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Time -frequency analysis of high power microwave sources

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TIM E-FREQUENCY ANALYSIS OF HIGH POW ER MICROW AVE SOURCES
by
Christopher Wayne Peters
A dissertation submitted in partial fulfillment
o f the requirements for the degree o f
Doctor o f Philosophy
(Nuclear Engineering)
in The University o f Michigan
2001
Doctoral Committee:
Professor Ronald M. Gilgenbach. Co-Chair
Professor William J. Williams, Co-Chair
Associate Professor Mary Brake
Professor Yue Y. Lau
Dr. Thomas A. Spencer. Nuclear Engineer, USAF
Air Force Research Lab,
Phillips Research Site
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UMI Number: 3001028
Copyright 2001 by
Peters, Christopher Wayne
All rights reserved.
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C h risto p h er W a y n e P eters
2001
A ll R ig h ts R ese r v e d
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This is dedicated to my family, friends, and colleagues who have helped me stay in
school through all the good times and bad
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ACKNOWLEDGMENTS
The person I would like to acknowledge first is my advisor, Dr. Ronald
Gilgenbach. His dedication to the educational and professional development o f his
students is unbounded.
While the research in the lab is geared towards specific areas, I
thank him for giving me the freedom to pursue a road o f research which was not expected
at the forefront o f the experiment. I also would like to thank him for his tolerance with
my struggle trying to understand microwave physics and signal processing
simultaneously. W ithout his understanding, I would not have been able to write this
dissertation.
I thank Professor William J. Williams for his insight in signal processing. His
patience in teaching me spectral analysis will not go unremembered. His understanding
and patience that learning topics outside a student’s concentration is limitless. I wish to
thank Professor Y. Y. Lau for helping me understand plasma physics and
electromagnetics inside and outside the classroom. I also thank him for making me
understand how important my research is to the high power microwave community. My
gratitude also is extended to Professor Mary Brake, whose optimism has helped me make
it through my school years. I thank Dr. Thomas Spencer for being the person who
assigned me this project during the summer I worked at AFRL-Phillips Site (even though
I failed miserably on this project while out there) and for his insight about the physics
related to the signal processing results.
I thank all o f the staff and faculty in the Department o f Nuclear Engineering and
Radiological Sciences, University o f Michigan. Their dedication to the NERS students
cannot be compared anywhere.
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I would also like to acknowledge all o f the students in the lab whom I have been
privileged to work with. Dr. Jonathan Hochman for his constant jokes about our common
past experiences; Dr. Reginald Jaynes for his insight on electromagnetism; Dr. William
Cohen for his support in the time I questioned the validity o f my research; Dr. Josh
Rintamaki for ju st being a heck o f a nice guy; Dr. Scott Kovaleski for his positive
annoyances; Mike Lopez for his philosophical discussions; Mark Johnston for his
reminiscing about his life as a sailor; Bo Qi for playing his D ef Leppard music and his
comical relief; Scott Anderson for ju st being there whenever I needed support in any
fashion; Rex Anderson for being another heck o f a nice man; and Hiroto Miyake for his
com puter knowledge.
I would like to acknowledge the undergraduates who contributed to the lab
greatly, especially Antwan Edson and Nick Eidietis. Your help with obtaining and
scanning my references is greatly appreciated.
Finally, I thank my family and close friends. To my wife, Allison, thank you for
your infinite patience. The ride o f marriage has been wonderful so far, and the future
looks bright. To my family, I extend the utmost o f my appreciation for their love and
support in every endeavor I have taken. To Jason and Trinity Justian, Francesco Leonini,
and Kevin Rogovin thank you for being there from day one in my ups and down in life.
This work has been supported by the Air Force Office o f Scientific Research, the
M ultidisciplinary University Research Initiative (MURI) through a Texas Tech
subcontract, AFOSR/AASERT, Air Force Research Lab, and the Northrop Grumman
Corporation.
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TABLE OF CONTENTS
DEDICATION...................................................................................................................................ii
ACK NO W LEDG M EN TS............................................................................................................iii
LIST OF FIG U R E S.....................................................................................................................viii
LIST OF T A B L E S........................................................................................................................ xii
LIST OF A P P E N D IC E S............................................................................................................xiii
CHAPTER
1. INTRODUCTION...............................................................................................................1
2. GYROTRON BACKGROUND T H E O R Y ................................................................5
2.1 Fundamental Electromagnetic C oncepts.....................................................5
2.1.1 The Generalized Wave
Equation...............................................5
2.1.2 Q and Bandw idth...............................................................................6
2.1.3 Plasma Terminology and Effects.................................................. 7
2.2 The G yrotron....................................................................................................... 8
2.2.1 The Dispersion Relation Curve for the G yrotron....................9
2.3 M icrowave Pulse Shortening........................................................................ 10
2.3.1 Voltage Fluctuations o f the Electron B eam ............................. 11
2.3.2 Mode C om petition...........................................................................11
3. SIGNAL PROCESSING TH EO R Y ............................................................................16
3.1 Signal Processing Fundam entals..................................................................16
3.1.1 Model and Basic Terminology o f Digital Signal
Processing.......................................................................................... 16
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3.1.2
The Sampling T heorem ............................................................... 17
3.1.3
The Autocorrelation F unction................................................... 19
3.2 The Fourier T ransform .................................................................................... 20
3.2.1 Some Properties of the Discrete Fourier Transform ............21
3.3 Signal and Spectrum Statistics...................................................................... 23
3.3.1
Mean Time and Duration............................................................ 23
3.3.2 Mean Frequency and B andw idth..............................................24
3.3.3 The Uncertainty Principle........................................................... 25
3.3.4 The Analytic Signal........................................................................26
3.3.5
Inherent Problems With the Discrete Fourier
Transform ........................................................................................ 27
3.3.5.1 Spectrum B iasing........................................................27
3.3.5.2 Variance R eduction ................................................... 30
3.3.5.3 D iscontinuities............................................................. 33
3.3.5.4 Discrete Prolate Spherical (Slepian)
Sequences...................................................................... 33
3.4 Spectral Analysis of Nonstationary Signals........................................37
3.4.1
Instantaneous F requency..................................................... 38
3.4.2
Properties of Tim e-Frequency Distributions................... 39
3.4.3 The Short-Time Fourier Transform and
Spectrogram .............................................................................. 40
3.4.4 The Symmetric Ambiguity Function.................................. 42
3.4.5 The W igner D istribution...................................................... 44
3.4.5.1 Properties o f the W igner D istribution............... 44
3.4.5.2 Reduced Interference K ernels..............................45
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3.4.5.3 Instantaneous Bandwidth......................................51
4. SYSTEM C O N FIG U R A TIO N .................................................................................... 53
4.1 The Coaxial G yrotron ...................................................................................... 53
4.1.1 Electron Beam Current Diagnostics.......................................... 56
4.1.2 Heterodyne M ixing......................................................................... 60
4.2 M icrowave Oven M agnetron.......................................................................... 60
5. EXPERIM ENTAL ANALYSIS AND RESULTS.................................................. 63
5.1 Gyrotron A nalysis.............................................................................................. 63
5.1.1 Fourier Transform and Spectrogram ........................................ 63
5.1.2 Tim e-Frequency Analysis Utilizing Reduced
Interference Distributions.............................................................65
5.1.3 Instantaneous Bandwidth............................................................... 70
5.1.4 Current M odulations.......................................................................73
5.2 Tim e-Frequency Analysis o f Other High Power Microwave
S ou rces...................................................................................................................82
5.2.1
Air Force Research Laboratory HPM Sources....................82
5.2.2
Microwave Oven M agnetron.....................................................85
5.3 The Use o f Discrete Prolate Spherical Sequences.....................................90
6. C O N C L U SIO N S.............................................................................................................. 98
A PPEN D IC ES...............................................................................................................................100
BIBLIO G RAPH Y........................................................................................................................146
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LIST OF FIGURES
Figure
2.1
Example o f two modes a) widely separated b) closely separated....................7
2.2
Dispersion relation curves for the electron beam and the TEj i
waveguide mode. The dashed line is the dispersion relation that
results in backward wave oscillation ....................................................................10
2.3
Simulation for an oscillator with stable two-mode operation......................... 14
2.4
Simulation for mode competition in which the second mode eventually
dom inates....................................................................................................................15
3.1
Comparison o f three signals, each with a normalized sampling
frequency o f a) 0.3, b) 0.5, c) 0.7, respectively..................................................19
3.2
Results for a discrete signal consisting o f 64 points, all equal to the
value o f 1. a) Biased autocorrelation, b) unbiased autocorrelation, c)
spectral energy densities o f the autocorrelations............................................... 22
3.3
Magnitude o f the Dirichlet kernel (d B ).............................................................. 28
3.4
Example spectrum magnitude for a signal with two frequencies a)
N=1024 points, b) N=64 p o in ts............................................................................ 29
3.5
Commonly used w indow fu n ctio n s..................................................................... 32
3.6
First four Slepian sequences with a time-bandwidth product o f 3................. 34
3.7
a) Generated signal with Gaussian distributed noise, b) histogram o f
noise values, c) energy density spectrum using the F F T ................................. 36
3.8
Energy density spectrum for data used in Figure 3.7 using a) Bartlett
method (5 segments), b) DPSS (NW=4), and c) filter bank method
(filter bandwidth o f 0.01)........................................................................................37
3.9
Spectrograms o f w indow sizes 17, 33, and 65 points....................................... 41
3.10 Magnitude o f the am biguity function for a) a two component signal
and b) its analytic version. The values o f 0 are norm alized...........................43
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3.11 Time-frequency analysis o f a multicomponent signal using the a)
W igner, b) binomial, and c) Bom-Jordan distributions................................... 49
3.12 Time-frequency analysis for a signal with phase jum ps o f a) rc/4, b)
ti/2, c ) 3tt/4, and d) n .............................................................................................. 50
4.1
Experimental setup o f coaxial gyrotron..............................................................54
4.2
Typical MELBA experimental shot, a) Cathode voltage, mixer signal,
and rf power, b) diode current, aperture current, cavity entrance
current, and cavity exit current (integrated).......................................................55
4.3
Simple RC integrator/low-pass filter circuit......................................................56
4.4
a) Magnitude response and b) phase response for an integrator with
RC=20 p s .................................................................................................................. 57
4.5
a) M agnitude response and b) phase response for a periodic sine
function o f 4001 points.......................................................................................... 59
4.6
Microwave oven magnetron experimental se tu p .............................................. 61
4.7
M easurements from the voltage doubler in the oven magnetron, a)
Voltage, and b) cu rren t...........................................................................................62
5.1
a) Example o f heterodyned microwave signal (LO=2.3 GHz),
b)Fourier transform o f heterodyned microwave sig n al................................... 64
5.2
STFT o f the mixer signal in Figure 5.1a (LO=2.3 G H z).................................64
5.3
Time-frequency analysis utilizing RIDs o f the mixer signal in Figure
5.1a (LO=2.3 GHz) (MELBA shot 7 2 1 9 ).......................................................... 66
5.4
Overlay o f cathode voltage with the TFA o f the mixer signal (LO=2.3
GHz) (M ELBA shot 7 219)....................................................................................67
5.5
a) Maximum intensity o f TFD in mixer signal from Figure 5.1a, b)
detector signal shot number 72 1 9.........................................................................68
5.6
a) Signal with mode hopping properties, b) TFD o f signal in a), c)
maximum intensity o f each mode from the TFD in b). The local
oscillator was set at 2.2 GHz. The lower mode is the T E i 11 mode, and
the upper mode is the TEuz m o d e........................................................................69
5.7
a) Signal with multi-moding properties, b) TFD o f signal in a), c)
maximum intensity o f each mode from the TFD in b). The local
oscillator was set at 2.2 GHz. The lower mode is the TEi 11 mode,
while the upper mode is the T E 112 m o d e............................................................ 70
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5.8
Filtered heterodyned m ixer signal and envelope (top), spectral density
(middle), instantaneous bandwidth (bottom). The large instantaneous
bandwidths before and after the main part o f the pulse are numerical
artifacts....................................................................................................................... 72
5.9
Signals from a singe mode shot, a) Mixer, b) unintegrated entrance
current, c) unintegrated exit current..................................................................... 75
5.10 TFAs from signals in Figure 5.9. a) Mixer, b) entrance current, c) exit
cu rren t........................................................................................................................ 76
5.11 Signals from a multimode shot, a) Mixer, b) unintegrated entrance
current, c) unintegrated exit current..................................................................... 77
5.12 TFAs from signals in Figure 5.11. a) Mixer, b) entrance current, c) exit
cu rren t........................................................................................................................ 78
5.13 Example frequency response magnitude for numerical in tegration.............. 79
5.14 Power Spectral Density for the integrated entrance current, single
mode case...................................................................................................................80
5.15 Power Spectral Density for the integrated exit current, single mode
case.............................................................................................................................. 80
5.16 Power Spectral Density for the integrated entrance current, multi mode
case.............................................................................................................................. 81
5.17 Power Spectral Density for the integrated exit current, multi mode case
81
5.18 a) Mixer from the AFRL MILO, b) TFD o f the signal in a )............................82
5.19 a) Mixer signal o f a conventional magnetron, and b) TFD o f the mixer
signal...........................................................................................................................83
5.20 a) M ixer signal o f the AFRL RKO and b) the TFD o f the mixer signal
84
5.21 Cold test results for the microwave oven magnetron........................................ 85
5.22 Typical microwave oven magnetron p u lse......................................................... 86
5.23 Microwave oven magnetron signal pulse rise: a) signal, b) Fourier
transform, c) TFA. The local oscillator was set at 2.35 G H z ........................ 87
5.24 Microwave oven magnetron signal pulse peak: a) signal, b) Fourier
transform, c) TFA. The local oscillator was set at 2.35 G H z ........................ 88
5.25 Microwave oven magnetron signal pulse fall: a) signal, b) Fourier
transform, c) TFA. The local oscillator was set at 2.35 G H z ........................ 89
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5.26 Generated window function used (NP=3) (top) and its power spectral
density (bottom )....................................................................................................... 90
5.27 TFA o f single com ponent signal using DPSS (top) and marginals
(b o tto m )..................................................................................................................... 91
5.28 TFA o f multicomponent signal using DPSS (top) and frequency
marginals o f the TFA (bottom )............................................................................. 92
5.29 TFA o f signal with two Kronecker delta functions in time using a)
Hanning window' and b) DPSS applied to the local autocorrelation
fu n ctio n...................................................................................................................... 93
5.30 TFA o f experimental data using DPSS (Np=3): (a) one window, (b)
two windows, (3) three window (LO=2.3 G H z)................................................ 94
5.31 M axim um intensity comparison using one, two, three, and four
windows (N p = 3 )...................................................................................................... 95
5.32 a) TFA o f signal using DPSS (3 windows, Np=3), b) maximum
intensity comparison with power signal. LO=2.3 G H z .................................. 96
5.33 TFA o f signal using a data window o f 512 data points and a) Hanning
window b) DPSS generated window applied to the local
autocorrelation function. The local oscillator was set at 2.3 G H z................ 97
B1
Six common window functions.......................................................................... 121
B2
Chebyshev window function for various values o f b eta................................ 122
B3
Kaiser window function for various values o f b e ta .........................................123
B4
Slepian sequences for a time-bandwidth product o f 3 .................................... 124
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LIST OF TABLES
Table
3.1
Characteristics o f commonly used window functions [S T 0 9 7 ].....................32
3.2. Some common kernel distributions [C O H 95]................................................... 47
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LIST OF APPENDICES
Appendix
A.
MATLAB Plasma Bay Toolbox...........................................................................101
B.
MELBA Time-Frequency Analysis Program (T FPR O G )...............................117
C.
Time-Frequency Analysis Using the Binary Distribution Source C ode...... 130
D.
Time-Frequency Analysis Using the Bom-Jordan Distribution Source
Code...........................................................................................................................132
E.
Time-Frequency Analysis Using Slepian Sequences Source C o d e.............. 134
F.
Time-Frequency Analysis Using the Wigner Distribution Source Code. ..136
G.
Time-Frequency Analysis Using the Zhao-Atlas-M arks Distribution
Source C o d e.............................................................................................................138
H.
Source Code for TFPR O G ..................................................................................... 140
I.
Derivation o f M icrowave Frequency Modulation Due to Cathode
Voltage Fluctuations..............................................................................................143
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CHAPTER 1
INTRODUCTION
High power microwave (HPM) devices (>1 MW) have been used to heat plasmas
in tokamak fusion reactors [GIL80]. As demand for newer technologies increases, they
are expected to provide a larger role in such areas as plasma processing o f materials, high
resolution radar [GAP94, BEN92] and neutralization o f hostile electronic devices
[CAR94]. But as the power level o f these devices is increased, the pulse length is found
to decrease correspondingly such that the total energy content in the HPM pulse is
roughly a constant. This phenomenon is called microwave pulse shortening, and is the
subject o f intense current interest [IEE98]. Some physical causes for this effect have
been identified [BEN97] since improvements in pulse length have been accomplished,
but removal o f all pulse shortening contributors is yet to be accomplished.
Along with the progression o f technology, newer methods to analyze data are
required for better understanding the behavior o f devices.
Perhaps the most popular
method is the Fast Fourier Transform (FFT), a technique used to rapidly obtain the
frequency content o f a signal. Before the development o f the FFT, large data sets could
not be analyzed within an acceptable length o f time.
The development o f the FFT
[C 0 0 6 5 ] has contributed to the overall technology explosion over the last 30 years.
However, the major deficiency with the Fourier transform/FFT is the inability to timeresolve the frequency content o f a signal. Some techniques to improve this situation have
been implemented, but have not been fully successful due to the existing uncertainty
principle (spectrogram) or the use o f different basis vectors that can make interpretation
o f the spectrum difficult (wavelets)[NAS99].
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This dissertation reports the results o f applying a new signal processing technique
to HPM devices, with emphasis on a large orbit, axis encircling, coaxial gyrotron. This
technique, called time-frequency analysis (TFA) using reduced interference distributions
(RIDs)[COH95], has proven very successful in analyzing heterodyned microwave signals
from gyrotrons. The FFT has been the predominant method o f analyzing heterodyned
microwave signals from all HPM devices prior to our present works [PET98]. While this
dissertation discusses the use o f TFA to experimental data, TFA can also be used on
computationally generated data with equal success.
Coaxial gyrotron experiments were conducted at the Intense Energy Beam
Interaction Laboratory at the University o f Michigan [JAY99, JAYOO]. The gyrotrons
were driven by the Michigan Electron Long Beam Accelerator (MELBA), a Marx
generator, with the following parameters: Vcathodc = -800 kV,
kA, and pulse lengths o f 0.5 - 1.5 ps.
I c a th o d c
= 6 kA, Itube = 0.8
An Abramyan circuit [ABR.77] is installed on
MELBA to flatten the output voltage.
All HPM devices, when long pulses are desirable, suffer from pulse shortening in
one form or another. Recently, optical emission spectroscopy was used to examine the
effects o f plasma in the coaxial gyrotron at the University o f Michigan [COHOO]. In the
present dissertation the heterodyned microwave signal o f the coaxial gyrotron is
rigorously analyzed using advanced signal processing techniques with emphasis on using
TFA using RIDs. A strong correlation between microwave frequency modulation and
cathode voltage fluctuations has been established. Mode competition (mode hopping and
multi-moding) has been observed.
similar to the detector/power signal.
Time-evolved maximum intensity plots are very
Therefore it is now possible to track the power
evolution o f each mode with adequate resolution.
For the first time pulse shortening
mechanisms are traced from the temporal evolution o f the individual modes.
In addition to analyzing the frequency spectrum o f a coaxial gyrotron, this
dissertation also discusses the preliminary results o f analyzing a microwave oven
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3
magnetron.
While on the surface it may seem unnecessary to analyze a piece of
equipment that has been a standard household item for three decades, a great deal of
insight can still be obtained by applying the TFA technique. The magnetron is one o f the
oldest, least expensive, and most efficient (85 percent) microwave devices produced and
yet there is no adequate analytic theory to describe its operation. There is another reason
to analyze the spectrum o f microwave oven magnetrons. The operating frequencies o f
wireless communication devices are increasing as the bandwidth is starting to saturate.
Eventually the operating frequency o f these devices will be near that o f the microwave
oven magnetron (2.45 GHz). The bandwidth o f the hundreds o f millions o f microwave
oven sold throughout the world may be wide enough to overlap the operating frequencies
with that o f the communications device, thereby causing interference [YAM95]. Finally,
the time-frequency analysis is being applied to a relativistic magnetron which operates
with an efficiency rarely exceeding 30 percent, much lower than conventional
magnetrons [BEN92].
Comparing the time-evolved spectra o f relativistic magnetrons
with that o f microwave oven magnetrons may provide some insight into the low
efficiency observed in the relativistic magnetron.
Spectral estimation can be difficult.
High variances can occur, producing
unstable spectra. Autocorrelation functions and window functions are commonly used in
signal processing to help make the spectrum interpretable. However, window functions
bias spectra by altering the data to be analyzed. The use o f multitapering data to provide
spectra with reduced bias and lower variance has been previously used on the Fourier
transform. This dissertation discusses the extension o f multitapering to time-frequency
analysis. Results have been useful when the time-evolved spectra has only one dominant
frequency component at any time. The time-evolved power o f a mode (eigenfunction
solution o f the Helmholtz equation) is better approximated using this method. Finally,
when using large data windows to estimate the instantaneous spectrum, cross-terms in the
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4
time-frequency plane can be reduced significantly.
The result is a TFA that is
understandable.
This dissertation is divided into six chapters. Chapter 2 supplies the background
theory for the coaxial gyrotron.
processing theory and methods.
Chapter 3 provides an extensive discussion on signal
Chapter 4 explains the system configuration and
diagnostics used for this dissertation. Chapter 5 provides the experimental methods and
results. Chapter 6 is the summary o f the experiment and conclusions. This dissertation
not only discusses the novel results o f TFA, but also introduces the reader to basic and
advanced signal processing techniques.
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CHAPTER 2
GYROTRON BACKGROUND THEORY
2.1 Fundamental Electromagnetic Concepts
2.1.1 The Generalized Wave Equation
The electric field vector E inside an empty wave structure satisfies the Helmholtz
equation [HAR61]:
(v 2 +A': )e = 0
where k=co/c is the wave number in free space.
(2. 1)
The coaxial gyrotron interacts
predominantly with transverse electric (TE) waves, that is, electromagnetic waves which
have no electric field along the axis o f propagation.
The radial electric field Er and
azimuthal electric field Eo inside a cylindrical waveguide are calculated by [RAM84]:
Er = i ^ - B J n {kcr)sm{md)
k'r
(2.2a)
Eo = ^ - B J m{kcr)cos{m9)
(2.2b)
where p is the magnetic permeability, co is the radian frequency o f the electromagnetic
radiation, kc is the wave number propagation cutoff, r is the radius, B is an amplitude
constant, m is the azuimuthal mode number, Jm is the Besselfunction o f the mlh kind, and
Jm’ isthe derivative o f the Bessel function o f the mth kind.
Forcylindrical cavities o f
length L the electric fields also vary with axial position. Thus,
Er = ^ T ^ - B J m{kcr)s\n{nid)sm\
k ;r
\ L )
\
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(2.2c)
6
The modes o f a cavity are defined by three numbers: m, n, and p. The resonant
frequencies (TEmnp) o f a cylindrical cavity o f length L and radius R is given by:
»“' m n p
=c.£k +£!£l
t-W
2
VR
r 2
1
(2.3)
where x mn is the nth root o f the equation J m(x)=0. Determination o f the mode cutoff
frequency coc can be determined by setting p=0.
