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Fast Fourier Transform

Fast Fourier Transform. Theory. Spectral Analysis of Arbitrary Functions. In general, there is no requirement that f(t) be a periodic function We can force a function to be periodic simply by duplicating the function in time (text fig 5.10)

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Fast Fourier Transform

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  1. Fast Fourier Transform Theory

  2. Spectral Analysis of Arbitrary Functions • In general, there is no requirement that f(t) be a periodic function • We can force a function to be periodic simply by duplicating the function in time (text fig 5.10) • We can transform any waveform to determine it’s Fourier spectrum • Computer software has been developed to do this as a matter of routine. - One such technique is called “Fast Fourier Transform” or FFT- Excel has an FFT routine built in

  3. Fourier Transform Theory harmonic frequency terms fundamental frequency term (after some algebra)

  4. Fourier Transform Theory, cont’d f(t) equation is known as the “Inverse Fourier Transform”

  5. Fourier Transform Pair

  6. is called the “discrete” Fourier Transform. - A sophisticated algorithm called the Fast Fourier Transform or FFT has been developed to compute discrete Fourier transforms very rapidly. - Only restriction is that the value of N be a power of 2 (i.e. 128, 512, 1024, etc.) - Also note that the sampling rate is N/T. So the maximum frequency that we can get meaningful (non aliased) results for will be Thus, k can only run up to (N-1)/2 Sample N (n=N-1) Sample 1 (n=0) 0 T=(N-1) t t=nt

  7. Let’s Clarify with a Fast Fourier Transform Example Let f(t) be the sum of two sinusoids with f1 = 10 Hz f2 = 15 Hz f(t) = A1sin2f1t + A2sin2 f2t = 2 sin210t + 1sin2 15t Let’s transform this.

  8. Use FFT Routine Organic to Excel Let N=128 (=27), T= 1 second, t= T/(N-1) = 1/127 sec. t Start by laying out n index, corresponding time, and f(t) in columns.

  9. FFT Example Then click “analysis tools” under Tools menu to find “Fourier Analysis” function. n t f(t) last values

  10. FFT Example, cont’d The Fourier Analysis Screen will ask for 1. input range. Here enter column containing f(t). 2. output range Here enter column where you want the program to display the transform data. Click OK and you’ll get a column of data.

  11. Here’s our result highlighted in yellow But still need to determine frequency and magnitude

  12. Magnitude is found using the function IMABS Frequency starts at 0 and goes up in increments of 1/Ttrace = 1/1 Hz. Legitimate FFT values only up to f = N/(2T)= 128 Hz/2 = 64 Hz After adding the corresponding frequency column, we notice that the maximum magnitudes correspond to 10 Hz and 15 Hz. We now plot the magnitude column vs frequency out to 64 Hz.

  13. Final Result Time Domain FFT Frequency Domain

  14. Voice Recognition The “ee” sound

  15. Voice Recognition (continued) The “eh” sound

  16. Voice Recognition The “ee” sound

  17. Voice Recognition (continued) The “eh” sound

  18. Voice Recognition (continued) The “ah” sound

  19. Voice Recognition (continued) The “oh” sound

  20. Voice Recognition (continued) The “oo” sound

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