Asymptotically optimal low-complexity estimation of sampled improper-complex second-order cyclostationary random process

In this paper, a low-complexity widely linear minimum mean-squared error (WLMMSE) estimator is proposed for a sampled improper-complex second-order cyclostationary (SOCS) random process. The conventional WLMMSE estimator that directly processes the sampled vector and its complex conjugate suffers from high computational complexity in matrix inversion and multiplication when the sample size is large. To overcome this shortcoming, the proposed estimator approximates the frequency-domain covariance and complementary covariance matrices by a block matrix with diagonal blocks and that with anti-diagonal blocks, respectively, which is motivated by the fact that both the two-dimensional (2-D) power spectral density (PSD) and the 2-D complementary PSD of an SOCS random process consist of equally spaced impulse fences. Moreover, unlike the conventional WLMMSE estimator, the proposed estimator processes the sampled vector and its complex conjugate after removing redundancies, which further reduces the complexity. It is shown that, for sufficiently large observation period, the redundancy-removing procedure results in a statistic that is approximately proper and sufficient. It is also shown that the proposed estimator is asymptotically optimal in the sense that the average mean-squared error converges to that of the WLMMSE estimator as the observation period tends to infinity.