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The Impulse response from a simple audio system. Showing the original impulse, with high frequencies boosted, then with low frequencies boosted.

In simple terms, the impulse response of a system is its output when presented with a very brief signal, an impulse. While an impulse is a difficult concept to imagine, and an impossible thing in reality, it represents the limit case of a pulse made infinitely 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 concept as an idealization. Image File history File links Broom_icon. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... 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. ...

Mathematical basis

Mathematically, an impulse can be modeled as a Dirac delta function. Suppose that T is a (discrete) system, i.e. something that takes an input x[n] and produces an output y[n]: The Dirac delta function, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere. ...

So T is an operator acting on sequences (over the integers) and producing sequences. Beware that T is not the system but a mathematical representation of the system. T can be non-linear, e.g.

or linear e.g.

.

Suppose that T is linear. Then

and

Suppose also that T is invariant under translation i.e. if then . In such a system any output can be calculated in terms of the input and a very special sequence called impulse response which characterizes the system completely. This can be seen as follows: Take the identity

and take the T of both sides

Of course this has a meaning only if

lies in the domain of T. Now, since T is linear and invariant under translation we may write

Since the output y[k] is given by

we may write

Putting

we have finally

The sequence is the impulse response of the system represented by T. As can be seen from the above, h[n] is the output of the system when its input is the discrete Dirac delta. Similar results hold for continuous time systems.

As a conceptual example consider a room and a balloon in it at point p. The balloon pops and makes a "pow" sound. Here the room is a system T which takes the "pow" sound and diffuses it through multiple reflections. The input δ_p[n] is the "pow", which is similar (due in part to its short duration) to a Dirac delta, and the output h[n,p] is the sequence of the damped sound. Here h[n,p] depends on the location (point p) of the balloon. If we know h[n,p] for every p of the room, then we actually know the impulse response of the room. It is then possible to predict its response to any sound produced in it.

Mathematical applications

In the language of mathematics, the impulse response of a linear transformation is the image of Dirac's delta function under the transformation. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... The Dirac delta function, sometimes referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere such that the total integral...

The Laplace transform of the impulse response function is known as the transfer function. 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, also known as the frequency domain. An inverse Laplace transform of this result will yield the output function in the time domain. In mathematics, the Laplace transform is a technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems. ... A transfer function is a mathematical representation of the relation between the input and output of a linear time-invariant system. ... In mathematics, the Laplace transform is a technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... Frequency domain is a term used to describe the analysis of mathematical functions with respect to frequency. ... In mathematics, the Bromwich integral or inverse Laplace transform of F(s) is the function f(t) which has the property where is the Laplace transform. ... Time-domain is a term used to describe the analysis of mathematical functions, or real-life signals, with respect to time. ...

To determine an output function directly in the time domain requires the convolution 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. In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ... Frequency domain is a term used to describe the analysis of mathematical functions with respect to frequency. ...

Practical applications

In real, practical systems, it is not possible to produce a perfect impulse to serve as input for testing. Therefore, a brief pulse is used as an approximation of an impulse. Provided that the pulse is short compared to the impulse response, the result will be near enough to the true, theoretical, impulse response.

Loudspeakers

A very useful real application that demonstrates this idea was the development of impulse response loudspeaker testing in the 1980s which led to big improvements in loudspeaker design. Loudspeakers suffer from colouration, a defect unlike normal measured properties like frequency response. Colouration 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. Colouration 'smears' the sound which reduces the 'clarity' and 'transparency.' 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. Initially, short pulses were used, but the need to limit their amplitude to maintain the linearity of the system meant that the resulting output was very small and hard to distinguish from the noise. Later techniques therefore moved towards the use of other types of input, like maximal length sequences, and using computer processing to derive the impulse response. Recently this has led to graphical spectrogram plots that show delayed response against time for each frequency. “Loudspeaker” redirects here. ... Frequency response is the measure of any systems response to frequency, but is usually used in connection with electronic amplifiers and similar systems, particularly in relation to audio signals. ... For the Irish mythological figure, see Naoise. ... A maximum length sequence (MLS) is a type of pseudorandom binary sequence. ... It has been suggested that this article or section be merged with periodogram. ...

Digital filtering

Impulse response is a very important concept in the design of digital filters for audio processing, because these differ from 'real' filters in often having a pre-echo, which the ear is not accustomed to. An FIR filter In electronics, a digital filter is any electronic filter that works by performing digital mathematical operations on an intermediate form of a signal. ... Pre-echo is a psychoacoustic phenomenon where an unusually noticeable artifact is heard in a sound recording from the energy of time domain transients smeared backwards in time after processing in the frequency domain due to the Gibbs phenomenon. ...

Electronic processing

Impulse response analysis is a major facet of radar, ultrasound imaging, and many areas of digital signal processing. An interesting example would be broadband internet connections. Where once it was only possible to get 4 kHz speech signal over a local telephone wire, or data at 300 bit/s using a modem, it is now commonplace to pass 2 Mb/s over these same wires, largely because of 'adaptive equalisation' which processes out the time smearing and echoes on the line. This long range radar antenna, known as ALTAIR, is used to detect and track space objects in conjunction with ABM testing at the Ronald Reagan Test Site on the Kwajalein atoll. ... Medical ultrasonography is an ultrasound-based imaging diagnostic technique used to visualize internal organs, their size, structure and their pathological lesions. ... Digital signal processing (DSP) is the study of signals in a digital representation and the processing methods of these signals. ... Broadband in telecommunications is a term which refers to a signaling method which includes or handles a relatively wide range of frequencies which may be divided into channels or frequency bins. ... An adaptive filter is a digital filter that performs digital signal processing and can adapt its performance based on the input signal. ...

Control systems

In control theory 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. In engineering and mathematics, control theory deals with the behavior of dynamical systems. ... In engineering and mathematics, a dynamical system is a deterministic process in which a functions value changes over time according to a rule that is defined in terms of the functions current value. ... In mathematics, the Laplace transform is a technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems. ... In mathematics, the Bromwich integral or inverse Laplace transform of F(s) is the function f(t) which has the property where is the Laplace transform. ... A transfer function is a mathematical representation of the relation between the input and output of a linear time-invariant system. ...

See also

• Kronecker delta
• Dirac delta function
• Unit impulse function
• Green function
• frequency response
• LTI system theory
• system analysis
• transfer function
• Convolution reverb
• transient
• transient response