AN INVESTIGATION OF DIGITAL SPECTRAL ANALYSIS PROGRAMS AND COMPUTER METHODS UTILIZED AT THE NAVAL POSTGRADUATE SCHOOL IN THE ANALYSIS OF HIGH FREQUENCY RANDOM SIGNALS John DeMille McKendrick NAVAL POSTGRADUATE SCHOOL Monterey, California THESIS AN INVESTIGATION OF DIGITAL SPECTRAL ANALYSIS PROGRAMS AND COMPUTER METHODS UTILIZED AT THE NAVAL POSTGRADUATE SCHOOL IN THE ANALYSIS OF HIGH FREQUENCY RANDOM SIGNALS by John DeMille McKendrick Thesis Advisor: N . E. J. Boston March 1972 Approved ^oh. pubtic kqJLqjo&z; dibtxibuXion untunltzd. An Investigation of Digital Spectral Analysis Programs and Computer Methods Utilized at the Naval Postgraduate School in the Analysis of High Frequency Random Signals by John DeMille McKendrick Lieutenant, United States Navy B.S., United States Naval Academy, 1966 Submitted in partial fulfillment of the requirements for the degree of MASTER OF- SCIENCE IN OCEANOGRAPHY from the NAVAL POSTGRADUATE SCHOOL March 1972 ABSTRACT The digitizing procedure used at the Naval Postgraduate School was investigated for possible sources of noise and other errors. Signals of known form were digitized through the Analog-to Digital Hybrid computer system (Ci 5000/XDS9300) . Similar signals were generated by digital programs on the IBM 36O/67. The resultant signals were analyzed by the com- puter programs UBCFTOR, which computed the Fourier coefficients of each block of data, and by UBCSCOR, which computed the power spectra of the signals. The power-spectral plots of the computer-generated signals were compared with the power- spectral plots of digitized signals. The analog-to-digitital process appeared to be relatively noise free. To further test the system, atmospheric temperature and wind velocity signals were digitized and analyzed under UBCFTOR and UBCSCOR. Plots of the time-varying spectra of these signals compared favorable with results obtained at other digitizing facilities. TABLE OF CONTENTS J. INTRODUCTION 12 A. PROBLEM 12 B. OBJECTIVE ™ , 12 II. THEORY , 13 A. DIGITAL REPRESENTATION OF CONTINUOUS TIME VARY- ING PROCESSES 13 1. Analog-to-Digital Conversion 13 2. Digital Sampling Theory ' 13 3. Limitations on Frequency Resolution 15 a. Low Frequency Limitations 15 b. High Frequency Limitations 16 c. Aliasing « ~ 16 B. FOURIER TRANSFORMATION : 17 1. Fourier Transformation of a Continuous Signal: Fourier Integral 17 2. Fourier Transformation of Discreet Data Signal 19 3. Fourier Transform of Truncated Continuous Wave Form 21 4. Convolution of Continuous Signals ■ — 21 C. POWER-SPECTRAL-DENSITY FUNCTION — 22 1. Methods of Computing Power-Spectral-Density 22 a. Direct Fourier Transform Method — -22 b. Analog or Bandpass Filter Method 22 c. Auto-Correlation Method — * — — « 22 2. Typical Power-Spectral-Density Functions 2k 3. Problem of Single-Sided Spectrum 26 3 4.. Power and Energy Signals -ww^^^wwt»^- 26 5. Power from the Fourier Coefficients -—r^-^- ' 27 D. PAST FOURIER TRANSFORM ™ — ^m^-^_m 29 1. Computational Economy Afforded by FFT 29 2. Importance of FFT in Turbulence Analysis 30 III. NAVAL POSTGRADUATE SCHOOL DIGITIZING FACILITY AND POWER SPECTRAL ANALYSIS PROGRAMS ~ 31 A. NPS COMPUTER FACILITIES 31 1. Hybrid Computer 31 2. IBM 360/67 Computer 31 B. HYBRID COMPUTER SYSTEM ' 33 1. Analog Computer (Ci 5000) 33 2. Digital Computer (XDS 9300) 36 3. Multi-Channel Analog-to-Digital Progarm 39 C. ANALOG-TO-DIGITAL OPERATIONS 44 1. Equipment Set-up 44 a. Analog Tape Recorder 44 b. Filters 44 c. Analog Patchboard 44 d. Logic Patchboard 46 e. Oscilloscope 46 2.- Energizing the Ci-5000 Computer -- -< 46 3. Energizing XDS-9300 46 4. Mounting Magnetic Tapes 5^2 5. Variable Tape Digitizing Parameters — < — •■ 56 D. CONVERT PROGRAM * 57 E. NAVAL POSTGRADUATE SCHOOL PSD COMPUTER PROGRAM - 6l 1. Fourier Coefficient Program UBCFTOR " 63 2. Spectral Analysis Program UBCSCOR — < 64 P. IBM 360/67 TAPE OPERATIONS * 66 1. Job Control Language « 66 2. Multi-File Tape Operations — ^~^.-^^__ 68 3. Multi-Volume Tapes Operations « — ^ 69 G. PREPARATION OF CARDS AND TAPES FOR PSD ANALYSIS- 69 1. JCL Cards for FTOR 69 2. Modification to FTOR 71 3. JCL for SCOR 72 4. JCL Cards for FCPLT 73 5. Suggestions for Efficient Tape Processing - 74 a. Program Submission 76 b. Stacking Programs — 77. IV. EXPERIMENTAL PROCEDURE 78 A. ANALYSIS OF PURE SIGNALS 78 1. Computer Genreated Digital Sine Function — 78 2. Computer Generated Digital Random Signal — 79 B. A/D CONVERSION AND PSD OF LABORATORY SIGNALS — 79 1. Random Signal 80 a. Single Channel Digitization 80 b. Dual Channer Digitization 80 2. Random Signal and Sine Signal -8l C. DATA FROM GEOPHYSICAL SIGNALS — wr™^™ 82 V. ANALYSIS OF RESULTS — ~ w™ ~, 83 A. PSD OF COMPUTER GENERATED SIGNALS 83 1. Sine Wave -T-< 83 2. Gaussian Noise '83 B. PSD OF LABORATORY SIGNALS — — 86 1. Single Channel Sine -< 86 a. Signal Leakage into Open Amplifier 86 b. Effect of Increasing Signal Amplitude- 90 2. Single Channel Gaussian Signals 93 3. Two Channel PSD of Gaussian Signals 95 C. PSD OF TURBULENCE SIGNALS 95 1. General Signal Characteristics Found; Com- parison with Presious Results 95 a. Temperature Signal 95 b. Differentiated Temperature Signal 98 c. Velocity Signal 101 2. New Results Obtained 104 a. Temporal Variations in the PSD 104 b. PSD Values for Five Minutes of Data — 111 VI. CONCLUSIONS AND RECOMMENDATIONS 117 A. CONCLUSIONS . 117 1. PSD Programs 117 2. Analog-to-Digital Conversion 117 3. PSD Analysis Procedures 118 4. PSD Analysis of Turbulence Signals 118 B. RECOMMENDATIONS FOR FUTURE WORK -™ 119 APPENDIX A. Program to Generate Digital 10 Hz Sine Wave Samples and Write onto 9-Track Tape 120 APPENDIX B. Program to Generate Random Signal and Write onto 9-Track Tape •»— . _^-^-^^~^-^_, 121 APPENDIX C. Equipment Specifications 122 APPENDIX D. PSD Yalues from Computer Generated 1Q Hz Sine Wave . 123 APPENDIX E. PSD Values from Computer Generated Random Signal. Sampling Rate=5Q0Q SPS 124 APPENDIX P. PSD Values from Computer Generated Random Signal. Sampling Rate=1000 SPS 125 APPENDIX G. PSD Values for Signal Leakage into Open Amplifier — — — . __^_^._..„ „ , 126 APPENDIX H. PSD Values for Signal Leakage into Open Amplifier. Increased Signal Amplitude 127 APPENDIX I. PSD Values for a Real 1000 Hz Sine Wave Amplitude ±20 volts 128 APPENDIX J. PSD Values for a Real 1000 Hz Sine Wave Amplitude ±30 volts 129 APPENDIX K. PSD Values for Real Gaussian Signal; Elgenco Signal Generator 130 APPENDIX L. PSD Values for Feal Gaussian Signal; Ci 5000 Random Signal Generator (HF) 131 APPENDIX M. PSD Values for Temperature Signal: 57.12 seconds of 203(1) E1(A) 132 APPENDIX N. PSD Values for NPS Analysis of Differentiated Temperature Signal: 5 Minutes of 203(1) El'- 133 APPENDIX 0. PSD Values for NPS Analysys of Velocity Signal: 51.2 Seconds of 203(1) U(A) 134 APPENDIX P. PSD Values for NPS Analysis of Differentiated Velocity Signal: 5 Minutes of 203(1) U 135 APPENDIX Q. PSD Values for NPS Analysis of Temperature Signal: 306 Seconds of 203(1) U 136 APPENDIX R. Temperature and Velocity Signals: Boston 203(1) El, El', U and U' < < 137 REFERENCES * _„™-™™,._„™_„.-.^-*™-, 14 2 BIBLIOGRAPHY ™ m__^_m__„ _.— 143 INITIAL DISTRIBUTION LIST , — 144 FORM DD 1473 < 146 LIST OF FIGURES 1. Analog and Digital Signals 14 2. Fourier Transformations — * — * 18 3. Flow Chart Showing Fourier Transform Procedure for Use with Discrete Data 20 4. Effect of Truncation of Sine Function 23 5. Characteristic PSD Plots 25 6. Power Spectrum of I Volt 60 Hz Sine Signal • 28 7. Power Spectrum of 1 Volt 60 Hz Sine Signal Analysis Procedure 28 8. Hybrid Computer Facility 32 9a. Ci 5000 Patchboards and Keyboard 34 9b. Ci 5000 Keyboard and Power on Switch 35 10. Ci 5000 Keyboard ■ 37 11. Digitizing Facility 38 12. Block Diagram of Analog-to-Digital Conversion 40 13. Teletype 4l 14. Typical Teletype Inputs for Analog-to-Digital Program Control . 43 15. Ana'log Patchboard Used for the Analog-to-Digital Con- version of Electrical Signals 45 16. Ci 5000 Oscilloscope 47 17. Ci 5000 Logic Board and Operating Switches 48 18. XDS 9300 Control Console ,™-^ , :' 50 19. XDS 9300 Card Reader _m_^_m „-„,. _^— 51 20. 7-Track Magnetic Tape Drive Unit *—r*-, — 53 21. 7-Track Tape Operating Dials and Buttons 55 22. 7-Track Tape Formatting for Single and Dual Channel Digitization 58 23. 9-Track Tape Formatting for FTOR Program — .-^^^-^ 59 24. Convert Flow Chart ——- ^_^^_^^^.^-^_^^_, 60 25. Schematic Diagram of Complete Digitization and PSD Analysis Procedure -— — •■« — -■ — - — *«->■< 62 26. Program Sequencing and JCL Cards Needed in PSD Analysis 75 27. PSD Obtained from Computer-Generated 10 Hz Sine Wave - 84 28. Spectral Plot of Computer-Generated Random Signal (Sampling Rate Equals 5000 SPS) 85 29. Spectral Plot of Computer-Generated Random Signal U000 SPS) ' 87 30. Signal Leakage into Open Amplifier 88 31. Two Channel Digitization of Random Noise and 1000 Hz Sine 91 32. Spectrum of Gaussian Signal with Analog Filter Set at 2.0KHz Low-Pass Max Flat 94 33- Spectral Plot of Ci 5000 Random Noise Generator 96 34. Comparison of Magnitude of Temperature PSD Results Obtained at UBC and NPS 97 35. Comparison of Slopes of Temperature PSD 99 36. Comparison of NPS Analysis of El' with UBC Analysis — 100 37. Comparison of UBC and NPS Analysis of differentiated Temperature. 