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 (2π4t)
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(2π4t)
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(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!

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
( ) ( ) 




 −
=
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