1. Step Response g(t)
g(-1.00) = 0.00
2. Impulse Response h(t)
h(-1.00) = 0.00
3. Local Derivative Visualization
Slope = dg/dt
4. Mathematical Differentiation
g(t) = t [u(t) - u(t-2)]
h(t) = 1·[u(t) - u(t-2)] + t [ δ(t) - δ(t-2) ]
h(t) = [u(t) - u(t-2)] - 2δ(t-2)
h(t) = 1·[u(t) - u(t-2)] + t [ δ(t) - δ(t-2) ]
h(t) = [u(t) - u(t-2)] - 2δ(t-2)
What to notice
- By definition, the step response g(t) = h(t) * u(t). Differentiating both sides yields dg(t)/dt = h(t) * δ(t) = h(t).
- Drag the time cursor. The slope (derivative) of g(t) exactly forms the continuous part of h(t).
- Crucial: If g(t) has an abrupt jump (discontinuity), the derivative is infinite at that instant. This produces a Dirac delta impulse δ(t) in h(t) with an area equal to the jump height.
- Try turning off "Include Impulses at Jumps". The resulting function is just the ordinary derivative and fails to correctly describe the system's full impulse response.