10/09 - Fourier Transform

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ej2πnt/Tej2πmt/T =ej2πnt/Tej2πmt/Tdt
=ej2π(nm)t/Tdt
=T/2T/2ej2π(nm)t/Tdt Assuming the function is perodic with the period T
=Tδm,n

Fourier Transform

Remember from 10/02 - Fourier Series

  • αm=1TT/2T/2x(t)ej2πmt/Tdt
  • x(t)=x(t+T)=n=αmej2πm/T

If we let T

1T df
nT f Remember f=2πnT
T
n=1T ()df

Definitions

F[x(t)] =X(f) =x(t)ej2πftdt =x(t)ej2πftt
F1[x(t)] =x(t) =X(f)ej2πftdf =X(f)ej2πftf

Examples

ej2πftej2πfλdf =ej2πftej2πftf =δ(tλ)
ej2πtfej2πtf0dt =ej2πtfej2πtf0t =δ(ff0)
F1[F[x(t)]] =[x(λ)ej2πfλdλ]ej2πftdf =X(f)ej2πftdf =x(t)
=x(λ)ej2πf(tλ)dfdλ =x(λ)δ(tλ)dλ =x(t)
=[x(λ)ejωλdλ]ejωt12πdω =x(λ)[12πej(tω)λdω]dλ =x(λ)δ(tω)dλ =x(t)

Sifting property of the delta function

The dirac delta function is defined as any function, denoted as δ(tu), that works for all variables that makes the following equation true: x(t)=x(u)δ(tu)du

  • When dealing with ω, it behaves slightly different than dealing with f.