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Aristotle University of Thessaloniki School of Rural & Surveying Engineering

Learn about Fourier Analysis, Signals, Information Transmission, Data Analysis, and Signal Processing in Geoinformatics in this comprehensive lecture series from Aristotle University of Thessaloniki.

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Aristotle University of Thessaloniki School of Rural & Surveying Engineering

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  1. Aristotle University of Thessaloniki School of Rural & Surveying Engineering Department of Geodesy & Surveying 3rdSemester SignalsandSpectral Methods in Geoinformatics Lecture Presentations prepared by Prof. Athanasios Dermanis

  2. Lecture 1: Introduction

  3. (1) The mathematical tool: «Spectral Methods» or «Fourier Analysis» or «Harmonic Analysis» 2 parts: (2) The physical application: «Signals» Two meanings

  4. (1) The mathematical tool: «Spectral Methodsς Μέθοδοι» or «Fourier Analysis» or «Harmonic Analysis» 2 parts: (2) The physical application: «Signals» Communication Engineering: Information transmmittedthroughelectromagnetic radiation (e,g,audio-visualsignal)

  5. (1) The mathematical tool: «Spectral Methodsς Μέθοδοι» or «Fourier Analysis» or «Harmonic Analysis» 2 parts: (2) The physical application: «Signals» Communication Engineering: Information transmmittedthroughelectromagnetic radiation (e,g,audio-visualsignal) Data Analysis:The part of an observations contatining information about the physical reality Observation = Signal + Noise

  6. (1) The mathematical tool: «Spectral Methodsς Μέθοδοι» or «Fourier Analysis» or «Harmonic Analysis» 2 parts: (2) The physical application: «Signals» Communication Engineering: Information transmmittedthroughelectromagnetic radiation (e,g,audio-visualsignal) Data Analysis:The part of an observations contatining information about the physical reality Observation = Signal + Noise Signal = information !

  7. (1) The mathematical tool: «Spectral Methodsς Μέθοδοι» or «Fourier Analysis» or «Harmonic Analysis» 2 parts: (2) The physical application: «Signals»

  8. (1) The mathematical tool: «Spectral Methodsς Μέθοδοι» or «Fourier Analysis» or «Harmonic Analysis» 2 parts: (2) The physical application: «Signals» ModernSurveying and Geodesy relay on signal analysis. Artifisial: ΕDM, GPS, Laser Natural: Gravity Field, Radiosignals Target:

  9. Development of a function defined in an interval intoFourier Series Jean Baptiste Joseph Fourier

  10. (real) numbers THE ROLE OF APPLIED SCIENCES Mathematical-Physical model Euclidean Space Point Coordinates Newtonian Time NumericalTime Natural World Mathematical Objects: Vectors Functions Vector Components Numerical Coefficients  Algorithms Computations Results Applications

  11. TURNINGMATHEMATICAL OBJECTS INTO NUMBERS mathematical objects numbers

  12. TURNINGMATHEMATICAL OBJECTS INTO NUMBERS mathematical objects numbers coordinates q1(Ρ),q2 (Ρ), q3 (Ρ) coordinate systemq1, q2, q3 pointΡ

  13. TURNINGMATHEMATICAL OBJECTS INTO NUMBERS mathematical objects numbers coordinates q1(Ρ),q2 (Ρ), q3 (Ρ) coordinate systemq1, q2, q3 pointΡ componentsu1, u2, u3 ofu=u1e1+u2e2+u3e3 local vectorial basis e1, e2, e3 localvector u

  14. TURNINGMATHEMATICAL OBJECTS INTO NUMBERS mathematical objects numbers coordinates q1(Ρ),q2 (Ρ), q3 (Ρ) coordinate systemq1, q2, q3 pointΡ componentsu1, u2, u3 ofu=u1e1+u2e2+u3e3 local vectorial basis e1, e2, e3 localvector u coefficientsα1, α2, ... of the function f = a1φ1+ a2 φ2 + ... known base functions φ1, φ2, ... function f

  15. REPRESENTING A FUNCTION BY NUMBERS f(t) t 0 Τ coefficientsα1, α2, ... of the function f = a1φ1+ a2 φ2 + ... known base functions φ1, φ2, ... functionf

