Capacity Analysis of Index Modulations Over Spatial, Polarization, and Frequency Dimensions

Abstract

Abstract-Determining the capacity of a modulation scheme is a fundamental topic of interest. Index Modulations (IM), such as Spatial Modulation (SMod), Polarized Modulation (PMod) or Frequency Index Modulation (FMod), are widely studied in the literature. However, finding a closed-form analytical expression for their capacity still remains an open topic. In this paper, we formulate closed-form expressions for the instantaneous capacity of IM, together with its 2nd and 4th order approximations. We show that, in average, the 2nd approximation error tends to zero for low Signal to Noise Ratio (SNR) and is o (SNR). Also, a detailed analysis of the ergodic capacity over Rayleigh, Rice and Nakagami-m channel distributions is provided. As application of the capacity analysis, we leverage the proposed expressions to compute the ergodic capacities of SMod for different antenna configuration and correlations, PMod for different channel components and conditions, and FMod for different frequency separations. Although V-BLAST offers higher capacity, it requires high power budget. On the contrary, OSTBC exploits the full channel diversity at the expense of sacrificing multiplexing gain. Note also that OSTBC can only provide full diversity with 2 transmitting antennas. SMod strikes a balance between V-BLAST and OSTBC by increasing the channel capacity that OSTBC offers with less power requirements and is a trade-off between V-BLAST and OSTBC. In addition, it also provides more flexibility than OSTBC in the achievable rate that is obtained by increasing the number of transmitting antennas.SMod can be classified in to the general modulation strategy of Index Modulations (IM), term employed by [23], [30].

Channel Dependent Mutual Information in Index Modulations

“…) dx is the differential entropy of X . Note that, in contrast to [11], where the capacity is obtained, in our case the symbol s is not maximized and belongs to a particular constellation.…”

“…are the arithmetic and geometric means over s ′ and l ′ by keeping s and l fixed. Hence, by plugging (11) in (10), the second order approximation of MI is described by (12).…”

“…This paper presents closed-form expressions based on different order approximations of the MI of IM. Based on the works [9]- [11], which compute the capacity of IM, we aim at solving the difficulty of finding a closed-form expression of MI. Thanks to this expression, we are able to compute the MI at each time instant with much less computational complexity and making the problem of adaptive IM affordable.…”

Beam Index Modulation Wireless Communication With Analog Beamforming

“…In [41], [42], the integral-based expressions of the instantaneous SM capacity, for ZMCG signaling in IQ domain and uniform selection in index domain, were presented. Later, its closed-form approximation was derived in [43]. In this paper, we adopt the 4 th order approximation, i.e., (29) in [43].…”

Practical Implementation of Link Adaptation with Dual Polarized Modulation

“…where the MI is expressed as the sum of two components: the capacity of the symbols I S and the polarization bit capacity I P . The capacity of IM is studied in [6] and [8], among others. In [6] a first order approximation of (2) for a general Index Modulation is obtained.…”