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Exploring Fourier Analysis and FFT: A Comprehensive Guide for Physics 434

This module delves into the principles of Fourier Analysis and the Fast Fourier Transform (FFT), essential for understanding resonance in physical systems. Cover key concepts like the resonance function, input sequences, and how to execute FFT routines using real-valued inputs. Learn about frequency measurement, Nyquist theorem, and the implications of phase adjustments. This week emphasizes thorough analysis of FFT outputs, converting to amplitude for resonance fits, and additional insights from recent experimental results in particle physics.

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Exploring Fourier Analysis and FFT: A Comprehensive Guide for Physics 434

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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 • 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. 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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