Impulse response

# Impulse response

Overview
In signal processing
Signal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...

, 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
Dirac delta function
The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...

. 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 function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

of time (or possibly as a function of some other independent variable
Independent variable
The 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).
Discussion

Encyclopedia
In signal processing
Signal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...

, 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
Dirac delta function
The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...

. 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 function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

of time (or possibly as a function of some other independent variable
Independent variable
The 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 planetary system
Planetary system
A planetary system consists of the various non-stellar objects orbiting a star such as planets, dwarf planets , asteroids, meteoroids, comets, and cosmic dust...

in orbit around a star
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...

; 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 function
Dirac delta function
The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...

for continuous-time systems, or as the Kronecker delta for discrete-time systems. The Dirac delta represents the limiting case of a pulse
Pulse (signal processing)
In 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 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 function, the output function can be calculated in terms of the input and the impulse response. (See LTI system theory
LTI system theory
Linear time-invariant system theory, 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 time-invariant...

.) The impulse response of a linear transformation
Linear transformation
In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...

is the image of Dirac's delta function under the transformation, analogous to the fundamental solution
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...

of a partial differential operator.

The Laplace transform of the impulse response function is known as the transfer function
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 linear time-invariant 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 transform of a system's output may be determined by the multiplication of the transfer function with the input function in the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

, also known as the frequency domain
Frequency domain
In electronics, control systems engineering, and statistics, frequency domain is a term used to describe the domain for 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 domain
Time domain
Time domain is a term used to describe 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 case of continuous time, or at various...

.

To determine an output function directly in the time domain requires the convolution
Convolution
In 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 domain
Frequency domain
In electronics, control systems engineering, and statistics, frequency domain is a term used to describe the domain for analysis of mathematical functions or signals with respect to frequency, rather than time....

.

The impulse response, considered as a Green's function
Green's function
In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions...

, 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
Loudspeaker
A loudspeaker is an electroacoustic transducer that produces sound in response to an electrical audio signal input. Non-electrical loudspeakers were developed as accessories to telephone systems, but electronic amplification by vacuum tube made loudspeakers more generally useful...

testing in the 1970s. Loudspeakers suffer from phase inaccuracy, a defect unlike other measured properties such as frequency response
Frequency response
Frequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system. It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input...

. 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
Maximum length sequence
A 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...

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-echo
Pre-echo
Pre-echo is a digital 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 radar
Radar is an object-detection system which uses radio waves to determine the range, altitude, direction, or speed of objects. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weather formations, and terrain. The radar dish or antenna transmits pulses of radio...

, ultrasound imaging, and many areas of digital signal processing
Digital signal processing
Digital signal processing is concerned with the representation of discrete time 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 broadband
The term broadband refers to a telecommunications signal or device of greater bandwidth, in some sense, than another standard or usual signal or device . Different criteria for "broad" have been applied in different contexts and at different times...

An adaptive filter is a filter that self-adjusts its transfer function according to an optimization algorithm driven by an error signal. Because of the complexity of the optimization algorithms, most adaptive filters are digital filters. By way of contrast, a non-adaptive filter has a static...

techniques to help compensate for signal distortion and interference introduced by the copper phone lines used to deliver the service.

### Control systems

In control theory
Control theory
Control 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 transform of the delta function is 1, so the impulse response is equivalent to the inverse Laplace transform of the system's transfer function
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 linear time-invariant system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...

.

### 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 commercial 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
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...

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

### Economics

In economics
Economics
Economics is the social science that analyzes 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 modeling
Model (macroeconomics)
A macroeconomic model is an analytical tool designed to describe the operation of 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 level of...

, impulse response functions describe how the economy reacts over time to exogenous
Exogenous
Exogenous 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 'shocks
Shock (economics)
In 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 autoregression
Vector autoregression
Vector autoregression is a statistical model used to capture the linear interdependencies among multiple time series. VAR models generalize the univariate autoregression models. All the variables in a VAR are treated symmetrically; each variable has an equation explaining its evolution based on...

. Impulses that are often treated as exogenous from a macroeconomic point of view include changes in government spending
Government spending
Government spending includes all government consumption, investment but excludes transfer payments made by a state. Government acquisition of goods and services for current use to directly satisfy individual or collective needs of the members of the community is classed as government final...

, tax rates, and other fiscal policy
Fiscal policy
In economics and political science, fiscal policy is the use of government expenditure and revenue collection to influence the economy....

parameters; changes in the monetary base
Monetary base
In economics, the monetary base is a term relating to the money supply , the amount of money in the economy...

or other monetary policy
Monetary policy
Monetary policy is the process by which the monetary authority of a country controls the supply of money, often targeting a rate of interest for the purpose of promoting economic growth and stability. The official goals usually include relatively stable prices and low unemployment...

parameters; changes in productivity
Total factor productivity
In economics, total-factor productivity is a variable which accounts for effects in total output not caused by inputs. If all inputs are accounted for, then total factor productivity can be taken as a measure of an economy’s long-term technological change or technological dynamism.If all inputs...

or other technological
Production function
In microeconomics and macroeconomics, 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 preferences
Utility
In economics, utility is a measure of customer satisfaction, referring to the total satisfaction received by a consumer from consuming a good or service....

, such as the degree of impatience. Impulse response functions describe the reaction of endogenous
Endogeneity (economics)
In an econometric model, a parameter or variable is said to be endogenous when there is a correlation between the parameter or variable and the error term. Endogeneity can arise as a result of measurement error, autoregression with autocorrelated errors, simultaneity, omitted variables, and sample...

macroeconomic variables such as output, consumption
Consumption (economics)
Consumption is a common concept in economics, and gives rise to derived concepts such as consumer debt. Generally, consumption is defined in part by comparison to production. But the precise definition can vary because different schools of economists define production quite differently...

, investment, and employment
Employment
Employment is a contract between two parties, one being the employer and the other being the employee. An employee may be defined as:- Employee :...

at the time of the shock and over subsequent points in time.

• Convolution reverb
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
Dirac delta function
The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...

• Dynamic stochastic general equilibrium
Dynamic stochastic general equilibrium
Dynamic stochastic general equilibrium modeling is a branch of applied general equilibrium theory that is influential in contemporary macroeconomics...

• Frequency response
Frequency response
Frequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system. It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input...

• Gibbs phenomenon
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...

• LTI system theory
LTI system theory
Linear time-invariant system theory, 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 time-invariant...

• Pre-echo
Pre-echo
Pre-echo is a digital 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
System analysis in the field of electrical engineering characterizes electrical systems and their properties. System Analysis can be used to represent almost anything from population growth to audio speakers, electrical engineers often use it because of its direct relevance to many areas of their...

• Step response
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...

• Time constant
Time constant
In physics and engineering, the time constant, usually denoted by the Greek letter \tau , is the risetime characterizing the response to a time-varying input of a first-order, linear time-invariant system.Concretely, a first-order LTI system is a system that can be modeled by a single first order...

• Linear response function
Linear response function
A linear response function describes the input-output relationship of a signal transducer such as a radio turning electromagnetic waves into music or a neuron turning synaptic input into a response...

• Transient
Transient (oscillation)
A transient event is a short-lived burst of energy in a system caused by a sudden change of state.The source of the transient energy may be an internal event or a nearby event...

• Transient response
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. The transient response is not necessarily tied to "on/off" events but to any event that affects the equilibrium of the system...

• Unit impulse function