102 38. Comparison of Magnitude of Velocity PSD Results Ob- tained at UBC and at NPS 103 39. Comparison of Slopes of Velocity PSD 105 40. NPS Analysis of Differentiated Velocity Signal 106 41. Ten Second Brush Record of Temperature Fluctuation 108 42. Temporal PSD Variations of Atmospheric Temperature Signal 109 43. Effect of Increasing Number of Records in PSD Analysis of Temperature Signal : 110 44 . Temporal PSD Variations of Atmospheric Wind Velocity Signal — ~ 112 45. Effect of Increasing Number of Records in PSD Analysis of Velocity Signal — - — 113 46. Comparison of 56 Seconds and 5 Minutes of Temperature Signal — 114 47. PSD for 600 Blocks (5 Minutes) of Velocity Signal 116 10 ACKNOWLEDGEMENT The author would like to acknowledge the invaluable advice on all phases of this research offered by Dr. Noel E. J. Boston. Special thanks are offered to Mr. Robert Limes for advice on the Hybrid Computer System, and Miss Sharon Raney for assistance in programming and in IBM tape utilization procedures. Finally, I wish to express appreciation to my wife for her patience with long hours- of typing and checking the manuscript. J.1 I. INTRODUCTION A. PROBLEM Random processes are often recorded in a fluctuating yol- tage in analog form. However, it is frequently more convenient to analyze the signal digitally (on large digital computers). Thus, investigators are faced with the problem of converting a continuous signal into discrete data samples, and with sub- sequent analysis of these digitized samples. There are several steps in the analog-to-digital conversion procedure, and in the digital analysis procedure, which allow for errors and for possible contamination of a signal with noise from external, sources. B. OBJECTIVE The main goal of this study was to investigate digitization and analysis procedures for possible sources of noise which may be introduced to the true signals. The overall procedure used at the Naval Postgraduate School, from digitaization of the actual signal to the computation of the Power-Spectral- Density (PSD), is quite complex and requires four computer programs. The next objective was to improve the routine procedures required in this particular time series analysis technique. The final objective was to digitize actual geophysical signals and compare the results with similar analyses of these signals undertaken at the University of British Columbia, by Boston in 1970 [Ref. 1]. 12 II. THEORY A. DIGITAL REPRESENTATION OF CONTINUOUS, TIME-VARYING SIGNAL Random geophysical processes are often studied by re- cording a continually changing event as a continuous, fluctuat- ing voltage, which corresponds linearly with the original process. The actual geophysical variables considered later in this study were small-scale fluctuations of air temperature, wind velocity, and time derivatives of both. 1. Analog-to-Digital Conversion A randomly fluctuating voltage signal might look like the one in Figure 1(a). In order to analyze the signal by digital techniques, discrete samples of the fluctuating voltage must be1 taken, and these are referred to as sequential digital samples (vi). The requirement for sampling at equal intervals of time is set by the assumption, in most analyses, that they are equal time interval samples. The digitized samples would look like Figure 1(b). 2 . , Digital Sampling Theory Implied, in the digitization problem, is ttie question of how well the sequence represents the original voltage. In turning to Sampling Theory for the answer, three hypotheses are made: a. V(t) is a random variable defined for -°° > i > > t > « i > i i i ■■— * • i h* b) Digital representation of analog signal Figure 1. Analog and Digital Signals 14 c. G(f) equals zero for frequencies equal to, or greater than B. B is the highest frequecy which can be re- solved and f is defined as frequency. Further, if we let At be the sampling interval in seconds, so that At *, <* K <4 ^ c) Complex periodic Signal d) Fourier coefficients of transformed signal Figure 2. Fourier Transformations 18 This transformation is shown graphically in Figure 2b). A complex, almost periodic signal composed of several sine waves of differing amplitudes and differing frequencies would be similarly transformed into the frequency-domain, as shown in Figure 2d) . 2. Fourier Transform of Discrete Data Signal When dealing with digitized signals or discrete data the finite form of the Fourier-Transform must be used: N-l -j2TTfiAt V(fK) .= AtE vi e _oo f - >' Co*^pu-Ve s\« e Co%© ' -u; to«,© ' I Terms <^^-\X , T -r i - i jl ' r A- A.-V \ STOP Figure 3- Flow Chart Showing Fourier Transform Procedure for use with Discrete Data 20 The number of calculations required increases as the square of the number of data points increases. The integral trans- form, in the past, was limited to studying theoretical functions, such as sine waves and square waves, which were relatively well-behaved functions. For cases such as these, even the discrete transform could be used to readily obtain the Fourier-transform of a signal. With the introduction of the Fast Fourier Transform (FFT), algorithm, computation time has been greatly reduced. 3 . Fourier Transform of Truncated Continuous wave Form If a signal v(t) exists only for the time interval from 0 to T seconds, and is zero at all other times, its Fourier-transform is: T -j2irft V(f) = / v(t)e dt (5) 0 If v(t) is repeated in intervals of period T seconds, the frequency difference between coefficients will be 1/T Hz. The Kth coefficient will be: T - j 2TT^t VK =i / v(t) e T dt (oa) and for the discrete case: ; N/2 -J2ir-|i V = L.Z v* e T Cob) K T i=1 1 4 . Convolution of Continuous Signals The finite-transform of a finite-length time series can be viewed as the product of a finite-length rectangular function C (t), times an infinitely long-time history v(t). The finite transform of v(t) becomes: T -j2TTft V(f) = / CT/?(t) v(t) e dt (7) 0 ' - . ■ 21 Since products transorm into convolutions, the convoluted Fourier-transform becomes Vc(f) = V(f)x(JT/2(f) (8) where T . -j2irft C|T/2 = / CT/2(t) e dt (9) Figure 4(c) shows the Fourier-transformation of the rectangular function and Figure 4(d) shows the convolution of a sine wave with the rectangular function. This convolution problem arises when switches are opened and closed while recording the data. The side lobes of the convolved function can be minimized by allowing the time length of the record to be sufficiently large. This decreases the interval 0 to 1/T,. C. POWER SPECTRAL DENSITY FUNCTION 1 . Methods of Computing Power Spectral Density There are three methods which may be used to compute the power spectral density of a signal. They are: a. Direct Fourier Transform Method The Fourier transform of the signal is computed and from this the mean value squared is determined. b. Analog or Bandpass Filter Method The signal is put through a bank of bandpass filters and each filter output is squared and integrated. c. Auto-correlation Function Method The Auto-correlation function of the time series is computed, and then its Fourier-transform is computed. 22 a) Rectangular function (WW b) Truncated sine function c) Fourier transform of rectangular function ^=7" d) Fourier transform of truncated sine function Figure 4. Effect of Truncation of Sine Function 23 Historically, the direct method was developed first, but could be used only on nicely behayed theoretical signals. When less well behaved signals where analyzed, smooth (deterministic) theoretical functions were replaced by discrete data points representing the signal. This necessitated using the discrete Fourier-transform; however, for large data-sets, the time for calculating the Fourier-transform was prohibitive. The computer program FTOR utilized a variation of the direct method employing the Fast Fourier Transform (FFT). Using the sampling rates 1000-5000 samp/sec, (SPS), and having the ability to select the block sample length and the number of blocks desired for a particular run, no pro- blems were encountered in which filler data, usually zeros, had to be inserted into a. block. The block size was always chosen as an integral power of two. 2 . Typical Power Spectral Density Functions The Power Spectral Density (PSD) plot for random data shows the distribution of electrical power within the signal as a function of frequency. Several characteristic PSD plots are encountered. Figure 5(a) is a PSD plot of a pure sine wave. It gives the Dirac-delta function which implies that the power at the sine frequency is infinitely large, and zero at all other frequencies. Figure 5(a) is a PSD of a Gaussian' random signal. The PSD of this signal is constant. Figure 5(c) shows the PSD of a random' signal carrying a sine function on it. The PSD plot is the sum of the power spectra of the random signal and sine figured separately. 24 PSD (voW/vw^ a) Sine Wave F5D l*o\WV«0 b) Gaussian Noise ^■rtc\oe.ACv| |^\A^ 4ree«M*(ltt-^ ■Weq^e^M^5' c) Gaussian Signal Plus Sine Wave Figure 5. Characteristic PSD Plots 25 3. Problem of Single-Sided Spectrum The Fourier coefficients under the transformation -J2itft' Vlf.) =/_«> v(t)e dt (10) exist for both positive and negative frequencies. The co- efficients can be transferred to a single-sided frequency plot by simply doubling each coefficient as it is plotted only a- long the positive frequency axis. If S(f) represents the two- sided spectral density function and G(f) represents the single- sided spectral density function, the single-sided spectral density function equals twice the double-sided density function G(f)=2S(f). Likewise, if a watt meter was used to find the power in a signal as a function of frequency, the values ob- tained would have to be divided by two before plotting on a two-sided spectrum. 