Plane-wave decomposition of spatially random fields

We investigate the uniqueness of the plane-wave decomposition of temporally deterministic, spatially random fields. Specifically, we consider the decomposition of spatially ergodic and, thus, statistically homogeneous fields. We show that when the spatial power spectrum is injective, the plane waves are the only possible coherent modes. Furthermore, the randomness of such fields originates in the spatial spectral phase, i.e., the phase associated with the coefficients of each plane wave in the expansion. By contrast, the spectral amplitude is deterministic and is specified by the spatial power spectrum. We end with a discussion showing how the results can be translated in full to the time domain.