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SPECTRAL ANALYSIS OF FIRST AND SECOND ORDER SIDEBANDS FROM VLF TRIGGERED EMISSIONS by RANDALL EVANS WALL JR. B.S., University of Denver, 2011 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Masters of Science Electrical Engineering Program 2014
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ii This thesis for the Master of Science degree by Randall Evans Wall Jr. Has bee n approved for the Electrical Engineering Program By Mark Golkow s ki, Chair Titsa Papantoni Yiming Deng November 20, 2014
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iii Wall, Randall Evans Jr. ( M.S., Electrical Engineering) Spectral Analysis of First and Second Order Sidebands from VLF Triggered Emissions Thesis dire c ted by Assistant Professor Mark Golkowski ABSTRACT a occur, including the modulation of transmitted signals which pass through its interaction region along the L = field Th e s e interactions create A theoretical modulation model is created which combines Amplitude M odulation (AM) and Frequency Modulation (FM) to m imic the modulation found in the magnetosphere. A separate parameter named is created which measures relative AM and FM contributions to the signal from the magnetosphere This parameter may be used to theoretically interpret the physical processes c ausing the modulation. Arion Systems, Inc. produced a tool called ReFOCUSD that was used in this thesis to denoise the signals so that both orders of sidebands could be resolved Generally, indicating the signal encompasses larger A M contributions, while F M effects cannot be dismissed. shown that together with the other parameters of the signal, evolve over time, and that side bands appear starting a round 25 Hz removed from the transmitted frequency 0.3 ms after the t ransmitted signal enters the interaction region, and peaking at 35 Hz 0.5 ms after the sidebands first show up The form and content of this abstract are approved. I recommend its publication. Approved: Mark Gol kowski
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iv ACKNOWLEDGEMENTS This thesis is de dicated to my wife, Kathryn, who supported me though the entirety of its writing and my parents, who have supported everything that I have ever done and wanted to do For my wife, writing this has taken many long hours and late nights and we have both sacrifice d time together for its completion. And my parents, who I have also sacrificed time together for this thesis, have always supported my goals and dreams I am eternally grateful appreciative and thankful for my wife and parents love and support during the completion of this thesis, and really all other times of my life. I would also like to thank Dr. Andrew Gibby and the engineers from Arion Systems, Inc. for the instrumental use of ReFOCUSD in this thesis Lastly, I would like to thank my Adv isor, Dr. advisor; I have certainly learned a lot from him as a with him as well as gained experience and research skills from him in his role as my advisor. Thank you everyone!
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v TABLE OF CONTENTS CHAPTER I. INTRODUCTION ................................ ................................ ................................ ................... 1 II. MATHEMATICAL ANALYSIS OF S IDEBANDS ................................ ................................ ........ 8 Amplitude Modulation (AM) ................................ ................................ ................... 9 Frequency Modulation (FM) ................................ ................................ ................. 1 2 Amplitude Frequency Modulation ................................ ................................ ....... 1 5 III. REFOCUSD ................................ ................................ ................................ ......................... 24 IV. RESULTS ................................ ................................ ................................ ............................. 28 V. CONCLUSIONS ................................ ................................ ................................ ................... 3 6 REFERENCES ................................ ................................ ................................ ...................... 37 APPENDIX A ................................ ................................ ................................ ............................ 3 8 B ................................ ................................ ................................ ............................ 4 8
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vi LIST OF TABLES TABLE 1. Signal Parameters Determined from Eq. 22 ................................ ................................ ..... 29 2. Signal Parameters Determined from Eq. 22 for Successive Time Instances in a Single Pulse ................................ ................................ ................................ ................................ ........... 35
