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Fourier theory

Fourier theory. 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. The Nyquist Frequency.

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Fourier theory

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  1. Fourier theory

  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 are expanded into sines and cosines

  7. What is a Transform? • A transform inputs one function and outputs another.

  8. The Fourier Transform

  9. DFT

  10. DFT • 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!

  11. Fast Fourier Transform • an efficient DFT algorithm • used by Gauss in 18?? • published by Cooley & Tukey in 1965

  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

  18. FFT

  19. FFT

  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

  23. Effect of changing sampling duration

  24. Effect of changing sampling duration

  25. 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

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