# Information-Theoretic Analysis of a Family of Improper Discrete Constellations
## 1 Introduction  
In the theoretical analysis of wireless communication systems, the transmitted symbols are typically assumed to be proper complex Gaussian random variables, i.e., the transmitted symbols are complex Gaussian and uncorrelated with their complex conjugate. This is motivated by the fact that such an input distribution achieves the capacity of the point-to-point, broadcast and multiple-access channels [1]. In more complex interference-limited scenarios, such as the interference channel, transmitting proper Gaussian signals is in general suboptimal, so assuming this input distribution in such cases is not theoretically justified. As a matter of fact, recent works have shown that, when treating interference as noise, a higher transmitted rate may be achieved in interference-limited networks by transmitting improper Gaussian signals, which are correlated with their complex conjugate [2; 3; 4; 5; 6; 7; 8]. Improper signals are common in communications since many important digital modulation schemes yield improper signals. Examples are binary phase shift keying (BPSK) or Gaussian minimum shift keying (GMSK) [9]. Additionally, power imbalance between the in-phase and in-quadrature components also yields an improper signal. A substantial body of work has focused on the design of widely linear receivers to exploit the impropriety of a received improper signal (we refer the interested reader to [10; 11; 12; 13] and the references therein). While these works devise techniques to deal with signals whose impropriety was not intentionally introduced, recent works consider the deliberate transmission of improper signals as a novel way to handle interference in a multiuser channel. [PERSON] et al. showed the benefits of improper signaling for the first time in their pioneering work [2]. They considered a single-antenna three-user interference channel with constant channel extensions and showed that the number of degrees of freedom (DoF), which is the number of interference-free signals that can be transmitted, was larger when users transmit improper signals as opposed to proper ones. Similar DoF results were derived for the four-user interference channel in [14]. After that, a number of new works has emerged, and the payoffs of improper signaling have been revealed for different multiuser scenarios such as the interference channel [3; 4; 5; 15], \(Z\)-interference channel [6; 16; 17; 18], broadcast channels with linear precoding [19; 20], underlay and overlay cognitive radio networks [7; 8; 21] and relay channels [22; 23]. More recently, in [24], the authors showed the advantages of improper signaling in a multiuser scenario where a point-to-point link coexists with a multiple access channel. Even though these works present new and intriguing results that pave the way for more efficient interference management schemes, the vast majority of them assume Gaussian codebooks (the only exception is [15], where the symbol error rate is minimized for arbitrary digital modulation schemes) and, hence, can never be realized in practice since practical codes use non-Gaussian finite signaling constellations. It is well known that the penalty in signal-to-noise ratio (snr) paid by constraining the complex symbols to a uniform \(M\)-ary quadrature amplitude modulation (QAM) constellation is given by the shaping loss, which for large \(M\) is 1.53 dB in the proper case [25]. When dealing with proper signals, this shaping loss is due to enclosing the uniformly-transmitted codewords inside a hypercube rather than a hypersphere, which would be the optimal region that minimizes the transmitted power. When using non-Gaussian improper signals, different questions arise: What is the optimal way to generate an improper constellation with a given circularity coefficient (i.e., degree of impropriety)? In coded communication systems, what is the reduction in rate compared to the improper Gaussian capacity when the coded channel symbols are constrained to be in a given discrete improper constellation?