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vii LIST OF FIGURES FIGURE 1. Large Scale Experimental Setup ................................ ................................ .......................... 3 2. Time Domain of (a) Several Pulses and (b) Within a Signal Pulse ................................ ...... 5 3. Spectrogram of (a) Several Pulses and (b) Within a Signal Pulse ................................ ....... 6 4. Fourier Transform of Signal between 57.5 ms and 57.9 ms ................................ ............... 7 5. Non Modulated Sinusoidal Signal ................................ ................................ ....................... 8 6. Time Domain AM Signal ................................ ................................ ................................ ...... 9 7. Fourier Transform of (a) AM and (b) FM Signals ................................ .............................. 11 8. Time Domain of FM Signal ................................ ................................ ................................ 1 3 9. Fourier Transform of (a) AM and (b) FM Signals at ................................ ......... 1 4 10. Band Configuration ................................ ................................ ................................ ........... 1 5 11. Fourier Transform of Eq. 13 ................................ ................................ .............................. 2 0 12. (a) simple AM and (b) Data Comparison with ReFOCUSD ................................ ................ 26 13. ReFOCUSD Fourier Transform of (a) Case 1 and (b) Case 2 ................................ .............. 31 14. ReFOCUSD Fourier Transform of (a) Case 3 and (b) Case 4 ................................ .............. 32 15. ReFOCUSD Four Table 1 (Green) ................................ ................................ ................................ ................. 33 16. ReFOCUSD Sign al f or (a) 21.80 22.05 s, (b) 22.05 22.30 s, (c) 22.30 22.55 s, (d) 22.55 22.80 s, (e) 22.80 22.95 s, and (f) 22.95 23.20 s after 14:25 UT on December 8, 1986 .. 34 17. Spectrogram Representation of Signal Pulse with each letter correspond ing to the same letter in Fig 16 ................................ ................................ ................................ ................... 35
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1 CHAPTER I INTRODUCTION The near Earth space environment hosts a rich array of electromagnetic phenomenon, a product of the diverse plasma populations cr eated from ionization from the S un [ 1 ] Starting at where the atmosphere is partially ionized and electron densities exceeding 10 12 m 3 are observed. At even higher altitudes (>1000 km) the ambient background is rarefied enough so that it becomes almost complete ly ionized; thi s occurs in magnetosphere name d force. The plasma in the magnetosphere can be divided into two populations. There is the background cold plas ma with densities in the range of 10 7 10 10 m 3 and low energies of less than 1 eV and there is the hot plasma population with low densities (< 10 6 m 3 ) and very high energies of up to 1 MeV. While the cold plasma determines the main propagation characte ristics of electromagnetic waves such as the propagation velocity and wavelength, the hot plasma electrons can take part in resonant interactions that lead to energy exchange between the particles and the wave and even the generation of new waves. One su ch resonant interaction phenomen on is that known as VLF triggered emissions or the coherent wave instability [ 2 ] This occurs when waves with frequencies below the electron gyrofrequency are injected into the magnetosphere. These waves propagate in the whistler mode a special mode existing only in magnetized plasmas and characterized by circular polarization and a propagation velocity much lower than the
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2 speed of light. It has been observed that such signals can be amplified by as much as 30 dB and new frequencies called triggered emissions can be generated. VLF triggered emissions were investigated extensively in an experiment operated in Antarctica from 1973 1988 [2] A base station in Siple Station Antarctica transmitted VLF waves along t he L = 4 near the plasmapause boundary The L shell is a parameter used to describe the altitude, in terms of Earth radii from its center at which a magnetic field line passes over Earth s equator ; for this experiment, the lines travel up to 4 Earth rad ii from center, or 3 Earth radii surface. These waves would travel s in the magnetosphere and back to Earth a t the magnetic conjugate point, where they were observed at two station s in Canada located in Roberval and Lake Mistissini. During their traverse through the magnetosphere, the waves would undergo nonlinear interactions with hot plasma electrons leading to amplification, triggering of emissions, and generation of sidebands. The location of both the transmitter and the receivers provided several advantages for the experiment. Siple Station is located in Antarctica atop a 2 km thick sheet of ice, which allows for finite elevation above the uctor at VLF frequencies. A t the operating frequency of 3 kHz the free space wavelength is 100 km Other reasons include: 1) located near the desired L = 4 shell, 2) the conjugate stations are accessible in Canada, 3) Siple Station and Lake Mistissini were mostly void of industrialization (two power grid extensions were later built in areas that prevent Roberval from having a good SNR at the desired frequencies ) Figure 1 shows the large scale experiment set up.
