(Signal-to-Frequency ／ Fourier-Transform) Algorithms

• is a way of representing a periodic function as a (possibly infinite) sum of sine and cosine functions
  • for functions that are not periodic, the Fourier transform is used in place of the Fourier series
  • for functions of two variables that are periodic in both variables, the trigonometric basis in the Fourier series is replaced by the spherical harmonics
• is a continuous transformation of a continuous periodic signal
• is analogous to a Taylor series, which represents functions as possibly infinite sums of monomial terms
• cases:
  • periodic function → converts into a discrete exponential function or sine and cosine function
  • non-periodic function → not applicable

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• takes a DISCRETE signal as input and outputs a DISCRETE frequency spectrum
• transforms data in the time or spatial domain into the frequency domain
• should technically be called a Discrete Fourier Series
• is like a Fourier Series on data instead of analytic functions
• converts a finite sequence of equally-spaced samples of a function 𝑓(𝑥) into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency
• is a unitary, invertible, linear transformation
  $\mathcal{F}:C^N\to C^N$
• most often uses the FFT algorithm to compute the DFT
• is the inverse of the Inverse Discrete Fourier Transform (IDFT)

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• takes DISCRETE signal as input and outputs a CONTINUOUS frequency spectrum
• it’s defined by an infinite summation over the discrete-time signal
• while mathematically powerful, the DTFT is not directly computable on a computer because of its continuous nature and infinite summation
• thus Discrete Fourier Transform (DFT) - Discrete Fourier Series is a computationally efficient way to approximate the DTFT

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• takes CONTINUOUS time signal as input and outputs a CONTINUOUS frequency spectrum
• is a continuous transformation of a continuous function into its constituent frequencies
• cases:
  • periodic function → converts its Fourier series in the frequency domain
  • non-periodic function → converts it into a continuous frequency domain
• can ONLY transform nice well-behaved functions that go to 0 at ±∞; otherwise use Laplace Transform

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• is a mathematical method that converts a function of a real variable to a function of a complex variable
• a function 𝑓(𝑡) that can’t be Fourier Transformed can be Laplace Transformed by multiplying the function by a decaying exponential 𝑒^-𝜁𝑡 and a heavyside function 𝐻(𝑡) where 𝜁 is a constant:
  • 𝐹(𝑡) = 𝑓(𝑡)𝑒^-𝜁𝑡𝐻(𝑡)
  • thus:
    • the Laplace transform of 𝑓(𝑡) is the Fourier transform of 𝐹(𝑡)
    • the Laplace transform is a one-sided weight Fourier transform
• a discrete version is Z-Transform

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• converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex valued frequency-domain representation
• it can be considered a discrete-time equivalent of the Laplace transform

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