A Unified Precoding Scheme for Generalized Spatial Modulation

Abstract

Generalized spatial modulation (GSM) activates nt out of Nt (1 ≤ nt < Nt) available transmit antennas, and information is conveyed through nt modulated symbols as well as the index of the nt activated antennas. GSM strikes an attractive tradeoff between spectrum efficiency and energy efficiency. Linear precoding that exploits channel state information at the transmitter enhances the system error performance. For GSM with nt = 1 (the traditional SM), the existing precoding methods suffer from high computational complexity. On the other hand, GSM precoding for nt ≥ 2 is not thoroughly investigated in the open literature. In this paper, we develop a unified precoding design for GSM systems, which universally works for all nt values. Based on the maximum minimum Euclidean distance criterion, we find that the precoding design can be formulated as a large-scale non-convex quadratically constrained quadratic program (QCQP) problem. Then we transform this challenging problem into a sequence of unconstrained subproblems by leveraging augmented Lagrangian and dual ascent techniques. These subproblems can be solved in an iterative manner efficiently. Numerical results show that the proposed method can substantially improve the system error performance relative to the GSM without precoding, and features extremely fast convergence rate with a very low computational complexity.

Signal Shaping for Generalized Spatial Modulation and Generalized Quadrature Spatial Modulation

“…2) Signal Shaping with CSIT With instantaneous/statistical CSIT, GenSM/GenQSM systems can benefit from adaptive signal shaping. For instance, adaptive signal constellation design was investigated for SM in [18]; [19]- [21] studied antenna selection for SM systems, which can be regarded as the adaptive spatial constellation optimization; A large body of literature such as [22]- [34] probed into precoding schemes as well as their combinations with the adaptive signal or spatial constellation optimization. However, most literature considered SM or single-data-stream GenSM.…”

“…However, such precoding-aided signal shaping methods in [34] are suboptimal. The reasons are twofolds: First, each TAC carries the same number of data symbol vectors in [34], while due to the random nature of wireless channels, different TACs corresponding to separate channels have distinct information-carrying capabilities; Second, the data symbol vectors in the signal constellation of an activated TAC are modified by the same precoder, which can not guarantee the global signal shaping optimality, because the performance also highly depends on the previously given signal constellation. In addition, few work was dedicated to the adaptive shaping for GenQSM, except [6] which adjust the precoding process for QSM systems with a single IQ RF chain.…”

“…• Numerical comparisons with existing designs are presented to verify the superiority of our designs. In particular, we compare the proposed joint constellation design with the sole signal constellation design in open-loop systems without CSIT and closed-loop systems with statistical CSIT, where the well-recognized best signal constellations (e.g., the binary phase shift keying, BPSK) are included in comparisons; we compare the proposed adaptive signal shaping methods with precoding-aided signal shaping methods in [34]. Comparison results show that the proposed optimization-based signal shaping (OBSS) approach considerably outperforms existing designs in literature and the proposed codebook-based signal shaping (CBSS) approach shows comparable performance with much lower complexity.…”

“…where d is a target minimum distance. The problem (S-OP-b) can be solved by the Lagrangian method developed in [34]. The complexity is also around O(N 4 N 2 RF ) in each iteration 1 but fortunately the algorithm in [34] converges faster than that in [27].…”

“…The proposed OBSS approach aims to directly optimize the transmit vectors, which is different from the precodingaided signal shaping in [34] that modifies a given signal constellation of size M N R F via a diagonal preceding matrix or a full precoding matrix for each TAC, where they assume Mary modulation is adopted for each data stream transmission. Even though the diagonal precoder and the full precoder can be designed by solving a similar QCQP problem as (S-OP), there exists a difference in the number of optimization variables.…”