplifier circuit boards. . . . . . . . . . . . . . . . . . . 179F.2 Logarithmic amplifier test fixture. . . . . . . . . . . . . . . . . . . . . 180

G.1 10 Gb/s optical experiment setup using only HT, Part 1 of 3. . . . . 186G.2 10 Gb/s optical experiment setup using only HT, Part 2 of 3. . . . . 186G.3 10 Gb/s optical experiment setup using only HT, Part 3 of 3. . . . . 187G.4 5 Gb/s optical experiment setup using HT and logarithmic amplifier,

Part 1 of 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187G.5 Logarithmic amplifier DC offset tuning circuit. . . . . . . . . . . . . . 191

xv

List Of Symbols and Abbreviations

5AM Five Analog Metal, an IBM SiGe technology

5HP Five High Performance, an IBM SiGe technology

A Amperes, the unit of current

A Factor difference between the input voltages which causetwo logarithmic amplifier gain paths to limit

A1 Gain across Cµ1 in CHEF amplifier

A2 Gain across Cµ2 in CHEF amplifier

AE Cross sectional area of the base-emitter junction

A(f) Frequency domain bandpass signal containing only positive frequencies

AC Alternating current

AGC Automatic gain control

a(t) Time domain bandpass signal containing only positive frequencies

aL(t) Time domain low pass signal

aLSSB(t) Time domain low pass signal containing only positive or negative

frequencies

a′LSSB(t) Detectable time domain low pass signal containing only positive or

negative frequencies

B Material parameter =5.4 ×1031 for silicon

BER Bit error rate or ratio

BJT Bipolar junction transistor

C Capacitance

C Constant number

C Coulombs, the unit of charge

C12 Noise voltage correlation coefficient

CL Capacitance which loads a CHEF amplifier output terminal

Csubk Collector-substrate parasitic capacitance, k = 1, 2, 3, ...

Cµk Base-collector parasitic capacitance, k = 1, 2, 3, ...

xvi

Cπk Base-emitter parasitic capacitance, k = 1, 2, 3, ...

CHEF Cherry-Hooper amplifier with emitter follower feedback

CMOS Complimentary metal oxide semiconductor

COSSB Compatible optical single sideband

c Speed of light in a vacuum

D Dispersion parameter

Dn Concentration of ‘donor’ or phosphorous atoms

D(s) Laplace domain transfer function denominator

DC Direct current

DCF Dispersion compensating fiber

DP Differential pair

DR Dynamic range

DSB Double sideband

d(t) Detected signal in an optical receiver (generic)

dB Decibels, a logarithmic ratio of power

dBv Logarithmic ratio of voltage referenced to one volt

dBm Logarithmic ratio of power referenced to one milliwatt

EG Bandgap energy =1.12 electron volts for silicon

Ein Mach-Zehnder input signal envelope electric field

Eout Mach-Zehnder output signal envelope electric field

EF Emitter follower

e2R Mean square noise voltage due to resistor R

F Fourier transform operator

F Farads, the unit of capacitance

FFP Fiber Fabry-Perot

f Frequency in hertz

fc Carrier frequency of a bandpass electrical signal

fo Carrier frequency in an optical system

xvii

fT Frequency where bipolar transistor common emittercurrent gain becomes unity

fβ Frequency at which β decreases to 3 dB below βo

fF Femtofarads

GA Gain of an amplifier within a logarithmic amplifier

Gb/s Gigabits per second

Ge Germanium

GN Sum of logarithmic amplifier path gains one to N

Gpk Gain of kth logarithmic amplifier path, k = 1, 2, 3, ...

GHz Gigahertz

GVD Group velocity dispersion

GS/s Gigasamples per second

gm Transconductance (general)

H(f) Frequency domain transfer function (generic)

HDn(f) Transfer function of nth Hilbert transformer delay element , n = 1, 2, 3, ...

HL[ ] Indicates the Hilbert transform of the logarithm of a variable

H Henrys, the unit of inductance

HBT Heterojunction bipolar transistor

HP Hewlett Packard

HSPICE A brand name of SPICE

HT Hilbert transformer

hfe Common emitter current gain of a bipolar transistor

Ik Current (generic) , k = 1, 2, 3, ...

IBk DC base current, k = 1, 2, 3, ...

ICk DC collector current, k = 1, 2, 3, ...

IEk DC emitter current, k = 1, 2, 3, ...

IEEk DC bias current (generic), k = 1, 2, 3, ...

IESk Scaling current proportional to base-emitter junction area, k = 1, 2, 3, ...

IL Limiting current of transconductance element

xviii

IS Output current step for a parallel summation logarithmic amplifier

Ihigh Bias current of a high gain amplifier

Ilow Bias current of a low gain amplifier

Iout Logarithmic amplifier output current

IBM International Business Machines

IC Integrated circuit

IF Intermediate frequency

ISI Inter-symbol interference

IV Current voltage

In Indium

ib AC base current

i2c Mean square collector shot noise current

ip Signal current of transformer primary

is Signal current of transformer secondary

J Joules, the unit of energy

j Imaginary number

K Kelvin, the unit of absolute temperature

k Counting index

k Coupling coefficient between the coils of a transformer

kb Boltzmann’s constant (=1.38 ×10−23J/K)

km Kilometers

L Length of fiber

L Inductance

LE Emitter length

Lp Inductance of transformer primary

Ls Inductance of transformer secondary

LED Light emitting diode

LO Local oscillator

xix

Li Lithium

logA Logarithm with base A

M Scaling factor

M12 Mutual inductance of two transformer coils

MHz Megahertz

MPC Minimum phase compensator

m Meters, the unit of distance

m(t) Optical signal complex envelope, also referred to as simply m

mA Milliamperes

mHz Millihertz

mV Millivolts

mW Milliwatt

N Counting index

NA Concentration of ‘acceptor’ or boron atoms

Nb Niobate

NF Noise figure

NPN P-type silicon between two sections of n-type silicon

NRZ Non-return-to-zero

n Counting index

ni Concentration of holes or electrons in silicon at a given temperature

nr Fiber refractive index

nm Nanometers

ns Nanoseconds

nV Nanovolts

O Oxygen

OC Optical communications

OH Oxygen hydrogen

OSSB Optical single sideband

xx

P Phosphide

PD Detected power

PLL Phase locked loop

PN Junction of p-type and n-type silicon

PNP N-type silicon between two sections of p-type silicon

PRBS Pseudo random bit sequence

PSPL Picosecond Pulse Labs

PTAT Proportional to absolute temperature

pL A pole in the CHEF amplifier transfer function

ps Picoseconds

Q Pole quality factor

Qk kth bipolar transistor; k = 1, 2, 3, ...

QAM Quadrature amplitude modulation

q Electron charge (=1.60 ×10−19C)

< Real operator

R Resistor (generic)

RC Resistor connected to a transistor collector

RE Resistor connected to a transistor emitter

Rf Feedback resistor in the CHEF amplifier

R′

f Rf + re5 in the CHEF amplifier

Rmk Current mirror resistor, k = 1, 2, 3, ...

Ro Output impedance of the CHEF amplifier

Rp Resistance of transformer primary

RS Resistor connected to a voltage source

R′

S Rs + rb1 where rb1 is part of the CHEF amplifier

RF Radio frequency

RMS Root mean square

RSSI Receive strength signal indicator

xxi

RZ Return-to-zero

rbk Parasitic base resistance of kth transistor, k = 1, 2, 3, ...

rc Parasitic collector resistance

rce Collector-emitter resistance

rdk Inverse of transconductance of kth transistor, k = 1, 2, 3, ...

r′dk Equal to 1gmk

+ rek, k = 1, 2, 3, ...

rek Parasitic emitter resistance of kth transistor, k = 1, 2, 3, ...

rπk Intrinsic base-emitter resistance of kth transistor, k = 1, 2, 3, ...

SB(f) Frequency domain bandpass signal

S11 Input reflection coefficient s-parameter

S21 Forward transmission coefficient s-parameter

S22 Output reflection coefficient s-parameter

SICS Supply voltage independent current source

Si Silicon

SMA Subminiature Version A

SNR Signal to noise ratio

SONET Synchronous optical network

SPICE Simulation Program with Integrated Circuit Emphasis

SPM Self-phase modulation

SSB Single sideband

s Laplace domain variable

s(t) Information signal

s(t) Hilbert transform of information signal

sB(t) Time domain bandpass signal

sBSSB(t) Time domain bandpass single sideband signal

T Temperature in kelvins

THz Terahertz

Tb/s Terabits per second

xxii

t Time

U(f) Frequency domain unit step frequency

V Volts, the unit of voltage

VBE DC base-emitter voltage

VCC Positive DC supply voltage

VCE DC collector-emitter voltage

VDC DC offset voltage

VEE Negative DC supply voltage

VL Limiting amplifier maximum output voltage

VT Thermal voltage

Va Mach-Zehnder contact voltage

Vb Mach-Zehnder contact voltage

Vg Propagation velocity of an optical signal

Vin Input DC voltage (generic)

Vintercept Logarithmic transfer function intercept voltage

Vmin Lowest input voltage in logarithmic amplifiertheoretical dynamic range

Vout Output DC voltage (generic)

Vπ Mach-Zehnder modulator bias parameter

V BCE Collector-emitter breakdown voltage

VBIC Vertical Bipolar Intercompany Model

vbe AC voltage across rπ

vin AC input voltage (generic)

vo AC output voltage (generic)

vp Voltage across transformer primary

vπ AC voltage across rπ, the same as vbe

W Effective width of the base

WDM Wavelength division multiplexing

ZinEF Input impedance of an emitter follower

xxiii

ZL Impedance which loads a CHEF amplifier output terminal

Zo Characteristic impedance

z1 A zero in the CHEF amplifier transfer function

α Ratio of collector to emitter DC current

β AC common emitter current gain of a bipolar transistor

βDC DC common emitter current gain of a bipolar transistor

βo Low frequency common emitter current gain of a bipolar transistor

∆ Change operator

δ Partial derivative operator

δ(t) Impulse response function

λ Wavelength (generic)

λo Optical system carrier wavelength

µV Microvolts

Ξ(f) Frequency domain dispersive fiber transfer function

Π Propagation constant

Π2 Group velocity dispersion parameter

ρ Wave number

τ Time variable

Υ Hilbert transformer tap delay

Φ Phase of an optical signal envelope

χ Discrete time index

ψ(f) Phase response of a system as a function of frequency (generic)

Ω Ohms, the unit of resistance

ω Frequency in radians/s

ωo Pole frequency

xxiv

Chapter 1

Introduction

The development of low loss silica fiber in the 1960s and 70s enabled fiber-optic

communications to become commercially viable. When it was first developed, the

bandwidth-distance product of fiber was so enormous compared to copper wire, that

scientists were compelled to pour effort into improving the other two components

required in a fiber optic link, light emitting elements and photo-diodes. As of 2004,

fiber-optic systems span the entire earth, both across land and under the oceans.

Commercial fiber-optic systems operate at a data rate of over 1 Tb/s, and 10 Tb/s

systems have been demonstrated in the laboratory [1].

These data rates are possible because of a number of important technologies,

including Wavelength Division Multiplexing (WDM) and optical amplifiers. WDM

is the technique whereby information is modulated onto several wavelengths of light

simultaneously and is transmitted over a single fiber. The development of commercial

WDM systems with as many as 120 channels in the last decade has increased the

capacity of fiber-optic systems by a similar factor. The impact of optical amplifiers on

long-haul fiber-optic systems may be seen using Figure 1.1. Figure 1.1(a) shows the

architecture of long-haul fiber-optic systems which was used before the proliferation

of fiber-optic amplifiers. An intensity modulated laser or light emitting diode (LED)

inside the transmitter sends information on a fiber-optic cable. After the optical

signal is transmitted approximately 50 km, the loss and the non-linearities in the

fiber distort the signal and a regenerator is used. The regenerator receives the

1.0 Introduction 2

Tx

Transmitter

TxRx TxRx Rx

Regenerator Regenerator

...

Receiver

(a) System using regeneration.

Rx

ReceiverTransmitter

Tx

Amplifier Amplifier

...

(b) System using optical amplification.

Figure 1.1: Long-haul optical system architectures.

information using a photodiode, the output current of which is processed into ones

and zeros in the electrical domain. The data is then re-timed and is again modulated

onto a light emitting element and transmitted to the next regenerator. This process

of reception and transmission occurs at each regeneration node until the data is

received at the final destination.

The regeneration nodes in Figure 1.1(a) become quite complex and expensive in

the case of WDM systems. In fact, WDM was not widely used until the proliferation

of optical amplifier technology around 1995 [2]. Optical amplifiers are devices which

may amplify an entire range of light wavelengths at once. This makes it possible to

amplify a WDM signal without having to regenerate the data. Figure 1.1(b) shows

a system which uses this technology. As the optical signal propagates and loses

power due to losses in the fiber, it may be repeatedly amplified as long as system

performance is not limited by amplifier noise, nonlinearities in the fiber and the

amplifiers, and a problem known as chromatic dispersion, which will be described

1.0 Introduction 3

shortly. Most modern long-haul commercial fiber systems use a combination of the

two techniques shown in Figure 1.1, with optical amplifiers used approximately every

50 km and regeneration used after several optical amplifiers [2].

One of the most significant challenges in designing WDM systems is overcom-

ing the effects of chromatic dispersion. Chromatic dispersion is characterized by a

wavelength dependent propagation velocity in the fiber. When a light signal with

information contained on a range of wavelengths is transmitted on fiber, different

wavelengths propagate at different speeds, and so they arrive at the detector at

different times. One particularly harmful form of distortion caused by chromatic

dispersion is the power penalty incurred by transmission of double sideband signals.

When a double sideband signal, which is the usual form of a signal, is transmitted,

the upper and lower sidebands of information arrive at the receiver at different times.

These sidebands may interfere destructively, causing loss of power.

Unless it is compensated, dispersion severely limits system performance above

2 Gb/s [2]. Cascading optical amplifiers as in Figure 1.1(b) solves the loss problem.

However, since an amplifier does not restore the signal to its original state, dispersion

induced degradation of the signal is allowed to accumulate over several amplifiers.

For this reason, all commercial long-haul fiber systems operating at 10 Gb/s or

higher use some form of dispersion compensation. The dispersion is compensated

by adding a device in series with the fiber optic cable which has a frequency de-

pendent delay profile opposite to that of the fiber, so that the overall system delay

becomes approximately wavelength independent. However, it is difficult to attain

full compensation for all channels in a WDM system. A small amount of residual

dispersion usually remains and may become a problem for transmission distances of

a few hundred kilometers or more. Further complicating the problem is that the

approximate amount of dispersion in a WDM system must be known before it can

1.1 Research Objective and Scope 4

be compensated. However, in reconfigurable networks, where entire spans of fiber

may be added or dropped from the network during operation, the amount of over-

all dispersion may vary significantly. Furthermore, at bit rates of 40 Gb/s, even

temperature induced changes in the fiber delay characteristics become of concern.

The goal of this work is to help overcome the negative effects of chromatic dis-

persion using optical single sideband (OSSB). It is widely known that by only trans-

mitting one sideband of the electrical information on the fiber, the problem of two

sidebands interfering with each other is overcome. OSSB could be used with either

no optical dispersion compensation or with a reduced amount of optical compensa-

tion. OSSB is already widely used for bandpass electrical signals, such as a 1 Gb/s

signal centered at 20 GHz [3, 4]. However, a limited number of experiments have

been performed using OSSB with baseband signals.

1.1 Research Objective and Scope

The objectives of this thesis are to develop an integrated circuit logarithmic ampli-

fier and Hilbert transformer for the Compatible Optical Single Sideband (COSSB)

system. A further objective is to quantify the performance of these circuits in the

COSSB system.

1.2 Thesis Outline

In Chapter 2, chromatic dispersion is examined along with a discussion of its undesir-

able effects on system performance and methods of compensating for it. One of these

methods is COSSB transmission, and the COSSB system architecture is introduced.

Chapter 3 describes a novel design procedure for a necessary logarithmic amplifier

building block, the Cherry-Hooper amplifier with emitter follower feedback. An em-

1.2 Thesis Outline 5

phasis is placed on low distortion magnitude and group delay frequency responses,

which are important for broadband operation. Chapter 4 describes the development

of a novel DC-4 GHz Si BJT logarithmic amplifier, and of a DC-6 GHz SiGe HBT

logarithmic amplifier. The latter implementation uses the same parallel summation

architecture as the first, but makes use of the Cherry-Hooper amplifiers developed

in Chapter 3. Chapter 5 describes the implementation of the first fully integrated

10 Gb/s Hilbert transformer. In Chapter 6, simulations of the SiGe HBT log ampli-

fier and the Hilbert transformer in the COSSB system are described. This chapter

lays the foundation for Chapter 7, where a COSSB transmitter is constructed, and

the sideband suppression is quantified. Finally, Chapter 8 concludes the thesis and

provides recommendations.

Chapter 2

Background

2.1 Introduction

The target application of the circuits in this work is COSSB, which is described in

Section 2.4. However, in order to understand why COSSB is desirable, it is essential

to understand chromatic dispersion. Section 2.2 describes chromatic dispersion, and

Section 2.3 describes methods of compensating for it.

2.2 Chromatic Dispersion

To understand chromatic dispersion, the distinction should be made between single

mode and multi-mode fiber. When an optical signal is launched into fiber, the exact

mode in which the optical wave propagates depends on the dimensions of the fiber.

Silica fiber has a core surrounded by cladding in order to form a waveguide. As

with any waveguide, if the diameter of the core is large compared to the wavelength

of the signal, the signal may move down the fiber using multiple modes of wave

propagation. The core diameter may be reduced until only a single mode may

propagate, in which case the fiber is called ‘single mode’. Commercial long-haul

systems with an information capacity of 10 Gb/s or higher use single mode fiber

almost exclusively. For this reason, single mode fiber will be assumed throughout

this work.

If a short pulse of light, such as a ‘one’ signal in a digital system, is launched

into the fiber and allowed to propagate many tens of kilometers, that pulse will

2.2 Chromatic Dispersion 7

Time

Distance

Time

Figure 2.1: Pulse spreading due to chromatic dispersion.

spread out in time. Chromatic dispersion is the pulse spreading that occurs within

a single mode [5]. Figure 2.1 shows how two pulses may begin to overlap each other

as they propagate down the fiber. This phenomenon is detrimental to the operation

of high data rate communication systems because overlapping pulses cause inter-

symbol interference and bit errors at the receiver. Chromatic dispersion may also be

explained in terms of group delay, defined as

group delay = −dψ(f)

df

1

2π(2.1)

where ψ(f) is the phase response of that system, and f is frequency. Chromatic

dispersion is also known as Group Velocity Dispersion (GVD), because it may be

characterized by a wavelength dependence of group delay.

2.2.1 Types of Dispersion

Three different types of dispersion are waveguide dispersion, nonlinear dispersion,

and material dispersion. Waveguide dispersion occurs because only about 80 percent

of the optical power is confined to the core of the fiber, and about 20 percent of the

power propagates in the cladding. The optical signal in the cladding travels faster

than the signal in the core, causing dispersion [5]. Nonlinear dispersion is caused

by the dependence of the fiber refractive index on the optical signal intensity. The

physical origin of this effect may be traced to the nonlinear response of electrons

to optical fields [2]. Nonlinear dispersion may be mitigated by avoiding high power

2.2 Chromatic Dispersion 8

levels in optical systems. Material dispersion is due to the wavelength dependence of

the refractive index of fiber, which causes a wavelength dependence of propagation

velocity. The material and waveguide dispersion may be added together to obtain an

estimate of the dispersion for low to medium power pulses [2]. In the next section,

a mathematical description of dispersion is developed.

2.2.2 Mathematical Definition of Dispersion

As a signal propagates down a fiber, each wavelength requires a certain amount of

time or group delay per unit length of travel. This delay τ is given by [5]

τ =L

Vg

(2.2)

where L is the distance traveled, and the group velocity Vg is given by

Vg =

(

dΠ

dω

)

−1

(2.3)

where Π is the propagation constant, λ is wavelength, and c is the speed of light in

a vacuum. Furthermore, the propagation constant is given by

Π =2πnr

λ(2.4)

where nr is the fiber refractive index. For a signal with spectral width ∆λ the amount

of pulse broadening over a distance L is given by [2]

∆τ =d

dλτ∆λ. (2.5)

In terms of angular frequency this is given by [5]

∆τ =d

dωτ∆ω =

d

dω

(

L

Vg

)

∆ω = L

(

d2Π

dω2

)

∆ω. (2.6)

2.3 Methods to Compensate for Dispersion 9

The factor d2Π/dω2(= Π2) is the group velocity dispersion (GVD) parameter. By

using ω = 2πc/λ and ∆ω = (−2πc/λ2)∆λ, equation (2.6) may then be rewritten as

∆τ =d

dλ

(

L

Vg

)

∆λ = DL∆λ (2.7)

where

D =d

dλ

(

1

Vg

)

= −2πc

λ2Π2. (2.8)

In this case, D is the dispersion parameter and is typically expressed in units of

ps/(nm · km). The meaning of D is best understood by equation (2.7), in relation

to the amount of pulse broadening that it causes. If D is slightly negative, then the

pulse may actually compress, which is the principle of dispersion compensating fiber

to be discussed in Section 2.3.1.

Figure 2.2 shows a plot of typical values for the parameter D and the fiber loss,

also called attenuation, versus wavelength for single mode fiber. The relatively high

loss at 1390 nm is due to signal absorption by small concentrations of the OH ion in

fiber [2]. The wavelength which experiences the least amount of loss is at approx-

imately 1550 nm. Unfortunately, the dispersion at this wavelength is significant,

typically 15-20 ps/(nm · km). The dispersion is zero near 1350 nm, however, the loss

is prohibitively high at this wavelength for systems spanning many tens of kilometers

or more. For this reason, it is common practice for long-haul fiber systems to oper-

ate at 1550 nm, and the dispersion is compensated for. The next section describes

dispersion compensation techniques.

2.3 Methods to Compensate for Dispersion

Chromatic dispersion is a major problem in optical systems, and several techniques

exist to deal with it. For a dispersion compensation technique to be useful in com-


1.2

0.3

0.9

0.6

0

10

0

−10

20

1250 1300 1350 1400 1450 1500 1550 1600

Wavelength (nm)

Dis

pers

ion

para

met

er D

Atte

nuat

ion

(dB

/km

)

(ps/

(nm

−km

))

Figure 2.2: A typical loss and dispersion profile for single mode fiber.Adapted from [2].

mercial systems, it should be capable of compensating the dispersion in all channels

of a WDM system simultaneously. In this section, three techniques which meet

this criteria are described. One of the techniques, post detection compensation, is

uniquely suited to the COSSB system described in Section 2.4.

2.3.1 Optical Techniques

Dispersion Compensating Fiber

One broadband dispersion compensation technique involves the use of Dispersion

Compensating Fiber (DCF). In Figure 2.2, it was shown that the dispersion param-

eter of standard single mode fiber increases with wavelength. If it were possible to

design another type of fiber that had a large negative dispersion parameter, then

adding this fiber to a system would compensate for the dispersion in standard fiber.

Single mode fiber with a negative dispersion parameter may be fabricated, how-

ever it may only support relatively low levels of optical power with acceptable linear-

ity. Single mode fiber has a relatively small core diameter, and if the core diameter

is increased just enough to allow a second mode to propagate, the second mode will

exhibit a negative dispersion parameter and the fiber can support higher power levels


with less nonlinearity [2]. The dispersion of the second mode can be as large as -770

ps/(nm · km).

Dispersion compensation is achieved by using approximately 2-5 km of DCF fiber

for every 50 km of standard single mode fiber. Furthermore, a mode coupling device

is required at each interface of the DCF fiber and the standard fiber, in order to

convert between the standard propagation mode to the second order mode with the

negative dispersion.

Dispersion management using DCF is practical and effective in dense WDM

systems, and it is used in virtually all systems with a spectral width of 30 nm or

more. In one of the highest capacity experiments to date, DCF was used to transmit

40 Gb/s on each of 273 channels over 117 km, resulting in a total bandwidth of

11 Tb/s and a spectral width of more than 100 nm [1]. The disadvantage of DCF

is its high loss, which can be 5 dB for a 5 km length. This can be compensated

for by increased optical amplifier gain, however the resulting amplifier noise may

corrupt the signal to an unacceptable level. For this reason, other optical dispersion

compensation schemes have been developed. One of the more successful technologies

is described in the next section.

Fiber Bragg Gratings

Fiber Bragg gratings are optical filters approximately 10 cm in length that may

compensate for the dispersion of approximately 100 km of fiber. As its name implies,

a grating is a periodic change in the material inside an optical transmission medium.

The medium commonly used is simply optical fiber. Its refractive index may be

changed at small spacings through a photo-imprinting process [5]. The result is that

certain wavelengths of light are transmitted, whereas others are reflected. At the

Bragg wavelength, the light is almost completely reflected, and so there is a type of


stop-band at this frequency. At this wavelength, the phase response of the grating is

almost linear, and so it could not be used to compensate for dispersion. However, at

wavelengths slightly above the Bragg wavelength, most of the light is allowed to pass

and it undergoes a negative dispersion. The mathematics behind Bragg gratings

are somewhat involved, and will not be shown here. They involve an analysis of

the coupling between forward and backward waves. The situation becomes more

complicated for gratings whose refractive index is linearly increased over the length

of the grating in order to achieve an even larger negative dispersion parameter [2].

Fiber Bragg gratings are advantageous because they are physically small. Fur-

thermore, they only pass signals at periodic, narrow regions of spectrum, and so

they filter out some optical amplifier noise. However, in order to compensate the

dispersion of more than one channel in a WDM system, gratings which are centered

at different wavelengths must be cascaded with optical isolators between each grat-

ing. As the number of channels increases to 10 or more, it becomes very difficult to

compensate the dispersion of all of the channels at once. For this reason, DCFs are

preferred over Bragg gratings in dense WDM systems.

2.3.2 Post Detection Compensation

Another general dispersion compensation technique is electrical compensation used

after the signal is detected. The type of detection most widely used in optical systems

is square law detection with a photodiode, also known as direct detection. The output

current of the photodiode is proportional to the optical power or intensity. As part

of this process, the information on the optical signal is converted directly down to

baseband. This is known as homodyne detection, as opposed to heterodyne detection

where an intermediate frequency is used. Furthermore, since no local oscillator (LO)

is used, direct detection is also referred to as self-homodyning detection in this thesis


and in [6, 7]. The dispersion incurred by the optical signal during transmission

imposes a different delay on the upper and lower sidebands of the signal. Once the

signal is detected using self-homodyning detection, the upper and lower sidebands

fold onto each other and the dispersion may no longer be compensated [6, 8]. There

are two ways that the dispersion information may remain intact so that it may be

compensated post detection; heterodyne detection and optical single sideband. Each

of these will be discussed in turn.

In heterodyne detection, the main optical signal must be combined with an LO

optical signal at a slightly different wavelength prior to detection. The photodiode

output then contains an intermediate frequency (IF) containing the data. The dis-

persion distortion inherent in the data may then be electrically equalized at the IF,

and the resulting signal may then be converted down to baseband using a mixer [7].

Furthermore, since the propagation delay of fiber increases with wavelength, it de-

creases with increasing frequency since c = λf . As a result, all that is required to

equalize the dispersion in the IF signal is an electrical structure whose delay increases

with frequency. Microstrip line is most commonly used for this purpose [2, 7, 8, 9, 10].

The disadvantage of heterodyne systems is that for 10 or 40 Gb/s signals, the re-

quired mixers, which would have minimum RF frequencies at approximately 20 and

80 GHz respectively, would be expensive if not impractical.

If optical single sideband (OSSB) is used, the dispersion characteristic in the

signal remains intact during self-homodyning detection. Hence, the signal may be

detected using a photodiode and the requirement for broadband mixers is elimi-

nated, and the signal may be equalized post detection. In 1998, Sieben demon-

strated 10 Gb/s transmission over 320 km using OSSB transmission and post detec-

tion compensation using microstrip [7]. Furthermore, Winters described a tunable

analog tapped delay line for dispersion compensation [11]. It had fractionally spaced

2.4 Compatible Optical Single Sideband 14

weights that could be tuned based on the amount of dispersion. The disadvantage of

these methods is that long lengths of microstrip are needed in order to compensate

long lengths of fiber, such as 32 cm for 320 km of fiber. The loss of microstrip typi-

cally increases with frequency, and this may become problematic. Nevertheless, this

problem may be mitigated through the use of low loss microstrip substrates. As well,

if OSSB is used along with a reduced amount of optical dispersion compensation, the

lengths of the required microstrip and optical compensators are reduced compared

to the case where only OSSB or optical compensation is used.

