Indirect Fourier transformation

In a Fourier transformation (FT), the Fourier transformed function f ^ ( s ) {\displaystyle {\hat {f}}(s)} is obtained from f ( t ) {\displaystyle f(t)} by: f ^ ( s ) = ∫ − ∞ ∞ f ( t ) e − i s t d t {\displaystyle {\hat {f}}(s)=\int _{-\infty }^{\infty }f(t)e^{-ist}dt} where i {\displaystyle i} is defined as i 2 = − 1 {\displaystyle i^{2}=-1} . f ( t ) {\displaystyle f(t)} can be obtained from f ^ ( s ) {\displaystyle {\hat {f}}(s)} by inverse FT: f ( t ) = 1 2 π ∫ − ∞ ∞ f ^ ( s ) e i s t d t {\displaystyle f(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\hat {f}}(s)e^{ist}dt} s {\displaystyle s} and t {\displaystyle t} are inverse variables, e.g.

Source: Wikipedia — Indirect Fourier transformation (CC BY-SA 4.0)

Indirect Fourier transformation

In a Fourier transformation (FT), the Fourier transformed function f ^ ( s ) {\displaystyle {\hat {f}}(s)} is obtained from f ( t ) {\displaystyle f(t)} by: f ^ ( s ) = ∫ − ∞ ∞ f ( t ) e − i s t d t {\displaystyle {\hat {f}}(s)=\int _{-\infty }^{\infty }f(t)e^{-ist}dt} where i {\displaystyle i} is defined as i 2 = − 1 {\displaystyle i^{2}=-1} . f ( t ) {\displaystyle f(t)} can be obtained from f ^ ( s ) {\displaystyle {\hat {f}}(s)} by inverse FT: f ( t ) = 1 2 π ∫ − ∞ ∞ f ^ ( s ) e i s t d t {\displaystyle f(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\hat {f}}(s)e^{ist}dt} s {\displaystyle s} and t {\displaystyle t} are inverse variables, e.g.

Source: Wikipedia "Indirect Fourier transformation" · CC BY-SA 4.0

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