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3 The experiment yielded a large quantity of data most of which continues to lack satisfactory theoretical explanation. The original data w ere recorded on magnetic tapes, but parts of the data archive were digitized in the early 2000s to facilitate continued analysis. An example of the d ata received at Lake Mistissini is shown in Fig 2 (a) which is the time domain representation of the signal in December 198 6. The large amplitude impulses observed in Fig. 2(a) are from electromagnetic radiation from lightning feature of observations in this frequency band. The signal being transmitted is a 2.7 kHz pulse, where Fig. 2 (b) shows the time domain representation of a single pulse. Generally not much information is seen in the time domain, so the spectrogram (short time Fourier transform) of the signal is calculated as shown in Fig 3(a), and a signal pulse, shown in Fig 3(b) Once this is done, several interesting characteristics can be seen. As noted above the signal can be amplified by as much as 30 dB which is s een in the spectrogram starting as soon as the pulse starts Another interesting feature is that after a certain amount of time hovering around 75 ms triggered emissions are produced, creating two obvious Figure 1 : Large Scale Experimental Setup
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4 sidebands around 2670 Hz and 2730 Hz In t he Fourier transform of the signal between 57.5 ms and 57.9 ms the side band s are seen to be different amplitud es with the sideband located at 2730 Hz ha ving a larger amplitude than the sideband located at 2670 Hz by Figure 4 shows the Fourier transform. The magnetospherically generated sidebands of the observed signals are the feature that we focus on in this work. Mechanisms for the production of these sidebands have been proposed by [ 2 ] [ 3 ], but s e l f consistent p hysical models describing amplitude and phase of the induced modulation have yet to be developed. Likewise, little work has been done on the relative phase of the observed sidebands. In this work we analyze the amplitude and phase of sidebands of trigger ed emissions observed in 1986. We use advanced spectral techniques to identify the amplitude and relative phase of higher order sidebands. The observations are compared to a hybrid modulation scheme of AM and FM modulation.
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5 Figure 2 : Time Domain of (a) S everal P ulses and (b) W ithin a S ignal P ulse. (a) (b)
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6 (a) (b) Figure 3 : Spectrogram of (a) S everal P ulses and (b) W ithin a S ignal P ulse.
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7 Figure 4 : Fourier Transform of S ignal between 57.5 ms and 57.9 ms.
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8 CHAPTER II MATHEMATICAL ANALYSIS OF SIDEBANDS The spectrogram is Fig. 3 shows significant distortio n to the signal starting around 57.3 ms It is seen that in addition to the transmitted frequency at 2700 Hz, two sidebands occurring on either side at 2700 30 Hz are observed The sidebands are generated by resonant interactions i n the magnetosphere that are not fully understood. Nevertheless, the creation of sidebands by any process is akin to signal modulation which is a common technique in communication systems. To test the modulation processes occurring in the magnetospher e, a theoretical model is built here in order to investigate if standard modulation techniques from communication theory can be used to explain the observed modulations Specifically, we investigate AM and FM type modulation, which is reviewed below Figure 5 : Non M odulated S inusoidal S ignal. Carrier Wave
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9 Amplitude Modulation (AM) Amplitude modulation is a signal process in which a message wave is transmitted on a carrier wave by a changing amplitude, taking the form (1) where is the carrier wave equation and is the message wave equation [ 4 ] As seen in Eq. 1 AM is accomplished by multiplying the carrier signal by the message signal. If both waves are sinusoidal they can be expressed as where and are the angular frequencies and and are the phase s of the carrier and message waves, respectively. Now, Eq. 1 becomes (2) Figure 6 : Time Domain AM S ignal.
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10 The carrier signal can be seen in Fig. 5 and the resulting AM wave in Fig. 6 For simplicity, assume Using this assumption and some trigonometric identities, Eq. 2 can be expanded to produce ( 3 ) Let the Fourier transform of be denoted by with the absolute, real, and imaginary spectrum plot s of being shown in Fig 7(a) This figure shows that the amplitude of each band in is equivalent to the amplitude of the sinusoid with the corresponding frequency in This unique property between the time and frequency domain representations of the same s ignal will be used as a basis for the rest of our analysis
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11 Figure 7 : Fourier Transform of (a) AM and (b) FM S ignals. (a) (b)
