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S9.


Solution:

 
It is known from the relation between the step response function and the weighting function of a linear system:
$                  w(t)=\frac{\delta h(t)}{\delta t} , i.e. the weak derivative of the step response function gives the weighting function.

$                  w(t)=\frac{\delta h(t)}{\delta t}= h(0)\delta(t)+1(t)\dot{h}(t)

Consequently,
$                  w(t)= \left [ 2-e^{-5t}+2e^{-10t} \right ]_{t=0}\delta(t)+1(t) \frac{d}{dt} \left [ 2-e^{-5t}+2e^{-10t} \right ]

$                  w(t)=3\delta(t)+1(t) \left [ 5 e^{-5t}-20e^{-10t} \right ]

 
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