2.4 Compatible Optical Single Sideband

In this section, the COSSB system is introduced as a spectrally efficient and generally

desirable system architecture. It is described how chromatic dispersion causes power

fading in double sideband (DSB) signals, and how OSSB signals are immune to this

problem. Before describing the COSSB system, it is worthwhile to introduce the

idea of complex envelope representation of signals. This will greatly simplify the

mathematics in the rest of this section. Some of the mathematical development in

this section is paraphrased from [6], and it is reprinted with permission.

2.4.1 Complex Envelope Representation of Bandpass Signals

We begin by defining a low pass, also called a baseband, information signal s(t),

such as a 10 Gb/s signal. In an optical system, this information is modulated onto

a light signal or carrier. Since c = fλ, the typical carrier wavelength of 1550 nm

corresponds to a carrier frequency of 200 THz. As a result, the 10 Gb/s information

becomes a narrow-band bandpass signal on the fiber. The resulting optical signal

sB(t) has the frequency spectrum SB(f) as shown in Figure 2.3, where fo is the


fo fof

+−

SB

Figure 2.3: Narrow-band bandpass signal.

optical carrier frequency.

The bandpass signal SB(f) has two copies of the same information at −fo and

+fo. In order to obtain the complex envelope of SB(f), we first construct a signal

that has only the positive frequencies of SB(f) according to [6, 12]

A(f) = 2U(f)SB(f) (2.9)

where U(f) is the frequency domain unit step function. Since multiplication in

the frequency domain involves convolution in the time domain, the time domain

expression for (2.9) is given by

a(t) = 2F−1[U(f)] ∗ sB(t) (2.10)

where F−1 denotes the inverse Fourier transform. The inverse Fourier transform of

U(f) is given by

F−1[U(f)] =1

2δ(t) +

j

2πt(2.11)

where δ(t) is the impulse function. a(t) may then be expressed as [12]

a(t) = 2sB(t) ∗ 1

2δ(t) + 2sB(t) ∗ j

2πt

= sB(t) +j

π

∫

∞

−∞

sB(τ)

t− τdτ. (2.12)


The integral on the right hand side of (2.12) is defined as

sB(t) =1

π

∫

∞

−∞

sB(τ)

t− τdτ (2.13)

where sB(t) is called the Hilbert transform of the signal sB(t). A Hilbert transformer

may be represented as a filter with frequency response [12]

H(f) = −j · sgn(f). (2.14)

where sgn indicates the signum function. The signal A(f) in (2.9) is still bandpass

in nature, and a low-pass equivalent may be created as

AL(f) = A(f + fo). (2.15)

In the time domain this amounts to multiplication of a(t) with a complex sinusoid

as in:

aL(t) = a(t)exp(−j2πfot). (2.16)

The low pass equivalent signal aL(t) is also called the complex envelope of sB(t).

Conversely, the bandpass signal sB(t) is related to aL(t) by

sB(t) + jsB(t) = aL(t)exp(j2πfot). (2.17)

The left hand side of this equation contains what is known as a Hilbert transform

pair.

Before moving on to the description of the COSSB system, it will be shown how

a single sideband signal may be generated. A single sideband signal is a bandpass

signal with either the upper or lower half of its spectrum removed. An upper single

sideband signal has zero magnitude for 0 < f < fo and a lower single sideband


signal has zero magnitude for f > fo. The complex envelope of a SSB signal, defined

as aLSSB(t), may be acquired starting with the baseband information signal s(t)

according to

aLSSB(t) = s(t) ± js(t). (2.18)

The real bandpass signal associated with aLSSB(t) is given by

sBSSB(t) = < [aLSSB

(t)exp(±j2πfot)]

= s(t)cos(2πfot) ± s(t)sin(2πfot) (2.19)

with the lower single sideband signal corresponding to the plus sign and the upper

single sideband signal corresponding to the negative sign.

2.4.2 COSSB Modulation

We now consider the process of optical detection as it relates to optical single side-

band. The output current of a photodiode is proportional to the optical power or

intensity, which, in turn, is proportional to the information envelope squared. As

a result, any phase information in the optical signal is discarded during detection.

What is of interest in this thesis is what happens when square law detection is ap-

plied to a single sided bandpass signal. The answer to this question may be found

more readily by applying square law detection to the complex envelope of a SSB

signal, aLSSB(t). With aLSSB

(t) being complex, its real and imaginary parts may be

expressed as

aLSSB(t) = s(t) ± js(t). (2.20)

Furthermore, the polar form of aLSSB(t) is defined as


aLSSB(t) = m(t)exp (jφ(t)) (2.21)

where m(t) is given by

m(t) =√

s2(t) + s2(t) (2.22)

and

φ(t) = tan−1

(

s(t)

s(t)

)

. (2.23)

Using these definitions, square law detection may be applied to the complex envelope

aLSSB(t) according to

d(t) = |aLSSB(t)|2

= s2(t) + s2(t). (2.24)

Unfortunately this result is not proportional to the original information s(t), and

so the data will not be recovered. We now derive the complex envelope of a SSB

signal that may be square law detected without loss of information, which we will

denote as a′LSSB(t). We begin by making the important observation that since the

information obtained after square law detection is the square of the magnitude of the

signal, the magnitude of the signal must be equal to the data s(t). In addition, the

information should always be greater than zero, meaning that a DC offset should be

added if needed. However, the phase of the signal, which is discarded during square

law detection, may be modulated in a way so that the signal is single sideband.

Expressing a′LSSB(t) in polar form with the magnitude equal to s(t),

a′LSSB(t) = s(t)exp (jφ(t)) . (2.25)


Taking the natural logarithm of both sides of this equation gives

lna′LSSB(t) = lns(t) + jφ(t). (2.26)

Let us force this signal to consist of a Hilbert transform pair. In this case, φ(t) is

found to be

φ(t) = ±HL [s(t)] (2.27)

where HL [s(t)] indicates the Hilbert transform of the logarithm of s(t). This will

ensure that lna′LSSB(t) contains only positive or negative frequencies, as is required

of the complex envelope of a single sideband signal. However, if lna′LSSB(t) meets

this condition, then a′LSSB(t) does as well [6]. Hence, a′LSSB

(t) may be expressed as

a′LSSB(t) = s(t) · exp (jHL [s(t)]) . (2.28)

It is now possible to use square law detection to recover s(t) as in

d(t) = |s(t) · exp (jHL [s(t)])|2

= s2(t). (2.29)

Thus, compatibility with direct detection is achieved in theory [13]. In practice,

the type of modulation suggested by (2.28) is difficult to perform. It is difficult to

take the logarithm of s(t), especially if it is at a data rate of 10 Gb/s or higher.

Furthermore, designing a Hilbert transformer is difficult because of the abrupt tran-

sition in its phase response at DC, as indicated by its −j · sgn(f) phase response.

Building suitable approximations to a broadband logarithmic converter and Hilbert

transformer in integrated circuit form is the main goal of this thesis.


2.4.3 COSSB Implementation: The Ideal Minimum Phase Modulator

The structure of the modulator suggested by (2.28) is shown in Figure 2.4. The

term ‘minimum phase’ relates to the trajectory of the analytic signal in the complex

plane, and the reader is referred to Reference [6] for further details on minimum

phase signals. In Figure 2.4, M is a scaling factor applied to s(t) and a DC offset

VDC is added to ensure that the resulting signal is strictly greater than zero. It is

instructive to view the electrical waveform before and after applying the logarithm,

and after it has been Hilbert transformed. These waveforms are shown in Figure 2.5

for a 10 Gb/s 211−1 length pseudo random bit sequence (PRBS) scaled with M=0.6

and VDC=1. The PRBS sequence is filtered with a fifth order Butterworth filter with

a cutoff frequency of 5 GHz immediately after it is scaled in order to avoid aliasing.

Furthermore, filtering at 5 GHz removes higher frequency components that would

be the most distorted by dispersion.

It is seen in Figure 2.5(b) that the effect of the logarithm is to stretch out the

parts of the waveform that are closest to zero amplitude. If a signal consists of two

perfect digital levels, and if it is scaled by M and VDC and then logged, the logarithm

of this waveform also consists of two perfect signal levels. Hence, for perfect binary

signals that are strictly positive, a level shifter and an amplifier may act in place

of a logarithmic converter. The waveform at the output of the Hilbert transformer,

on the other hand, is very different, and appears to be a series of amplitude spikes

corresponding to each bit transition. This demonstrates the unique response of the

Hilbert transformer. The −j · sgn(f) frequency response indicates that all positive

frequencies undergo a constant -90 phase shift, resulting in the unusual waveform

in Figure 2.5(c). This is in contrast to the phase shift imposed on a sinusoid passing

through a wire, for example, which increases with increasing frequency.


AmplitudeModulator

PhaseModulator

To Fiber

exp(j t)ω

+

VDC

HilbertTransformer

Laser

ConverterLogarithmic

.M s(t)

Figure 2.4: Ideal minimum phase modulator.Adapted from [6].

Figure 2.6 shows the optical spectra of the original DSB information and of the

COSSB signal created using ideal amplitude and phase modulation. Note that the

mild distortion in the OSSB signal spectrum is due to the choice of filter used on

the data, and is also effected by the length of the PRBS sequence. However, the eye

diagram of the detected COSSB signal in Figure 2.7 shows that the information is

recovered without error. It is noted that optical USB, which is LSB in the frequency

domain, must be transmitted if microstrip post detection equalization is to be used.

This ensures that the microstrip will equalize the phase distortion caused by fiber

dispersion [7]. If optical LSB is transmitted and microstrip equalization is attempted,

the phase distortion due to fiber dispersion will be made worse, and the equalization

will fail.

2.4.4 Dispersion Effects on Double and Single Sideband Signals

In this section, the distortion of DSB signals caused by dispersion is demonstrated,

and it is shown how single sideband signals defeat the distortion mechanism. To be-


5 6 7 8 9 10 11 12 13 14 150

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Li

near

Am

plitu

de

Time (ns)

(a) Raw waveform.

5 6 7 8 9 10 11 12 13 14 15−2

−1.5

−1

−0.5

0

0.5

1

Line

ar A

mpl

itude

Time (ns)

(b) Logged waveform.

5 6 7 8 9 10 11 12 13 14 15−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Line

ar A

mpl

itude

Time (ns)

(c) Hilbert transformed waveform.

Figure 2.5: Electrical signals at points throughout the COSSB system.

gin, a mathematical representation is needed of the way in which dispersion increases

with wavelength, as in Figure 2.2. This characteristic may be represented by defining

the complex envelope of the frequency domain transfer function of dispersive fiber,

given by

Ξ(f) = exp

(

jπDλ2of

2L

c

)

(2.30)


−10 −8 −6 −4 −2 0 2 4 6 8 10−80

−70

−60

−50

−40

−30

−20

−10

0

Normalized Frequency (GHz)

Det

ecte

d P

ower

(dB

)

(a) Filtered 10 Gb/s DSB signal.

−10 −8 −6 −4 −2 0 2 4 6 8 10−80

−70

−60

−50

−40

−30

−20

−10

0


Det

ecte

d P

ower

(dB

)

(b) Filtered 10 Gb/s SSB signal.

Figure 2.6: Filtered 10 Gb/s DSB and SSB signals plotted against frequency nor-malized to the carrier.

−100 −50 0 50 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time (ps)

Am

plitu

de

Figure 2.7: Eye diagram of detected SSB signal.

where, as before, D is the dispersion parameter, λo is the optical wavelength, and L

is the length of fiber [6]. The group delay of Ξ(f) is proportional to the derivative of

its phase with respect to frequency. The phase of Ξ(f) is contained in the argument

of the exponential, whose derivative is proportional to f . Hence, the group delay of

Ξ(f) is linearly proportional to frequency, and so it models first order dispersion.


Consider the case where a sinusoid defined by

r(t) = cos(2πfct) (2.31)

propagates in dispersive optical fiber. The upper and lower sidebands of that signal

will experience different group delays. Assume that the signal is then recovered

using direct detection. It was mentioned in Section 2.3.2 that the information on

the upper sideband is combined with the information in the lower sideband during

direct detection. It is easy to imagine that if two corresponding frequencies in each

sideband were out of phase due to dispersion, they may cancel each other out. In

fact, using (2.30) it may be shown that dispersion causes fading in the detected

power PD as a function of frequency and fiber length according to [6, 14]

PD ∝ cos2

(

jπDλ2of

2c L

c

)

. (2.32)

From this equation, it is seen that the detected power reduces to zero for arguments

corresponding to ±π/2± kπ, k = 1, 2, 3, ... . Hence, for a given frequency, the power

will fade to zero at certain lengths of fiber. Alternatively, for a given length of

fiber, the power at certain frequencies will fade to zero. Although this equation

is for the case of a sinusoid at frequency fc, the same fading mechanism will be

present for broadband signals, such as 10 Gb/s signals. Intuitively, we would expect

a frequency selective nulling of broadband signals at certain lengths of fiber. This

hypothesis may be verified through the simulation of a DSB 10 Gb/s signal through

dispersive fiber. Figure 2.8(a) shows the detected spectrum of the 10 Gb/s DSB

from Section 2.4.3 after transmission on 200 km of fiber with a dispersion parameter

of D = 18 ps/(nm · km). The detected signal shows significant power fading in the

4 GHz region of the spectrum.


−10 −8 −6 −4 −2 0 2 4 6 8 10−80

−70

−60

−50

−40

−30

−20

−10

0


Det

ecte

d P

ower

(dB

)

(a) Spectrum of recovered DSB signal.

−100 −50 0 50 1000

0.2

0.4

0.6

0.8

1

Time (ps)

Am

plitu

de

(b) Eye diagram of recovered DSB signal.

Figure 2.8: Detected 10 Gb/s DSB signal after 200 km of dispersive fiber.

Based on the above discussion, it may be realized that if one of the sidebands in a

signal is removed, then there is no second sideband to cause destructive interference

upon detection, and power fading is avoided. In fact, the immunity of OSSB signals

to chromatic dispersion induced power fading is well established in the optical com-

munity [3, 4, 8, 15]. This notion may readily be verified through a simulation of the

10 Gb/s COSSB signal from Section 2.4.3 over 200 km of dispersive fiber and using

self-homodyning post detection dispersion compensation. Figure 2.9(a) shows the

detected spectrum of the COSSB signal where it is observed that the power fading

problem is gone. The small ripples that are present in the spectrum do not cause

a serious degradation in the signal. This fact makes post detection dispersion com-

pensation possible in COSSB systems. If the detected power had nulls in it, then

there would be no way to compensate its distorted phase. This also explains why

the dispersion in DSB signals must be compensated prior to direct detection.


−10 −8 −6 −4 −2 0 2 4 6 8 10−80

−70

−60

−50

−40

−30

−20

−10

0


Det

ecte

d P

ower

(dB

)

(a) Spectrum of recovered SSB signal.

−100 −50 0 50 100−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time (ps)

Am

plitu

de

(b) Eye diagram of recovered SSB signal.

Figure 2.9: Detected 10 Gb/s SSB signal after 200 km of dispersive fiber.

2.4.5 Minimum Phase Dispersion Compensation

In Section 2.3.2, it was described how microstrip line may be used to equalize a SSB

signal post detection. At least one other post detection compensation scheme has

been developed for OSSB systems [6]. Davies showed that the optimal post detection

dispersion compensator is of the form shown in Figure 2.10. The compensator is best

described by Davies as follows [6]:

The minimum phase compensator (MPC) structure could be thought of

as a ‘mirror’ of the minimum phase modulator in which the optical elec-

tric field linear envelope (m) is recovered and preserved as the envelope

portion of the detected signal. Under the minimum phase assumption the

phase of the optical electric field is recovered by the successive operation

of the natural log and Hilbert Transform of the linear envelope (Φ). The

phase and amplitude signal are then combined in a polar representation


of the complex predetection electric field, given by

a(t) = m(t) · exp jΦ(t) . (2.33)

The complex signal is then passed through a complex anti-dispersive filter

to remove the fiber induced distortion. The resulting signal, which is still

complex, will be the corrected signal in which the original information

is contained in the envelope. Since the recovered signal in this case is

single sideband, the anti-dispersive filter transfer function is simply the

inversion of the fiber transfer function:

Cmp(f) = exp

(

−jπDλ2f 2L

c

)

; f ≥ 0. (2.34)

Simulations by Davies and in Chapter 6 of this thesis show that this type of

compensation, while requiring more processing steps, may double the transmission

distance in a fiber optic system. The most significant challenge to implementing

this compensator for a 10 Gb/s system remains the design of the amplitude and

phase combiner. The minimum phase compensator in Figure 2.10 also requires a

logarithmic function and Hilbert transformer, similar to the minimum phase OSSB

modulator. This provides a further motivation for the development of the logarithmic

amplifiers and the Hilbert transformer in this thesis.

2.4.6 Previous Experiments Using COSSB

OSSB transmission is popular for systems where the electrical information is a radio

signal above 15 GHz [3, 4]. The reason is that since these signals are at such high

frequencies, the dispersion induced power fading described in the last section occurs

even with short lengths of fiber, such as 10 km. For these signals, single sideband may

be generated using a narrow-band -90 phase shift and a baseband gigabit per second


2

CompensatedSignal

Φ

Combiner

Complex Anti−dispersiveFilter

Phase (Φ)

.m exp(j )

Amplitude (m)

Converter TransformerHilbertLogarithmic

AbsoluteValue

Photodiode RootSquare

Fiber

Figure 2.10: Ideal minimum phase dispersion compensator.Adapted from [6].

Hilbert transformer is not required [3, 4]. Previous OSSB experiments where the

signal was a baseband, multi-gigabit per second data signal include Yonenaga’s work

where he demonstrated 6 Gb/s OSSB transmission of a 27−1 length PRBS sequence

over 270 km of fiber in 1993 [8]. In his work, an optical filter was used to remove

the unwanted sideband. In contrast, Sieben implemented OSSB using the method

described in Section 2.4.3 in 1997. He demonstrated transmission of a 10 Gb/s PRBS

sequence up to pattern lengths of 214−4 over 320 km of fiber and achieved a bit error

ratio (BER) of 10−9 [7, 16, 17]. He used post detection dispersion compensation

consisting of a 32 cm length of microstrip. In that experiment, the logarithmic

converter required to implement the system in Figure 2.4 was not included, because

no such converter existed. It was described in Section 2.4.3 how the logarithm

of digital waveforms with only two levels is also a waveform with only two levels.

Hence, by providing the proper scaling and DC offset, the logarithm function was

approximated in Sieben’s experiment.

As part of Sieben’s research, he investigated ways to generate the Hilbert trans-

form of a 10 Gb/s signal [7]. One of the methods that he discussed was to implement


a Hilbert transformer using a tapped delay filter. This technique will be described

further in Section 5.1, when the fully integrated Hilbert transformer in this thesis is

presented.

2.4.7 The Mach-Zehnder Modulator

For the simulations in Section 2.4.3, an ideal amplitude and phase modulator were

used. This section describes a realistic amplitude modulation device. It is not

advisable in optical systems to amplitude modulate optical lasers directly. This

is because amplitude modulation in lasers is accompanied by an inherent phase

modulation process, known as chirp. Instead of modulating the laser, an external

modulator is used, such as the Mach-Zehnder modulator. This type of modulator

will be used for the experimental work in this thesis, so it is introduced here.

The Mach-Zehnder modulator is a type of interferometer, where two signals inter-

fere constructively or destructively. Figure 2.11 shows a diagram of a Mach- Zehnder

modulator. It consists of two waveguides which are, for example, formed by diffusing

titanium on a LiNbO3 substrate [2]. The refractive index of LiNbO3 may be changed

by applying an electric field. There is an electrical contact on each waveguide path

of the modulator for applying an electrical signal, Va or Vb. With no applied signals,

the refractive index and hence the phase shift through each path is the same and

the output signals of each path simply add. If a voltage is applied to the contact on

one arm, the phase shifts will differ and the optical signals in each arm will interfere

when they combine at the output. A phase difference of π between the two arms

occurs at a voltage difference Vπ between Va and Vb. Simply stated, Vπ is the voltage

change that causes the modulator to go from full optical output power to minimum

optical output power. Some Mach-Zehnder modulators have electrical bandwidths in

excess of 40 GHz, making them suitable for amplitude modulation of optical signals


Waveguide

Substrate

Contact

Figure 2.11: Dual arm Mach-Zehnder modulator.

with 10 or 40 Gb/s data streams.

The electric field of the data contained in the output signal of the Mach-Zehnder,

Eout, may be represented in terms of the electric field of the data on the input signal,

Ein, according to

Eout =Ein

2exp

(

jπVa

Vπ

)

+Ein

2exp

(

jπVb

Vπ

)

. (2.35)

The power in the electrical signal is found by taking the square of (2.35). Figure 2.12

shows the output signal power characteristic for a modulator with Vπ=1. In order to

obtain the most linear modulation possible, the modulator is typically biased with

Va − Vb=3Vπ/2 ± k · 2Vπ, k = ...,−2,−1, 0, 1, 2, ... . Even at this bias point, the

amplitude modulation will induce some nonlinearity. The modulator may also be

biased at Va − Vb=Vπ/2 ± k · 2Vπ, k = ...,−2,−1, 0, 1, 2, ..., but then any applied

amplitude modulation is inverted when changed into optical intensity.

When the external bias voltages Va and Vb are applied to the modulator, the

change in the material refractive index may result in there being some amplitude

dependent phase shift or chirp in Eout. However, the total chirp may be eliminated

by having Va = −Vb, and this is usually done in practice [6, 7, 18].

The above discussion describes that a Mach-Zehnder modulator contains two

phase modulation paths whose outputs are combined. The minimum phase COSSB


0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

a

out

E

(

V)

2

(V −V ) x V (V)πb

Figure 2.12: Transfer function of a Mach-Zehnder with Vπ=1.

modulator structure described in Section 2.4.3 consists of an intensity modulator,

usually implemented with a Mach-Zehnder, followed by a phase modulator. It is

possible to perform the amplitude modulation and phase modulation using only one

Mach-Zehnder by superimposing a common mode signal onto both Mach-Zehnder

control voltages [7]. This common mode voltage would be the Hilbert transform of

the logarithm of the information. Although this reduces the amount of hardware

required, the Mach-Zehnder control voltages become prohibitively large, and so this

method was not explored further in this thesis.

2.4.8 Competing Technologies: Solitons, Coherent Detection Systems,

and Duobinary Transmission

Some time has passed since Sieben and Davies conducted their experiments on

COSSB. In that time, significant research has taken place worldwide on different

fiber system architectures. It is important to establish whether COSSB is still rel-

evant. Two areas which are a current topic of fiber systems research are soliton

transmission and coherent detection. Each of these will be discussed in turn.


Solitons

Solitons are pulses that propagate in a nonlinear medium. In the case of fiber sys-

tems, soliton transmission occurs when dispersion is canceled by self-phase modula-

tion (SPM) of the optical pulse. SPM occurs as the result of the nonlinear refractive

index of optical fiber. Hence, soliton propagation relies on the balancing of two

effects which, on their own, are detrimental to fiber systems.

Since solitons rely on dispersion to propagate, their advantage is that they elim-

inate the need for exact dispersion compensation. One disadvantage of soliton com-

munications is that the optical power must return to zero after each pulse, known

as return-to-zero (RZ) transmission, or else the solitons will interact with each other

and they will no longer propagate undistorted [2]. This occupies considerably more

bandwidth than the non-RZ (NRZ) transmission format that is typically used. Fur-

thermore, optical amplifiers must be still be used for long-haul systems, and these

amplifiers introduce considerable noise. This noise results in frequency fluctuations

in the solitons, and this results in timing jitter which limits the transmission distance

in a soliton link [2]. Due to these and other issues, soliton systems operate at lower

data rates than non-soliton systems. For this reason, improvements to non-soliton

systems such as COSSB are still needed.

Coherent Detection Systems

A different set of system architectures that are growing in popularity are coherent

detection systems. As an analogy, consider that many radio systems modulate the

data onto an RF signal using the phase and amplitude of that data. This requires

that a local oscillator be present at the receiver so that the LO may be mixed or

multiplied with the received signal. Coherent optical systems operate the same way.

A laser must be present at the receiver so that it is combined with the received signal


prior to detection. If homodyne detection is used, the laser at the receiver must be

locked in phase to the optical carrier in the received signal using an optical phase-

locked loop (PLL), the design of which is not trivial and which remains a new area

of research. The need for an optical PLL may be eliminated if heterodyne detection

is used. In a heterodyne system the receive laser is at a slightly different frequency

than the main frequency of the received signal, so that the output current of the

photodiode is at an IF. It was also mentioned in Section 2.3.2 that dispersion may

be compensated post detection at an IF frequency, eliminating the need for optical

compensation techniques. Another advantage of coherent detection has to do with

the fact that the received signal in direct detection fiber systems contains a lot of

carrier power compared to the signal power. During coherent detection, the carrier

power is effectively stripped away leaving only the signal. This can increase the signal

to noise (SNR) ratio at the receiver by up to 20 dB compared to self-homodyning

transmission [2]. Furthermore, coherent detection allows for the use of modulation

techniques that use the phase and amplitude of the optical signal, allowing for the

use of more spectrally efficient modulation techniques. This is important in dense

WDM systems.

Research continues on the development of coherent detection optical systems,

particularly on ways of integrating coherent optical receivers. They have yet to be

adopted in commercial systems, partly because of the success of WDM technology

using optical amplifiers [2]. So, although research will continue on coherent detection

systems, direct detection systems, to which the work in this thesis belongs, will be

around for a long time.

2.5 Conclusion 34

Duobinary Transmission

In a typical NRZ system, an optical intensity modulator, such as a Mach-Zehnder,

is biased in its linear range and is used to modulate the optical signal with the data.

The disadvantage of this technique is that the bias voltage used on the intensity

modulator causes a large DC content in the electrical data. This DC offset increases

the quiescent optical power and reduces the linearity, and decreases the SNR ratio.

The SNR may be improved using duobinary transmission, where the electrical signal

swings positive and negative. In order to perform duobinary modulation and main-

tain linearity, the conventional NRZ signal is encoded into two signals, one positive

and one negative. These encoded signals are used to separately modulate two optical

streams, which are then combined and transmitted. Lee et al. recently demonstrated

that duobinary signals are somewhat more tolerant to chromatic dispersion than bi-

nary signals [19]. However, Davies recently patented a technique for performing

single sideband modulation of duobinary signals [20]. A duobinary single sideband

signal, even with modest 15 dB sideband cancellation, would have extremely high

tolerance to dispersion, and a high SNR. Davies’ method still requires an integrated

Hilbert transformer in the case of digital signals, providing still further motivation

for the development of the Hilbert transformer in this thesis. The only disadvantage

of duobinary transmission is its complexity, such as the requirement for a duobinary

encoder.

2.5 Conclusion

The problem of chromatic dispersion was introduced in this chapter along with com-

monly used techniques used to solve it. COSSB transmission was described and

it was shown that COSSB signals are immune to dispersive power fading, and so

2.5 Conclusion 35

the dispersion in these systems may be compensated post detection. The challenges

of how to generate the logarithm of a broadband signal, and how to integrate a

broadband Hilbert transformer for the COSSB system have been posed.

Chapter 3

Analysis and Design of HBT Cherry-Hooper

Amplifiers with Emitter Follower Feedback for

Optical Communications

In this chapter, the design of Cherry-Hooper amplifiers with emitter follower feedback

(CHEF) is described. This amplifier is required in order to design the SiGe HBT

logarithmic amplifier in the next chapter.

3.1 Introduction

The CHEF amplifier, shown in Figure 3.1, is widely used in limiting amplifiers and

decision circuits in fiber-optic receivers [21, 22, 23, 24, 25, 26]. The use of these ampli-

fiers operating at 40 Gb/s in InP technology has recently been demonstrated [27, 28].

