Introduction / Context:
Time shifting is one of the most frequently used properties in Laplace-domain analysis. It enables modeling of delayed inputs and piecewise signals via step functions. The unit step u(t − T) ensures causality of the shifted segment starting at t = T.
Given Data / Assumptions:
- £ denotes the unilateral Laplace transform.
- F(s) = £{f(t)} for a causal f(t) (or one for which the unilateral transform exists).
- We apply a shift by T > 0 with gating u(t − T).
Concept / Approach:
The standard time-shift property states: £{f(t − T)u(t − T)} = e^{-sT} F(s). The exponential factor in s-domain encodes the delay in time-domain.
Step-by-Step Solution:
Definition: £{f(t − T)u(t − T)} = ∫_0^∞ f(t − T)u(t − T) e^{-s t} dt.Let τ = t − T, with t = τ + T; lower limit becomes 0.Integral becomes e^{-sT} ∫_0^∞ f(τ) e^{-s τ} dτ = e^{-sT} F(s).
Verification / Alternative check:
Check with f(t) = u(t): F(s) = 1/s; delay gives e^{-sT}/s, matching tables.
Why Other Options Are Wrong:
e^{+sT}F(s): corresponds to advance, not delay.F(s − T): not a Laplace time-shift rule (that is a frequency shift, not time shift).sF(s), F(s)/s: relate to differentiation/integration in time, not shift.
Common Pitfalls:
Forgetting the gating step u(t − T); without it, the transform may not be unilateral-causal.
Final Answer:
e^{-sT} F(s)
Discussion & Comments