In
signal processingSignal processing is an area of electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time to perform useful operations on those signals...
, the
impulse response, or
impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an
impulseThe Dirac delta or Dirac's delta is a mathematical construct introduced by theoretical physicist Paul Dirac. Informally, it is a generalized function representing an infinitely sharp peak bounding unit area: a 'function' δ that has the value zero everywhere except at x = 0 where its value is...
. More generally, an impulse response refers to 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
functionIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
of time (or possibly as a function of some other
independent variableThe terms "dependent variable" and "independent variable" are used in similar but subtly different ways in mathematics and statistics as part of the standard terminology in those subjects...
that parameterizes the dynamic behavior of the system).
For example, the dynamic system might be a
televisionTelevision is a widely used telecommunication medium for transmitting and receiving moving images, either monochromatic or color, usually accompanied by sound. "Television" may also refer specifically to a television set, television programming or television transmission...
; then the external influence may be an electronic signal, and the output may be the image produced on the screen. In this case, the impulse response refers to the change in the image, over time, in response to the initial signal.
Alternatively, the dynamic system might be a
planetary systemA planetary system consists of the various non-stellar objects orbiting a star such as planets, dwarf planets, moons, asteroids, meteoroids, comets, and cosmic dust...
in orbit around a
starA star is a massive, luminous ball of plasma that is held together by gravity. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth. Other stars are visible in the night sky, when they are not outshone by the Sun...
; the external influence in this case might be another massive object arriving from elsewhere in the galaxy; the impulse response is the change in the motion of the planetary system caused by interaction with the new object.
In all these cases, the 'dynamic system' and its 'impulse response' may refer to actual physical objects, or to a mathematical system of equations describing these objects.
Mathematical considerations
Mathematically, how the impulse is described depends on whether the system is modeled in discrete or continuous time. The impulse can be modeled as a
Dirac delta functionThe Dirac delta or Dirac's delta is a mathematical construct introduced by theoretical physicist Paul Dirac. Informally, it is a generalized function representing an infinitely sharp peak bounding unit area: a 'function' δ that has the value zero everywhere except at x = 0 where its value is...
for continuous-time systems, or as the
Kronecker deltaIn mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker , is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise...
for discrete-time systems. The Dirac delta represents the limiting case of a
pulseIn signal processing, the term pulse has the following meanings:#A rapid, transient change in the amplitude of a signal from a baseline value to a higher or lower value, followed by a rapid return to the baseline value....
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 (LTI) is completely characterized by its impulse response. That is, for any input function, the output function can be calculated in terms of the input and the impulse response. (See
LTI system theoryLinear time-invariant system theory, most commonly known as LTI system theory, comes from applied mathematics and has direct applications in NMR spectroscopy, seismology, circuits, signal processing, control theory, and other technical areas. It investigates the response of a linear and...
.) The impulse response of a
linear transformationIn mathematics, a linear map is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. The expression "linear operator" is commonly used for linear maps from a vector space to itself...
is the image of Dirac's delta function under the transformation, analogous to the
fundamental solutionIn mathematics, a fundamental solution for a linear partial differential operator L is a formulation in the language of distribution theory of the older idea of a Green's function...
of a partial differential operator.
The
Laplace transformIn mathematics, the Laplace transform is a widely used integral transform. It has many important applications in mathematics, physics, optics, electrical engineering, control engineering, signal processing, and probability theory....
of the impulse response function is known as the
transfer functionA transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...
. It is usually easier to analyze systems using transfer functions as opposed to impulse response functions. The
Laplace transformIn mathematics, the Laplace transform is a widely used integral transform. It has many important applications in mathematics, physics, optics, electrical engineering, control engineering, signal processing, and probability theory....
of a system's output may be determined by the multiplication of the transfer function with the input function in the
complex planeIn mathematics, the complex plane or z-plane is a geometric representation of the complex numbersestablished by the real axis and the orthogonal imaginary axis...
, also known as the
frequency domainIn electronics, control systems engineering, and statistics frequency domain is a term used to describe the analysis of mathematical functions or signals with respect to frequency, rather than time....
. An inverse Laplace transform of this result will yield the output function in the
time domainTime domain is a term used to describe the analysis of mathematical functions, or physical signals, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the case of continuous time, or at various separate instants in the case of discrete time....
.
To determine an output function directly in the time domain requires the
convolutionIn mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
of the input function with the impulse response function. This requires the use of integrals, and is usually more difficult than simply multiplying two functions in the
frequency domainIn electronics, control systems engineering, and statistics frequency domain is a term used to describe the analysis of mathematical functions or signals with respect to frequency, rather than time....
