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/non-circularity and then surveys majority of the contributions in this enormous journey.

中文翻译：

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从不正确的高斯信号到不对称信号的旅程


连续和离散复数随机变量分别从传统的正确和对称假设到广义的不正确和不对称表征（考虑随机实体与其复数共轭之间的相关性）的偏差，引入了新的设计自由度和各种潜在优点。因此，不当理论在医学、地质学、声学、光学、图像和模式识别、计算机视觉以及其他众多研究领域（我们主要关注通信系统）有着广泛的应用。这个旅程从干扰限制通信中不正确的高斯信令的设计开始，并导致更复杂和实际可行的非对称离散调制设计。这种不对称整形弥补了编码/非编码无线/光通信系统中复杂的收发器和检测方案在理论和实际可实现的限制之间的差距。有趣的是，引入不对称性并根据某些设计标准调整传输参数可以在不影响系统带宽或功率要求的情况下实现最佳性能。这篇双重风格的文章首先介绍了教程基础内容，涵盖现实/复杂性、适当性/不适当性和循环性/非循环性之间的相互作用，然后调查了这一巨大旅程中的大部分贡献。