Fourier theory made easy (?)
A sine wave
8
5*sin (24t)
6
Amplitude = 5
4
Frequency = 4 Hz
2
0
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-6
-8
0
0.1
0.2
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0.4
0.5
seconds
0.6
0.7
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0.9
1
A sine wave signal
8
5*sin(24t)
6
Amplitude = 5
4
Frequency = 4 Hz
2
Sampling rate = 256
samples/second
0
-2
Sampling duration =
1 second
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-8
0
0.1
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0.5
seconds
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1
An undersampled signal
sin(28t), SR = 8.5 Hz
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
0
0.2
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0.8
1
1.2
1.4
1.6
1.8
2
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.
Fourier series
• Periodic functions and signals may be
expanded into a series of sine and cosine
functions
http://www.falstad.com/fourier/j2/
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:

H  f    h t e 2ift dt


h t    H  f e 2ift df

The Fourier Transform
• A transform takes one function (or signal)
and turns it into another function (or signal)
• The Discrete Fourier Transform:
N 1
H n   hk e 2ikn N
k 0
1 N 1
hk   H n e 2ikn N
N n 0
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!
Famous Fourier Transforms
2
1
Sine wave
0
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-2
0
0.2
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0.8
1
1.2
1.4
1.6
1.8
2
300
250
200
Delta function
150
100
50
0
0
20
40
60
80
100
120
Famous Fourier Transforms
0.5
0.4
0.3
Gaussian
0.2
0.1
0
0
5
10
15
20
25
30
35
40
45
50
6
5
4
Gaussian
3
2
1
0
0
50
100
150
200
250
Famous Fourier Transforms
1.5
1
Sinc function
0.5
0
-0.5
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
6
5
4
Square wave
3
2
1
0
-100
-50
0
50
100
Famous Fourier Transforms
1.5
1
Sinc function
0.5
0
-0.5
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
6
5
4
Square wave
3
2
1
0
-100
-50
0
50
100
Famous Fourier Transforms
1
0.8
0.6
Exponential
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
30
25
20
Lorentzian
15
10
5
0
0
50
100
150
200
250
FFT of FID
2
1
0
f = 8 Hz
SR = 256 Hz
T2 = 0.5 s
-1
-2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
70
60
50
40
30
20
10
0
0
20
40
60
80
100
t
F t   sin 2ft  exp  
T 2
120
FFT of FID
2
f = 8 Hz
SR = 256 Hz
T2 = 0.1 s
1
0
-1
-2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
14
12
10
8
6
4
2
0
0
20
40
60
80
100
120
FFT of FID
2
1
0
-1
-2
f = 8 Hz
SR = 256 Hz
T2 = 2 s
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
200
150
100
50
0
0
20
40
60
80
100
120
Effect of changing sample rate
2
1
0
-1
-2
f = 8 Hz
T2 = 0.5 s
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
70
35
60
30
50
25
40
20
30
15
20
10
10
5
0
0
10
20
30
40
50
60
0
Effect of changing sample rate
2
SR = 256 Hz
SR = 128 Hz
1
0
-1
-2
f = 8 Hz
T2 = 0.5 s
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
70
35
60
30
50
25
40
20
30
15
20
10
10
5
0
0
10
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40
50
60
0
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
2
1
0
-1
-2
f = 8 Hz
T2 = .5 s
0
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1
1.2
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1.8
2
0
2
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8
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12
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18
20
70
60
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40
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20
10
0
Effect of changing sampling duration
2
1
ST = 2.0 s
ST = 1.0 s
0
-1
-2
f = 8 Hz
T2 = .5 s
0
0.2
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1
1.2
1.4
1.6
1.8
2
0
2
4
6
8
10
12
14
16
18
20
70
60
50
40
30
20
10
0
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
2
1
0
-1
-2
0
0.2
0.4
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1
1.2
1.4
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1.8
2
200
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50
0
f = 8 Hz
T2 = 2.0 s
0
2
4
6
8
10
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20
Effect of changing sampling duration
2
ST = 2.0 s
ST = 1.0 s
1
0
f = 8 Hz
T2 = 0.1 s
-1
-2
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12
10
8
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4
2
0
Measuring multiple frequencies
3
2
f1 = 80 Hz, T21 = 1 s
f = 90 Hz, T2 = .5 s
1
f3 = 100 Hz, T23 = 0.25 s
2
2
0
-1
-2
-3
SR = 256 Hz
0
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0.8
1
1.2
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1.8
2
120
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20
0
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120
Measuring multiple frequencies
3
2
f1 = 80 Hz, T21 = 1 s
f = 90 Hz, T2 = .5 s
1
f3 = 200 Hz, T23 = 0.25 s
2
2
0
-1
-2
-3
SR = 256 Hz
0
0.2
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0.6
0.8
1
1.2
1.4
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1.8
2
120
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60
40
20
0
0
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100
120
Some useful links
•
•
•
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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