Impulse response
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225879320
2008-07-15T20:50:14Z
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robot Adding: [[su:Respon Impuls]]
[[Image:Impulse.png|thumb|300px|right|The Impulse response from a simple audio system. Showing the original impulse, with high frequencies boosted, then with low frequencies boosted.]]
The '''impulse response''' of a system is its output when presented with a very brief input signal, an impulse. Mathematically, an impulse can be modeled as a [[Dirac delta function]] for [[continuous-time]] systems, or as the [[Kronecker delta]] for [[discrete-time]] systems. The Dirac delta represents the limiting case of a [[pulse (signal processing)|pulse]] 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 theory]].)
== 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.
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]].
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]].
== 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 [[phase inaccuracy]], a defect unlike normal measured properties like [[frequency response]]. 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. Phase inaccuracy '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, as well as changes to the speaker crossover. 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 [[maximum length sequence]]s, 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.
=== 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.
=== 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 filter|adaptive equalisation]]' which processes out the time smearing and echoes on the line.
=== 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]].
== See also ==
*[[Convolution reverb]]
*[[Dirac delta function]]
*[[Green's function]]
*[[Frequency response]]
*[[Kronecker delta]]
*[[LTI system theory]]
*[[System analysis]]
*[[Step response]]
*[[Transfer function]]
*[[Transient (oscillation)|Transient]]
*[[Transient response]]
*[[Unit impulse function]]
[[Category:Signal processing]]
[[Category:Control theory]]
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