In this work, a power efficient THP scheme has been proposed. Based on a power
consumption analysis, it has been concluded that it is possible to improve the results of
the previous approaches in terms of power loss by scaling the original user symbols. It
has been confirmed that the proposed technique can maintain the error rates of THP and
CIO-THP while, at the same time, decreasing their power loss. In this line, the benefits
of both CIO-THP and PE-THP have been shown to be particularly significant at large
SNRs, where transmission power savings larger than 50% have been demonstrated.
Critique: The above-mentioned enhancements come at the cost of a higher com-
putational complexity. Indeed, there exists a trade-off between performance and com-
plexity: while complexity increases when a higher number of users are scaled, this also
entails that the known interference can be better aligned with the symbols to transmit,
hence further reducing the transmission power. This suggests that the proposed scheme
is particularly suitable for MIMO setups with a small number of antennas, where the
signal processing complexity increase is constrained. Moreover, note that the optimal
scaling factors might be re-utilized without the need of solving again the optimization
problem if the channel remains constant, e.g., by implementing a lookup table for the
possible input symbol combinations. This aspect promotes the application of the pro-
posed scheme in slowly varying channels and in small scale MIMO systems, where the
Pre-Scaling for Space Shift
Keying via Semidefinite
Programming
4.1
Introduction
Both space shift keying (SSK) and spatial modulation (SM) aim for reducing the
analog hardware complexity of conventional spatial multiplexing [18,19,158,167]. Specif-
ically, SSK and SM rely on encoding information into the active antenna indices, which
allows reducing the number of radio frequency (RF) chains employed for transmission,
when compared to the family of classic spatial multiplexing schemes [19, 158, 167]. The
essential objective behind reducing the number of RF chains is to enhance the system’s
energy efficiency by lowering down the total power consumption [2].
While the development of strategies for improving the attainable performance has
been mostly concentrated on techniques at the receiver side [168–171], a parallel line
of research proposes to exploit the availability of channel state information (CSI) at
the transmitter for devising constellation shaping schemes [163, 174–177]. In this con-
text, [174] analyzes the design of amplitude and phase constellations for minimizing the
transmit diversity order under different design conditions. SSK’s particular character-
istic of solely carrying information in the spatial domain has also been exploited for the
design of constellation shaping strategies in [163, 176, 177].
The maximization of the minimum Euclidean distance (MED) in the resultant SSK
and SM constellations via symbol pre-scaling has been the focus of [178–180]. In par-
ticular, the pre-scaling strategies developed in [178, 179] rely on forcing the received
SM constellation to resemble a classic quadrature amplitude modulation (QAM) con-
stellation from an inter-symbol distance perspective. However, the employment of the
regimes in [178, 179] may severely affect the system’s signal-to-noise ratio (SNR) due
to the stringent requirement of inverting the channel coefficients, which may become
critical for ill-conditioned channels. The scheme introduced in [180] mitigates this prob-
lem by solely applying a phase shift by the pre-scaling procedure. However, the above
designs only consider a single antenna at the receiver, which in turn simplifies both the
characterization and shaping of the received SM constellation.
The application of pre-scaling strategies to the more intricate systems with multiple
antennas at both communication ends has been promulgated in [181–184,208]. In partic-
ular, the schemes of [181, 182] propose opportunistic power allocation methods for both
SSK and generalized SSK for the sake of improving their performance, which implies
that only the amplitude of the transmit signals is modified. By contrast, simultaneous
phase and amplitude pre-scaling is considered in the constellation randomization (CR)
technique of [183]. This low-complexity scheme relies on generating Ds complex-valued
scaling factors off-line, and subsequently employing those specific scaling factors that
maximize the MED. The schemes introduced in [184], which were developed in parallel
to the work described in this chapter [209], further improve the performance by employ-
ing a successive convex approximation technique for solving the resultant optimization
problems for maximizing the MED. Subsequent to this work, [208] has proposed algo-
rithms with the similar objectives of maximizing the MED and minimizing the bit error
rate (BER) of SM systems. However, the schemes devised in [208] for maximizing the
MED only concentrate on 2 × K MIMO systems. Interestingly, the authors in [208]
improvements over schemes that solely aim at maximizing the MED.
Against the above contributions, this chapter considers the optimization of the pre-
scaling factors for SSK transmission via semidefinite programming. Specifically, the
original non polynomial (NP)-hard optimization problems are recast for the sake of
maximizing the performance of SSK transmissions via semidefinite relaxation (SDR).
This guarantees the applicability of the proposed pre-scaling designs to multi-antenna
aided receivers by carefully adapting the schemes introduced in [178–180]. Additionally,
this contribution aims at improving the performance of the strategy developed in [183] by
taking into account the channel conditions in the design of the pre-scaling vectors, while
reducing the signal processing complexity of the algorithms advocated in [184], where
multiple convex optimization problems have to be solved before reaching convergence.