The amplifier shown includes resistor R2, an addition suggested by Greshishchev and

Schvan in order to raise the gain [29]. Although this circuit is useful as a high per-

formance broadband amplifier, it can have excessive gain and group delay peaking

for certain choices of the component values. Designing the gain to peak with fre-

quency may give the highest bandwidth, but this results in the group delay peaking

with frequency as well and leads to a distorted eye pattern [24]. For this reason, it

is necessary to strike a balance between gain and delay flatness and bandwidth in

transceiver amplifiers.

Ohhata et al. presented an analysis of the CHEF amplifier for an implemen-

tation using selective-epitaxial SiGe HBTs for which the parasitic capacitances are

3.1 Introduction 37

R1

R2

Rf

Q5

Q3

Q1

Q4

R2

R1

Rf

Q2

Q6

IEE2

IEE1

Vo1,

gm1vin2

gm3(v1-v2)V1,v1

VEE

Vin

vin2

2

-Vin

-vin2

2

V2,v2

vo1

Vo2,vo2

2

Figure 3.1: Cherry-Hooper amplifier with emitter follower feedback.

relatively small and the base resistance is relatively large [24]. In that work, the base

resistance of the small feature size HBTs had a significant effect on the small signal

transfer function of the amplifier. In this chapter, amplifiers that are designed with

SiGe HBTs having base resistances less than 100 Ω are examined [30]. To the au-

thor’s knowledge, the small signal behavior of the CHEF amplifier using such devices

and using R2 has not been characterized in a way that would allow designers to op-

timize group delay and bandwidth. Also in this chapter, equations for the frequency

response, DC transfer characteristic, and output noise of the amplifier are given and

are used to develop design guidelines. Using these guidelines, the amplifier may be

designed as a second order all pole system to have a Bessel transfer function.

This chapter is arranged as follows. In Section 3.2 the large signal performance

of the amplifier is considered. In Section 3.3, a small signal high frequency model of

the amplifier is presented. Suggestions for low noise design are given in Section 3.4.

Section 3.5 uses the equations that are presented to design a 13.7 GHz bandwidth,

19.7 dB gain implementation of the amplifier in a 47 GHz fT SiGe HBT technology.

Measurement results are presented to confirm the theory.

3.2 Large Signal Performance 38

3.2 Large Signal Performance

The large signal performance of the amplifier may be understood by first consider-

ing the DC transfer characteristic. In Figure 3.1, uppercase variables are used to

represent DC voltages and currents. In Appendix A, equations (3.1) and (3.2) are

derived for the amplifier in Figure 3.1. In these equations, βDC is the DC common

emitter current gain of the transistors and VT is the thermal voltage of approximately

26 mV at room temperature. These expressions show that the DC voltage difference

V1 − V2 for a given Vin may be calculated through iteration, and then Vo1 − Vo2 may

be calculated.

V1 − V2∼= R1IEE2 · tanh

(

V2 − V1

2VT

)

+RfIEE1 · tanh(−Vin

2VT

)

+VT · ln(

IEE1

1 + eVin/VT+

IEE2

βDC(1 + e(V1−V2)/VT )

)

(3.1)

−VT · ln(

IEE1

1 + e−Vin/VT+

IEE2

βDC(1 + e(V2−V1)/VT )

)

Vo1 − Vo2∼= (R1 +R2)IEE2 · tanh

(

V2 − V1

2VT

)

(3.2)

A first observation from equations (3.1) and (3.2) is that R2 will only scale the

output voltage, without affecting its basic shape. Hence, increasing R2 is an ef-

fective means of increasing the output voltage swing of the amplifier. However,

equation (3.2) also shows that in order for Vo1 − Vo2 to reach the maximum output

swing of (R1 + R2)IEE2, tanh must reach its full value of ±1. This occurs when

|V1 − V2| >> VT . Having Vo1 − Vo2 reach its approximate full output swing is desir-

able in the presence of large high frequency signals, because the resulting voltage will

then be ±(R1 + R2)IEE2, which is known and well defined. In contrast, the voltage

V1 − V2 will not be well defined, since it depends on the emitter voltages of Q5 and

3.2 Large Signal Performance 39

Q6, which will suffer from amplitude overshoot due to capacitive feed through at the

emitters of Q5 and Q6.

As a further consideration, IEE1 and IEE2 should be large enough to achieve a

high fT for the HBTs in the amplifier. However, there is a range of bias current

for certain SiGe HBT technologies where the current may be changed, for example

by a factor of two, without significantly affecting fT [31]. This gives the designer

the freedom to choose the bias currents at or somewhat lower than that required for

peak fT , and near peak fT may still be achieved.

3.2.1 HBT β

It is instructive to note that if the CHEF amplifier bandwidth reaches nearly one

quarter of fT , then β will be significantly lower at this frequency than its low fre-

quency value. The AC β is frequency dependent, and is given by [32]

β =βo

1 + j ffβ

. (3.3)

The frequency at which β is 3 dB down from βo is called fβ and is given by

fβ∼= 1

2πrπ(Cπ + Cµ). (3.4)

The frequency at which β becomes unity, which is fT , is approximately expressed as

fT∼= gm

2π(Cπ + Cµ). (3.5)

From these relations, it is observed that fT = βofβ. Figure 3.2 shows a rough plot

of β versus frequency and is marked with βo, fβ, and fT .

3.3 Small Signal Analysis 40

fβ T

βo

f

−20 dB/decade

Figure 3.2: Plot of AC β.

3.3 Small Signal Analysis

Figure 3.3(a) shows the differential mode half circuit of the amplifier and Fig-

ure 3.3(b) shows the small signal equivalent with only the dominant parasitics. It

is assumed that the circuit is symmetrical, so that the small signal parameters of

Q1 and Q2, for example, are equal. Capacitance C1 represents the sum of Csub1,

and Cµ1 and Cµ3 reflected to node v1 using Miller’s theorem. The fact that Cπ3

connects to ground through re3 will be neglected and Cπ3 will simply be added to

C1. Specifically,

C1 = Cπ3 + (1 − 1/A1)Cµ1 + Csub1 + Cµ3(1 − A2) (3.6)

where A1 and A2 are the gains across Cµ1 and Cµ3 respectively and are given by

A1∼= −

r′d3(R′

f + rd5)

r′d1(r′

d3 +R1)(3.7)

A2∼= −R1 +R2

r′d3

(3.8)

where rdk = 1/gmk, r′

dk = 1/gmk + rek, gmk is the transconductance of transistor

k, rek is the parasitic emitter resistance of transistor k, and R′

f = Rf + re5. It is

noted that Cπ3 is shown explicitly in Figure 3.3(b) but was added to C1 during for


vin

2

vo1

R1

R2

RfQ3

Q1

Q5

Rs

Vin

2

ZinEF

(a)

R1

C1vin

rd5

vo1

β

rd3β

R2

ZLrd1β

Cπ1

rb1Rs

Rs

re1

vbe5rd5

vbe5

rd3

vbe1

vbe1

Rf’

2rd1

ic5

ic1=

=

rb5

re3

Cπ3

Cπ5

vbe3vbe3

ic3 =

’

v1

(b)

Figure 3.3: CHEF amplifier (a) differential mode half circuit and (b) small signalequivalent including source and load impedances.

the computations in this chapter. The component of Cµ1 reflected to the base of

Q1 through Miller effect has been ignored, since the gain across Q1 is usually small

and so the Miller capacitance will be much less than Cπ1. Also in Figure 3.3(b), the

base-emitter resistance rπk for a transistor k has been rewritten using the identity

rπk = βrdk. These approximations facilitate analysis without introducing significant

error, as will be shown.

The small signal transfer function may be broken into two parts, vo1/vin =

vbe1/vin × vo1/vbe1. The transfer function vbe1/vin is calculated to be

vbe1

vin

∼= rd1

2r′d1 + rd1sCπ1(R′

s + re1). (3.9)

This expression has a single pole resulting from Cπ1 and mainly from R′

s, which is

the resistance of the input signal source plus the base resistance of the input HBT.

In practice, the amplifier is usually driven by an emitter follower, which has a very

low output impedance, making R′

s small. As a result, this pole will be at a frequency

significantly higher than the bandwidth of the overall amplifier.


The stage following the amplifier is usually an emitter follower output buffer, and

the amplifier is loaded by the input impedance of this emitter follower, ZinEF . This

impedance, along with Csub3 and the Miller capacitance of Cµ3 reflected to node Vo1,

form the load impedance ZL, according to

1

ZL

=1

ZinEF

+ sCsub3 + sCµ3(1 − 1/A2). (3.10)

The magnitude of the impedance ZinEF decreases with frequency, and so ZL may be

modeled to first order by a capacitor CL. A more accurate prediction of the amplifier

response may be obtained with a more detailed formulation of ZinEF , as described

in Appendix B.

Nodal analysis was used to find the transfer function vo1/vbe1, which is derived in

Appendix C. With the assumption that ZL∼= 1/sCL, vo1/vbe1 may be approximately

expressed as

vo1

vbe1

∼= 1

rd1r′

d3C1CL

[

s2 +ωo

Qs+ ω2

o

] (3.11)

where Q and ωo are the pole quality factor and pole frequency given by

Q ∼=[

C1CL(R1 +R2)(R′

f + rd5)(R1 + r′d3)

r′d3

C1(R′

f + rd5) + CL(R1 +R2)2

]1/2

(3.12)

ωo∼=

[

R1 + r′d3

r′d3C1CL(R1 +R2)(R′

f + rd5)

]1/2

. (3.13)

Equations (3.12) and (3.13) are valid for R′

f/R1 > 2, which approximately corre-

sponds to the bandwidth where the equations are reasonably accurate. The lat-

ter equation indicates that the pole frequency ωo, which is one indication of am-

plifier bandwidth, is inversely proportional to the square root of C1, CL, and R′

f


if R′

f >> rd5. This indicates that although increasing R′

f raises the amplifier

gain, it decreases the bandwidth. Equation (3.12) may be further simplified if

C1(R′

f + rd5) >> CL(R1 +R2) to

Q ∼=[

CL(R1 +R2)(R1 + r′d3)

r′d3C1(R′

f + rd5)

]1/2

. (3.14)

This shows that increasing R1 raises Q, and increasing R′

f or C1 lowers Q. Further-

more, increasing R2 will result in modest increases in Q, and a comparable decrease

in ωo, two effects which may be expected to leave the bandwidth roughly unchanged.

This is consistent with the observation of Greshishchev and Schvan, who noted that

increasing R2 in the range 0 < R2/R1 < 2.5 had little effect on the bandwidth but

increased the gain significantly [29].

The two poles in (3.11) located at ωo are a complex conjugate pair for Q > 0.5

and they dominate the frequency response of the CHEF amplifier. Figure 3.4 shows

the magnitude and group delay responses for such a system for different values of Q

and for ωo = 1 rad/s. The case where Q = 1/√

2 ∼= 0.707 corresponds to a second

order Butterworth response, where the magnitude response is maximally flat. The

case where Q = 1/√

3 ∼= 0.58 corresponds to a second order Bessel response, where

the group delay is maximally flat [33]. Figure 3.5 shows eye diagrams of 10 Gb/s

signals which have passed through second order systems with Q values of 1/√

3 and

1.0 respectively, and with ωo = 2π · 15 × 109 rad/s. The eye diagram for the system

with Q = 1.0 is distorted due to the gain peaking and group delay distortion of that

system. To avoid this type of distortion, it is always desirable to have a Q factor of

approximately 1/√

3 for broadband optical applications. The disadvantage of such a

design is that it has less bandwidth than a design with a higher Q factor. It should

be noted that when this amplifier is viewed as a lowpass system, a higher Q increases


100

101

0

10

20

30

40

50

60

70

80

90

Frequency (rad/s)

Gro

up D

elay

(s)

100

101

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Frequency (rad/s)

Mag

nitu

de

Q=0.5

Q=0.85

Q=1/ 3

Q=1/ 2

Q=1

Q=0.5

Q=1

Q=0.85

Q=1/ 3

Q=1/ 2

Figure 3.4: Magnitude and group delay responses for a second order system.

−150 −100 −50 0 50 100 150−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (ps)

Am

plitu

de

(a)

−150 −100 −50 0 50 100 150−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (ps)

Am

plitu

de

(b)

Figure 3.5: Eye diagrams for a 10 Gb/s signal filtered with second order systemshaving (a) Q = 1/

√3 and (b) Q = 1.0.

the bandwidth. However, if this amplifier is viewed as a bandpass system, a higher

Q decreases the bandwidth.

3.3.1 Design Example

The method in which the amplifier may be designed to have Q = 1/√

3 can be

illustrated through an example. Bias currents are first chosen based, for instance,

on the desired amount of power dissipation. The value of IEE2 will also partially

determine the output voltage swing. For the purpose of illustration, one amplifier

with two different current bias levels will be considered here. One bias level is


IEE1 = IEE2 = 1.0 mA, and the other bias level is IEE1 = 3.0 mA and IEE2 = 3.2 mA,

chosen based on output swing. Higher currents than these may cause a biasing

problem when a 3.3 V or lower supply voltage is used.

When designing the amplifier, the emitter length LE of the HBTs should be

chosen to provide peak or near peak fT , assuming that such an LE is large enough

to safely handle the desired amount of bias current. The emitter length of Q1 and

Q2 in the first stage of a limiting amplifier chain should be chosen based on noise

considerations as well, as will be shown in Section 3.4. For the present circuit, dual

stripe emitters are used for all transistors, each emitter stripe width is 0.5 µm, and

the emitter length of each stripe is 5.0 µm for Q1, Q2, Q5, and Q6; and 2.5 µm for Q3

and Q4. Once the bias currents and LE of the HBTs are chosen, the only unknowns

in equation (3.11) are R1, R2, R′

f , and ZL. ZL may be calculated by equation (3.10)

with a capacitance chosen to model ZinEF . The choice of the resistors R1, R2,

and R′

f will determine the gain and bandwidth of the amplifier, and the Q factor

of the two dominant poles. R1 may be chosen based on the desired amount of

output swing, and is chosen as 55 Ω for this example. The values of R′

f and R2

may then be varied in order to observe the obtainable performance. Figs. 3.6, 3.7,

and 3.8 show plots of Q, 3 dB bandwidth, and low frequency gain respectively versus

R′

f/R1 for different values of R2. At the lower current bias level, the circuit has

reduced bandwidth because the fT of the HBTs is reduced to 50-75% of the peak

value. These plots were generated using the complete expression for vo1/vbe1 in

Appendix C, equation (C.2), with ZL = 1/sCL and CL = 40 fF. Of this capacitance,

approximately 15 fF represents Cµ3 reflected to node vo1 and Csub3, and 25 fF was

used to model ZinEF .

Using these plots, consider the amplifier when biased with IEE1 = 3.0 mA and

IEE2 = 3.2 mA. From Figure 3.6, it is seen that in order to have Q ∼= 1/√

3,


100

101

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Q

Rf /R

1

R2=2*R1R2=R1R2=R1/2R2=0

1/ 3

LC =40 fF

I =1.0mAEE1

I =1.0mAEE2

I =3.0mA, I =3.2mAEE2EE1

’

Figure 3.6: Plot of Q factor for different values of R1,R2, and R′

f .

100

101

0

5

10

15

20

25

30

35

Rf /R

1

Ban

dwdi

th (

GH

z)

R2=2*R1R2=R1R2=R1/2R2=0

C =40 fFL

I =1.0mAI =1.0mAEE1

EE2

I =3.0mA, I =3.2mAEE2EE1

’

Figure 3.7: Plot of 3 dB bandwidth for different values of R1,R2, and R′

f .

100

101

0

5

10

15

20

25

30

35

Rf /R

1

Gai

n (d

B)

R2=2*R1R2=R1R2=R1/2R2=0

I =3.0mA, I =3.2mA EE2EE1

I =1.0mAI =1.0mA

EE1

EE2

C =40 fFL

’

Figure 3.8: Plot of low frequency gain for different values of R1,R2, and R′

f .

3.4 Amplifier Noise Performance 47

R′

f/R1 should be in the range from 2.3 to 2.7. This shows that the ratio R′

f/R1 is

highly constrained for a fixed set of bias currents. For this example, Rf = 160 Ω

gives Rf/R1 = 2.9. This gives a Q factor slightly lower than 1/√

3 and will allow

for some variation in the load impedance ZL without introducing significant group

delay peaking. From Figs. 3.7 and 3.8, it may be seen that for R′

f/R1∼= 2.9 the

bandwidth ranges from 11.5 GHz to 14.5 GHz and the gain ranges from 14.4 dB to

24.0 dB for different values of R2. Choosing R2 = R1 = 55 Ω gives a reasonable

compromise, leading to a theoretical bandwidth of 12.9 GHz and a gain of 20.5 dB.

It should be noted that IEE1 and IEE2 need not be equal. However, with LE of

Q1 and Q2 chosen to be 5 µm to lower rb, IEE1 must be 3.0 mA to obtain near peak

fT . With IEE2 = 3.2 mA chosen to obtain the desired output swing and peak fT ,

the currents end up being nearly equal.

3.4 Amplifier Noise Performance

The CHEF amplifier often follows the transimpedance amplifier (TIA) in a fiber-

optic receiver. Hence, in addition to having a good frequency response, low noise

operation is desired in the first amplifier following the TIA. In this section, a simple

expression is given for the output noise power spectral density (PSD) of the amplifier.

In the following analysis, the mean square thermal noise voltage due to a given

resistor R will be expressed as e2R, where e2

R = 4kTR∆f and where k is Boltzmann’s

constant, T is temperature in Kelvins, and ∆f is equivalent noise bandwidth. For

a DC collector current IC , the mean square collector shot noise current will be

expressed as i2ck for the kth transistor, where i2ck = 2qICk∆f and where q is the

electron charge.

Using the above notation, an example calculation of the differential output noise

3.4 Amplifier Noise Performance 48

due to i2c1 is given in Appendix D. In the same way, the differential noise PSD

between vo1(t) and vo2(t) due to all noise sources in the amplifier is calculated as

v2o

∆f∼= 2(R1 +R2)

2

(R1 + r′d3)2

[

e2R′

f

∆f+

(R′

f + rd5)2

r′ 2d1

×(

e2rb1,2

∆f+e2re1,2

∆f+ r 2

d1

i2c1,2

∆f

)]

(3.15)

where, for example, e2rb1,2

is the noise of either rb1 or rb2. The input referred differential

noise PSD may be obtained if desired by dividing this equation by the square of the

low frequency differential amplifier gain, given by the product of (3.7) and (3.8).

From equation (3.15), it is seen that if (R′

f + rd5)2/r′ 2

d1 >> 1 and rb1 >> re1, then

the most important amplifier noise sources will be the collector current shot noise and

base resistance thermal noise of Q1 or Q2, i2c1,2 and e2rb1,2respectively. As an example,

consider the amplifier with IEE1 = 3.0 mA, IEE2 = 3.2 mA, R1 = R2 = 55 Ω, and

Rf = 160 Ω. Table 3.1 gives the values predicted by the terms in equation (3.15)

with the values simulated at 1 GHz using 47 GHz fT SiGe HBTs. This table shows

only the noise of the amplifier. The noise equations assume zero source impedance

because the amplifier is usually driven by emitter followers. In Table 3.1, it is clear

that the parasitic base resistance and collector shot noise of Q1,2 are the dominant

noise sources. The collector current shot noise of Q1,2 is difficult to reduce, since

a certain amount of current is required for high fT and gain. However, the base

resistance may be minimized by increasing the emitter length at a fixed emitter

width while keeping the bias current constant. The effective emitter length may

also be increased by using multiple transistors in parallel. This will result in some

decrease in fT , but a greater decrease in noise than in bandwidth. Using multiple

transistors also reduces self heating in these HBTs [29].

3.5 Experimental Results 49

Table 3.1: Differential output noise of the CHEF amplifier at 1 GHz.

Simulated Simulated Calculated

Noise Noise % of Noise % Error

Source (V 2/Hz × 10−17) Total (V 2/Hz × 10−17)

Q1,2 : rb 13.6 65.1% 15.4 13.2%Q1,2 : Ic 3.83 18.3% 4.11 7.3%Q1,2 : re 1.23 5.9% 1.39 13.0%Rf × 2 0.89 4.3% 1.01 13.5%Totals 19.6 93.6% 21.9 11.7%

3.5 Experimental Results

The design described in Section 3.3.1 was fabricated in IBM’s 5HP (Five High Perfor-

mance) SiGe process with 47 GHz fT HBTs and 0.5 µm feature sizes. This technology

was made available to the author through the Canadian Microelectronics Corpora-

tion. The schematic diagram of the amplifier is shown in Figure 3.9. The tail current

sources of the amplifier are provided by a modified Widlar biasing scheme, described

in Appendix E. The input of the circuit is an emitter follower input buffer, with

resistors and a capacitor arranged to maintain an impedance match up to 10 GHz

when the input leads are wire bonded. It was found that if a simple 50 Ω resistor was

used, the parasitic capacitance at the input node of the HBTs and approximately

1 nH of bond wire inductance would unacceptably degrade the input match if the

chip was wire bonded, as is typically required for packaging. By having the approx-

imate 70 Ω input impedance at low frequencies, it was found that the effects of the

300 fF capacitor and the bond wire inductance combine to achieve a better 50 Ω

match above 5 GHz [29]. However, for the measurements presented in this section,

wafer probing of RF signals was used to provide a characterization of the amplifier

that is unobscured by bond wire parasitics. Wafer probing is the technique whereby

microscopic metal probes directly contact the square metal pads on the unpackaged


300fF

300fF

45Ω

34Ω

550Ω

45Ω

34Ω

550Ω

55Ω

55Ω 55Ω

55Ω

160Ω 160Ω

45Ω

34Ω 34Ω

45Ω300fF 300fF

3.0mA 3.2mA

-3.3V

Vchip_in1

Vchip_in2

Vchip_out1

Vchip_out1

3.2mA3.2mA 1.5mA 1.5mA 11.3mA

300fF

Input Buffer Output Buffer

0.5µm10.0µm

x

0.5µm10.0µm

x

0.5µm15.0µm

x

each

0.5µm5.0µm

x

each

0.5µm10.0µm

x 0.5µm10.0µm

x

0.5µm10.0µm

x 0.5µm10.0µm

x

0.5µm8.0µm

x

0.8mA/µm2

0.7mA/µm2

0.7mA/µm2

0.3mA/µm2

0.3mA/µm2

0.3mA/µm2 0.3mA/µm2

0.4mA/µm2

each

0.7mA/µm2

1.0mA

Figure 3.9: Schematic diagram of the CHEF amplifier test circuit.

integrated circuit. Furthermore, the power supply and ground were wire bonded

to the chip. All of the measurements were obtained by wafer probing the amplifier

with ground-signal-ground probes on Cascade Microtech REL 4800 and Summit 11K

probe stations.

The output buffer is a differential pair with an impedance match similar to that

used in the input buffer. The important component values of the amplifier are

shown in the schematic diagram. The 300 fF capacitor in the biasing reference was

restricted in size by the available chip area, or else it would have been increased in

value. Furthermore, it was suggested this capacitor could be connected from the

base of the β helper transistor to VEE in the current mirror reference for improved

stability [34]. The amplifier core draws 10.2 mA from a -3.3 V supply. A micro-

photograph of the integrated circuit is shown in Figure 3.10.

Figure 3.11 shows the theoretical transfer function of the amplifier predicted by

equations (3.9) and (C.2), the simulated transfer function with and without resistive


700 µm

µm700

CHEF Amplifier

Input Buffer

Output Buffer

Figure 3.10: CHEF amplifier IC microphotograph.

and capacitive layout parasitics, and the deembedded measured transfer function.

The measured gain was deembedded from a measurement of S21 taken with a -

20 dBm input signal using a Hewlett Packard (HP) 8510C network analyzer. The

accuracy of the calibrated network analyzer measurement is estimated to be better

than 0.1 dB. For the theoretical transfer function calculation, the load impedance was

modeled using equations (3.10) and (B.1). All of the simulated curves in this section

were generated using Cadence Spectre software. The error between the theoretical

and deembedded measurement is less than 1.2 dB between 0-5 GHz, and is less than

0.8 dB between 5-15 GHz, which is sufficiently small for design purposes. In the

layout, multiple resistors in parallel were used to implement R1 and R2, leading

to increased metalization area connecting to these resistors. The difference in the

simulated response with parasitics is partially attributed to the capacitance of this

metalization to the substrate and to surrounding nodes.


108

109

1010

1011

5

10

15

20

25

Frequency (Hz)

AC

Gai

n (d

B)

TheoreticalDeembedded MeasurementSimulation − Without Layout ParasiticsSimulation − With Layout Parasitics

Figure 3.11: Comparison of theoretical gain based on equations (3.9) and (C.2),simulated gain, and measured gain.

2 4 6 8 10 12 140

10

20

30

40

50

60

70

80

90

100

Frequency (GHz)

Gro

up D

elay

(ps

)

TheoreticalDeembedded MeasurementSimulation − Without Layout ParasiticsSimulation − With Layout Parasitics

Figure 3.12: Comparison of theoretical group delay based on equa-tions (3.9) and (C.2) , as well as simulated and measured group delay.

Figure 3.12 shows a comparison of the theoretical group delay, the simulated

group delay with and without resistive and capacitive layout parasitics, and the

measured group delay. Note that the measured phase response was closely fit to a

seventh order polynomial and the group delay was obtained using the derivative of

this polynomial curve. It is estimated that the accuracy of the group delay obtained

with this technique is better than ±5 ps. Since the phase response is roughly a

straight line up to the frequencies of interest, using a polynomial results in negligible


error in representing the phase response, but removes the 50 ps of noise which would

be present if the raw measured phase response were used to obtain group delay.

This is true even if averaging is turned on in the network analyzer. The group delay

distortion up to 10 GHz is approximately ±10 ps theoretical, ±6 ps simulated without

layout parasitics, ±12 ps simulated with layout parasitics, and ±10 ps measured.

In this case, the theory provides a reasonable prediction of group delay for design

purposes.

The noise figure (NF) was measured at one output terminal with the signal

applied to one input terminal, and the other input and output terminals were

terminated in 50 Ω. The noise figure was measured using a HP8970B noise fig-

ure meter with a HP346B noise source. The measured NF is 14.7 dB at 1 GHz.

The equivalent deembedded differential noise at the output of the amplifier core is

2.36 × 10−16 V 2/Hz. Comparing this with the theoretical output noise from Ta-

ble 3.1, it is observed that equation (3.15) predicts the measured noise with −8%

error.

Figure 3.13 shows measured eye diagrams for a data rate of 10 GB/s. A PRBS

signal with a pattern length of 231 − 1 was used for all measurements. The PRBS

signal was generated using the Anritsu MP1763B pattern generator and the eye

diagram was measured on a HP 54750A oscilloscope with a 54754A 18 GHz sampling

module. Figure 3.13(a) shows the signal at the output of the pattern generator, and

Figs. 3.13(b),(c), and (d) show the single ended amplifier output eye for a differential

input signal with amplitudes of 7 mVpp, 20 mVpp, and 400 mVpp respectively. The

data eye has good opening and low overshoot for all cases. In order to have a more

square eye, which is desirable, the required bandwidth is approximately 1.5 times

the clock rate, or 15 GHz for 10 Gb/s. Although the bandwidth of the amplifier is

13.7 GHz, the inclusion of the input and output buffers reduces the bandwidth to

3.6 Conclusion 54

20 mV

100 ps

(a)

100 ps

30 mV

(b)

80 mV

100 ps

(c)

100 ps

220 mV

(d)

Figure 3.13: Measured eye diagrams at 10 Gb/s: (a) Through measurement at20 mVpp and single-ended CHEF amplifier output for differential input signals ofamplitude (b) 7 mVpp, (c) 20 mVpp, and (d) 400 mVpp.

11.2 GHz. A bandwidth of 15 GHz could be achieved in this technology by sacrificing

some core amplifier gain, and by reducing the gain of the output buffer.

3.6 Conclusion

When used for optical transceiver applications, the CHEF amplifier must have high

gain and bandwidth, low noise, and high output swing. Design techniques for achiev-

ing all of these goals simultaneously were given in this chapter. A pair of complex

poles was found to dominate the amplifier frequency response. The relative values

of resistors which result in optimum pole quality factor were described. Further-

more, the dominant noise sources in the amplifier were identified. A 19.7 dB gain,

13.7 GHz bandwidth implementation of the circuit verified the small and large signal

behavior. The CHEF amplifier from this chapter is used in the DC-6 GHz SiGe HBT

logarithmic amplifier in the next chapter.

