fourier theory made easy (?)
DESCRIPTION
Fourier theory made easy (?). 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. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Fourier theory made easy (?)](https://reader035.vdocument.in/reader035/viewer/2022062314/56813270550346895d99093e/html5/thumbnails/1.jpg)
Fourier theory made easy (?)
![Page 2: Fourier theory made easy (?)](https://reader035.vdocument.in/reader035/viewer/2022062314/56813270550346895d99093e/html5/thumbnails/2.jpg)
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 (24t)
Amplitude = 5
Frequency = 4 Hz
seconds
A sine wave
![Page 3: Fourier theory made easy (?)](https://reader035.vdocument.in/reader035/viewer/2022062314/56813270550346895d99093e/html5/thumbnails/3.jpg)
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(24t)
Amplitude = 5
Frequency = 4 Hz
Sampling rate = 256 samples/second
seconds
Sampling duration =1 second
A sine wave signal
![Page 4: Fourier theory made easy (?)](https://reader035.vdocument.in/reader035/viewer/2022062314/56813270550346895d99093e/html5/thumbnails/4.jpg)
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
2sin(28t), SR = 8.5 Hz
An undersampled signal
![Page 5: Fourier theory made easy (?)](https://reader035.vdocument.in/reader035/viewer/2022062314/56813270550346895d99093e/html5/thumbnails/5.jpg)
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.
![Page 6: Fourier theory made easy (?)](https://reader035.vdocument.in/reader035/viewer/2022062314/56813270550346895d99093e/html5/thumbnails/6.jpg)
http://www.falstad.com/fourier/j2/
Fourier series
• Periodic functions and signals may be expanded into a series of sine and cosine functions
![Page 7: Fourier theory made easy (?)](https://reader035.vdocument.in/reader035/viewer/2022062314/56813270550346895d99093e/html5/thumbnails/7.jpg)
The Fourier Transform
• A transform takes one function (or signal) and turns it into another function (or signal)
![Page 8: Fourier theory made easy (?)](https://reader035.vdocument.in/reader035/viewer/2022062314/56813270550346895d99093e/html5/thumbnails/8.jpg)
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
![Page 9: Fourier theory made easy (?)](https://reader035.vdocument.in/reader035/viewer/2022062314/56813270550346895d99093e/html5/thumbnails/9.jpg)
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
![Page 10: Fourier theory made easy (?)](https://reader035.vdocument.in/reader035/viewer/2022062314/56813270550346895d99093e/html5/thumbnails/10.jpg)
• 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
Niknnk
N
k
Niknkn
eHN
h
ehH
![Page 11: Fourier theory made easy (?)](https://reader035.vdocument.in/reader035/viewer/2022062314/56813270550346895d99093e/html5/thumbnails/11.jpg)
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!
![Page 12: Fourier theory made easy (?)](https://reader035.vdocument.in/reader035/viewer/2022062314/56813270550346895d99093e/html5/thumbnails/12.jpg)
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 1200
50
100
150
200
250
300
Famous Fourier Transforms
Sine wave
Delta function
![Page 13: Fourier theory made easy (?)](https://reader035.vdocument.in/reader035/viewer/2022062314/56813270550346895d99093e/html5/thumbnails/13.jpg)
Famous Fourier Transforms
0 5 10 15 20 25 30 35 40 45 500
0.1
0.2
0.3
0.4
0.5
0 50 100 150 200 2500
1
2
3
4
5
6
Gaussian
Gaussian
![Page 14: Fourier theory made easy (?)](https://reader035.vdocument.in/reader035/viewer/2022062314/56813270550346895d99093e/html5/thumbnails/14.jpg)
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 1000
1
2
3
4
5
6
Sinc function
Square wave
![Page 15: Fourier theory made easy (?)](https://reader035.vdocument.in/reader035/viewer/2022062314/56813270550346895d99093e/html5/thumbnails/15.jpg)
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 1000
1
2
3
4
5
6
![Page 16: Fourier theory made easy (?)](https://reader035.vdocument.in/reader035/viewer/2022062314/56813270550346895d99093e/html5/thumbnails/16.jpg)