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12 Frequency Modulation (FM) Using the same nomenclature as in the AM s ection, the equation for an FM signal is ( 4 ) where is the frequency deviation and is the maximum amount that the cosine argument in Eq. 4 will deviate from [ 4 ] Solving the integral for with the used in the AM s ection yields ( 5 ) where the modulation index is For narrowband FM, the condition which amounts to must hold With this assumption, the wave equation for FM becomes ( 6 ) The time domain of the FM signal is shown in Fig. 8 and the Fourier transform is in Fig. 7(b) Like in the case for AM, the peaks in the Fourier Transform correspond to the amplitude of the waves at the same frequency. The subtraction of the term is seen in the real part as the peak located at and is equal but opposite to the p eak at This asymmetr y around in the real spectrum is what differentiates AM and FM that is AM will be symmetric and FM will be asymmetric around
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13 This property holds when but As and changes, so will the frequency spectrum of both AM and FM. When and the message signal is effectively a sine wave, the frequency spectrum becomes real and imaginary which is seen in Fig. 9 Now If then the frequency spectrum is all real, but each peak is opposite that when Generally, is not an even multiple of so will have both real and ima ginary parts. Similarly variations occur ring in will produce similar results, except the center frequency can also be effected but since is measured with respect to the carrier phase is assumed to be zero radians Figure 8 : Ti me Domain of FM S ignal. Frequency Modulation
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14 Another difference between AM and FM in the frequency domain is that FM can develop extra side bands. [ 4 ] shows that the number of sidebands depends on ; as increases, the number of sidebands increase. Here, only two orders of sidebands are considered and are labeled in Fig 10 where each band is denoted by a subscript These two models are now combined to form the ba sis for matching the experimental data Figure 9 : Fourier T ransform of (a) AM and (b) FM S ignals at (a) (b)
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15 Amplitude Frequency Hybrid Modulation Amplitude Modulation and Frequency Modulation by themselves fail to replicate features of the observed data and an adequate description of the magnetospher ic modulation process For example, t he frequency show up in AM analysis. There is also disc repancy in the amplitude of the first order sidebands, and as defined in Figure 10 In the observed Fourier transform like Fig. 4 t he second band, is larger types where there is symmetry in the magnitude of the bands around the carrier A combination of the two modulation schemes is proposed here. Let be the transmitted signal taking on the form of a sinusoidal at angular frequency and let the effects of the magnetosphere be de noted by with angular frequency the frequency induced by the magnetosphere The wave equation due to magnetospheric effects will be ( 7 ) Figure 10 : Band C onfiguration.
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16 where is the amplitude of the induced signal and is the phase induced by the magnetosphere Since there will be both AM and FM effects, th e amplitude and phase will be and when referring to the A M part and and for the F M part. The AM contribution will be denoted as ( 8 ) and the FM contribution as ( 9 ) where is the amplitude of the transmitted signal and is the frequency deviation and is the maximum amount that the argument in Eq. (3) will deviate from The proposed wave equation for the signal after excitation in the magnetosphere is then ( 10 ) which expands to ( 11 ) If then only FM contributions will be induced; if then only AM contributions are included ; finally, if y the magnetosphere at all. Inserting the modulation equations into Eq. 11 yields ( 12 ) The modulation index is Now when Eq. 12 is expanded assuming narrowband, the modulated signal becomes
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17 ( 1 3) The full derivation of Eq. 13 can be found in Appendix A and it s Fou rier transform is seen in Fig. 11 For convenience, Eq. 13 is rewritten as ( 14 ) where
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18 Taking real and imaginary parts, each band can be spl it up into equations describing the amplitude of each ban d in the complex frequency domain s The equations are derived at the end of Appendix A and produce the following results: a) Center Band (1 5 ) b) Band 1 (1 6 )
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19 c) Band 2 (1 7 ) d) Band 3 (1 8 ) e) Band 4 (1 9 )
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20 Figure 11 : Fourier Transform of Eq. 13
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21 Equation 15 19 show amplitudes for each band in both the real and imaginary spectra, along with the absolute spectrum. The equations b ring up some noteworthy results deserving discussion. While in both AM and FM the absolute amplitude of the center band in the frequency domain was equal in this model the real part of the center band amplitude is equal to while now This model also explains the difference in absolute amplitude between and ; the equations are almost identical except the is subtracted in and added in If such that then as the will vanish. Also, if such that then It can be see n that g enerally so that Lastly, it can be observed that and the absolute amplitudes are independent of the phases. The imaginary parts are also equal, that is but the real parts are equal but opposite, There are fifteen equations and five unknowns, so now the variables can be fou nd based on measured values. The derivation for these variables are shown in Appendix B and produce : ( 2 0 a ) ( 2 0 b )
