Impulse response

In signal processing, the **impulse response**, or **impulse response function (IRF)**, of a dynamic system is its output when presented with a brief input signal, called an impulse. More generally, an impulse response is the reaction of any dynamic system in response to some external change. In both cases, the impulse response describes the reaction of the system as a function of time (or possibly as a function of some other independent variable that parameterizes the dynamic behavior of the system).

In all these cases, the dynamic system and its impulse response may be actual physical objects, or may be mathematical systems of equations describing such objects.

Since the impulse function contains all frequencies, the impulse response defines the response of a linear time-invariant system for all frequencies.

## Mathematical considerations

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 or continuous time. The impulse can be modeled as a Dirac delta function for continuous-time systems, or as the Kronecker delta for discrete-time systems. The Dirac delta represents the limiting case of a 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 idealisation. 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* (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.)

To determine an time-domain output in response to an arbitrary input function requires the convolution of the input function with the impulse response. The reason this works is because one can think of the input function as an infinite number of impulses and the response is simply the sum of the individual impulse responses.

It is usually easier to analyze systems using transfer functions 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.

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.

## Convolution

τ
.

{\displaystyle \tau.}
2. Reflect one of the functions:

g
(
τ
)

{\displaystyle g(\tau)}

→

g
(
−
τ
)
.

{\displaystyle g(-\tau).}
3. Add a time-offset, t, which allows

g
(
t
−
τ
)

{\displaystyle g(t-\tau)}

 to slide along the

τ

{\displaystyle \tau}

-axis.
4. Start t at −∞ and slide it all the way to +∞. Wherever the two functions intersect, find the integral of their product. In other words, compute a sliding, weighted-sum of function

f
(
τ
)
,

{\displaystyle f(\tau),}

 where the weighting function is

g
(
−
τ
)
.

{\displaystyle g(-\tau).}

The resulting waveform (not shown here) is the convolution of functions f and g.
If *f*(*t*) is a unit impulse, the result of this process is simply *g*(*t*). Formally:

∫

−
∞

∞

δ
(
τ
)
g
(
t
−
τ
)

d
# τ
g
(
t
)

{\displaystyle \int\_{-\infty}^\infty \delta(\tau) g(t - \tau)\, d\tau = g(t)}

g
(
τ
)
,

{\displaystyle \ g(\tau),}

 is an even function

(

g
(
−
τ
# )
g
(
τ
)

)
,

{\displaystyle (\ g(-\tau)=g(\tau)\ ),}

 so convolution is equivalent to correlation. A snapshot of this "movie" shows functions

g
(
t
−
τ
)

{\displaystyle g(t-\tau)}

 and

f
(
τ
)

{\displaystyle f(\tau)}

 (in blue) for some value of parameter

t
,

{\displaystyle t,}

 which is arbitrarily defined as the distance from the

# τ
0

{\displaystyle \tau=0}

 axis to the center of the red pulse. The amount of yellow is the area of the product

f
(
τ
)
⋅
g
(
t
−
τ
)
,

{\displaystyle f(\tau)\cdot g(t-\tau),}

 computed by the convolution/correlation integral. The movie is created by continuously changing

t

{\displaystyle t}

 and recomputing the integral. The result (shown in black) is a function of

t
,

{\displaystyle t,}

 but is plotted on the same axis as

τ
,

{\displaystyle \tau,}

f
(
τ
)

{\displaystyle f(\tau)}

 could represent the response of an RC circuit to a narrow pulse that occurs at

# τ

{\displaystyle \tau=0.}

 In other words, if

g
(
τ
# )
δ
(
τ
)
,

{\displaystyle g(\tau)=\delta(\tau),}

 the result of convolution is just

f
(
t
)
.

{\displaystyle f(t).}

 But when

g
(
τ
)

{\displaystyle g(\tau)}

 is the wider pulse (in red), the response is a "smeared" version of

f
(
t
)
.

{\displaystyle f(t).}

 It begins at

# t
−
0.5
,

{\displaystyle t=-0.5,}

 because we defined

t

{\displaystyle t}

 as the distance from the

# τ
0

{\displaystyle \tau=0}