HW 05: Difference between revisions

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|<math>=\sum_{-\infty}^{\infty}\alpha_n \delta_{\frac{n}{T}-f} </math>
|<math>=\sum_{-\infty}^{\infty}\alpha_n \delta\left(\frac{n}{T}-f\right) </math>
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|<math>=\alpha_{fT}\,\!</math>
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Is the last problem done correctly?

Revision as of 21:25, 23 November 2008

Find the following Fourier Transforms

  • F[ejω0t]
  • F[cosω0t]
  • F[αnej2πnt/T]
  • F[sinω0t]

Solutions

F[ejω0t] =ejω0tejωtdt
=ej(ω0ω)tdt
=2π[12πej(ω0ω)tdt]
=2πδ(ω0ω)
F[cosω0t] =ejω0t+ejω0t2ejωtdt
=12(ejω0t+ejω0t)ejωtdt
=12[ej(ω0ω)t+ej(ω0+ω)t]dt
=π[12π(ej(ω0ω)t+ej(ω0+ω)t)dt]
=πδ(ω0ω)+πδ(ω0+ω)
F[sinω0t] =ejω0tejω0t2jejωtdt
=12j(ejω0tejω0t)ejωtdt
=12j(ej(ω0ω)tej(ω0+ω)t)dt
=πj[12π(ej(ω0ω)tej(ω0+ω)t)dt]
=jπδ(ω0ω)+jπδ(ω0+ω)
F[αnej2πnt/T] =(αnej2πnt/T)ejωtdt
=αn(ej2πnt/Tej2πftdt)
=αn(ej2πt(nTf)dt)
=αnδ(nTf)