| Line 4: | Line 4: | ||
*<math>X(w) = \int{x(t)*e^{-jwt} dt }</math> | *<math>X(w) = \int{x(t)*e^{-jwt} dt }</math> | ||
** <span style="color:green">Careful here: the symbol <math>~_*</math> is for convolution, not multiplication.</span>--[[User:Mboutin|Mboutin]] 20:18, 1 September 2009 (UTC) | ** <span style="color:green">Careful here: the symbol <math>~_*</math> is for convolution, not multiplication.</span>--[[User:Mboutin|Mboutin]] 20:18, 1 September 2009 (UTC) | ||
| − | *<math>x(t) = | + | *<math>x(t) = \frac{1}{2\pi}\int{X(w)*e^{jwt} dw }</math> |
Duality Property | Duality Property | ||
* <math>'''{x(t)\stackrel{\text{CTFT}}{\longrightarrow}X(f)}'''</math> | * <math>'''{x(t)\stackrel{\text{CTFT}}{\longrightarrow}X(f)}'''</math> | ||
| − | * <math>'''{X(t) | + | * <math>'''{X(t)\stackrel{\text{CTFT}}{\longrightarrow}x(-f)}'''</math> |
Example | Example | ||
| Line 18: | Line 18: | ||
Cosine and Sine Functions | Cosine and Sine Functions | ||
| − | *<math>cos(t) = 0.5 . ( delta(f - f0) + delta(f + f0) )</math> | + | *<math>\cos(t) = 0.5 . ( \delta(f - f0) + \delta(f + f0) )</math> |
*<math>sin(t) = 0.5 i .( delta(f + f0) - delta(f - f0))</math> | *<math>sin(t) = 0.5 i .( delta(f + f0) - delta(f - f0))</math> | ||
Rept and Comb Functions | Rept and Comb Functions | ||
| − | * <math>Rept(x(t)) = x(t) * | + | * <math>Rept(x(t)) = x(t) * \sum_{k=-\infty}^\infty(\delta(t-kT))</math> |
*<math>Comb(x(t)) = x(t) . sum(delta(t-kT))</math> | *<math>Comb(x(t)) = x(t) . sum(delta(t-kT))</math> | ||
Revision as of 16:21, 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.
