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Revision as of 17:15, 26 September 2008
A continuous-time Linear Time-Invariant (LTI) system defined for the purpose of this page will be
$ \ w(t) = 5v(t) $
where v(t) is an input signal dependent on the parameter of time.
The unit impulse response of the system would then simply be
$ \ w(t) = 5\delta(t) $
and the system function H(s) of the system, where
$ \ s = j\omega $
can be determined by taking the Laplace Transform of the system's unit impulse response, h(t).
$ \ L[h(t)] = \int_{-\infty}^{+\infty} h(t)e^{-st}\, dt $
$ \ L[h(t)] = \int_{-\infty}^{+\infty} 5\delta(t)e^{-st}\, dt $
$ \ L[h(t)] = 5\int_{-\infty}^{+\infty} \delta(t)e^{-st}\, dt $
$ \ L[h(t)] = 5\int_{-\infty}^{0^-} \delta(t)e^{-st}\, dt + 5\int_{0^-}^{+\infty} \delta(t)e^{-st}\, dt $
$ \ L[h(t)] = 5\int_{0^-}^{+\infty} \delta(t)e^{-st}\, dt $
$ \ L[h(t)] = \left [ \frac{a}{b} \right ] \quad \left \lbrack \frac{a}{b} \right \rbrack $