4 . Power and Energy Signals The average ■ electrical power in a fluctuating voltage signal v(t) is given by P = lim _l/a |v(t)|2 dt (11) a-*-* 2 a Mix [Ref. 3] defined a power signal as one which is, for all practical purposes, infinitely continuous. Energy signals are pulse-like in form and are given by: E = /" |v(t)|2 dt . (12) — 00 26 The power in a periodic signal is shown from Parseval's Theorem to be: P = i^;+T ivct)i2 dt (13) Thus, the power in a continuous sine function, v(t)=sin 377t, would be P =_1 /T Sin2 377t dt = 1/2 watt (14) T 0 If a real wide band signal were to be analyzed by using a tun- able filter of band width of 6 f , a plot of power versus frequency as shown in Figure 6 might result. Here, the real power from zero to an upper frequency was determined, divided by two and the values folded back into negative frequencies. 5. Power from the Fourier Coefficients Parseval's Theorem leads to the definition of power in terms of the Fourier coefficients p = ^/*:+T Mt)i2 dt = i jvKiz d5) Thus, knowing v(t), the average power can be computed. The average power in a one volt, 60Hz sine wave is: v(t) = Sin2 60t P = i/^ Sin2 377t dt = 1/2 watts (16) TO Using the Fourier coefficients: V.-l = 1/2 V-l = 1/2 VK>1 = 0 . P = Z |VJ2 = 1/4 + 1/4 = 1/2 (17) K=-«> K 27 Power (watts) Figure 6. Two -Sided Power Spectrum of Wide Band Signal Power (watts) iVa TV4 Figure 7. Power Spectrum of 1 volt 60 Hz Sine Signal 28 If a watt meter was used to check the power of this sine signal, one-half watt would be found at 60Hz, and zero watts at other frequencies. If a plot of power versus fre- quency were made, one similar to Figure 7 would result. Since the values have been plotted for positive and negative fre- quencies, the total power has been reduced by one-half and the values have been plotted. D. FAST FOURIER TRANSFORM 1 . Computational Economy Afforded, by FFT The Fast Fourier Transform (FFT) algorithm is a fast method for computing the finite Fourier transform of a series of N data points. The FFT computation requires N log? N computations, which leads to a substantial saving in 2 computer time over the old method which required N operations. The FFT economy of computational effort is greatest for large values of N. Reference [4] gives the general computational routine for figuring the FFT. It is pointed out that the FFT is general in that N is not necessarily a power of two; however, by selecting N to be a power of two, further computa- tional savings result. When this requirement is met, the FFT algorithm is essentially a successive doubling operation. Thus for N=2J , only Nj multiply-add operations would be re- quired under the FFT assuming the necessary complex exponential table of values has been computed in advance. 29 2. Importance of FFT in Turbulence Analysis Computationally, the FFT is an indispensable aid in the PSD investigation of turbulence. Due to the extremely high sampling rates necessary for studying high wave number processes, enormous amounts of data are collected. A five minute section of a fluctuating temperature signal sampled at a rate of 4000 samples/second (SPS) generated almost a million and a half data samples. Using the old method for computing the Fourier transform for this quantity of data, several thousand billion operations would be required. Clearly the time element in this procedure would prohibit the calculation on any but the fastest computers. Turning to the FFT the operation would only take about 21 million operations or an improvement in excess of 5 orders of magnitude in computational time. The FFT thus brings the study of turbulence signals within the bounds of computational feasibility . The ability to determine the power spectral density function which is computed from the Fourier-transform and which is such an important aspect of turbulence analysis is made possible through the use of the FFT algorithm. 30 III. NAVAL POSTGRADUATE SCHOOL DIGITIZATION FACILITY AND SPECTRAL ANALYSIS PROGRAMS A. NPS COMPUTER FACILITIES FOR DIGITIZATION AND PSD ANALYSIS The Naval Postgraduate School has two sophisticated computer systems which are used in the digitization and PSD analysis procedures. One is a Hybrid system which consists of an analog computer, COMCOR Ci 5000, which is electrically inter- faced with a digital computer, XDS 9300. The XDS 9300 con- tains 3^K bytes of core storage and uses an octal number base. This base consists of binary numbers made up of 3-bit digits. The second system is an IBM 360/67 which has a core storage of 762K bytes. It uses a hexidecimal number base which consists of binary numbers of 4-bit digits. 1 . Hybrid Computer The Hybrid computer facility is located on the fifth floor of Spangel Hall. The equipment disposition is shown in Figure 8. Though the individual user performs all computer operations, technical assistance is availabe from the Electrical Engineering Computer Laboratory staff. This facility is used for the analog-to-digital- conversion of signals. ....... * 2. IBM 360/67 Computer The large, digital computation facility is located on the ground floor of Ingersol Hall. Computer Center personnel handle all computer operations. Individual programs are input 31 ■p •H rH •H a fc CD ■P E o o T5 •H JO oo (1) in bO •H 32 to the computer through a card reader and requests for the subsequent input of programs, tapes, disk memory, etc. are submitted "over the counter" to computer personnel. A 24- hour operating schedule is usually maintained. All PSD analyses of digital tapes were performed on this computer. B. HYBRID COMPUTER SYSTEM Computer time on the Hybrid computer system is signed for in advance in the Electrical Engineering Computer Laboratory. The whole system is operated on a self-service basis; however, the computer center staff is available to assist during working hours. ■ Computer time was easiest to obtain early in the term or between terms when the work-load was lightest . The facility is available 24 hours a day and as one becomes more proficient in equipment operations, night or week-end operations can be scheduled if week-day operations are precluded 1. The Analog Computer (Ci 5000) The Analog Computer contains the input points for raw signals input. It also has control features for signal amplification, selection of sampling rate and starting and stopping of the digitizing procedure. The two removable patch boards are the heart of the system. Individual patchboards have been reserved solely for analog-to-digital conversion (board #24). 33 T3 Ih cd O £1 >3 ^; c cd m t3 Jh cd o £) x: o -p cd o o o in o cd CTn cd •H 34 o -p •H m c o 0) o c o .a >i (D o o o m o X3 ■H 3 3 The logic board, the smaller of the two patchboards, was utilized to, set the sampling rate of the analog computer. The logic board with its patch is shown in Figures 9 and 17. The sampling rate was changed by turning a counter-like number display, below and to the left of the logic board, as shown in Figure 17. The counter operated a voltage divider which was electrically connected to a resistance input on the logic board. The resistance value at this point was changed by inserting a resister in one of 4 positions. The resistance value was then divided by the counter setting plus one to give the sampling rate (in samples per second) desired. Example: lOOkc = 4Kc or 4000 samples per second. (24+1) No external leads are connected to the logic board. . The analog patchboard is the input point for all external signals entering the analog computer. The input signal from a tape recorder or signal generator, after passing through signal conditioning equipment is input to one of the analog board amplifiers. (Figure 15 shows inputs going into amplifier A001). Various gain factors can be applied to the input signal to bring it up to a value optimum for utilization of the analog computers dynamic range. Figure 10 shows the keyboard of the analog computer which is used to control the desired operational mode of the analog computer. . ... x 2. Digital Computer QCSD 9300) The heart of the digital computer system is the Xerox Data System 9300 Central Processor Unit (CPU). Two tape drive units, line printer, card reader and teletype unit are inter- faced with the CPU. Figure 11 shows, diagrammatically , the 36 T3 £h O CD « o o o LO •H o o in •H 37 38 equipment used in digitization. The tape driye units are used for data input from tape to computer or for data out- put from computer to tape; the line printer can be used for program listings and printing results; the card reader as pro- gram input for compiling programs; the teletype unit for input of parameters necessary to control the "multi-channel analog- to-digital Program." The CONTROL CONSOLE of the XDS 9300 is used to compile and to initialize the program. After the pro- gram has been compiled, the teletype is used to select one of seven digitizing options. The digital and analog computers are connected through the Analog-to-Digital Converter (ADC), (Fig. 12). 