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Many of the standard properties of the Fourier transform are immediate consequences of this more general framework. [33] For example, the square of the Fourier transform, W 2, is an intertwiner associated with J 2 = −I, and so we have (W 2 f)(x) = f (−x) is the reflection of the original function f.
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FFT – fast Fourier transform. FIP – finite intersection property. FOC – first order condition. FOL – first-order logic. fr – boundary. (Also written as bd or ∂.) Frob – Frobenius endomorphism. FT – Fourier transform. FTA – fundamental theorem of arithmetic or fundamental theorem of algebra.
Fourier transform on finite groups. Discrete Fourier transform (general). The use of all of these transforms is greatly facilitated by the existence of efficient algorithms based on a fast Fourier transform (FFT). The Nyquist–Shannon sampling theorem is critical for understanding the output of such discrete transforms.
List of Fourier-related transforms; Fourier transform on finite groups; Fractional Fourier transform; Continuous Fourier transform; Fourier operator; Fourier inversion theorem; Sine and cosine transforms; Parseval's theorem; Paley–Wiener theorem; Projection-slice theorem; Frequency spectrum
The discrete version of the Fourier transform (see below) can be evaluated quickly on computers using fast Fourier transform (FFT) algorithms. [8] In forensics, laboratory infrared spectrophotometers use Fourier transform analysis for measuring the wavelengths of light at which a material will absorb in the infrared spectrum.
Fig 2: Depiction of a Fourier transform (upper left) and its periodic summation (DTFT) in the lower left corner. The spectral sequences at (a) upper right and (b) lower right are respectively computed from (a) one cycle of the periodic summation of s(t) and (b) one cycle of the periodic summation of the s(nT) sequence.
The 2D Z-transform, similar to the Z-transform, is used in multidimensional signal processing to relate a two-dimensional discrete-time signal to the complex frequency domain in which the 2D surface in 4D space that the Fourier transform lies on is known as the unit surface or unit bicircle.