Redesigned Spatial Modulation for Spatially Correlated Fading Channels

Abstract

In this paper, a new variant of Spatial Modulation (SM) Multiple-Input Multiple-Output (MIMO) transmission technique, designated as Redesigned Spatial Modulation (ReSM) has been proposed. In ReSM scheme, a dynamic mapping for antenna selection is adopted. This scheme employs both single antenna as well as double antenna combinations depending upon channel conditions to combat the effect of spatial correlation. When evaluated over spatially correlated channel conditions, for a fixed spectral efficiency and number of transmit antennas, ReSM exhibits performance improvement of at least 3 dB over all the conventional SM schemes including Trellis Coded Spatial Modulation (TCSM) scheme. Furthermore, a closed form expression for the upper bound on Pairwise Error Probability (PEP) for ReSM has been derived. This has been used to calculate the upper bound for the Average Bit Error Probability (ABEP) for spatially correlated channels. The results of Monte Carlo simulations are in good agreement with the predictions made by analytical results. The relative gains of all the comparison plots in the paper are specified at an ABER of 10⁻⁴.

Appendix 1

Proof of Eq. (24)

$$ABER_{ReSM} \le \frac{1}{{2^{\eta } }}\mathop \sum \limits_{{n_{t} s_{t} }} \mathop \sum \limits_{n,s} \frac{{N\left( {x_{{n_{t} s_{t} ,}} x_{n,s} } \right)}}{\eta }E_{H} \left\{ {PEP} \right\}$$

(36)

In this equation, $n_{t}$ is the active transmit antenna, $s_{t}$ is the transmit symbol, η is the spectral efficiency. $N\left( {x_{{n_{t} ,s_{t} ,}} x_{n,s} } \right)$ is the total number of bits in error between $x_{{n_{t} s_{t} ,}} x_{n,s}$. $E_{H} \left\{ \cdot \right\}$ is the expectation across the channel matrix $\varvec{H}$.

PEP is given by

(37)

where

Following from [3] an alternative integral expression of the Q-function and taking the expectation of Eq. (37) we arrive at

$$E_{H} \left\{ {PEP} \right\} = \frac{1}{\pi }\mathop \int \limits_{0}^{{\frac{\pi }{2}}} \varphi \left( { - \frac{1}{{4\sigma_{n}^{2} sin^{2} \left( \theta \right)}}} \right)d\theta$$

(38)

where $\varphi \left( \cdot \right)$ is the moment generating function (MGF) of the random variable . From [3] it is clear that any i.i.d. Gaussian random variable with mean $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{v}$ and any Hermitian matrix . The MGF then can be written as

(39)

$L_{v}$ is the covariance matrix of v. From (38) invoking MGF of we can write

(40)

$I_{n}$ is an $n \times n$ identity matrix, ${\mathbf{vec}}\left( \varvec{H} \right)$ is the column of matrix $\varvec{H}$ into a column vector.  ${\mathcal{L}}_{H}$ is the covariance matrix, $\varvec{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} }$ is the mean matrix, $\otimes$ is the Kronecker product and $\left( \cdot \right)^{H}$ is the Hermitian.

Making use of the Chernoff bound, PEP then can be written as,

(41)

Finally the upper bound is given by

(42)

1.1 Appendix 2

See Table 6.

Table 6 Mapping followed in ReSM for 6 bpcu employing 16QAM Constellation