signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...

and

control theory Control theory is a field of mathematics that deals with the control system, control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive ...

, 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 Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or dema ...

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 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 In system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly defin ...

for all frequencies.

Mathematical considerations

Mathematically, how the impulse is described depends on whether the system is modeled in discrete or

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...

time. The impulse can be modeled as a

Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the enti ...

for

continuous-time In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "po ...

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 LTI can refer to: * '' LTI – Lingua Tertii Imperii'', a book by Victor Klemperer * Language Technologies Institute, a division of Carnegie Mellon University * Linear time-invariant system, an engineering theory that investigates the response o ...

.) 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 functions as opposed to impulse responses. The transfer function is the

Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in 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

complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...

, 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 In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''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 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

loudspeaker A loudspeaker (commonly referred to as a speaker or speaker driver) is an electroacoustic transducer that converts an electrical audio signal into a corresponding sound. A ''speaker system'', also often simply referred to as a "speaker" or ...

testing in the 1970s. Loudspeakers suffer from phase inaccuracy, a defect unlike other measured properties such as frequency response. 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 maximum length sequences, and to the use of computer processing to derive the impulse response.

Electronic processing

Impulse response analysis is a major facet of

radar Radar is a detection system that uses radio waves to determine the distance (''ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, Marine radar, ships, spacecraft, guided missiles, motor v ...

, ultrasound imaging, and many areas of

digital signal processing Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are ...

. An interesting example would be

broadband In telecommunications, broadband is wide bandwidth data transmission which transports multiple signals at a wide range of frequencies and Internet traffic types, that enables messages to be sent simultaneously, used in fast internet connections. ...

internet connections. DSL/Broadband services use adaptive equalisation techniques to help compensate for signal distortion and interference introduced by the copper phone lines used to deliver the service.

Control systems

the impulse response is the response of a system to a Dirac delta input. This proves useful in the analysis of

dynamic systems In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a ...

; the

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 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

convolution reverb A reverb effect, or reverb, is an audio effect applied to a sound signal to simulate reverberation. It may be created through physical means, such as echo chambers, or electronically through audio signal processing. Echo chambers The first re ...

applications to enable the acoustic characteristics of a particular location to be applied to target audio.

Economics

economics Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics anal ...

, and especially in contemporary macroeconomic modeling, impulse response functions are used to describe how the economy reacts over time to

exogenous In a variety of contexts, exogeny or exogeneity () is the fact of an action or object originating externally. It contrasts with endogeneity or endogeny, the fact of being influenced within a system. Economics In an economic model, an exogen ...

impulses, which economists usually call shocks, and are often modeled in the context of a

vector autoregression Vector autoregression (VAR) is a statistical model used to capture the relationship between multiple quantities as they change over time. VAR is a type of stochastic process model. VAR models generalize the single-variable (univariate) autoregres ...

. Impulses that are often treated as exogenous from a macroeconomic point of view include changes in

government spending Government spending or expenditure includes all government consumption, investment, and transfer payments. In national income accounting, the acquisition by governments of goods and services for current use, to directly satisfy the individual ...

, tax rates, and other

fiscal policy In economics and political science, fiscal policy is the use of government revenue collection ( taxes or tax cuts) and expenditure to influence a country's economy. The use of government revenue expenditures to influence macroeconomic variabl ...

parameters; changes in the

monetary base In economics, the monetary base (also base money, money base, high-powered money, reserve money, outside money, central bank money or, in the UK, narrow money) in a country is the total amount of money created by the central bank. This include ...

or other

monetary policy Monetary policy is the policy adopted by the monetary authority of a nation to control either the interest rate payable for federal funds, very short-term borrowing (borrowing by banks from each other to meet their short-term needs) or the money s ...

parameters; changes in productivity or other technological parameters; and changes in

preferences In psychology, economics and philosophy, preference is a technical term usually used in relation to choosing between alternatives. For example, someone prefers A over B if they would rather choose A than B. Preferences are central to decision t ...

, such as the degree of

impatience (or forbearance) is the ability to endure difficult circumstances. Patience may involve perseverance in the face of delay; tolerance of provocation without responding in disrespect/anger; or forbearance when under strain, especially when faced ...

. Impulse response functions describe the reaction of

endogenous Endogenous substances and processes are those that originate from within a living system such as an organism, tissue, or cell. In contrast, exogenous substances and processes are those that originate from outside of an organism. For example, ...

macroeconomic variables such as output, consumption,

investment Investment is the dedication of money to purchase of an asset to attain an increase in value over a period of time. Investment requires a sacrifice of some present asset, such as time, money, or effort. In finance, the purpose of investing is ...

, and

employment Employment is a relationship between two parties regulating the provision of paid labour services. Usually based on a contract, one party, the employer, which might be a corporation, a not-for-profit organization, a co-operative, or any o ...

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.

References

{{Authority control Control theory Time domain analysis Analog circuits