A Linear Time Invariant system with an impulse response

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A Linear Time Invariant system with an impulse response h(t) produces output y(t) when input x(t)is applied. When the input x(t - τ) is applied to a system with impulse response h(t - τ),then output will be

1. y(t)
2. y(2(t - τ))
3. y(t - τ)
4. y(t - 2τ)

Correct Option: D

Laplace t ransform of impulse response of a system is its transfer function.
∴ G₁ (s) = L[h(t)] = H(s)
⇒ y₁ (s) = H(s) X(s)
G₁ (s) = L[h(t – τ)] = e^{– sτ}H(s)
⇒ y₂ (s) = G₂(s) L[x(t – τ)]
= G₂ (s) e^{– sτ} X(s)
= e^–2sτ H(s)X(s)
= e^–2sτ Y(s)
∴ y₂ (t) = y(t – 2τ)
Alternately
x (t) → h (t) → y(t) = x(t). h(t) ...(i)
x (t – τ) → h(t – z) → y' ...(ii)
For system 1.
y(t) = x(t) h(t)
Z-transform y(z) = x(z) H(z) ...(iii)
For system 2.
y'(t) = x(t – τ) h(t – τ)
Z-transform y'(τ) = z^–1 x(z) z^–1
H(z) = z^–2 x(z) H(z) ...(putting from (iii)
y'(z) = z^–2y(z)
∴ Inverse Z - transform = y'(t) = y(t– 2τ)