Fourier Transforms & Series Flashcards
FT Definition & Purpose
F(ω) = ∫[∞,-∞] f(t)*e^(-iωt) dt
Tool in signal analysis as it takes function of time and converts it into function of frequency.
Linearity Property
αf(t) + βg(t) -> αF(ω) + βG(ω)
Differentiation Property
df/dt =iωF(ω)
dªf/dtª = (iω)ªF(ω)
Time Shift Property
f(t-τ) = e^(-iωτ)*F(ω)
Symmetry Property
F{f(t)} = 2πf(-ω)
Scaling Property
f(bt) = (1/mag(b))*F(ω/b)
Frequency Shift Property
e^(iω0t) = F(ω - ω0)
FT’s of Even Functions
∫[∞,-∞] f(t)*cos(ωt) dt
FT’s of Odd Functions
-i∫[∞,-∞] f(t)sin(ωt) dt
Fourier Series Definition
Periodic function that can be expressed as an infinite series of sine & cosine functions.
Fundamental Period of a Function
Smallest value of T such that f(t +T) = f(t).
T = Fundamental Period
e.g f(t + 2) = f(t) -> T = 2
Natural Frequency of Fourier Series
ωn = (2πn)/T
Fourier Coefficients (an,bn,a0)
a0 = (2/T)*∫[T/2,-T/2] f(t) dt
an = (2/T)∫[T/2,-T/2] f(t)cos(ωnt) dt
bn = (2/T)∫[T/2,-T/2] f(t)sin(ωnt) dt
Fourier Cosine Series + Use
When f(t) = even, bn = 0
∴ S[f] = (a0/2)Σancos(ωnt)
where:
a0 = (4/T)*∫[T/2,0] f(t) dt
an = (4/T)*∫[T/2,0] f(t)cos(ωnt) dt
Fourier Sine Series + Use
When f(t) = odd, an = 0
∴ S[f] = (a0/2)Σancos(ωnt)
where:
a0 = (4/T)*∫[T/2,0] f(t) dt
bn = (4/T)*∫[T/2,0] f(t)sin(ωnt) dt
