Question

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-$\tau$) is applied to a system with impulse response h(t-$\tau$) the output will be

Answer 1

$\text{Case 1: Y(s) = H(s) . X(s)}$

$\text{Case 2: input = x(t-}\tau \text{)}\Rightarrow \text{X(s) . e}{}^{-s\tau }$

$\text{impulse response = h(t -}\tau )\Rightarrow \text{H(s) . e}{}^{-s\tau }$

$\text{Output: Y'(s) = X(s) . e}{}^{-s\tau }\text{. H(s) e}{}^{-s\tau }$

$\text{Y'(s) = X(s) . H(s) . e}{}^{-s2t}\text{= Y(s) e}{}^{-s2T}$

$\Rightarrow \text{y'(t) = y(t - 2}\tau \text{)}$