2.1.2 Q and Bandwidth
Consider a cavity as a standard linear, time-invariant system. Excitation o f the
cavity with a delta function in time will yield all possible modes. The electric field inside
the cavity is a superposition o f the electric fields for all existing modes. The total electric
field inside the cavity can be modeled by the equation
£ ( 0 = J £<>..* ^ c o s ( a v )
(2.4)
n= l
where Eon is the initial electric field, Qn is the quality factor, and con=27tfn is the frequency
o f emitted radiation for the nth mode. The quality factor is a measure o f the energy stored
to the power dissipated, and is defined as
a _
n
EnergyStored _ f„
nPowerDissipated Afn
(2.5)
where Afn is the frequency spread (bandwidth) representing the full width half max
(FWHM) on the power vs. frequency response curve for the nlh mode. Large values o f
Qn indicate that the nlh mode oscillates for a long time, while small values o f Qn indicate
that the nth mode decays rapidly. The bandwidth o f the nlh mode is proportional to 1/Qn.
Consider the case where two modes are distant and have small resonant
bandwidths (high Qs), as shown in Figure 2.1a. Here the modes are well separated. Now
consider the
case where there are
bandwidths),
Figure 2.1b.
two modes closely separated with low
Mode determination can become more difficult.
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Qs (high
The
7
determ ination o f existing modes is o f utmost concern in understanding oscillator
behavior, and is the focus o f this dissertation.
ru3
■8 0.4
2
|
0-2
<
0
10
20
30
40
50
60
70
80
90
100
60
70
80
90
100
Frequency (Hz)
1.5
'5
s
S3
.*J
XI
h»
e
<
0
10
20
30
40
50
Frequency (Hz)
Figure 2.1 Example o f two modes a) widely separated b) closely separated.
2.1.3 Plasma Terminology and Effects
Plasmas are gases that are at least partially ionized and are dominated by
Coulom b forces.
Consider a cold, collisionless plasma. Displacing the electrons from
the ions results in a net electric field. Assume the ions do not move due to the large mass
com pared with the mass o f the electrons. The electric field will produce a force on the
electrons such that the electrons will move towards and away from the ion sheet in a
harmonic fashion with a plasma frequency cop [CHE84] o f
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where n is the plasma density, e is the charge o f the electron, me is the electron mass, and
So is the free space permittivity.
Plasmas interact in many ways with electromagnetic waves, depending on the
frequency and polarization o f the electromagnetic waves. For a plasma with no external
electric or magnetic fields, the resulting dispersion relation is
co2 = eo2
p + c 2k 2
(2.7)
If the propagation constant k is real, then the electromagnetic wave will propagate
through the plasma. Therefore the plasma frequency has to be less than the frequency of
the electromagnetic wave for the electromagnetic wave to propagate in the plasma.
Reflection o f the electromagnetic wave occurs when co<cop. In waveguides and cavities,
the presence o f plasmas can alter the propagation constant and the resonance frequency.
2.2 The Gyrotron
The gyrotron is a high power microwave source in which the driver is a rotating
electron beam about an external magnetic field. The gyrotron in this dissertation uses a
large orbit, axis encircling, relativistic electron beam. The high voltage applied to the
cathode tip o f the Marx bank is the source o f electrons. The electron beam is bunched
azimuthally as it travels through the cavity while at the same time rotating at the electron
cyclotron frequency.
The rotational energy o f the electron beam is transferred to the
microwave cavity, setting up transverse electric modes, and thus producing microwave
radiation.
Radiation is generated when there is synchronism between the cyclotron
frequency and the transverse mode frequency [LAU82].
Operation o f the gyrotron is
extensively detailed in several dissertations [HOC98, JAYOO, COHOO].
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9
2.2.1 The Dispersion Relation Curve for the Gyrotron
The dispersion relation [DOH88] is a means to represent the coupling o f the
electron beam to the structure. The dispersion equation for the e-beam's Doppler-shifted
cyclotron wave is:
a) = 5Qc + k,v.
(2.8)
ar=a>;+k:cl
(2.9)
The waveguide dispersion relation is:
where s is the harmonic number, Qc is the relativistic electron cyclotron frequency, kz is
the axial propagation constant, vz is the electron axial velocity, and coc is the waveguide
cutoff frequency. Microwave generation can occur when Equations 2.8 and 2.9 are both
approximately satisfied.
Physically, this means that the phase velocity o f the wave
matches that o f the electron beam. Figure 2.2 displays the gyrotron dispersion relation o f
the TE n waveguide mode for the following parameters:
Vcathodo= -750 kV, magnetic
field o f 1.6 kG, and a beam a (defined as the ratio o f the perpendicular-to-parallel
velocity with respect to the external magnetic field) o f unity [JAY99]. If the slopes o f the
waveguide and beam dispersion curves at the point o f intersection are both
positive, this
device operates on the forward wave. Further inspection o f the dispersion curves reveals
the phase velocity, v^, = co/kz, o f each equation is greater than the speed o f light, resulting
in a fast wave [FEL99]. Backward wave oscillation can also result, leading to tunability
o f the gyrotron, shown by the dashed line in Figure 2.2.
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10
3.5
(Qvvg= coc+ k zc
a
3
s« 2.5
U
2
o
fH
X
3
2
1.5
v=c
'earn
0.5
-100
-80
-60
-40
-20
0
m m " 1]
20
40
60
80
100
Figure 2.2 Dispersion curves for the electron beam and the T E n waveguide mode. The
dashed line is the dispersion relation that results in backward wave oscillation.
2.3 M icrowave Pulse Shortening
Pulse shortening is the phenomenon in which the desired duration o f the
microwave pulse is smaller than the voltage and current pulses.
This effect is very
undesirable in HPM [AGE98]. Causes for pulse shortening are:
1) electron beam voltage fluctuations
2) mode competition
3) beam loading
4) plasma generation
5) electron streaming
6) breakdown
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11
Benford [BEN97] reported plasma generation as the primary contributor to pulse
shortening in the relativistic magnetron.
While plasma effects on microwave pulse
shortening have been studied at the University o f Michigan [GIL98,COHOO], this
dissertation discusses electron beam voltage fluctuations and mode competition.
2.3.1 Voltage Fluctuations o f the Electron Beam
Voltage fluctuation o f the electron beam occurs for many reasons, therefore it is
important to understand the response o f the system when the electron beam voltage
changes. Both terms in the electron beam dispersion equation (Equation 2.8) change due
to voltage fluctuations, as the relativistic cyclotron frequency changes by changing the
relativistic factor y: increasing voltage reduces the relativistic cyclotron frequency Qc,
whereas the relativistic cyclotron frequency increases with a reduction in voltage.
Secondly, the axial velocity vz changes with a voltage change. These changes in beam
voltage leads to the following changes in operation frequency:
Sco
co
5y V j
v M .-
V
P^c
I
^
( 2 . 10 )
where p2= l-y ‘2 (see Appendix I for the derivation). Large changes in voltage result in
large frequency shifts, increasing the bandwidth. Voltage fluctuations can also change
the coupling between the beam and the structure, affecting the microwave power output
[GOE98].
2.3.2 Mode Competition
When more than one mode exists in a given time interval, mode competition is
said to occur.
We identify two types o f mode competition: mode hopping and multi-
moding. Mode hopping occurs when the oscillator transitions to another mode from the
currently existing mode.
Multi-moding is the phenomenon where two or more modes
exist simultaneously.
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12
The effects o f multi-moding in a system designed for operation in one mode are
detrimental. Unnecessary modes cause adverse bunching o f the beam, resulting in HPM
disruption in the worst scenario [KRE84].
Multi-moding reduces the coupling o f the
beam to the desired mode, reducing efficiency [GUS92].
Many factors determine the gyrotron mode that oscillates in a cavity. A large
cavity size may result in a spectrum in which the density o f modes supported is large.
The geometry o f the electron beam is also important. For a fixed beam radius, the cavity
radius may be large if wall loading is to be reduced.
excite longitudinal or transverse modes [MUG90].
Reflections o f microwaves can
When the Q o f the desired mode
reaches a minimum, other modes may have a comparable or higher Q and can therefore
be excited. Starting currents o f the various modes are also important [WHA94]. The ebeam velocity spread, pitch, and current can vary during the startup. Thus, during the
startup phase, several modes can oscillate when the e-beam a and energy changes.
M ulti-moding reduces the efficiency o f an oscillator because energy is drained
from the desired mode.
Extensive general theory on the coupling o f modes has been
conducted by Haus [HAU91]. M odeling o f mode competition was conducted for lasers
[LAM64] and was later interpreted for masers [MCC88, NUS99], For a system operating
with only two modes, the predator-prey equations, which have often been used to model
mode competition, are
—-j— = A l [o'l —/? \A \—y l2 A2]
(2.11a)
^ L = gA2[o-2 - / f f 2A2 - r 2,A,]
at
(2.11b)
where q isthe ratio o f the coupling impedances o f the beam tothe two modes, Ai is the
mode amplitude, a, isthe linear growth rate, Pi is the self-saturation coefficient, and y-, is
the cross-saturation coefficient for mode i, i= l, 2. The parameters are assumed to remain
constant over time.
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13
Equations 2.11a and 2.11b reveal when a two-mode operation is stable and
unstable. For stable two-mode operation to occur, the weak coupling relation
( 2 . 12 )
P \ P i > Y \iY 21
is satisfied. Dominant single mode operation takes place when strong coupling occurs,
satisfied by
P \P i< r n Y i\
(2-13)
Figure 2.3 shows a case when there is stable two-mode operation, while Figure 2.4 shows
an exam ple when a single mode dominates at any given time. The switching o f mode
domination in Figure 2.3 shows the second mode eventually growing and then saturating,
while the initial mode intensity is reduced to zero over time. The initial amplitude o f the
second mode was 1/10000 o f the first mode initial amplitude.
For the case when the
initial amplitude o f the second mode was zero, no growth for the second mode was
observed.
Therefore, for mode competition to occur, there must be some initial
am plitude in the second mode [see Equation (2.11b)].
Intense electron beams always
generate incoherent noise, providing the source for the second mode initial amplitude.
The stability o f a mode is very important. Parasitic modes will not grow in the
presence o f a stable mode, even if the system lies in a region where the parasitic mode
can oscillate [WHA94].
Unwanted modes can be suppressed by the stable mode. The
stable mode will raise the threshold e-beam currents o f nearby modes such that they will
not start.
The e-beam parameters determine if a mode is stable, and can cause mode
stability to change from stable to unstable.
If the modes are unequally spaced in
frequency, the interaction between the modes will primarily be determined through their
amplitudes [MCC92],
The relative phase between modes plays an important role for
equally spaced modes.
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14
Amplitude (arbitrary units)
9.5
M ode 1
8.5
M ode 2
q=2
7.5
0
5
10
15
20
25
30
35
40
45
Tim e Step (arbitrary units)
Figure 2.3 Simulation for an oscillator with stable two-mode operation.
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50
15
45
M o d el
A
40
Mode 2
=10
a 2Q=0.001
a =1000
Amplitude (arbitrary units)
35
30
25
20
15
10
5
0
0
5
10
15
20
25
30
35
40
45
Time Step (arbitrary units)
Figure 2.4 Simulation for mode competition in which the second mode eventually
dominates.
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50
CHAPTER 3
SIGNAL PROCESSING THEORY
3.1 Signal Processing Fundamentals
Analysis o f sampled data o f the high power microwave sources benefits from the
use o f digital signal processing techniques. When properly applied, these techniques can
reveal information not readily identifiable by visual inspection o f the signal as a function
o f time.
3.1.1 Model and Basic Term inology o f Digital Signal Processing
The model considered is the oscillating function
.v(/) = c o s ( 2 ^ + ^)
where
x(t) isthe
derivations
signal attime
(3.1)
t with frequency F and phase 4>.Throughout all
and explanations,uniform sampling is assumed.
Let T
denote the time
difference between samples; then 1/T is the sampling frequency, Fs. The data is then
sampled at times 0, T, 2T, 3T, ....nT , where n is an integer.
Ignoring the phase, the
sampled model becomes
x(nT ) = c o s(2 xfn T )
(3.2)
Replacing x(nT) with x(n) for notational purposes only and substituting 1/FS for T, the
model appears as
x(n) = cos{2xfn)
16
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(3.3)
17
where f =F/FS and is called the normalized frequency.
Noting the radian frequency
co=2jrf, the final model becomes
-v(«) = cos(ry «)
(3.4)
which is commonly used for analysis.
3.1.2 The Sam pling Theorem
Consider Equation 3.4.
The model will have the same values for x(n) for all
values o f co=coo+2nk, where k is an integer. The restriction 0< co<2 tt is applied to assure
proper reconstruction/evaluation o f the signal. Shifting the frequency range to -ji<co<7c
allows for representation o f both positive and negative frequencies.
However, cither
range representation is fine as the results will be the same. The shifting o f the frequency
range provides for understanding the requirements on sampling:
- n < 2 / f <.7
(3.5)
Solving for f,
1 ^
~~2~
1
~2
^
which yields the restriction for F in terms o f Fs:
- — <F <—
2
2
(3.7)
The limits for F indicate the maximum resolvable frequency is one half o f the sampling
frequency and is called the Nyquist limit or folding frequency.
Sampling data with a frequency content greater than the Nyquist limit results in
aliasing.
Take for example a signal with a frequency content o f 600 MHz and sampled at
1 GHz. Plugging the data into the model equation, the signal is represented as
.t(n) = c o s ( 2 r a - 6 0 0 /1 0 0 0 ) = cos(l.27Tn) = cos(0.8jtrt)
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(3 .8 )
18
which represents a signal with a frequency content o f 400 MHz. The frequency content
is “ folded” about the Nyquist limit by the difference between the actual frequency and the
Nyquist limit.
Figure 3.1 is a graph o f three signals, each with a normalized sampling frequency
o f 1, and therefore a Nyquist limit o f 0.5. Figure 3. la is the graph for s(n)=cos(27tn-0.3).
Because the signal frequency is below the Nyquist limit, the signal is said to be
oversampled.
Figure 3.1b displays the signal s(n)=cos(27m-0.5), and is considered
critically sampled because the signal frequency is equal to the Nyquist limit. Figure 3. lc
displays the signal s(n) = cos(2rcn-0.7), and is undersampled because the signal frequency
is greater than the Nyquist limit. Inspection reveals the signal with a frequency o f 0.7
looks exactly like the signal with a frequency o f 0.3. Figure 3.1c is a signal in which
aliasing has occurred.
Aliasing is a very important phenomenon that m ust be understood. One example
o f aliasing in everyday life is a video cam era observing the wheels o f a car. Accelerating
from stop, the car wheels turn the proper way. However, when the car reaches a speed
such that the frequency o f rotation is the same as the sampling frequency o f the camera,
the wheels appear stationary. Finally, if the car goes faster, the wheels appear to turn the
opposite way. This effect is due to aliasing.
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Figure 3.1 Comparison o f three signals, each with a normalized sampling frequency o f
a) 0.3, b) 0.5, c) 0.7, respectively.
3.1.3 The Autocorrelation Function
Sometimes data is noisy and is very difficult to determine if any periodicities
occur. The continuous autocorrelation function R is widely used in signal processing to
detect hidden periodicities within the signal, and is defined as
(3.9)
where x denotes the lag (signal delay). The discrete autocorrelation function with length
N is defined in two ways:
* * 0 ") = — £ jr(w + /w)r’ (/i)
/i*0
1
(3.10a)
.V - m - l
(3.10b)
w -H
^o
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20
where N is the recorded data length, m is the lag (delay), Rb is the biased estimate and Ru
is the unbiased estimate. The discrete form o f the autocorrelation function is not over all
lags, but over a range o f lags. The differences between the biased and unbiased estimates
are the weight functions for each lag. The biased autocorrelation function weighs each
lag value equally, while the lag values for the unbiased case are weighed more for larger
lag values. Unbiased estimates try to approach the continuous autocorrelation function.
Take for example a signal with x(n)=l for all n and with length N—>0 0 . The continuous
autocorrelation function would be ones for all lag values. The biased estimate imposes a
triangular weighting function to the recorded signal such that the zero lag value is
maximum, and therefore the estimate will be incorrect. However, the unbiased estimate
will yield all ones.
The importance o f the autocorrelation function as pertaining to
spectral analysis is explained in section 3.2.1.
3.2 The Fourier Transform
Frequency is the number o f sinusoidal oscillations per second. Understanding the
frequency content is o f paramount importance in many experiments.
Integrable
transforms [ARF85] provide for a mapping o f a function from one space to a desired
space. The Fourier transform is an obvious choice to understand frequency content. The
continuous one dimensional Fourier transform X(co), along with the inverse Fourier
transform, are defined as
(3.11a)
and
(3.11b)
and the discrete Fourier transform (DFT), along with its inverse, defined as
(3.12a)
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21
.
2 it k n
iV - l
=
‘v
(3.12b)
' V *=0
where co=27tk/N. The Fourier transform properly represents the signal if the Dirichlet
conditions are met [PR 096]:
1.
The
signal x(t) has a finite number o f discontinuities
2.
The
signal x(t) has a finite number o f minimums and maximums
3.
The
signal x(t) is absolutely integrable, satisfying
*
dt < oo
(3.13)
-3 C
3.2.1 Some Properties o f the Discrete Fourier Transform
Most transforms conserve some parameter.
The DFT conserves energy.
The
energy density o f a signal x(n) at time n is |x(n)|2, whereas the energy density o f the
spectral coefficients X(k) is |X(k)|2. Thus, the following equation
n=0
' v *=0
(3 1 4 )
is the mathematical expression for conservation o f energy and is called Parseval's
theorem.
There are two approaches to calculating the spectral energy density.
method is to square the magnitude o f the Fourier transform o f the signal.
The first
The other
method is to simply Fourier transform the autocorrelation (discrete autocorrelation
function) o f the signal.
The second method is widely used when significant noise is
present. Care must be taken when Fourier transforming the autocorrelation o f the signal.
Figure 3.2 is an example o f the autocorrelations o f a signal consisting o f 64 points, each
point with a value o f one. Figure 3.2a is the biased autocorrelation, while Figure 3.2b is
the unbiased autocorrelation.
autocorrelation.
Figure 3.2c is the spectral energy density for each
The energy density o f the unbiased autocorrelation has a large, thin
peak, while the energy density o f the biased autocorrelation has a smaller, wider peak.
The effect o f the imposed triangular window on the biased estimate is discussed in
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22
section 3.3.51.
The biased estimate is useful when reduction in statistical spectral
variance is important.
The Fourier transform is a linear transform; the transform o f the sum o f two
signals is the same as the sum o f the two transforms. However, the energy density o f the
sum o f two signals is not the sum o f the energy densities. Let Xi(co) and X 2 (co) be the
Fourier transform o f signals xj(t) and X2 (t). The energy density |Xtol(co)|2 is calculated by
H * , H 2 + |* 2 M l2 + 2 Re{*,> ) * > ) }
(3.15)
where the last term is an interference/cross-term effect.
vT
' 1• «V • •.
S i
‘^ v , .
1 1 05
" I
“ V ;- .
o>~'............................. -.................... .................... .........
■60
-40
-20
-40
-20
0
Lag Value
20
40
0
Lag Value
20
40
60
15
b)
S1
15
f2
< § 05
<
0
-60
-
.............
60
150
C)
'100
•
0.1
unbiased
-
0.08
-
0.06
-
0.04
-
biased
0.02
0
0.02
Normalized Frequency
0.04
0.06
0 08
0.1
Figure 3.2 Results for a discrete signal consisting o f 64 points, all equal to the value o f 1.
a) Biased autocorrelation, b) unbiased autocorrelation, c) spectral energy
densities o f the autocorrelations.
One property o f the DFT is that the magnitude o f the transform is invariant with
time.
This means the spectrum magnitude will be the same whether the signal was
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23
started five seconds or five years from now. Another property o f the DFT is the realsymmetric property. Real (not complex) data will yield a conjugate symmetric spectra,
and vice versa. The real-conjugate symmetry property plays a role later in the discussion
o f time-frequency analysis.
Computations can be reduced significantly if the data is
symmetric.
Frequency modulation o f a signal occurs when the signal x(n) is multiplied by
cos(coon). The result is a spectrum with frequency shifts o f ±coo- This property forms the
basis o f heterodyning signals with a local oscillator, which is used in the microwave
experiments described in this dissertation.
3.3
As
Signal and Spectrum Statistics
previously noted, both |x(t)|2 and |X(co)|2 are energy density
functions.
Particular information, mainly means and standard deviations, can be calculated from
density functions. The mean provides an estimate o f the location about which the data is
centered, whereas, the standard deviation describes the spread o f the data about the mean.
Important information derived from the density function are mean time, duration, mean
frequency, and bandwidth.
3.3.1 Mean Time and Duration
The total energy E o f a signal x(t) is calculated by
x
E= j|.x(r)|2 dt
(3.16)
—x
The mean time o f a signal,<t>, describes the time about which the signal is concentrated
and is calculated by
) '\ - M 2 dt
=
where E is used as a normalizing factor.
(3-17)
Another important statistic o f time is the
duration, a t, which is the standard deviation o f time. The duration describes the time
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24
length over which most o f the signal has passed (2 standard deviations) and is calculated
by
CT, = -
E
(3.18)
A signal with a short duration is concentrated around the mean, whereas a signal with a
long duration is widely spread around the mean. These parameters can be important in
high power microwave experiments because it allows some description o f pulse length
and how much energy is contained in the duration.
3.3.2 Mean Frequency and Bandwidth
Along with the time statistics, frequency statistics are important.
The mean
frequency <co> describes the frequency about which the spectral energy is centered. The
frequency domain version o f duration is bandwidth, denoted as B or ctC), and gives the
spread o f frequencies about the mean frequency. A small bandwidth means the spectrum
o f the signal is very narrow, whereas a large bandwidth indicates a very broad spectrum.
The mean frequency and bandwidth are calculated by
Jco|A"(co)j~ c/co
B 2 = < *1= ------- ------------(to)2
b
(3.20)
Bandwidth has two contributors: amplitude modulation (AM) and frequency
modulation (FM).
A signal with constant amplitude but varying frequency (FM) can
have the same bandwidth as a signal with constant frequency but varying amplitude
(AM). For example, a signal with the form
J “oO
s(t) = e~^2Q' sin((oQt)
has the same "bandwidth" as the signal
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(3 .2 1 )
25
s(f)=.sz>i(<»0/ + Aco(r) r)
(3.22)
where the former signal is an exponentially decaying sinusoid while the other signal is
frequency modulated with essentially a constant amplitude. How can we determine if a
signal is frequency modulated, amplitude modulated, or both? Consider the signal
x{t)= A{t]eJ *(,)
(3.23)
where A is the signal amplitude and <j> is the phase o f the signal. Cohen [COH90] defines
the AM and FM bandwidth contributions as
x
Bis, = J(^ '(0 )2^
(3.24)
-x
X
B h, = j(<i>'(')-<®))2 -42(0 dt
(3.25)
—x
where A'(t) and <J>'(t) are the derivatives o f the signal amplitude and phase respectively.
The total bandwidth is defined as
B = B as, + BFU
(3.26)
By examining Equations 3.22 and 3.23, one can see the AM bandwidth calculations only
include the amplitude term,whereas the FM bandwidth equation contains both frequency
and amplitude.
