10/09 - Fourier Transform: Difference between revisions

From Class Wiki
Jump to navigation Jump to search
Fonggr (talk | contribs)
Fonggr (talk | contribs)
No edit summary
Line 46: Line 46:
|<math>=X(f)\,\!</math>
|<math>=X(f)\,\!</math>
|<math>=\int_{-\infty}^{\infty} x(t) e^{-j2\pi ft}dt</math>
|<math>=\int_{-\infty}^{\infty} x(t) e^{-j2\pi ft}dt</math>
|<math>=\left \langle x(t) \mid e^{j2\pi ft}\right \rangle</math>
|<math>=\left \langle x(t) \mid e^{j2\pi ft}\right \rangle_t</math>
|-
|-
|<math>F^{-1}[x(t)]\,\!</math>
|<math>F^{-1}[x(t)]\,\!</math>
|<math>=x(t)\,\!</math>
|<math>=x(t)\,\!</math>
|<math>=\int_{-\infty}^{\infty} X(f) e^{j2\pi ft}dt</math>
|<math>=\int_{-\infty}^{\infty} X(f) e^{j2\pi ft}df</math>
|<math>=\left \langle X(f) \mid e^{-j2\pi ft}\right \rangle</math>
|<math>=\left \langle X(f) \mid e^{-j2\pi ft}\right \rangle_f</math>
|}
|}
==Examples==
==Examples==
Line 57: Line 57:
|-
|-
|<math>F^{-1}[F[x(t)]]\,\!</math>
|<math>F^{-1}[F[x(t)]]\,\!</math>
|<math>=</math>
|<math>=\int_{-\infty}^{\infty}\left [ \int_{-\infty}^{\infty} x(t) e^{-j2\pi ft}dt\right ]e^{j2\pi ft} df</math>
|-
|-
|
|
|<math>=n^2 + 2n + 1</math>
|<math>=\int_{-\infty}^{\infty}X(f) e^{j2\pi ft} df</math>
|-
|
|<math>=x(t)\,\!</math>
|}
|}

Revision as of 15:10, 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)