  The periodization of a stationary Gaussian random field on a sufficiently
large torus comprising the spatial domain of interest is the basis of various
efficient computational methods, such as the classical circulant embedding
technique using the fast Fourier transform for generating samples on uniform
grids. For the family of Mat\'ern covariances with smoothness index $\nu$ and
correlation length $\lambda$, we analyse the nonsmooth periodization
(corresponding to classical circulant embedding) and an alternative procedure
using a smooth truncation of the covariance function. We solve two open
problems: the first concerning the $\nu$-dependent asymptotic decay of
eigenvalues of the resulting circulant in the nonsmooth case, the second
concerning the required size in terms of $\nu$, $\lambda$ of the torus when
using a smooth periodization. In doing this we arrive at a complete
characterisation of the performance of these two approaches. Both our
theoretical estimates and the numerical tests provided here show substantial
advantages of smooth truncation.
