Bujosa Brun, MarcosBujosa Brun, AndrésGarcía Ferrer, Antonio2023-06-192023-06-192013https://hdl.handle.net/20.500.14352/41468This working paper has been accepted for publication in a future issue of IEEE Transactions on Signal Processing. Content may change prior to final publication. Citation information: DOI:10.1109/TSP.2015.2469640. 1053-587X copy right 2015 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs-permissions@ieee.org See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.Although spectral analysis of stationary stochastic processes has solid mathematical foundations, this is not always so for some non-stationary cases. Here, we establish a rigorous mathematical extension of the classic Fourier spectrum to the case in which there are AR roots in the unit circle, ie, the transfer function of the linear time-invariant filter has poles on the unit circle. To achieve it we: embed the classical problem in a wider framework, the Rigged Hilbert space, extend the Discrete Time Fourier Transform and defined a new Extended Fourier Transform pair pseudo-covariance function/pseudo-spectrum. Our approach is a proper extension of the classical spectral analysis, within which the Fourier Transform pair auto-covariance function/spectrum is a particular case. Consequently spectrum and pseudo-spectrum coincide when the first one is defined.engtechnical reporthttps://www.ucm.es/icaeopen accessC00C22Spectral analysistime seriesnon-stationarityfrequency domainpseudo-covariance functionlinear stochastic difference equationsRigged Hilbert spacepartial inner productExtended Fourier Transform.Econometría (Economía)5302 Econometría