Impulse response

The impulse response , also the weight function or impulse response called, is the output signal to the input a of a system Dirac pulse is supplied. It is used in systems theory to characterize linear, time-invariant systems . The (ideal) Dirac impulse is often used for theoretical considerations because it has an infinitely wide frequency spectrum and represents the invariant element of the convolution .

In the experimental analysis, on the other hand, LZI systems are often excited with the step function and the step response measured, which also fully describes the transfer behavior of such a system. This avoids generating a Dirac impulse in good approximation, for which the input signal has to briefly assume a very high value.

General

Impulse responses from PTn elements

The impulse response is the derivation of the step response with respect to time: ${\ displaystyle g (t)}$ ${\ displaystyle h (t)}$

{\ displaystyle g (t) = {\ frac {\ mathrm {d} h (t)} {\ mathrm {d} t}}}

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 it represents the frequency range [-π, π], corresponding to the Nyquist frequency .

With the help of the impulse response, a linear, time-invariant system (LTI system) can be characterized and z. B. determine its frequency response or transfer function. In strictly stable systems, this is the Fourier transform of the impulse response.

If a Dirac impulse is given to an unknown LTI system , the frequency response of the unknown system can be determined from the impulse response by Fourier analysis , specifically by the Laplace transform . Conversely, the effect of the LTI system can be determined by convolution with the impulse response in the time domain or by multiplication with the transfer function in the frequency domain .

This principle is practically applied recently in some DirectX - and VST - Plug-ins (see convolution reverb ), the acoustic LTI systems (rooms, microphones , ...) virtually replicate.

detection

Measurement using a Dirac pulse

Theoretically, the impulse response of a system can be determined by applying a Dirac impulse. However, it is practically impossible to generate such a pulse (infinite amplitude in an infinitesimally short time), it can only be approximated to a limited extent. To do this, the system would have to be given the shortest possible, strong “bang” or current surge and its response measured using a microphone or similar. The frequency response determined in this way can lead to distortions , mainly due to nonlinearities of the components ( distortion factor ), noise , measurement inaccuracies and limited load capacity.

The impulse response provides information about the impulse fidelity for loudspeaker boxes and the time and frequency behavior of the reverberation for rooms .

Determination by means of step response

The impulse response is obtained from the step response of a system by differentiation . Due to the sudden increase in the step function, however, there are similar problems when measuring it as when measuring the impulse response directly.

Determination by means of a broadband signal

The impulse response can also be determined with a broadband noise signal such as white noise . To do this, you send the noise signal into the system (e.g. into a room via a loudspeaker ) and at the same time measure the response of the system for a while (e.g. record with a microphone for a while). Then the cross-correlation of the transmitted and received signals is calculated ; in this case it is directly the impulse response of the system.

A great advantage of this method is that, in addition to the test signal, other signals can also be present on the system. E.g. It does not have to be quiet in a room for the measurement as long as the background noises (e.g. conversations) are uncorrelated to the test signal, because they are then eliminated due to the cross-correlation.

Web links

Commons : impulse response - collection of images, videos and audio files

Application example: impulse response for loudspeakers (including graphics)

Individual evidence

↑ B. Girod, R. Rabenstein, A. Stenger: Introduction to the system theory . Ed .: Teubner. 2nd Edition. Teubner, Stuttgart, Leipzig, Wiesbaden 2003, ISBN 978-3-519-16194-3 , pp. 434 .

[1] B. Girod, R. Rabenstein, A. Stenger: Introduction to the system theory . Ed .: Teubner. 2nd Edition. Teubner, Stuttgart, Leipzig, Wiesbaden 2003, ISBN 978-3-519-16194-3 , pp. 434 .