  The spectral proper orthogonal decomposition (SPOD) is a newly introduced
extension of snapshot POD that recently gained attention but also brought up
controversial issues. Within the first proposition, the approach was mainly
presented in a methodological and phenomenological way. The present paper will
detail the relations between SPOD and related POD approaches from an analytical
point of view. To allow for a better grasp of the approach, an alternative
formulation is given that is based on the classic idea from Lumley that was
carried on by George. As will be shown, SPOD is closely related to POD with a
prior segmentation and Fourier transformation in time. Moreover, the SPOD is
shown to be equivalent to snapshot POD combined with time delay embedding.
