Statistical Signal Processing of Complex-Valued Data

Abstract

Complex-valued random signals are embedded in the very fabric of science and engineering, yet the usual assumptions made about their statistical behavior are often a poor representation of the underlying physics. This book deals with improper and noncircular complex signals, which do not conform to classical assumptions, and it demonstrates how correct treatment of these signals can have significant payoffs. The book begins with detailed coverage of the fundamental theory and presents a variety of tools and algorithms for dealing with improper and noncircular signals. It provides a comprehensive account of the main applications, covering detection, estimation, and signal analysis of stationary, nonstationary, and cyclostationary processes. Providing a systematic development from the origin of complex signals to their probabilistic description makes the theory accessible to newcomers. This book is ideal for graduate students and researchers working with complex data in a range of research areas from communications to oceanography.

Tapering Promotes Propriety for Fourier Transforms of Real-Valued Time Series

“…Since it is a covariance matrix it is positive semidefinite. Additionally, Z has a complementary covariance matrix ccov{Z, Z} [17], defined as R = ccov{Z, Z} def = cov{Z, Z ⇤ } = E{ZZ T }, which is complex-valued and symmetric, but not in general positive semidefinite. (The complementary covariance matrix is also called the pseudo-covariance matrix, e.g., [17].…”

“…The two matrices ⌃ and R both appear in the so-called augmented covariance matrix [17]. Suppose we form the augmented vector U by joining Z and Z ⇤ , i.e.,…”

“…If Z is proper, it is uncorrelated with its complex conjugate Z ⇤ . When R 6 = 0, the vector is an improper complex-valued random vector [17].…”

“…In recent years there has been a great deal of interest in the concept of improper signals, for which this second-order circularity does not hold, since performance advantages accrue by processing such signals in a way which exploits the extra information contained in the signal [2], [7], [17].…”

Convergence Comparison of the CMA and ICA for Blind Polarization Demultiplexing

“…Assuming that the distribution p S (s k ) at a generic time k is known and independent of k, the distribution of the received symbol, x k = A k s k , is [16] …”