fourier theory
DESCRIPTION
Fourier theory. A sine wave. 5*sin (2 4t). Amplitude = 5. Frequency = 4 Hz. seconds. A sine wave signal. 5*sin(2 4t). Amplitude = 5. Frequency = 4 Hz. Sampling rate = 256 samples/second. Sampling duration = 1 second. seconds. An undersampled signal. The Nyquist Frequency. - PowerPoint PPT PresentationTRANSCRIPT
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-8
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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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-2
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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
0
0.5
1
1.5
2sin(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.
DFT
• 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!
Fast Fourier Transform
• an efficient DFT algorithm
• used by Gauss in 18??
• published by Cooley & Tukey in 1965
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Famous Fourier Transforms
Sine wave
Delta function
Famous Fourier Transforms
0 5 10 15 20 25 30 35 40 45 500
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0.2
0.3
0.4
0.5
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1
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Gaussian
Gaussian
Famous Fourier Transforms
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1
1.5
-100 -50 0 50 1000
1
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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
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0.5
1
1.5
-100 -50 0 50 1000
1
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5
6
Famous Fourier Transforms
Exponential
Lorentzian
0 50 100 150 200 2500
5
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25
30
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
FFT
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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
Tt
fttF
FFT
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2
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14
f = 8 HzSR = 256 HzT2 = 0.1 s
FFT
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-1
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1
2
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f = 8 Hz SR = 256 HzT2 = 2 s
Effect of changing sample rate
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10
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60
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
0 10 20 30 40 50 600
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70
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-1
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5
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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
Effect of changing sampling duration
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-1
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1
2
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70
f = 8 Hz T2 = .5 s
Effect of changing sampling duration
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-1
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1
2
0 2 4 6 8 10 12 14 16 18 200
10
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70
ST = 2.0 sST = 1.0 s
f = 8 HzT2 = .5 s
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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f = 8 Hz T2 = 2.0 s
Effect of changing sampling duration
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-1
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ST = 2.0 sST = 1.0 s
f = 8 Hz T2 = 0.1 s
Measuring multiple frequencies
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-1
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3
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f1 = 80 Hz, T21 = 1 s
f2 = 90 Hz, T22 = .5 s
f3 = 100 Hz, T2
3 = 0.25 s
SR = 256 Hz