# Impulse response

Consider a linear-time-invariant system*H*:[

*Integers*→

*Reals*] → [

*Integers*→

*Reals*]. Define its

**impulse response**to be the output when the input is the Kronecker delta function (an impulse). That output is a signal that we call

*h*. The impulse response of a continuous-time system is similarly defined to be the output when the input is the Dirac delta function. Again, the impulse response is a signal that we call

*h*.

If the input is a time-shifted delta function *x*(*k*)δ
(*n* − *k*), then by linearity and time-invariance,
the output is *x*(*k*)* h*(*n* −
*k*), a time-shifted and scaled version of the impulse response.

Since the impulse contains all frequencies in equal amounts, the impulse response represents the response to all frequencies. It has the same information about the system as the frequency response..