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Physics 434 Module 4 week 2: the FFT

Physics 434 Module 4 week 2: the FFT. Explore Fourier Analysis and the FFT. Exploration VI. The resonance function. Note that this is the response function to driving the system at a frequency . Now, go discrete!.

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Physics 434 Module 4 week 2: the FFT

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  1. Physics 434 Module 4 week 2: the FFT Explore Fourier Analysis and the FFT Physics 434 Module 4-FFT - T. Burnett

  2. Exploration VI Physics 434 Module 4-FFT - T. Burnett

  3. The resonance function • Note that this is the response function to driving the system at a frequency . Physics 434 Module 4-FFT - T. Burnett

  4. Now, go discrete! • Parameters: total digitizing time T, sample frequency fs implies time interval t = 1/ fs, number of samples n = T fs Physics 434 Module 4-FFT - T. Burnett

  5. 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 FFT[X] is complex and returns in one complex array • Y = YRe + jYIm Physics 434 Module 4-FFT - T. Burnett

  6. Comments • There are n real numbers input, but ncomplex 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  f=1/T. all are harmonics of this • Largest: period is 2 t  fN=n f/2.(This is the Nyquist frequency!) • How many are there? 0,f, 2f, 3f … (n/2)f or 1+n/2 different frequencies (assume m is even). That is, for n=4, there are 3 different frequencies. What is missing? Physics 434 Module 4-FFT - T. Burnett

  7. 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 Physics 434 Module 4-FFT - T. Burnett

  8. Table from the help Phase information for each of these Negative frequencies! If h(f) is real, then H(f)=H(-f) Physics 434 Module 4-FFT - T. Burnett

  9. Plot from the help Physics 434 Module 4-FFT - T. Burnett

  10. Study of the demo VI • Verify negative frequencies • See if the phase at zero and Nyquist frequency is 0. • If not enough samples (Nyquist <= 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) Physics 434 Module 4-FFT - T. Burnett

  11. Don’t 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 Physics 434 Module 4-FFT - T. Burnett

  12. A little bonus-time vs frequency in the news • Results from the CDF experiment at the Tevatron • Bs mixing requires measuring a damped sine wave. Physics 434 Module 4-FFT - T. Burnett

  13. Physics 434 Module 4-FFT - T. Burnett

  14. Physics 434 Module 4-FFT - T. Burnett

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