Chapter 4

A Novel Parallel Summation Logarithmic

Amplifier

4.1 Introduction

In Section 2.4.3, it was shown how the implementation of a COSSB modulator re-

quires a logarithmic conversion of data. In this chapter, logarithmic amplifiers suit-

able for the OSSB application are described. The parallel summation logarithmic

amplifier is described in Section 4.3 in a way that allows logarithmic converters with

or without gain to be designed.

Logarithmic amplifiers which are suitable for use in fiber optic applications should

meet a unique set of requirements. In the COSSB application, the input signal is

a baseband signal which is strictly positive, and so it has DC and low frequency

components. Hence, the logarithmic amplifier should be DC-coupled throughout. As

well, maximizing the bandwidth of the logarithmic amplifier is critical, because of

the high data rates. However, low group delay distortion is important for broadband

operation. This chapter presents logarithmic amplifiers that feature low group delay

distortion.

In Section 4.2, the various types of logarithmic amplifiers are considered. A band-

width limitation of the series linear-limit logarithmic amplifier topology is considered,

and parallel summation logarithmic amplifiers, which overcome this limitation, are

described. In Section 4.3, a unified design procedure for parallel summation logarith-

mic amplifiers is given. In Section 4.4, the design and operation of a novel DC-4 GHz

4.2 Distinction and Comparison of Logarithmic Amplifiers 56

Si bipolar logarithmic amplifier are described. In Section 4.5, a DC-6 GHz SiGe HBT

logarithmic amplifier that uses the CHEF amplifier from the last chapter is described.

4.2 Distinction and Comparison of Logarithmic Amplifiers

There are two basic types of logarithmic amplifiers, the true and the demodulating

types. True logarithmic amplifiers, also known as ‘baseband’ or ‘video’ logarithmic

amplifiers, provide the logarithm of the signal without detecting or demodulating

the signal. In contrast, demodulating logarithmic amplifiers provide the logarithm

of the envelope of a signal. Exceptions are those circuits that may operate in the true

and demodulating modes through the use of Gilbert cell multipliers [35]. Demodu-

lating logarithmic amplifiers are widely used in radar and radio receivers as received

strength signal indicators (RSSI). However, for the COSSB application, demodula-

tion is not wanted, and implementations of true logarithmic amplifiers are described.

A list of prior publications describing demodulating logarithmic amplifiers that are

not related to this thesis was compiled by the author, and is listed in [36].

Logarithmic amplifiers may be further subdivided into single stage and piece-

wise approximate types. Transconductance feedback log converters are based on

an amplifier with a PN junction or a MOSFET device in subthreshold in feedback

around the amplifier [37]. These converters provide an excellent logarithmic response

in low frequency applications. A technique that has been more successful at high

frequencies is the piecewise approximation of a logarithm and this is the technique

considered in this work.


1

1x

2 k N

Vin Vout

1x1x1x

GgainVin

Vout

VL

Σ Σ Σ Σ

Figure 4.1: Series linear-limit logarithmic amplifier.

4.2.1 The Series Linear-Limit Logarithmic Amplifier

Figure 4.1 shows the most widely used high frequency, true logarithmic amplifier

topology. The amplifier consists of a cascade of dual gain cells, with each cell having

a high gain, limiting amplifier in parallel with a unity gain buffer. For small signals,

this structure will simply amplify. However, as the signal becomes larger a point

will be reached at which the limiting amplifier in the last stage ceases to amplify

and provides a constant voltage VL. As the input signal becomes larger, the limiting

amplifiers will successively reach their upper bound, starting with the second last

stage and progressing toward the input. Meanwhile, the buffer amplifiers in all stages

will continue to pass the signal. This response, shown in Figure 4.2, approximates a

straight line when plotted on a semilogarithmic axis. A mathematical description of

this amplifier’s operation is given in [38].

The series linear-limit topology is attractive because process variations, such as a

low current gain for the transistors on a given wafer, will likely affect all stages more

or less equally. If the gain of all of the stages is lower than expected, the only result

is a scaling of the overall response, without affecting the logarithmic characteristic.

The linear-limit topology is also simple, since more stages may be added in cascade

if increased dynamic range is required, provided that the low gain path in each stage


10−5

10−4

10−3

10−2

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Ou

tpu

t V

olta

ge

(V

)

Input Voltage (V)

Ideal Logarithm Linear−Limit Log Amp

0−1−2−310 10 1010

Out

put V

olta

ge (

mV

)

Input Voltage (V)

log error

300

100

0

200

−20−40−60 0Input Voltage (dBv)

Figure 4.2: Linear-limit logarithmic amplifier response.

does not saturate.

The requirement that the low gain buffer amplifiers not saturate presents a chal-

lenge in designing the twin-gain stages. A common implementation consists of two

differential pairs in parallel with shared collector nodes, as shown in Figure 4.3

[38, 39, 40]. One differential pair uses emitter degeneration to provide low gain and

the other differential pair is undegenerated to provide high gain.

Since the buffer amplifier in the N th stage must buffer the signal generated by

stages 1 to N − 1, the requirement on Ilow with respect to Igain may be found to

be [38]

Ilow

Igain> N. (4.1)

However, Igain needs to be somewhat large in order to achieve the desired gain.

Therefore, (4.1) indicates that Ilow will be large, necessitating large devices with high

parasitic capacitance. These capacitances, in turn, will place bandwidth limitations


Ilow

Igain

in1V in2V

out2Vout1VRcRc

ReRe

Figure 4.3: High gain limiter and unity gain buffer in parallel.

on the series linear-limit logarithmic amplifier. One way to mitigate this difficulty is

to lower the gain of the buffer amplifiers below unity so that they require a higher

input voltage in order to limit, and so their current may be reduced. The gain may

be lowered using increased emitter degeneration or a series resistor connected to the

base of the buffering transistors [39].

Even if the size of the transistors in the buffer amplifier is reduced, the parasitic

capacitance of the buffer amplifier will still load the gain amplifier in parallel with

it. There is another class of log amplifiers where the high and low gain paths are

separate. These are known as parallel summation log amplifiers, which are described

in the next section.

4.2.2 Parallel Summation Logarithmic Amplifiers

The second class of log amplifiers is the parallel summation type. Parallel summation

log amplifiers may be divided into the progressive compression type and the parallel

amplification type. Each of these will be discussed in turn.

Figure 4.4 shows a progressive compression log amplifier [41]. Functionally it is

exactly the same as the series linear-limit amplifier, except that instead of sequen-


Σ Σ ΣΣ

gm

gm1 gm gm gm gm

gmgm1

gm

outI

Vin

p1G p2G pkG pNGp3G

A1G A2G Ak-1G AN-1G

Vin

Iout

IL

Vin

Iout

IL1

Figure 4.4: Progressive compression, parallel summation logarithmic amplifier.

tially summing and buffering the clipped outputs of each stage, the stage outputs

are summed in parallel. In the progressive compression amplifier, each component

amplifier output voltage is converted to a current using a transconductance element.

The transconductance elements provide current up to a maximum level, at which

point their output current limits. Another type of parallel summation log amplifier

is shown in Figure 4.5, where it is seen that the component signals are generated in

parallel. This topology exhibits high symmetry among the different paths. Hence, al-

though the logarithmic dynamic range is lower, the phase and group delay matching

are inherently improved compared to the progressive compression topology.

Comparing the two parallel summation topologies, it is seen that the progressive

compression structure has the advantage of using multiple cascaded amplifiers to

achieve high logarithmic dynamic range. As a result, the progressive compression

structure is widely used in log amplifier designs, for example in [35] and [42]. The

parallel amplification scheme is efficient in low dynamic range applications, and has

been used in works such as [43] and [44]. However a mathematical design procedure

has yet to be presented for parallel summation logarithmic amplifiers. Such a design

procedure is given in the next section, with proof of a logarithmic result. Some

4.3 Design Procedure 61

IoutΣ Σ Σ

gm gm gmgm1

Vin

p1G p2G pkG pNG

Figure 4.5: Parallel amplification, parallel summation logarithmic amplifier.

unique and efficient delay matching structures are then given for these circuits to

allow for ultra broadband operation.

4.3 Design Procedure

A mathematical model of parallel summation logarithmic amplifiers is now presented

which may be used for design purposes. It will be proven that this method yields an

exact logarithmic relationship between the output current and the input voltage of

the amplifier at the breakpoints of the approximation.

Considering the parallel summation logarithmic amplifiers in Figs. 4.4 and 4.5,

the desired transfer function is shown in Figure 4.6. The constant A is defined

as the factor increase in the input voltage between the cusps of the logarithmic

approximation. The current IS is defined as the step in output current between the

cusps of the approximation. The dynamic range of the logarithmic amplifier will be

defined as AN , so for a dynamic range DR the constant A is chosen as DR1N .

The gains through the kth paths in Figs. 4.4 and 4.5 are labeled as Gpk, where

k = 1, 2, ...N . This is because the same signals are being generated in two different

ways, using series and parallel amplifiers. For either topology there are N discrete

states corresponding to the cases where N, N-1, ... 1 paths are contributing linearly


Vin

C

TotalCircuitGains

GN

GN-1

Iout

G1

Gk

A VminN-1

A VminNAVmin

A Vmin2Vmin

SI +C

S2I +C

(N-k)I +CS

S(N-1)I +C

NI +CS

A VminN-k

GN-2

Figure 4.6: Parallel summation logarithmic amplifier transfer function.

to the output current. A path ceases to contribute linearly once its output current

limits at IL. As the input voltage increases, the logarithmic structure passes through

the N states where gain decreases and follows the series

GN = Gp1 +Gp2 + ...GpN

GN−1 = Gp1 +Gp2 + ...Gp(N−1)

...

Gk = Gp1 +Gp2 + ...+Gpk

...

G1 = Gp1. (4.2)

These state gains are labeled on the right side of Figure 4.6 at the signal levels at


which they occur.

In any given state, the output current of the overall amplifier consists of two

parts; the current that is proportional to the input voltage, and the fixed current

supplied by the gain paths which have already limited. The output currents shown

on the y-axis of Figure 4.6 may be expressed as a sum of these two components, as

in

C = VminGN

IS + C = AVminGN−1 + IL

2IS + C = A2VminGN−2 + 2IL

...

(N − k)IS + C = AN−kVminGk + (N − k)IL

...

(N − 2)IS + C = AN−2VminG2 + (N − 2)IL

(N − 1)IS + C = AN−1VminG1 + (N − 1)IL

NIS + C = VminGN +NIL. (4.3)

A straightforward solution to these equations may be found by allowing IS = IL.

As well, since G1 = Gp1 from (4.2), the overall amplifier gains, Gk, may be found

from (4.3) by assuming that the gain of the lowest gain path Gp1 is simply gm, the

gain of the first transconductance element. This yields the logarithmic amplifier

gains


G1 = gm

G2 = gmA

...

Gk = gmAk−1

...

GN−1 = gmAN−2

GN = gmAN−1. (4.4)

Using this knowledge of the gains in each state of the overall parallel summation

amplifier, the gains of the component amplifiers in both the progressive compression

and parallel amplification structures may be derived.

Solving (4.2) and (4.4) yields the gains of the paths through the parallel summa-

tion amplifiers

Gp1 = gm

Gp2 = gm(A− 1)

Gp3 = gmA(A− 1)

...

Gpk = gmAk−2(A− 1)

...

GpN = gmAN−2(A− 1). (4.5)

The path gains in equation (4.5) correspond directly to the amplifier gains in the

parallel amplification, parallel summation topology in Figure 4.5, multiplied by the


gm of the transconductance elements. Note that these gains may be increased or

decreased by any factor as long as their respective ratios stay the same.

Applying equation (4.4) to the progressive compression topology, the gain of the

first path is just gm from the first transconductance element. The amplifier gains in

Figure 4.4 are

GA1 = A− 1

GA2 = A

...

GAk−1 = A

...

GAN−1 = A. (4.6)

In practice, it is more convenient in the progressive compression structure to make

the gain of amplifier GA1 equal to A so that all of the amplifiers are the same. This

may be done provided that the first transconductance element is also scaled from

gain gm to gmA/(A− 1).

Although a logarithmic amplifier with loss in some paths could be constructed,

the limiting transconductance elements would still be required. The input signal of

the amplifier would have to be large enough to cause the transconductance elements

to limit even after the signal is attenuated. It is impractical to use attenuation larger

than 20 dB in the paths, because the input signal voltage would have to be more

than ten times IL/gm. Hence, in a logarithmic converter with 40 dB of dynamic

range or more, some paths must use gain.

Having chosen the gains, it may be shown that the breakpoints are logarithmically

related to the input voltage. The proof bears some resemblance to the description


of the linear-limit amplifier given in [38]. Assuming that the kth path in Figure 4.4

or 4.5 is just on the point of limiting, then the input is

Vink= Vin =

ILGpk

. (4.7)

However, Gpk is known from (4.5) to be Gpk = gmAk−2(A− 1), so that

Vin =IL

gmAk−2(A− 1)k ≥ 2. (4.8)

Additionally, if the kth path is limiting, then there are N −k paths with higher gains

which are also limiting, and k − 1 more paths which are still amplifying linearly.

Thus, the output current is

Iout = (N − k)IL + [Gp1 +Gp2...+Gpk]Vin. (4.9)

Using (4.2) and then (4.4),

Gk = Gp1 +Gp2 + ... +Gpk

= gmAk−1. (4.10)

Substituting (4.8) and (4.10) into (4.9) yields

Iout = (N − k)IL +AILA− 1

. (4.11)

Additionally, (4.8) is rewritten as

k = logA

[

A2ILgmVin(A− 1)

]

. (4.12)

Finally, substituting (4.12) into (4.11) gives


Iout = IL

[

N +A

A− 1+ logA

[

gmVin(A− 1)

A2IL

]]

(4.13)

which is the desired logarithmic relationship between Iout and Vin.

A check of (4.13) may be made by substituting values for Vin and verifying that

the output current behaves as shown in Figure 4.6. Consider the case where the kth

amplifier path is just on the point of limiting. The input voltage for this case is given

in equation (4.8). Substituting this into (4.13) for Vin and simplifying gives

Iout = IL

[

N − k +A

A− 1

]

. (4.14)

For different values of k, this equation evaluates to

k = N Io = IL

[

A

A− 1

]

= IL

[

1 +1

A− 1

]

k = N − 1 Io = IL

[

1 +A

A− 1

]

= IL

[

2 +1

A− 1

]

...

k = 2 Io = IL

[

N − 1 +1

A− 1

]

(4.15)

which confirms that as each gain path limits, the output current increases by a fixed

step as shown in Figure 4.6. The constant C in Figure 4.6 is identified from (4.15)

to be IL/(A− 1). As well, Vmin from Figure 4.6 is identified from equation (4.3) as

Vmin =C

GN=

IL(A− 1)gmAN−1

. (4.16)

There is one final consideration regarding the case of k = 1 not considered in (4.8),

which is the case where the lowest gain path limits. The highest output current


considered in (4.15) is the case where the second lowest gain path, whose gain is

Gp2 = gm(A−1), limits and provides a current of IL. The input voltage at this point

is

Vin =ILGp2

=IL

gm(A− 1). (4.17)

At this input voltage, the current provided by the lowest gain path is

I = gmVin =IL

A− 1. (4.18)

This point occurs at the total system output current of (N − 1)IS +C in Figure 4.6,

and in order for the logarithmic slope of the output to continue, the lowest gain path

must provide another IS of current before it limits. However, IS = IL, and so adding

this to (4.18) yields

IL1 =IL

A− 1+ IL =

A

A− 1IL, (4.19)

which represents the value of the limiting current required in the lowest gain path.

Thus, the lowest gain path provides a maximum current that is A/(A − 1) times

higher than the other paths.

4.3.1 Logarithmic Slope and Intercept

The response of logarithmic amplifiers may also be characterized in terms of the

logarithmic slope and intercept of the transfer characteristic, as in the equation

Iout = Islope 20 log10

(

Vin

Vintercept

)

. (4.20)

The amplifier transfer function in equation (4.13) may be used to calculate these

parameters for the given model. Solving (4.13) for the intercept yields


Vintercept =A2−N−A/(A−1)ILgm(A− 1)

. (4.21)

The logarithmic slope may be found from (4.13) to be

Islope =dIout

d [20 log10(Vin)]=

IL20 log10(A)

Amperes/dBv. (4.22)

Hence, the logarithmic slope is directly proportional to IL and is inversely propor-

tional to the logarithm of A.

4.3.2 The Delay Matched Progressive Compression Amplifier

Having described a model of parallel summation amplifiers, it is worth considering

a disadvantage of the progressive compression amplifier at high frequencies. In the

progressive compression circuit, the phase delays of each signal undergoing parallel

summation will be different. This is because the first component signal in Figure 4.4

does not pass through any amplifiers and has zero phase shift, and this signal must

be added to the Nth parallel signal, which will have significantly higher phase shift

from having propagated through N amplifiers. A method proposed by the author of

extending the bandwidth is shown in Figure 4.7, where the group delay and phase

shift of each gain path are matched using delay amplifiers. It has been recognized

that the signals in the lowest gain paths may share delay amplifiers after they have

been limited and summed. Any number of paths may be combined and delayed using

this method, provided that the output voltage swing requirement of the shared delay

amplifiers does not lower their bandwidth below that of the highest gain path.

4.4 Implementation 1: A DC-4 GHz Si BJT Logarithmic Amplifier 70

Σ Σ

∆

∆∆

Σ

gm1 gm gmgm

∆Delay

Amplifiers

outI

Vin

gm

A1G A2G A3G

Figure 4.7: An example of a three stage delay matched progressive compression logamplifier.

4.4 Implementation 1: A DC-4 GHz Si BJT Logarithmic

Amplifier

4.4.1 Design of Implementation 1

In the present design a novel hybrid series/parallel topology, as shown in Figure 4.8,

was chosen for implementation [36]. A logarithmic voltage dynamic range of 50 dB

was targeted, corresponding to a factor A of 32014 = 4.23 in a four branch amplifier.

The scaled gains of the four paths are thus 1.00, 3.23, 13.7, and 57.8. However

in practice, some logarithmic dynamic range will be lost when DC-coupling is used

because the DC offsets in the component amplifiers will be amplified and will reduce

the available signal swing.

The cascode long-tail differential pair circuit, which is shown in Figure 4.9, was

chosen for both the gain and delay amplifiers. Using the CHEF amplifier as a building

block in the logarithmic amplifier would have resulted in more bandwidth. However,

the first logarithmic amplifier was designed long before the CHEF amplifier was fully

understood by the author and before the integrated circuit in the previous chapter


I log1

Vin1 Vin2

I log2

ΣΣ

ΣΣ

ΣΣ

∆

∆

gm1 gm gm gm

p1G p2G p4Gp3G

Figure 4.8: Parallel summation logarithmic amplifier implementation.

was designed. The second logarithmic amplifier implementation will be described in

Section 4.5, and it makes use of CHEF amplifiers.

The voltage outputs of the four amplification paths were each converted to cur-

rents and summed using the amplifier shown in Figure 4.10. This summing/limiting

circuit consists of four differential pairs in parallel. The circuit will sum an input

signal up to the point where that signal is large enough to steer all of the current in

one of the differential pairs to one side, at which point it limits that input’s contri-

bution. The bias current in each amplifier in Figure 4.10 is chosen as IL, except the

current in the lowest gain path, which is chosen as IL1 = ILA/(A − 1). This also

increases the gain of that differential pair, however, the gain of the delay amplifiers

in the lowest gain path may be lowered to compensate.

In addition to summing and limiting, the summing amplifier has a collector

impedance of 50 Ω, so as to allow for direct DC coupling of the output to a 50 Ω

system. The inputs of the chip are matched to 50 Ω using the circuit shown in

Figure 4.11.

For the first implementation, the best available technology was a silicon bipolar

process with fT values of 35 GHz and a 0.35 µm minimum feature size. This fabri-


Vee

+

-Vin Re Re

Rc Rc

+

-Vout

Q1 Q2

Q3 Q4

Figure 4.9: Amplifier used as a gain or delay cell.

in3

out1 out2

in3in4 in4 in1in2in2in1

5050

Vee

IL

IL

IL

IL1

Figure 4.10: Summing/limiting amplifier.

50

in1

out1in2

out2

50

Vee

Vee

Figure 4.11: Input matching circuit.


Output

Summer1 Gain

Stages

st

2 Gain

Stages

nd

Input

Buffer

2 mm

2 mm

Figure 4.12: Microphotograph of the Si logarithmic amplifier integrated circuit.

cation process was made available to the author through TRLabs’ association with

Nortel. Simulations consisted of transient, AC, and s-parameter analysis using the

Cadence Spectre and Avanti HSPICE simulators within the Cadence design software,

made available to the author by the Canadian Microelectronics Corporation. A die

microphotograph of the 2×2 mm2 parallel logarithmic amplifier chip is shown in

Figure 4.12. It has a single supply voltage of -5 Volts and draws 150 mA of current.

A negative supply voltage was used so that the circuit’s input and output common

mode voltage would be close to zero.

4.4.2 Measurements of Implementation 1

The small signal gain and reflection coefficients of the chip, obtained using using

an HP 8510C network analyzer, are shown in Figure 4.13. Similar to in the last


100

101

−30

−20

−10

0

10

20

30

40

dB

f (GHz)

S11S22S21

27

4 GHz

Figure 4.13: Measured return loss and gain.

chapter, measurements or RF signals were obtained using wafer probing, and the

power supply and ground connections were wire bonded. The small signal gain is

30 dB with a 3 dB bandwidth of 4.0 GHz. The circuit is impedance matched at its

input and output terminals below -10 dB for S11 and S22 up to 5 GHz.

Variations in the group delay versus frequency are also of prime importance in the

COSSB application. Figure 4.14 shows the measured small signal group delay of the

amplifier versus frequency. The maximum deviation of the group delay within the

4 GHz passband is 35 ps, which represents approximately one tenth of the period of a

2.5 Gb/s signal. The low group delay distortion is a direct consequence of the parallel

summation topology used, which achieves low delay by using only two amplifiers in

series to generate a four segment logarithmic response.

The measured one-tone response of the chip is shown in Figure 4.15 (a) for fre-

quencies from DC to 4 GHz. To make this measurement, a microwave signal source

was ramped in power, and a HP8563A spectrum analyzer was used to measure the


0 1 2 3 4 5150

160

170

180

190

200

210

220

dela

y (S

21)

(pic

osec

onds

)

f (GHz)

Group Delay

Figure 4.14: Measured group delay response.

logarithmic amplifier output signal. The loss in the measurement system has been

calibrated out of the measurements that are shown here. The single-ended loga-

rithmic slope is approximately 1.2 mV/dB and the logarithmic intercept is 167 µV.

Regarding the effect of process variations on the logarithmic response, Figure 4.15 (b)

shows the measured amplitude response of eight die samples at 1 GHz. The loga-

rithmic slope is quite constant among the samples, indicating that the logarithmic

response of the chosen topology may be designed to be robust. As a metric of

logarithmic linearity, the logarithmic error was calculated for the responses in Fig-

ure 4.15 (a) at each frequency and also for a broadband logarithmic fit. The definition

of the log error is shown graphically in Figure 4.2, where the definition of logarithmic

error indicated in [37] and [45] has been used. This log error was computed at each

1 GHz interval, and is plotted in Figure 4.16 (a). It is seen that the log conformity

is ± 2dB over the 4 GHz interval. The broadband log error of the data fit to a single

logarithmic line for a DC-3 GHz bandwidth is shown in Figure 4.16 (b), where the


error is kept to a maximum of ±5dB.

Figure 4.17 shows the normalized measured and ideal frequency domain spectrum

of the logarithmic amplifier output to a 1.8 GHz, -10 dBm signal with a 90 mV DC

offset added in order to make the input signal greater than zero. The measured

waveform was obtained with the HP 54750A oscilloscope with a HP 54751A 20 GHz

sampling module, and was Fourier transformed in Matlab. The amplitude of the

first three measured harmonics matches the ideal response well. The errors at the

fundamental and first two harmonics are 0.5, 0.1, and 2.5 dB respectively.

A real time oscilloscope plot of the log amplifier’s response to a 40 dB range of

input power at 1.8 GHz is shown in Figure 4.18. The HP 54750A oscilloscope with

a HP 54751A 20 GHz sampling module was also used to measure this signal. The

rise and fall times of this circuit are 100 ps each.

In Figure 4.18, it is apparent that the circuit contributes substantial noise for

small signals, with the amount of noise decreasing as the higher gain amplification

paths begin to limit. The noise figure of the amplifier was measured to be 20 dB

using a HP 8970B noise figure meter with a HP 346B noise source. This corresponds

to a maximum input referred noise density of 9 nV/√Hz, which is undesirably high.

Included in this version of the design was a unity gain buffer at the input of the chip,

which was removed in the second implementation described in Section 4.5. Another

significant noise component in the logarithmic amplifier is the thermal noise arising

from the parasitic base resistance of the transistors in the input matching circuit, and

from Q1 and Q2 in the first amplifier (see Figure 4.9) in the highest gain path only.

In the implementation to be presented in the next section, this noise was reduced by

using transistors with larger emitter areas.

In the real time oscilloscope plot of Figure 4.18 the log signal is also observed

to peak or overshoot in response to the input sinusoid. In designing this chip, the


10−3

10−2

10−1

0

0.01

0.02

0.03

0.04

0.05

0.06DC1 GHz2 GHz3 GHz4 GHz

10 10 10−3 −2 −1

0

10

20

30

40

50

60

Input Voltage (V)

|Out

put V

olta

ge| (

mV

)

Input Voltage (dBv)40 20− −−60

(a) Measured one tone response at different frequen-cies.

10−3

10−2

10−1

0

10

20

30

40

50

60

70

10−3 10−2 10−1

|Out

put V

olta

ge| (

mV

)

10

20

30

40

50

60

70

Input Voltage (V)

0

Input Voltage (dBv)60− 40− 20−

(b) Measured one tone response at 1 GHz of eight diesamples.

Figure 4.15: Measured logarithmic responses, peak voltages shown.


10−3

10−2

10−1

−5

−4

−3

−2

−1

0

1

2

3

4

5DC1 GHz2 GHz3 GHz4 GHz

− −−

10 10 10−3 −2 −1

1

−1

−5

−4

0

5

4

3

2

−3

−2

Log

Err

or (

dB)

Input Voltage (V)

Input Voltage (dBv)

60 40 20

(a) Separate fit error.

10−3

10−2

10−1

−8

−6

−4

−2

0

2

4

6

8DC1 GHz2 GHz3 GHz

10 10 10−3 −2 −1

−8

4

−4

2

−2

−6

6

8

0

− −−

Input Voltage (V)

Log

Err

or (

dB)

Input Voltage (dBv)60 40 20

(b) Broadband fit error.

Figure 4.16: Logarithmic error for separate and broadband line fits.


−1 0 1 2 3 4 5 6−60

−50

−40

−30

−20

−10

0

f (GHz)

dB

Ideal Logarithm of InputMeasured

Figure 4.17: Measured and ideal logarithmic amplifier output spectrum for a 1.8 GHzinput tone.

−45 dBm

−32 dBm

−19 dBm

−5 dBm

Input Power

Figure 4.18: Real time oscilloscope plot of single ended output voltage.

delay through the two lower gain paths was set using delay amplifiers of the topology

shown in Figure 4.9. However, it was later determined that the delay of the buffer

amplifiers could be increased to better match the high gain path. The resistive loads

Rc interact with the parasitic capacitance of transistors Q3 and Q4 in the amplifier

in Figure 4.9 and form the dominant pole that limits the amplifier’s bandwidth.

4.5 Implementation 2: A DC-6 GHz SiGe HBT Logarithmic Amplifier 80

The phase response of the delay amplifiers is approximately −45 degrees near the

pole frequency. The lower the dominant pole frequency, the more phase shift the

amplifier contributes at lower frequencies. The derivative of the phase with respect

to frequency defines the group delay, and so a lower 3 dB bandwidth corresponds to a

higher group delay. Hence, amplifiers with increased group delay may be constructed

by using differential amplifiers with sufficient capacitive loading to increase the delay

as needed. The error in the phase shift through each gain path may be as high

as approximately 20 at the log amplifier’s highest frequency of operation without

introducing significant distortion. Figs. 4.19 (a) and (b) show the simulated response

of the amplifier with and without capacitive loading respectively. It is seen that by

including enough capacitance in the load of the delay amplifiers, the response appears

more like a compressed sinusoid as desired. Care must be taken, however, not to

lower the bandwidth of the delay amplifiers below that of the highest gain path.

