Second-order improperness in frequency-domain colored signal model

This study discusses the second-order improperness in the frequency-domain colored signal model. After the real-valued and colored signal in the time-domain is transformed into the frequency domain, the convolutive blind source separation (BSS) methods may generate a window indexed vector, and minimum variance distortionless response (MVDR) beamforming may produce an array sensor indexed vector, and the trial structured EEG data set may provide a trial-indexed vector. These three complex-valued vectors have such similarities: 1) despite of resulting from the Fourier transformation, they are not indexed by the frequency bins; 2) its real part and image part are proved to be correlated. Hence, such a complex-valued vector is improper, and the pseudo-autocorrelation matrix of this improper vector exists. Moreover, if components of an improper complex-valued vector are uncorrelated, except the autocorrelation matrix, the pseudo-autocorrelation matrix should also be diagonal. This one more statistics could be useful to design better the complex-valued BSS algorithm and MVDR beamforming algorithm.