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Revision as of 16:31, 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.