3.3.3 The Uncertainty principle
In 1927 W erner Heisenberg published the uncertainty principle in quantum
mechanics. The uncertainty principle states that
crpa s > h
(3.27)
where ctp and cts are, respectively, the uncertainties in a particle's momentum and
position, and h-bar is Planck's constant. Time and frequency are a Fourier transform pair,
and they also obey an equivalent uncertainty principle [COH94]:
(3.28)
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26
which means if the duration o f a signal is reduced, the bandwidth will increase and vice
versa.
A purely oscillating signal (containing one frequency) will yield an infinite
duration but zero bandwidth. Likewise, a delta function in time has infinitesimal duration
but infinite bandwidth.
3.3.4 The Analytic Signal
The signals used for analysis in this dissertation are not complex. This leads to a
problem because the Fourier transform o f a non-complex signal results in a symmetric
spectrum magnitude. Calculation o f the mean frequency from real data o f a symmetric
spectrum will always yield a mean frequency o f zero. Another problem with real signals
is the determination o f the instantaneous values o f phase and frequency.
The analytic signal [REI94, WIL98] provides for a way to solve these problems.
For a signal x(t) the analytic transform z(t) is calculated by
(3.29)
-x
where P is the principal value.
The second part o f the right hand side o f the above
equation is the Hilbert transform H[x(t)], which is the convolution o f the signal with 1/rct:
(3.30)
The analytic signal does not conserve energy because the real part o f the analytic
signal is the signal itself (assuming the original signal is not complex).
However, the
Hilbert transform does yield the same energy content o f the original signal. Therefore,
the analytic signal has twice the energy o f the original signal (the energies o f the real and
imaginary parts o f the analytic signal are equal). The analytic signal should not be used
for signals with energy concentration near DC, as distortion in the spectrum will occur
[WIL90].
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27
The analytic signal has three benefits. First, the Fourier transform o f an analytic
signal will only have frequency content in the positive frequencies, providing a means to
calculate average frequency.
Second, the amplitude and phase o f the signal is
unambiguously determined [PIC97].
Third, since there is only positive frequency
content, there is no cross-term effect at the DC frequency when using time-frequency
analysis, which in section 3.4.4.
3.3.5 Inherent Problems With The Discrete Fourier Transform
The DFT works well in many cases. However, there are three problems with the
calculation o f the Fourier transform: spectrum biasing, large spectral variance, and
discontinuities.
Extensive research has been conducted to remove or reduce these
problems with some success.
3.3.5.1 Spectrum Biasing
The problem with spectrum biasing is straightforward to understand. The lengths
o f recorded data are finite, and are only a part o f the true function. Assume x(n) to be the
true function. The data recorded, xr(n), will have the functional form o f
X,
(n) = .r(/i)vv{«)
(3.31)
where
J
1, 0 < n < jV -1
| 0,
elsewhere
(3.32)
where N is the number o f recorded data points. Therefore the recorded data is the true
function multiplied by a rectangular window. One property o f the Fourier transform is
the multiplication/convolution property.
The multiplication o f two functions in the
time/frequency domain results in convolution in the frequency/time domain. Extending
this property to the current problem, the DFT o f the rectangular window is
(3.33)
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28
with a magnitude response o f
(3.34)
CO
Slnl l .
and is called the Dirichlet kernel. Convolution o f the true spectrum with the spectrum o f
the Dirichlet kernel results in two major effects: smearing and leakage. Smearing causes
broadening o f the spectrum around the actual peaks.
density for the Dirichlet kernel (N=31).
Figure 3.3 is the energy spectral
The main peak, centered on a normalized
frequency o f zero, is called the main lobe. The other ripples are called side lobes. The
main lobe is responsible for smearing and spectral variance reduction. Leakage occurs
due to side lobes in the Dirichlet kernel located throughout the spectrum. The effect o f
side lobes is the placement o f energy in frequencies that do not exist. Increasing the data
size reduces smearing but not leakage; the side lobes move closer to the main lobe as the
size o f the data sample is increased while maintaining their amplitude.
m ain lobe
-10
-20
-50
-60
side lobes
-70
-
0.5
-
0.4
-
0 .3
side lobes
-
0.2
-
0.1
0.1
Normalized Frequency
0.2
0.3
0.4
Figure 3.3 Magnitude o f the Dirichlet kernel (dB).
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0.5
29
Figure 3.4 displays an example o f broadening from the smearing effect. A signal
(not converted to its analytic form) was generated with normalized frequencies fi=0.1 and
f2=0.1l.
Figure 3.4a is the spectrum magnitude when using 1024 points.
The
frequencies are easily resolved. Figure 3.4b is the spectrum magnitude when using only
64 points. There is apparently only one frequency peak, although this is not true. The
effects o f broadening can be reduced by either increasing the number o f data points used
or calculating the spectrum using parametric analysis (requires prior knowledge o f the
quantity o f spectral peaks).
When the biased autocorrelation o f a signal is applied to
calculate the spectrum, the window becomes triangular (also known as a Bartlett
window):
>*'«(*)=
(3.35)
0
,
elsewhere
with the respective spectrum magnitude, which is known as the Fejer kernel:
sin(<y/V / 2 )
(3.36)
sin (< y / 2 )
N=1024
g 300
E
2 200
TJ
g. 100
VJ
•c
a
0.2
0.25
0.3
Normalized Frequency
0.35
0.25
0.3
0.2
Normalized Frequency
0.35
30
20
s3
h
o
8.
5/3
10
0
0
0.05
0.1
0.15
0.4
0.45
0.5
Figure 3.4 Example spectrum magnitude for a signal with two frequencies a) N=1024
points, b) N=64 points.
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30
3.3.5.2 Variance Reduction
While the Fourier transform has enjoyed some success in estimating spectral
peaks, it does suffer from a statistical standpoint.
Recorded data is always finite in
length, yet the estimation method (continuous Fourier transform) requires an infinite
number o f data points.
Further, the energy spectral density o f is from one
realization/signal only, leading to a high variance in the spectrum. Modifications to the
energy spectral density calculations are imperative to reduce variance.
Three
nonparametric methods used to reduce the variance o f the spectrum are averaging pieces
o f a signal, filter banks, and window functions.
One approach to reducing variance and increasing statistical stability with only
one realization is to divide the one realization into multiple, smaller length realizations.
Operations would be conducted on the segments and then averaging the spectrums o f the
segments is conducted.
Two commonly used techniques are the Bartlett and Welch
methods [ST097],
The Bartlett method involves dividing the data into M segments o f length N with
no overlap. Then the DFT would be applied to each segment.
yields the new, reduced variance results.
Averaging the spectra
The Welch method is very similar to the
Bartlett method, except overlapping occurs.
Maximum performance o f the Welch
method occurs at an overlap o f half the signal length.
Problems do occur with the Bartlett and Welch method. Shorter segments lead to
a broader spectrum due to biasing.
The assumption behind dividing the signal into
segments is to provide for several uncorrelated samples.
However, when using the
Welch method, a large overlap results in a large correlation between two segments.
Large variance will occur, resulting in a statistically unstable spectrum.
A nother approach to reducing spectral variance is to apply a bandwidth filter.
Consider a bandpass filter o f width Aco, centered about frequency coc and with unity
magnitude. Applying the bandpass filter to a signal, only components with frequency
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31
content coc-A/2< co <coc+A/2 will exist.
From Parseval’s theorem, the energy o f the
filtered signal is equal to the total energy in the frequency band o f interest. The energy
density for each frequency within the band is equal to energy in the band divided by the
filter bandwidth. Sweeping the filter through all frequencies, a bank o f filters is formed.
The result is a spectrum with reduced variance.
Before applying the filter bank
technique, three assumptions are made:
1. The spectrum is nearly constant over the filter passband
2. The filter magnitude response is one over the passband and approximately
zero outside o f the passband
3. Consistent power estimation o f the filtered signal is satisfied
Problems do occur using the filter bank technique.
If a very narrow bandpass
filter is used, the impulse response o f the filter is very long.
The results are a data
sequence with very few data points, violating assumption 3. However, a wide bandpass
filter results in a short impulse response. Violation o f assumption 1 can occur in this
case. Application o f the filter bank technique must be carefully implemented.
The most commonly used method to reduce spectral variance is the application of
window functions.
These window functions act as weighting functions on the data.
Conventional data windows are symmetric. A peak value o f unity is located at the center
o f the window, and the window function gradually tapers to a non-negative value (usually
zero). Figure 3.5 displays some o f the commonly used window functions: rectangular,
Bartlett, triangular, Hamming, Hanning, and Blackman.
Table 3.1 displays the
characteristics o f some common window functions.
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32
Table 3.1 Characteristics o f commonly used window functions[ST 097|
Window
Function
Main Lobe
Peak Sidelobe
Width (radians)__________ (dB)_____
Rectangular
Bartlett
Hanning
Hamming
Blackman
2n/N
4rc/N
4rc/N
4n/N
6roW
-13
-25
-31
-41
-58
Rectangular
0 .2
0 .4
0.6
Hanning
0.8
0.2
Normalized Time
Hamming
0.4
0.6
0.8
Normalized Time
Bartlett
JZ
" 0.5
« 0.5
0 .2
0 .4
0.6
0.8
0.2
Normalized Time
Blackman
0 .2
0 .4
0.6
0.8
Normalized Time
0.4
0.6
0.8
Normalized Time
Triangular
0 .2
0.4
0 .6
0.8
Normalized Time
Figure 3.5 Comm only used window functions.
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33
3.3.5.3 Discontinuities
Use o f the Fourier transform is optimal when the data is oscillatory.
Fourier
analysis has a problem when analyzing data with sharp jum ps or discontinuities. When
reconstructing the signal using the inverse Fourier transform, large overestimations
and/or underestimations can occur at the discontinuities.
Gibbs phenomenon.
This behavior is called the
If a window function (such as a Blackman window) that tapers
gently to zero is used, reduction o f overestimation can be accomplished.
3.3.5.4 Discrete Prolate Spherical (Slepian) Sequences
The previous methods for reducing variance either divide the data into smaller
segments or use a window function. What if there were a way to combine the best o f
both methods and keep the negative effects from being implemented?
One method
w ould be to use multiple window functions on the whole data segment. Multiple data
sets would be created with the same length o f the original data. If the window functions
were orthogonal, then the data sets would be independent (uncorrelated). Analysis could
be conducted on each data set and then the mean o f the parameter in question could be
calculated.
Such a method, called discrete prolate spherical, or Slepian, sequences (DPSS)
[T H 082], exists. The windows are nonconventional because some o f the windows are
odd (nonsymmetric) functions instead o f even. Calculation o f the window functions can
be conducted by taking the eigenvalue decomposition o f the Toeplitz matrix
sinftfc - p )fi) t\
k'P ~
(3 2 1 )
( k - P }r
where k and p are integers, and P is the param eter that determines the baseband filter
cutoff [-P ji,P tc]. The eigenvectors o f T are the window functions (the DPSS) that are to
be applied to the data. The eigenvalues o f this matrix describe the fraction o f energy each
window function keeps in the baseband [XU99]. Let K. be the time-bandwidth product
K = N fi> \
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(3.38)
34
where N is the number o f data points in the data segment. The K largest eigenvalues are
close to one, while the next (N-K) eigenvalues are close to zero. One attractive feature o f
this method is that the first K. eigenvectors are orthonormal to each other and form a set
o f baseband filters restricted to the region [-Pit,pit].
Figure 3.6 shows the first four
Slepian sequences for a time-bandwidth product o f 3. Let wp(n) denote the pth Slepian
sequence. The spectrum approximation is obtained by
^ Z Z x(n)wp{n)e'
&
jcj\
(3.39)
p ~ \ n=0
Some properties o f using DPSS should be noted [TH 082]. First, the variance of
the spectrum estimate is reduced by a factor o f K.
Second, the bias is reduced
significantly, resulting in a “truer” representation o f the signal and spectrum. There are
no arbitrary windows, keeping the estimate consistent. Finally, the use o f DPSS provides
for an analysis o f variance test for spectra with line components.
W indow N um ber 1
W indow N um ber 2
0.2
0.2
0.15
0.1
x
w
"5
5
0.05
0
-
0.2
0.6
0.8
0.4
N orm alized T im e
1
0.1
-0 . 2:
0
0.2
W indow N um ber 3
-
0.2i
0.1
0.1
0.1
0
-
0.2
0.4
0.6
0.8
N orm alized T im e
1
W indow N um ber 4
0.2
- 0 . 2:
0.3
0.4
0.6
N orm alized T im e
1
0.1
-0 .2:
0
0.2
0.8
0.4
0.6
N orm alized T im e
Figure 3.6 First four Slepian sequences with a time-bandwidth product o f 3.
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1
35
To show the effectiveness o f each method, a signal with normalized frequency of
0.2 with Gaussian distributed noise was generated.
Four methods were compared:
Fourier transform, Bartlett method, DPSS, and filter banks.
generated signal.
Figure 3.7a shows the
The noise distribution is displayed in Figure 3.7b.
energy density o f the signal is displayed in Figure 3.7c.
The spectrum
There is a sharp peak at a
normalized frequency o f 0.2, and some energy distributed throughout the other
frequencies, but at a significantly smaller contribution.
Figure 3.8a shows the results
when using the Bartlett method (5 segments used). A wide frequency spread in the main
component verifies reduction in variance has been accomplished. Figure 3.8b displays
the results using Slepian sequences. The rounded peak is due to the window constructed
by the DPSS, creating a more “rectangular” pulse. The DPSS result does not look similar
to the Bartlett method because the bias is also reduced. Figure 3.8c shows the results
using the filter bank method (filter width o f 0.01). The filter bank method looks very
similar to the DPSS results.
Recall from earlier discussion Slepian sequences contain
most o f the energy in the bandwidth o f choice.
considered a refined filte r bank method.
Therefore, the DPSS method can be
Differences in amplitude between the
comparisons are from the functions used in the analysis. The fractional variance is the
parameter o f concern.
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36
e
3
au
Si
3
■a
>
0
100
200
300
400
500
600
700
800
900
Time Step
S 100-
-1
0
1
Value (arbitrary units)
0.05
0.2
0.25
0.3
0.35
Normalized Frequency
F igure 3.7 a) Generated signal with Gaussian distributed noise, b) histogram o f noise
values, c) energy density spectrum using the FFT.
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1000
37
Q
20
U) Q
J
0.2
0.25
03
N orm alized Frequency
0.2
0.25
03
035
N orm alized Frequency
0.2
0.25
03
035
N orm alized Frequency
Figure 3.8 Energy density spectrum for data used in Figure 3.7 using a) Bartlett method
(5 segments), b) DPSS (Nfi=4), and c) filter bank method (filter bandwidth o f
0 .01 ).
3.4 Spectral Analysis o f Nonstationary Signals
The previous sections o f this chapter discuss methods to analyze stationary
signals, that is, the properties o f that signal do not change statistically with time.
However, many natural phenom ena do change properties over time, making the signal
nonstationary.
Examples o f occurrences with nonstationary frequency content are
whistler waves, music, bird chirps, and even electromagnetic waves in a plasma with
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38
changing densities. The rest o f this chapter discusses methods used to analyze data with
nonstationary frequency content.
3.4.1 Instantaneous Frequency
Let x(t) be a signal (which can be its analytic version) o f the form
x{t)= A {t)eJ *(,)
(3.40)
where the phase <{> can change over time. Cohen [COH93] derives a simple formula for
finding the instantaneous value for any quantity. The instantaneous frequency, o)j o f x(t),
is governed by the equation [LOV93]
a,. = M 1
dt
(3.41)
The instantaneous frequency is actually a conditional average o f frequency for a given
time [COH92, JON90]. Equation 3.41 yields the actual frequency in only one situation,
and that
isif there actually exists only one frequency component at each
time.
If, for
example, the signal consists o f two sinusoids simultaneously existing with amplitudes Ai
and A 2 and phases <j>i(t) and <j>i(t), the overall phase o f the signal is
= arc tan
A, sin(<y,r)+ A , sin(<y,/)
^ At cos((Vlt) + A 2 cos{ a 2t)
(3.42)
and the instantaneous frequency is
_
C0‘
+OJ2
________(&>2 ~ 0J\ ) ( ^ 2 ~ A \ )________
2
2(/If + Aj + 2A ,A 2 cos((&2 - <y, ]t)j
(3.43)
which oscillates due to the interaction between frequencies and can cause difficulty when
interpreting results [LOU99].
Concerns over the interpretation o f instantaneous frequency have been expressed
[LOU97b], and only a few o f those concerns are discussed here. Negative instantaneous
frequencies can occur in analytic signals, even though the frequency content for all
negative frequencies is zero. Another problem is that the instantaneous frequency may be
outside the range o f a bandlimited signal [LOU97].
These shortcomings make the
instantaneous frequency calculation unattractive for multicomponent signals.
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39
3.4.2 Properties o f Time-Frequency Distributions
The functions |x(t)|2 and |X(co)|2 are energy density functions in the one­
dimensional case.Extension to a time-frequency distribution
requires a two-dimensional
(or joint) energy distribution. Let P(t,co) be the density distribution.
The distribution
should satisfy [LOU92]
x
\P [t,a)) dtu = \ x { t f
(3.44)
—x
X
( 3 .4 5 )
\ p (t , a j ) d t = \ x { c o f
-X
which are called the time and frequency marginals, respectively. A density distribution
satisfying the marginals also satisfies the total energy requirements (which is a twodimensional form o f Parseval’s Theorem):
X X
X
X
E = J | P (t,a>) da) dt = J|.r(r))" dt =
- X —X
- X
dco
(3.46)
-X
where E is the signal energy.
Three important properties aside from marginals and energy conservation are time
shift, frequency shift, and scaling. Time shift is desirable because it provides the same
spectrum regardless o f when the signal actually starts. One important example o f a time
shift invariant density distribution is the magnitude o f the Fourier transform. Frequency
shift is desirable in the sense that if a signal is modulated by a constant frequency, the
distribution should look the same as the original distribution only shifted by the
modulation frequency. The scaling property is desirable in a density function when a
signal is compressed or expanded. The density function o f the scaled signal should be
identical to the original density function only compressed or expanded.
The support properties o f a joint distribution are also important.
If a joint
distribution is zero before the signal starts and after the signal stops, then the distribution
has weak fin ite time support.
Similarly, if the joint distribution is zero outside the
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40
frequency band, then the joint distribution has weak spectral support. If the signal stops
for a while and starts later, it is desirable for the joint distribution to be zero whenever the
signal is zero or when there is no spectral content. This distribution is said to have strong
finite support.
3.4.3 The Short Tim e Fourier Transform and Spectrogram
The most logical idea to analyzing nonstationary frequency content is to break up
the data into segments and then calculate the Fourier transform for each segment. The
spectra represent the frequency content over that time period. This idea has been widely
used over the years and is called the short time Fourier transform (STFT), and can be
mathematically described as
=
dr
(3-47)
where t is the running time and h is a weighting function used to taper the data. The
spectrogram is the energy density spectrum o f the STFT and is simply |Xt(co)|2.
Problems do exist when using the STFT/spectrogram.
window length is one.
Determination o f the
Short windows yield good time resolution but poor frequency
resolution. Large windows provide good frequency resolution but poor time localization.
Figure 3.9 shows a signal with a normalized frequency o f 0.1 at the start o f the signal
which jum ps to a normalized frequency o f 0.3 sometime during the duration o f the signal,
along with three spectrograms (window lengths 17, 33 and 65 points).
The smallest
window length (17 points) provides a good temporal localization o f the frequency jump.
However, the largest data window (65 points) displays a frequency o f 0.1 after the signal
jum ped frequency. The poor time localization can misinform the observer o f existing
frequencies. At the same time, poor frequency localization can be misinterpreted as a
signal with a wide frequency band.
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0
50
100 150 200 250 300 350 400 450 500
Time Step (arbitrary)
17 point window
50
100 150 200 250 300 350 400 450 500
Time Step (arbitrary)
33 pomt window
0
50
100 150 200 250 300 350 400 450 500
Time Step (arbitrary)
65 pomt window
50
100 150 200 250 300 350 400 450 500
Time Step (arbitrary)
Figure 3.9 Spectrograms o f window sizes 17, 33, and 65 points.
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42
3.4.4 The Symmetric Ambiguity Function
Section 3.1.3 introduced the autocorrelation function, which is a one-dimensional
function o f lag. Recall Equation 3.9:
(3.48)
—
fi
Applying the Fourier transform o f R(x) yields |X(co)|2, the energy spectral density
[COH96b]. The autocorrelation function is sometimes used instead o f the actual signal
due to the increased resolution (almost twice).
It is reasonable to conceive o f a two-
dimensional autocorrelation function such that applying the two-dimensional Fourier
transform to the two-dimensional autocorrelation function would result in an energy
density distribution in time and frequency.
The symmetric ambiguity function [COH96], denoted as A (0,t), is a twodimensional autocorrelation function and is calculated by
(3.49)
where 0 is the Doppler shift o f the signal. Examination o f the ambiguity function reveals
an inverse Fourier transform o f past and present values o f data.
But what does the
ambiguity function tells us about the signal? The area situated on and near the axes (x=0
and 0=0) describes the auto terms, which are from the actual signal components. The
area outside o f this region describes the cross terms, which are from the interaction
(beating) o f the signal components caused by using the signal twice in Equation 3.49.
Figure 3.10a is the magnitude o f the ambiguity function for the signal
.v(rt) = sin(2;r • 0.1 • n) + sin(2;r • 0.25 • n)
(3.50)
while Figure 3.10b is the magnitude o f the ambiguity o f the analytic version o f the same
signal. Observation reveals many vertical lines in the ambiguity plane. The line centered
at 0=0 represents the auto-terms o f the signal. All other vertical lines represent the cross
terms o f the signal. For this example, the cross terms in Figure 3.9a occur at ±2fi, ±2fi,
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43
±(fi+f2), and ±(fi-f2). However, the ambiguity function o f the analytic signal yields cross
terms only at ±(fi-f2).
It should be noted that the original signal has more than two
frequencies because the sine function can be described from two exponential functions
containing positive and negative frequencies, but the analytic signal only has positive
frequencies, thereby greatly reducing the number o f cross terms. The reduction in cross
terms is a m ajor reason why the analytic signal is often used in time-frequency analysis.
-100
■50
0
50
100
-
0.5
-
0.4
-
0.5
-
0.3
-
0.4
-
0.2
-
0.3
-
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.2
-
0.1
0
0
0.1
0.2
0.3
0.4
0.5
-100
■50
50
100
•
Figure 3.10 Magnitude o f the ambiguity function for a) a two component signal and b)
its analytic version. The values o f 0 are normalized.
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44
3.4.5 The W igner Distribution
As previously stated, the ambiguity function is a two-dimensional autocorrelation
function with respect to time lag and Doppler shift.
The two-dimensional Fourier
transform o f the ambiguity function yields the Wigner distribution [BIZ99]:
(3.51)
which is the same distribution commonly used in quantum mechanics.
3.4.5.1 Properties o f the W igner Distribution
Consider a signal that starts at time ti, stops at time tj, and starts again at time t3.
Strong finite support would require no spectral content between ti and t3. The Wigner
distribution satisfies weak finite support.
However, the W igner distribution is highly
nonlocal, using both past and present data at the same time, placing products at times
where no signal occurs. Thus a small region o f noise is introduced everywhere in the
calculations [COH89]. Discrete analysis o f finite data segments reduces this effect due to
finite length data windows used.
Take for example the following signal
x{t)= Axe i ‘u" + A2e JaM-'
(3.52)
The Wigner distribution is
The first two terms are the auto-terms, the terms lying on the axes o f the ambiguity
function. The last term is the cross term, data lying away from the axes o f the ambiguity
function. Examination o f the cross terms reveals two important features. First, the cross
term is located h alf the distance between the two frequencies.
oscillates at the difference between the two frequencies.
