Revision as of 19:12, 2 September 2009 by Zhang253 (Talk | contribs)

CTFT ( Continuous Time Fourier Transform )

Equations**

  • $ X(w) = \int{x(t)*e^{-jwt} dt } $
    • Careful here: the symbol $ ~_* $ is for convolution, not multiplication.--Mboutin 20:18, 1 September 2009 (UTC)
  • $ x(t) = \frac{1}{2\pi}\int{X(w)*e^{jwt} dw } $

Duality Property

  • $ '''{x(t)\stackrel{\text{CTFT}}{\longrightarrow}X(f)}''' $
  • $ '''{X(t)\stackrel{\text{CTFT}}{\longrightarrow}x(-f)}''' $

Example

  • $ delta(t-t0) ->CTFT-> exp(-j2pi.f.t0) $
  • $ exp(j.2pi.f0t) -> CTFT -> delta(f-f0) $

Another Example:

  • $ rect(t) -> CTFT -> sinc(f) $
  • $ sinc(t) -> CTFT -> (rect(-f) = rect(f)) $

Cosine and Sine Functions

  • $ \cos(t) = 0.5 . ( \delta(f - f0) + \delta(f + f0) ) $
  • $ sin(t) = 0.5 i .( delta(f + f0) - delta(f - f0)) $

Rept and Comb Functions

  • $ Rept(x(t)) = x(t) * \sum_{k=-\infty}^\infty(\delta(t-kT)) $
  • $ Comb(x(t)) = x(t) . sum(delta(t-kT)) $



DTFT ( Discrete Time Fourier Transform )

  • $ X(w) = \sum{x(n)*exp(-jwn) dn } $
  • $ x(t) = (1/2pi)\int{X(w)*exp(jwt) dw } $
  • Note that x[n] is always periodic with 2pi

I will add more later.


TA comments: Hamad did a very good job to lead the first recitation. He especially had deep understanding in Duality property of the CTFT/DTFT. He used it to easily solve the CTFT of rect(t) and sinc(t). Furthermore, he handled well with the rept/comb function. This recitation impressed me very much. Thanks, Hamad.


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