Fourier%20theory%20made%20easy%20(?)

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Fourier theory made easy (?)
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A sine wave
5sin (2?4t)
Amplitude 5
Frequency 4 Hz
seconds
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A sine wave signal
5sin(2?4t)
Amplitude 5
Frequency 4 Hz
Sampling rate 256 samples/second
Sampling duration 1 second
seconds
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An undersampled signal
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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.

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Fourier series

• Periodic functions and signals may be expanded
  into a series of sine and cosine functions

http//www.falstad.com/fourier/j2/
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The Fourier Transform

• A transform takes one function (or signal) and
  turns it into another function (or signal)

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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 dont like integrals
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The Fourier Transform

• A transform takes one function (or signal) and
  turns it into another function (or signal)
• Continuous Fourier Transform

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The Fourier Transform

• A transform takes one function (or signal) and
  turns it into another function (or signal)
• The Discrete Fourier Transform

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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
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Famous Fourier Transforms
Gaussian
Gaussian
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Famous Fourier Transforms
Sinc function
Square wave
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Famous Fourier Transforms
Sinc function
Square wave
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Famous Fourier Transforms
Exponential
Lorentzian
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FFT of FID
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FFT of FID
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FFT of FID
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Effect of changing sample rate
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Effect of changing sample rate
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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

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Effect of changing sampling duration
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Effect of changing sampling duration
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Effect of changing sampling duration

• Reducing the sampling duration
• Lowers the frequency resolution
• Does not affect the range of frequencies you can
  measure

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Effect of changing sampling duration
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Effect of changing sampling duration
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Measuring multiple frequencies
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Measuring multiple frequencies
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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/McadI
  nChem/mcad008/FT4FreeIndDecay.pdf
• Mathcad tutorial for exploring Fourier transforms
  of free-induction decay
• http//lcni.uoregon.edu/fft/fft.ppt
• This presentation