Fourier

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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 N complex numbers x_0, \ldots, x_{N-1}
  2. into another sequence of complex numbers X_0, \ldots, X_{N-1}
  3. defined by X_k = \sum_{n=0}^{N-1} x_n e^{-2\pi i p_n \omega_k}
  4. for \quad 0 \leq k \leq N-1,

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