Mathematical Framework for Pseudo-Spectra of Linear Stochastic Difference Equations

作者：Bujosa Marcos^*; Bujosa Andres; Garcia Ferrer Antonio

来源：IEEE Transactions on Signal Processing, 2015, 63(24): 6498-6509.

DOI：10.1109/TSP.2015.2469640

• Spectral analysis
• time series
• non-stationarity
• frequency domain
• pseudo-covariance function
• linear stochastic difference equations
• partial inner product
• Extended Fourier Transform

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, i.e., 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, 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.

• 出版日期2015-12-15