5.2 Sampling (Interpolation Theory)

lf f _T ( x, y) is band limited in the x- and y-axes to k _{x 0} and k _{y 0}, then a sample spacing of ? x, ? y, as per (5.1) and (5.2) will be sufficient to allow the entire function to be reconstructed from sampling theory [1]. That is, F _T( k _x, k _y, z = 0) = 0 when k _x ? k _{x 0} and k _y ? k _{y 0}. Hence the limits of the integration become finite and the continuous field can be reconstructed from the samples,

(5.1)

The lattice spacing in each axis may be determined from

(5.2)

and

(5.3)

where

(5.4)

Hence, the sample spacing required to guarantee this, when related to the wavelength is given by

(5.5)

Therefore, for a band-limited function, the conventional sampling criterion is sufficient for the case of those measurements that are taken over non-tangential planar surfaces. If the maximum angle of coverage is less than 90 then the expressions are modified as

(5.6)

In one dimension, the ideal band-limited interpolation procedure required to reconstruct the continuous function from the samples taken at a set of grid points can be expressed as

(5.7)

Here, the series is convergent and the sine function is defined as

(5.8)

The value of the sine function in the limit as x tends to zero has...