Skip to main content

Unit step response for h(t) = exp(-at)

Short answer

The step response is the integral of the impulse response: s(t) = (1/a)(1 - e^(-at)) u(t).

Find the unit step response of a linear time invariant system having impulse response h(t) = exp(-at), for a>0.

The step response is the output when x(t) = u(t):

s(t) = h(t) * u(t) = ∫ h(τ) u(t - τ) dτ = ∫_{-∞}^{t} h(τ) dτ

The given impulse response is e^(-at) for a > 0. For a physical (causal) system, this means h(t) = e^(-at) u(t), which I assume; if it were non-zero for t < 0, the integral would diverge.

t < 0: s(t) = 0.

t ≥ 0: s(t) = ∫₀^t e^(-aτ) dτ = [-e^(-aτ)/a]₀^t = (1 - e^(-at))/a

Answer: s(t) = (1/a)(1 - e^(-at)) u(t)

s(t)
1/a |        ________
    |     _-'
    |   /
    +-----------------> t

The response starts at 0 and approaches 1/a, the DC gain H(0) = ∫h dt = 1/a, with a time constant 1/a.

✓ Answer checked by Claude Opus 5.5 (high effort)

Related questions

All Data Communication chapter 2 questions →