Second, the cross term
While the cross term is not
desirable in many applications due to lack o f physical meaning, it is very desirable in
applications such as radar, sonar, and communications as the cross terms hold
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45
information regarding the relative phase difference between two frequencies. So, what if
a signal consists o f three frequencies at the same time, such as normalized frequencies
0.1, 0.2. and 0.3? There will be cross-terms at a frequency o f 0.2. However, examination
o f the amplitude o f the Wigner distribution shows the spectral amplitude at 0.2 is greater
than twice the product o f amplitudes o f the frequencies 0.1 and 0.3. For the purposes of
this experimental analysis, the cross term is undesirable. This problem is o f great interest
and can be reduced using techniques discussed later.
The W igner distribution yields a real energy distribution because the ambiguity
function is conjugate symmetric. However, the energy density can be negative. This can
be explained by the one-dimensional autocorrelation function, a conjugate symmetric
function.
The Fourier transform o f the autocorrelation function yields the spectrum
energy density.
Symmetric functions yield real spectra, which can be negative or
positive. One example is the autocorrelation function
R ( t ) = - c o s ( coot),
which will have a
negative density function. The only W igner distribution that is positive for all time is a
Gaussian with a linear chirp.
Three other properties the Wigner distribution satisfies are marginal satisfaction
(see section 3.4.2) [CLA80, CLA80b], energy conservation, and shifts in time and
frequency.
If the signal is shifted in time or frequency, so is the Wigner distribution.
However, one noted negative feature o f this distribution is the inability to totally recreate
the signal from the inverse W igner distribution. The signal can be recovered up to a
constant, which is usually acceptable for most analysis.
3.4.5.2 Reduced Interference Kernels
W hen a signal is filtered, the spectrum o f that signal is modified to suit the user's
needs.
The inverse Fourier transform is then applied to get the filtered signal.
M athematically, it is described as
(3.54)
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46
where F(co) is the filter function distribution. An example o f a filter function distribution
is an ideal low pass filter.
The value o f F(co) will be 1 between ±coip (the cutoff
frequency) and zero everywhere else.
Cohen's class o f time-frequency distributions are calculated by
[ [ f.t*f« - — u + —)<t>(6,t) e ' J 0
-i-t-t v
2J v
2)
C(t.co) = —^
where <j) is
9
“ du dx dQ
the kernel function [KHA98] and u is time. The kernel
frequency distribution.
A kernel value
(3.55)
determines the time-
o f 1 for all values o f <j> yields the
Wigner
distribution. Rearranging o f Equation 3.53 reveals the time-frequency calculations are a
two-dimensional Fourier transform o f the products o f the ambiguity function and the
kernel [CLA80c]:
C(t.co) = —
e~J 0
J
“ dQ dx
( 3 .5 6 )
—X — X
which, when compared to Equation 3.54, reveals a two-dimensional filter function
applied to the ambiguity function.
Recall that the terms from the ambiguity function
away from the axes are cross terms. Reduction o f cross terms can occur if the kernel is a
two-dimensional low-pass filter.
Kernels with these properties are called reduced
interference kernels. Table 3.2 displays some common kernels.
Another way to look at Cohen's equation is to recall the autocorrelation function.
Recall, the Fourier transform o f the autocorrelation function yields the energy density. In
terms o f time-frequency, a local autocorrelation function Rt(t) is derived:
= — | j V f t / - — x u + —l<t>(6.x) eJ
•J \
dQ du
(3-57)
from which the time-frequency distribution is the Fourier transform o f the local
autocorrelation function:
" 1 dx
-X
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(3.58)
47
T able 3.2 Some common kernel distributions [COH951
Name
Kernel <t>(9,x)
Wigner
1
c o s (# r/2 )
Margenau-Hill
e jo<n-
Kirkwood/Rihaczek
s in (# r/2 )
Bom-Jordan
evti
Page
£ -(0 zfia
Choi-W illiams
i \,js in ( a # r )
Zhao-Atlas-Marks
adz
Certain conditions must be adhered to in order for the kernel to be a reduced
interference kernel.
Jeong [JE092] discusses many properties for designing reduced
interference distributions, but only a few are discussed here. The kernel must be real and
symmetric if the energy distribution is to be real. Another important requirement is finite
support [ZHA90]. For weak finite support, the kernel must satisfy
x
e-j 6 ' dQ = 0
|x| < 2|/|
(3.59)
|e| < 2|co|
(3.60)
—x
x
J^ e .x ) e ' J " ' dx = 0
-X
where Equation 3.57 describes weak finite time support and Equation 3.58 describes
weak finite frequency support.
Kernels satisfying this requirement are called cone-
shaped kernels [OH92]. For strong finite support, which is harder to obtain, the kernel
must satisfy
X
J<t)(e,t) e-JBl dQ = 0
|x|*2|r|
(3.61)
|e|*2|co|
(3.62)
-x
X
|<()(e.t)
* =0
-X
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48
Reduced interference distribution kernels w ork by attenuating the cross-term effect.
Observation in the ambiguity domain reveals the kernel distribution must be unity along
the axes to satisfy the time and frequency marginals, but rapidly decreasing away from
the axes.
The effectiveness o f the kernel depends on the signal itself. Signals with a large
signal-to-noise ratio (SNR) might be analyzed best with a kernel that tapers gently, such
as the Bom-Jordan. On the other hand, a noisy signal (low SNR) might be analyzed best
with a rapidly decreasing kernel, such as the Choi-W illiams kernel (for small a). Figure
3.11 displays the time-frequency analysis o f a multicomponent signal (normalized
frequencies 0.1 and 0.3) using three different kernels: Wigner, binomial (a discrete
distribution calculated on the time-lag plane), and Bom-Jordan. Observation reveals the
W igner distribution has an oscillating term (cross term) at a normalized frequency o f 0.2,
which is expected.
However, the binomial and Bom-Jordan do not seem to have the
strong cross-term between the two frequency components.
The cross-terms still exist
using these distributions, but have been reduced. Most reduced interference distribution
kernels yield negative energy densities somewhere in the time-frequency distribution.
Kernels that provide positive energy densities are possible [PIT98], but are dependent on
the characteristics o f the signal itself.
Phase jum ps occur in many phenomenon.
One advantage o f time-frequency
analysis using bilinear distributions, which uses the signal twice in each calculation, is
the ability to observe phase jum ps. Figure 3.12 displays the time-frequency analysis o f a
signal with different phase jum ps. Small phase jum ps look like a chirping in frequency,
while large phase jum ps look like a bifurcation/forking o f frequencies.
Larger phase
jum ps also include a high negative energy density/pole at the phase discontinuity. Use o f
the Fourier transform in signals with a phase jum p would yield a broad spectrum with
possible wide sideband contribution. This is another advantage time-frequency analysis
has over the standard Fourier transform.
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49
100
150
200
Tim e Step (a rb itra ry units)
►>
| 0.4
3
cr
QJ
£ 0 .3 - m *
73
b)
wag-
1 02
a
1 0.1 - w m
O
z
50
100
150
200
Tim e Step (arb itra ry units)
250
1 0.4
cr
OS
£ 0 .3 - m
73
lo .2
b
«N r
0
50
100
150
200
250
Tim e Step (arb itra ry units)
Figure 3.11 Time-frequency analysis o f a multicomponent signal using the a) Wigner, b)
binomial, and c) Bom-Jordan distributions.
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50
■o 0.3
0
50
100 150 200
Time Step (arbitrary units)
0
50
100 150 200
Time Step (arbitrary units)
>£Wi*
S?4lS.I*?*'
isqfafCiW
W&r^rOi**' •<!V^>fi5*
0
50
100 150 200
Time Step (arbitrary units)
0
50
100 150 200
Time Step (arbitrary units)
F igure 3.12 Time-frequency analysis for a signal with phase jum ps o f a) tt/4. b) n/2, c)
37t/4, and d) k
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51
3.4.5.3 Instantaneous Bandwidth
The instantaneous frequency is a measure o f the average frequency at any given
time in a signal. Along with the instantaneous frequency is the instantaneous bandwidth,
Bt, which Lee [LEE88] described as
(3.63)
where A is the amplitude o f the signal and A'(t) is the derivative o f A. The instantaneous
bandwidth describes the spread in frequencies at a given time.
Observation o f the
instantaneous bandwidth equation reveals no signal phase contribution, only signal
amplitude. A signal with a rapidly changing amplitude will have a high instantaneous
bandwidth, regardless if the signal is increasing or decreasing.
The instantaneous bandwidth is intuitive for monocomponent signals, but
consider the m ulticomponent signal
.vl{ t) = x i{t) + x 2{t)= A{[t)eJ*{n + A2(t]eJ* {,)
(3.64)
The instantaneous bandwidth for each component can be resolved if and only if the signal
truly has multiple components [COH92b].
The condition
for a signal to be
multicomponent is
(3.65)
which simply states the instantaneous bandwidth for each component has to be much less
than the frequency difference o f the components.
oscillators.
This is analogous to resonances in
Two closely spaced resonances are difficult to distinguish if the quality
factors o f one or both resonances are small enough such that the bandwidth o f one
resonance overlaps the other resonance.
Because time-frequency distributions are considered to be density functions, the
first and second conditional moments o f the distribution should allow for easy calculation
o f instantaneous frequency and instantaneous bandwidth. However, due to the design of
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52
some kernels, complex bandwidths can occur and therefore the kernel must be carefully
designed if instantaneous bandwidth is o f concern [LOU98].
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CHAPTER 4
SYSTEM CONFIGURATION
This chapter deals with the experimental configurations o f the coaxial gyrotron
and microwave oven magnetron.
While this dissertation emphasizes the use o f an
analytic technique for interpreting microwave signals, the experimental configurations
are necessary to demonstrate how the data to be analyzed was obtained.
4.1 The Coaxial Gyrotron
The system configuration for the coaxial gyrotron has been extensively described
in detail [HOC98, JAYOO, COHOO], therefore this dissertation will briefly describe the
coaxial gyrotron system parameters and then discuss current measurements and
heterodyne mixing.
Figure 4.1 displays the experimental configuration o f the coaxial gyrotron.
Approximately -800 kV is applied to the cathode tip.
The emitted current from the
cathode tip ranges from 1 - 1 0 kA. In most experiments, the current entering the tube
region is approximately 0.8 kA as the anode has a very narrow entrance port.
The
voltage pulse length is usually between 0.5 - 1.5 ps. The magnetic field in the microwave
cavity region is about 1.5 kG.
Figure 4.2 is a typical set o f signals from a shot for a coaxial gyrotron driven by
the Michigan Electron Long Beam Accelerator (MELBA).
Figure 4.2a displays the
cathode voltage, mixer, and microwave power signals. Figure 4.2b displays the diode,
aperture, cavity entrance, and cavity exit currents. The aperture, entrance, and exit
currents are integrated with respect to time.
53
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
H orizontal Polarization M icrow ave Pow er Signal
(to attenuators and crystal detector)
C oaxial W aveguide
C oaxial C avity
M ascr
Fiber O ptic lines to
Solenoid Coils
0.75 m Spectrograph and
0.275m M onochrom ator
C usp
D um ping
m agnets
A lum inum
C athode
c/>
-p.
C athode
S ta lk -
S-B and R ectangular
W aveguide
Carbon^
A node
Rogow ski C oil \
(E ntrance Current.
Rogow ski Coil
(E xit C urrent)
R ogow ski Coil
(A perture C urrent)
V ertical Polarization M icrow ave Pow er Signal
(to attenuators and crystal detector)
D iode M agnetic
Field Coils
F igure 4. 1 Experimental Setup o f coaxial gyrotron.
To H eterodyne
M ixer
55
'
a)
Cathode Voltage
(169 kV/div)
Mixer (0.1 V/div)
Microwave Power
(0.2 MW/div)
|
500
1000
2000
1500
2500
Time ( ns )
Diode
div = 1kA
Aperture
Entrance
Exit
0
500
1000
1500
2000
2500
Time ( ns)
F igure 4.2 Typical MELBA experimental shot, a) Cathode voltage, mixer signal, and
rf power, b) diode current, aperture current, cavity entrance current, and cavity
exit current (integrated).
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56
4.1.1 Electron Beam Current Diagnostics
M easurement o f the aperture, entrance, and exit currents were conducted by using
Rogowski coils. As the electron beam travels, an azimuthal magnetic field is created.
The induced voltage o f the Rogowski coil is proportional to the time rate o f change o f the
magnetic flux through the coils [GRI89]. Thus,
V =-K —
dt
(4.1)
where V is the induced voltage, O is the magnetic flux, and K is a proportionality
constant. Integration o f Equation 4.1 yields the current. The initial form o f integration
used RC integrating circuits [K N 089]. Figure 4.3 displays a basic RC integrator design.
a -
T
v
a -
AAAA
T
v
in
j
!
O
out
■o
Figure 4.3 Simple RC integrator/low-pass filter circuit.
The relationship between the input voltage and the output voltage is
+
(4.2)
where x = RC and is the time constant. Proper integration will only occur if x is large
with respect to the duration o f the input pulse. The problem with using analog integrators
can be understood by analyzing the frequency response o f Equation 4.2. Consider an
input signal o f the form
Vm =sin(2/z/?)
The output signal will be
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(4.3)
57
Voul =\A\s\n(l7rft + d)
(4.4)
M = :------------- jr
l + (2nrf)l \-
(4.5)
9 = - t a n 'l (2/rz/’)
(4.6)
with
Figure 4.4 displays the magnitude and frequency response curves for a simple integrator
with x= 20 ps.
K
e
0.8
s.
|SC - 0.6
V oVI
"O “
.= 5 0.4
3
M
a
c
s
0.2
0
10
20
30
40
50
60
70
80
90
100
90
100
Frequency (kHz)
•=• -0.5
c.
A sym ptote at -7t/2
-1.5
0
10
20
30
40
50
60
70
80
Frequency (kHz)
Figure 4.4 a) Magnitude Response and b) Phase response for an integrator with
RC = 20 ps.
From Figure 4.4 it is evident that the simple RC integrator circuit also acts as a
low pass filter.
The 3 dB point for an integrator with a time constant o f 20 ps is
approximately 7.96 kHz.
For frequencies above 1 MHz, there is almost complete
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58
attenuation. This makes it difficult to determine if there exist any high frequency current
modulations.
In signal analysis, not only is the magnitude response o f the filter important, but
also the phase response. The group delay is the time lag between the input signal and the
output signal, and is defined as
d9
1
where 0 is the phase response.
w
Thus, if the phase response is linear, all frequency
components travel with the same group delay and there is no dispersion o f the signal.
Figure 4.4b shows the phase response is very nonlinear, indicating a broad range of
frequencies in which there are many different group delays, and therefore dispersion of
the signal can occur. Group delay in the signal processing aspect is analogous to group
delay/dispersion in electromagnetics. The other problem with analog filters is that if the
time constant is too small, unphysical spikes can occur, since Vout becomes comparable in
size to the T(dV0Ui/dt) term.
The more recent method o f integrating the Rogowski coil signals from the coaxial
gyrotron experiment utilizes digital signal processing techniques.
Instead o f using an
analog integrator, digital filtering using linear phase filters on the unintegrated signal was
conducted. A periodic sine function was used to convolve the data, therefore low pass
filtering the signal. Figure 4.5 shows the magnitude and phase responses for a periodic
sine function o f 4001 points designed for low pass filtering with a normalized cutoff
frequency o f 0.25. The periodic sine function was modified by a Blackmann window to
minimize the ringing effect at the edges o f the magnitude response.
Instead o f a
nonlinear magnitude response as in the case o f the analog filter, a very flat magnitude
response o f approximately unity is shown for all frequencies below the cutoff frequency.
The phase response outside o f the cutoff frequency is relatively constant, while the phase
response inside the cutoff frequency is piecewise linear.
The result yields a constant
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59
group delay o f all desired frequencies. After filtering, integration o f the new signal was
conducted using the trapezoidal method [YAK89],
The average DC offset was
subtracted before integration to prevent artificial sloping.
(/)
e
3
0.8
-0.5
-0.4
-0.3
-
0.2
-
0.1
0
0.1
0.2
0.3
0.4
0.5
0.2
0.3
0.4
0.5
Normalized Frequency
0
.2 -2000
■o
o -4000
a.
2 -6000
-0.5
-0.4
-0.3
-
0.2
-
0.1
0
0.1
Normalized Frequency
F igure 4.5 a) Magnitude response and b) phase response for a periodic sine function o f
4001 points.
Analog filters should not be completely discredited. The advantage analog filters
have over digital filters is in the area o f band limitation. There is no band limitation for
analog filters.
Before numerically integrating and digital filtering, the signal must be
band limited to prevent aliasing effects, limiting the maximum frequency resolution that
can be measured.
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60
4.1.2 Heterodyne Mixing
Heterodyne mixer diagnostics (hereafter referred to as mixers) [POZ98] are very
useful to determ ine spectral properties o f a microwave signal. Two inputs are required: a
local oscillator (LO) and an (unknown) radio frequency (RF) signal (the microwave
signal in this case). The result is the intermediate frequency (IF), which has a spectrum
based on the difference o f the frequency content between the LO and RF.
f[F=fRF±fLo-
Hence,
The local oscillator was set below the microwave cavity cutoff frequency
to always have the intermediate frequency consist o f the difference between the RF and
LO signals.
The IF signal was analyzed using time-frequency analysis with reduced
interference distributions.
The
advantage
o f using mixers
is the
increased
sensitivity
and noise
characteristics com pared to direct measurement (using only the microwave signal).
Heterodyne systems also have the advantage o f band tuning by changing the local
oscillator frequency. Standard detection methods which do not use mixers would have to
have high-gain, wideband RF amplifiers.
4.2 Microwave Oven M agnetron
Figure 4.6 displays the system configuration for the microwave oven magnetron
experiment. The power meter (Hewlett Packard Model E4418B) was used to calibrate
the local oscillator (General Radio Model 1360-B) to 13 dBm. The local oscillator was
protected from reflections by an isolator (FXR, Inc. Model N157F). A frequency counter
(Hewlett Packard Model 5 3 6 IB) was used to ensure that the local oscillator was
generating a signal with the desired frequency.
The microwave oven used was a Panasonic Genius Premier, rated at 1.1 kW.
Leakage from the microwave oven was collected by a horn and then sent into a mixer
(M ini-Circuits M odel ZEM-4300), where the local oscillator was the second input signal.
The intermediate frequency (IF) signal was generated and displayed onto a Tektronix
3052 digital oscilloscope. Data was recorded from the scope for analysis.
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Hewlett
Packard
Frequency
Counter M odel
5361B
M icrow ave collecting horn
G eneral
C ommunic ations C o.
G enius Premier
1100W
M icrow ave Oven
T ektronix
TDS 3052
Oscilloscope
M odel 3N60MC
Hewlett
Packard
Power
Sensor
M odel
E4412A
M iniC ircuits
ZEM -4300
M ixer
Panasonic
RF Coax Switch
FXR, Inc.
Isolator
Model
N157F
G eneral Radio
Local
Oscillator
M odel 1360-B
F igure 4.6 M icrowave oven magnetron experimental setup.
Hewlett
Packard
RF Power
Meter
M odel
E4418B
62
The voltage across the voltage doubler in the oven magnetron and ground was
measured using a high voltage probe. The current o f the voltage doubler was measured
using a Pearson coil.
Figure 4.7 displays the voltage and current traces o f a typical
measured oven magnetron signal. The traces are results during the (half-cycle) time RF
power is emitted.
> -1
> -3
0
5
10
15
5
10
15
20
25
T im e (ms)
30
35
40
1.2
1.0
~
<
0.8
■S 0.6
o
§ 0.4
U
0.2
-0.20
20
25
30
35
40
T im e (m s)
Figure 4.7 M easurements from the voltage doubler in the oven magnetron, a) Voltage,
and b) current.
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C H A PTER S
EXPERIM ENTAL ANALYSIS AND RESULTS
This chapter discusses the utilization and results o f advanced signal processing
techniques to analyze a few microwave devices, with the main emphasis on the
relativistic gyrotron. The diagnostics used for analysis are the Fourier transform (FT),
short time Fourier transform (STFT), and time-frequency analysis utilizing Reduced
Interference Distributions (RJDs). The theory behind these diagnostics is described in
Chapter 3.
5.1 Gyrotron Analysis
5.1.1 Fourier Transform and Spectrogram
The first method used to analyze the heterodyned microwave signal was the
Fourier transform. This technique has been the standard o f spectral analysis for many
years. Figure 5.1a shows the heterodyned microwave signal (local oscillator set at 2.3
GHz), and Figure 5.1b shows the Fourier transform o f the signal (the anaytic form o f the
signal).
The results show a rather noisy spectrum with one strong peak and possible
multiple smaller peaks.
Observation o f the heterodyned microwave signal reveals
obvious changes in frequency over time. The first method used to analyze the frequency
evolution o f the mixer signal over time was the spectrogram. The time window used to
conduct the spectrogram was o f moderate length (50 points) to obtain a decent time
resolution while maintaining an acceptable bandwidth.
Figure 5.2 displays the
magnitude o f the STFT o f the heterodyned microwave signal in Figure 5.1a . The time
resolution capability o f the STFT/spectrogram indicates a behavior in which the
(intermediate) frequency changes from approximately 75 MHz to approximately 250
63
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64
MHz over a span o f 275 ns. From this result, it is evident that a technique having better
time and frequency resolution than the spectrogram would provide more information.
0.4
0.2
- 0.2
£
-0 .4
700
800
900
1000
T im e (
iu
1100
1200
1300
)
1000
£•
I £
Q >
« ?
u u
t W
cn"
500
0
-5 0 0
- 400
- 300
-200
-100
0
100
200
300
400
Frequency (M H z)
F igure 5.1 a) example o f heterodyned microwave signal (LO = 2.3 GHz), b) Fourier
transform o f heterodyned microwave signal.
500
450
~
400
2
350
300
I
250
£
200
150
100
50
0 750
800
850
900
950
1000
1050
1100
1150 1200
1250
Time ( as )
F igure 5.2 STFT o f the heterodyned microwave signal in Figure 5. la (LO=2.3 GHz).
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65
5.1.2 Time-Frequency Analysis Utilizing Reduced Interference Distributions
The method which yielded the best time and frequency resolution for spectral
analysis o f high power microwaves was that o f an alias-free time-frequency analysis
utilizing reduced interference distributions approach [JE092b].
This allowed the
frequency content o f the mixer signal to be calculated at each time step while maintaining
an excellent frequency resolution. Parameters o f the time-frequency program were kept
constant to maintain a standard for comparing spectra.
A data window length o f 128
points and a frequency resolution o f 1024 points were used to provide for fast
computation while maintaining a good approximation to the time-frequency plane. The
signal was converted to its analytic form to provide for a spectrum with only positive
frequencies, eliminating cross-term interaction at DC between the positive and negative
frequencies.
A Hanning window was applied to the local autocorrelation function for
easier interpretation o f the TFA by smoothing the spectrum (result o f the broad main lobe
o f the Hanning window) and reducing the spectral leakage (result o f the side lobes o f the
Hanning window). The full outer product (VTV, where V is a row vector and T is the
conjugate transpose) o f the data was used to provide for alias-free spectra; otherwise the
maximum frequency resolved wouid be 1/4 o f the sampling rate. Finally, the binomial
distribution was applied as the kernel function to provide for reduced interference.
