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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