Physics 434 Module 4 week 2: the FFT

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Physics 434 Module 4 week 2 the FFT

• Explore Fourier Analysis and the FFT

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Exploration VI
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The resonance function

• Note that this is the response function to
  driving the system at a frequency ?.

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Now, go discrete!

• Parameters total digitizing time T, sample
  frequency fs implies time interval ?t 1/ fs,
  number of samples n T fs

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Details from FFT help

• The input sequence is real-valued.
• The Real FFT VI executes fast radix-2 FFT
  routines if the size of the input sequence is a
  valid power of 2
• size 2m.
• m 1, 2,, 23.
• If the size of the input sequence is not a power
  of 2 but is factorable as the product of small
  prime numbers, the VI uses a mixed radix
  Cooley-Tukey algorithm to efficiently compute the
  DFT of the input sequence.
• Refer to the Fast FFT Sizes section of Chapter 4,
  Frequency Analysis in the LabVIEW Analysis
  Concepts manual for more information about fast
  FFT input sequence sizes.
• The output sequence Y Real FFTX is complex
  and returns in one complex array
• Y YRe jYIm

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Comments

• There are n real numbers input, but n complex
  numbers output, twice as many real numbers. They
  cannot all be independent!
• Think about which frequencies can be measured,
  from smallest to largest.
• Smallest DC, or average! Frequency is 0
• Next period is T ? ?f1/T. all are harmonics of
  this
• Largest period is 2 ?t ? fNn ?f/2.(This is the
  Nyquist frequency!)
• How many are there? 0,?f, 2?f, 3?f (n/2)?f or
  1n/2 different frequencies (assume m is even).
  That is, for n4, there are 3 different
  frequencies. What is missing?

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Counting frequencies, cont.

• The FT is complex to keep track of two integrals
  sine and cosine! Remember
• Only one component for zero frequency since
  sin(0)0. (No phase if no wiggles)
• The sine also vanishes for the Nyquist
  frequency! Plot is for 4 measurements red for
  ?f, blue 2?f (Nyquist)
• The linear combinations for the 4 frequency
  components

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Table from the help
Phase information for each of these
Negative frequencies! If h(f) is real, then
H(f)H(-f)
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Plot from the help
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Study of the demo VI

• Verify negative frequencies
• See if the phase at zero and Nyquist frequency is
  0.
• If not enough samples (Nyquist lt actual
  frequency, get aliasing
• What determines the spacing of frequencies around
  the resonance? (I.e., ?f)
• What happens when you adjust the phase of the
  input signal? What are reasonable limits for Q
  (especially, small)

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Dont forget that

• This Module is due next week at class time
• We expect extensive analysis in your document
  section.
• You need to convert your FFT output to amplitude
  for the resonance fit, to compare with the Module
  3 results

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A little bonus-time vs frequency in the news

• New results from the CDF experiment at the
  Tevatron, presented at the American Physical
  Society meeting in Hawaii 2 weeks ago
• Bs mixing requires measuring a damped sine wave.

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