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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