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

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

3. 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 .

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

5. 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

6. Physics 434 Module 4-FFT - T. Burnett 6 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   f=1/T. all are harmonics of this Largest: period is 2  t  f N =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?

7. 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

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

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

10. 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)

11. 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

12. 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.

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

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