.
The impulse response, considered as a
Green's functionIn mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to boundary conditions. The term is also used in physics, specifically in quantum field theory, electrodynamics and statistical field theory, to refer to various types of correlation...
, 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
loudspeakerA loudspeaker is an electroacoustic transducer that converts an electrical signal into sound. The speaker pulses in accordance with the variations of an electrical signal and causes sound waves to propagate through a medium such as air or water.Loudspeakers are the most variable elements in a...
testing in the 1970s. Loudspeakers suffer from phase inaccuracy, a defect unlike other measured properties such as
frequency responseFrequency response is the measure of any system's output spectrum in response to an input signal. In the audible range it is usually referred to in connection with electronic amplifiers, microphones and loudspeakers. Radio spectrum frequency response can refer to measurements of coaxial cables,...
. Phase inaccuracy is caused by small delayed sounds that are the result of 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 sequenceA maximum length sequence 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 that can be reproduced by the shift registers A maximum length sequence...
s, and to the use of computer processing to derive the impulse response.
Digital filtering
Impulse response is a very important concept in the design of digital filters for audio processing, because digital filters can differ from 'real' filters in often having a
pre-echoPre-echo is an audio compression artifact where a sound is heard before it occurs . It is most noticeable in impulsive sounds from percussion instruments such as castanets or cymbals....
, which the ear is not accustomed to.
Electronic processing
Impulse response analysis is a major facet of
radarRadar is an object detection system that uses electromagnetic waves to identify the range, altitude, direction, or speed of both moving and fixed objects such as aircraft, ships, motor vehicles, weather formations, and terrain. The term RADAR was coined in 1941 as an acronym for RAdio Detection And...
, ultrasound imaging, and many areas of
digital signal processingDigital signal processing is concerned with the representation of the signals by a sequence of numbers or symbols and the processing of these signals. Digital signal processing and analog signal processing are subfields of signal processing...
. An interesting example would be
broadbandThe term broadband can have different meanings in different contexts. The term's meaning has undergone substantial shifts.-In telecommunication:...
internet connections. DSL/Broadband services use
adaptive equalisationAn adaptive filter is a filter that self-adjusts its transfer function according to an optimizing algorithm. Because of the complexity of the optimizing algorithms, most adaptive filters are digital filters that perform digital signal processing and adapt their performance based on the input signal...
techniques to help compensate for signal distortion and interference introduced by the copper phone lines used to deliver the service.
Control systems
In
control theoryControl theory is an interdisciplinary branch of engineering and mathematics, that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
the impulse response is the response of a system to a Dirac delta input. This proves useful in the analysis of dynamic systems: the
Laplace transformIn mathematics, the Laplace transform is a widely used integral transform. It has many important applications in mathematics, physics, optics, electrical engineering, control engineering, signal processing, and probability theory....
of the delta function is 1, so the impulse response is equivalent to the inverse Laplace transform of the system's
transfer functionA transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...
.
Economics
In
economicsEconomics is the social science that studies the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
, and especially in contemporary
macroeconomic modelingA macroeconomic model is a model or framework designed to describe the operation and activity in the economy of a country or a region. These models are usually designed to examine the dynamics of aggregate quantities such as the total amount of goods and services produced, total income earned, the...
, impulse response functions describe how the economy reacts over time to
exogenousExogenous refers to an action or object coming from outside a system. It is the opposite of endogenous, something generated from within the system....
impulses, which economists usually call '
shocksIn economics a shock is an unexpected or unpredictable event that affects an economy, either positively or negatively. Technically, it refers to an unpredictable change in exogenous factors -- that is, factors unexplained by economics -- which may have an impact on endogenous economic variables.The...
', and are often modeled in the context of a
vector autoregressionVector autoregression is an econometric model used to capture the evolution and the interdependencies between multiple time series, generalizing the univariate AR models. All the variables in a VAR are treated symmetrically by including for each variable an equation explaining its evolution based...
. Impulses that are often treated as exogenous from a macroeconomic point of view include changes in
government spendingGovernment spending or government expenditure is classified by economists into three main types. Government purchases of goods and services for current use are classed as government consumption. Government purchases of goods and services intended to create future benefits, such as infrastructure...