3 . Multi-Channel Analog-to-Digital Program Several programs have been developed by the computer laboratory staff to control the XDS 9300 system during multi- channel digitizing operations. A card deck of the latest re- vision of the program is maintained in the computer lab. This program can be input from either cards or from a tape which has had the machine language program stored on it . The com- piling sequence takes about five minutes using the card input and only about 30 seconds using the tape input. Once the program has been compiled, one of the seven Program Control Options can be selected by typing one of the , options followed by carriage RETURN (C/R) on the teletype (see Figure 13.)- The Program. Control Options are: 1. Enter new parameters. (NSAMP, NCHAN, NREC, ITAPE) 2. Start digitizing the analog input signals (Digitization actually starts when manual switch DSI on Ci 5000 is thrown to up position) . 39 UPL UMjT F\lTfcR C;5ooo ADC UU\T XDSWO CPU TTRACK TM>£ UK\T Figure 12. Block Diagram of Analog-to-Digital Conversion 40 Figure 13- Teletype 41 3. Write End-of-File on tape. 4. Rewind tape to load point. 5. Skip files (the number of files to be skipped specified by teletype input of a four digit number, i.e. skip five files 000 5 as shown in Figure 14). 6. Print digitized data (the number of lines to be printed specified by teletype input of a four digit number, i.e. print ten lines of digitized data 0010. 7. Actuate the Digital-to-Analog subroutine for the next single block of data encountered and input the data into the strip chart recorder (a subsequent computer comment types "start strip recorder and type C/R - carriage return"). The strip recorder must first be connected to the digital-to-analog output on the analog patchboard on the Ci 5000. After the multi-channel digitization program has been compiled and the input light on the teletype is on, tape parameters input through the teletype are: NREC - ' sets the maximum number of records to be digitized (see Figure 14). ' NSAMP - sets the number of digitized samples contained in each record on the seven-track tape ( see Figure 14 ) . ITAPE - sets the magnetic tape unit into which the ' digitized samples are sent. This number must be the same as the dial number selected on the front of the tape drive unit (see Figure 14) NCHAN - sets the number of analog channels to be digitized (see Figure 14). 42 a} -p •H hO •H Q I o -p I hO O i— I CC C < O H c cd O c o < m 0) hD •H 45 The outputs of each amplifier go to both the oscilloscope input points and the ADC input points (T500 and T501). If an analog playback of the digitized signal is desired, inputs from T420 and T^21 (see Figure 15) must be made into the recorder inputs (1-8). d. Logic Patchboard The patch for the logic board is shown in Figure 17. This is for continuous mode digitizing in which digitization is started by throwing switch DS1 . The only logic setting necessary is the selection of the required resistance value (Figure 17) which in conjuction with the wheel counter sets the sampling rate. e. Oscilloscope The "scale illumination" dial energizes the oscil- loscope (see Figure 16). The sampling pulse was usually patched into "channel 1" by throwing switch Yl to the left. This permitted continuous monitering of the sample pulse fre- quency and width. Dial DFOO (see Figure 17) should be set to Ms. 1 and the width of the sample pulse adjusted with this dial. 2 . t Energizing the Ci 5000 Computer If the power on the CI 5000 has been secured (see power light in Figure 9b) it can be turned on by depressing the "on" switch. Next, the buttons KEYBOARD, POTSET and RESET are depressed in that order (see Figure 10). 3. Energizing the JSDS 9300 The step by step procedure below should be followed in energizing the Hybrid computer system. The latest update to this procedure is kept on the XDS 9300 Control Console. 46 o o m O o ra o o o o in •H o U3 CD •H 47 Figure 17. Ci 5000 Logic Board and Operating Switches 48 a. XDS 9300 Power on (if power is off), see Figure 18 1) Depress IDLE 2) With RESET depressed press POWER 3) Turn teletype switch to ON 4) Press CLEAR and CLEAR FLAGS simultaneously b. Energizing the line Printer (if the READY light isn't on) 1) Insure that POWER button on line printer is on. If not, press POWER button. 2) When lower half of POWER is lit (within one min. after POWER pressed) press READY, NOTE: READY must be off to ad- vance paper. A whole page is advanced with TOP of FORM and the paper is advanced a single line with SINGLE SPACE. c. Card Reader (if A/D program being input from cards) 1) Place BOOT card in front of the JOB card in the A/D deck. 2) Put cards into card reader (see Figure 19) face down, top outward (toward you) so that column 1 is first into reader. 3) Put card reader press in order: POWER and START . d.. Program Compile and Load 1) Ready the Line printer (sequence b above). 2) With cards in place ready the card reader (sequence c above). 3) At 9300 console turn off (by depressing) all SENSE switches which are lit . 4) Press in order: IDLE; RESET: RUN: CARDS. 5) A JOB is printed out by the teletype indicat- ing compiling has begun. "' ^9 CD r-\ O W C o o o u -p c o o o o on co Q oo bO •H 50 u CD t3 cti CQ cd cd «H P hO C •H +3 cd (D o CD cd Eh M o cd !h Eh I H c\J CD !m hO •H Eh 55 5. Variable Tape Digitizing Parameters Once the energizing and sampling sequences have been completed (the INPUT light on the teletype should be on), the tape options are entered through the teletype by typing a single digit number and then depressing the CARRIAGE RETURN., C/R button. In order to input new parameters, "1 C/R" would be typed. Then the following example parameters could be typed in: NSAMP = 2048 (C/R) NREC = 100 (C/R) NCHAN = 2, ITAPE = 1, NDEL = 2000 (C/R) The computer would then type OPTION (II), soliciting a further response from the user. If analog-to-digital collection was to commence, the programmer would type "2 (C/R)." If the Analog computer switch DS0 was in the UP position, digitization would begin when DSI was thrown to the UP position. As would be expected, a high sampling rate would fill up 100 records faster than a lower sampling rate. Thus, if a sampling rate of 2000 SPS had been selected, and the above tape parameters had been selected, the digitization would result in 204, 800 samples being collected and a total signal length of 102.4 seconds. 2048 SAMPLES x ioo RECORDS = 204,800 SAMPLES 204,800 SAMPLES , e-1 ? nr?nrmuc, 2000 SAMPLES/SEC. x 2 CHAN 0± ' d ^ouMJb 5'6 There would only be 102,400 samples of each channel; however, there were two channels being digitized simultaneously. Figure 22 shows how digitized data for one and two channel digitization would be formatted on a seven-track tape. D. CONVERT PROGRAM The CONVERT PROGRAM converts the seven-track octal base data samples into nine-track hexidecimal samples. The basic character of the data samples being in blocks on the magnetic tape is maintained; however, two additional parameters (see Figure 23) are affixed to the beginning of each block by the CONVERT program. These are the numerical values KMAX which is set equal to the maximum number of samples per block and NCHAN, the number of channels of analog data digitized on the seven-track tape. This change in the block format, the change in number base and the addition of two new parameters is necessary to put the tape in the proper format for input into the FTOR program. Figure 24 shows the CONVERT procedure in flow-chart form. The program processes all of the records in a file and the number of files to be converted is specified in the IF state- ment which compares the number of files completed with the number specified for processing. The number in parenthesis is set to the number of files to be converted. The JOB CONTROL LANGUAGE (JCL) cards which follow the CONVERT PROGRAM specify which files on a multi-file tape are to be processed. Jones [Ref. 5 Appendix D] gives the typical JCL cards used to con- vert five files. 57 single channel input All data from single channel digitization is storred secuentially one sample following next. I I I l I 1 I I ■■>■ _ . a) Single Channel Tape Formatting per Record (N=23) - 3 Records with 8 samples two channe incut Data from two channels digitized simultaneousl from alternate channel Number represents chan number of data sample. z 12 \i \ Zl 3 RECORDS OWE FILE t b) Dual Channel Tape Formatting - Records with 8 samples per Record y Figure 22. 7-Track Tape Formatting for Single and Dual Channel Digitization 58 7-TRACK OCTAL TAPE 1 1 1 1 -!• 1 1 1111 CONVERT inn n i - 7— Ik 11 1,1 TT71 11 T2T58 9-TRACK HEXIDECIMAL TAPE Figure 23. 9-Track Tape Formatting for FTOR Program 59 '31 DIPENSICm IOaTC2C6=),BiT(2045) ! REuI '.D 2 REWIND 4 IEND=Q FACTQR=100. 0/(2**23) lrecl=2048 NCHA\=1 KMAX=2048 J=n * st A0 ' 2 , 3E \D= 4 C1 , E P 9= 5 0 ) I -)* t FOR FAT (32 (6 4^4) ) / i J = J + 1 / / i CALL rORfl"(iDAT.XRt:CL) .70)J 1 1 ' ,10X, RECORD N0.= ,15) ,7OJ- ' ,10X, RECORD N0.= ,15) ,66)(dat(l),I=1 ,2048) 1X.8E16.8) -*3>.51 F0RPrAT(x0',5X,READ ERROR, RECORD NO. = ', 15 ) 40|uiRITE(6,4i ) J--|>4,1 FORFAT( v 0 ' , 5X, END OF TAPE, RECORD N0.