  16. +1 +1 +1 +1 0 0 0 0 –1 –1 –1 –1 0 Τ 0 Τ 0 Τ 0 Τ +1 +1 +1 +1 0 0 0 0 –1 –1 –1 –1 0 Τ 0 Τ 0 0 Τ Τ The base functions ofFourier series +1 0 –1 0 Τ

  17. Development of a real functionf(t)defined in the interval[0,T]intoFourier series 3 alternative forms: Everybase functionhas: period: frequency: angular frequency: fundamentalperiod fundamental frequency fundamental angular frequency term periods term frequencies term angular frequencies

  18. From the “language” of numbers to the “language of frequencies” Signalf(t) with a single frequency (monochromatic): t

  19. From the “language” of numbers to the “language of frequencies” Signalf(t) with a single frequency (monochromatic): t sin cos Repetition of sines or cosines

  20. From the “language” of numbers to the “language of frequencies” Signalf(t) with a single frequency (monochromatic): t sin cos Repetition of sines or cosines cycle (= 1 repetion) for cos cycle (= 1 repetion) for sin

  21. From the “language” of numbers to the “language of frequencies” Signalf(t) with a single frequency (monochromatic): Τ Τ t 1 sec (s = 2.5 cycles per second) Period Τ:time needed for the repetition of a cycle Frequencys : number of cycles in a unit of time (cyclesper second) Wavelength λ =c T : intervalcovered by a signal travellingwith velocity c within one period c : velocity of light (electromagnetic radiation) in vacuum

  22. From the “language” of numbers to the “language of frequencies” low frequency high frequency

  23. From the “language” of numbers to the “language of frequencies” large period small period

  24. From the “language” of numbers to the “language of frequencies” large wavelength small wavelength

  25. Three equivalent names: «Spectral Methods» «Fourier Analysis» «Harmonic Analysis» Fourier Analysis Fourier has been the first to realize that “ALL”functions can be expresses through linear combinations of trigonometric functions

  26. Three equivalent names: «Spectral Methods» «Fourier Analysis» «Harmonic Analysis» Spectral Methods in relation to the spectrum of light (visible part of the electromagnetic radiation) or more generally in relation to the spectrum ofelectromagnetic radiation

  27. The spectral approach The (white) sunlight is composed by components having different wavelenghts. Reflection of each components depends on its wavelenght. White light is spread out into the rainbow colors. Color = mental perception of the frequency of incoming light The light spectrum (rainbow colors)

  28. The spectral approach The electromagnetic spectrum MICROWAVES ULTRAVIOLET INFRARED VISIBLE

  29. Three equivalent names: «Spectral Methods» «Fourier Analysis» «Harmonic Analysis» Harmonic Analysis in relation to the role of frequencies in the harmony of musical sounds

  30. Harmonic Analysis (Musical harmony and frequencies) FREQUENCY vibrating cord Note Α ♪ s = 440 cycles per second L ♪ 2s = 880 cycles per second L / 2 Note Α 1 octavehigher Frequency = inversely proportional to the length of the vibrating cord

  31. Harmonic Analysis (Musical harmony and frequencies) 1 soundwave (= 1 cycle) SOUNDWAVE: sequence of large and small densities of the air (arriving to the ear) FREQUENCY OF SOUND WAVE: number of consequent large-small air densities in a time unit DIFFERENT FREQUENCIES: different musical notes

  32. Harmonic Analysis (Musical harmony and frequencies) Pythagoras:2 notes sound together harmoniously when the ratio of the cord lengths (and hence of the frequencies) is equal to the ratio of two integers. harmonic ratios 6/5 5/4 4/3 3/2 8/5 5/3 2/1

  33. Harmonic Analysis (Musical harmony and frequencies) Pythagoras:2 notes sound together harmoniously when the ratio of the cord lengths (and hence of the frequencies) is equal to the ratio of two integers. Examples and names of intervals in western music harmonic ratios C  E flatsmall third C  E large third C  F fourth C  G fifth C  A flatsmall sixth C  A large sixth C  C octave 6/5 5/4 4/3 3/2 8/5 5/3 2/1

