System Selection
System Context (Circuit Domain)
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Why do these relate? Because the input \(u(t)\) is the integral of \(\delta(t)\), the step response \(y_s(t)\) is the integral of the impulse response \(h(t)\). Therefore, \(h(t) = \frac{d}{dt}y_s(t)\).
The Core Concept
\( h(t) = \underbrace{y'_{cont}(t)}_{\text{Continuous Derivative}} + \sum \underbrace{\left[y_s(t_k^+) - y_s(t_k^-)\right]}_{\text{Jump Size}} \delta(t - t_k) \)