| (2 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
== 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> | ||
| Line 12: | Line 14: | ||
importance | importance | ||
| − | 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 for easier Fourier transform enalysis |
| + | |||
| + | example | ||
| + | |||
| + | x(t)=u(t) | ||
| + | |||
| + | <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> | ||
Latest revision as of 04:45, 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 for easier Fourier transform enalysis
example
x(t)=u(t)
$ X(j*w)=G(j*w)*(1/jw)+\pi*G(0)*\delta(w) $
$ X(j*w)=(1/(j*w))+\pi*G(0)*\delta(w) $
