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