Famous Fourier Transforms
Exponential
Lorentzian
0 50 100 150 200 2500
5
10
15
20
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
![Page 17: Fourier theory made easy (?)](https://reader035.vdocument.in/reader035/viewer/2022062314/56813270550346895d99093e/html5/thumbnails/17.jpg)
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 1200
10
20
30
40
50
60
70
f = 8 Hz SR = 256 HzT2 = 0.5 s
2exp2sin
Tt
fttF
![Page 18: Fourier theory made easy (?)](https://reader035.vdocument.in/reader035/viewer/2022062314/56813270550346895d99093e/html5/thumbnails/18.jpg)
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 1200
2
4
6
8
10
12
14
f = 8 HzSR = 256 HzT2 = 0.1 s
![Page 19: Fourier theory made easy (?)](https://reader035.vdocument.in/reader035/viewer/2022062314/56813270550346895d99093e/html5/thumbnails/19.jpg)
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 1200
50
100
150
200
f = 8 Hz SR = 256 HzT2 = 2 s
![Page 20: Fourier theory made easy (?)](https://reader035.vdocument.in/reader035/viewer/2022062314/56813270550346895d99093e/html5/thumbnails/20.jpg)
Effect of changing sample rate
0 10 20 30 40 50 600
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 50 600
5
10
15
20
25
30
35
f = 8 Hz T2 = 0.5 s
![Page 21: Fourier theory made easy (?)](https://reader035.vdocument.in/reader035/viewer/2022062314/56813270550346895d99093e/html5/thumbnails/21.jpg)
Effect of changing sample rate
0 10 20 30 40 50 600
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 50 600
5
10
15
20
25
30
35
SR = 256 HzSR = 128 Hz
f = 8 HzT2 = 0.5 s
![Page 22: Fourier theory made easy (?)](https://reader035.vdocument.in/reader035/viewer/2022062314/56813270550346895d99093e/html5/thumbnails/22.jpg)
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
![Page 23: Fourier theory made easy (?)](https://reader035.vdocument.in/reader035/viewer/2022062314/56813270550346895d99093e/html5/thumbnails/23.jpg)
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 18 200
10
20
30
40
50
60
70
f = 8 Hz T2 = .5 s
![Page 24: Fourier theory made easy (?)](https://reader035.vdocument.in/reader035/viewer/2022062314/56813270550346895d99093e/html5/thumbnails/24.jpg)
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 18 200
10
20
30
40
50
60
70
ST = 2.0 sST = 1.0 s
f = 8 HzT2 = .5 s
![Page 25: Fourier theory made easy (?)](https://reader035.vdocument.in/reader035/viewer/2022062314/56813270550346895d99093e/html5/thumbnails/25.jpg)
Effect of changing sampling duration
• Reducing the sampling duration:– Lowers the frequency resolution– Does not affect the range of frequencies you
can measure
![Page 26: Fourier theory made easy (?)](https://reader035.vdocument.in/reader035/viewer/2022062314/56813270550346895d99093e/html5/thumbnails/26.jpg)
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 18 200
50
100
150
200
f = 8 Hz T2 = 2.0 s
![Page 27: Fourier theory made easy (?)](https://reader035.vdocument.in/reader035/viewer/2022062314/56813270550346895d99093e/html5/thumbnails/27.jpg)
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 18 200
2
4
6
8
10
12
14
ST = 2.0 sST = 1.0 s
f = 8 Hz T2 = 0.1 s
![Page 28: Fourier theory made easy (?)](https://reader035.vdocument.in/reader035/viewer/2022062314/56813270550346895d99093e/html5/thumbnails/28.jpg)
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 1200
20
40
60
80
100
120
f1 = 80 Hz, T21 = 1 s
f2 = 90 Hz, T22 = .5 s
f3 = 100 Hz, T2
3 = 0.25 s
SR = 256 Hz
![Page 29: Fourier theory made easy (?)](https://reader035.vdocument.in/reader035/viewer/2022062314/56813270550346895d99093e/html5/thumbnails/29.jpg)
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 1200
20
40
60
80
100
120
f1 = 80 Hz, T21 = 1 s
f2 = 90 Hz, T22 = .5 s
f3 = 200 Hz, T2
3 = 0.25 s
SR = 256 Hz
![Page 30: Fourier theory made easy (?)](https://reader035.vdocument.in/reader035/viewer/2022062314/56813270550346895d99093e/html5/thumbnails/30.jpg)
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