signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...

and

control theory Control theory is a field of mathematics that deals with the 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 system to a ...

, the impulse response, or impulse response function (IRF), of a

dynamic system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...

is its output when presented with a brief input signal, called an

impulse Impulse or Impulsive may refer to: Science * Impulse (physics), in mechanics, the change of momentum of an object; the integral of a force with respect to time * Impulse noise (disambiguation) * Specific impulse, the change in momentum per uni ...

(). 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 Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

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 demand ...

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 define ...

for all frequencies.

Mathematical considerations

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

discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory * Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a ...

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 ...

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 entire ...

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 In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...

for

discrete-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. The Dirac delta represents the limiting case of a

pulse In medicine, a pulse represents the tactile arterial palpation of the cardiac cycle (heartbeat) by trained fingertips. The pulse may be palpated in any place that allows an artery to be compressed near the surface of the body, such as at the nec ...

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 In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Josep ...

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 of ...

.) The impulse response of a

linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...

is the image of

Dirac's 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 ...

under the transformation, analogous to the

fundamental solution In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not ad ...

of a

partial differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...

. It is usually easier to analyze systems using

transfer function In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, theoretically models the system's output for ...

s 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 In mathematics, an integral transform maps a function from its original function space into another function space via integra ...

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 In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a signa ...

. An

inverse Laplace transform In mathematics, the inverse Laplace transform of a function ''F''(''s'') is the piecewise-continuous and exponentially-restricted real function ''f''(''t'') which has the property: :\mathcal\(s) = \mathcal\(s) = F(s), where \mathcal denotes the La ...

of this result will yield the output in the

time domain Time domain refers to the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the cas ...

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

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

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

Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differential ...

, 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 In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of sy ...

. 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 sequence A maximum length sequence (MLS) is a type of pseudorandom binary sequence. They are bit sequences generated using maximal linear-feedback shift registers and are so called because they are periodic and reproduce every binary sequence (except the ...

s, 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, ships, spacecraft, guided missiles, motor vehicles, w ...

ultrasound imaging Medical ultrasound includes diagnostic techniques (mainly medical imaging, imaging techniques) using ultrasound, as well as therapeutic ultrasound, therapeutic applications of ultrasound. In diagnosis, it is used to create an image of internal ...

, 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 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 entire ...

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 p ...

; 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

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 r ...

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 (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and intera ...

, 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 exogeno ...

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 o ...

tax rate In a tax system, the tax rate is the ratio (usually expressed as a percentage) at which a business or person is taxed. There are several methods used to present a tax rate: statutory, average, marginal, and effective. These rates can also be p ...

s, 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 variables ...

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 very short-term borrowing (borrowing by banks from each other to meet their short-term needs) or the money supply, often a ...

parameters; changes in

productivity Productivity is the efficiency of production of goods or services expressed by some measure. Measurements of productivity are often expressed as a ratio of an aggregate output to a single input or an aggregate input used in a production proces ...

or other

technological Technology is the application of knowledge to reach practical goals in a specifiable and reproducible way. The word ''technology'' may also mean the product of such an endeavor. The use of technology is widely prevalent in medicine, science, ...

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 theo ...

, such as the degree of impatience. 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, es ...

macroeconomic variables such as

output Output may refer to: * The information produced by a computer, see Input/output * An output state of a system, see state (computer science) * Output (economics), the amount of goods and services produced ** Gross output in economics, the value of ...

consumption Consumption may refer to: *Resource consumption *Tuberculosis, an infectious disease, historically * Consumption (ecology), receipt of energy by consuming other organisms * Consumption (economics), the purchasing of newly produced goods for curren ...

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 i ...

, 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 othe ...

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