=',I5) 1 — END FILE 4 I IEND=IEND+1 IEND samples. This restriction was imposed by PKFORT, the subroutine which computes the FFT . The details of this subroutine are listed under PKFORT in the computer library. For optimal efficiency PKFORT has been designed to read in data blocks which con- tain an integral power of two (i.e. 2^ ) samples. PKFORT is then called and the Fourier transform is computed, which re- sults in 2^~ Fourier coefficients. Since only one block of data is analyzed at a time, the original signal is divided into short, sequential sections of signal and a new length of record T transpires. The resulting coefficients are stored on another magnetic tape and they become the basis for the 63 spectra. For this block of data, the highest frequency for which a coefficient has been computed is the Nyquist fre- quency fs/2, where fs is the sampling frequency. The lowest frequency which can be resolyed for this data block is 1/T or 2^/fs. The Fourier coefficients are written into blocks on a "coefficient" tape which also contains identifying information such as the block number, sampling frequency, number of samples in each block, etc. Upon completion of the transformation and writing sequence for one block, the next block of data is read from the digitized tape. The sequence is repeated for as many blocks as specified on the input data cards, until an end-of- file mark is sensed on the digitized tape or until a blank card is . encountered in the input data cards. 2. Spectral Analysis Program - SCOR' Once the Fourier coefficients have been computed, the next step is to convert these values into spectral values. The program SCOR reads the coefficient tape generated by FTOR and averages the PSD values over 32 bandwidths.. The data card section of the SCOR program specifies the number of channels, the number of blocks to be analyzed within a file, the block number where the analysis is to begin, whether the spectra for a single channel or the cross-spectra between channels is to be computed, the type of bandwidth desired (constant or logarithmic), etc. Just as with FTOR procedures, SCOR reads and analyzes only one block of coefficients at a time . 64 If smoothing of the coefficients is desired, various smoothing functions may be selected. The "harming" option performs a three point running average on the data with weights of -1/4, 1/2, -1/4. References 2 and 6, contain further de- tails of the smoothing functions. If individual Fourier coefficients of two channels are 1/2 R-, + il, and R + ilp, where i= (-1) and x equals the block length in seconds ( and 6F=l/x where 1 ...... File for which conversion is desired. 9-Track Digitized Data Tape FTOR PRUCRAM DEC" 0.FT)BF00O DD UN I T* ?4nn,uTIL = SE R = Nf ,1 t'S , DI 5P = 0Ln LAbEL-(l Si) OSN*("CKEni,DCB=(OEN=?,RECFC=;S,SLKSIZE = ??n«) ' Additional JCL cards For eac- rilp on NPSiiS Because of SKIPFILE routing :l card 'or each file on mPSlPS must De included. O.FT03F001 DD UNI T=2400,\/OL = SER=:\PS2??. ,DI 3P=(tli£UI KEFP) LABEL-M Sl^ DSN=C0Er-,DC8«(DEN=2,RECFM=y3,-LK5IZi> :CA) ' ' Additional JCL for each file to oe written on NPS223. O.SYSIN DD • Four CONTROL CARDS for each file ror which Fourier coefficients are to be computed. ( BLANK CARD TERMINATES COMPUTATIONS ) 9-Track Fourier Coefficient Tape //FORT.SVSIN DD UNI T= 2400, V0L=SERrNPS21 6 , DI SP=0LD. L AbEL* f 2. SL ) // DSN=UBCSC0R //G0.FT03F001 DD UNIT=2400,U0L = SER=NPS1B5, DISP= (OLD , KEEP ) LABEL=(1 // DSN=C0EF,DC8=(DEN=2,RECFMsyS,6LKSIZE=e204) Additional JCL cards for each file on NPS18S. Because of SKIPFILE routine JCL card for every must be Included. //GO.SYSIN 0D • ...POWER SPECTRUM PLOT OF TAPE 001 -----------Three cards minimum for which spectrum is desired ( BLANK CARD TERMINATES /• file on NPS1P5 j.d.mekendri; each file for- K. .GCEAN0. , :0MPJTATI0NS ) PSD G^iplns 1>5D Valuts Figure 26. Program Sequencing and JCL Cards Needed for PSD Analysis 75 a. Program Submission Submission The best time for processing digital tapes and debugging particular problems was found to be during the summer and winter break when the work load at the computing center was very light. During that time, generally, about ten runs per day, depending on the length of the JOB, could be achieved. If all was going well only one run per day was often adequate; however, when problems developed, an immediate debugging run was essential. It was found that oversights on the part of the programmer, in the form of errors in JCL and control cards, the "dumb" computer was only doing what it had been instructed to do. However, occassional computer malfunctions did occur. The next best time for program processing was on weekends at the beginning of the quarter. As the quarter pro- gressed, the weekend would offer several runs per day. Toward the end of the quarter, due to heavy work load, only one run1 at around 3 A.M. resulted, if the job had been input prior to 10 A.M. of the previous day. In general, turn around averaged 24 hours. The most CPU used for any program sequence was ten minutes. Due to the fact that each analysis uses different parameters, the only rule of thumb which was found useful was that the FTOR procedure took about 22.5 seconds per record (204 8 samples per record) to compute the Fourier coefficients. Thus, 400 records required 490 seconds of CPU time. The FTOR pro- gram was the most time consuming thus this figure (22.5 seconds) can be used as an upper limit in estimating CPU time for other programs . 76 b. Stacking Programs Another technique used to affect faster PSD analysis involved submitting CONVERT, ETOR and SCOR under one JOB card. The reason the programs were "stacked" was to per- mit CONVERT, FTOR and SCOR programs to run sequentially. Btherwise, it was necessary to get the results of each program before the next program could be input . If no mistakes were made in the JCL cards and if no problems existed with the tapes, the complete analysis would be made in one run. If a problem in any step was encountered, corrective action was taken and the remaining programs were run individually. The option as to when to stack the programs and when to run them individually varied with the number of files on the tapes. The higher the number of files, the more JCL and control cards required, and the greater the chance for errors. 77 IV. EXPERIMENTAL PROCEDURE The initial plan for the investigation of the noise pro- blem, involved digitization and spectral-analysis of real signals. These were to be sine waves, square waves, ramps, and random noise. Since these input-signal characteristics were readily known, the Fourier-transform and, the power sectra were known functions. The first signal, a 10HZ sine wave, did not give the expected spectral values. The digital program was checked to see if its calculations were valid and then the A/D procedure was checked. For purposes of clarity, a chronological discussion of the experimental procedure will follow. A. ANALYSIS OF PURE SIGNALS Though the first series of 10HZ sine waves showed an expected energy peak at 10HZ, the exponential decrease of energy with increasing frequency was not expected for a sine function. An assessment was made that the problem could be either in the actual A/D step or in the spectral-analysis programs. Before definite conclusions concerning this pro- blem and the noise sources could be made, it was necessary to gather baseline information on computer analysis of theoretically pure signals. 1 . Computer Generated Digital Sine Function The program listed in Appendix I was used to generate a simulated digital sine signal and the computed samples 78 were stored on a digital nine-track tape. The format of the tape was made to be compatable for input of this data to the FTOR program. In the sine generation program, the peak voltage could be varied by changing the sine amplitude, and the the sampling rate could be changed by varying At. A standard block size of 2048 samples per block was maintained throughout this study. The CONVERT procedure was not needed because the data was already in a hexidecimal format. 2 . Computer-Generated Digital, Random Signal The next step was to test the FTOR and SCOR spectral density analysis of a signal with a wide range of frequencies. To do this, a Gaussian signal, which has a flat spectral density function, was used. The program listed in Appendix II generated a simulated digital random signal, by "calling" the computer sub-routing RANDU . The peak voltage values could be changed by altering the constant multiplier of YFL; the sampling rate could be changed by altering the sampling rate specified on the FTOR input data deck. The peak amplitude was maintained at 10 volts; however, the different sampling rates were investigated. Since the data was stored on a nine- track tape with a format compatible with FTOR, the CONVERT program was not used. B. A/D CONVERSION AND PSD OF LABORATORY SIGNALS Once characteristics had been established for computer processing of pure control signals, the next step was to digitize actual signals. It was decided that a random or Gaussian signal would give all the information required, and 79 thus, the sine, ramp, and square wave signals could be by- passed. The random signal was expected to give optimum power- spectral density information for purposes of the study. 1 . Random Signal a. Single Channel Digitization An Elgenco model 603 A, Gaussian noise generator was used to give a random noise output. Its characteristics are listed in Appendix III. A frequency setting of 5Hz to 20 KHz, attenuation schale XI . 