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.


I think this is a pretty good beginning. But in preparation for the test, I would like students to go even further and try to understand the relationship between the CT and the DT Fourier transforms. More precisely, given a CT signal and a discretization of this signal, "how do their respective Fourier transforms compare?" and "What should the DT Fourier transform look like if the discretization represents the CT signal well?". Should we organize another recitation on that topic?--Mboutin 08:11, 3 September 2009 (UTC)


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Ph.D. on Applied Mathematics in Aug 2007. Involved on applications of image super-resolution to electron microscopy

Francisco Blanco-Silva