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Comment: Did you mean to put this on the page itself? Right now, it is on the "talk" page associated with it. --[[User:Mboutin|Mboutin]] 15:09, 1 September 2009 (UTC) | Comment: Did you mean to put this on the page itself? Right now, it is on the "talk" page associated with it. --[[User:Mboutin|Mboutin]] 15:09, 1 September 2009 (UTC) | ||
| − | + | CTFT ( Continuous Time Fourier Transform ) | |
| − | + | Equations** | |
| − | <math>X(w)= | + | *<math>X(w) = \int{x(t)*exp(-jwt) dt }</math> |
| − | <math> | + | *<math>x(t) = (1/2pi)\int{X(w)*exp(jwt) dw }</math> |
| + | |||
| + | Duality Property | ||
| + | * <math>'''{x(t)-CTFT->X(f)}'''</math> | ||
| + | * <math>'''{X(t)-CTFT->x(-f)}'''</math> | ||
| + | |||
| + | Example | ||
| + | *<math>delta(t-t0) ->CTFT-> exp(-j2pi.f.t0)</math> | ||
| + | *<math>exp(j.2pi.f0t) -> CTFT -> delta(f-f0)</math> | ||
| + | Another Example: | ||
| + | *<math>rect(t) -> CTFT -> sinc(f)</math> | ||
| + | *<math>sinc(t) -> CTFT -> (rect(-f) = rect(f))</math> | ||
| + | |||
| + | Cosine and Sine Functions | ||
| + | *<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> | ||
| + | |||
| + | Rept and Comb Functions | ||
| + | * <math>Rept(x(t)) = x(t) * sum(delta(t-kT))</math> | ||
| + | *<math>Comb(x(t)) = x(t) . sum(delta(t-kT))</math> | ||
| + | |||
| + | |||
| + | |||
| + | ---------------------------------------- | ||
| + | DTFT ( Discrete Time Fourier Transform ) | ||
| + | |||
| + | * <math>X(w) = \sum{x(n)*exp(-jwn) dn }</math> | ||
| + | * <math>x(t) = (1/2pi)\int{X(w)*exp(jwt) dw }</math> | ||
| + | * Note that x[n] is always periodic with 2pi | ||
| + | |||
| + | I will add more later. | ||
Revision as of 11:46, 1 September 2009
Comment: Did you mean to put this on the page itself? Right now, it is on the "talk" page associated with it. --Mboutin 15:09, 1 September 2009 (UTC)
CTFT ( Continuous Time Fourier Transform )
Equations**
- $ X(w) = \int{x(t)*exp(-jwt) dt } $
- $ x(t) = (1/2pi)\int{X(w)*exp(jwt) dw } $
Duality Property
- $ '''{x(t)-CTFT->X(f)}''' $
- $ '''{X(t)-CTFT->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(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.