In the next section, an improved SiGe HBT implementation is described.

4.5 Implementation 2: A DC-6 GHz SiGe HBT Logarithmic

Amplifier

4.5.1 Design of Implementation 2

For the second implementation of the logarithmic amplifier, IBM’s 5HP SiGe process

was used, similar to Chapter 3. Using this technology, a logarithmic amplifier was

fabricated which has 50% higher bandwidth, 5.5× higher logarithmic slope, 10 dB

higher logarithmic dynamic range, significantly lower noise figure, half of the chip

area, and consumes 43% less power than the amplifier in the last section.

Figure 4.20 shows a block diagram of the second logarithmic amplifier implemen-

tation. This architecture is the same as for the first implementation. The logarithmic


(a) Without capacitance.

(b) With capacitance.

Figure 4.19: Simulated log amplifier differential responses to sinusoidal inputs withand without capacitive delay tuning.

amplifier was again designed using DC coupled amplifier stages, as required by the

optical application. However, in order to improve the dynamic range, it is critical

that some form of DC offset cancellation be used. In the circuit in Figure 4.20, only

the DC offset errors in the highest gain path Gp4 are large enough to cause significant

performance degradation. For this reason, an amplifier and a low pass filter network

were used in negative feedback around path Gp4 in order to reduce the DC offset

error.

Figure 4.21 shows the schematic diagram of the input buffer, as well as the


gm1

gm

gm

gm Vout1Vout2

RLRL

gm

DC offset reductionamplifier

Vin1Vin2

bufferInput

Gp2

Gp1

Gp3

Gp4

Linear amplifier

Limiting transconductance element

Figure 4.20: Logarithmic amplifier block diagram.

first stage in the highest gain path Gp4. The input match uses resistors as well

as a capacitor to lower the input impedance at high frequencies. This capacitor

counteracts the effect of bond wire inductances and so the input remains impedance

matched to 50 Ω within 10 dB up to approximately 9 GHz. Large emitter lengths

were used in the emitter follower input buffers in order to achieve low base resistance

and low noise. As well, the bias currents of these buffers were optimized for low

noise. Following the input buffer in Figure 4.21 is the first stage of the highest gain

path Gp4. The noise figure of the logarithmic amplifier is completely dominated by

the input buffer and this stage. This stage is a CHEF amplifier, and it uses the

component values from the design example in Section 3.3.1. The emitter lengths of

the two input transistors to this stage were also chosen relatively large in order to

achieve low noise. All of the amplifier gain stages use CHEF amplifiers. Both of

the gain stages in path Gp4 are the same and were designed with a voltage gain of

approximately 10. Furthermore, it should be noted that by using the Cherry-Hooper


First Stage of Gp4

300fF300fF

45Ω

34Ω

550Ω

45Ω

34Ω

550Ω

55Ω

55Ω 55Ω

55Ω

160Ω 160Ω

3.0mA3.2mA

-3.3V

Vin1

Vin2

Vout_high1

Vout_high22.9mA

1.5mA2.9mA

1.5mA

To Gp1

Input Buffer

Figure 4.21: Schematic diagram of the input impedance match circuit and first highgain stage.

stage in Figure 4.21 for all gain stages, the delay through each gain path was designed

to be approximately the same, and no capacitive compensation was needed. Emitter

degeneration was used in the input emitter coupled pair of each Cherry-Hooper stage

in the lowest gain path to achieve low gain while maintaining high bandwidth. Both

stages in the lowest gain path Gp1 are exactly the same.

Figure 4.22 shows the amplifier that was used in negative feedback around path

Gp4 in order to reduce the DC offsets. In order to achieve a high pass corner frequency

of 500 kHz for the offset cancellation, a 1 nF off-chip capacitor was used.

The four limiting transconductance elements in Figure 4.20 were all integrated

into a single amplifier, shown in Figure 4.23. When an input signal from one of the

gain paths is applied to one of the degenerated emitter coupled pairs, it steers the bias

current of that pair to the side with the highest applied voltage. When the applied

voltage becomes large enough, the amount of current steered limits at IL=6.7 mA,


550Ω 550Ω

1.9mA

-3.3V

Vfin1

Vfout1 Vfout23.2 pF

6.4 pF

1 nFOff Chip

Vfin2

Figure 4.22: Schematic diagram of the DC offset error reduction circuit.

73Ω

-3.3V

Vin1_L

Vout2

9.0mA6.7mA

73Ω

6.7mA6.7mA

5Ω 5Ω

5Ω5Ω

5Ω

Vout1

Vin2_L

Vin3_L

Vin4_L

Vin1_R

Vin2_R

Vin3_R

Vin4_R

5Ω5Ω

5Ω

Figure 4.23: Schematic diagram of the output summation circuit.

or IL1=9.0 mA for the pair connected to the output of path Gp1. The load resistor

was chosen as 73 Ω so that when it is combined with the parasitic capacitance of

the transistors and the inductance of an output bond wire, the impedance remains

matched to 50 Ω within 10 dB to approximately 8 GHz. A negative power supply

of -3.3 V was used so that the amplifier inputs and outputs may be directly coupled

to a 50 Ω load. Emitter degeneration was used to reduce the gain of the summing

circuit, thereby reducing DC offset errors at the output.


Figure 4.24 shows the simulated DC transfer characteristic of the logarithmic

amplifier, including the effect of parasitic capacitive and resistive layout parasitics,

for one hundred Monte Carlo iterations. This simulation uses information about

typical process variations in the HP5 manufacturing process, and the variables and

probabilities are set by IBM. Although the logarithmic intercept varies somewhat,

the logarithmic linearity is acceptably reliable for the OSSB application. The fact

that the intercept point varies is attributed to the design of the summation amplifier.

The gain of this amplifier is expected to vary somewhat since Re=5 Ω, which is not

large enough to fix the gain in the presence of process variations. A larger Re was not

used because the amplifier gain would then be quite low, and the voltage headroom

would be reduced. The fact that the logarithmic linearity remains approximately

constant is attributed to the Widlar biasing scheme, described in Appendix E, used

in the CHEF amplifier stages. The stability of the logarithmic response could be

further improved using a more process independent current mirror reference than

the resistor to ground shown in the current mirrors in Figure 4.21.

Figure 4.25 shows the simulated logarithmic response of the amplifier at 4 GHz

for three different temperatures. The observed change in the logarithmic intercept

and slope over the 120 degree Celsius range has no negative impact in the OSSB

application. This is because a small change in intercept may be corrected by changing

the amount of level shifting applied to the input signal, and a change in slope can

be corrected by changing the attenuation of the input signal. The robustness of the

log amplifier signal despite temperature changes is expected from the Widlar biasing

scheme used in the CHEF amplifier stages, described in Appendix E.


Figure 4.24: Simulated DC transfer characteristic of logarithmic amplifier over onehundred Monte Carlo iterations.

10−3

10−2

10−1

100

0

0.2

0.4

0.6

Input Peak Voltage (V)

Out

put P

eak

Vol

tage

(V

)

−40 degrees Celsius+27 degrees Celsius+80 degrees Celsius

Figure 4.25: Simulated logarithmic response of SiGe logarithmic amplifier at 4 GHzfor three different temperatures.


DC OffsetFeedback

BranchLow

BranchHigh

Output Buffer

Input Buffer

High1 Low1High2 Low2

1.50 mm

1.33 mm

Figure 4.26: Microphotograph of the SiGe logarithmic amplifier integrated circuit.

4.5.2 Measurements of Implementation 2

A microphotograph of the integrated circuit, fabricated in IBM’s 5HP technology, is

shown in Figure 4.26. The circuit draws 130 mA from the -3.3 V supply. Measure-

ments were performed with the same equipment used to measure the Si logarithmic

amplifier. A connectorized test fixture was designed for the log amplifier, and is

described in Appendix F. The test fixture was used for the OSSB experiments in

Chapter 7. However, for simplicity, wafer probing was used for the measurements

given in this section. The S21 or gain of the amplifier is equal to 39 dB, compared

to a simulated S21 of 40 dB. The measured small signal bandwidth of the amplifier

is 6 GHz, compared to 8 GHz simulated. It is expected that the close proximity

of some circuits created unintentional feedback loops, and this may account for the

lower measured bandwidth. The measured noise figure at 1 GHz is 12.6 dB, which

is the same as the simulated noise figure.

Figure 4.27 shows the measured one-tone response of the amplifier for frequencies


−60 −50 −40 −30 −20 −10 00

0.1

0.2

0.3

0.4

Input Power (dBm)

Pea

k O

utpu

t Vol

tage

(V

)

100 MHz1 GHz2 GHz3 GHz4 GHz5 GHz6 GHz

Figure 4.27: Measured single ended logarithmic response from 100 MHz to 6 GHz.

up to 6 GHz. The high logarithmic slope of 6.5 mV/dB was achieved by using

relatively large currents in the output summing stage in Figure 4.23. The logarithmic

response error was calculated using the definition given in Section 4.4.2. The log error

at individual frequencies from 100 MHz to 6 GHz was less than 2.5 dB from an input

power level of -52 dBm to -2 dBm. The error for frequencies from 100 MHz to 4 GHz

when fit to a single line was less than 4.5 dB over the same input power range.

Figure 4.28 shows the output waveforms for one of the two log amplifier outputs

for frequencies of 100 MHz and 4 GHz. The rise and fall times of the amplifier are

50 ps. The logarithmic amplitude compression in the waveforms in Figure 4.28 is

evident. The observed noise at lower amplitude levels is partly due to amplifier noise

and partly due to timing noise inherent in the sampling oscilloscope measurement.

Table 4.1 compares the two log amplifier implementations described in this chap-

ter with three other high frequency true logarithmic amplifiers. The second imple-

mentation in this work has the highest bandwidth, and has excellent logarithmic

dynamic range and slope. It is disappointing that the SiGe logarithmic amplifier in


Input Power

Input Power

− 2.00 dBm− 15.0 dBm

− 40.0 dBm

− 2.00 dBm

− 15.5 dBm

− 42.0 dBm

(a) Logarithmic response at 100 MHz

(a) Logarithmic response at 4 GHz

- 29.0 dBm

- 27.5 dBm

Figure 4.28: Measured real time logarithmic amplifier single ended output wave-forms.

this chapter did not achieve 8 GHz bandwidth as the simulation predicted. The high

gain CHEF amplifiers are what limited the log amplifier bandwidth in simulation.

However, since these amplifiers had a bandwidth of 11.2 GHz when measured sep-

arately on the IC from the last chapter, it is definitely feasible to obtain an 8 GHz

logarithmic amplifier using the same basic design in the same technology used here.

After trying a more compact layout for the second log amplifier, the author suggests

going back to a less crowded layout, as was used in the first log amplifier implemen-

tation.


Tab

le4.

1:C

ompar

ison

ofhig

hfr

equen

cytr

ue

log

amplifier

s.C

ircu

itTop

olog

yTec

hnol

ogy

Pow

erG

ain

Ban

d-

Ris

e/D

yna-

Log

Supply

wid

thFal

lm

icSlo

pe

Tim

eR

ange

Acc

iari

Tw

in-G

ain

GaA

sFE

T-

36dB

0.3-

-40

dB

10m

V/d

Betal.[4

5]

Sta

ge-

5G

Hz

Sm

ith

Tw

in-G

ain

GaA

sFE

T-8

V,+

8V70

dB

0.5-

-70

dB

6.3

mV

/dB

[46]

Sta

geH

ybri

d5.

2W

4G

Hz

Cir

cuit

Oki

Tw

in-G

ain

GaA

sH

BT

-8V

48dB

DC

-40

0ps/

40dB

3.3

mV

/dB

etal.[4

0]

Sta

ge1.

06W

3G

Hz

400

ps

Zo

=10

0Ω

Imple

men

t-B

ranch

Silic

on-5

V30

dB

DC

-10

0ps/

40dB

1.2

mV

/dB

atio

n1

Par

alle

lB

ipol

ar0.

75W

4G

Hz

100

ps

Zo

=50

ΩSum

mat

ion

fT

=35

GH

z

Imple

men

t-B

ranch

SiG

e-3

.3V

39dB

DC

-50

ps/

50dB

6.5

mV

/dB

atio

n2

Par

alle

lH

BT

0.43

W6

GH

z50

ps

Sum

mat

ion

fT

=47

GH

z

4.6 Conclusion 91

4.6 Conclusion

In this chapter, two high performance parallel summation logarithmic amplifiers

were presented. It was demonstrated how the branch parallel summation architecture

provides high bandwidth and logarithmic slope while consuming relatively low power.

In the next chapter, the other component required in the COSSB application, a

Hilbert transformer is described. The measured and simulated performance of these

two components is described in later chapters.

Chapter 5

A 10 Gb/s Hilbert Transformer with Q-Enhanced

LC Transmission Lines

5.1 Introduction

The previous chapter described the development of logarithmic amplifiers suitable

for use in the COSSB system. This chapter presents the development of the other

key component for COSSB, a fully integrated 10 Gb/s Hilbert transformer (HT).

The frequency response of an ideal HT is −j · sgn(ω), which indicates an abrupt

transition in the phase response at DC, as shown in Figure 5.1. The impulse response

is given by

h(t) =1

πt. (5.1)

This impulse response exists over an infinite range of t, but for practical purposes

the response is truncated at t =-NΥ to +NΥ, where N is an integer and Υ is a time

step. Since there are negative values of t, this response cannot be implemented using

delays. This can be avoided by shifting h(t) by a time NΥ so that all of h(t) is in

positive time. The time shifted impulse response may then be realized as a direct

form continuous time FIR filter. The tap weights are chosen to be equal to values of

the time shifted impulse response at integer values of Υ, as shown in Figure 5.2.

The structure of the filter is such that it has perfectly linear phase if two or

more taps are used, with the phase response approaching -90 toward DC and the

delay equal to half the length of the delay line. The RMS amplitude ripple decreases

5.1 Introduction 93

ω

ω

ω

ω

ω

ω

ω

+90

−90

|H( )|ω

H( )ω

1

Figure 5.1: Response of filter with an infinite number of taps.

π−1π π π3

......

Summer

Weight Weight Weight Weight

Time Delay Time Delay Time Delay

Σ

1 1

Input

Output

2 2 2Υ Υ Υ

3−1

Figure 5.2: Tapped delay implementation of an HT.Adapted from [47].

with an increasing number of filter taps. This approach of implementing a HT is

uniquely well suited to 10 Gb/s signals because it only requires delay, weighting,

and summation, all of which may be achieved in a broadband fashion. In Sieben’s

experiment on COSSB described in Section 2.4.6, discrete parts were used including

packaged amplifiers, splitters, attenuators, and delay lines to piece together the HT.

Since a four tap HT is not perfect, part of the unwanted sideband or a vestige

remained in the COSSB signal. However, the performance of the COSSB system

in Sieben’s experiment was not limited by this vestige, but by distortion caused by

the Mach-Zehnder modulator. His conclusion, supported by simulations described in

5.1 Introduction 94

the next chapter, is that having more than four taps yields little or no improvement

because the OSSB system performance is limited by the nonlinearity of the Mach-

Zehnder optical intensity modulator [7]. Furthermore, Sieben investigated the effect

of varying the time delay Υ in the filter. The tapped delay HT has a high pass

corner frequency which is ideally at DC, as indicated by the −j · sgn(ω) frequency

response of an ideal HT. This high pass corner frequency may be designed to be

as close to DC as possible by maximizing the delay through the filter, either by

increasing the number of taps at a fixed tap delay, or by increasing the delay Υ

for a fixed number of taps. Minimizing the high pass corner frequency ensures that

cancellation of the unwanted sideband will occur as close to the carrier wavelength

as possible. However, having excessive magnitude ripple in the passband of the HT

is detrimental to the signal. Figure 5.3 shows the magnitude responses of four tap

HTs for different values of the delay Υ. In each case in Figure 5.3, a value of Υ

that is a fraction of the period of a 10 Gb/s signal, or 100 ps, was used. A value of

Υ=55 ps leads to the lowest high pass corner frequency, but places magnitude ripple

in the 1-5 GHz high power region of a 10 Gb/s signal. For this reason, Sieben chose

to use a filter with Υ=37.5 ps instead. Figure 5.4 shows the spectrum of a simulated

10 Gb/s COSSB signal generated without the logarithmic conversion and using a

four tap HT with Υ=37.5 ps as per Sieben’s experiment. As expected, a vestige of

the unwanted sideband is observed. For the present design, the tap delay Υ was

chosen to be 30 ps, slightly less than in Sieben’s experiment. Having Υ=30 ps results

in a total delay of 180 ps in the four tap filter in Figure 5.2, and a 3 dB passband

of 1.7-12.6 GHz. The ideal four tap HT has a nominal group delay that is equal to

half of the total length of the delay line, 90 ps in this case. This is compared to the

theoretical −j · sgn(ω) frequency response, which indicates zero group delay at all

frequencies except DC.

5.2 HT Transfer Function 95

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

Frequency (GHz)

Nor

mal

ized

Lin

ear

Mag

nitu

de

Tap Delay=20 psTap Delay=37.5 psTap Delay=55 ps

Figure 5.3: Magnitude responses of four tap HTs for three different values of Υ.

−10 −8 −6 −4 −2 0 2 4 6 8 10−80

−70

−60

−50

−40

−30

−20

−10

0


Pow

er (

dB)

Figure 5.4: Spectrum of a COSSB signal generated with a four tap HT.

5.2 HT Transfer Function

Delay lines with gigahertz of bandwidth and 180 ps of delay may be fabricated using

on-chip LC transmission lines. It is useful for design purposes to model the effect

of delay distortion and loss in the LC lines on the transfer function of the HT.

Considering the performance of the filter structure in continuous time, the transfer

5.3 Design of LC Transmission Lines 96

function of the nth delay element is generalized to be

HDn(ω) = |HDn(ω)|e−jω∆HDn(ω). (5.2)

where ideally |HDn(ω)|=1 and ∆HDn(ω)=2Υ. The transfer function of the four tap

HT is then found to be

H(ω) = − 1

3π− 1

π|HD1|e−jω∆HD1

+1

π|HD1||HD2|e−jω(∆HD1+∆HD2) (5.3)

+1

3π|HD1||HD2||HD3|e−jω(∆HD1+∆HD2+∆HD3)

where the (ω) is not shown for HDn(ω) but is understood.

5.3 Design of LC Transmission Lines

One method of fabricating a 180 ps delay line on-chip is to use a metal transmission

line. Simulations performed in Agilent ADS indicate that such a 50 Ω line would

be 31.0 mm long if fabricated using an upper metal layer for the signal and a lower

metal layer 5 µm below as the ground plane, with SiO2 between with a dielectric

constant of 3.9. The length of this line may be reduced dramatically if on-chip LC

transmission lines are used. An LC line with 180 ps of delay may be constructed using

the circuit in Figure 5.5. The characteristic impedance of the line is related to the

inductor and capacitor values according to Zo =√

L/C. Furthermore, the structure

in Figure 5.5 has a cutoff frequency which decreases with increasing L and C. If a

capacitance of 250 fF is chosen, L=625 pH for Zo =50 Ω, and the cutoff frequency

will be greater than 10 GHz. The four tap HT requires three separate delay elements

with a delay of 60 ps each. During preliminary design, each section was designed


...

VEE VEE VEE VEE

C

L/2

C C C

L L L/2

Figure 5.5: Schematic diagram of the LC transmission line used in the HT.

using the structure in Figure 5.5 with four 625 pH inductors, a 312 pH inductor

on each end, and five 250 fF capacitors. The layout of the inductors only is shown

in Figure 5.6(a). Version 2003A of the Agilent Momentum 2.5D electromagnetic

simulator was used to simulate the inductors. The results of the electromagnetic

simulation were then used to simulate the LC transmission line.

The simulated loss of the 180 ps transmission line was 2.1, 5.8, and 7.6 dB at 2.0,

7.0, and 10.0 GHz respectively, as shown in Figure 5.6(b). The high loss of the passive

line was found to significantly degrade the operation of the HT, and the simulated

performance of the OSSB system in which the HT is used. This loss is due to the

poor quality factor of the on-chip inductors. Fortunately, techniques are available

for canceling the loss of on-chip inductors using negative resistance circuits [48, 49].

In the next section, it is demonstrated how the coupled inductor technique of Q-

enhancement may be used to fabricate reduced loss, on-chip LC transmission lines.

5.3.1 Q-Enhanced LC Transmission Lines

Recently, Georgescu et al. presented a method for Q-enhancement, where a com-

pensating current is applied to a secondary inductor that is coupled to the primary

inductor, i.e. using a transformer, and the loss is canceled by the induced voltage in

the primary [48]. Figure 5.7 shows a realization of the circuit presented in [48]. The


(a) Layout of passive LC transmission line.

(b) Loss of passive LC line.

Figure 5.6: Layout and loss of passive LC transmission line.

voltage vp across the primary of the transformer is expressed as

vp = iiRp + ipsLp + issM12 (5.4)

where M12 = k√

LpLs, and k is the coupling coefficient of typical value 0.5-0.75. In

order to cancel the inductor loss, ipRp = −issM12 resulting in ip = vp/sLp. The

required secondary current is [48]

is =−ipRp

sM12

=−vpRp

s2M12Lp

. (5.5)


isL2 Rs

RpL1 sMis

IEE

VEE

VCC

VCC

vp

vpgm

2

ip

Figure 5.7: Transformer based Q-enhanced floating inductor.

Hence, optimum loss compensation occurs when the current through the secondary

is in phase with the voltage across the primary. A differential pair is used to generate

the required current in the circuit in Figure 5.7.

A transmission line with a delay of 180 ps in the 2-7 GHz range was designed

using the Q-enhanced inductors of the form in Figure 5.7, and is shown in Figure 5.8.

A bias current of 1 mA was used for each of the nine Q-enhanced inductors, and the

overall LC line consumes 30 mW. The loss of the transmission line simulated using

ADS Momentum 2003a is 1.9, 3.2, and 3.4 dB at 2.0, 7.0, and 10.0 GHz respectively.

This is approximately half the loss of the passive line at 5 GHz and higher, and the

loss of the Q-enhanced line has significantly reduced frequency dependence in the

4-10 GHz range. It is possible to further decrease the amount of loss using more

bias current in the Q-enhancement circuit, however, it may then be necessary to

stabilize the circuit above 10 GHz using capacitors across the secondaries of the

transformer, as described in [48]. This was not done here since the loss was already

acceptable with a modest amount of Q-enhancement, and the circuit is stable under

these conditions.


70ps

VCC

V2L V2R

IEF

VCCVCC

50Ω70ps 70ps

Each 70ps Section

VCC

V3R V3L

VCCVCC

V4R

IEF

V4L

VCC

VCC

V1L

IEF

V1R

IEF

VCC

IEF

IEFIEF

IEF

VEE

Vin

VEE

C C C C

L/2 L/2L L L

C=190fFL=470pH

Three

InductorsQ-Enhanced

Figure 5.8: Delay line with emitter follower tap buffers.

After the circuits designed in this chapter were already submitted for fabrication,

the author discovered a relatively simple method of laying out the Q-enhancement

circuit symmetrically [50]. Figure 5.9(a) shows the layout of a symmetrical trans-

former with a center tapped secondary, and Figure 5.9(b) shows the schematic di-

agram of how it is connected. Using this structure doubles the efficiency of the

Q-enhancement by using the fully differential signal current. Furthermore, with the

transformer layed out in this way, all four transformer terminals are close together,

making the metal interconnect to the differential pair short.

Also after the circuits designed in this chapter were already submitted for fab-

rication, the author discovered a much simpler way to modestly Q-enhance an LC

transmission line. Figure 5.10 shows a circuit in which the inductor Q is enhanced

using a simple cross coupled pair, used for example in [49]. The advantage of this

circuit over the transformer based application is that the starting Q of the inductor

is higher than the starting Q of a transformer primary. This circuit is recommended

for use in future implementations of the HT. The transformer based circuit is recom-

5.4 Circuit Implementation 101

PrimaryInputs

SecondaryCenter Tap

SecondaryInputs

(a) Symmetrical transformer with centertapped secondary.

LS1 RS1

RPLP

IEE

VEE

vp

vpgm

2

LS2 RS2

VCC

(b) Schematic diagram.

Figure 5.9: Efficient Q-enhancement circuit using both signal currents.

mended for filters where Q is enhanced into the hundreds, however. Soorapanth and

Wong demonstrated how the use of the circuit in Figure 5.10 in a bandpass filter de-

sign leads to a highly distorted passband [51]. The transformer based Q-enhancement

circuit may be used to fabricate bandpass filters with relatively flat pass-bands, as

research in progress on filters by the author’s colleagues demonstrates.

5.4 Circuit Implementation

As shown in Figure 5.8, signals are taken from the LC delay lines at four locations.

These signals are connected to a summing amplifier through emitter followers, which

provide a DC level shift. The summing amplifier, shown in Figure 5.11, consists


RPLP

IEE

VEE

vp

vpgm

2

Figure 5.10: A floating inductor which is Q-enhanced using a simple cross coupledpair .

of four differential pairs in parallel with their collectors tied together so that their

currents add. Each pair of emitter followers in Figure 5.8 drives one of the differ-

ential pairs in the summing amplifier. The 50 Ω load allows the amplifier output

to be impedance matched and DC coupled to a 50 Ω system. A more complicated

impedance match, such as that used in the logarithmic amplifier, was not used be-

cause the parasitic capacitance at the output node was designed to be smaller by

using smaller currents and HBT devices. Emitter degeneration resistors are used in

each of the four differential pairs in the summing amplifier in order to realize the

tap weights. The two negative tap weights indicated in Figure 5.2 were achieved by

reversing the input connections to the summing amplifiers for these weights. Fur-

thermore, an adjustment was made to the fourth tap weight through simulation to

compensate for the loss in the LC line. Each of the four differential pairs in the

summing amplifier is biased with a current of 5 mA, and so the summing amplifier

consumes 70 mW. This relatively high current is needed to drive the 50 Ω load. The

basic design of the summing amplifier is such that loss is incurred through the HT.

However, a compromise between this loss and the linearity of the HT was chosen to

achieve acceptable performance in the OSSB application. The loss may be reduced


50Ω

Vin1_left

Vout2

2.5mA2.5mA

Vout1

Vin2_left

Vin3_left

Vin4_left

Vin1_right

Vin2_right

Vin3_right

Vin4_right

2.5mA2.5mA

VCC

VCC

VSICS

50Ω

2.5mA2.5mA

2.5mA2.5mA

300Ω

50Ω

50Ω

215Ω

VEE

Figure 5.11: HT summing amplifier.

by 6 dB where possible by using the differential output signal. A microphotograph

of the HT integrated circuit is shown in Figure 5.12. It was fabricated in IBM’s 5AM

technology, which has 47 GHz fT HBTs and 0.5 µm feature sizes. The circuit mea-

sures 1.70 x 1.25 mm and consumes 43 mA from the -3.3 V supply for a total power

consumption of 142 mW. With the Q-enhancement circuits turned off, the circuit

draws 34 mA and consumes 112 mW. The delay line was arranged so that there is

equal length from each tap point to the inputs of the summing amplifier, which is

in the center of the chip. Furthermore, a 50 pF on-chip power supply decoupling

capacitor was chosen so that it would resonate below 500 MHz with the inductance

of the power supply bond wires.

A CMOS supply voltage independent current source (SICS), shown in Figure 5.13,

was used to generate the reference voltages for the Q-enhancement circuits around


VCC

VCC

VCC

VCC

VCC VCC VCC VCC VCC VCC VCC

VCC

EEV

EEV

EEV

EEV

REFR EEV

OUT INROUT L TEST

VEnhance

1.25 mm

1.70 mm

Figure 5.12: Integrated HT microphotograph.

SICSStart−Up Circuit

VEE

VEE

VEE

V = -3.3 VEE

V =0 Vcc

M1 M2

M3 M4

1 mA1 mA

M7

Ibias

255 Ω

VEE V

50 k Ω

40 k Ω

M5 M6

0.4 mA

W=800 m/µL=2 mµ

each

W=60 m/µL=2 mµ

W=180 m/µL=2 mµ

W=200 m/µL=0.5 mµ

W=500 m/µL=0.5 mµ

M8 Mn

...

