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Physics 434 Module 4 week 2: the FFT Explore Fourier Analysis and the FFT 1 Physics 434 Module 4-FFT - T. Burnett Exploration VI 2 Physics 434 Module 4-FFT - T. Burnett The resonance function The solution t o the damped harmonic oscillator differenti al equation d 2 r d 2 2 r h(t ) 0 Q dt dt is (A is an arbitrary constant) 1 1 h(t ) A exp r t i 1 2 4Q 2Q Its Fourier tr ansform is : H ( ) A 1 iQ r r Note that this is the response function to driving the system at a frequency . 3 Physics 434 Module 4-FFT - T. Burnett Now, go discrete! 4 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 Details from FFT help 5 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 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. 6 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, 2f, 3f … (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 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 2f (Nyquist) The linear combinations for the 4 frequency components eit cos(t ) i sin( t ) 1 0.8 0.6 0.4 0.2 0 -0.2 0 1 2 3 4 -0.4 -0.6 -0.8 -1 sin(ft) cos(ft) sin(2ft) cos(2ft) measure here 0 : h0 h1 h2 h3 cos ft : h0 h2 sin ft : h1 h3 cos( 2ft ) : h0 h1 h2 h3 7 Physics 434 Module 4-FFT - T. Burnett Table from the help Phase information for each of these Negative frequencies! If h(f) is real, then H(f)=H(-f) 8 Physics 434 Module 4-FFT - T. Burnett Plot from the help 9 Physics 434 Module 4-FFT - T. Burnett Study of the demo VI 10 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 Don’t forget that… 11 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 A little bonus-time vs frequency in the news 12 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