User talk:Mct mht/Wide-sense stationary time series

statisticians don't study complex-valued random variables, because data is always real. In electrical engineering, there are some conveniences to using complex numbers and the complex form of the Fourier transform. Basically, two: a) phase and amplitude can be combined into one number; b) the operational properties of the Fourier transform are clarified by the use of the complex exponential instead of sines and cosines: products go to convolutions, and vice versa.

Now, would you believe it, both of these advantages are nugatory in the topic of the power spectral density. a) The auto-correlation function throws away all the information about the phases. b) The sample paths studied are so remote from having limited total variation in any interval, that their Fourier transforms only exist as distributions, and distributions cannot be multiplied together, so the operational properties do not hold in this context. The usual hand-waving proofs of the Wiener-Khintchine theorem, which use how products go into convolutions, are completely invalid if the spectrum is continuous. They only apply to the case of discrete time or a discrete line spectrum.66.167.204.242 (talk) 06:53, 18 April 2014 (UTC)