S9.

Book for Physicists » Introduction to the mathematics of medical imaging » Mathematical Methods of Linear Model Based Image Processing Procedures » Appendix. » Solution of problems » 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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