0 , output voltage reading 2.62 Vrms was input to a Khron-Hite filter model 3321 set at 2000 Hz, low pass max. flat and Odb gain. The filter output was input to the Hybrid computer through an operational amplifier with a ten volt gain. A sampling rate of 5000 SPS was selected, A total of 4l seconds of the signal was digitized onto a seven-track tape. An End-of-File mark was written onto the tape by typing the EOF option on the teletype keyboard. b. Dual Channel Digitization The second file on the tape mentioned above was filled with data from two input channels which were digitized simultaneously. The Elgenco noise generator described above was used as one input, and the other input was from the random- noise generator which is built into the Ci 5000. The noise-generator was utilized with the same settings as above; however, the filter was reset to 1880Hz. The random-noise generator from the Ci 500Q was input to a Krohn-Hite filter also set at 1880 Hz. The output of this filter was fed into a different operational amplifier with 80 a gain of 10. The sampling rate was lowered to 4000 SpS. A total of 25.5 seconds of signal was digitized. Though the single and dual channel cases both considered 100 records, this shorter digitizing time occurred because, two samples were being taken, simultaneously, at the rate of 4000 SPS: 2048 Samp x 100 Blks 4000 Samp ~ = 25*5 Sec * Sec x d An End-of-File mark was again written on the tape to end this file of data. The CONVERT program as used to convert from seven- to nine-track tape. The program SCOR allowed the computation and plotting of the spectrum of each channel and the eo- spectrum and the quad-spectrum of one channel with another. 2 . Random Signal and Sine Signal To determine whether a sine wave could be picked out of the random noise, a 1000 Hz sine and random signal were digitized. An attempt to digitize a single channel of the sine combined with the random signal failed due to a faulty patch on the analog board. PSD values were obtained, seemingly because the open amplifier actually picked up stray signal. The sine amplitude was increased and the a second file was digitized. The signals in these two files were digitized at 5000 SPS and filtered at 2000 Hz. Dual channel digitized samples of the 1000 Hz sine with amplitude ±20 volts and Gaussian signals filled the third file. The amplitued was increased to ±30 volts and the two separate sine and Gassian signals digitized into the 81 fourth file. The signals in these third and fourth files were digitized at 4000 SPS and filtered at 2000 HZ. C. DATA FROM GEOPHYSICAL SIGNALS Atoraspheric-temerature and velocity signals have pre- viously been recorded on one-inch magnetic tape by Boston [Ref. 1]. These signals were used as a final check on the system to determine if correct spectral values could be achieved. The signals were reproduced on a Sangano Model 3562 FM tape recorder at 60 ips. The tape playback output was filtered at 1000 H'z for the temperature signal, and at 2000 Hz for velocity, the differentiated velocity and temperature signals. The sampling rate for the temperature-signal digitization was 2000 SPS and the other three signals were sampled at 4000 SPS. The filter setting was low-pass max, flat, Odb gain. 82 V. ANALYSIS OF RESULTS A. PSD OF COMPUTER GENERATED SIGNALS 1 . Sine Wave Figure 27 was the spectral plot of a computer- generated sine wave. The PSD values were computed for 24 re- cords of signal giving a total signal length of 11.9 seconds. The total integrated power was .499 V ; However, the power in the 8.12 Hz band centered about 7.11 Hz had a total of .495 V2 (8.13 Hz X 6.09 X 10"2V2/Hz). Essentially, all the power was contained in the band between 2.05 Hz and 11.17 Hz. Though this appeared to be a wide band, the whole region from 11.7 Hz to 263.1 Hz had less than .8 percent of the total power: •4".499495 x 100 -,8* Another test run with a 200 Hz sine wave with a peak-to-peak amplitude of 10 volts, proved inconclusive due to an error in specifying the sampling rate on the FTOR data- card input. A sampling rate of 400 SPS was specified, rather than 500 SPS, which was the actual rate used. The expected total power value of 50. V"- was achieved; however, the error with the sampling rate shifted the frequency peak from 200 Hz to 159 Hz. Though the error produced erroneous results, the effect of not specifying the correct sampling rate was observed 2 . Gaussian Noise Figure 28 was the PSD plot of the computer-generated random signal. The digital samples of the signal simulated 83 -1.0 , -2.0 -3.0- o -4.0 s CD K Eh O W Ph CO § -5-0 -6.0. -7.0 -1.0 Figure 2 7. 7 1.0 71) L0G1Q FREQUENCY f(Hz) PSD Plot obtained from Computer Generated 10 Hz Sine Wave 84 in ■H I o\ I 5 cti » P ctf -P CQ O O O I 4 © 0 C\J p C\J LTv -r* o C\J I — r • CM I 01 LA O on I CO G faO •H CO S o N T5 ffi G s-' cti Ph K >^ T3 o CD s P w cd D £h c? CD w g cc CD fc CJ o in rH CD C3 -P o d J P, £ o o <*H o •p o iH Oh Q CO CU • co C\J CD U 3 bO fe (08s-sa) ^naioads ooi 85 a random signal of 9.8 seconds in length, sampled at 5000 SPS. The spectral level was found to be very flat with increasing -3 2 frequency. Average spectral density of 3.30 X 10 V /Hz -3 2 varied from a high value of 3 .45 X 10 V /Hz to a low value of 3.25 X 10""-* V /Hz . Actual variance was very low and quite uniform. No frequency spikes were observed and the conclusion was drawn that the random number generating sub-routine RANDU produced a true random series for at least the first 50,000 numbers. It was noted that the random-number generating 39 capacity of this sub-routine is 2 numbers before the series 17 repeats itself. Since only 2 ' numbers were used, the full potential of the number generator was not fully tested. Figure 29 is the SPD plot of the same random series sampled at 1000 SPS rather than 5000 SPS. This changed the record length to 2.05 seconds, and since the results for 100 records was computed, the total length of signal sampled was 205 seconds. Although the sampling rate-change produced a higher PSD value, the mean showed little variation. The mean "2 2 ? level was about 1.57 X 10 V /Hz with a low of 1.62 X 10"^ 2 2 V /Hz and a high of 1.72 V /Hz. The variance around the mean level was very small. As expected, averaging over a fewer number of records (24 versus 100), the variance was higher. B. PSD LABORATORY SIGNALS 1 . Single Channel Sine a. Signal Leakage Into Open Amplifier Figure 30 was the PSD plot of the Spectrum of signals with peak voltages of ±2.0 and ±3.0. The signals 86 in I I o C\J I "3T CM I • ^sj on m Signal Pea Seconds , LTt O LT\ • • 'Ci O C\J erated Ran ength = 2 . >H C J o 0) s o T3 w ^ -o ^ o CD O CM cr; -P CD CO 3 cr; p., Ph p, co o O w o H O -P o o rH O o In OH t-3 o > II -P o OH (D IT\ rH -P » • p, cd cd r-i T3 K H 3 cd -P CD Spectr Amp 11 Sampl O CJN • CM r-1 o CD • £h. m d 1 bO (oas-^A) wnHioads 01 fe 87 -4.0, N OJ > O W IX, CO o -5.0. . Peak Voltage ±2 q Peak Voltage ±3 -6.0 1.0 0 $ © 0 © o© o©<3s> ^&&&%$qPs> TTo^ To" LOG _ FREQUENCY Figure 30. Signal Leakage into Open Amplifier 88 were picked up by an open input amplifier, which was next to the amplifier into which the signals were actually input. The open amplifier, whose output was being digitized, acted like an antenna in picking up these stray signals. The plots show considerable consistency and several conclusions can be made. The spectral peak of the ±3 volt signal was higher than that of the ±2 volt signals. The peak PSD was 2.68 X 10~5 2 V /Hz for the lower and 6.58 X 10"5 V2/Hz for the higher signal These values multiplied by the band width, 78. 1 Hz in both cases, gave power of 2.09 X 10"3 V2 and 5.12 X 10~3 V2 respectively. The power ratio produced by this difference in output voltage was Power Ratio = 6.56 X 10~5 2.68 X 10-3 - 2.45. Since this was a power ratio, the voltage ratio" is the square root of the power ration, or Voltage Ratio = 2.45 = I.56 The actual voltage input into the open amplifier was computed from the power of each signal. The lower power -3 2 2.09 X 10 V resulted from an input voltage of ±4.5 X 10~2V, ~3 2 and 5.12 X 10 V resulted from an input of ±7.2 X 10~2V. The expected voltage ratio was computed using the observed input voltage values: Voltage Ratio = 3.0 2.0 1,& The power ratio was; Power Ratio = (1.5)2 = 2.25 89 The ratio of the signal voltage in the input line to the actual computed signal voltage gave the actual percent of signal picked up by the open amplifier. _2 4^2^Q-1Q — X 100 = 2.3$ for the 2V signal 7,2 X 10 — X 100 = 2.4$ for the 3V signal Assuming the leakage was coming from the voltages in the input lead to the open amplifier, the percent of signal leakage would be the ratio of the voltage in the input lead to the signal voltage computed from the PSD. For the ±2 V and ±3 V signals the percent of signal picked up by the open amplifier was 2.3 percent and 2.4 percent respectively: 4.6 X 10~2 2.0 X 100 = 2.3$ 7t2 X 10~ X 100 = 2A% j . u If the signal was leaking from the closed amplifier, the leakage was .2 percent for both cases. 4.6 X 10~2 2.0 x ioo = .2: _2 7-2 x 1Q X 100 = .2% 3.0 Thus, only a small signal leakage was observed and it was independent of signal amplitude. b. Effect of Increasing Signal Amplitude Figure 31 is a PSD plot showing the effect of in- creasing the voltage of the signal going into the data taking amplifier. The input signals had peak-to-peak voltages of ±20 volts and ±30 volts. 