1. Fourier theory http://lcni.uoregon.edu/fft/fft.ppt

2. Fourier series
Periodic functions and signals may be expanded into a series of sine and cosine functions

3. 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.

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

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

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

7. Famous Fourier Transforms
Sinc function
Square wave

8. The Fourier Transform

9. ,

11. ,and

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

15. Fast Fourier Transform
The Fast Fourier Transform (FFT) is a very efficient algorithm for performing a discrete Fourier transform
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!

16. Famous Fourier Transforms
Sine wave
Delta function

17. Famous Fourier Transforms
Gaussian
Gaussian

18. Famous Fourier Transforms
Sinc function
Square wave

19. Famous Fourier Transforms
Exponential
Lorentzian

20. FFT of FID

21. FFT of FID

22. FFT of FID

23. Effect of changing sample rate

24. Effect of changing sample rate

25. 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

26. Effect of changing sampling duration

27. Effect of changing sampling duration

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

29. Effect of changing sampling duration

30. Effect of changing sampling duration

31. Measuring multiple frequencies

32. Measuring multiple frequencies

33. Some useful links http://www.falstad.com/fourier/
Fourier series java applet
Collection of demonstrations about digital signal processing
FFT tutorial from National Instruments
Dictionary of DSP terms
Mathcad tutorial for exploring Fourier transforms of free-induction decay
This presentation

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

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

36. An undersampled signal