Figure 5.3 displays the time-frequency analysis via RIDs for the mixer signal in
Figure 5.1a. The results show a time-frequency spectrum which looks very similar to the
spectrogram (which is itself a time-frequency distribution using the W igner distribution
modified by a kernel), but with three major differences. First, the time and frequency
resolutions are better than the spectrogram. Second, some o f the energy densities using
TFA are negative (in blue in Figure 5.3), whereas all energy densities o f the spectrogram
are nonnegative. Last, the marginals o f the spectrogram are not satisfied, whereas the
TFA using RIDs does satisfy the marginals.
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66
450
400
~
350
£
§, 34)0
>■»
g 250
3
2 200
Cb
150
100
50
0
70 0
800
90 0
1000
Time (ns)
1100
1200
1300
Figure 5.3 Time-frequency analysis utilizing RIDs o f the mixer signal in Figure 5.1a
(LO=2.3 GHz) (M ELBA shot 7219).
Figure 5.3 reveals a complex frequency spectrum, with the frequencies chirping
up and down at various points. The next investigation was to identify the cause o f these
chirps.
A com parison o f the TFA with cathode voltage was conducted.
Figure 5.4
displays the cathode voltage signal placed on top o f the TFA o f the mixer signal. Results
clearly show a definite correlation: electron beam voltage fluctuations directly modulate
the microwave frequency. The departure from the calculation (Equation 2.10) is due to
the synchronism condition approaching the cavity cutoff frequency.
W hile frequency
modulation due to cathode voltage fluctuations has been suggested [REA80], the
application o f this new diagnostic technique provides conclusive evidence that this occurs
in the coaxial gyrotron [PET98].
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67
300 -i
700
800
900
1000
1100
1200
Tim e (ns)
F igure S.4 Overlay o f cathode voltage (in black) with the TFA o f the mixer signal
(LO=2.3 GHz) (MELBA shot 7219).
The
utility
o f using
time-frequency
analysis using
reduced
interference
distributions does not stop at the cathode voltage-frequency modulation correlation. The
time-resolved maximum intensity o f the time-frequency distribution (TFD) provides
information on the maximum energy density o f a signal.
Figure 5 5a shows the time-
resolved maximum intensity o f the TFD, and Figure 5.5b shows the signal from the
power detector. The detector behaves as a square-law detector, measuring the square o f
the electric field o f the mixer signal, and should be roughly representative o f the signal
power. Similarities between the maximum intensity o f the TFD and the power detector
suggest TFA is a valid method to infer the time-resolved frequency spectrum and the
power distribution.
The explanation lies in the time marginal calculations.
The time
marginal o f the TFD must be the same as |x(t)|2. Integration o f the time marginal with
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68
respect to time yields the total signal energy, while integration o f microwave power with
respect to time yields total microwave energy. Since microwave power is proportional to
|E(t) | 2 (E is the electric field), which can be considered a density function, there is a
correlation between time marginal and microwave power. W hile Figure 5.5a displays
only the maximum intensity o f the TFD and not the time marginal o f the TFD, it is
strongly indicative o f the power o f a signal. Therefore it is possible to determine the
time-evolution o f power for each existing mode.
The combination o f time-frequency
analysis, cathode voltage, and detector signal strongly suggests the following: cathode
voltage fluctuations lead to fluctuations in interaction frequencies and therefore variations
in output RF pow er for the coaxial gyrotron experiment.
6
>■»
• mm
8
c
=
4
3
SB
.§ w
X c
a uj
s -
2
o 1-
700
800
900
1000
Time (ns)
1100
1200
1300
800
900
1000
Time (ns)
1100
1200
1300
0.06
0.04
*
o
>
0.02
o
>
-
0.02
700
Figure 5.5 a) M aximum intensity o f TFD o f mixer signal from Figure 5.1a, b) detector
signal from MELBA shot 7219.
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69
As stated before, the intensity o f the TFD is related to the power o f a signal, and
therefore the time-resolved relative power o f each mode can be approximated.
Figure
5.6a is a signal in which mode hopping takes place from the T E 112 mode to the T E m
mode (cold test and analytic results are presented in previous dissertations [JAYOO,
COHOO]). Figure 5.6b is the TFD o f the signal, and Figure 5.6c is the maximum intensity
o f the TFD for each individual mode.
Figure 5.7a is a signal in which multi-moding
occurs simultaneously, with Figure 5.7b and Figure 5.7c being the TFD o f the signal and
maximum intensity o f the TFD for each mode respectively. For the first time the power
evolution o f each mode has been obtained from coaxial gyrotron experimental data with
unprecedented clarity.
1000
1300
Time (is)
b£ 100
1000
Time (is)
15
Upper mode
10
I*
“ e
Cil
' Lower mode
5
o 1700
800
900
1000
1100
1200
1300
Time (is)
F igure 5.6 a) Signal with mode hopping properties, b) TFD o f signal in a), c) maximum
intensity o f each mode from the TFD in b). The local oscillator was set at
2.2 GHz. The lower mode is the T E m mode, and the upper mode is the
T E m mode.
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70
o
>
700
800
900
1000
Time (ns)
1100
1200
1300
1300
1000
Time (ns)
N
Upper Mode
Lower Mode
s
>■'5
35
e >
i g2
5 t
u
s
r0 0
800
900
1000
Time (ns)
1100
1200
1300
Figure 5.7 a) Signal with multi-moding properties, b) TFD o f signal in a), c) maximum
intensity o f each mode from the TFD in b). The local oscillator was set at
2.2 GHz. The lower mode is the T E m mode, while the upper mode is the
T E 112 mode.
5.1.3 Instantaneous Bandwidth
The instantaneous bandwidth o f a signal describes the frequency spread about a
central frequency for any given time.
While the value o f Q can be calculated by
1/bandwidth, the concept o f obtaining an instantaneous Q from the instantaneous
bandwidth would seem logical. However, as pointed out earlier, a signal o f the form
'V
20
y(r) — o ■ * ' sin{a0t)
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71
would have the same "bandwidth" as the signal
s(t) = sin(a>Qt + Aco(r )t)
if |Aco(t)|/coo is o f order 1/Q. The former signal designates decay o f signal energy in time
while the latter represents a signal with essentially a constant amplitude [COH95], Thus,
inferring the value o f the quality factor Q from the instantaneous bandwidth is uncertain.
However, calculations o f instantaneous bandwidth from the mixer signal o f the coaxial
gyrotron have been attempted.
Isolation o f a mode was conducted by bandpass digital filtering o f the mixer
signal.
The calculation o f instantaneous bandwidth relied upon the envelope o f the
filtered mixer signal, that was constructed from the analytic version o f the signal.
A
three-point differentiation method was used to find the derivative o f the envelope. The
instantaneous
bandwidth
results
were
smoothed
using
three
points
(0.25x(n-
l)+0.5x(n)+0.25x(n+l)) to remove any spurious trends. Figure 5.8 displays the filtered
m ixer signal,
envelope,
spectral
energy
density,
and
instantaneous
bandwidth.
Observation shows the mean instananeous bandwith is approximately 3 MHz, and
oscillates between 2-10 MHz. The oscillation behavior can be attributed to the existence
o f multiple frequencies within the bandpass. While there is only one peak, the spread in
frequencies contribute to a nonlinear term in the instantaneous bandwidth calculation.
The local oscillator was set at 2.25 GHz, resulting in an extrapolated value o f "Q" of
about 850 for this mode. The tail o f the pulse shows an instantaneous bandwidth much
higher than the middle part o f the pulse. From this analysis, it seems that the microwave
device might be 'de-Qing' rapidly.
W hile this method for calculating instantaneous Q might seem appropriate, there
is skepticism as noted two paragraphs earlier. The definition o f an instantaneous Q is of
concern. Equation 2.5 is one definition o f Q. Another definition o f Q is the damping
factor in a damped harmonic oscillator. Therefore, Q is defined only when the envelope
o f the signal is being reduced. Part o f the signal has to rise, but then the damping factor
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72
has to be negative. From Equation 2.5 this would mean a negative stored energy or a
negative RF power radiated, which would have questionable physicality. The Q o f the
signal can also be calculated by squaring the Fourier transform o f the signal. The Fourier
transform is defined with limits from
to
-oo
qo,
and therefore the concept o f an
instantaneous Q is trite. The conclusion is that instantaneous bandwidth may not have a
direct relationship with "instantaneous Q."
0.5
200
250
T im e (ns)
260
270
280
290
310
300
320
330
340
350
360
Frequency (MHz)
XI 40
T3 20
0
50
100
150
200
250
300
350
400
450
Tim e (ns)
F igure 5.8 Filtered heterodyned mixer signal and envelope (top), spectral density
(middle), instantaneous bandwidth (bottom). The large instantaneous
bandwidths before and after the main part o f the pulse are numerical artifacts.
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73
5.1.4 Current Modulations
Time-frequency analysis has shown to be a great asset in analyzing heterodyned
microwave signals. In addition to mixer signals, TFA has been applied on the entrance
and exit currents with some unexpected results. In this case, the current signals used are
unintegrated. Figure 5.9 shows the mixer, entrance current, and exit current signals from
a single-mode shot. Figure 5.10 displays the logio(|TFD|) o f those signals to observe low
level modulations.
The absolute value was considered because the TFA results have
negative energy density somewhere on the T-F plane.
shows one strong frequency component.
The TFA o f the mixer signal
The wavelike features above and below the
frequency component are due to spectral leakage, and are not o f importance. The TFAs
o f the entrance and exit currents show a noisy spectrum at the lower frequencies and a
generally clean spectrum at the upper frequencies. Figure 5.11 shows the mixer, entrance
current, and exit current signals for a shot which is multi-moding. Visual inspection of
the entrance current signal shows a high frequency oscillation on the signal between 1050
ns and 1250 ns.
Figure 5.12 displays the TFAs o f the signals in Figure 5.11.
heterodyned microwave signal shows the existence o f multi-moding.
The
The unexpected
part o f the current analysis is the results o f the TFA o f the unintegrated currents.
A
frequency component with a significant amplitude at approximately 235 MHz exists.
This component only existed when multimoding occurred and a long center conductor,
which extended before the cavity, was used. The long center conductor was used in RF
processing experiments [COHOO] to increase space charge limiting current.
The
frequency o f current modulation always occurred at approximately 235 MHz in all
instances in which it existed. In all cases where this modulation occur, the T E 112 and
TEi 13 modes were present. The frequencies o f the T E 112 and T E m were calculated by
Cohen [COHOO] to be 2.444 GHz and 2.703 G Hz respectively. The differences in the
frequencies o f the modes are 259 MHz. The sampling rate o f the current signals is 500
MHz, resulting in a Nyquist limit o f 250 MHz. An aliased version o f a signal with 259
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74
MHz would show up as a signal o f 241 MHz, only
approximated 235 MHz.
6
MHz difference from the
It should also be noted the current signals had no local
oscillator input, therefore the frequencies are from raw signals. In addition, there should
not be any other possible frequencies existing due to aliasing because o f the bandwidth o f
the scope. The overall conclusion is that mode interaction inside the cavity is creating a
nonlinear effect in which the frequency component is the difference between the two
modes. The wave o f the nonlinear term travels back through the entrance o f the cavity.
The extended part o f the center conductor acts as an antenna, radiating the wave. The
effect would be an axial pre-bunching o f the electron beam. This is perhaps the first
evidence/observation o f intermodulation products inside a coaxial gyrotron.
The observation o f the current modulation at 235 MHz was apparent only when
the analysis was conducted on the raw, unintegrated signal.
Explanation o f why the
integrated current signal did not reveal the current modulation lies in the frequency
response o f integration. For simplicity, let y(n)=x(n)+y(n-l), where x(n) is the input and
y(n) is the output. The equation is the basic form o f integration. Fourier transforming
this equation yields a transfer function H (co)=l/(l-e,tu). The magnitude response, |H(co)|,
describes the attenuation each frequency component o f the input signal undergoes in
relation to the output signal (the integrated signal in this case). Figure 5.13 shows the
magnitude response energy density
|H(co) | 2 on a dB scale in which the sampling
frequency is 500 MHz. Higher frequency components are attenuated more than lower
frequency components. The scale o f attenuation is approximate due to the pole at DC.
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75
A /*
' ' 700
1
800
900
1000
1100
1
1200
1300
1400
i
i
r
1500
Time (ns)
-1 0 ,
700
800
900
Time (ns)
6
i
i
i
i
Time (ns)
F ig u re 5.9 Signals from a single mode shot, a) Mixer, b) unintegrated entrance current,
c) unintegrated exit current.
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76
.1000
m
^
0
•
k
;•: :l 'L
§00
I’-i}
.. .t; > .-< }
b
'VAfcr^vu^-r.: ».:•■■-’*I
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700
800
i.
1000
1100
1200
1400
1500
Time(ns)
^ 2 0 0
.V ? ? . • ? : • : *
S150
..>
00
1000
1100
1200
1300
1400
1500
1100
1200
Time(ns)
1300
1400
1500
Time (as)
_200
100
700
800
900
1000
Figure 5.10 TFAs from signals in Figure 5.9. a) Mixer, b) unintegrated entrance current,
c) unintegrated exit current.
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( Volts (V)
f
900
1000
. 1100v
T une (ns)
1200
1300
1400
1200
1300
1400
1200
1300
1400
(A/s)
1 0 - - - - - - - - - - - - - - - - - - 1- - - - - - - - - - - - - - - - - 1- - - - - - - - - - - - - - - - - - J- - - - - - - - - - - - - - - - - - r
(
Unsealed Current
0,4800
900
1000
900
1000
1100
i u n e in s;
5
0
(A/s)
Unsealed Current
10800
•5
800
1100
Time (ns)
F igure 5.11 Signals from a multimode shot, a) Mixer, b) unintegrated entrance current,
c) unintegrated exit current.
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78
800
900
1000
1100
1200
1300
1400
1200
1300
1400
Time (ns)
800
900
1000
1100
Time (ns)
Time (ns)
F igure 5.12 TFAs from signals in Figure 5.11. a) Mixer, b) entrance current, c) exit
current
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79
-1 0
-60
-70
150
F requency (MHz)
100
200
250
F igure 5.13 Example frequency response magnitude for numerical integration
Power spectral density (PSD) calculations were conducted on the integrated
current signals during the same duration as the unintegrated signals. Figures 5 .1 4 -5 .1 7
displays the PSD calculations for the integrated signals for the single mode and multimode case.
Examination o f the power spectral densities show an approximate
monotonically decreasing magnitude spectrum, attributed by the frequency response of
integrating. The PSDs for the integrated entrance and exit currents in the single mode
case (Figures 5.14 and 5.15 respectively) show no peak at 235 GHz, which is expected.
However, the multi-mode case shows a very small contribution for both integrated
entrance and exit currents (Figures 5.16 and 5.17 respectively), approximately 65 dB
down from DC.
The small peaks at 235 MHz might have been overlooked and
considered as noise in this case due to the small contribution. The unintegrated signals
seem to work better in this case when analyzing small contributors.
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80
80
70
£
60
2
I 50
S 40
E
£ 30
$■20
£
O
1 0
ft.
-1 0 ,
50
150
Frequency (MHz)
1 0 0
2 0 0
250
Figure 5.14 Power Spectral Density for the integrated entrance current, single mode case.
70
SO
3 . 60
V
•a
i
1 50 \
S>
2 40 £
^ 30 I
cn
| 20 o
10
-
50
100
150
Frequency (M Hz)
2 0 0
250
F igure 5.15 Power Spectral Density for the integrated exit current, single mode case.
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81
CQ
3
01
T3
3
a
«
S
E
3
u
4*
u
S.
CD
u
I
o
c-
-1 0 .
100
150
200
250
Frequency (MHz)
F igure 5.16 Power Spectral Density for the integrated entrance current, multi-mode case.
■a
CD
-1 0
-2 0
-30
-40
-50
150
Frequency (MHz)
100
200
250
F ig u re 5.17 Power Spectral Density for the integrated exit current, multi-mode case.
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82
5.2 Time-Frequency Analysis o f Other High Power M icrowave Sources
5.2.1 Air Force Research Laboratory HPM Sources
Three high power microwave devices from the Air Force Research Lab at
Kirtland AFB, New M exico have been analyzed with time-frequency analysis.
Figure
5.18a shows the heterodyned microwave signal from a magnetically insulated line
oscillator (MILO) [HAW98], and Figure 5.18b displays the TFA o f the mixer signal.
The frequency chirping oberserved in Figure 5.18 was not expected. TFA, in conjunction
with other diagnostics, has increased the understanding o f the operation o f the MILO
[HAWOO] as the new theory postulates anode plasma formation on the third choke vane
o f the MILO.
a 0.2
0 .1
0
■g -0 . 1
a.
g -0.2 L<
100
150
200
300
350
250
Time Step (arbitrary units)
400
450
500
100
150
200
250
300
350
Time Step (arbitrary units)
400
450
500
Figure 5.18 a) Mixer signal from the AFRL MILO, b) TFD o f the signal in a).
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83
The next two devices that were analyzed were a conventional magnetron and an
injection locked relativistic klystron oscillator (RKO) [HEN96],
The conventional
magnetron is the source for the RF input to the RKO. Figure 5.19 shows the heterodyned
m icrowave signal for the conventional magnetron as well as the signal’s TFD.
The
spectrum is very clean until the end o f the signal, for which a wide spectrum is indicated.
Figure 5.20 shows the mixer signal and TFD for the RKO.
The TFD shows a low
frequency component for most o f the duration o f the signal, with the bandwidth
increasing over time. A sudden jum p into a higher mode occurs very late in the signal.
0.4
0 .2
2 0 0
300
400
500
600
700
800
900
1000
800
900
1000
T im e Step (a rb itra ry units)
200
300
400
500
600
700
T im e Step (a rb itra ry units)
Figure 5.19 a) M ixer signal o f a conventional magnetron, and b) TFD o f the mixer
signal.
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Amplitude (arbitrary units)
84
0 .2
0 .1
-0 . 1
-0 . 2
100
200
300
400
500
600
700
500
600
700
Frequency (MHz)
T im e (ns)
100
200
300
400
Tim e (ns)
Figure 5.20 a) M ixer signal o f the AFRL RKO and b) the TFD o f the mixer signal.
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85
5.2.2 M icrowave Oven Magnetron
The microwave signal from the microwave oven magnetron was studied with
TFA. It is half-wave rectified with 16.7 ms between pulses (-6 0 Hz) [See also Figure
4.7]. Cold tests o f the magnetron were conducted using an HP8722D network analyzer,
and the cold test results are displayed in Figure 5.21.
•a -10
3.08
2.46
-15
4.50
-2 0
3.96
-25
1.5
1
2
3
4
4.5
5
Frequency (GHz)
Figure 5.21 Cold test results for the microwave oven magnetron.
The 60 Hz oven magnetron power supply is half-wave rectified (16.7ms between pulses):
the microwave pulse is divided into three phases, defined by Figure 5.22: pulse rise, pulse
peak, and pulse fall. The pulse peak is the phase in which the amplitude o f the peak is
constant.
The pulse rise and pulse fall are the phase before and after the pulse peak
respectively.
Figures 5.23-5.25 show the heterodyned signal, Fourier transform, and
time-frequency analysis for all three pulse phases. The Fourier transform o f the pulse
rise seems to have some noise present. However, the TFA o f the pulse rise indicates
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86
frequency chirping and then eventual locking into a single mode. The pulse peak region
has a clean Fourier and TFA spectra, which is expected. The pulse fall shows the same
phenomena as the pulse peak, that is, chirping and eventually locking into a mode.
W hether the signal is in the same mode at all time or in different modes is still being
investigated. The periodic oscillations during the rise and fall o f the pulse are also under
current investigation.
HJL
Time (ms)
Figure 5.22 Typical microwave oven magnetron pulse.
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87
0.3 ------------ 1------------1------------1------------1------------ 1------------ \------------1------------ 1------------ r
Time (microseconds)
*T
110 MHz
e
M
0.5
100
150
250
300
Frequency (MHz)
200
350
400
450
500
Time (microseconds)
F igure 5.23 Microwave oven magnetron signal pulse rise: a) signal, b) Fourier transform,
c) TFA. The local oscillator was set at 2.35 GHz.
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^
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1 .8
350
400
450
1.4
1 .6
1 .8
Time (microseconds)
*r.
©
6
1
&
4
41
S
Cxi
0
0
50
100
150
200
250
300
Frequency (MHz)
>> 300
S 200
0.2
0.4
0.6
0.8
1
1.2
Time (microseconds)
Figure 5.24 Microwave oven magnetron signal pulse peak: a) signal, b) Fourier
transform, c) TFA. The local oscillator was set at 2.35 GHz.
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500
89
a 0.2
>
0.8
1
1.2
Time (microseconds)
15
10
s^
& €s
ZJ
5
0
fad
0
50
100
150
200
250
300
350
400
450
1.4
1.6
1.8
Frequency (MHz)
S
400
1
200
0.8
1
1.2
Time (microseconds)
F igure 5.25 Microwave oven magnetron oven signal pulse fall: a) signal, b) Fourier
transform, c) TFA. The local oscillator was set at 2.35 GHz.
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500
90
5.3 The Use o f Discrete Prolate Spherical Sequences
When the data window used for TFA is increased, more cross terms can exist and
produce a spectrum that is very difficult to interpret.
Slepian sequences were used to
provide a means o f improving the readability o f the time resolved spectra.
The window function, formed by summing the autocorrelations o f three Slepian
sequences, is applied to the local autocorrelation function; this was based on a timebandwidth product (N(3) o f three. Figure 5.26 shows the constructed window function
and its power spectral density. The generated window function looks very similar to a
sine function. This makes sense because this will produce a spectrum in which the “true”
spectrum is convolved with more or less a rectangular function.
The power spectral
density reveals a very wide main lobe, indicating the spectrum will have reduced
variance.
0 .8
0 .6
£BC
0.4
0 .2
-0 .2
-0.5
-0.4
-0.3
-0.2
•
0.1
0
0 .2
0.3
0.4
0.5
0.35
0.4
0.45
0.5
Normalized Lag
C
■fio
39
-2 0
-40
-60
5
-80
I
-100
U
-120
1
-140
4*
a.
0.05
0.1
0.15
0 .2
0.25
0J
Normalized Frequency
Figure 5.26 Generated window function used (N(3=3) (top) and its power spectral
density (bottom).
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91
Three tests were conducted to verify the validity o f using Slepian sequences: a
monocomponent signal, multi-component signal, and a signal composed o f two
Kronecker delta functions.
There are some expectations to these results.
First, the
window function is applied to the local autocorrelation function, providing a broader
spectrum. For a single component signal, the TFA should still show a monocomponent
signal but with a very broad spectrum.
M ulticomponent signals should suffer stronger
frequency cross term interaction because the two components are broader, reducing the
interaction “distance” between the components. Lastly, since the frequency spectrum is
broadened, the averaging o f cross terms in time should be reduced.
Figure 5.27 shows the TFA o f a generated single component signal.
The
bandwidth is definitely broad. The frequency marginal o f this TFA spectrum, along with
the energy density o f the Fourier transform o f the signal, is displayed. The results from
the Fourier transform show a very sharp peak, while the marginals o f the DPSS case are
lower and broader than the Fourier transform case.
0.5
|
|
0.4
0.3
“3
%
|
z
0 .1
0
0
SO
100
ISO
200
Tim e Step (a rb itra ry units)
250
300
5
DPSS
4
i f
s &
O s
3
i
1
s
4
u
Q
0
0
0.05
0.1
0.1S
D
0.2
0.2S
0.3
N orm alized Frequency
0.3S
0.4
0.4S
0.5
F igure 5.27 TFA o f signal component signal using DPSS (top) and marginals (bottom).
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92
Figure
5.28
shows
the
s(t)=sin(27t-0.1-n)+sin(27i-0.3-n).