, tax rates, and other
fiscal policyIn economics, fiscal policy is the use of government spending and revenue collection to influence the economy.Fiscal policy can be contrasted with the other main type of economic policy, monetary policy, which attempts to stabilize the economy by controlling interest rates and the supply of money....
parameters; changes in the
monetary baseIn economics, the monetary base is a term relating to the money supply, the amount of money in the economy. The monetary base comprises only coins, paper money, and commercial banks' reserves with the central bank...
or other
monetary policyMonetary policy is the process by which the government, central bank, or monetary authority of a country controls the supply of money, availability of money, and cost of money or rate of interest, in order to attain a set of objectives oriented towards the growth and stability of the economy...
parameters; changes in
productivityIn economics, total-factor productivity is a variable which accounts for effects in total output not caused by inputs. For example, a year with unusually good weather will tend to have higher output, because bad weather hinders agricultural output...
or other
technologicalIn economics, a production function is a function that specifies the output of a firm, an industry, or an entire economy for all combinations of inputs...
parameters; and changes in
preferencesIn economics, utility is a measure of the relative satisfaction from, or desirability of, consumption of various goods and services. Given this measure, one may speak meaningfully of increasing or decreasing utility, and thereby explain economic behavior in terms of attempts to increase one's utility...
, such as the degree of impatience. Impulse response functions describe the reaction of
endogenousIn an economic model, a parameter or variable is said to be endogenous when there is a correlation between the parameter or variable and the error term...
macroeconomic variables such as output,
consumptionConsumption is a common concept in economics, and gives rise to derived concepts such as consumer debt. Generally consumption is defined by opposition to production. But the precise definition can vary because different schools of economists define production quite differently...
, investment, and
employmentEmployment is a contract between two parties, one being the employer and the other being the employee. An employee may be defined as: "A person in the service of another under any contract of hire, express or implied, oral or written, where the employer has the power or right to control and direct...
at the time of the shock and over subsequent points in time.
See also
- Convolution reverb
In audio signal processing, convolution reverb is a process for digitally simulating the reverberation of a physical or virtual space. It is based on the mathematical convolution operation, and uses a pre-recorded audio sample of the impulse response of the space being modelled...
- Dirac delta function
The Dirac delta or Dirac's delta is a mathematical construct introduced by theoretical physicist Paul Dirac. Informally, it is a generalized function representing an infinitely sharp peak bounding unit area: a 'function' δ that has the value zero everywhere except at x = 0 where its value is...
- Dynamic stochastic general equilibrium
Dynamic stochastic general equilibrium modeling is a branch of applied general equilibrium theory that is increasingly influential in contemporary macroeconomics...
- Green's function
In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to boundary conditions. The term is also used in physics, specifically in quantum field theory, electrodynamics and statistical field theory, to refer to various types of correlation...
- Frequency response
Frequency response is the measure of any system's output spectrum in response to an input signal. In the audible range it is usually referred to in connection with electronic amplifiers, microphones and loudspeakers. Radio spectrum frequency response can refer to measurements of coaxial cables,...
- Fundamental solution
In mathematics, a fundamental solution for a linear partial differential operator L is a formulation in the language of distribution theory of the older idea of a Green's function...
- Gibbs phenomenon
In mathematics, the Gibbs phenomenon, named after the American physicist J. Willard Gibbs, is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity: the nth partial sum of the Fourier series has large...
- Kronecker delta
In mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker , is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise...
- LTI system theory
Linear time-invariant system theory, most commonly known as LTI system theory, comes from applied mathematics and has direct applications in NMR spectroscopy, seismology, circuits, signal processing, control theory, and other technical areas. It investigates the response of a linear and...
- Pre-echo
Pre-echo is an audio compression artifact where a sound is heard before it occurs . It is most noticeable in impulsive sounds from percussion instruments such as castanets or cymbals....
- System analysis
System analysis is the branch of electrical engineering that characterizes electrical systems and their properties. Although many of the methods of system analysis can be applied to non-electrical systems, it is a subject often studied by electrical engineers because it has direct relevance to...
- Step response
The step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory, step response is the time behaviour of the outputs of a general system when its inputs change from...
- Transfer function
A transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...
- Transient
A transient event is a short-lived oscillation in a system caused by a sudden change of voltage, current or load.-Electrics:In electrical and electronic engineering these sudden changes are mostly found as the result of the operation of switching devices...
- Transient response
In Electrical Engineering and Mechanical Engineering, a transient response or natural response is the response of a system to a change from equilibrium. Specifically, transient response in Mechanical Engineering is the portion of the response that approaches zero after a sufficiently long time...
- Unit impulse function