Data Processing and Analysis

F(φ(t)) = Φ(f) = ∫ ∞ −∞ f(t)e−2πiftdt F−1(Φ(f)) = φ(t) = ∫ ∞ −∞ Φ(f)edf The notation φ(t) ⊃ Φ(f) is also used to indicate that Φ(f) is the Fourier transform of φ(t). Another common notational convention is that capital letters are used for the names of functions in the frequency domain, and corresponding lower case letters are used for functions in the time domain. In this notation, s is often used in place of f for frequency. Thus F (s) denotes the Fourier transform of f(t). Because these two definitions are very nearly symmetric (note the sign change in the exponent of e), it is possible to use a Fourier transform pair φ(t) ⊃ Φ(f) to construct a second Fourier transform pair Φ(t) ⊃ φ(−f). For example, the transform F(sgn t) = 1 πif translates into F (