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Fast Fourier Transforms

Fast Fourier Transforms. A real sinusoid and its FFT. Here is the storage scheme for an 8 cell fft. Remember that the upper half stores the negative frequencies. This is the highest frequency according to the Nyquist criteria.

andrew-livingston
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Fast Fourier Transforms

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  1. Fast Fourier Transforms

  2. A real sinusoid and its FFT

  3. Here is the storage scheme for an 8 cell fft. Remember that the upper half stores the negative frequencies.

  4. This is the highest frequency according to the Nyquist criteria.

  5. The broader it is in the time domain, the narrower it is in the frequency domain, confirming uncertainty.

  6. A wavefrom whose wavelength is NOT and integer fraction of the size of the buffer, which is 64. The truncation at the end introduces other frequency components.

  7. I can improve this somewhat by doing the FFT in a 4x64 = 256 cell buffer.

  8. FFT of the rectangular window (left) and the Hanning window (right).

  9. By multiplying the space domain data by a Hanning window (above) we get a FFT with less noise (below).

  10. The complex waveforms of quantum mechanics allow us to tell the direction that the pulse is propagating throught the FFT

  11. Positive wavelengths Negative wavelengths

  12. Using this, I can scale the axis of my frequency domain plot to tell me the wavelength content.

  13. The negative going pulse is actually at 0.9 – 1 = -.1

  14. Of course, everywhere in quantum mechanics, the frequency domain is written as a function of k. .

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