Editing Impulse response

{{Short description|Output of a dynamic system when given a brief input}}
[[File:Impulse.png|thumb|300px|right|The impulse response from a simple audio system. Showing, from top to bottom, the original impulse, the response after high frequency boosting, and the response after low frequency boosting.]]

In [[signal processing]] and [[control theory]], the '''impulse response''', or '''impulse response function''' ('''IRF'''), of a [[dynamic system]] is its output when presented with a brief input signal, called an [[Dirac delta function|impulse]] ({{math|δ(''t'')}}). 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 (mathematics)|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 (see [[Dirac delta function#Fourier transform|the Fourier transform of the Dirac delta function]], showing infinite frequency bandwidth that the Dirac delta function has), the impulse response defines the response of a [[linear time-invariant system]] for all frequencies.

==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.

==Practical applications==
In practice, it is not possible to perturb a system with a perfect impulse. One can use a brief pulse as a first approximation. Limitations of this approach include the duration of the pulse and its magnitude. The response can be close, compared to the ideal case, provided the pulse is short enough. Additionally, in many systems, a pulse of large intensity may drive the system into the nonlinear regime. Other methods exist to construct an impulse response. The impulse response can be calculated from the input and output of a system driven with a pseudo-random sequence, such as [[Maximum length sequence|maximum length sequences]].<ref>{{cite book | title = Master Handbook of Acoustics | author = F. Alton Everest | author-link = F. Alton Everest | publisher = McGraw-Hill Professional | year = 2000 | isbn = 0-07-136097-2 | edition = Fourth | url = https://books.google.com/books?id=sgwg1Vwm9VUC&q=%22impulse+response%22+loudspeaker+testing&pg=RA1-PA510 }}</ref> Another approach is to take a sine sweep measurement and process the result to get the impulse response.<ref>{{cite journal |last1=Stan |first1=Guy-Bart |date=April 2002 |title=Comparison of Different Impulse Response Measurement Techniques |url=https://aes2.org/publications/elibrary-page/?id=11083 |journal=Journal of the Audio Engineering Society |volume=50 |issue=4 |page= |pages=249-262 |access-date=2 May 2025}}</ref>

===Loudspeakers===
Impulse response [[loudspeaker]] testing was first developed in the 1970s. Loudspeakers suffer from phase inaccuracy (delayed frequencies) which can be caused by passive [[Audio crossover|crossovers]], resonance, cone momentum, the internal volume, and vibrating enclosure panels.<ref>{{cite journal |title=Modeling and Delay-Equalizing Loudspeaker Responses |journal=Journal of the Audio Engineering Society |last1=Mäkivirta |first1=Aki |last2=Liski |first2=Juho |last3=Välimäki |first3=Vesa |volume=66 |issue=11 |year=2018 |pages=922–934 |doi=10.17743/jaes.2018.0053 |url=https://aes2.org/e-lib/download.cfm/pg922.pdf?ID=19869 }}</ref> The impulse response can be used to indicate when such inaccuracies can be improved by different materials, enclosures or crossovers.

Loudspeakers have a physical limit to their power output, thus the input amplitude must be limited to maintain linearity. This limitation led to the use of inputs like [[maximum length sequence]]s in obtaining the impulse response.<ref>{{cite web|url=https://books.google.com/books?id=q6w7AAAAMAAJ&q=%22impulse+response%22+%22loudspeaker+testing%22|title=Monitor|date=9 April 1976|access-date=9 April 2018|via=Google Books}}</ref>

===Electronic processing===
Impulse response analysis is a major facet of [[radar]], [[ultrasound imaging]], and many areas of [[digital signal processing]]. An interesting example is found in [[broadband]] internet connections. [[Digital subscriber line]] service providers use [[Adaptive filter|adaptive equalization]] to compensate for signal distortion and interference from using copper phone lines for transmission.

===Control systems===
In [[control theory]] the impulse response is the response of a system to a [[Dirac delta function|Dirac delta]] input. This proves useful in the analysis of [[dynamic systems]]; the [[Laplace transform]] of the delta function is 1, so the impulse response is equivalent to the [[inverse Laplace transform]] of the system's [[transfer function]].

===Acoustic and audio applications===
In acoustic and audio settings, impulse responses can be used to capture the acoustic characteristics of many things. The reverb at a location, the [[Acoustic guitar#Body shape|body]] of an instrument, certain analog audio equipment, and [[Amplifier modeling|amplifiers]] are all emulated by impulse responses. The impulse is convolved with a dry signal in [[convolution reverb|software]], often to create the effect of a physical recording. Various packages containing impulse responses from specific locations are available online.<ref>http://www.acoustics.hut.fi/projects/poririrs/ the Concert Hall Impulse Responses from Pori, Finland</ref>

===Economics===
In [[economics]], and especially in contemporary [[Model (macroeconomics)|macroeconomic modeling]], impulse response functions are used to describe how the economy reacts over time to [[exogenous]] impulses, which economists usually call [[Shock (economics)|shocks]], and are often modeled in the context of a [[vector autoregression]]. Impulses that are often treated as exogenous from a macroeconomic point of view include changes in [[government spending]], [[tax rate]]s, and other [[fiscal policy]] parameters; changes in the [[monetary base]] or other [[monetary policy]] parameters; changes in [[total factor productivity|productivity]] or other [[production function|technological]] parameters; and changes in [[Utility#Preference|preferences]], such as the degree of [[discount factor|impatience]]. Impulse response functions describe the reaction of [[Exogenous and endogenous variables|endogenous]] macroeconomic variables such as [[GDP|output]], [[Consumption (economics)|consumption]], [[Investment#Economics|investment]], and [[employment]] at the time of the shock and over subsequent points in time.<ref>{{cite book |author-link=Helmut Lütkepohl |first=Helmut |last=Lütkepohl |year=2008 |chapter=Impulse response function |title=The New Palgrave Dictionary of Economics |edition=2nd }}</ref><ref>{{cite book |author-link=James D. Hamilton |first=James D. |last=Hamilton |year=1994 |title=Time Series Analysis |chapter=Difference Equations |page=5 |publisher=Princeton University Press |isbn=0-691-04289-6 }}</ref> Recently, asymmetric impulse response functions have been suggested in the literature that separate the impact of a positive shock from a negative one.<ref>{{cite journal |last=Hatemi-J |first=A. | year=2014 |title= Asymmetric generalized impulse responses with an application in finance |journal=[[Economic Modelling]] |volume=36 |pages=18–2 |doi=10.1016/j.econmod.2013.09.014 |url=https://ideas.repec.org/a/eee/ecmode/v36y2014icp18-22.html|url-access=subscription }}</ref>

==See also==
{{Div col}}
* [[Convolution reverb]]
* [[Duhamel's principle]]
* [[Dynamic stochastic general equilibrium]]
* [[Frequency response]]
* [[Gibbs phenomenon]]
* [[Küssner effect]]
* [[Linear response function]]
* [[LTI system theory]]
* [[Point spread function]]
* [[Pre-echo]]
* [[Step response]]
* [[System analysis]]
* [[Time constant]]
* [[Transient (oscillation)]]
* [[Transient response]]
* [[Variation of parameters]]
* [https://www.roomeqwizard.com/ Room EQ Wizard]: Software for creating audio impulse responses
{{div col end}}

==References==
{{Reflist}}

==External links==
* {{Commons category-inline}}

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[[Category:Control theory]]
[[Category:Time domain analysis]]
[[Category:Analog circuits]]