Transform´ ee de Fourier
D´efinitions : F(ω) =R+∞
−∞ f(t)e−jω(t)dt f(t) = 2π1 R+∞
−∞ F(ω)e+jωtdω Propri´et´es :
inverse F(t) 2πf(−ω)
lin´earit´e af1(t) +bf2(t) aF1(ω) +bF2(ω)
homoth´etie f(aT) |a|1 F(ωa)
d´ecalage (temporel) f(t−T) F(ω)e−jωT
d´ecalage (fr´equentiel) f(t)ejΩt F(ω−Ω)
modulation f(t) cos Ωt 12(F(ω+ Ω) +F(ω−Ω))
d´erivation (temporelle) dtdf(t) jω F(ω)
d´erivation (fr´equentielle) −j t f(t) dωd F(ω)
int´egration Rt
−∞f(u)du F(ω)jω +F(0)πδ(ω) convolution (temporelle) f1(t)∗f2(t) F1(ω)F2(ω) convolution (fr´equentielle) f1(t)f2(t) 2π1F1(ω)∗F2(ω)
fonction conjugu´ee f∗(t) F∗(−ω)
fonction r´eelle f(t)r´eelle <[F(ω)]paire et=[F(ω)]impaire fonction imaginaire f(t)imaginaire <[F(ω)]impaire et=[F(ω)]paire
fonction paire f(t)paire F(ω)paire
fonction impaire f(t)impaire F(ω)impaire
Identit´e de Parseval : R+∞
−∞|f(t)|2dt= 2π1 R+∞
−∞|F(ω)|2dω . . . et une identit´e similaire : R+∞
−∞ f(t)g∗(t)dt=2π1 R+∞
−∞ F(ω)G∗(ω)dω
Dictionnaire de transform´ees :
f(t) = 1 F(ω) = 2π δ(ω)
f(t) =δ(t) F(ω) = 1
f(t) = sign(t) F(ω) = jω2
f(t) = 1+(t) F(ω) = jω1 +πδ(ω)
f(t) =sin(πtB)πtB F(ω) = B1(1+(ω+πB)−1+(ω−πB)) f(t) =hsin(πtB)
πtB
i2
F(ω) = B1 h
1−2πB|ω|i
·(1+(ω+ 2πB)−1+(ω−2πB)) f(t) =Pδ(t−nT) F(ω) = 2πT Pδ(ω−n2πT )
f(t) =ejω0t F(ω) = 2π δ(ω−ω0)
f(t) = cosω0t F(ω) =π(δ(ω+ω0) +δ(ω−ω0)) f(t) = sinω0t F(ω) =j π(δ(ω+ω0)−δ(ω−ω0))