TFA
o f the
generated
multicomponent
signal
The frequency marginal o f this TFA spectrum, along
with the energy density o f the Fourier transform o f the signal, are also displayed. The
TFA indicates a very strong presence o f frequency cross terms.
i l • i ! i i > 11 * i i | i i i i n i i i M ' i i 11 i; 1. 1 i i i u i * i H i i i i M 11 i U u i
I j i • t i l ' 1 ! ! l I ' i 1 1 ! ! ! ( f I1M j ! I ( ! H f I j I ' M j
l i [ f l ! ! 1111 H. t j ! •
50
100
150
200
Time Step (arbitrary units)
300
250
8000
•I* — 6000
2
N
® & 4000
^ a
J “ 2000
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Normalized Frequency
0.4
0.45
0.5
F igure 5.28 TFA o f multicomponent signal using DPSS (top) and frequency marginals o f
the TFA (bottom).
Figure 5.29 shows the TFA o f a signal composed o f two delta functions in time.
The top figure is the TFA o f the signal when a Hanning window is applied to the local
autocorrelation function. Strong time cross terms are present. The bottom figure is the
TFA o f the signal using DPSS. There is a marked reduction in time cross terms. Overall,
the use o f DPSS behaves very except in the case in which the signal is multi-component
and has the components closely spaced.
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0
50
100
150
200
250
300
350
400
350
400
Time Step (arbitrary units)
0
50
100
150
200
250
300
Time Step (arbitrary units)
F igure 5.29 TFA o f a signal with two Kronecker delta functions in time using a) Hanning
window and b) DPSS applied to the local autocorrelation function.
Slepian sequences were used on experimental data. Figure 5.30 shows the TFA
results for a time-bandwidth product o f three when using 1, 2, and 3 windows.
The
results using one window were similar to the results when using a Hanning window. This
makes sense because the first Slepian sequence looks very similar to a Hanning window.
The spectra converged very well after only using three windows.
densities were reduced when using more than one sequence.
Negative energy
However, the spectrum
became broader as the number o f sequences used increased. The result when using three
windows is best as three baseband filters were used instead o f one to obtain a better
confidence in the time-frequency analysis.
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94
S
§•
<
uU
fa
1000
1300
Time (ns)
w
i
3
1000
1300
Time (ns)
M
X
C
J
tm
fa
1000
1200
1300
Time (ns)
F igure 5.30 TFA o f experimental data using DPSS (Np=3): (a) one window, (b) two
windows, and (c) three windows (LO=2.3 GHz).
Previously the maximum intensity as a function o f time was used as a comparison
to the power/detector signal. While the power o f a mode is actually linked to the sum
over a band o f energies, the maximum intensity is a good representation o f the power
evolution o f a signal. Figure 5.31 displays the maximum intensity for TFAs o f one, two,
three, and four windows. The result for using four windows was displayed to compare
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95
convergence. The second and third peaks changed shape and stayed approximately the
same after using only tw o windows.
Figure 5.32 displays the TFA o f the signal after
using three windows (NW=3), as well as the power signal and intensity comparisons
when using a Hanning window and Slepian sequences.
The fidelity o f DPSS is more
representative o f the pow er signal than using the Hanning window, possibly providing a
better means to calculate time-resolved power.
3.5 '
~
N
=
3 ‘
|
2.5 -
1 Window
2 W indows
3 W indows
4 W indows
on
w
£
a
2 1.5 -
s
0.5
700
800
900
1000
1100
1200
1300
Tim e (ns)
Figure 5.31 Maximum intensity comparison using one, two, three, and four windows
(NP=3).
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700
800
900
1000
Time (ns)
1100
1200
1300
Power Signal
Hanning
DPSS
Time (ns)
F igure 5.32 a) TFA o f signal using DPSS (3 windows, N|3=3), b) maximum intensity
comparison with power signal. LO=2.3 GHz.
The last result using DPSS concerned the case when a larger data window was
used to calculate the TFA. The window resolution was increased to 512 points (vice 128
points).
Figure 5.33 shows the TFA when using the Hanning window and DPSS
generated window. The result when using the Hanning window reveals many cross terms
and is difficult to interpret around 1000 ns. However, the results when using the DPSS
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97
method showed a TFA that is much easier to understand. Cross terms are reduced. The
results using DPSS (data window o f 512 points) look very similar to the results using a
Hanning window (data window o f 128 points).
The reason one might use longer
sequences is to get a better representation o f the signal. While smaller data windows are
good for computational speed, larger windows are better for improved spectral
estimation.
500
400
2 , 300
<-»
S
4*
3
3*
X
wi
200
100
700
800
900
1000
1100
1200
1300
1100
1200
1300
Time (ns)
700
800
900
1000
Time (ns)
Figure 5.33 TFA o f signal using a data window o f 512 data points and a) Hanning
Window b) DPSS generated window applied to the local autocorrelation
function. The local oscillator was set at 2.3 GHz.
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CHAPTER
6
C O N C LU SIO N S
Rigorous signal processing techniques have been applied to heterodyned
microwave signals from a coaxial gyrotron. While the Fast Fourier Transform is useful
in determining what modes exist in the signal, it cannot reliably determine when each
mode exists. Conducting Fourier analysis on segments o f the signal has proved to be a
logical approach to determining the time evolution o f the spectrum, but still does not
address the problem with time and frequency resolution. Time-frequency analysis using
reduced interference distributions has been used to obtain unprecedented time and
frequency resolution. Cathode voltage/ electron beam fluctuations have been definitively
linked to the modulation o f the microwave frequency directly. Mode hopping and multimoding have also been identified. The time-resolved maximum intensity o f the TFA
compares well with the detector/power signal. This results in the ability to resolve the
time-evolution o f each mode. The conclusion is that electron beam voltage fluctuations
modulate the power by changing the synchronism and coupling conditions between the
electron beam and the structure for the coaxial gyrotron.
Time-frequency analysis has also been applied to heterodyned microwave signals
from a microwave oven magnetron. Results have shown frequency changes during the
rise and fall o f the pulse, while the peak o f the pulse is sharp and flat. The cause o f this
effect is still being researched.
The use o f discrete prolate spherical sequences has been extended to TFA. The
analysis works best for signals with one frequency component existing at any time.
98
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99
Power approximations are improved. Finally, when using large data windows to obtain a
better time-frequency spectrum, improvements in reducing interference have been
obtained, providing a more interpretable spectrum.
Future work in analyzing signals from HPM devices would be implementing TFA
programs that have adaptable kernels, providing a method based on the properties o f the
signal.
99
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A PPEN D IC ES
100
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101
APPENDIX A
MATLAB PLASMA BAY TOOLBOX
Commands and Descriptions
Programs
analgui
Analyzes the mixer signal TFA with other signals
curgui2
powgui
Calculates APER, ENTC, and EXIT current signals
Power Signal analysis program
tfgui
Latest version o f Time-Frequency Analysis GUI
Functions
Signal Processing
apfilter
binlac
bpfilta
bpfilter
bsfilta
bsfilter
cycshift
dishift
fpeak
fmean
f 2 mean
gray 2 bin
gsorth
hadmat
hilstats
hpfilta
hp ft Iter
iwhth
All pass filtering o f signal
Local autocorrelation using binomial distribution
Band pass filter o f signal (leading and trailing zeros
removed)
Band pass filtering o f signal
Band stop filter o f signal (leading and trailing zeros
removed)
Band stop filtering o f signal
Cyclically shifts data to the left
Diadically shifts data
Peak frequency using a time-frequency distribution
Mean frequency using a time-frequency distribution
Mean (freuqencyA2) using a time-frequency distribution
Gray code to binary converter
Gram-Schmidt orthogonalization o f a basis
Constructs the Hadamard ordered W alsh Hadamard matrix
o f order N
Hilbert transform information
High pass filter o f signal (leading and trailing zeros
removed)
High pass filtering o f signal
Inverse Hadamard ordered Walsh Hadamard Transform
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102
iwhtw
lpfilta
lpfilter
marginal
tfdbin
tfddpss
walshmat
whth
whthspec
whtw
whtwspec
Inverse Walsh ordered Walsh Hadamard Transform
low pass filter o f signal (leading and trailing zeros removed)
Low pass filtering o f signal
Calculates the time and frequency marginals
Time-frequency analysis using the binomial distribution
Time-frequency analysis using Slepian sequences
Constructs the Walsh ordered Walsh Hadamard matrix o f
order N
Hadamard ordered Walsh Hadamard Transform
Hadamard ordered Walsh Hadamard Transform Power
Spectrum
Walsh ordered Walsh Hadamard Transform
Walsh ordered Walsh Hadamard Transform Power
Spectrum
Plotting
fftplot
stftplot
Conducts FFT o f signal and plots results
Conducts and plots Short Time Fourier Transform
Other Functions
difO
fileconv
linterp
Conducts 3 point derivative o f data
Converts data from DSA format to Time/Volts format
linear interpolation between two points
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103
APFILTER
This function conducts an all-pass filter o f a signal.
Syntax: y=apfilter(X,winsize)
X is the data to be filtered, where winsize is the size o f the rectangular window to be
used. Winsize must be an odd value.
Example: all pass filtering o f a signal G with a rectangular window o f 25 points.
y=apfilter(G,25)
BINLAC
This program calculates the local autocorrelation matrix using the binomial distribution.
Syntax: lac=binlac(S,winres)
S is the data (in a row vector format) and winres is the window resolution imposed to
calculate the local autocorrelation matrix.
BPFILTA
This function conducts a band pass filtering o f a signal and removes the zeros due to
convolution (unlike BPFILTER).
Syntax: y=bpfilta(X,fs,fc 1,fc2,wintype,winsize)
X is the data to be filtered, fs is the sampling frequency, in Hertz, fcl is the lowest
frequency and fc2 is the highest frequency, in Hertz, to allow, wintype is the type o f
window to be used for filtering, while winsize is the size o f the window being used,
winsize must be o f odd size. The options for wintype is as follows:
1=
2=
3=
4=
5=
6 =
Rectangular
Bartlett
Blackman
Hamming
Hanning
Triangular
Example: Band Pass Filtering o f a signal G with sampling frequency o f lK H z. Lowest
frequency to allow is 100 Hz. Highest frequency to allow is 200 Hz. A
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104
Hanning window o f 25 points is to be used. The command will be
y=bpfilta(G, 1000,100,200,5,25)
BPFILTER
This function conducts a band pass filtering o f a signal.
Syntax: y=bpfilter(X,fs,fc I ,fc2,wintype,winsize)
X is the data to be filtered, fs is the sampling frequency, in Hertz, fcl is the lowest
frequency and fc2 is the highest frequency, in Hertz, to allow, wintype is the type o f
window to be used for filtering, while winsize is the size o f the window being used,
winsize m ust be o f odd size. The options for wintype is as follows:
1=
2=
3=
4=
5=
6 =
Rectangular
Bartlett
Blackman
Hamming
Hanning
Triangular
Example: Band Pass Filtering o f a signal G with sampling frequency lKHz. Lowest
frequency to allow is 100 Hz. Highest frequency to allow is 200 Hz. A
Hanning window o f 25 points is to be used. The command will be
y=bpfilter(G, 1000,100,200,5,25)
BSFILTA
This function conducts a band stop filtering o f a signal and removes the zeros due to
convolution (unlike BSFILTER).
Syntax: y=bsfilta(X,fs,fc I ,fc2,wintype,winsize)
X is the data to be filtered, fs is the sampling frequency, in Hertz, fc 1 is the lowest
frequency and fc2 is the highest frequency, in Hertz, to stop, wintype is the type o f
window to be used for filtering, while winsize is the size o f the window being used,
winsize must be o f odd size. The options for wintype is as follows:
1 = Rectangular
2 = Bartlett
3 = Blackman
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105
4 = Hamming
5 = Hanning
6 - Triangular
Example: Band Pass Filtering o f a signal G with sampling frequency I KHz. Lowest
frequency to stop is 100 Hz. Highest frequency to stop is 200 Hz. A Hanning
window o f 25 points is to be used. The command will be
y=bsfilta(G, 1000,100,200,5,25)
BSFILTER
This function conducts a band stop filtering o f a signal.
Syntax: y=bsfilter(X,fs,fc 1,fc2,wintype,winsize)
X is the data to be filtered, fs is the sampling frequency, in Hertz, fc 1 is the lowest
frequency and fc2 is the highest frequency, in Hertz, to stop, wintype is the type o f
window to be used for filtering, while winsize is the size o f the window being used,
winsize must be o f odd size. The options for wintype is as follows:
1=
2=
3=
4=
5=
6 =
Rectangular
Bartlett
Blackman
Hamming
Hanning
Triangular
Example: Band Pass Filtering o f a signal G with sampling frequency I KHz. Lowest
frequency to stop is 100 Hz. Highest frequency to stop is 200 Hz. A Hanning
window o f 25 points is to be used. The command will be
y=bsfilter(G, 1000,100,200,5,25)
C Y C S H IF T
Takes a signal X (column vector) and cyclically shifts it by L to the upward. L must be
less than or equal to the number o f entries in X.
Syntax: y=cycshift(X,L)
Example: cyclically shift the column vector A upwards by 5, where
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106
1
4
23
5
3
7
2
Matlab code: a= [l 4 23 5 3 7 2].’;y=cycshift(a,5)
Output:
y= 7
2
1
4
23
5
3
DIFF5
Calculates the derivative o f data f using the 3-point method
Syntax: y=diff3(f)
DISHIFT
Calculates the diadic shift signal S by N points.
Syntax: Y=dishift(S,N)
Note the signal S must have a length equal to a power o f two. The maximum shift can be
no more than the logi o f the data length.
Example: dyadically shift the column vector A upwards by 2, where
/l = [l
4
23
5
3
7
2
10]
Matlab code: a= [l 4 23 5 3 7 2 10];
y=dishift(a,3)
Output:
y = 5 23 4 1 10 2 7 3
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107
FILECONV
This function converts a DSA file from DSA (UMICH) format to TIME,VOLTS data
format for easier use. The only thing the user needs to know is the file name. If the user
wants to print the data into a file with extension '.asc', then prch='y'. Note the path o f the
file is needed if the desired file is not in the current directory in MATLAB.
Syntax: y=fileconv(filename,prch)
Example: convert the DSA file MIXR7445 to TIM E,VOLTS format and place
the result into array VALUE and save it to the file MIXR7445.ASC
Matlab Code: value=fileconv('mixr7445',ly')
GRAY2BIN
Converts a column vector G o f gray codes to binary values and then to the decimal values
o f the binary code.
Syntax: y=gray2bin(G)
Example: find the decimal version o f the following Gray code matrix:
'
G—
0000
'
0001
0011
0010
0110
0111
Matlab code: G^'OOOOVOOOl’j'OOl IVOOIO’j'OI 10';'0111’];
y=gray2bin(G)
Output:
y=0
1
2
3
4
5
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108
G SO R T H
Conducts the Gram-Schmidt Orthogonalization process on matrix V. The rows o f V are
the basis vectors to be orthogonalized. The new vectors will be orthonormal. This
method is useful for determining a basis set for a group o f data. However, the problem
with this method is that the basis vector calculation depends on the ordering o f the data.
Syntax: a=gsorth(V)
Example: Conduct the Gram-Schmidt orthogonalization on the matrix
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
V=
Matlab Code: V =[l 2 3 4 5;6 7
a=gsorth(V)
Output:
a=
0.1348
0.7628
0.7136
8
9 10;11 12 13 14 15];
0.2697
0.4767
0.4757
0.4045
0.1907
0.3865
0.5394 0.6742
-0.0953 -0.3814
0.1635 -0.2973
HADM AT
Creates the Hadamard ordered Hadamard matrix W with order N. The matrix will be
square with size 2AN. This matrix is used to create an orthonormal basis set for the
Hadamard ordered Walsh transform
Syntax: H=hadmat(N)
Example: Construct the Hadamard Matrix o f order 3
Matlab Code: H=hadmat(3)
Output:
H= 1
1
1
1
1
1
1
1
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109
H IL ST A T S
Calculates the Amplitude, instantaneous frequency, and instantaneous bandwidth o f a
signal s using its Hilbert transform, fs is the sampling frequency (Hz).
Syntax: [A,ifr,ibw]=hilstats(s,fs)
Where
A=amplitude
ifr=instantaneous frequency
ibw=instantaneous bandwidth
H P F IL T A
This function conducts a high pass filtering o f a signal and removes the zeros due to
convolution (unlike HPFILTER).
Syntax: y=hpfilta(X,fs,fc,wintype,winsize)
X is the data to be filtered, fs and fc are the sampling frequency and cutoff frequency, in
Hertz,respectively, wintype is the type o f window to be used for filtering, while winsize
is the size o f the window being used, winsize must be o f odd size. The options for
wintype is as follows:
1 = Rectangular
2 = Bartlett
3 = Blackman
4 = Hamming
5 = Hanning
6 = Triangular
Example: High Pass Filtering o f a signal G with sampling frequency lKHz. Minimum
frequency to allow is 100 Hz. A Hannin window o f 25 points is to be used.
The com mand will be
y=hpfilta(G, 1000,100,5,25)
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110
HPFILTER
This function conducts a high pass filtering o f a signal.
Syntax: y=hpfilter(X,fs,fc,wintype,winsize)
X is the data to be filtered, fs and fc are the sampling frequency and cutoff frequency, in
Hertz, respectively, wintype is the type o f window to be used for filtering, while winsize
is the size o f the window being used, winsize must be o f odd size. The options for
wintype is as follows:
1=
2=
3=
4=
5=
6 =
Rectangular
Bartlett
Blackman
Hamming
Hanning
Triangular
Example: High Pass Filtering o f a signal G with sampling frequency lKHz. Minimum
frequency to allow is 100 Hz. A Hanning window o f 25 points is to be used.
The command will be
y=hpfilter(G, 1000,100,5,25)
IWHTH
Calculates the inverse Hadamard-ordered Walsh-Hadamard Transform o f a
signal S. If S is a matrix, the transform will be conducted for each row. Note the length
o f the rows o f S must be a power o f 2. The result X is a matrix the size o f S where the
rows are the transforms o f the rows o f S respectively.
Syntax: X=iwhth(S)
Example: let
S = [1 2 3 4 5 6 7
8
]
To find X, the inverse Hadamard ordered Walsh-Hadamard Transform, the procedure is:
Matlab Code: S=[l 2 3 4 5
X=iwhth(S)
Output:
6
7 8]
X= 4.5000 -0.5000 -1.0000 0 -2.0000 0 0 0
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Ill
IW H T W
Calculates the inverse Walsh-ordered W alsh-Hadamard Transform o f signal S. If S is a
matrix, the transform will be conducted for each row. Note the length o f the rows o f S
must be a pow er o f 2. The result is a matrix the size o f S where the rows are the
transforms o f the rows o f S respectively.
Syntax: X=iwhtw(S)
Example: let
S = [1
2 3 4 5 6 7
8
]
To find X, the inverse Walsh ordered W alsh-Hadamard Transform, the procedure is:
Matlab Code: S =[l 2 3 4 5
X=iwhtw(S)
Output:
6
7 8]
X= 4.5000 -2.0000 0 -1.0000 0 0 0 -0.5000
L IN T E R P
Syntax: y2=linterp(x 1,y 1,x3,y3,x2)
This function calculates the value o f y2 via linear interpolation given points (x l,y l),
(x3,y3), and x2, where the value o f x2 is between x l and x3.
Example: find the value o f y2 for between the points (1,2) and(4,7) when x2=3
Matlab code: y2=linterp(l,2,4,7,3)
Output:
y2=5.333
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112
L P F IL T A
This function conducts a low pass filtering o f a signal and takes away the beginning and
ending zeros due to convolution (unlike the LPFILTER command).
Syntax: y=lpfilta(X,fs,fc,wintype,winsize)
X is the data to be filtered, fs and fc are the sampling frequency and
cutoff frequency, in Hertz, respectively, wintype is the type o f window to be used for
filtering, while winsize is the size o f the window being used, winsize m ust be o f odd
size. The options for wintype is as follows:
1=
2=
3=
4=
5=
6 =
Rectangular
Bartlett
Blackman
Hamming
Hanning
Triangular
Example: Low Pass Filtering o f a signal G with sampling frequency I KHz. Maximum
frequency to allow is 100 Hz. A Hanning window o f 25 points is to be used.
The command will be
y=lpfilta(G, 1000,100,5,25)
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113
L P F IL T E R
This function conducts a low pass filtering o f a signal.
Syntax: y=lpfilter(X,fs,fc,wintype,winsize)
X is the data to be filtered, fs and fc are the sampling frequency and cutoff frequency, in
Hertz, respectively, wintype is the type o f window to be used for filtering, while winsize
is the size o f the window being used, winsize must be o f odd size. The options for
wintype is as follows:
1=
2=
3=
4=
5=
6 =
Rectangular
Bartlett
Blackman
Hamming
Hanning
Triangular
Example: Low Pass Filtering o f a signal G with sampling frequency lKHz. Maximum
frequency to allow is 100 Hz. A Hanning window o f 25 points is to be used.
The command will be
y=lpfilter(G, 1000,100,5,25)
M A R G IN A L
This function performs normalized marginal calculations for a time-frequency
distribution tfd. Integrating along time (frequency marginal) gives the power density
spectrum (the square o f the absolute value o f the FFT o f the signal), while integrating
along frequency (time marginal) gives the energy o f the signal at that time.
Syntax: [tmar,fmar]=marginal(tfd)
T FD B IN
This function calculates the time-frequency distribution using the
binomial distribution.
Syntax: tfd=tfdbin(S,winres,freqres,windata)
S is the data (in a row vector format), winres is the window resolution, freqres is the
frequency resolution, and windata is the window function values. Note that windata is a
column vector, and the length o f windata must be 2*w inres+l. The output is tfd (the
time-frequency distribution).
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114
Example: find the tfd o f signal s with a window resolution o f 128, and
a frequency resolution o f 1024
Matlab Code: tfd=tfdbin(s, 128,1024,hanning(257))
TFD D PSS
This program calculates the time-frequency distribution using the
binomial distribution, but with Slepian Sequences used to create the window function
used to modify the local autocorrelation.
Syntax: tfd=tfddpss(S,winres,freqres,NW ,numwins)
S is the data (in a row vector format), winres is the window resolution, freqres is the
frequency resolution, NW is the time-bandwidth product, and numwins is the number o f
Slepian sequences used to create the window function. The output is tfd (the timefrequency distribution).
Example: find the tfd o f signal s with a window resolution o f 128, a frequency resolution
o f 1024, at time-bandwidth product o f 4, and using 5 Slepian sequences
tfd=tfddpss(s, 128,1024,4,5)
W A LSH M A T
Creates the Walsh ordered Hadamard matrix o f order N. The matrix will be a square
matrix o f size 2AN.
Syntax: W=Walshmat(N)
Example: Create a Walsh ordered Hadamard matrix o f order 3
Matlab code: W=Walshmat(3)
Output: W =
1
1
1
1
-1
-1
1
1
-1
-1
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115
WHTH
Calculates the Hadamard-ordered Walsh Hadamard Transform o f signal S. If S is a
matrix, the transform will be conducted for each row. Note the length o f the rows o f S
must be a power o f 2. The result is a matrix the size o f S where the rows are the
transforms o f the rows o f S respectively.
Syntax: B=whth(S)
Example: calculate the WHTH o f
S = [1
2 3 4 5 6 7
Matlab code: S=[l 2 3 4 5
B=whth
Output: B = 36
-4
-8
6
8
]
7 8 ];
0 -16
0
0
0
W HTHSPEC
Syntax: P=whthspec(S)
Calculates the Hadamard-ordered Walsh Hadamard transform power spectrum. If S is a
matrix, the power spectrum will be conducted along the rows o f S.
