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Properties of the Fourier transform

Fourier transform. The Fourier transform of the function f(t) is F(ω). We can write F[f(t)] = F(ω) and F-1[F(ω)]= f(t).



Magnitude spectrum and phase spectrum of a function f(t). Let F(ω) be the Fourier transform of a function f (t). The function F(ω) is, in general, complex. Let us write it as


            F(ω) = R(ω) + i X(ω) = |F(ω)| e i φ(ω)

where |F(ω)| is called the magnitude spectrum of f (t) and φ(ω) is called the phase spectrum of f (t). Since for any complex number x + iy, x + iy = re, the magnitude spectrum is given by


and the phase spectrum is given by


In the table below the functions are periodic with period T, a > 0. Moreover, b, t0, and ω0 =2π/T are real constants and n = 1, 2, .... Furthermore, fe(t) is an even function, fo(t) is an odd function, and u(t) is the step function.

                        f(t)                                                                             F(ω) 


            a1f1(t) + a2f2(t)                                                                   a1F1(ω)+ a2F2(ω)

            f(at)                                                                                         ole4.gif

            f(-t)                                                                                        F(-ω)

            f(t-t0)                                                                                      ole5.gif


             ole6.gif                                                                                     F(ω - ω0)

            f(t) cos ω0t                                                                 ole7.gif

            f(t) sin ω0t                                                                  ole8.gif


            fe(t) = ½[f(t) + f(-t)]                                                           R(ω)

            fo(t) = ½[f(t) - f(-t)]                                                            iX(ω)

            f(t) = fe(t) + fo(t)                                                     F(ω) = R(ω) + iX(ω)


            F(t)                                                                                        2π f(-ω)

             ole9.gif                                                                                     iω F(ω)

            f(n)(t)                                                                                      (iω)nF(ω)

             ole10.gif                                                                          ole11.gif

            -itf(t)                                                                                       ole12.gif

            (-it)nf(t)                                                                                 F(n)(ω)

             ole13.gif                                                                         F1(ω)F2(ω)


            f1(t)f2(t)                                              ole14.gif

            e-atu(t)                                                                                                ole15.gif

            e-a |t|                                                                                                    ole16.gif

             ole17.gif                                                                                                  ole18.gif

             ole19.gif                                                                                      ole20.gif  

             ole21.gif                                                                                                 P2a (ω)

            te-at u(t)                                                                                 ole22.gif

             ole23.gif                                                                                      ole24.gif

            e-atsin bt u(t)                                                                                    ole25.gif


            e-atcos bt u(t)                                                                        ole26.gif

             ole27.gif                                                                                                  ole28.gif

             ole29.gif                                                                                                  ole30.gif  

             ole31.gif                                                                                                  ole32.gif

            δ(t)                                                                                                     1

            δ(t - t0)                                                                                               ole33.gif


            δ(n)(t)                                                                                                 (iω)n 

            u(t)                                                                                                     πδ(ω) + 1/(iω)

            u(t - t0)                                                                                   ole34.gif

            1                                                                                                         2π δ(ω)

            t                                                                                                          2πi δʹ(ω)

            tn                                                                                                        2πin δ(n)(ω)

             ole35.gif                                                                                                 2π δ(ω - ω0)

            cos ω0t                                                                                   π [δ(ω - ω0) + δ(ω + ω0)]

            sin ω0t                                                                                    -iπ [δ(ω - ω0) - δ(ω + ω0)]

            sin ω0t u(t)                                                      ole36.gif

            cos ω0t u(t)                                         ole37.gif

            t u(t)                                                                                      iπ δʹ(ω) - 1/ω2

            1/t                                                                                           πi - 2πi u(ω)


            1/tn                                                                               ole38.gif


            sgn t                                                                                                                2/iω

ole39.gif                                                                                      ole40.gif

Other properties:





  Hsu. Fourier Analysis

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