Editing Impulse response (section)

==Mathematical considerations==
[[Image:unit impulse.gif|thumb|right|Unit sample function]]

{{See also|Vector autoregression#Impulse response|Moving average model#Interpretation}}
Mathematically, how the impulse is described depends on whether the system is modeled in [[discrete-time|discrete]] or [[continuous-time|continuous]] time. The impulse can be modeled as a [[Dirac delta function]] for [[continuous-time]] systems, or as the [[Kronecker delta|discrete unit sample function]] for [[discrete-time]] systems. The Dirac delta represents the limiting case of a [[pulse (signal processing)|pulse]] made very short in time while maintaining its area or integral (thus giving an infinitely high peak). While this is impossible in any real system, it is a useful idealization. In [[Fourier analysis]] theory, such an impulse comprises equal portions of all possible excitation frequencies, which makes it a convenient test probe.

Any system in a large class known as ''linear, time-invariant'' ([[Time-invariant system|LTI]]) is completely characterized by its impulse response. That is, for any input, the output can be calculated in terms of the input and the impulse response. (See [[LTI system theory]].) The impulse response of a [[linear transformation]] is the image of [[Dirac's delta function]] under the transformation, analogous to the [[fundamental solution]] of a [[partial differential operator]].

It is usually easier to analyze systems using [[transfer function]]s as opposed to impulse responses. The transfer function is the [[Laplace transform]] of the impulse response. The Laplace transform of a system's output may be determined by the multiplication of the transfer function with the input's Laplace transform in the [[complex plane]], also known as the [[frequency domain]]. An [[inverse Laplace transform]] of this result will yield the output in the [[time domain]].

To determine an output directly in the time domain requires the [[convolution]] of the input with the impulse response. When the transfer function and the Laplace transform of the input are known, this convolution may be more complicated than the alternative of multiplying two functions in the [[frequency domain]].

The impulse response, considered as a [[Green's function]], can be thought of as an "influence function": how a point of input influences output.