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System

y(t) = 5x(t)

Unit Impulse Response

$ x(t) = \delta(t) $ $ h(t) = 5\delta(t) $

System Function

$ y(t) = \int^{\infty}_{-\infty} h(\tau) * x(\tau) d\tau, $
$ y(t) = \int^{\infty}_{-\infty} 5\delta(\tau) * e^{-jw(t -\tau)} d \tau, $
$ y(t) = e^{jwt} \int^{\infty}_{-\infty} 5 \delta(\tau) e^{-jw\tau} d\tau $
$ H(s) = \int^{\infty}_{-\infty} 5 \delta(\tau) e^{-jw\tau} d\tau $
$ H(s) = 5 e^{-jw0} $
$ H(s) = 5\, $

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