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.

1. Fourier theory made easy (?)

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-8
-6
-4
-2
0
2
4
6
8
5*sin (2π4t)
Amplitude = 5
Frequency = 4 Hz
seconds
A sine wave

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-8
-6
-4
-2
0
2
4
6
8
5*sin(2π4t)
Amplitude = 5
Frequency = 4 Hz
Sampling rate = 256
samples/second
seconds
Sampling duration =
1 second
A sine wave signal

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-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
sin(2π8t), 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
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:
( ) ( )
( ) ( )∫
∫
∞
∞−
−
∞
∞−
=
=
dfefHth
dtethfH
ift
ift
π
π
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
π
π

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!

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-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
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-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
( ) ( ) 




 −
=
2
exp2sin
T
t
fttF π

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