Fourier theory made easy (?)
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 (2?4t)
Amplitude = 5
Frequency = 4 Hz
seconds
A sine wave
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(2?4t)
Amplitude = 5
Frequency = 4 Hz
Sampling rate = 256
samples/second
seconds
Sampling duration =
1 second
A sine wave signal
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
s i n ( 2? 8 t ),S R = 8,5 H z
An undersampled signal
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.
http://www.falstad.com/fourier/j2/
Fourier series
Periodic functions and signals may be
expanded into a series of sine and cosine
functions
The Fourier Transform
A transform takes one function (or signal)
and turns it into another function (or signal)
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
The Fourier Transform
A transform takes one function (or signal)
and turns it into another function (or signal)
Continuous Fourier Transform:




dfefHth
dtethfH
i f t
i f t
2
2
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
Ni k n
nk
N
k
Ni k n
kn
eH
N
h
ehH
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? 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 1 0 0 1 2 0
0
50
1 0 0
1 5 0
2 0 0
2 5 0
3 0 0
Famous Fourier Transforms
Sine wave
Delta function
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 1 0 0 1 5 0 2 0 0 2 5 0
0
1
2
3
4
5
6
Gaussian
Gaussian
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
- 1 0 0 - 5 0 0 50 1 0 0
0
1
2
3
4
5
6
Sinc function
Square wave
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
- 1 0 0 - 5 0 0 50 1 0 0
0
1
2
3
4
5
6
Famous Fourier Transforms
Exponential
Lorentzian
0 50 1 0 0 1 5 0 2 0 0 2 5 0
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
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 1 0 0 1 2 0
0
10
20
30
40
50
60
70
f = 8 H z
S R = 2 5 6 H z
T 2 = 0,5 s
2e x p2s in T tfttF?
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 1 0 0 1 2 0
0
2
4
6
8
10
12
14
f = 8 H z
S R = 2 5 6 H z
T 2 = 0,1 s
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 1 0 0 1 2 0
0
50
1 0 0
1 5 0
2 0 0
f = 8 H z
S R = 2 5 6 H z
T 2 = 2 s
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
5
10
15
20
25
30
35
f = 8 H z
T 2 = 0,5 s
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
5
10
15
20
25
30
35
S R = 2 5 6 H z
S R = 1 2 8 H z
f = 8 H z
T 2 = 0,5 s
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
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 H z
T 2 =,5 s
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
S T = 2,0 s
S T = 1,0 s
f = 8 H z
T 2 =,5 s
Effect of changing sampling duration
Reducing the sampling duration:
– Lowers the frequency resolution
– Does not affect the range of frequencies you
can measure
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
1 0 0
1 5 0
2 0 0
f = 8 H z
T 2 = 2,0 s
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
S T = 2,0 s
S T = 1,0 s
f = 8 H z
T 2 = 0,1 s
Measuring multiple frequencies
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-3
-2
-1
0
1
2
3
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0
20
40
60
80
1 0 0
1 2 0
f
1
= 8 0 H z,T 2
1
= 1 s
f
2
= 9 0 H z,T 2
2
=,5 s
f
3
= 1 0 0 H z,T 2
3
= 0,2 5 s
S R = 2 5 6 H z
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 1 0 0 1 2 0
0
20
40
60
80
1 0 0
1 2 0
f
1
= 8 0 H z,T 2
1
= 1 s
f
2
= 9 0 H z,T 2
2
=,5 s
f
3
= 2 0 0 H z,T 2
3
= 0,2 5 s
S R = 2 5 6 H z
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.pdf
– Mathcad tutorial for exploring Fourier transforms of free-induction decay
http://lcni.uoregon.edu/fft/fft.ppt
– This presentation