5.1 Discretisation of Continuous Signals

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Fig. 5.6: Spectrum of the signal sampled with Dirac pulses with overlapping periodic spectrum of the

original signal.

Fig. 5.7: Interpolation between samples using the impulse response of an ideal low-pass filter.

has an attenuation due to the si function. This means that when sampling with square

pulses, the spectrum of the sampled signal is not simply repeated with the sampling

frequency, but its values are also reduced during the repetition. Furthermore, distor-

tions can also occur during reconstruction, since the si also affects the spectral values

of the signal in the passband of the reconstruction low-pass filter at the output.

These distortions are smaller the narrower the pulse width ∆T of the square

pulses. However, the spectral components then become smaller and smaller (cf. Equa-

tion 5.6). This could be compensated by the fact that the pulse amplitude A becomes

larger and overall according to Equation 5.6 the product A ⋅∆T is a constant. This is

in ideal way the case with the Dirac momentum. Its width approaches zero, its height

approaches infinity, but its area (product of width times height) is one. Sampling with

a Dirac pulse therefore produces no distortions of the spectrum.