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Geol 491: Spectral Analysis

Geol 491: Spectral Analysis. tom.wilson@mail.wvu.edu. Fourier said that any single valued function could be reproduced as a sum of sines and cosines. Introduction to Fourier series and Fourier transforms. 5*sin (2 4t). Amplitude = 5. Frequency = 4 Hz. seconds.

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Geol 491: Spectral Analysis

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  1. Geol 491: Spectral Analysis tom.wilson@mail.wvu.edu

  2. Fourier said that any single valued function could be reproduced as a sum of sines and cosines Introduction to Fourier series and Fourier transforms 5*sin (24t) Amplitude = 5 Frequency = 4 Hz seconds

  3. We are usually dealing with sampled data 5*sin(24t) Amplitude = 5 Frequency = 4 Hz Sampling rate = 256 samples/second Sampling duration = 1 second seconds

  4. Faithful reproduction of the signal requires adequate sampling If our sample rate isn’t high enough, then the output frequency will be lower than the input,

  5. The Nyquist frequency is equal to one-half of the sampling frequency. The Nyquist frequency is the highest frequency that can be measured in a signal. The Nyquist Frequency Where t is the sample rate Frequencies higher than the Nyquist frequencies will be aliased to lower frequency

  6. The Nyquist Frequency Thus if t = 0.004 seconds, fNy = Where t is the sample rate

  7. Periodic functions and signals may be expanded into a series of sine and cosine functions Fourier series: a weighted sum of sines and cosines

  8. This applet is fun to play with & educational too. Experiment with http://www.falstad.com/fourier/

  9. Try making sounds by combining several harmonics (multiples of the fundamental frequency) An octave represents a doubling of the frequency. 220Hz, 440Hz and 880Hz played together produce a “pleasant sound” Frequencies in the ratio of 3:2 represent a fifth and are also considered pleasant to the ear. 220, 660, 1980etc. Pythagoras (530BC)

  10. You can also observe how filtering of a broadband waveform will change audible waveform properties. http://www.falstad.com/dfilter/

  11. The Fourier series can be expressed more compactly using summation notation Fourier series You’ve seen from the forgoing example that right angle turns, drops, increases in the value of a function can be simulated using the curvaceous sinusoids.

  12. Try the excel file step2.xls Fourier series

  13. A transform takes one function (or signal) in time and turns it into another function (or signal) in frequency This can be done with continuous functions or discrete functions The Fourier Transform

  14. The general problem is to find the coefficients: a0, a1, b1, etc. The Fourier Transform Take the integral of f(t) from 0 to T (where T is 1/f). Note =2/T What do you get? Looks like an average! We’ll work through this on the board.

  15. Getting the other Fourier coefficients To get the other coefficients consider what happens when you multiply the terms in the series by terms like cos(it) or sin(it).

  16. Now integrate f(t) cos(it) This is just the average of i periods of the cosine

  17. Now integrate f(t) cos(it) Use the identity If i=2 then the a1 term =

  18. What does this give us? And what about the other terms in the series?

  19. In general to find the coefficients we do the following and The a’s and b’s are considered the amplitudes of the real and imaginary terms (cosine and sine) defining individual frequency components in a signal

  20. Arbitrary period versus 2 Sometimes you’ll see the Fourier coefficients written as integrals from - to  and

  21. Exponential notation cost is considered Re eit where

  22. A transform takes one function (or signal) and turns it into another function (or signal) Continuous Fourier Transform: The Fourier Transform

  23. A transform takes one function (or signal) and turns it into another function (or signal) The Discrete Fourier Transform: The Fourier Transform

  24. http://www.falstad.com/fourier/ Fourier series java applet http://www.jhu.edu/~signals/ Collection of demonstrations about digital signal processing http://www.ni.com/events/tutorials/campus.htm FFT tutorial from National Instruments http://www.cf.ac.uk/psych/CullingJ/dictionary.html Dictionary of DSP terms http://jchemed.chem.wisc.edu/JCEWWW/Features/McadInChem/mcad008/FT4FreeIndDecay.pdf Mathcad tutorial for exploring Fourier transforms of free-induction decay http://lcni.uoregon.edu/fft/fft.ppt This presentation Some useful links

  25. Meeting times? Other questions?

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