Abstract
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The deviation of continuous and discrete complex
random variables from the traditional proper and symmetric
assumption to a generalized improper and asymmetric characterization
(accounting correlation between a random entity
and its complex conjugate), respectively, introduces new design
freedom and various potential merits. As such, the theory of
impropriety has vast applications in medicine, geology, acoustics,
optics, image and pattern recognition, computer vision, and
other numerous research fields with our main focus on the
communication systems. The journey begins from the design of
improper Gaussian signaling in the interference-limited communications
and leads to a more elaborate and practically
feasible asymmetric discrete modulation design. Such asymmetric
shaping bridges the gap between theoretically and practically
achievable limits with sophisticated transceiver and detection
schemes in both coded/uncoded wireless/optical communication
systems. Interestingly, introducing asymmetry and adjusting the
transmission parameters according to some design criterion
render optimal performance without affecting the bandwidth or
power requirements of the systems. This dual-flavored article
initially presents the tutorial base content covering the interplay
of reality/complexity, propriety/impropriety and
circularity/noncircularity
and then surveys majority of the contributions in this
enormous journey.
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