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Fourier theory made easy (?)

Fourier theory made easy (?). A sine wave. 5*sin (2 4t). Amplitude = 5. Frequency = 4 Hz. seconds. A sine wave signal. 5*sin(2 4t). Amplitude = 5. Frequency = 4 Hz. Sampling rate = 256 samples/second. Sampling duration = 1 second. seconds. An undersampled signal.

jacob-gross
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Fourier theory made easy (?)

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  1. Fourier theory made easy (?)

  2. A sine wave 5*sin (24t) Amplitude = 5 Frequency = 4 Hz seconds

  3. A sine wave signal 5*sin(24t) Amplitude = 5 Frequency = 4 Hz Sampling rate = 256 samples/second Sampling duration = 1 second seconds

  4. An undersampled signal

  5. The Nyquist Frequency • 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.

  6. Fourier series • Periodic functions and signals may be expanded into a series of sine and cosine functions http://www.falstad.com/fourier/j2/

  7. The Fourier Transform • A transform takes one function (or signal) and turns it into another function (or signal)

  8. The Fourier Transform • A transform takes one function (or signal) and turns it into another function (or signal) • Continuous Fourier Transform: close your eyes if you don’t like integrals

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

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

  11. Fast Fourier Transform • The Fast Fourier Transform (FFT) is a very efficient algorithm for performing a discrete Fourier transform • FFT principle first used by Gauss in 18?? • FFT algorithm published by Cooley & Tukey in 1965 • In 1969, the 2048 point analysis of a seismic trace took 13 ½ hours. Using the FFT, the same task on the same machine took 2.4 seconds!

  12. Famous Fourier Transforms Sine wave Delta function

  13. Famous Fourier Transforms Gaussian Gaussian

  14. Famous Fourier Transforms Sinc function Square wave

  15. Famous Fourier Transforms Sinc function Square wave

  16. Famous Fourier Transforms Exponential Lorentzian

  17. FFT of FID

  18. FFT of FID

  19. FFT of FID

  20. Effect of changing sample rate

  21. Effect of changing sample rate

  22. Effect of changing sample rate • Lowering the sample rate: • Reduces the Nyquist frequency, which • Reduces the maximum measurable frequency • Does not affect the frequency resolution

  23. Effect of changing sampling duration

  24. Effect of changing sampling duration

  25. Effect of changing sampling duration • Reducing the sampling duration: • Lowers the frequency resolution • Does not affect the range of frequencies you can measure

  26. Effect of changing sampling duration

  27. Effect of changing sampling duration

  28. Measuring multiple frequencies

  29. Measuring multiple frequencies

  30. Some useful links • 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

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