Difference between revisions of "Fourier"

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* [https://github.com/jakevdp/nfft nfft python package] Jake VDP pure python implementation  
 
* [https://github.com/jakevdp/nfft nfft python package] Jake VDP pure python implementation  
  
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==Harmonics and Harmonic Analysis==
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* [https://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem Nyquist sampling theorem] - can only measure frequencies that are half the sampling rate. E.g., to detect a 10 kHz signal you have to sample at 20kHz or else you'll get aliasing/Moire pattern/beat phenomenon in wave interference.
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* [https://www.dspguide.com/ch11/5.htm Harmonics] - The Scientist and Engineer's Guide to Digital Signal Processing, chapter 11 - NICE FREE BOOK!
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* Signal is periodic with frequency f
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* The frequencies composing the signal are integer multiples of f, i.e., f, 2f, 3f, etc.
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* The first harmonic is f, the second harmonic is 2f
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* The first harmonic is given a special name - the fundamental frequency
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* If the signal is symmetrical (Peak and trough of sine wave are mirror images of each other across horizontal axis) THE EVEN HARMONICS WILL HAVE A VALUE OF ZERO. In other words, ASYMMETRICAL distortion of signal produces fundamental plus BOTH even and odd harmonics.
  
 
==See Also==
 
==See Also==

Latest revision as of 18:10, 13 September 2019

Vocab

  • signal - function in time
    • musical chord can be expressed in terms ofthe volume and frequencies of constituent notes
  • magnitude - amount of a given frequency present in the original signal
  • phase offset of the basic sinusoid

General

  • Some differential equations are easier to analyze in the frequency domain. After performing the desired operations, transformation of the result can be made back to the time domain.
    • Differentiation in time domain corresponds to multiplication in the frequency domain.
    • Convolution is also a multiplication operation
  • Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa
  • The Fourier transform of a Gaussian function is another Gaussian function.
  • The fourier transform of a function f is denoted by f hat.

Types

NFFT

  1. Transforms a sequence of complex numbers
  2. into another sequence of complex numbers
  3. defined by
  4. for

Implementations

Harmonics and Harmonic Analysis

  • Nyquist sampling theorem - can only measure frequencies that are half the sampling rate. E.g., to detect a 10 kHz signal you have to sample at 20kHz or else you'll get aliasing/Moire pattern/beat phenomenon in wave interference.
  • Harmonics - The Scientist and Engineer's Guide to Digital Signal Processing, chapter 11 - NICE FREE BOOK!
  • Signal is periodic with frequency f
  • The frequencies composing the signal are integer multiples of f, i.e., f, 2f, 3f, etc.
  • The first harmonic is f, the second harmonic is 2f
  • The first harmonic is given a special name - the fundamental frequency
  • If the signal is symmetrical (Peak and trough of sine wave are mirror images of each other across horizontal axis) THE EVEN HARMONICS WILL HAVE A VALUE OF ZERO. In other words, ASYMMETRICAL distortion of signal produces fundamental plus BOTH even and odd harmonics.

See Also

  • Harmonic analysis - a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves.
  • Convolution - a mathematical operation on two functions (f and g) that produces a third function expressing how the shape of one is modified by the other.
  • Z-transform - Not to be confused with a z-score from statistics