1. Fourier theory made easy (?)

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

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

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

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

20. Effect of changing sample rate

22. 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

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

28. 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/FT 4FreeIndDecay.pdfhttp://jchemed.chem.wisc.edu/JCEWWW/Features/McadInChem/mcad008/FT 4FreeIndDecay.pdf –Mathcad tutorial for exploring Fourier transforms of free-induction decay http://lcni.uoregon.edu/fft/fft.ppt –This presentation