Fourier theory made easy (?)
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-2
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5*sin (24t)
Amplitude = 5
Frequency = 4 Hz
seconds
A sine wave
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5*sin(24t)
Amplitude = 5
Frequency = 4 Hz
Sampling rate = 256
samples/second
seconds
Sampling duration =
1 second
A sine wave signal
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-1.5
-1
-0.5
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0.5
1
1.5
2
sin(28t), SR = 8.5 Hz
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
ift
ift
2
2
• A transform takes one function (or signal)
and turns it into another function (or signal)
• The Discrete Fourier Transform:
The Fourier Transform
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1 N
n
Nikn
nk
N
k
Nikn
kn
eHN
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 ½ hours. Using the FFT, the same task on the same machine took 2.4 seconds!
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Famous Fourier Transforms
Sine wave
Delta function
Famous Fourier Transforms
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0.2
0.3
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0.5
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1
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Gaussian
Gaussian
Famous Fourier Transforms
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0.5
1
1.5
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1
2
3
4
5
6
Sinc function
Square wave
Famous Fourier Transforms
Sinc function
Square wave
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0.5
1
1.5
-100 -50 0 50 1000
1
2
3
4
5
6
Famous Fourier Transforms
Exponential
Lorentzian
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5
10
15
20
25
30
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0.2
0.4
0.6
0.8
1
FFT of FID
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-1
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2
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70
f = 8 Hz SR = 256 HzT2 = 0.5 s
2exp2sin
T
tfttF
FFT of FID
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-1
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14
f = 8 HzSR = 256 HzT2 = 0.1 s
FFT of FID
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-1
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1
2
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200
f = 8 Hz SR = 256 HzT2 = 2 s
Effect of changing sample rate
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70
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-1
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1
2
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5
10
15
20
25
30
35
f = 8 Hz T2 = 0.5 s
Effect of changing sample rate
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10
20
30
40
50
60
70
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-1
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2
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5
10
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30
35
SR = 256 HzSR = 128 Hz
f = 8 HzT2 = 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
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-1
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1
2
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60
70
f = 8 Hz T2 = .5 s
Effect of changing sampling duration
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-1
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2
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ST = 2.0 sST = 1.0 s
f = 8 HzT2 = .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
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-1
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1
2
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200
f = 8 Hz T2 = 2.0 s
Effect of changing sampling duration
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-1
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1
2
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14
ST = 2.0 sST = 1.0 s
f = 8 Hz T2 = 0.1 s
Measuring multiple frequencies
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-2
-1
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3
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120
f1 = 80 Hz, T2
1 = 1 s
f2 = 90 Hz, T2
2 = .5 s
f3 = 100 Hz, T2
3 = 0.25 s
SR = 256 Hz
Measuring multiple frequencies
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-2
-1
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1
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3
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20
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f1 = 80 Hz, T2
1 = 1 s
f2 = 90 Hz, T2
2 = .5 s
f3 = 200 Hz, T2
3 = 0.25 s
SR = 256 Hz
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/FT4FreeIndDecay.pdf
– Mathcad tutorial for exploring Fourier transforms of free-induction decay
• http://lcni.uoregon.edu/fft/fft.ppt
– This presentation