Impulse response

The impulse response, also called weight function or impulse response, the output signal of a system in which on receipt of a Dirac pulse is supplied. It is used in systems theory for the characterization of linear, time-invariant systems. The ( ideal ) Dirac impulse therefore is used for theoretical considerations, because it has an infinitely wide frequency spectrum and represents the invariant element of folding. In the experimental analysis systems are, however, often stimulated with the step function and measured the step response, which is also fully describes the response of such a system. This avoids to have to generate a Dirac pulse with a good approximation, for which the input signal momentarily would take a very high value.

General

The impulse response is the derivative of the step response according to the time:

In the case of discrete signals, the system is a linear digital filter. The Dirac - pulse signal is also the one element of the discrete convolution, but here representing the frequency range [- π, π ], corresponding to the Nyquist frequency.

Using the impulse response can be a linear, time-invariant system (LTI ) system characterize and determine its frequency response or transfer function, for example. This is the Fourier transform of the impulse response at strictly stable systems.

Thus, if a Dirac pulse, where an unknown LTI system, so can be derived from the impulse response by means of Fourier analysis, specifically determined by the Laplace transform, the frequency response of the unknown system. Conversely, the effect of the LTI system is determined in the frequency domain by convolution with the impulse response in the time domain or by multiplication by the transfer function.

Practical application of this principle is in recent times in some DirectX and VST plug-ins (see convolution reverb ), the acoustic LTI systems (rooms, microphones, ...) can reproduce virtually. For the recovery of the impulse response, there are various methods:

Measurement by means of Dirac pulse

Theoretically, the impulse response of a system is determined by the supply of a Dirac pulse. However, it is virtually impossible to produce such a pulse (infinite amplitude in an extremely limited time ), it can be approximated only to a limited extent. This would require the shortest possible, strong "pop" or surge on the system are given and his answer through a microphone, etc. are measured. In this way ermitteltem frequency response can result in distortions, mainly because of non-linearities of the components ( THD ), noise, measurement inaccuracies and limited capacity.

With speakers The impulse response provides an indication of the impulse response, for rooms on the time and frequency response of the reverberation.

Determination by means of step response

From the step response of a system is obtained by differentiating the impulse response. Due to the sudden increase of the step function, there is in their measurement but similar problems as the direct measurement of the impulse response.

Determination by means of broad-band signal

The impulse response can also be using a broadband noise signal, such as white noise, determined. But one sends the noise signal into the system (eg, via a loudspeaker in a room ) and simultaneously measures the response of the system for a while ( for example, records with a microphone for a while on ). Then one calculates the cross-correlation of the transmitted and the received signal, it is in this case directly the impulse response of the system.

A major advantage of this method is that in addition to the test signal nor may bear further signals to the system. For example. it must be in a room measuring not be quiet as long as the noise ( eg calls) are uncorrelated for the test signal, because they fall out by the cross-correlation of the connection.