  34. 0 1 2 3 4 5 6 7 8 9 10 11 12 C D E F G A B C+ Harmonic Analysis (Musical harmony and frequencies) In western “well-tempered” music: The octave is separated into 12 “equal”intervals (semitones) 2 notes differing by 1 semitone have constant frequency ratio C

  35. 0 1 2 3 4 5 6 7 8 9 10 11 12 C D E F G A B C+ Harmonic Analysis (Musical harmony and frequencies) In western “well-tempered” music: The octave is separated into 12 “equal”intervals (semitones) 2 notes differing by 1 semitone have constant frequency ratio C

  36. 0 1 2 3 4 5 6 7 8 9 10 11 12 C D E F G A B C+ Harmonic Analysis (Musical harmony and frequencies) In western “well-tempered” music: The octave is separated into 12 “equal”intervals (semitones) 2 notes differing by 1 semitone have constant frequency ratio C

  37. 0 1 2 3 4 5 6 7 8 9 10 11 12 C D E F G A B C+ Harmonic Analysis (Musical harmony and frequencies) In western “well-tempered” music: The octave is separated into 12 “equal”intervals (semitones) 2 notes differing by 1 semitone have constant frequency ratio C

  38. 0 1 2 3 4 5 6 7 8 9 10 11 12 C D E F G A B C+ Harmonic Analysis (Musical harmony and frequencies) In western “well-tempered” music: The octave is separated into 12 “equal”intervals (semitones) 2 notes differing by 1 semitone have constant frequency ratio C TEMPERED !

  39. Harmonic Analysis (Musical harmony and frequencies) ANCIENT GREEK MUSIC WESTERN WELL-TEMPERED MUSIC Examples and names in western music of intervals which can be harmoniously heard together harmonic frequency ratios C  E flatsmall third C  E large third C  F fourth C  G fifth C  A flatsmall sixth C  A large sixth C  C octave 6/5 5/4 4/3 3/2 8/5 5/3 2/1 = 1,20 = 1,25  1,33 = 1,50 = 1,60  1.67 = 2 1,189207115 1,259921050 1,334839854 1,498307077 1,5874010523 1,6817928305 2

  40. Theory of Signals

  41. Observations and signals f(t) Physical quantity (smooth = low frequencies) t

  42. Observations and signals f(t) Physical quantity (smooth = low frequencies) t y(t) = f(t) + n(t) Observation with additional noise (highfrequencies) t

  43. f(t) f(t) Observations and signals f(t) Physical quantity (smooth = low frequencies) t y(t) = f(t) + n(t) Observation with additional noise (highfrequencies) t DATA ANALYSIS Estimation = noise removal (filtering of high frequencies) t

  44. f(t) Observations and signals f(t) Physical quantity (smooth = low frequencies) t y(t) = f(t) + n(t) Observation with additional noise (highfrequencies) t DATA ANALYSIS Estimation = noise removal (filtering of high frequencies) t

  45. f(t) f(t) Observations and signals f(t) Physical quantity (smooth = low frequencies) t y(t) = f(t) + n(t) Observation with additional noise (highfrequencies) t DATA ANALYSIS Estimation = noise removal (filtering of high frequencies) t

  46. f(t) f(t) Observations and signals f(t) Physical quantity (smooth = low frequencies) t y(t) = f(t) + n(t) Observation with additional noise (highfrequencies) t DATA ANALYSIS Estimation = noise removal (filtering of high frequencies) t estimation error

  47. Signal Applications (General) RADIO – TELEVISION – COMMUNICATIONS TRANSMITTER Signal production MODULATION placing the signal on a carrierfrequency LINK OSCILATOR production of carrier frequency

  48. Signal Applications (General) RADIO – TELEVISION – COMMUNICATIONS TRANSMISSION RECEIVER   DEMODULATION separation of signal from the carrier frequency  

  49. Signal Applicationsto Surveying and Geodesy ΕDM = ELECTROMAGNETIC DISTANCE MEASUREMENT λ s transmitted signal reflected signal reflector EDM

  50. Signal Applicationsto Surveying and Geodesy ΕDM = ELECTROMAGNETIC DISTANCE MEASUREMENT λ s s 2s transmitted signal reflected signal

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