Example: Calculate the WHTH Power Spectrum o f
S = [1
2 3 4 5 6 7
8
]
Matlab code: S=[l 2 3 4 5 6 7 8 ];
P=whthspec(S)
Ouput: P = 1296
16 64 256
WHTW
Calculates the Walsh-ordered Walsh Hadamard Transform o f signal S. If S is a matrix,
the transform will be conducted for each row. Note the length o f the rows o f S must be a
power o f 2. The result is a matrix the size o f S where the rows are the transforms o f the
rows o f S respectively.
Syntax: B=whtw(S)
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11 6
Example: Calculate the WHTW o f the signal
5 = [1 2 3 4 5 6 7
M atlab code: S =[l 2 3 4 5
B=whtw(S)
Output: B = 36 -16
0
6
-8
8
]
7 8 ];
0
0
0 -4
W H TW SPEC
Calculates the Walsh-ordered Walsh Hadamard transform power spectrum. If S is a
matrix, the pow er spectrum will be conducted along the rows o f S.
Syntax: P=whtwspec(S)
Example: Calculate the WHTW o f the signal
5 = [1 2 3 4 5 6 7
8
]
Matlab code: S=[ 1 2 3 4 5 6 7 8 ];
P=whtwspec(S)
Output: P = 1296 256 64 0
16
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11 7
APPENDIX B
M ELBA Time-Frequency Analysis Program (TFPROG)
1. Introduction
The MELBA Time-Frequency analysis program is a tool for determining the
time-evolved frequency content o f a nonstationary signals (signals in which the
frequency content and possibly the amplitude o f the frequency content changes with
time). It is currently designed as an auxiliary program, and all parameters must be
entered in the configuration file.
2. Requirements
IBM PC or compatible
DLL files (to be placed in a path the computer knows (e. g., c:\windows)
-
ago4510.dll
-
c4510v.dll
-
glu32.dll
-
mmatrix.dll
-
msvbvm60.dll
-
msvcirt.dll
mxvcrt.dll
-
opengl32.dll
-
v4510v.dll
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118
3. Limitations
a) The program works for well up to 2000 points
b) The input data must be in 1-column MATLAB or ASCII format.
c) The program will terminate if the configuration file is not properly constructed
4. Understanding Time-Frequency Analysis
The Cohen’s class distribution - general equation
The time-frequency analysis program is the alias-free discrete version o f Cohen’s
class distributions:
C (t, a>)= —— f f f ^ (w + r / 2 )s' (u - r /
2 x JJJ
r ) e ' J^ ,+ f a~° “) du d r d d
Where
s = signal
u,t = time
x = lag
co = radian frequency
0 = Doppler shift
<J) = kernel
Alternatively, the above equation can be rewritten for ease o f understanding. If
we define the local autocorrelation as
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119
/? ,(r ) = —
2k
\ s ’ { u - t I 2 ) s { u + t I 2 ) ^ { 0 , A e 10^
d d du
j
then the time-frequency distribution can be calculated by taking the Fourier transform o f
the local autocorrelation function:
c M
5.
= i - f A ,( r > - '" r f r
The Kernel Function
The art in reducing interference terms is in the construction o f the kernel <j). The kernel
acts as a two-dimensional low-pass filter when applied to the am biguity function. If <j)=l
for all values, the distribution is called the Wigner distribution, and no cross-term
reduction occurs. Another kernel is the Bom-Jordan distribution, defined as
(Z*(0,r) =
6.
s in ( # r / 2 )
Or! 2
The Analytic Signal
The analytic signal is the Hilbert transform o f the signal. It is used to remove the spectral
energy from the negative frequencies and place it in the positive frequencies.
spectral energy is conserved.
Thus,
The analytic signal is used when only the positive
frequencies are concerned and to reduce cross-term interaction near DC. Note that the
analytic function should not be used if the original signal has a component near DC, as
the signal will be warped.
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120
7. W indow Functions
The Fourier transform is calculated for a signal extending from -oo to oo. However, it is
impossible to sample an infinite number o f data points. This limitation is important as it
invokes a bias on the spectrum calculations. The bias allows for ringing in the spectrum,
creating false frequencies at very low levels o f power. One way to overcome the severity
o f ringing is to use window functions. The window function tapers the data before the
Fourier transform is applied. The tradeoff is between the severity o f false frequencies
and the spreading o f actual frequencies. Thus, if a signal has two sinusoidal components
existing simultaneously and at frequencies very close to each other, the Fourier transform
would smooth the peaks together and only one peak would exist. Figure B1 displays six
very com mon window functions applied to data.
Figure B2 displays the Chebyshev
window for six values o f p. Figure B3 displays the Kaiser window for six values o f p.
Figure B4 shows the Slepian sequences for at time-bandwidth product o f 3.
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121
Rectangular
Bartlett
J .5 -
0
0 .2
0 .4
0 .6
0
0 .2
Blackman
0.6
Hamming
J=33 .5 r
J .5 ca
0
0 .4
'- ^ 0
0 .2
^
0 .4
0 .6
0 .8
0
1
0
■
0 .2
Hanning
X/
/
■
0 .4
■
0 .6
0.8
1
Triangular
1--------- .-------------------
X
\
-
.-------- :
\
\
!
\
'
J .5 -
/
\
\
'
\
/'
\
. y
0
:
:
0 .2
0 .4
0 .6
0 .8
i
V !
0^
1
0
--
0 .2
0 .4
0 .6
F igure B1 Six common window functions
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0 .8
1
122
,=100
,=80
1r
afl.5
0
1
0
0
0 .2
0 .4
3=200
0 .6
0.8
^
0
0 .2
0 .4
0 .6
0 .8
1
8 = 50 0
-
g £ .5 -
co
0
0 1
0
0 .2
0 .4
=
1
0
1000'
1
-
d3.5r
0 .4
0 .6
0 .8
0 .2
0 .4
0 .6
0 .8
3=2000
:
g £ .5 r
0
0
0 .2
0 .2
0 .4
L
0
F igure B2 Chebyshev window function for various values o f beta.
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1
123
P = 10
P = 20
- 0 .5 -
^ 0 .5 -
00
0
0 .2
0 .4
0 .6
0.8
1
0
0 .2
0 .4
P = 50
P = 100
1r
1
- 0 .5 -
j0.5^
CT3
0
0
0 .2
0.8
0 .4
1
0
0 .2
P = 500
P = 600
- 0 .5 r
0
0
0.4
s> 0.5 -
0
0 .2
0 .4
0.8
1
0
0 .2
0 .4
0 .6
F igure B3 Kaiser window function for various values o f beta
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0 .8
1
124
Window 1
Window 2
0.2 r
02
.
gjQ.11
-
<L> Or
“t o
0
0
02
.
gj
0
02
- . -
-
0.2
0.4
0.6
0.8
1
0
02
-
.
"
gg
“oo
0
0.2
0.4
0.6
Window 4
0.8
0.2
0.4
0.6
Window 6
0.8
0.2
0.4
0.8
-
L
" to
0 20
02
- .
- .
0.2
0.4
0.6
Window 5
0. 2 -
0.8
-
0
0.2 r
o
“to
Or
-0 .2 L
0
0 21
0
• .
0.2
0.4
0.6
F igure B4 Slepian sequences for a time-bandwidth product o f 3.
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1
125
8. PROGRAM PARAMETERS
D ata file s
The input file must be single column and in ASCII or MATLAB format. The
program automatically detects the format o f the input file.
The output file can be either in MATLAB (0) or ASCII (1) format. For the output
file, do not add any extensions. The file will be generated with a .TXT extension.
Window functions
The program supports the following window functions to be applied to the local
autocorrelation function:
(0) Rectangular (no weighing)
(1) Bartlett
(2) Blackman
(3) Hamming
(4) Hanning
(5) Triangular
( 6 ) Chebyshev Window (must also enter value o f P)
(7) Kaiser Window (must also enter value o f P)
( 8 ) Discrete Prolate Spherical (Slepian) Sequences (must also enter timebandwidth product, number o f sequences to use, and the type o f weighing on
the autocorrelation o f the windows:
( 0 ) biased
( 1 ) unbiased
(2) Normalize such that the autocorrelation at zero lag is 1
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126
(3)
No weighing
Analytic Function
Enter (1) to convert the signal to its analytic form, or (0) to keep the signal as it is.
Kernel Distributions
The following alias-free kernels are available:
(0) Wigner Distribution
(1) Binomial Distribution
(2) Bom-Jordan Distribution
(3) Zhao-Atlas-M arks (ZAM) distribution (with a= l/2)
Output M atrix Format
Enter (0) if all frequencies or (1) only positive frequencies are to be saved.
Plot Option
Enter (0) to skip plotting or (I) to plot the t-f distribution
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127
9. C O N F IG U R A T IO N F IL E E X A M PLE S
N O T E : The configuration file must be c:\windows\tfdconf
EX A M PLE 1
c:\data\infile
(input file)
c:\data\outfile
(output file)
0
(output file in ASCII format)
64
(window resolution)
1024
(frequency resolution)
4
(Hanning window)
1
(Convert to analytic signal)
2
(Bom-Jordan distribution)
1
(only positive frequencies)
1
(plot the t-f distribution)
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128
EX A M PLE 2
c:\data\infile
(input file)
c:\data\outfile
(output file)
1
(output file in MATLAB format)
64
(window resolution)
1024
(frequency resolution)
6
(Chebyshev Window')
100
(beta)
0
(do not convert to analytic form)
1
(binomial distribution)
0
(keep all frequencies)
1
(plot the t-f distribution)
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129
EX A M PLE 3
c:\data\infile
(input file)
c:\data\outfile
(output file)
1
(output file in MATLAB format)
64
(window resolution)
1024
(frequency resolution)
8
(DPSS windows)
3
(Time-Bandwidth product)
2
(Number o f windows to use)
I
(unbiased window autocorrelation)
1
(convert to analytic form)
1
(binomial distribution)
0
(keep all frequencies)
0
(do not plot the t-f distribution)
Execution o f the Program : Type TFPROG in the appropriate directory
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130
APPENDIX C
Time-Frequency Analysis Using the Binomial Distribution Source Code
NOTE: This program is proprietary, property of the
University of Michigan. Use of this source code is
authorized only for educational purposes.
function tfd=tfdbin(s,winres,freqres,windata)
% Syntax:
tfd=tfdbin(s,winres,freqres,windata)
%
%
%
%
%
%
%
%
This program calculates the time-frequency distribution using the
binomial distribution,
s is the data (in a row vector format),
winres is the window resolution, freqres is the frequency resolution,
and windata is the
window function values.
Note that windata isa
column vector,
and
the length of windata must be 2*winres+l. The
output is tfd (the time-frequency distribution).
%
% Example:
%
find
128,
the
and
tfd of signal s with a window resolution of
a frequency resolution of 1024
%
%
tfd=tfdbin(s,128,1024,hanning(257))
%
% Christopher Peters
% October 19,1999
N=length(s);
% Create tfd and autocorrelation matrices
tfd=zeros (freqres, N) ,R=zeros(1,winres+1);
RR=zeros(winres+1,N ) ;
%
First the data must be padded with zeros
y=[zeros(l,winres/2)
s zeros(l,winres/2) ] ;
% Determine the starting and ending points of the analysis
start=l+winres/2;
fin=N+winres/2;
%
%
Now that we're all set up and defined, begin the actual
analysis
for t=start:fin
%
Define the range
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131
d a t a = y ( ( t - w i n r e s / 2 ) : ( t + w i n r e s / 2 ) );
%
Define the outer product
op=data'*data;
%
%
Now we have to convolve the diagonals of the outer
product with that of the binary distribution
binom=l;
for k=l:(winres+1)
sl=binom;
s2=diag(op,(k-1)) . 1;
LL=length(si);
LLL=length(s2);
if (LLLcLL)
temp=sl;
S1 = S 2 ,*
s2=temp;
LL=length(si);
end
R(ki =sum(si.*fliplr(s2( (winres/2-LL+2) :(winres/2 + 1))));
binom=conv([0.5,0.5],binom);
end
R R (1:end,(t-start+1))= R .';
end
R R (1,:)= R R (1,:)/ 2 ;
tfd=2*real(fft(RR.* (windata((winres+1):end)*ones(1,N)),freqres));
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132
APPENDIX D
Time-Frequency Analysis Using the Born-Jordan Distribution Source Code
NOTE: This program is proprietary, property of the
University of Michigan.
Use of this source code is
authorized only for educational purposes.
function tfd=tfdbj(s,winres,freqres,windata)
% Syntax:
tfd=tfdbj(s,winres,freqres,windata)
%
%
%
%
%
%
%
%
%
%
%
%
%
This program calculates the time-frequency distribution using the
B o m - J o r d a n distribution,
s is the data (in a row vector format) ,
winres is the window resolution, freqres is the frequency resolution,
and windata is
the window function values.
Note
that windata isa
column vector,
and the length of windata must be
2*winres+l. The
output is tfd (the time-frequency distribution).
Example:
find the
128, and
tfd of signal s with a window
a frequency resolution of 1024
resolution of
tfd=tfdbj(s,128,1024,hanning(257))
%
% Christopher Peters
% October 19,1999
N=length(s);
% Create tfd and autocorrelation matrices
tfd=zeros(freqres,N ) ;
R=zero3(1,winres+1) ;
RR=zeros(winres+1,N);
%
First the data must be padded with zeros
y=[zeros(1,winres/2) s zeros(l,winres/2) ] ;
% Determine the starting and ending points of the analysis
start=l+winres/2;
fin=N+winres/2;
%
%
Now that we're all set up and defined, begin the actual
analysis
for t=start:fin
%
Define the range
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133
d a t a = y ( ( t - w i n r e s / 2 ) : (t+ w in re s/2 ) ) ;
%
Define the outer product
op=data'*data;
%
%
Now we have to convolve the diagonals of the outer
product with that of the binary distribution
for k = l :(winres+1)
sl=ones(1,k)/k;
s2=diag(op,(k-1)).';
LL=length(si);
LLL=length(s2);
if (LLL<LL)
temp=sl;
sl=s2 ;
s2=temp;
LL=length(si);
end
R(k)=sum(si.*fliplr(s2((winres/2 -LL+2) : (winres/2+1))));
end
R R (1:e n d , (t-start+1))= R . ';
end
R R (1 1 :)= R R (1, :)/2 ;
tfd=2*real(fft(RR.* (windata((winres+1):end)*ones(1,N)),freqres));
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134
APPENDIX E
Time-Frequency Analysis Using Slepian Sequences Source Code
NOTE: This program is proprietary, property of the
University of Michigan. Use of this source code is
authorized only for educational purposes.
function tfd=tfddpss(s,winres,freqres,NW,numwins)
% Syntax:
tfd=tfddpss(s,winres,freqres,NW, numwins)
%
% This program calculates the time-frequency distribution using the
% binomial distribution and Slepian Sequences.
s is the data (in a
% row vector format), winres is the window resolution, freqres is the
frequency
% resolution, NW is the time-bandwidth product, and numwins is the
number of
% Slepian Sequences to use.
The output is tfd (the time-frequency
distribution).
%
% Example:
%
%
find the tfd of signal s with a window resolution
of
128, a frequency resolution of1024, time bandwidth of 3,
and using 2 windows.
%
%
tfd=tfddpss(s,128,1024,3,2)
%
% Christopher Peters
% October 19,1999
N=length(s);
windata=0;
[e,v]=dpss(winres+1,N W ) ,•
for nw=l:numwins
windata=windata+xcorr(e(1:end,nw));
end
windata=windata/numwins;
% Create tfd and autocorrelation matrices
tfd=zeros(freqres,N ) ;
R=zeros(1,winres+1);
RR=zeros(2*winres+l,N ) ;
%
First the data must be padded with zeros
y=[zeros(l,winres/2)
s zeros(l,winres/2) ] ;
% Determine the starting and ending points of the analysis
start=l+winres/2;
fin=N+winres/2 ;
%
Now we create a phase shift function
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135
phshift=exp(j *2*pi*(0:(freqres-1))/freqres*winres) .';
%
%
Now that we're all set up and defined, begin the actual
analysis
for t=start:fin
%
Define the range
data=y((t-winres/2): (t+winres/2));
%
Define the outer product
op=data1*data;
%
%
Now we have to convolve the diagonals of the outer
product with that of the binary distribution
binom=l;
for k = l :(winres+1)
R(k)=pconv(binom,diag(op,(k-1)) . 1,winres/2);
binom=conv([0.5,0.5],binom);
end
%
%
%
%
%
Now that we have the new matrix, we need to flip R to
get the proper sequence.
We have only winres+1 data,
and since the lower diagonal of the matrix is the
conjugate of the upper diagonal, we have to add the
"mirror conjugate", creating symmetric data.
R R (1:en d , (t-start+1)) = [conj(fliplr(R(2:end))) R] . ';
%
%
%
%
Now finally, the vector RR is symmetric, which means
the spectrum should be real. In order to properly
get the spectrum, we have to multiply by the phase
shift function,
end
tfd=real (f ft (RR. * (windata*ones (1, N) ) ,freqres) .* (phshift*ones (1, N) ) ) ,function y=pconv(si,s 2 ,p)
%
% syntax y=pconv(si,s 2 ,p)
% Gives the pth point of a convolution between si and s2.
%
N=length(sl);
N2=length(s2);
if (N2<N)
temp=sl;
sl=s2;
s2=temp;
N=length(sl);
end
y=sum(sl(1:end).*fliplr(s2(p-N+2:p+l)));
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136
APPENDIX F
Time-Frequency Analysis Using the Wigner Distribution Source Code
NOTE:
This program is proprietary, property of the
University of Michigan.
Use of this source code is
authorized only for educational purposes.
function tfd=tfdwig(s,winres,freqres,windata)
%
Syntax:
tfd=tfdwig(s,winres,freqres,windata)
%
%
%
%
%
%
%
%
This program calculates the time-frequency distribution using the
Wigner distribution.
s is the data (in a row vector format),
winres is the window resolution, freqres is the frequency resolution,
and windata is the
window function values.
Note that windata isa
column vector,
and
the length of windata must be 2*winres+l.The
output is tfd (the time-frequency distribution).
%
% Example:
%
find
128,
the
and
tfd of signal s with a window resolution of
a frequency resolution of 1024
%
%
tfd=tfdwig(s,128,1024,harming(257))
%
% Christopher Peters
% October 19,1999
N=length(s);
% Create tfd and autocorrelation matrices
tfd=zeros(freqres,N);
R=zeros(1,winres+1);
RR=zeros(winres+1,N ) ;
%
First the data must be padded with zeros
y= [zeros(l,winres/2)
s zeros(l,winres/2) ] ;
% Determine the starting and ending points of the analysis
start=l+winres/2;
fin=N+winres/2;
%
%
Now that we're all set up and defined, begin the actual
analysis
for t=start:fin
%
Define the range
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137
d a t a = y ( ( t - w i n r e s / 2 ) : ( t + w i n r e s / 2 ) );
%
Define the outer product
op=data'*data;
%
%
Now we have to convolve the diagonals of the outer
product with that of the wigner distribution
for k = l :(winres+1)
sl=zeros(1,k ) ;
mnm=mod(k ,2);
if (mnm-=l)
si (ceil(k/2))=1;
else
si(k/2)=0.5;
Sl(k/2+1)=0.5;
end
s2=diag(op,(k-1)).1;
LL=length(sl);
LLL=length(s2);
if (LLL<LL)
temp=sl;
sl=s2;
s2=temp;
LL=length(sl);
end
R (k)=sum(sl.*fliplr(s2((winres/2-LL+2):(winres/2+1))));
end
R R (1:end,(t-start+1))=R .1;
end
R R (1,:)= R R (1,:)/2;
tfd=2*real(fft(RR.* (windata((winres+1):end)*ones(1,N ) ),freqres));
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138
APPENDIX G
Time-Frequency Analysis Using the Zhao-Atlas-Marks Distribution Source Code
NOTE:
This program is proprietary, property of the
University of Michigan.
Use of this source code is
authorized only for educational purposes.
function tfd=tfdzam(s,winres,freqres,windata)
% Syntax:
tfd=tfdzam(s,winres,freqres,windata)
%
%
% This program calculates the time-frequency distribution using the
% Zhao-Atlas-Marks distribution.
s is the data (in a row vector
format),
% winres is the window resolution, freqres is the frequency resolution,
% and windata is the window function applied to the local
autocorrelation
% function.
Note that windata is a column vector, and the length of
% windata must be 2*winres+l.
The output is tfd (the time-frequency
% distribution).
