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, Scalar potential, potential fields, Seismic tomograph ...

and

control theory Control theory is a field of control engineering and applied 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 applic ...

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

dynamic system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock ...

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 unit ...

(). 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-orie ...

of time (or possibly as a function of some other

independent variable A variable is considered dependent if it depends on (or is hypothesized to depend on) an independent variable. Dependent variables are studied under the supposition or demand that they depend, by some law or rule (e.g., by a mathematical function ...

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 Linear system#Definition, linearity and Time-invariant system, ...

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, ...

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 mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...

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 "poi ...

systems, or as the discrete unit sample function 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 "poi ...

systems. The Dirac delta represents the limiting case of a

pulse In medicine, the pulse refers to the rhythmic pulsations (expansion and contraction) of an artery in response to the cardiac cycle (heartbeat). The pulse may be felt ( palpated) in any place that allows an artery to be compressed near the surfac ...

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 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 Joseph Fo ...

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 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 ...

.) 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 pr ...

is the image of

Dirac's delta function In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...

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 ...

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 return ...

. 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, models the system's output for each possible ...

s as opposed to impulse responses. The transfer function is the

Laplace transform In mathematics, the Laplace transform, named after Pierre-Simon Laplace (), is an integral transform that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time domain'') to a f ...

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 (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...

, also known as the

frequency domain In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency (and possibly phase), rather than time, as in time ser ...

. An

inverse Laplace transform In mathematics, the inverse Laplace transform of a function F(s) is a real function f(t) that is piecewise- continuous, exponentially-restricted (that is, , f(t), \leq Me^ \forall t \geq 0 for some constants M > 0 and \alpha \in \mathbb) and h ...

of this result will yield the output in the

time domain In mathematics and signal processing, the time domain is a representation of how a signal, function, or data set varies with time. It is used for the analysis of mathematical functions, physical signals or time series of economic or environmental ...

. 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 f and g that produces a third function f*g, as the integral of the product of the two ...

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 (or Green 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 L is a linear dif ...

, 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 sequences. Another approach is to take a sine sweep measurement and process the result to get the impulse response.

Loudspeakers

Impulse response

loudspeaker A loudspeaker (commonly referred to as a speaker or, more fully, a speaker system) is a combination of one or more speaker drivers, an enclosure, and electrical connections (possibly including a crossover network). The speaker driver is an ...

testing was first developed in the 1970s. Loudspeakers suffer from phase inaccuracy (delayed frequencies) which can be caused by passive crossovers, resonance, cone momentum, the internal volume, and vibrating enclosure panels. 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 sequences in obtaining the impulse response.

Electronic processing

Impulse response analysis is a major facet of

radar Radar is a system that uses radio waves to determine the distance ('' ranging''), direction ( azimuth and elevation angles), and radial velocity of objects relative to the site. It is a radiodetermination method used to detect and track ...

ultrasound imaging Medical ultrasound includes diagnostic techniques (mainly imaging) using ultrasound, as well as therapeutic applications of ultrasound. In diagnosis, it is used to create an image of internal body structures such as tendons, muscles, join ...

, 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 a ...

. An interesting example is found in

broadband In telecommunications, broadband or high speed is the wide-bandwidth (signal processing), bandwidth data transmission that exploits signals at a wide spread of frequencies or several different simultaneous frequencies, and is used in fast Inter ...

internet connections.

Digital subscriber line Digital subscriber line (DSL; originally digital subscriber loop) is a family of technologies that are used to transmit digital data over telephone lines. In telecommunications marketing, the term DSL is widely understood to mean asymmetric dig ...

service providers use

adaptive equalization Adaptation, in biology, is the process or trait by which organisms or population better match their environment Adaptation may also refer to: Arts * Adaptation (arts), a transfer of a work of art from one medium to another ** Film adaptation, ...

to compensate for signal distortion and interference from using copper phone lines for transmission.

Control systems

the impulse response is the response of a system to a

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

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, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock 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 settings, impulse responses can be used to capture the acoustic characteristics of many things. The reverb at a location, the

body Body may refer to: In science * Physical body, an object in physics that represents a large amount, has mass or takes up space * Body (biology), the physical material of an organism * Body plan, the physical features shared by a group of anim ...

of an instrument, certain analog audio equipment, and

amplifiers An amplifier, electronic amplifier or (informally) amp is an electronic device that can increase the magnitude of a signal (a time-varying voltage or current). It is a two-port electronic circuit that uses electric power from a power suppl ...

are all emulated by impulse responses. The impulse is convolved with a dry signal in

software Software consists of computer programs that instruct the Execution (computing), execution of a computer. Software also includes design documents and specifications. The history of software is closely tied to the development of digital comput ...

, often to create the effect of a physical recording. Various packages containing impulse responses from specific locations are available online.

Economics

economics Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interac ...

, 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 is the opposite of endogeneity or endogeny, the fact of being influenced from within a system. Economics In an economic model, an ...

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 or ...

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. The tax rate that is applied to an individual's or corporation's income is determined by tax laws of the country and can be in ...

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 variab ...

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 includ ...

or other

monetary policy Monetary policy is the policy adopted by the monetary authority of a nation to affect monetary and other financial conditions to accomplish broader objectives like high employment and price stability (normally interpreted as a low and stable rat ...

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 proce ...

or other

technological Technology is the application of conceptual knowledge to achieve practical goals, especially in a reproducible way. The word ''technology'' can also mean the products resulting from such efforts, including both tangible tools such as ute ...

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 the ...

, such as the degree of

impatience or forbearance, is the ability to endure difficult or undesired long-term circumstances. Patience involves perseverance or tolerance in the face of delay, provocation, or stress without responding negatively, such as reacting with disrespect ...

. Impulse response functions describe the reaction of

endogenous Endogeny, in biology, refers to the property of originating or developing from within an organism, tissue, or cell. For example, ''endogenous substances'', and ''endogenous processes'' are those that originate within a living system (e.g. an ...

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 valu ...

consumption Consumption may refer to: * Eating *Resource consumption *Tuberculosis, an infectious disease, historically known as consumption * Consumer (food chain), receipt of energy by consuming other organisms * Consumption (economics), the purchasing of n ...

investment Investment is traditionally defined as the "commitment of resources into something expected to gain value over time". If an investment involves money, then it can be defined as a "commitment of money to receive more money later". From a broade ...

, and

employment Employment is a relationship between two party (law), parties Regulation, regulating the provision of paid Labour (human activity), labour services. Usually based on a employment contract, contract, one party, the employer, which might be a cor ...

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

External links

* {{Authority control Control theory Time domain analysis Analog circuits