V

5 pF

5 pF

EEEE

Off Chip

Figure 5.13: Supply independent current source.


the chip [52]. This circuit allows the metal interconnect to the gates of M7 to

Mn to be relatively long without voltage drop. Unfortunately, this circuit would

not start up properly when the chip was returned from the foundry and turned on,

even though the simulation did not indicate any problems. The author had used

fully CMOS implementations of this bias circuit before with complete success in a

previous amplifier design [26]. In order to get the bias circuit to start up without

oscillating, M5 was disconnected using laser cutting from M4, M1 was disconnected

from M2 and M3 using laser cutting, and the 255 Ω resistor to Vee was replaced by

a 47 Ω resistor connected to a voltage a few hundred millivolts below VEE, and this

voltage was tuned to set the HT current to the simulated level. The author is not

certain what caused the start-up problem. It may have to do with the much higher

output resistance of the HBTs compared to the NMOS devices.

Similar to the last chapters, wafer probing of RF signals was used and the VEE,

VCC , Venhance, and RREF connection were wire bonded. An Agilent PSA network

analyzer was used to measure the HT’s small signal parameters.

Figure 5.14 shows S11 and S22 for three measured HT dice. S11 is measured

looking into one end of the LC transmission line with the other end terminated in

50 Ω on-chip, and is lower than -10 dB up to 9 GHz. This indicates that Zo of the

transmission line is close to 50 Ω. S22 is lower than -10 dB up to 8 GHz.

Figures 5.15(a) and 5.15(b) show the simulated S21 and group delay of the

HT including layout resistive and capacitive parasitics for the cases where the Q-

enhancement is turned on and off. When the Q-enhancement is turned off, the gain

above three gigahertz decreases by 1-2 dB. The Q-enhancement has little effect on

the group delay.

Figures 5.16(a) and 5.16(b) show the measured S21 and group delay of the HT

for three different die for the cases where the Q-enhancement is turned on and off.


0 2 4 6 8 10 12 14 16 18 20−30

−25

−20

−15

−10

−5

0

Frequency (GHz)

Mag

nitu

de (

dB)

Chip C S11Chip D S11Chip F S11Chip C S22Chip D S22Chip F S22

Figure 5.14: Measured S11 and S22 of three dice with Q-enhancement on.

Although six die were prepared for test, three die were severely damaged before they

could be measured. The dip in S21 was simulated to be at 5 GHz in Figure 5.15,

and was measured at 7 GHz. This difference is attributed to the failure to simulate

the inductance of on-chip interconnect of the power supply, since no straightforward

means to simulate this inductance was available using the version of the Diva ex-

tractor used. Fortunately, the power supply interconnect was kept short, and the

measured S21 and group delay is relatively close to the simulated response from Fig-

ure 5.15. Figure 5.16(a) shows that the measured gain exhibits amplitude ripple,

similar to the ideal gain, but shows a lower high pass corner frequency. This is

attributed to the fact that although the tap delay of the ideal four tap HT is con-

stant with frequency at T=30 ps, the measured group delay and hence tap delay are

actually frequency dependent. For a better theoretical estimate, the measured gain

could be compared against equation (5.3).

Figure 5.16(b) indicates that the measured average group delay of the HT is


0 2 4 6 8 10 12 14 16 18 20−20

−18

−16

−14

−12

−10

−8

−6

−4

Frequency (GHz)

S21

(dB

)Simulated HT with Parasitics − Q−enhancement onSimulated HT with Parasitics − Q−enhancement off

(a) S21.

1 2 3 4 5 6 7 8 9 1050

100

150

200

250

300

Frequency (GHz)

Gro

up D

elay

(ps

)

Simulated HT with Parasitics − Q−enhancement onSimulated HT with Parasitics − Q−enhancement off

(b) Group delay.

Figure 5.15: Plot of (a) S21 and (b) group delay simulated with resistive and capac-itive layout parasitics.

approximately 120 ps between 2.0 and 7.0 GHz, and the group delay distortion is

±30 ps from 2.0 to 8.0 GHz. Approximately 90 ps of the group delay comes from

the delay line, and 30 ps comes from the summing amplifier. Fig. 5.17 shows the

deembedded measured and theoretical phase response, where ±10 of phase error is

observed from 2.0 to 7.0 GHz.

The simulated and measured noise figure (NF) of HT die C with Q-enhancement

turned on are given in Table 5.1. This table shows that the simulator underestimates

the noise of the HT by 5-7 dB at some frequencies, although the simulation correctly

predicts the noise at 4.0 GHz. The NF was measured using a Rhode and Schwarz

FSEK 30 spectrum analyzer with a NoiseCom NC346C noise source. The measured

NF with the Q-enhancement turned off was the exactly the same as that shown in

Table 5.1, except from 6-8 GHz where it was 2 dB higher with Q-enhancement turned

off because of the increased loss. The noise at the output of the HT consists of noise

from the emitter followers, the summing amplifier, and the ohmic losses in the LC


0 2 4 6 8 10 12 14 16 18 20−20

−18

−16

−14

−12

−10

−8

−6

−4

Frequency (GHz)

S21

(dB

)Chip CChip DChip FIdeal 4 Tap HT

Q−Enhancement Off

Q−Enhancement On

(a) S21.

1 2 3 4 5 6 7 8 9 1080

100

120

140

160

180

200

220

240

Frequency (GHz)

Gro

up D

elay

(ps

)

Chip CChip DChip F

Q−Enhancement On or Off

(b) Group delay.

Figure 5.16: Plot of (a) measured S21 and normalized theoretical S21 and (b) groupdelay for three die with Q-enhancement on and off.

1 2 3 4 5 6 7 8 9−105

−100

−95

−90

−85

−80

−75

Frequency (GHz)

Pha

se (

Deg

rees

)

Chip 1Chip 2Chip 3

Ideal Four TapHT Phase

Figure 5.17: Measured phase of three die with a phase shift corresponding to 120 psof delay subtracted, and theoretical four tap HT phase response with a phase shiftcorresponding to 90 ps of delay subtracted.

line. The base resistance thermal noise from the emitter followers in Figure 5.8 and

of the differential pair transistors in the summing amplifier was kept low in the design

by using larger than minimum length emitters in these HBTs. The noise of the Q-

enhancement circuits constituted less than 5% of the total HT noise in simulation.

Turning the Q-enhancement on or off during the measurement had a negligible effect


Table 5.1: Noise figure of HT die C with Q-enhancement turned on.

Frequency (GHz) Simulated (dB) Measured (dB)

2.0 26.5 30.0

2.5 24.5 29.0

3.0 24.5 27.0

4.0 26.0 26.0

8.0 22.0 28.5

on the output noise power of the HT, which agrees with the simulation.

As a further verification of the HT’s performance, Figure 5.18(a) shows the re-

sponse of the HT to the repeated binary pattern 01001000. The signal was gen-

erated using the Anritsu MP1763B pattern generator and measured using a HP

54750A oscilloscope with a HP 54751A 20 GHz sampling module. For this trace,

the Q-enhancement is turned on. Figure 5.18(b) shows the same trace with the Q-

enhancement turned off. There is little difference in the shape of the signal, and the

noise appears to be less with the Q-enhancement turned off, a surprising result which

appears to contradict the measurements of NF alone. Since the Q-enhancement cir-

cuits are a type of feedback loop, it is possible that some ringing occurs in the

response of these circuits, which would account for the noise at the HT output. The

simulated response of the transmission line in the HT to a sequence of pulses each

with 25 ps rise/fall times and pulse widths of 75 ps is shown in Figure 5.19 for the

cases where Q-enhancement is turned on and off. This simulation indicates that the

Q-enhancement results in some increased ringing. It is important to note that even

if the LC transmission line were designed with ideal inductors and capacitors, it will

still ring in response to a pulse. This is because an LC line with a finite number of

sections is only an approximation to a transmission line. It may be shown that for

an LC transmission line with a fixed delay, the amount of ringing in response to a


pulse decreases as the line is broken into smaller and smaller sections.

The fact that the circuit performs so well even when the Q-enhancement is turned

off is partially attributed to the way in which the fourth tap weight was raised in order

to compensate for loss in the line. Figure 5.18(c) shows the response of an ideal four

tap HT to the same 01001000 pattern. The measured responses are relatively close

to the ideal waveform. The author observed little or no difference in the shape of the

waveform when the Q-enhancement was turned on and off for several other patterns,

and leaving the Q-enhancement turned off made the waveform appear less noisy.

Figures 5.20, 5.21, 5.22 show the measured response of the HT with Q-enhancement

turned off for the binary patterns 10, 1000, and 0111 respectively, along with the

outputs of an ideal four tap HT to these patterns. The response of the HT to these

patterns is similar to the ideal four tap output in each case.

The fact that the HT appears to work without the Q-enhancement is surpris-

ing, but the development of the Q-enhanced transmission line is still useful. The

technology used had a relatively thick top layer of metal for the on-chip spirals. It

is expected that for other technologies where the metal is thinner, Q-enhancement

would be more important. As well, although the weights in the filter may be adjusted

to compensate for loss, this does not compensate for the frequency-dependent loss

of the line. Furthermore, the weights in the summing amplifier can not be adjusted

to account for more than a few decibels of loss without sacrificing bandwidth and

linearity.

The 1 dB compression point of HT die C was obtained by performing power

sweeps using an Agilent 83650L signal generator and Agilent E4417A power meter

with a E9327A power sensor to measure the HT output power. With Q-enhancement

turned off, the measured input referred 1 dB compression point is +5 dBm at 2.0

GHz and +4 dBm at 5.0 GHz. With Q-enhancement turned on, the compression


(a) Upper trace is response of HT IC to re-peated DATA pattern 01001000. Lower traceshows DATA output from the pattern gener-ator. Q-enhancement is turned on.

(b) Same as (a), with Q-enhancement turnedoff.

0 200 400 600 800 1000 1200 1400 1600 1800−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Am

plitu

de (

Line

ar U

nits

)

Time (ps)

(c) Response of ideal four tap HT with Υ =0.30 from Matlab.

Figure 5.18: Responses of four tap HTs to the repeated binary pattern 01001000.

5.5 Conclusion 112

Q−Enhancement Turned Off

Q−Enhancement Turned On

Figure 5.19: Output of transmission line to a sequence of pulses.

point was slightly higher, +7 dBm at 2.0 GHz, and +6.0 dBm at 5.0 GHz. The

measured results for HT die F were very similar. The author is unsure why the

compression point was higher with Q-enhancement turned on. The circuit has the

advantage that only a fraction of the applied voltage to the HT appears across each

inductor. The linearity of the Q-enhancement circuits may be further increased if

needed using the multi-tanh technique [53].

5.5 Conclusion

In this chapter, it was described how the idea of a tapped delay HT, developed by

previous authors, was fabricated as an integrated circuit with enough bandwidth

for 10 Gb/s operation. Methods of Q-enhancing LC transmission lines were also

described. Measurements show that the HT implementation works correctly, and

should be capable of generating single sideband signals. In the next chapter, the

5.5 Conclusion 113

(a) Upper trace is response of HT IC to repeated DATA pat-tern one-zero (10) with Q-enhancement off. Lower trace showsDATA output from the pattern generator.

0 50 100 150 200 250 300 350 400−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Am

plitu

de (

Line

ar U

nits

)

Time (ps)

(b) Response of ideal four tap HT with Υ =0.30 from Matlab.


simulated performance of the HT and the log amplifier in the COSSB system are

examined in simulation.

5.5 Conclusion 114

(a) Upper trace is response of HT IC to repeated DATA pat-

tern 1000 with Q-enhancement off. Lower trace shows DATA

output from the pattern generator.

0 100 200 300 400 500 600 700 800 900−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Am

plitu

de (

Line

ar U

nits

)

Time (ps)



5.5 Conclusion 115

(a) Upper trace is response of HT IC to repeated DATA pat-

tern 0111 with Q-enhancement off. Lower trace shows DATA

output from the pattern generator.

0 100 200 300 400 500 600 700 800 900−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Am

plitu

de (

Line

ar U

nits

)

Time (ps)



Chapter 6

Simulations of COSSB System Implementations

6.1 Introduction

In previous chapters, measurements of the logarithmic amplifier and HT were de-

scribed and were shown to agree reasonably well with their simulated performance.

This chapter is devoted to simulation of the COSSB transmitter. Sections 6.2 and 6.3

describe the individual performance of the logarithmic amplifier and the HT in the

COSSB system respectively. Section 6.4 then describes the performance obtainable

using both circuits together. The results indicate what to expect when a COSSB

transmitter is pieced together in the laboratory.

6.2 Performance of the Logarithmic Amplifier

When a digital signal with two perfect signal levels is used, it is not necessary to use a

logarithmic amplifier in the COSSB system. This is because the logarithm of a signal

with two discrete levels will also be a signal with two discrete levels, and so scaling

and level shifting may be used in place of a logarithm. To gage the performance of

the logarithmic amplifier, a better signal would be analog with at least two decades,

or 40 dB of dynamic range. One such signal is a 5 Gb/s 211 − 1 length PRBS signal

filtered with a fifth order Butterworth filter with a cutoff frequency of 2.75 GHz.

To transmit this signal over the COSSB system, the electrical data is first used

to amplitude modulate the optical signal, and then this signal is phase modulated,

as shown in Figure 6.1. In order to design the phase modulation to generate the

6.2 Performance of the Logarithmic Amplifier 117

AmplitudeModulator

PhaseModulator

To Fiber

exp(j t)ω

+Ms(t)

VDC

HilbertTransformer

Laser

LogarithmicAmplifier

Figure 6.1: Minimum phase COSSB transmitter.

optimal COSSB signal, it is necessary to know any imperfections that occurred while

performing the amplitude modulation. For the present simulations, it is assumed

that a Mach-Zehnder amplitude modulator modulates a 1550 nm optical signal at

a modulation depth of 0.25 of the modulator Vπ. That is, the signal applied to

one electrode of the Mach-Zehnder modulator swings above the bias point on that

electrode by 0.25·Vπ, and below the bias point on that electrode by 0.25·Vπ, and the

complementary signal applied to the other electrode swings by the same amount. The

amplitude of the signal on the fiber will have some distortion because of the non-

ideal amplitude modulation characteristic of the Mach-Zehnder modulator, which

was shown in Figure 2.12. This knowledge may be used to ensure that the phase

modulation is optimal as follows. The input signal to the logarithmic amplifier will

be scaled or linearly predistorted by a factor M and have an offset VDC added so that

the log amplifier output looks like the logarithm of the envelope of the amplitude

modulated signal on the fiber.

For the present simulations, the predistorted information signal was saved in

Matlab and loaded into the Cadence Spectre circuit simulator, where it was passed


through the SiGe HBT logarithmic amplifier using transient simulation for a dura-

tion of 200 ns, and the effect of parasitic layout capacitances and resistances were

included. Once the logarithmic amplifier output signal was obtained, it was loaded

back into Matlab. Since transient simulation of the logarithmic amplifier does not

include the amplifier noise, a random noise signal with the same bandwidth as the

logarithmic amplifier, 6 GHz, and the same RMS noise amplitude as the amplifier

output noise, 10.6 mV, was added to the output of the logarithmic amplifier. The

Spectre simulator provides the RMS noise voltage at one terminal of the log amplifier

output. If the noise is assumed to have a Gaussian probability density function, the

standard deviation of the noise will be the same as its RMS value. This is useful

when constructing the noise signal in Matlab. Furthermore, it is important to get

the correct RMS noise value for the log amplifier when the appropriate DC level of

the signal is applied. Without the DC offset the gain of the log amplifier is higher

and the output noise is also higher, 16.0 mV RMS.

After the noise was added, the resulting signal was passed through one of four

HTs. The four HTs are an ideal Hilbert transform computed by Matlab, as well as

ideal four, six, and eight tap HTs implemented in Cadence. For the three tapped

delay HTs, various values of the time delay parameter Υ were investigated, and

the values of Υ=0.45, 0.27, and 0.19 were found to give acceptable high pass cor-

ner frequencies and sufficient bandwidth at 5 Gb/s for the four, six, and eight tap

transformers respectively. The value of Υ=0.45 for the 5 Gb/s four tap HT is in

contrast to the value of 0.35 chosen in the previous chapter for the 10 Gb/s case.

The Cadence HTs used ideal delay elements and an ideal adder. The outputs of the

Cadence HTs were streamed out of Cadence, loaded into Matlab, and used to phase

modulate the simulated optical signal. Figure 6.2 shows the spectra of the signals

at the COSSB transmitter output. The spectra have been normalized in frequency


to the carrier frequency. As well, the DC content of the signals has the greatest

amplitude and is normalized to 0 dB, and the information is then contained in the

range of approximately -70 dB to -20 dB.

The best suppression of the unwanted sideband is obtained with the ideal Matlab

HT, and the performance improves slightly with an increasing number of taps for

the tapped delay HTs. However, the length of the required delay line within the

HT increases significantly using six or eight taps compared to only four, illustrating

why a four tap HT remains the most appropriate for integration. Figure 6.2(e)

shows the eye diagram for the ideal HT case, and the eye diagrams for the other

configurations are identical, since only the phase of the signal is being used to achieve

single sideband, and the phase is thrown away during direct detection. The inter-

symbol interference (ISI) in Figure 6.2(e) is a result of band-limiting the signal to

2.75 GHz at the start of the simulation in order to obtain an analog signal to test the

logarithmic amplifier. Since the phase of the optical signal is thrown away during

direct detection, the ISI has nothing to do with the phase modulation at zero fiber

length.

Figure 6.2(b) indicates that 15-20 dB of sideband suppression is possible using a

four tap HT. In Sieben’s experiments, he obtained 15-20 dB of measured sideband

cancellation for the case of relatively square, digital pulses [7]. For the simulation to

predict a comparable amount of sideband cancellation for analog signals is encour-

aging for this experiment, which relies on the logarithmic amplifier IC. However, so

far this simulation assumes an ideal four tap HT. The COSSB performance of the

four tap HT IC described in the last chapter will be considered in the next section.

In Section 6.4, the combined COSSB performance of the logarithmic amplifier IC

with the HT IC will be considered.


−10 −8 −6 −4 −2 0 2 4 6 8 10−70

−65

−60

−55

−50

−45

−40

−35

−30

−25

−20

Frequency (GHz)

Mag

nitu

de (

dB)

(a) With ideal Hilbert transform.

−10 −8 −6 −4 −2 0 2 4 6 8 10−70

−65

−60

−55

−50

−45

−40

−35

−30

−25

−20

Frequency (GHz)

Mag

nitu

de (

dB)

(b) With ideal four tap Hilbert transform.

−10 −8 −6 −4 −2 0 2 4 6 8 10−70

−65

−60

−55

−50

−45

−40

−35

−30

−25

−20

Frequency (GHz)

Mag

nitu

de (

dB)

(c) With ideal six tap Hilbert transform.

−10 −8 −6 −4 −2 0 2 4 6 8 10−70

−65

−60

−55

−50

−45

−40

−35

−30

−25

−20

Frequency (GHz)

Mag

nitu

de (

dB)

(d) With ideal eight tap Hilbert trans-form.

(e) Eye diagram with ideal HT after zerofiber length.

Figure 6.2: 5 Gb/s COSSB signals obtained through transient simulation of variousHTs and of the logarithmic amplifier IC.

6.3 Performance of the HT 121

6.3 Performance of the HT

In order to simulate the performance of the HT without the logarithmic amplifier, a

211−1 length 10 Gb/s unfiltered PRBS signal was used. A data rate of 10 Gb/s may

be used, because the HT has more bandwidth than the logarithmic amplifier. Similar

to the last section, a 1550 nm optical signal was amplitude modulated using a Mach-

Zehnder in simulation, this time at a modulation depth of 0.20. The COSSB signal

is generated as in Figure 6.1, however, the logarithmic amplifier was not used and

it was replaced by a short circuit. The scaling and offset applied to the information

were optimized so that the input to the HT looked as similar as possible to the

logarithm of the signal envelope on the fiber, as required by COSSB theory. Since

the 10 Gb/s data was not filtered and consists of relatively square bits, scaling and

level shifting will generate a signal that looks relatively close to the logarithm of the

signal envelope on the fiber.

Using the scaled and level shifted data signal as an input, transient simulation

of the HT over a 200 ns duration was performed in Cadence including the effect of

layout capacitances and resistances. Once the output of the HT was obtained and

loaded into Matlab, a noise signal with approximately the same noise bandwidth,

10 GHz, and RMS value, 590 µv, as the noise of the HT was added to the HT

output. The resulting electrical signal was filtered at 13 GHz before it was used to

phase modulate the optical signal at the transmitter. Filtering the HT output helped

remove unwanted energy and was found to improve the transmission characteristics.

Figure 6.3(a) shows the spectrum of the transmitted COSSB signal and Fig-

ure 6.3(b) shows the corresponding eye diagram. Figure 6.3(a) indicates that be-

tween 15 and 20 dB of sideband cancellation is possible. This is approximately the

same as what Sieben observed for a 10 Gb/s signal, except in this experiment the


−20 −15 −10 −5 0 5 10 15 20−70

−65

−60

−55

−50

−45

−40

−35

−30

−25


Mag

nitu

de (

dB)

(a) Spectrum. (b) Eye diagram.

Figure 6.3: Transmitted 10 Gb/s COSSB spectrum and eye diagram with transientsimulation of HT IC without logarithmic amplifier.

HT has been successfully integrated.

The signal in Figure 6.3(a) requires only the HT IC, an amplitude modulator,

and a phase modulator. Since these parts are all available, this signal should be

obtainable experimentally. It is instructive to consider the transmission properties

of this signal over long lengths of fiber. Figure 6.4 shows the eye diagrams of the

detected signal from Figure 6.3(a) after transmission over various distances for the

case where only self-homodyning post detection equalization is used. Recall from

Section 2.3.2 that this type of equalization is typically performed by passing the

detected signal through a microstrip line, the delay of which increases with increasing

frequency. To simulate dispersion, the fiber transfer function from equation (2.30)

was used with a dispersion parameter of D = 18 ps/(nm · km). Figure 6.4 indicates

that the eye stays open after 200 km, even though no optical dispersion compensation

is being used. Figure 6.5 shows similar eye diagrams for the case where the minimum

phase dispersion compensation technique from Section 2.4.5 is used. For this case,


(a) After 100 km. (b) After 200 km.

(c) After 300 km. (d) After 500 km.

Figure 6.4: Eye diagrams of COSSB system using only the HT and using onlyself-homodyning post detection equalization.

the eye remains open after transmission over 700 km of dispersive fiber without

optical dispersion compensation. This indicates that construction of the minimum

phase dispersion equalizer from Figure 2.10, while beyond the scope of this thesis,

would be very useful. The logarithmic amplifier and HT developed in this thesis is

a necessary component of that equalizer.

Granted, the above simulations are not totally realistic because some important


(a) After 100 km. (b) After 200 km.

(c) After 300 km. (d) After 500 km.

(e) After 700 km. (f) After 900 km.

Figure 6.5: Eye diagrams of COSSB system using only the HT and using onlyminimum phase post detection equalization.

6.4 Combined Performance of Logarithmic Amplifier and HT Circuits 125

effects have been ignored. If a signal were transmitted over hundreds of kilometers of

fiber, some optical amplifiers would be required. These would contribute noise and

distortion. If these effects were taken into account, the transmission distance ob-

tained while maintaining comparable eye diagrams would decrease somewhat. Still,

the eye diagrams in Figure 6.9 give an idea of the performance degradation versus

distance due only to dispersion.

Furthermore, the above simulations indicate the transmission distances that are

possible when only post detection equalization is used. However, in this author’s

opinion, a more intriguing possibility is when the dispersion is partially compensated

using optical dispersion compensation and OSSB. As an example, consider the case

where, based on a rough estimate of the amount of dispersion in a fiber-optic link,

the system engineer inserted optical dispersion compensation to cancel out half of the

estimated dispersion. Normally, such imprecise compensation would give inadequate

performance. However, if this technique were used with OSSB and microstrip self-

homodyning post detection compensation, the system eye diagrams would be exactly

the same as those shown in Figure 6.4, however all of the distances will have doubled

since the optical dispersion has been halved. This approach combines the benefits

of both dispersion management techniques.

6.4 Combined Performance of Logarithmic Amplifier and

HT Circuits

6.4.1 Performance at a Mach-Zehnder Modulation Depth of 0.25

In this section, the performance of the logarithmic amplifier IC and the HT IC to-

gether in the COSSB system will be evaluated. To start with, a 1550 nm optical

signal was again amplitude modulated using a Mach-Zehnder modulator at a mod-


ulation depth of 0.25. The electrical data consists of a 211 − 1 PRBS signal filtered

at 2.75 GHz. This signal is simply an example, and it will serve to illustrate the

usefulness of the log amplifier. Signals for which the logarithmic amplifier are im-

portant include predistorted or QAM signals for broadband radio-on-fiber systems,

and multilevel baseband signals. Furthermore, the logarithmic amplifier is especially

important for binary signals that are band-limited or when they are transmitted at

modulation depths of 0.25 or higher, as will be shown.

Figure 6.6(a) shows the normalized envelope of the amplitude modulated optical

signal. The relatively high modulation depth of 0.25 results in the signal having high

extinction, meaning that the amplitude approaches zero. Figure 6.6(b) shows the

logarithm of this signal, which is the ideal output of the logarithmic amplifier. Since

the signal in Figure 6.6(a) contains information over the 40 dB of dynamic range

between 0.01 and 1.0, the logarithm of this envelope looks much different than the

envelope itself, and the use of the logarithmic amplifier is important in achieving

effective single sideband transmission.

The output of the logarithmic amplifier is exactly the same as that used in Sec-

tion 6.2, where an ideal HT was assumed. However, this time the output of the

logarithmic amplifier with the log amplifier noise added was then scaled and input

to the HT IC in simulation. The output of the HT IC then had the HT IC noise

added to it. For all 5 Gb/s simulations in this chapter, the signal at the output of

the HT IC was filtered at 6 GHz. This helped remove unwanted energy from the

phase modulation signal and was found to improve the transmission characteristics

of the signal. The resulting signal was used to phase modulate the optical signal.

For comparison purposes, a COSSB signal was also generated using only the HT IC,

and not the log amplifier IC. Figure 6.7 shows the spectra and eye diagram at the

COSSB transmitter output for the two cases. It is observed that the inclusion of the


30 31 32 33 34 35 36 37 38 39 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (ps)

Line

ar A

mpl

itude

(a) Optical signal envelope.

30 31 32 33 34 35 36 37 38 39 40−6

−5

−4

−3

−2

−1

0

Time (ps)

Line

ar A

mpl

itude

(b) Logarithm of optical signal envelope.

Figure 6.6: Scaled optical signal envelope and its logarithm for a modulation depthof 0.25.

logarithmic amplifier improves the sideband cancellation by approximately 5 dB.

Using the dispersive fiber transfer function from equation (2.30) with a dispersion

parameter of D = 18 ps/(nm · km), the effect of dispersion on various signals was

examined. Figure 6.8 shows the eye diagram of the comparable DSB signal after

propagation over 400 km of uncompensated dispersive fiber. This signal is badly

distorted because the dispersion-induced phase distortion of the signal results in

frequency selective power fading during detection. Figure 6.9 shows the eye diagram

after 400 km for the case where the only the HT IC was used, and the log amplifier IC

was not included. For this simulation, self-homodyning post detection equalization

was simulated for the received optical signal. A distance of 400 km was chosen,

because this is the approximate distance that a 5 Gb/s signal must propagate for

dispersion to be significant. A 10 Gb/s signal only has to travel one quarter of this

distance, or 100 km, for dispersion to be significant. The four-to-one relationship

arises from the fact that frequency is squared in the dispersive fiber transfer function


−10 −8 −6 −4 −2 0 2 4 6 8 10−70

−65

−60

−55

−50

−45

−40

−35

−30

−25

−20

Frequency (GHz)

Mag

nitu

de (

dB)

(a) With HT IC and without logarithmic am-plifier.

−10 −8 −6 −4 −2 0 2 4 6 8 10−70

−65

−60

−55

−50

−45

−40

−35

−30

−25

−20

Frequency (GHz)

Mag

nitu

de (

dB)

(b) With HT and logarithmic amplifier ICs.

