Fourier

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

• Discrete-time Fourier transform - DTFT
  • Discrete Fourier transform - DFT
    • Non-uniform discrete Fourier transform - NDFT

NFFT

• The nonuniform discrete Fourier transform transforms a sequence of $N$ complex numbers $x_{0},\ldots ,x_{N-1}$ into another sequence of complex numbers $X_{0},\ldots ,X_{N-1}$ defined by

{{NumBlk|:|$X_{k}=\sum _{n=0}^{N-1}x_{n}e^{-2\pi ip_{n}\omega _{k}},\quad 0\leq k\leq N-1,$

Implementations

• nfft python package Jake VDP pure python implementation

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