Impulse response

In, the impulse response, or impulse response function (IRF), of a is its output when presented with a brief input signal, called an. 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 of time (or possibly as a function of some other  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 for all frequencies.

Mathematical considerations
Mathematically, how the impulse is described depends on whether the system is modeled in or  time. The impulse can be modeled as a for  systems, or as the  for  systems. The Dirac delta represents the limiting case of a 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 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 .) The impulse response of a is the image of  under the transformation, analogous to the  of a.

It is usually easier to analyze systems using s as opposed to impulse responses. The transfer function is the 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, also known as the. An of this result will yield the output in the.

To determine an output directly in the time domain requires the 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.

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

Practical applications
In practical systems, it is not possible to produce a perfect impulse to serve as input for testing; therefore, a brief pulse is sometimes used as an approximation of an impulse. Provided that the pulse is short enough compared to the impulse response, the result will be close to the true, theoretical, impulse response. In many systems, however, driving with a very short strong pulse may drive the system into a nonlinear regime, so instead the system is driven with a pseudo-random sequence, and the impulse response is computed from the input and output signals.

Loudspeakers
An application that demonstrates this idea was the development of impulse response testing in the 1970s. Loudspeakers suffer from phase inaccuracy, a defect unlike other measured properties such as. Phase inaccuracy is caused by (slightly) delayed frequencies/octaves that are mainly the result of passive cross overs (especially higher order filters) but are also caused by resonance, energy storage in the cone, the internal volume, or the enclosure panels vibrating. Measuring the impulse response, which is a direct plot of this "time-smearing," provided a tool for use in reducing resonances by the use of improved materials for cones and enclosures, as well as changes to the speaker crossover. The need to limit input amplitude to maintain the linearity of the system led to the use of inputs such as pseudo-random s, and to the use of computer processing to derive the impulse response.

Electronic processing
Impulse response analysis is a major facet of, , and many areas of. An interesting example would be internet connections. DSL/Broadband services use techniques to help compensate for signal distortion and interference introduced by the copper phone lines used to deliver the service.

Control systems
In the impulse response is the response of a system to a  input. This proves useful in the analysis of ; the of the delta function is 1, so the impulse response is equivalent to the  of the system's.

Acoustic and audio applications
In acoustic and audio applications, impulse responses enable the acoustic characteristics of a location, such as a concert hall, to be captured. Various packages are available containing impulse responses from specific locations, ranging from small rooms to large concert halls. These impulse responses can then be utilized in applications to enable the acoustic characteristics of a particular location to be applied to target audio.

Economics
In, and especially in contemporary , impulse response functions are used to describe how the economy reacts over time to impulses, which economists usually call , and are often modeled in the context of a. Impulses that are often treated as exogenous from a macroeconomic point of view include changes in, s, and other parameters; changes in the  or other  parameters; changes in  or other  parameters; and changes in , such as the degree of. Impulse response functions describe the reaction of macroeconomic variables such as, , , and  at the time of the shock and over subsequent points in time. Recently, asymmetric impulse response functions have been suggested in the literature that separate the impact of a positive shock from a negative one.