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22 ( 2 0 c ) ( 2 0 d ) ( 2 0 e ) We have thus provided a set of narrowband equations that can be used to quantify amplitude and phase effects of magnetosphere induced processes The se equations can determine which process is influencing the observed modulation the most, either AM with o r FM with and can thus serve as a guide for theoretical interpretations of the physical processes generating the modulation Equation 2 0 is calculated by measuring the values of the peaks at each corresponding frequency in the absolute, real and imaginary frequency domains. A new parameter needs to be defined in order to describ e the effects of AM versus FM. The parameter will be defined as the following ratio : ( 2 1 ) which has units of radians per second Equation 2 1 allows us to quantify how much of each modulation effect is present in a signal observation Unfortunately in this analysis, we are not able to determine independently and since they are always coupled together in but if we remove the dependence in which is what is being
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23 done in Eq. 2 1 then we can isolate the two variables that solely determine FM dependence of the magnetosphere induced signal. When this is done, Eq. 2 1 turns into ( 2 2 ) We can see from Eq. 22 that if either the magnetosphere i have any F M effects or the AM effects are much larger than the FM effects then However, if either the magnetosphere induced signal is depleted of A M effects or the FM effects are much larger than the AM effects then Because our analysis ass um es we see that This allows us to approximate closer the degree by which a signal m ay be AM or FM. If the AM and FM contributions are equal, that is if then
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24 CHAPTER III REFOCUSD Analysis of the data from the Siple experiment, especially the rare cases of second order sidebands is hampered by the noise levels in the data In order to be able to accurately extract amplitudes and phases of the observed sidebands, we employ an advanced spectral analysis procedure known as ReFOCUSD, d eveloped by Arion Systems, Inc. [ 5 ] The program was originally developed to yield high resolution SONAR image s but can be applied to the one dimensional Fourier transformers used here. Arion Syst ems provided the algorithm in the form a proprietary protected MATLAB function file. ReFOCUSD follows the patterns of a Minimum Variance Distortionless Response (MVDR) algorithm [ 5 ] but does it recursively in a way that minimizes the noise level at each iteration b y re estimating the noise fi e l d strength distribution. MVDR is an adaptive technique used to find optimal frequency resolution [6] dynamic optimal window solely from the data. ReFOCUSD estimates the necessary matrix in order for it to use the MVDR technique and then reiteratively recalculates the matrix. This allows the estimation of the noise content to improve with each iteration so that ReFOCUS D produces the least noisy ve rsion of the data. The process here will be described assuming a single, one dimensional signal. Let this signal be ( 2 3 )
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25 so that the Fourier transform is and is the transpose Here, is a vector describing the direction of the signal and is the position of the sensor element. In general, ReFOCUSD is used for three dimensional signals with many sensor elements, but here only one element needs to be considered so the values of an d can be arbitrary. Now let the total response of the array be ( 2 4 ) where is a matrix of complex weighting factors and and H denotes the Hermitian of matrix The distortionless response of Eq. 2 4 is defined so that which, with some matrix algebra, we can find the weighting vector to be ( 2 5 ) This weighting factor matrix is then placed into Eq. 24 and squared to get (2 6 ) where factor is the covariance of the signal [5 ] shows that the optimal weighting factor matrix is then equal to ( 2 7 ) Often times, the covariance of the signal can be estimated using a finite s et of samples from the signal. Now recursively, the algorithm will repeat Eq. 2 4 2 7 so that where is the original signal and is the first iteration. Knowing this, the total response of the arr ay from
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26 Eq. 2 4 of the original signal, coupled with the optimal weighting factor matrix in Eq. 2 7 calculated from the original signal, leads to so that in general we have ( 2 8 ) This process continues for iterations when ( 29 ) Figure 12 : (a) simple AM and (b) Data Comparison with ReFOCUSD.
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27 To test the ReFOCUSD algorithm for accuracy, we used it on a simple AM signal. The results are shown in Fig. 1 2 (a) What it shows is that on a signal with no noise added, and one with a known analytical Fourier transform, ReFOCUSD reproduce s the frequency spectrum exactly. The utility of ReFOCUSD becomes apparent when we apply it to a noisy signal in this case the Siple experiment data as shown in Fig. 1 2 (b) Here ReFOCUSED captures the amplitude and phase of the carrier and sidebands but with much higher signal to noise ratio (SNR). The value o f the improved SNR becomes readily apparent when we look at the second order sidebands, which are readily visible in the ReFOCUSED spectrum but hard to discern from noise in the standard FFT. This improvement allows the equations derived in Chapter 2 to b e used effectively. When the spectrograms of signal is shown with and without ReFOCUSD, more details can be seen which allows for the evolution of each band to be seen.
