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== Differentiation == | == Differentiation == | ||
| + | def. | ||
| + | x'(t) = j*w*(j*w) | ||
x(t) = <math>\int\limits_{-\infty}^{\infty}X(jw)e^{(-\jmath wt)}dt</math> | x(t) = <math>\int\limits_{-\infty}^{\infty}X(jw)e^{(-\jmath wt)}dt</math> | ||
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replacing a differentiation operation in the time domain with a multiplication operation in the frequecy domain. | replacing a differentiation operation in the time domain with a multiplication operation in the frequecy domain. | ||
| + | |||
| + | example | ||
| + | |||
| + | x(t) = <math>\int\limits_{-\infty}^{t}g(\tau)d/tau</math> | ||
| + | |||
| + | <math>X(j*w)=G(j*w)*1/jw^+\pi*G(0)*\delta(w)</math> | ||
| + | |||
| + | <math>X(j*w)=1/(j*w)+\pi*G(0)*\delta(w)</math> | ||
Revision as of 04:40, 9 July 2009
Differentiation
def. x'(t) = j*w*(j*w)
x(t) = $ \int\limits_{-\infty}^{\infty}X(jw)e^{(-\jmath wt)}dt $
diffrentiate both sides
x'(t) = d($ \int\limits_{-\infty}^{\infty}X(jw)e^{(-\jmath wt)}dt $)
x'(t) = j*w*(j*w)
importance
replacing a differentiation operation in the time domain with a multiplication operation in the frequecy domain.
example
x(t) = $ \int\limits_{-\infty}^{t}g(\tau)d/tau $
$ X(j*w)=G(j*w)*1/jw^+\pi*G(0)*\delta(w) $
$ X(j*w)=1/(j*w)+\pi*G(0)*\delta(w) $