−200 −150 −100 −50 0 50 100 150 200−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (ps)

Am

plitu

de

(c) With HT and logarithmic amplifier ICs.

Figure 6.7: Spectra and eye diagram of 5 Gb/s COSSB signals filtered at 2.75 GHzfor a modulation depth of 0.25.

from equation (2.30). The signal in Figure 6.9 is badly distorted, and so using the HT

IC alone for such a signal gives poor performance. Minimum phase post-detection

compensation was also tried for this signal, however the eye diagram was still closed.

Figure 6.10 shows the eye diagram of the COSSB signal generated using both

the log amplifier and HT ICs and using self-homodyning post detection equalization.


Figure 6.8: Eye diagram of 5 Gb/s signal recovered from a DSB system after 400 kmof uncompensated dispersive fiber for a modulation depth of 0.25.

Figure 6.9: Eye diagram of 5 Gb/s signal recovered from COSSB system with HTIC, without logarithmic amplifier, and using only self-homodyning post detectionequalization after 400 km. The modulation depth is 0.25

In this case, the shape of the eye is improved, and there actually is an eye opening

up to 600 km. However, the signal is somewhat noisy and distorted. Some of this

distortion relates to the fact that the 5 Gb/s signal was band-limited to 2.75 GHz at

the transmitter in order to obtain an analog signal in order to test the logarithmic

amplifier. The band-limited signal had some ISI at the transmitter, and the imperfect

sideband cancellation allows for some dispersion induced degradation of the signal.

Still, the received signal at 400 km is unusable without the log amplifier, and adding


(a) 400 km. (b) 600 km.

(c) 800 km.

Figure 6.10: Eye diagrams of 5 Gb/s signals recovered from COSSB system with HTIC and with logarithmic amplifier and using only self-homodyning post detectionequalization. The modulation depth is 0.25.

the log amplifier opens the data eye.

Next, consider the eye diagrams for the case where the log amplifier IC is included

and where minimum phase post detection equalization is used at the receiver, shown

in Figure 6.11. In this case, the noise and distortion remain significant, and the

benefits of minimum phase compensation are less evident.

It was hypothesized that since the eye diagram was poor after 400 km for the


(a) 400 km. (b) 600 km.

(c) 800 km.

Figure 6.11: Eye diagrams of 5 Gb/s signals recovered from COSSB system withHT IC and with logarithmic amplifier and using only minimum phase post detectionequalization. The modulation depth is 0.25.

system without the logarithmic amplifier, that the recovered OSSB spectrum for

this system would have large power fades in it. However, the simulation indicated

that this was not the case. Figure 6.12 shows spectra recovered from OSSB systems

with the system containing the log amplifier plotted in black in the background,

and the spectra from the system with no log amplifier in yellow in the foreground,

which will show up as a light gray when viewed in black and white. There are


−10 −8 −6 −4 −2 0 2 4 6 8 10−70

−65

−60

−55

−50

−45

−40

−35

−30

−25

−20

Frequency (GHz)

Mag

nitu

de (

dB)

System without logarithmic amplifier.

System with logarithmic amplifier.

Figure 6.12: Spectra recovered from OSSB systems after 400 km with and withoutlogarithmic amplifier.

subtle differences between the two recovered spectra, however it is not obvious from

this plot that the system without the logarithmic amplifier yields a much worse eye

diagram when recovered. Filtering the recovered spectra at 4 GHz before plotting the

recovered signal eye diagrams was tried, however this had a negligible effect on the

eye diagrams, indicating that the problem with the spectrum for the system with no

log amplifier lies at frequencies below 4 GHz. Davies suggested that the distortion

in the signal from the system without the log amplifier may be contained in the

phase of the recovered signal [54]. This indicates that although the types of post

detection compensation discussed in this thesis fail to recover the signal, another

method which compensates for the particular distortion of this signal may succeed.

This possibility is beyond the scope of this thesis, and was not investigated further.


30 31 32 33 34 35 36 37 38 39 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (ps)

Line

ar A

mpl

itude

(a) Optical signal envelope.

30 31 32 33 34 35 36 37 38 39 40−2

−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

Time (ps)

Line

ar A

mpl

itude

(b) Logarithm of optical signal envelope.

Figure 6.13: Scaled optical signal envelope and its logarithm for a modulation depthof 0.20.

6.4.2 Performance at a Mach-Zehnder Modulation Depth of 0.20

In this section, we consider the obtainable performance when the system simulations

are identical to those in Section 6.4.1, but a Mach-Zehnder modulation depth of

0.20 is used instead of 0.25. As a result, it will be shown that the log amplifier is

unnecessary at this lower modulation depth.

Figure 6.13(a) shows the envelope of the amplitude modulated optical signal. The

modulation depth of 0.20 results in the amplitude reaching approximately 0.18 on

the lower end, instead of about 0.01 as in the last section. As a result, the logarithm

of this envelope, shown in Figure 6.13(b), looks more like a scaled version of the

envelope itself, and the logarithmic amplifier becomes less important.

Figure 6.14 shows the spectra and eye diagrams of the signals at the output of

the transmitter for the cases where the HT IC is used and the log amplifier is or is

not used. The improvement in sideband cancellation obtained with the log amplifier

is not as significant as in the last section, as expected.


−10 −8 −6 −4 −2 0 2 4 6 8 10−70

−65

−60

−55

−50

−45

−40

−35

−30

−25

−20

Frequency (GHz)

Mag

nitu

de (

dB)

(a) With HT IC and without logarithmic am-plifier.

−10 −8 −6 −4 −2 0 2 4 6 8 10−70

−65

−60

−55

−50

−45

−40

−35

−30

−25

−20

Frequency (GHz)

Mag

nitu

de (

dB)

(b) With HT IC and with logarithmic ampli-fier.

(c) With HT and logarithmic amplifier ICs.

Figure 6.14: Spectra and eye diagram of 5 Gb/s COSSB signals filtered at 2.75 GHzfor a modulation depth of 0.20.

Figure 6.15 shows the eye diagram of a DSB signal created with a modulation

depth of 0.20 after 400 km of uncompensated fiber. The signal is badly distorted,

again showing the motivation for performing single sideband modulation.

Figure 6.16 shows the eye diagrams at incremental distances for the COSSB

system where only the HT IC is used and self-homodyning post detection equalization


Figure 6.15: Eye diagram of 5 Gb/s signal recovered from a DSB system after 400 kmof uncompensated dispersive fiber for a modulation depth of 0.20.

is used. The eye remains open for longer distances than in the last section even

though the log amplifier is not being used, showing that the log amplifier is not

required here. As well, the lower modulation depth reduces nonlinear distortion

from the Mach-Zehnder and improves transmission distance. The price is reduced

signal-to-noise ratio in an optical system. Figure 6.17 shows the eye diagrams for the

case where minimum phase post detection equalization is used. These eye diagrams

are improved over the self-homodyning equalization case.

Figure 6.18 shows the eye diagrams when self-homodyning detection is used and

both the log amplifier and HT ICs are used. Comparing this to Figure 6.16, it is ob-

served that there is little improvement in the eye diagram, as expected. Figure 6.19

shows the eye diagrams for the case where the log amplifier is included and mini-

mum phase dispersion compensation is used. The minimum phase eye diagrams are

improved compared to the self-homodyning case, but are not substantially different

than for the case where the log amplifier was not used. In short, if a modulation

depth of 0.20 or lower is used, the HT alone should be sufficient to achieve good

COSSB transmission.

6.5 Conclusion 136

(a) 400 km. (b) 1000 km.

(c) 2000 km.

Figure 6.16: Eye diagrams of 5 Gb/s signals recovered from COSSB system with HTIC and without logarithmic amplifier and using only self-homodyning post detectionequalization. The modulation depth is 0.20.

6.5 Conclusion

In this chapter, simulations showed that the HT IC allowed for COSSB transmis-

sion at bit rates of 5 and 10 Gb/s over uncompensated dispersive fiber lengths of

600 and 200 km respectively using only post detection equalization. Furthermore,

the logarithmic amplifier was shown to improve performance for the case of a band-

6.5 Conclusion 137

(a) 400 km. (b) 1000 km.

(c) 2000 km. (d) 3000 km.

Figure 6.17: Eye diagrams of 5 Gb/s signals recovered from COSSB system withHT IC and without the logarithmic amplifier and using only minimum phase postdetection equalization. The modulation depth is 0.20.

limited 5 Gb/s signal at a modulation depth of 0.25. The next chapter contains actual

measurements of the SSB spectra generated at the output of a COSSB transmitter.

6.5 Conclusion 138

(a) 400 km. (b) 1000 km.

(c) 2000 km.

Figure 6.18: Eye diagrams of 5 Gb/s signals recovered from COSSB system with HTIC and with logarithmic amplifier and using only self-homodyning post detectionequalization. The modulation depth is 0.20.

6.5 Conclusion 139

(a) 400 km. (b) 1000 km.

(c) 2000 km. (d) 3000 km.

Figure 6.19: Eye diagrams of 5 Gb/s signals recovered from COSSB system withHT IC and with logarithmic amplifier and using only minimum phase post detectionequalization. The modulation depth is 0.20.

Chapter 7

Measurements of COSSB Transmitters

7.1 10 Gb/s COSSB Experiment Using the HT

Figure 7.1 shows a block diagram of the COSSB digital signal measurement system.

More details of the equipment are given in Appendix G. The key components are

the UTP phase modulator, which is driven by the amplified HT output, and the

Sumitomo intensity modulator, which is driven by the amplified digital input data.

A JCA amplifier drives the intensity modulator, and this amplifier is a digital limiting

amplifier, meaning that the output is a 7 Vpp square wave regardless of the input

amplitude. If the input signal is below approximately 100 mV, the output signal

simply switches between 0 and 7 Vpp at approximately 1 MHz. Since this digital

limiting amplifier was chosen, it was decided to put the phase modulator before the

intensity modulator and to delay the input signal to the intensity modulator. This

arrangement of optical modulators is contrary to all of the minimum phase COSSB

modulators shown so far in this thesis. The intensity modulator has two meters of

fiber between the body of the modulator and the input or output connector, and the

phase modulator has eighty centimeters of fiber between the body of the modulator

and the input or output connector. The length of the fiber leads on the modulators

should not be shortened, or inter-modal optical distortion may result. Hence, there

are 2.8 m of fiber between the optical modulators, and approximately the same length

of SMA cable is needed to delay the signal going to the second optical modulator,

as described in Appendix G. When the 10 Gb/s signal propagates through such a

long length of cable, there will be a low pass filter effect. However, if this signal goes

7.1 10 Gb/s COSSB Experiment Using the HT 141

to the JCA amplifier, it will be restored to a square wave. The only disadvantage

of doing this is that the optical intensity modulator is polarization sensitive, and a

polarization adjuster can not be placed before it as that would create a much longer

length of fiber between the modulators. Although having the phase modulator first

gave the best results, it made it difficult to reliably set the polarization during

the experiments. The polarization would change over periods of a few seconds.

Changes in polarization showed up when they caused a sub-spike in power beside

the carrier and clock spikes in the optical spectrum due to interactions with the

optical filter used. For this reason, some experiments in later sections in this chapter

were performed with the intensity modulator first and the input signal to the HT

was delayed by the required amount. These problems could be overcome by using

a cascaded amplitude and phase modulator on a single substrate, a device which

already exists but was not readily available at the bandwidth required. Furthermore,

it should be noted that the DATA signal served as the basis for one modulator

signal, and the DATA signal was the basis for the other modulator signal. It may

be argued that slight differences in the two signals will degrade the best obtainable

performance, and that one of the signals should be passed through a splitter and

used for both modulators. This was tried, and the results were no better because

the JCA limiting amplifier restores one signal to a square wave but the other signal

experiences the loss of the splitter. Sieben had to put up with similar imperfections

in his experiments [7].

A Micron Optics tunable Fiber Fabry-Perot (FFP) bandpass filter with 400 MHz

bandwidth was used to measure the the optical spectra. To view the optical spectra

in real time, a 1 Hz triangle wave was used to sweep the filter, and the analog output

of the optical power meter was connected to an oscilloscope. To record the optical

spectra, the FFP was swept at 1 mHz while a computer downloaded readings from


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the optical power meter. The fact that the FFP has only 400 MHz bandwidth is

amazing, since the center wavelength of 1550 nm corresponds to a center frequency

of approximately 200 THz. The filter is somewhat non-linear, so that the wavelength

does not increase linearly with increasing applied voltage, although it is fairly close.

Furthermore, it exhibits some hysteresis, so that the center frequency of the filter

does not alway correspond to a given voltage. It was necessary to use a modest

exponential distortion of the frequency before the data was plotted so that the clock

spikes on either side of the optical carrier were equidistant from the carrier. However,

the magnitude response of the filter was assumed perfectly flat, and no changes were

made to the spectral power levels prior to plotting.

Figure 7.2 shows the measured frequency spectrum and eye diagram for an optical

signal modulated with a 10 Gb/s signal using COSSB modulation at 16 dBm of

optical intensity power, and with 130 mVpp input to the HT and 40 dB of gain after

the HT. A 231 − 1 length PRBS data signal was used for all broadband experiments

in this chapter. The lower sideband has a measured average spectral power density

which is 7.5 dB higher than the upper sideband, compared to 11.6 dB simulated.

All measurements in this chapter were made with the Q-enhancement of the HT

turned off, and when it was turned on it had no noticeable impact. The input to the

phase modulator for this measurement was approximately 4.5 Vpp, and the power

of this signal measured on a power meter was 14.5 dBm. Figure 7.2(a) also shows

the laser spectrum, and it is observed that the upper sideband has been canceled

down to the laser power across the band within the accuracy of the measurement. No

changes have been made to the optical power, these spectra are at the absolute optical

power recorded. The intensity modulation power of 16 dBm corresponds to a 1.4 V

signal, and since the Vπ of the modulator is approximately 6 V, this modulation

power is 0.23·Vπ. This is approximately the same as the 0.20 modulation depth


−10 −5 0 5 10−60

−55

−50

−45

−40

−35

−30

−25

−20

−15


Opt

ical

Pow

er (

dBm

)COSSB SignalLaser Only

(a) COSSB Spectrum.

7 mV/div25 ps/div

(b) Eye diagram.

Figure 7.2: Spectrum and eye diagram of 10 Gb/s COSSB signal for 16 dBm ofintensity modulation power.

used in the simulations in the last chapter. Data was also recorded for intensity

modulation powers which are 5 dB lower and 4 dB higher than this. When the

intensity modulation is lower than 16 dBm, there is less than 10 dB of signal power

above the laser spectral floor and there is less signal to noise ratio in the eye diagram.

Figure 7.3 shows the optical spectrum and eye diagram for an intensity modulation

power of 20 dBm or 0.375 ·Vπ. Here, the lower sideband has an average spectral

power density which is 6.8 dB higher than the upper sideband. This is comparable

to the performance at 16 dBm, except now the signal has a better SNR. However, at

this higher intensity modulation power some distortion in the optical signal envelope

is expected from the Mach-Zehnder which is not accounted for by the HT.

The fact that approximately 7 dB of average spectral suppression was obtained

instead of the 11.6 dB simulated suppression is partially attributed to the failure of

the simulation of the HT to account for substrate coupling on the IC. Although the

time domain measurements shown in Chapter 5 did not indicate a serious problem,

7.2 COSSB Experiments Using the HT and the Logarithmic Amplifier 145

−10 −5 0 5 10−60

−55

−50

−45

−40

−35

−30

−25

−20

−15


Opt

ical

Pow

er (

dBm

)COSSB SignalLaser Only

(a) COSSB Spectrum.

7 mV/div25 ps/div

(b) Eye diagram.

Figure 7.3: Spectrum and eye diagram of 10 Gb/s COSSB signal for 20 dBm ofintensity modulation power.

substrate coupling could be a problem with certain data patterns. Other factors

such as mismatches in the phase and intensity modulation signals, distortion in the

modulator driver amplifiers, frequency dependent loss in the modulators, and any

extra group delay distortion in the measurement setup contribute to the discrepancy.

7.2 COSSB Experiments Using the HT and the Logarithmic

Amplifier

7.2.1 Experiment Using a 1.9 GHz Sinusoid

In order to quantify the performance of the COSSB system when the logarithmic

amplifier is added, a sinusoid was first used as the data signal. Sinusoids at 1.9 GHz

were generated by two different signal sources, and were filtered at the output of

the sources using Lorch 8BP8-1800/300-S bandpass filters to remove any harmonics

at the input of the system. The rest of the test system was similar to that shown


in Figure 7.1, except that the JCA amplifier was not used since it is nonlinear. An

intensity modulation power of 23 dBm was used. The resulting COSSB spectra were

then measured for the case where the logarithmic amplifier is or is not used. Fig-

ure 7.4 shows the spectra for the two cases. There are harmonics present because

of the nonlinearity of the Mach-Zehnder amplitude modulator, and the best an ex-

perimenter can hope to do is to cancel all of the tones in the unwanted sideband,

and leave the desired sideband unchanged. In either case in Figure 7.4, better than

20 dB suppression of the fundamental tone in the lower sideband was obtained. The

fact that this signal is upper sideband is simply because the signal polarization was

such that the phase and intensity modulation inputs were 180 out of phase. The

next highest power tone is the second harmonic. Note that the optical envelope

contains even harmonics, but the detected optical signal does not. Essentially no

suppression of the second harmonic was obtained using only the HT, as expected.

However including the logarithmic amplifier yielded 10 dB of second harmonic sup-

pression, which is a strong indication of the value of the logarithmic amplifier. The

third and higher harmonics are very low power. The third harmonic in the HT only

signal for the upper sideband is located just below 6 GHz, and is barely noticeable,

and is smaller than the third harmonic in the lower sideband. Simulations of a

sinusoidal signal in a COSSB system without a logarithmic amplifier confirm that

the harmonics in the desired sideband are sometimes suppressed. In the spectrum

of the signal with the logarithmic amplifier, the third and fourth harmonics in the

lower sideband are suppressed by approximately 4 dB each. The frequencies of the

second and higher harmonics for the two signals don’t line up perfectly because not

all of the nonlinear frequency response of the filter could be equalized. However, the

power levels are unaltered and directly from the optical power meter. The intensity

modulator had to be placed first for this experiment for precise optical polarization


−10 −8 −6 −4 −2 0 2 4 6 8 10−55

−50

−45

−40

−35

−30

−25

−20

−15

−10


Opt

ical

Pow

er (

dBm

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With HT OnlyWith HT and Logarithmic Amplifier

Figure 7.4: Spectrum of 1.9 GHz COSSB signal for 23 dBm of intensity modulationpower.

control.

It is also observed that the third harmonic of the upper sideband is suppressed

in the spectrum for only the HT in Figure 7.4. In fact, Davies showed that the odd

harmonics are suppressed in alternating sidebands, as shown in Figure 4.13 of his

Ph.D. dissertation [6].

7.2.2 Experiment Using Filtered 5 Gb/s Data

Figure 7.5 shows the measurement system used to quantify the performance of the

logarithmic amplifier for the case of a 5 Gb/s signal filtered at 2.9 GHz. This

setup is similar to the simulations in the last chapter. The filter cutoff of 2.9 GHz is

somewhat higher than the 2.75 GHz simulated cutoff frequency. However, the change

in performance that this causes should not invalidate the experiment. A data rate


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of 5.5 Gb/s was also tried, and no difference in the results was observed.

It is somewhat difficult to determine the DC offset that should be added to the

log amplifier input so that the output signal appears like the logarithm of the optical

signal envelope. Figure 7.6 shows the measured input and out waveforms for the

logarithmic amplifier for the repeated DATA sequence 01001111. It is necessary

that the bits of data applied to the intensity modulator that cause the optical signal

to extinguish are stretched out by the logarithmic amplifier. In the example shown,

the signal in Figure 7.6(b) has been inverted and then logged compared to the signal

in Figure 7.6(a). Nonetheless, this example shows the type of distortion that is per-

formed by the logarithmic amplifier. Due to the particular arrangement of inverting

modulator driving amplifiers, the inversion of the DATA signal which drives the log-

arithmic amplifier compared to the DATA signal for the intensity modulator, and

even the use of inverting modulation points on the Mach-Zehnder modulator, the

optical signal can be lower or upper sideband. Whether the signal was upper or lower

sideband was always carefully predicted before each experiment. It is also critical

to predict if the signal is upper or lower sideband in order to get the delay between

the modulator input signals correct, as described in Appendix G. Once the COSSB

signal optical spectrum was displayed on the oscilloscope, the logarithmic amplifier

offset and the delay could be optimized in real time. The intensity modulator could

even be switched from a non-inverting bias point to an inverting bias point in or-

der to view the degradation in sideband cancellation when the logarithmic amplifier

performs the logarithm with the wrong polarity due to an incorrect DC offset.

Figure 7.7 shows the measured optical spectrum for the 5 Gb/s COSSB signal for

systems with and without the logarithmic amplifier. These measurements were ac-

tually taken with the intensity modulator placed first in the OSSB system, followed

by the phase modulator. This data was used because the sideband cancellation from


(a) Input waveform. (b) Output waveform.

Figure 7.6: Logarithmic amplifier waveforms.

the system with the phase modulator placed first was actually 2-3 dB worse for this

power level. This could have been caused by sub-optimal optical polarization for the

intensity modulator for this configuration, a problem which was already discussed in

Section 7.1. The spectra for the COSSB system without the logarithmic amplifier

has 6.4 dB contrast in average spectral density from 0-2.9 GHz, whereas the spec-

trum for the system with the logarithmic amplifier has 6.1 dB contrast in spectral

sideband suppression from 0-2.9 GHz, slightly worse but approximately equal within

the accuracy and repeatability of the measurements. Furthermore, the maximum

sideband cancellation at any one frequency is approximately 10 dB for these signals.

Based on the simulations in Section 6.4, the sideband cancellation at 2 GHz is ex-

pected to go from approximately 10 dB to 15 dB when the logarithmic amplifier is

added. However, simulation from Section 6.2 earlier in that chapter also predicted

15-20 dB maximum sideband cancellation for the 10 Gb/s digital signal, and only

about 12 dB was actually achieved. For this reason, better than 12 dB maximum

sideband cancellation should not be expected for the experiment in this section, due

to limitations in the HT and in the optical components. So, it is not surprising that


−6 −4 −2 0 2 4 6−60

−55

−50

−45

−40

−35

−30

−25

−20

−15

−10

−5


Opt

ical

Pow

er (

dBm

)

With HT OnlyWith HT and Logarithmic AmplifierLaser Power Only

Figure 7.7: Spectrum of 5 Gb/s COSSB signal for 17 dBm of intensity modulationpower.

no significant improvement is obtained by adding the logarithmic amplifier for the

5 Gb/s signal. The addition of the logarithmic amplifier for the 1.9 GHz sinusoid in

Section 7.2.1 suppressed the second harmonic, but this harmonic was approximately

17 dB below the fundamental, well below the noise and distortion floor of the broad-

band spectra observed so far. In short, the addition of the logarithmic amplifier for

the 5 Gb/s, 17 dBm intensity modulation power signal does not improve the side-

band cancellation, but this appears to be only partially due to imperfections in the

logarithmic amplifier.

Figure 7.8 shows the COSSB spectra for systems with and without the logarithmic

amplifier and eye diagrams for the COSSB and DSB signals for a 20 dBm intensity

modulation signal for the 5 Gb/s signal filtered at 2.9 GHz. The DSB eye diagram

7.3 Conclusion 152

is shown just to prove that it is basically the same as the COSSB eye diagram,

demonstrating that the phase modulation does not degrade the eye diagram at the

output of the transmitter. The spectra for this power level are based on data taken

with the phase modulator placed first in the COSSB system followed by the intensity

modulator, since the data from this case was the best for this intensity modulation

level. Here, the contrast in the average spectral density between the sidebands

between DC and 2.9 GHz is approximately 5.7 dB without the logarithmic amplifier

and is 6.7 dB with the logarithmic amplifier. This shows a small improvement

when the logarithmic amplifier is added, but it is hardly worth the increased system

complexity in this case. The spectrum for the system with the logarithmic amplifier

is also observed to increase in power by a few decibels beyond approximately 2.7 GHz.

These frequencies lie near the cutoff of the 2.9 GHz low pass filter, and so the input

signal to the logarithmic amplifier may be somewhat distorted by the low pass filter.

The phase shift of the logarithmic amplifier may combine with this distortion to

impede the sideband cancellation at these frequencies.

7.3 Conclusion

In this chapter, measurements of a 10 Gb/s optical single sideband system indicate

that 7 dB of broadband sideband suppression is obtainable using the HT IC for the

case of a digital signal. Measurements of a COSSB system using a 5 Gb/s signal

filtered at 2.9 GHz show that approximately 7 dB of sideband suppression is also

available whether or not the logarithmic amplifier is used. However, measurements

of a COSSB system using a 1.9 GHz sinusoid clearly show improved suppression

of all harmonics when the logarithmic amplifier is added. The logarithmic amplifier

seems best suited for narrow-band COSSB applications, and the HT was proved very

7.3 Conclusion 153

−10 −8 −6 −4 −2 0 2 4 6 8 10−60

−55

−50

−45

−40

−35

−30

−25

−20

−15

−10

−5


Opt

ical

Pow

er (

dBm

)With HT OnlyWith HT and Logarithmic AmplifierLaser Power Only

(a) COSSB Spectrum.

7 mV/div50 ps/div

(b) COSSB eye diagram.

7 mV/div50 ps/div

(c) DSB eye diagram.

Figure 7.8: Spectrum of 5 Gb/s COSSB signal and eye diagrams of 5 Gb/s COSSBand DSB signals for 20 dBm of intensity modulation power.

7.3 Conclusion 154

successful in broadband applications.

Chapter 8

Conclusions

In this thesis a logarithmic amplifier and HT were developed for the COSSB appli-

cation. While developing the logarithmic amplifier, a mathematical characterization

and design procedure for CHEF amplifiers was developed. This characterization dif-

fers from the one presented by Ohhata et al. because it is developed for HBTs for

which the base resistance is less than approximately 100 Ω, which is more appro-

priate for modern HBT technologies [24, 30]. The equations by Ohhata et al. are

not valid for this case. Furthermore, the mathematics presented here is different in

that resistor R2 is included. In addition, the equations necessary to design for the

important condition Q ∼= 1/√

3 are described. Furthermore, the work here is novel

in that equations for the DC transfer characteristic and output noise of the CHEF

amplifier are developed and are used to form a comprehensive design procedure.

This procedure is useful because it gives the design engineer an understanding of the

effect of the numerous component values in the CHEF amplifier. This work has been

published in the IEEE Journal of Solid State Circuits [25].

While designing a logarithmic amplifier for the COSSB application, the exist-

ing series linear-limit and parallel amplification, parallel summation architectures

were evaluated. The series linear-limit amplifier was found to have insufficient band-

width, and the parallel amplification, parallel summation architecture was found to

have insufficient logarithmic dynamic range. Using existing descriptions of each ar-

chitecture [38, 41], a novel generalized mathematical analysis of parallel summation

logarithmic amplifiers was developed. This analysis was the first, to the author’s

knowledge, which described the exact path gains and limiting currents required to

156

obtain a logarithmic response and then proved the ensuing logarithmic relationship

between the amplifier output current and input voltage. Furthermore, the analysis

made it clear that the logarithmic slope is proportional to the limiting currents in

the summing amplifier. As well, this analysis was used to choose the gains and lim-

iting current in a hybrid parallel summation architecture, which was implemented

in two different technologies, the second of which achieved the highest bandwidth to

date for a true logarithmic amplifier, DC-6 GHz. This amplifier is novel in that it

used branching amplification paths in order to achieve 39 dB of gain, 6 GHz band-

width, and matched group delay among the gain paths in a 1.33 mm by 1.50 mm

integrated circuit, fabricated in a 47 GHz fT technology. Furthermore, a novel and

efficient design for implementing the progressive compression logarithmic amplifier

with matched group delay among the paths was suggested. Previously published pro-

gressive compression logarithmic amplifiers did not have matched group delay paths,

or they required more amplifiers to implement [55]. Finally, circuits for reducing DC

offset errors in DC coupled logarithmic amplifiers were described and implemented

successfully.