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28 CHAPTER IV RE SULTS Using ReFOCUSD to analyze the experimental data we can match observations to the modulation parameters of our AM FM hybrid model. We select f ive cases for our analysis all of which were observed on December 8, 1986, and described below as follows: Case 1: Taken betwe en 14:25: 0 7.85 and 14:25: 0 8.00 UT Case 2: Taken between 14:25: 10.13 and 14:25: 10 23 UT Case 3: Taken between 14:25:12.45 and 14:25:12.55 UT Case 4: Taken between 14:25:20.80 and 14:25:20.95 UT Case 5: Taken between 14:25:22.80 and 14:25:22.95 UT Each of t hese cases is initiated from a 2700 Hz si gnal se n t from Siple Station and then received at Lake Mistissini The frequency domain representations of each case is seen in Fig. 1 3 15 (Fig 15 shows the comparison between the ReFOCUSD signal and the model signal) Analysis of each case follows the methodology described in Chapter 2 and t he values for , , and the resulting are shown in Table 1. The para meters shown in Table 1 show us that the observed carrier frequency is 2701 .2 0 0.75 Hz a slight increase from the 2700 Hz transmitted, and that the sidebands induced by the magnetosphere are offset by 31. 4 0 1.85 Hz which corresponds to 197.3 11.7 radians The phases induced by A M and FM effects showed some similar characteristics. The AM phase shift was always negative while the FM
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29 phase shift was positive except for Case 3 T he AM and FM phases were less stable ow a specific pattern in value. It can be seen in Fig. 1 3 ( b ) that Case 2 is noisy in the real and imaginary frequency domains. We believe this is why their value is large, which would normally indicate a larger FM presence with very little AM influence. We do see from Cases 1 and Case 3 5 that their values range from 101 to 1 6 5 rad/s If then when the signal is less noisy, like Cases 1, 3 5, the magnetosphere shows more A M characteristic than F M, but F M is also present or else the values would be very small It must be noted that all of these variables are time dependent. This is seen in Fig. 1 6 and 17 What Fig. 1 6 shows is the Fourier transform of the same pulse but at different time instances between 0.15 seconds and 0.25 seconds long It shows that the Case (Hz) (Hz) (radians) (radians) (radians) ( ) 1 2702 31 21.78 0.3982 0.2856 1.0353 0.4068 140 2 2701 30 42.49 0.0425 0.3546 0.6162 0.4732 1,573 3 2700 33 32.2 0.6768 0.5371 2.285 0.7534 165 4 2702 29 27.96 0.7778 0.4308 0.2482 1.2353 101 5 2701 34 32 78 0.7815 0.4724 1.282 0.2454 133 Table 1 : Signal Parameters Determined from Eq. 22.
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30 parameters , , and vary drastically from each point in time. As mentioned in Chapter 1, the amplitude of can change by quite a bit, though here it changed by 1 7 7 dB instead of 30 dB values until 22.30 22.55 seconds which agrees with what the spectrograms show. We also see that reaches as high as 11,713 rad/s indicating a large amount of FM content while at the time instant right after it 1 33 rad/s which shows that the AM and FM effects are cl oser to being equal However, t he time instances of 22 30 22.55 seconds, 22.55 22.80 seconds, and 22.95 23.20 seconds each have the narrowband model used here. And lastly, we see that the sidebands change frequency and that and can change frequency by as much as 1 0 Hz. Table 2 shows the values for each parameters in each time instance.
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31 Figure 13 : ReFOCUSD Fourier Transform of (a) Case 1 and (b) Case 2.
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32 Figure 14 : ReFOCUSD Fo urier Trans form of (a) Case 3 and (b) Case 4.
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33 Figure 15 : ReFOCUSD Fourier Tra ns form of Case 5 Parameters in Table 1 (Green).
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34 Figure 16 : ReFOCUSD Signal for (a) 21.80 22.05 s, (b) 22.05 22.30 s, (c) 22.30 22.55 s, (d) 22.55 22.80 s, (e) 22.80 22.95 s, and (f) 22.95 23.20 s after 14:25 UT on December 8, 1986.
