LTI Systems: Time domain

$$h(t)=T[\delta(t)] \qquad \mbox{impulse response}$$

$$\mbox{Time Invariant}\qquad T[\delta(t-t_0)]=h(t-t_0)$$

$$x(t)=\int x(\sigma)\delta(t-\sigma)d\sigma$$

Integral is a sum of terms so is linear

$$T[\int]=\int T[\;]$$

$$\begin{eqnarray*}T[x(t)]&=&\int_{-\infty}^{\infty}T[x(\sigma)\delta(t-\sigma)]d\... ...d\sigma\\ &=&\int_{-\infty}^{\infty}x(t-\sigma)h(\sigma)d\sigma \end{eqnarray*}$$

This is the convolution integral and that's why impulse response is useful for LTI systems.