%
% Example:
%
find the tfd of signal s with a window resolution
128, and a frequency resolution of 1024
of
%
%
tfd=tfdzam(s,128,1024,hanning(257))
%
% Christopher Peters
% October 19,1999
N=length(s);
% Create tfd and autocorrelation matrices
tfd=zeros(freqres,N ) ;
R=zeros(1,winres+1);
RR=zeros(winres+1,N);
%
First the data must be padded with zeros
y=[zeros(l,winres/2)
s zeros(l,winres/2)];
% Determine the starting and ending points of the analysis
start=l+winres/2;
fin=N+winres/2;
%
%
Now that we're all set up and defined, begin the actual
analysis
for t=start:fin
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139
%
Define the range
data=y((t-winres/2):(t+winres/2));
%
Define the outer product
op=data'*data;
%
Now we have to create the local autocorrelation function
for k = l :(winres+1)
R (k) =sum(middats (diag (op, (k-1) ) ,k) )
end
RR(1:end,(t-start+1))= R . ';
end
R R (1,:)= R R (1,:)/2;
tfd=2*real(fft(RR.* (windata((winres+1):end)*ones(1,N ) ),freqres));
function mdats=middats(s,number)
N=length(s);
if (number>N)
mdats=s;
else
midpt=ceil(N/2);
range=floor(number/2);
if (N/2 -= ceil(N/2)) % odd length
mdats=s((midpt-range):(midpt+range));
else
% even case
m aats=s(floor(midpt-range+1):floor(midpt+range));
end
end
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140
APPENDIX H
Source Code for TFPROG
NO TE: This source code has been altered for compilation into an executable. The
program can be run in MATLAB with minor alterations
function tfprogO
fid=fopen('c :\windows\tfdconf' , 1r t ') ;
infile=deblank(fgetl(fid));
outfile=deblank(fgetl(fid));
outfiletype=str2num(deblank(fgetl(fid) ) ) ;
opts=fscanf(fid, '% i ');
st=fclose('a l l ') ;
winres=opts(1);
% Window Resolution
freqres=opts(2);
% frequency Resolution
wintype=opts(3);
% Window Type
if (wintype == 8)
NW=opts (4);
numwins=opts(5);
corrtype=opts(6);
analyt=opts (7) ;
kern=opts(8);
pos=opts(9);
plopt=opts(10);
windata=0;
switch corrtype
case {o}
corrstr='b iased';
case {l}
ccrrstr='unbiased';
case {2}
corrstr='coef f ';
case {3}
corrstr='n o n e ';
end
[e,v]=dpss(winres+1,N W ) ;
for nw=l:numwins
windata=windata+xcorr(e(:,nw),corrstr) ;
end
elseif (wintype >5 & wintype <8)
beta=opts (4) ;
analyt=opts(5);
kern=opts(6);
pos=opts(7);
plopt=opts(8);
if (wintype==6)
windata=Chebwin(2*winres+l,beta);
else
windata=Kaiser(2*winres+l,beta);
end
else
analyt=opts(4);
kern=opts(5);
pos=opts(6);
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141
plopt=OptS(7) ;
switch wintype
case {0 }
windata=Boxcar(2*winres+l);
case {l}
windata=Bartlett(2*winres + l) ;
case {2}
windata=Blackman(2*winres+l);
case {3}
windata=Hamming(2*winres+l) ;
case {4 }
windata=Hanning(2*winres + l) ;
case {5}
windata=Triang(2*winres + l) ;
end
end
% Now load the signal
s = (load(infile)) . 1;
% Check for Analytic Signal Conversion
if (analyt==l)
s=hilbert(s) ,end
% Conduct TFA using specific kernels
switch kern
case {0}
tfd=tfdwig(s,winres,freqres,windata); % Wigner distribution
case {1 }
tfd=tfdbin(s,winres,freqres,windata); % Binomial distribution
case {2 }
tfd=tfdbj(s,winres,freqres,windata); % Born-Jordan distribution
case {3}
tfd=tfdzam(s,winres,freqres,windata); % Zhao-Atlas-Marks
distribution
end
% check to save only positive frequencies
if (pos==l)
tfd=tfd(l:(freqres/2) , :) ;
end
% save data to output file
if (outfiletype==0)
save(outfile,'t f d 1,'-mat');
else
outfilel=[outfile '.txt'];
save(outfilel,'t f d 1,'-ascii');
end
if (plopt==l)
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14 2
[m,n]=size(tfd);
t = 0 :(n-1);
figure;
if (pos==l)
imagesc(t,(0:(freqres/2-1))/freqres,flipud(tfd));
axis xy;
else
imagesc(t,(0:(freqres-1))/freqres,flipud(tfd));
axis xy;
end
x l a b e K ' T i m e Step');
ylabel('Normalized Frequency');
title('Time-Frequency Analysis');
set(gcf,'NumberTitle','off ') ;
set(gcf,'N a m e ','Time-Frequency Analysis Plot - Program by
Christopher Peters - 11 May 2000');
colorbar('v e r t ');
end
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143
APPENDIX I
Derivation o f M icrowave Frequency Modulation Due to Cathode Voltage
Fluctuations
Begin with some basic definitions:
p2 = i — V
a -i)
r
where P=v/c (where v is the particle velocity), and y is the relativistic factor. The beam a
is the ratio o f the beam's perpendicular velocity, vp, to its parallel velocity, V||:
( 1-2 )
a = —
We can then note the relations:
V!l +VP = ( 1 + Ct2 Vl! = P 2 c 2 = 1
-
—
r j
(l + a 2> 2 =p 2c 2
(1-3)
(1-4)
with ( l + a 2 )Pn2 =p2. Applying perturbation theory to equation 1-4, we get:
(l + a 2 )i'ilSv!l = ~ j c 2
(1-5)
Y
6v» - r ^ r r r -
i1 + a h vn
<l-6 >
Perturbation o f the relativistic cyclotron frequency yields:
5Qc = 8 ^ = - ^ f 5 y
Y
(1-7)
Y‘
where Qo is the nonrelativistic cyclotron frequency. The electron beam dispersion
relation is:
co=£iiv, i + n c
where k|j is the axial wave number. Perturbation o f Equation 1-8 results in:
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(i-8)
144
8ra= ¥ vll+ 5 n
^ l{J ^
- £
H
1
0
It
Qq
, Y y
v
(l + a 2) /2^
V
-Q „
\ -
(l+cx jy 2p.
8co
knC
'b y '
V L .
5 g) « —
yJ ,(i + a 2 k
( i + (x 2 ) / 2p 11a c.
r by'
Vll +Qc
v i y
irii
i+-
Q„
k .V n
8
J _ '' ^
Q
(i+ a } rP j
co
CO
v A
i+-
'S y N
u
V i
a
2n2
a . vY P /
1+
V i!
-
Taylor expanding the above equation using the formula
1-e,
1
+ Si
= ( l — S , X l — e 2) ~ 1 — e l “ S i
(1-9)
The final estimation is accomplished:
8co
co
l - V ll /
y ,
fi.
•+ l
V iJ _
- I I 1 - Qc p< Y J|
( 1- 10)
The parameters for the coaxial gyrotron presented in this dissertation have the following
typical values:
Cathode Voltage V=700 keV => y=l+700 keV /511 keV = 2.3699
=> P = .9066
Beam a ~ 1 => Py - P(2 ) ' 1/2 = 0.6412 => vn= Pyc = 1.9236 x 108 m/s
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145
Considering only the fundamental mode, k y « rc/L, where L is the length o f the cavity, the
results indicate k y = 12.083 m '1. The relativistic cyclotron frequency Q c s (2 GHz)(27t) =
1.2566 x 10 10 radians/s. Applying these parameters to Equation I - 10 yields
o)
co
8
5V
1
511 keV
700 keV
t
700 keV
1
+
700 keV
(0.775) = - 0 . 4 4 8
8V
511 keV
V
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(Ml)
BIBLIOGRAPHY
146
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147
BIBLIOGRAPHY
ABR77
E. A. Abramyan, E. N. Efimov, and G. D. Kuleshov, "Energy Recovery
and Power Stabilization o f Pulsed Electron Beams in Marx Generator
Circuits," in Proceedings o f the 2nd International Topical Conference on
High Power Electron and Ion Beam Research and Technology, Vol. 2, pp.
755-760, 1977.
AGE98
F. J. Agee, "Evolution o f Pulse Shortening Research in Narrow Band,
High Power Microwave Sources," IEEE Transactions on Plasma Science,
Vol. 26 (3), pp. 235-245, 1998.
ARF85
G. Arfken, Mathematical Methods for Physicists. Academic Press, Inc.,
Florida, 1985.
BEN92
J. Benford and J. Swegle, High Power M icrowaves. Artech House, 1992.
BEN97
J. Benford, "Survey o f Pulse Shortening in High-Power Microwave
Sources," IEEE Transactions on Plasma Science, Vol. 25 (2), pp. 311-317,
1997.
BIZ99
J. P. S. Bizarro and A. C. Figueiredo, "The W igner distribution as a tool
for time-frequency analysis o f fusion plasma signals: application to
broadband reflectometry data," Nuclear Fusion, Vol. 39 (1), pp. 61-82,
1999.
CAR94
G. Caryotakis, " 'High Power M icrowave' Tubes: In the Laboratory and
On-Line," IEEE Transactions on Plasma Science, Vol. 22 (5), pp. 683691, 1994.
CHE84
F. F. Chen, Introduction to Plasma Phvsics and Controlled Fusion. Plenum
Press, New York, 1984.
CLA80
T. A. C. M. Claasen and W. F. G. Mecklenbrauker, "The Wigner
Distribution - A Tool for Time-Frequency Signal Analysis - Part I:
Continuous-Time Signals," Philips Journal o f Research, Vol. 35 (3), pp.
217-250, 1980.
CLA80b
T. A. C. M. Claasen and W. F. G. Mecklenbrauker, "The Wigner
Distribution - A Tool for Time-Frequency Signal Analysis - Part II:
Discrete-Time Signals," Philips Journal o f Research, Vol. 35 (4/5), pp.
276-300, 1980.
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
148
CLA80c
T. A. C. M. Claasen and W. F. G. M ecklenbrauker, "The Wigner
Distribution - A Tool for Time-Frequency Signal Analysis - Part III:
Relations W ith Other Time-Frequency Signal Transformations," Philips
Journal o f Research, Vol. 35 ( 6 ), pp. 372-389, 1980.
COH89
L. Cohen, "Time-Frequency Distributions - A Review," Proceedings o f the
IEEE, Vol. 77 (7), pp. 941-981, 1989.
COH90
L. Cohen, "Instantaneous Bandwidth For Signals and Spectrogram,"
Proceedings o f the IEEE ICASSP-90, Vol. 5, pp. 2451-2454, 1990.
COH92
L. Cohen, "Instantaneous Bandwidth and Formant Bandwidth,"
Conference Proceedings, IEEE Sixth SP Workshop on Statistical Signal
and Array Processing, pp. 13-17, 1992.
COH92b
L. Cohen, “ What is a M ulticomponent Signal?,” Proceedings o f the IEEE
ICASSP-92, Vol. 5, pp. 113-116, 1992.
COH93
L. Cohen, "Instantaneous 'Anything'," Proceedings o f the IEEE ICASSP93, Vol. 4, pp. 105-108, 1993.
COH94
L. Cohen, "The Uncertainty Principle in Signal Analysis," Proceedings of
the IEEE-SP International Symposium on Time-Frequency and TimeScale Analysis, pp. 182-185, 1994.
COH95
L. Cohen, Time-Frequencv Analysis. Prentice-Hall, New Jersey, 1995.
COH96
L. Cohen, "A General Approach for Obtaining Joint Representations in
Signal Analysis - Part I: Characteristic Function Operator Method," IEEE
Transactions on Signal Processing, Vol. 44 ( 6 ), pp. 1080-1090, 1996.
COH96b
L. Cohen, "A General Approach for Obtaining Joint Representations in
Signal Analysis - Part II: General Class, Mean and Local Values, and
Bandwidth," IEEE Transactions on Signal Processing, Vol. 44 ( 6 ), pp.
1091-1098, 1996.
COHOO
W. E. Cohen, “Optical Emission Spectroscopy and Effects o f Plasmas in
High Power Microwave Pulse Shortening Experiments,” Ph. D.
Dissertation, University o f Michigan, 2000.
C 0065
J. W. Cooley and J. W. Tukey, "An algorithm for the machine calculation
o f complex Fourier series," Math. Computation, Vol. 19, pp. 297-301,
1965.
DOH 8 8
G. Dohler, "The Small-Signal Theory o f the Cyclotron M aser and Other
Gyrotron-Type Devices," IEEE Transactions on Electron Devices, Vol. 35
(10), pp. 1730-1745, 1988.
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
149
FEL99
K. L. Felch, B. G. Danly, H. R. Jory, K. E. Kreischer, W. Lawson, B.
Levush, and R. J. Temkin, “Characteristics and Applications o f Fast-Wave
Gyrodevices,” Proceedings o f the IEEE, Vol. 87 (5), pp. 752-781, 1999.
GAP94
A. V. Gaponov-Grekhov and V. L. Granatstein, Applications o f High
Power M icrowaves. Artech House, 1994.
GIL80
R. M. Gilgenbach, M. E. Read, K. E. Hackett, R. Lucey, B. Hui, V. L.
Granatstein, K. R. Chu, A. C. England, C. M. Loring, O. C. Eldridge, A.
C. Howe, A. G. Kulchar, E. Lazarus, M. M urakami, and J. B. Wilgen,
“Heating at the Electron Cyclotron Frequency in the ISX-B Tokamak,”
Phys. Rev. Lett., Vol. 44, pp. 647-650, 1980.
GIL98
R. M. Gilgenbach, J. M. Hochman, R. L. Jaynes, W. E. Cohen, C. W.
Peters, D. E. Vollers, and T. A. Spencer, "Optical Spectroscopy o f Plasma
in High Power Microwave Pulse Shortening Experiments Driven by a psecond e-beam," IEEE Transactions on Plasma Science, Vol. 26 (3), pp.
282-289, 1998.
GOE98
D. M. Goebel, “ Pulse Shortening Causes in High Power BWO and TWT
M icrowave Sources,” IEEE Transactions on Plasma Science, Vol. 26 (3),
pp. 263-274, 1998.
GRA97
V. L. Granatstein, B. Levush, B. G. Danly, and R. K. Parker, "A Quarter
Century o f Gyrotron Research and Development," IEEE Transactions on
Plasma Science, Vol. 25 ( 6 ), pp. 1322-1335, 1997.
GRI89
D. J. Griffiths, Introduction to Electrodynamics. Prentice-Hall, Englewood
Cliffs, NJ, 1989.
GUS92
W. C. Guss, M. A. Basten, K. E. Kreischer, and R. J. Temkin, "Sideband
Mode Competition in a Gyrotron Oscillator," Phys. Rev. Lett., Vol. 69
(26), pp. 3727-3730, 1992.
HAR61
R. F. Harrington, Time-Harmonic Electromagnetic Fields. McGraw-Hill,
New York, 1961.
HAU91
H. A. Haus and W. Huang, "Coupled-M ode Theory," Proceedings o f the
IEEE, Vol. 79 (10), pp. 1505-1518, 1991.
HAW98
M. D. Haworth, G. Baca, J.N. Benford, T. Englert, K. Hackett, K. J.
Hendricks, D. Henley, M. LaCour, R. W. Lemke, D. Price, D. Ralph, M.
Sena, D. Shiffler, and T. A. Spencer, "Significant Pulse Lengthening in a
Multigigawatt Magnetically Insulated Transmission Line Oscillator,"
IEEE Transactions on Plasma Science, Vol. 26 (3), pp. 312-319, 1998.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
150
H A W 00
M. D. Haworth, T. J. Englert, K. J. Hendricks, R. W. Lemke, J. W.
Luginsland, D. S. Shiffler, and T. A. Spencer, "Comprehensive diagnostic
suite for a magnetically insulated transmission line oscillator," Review o f
Scientific Instruments, Vol. 71 (3), pp. 1539-1547, 2000.
HEN96
K. J. Hendricks, P. D. Coleman, R. W. Lemke, M. J. Arman, and L.
Bowers, "Extraction o f 1 GW o f rf Power from an Injection Locked
Relativistic Klystron Oscillator," Physical Review Letters, Vol. 76 (1), pp.
154-157, 1996.
HOC98
J. M. Hochman, "Microwave Emission o f Large and Small Orbit
Rectangular Gyrotron Devices," Ph. D. Dissertation, University o f
Michigan, 1998.
IEE98
See the IEEE Transactions on Plasma Science special issue on high-power
microwave generation, Vol. 26 (3), 1998.
JAY99
R. L. Jaynes, R. M. Gilgenbach, J. M. Hochman, N. Eidietis, J. I.
Rintamaki, W. E. Cohen, C. W. Peters, Y. Y. Lau, and T. A. Spencer,
"Velocity Ratio M easurement Diagnostics and Simulations o f a
Relativistic Electron Beam in an Axis Encircling Gyrotron," IEEE
Transactions on Plasma Science, Vol. 27 (1), pp. 136-137, 1999.
JAYOO
R. L. Jaynes, “Generation o f High Power M icrowaves in a Large Orbit
Coaxial Gyrotron,” Ph. D. Dissertation, University o f Michigan, 2000.
JE 092
J. Jeong and W. J. Williams, "Kernel Design for Reduced Interference
Distributions," IEEE Transactions on Signal Processing, Vol. 40 (2), pp.
402-412, 1992.
JE 092b
J. Jeong and W. J. Williams, "Alias-Free Generalized Discrete-TimeTimeFrequency Distributions," IEEE Transactions on Signal Processing, Vol.
40(11), pp. 2757-2765, 1992.
JON90
G. Jones and B. Boashash, "Instantaneous Frequency, Instantaneous
Bandwidth and the Analysis o f Multicomponent Signals," 1990
International Conference on Acoustics, Speech, and Signal Processing,
pp. 2467-2470, 1990.
KHA98
L. M. Khadra, J. A. Draidi, M. A. Khasawneh, and M. M. Ibrahim, "TimeFrequency Distributions Based on Generalized Cone-Shaped Kernels for
the Representation o f Nonstationary Signals," J. Franklin Institute, Vol.
335B (5), pp. 915-928, 1998.
K N 089
G. F. Knoll, Radiation Detection and M easurement. John W iley & Sons,
New York, 1989.
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
151
KRE84
K. E. Kreishcer, R. J. Temkin, H. R. Fetterman, and W. J. Mulligan,
"Multimode Oscillation and Mode Competition in High-Frequency
Gyrotrons," IEEE Transactions on M icrowave Theory and Techniques,
Vol. MTT-32 (5), pp. 481-490, 1984.
LAM64
W. E. Lamb, Jr., “Theory o f an Optical M aser,” Physical Review, Vol.
134 ( 6 A), pp. A1429-A1450, 1964.
LAU82
Y. Y. Lau, "Simple Macroscopic Theory o f Cyclotron Maser Instability,"
IEEE Transactions on Electron Devices, Vol. 29, pg. 320, 1982.
LEE 8 8
C. Lee and L. Cohen, "Instantaneous Mean Quantities In Time-Frequency
Analysis," Proceedings o fth e IEEE ICASSP- 8 8 , Vol. 4, pp. 2188-2191,
1988.
LOU92
P. J. Loughlin, J. W. Pitton, and L. E. Atlas, "Proper Time-Frequency
Energy Distributions and the Heisenberg Uncertainty Principle,"
Proceedings o f the IEEE-SP International Symposium on Time-Frequency
and Time-Scale Analysis, pp. 151-154, 1992.
LOU97
P. J. Loughlin and B. Tacer, "Instantaneous frequency and the conditional
mean frequency o f a signal," Signal Processing, Vol. 60, pp. 153-162,
1997.
LOU97b
P. J. Loughlin and B. Tacer, "Comments on the Interpretation o f
Instantaneous Frequency," IEEE Signal Processing Letters, Vol. 4 (5), pp.
123-125, 1997.
LOU98
P. J. Loughlin and K. L. Davidson, "Positive Local Variances o f TimeFrequency Distributions and Local Uncertainty," Proceedings o f the IEEESP International Symposium on Time-Frequency and Time-Scale
Analysis, pp. 541-544, 1998.
LOU99
P. J. Loughlin and K..L. Davidson, "Instantaneous Bandwidth of
M ulticomponent Signals," SPIE conference on Advanced Signal
Processing, Algorithms, Architectures, and Implementations IX, Denver,
Colorado, SPIE Vol. 3807, pp. 610-624, July 1999.
LOV93
B. C. Lovell, R. C. Williamson, and B. Boashash, "The Relationship
Between Instantaneous Frequency and Time-Frequency Representations,"
IEEE Transactions on Signal Processing, Vol. 41 (3), pp. 1458-1461,
1993.
M CC 8 8
A. H. McCurdy and C. M. Armstrong, "Mode Selection by Application o f
an External Signal in an Overmoded Gyrotron Oscillator," Physical
Review Letters, Vol. 61 (20), pp. 2316-2319, 1988.
R ep ro d u ced with p erm ission o f t h e copyright ow ner. Further reproduction prohibited w ithout p erm ission.
152
MCC92
A. H. McCurdy, “Mode Interaction Through Amplitudes and Phases in a
Two-Mode Gyrotron Oscillator,” IEEE Transactions on Plasma Science,
Vol. 20 (3), pp. 139-148.
MUG90
P. Muggli, M. Q. Tran, T. M. Tran, H. G. Mathews, G. Agosti, S. Albert,
and A. Perrenoud, "Effect o f Power Reflection on the Operation o f a LowQ 8 GHz Gyrotron," IEEE Transactions on M icrowave Theory and
Techniques, Vol. 38 (9), pp. 1345-1351, 1990.
NAS99
G. P. Nason and R. von Sachs, "Wavelets in Time-Series Analysis," Phil.
Trans. R. Soc. Lond. A, Vol. 357, pp. 2511-2526, 1999.
NUS99
G. S. Nusinovich, “Review o f the Theory o f Mode Interaction in
Gyrodevices,” IEEE Transactions on Plasma Science, Vol. 27 (2), pp.
313-326, 1999.
OH92
S. Oh and R. J. Marks II, "Some Properties o f the Generalized TimeFrequency Representation with Cone-Shaped Kernel," IEEE Transactions
on Signal Processing, Vol. 40 (7), pp. 1735-1745, 1992.
PET98
C. W. Peters, R. L. Jaynes, Y. Y. Lau, R. M. Gilgenbach, W. J. Williams,
J. M. Hochman, W. E. Cohen, J. I. Rintamaki, D. E. Vollers, and T. A.
Spencer, "Time-frequency analysis o f modulation o f high-power
microwaves by electron-beam voltage fluctuations," Physical Review E,
Vol. 58 (5), pp. 6880-6883, 1998.
PIC97
B. Picinbono, "On Instantanteous Amplitude and Phase o f Signals," IEEE
Transactions on Signal Processing, Vol. 45 (3), pp. 552-560, 1997.
PIT98
J. W. Pitton, "Positive Time-Frequency Distributions via Quadratic
Programming," Mulitdimensional Systems and Signal Processing, Vol. 9
pp. 439-445, 1998.
POZ98
D. M. Pozar, Microwave Engineering. Addison-Wesley, 1998.
PR 096
J. G. Proakis and D. G. Manolakis, Digital Signal Processing. Prentice
Hall, New Jersey, 1996.
RAM84
S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in
Communications Electronics. John Wiley & Sons, New York, 1984.
REA80
M. E. Read, R. M. Gilgenbach, R. F. Lucey, Jr., K. R. Chu, A. T. Drobot,
and V. L. Granastein, "Spatial and Temporal Coherence o f a 35-GHz
Gyromonotron Using the TEoi Circular Mode," IEEE Transactions on
Microwave Theory and Techniques, Vol. MTT-32 ( 8 ), pp. 875-878, 1980.
R ep ro d u ced with p erm ission o f t h e copyright ow ner. Further reproduction prohibited w ithout p erm ission.
153
REI94
A. Reilly, G. Frazer, and B. Boashash, "Analytic Signal Generation - Tips
and Traps," IEEE Transactions on Signal Processing, Vol. 42 (11), pp.
3241-3245, 1994.
S T 097
P. Stoica and R. Moses, Introduction to Spectral Analysis. Prentice Hall,
New Jersey, 1997.
T H 082
D. J. Thomson, "Spectrum Estimation and Harmonic Analysis,"
Proceedings o fth e IEEE, Vol. 70, pp. 1055-1096, 1982.
W HA94
D. R. W haley, M. Q. Tran, T. M. Tran, and T. M. Antonsen, Jr., "Mode
Competition and Startup in Cylindrical Cavity Gyrotrons Using High
Order Operating Modes," IEEE Transactions on Plasma Science, Vol. 22
(5), pp. 850-860, 1994.
WIL90
W. J. W illiams, J. Jeong, M. Driscol, and S. Krishnamachari,
"Applications and Interpretations o f Reduced Interference TimeFrequency Distributions," 24th Asilomar Conference on Signals, Systems,
and Computers, Nov. 5-7, pp. 1049-1054, 1990.
WIL98
W. J. W illiams, "Cross Hilbert time-frequency distributions," SPIE
Conference on Advanced Signal Processing Algorithms, Architectures,
and Implementations VIII, SPIE Vol. 3461, pp. 120-129, 1998.
XU99
Y. Xu, S. Haykin, and R. J. Racine, "Multiple W indow Time-Frequency
Distribution and Coherence o f EEG Using Slepian Sequences and Hermite
Functions," IEEE Transactions on Biomedical Engineering, Vol. 46 (7),
pp. 861-866, 1999.
YAK89
S. Yakowitz and F. Szidarovszky, An Introduction to Numerical
Com putations. Macmillan Publishing Company, New York, 1989.
YAM95
Y. Yamanaka and T. Shinozuka, "Statistical Parameter Measurement o f
Unwanted Emissions from Microwave Ovens," IEEE International
Symposium on Electromagnetic Compatability, pp. 57-61,1995.
ZHA90
Y. Zhao, L. E. Atlas, and R. J. Marks II, "The Use o f Cone-Shaped
Kernels for Generalized Time-Frequency Representations o f
Nonstationary Signals," IEEE Transactions on Acoustics, Speech, and
Signal Processing, Vol. 38 (7), pp. 1084-1091, 1990.
R ep ro d u ced with p erm ission o f t h e copyright ow ner. Further reproduction prohibited w ithout perm ission.
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