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35 Time (seconds) (Hz) (Hz) (radians) (radians) (radians) ( ) 21.80 22.05 2701 0 0.5555 0 0 0 0 IND 22.05 22.30 2701 0 4.0415 0 0 0 0 IND 22.30 22.55 2701 25 3.9455 0.0897 4.9 1.4298 0.196 8,581 22.55 22.80 2701 29 17.07 0.1502 9.6555 1.5435 0.0171 11,713 22.80 22.95 2701 35 32 78 0.7815 0.4724 1.282 0.2454 133 22.95 23.20 3696 28 2.7375 1.5 3.1315 1.0804 1.2246 367 Table 2 : Signal Parameters Determined from Eq. 22 for Successive T ime I nstances in a S ingle Pulse. (a) Figure 17 : Spectrogram Representation of Signal Pulse with each letter corresponding to the same letter in Fig 15. ( b ) ( d ) ( e ) (c) ( f )
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36 CHAPTER V CONCLUSION S The developed hybrid AM FM modulation model allows us to study the effects the m agnetospher ic signal modulation for signal s transmitted along the L = 4 geomagnetic field line The model allows for quantification of AM versus FM effects in the observations I t is seen that when sidebands are visible, both AM and FM effects are present ; when then those of A M are slightly more significant ; this phenomenon is seen in the parameter We also observe that these values change throughout the duration of the signal. We even see that the frequency induced by the m agnetosphere changes as much as 1 0 Hz t hrough the life of the signal and the AM and FM effects change drastically as well Modeling the observations using our system of equations is made possible by the ReFOCUSD tool provided by Arion Systems, Inc. which allows us to fully resolve the second order sidebands in the spectra.
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37 REFERENCES [1] U. S. Inan and M. Golkowski, Principles of Plasma Physics for Engineers and Scientists 1 st ed., New York, Cambridge University Press, 2011. [2] VLF Wave Stimulatio n Experiments in the Magnetosphere from Siple Station, Antarctica Reviews of Geophysics vol. 26, no. 3, pp. 551 578, 1988. [3] Journal of Geophysical Res earch vol. 93, no. A3, pp. 1987 1992, 1988. [4] A. B. Carlson, P. B. Crilly, and J. C. Rutledge, Communication Systems 4 th ed. New York, McGraw Hill, 2002, pp. 141 224. [5] R. Minniti, Time Enhancement of Low Frequency SONAR Im unpublished. [6] resolution Frequency Proceedings of the IEEE vol. 57, pp, 1408 1418, 1969.
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38 APPENDIX A This appendix will go into the development of Eq. 15 19. The equation that will be expanded then simplified for each band will be where and Let so that where This is the Frequency modulation part (FM). with the quadrature version: making so that, after expan sion using a Taylor series, we get:
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39 so that Now let the Amplitude Modulation (AM) part, so that Then
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40 so that By gathering like terms of , , and
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41 Let s o that The first term in is the central frequency, and rest of the number subscripts denote the band number found in Fig. 10. We now expand these equations.
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42 Expand ing we get: `Expand ing we get: Expand ing we get:
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43 Expand ing we get : Finally, e xpand ing we get :
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44 Now we want to develop equations that will determine the amplitude of each band in the Fourier tra nsf orm. Like we did in Chapter 2 with the AM and FM analysis, we can take the amplitude of the sinusoid with the same frequency to get the equation for the amplitude in the frequency domain. However, here we now have cosines and sines. This means that ther e will now naturally be real and imaginary parts, whereas in the AM section in Chapter 2, Fig. 7 shows the Frequency domain with only cosine terms for both AM and FM. Figure 7 shows that the frequency domain is only real. When the phase of the signal is offset by then only sine terms exist for they sidebands and the frequency domain for the sidebands is only imaginary Here we see that cosine terms will be associated with the real frequency domain and sine terms will associated with the imaginary frequency domain. the va lues for each band in the real, imaginary and absolute frequency domain. This allows us to develop the equations found in Eq. 15 19, that is: Central Freq uency :
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45 Band 1 Freq uency : Band 2 Freq uency :
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46 Band 3 Freq uency : Band 4 Freq uency :
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47 The equations above are the same as those found in Eq. 15 19 except more generalized. For our specific problem, let , , , and and the Eq. 15 19 will develop.
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48 APPENDIX B Appendix B will show the development of Eq. 21. We will start by defining two variables: so that The variable shows up in and square each one, we get and and From we get
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49 so which leads to So now we can get Finally, Now we need to solve for which can be done using and that is Using the same algebraic and trigonometric techniques as for we get
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50 Finally, Now we can solve for and and From here we can solve for and both using and : so that therefore
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51 Now for : so that And finally, is very easy since And now all the equations have been developed found in Eq. 21 in Chapter 2. These are once again generalized equations so when applying them to the magnetosphere , , , and
