Impulse response is a fundamental concept in linear systems and signal processing. It describes how a system responds to an input signal that is an impulse, typically represented as a Dirac delta function. The impulse response characterizes the behavior and characteristics of the system over time.
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The impulse response records the response to a unit impulse, with specified initial or boundary conditions. In continuous time the impulse is a Dirac delta function; in discrete time it is the Kronecker delta . For a linear time-invariant system with zero initial state, the response to a general input is its convolution with the impulse response. The impulse response of a causal filter is zero before the impulse.
For the five-sample moving average, a discrete impulse produces at and zero otherwise. The response lasts five samples, so this is a finite impulse response system. Its frequency response is the finite sum of the five delayed phase contributions: one short input experiment determines how every sustained sinusoid is filtered.