Consider a linear system where (by definition) the response to a general excitation can be obtained by a superposition of simple excitation responses.
One of the simplest excitations is the delta function (or impulse function) which has the important property:
This property states that a general forcing function defined in the interval (t1 , t2) can be expressed as the superposition (or integration) of many delta functions with magnitude positioned throughout the excitation time interval.
Hence, if we define our forcing function f(t) as equaling the sum of delta functions when inside the time interval t1 to t2,
and equaling zero otherwise, the displacement response x(t) of a linear SDOF system subjected to f(t) is then given by,
The function g(t) is the impulse response of the system. By definition, a system's impulse response is equal to x(t) when f(t) is just a single delta function,