Reconstruction

To reconstruct a sampled function the frequency response of the original function, which is centered at the origin, has to be extracted from the frequency response of the sampled function. This is easily achieved, if the replicas do not overlap, by multiplying in frequency domain with a box function. This means, that in spatial domain the set of samples has to be convolved with the function, which frequency response is a box. This is the sinc function, defined by

$$\mathrm{sinc}(x) = \left\{ \begin{array}{cl} \frac{\sin \p... ...x \neq 0$}\\ 1 & \textrm{if $x = 0$}\\ \end{array} \right.$$ (8.4)

The replicas now do not overlap if the original function is band-limited, i.e., it contains no frequencies outside a certain frequency interval, and if it was sampled above the Nyquist frequency, that is, twice the highest frequency in the function. This is known as Shannon's sampling theorem.