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##

The impulse response

A useful (although mathematically difficult) fiction
is the Dirac "needle" impulse ,
supposed to be an
infinitely short, infinitely high, positive
impulse at time 0 whose integral over time equals 1.
It cannot be realized, but the time-integrated signal,
the unit step function, can be approximated by
switching on or off a current or by suddenly
applying or removing a force. According to the
definitions of the Laplace and Fourier
transforms, both transforms of the Dirac
pulse have the constant value 1. In this case Eq. (11)
reduces to *G*(*s*)=*H*(*s*), which means that the
transfer function *H*(*s*) is the Laplace transform of the
impulse response *g*(*t*). Likewise, the complex frequency response
is the Fourier transform of the impulse response.
All information contained in these complex functions is also
contained in the impulse response of the system.
The same is true for the step response,
which is often used to test or calibrate seismic
equipment. The amplitude spectrum of the Dirac pulse is
``white'' like that of ``white noise''; either type of signal
is therefore suitable to calibrate broadband systems.

Explicit expressions for the response of a linear system to impulses,
steps, ramps and other simple waveforms can be obtained by evaluating
the inverse Laplace transform over a suitable contour in the complex *s*
plane, provided that the poles and zeros are known. The result,
generally a sum of decaying complex exponential functions, can then be
numerically evaluated with a computer or even a calculator. Although
this is an elegant way of computing the response of a linear system to
simple input signals with any desired precision, a warning is
necessary: the numerical samples so obtained are not the same as the
samples that would be obtained with an ideal digitizer. The digitizer
must limit the bandwidth before sampling, and does therefore not
generate instantaneous samples but some sort of time-averages. For
computing samples of bandlimited signals, different mathematical
concepts must be used [Schuessler 1981].

Specifying the impulse or step response of a a system in place of its
transfer function is not practical because the analytic expressions are
cumbersome to write down, and represent signals of infinite duration
that cannot be tabulated in full length.

** Next:** The convolution theorem
**Up:** Basic Theory
** Previous:** The Fourier transformation
*Erhard Wielandt *

2002-11-08