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I'm struggling to understand the following solution of an exercise:
"Let $$a(t)=(1-e^{-t})\sigma{(t)}$$ be the Step Response of an LTI-System. The Impulse Answer $h(t)$ of the system can be obtained using $$h(t)= \frac{d}{dt} a(t) = e^{-t}\sigma{(t)}$$."

My question is: Why does $$\frac{d}{dt}(\sigma(t)-\sigma(t)e^{-t})=e^{-t}\sigma{(t)}$$ and not $$\delta(t)-(\delta(t)e^{-t} + \sigma(t)e^{-t})$$ ?? Or are they the same expressions ?

Thanks for your help !

1 Answers1

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Expressions are identical. $\delta(t)-(\delta(t)e^{-t} + \sigma(t)e^{-t}) = \delta(t)-(\delta(t) + \sigma(t)e^{-t}) = \sigma(t)e^{-t}$. $\delta(t)e^{-t}$ is only not equal to zero for t=0 so you get $\delta(t)e^0=\delta(t)$