90 1.0 8 -i-° > 6 U -p o CO , one channel 1000 Hz Sine ±16.4 volts q one channel 1000 Hz Sine ±25.2 volts b0 o H-P 0 n — fc • VJ -3.0 -4.0 1.0 Figure 31 -i — 2.0 TTcT "O L°g 10 Frequency f (Hz) Effect of Increasing Amplitude on PSD Plots of Real Sine Signals 91 The PSD for the peaks 3-92 V2/Hz and 9.29 V2/Hz were computed for a bandwith of 68.4 Hz. The power was 268V and 635V respectively. This gave a power ratio of; Power Ratio =9.29 3732 = 2.37 The voltage ratio is the square root of the power ratio or Voltage Ratio = *\ 2.37 = 1.54 Using the power spectral density to compute the o 2 voltage ratio, 268V^ implied an input of 16.4 V and 635 V rms implied an input of 25-2 V . This implied peak-to peak rms voltages of ±23. 2V and ±35. 6V. Due to the inaccuracy involved in reading peak-to-peak voltages from the oscilloscope on the Ci 5000, the observed inputs of ±20V and ±30V could have been ±5V in error. Assuming, the inputs were of ±20V and ±30V, the expected voltage ratio would have been: Voltage Ratio = 30 - ] 5 20 and the expected power ratio would have been: Power Ration = (1.50)2 = 2.25 The observed and computed power ratios compared favorable, and the difference between ovserved and computed peak-to-peak voltages ( 20V Vs. ±23. 2V and ±35. 6V) were with- in acceptable limits. The theoretical power for sine waves of ±20V and ±30V was found from the formula; 92 p where V is peak-to-peak voltage. This gave power of 200 V 2 and 450V for the two voltage signals respectively. The difference between expected power level and the power level derived from the spectral plots was assumed to be due to the error in reading the input signal amplitudes. 2 . Single Channel Gaussian Signal Figure 32 was the spectral plot of 40. 06 seconds of a random signal sampled at 5,000 SPS . The spectrum level was very flat to about 1.5 KHZ. Beyond 1.5 KHz, rapid decrease in the power spectral density with increasing frequency was to be expected. The 3db down point occurred at 2.0 KHZ. The slope of the filter, as specified in the equipment characteristics, was -48db/octave . The observed slope was very close to -96db/octave . This value would be expected if two filters had been cascaded, but this was not the case. The spectrum was quite free of noise spikes and had no 60 Hz harmonics present. If 60 Hz noise was present, its level was well below -17.5db/Hz. -2 ? The spectral level of 1.73 X 10 V /Hz compared favorably with the input signal. Specifications for the Elgenco noise generator gave a spectral density of approximately -3 5 X 10 V/ Hz at IV . The input signal had a meter read- rms ing of 2.62V . The gain factor was 10. The computed spectral & rms & density was 2.69 X 10~2V2/Hz. /5 x 1Q-3V\2 00 _? ? \ Hz / (2.6r(10r = 2.69 x 10 "TVHz Since no accurate spectral density information on the noise generation was available, these values are considered to compare favorably . 93 I O CM I — V I c N •H K cti i< O o J3 • T3 C\J O -p •\ cti -P Cti -P rH <•> . s tsl w C\J o (1) W s o s D K Eh o w CO C3 O -3.0- -4.0'- -5.0 -6.0- \ ^ © © © • , — j j 1.0 2.0 LOG FREQUENCY 10 3.0 Figure 39 Comparison of Slopes of Velocity PSD NFS Results Reduced by Factor of 340 105 J, 4- } >- LJ j "UJ ,n «ID v a UJ ~^ cr 1-4 -4 u_ n o tr m— < -;o E? -»■ - T- O 1 — o CM I — 1— O I I- oas (oss) ,n ramoaas aro ^3- 1 01 I I DOT en o CM C o ■p co o m cd c hO ■H OQ >s •P •H O O H CD > CD •4^> cd •H 4^> G CU fn CD Cm Cm •H Q Cm O co •H CO >s r-i CCi C < 00 PL. o CD hO •H 106 investigator the opportunity to relate fluctuations of the PSD with signal fluctuations. A section of a signal which appeared to have a nominal signal amplitude, as in Figure 4la had a low PSD value (as seen in the section of PSD plot marked 10.24 in Figure 42. Though the characteristics of the PSD are in- deed determined by sampling rate, the number of samples in each digitized record, filter settings, background noise, etc., meaningful results can be achieved within these limitations. Since the identity of each digitized record (digitized block of data) was maintained throughout the analysis under UBC FTOR and UBC SCOR, the PSD of sequential sections of the signal could be computed. Temporal variations in the PSD of temperature and velocity signals were drawn from the results obtained by sequentially analyzing a constant number of records. Plots were also drawn from the results of analyzing an increasing number of records, beginning with the first record. These plots showed that the fluctuating PSD became more stationary when more and more data is analyzed. (1) Temperature. Figure 42 shows the temporal variation of the temperature PSD. A total of 300 records were analyzed to give a total time of 307.2 seconds. To compute the PSD values, thirty passes were made through the Fourier coefficients with the PSD being computed for each set of ten records. The time difference between. each PSD was 10.24 seconds. Figure 43 showed how the temporal variations in the PSD are smoothed out by taking successively longer 107 0) p cd u CD p. e cu Eh CD -P «J •H P c CL> H CD Cm Cm •H Q -O C cd (D -P Cd •H -P c CD m^ CD - Ch W •H c c I=> Cti Cm H O W faO^ C >H •H ^ 'Ci 00 M O O c\j o ^ CD K rH £ C CO hO 3 -H m CO CQ CD M hO •H 108 33S.1.3) 13 wnui33dS "'301 -a 0) -p 3 Cu 6 o o Q CO Ph • rH Cti £ hD •H CO 0) Ph 3 -P cd U - > Z .c =3 iH O a cd cd oc ?H W o o" ft ^ o S o CD » rH Cd C < Q CO (^ m O «m T3 CD CO >> rH cti c < CO M o o CD K Cm O M S 50 C •H CO CtJ CD M O C (/x5 -P •H O O rH CD > O •H M CD x; Q^rH co cd o c s fat -P-H - -H CD S mco « cd S >CM lu rH = .H • "- cdcn M „s o.c S ao _, S cc CDC31 &H ? • • ? 5 r (,.2Mt.3as,wa) n Wn«133dS "'90 1 CD M bO •H 112 • / / / / / / ) ' / ' / / lv 1* i o u - u. : U-*n •*-■»* v^o)' n wnvjx>3dS '°W"| " O w •H CO >> rH cd C < Q CO u o s iH ctf C < CO Pn O o (D K Cm O U CD ,Q S 3 hD G •H CO Ctj rH CD cti M C O bD C -H H CO tin >> O -P •H -P O o o CD rH Oh 0) ^ > LTV CD U hO fe 113 4.0i o OJ 2.0 o W K Eh o w CO o -2.0 o -4.0 -0.8 d> .56 Seconds 05 minutes 9> V. © ?>© © (3 0© 0.2 1.2 2.2 — ^ 3.2 LOG FREQUENCY f(Hz) Figure 46. Comparison of 56 Records and 5 Minutes of Temperature Signal 114 short section of signal. The two lengths compare very closely, implying that for this situation, the shorter length of a minute would haye giyen statistics representative of longer sections, (for the frequency range 1 Hz to lKHz). Figure 47 showed the slope characteristics for the slong record length (5 minutes). A definite increase (-7/3) in the slope was noted from about 70 Hz to about 300 Hz. 115 - 1 '.♦ — r o T5~ I CM I I I oas • o£s a Mimosas _LLTO 01 DOT cti m C bO •H CO >3 •P .1 ri O O H N- > zn > -5 i H cd i': < CL E O T j Cm -P — O — \ iH — P4 "O, r- Q co "" >- CD M faO •H 116 VI. CONCLUSIONS AND RECOMMENDATIONS A. CONCLUSIONS The following conclusions were reached concerning the problem of the possible input of noise into the Analog-to- Digital conversion procedure and into the PSD analysis pro- cedure with signal inputs of real data. 1 . PSD Programs No significant noise sources were found to exist with- in the computational program of the Naval Postgraduate School FFT package. The PSD computations for noise -free sine and random signals agreed very well with theory. The programs FTOR, SCOR and FCPLT were found to be excellent for obtaining power spectra from large quantities of time series data. In the CONVERT program used in a previous study, it was found that a factor was missing which changed the octal base number to the hexidecimal. A corrected version of the program was used and PSD results indicate the procedure is functioning correctly. PSD values from four atmospheric turbulence signals which were compared with results obtained from other computational facilities showed very close correlation. 2. Analog-to-Digital_ Conversion No significant noise sources were found to be inter- fering with the digitization procedure carried out on the 117 Hybrid Computer. Patchboard noise which had previously been reported as excessive by Jones , £Ref. 5J was noted; however, its presence in the PSD plots of sine and random signals were not detected. Even the noise picked up by open 10 gain -6 p -1 amplifier had a very low level, on the order of 10 V Hz Early in this study it was discovered that the digitization procedure was missing several data samples at the end of each record. The problem was due to internal delays within the XDS 9300 which caused several data samples to be, omitted. The computer laboratory staff revised the digiti- zation program to include a cross-check between each sample and lapsed time. It is now possible to sample at a rate of at least 5000 SPS and check each block for missing data. Results obtained from PSD analysis indicate the problem has been corrected . 3 . PSD Analysis Procedures The IBM 360/67 was found to be quite capable of pro- cessing large quantities of time series data. The operational routine in PSD analysis has been improved. Methods for affecting faster "turn around" have been developed. Several methods have been developed for statistically checking the digital values on tapes and plotting the values by the Calcomp plotter. 