Difference between revisions of "Student summary CT DT Fourier transform ECE438F09" - Rhea

Revision as of 15:31, 1 September 2009

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.

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