The logarithmic amplifier implementations presented in this thesis were specifi-

cally designed for the OSSB application and have a power consumption that is likely

too high for hand-held radios. This is unfortunate, because many radios use demod-

ulating logarithmic amplifiers as receive strength signal indicators (RSSI). However,

the mathematical description and procedure for choosing the amplifier gains and

limiting currents that was developed here is useful for engineers designing logarith-

mic amplifiers in any application. In many cases, the only difference between the

logarithmic amplifiers presented here and those used in RSSI applications is a re-

laxed bandwidth constraint for RSSI and the addition of envelope detectors on the

output of each path. When one considers that RSSI circuits are used extensively in

157

military radio transceivers, and that circuits in these applications must have excel-

lent logarithmic linearity, then it is critical to choose the logarithmic amplifier path

gains and limiting currents correctly. The mathematical description of logarithmic

amplifiers in this thesis shows how to achieve that goal. The work on logarithmic

amplifiers has also been published in the IEEE Journal of Solid State Circuits and

has been patented [36, 56].

The HT in this work is the first published fully integrated 10 Gb/s HT to the

author’s knowledge, which is a significant engineering achievement. This circuit

was used to generate a 10 Gb/s OSSB signal in the laboratory, only the second

such experiment in which a 10 Gb/s OSSB signal was created with an HT, to the

author’s knowledge [7]. The implementation of the HT described in this thesis serves

as a useful example for engineers working on tapped delay filters for equalization or

predistortion in fiber optic networks, and for engineers designing tapped delay filters

for ultra wideband radios, which have similar broadband requirements to circuits for

fiber optics. A letter on the HT is scheduled to appear in the May issue of IEEE

Microwave and Wireless Component Letters.

The logarithmic amplifier and HT were successfully tested in the COSSB system.

The tests with a 1.9 GHz sinusoid verified the complete COSSB theory, including the

requirement for a logarithm. All tests performed with the HT verified the operation

of and provided rare measurement data for the COSSB system.

8.0.1 Future Work

It should now be possible to integrate an entire COSSB transmitter. It would consist

of an optical phase modulator cascaded with an intensity modulator, and an HT and

a driver amplifier to drive the phase modulator. A logarithmic amplifier could also be

used in the case of narrow-band signals, such as optical subcarrier signals. If the two

158

optical modulators are physically very close together, it may be necessary to delay

the electrical signal used for intensity modulation until the electrical signal used

for phase modulation has passed through the logarithmic amplifier and HT. Such

a 10 Gb/s COSSB transmitter would be an effective way to deal with dispersion,

and could be used on its own or alongside existing optical dispersion compensation

methods.

Some further suggestions for future work include:

• The OSSB system simulations could be compared with the performance of

duobinary transmission to evaluate the comparative advantages and disadvan-

tages of each technique.

• The HT could be implemented in CMOS technology, to allow it to be integrated

with any other signal processing technology in optical transceivers.

• The HT could be implemented at 40 Gb/s. This would be interesting to try

because the delays required for the on chip transmission line will be one quarter

of that given in this thesis. A 40 Gb/s HT would be even more useful than

the 10 Gb/s one, because of the severe dispersion at 40 Gb/s. However, the

requirements on group delay flatness would also be more severe for a 40 Gb/s

HT.

• If inductors are used for the on chip transmission line in the HT instead of

transformers, then the higher Q of the inductors compared to a transformer

primary will reduce the loss of the line. Furthermore, if the problem of exces-

sive transient ringing could be overcome, the line could be Q-enhanced using

the simple cross coupled pair in Figure 5.10. These ideas could be implemented

with the aim of improving the HT performance and improving the average side-

159

band cancellation. If the transient ringing of the Q-enhanced transmission line

could be reduced, then the resulting Q-enhanced LC line would have applica-

tion in all tapped delay filters for equalization or predistortion in fiber optic

networks, and for tapped delay filters for ultra wideband radio.

• The design procedure for logarithmic amplifiers could be modified for envelope

detecting logarithmic amplifiers.

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[50] J. W. M. Rogers and C. Plett, “A 5-GHz Radio Front-End with Automatically

Q Tuned Notch Filter and VCO,” IEEE J. Solid-State Circuits, vol. 38, no. 9,

pp. 1547–1554, Sept. 2003.

[51] T. Soorapanth and S. Wong, “A 0-dB IL 2140 ± 30 MHz bandpass filter utilizing

Q-enhanced spiral inductors in standard cmos,” IEEE J. Solid-State Circuits,

vol. 37, no. 5, pp. 579–586, May 2002.

[52] B. Razavi, Design of Analog CMOS Integrated Circuits. Boston, MA: McGraw

Hill, 2001.

[53] B. Gilbert, “The Multi-tanh Principle: A Tutorial Overview,” IEEE J. Solid-

State Circuits, vol. 33, pp. 2–17, Jan. 1998.

[54] R. Davies, Personal Conversation, Dec. 2004.

[55] R. Hughes, “High speed logarithmic video amplifier,” U.S. Patent 3,745,474,

July 1973.

[56] C. Holdenried, J. Haslett, J. McRory, R. Beards, and A. Bergsma, “A DC-4GHz

true logarithmic amplifier: Theory and implementation,” IEEE J. Solid-State

Circuits, vol. 37, no. 10, pp. 1290–1299, Oct. 2002.

[57] P. Gray, P. Hurst, S. Lewis, and R. Meyer, Analysis and Design of Analog

Integrated Circuits, 4th ed. John Wiley and Sons Inc., 2001.

[58] A. Sedra and K. Smith, Microelectronic Circuits, 4th ed. New York, NY:

Oxford University Press, 1998.

Appendix A

Amplifier DC Transfer Characteristic

The notation labeled in Figure 3.1 is used throughout this appendix. The parasitic

emitter resistances of the HBTs will be ignored for simplicity. Using IC2 − IC1∼=

IC2 − (IEE1 − IC2) ∼= 2IC2 − IEE1 and IC4 − IC3∼= 2IC4 − IEE2,

V1 − V2∼= R1(2IC4 − IEE2) + VBE6 − VBE5

+ Rf (2IC2 − IEE1). (A.1)

Next, it is noted that for transistor k

VBEk∼= VT · ln

(

ICk

IES

)

(A.2)

where IES is the scaling current proportional to the base-emitter junction area for a

given transistor. Using this equation for Q1 and Q2 and solving for IC1 and IC2 and

comparing gives

IC2∼= IC1 · e−

VinVT ∼= IEE1

1 + eVinVT

(A.3)

IC4∼= IEE2

1 + e(V1−V2)

VT

. (A.4)

For purposes of simplification, it is noted that IC5∼= IC1 + IC3/βDC and IC6

∼=

IC2 + IC4/βDC . Equation (3.1) is derived using equations (A.1)-(A.4) and using the

tanh function expressed in exponential form.

Next, it is recognized that

Appendix A 168

Vo1 − Vo2∼= (IC4 − IC3)(R1 +R2)

∼= (2IC4 − IEE2)(R1 +R2). (A.5)

Equation (3.2) is derived by substituting equation (A.4) into (A.5) and again using

tanh expressed in exponential form. Equation (3.2) overestimates the slope of the

transfer characteristic somewhat for small applied voltages because the parasitic HBT

emitter resistances have been ignored here for simplicity. When these parasitics are

included, one of IC1 or IC2 and one of IC3 or IC4 must be found iteratively.

Appendix B

Analysis of the Emitter Follower Load

The CHEF amplifier is usually followed by an emitter follower output buffer. The

amplifier will be loaded by the input impedance of the emitter follower, ZinEF , which

in turn depends on the input impedance of the stage connected to the output of the

emitter follower. Figure B.1(a) shows one example, where the output of the emitter

follower is connected to a differential pair. The differential pair could represent the

input to another CHEF amplifier stage, or a buffer to the output of the integrated

circuit. Figure B.1(b) shows the high frequency small signal circuit of one emitter

follower and a differential mode half circuit of the differential pair input. Using this

circuit, ZinEF is given by

ZinEF∼= Z1

sCµEFZ1 + 1+ rbEF (B.1)

where

Z1∼=

[

s2(R3 + reDP )

+s

(

R3 + reDP + rdEF

rdEFCπEF+reDP/2 + rdDP

rdDPCπDP

)

+

(

reDP/2 + rdDP

rdEFrdDPCπEFCπDP

)]

/

s2 (B.2)

and where R3 = reEF +rbDP . The magnitude of ZinEF decreases with frequency, and

so it may be modeled by a capacitor to first order. However, at high frequencies, the

impedance predicted by (B.1) includes a frequency dependent negative resistance. A

more detailed prediction of the CHEF amplifier frequency response may be obtained

by using (B.1) in (3.10) to obtain ZL, and then using this ZL in (C.2) to obtain

Appendix B 170

CπEF

rbEF

reEF

vbe1vbeEF

rdEF

icEF =

vo2

QEF2

reDP

Q8

QEF1

Q7

Rc Rcvo1 QDP1 QDP2

CµEF

vbeDP

2rdDP

icDP =vbeDPCπDP

rbDP

IDPVBIAS

VEEVEE

vo1

ZinEF

(a)

(b)

Rc

Figure B.1: Schematic diagram of (a) emitter follower output buffers and differentialpair load and (b) high frequency small signal circuit of one emitter follower and adifferential mode half circuit of the differential pair.

the frequency response. When this was done, it was found that there are still two

complex poles which largely determine the amplifier frequency response and whose

pole quality factor behaves in roughly the same way as shown in Figure 3.6. The net

effect is to only modify the response in a modest way.

Appendix C

Derivation of Equation (3.11)

Using nodal analysis, vo1/vbe1 was found for the circuit in Figure 3.3(a) and is given

by equation (C.2). The zero in (C.2) is given by

z1 ∼=(R1 +R2)(R

′

f + rd5)

rd5Cπ5(R′

f + rb5)(R1 +R2) − R1r′d3 +R1R2. (C.1)

R2 is of the order of r′d3 and R′

f >> rd5, so that the zero will be at a relatively high

frequency. Furthermore, in Figure 3.3(a), Q5 forms an emitter follower feedback

circuit. As Ohhata et al. describe in [24], the bandwidth of this emitter follower

is much greater than that of the common emitter amplifier formed by Q3, R1, and

R2. As a result, the pole due to Cπ5 will be at a much higher frequency than the

amplifier bandwidth and may be neglected for simplicity. With Cπ5 set to zero, and

assuming r′d3 >> R′

f/β, equation (C.2) reduces to equation (3.11).

vo1

vbe1=

N(s)

D(s)∼= 1

rd1

[

srd5Cπ5(R′

f + rb5)(R1 +R2) − R1r′

d3 +R1R2 + (R1 +R2)(rd5 +R′

f )]

/

[

s2Cπ5C1rd5r′

d3(R1 + rb5 +R′

f ) + sCπ5rd5(r′

d3 +R1) + C1r′

d3(rd5 +R′

f )

+R1 + r′d3 +1

ZL

s2Cπ5C1r′

d3rd5

(R1 +R2)(rb5 +R′

f ) +R1R2

(C.2)

+sr′d3

Cπ5rd5(R1 +R2) + C1(R1 +R2)(rd5 +R′

f)

+ (r′d3 +R′

f/β)(R2 +R1)

]

Appendix D

Example Calculation of an Amplifier Noise

Contribution

Figure D.1 shows half of the small signal circuit of the amplifier including the dom-

inant noise sources. For example, consider the case where only i2c1 is calculated and

all of the thermal noise voltage sources are short-circuited and the remaining shot

noise current sources are open-circuited. Assuming β >> 1, the output PSDs at

nodes vo1 and vo2 due to i2c1 are given by

|v2o1(t)|∆f

=|v2

o2(t)|∆f

∼=i2c1r

2d1(R1 +R2)

2(R′

f + rd5)2

∆f4r′d12(R1 + r′d3)

2. (D.1)

The mean square differential noise voltage between terminals vo1 and vo2 due to a

given noise source is found for low frequencies to be

v2o = |vo1(t) − vo1(t)|2

= v2o1(t) + v2

o2(t) − 2vo1(t)vo2(t)

= v2o1(t) + v2

o2(t) − 2C12

∣

∣

∣v2

o1(t) · v2o2(t)

∣

∣

∣

1/2

(D.2)

where v2o1(t) and v2

o2(t) are the mean square noise voltages at nodes vo1 and vo2

respectively, and C12 is a measure of the correlation between vo1(t) and vo2(t) and

always lies in the range −1 ≤ C12 ≤ 1. Substituting equation (D.1) into (D.2)

with C = −1, since the noise voltages are anti-phase and fully correlated, results in

v2o(t) = 4v2

o1(t). The noise current source i2c2 contributes an equal amount of noise.

The contributions of the remaining noise sources may be found in the same way.

Appendix D 173

R1

rd5

vo1

β

rd3β

rb5

vbe5

rd3

vbe3

rd5

vbe5

R2Rf

eR2

re1

er2

ic12

er

rb1

2b1

e1

f

re3

rd1βvbe1rd1

vbe1vbe3

+−

+−

+−’

’

Figure D.1: Half of the amplifier small signal circuit including dominant noisesources.

Appendix E

Widlar Biasing

In Figure E.1, degeneration resistor Re is shown. For this amplifier, the gain is equal

to

vout

vin

∼= − Rc

Re + rd1(E.1)

where vin = vin1 − vin2 and vout = vout1 − vout2. It is assumed that the circuit is

symmetrical, so that the small signal parameters of Q1 and Q2, for example, are

equal. It is also assumed that if Re >> rd1, then vout/vin∼= −Rc/Re and the gain

is only dependent on the ratio Rc/Re. In this situation, the gain may be precisely

controlled using resistive matching layout techniques. However, using resistor Re is

undesirable for two main reasons. In a high gain amplifier where a large value of

Rc/Re is required, then Rc must be large compared to the case where Re is zero.

Larger resistances increase the time constants within the amplifier, and lower the

bandwidth. As well, Re generates substantial thermal noise. For these reasons, it is

desirable if Re is zero.

Figure E.2 shows the differential pair without Re and with a Widlar-type current

source for I2 with a β-helper transistor [57]. It is assumed that the base currents of

Q3 and Q4, supplied by Q5, are large enough to set the operating point of Q5 so that

it has β >50. If this is not true, then a resistor should be added from the emitter of

Q5 to VEE so that the emitter current of Q5 is sufficiently large to give it β >50.

The benefit of the Widlar bias configuration will now be explained. The following

relationships are observed for the circuit in Figure E.2:

Appendix E 175

ReRe

Rc Rc

Vout1,vout1 Vout2,vout2

Vin1,vin1 Vin2,vin2

VEE

IEE

Q1 Q2

VCC

Figure E.1: Differential pair.

I1 ∼= IES3eVBE3/VT (E.2)

I2 ∼= IES4eVBE4/VT (E.3)

VT ln

(

I1IES3

)

∼= VT ln

(

I2IES4

)

+Rm2I2 (E.4)

This equation may be rearranged to give

I2 ∼=VT

Rm2ln

(

I2IES4

I1IES3

)

(E.5)

This shows that I2 is proportional to absolute temperature (PTAT) to first order,

because it is dependent on VT (= kT/q). However, it should be noted that IES is

itself strongly temperature dependent. Specifically,

IES =AEqDnn

2i

NAW(E.6)

n2i = BT 3e−EG/kT (E.7)

Appendix E 176

Rc Rc

vout2

vin1 vin2Q1 Q2

VCC

vout1

VCC

VEE

Q4Q3

Q5

Rm1

Rm2

I1I2

Figure E.2: Differential pair with Widlar current biasing.

where AE is the cross sectional area of the base-emitter junction, Dn is the concen-

tration of ‘donor’ or phosphorous atoms, ni is the concentration of holes or electrons

in silicon at a given temperature, NA is the concentration of ‘acceptor’ or boron

atoms, W is the effective width of the base, B is a material parameter =5.4 ×1031

for silicon, and EG is the bandgap energy =1.12 electron volts for silicon [58]. How-

ever, the temperature dependence of IES is mainly due to n2i , and this term should

cancel in the ratio IES4/IES3.

The larger the ratio IES4/IES3 in equation (E.5) and hence the ratio of emitter

lengths of Q4 to Q3, the larger the current I2 is and the more steeply it increases

with increasing temperature. Half of I2 is the quiescent bias current of Q1 or Q2.

Hence the transconductance of Q1 or Q2 becomes

Appendix E 177

gm1,2 =IC1,2

2VT

∼= 1

2Rm2ln

(

I2IES4

I1IES3

)

. (E.8)

Using this, the differential gain in equation (E.1) with Re=0 becomes

vout

vin

∼= − Rc

2Rm2ln

(

I2IES4

I1IES3

)

. (E.9)

The significance of this equation is twofold. It is evident that the gain is no longer

directly proportional to temperature, since the PTAT current source counteracts the

inverse dependence of gm1 and gm2 on temperature. Furthermore, the gain is depen-

dent on the ratio Rc/Rm2, which may be accurately set despite variations in process

using resistive matching layout techniques. In this thesis, some emitter-coupled pairs

are biased using this scheme. This improves both the frequency response and the

noise performance of the amplifiers compared to the case where the amplifiers are

degenerated with Re, all while maintaining gain that is independent of temperature

and process to first order. This is all assuming that I1 is fixed in Figure E.2, which

it isn’t if it is set using Rm1 to VCC as shown. For a commercial product, it would

be desirable to fix I1 using a proper current reference, such as the supply voltage

independent current source used in Chapter 5 and described in [52].

Appendix F

Design of the Logarithmic Amplifier Test Fixture

Figures F.1(a) and F.1(b) show the circuit boards used to test the logarithmic am-

plifier. They were fabricated with Rogers Corporation 4003 circuit board material.

The main circuit board in has a 0.016 inch thick substrate and contains four 50 Ω

coplanar waveguide lines for connecting the two signal inputs and outputs. It also

contains traces for connecting VCC , VEE, the power connection for the DC offset error

reduction circuit, and one connection for each terminal of the off-chip 1 nF capacitor

for the offset reduction circuit. The power supply of the DC offset reduction circuit

is separate so that it may be turned on (VCC) for narrowband AC applications, and

may be turned off (VEE) for the OSSB application. The IC sits on a pad which is

connected to VEE (=-3.3 V). If the chip is epoxied onto the main circuit board and

the two AC inputs and two AC outputs are wire bonded to the circuit board, the

bond wires must traverse the 1 mm height of the IC, leading to at least 1 nH of bond

wire inductance. This amount of inductance makes the circuit difficult to impedance

match up to 10 GHz. For this reason, two of the small circuit board shown in F.1(b)

may be epoxied to the main circuit board, one for the AC input signals and one for

the AC output signals. These small circuit boards are made from a 0.008 inch thick

substrate, which yields a 0.012 inch thick board when the half ounce copper and

gold plating are added on each side of the substrate. This is approximately the same

height as the IC. These small circuit boards have two 50 Ω coplanar waveguide lines

each. This leads to shorter bond wires for the AC signals, and reduced inductance.

Since the logarithmic amplifier is a high gain amplifier, it should be placed in an

enclosed box to prevent high levels of radio signals in the 50 MHz to 5.1 GHz range

Appendix F 179

Front Side

Back Side

(a) The two sides of the main circuit board.

Front Side

Back Side

(b) Top-mount circuit board.

Figure F.1: Logarithmic amplifier circuit boards.

Appendix F 180

Figure F.2: Logarithmic amplifier test fixture.

from reaching the IC through the air and appearing at the logarithmic amplifier

output. Figure F.2 shows a picture of the enclosed test fixture used to test the

logarithmic amplifier. It uses 50 Ω SMA connectors whose dielectric extends through

the wall of the test fixture, and a Tusonix feed-through filter for VEE. The inside of

the lid of the test fixture was lined with material designed to absorb radio signals,

so that the no signals would resonate inside the box.

A through measurement of one input and one output of the test fixture was

taken using a circuit board with a 50 Ω transmission line connecting the input to

the output. The loss of the test fixture was 3 dB at 5 GHz, which is unexpectedly

high. It is unknown if this problem relates to the coplanar waveguide lines on the

circuit boards, or to the connectors used. Nevertheless, the approximate 5 GHz of

Appendix F 181

bandwidth of the test fixture was adequate for the 5 Gb/s tests in Chapter 7.

When wire bonding the logarithmic amplifier, there are several pitfalls which

can prevent the amplifier from having its approximate 39 dB gain up to the 4-5 GHz

bandwidth of the test fixture. In addition to keeping the AC bond wires short, which

was already discussed, the following points should be understood:

• The AC bond wires must not touch the edge of the IC. They should touch the

bond pads, and arc clear of the edge of the chip. If the bond wires touch the

edge of the chip, the amplifier gain will decrease by 10 dB.

• It was found that in some cases where both inputs are wire bonded, the two

inputs can couple together, causing an approximate 10 dB loss in gain. It is

has not been established if the fields of the wire bonds couple in the air, or if

the tails of the bond wires, caused by wedge bonding, couple through the ring

surrounding the chip. The simplest solution to this problem is to leave one of

the inputs unbonded. Since the input circuits are emitter followers, the circuit

will still work properly when one input is unconnected. This is what was done

for the amplifier used in the experiments in Chapter 7. When this is done,

if the full amplifier gain is desired, then the DC offset error reduction circuit

should be turned on, and AC coupling should be used when connecting to the

one input.

• VEE must be decoupled on the top layer of the board within a few millimeters

of the chip. When this is not done, the gain of the amplifier decreases by

several decibels above 500 MHz. An 0402 package style 1 nF capacitor was

used to decouple VEE for the amplifiers tested in this thesis.

Appendix G

Description of Equipment Used for COSSB

Experiments

Table G.1 lists the major equipment used in the COSSB experiment. It does not

include several DC power supplies that were needed to power all of the parts. It also

does not include several attenuators that were used to provide the right signal levels

to the modulators, the HT IC, and the logarithmic amplifier IC. Figures G.1- G.3

show pictures of the 10 Gb/s optical experiment setup.

It was necessary to align the delays of the signal going to the phase modulator

and through the fiber to the intensity modulator, and of the signal being applied to

the intensity modulator. The output of the phase modulator had eighty centimeters

of fiber, and the input to the intensity modulator had two meters of fiber. The

JCA amplifier had 400 ps of delay. As a first attempt at matching the delay of the

intensity and phase modulation paths, cables were chosen connecting to the input

of the JCA amplifier whose total length were equal to the length of the fiber whose

delay was attempting to be matched, 2.8 m. It was observed that when the delay

through the intensity modulation path was approximately 1 ns or larger either too

big or too small, the resulting modulated optical power spectrum had nulls in it at

several frequencies where the excess phase difference caused components of the two

sidebands to interfere destructively. Line stretchers were used to tune the delay in

600 ps ranges. As the delay of the two paths began to align, the nulls became fewer

and fewer, until single sideband modulation was achieved. The delay of the cables

connecting the pattern generator to the JCA amplifier were measured to be 15.34 ns.

Appendix G 183

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Appendix G 186

Polarization Adjuster forIntensity Modulator

Cable to IntensityModulator

APC/FC to PC/FC Patch Cable

1/64 Clock Output forTriggering HP Oscilloscope

Micropositioner HT in TestFixture

GSGSG WaferProbe

CapacitorsBlocking

Line Stretchersfor HT Input

PatternGenerator

Laser Microscope

Attenuator

Figure G.1: 10 Gb/s optical experiment setup using only HT, Part 1 of 3.

PSPL 5828Amplifier

PSPL 5865Amplifier

BlockingCapacitor Modulator

APE PhaseAttenuator


Appendix G 187

FFP TunableFilter

Optical Power MeterWith Analog Output

Polarization Adjuster forFFP Tunable Filter

Variable OpticalAttenuator

IntensityModulator

APC/FC to PC/FC Patch Cable

OpticalIsolator

LeCroy Oscilloscope(Not Shown)

Analog Output to

BiasTee

JCA Amplifier

Attenuator


Circuit Board for Setting Log AmplifierDC Offset

LogarithmicAmplifier

Mini−CircuitsFilter

Figure G.4: 5 Gb/s optical experiment setup using HT and logarithmic amplifier,Part 1 of 1.

Appendix G 188

Other delay figures include 400 ps from the PSPL 5865-107, 240 ps from a PSPL

5828, 200 ps from the probe tips on the input and output of the HT, and 200 ps

from the HT itself. Based on the above information, the delay through the 2.8 m of

fiber was approximately 14 ns, corresponding to a propagation velocity in the fiber

of 2 × 108 m/s.

In the above discussion, it was described how the phase of the intensity modu-

lation signal was adjusted until single sideband is achieved. It is critical to know

whether to expect an upper sideband and lower sideband signal. If the HT signal

is 180 degrees out of phase at mid-band, the resulting optical signal will have good

cancellation of the unwanted sideband at mid-band frequencies, but mediocre can-

cellation at frequencies above and below this. This will still look like pretty good

single sideband, and the only way to know that it is off by 180 degrees is if one

knows which sideband they are expecting to be suppressed. As mentioned in Sec-

tion 2.4.1, the signal will be lower sideband if the HT signal is in phase with the

intensity modulation signal, and will be upper sideband if it is 180 degrees out of

phase either due to excess phase shift or due to the use of inverting amplifiers. As

well, an upper sideband signal in the wavelength domain is a lower sideband signal

in the frequency domain. In fact, there are several things that can cause a signal to

change from lower to upper sideband, including:

• If one signal from the pattern generator is used to drive the intensity modulator,

and the logical complement from the pattern generator is connected to the HT.

• If any inverting amplifiers are used, such as the PSPL 5828.

• If the intensity modulator is biased for inverting modulation, such that when

the electrical signal swings high, it extinguishes the optical intensity.

Appendix G 189

• If any inverting combination of HT and logarithmic amplifier inputs and out-

puts are used.

• If the down slope of a triangle wave is used to sweep the FFP optical filter

instead of the up slope.

During the COSSB experiments in this thesis, the signal was always predicted to

be either lower or upper sideband before trying to equalize the delay of the phase

modulation signal, but either type of signal was generated depending on how many

PSPL 5828 inverting amplifiers were used. It is noted that optical USB, which

is LSB in the frequency domain, must be transmitted if microstrip post detection

equalization is to be used. This ensures that the microstrip will equalize the phase

distortion caused by fiber dispersion [7]. If optical LSB is transmitted and microstrip

equalization is attempted, the phase distortion due to fiber dispersion will be made

worse, and the equalization will fail.

In addition to having correct phasing, the question of how much voltage should

be used to drive the phase modulator arises. In general, as the voltage applied to

the phase modulator is increased from a low level to the optimal level, the signal

will go from DSB to SSB. If the voltage applied to the phase modulator is increased

even further, the desired sideband will appear distorted, and both sidebands will rise

in power with increasing applied voltage to the phase modulator. The optimal level

appeared to be in the 4-6 Vpp range for most tests. Since this signal is the amplified

HT output, however, the RMS voltage of the signal applied to the phase modulator

will only be approximately one volt or less. Ideally, it would be known how much

voltage is needed to achieve one radian of phase shift, but this was not known for

the modulator used.

Table G.2 shows the measured power readings versus applied voltage for the

Appendix G 190

Table G.2: Power characteristic of the Sumitomo intensity modulator.

Applied Voltage (V) Optical Output Power (dBm)

0.0 -27.6

1.0 -29.0

2.0 -31.4

3.0 -36.3

4.0 -45.8

4.3 -47.7

5.0 -39.6

6.0 -33.3

7.0 -30.0

8.0 -28.2

9.0 -27.2

10.0 -27.0

Sumitomo intensity modulator. This test was performed several times during the

OSSB experiment in order to check for the most linear bias points of the modulator.

For example, this table shows that approximately 1 V is a linear inverting bias point,

and 7.0 V is a linear non-inverting bias points. The choice of bias point was always

confirmed experimentally by measuring the detected eye diagram. The absolute

power readings in the table are the power of the laser after it passes through the

Sumitomo modulator, several connectors, and a 25 dB attenuator.

DC blocking capacitors were placed on the outputs of the pattern generator in

order to reduce the possibility of damaging it. A DC offset had to be added to

the logarithmic amplifier input through a bias tee so that the logarithmic amplifier

output waveform was similar to the logarithm of the optical signal envelope. A

schematic of the breadboard tuning circuit used to tune the DC offset is shown in

Figure G.5.

Appendix G 191

VEE

150Ω

5 kΩ

100Ω

5 kΩ

Vout

Figure G.5: Logarithmic amplifier DC offset tuning circuit.

the university of calgary a logarithmic amplifier and hilbert transformer for optical single

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