
4.2
Sampling Using Orthonormal Functions
103
curve corresponds to the normalized sampling error for this case. It is clear
from the figure that for a given number of samples, Fourier sampling is more
efficient than standard sampling.
4.2.4 Optimal Sampling
Optimal sampling of a random field defined over a region Sf of the xj-plane
is achieved by that set of orthonormal functions (p
mn
(x, y) which for every
Μ and
TV
yields the minimum value for the sampling error ε
ΜΝ
. Since the first
term in (46) is independent of the choice of the orthonormal basis, ε
ΜΝ
is
minimized when the second term, i.e., the nonnegative quantity
Λί-l N-l rB/2 ΓΑ/2 fB/2 ...