Let the Laplace transform of a function f(t) which exists

Let the Laplace transform of a function f(t) which exists for t > 0 be F₁ (s) and the Laplace transform of its delayed version f(t – π) be F₂ * (s). Let F₁ * (s) be the complex conjugate of F₁ (s) with the Laplace variable set as s = σ + jω.
G(s) = F₂(s).F₁*(s) , then the inverse Laplace transform of G(s) is
|F₁(s)|²

1. an ideal impulse δ(t)
2. an ideal delayed impulse δ(t – τ)
3. an ideal step function u(t)
4. an ideal delayed step function impulse u(t – τ)

F₂ (t) = L{f(t – τ)} = e^{– Sτ} F₁ (S)

G(s) =	e^{– sτ} F₁ (s) . F₁* (s)	= e^{– sτ}
	\|F₁ (s)\|²

G(t) = L^{– 1} {G(S)} = δ(t – τ)

G(s) =	F₂(s).F₁*(s)	, then the inverse Laplace transform of G(s) is
	\|F₁(s)\|²