4 . PSD Analysis of Turbulence Signals A definite correlation was noted between the temporal PSD plots of both temperature and velocity and the original 118 analog signal from which the PSD results were computed. This tended to further support the conclusion that the PSD pro- grams were working properly. B. RECOMMENDATIONS FOR FUTURE WORK Due to the fact that this series of programs has proved to be an extremely powerful tool for signal analysis, its future use should be vigourously persued. The programs ' use in tur- bulence anlysis has been well established; however, its ap- plication in any field which employs PSD techniques should be followed . The time-varying PSD analysis of turblence signal should be persued. 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C>>J-rv.c<- rvjf-- xn<\ — LT.C r»^ *. T N »-• fiTC h- i ocoooDca ! I C — _. .»..-._- f\- in: <\l ~- aoccooooqo oooc I I I I I I I II I I I I I U U.O.U-L-_U UJUUU.U'ULLU;u|uUUU.U. ITCrrrfT-^-— -CM-Trr OOiNif^ X — Ic^a »J «J — i CvClt. vtairotNlr^rvLno — f— x f rv — 1 0- r- -4- j-f\j.-Cl>Cr<- rv._4f-.j-.ra ^rA^aD r\.e1f-. cqe i i U'U|L_ ipf"-|_c rr rf j- OCO I I I -.in rvuMr\ c i L__" — — OCCDC OOOOCDCaOOCOOOCQC UJU. U.LLU U LL r- h-LT. irsr- •-•re r-f-a- c cai LlJuU-UJU LLU. «* c r- -o l~ <-<- .— xjr~- _lt»_ix f- u.ulu x -^cr ccc U LLU. r a — -.If — C II L5 or L' C coc J .— -.cgr->»r sdo-^— ■ rs.rv. c 1 X _ _-rj — u- U C 2tZ -t" _■ qo uiu O . i C"L • r - - r 13 i ' _ JJL . U U U l- L c^r xx m. >i c >r • •) • • J. _.r,< a r- r o c c c ; t |U L U L U U l He — -ccr ^.o IT. — fr,'' f U „|U r HO ^ r»1 — <\ r> r- CC o I U LL r — f^u l.: c c r 1 1 it n r j .- v» X ^ C •- • C _ «-l:l ii -l — u >_ a > (J X t_ I- — <'i'*«yj->ff^cc?q— fv. (» -:i.'l>cI'"- --(.; - -< — . — _._-_-_-_j_rrv.r. r.L-a. i ^ cxJ C bO •H CO CD U cd cu cu S cu in Cm O CO •H CO >> H cd C < co Pl. M o to cu H cd > Q CO PL, X M Q W CM 136 :Ss55s~FT—"3 LA&JgS*a& cd W o cm C o -p CO o 10 H cti C hO •H co >, -p •H O o CD > c nj CD U -P cd u CD e CD Eh X H Q W 137 138 139 140 141 REFERENCES 1. Boston, H. E. J., An Investigation of High Wave Number Temperature and Velocity Spectra in Air, Ph.D. Thesis University of British Columbia, Vancouver, 1970. 2. Enochson, L. D. and Otenes, R. K., Programming and Analysis for Digital Time Series Data, Shock and Vibration Center, Naval Research Laboratory, 1968. 3. Mix, D. F., Random Signal Analysis, Addison Wesley Publish- ing Co., 196"9^ 4. Cochran, W. T. and others, "What is the Fast Fourier Transform?" IEEE Transactions on Audio and Electro Acoustics, Au - 5 (2), pp. 45-55, 1967. 5. Jones, R. D., "Time Series Analysis of Analog Data by Analog-to-Digital and Digital Data Processing Methods at the Naval Postgraduate School," M. S. Thesis, Naval Postgraduate School Monterey, 1971. 6. Dobson, F. W., Observations of Normal Pressure on Wind- Generated Sea Waves, Ph.D. Thesis, University of British Columbia, Vancouver, 1969. 7. Wilson, J. R., Boston, N. E. J., Denner, W. W., "Digital Analysis of Turbulence Data on the IBM 360/67 at the Naval Postgraduate School," NPS-58 DW 9071A, 1969. 142 9" BIBLIOGRAPHY Bendat, J. S. and Piersol, A. G., Measurement and Analysis of Random Data, John Wiley and Sons, Inc., New York, 1966. Blackman, R. B. and Tukey, J. W., The Measurement of Power Spectra, Dover Press, 1959. Boston, H. E. J. An Investigation of High Wave Number Temperature and Velocity Spectra in Air, Ph. D. Thesis, University of British Columbia, Vancouver, 1970. Cochran, W. T. and others, "What is the Fast Fourier Transform IEEE Transections on Audio and Electro-Acoustics, Au -5(2) pp. 45-55, 1967. Cooper, G. R. and McGillem, CD. Methods of Signal and System Analysis , Holt, Rinehart and Winston Inc . , 1967 . Dobson, F. W. , Observations of Normal Pressure on Wind-Generated Sea Waves, Ph. D. Thesis, University of British Columbia Vancouver, 1969- Enochson, L. D. and Otenes, R. K., Programming and Analysis for Digital Time Series Data, Shock and Vibration Center, Naval Research Laboratory, 1968. Jones, R. D., "Time Series Analysis of Analog Data by Analog-to-Digital and Digital Data Processing Methods at the Naval Postgraduate School," M. S. Thesis, Naval Post- graduate School, Monterey, 1971- Mix, D. F., Random Signal Analysis, Addison-Wesley Publishing Co., 1969. Naval Postgraduate School Computer Facility Technical Note 0211-08, Procedures For Converting 7-Track Magnetic Tapes to 9-Track Magnetic Tape, by Sharon Ramey , June, 1970. Wilson, J. R., Boston, N. E. J., Denner, W. W., "Digital Analysis of Turbulence Data on the IBM 360/67 at the Naval Postgraduate School," NPS-58 DW 9071A, 1969 . 143 INITIAL DISTRIBUTION No. Copies 1. Defence Documentation Center 2 Cameron Station Alexandria, Virginia 22314 2. Library, Code 0212 2 Naval Postgraduate School Monterey, California 93940 3. Department of Oceanography 3 Naval Postgraduate School Monterey, California 93940 4. Professor N. E. Boston, Code 58Bb 5 Department of Oceanography U. S. Naval Postgraduate School Monterey, California 93940 5. Professor W. W. Denner, Code 58DW 2 Department of Oceanography U.S. Naval Postgraduate School Monterey, California 93940 6. Professor K. L. Davidson, Code 51Ds 1 Department of Meterology Naval Postgraduate School Monterey, California 93940 7. Professor Edward Thornton, Code 58TM 1 Department of Oceanography U.S. Naval Postgraduate School Monterey, California 93940 8. Oceanographer of the Navy 1 The Madison Building 732 N. Washington Street Alexandria, Virginia 22314 9. Dr. Ned A. Ostenso 1 Code 480D Office of Naval Research Arlington, Virginia 22217 10. Professor H. Medwin, Code 6l 2 Department of Physics Naval Postgraduate School Monterey, California 93940 144 11. Lieutenant John D. McKendrick 103 Dundee Avenue Richmond, Virginia 23225 12. Dr. John F. Garrett Department of Environment Marine Sciences Branch Pacific Region 1230 Government Street Victoria, British Columbia, Canada 13. Mr. J. R. Wilson Marine Sciences Branch Department of Energy, Mines and Resources 615 Booth Street Ottow 4, Canada 14. LCDR Robert D. Jones, U.S.N. 807 W. 3rd Street Lampasas, Texas 76550 15. Mr. Robert Limes, Code 52EC Department of Electrical Engineering Naval Postgraduate School Monterey, California 939^0 16. Miss Sharon D. Raney, Code 0211 Computer Center Naval Postgraduate School Monterey, California 939^0 17. Professor D. G. Williams, Code 0211 Director Computer Center Naval Postgraduate School Monterey, California 939^0 18. Mr. R. R. Hilleary, Code 0211 Computer Center Naval Postgraduate School Monterey, California 939^0 19- Professor H. Titus Department of Electrical Engineering Naval Postgraduate School Monterey, California 939^0 145 UNCLASSIFIED Security Classification DOCUMENT CONTROL DATA -R&D [Security classification of title, body of abstract and indexing annotation must be entered when the overall report is classified) I. ORIGINATING ACTIVITY (Corporate author) Naval Postgraduate School Monterey, California 939^0 Jl. REPORT SECURITY CLASSIFICATION Unci 3ssi f i pd 2b. GROUP 3. REPORT TITLE An Investigation of Digital Spectral Analysis Programs and Computer Computer Methods Utilized at the Naval Postgraduate School in the Analysis of His-h Frequency Random Signals. 4. DESCRIPTIVE NOTES (Type ol report and, inclusi ve dates) Master's Thesis; (March 1972) S. au THORISI (First name, middle initial, last name) John DeMille McKendrick 6. REPOR T D A TE March 1972 7a. TOTAL NO. OF PAGES 147 7b. NO. OF REFS •a. CONTRACT OR GRANT NO. b. PROJEC T NO. 9a. ORIGINATOR'S REPORT NUMBER<5) 9b. OTHER REPORT NO(S) (Any other numbers that may be assigned this report) 10. DISTRIBUTION STATEMENT Approved for public release; distribution unlimited II. SUPPLEMENTARY NOTES 12. SPONSORING MILI TAR Y ACTIVITY Naval Postgraduate School Monterey, California 939^0 13. ABSTR AC T The digitizing procedure used at the Naval Postgraduate School was investigated for possible sources of noise and other errors. Signals of known form were digitized through the Analog-to-Digital Hybrid computer system (Ci 5000/XDS9300 ) . Similar signals were generated by digital programs on the IBM 3 6 0/67 • The resultant signals were analyzed by the computer programs UBCFTOR, which computed the Fourier coefficients of each block of data, and by UBCSCOR, which computed the power spectra of the signals. The power-spectral plots of the computer-generated signals were compared with the power-spectral plots of digitized signals. The analog-to-digital process appeared to be relatively noise free. To further test the system, atmospheric temperature and wind velocity signals were digitized and analyzed under UBCFTOR and UBCSCOR Plots of the time-varying spectra of these signals compared favorably with results obtained at other digitizing facilities. )D,r»M..1473 /N 0101 -807-681 1 (PAGE 1) 146 UNCLASSIFIED Security Classification A- 31408 UNCLASSIFIED Security Classification KEY WO ROS Turbulence analysis ftnalog-to-Digital Data Conversion Digital Data Processing Fast-Fourier Transform Power Spectral Density Spectra Plotting Cross Spectral Density Time Series Analysis LINK C DD,Fr:„1473 <3ack, 5/N 0101 -807-6821 147 UNCLASSIFIED Security Classification A- 31 409 DUPLICATE Thesis 134596 M223 McKendrick c.l An investigation of digital spectral analy- sis programs and computer methods utilized at the Naval Postgraduate School in the analysis of high frequency random signals. Thesis M223 c.l 134596 McKendrick An investigation of digital spectral anal- ysis programs and com- puter methods utilized at the Naval Postgrad- uate School in the anal- ysis or high frequency random signals. thesM223 /i.nVeSt'9ati0n °f d'9ltal SPeCtral a"£ 3 2768 001 88215 2 -, DUDLEY KNOX LIBRARY