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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 R ep ro d u ced with p erm ission of the copyright ow ner. Further reproduction prohibited w ithout p erm ission. UMI Number: 3001028 Copyright 2001 by Peters, Christopher Wayne All rights reserved. ___ ® UMI UMI Microform 3001028 Copyright 2001 by Bell & Howell Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. Bell & Howell Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. This is dedicated to my family, friends, and colleagues who have helped me stay in school through all the good times and bad ii R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. iii R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. iv R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 v R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 vi R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 vii R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 viii R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 ix R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 x R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 xi R ep ro d u ced with p erm ission of the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 xii R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 xiii R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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]. 1 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 2 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 ) \ 5 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (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. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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]. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. (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) R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. (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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission of the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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) R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. (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) R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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]. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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) R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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). R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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> \ R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. R ep ro d u ced with p erm ission of th e copyright ow ner. Further reproduction prohibited w ithout p erm ission . 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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, R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission . 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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) R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (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 R ep ro d u ced with p erm ission of th e copyright ow ner. Further reproduction prohibited w ithout p erm ission . 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. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 52 some kernels, complex bandwidths can occur and therefore the kernel must be carefully designed if instantaneous bandwidth is o f concern [LOU98]. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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). R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. (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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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). R ep ro d u ced with p erm ission of th e copyright ow ner. Further reproduction prohibited w ithout p erm ission . 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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]. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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) R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission. 76 .1000 m ^ 0 • k ;•: :l 'L §00 I’-i} .. .t; > .-< } b 'VAfcr^vu^-r.: ».:•■■-’*I 0 700 800 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. ( 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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). R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited without p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. ^ 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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). R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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). R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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). R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. A PPEN D IC ES 100 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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) R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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) R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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). R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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) R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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) R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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) R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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)); R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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)); R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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))); R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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)); R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission . 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); R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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) R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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: R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission . (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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. (Ml) BIBLIOGRAPHY 146 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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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