The paper deals with the discrete spectral-orthogonal decompositions of centered Gaussian random processes for two cases. In the first case, the process implementations are a sequence of pulses that are short in comparison with the observation time. The process decomposition was obtained as a generalized Fourier series on the basis of the delta-function formalism, and the variances of the coefficients (random values) of this series were found as well. The resulting expressions complement Kotelnikov’s formula because they cover both the highfrequency and the low-frequency regions of the canonical-decomposition spectrum. In the second case, a random process is a superposition of narrow-band Gaussian random processes, and its implementations are characterized by oscillations. For such a process the canonical decomposition in terms of Walsh functions was obtained on the basis of the generalized function formalism. Then this decomposition was re-decomposed in terms of trigonometric functions; it follows from the resulting series that the canonical decomposition spectrum is not uniform since a pedestal is formed in the constant component region.