Overview
A linear time-invariant system is a system which is linear and time-invariant
Impulse Response
The response of a system to an impulse, is the the following
{% h(t) = H(\delta(t)) %}
Given that
{% x(t) = \int_{-\infty}^{\infty} x(\tau) \delta(t-\tau) d \tau %}
{% y(t) = H[\int_{-\infty}^{\infty} x(\tau) \delta(t-\tau) d \tau] %}
{% y(t) = \int_{-\infty}^{\infty} x(\tau) H(\delta(t-\tau)) d\tau = \int_{-\infty}^{\infty} x(\tau) h(t-\tau) d \tau %}
That is, the response of an LTI system to any signal, can be computed from the impulse response.
The integral given is known as the
convolution.