Fft
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This document provides an introduction to Fourier theory and Fourier transforms. It begins by showing examples of sine waves and how they are sampled. It then defines key concepts like the Nyquist frequency and discrete Fourier transform. The document explains how the fast Fourier transform works and provides examples of famous Fourier transforms. It demonstrates how changing parameters like the sampling rate and duration impact the Fourier transform. Finally, it shows an example of measuring multiple frequencies simultaneously and provides some additional resources on Fourier theory.
Fft
- 1. Fourier theory made easy (?)
- 2. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -8 -6 -4 -2 0 2 4 6 8 5*sin (24t) Amplitude = 5 Frequency = 4 Hz seconds A sine wave
- 3. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -8 -6 -4 -2 0 2 4 6 8 5*sin(24t) Amplitude = 5 Frequency = 4 Hz Sampling rate = 256 samples/second seconds Sampling duration = 1 second A sine wave signal
- 4. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 sin(28t), SR = 8.5 Hz 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. http://www.falstad.com/fourier/j2/ Fourier series Periodic functions and signals may be expanded into a series of sine and cosine functions
- 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 dont like integrals
- 9. The Fourier Transform A transform takes one function (or signal) and turns it into another function (or signal) Continuous Fourier Transform: ( ) ( ) ( ) ( ) = = dfefHth dtethfH ift ift 2 2
- 10. A transform takes one function (or signal) and turns it into another function (or signal) The Discrete Fourier Transform: The Fourier Transform = = = = 1 0 2 1 0 2 1 N n Nikn nk N k Nikn kn eH N h ehH
- 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 遜 hours. Using the FFT, the same task on the same machine took 2.4 seconds!
- 12. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -2 -1 0 1 2 0 20 40 60 80 100 120 0 50 100 150 200 250 300 Famous Fourier Transforms Sine wave Delta function
- 13. Famous Fourier Transforms 0 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0 50 100 150 200 250 0 1 2 3 4 5 6 Gaussian Gaussian
- 14. Famous Fourier Transforms -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -0.5 0 0.5 1 1.5 -100 -50 0 50 100 0 1 2 3 4 5 6 Sinc function Square wave
- 15. Famous Fourier Transforms Sinc function Square wave -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -0.5 0 0.5 1 1.5 -100 -50 0 50 100 0 1 2 3 4 5 6
- 16. Famous Fourier Transforms Exponential Lorentzian 0 50 100 150 200 250 0 5 10 15 20 25 30 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1
- 17. FFT of FID 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -2 -1 0 1 2 0 20 40 60 80 100 120 0 10 20 30 40 50 60 70 f = 8 Hz SR = 256 Hz T2 = 0.5 s ( ) ( ) 錚 錚 錚 錚 錚 錚 = 2 exp2sin T t fttF
- 18. FFT of FID 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -2 -1 0 1 2 0 20 40 60 80 100 120 0 2 4 6 8 10 12 14 f = 8 Hz SR = 256 Hz T2 = 0.1 s
- 19. FFT of FID 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -2 -1 0 1 2 0 20 40 60 80 100 120 0 50 100 150 200 f = 8 Hz SR = 256 Hz T2 = 2 s
- 20. Effect of changing sample rate 0 10 20 30 40 50 60 0 10 20 30 40 50 60 70 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -2 -1 0 1 2 0 10 20 30 40 50 60 0 5 10 15 20 25 30 35 f = 8 Hz T2 = 0.5 s
- 21. Effect of changing sample rate 0 10 20 30 40 50 60 0 10 20 30 40 50 60 70 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -2 -1 0 1 2 0 10 20 30 40 50 60 0 5 10 15 20 25 30 35 SR = 256 Hz SR = 128 Hz f = 8 Hz T2 = 0.5 s
- 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 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -2 -1 0 1 2 0 2 4 6 8 10 12 14 16 18 20 0 10 20 30 40 50 60 70 f = 8 Hz T2 = .5 s
- 24. Effect of changing sampling duration 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -2 -1 0 1 2 0 2 4 6 8 10 12 14 16 18 20 0 10 20 30 40 50 60 70 ST = 2.0 s ST = 1.0 s f = 8 Hz T2 = .5 s
- 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 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -2 -1 0 1 2 0 2 4 6 8 10 12 14 16 18 20 0 50 100 150 200 f = 8 Hz T2 = 2.0 s
- 27. Effect of changing sampling duration 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -2 -1 0 1 2 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 ST = 2.0 s ST = 1.0 s f = 8 Hz T2 = 0.1 s
- 28. Measuring multiple frequencies 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -3 -2 -1 0 1 2 3 0 20 40 60 80 100 120 0 20 40 60 80 100 120 f 1 = 80 Hz, T2 1 = 1 s f 2 = 90 Hz, T2 2 = .5 s f 3 = 100 Hz, T2 3 = 0.25 s SR = 256 Hz
- 29. Measuring multiple frequencies 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -3 -2 -1 0 1 2 3 0 20 40 60 80 100 120 0 20 40 60 80 100 120 f 1 = 80 Hz, T2 1 = 1 s f 2 = 90 Hz, T2 2 = .5 s f 3 = 200 Hz, T2 3 = 0.25 s SR = 256 Hz
- 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/FT4FreeInd Mathcad tutorial for exploring Fourier transforms of free-induction decay http://lcni.uoregon.edu/fft/fft.ppt This presentation