10/09 - Fourier Transform: Difference between revisions

From Class Wiki
Jump to navigation Jump to search
Fonggr (talk | contribs)
No edit summary
Fonggr (talk | contribs)
Line 64: Line 64:
|
|
|<math>=x(t)\,\!</math>
|<math>=x(t)\,\!</math>
|}
{| border="0" cellpadding="0" cellspacing="0"
|-
|<math>\int_{-\infty}^{\infty}e^{j2\pi ft}e^{-j2\pi f\lambda}df</math>
|<math>=\left\langle e^{j2\pi ft}\mid e^{j2\pi ft}\right\rangle_f</math>
|<math>=\delta(t-\lambda)\,\!</math>
|-
|<math>\int_{-\infty}^{\infty}e^{j2\pi tf}e^{-j2\pi tf_0}dt</math>
|<math>=\left\langle e^{j2\pi tf}\mid e^{j2\pi tf_0}\right\rangle_t</math>
|<math>=\delta(f-f_0)\,\!</math>
|}
|}

Revision as of 15:28, 17 November 2008

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

F1[F[x(t)]] =[x(t)ej2πftdt]ej2πftdf
=X(f)ej2πftdf
=x(t)
ej2πftej2πfλdf =ej2πftej2πftf =δ(tλ)
ej2πtfej2πtf0dt =ej2πtfej2πtf0t =δ(ff0)