  Imagine an array with a massive (possibly uncountably infinite) number of
antennas in a compact space. We refer to a system of this sort as Holographic
MIMO. Given the impressive properties of Massive MIMO, one might expect a
holographic array to realize extreme spatial resolution, incredible energy
efficiency, and unprecedented spectral efficiency. At present, however, its
fundamental limits have not been conclusively established. A major challenge
for the analysis and understanding of such a paradigm shift is the lack of
mathematically tractable and numerically reproducible channel models that
retain some semblance to the physical reality. Detailed physical models are, in
general, too complex for tractable analysis. This paper aims to take a closer
look at this interdisciplinary challenge. Particularly, we consider the
small-scale fading in the far-field, and we model it as a zero-mean,
spatially-stationary, and correlated Gaussian scalar random field.
Physically-meaningful correlation is obtained by requiring that the random
field be consistent with the scalar Helmholtz equation. This formulation leads
directly to a rather simple and exact description of the three-dimensional
small-scale fading as a Fourier plane-wave spectral representation. Suitably
discretized, this leads to a discrete representation for the field as a Fourier
plane-wave series expansion, from which a computationally efficient way to
generate samples of the small-scale fading over spatially-constrained compact
spaces is developed. The connections with the conventional tools of linear
systems theory and Fourier transform are thoroughly discussed.
