Space-Polarization Shift Keying Modulation for MIMO Channels

Abstract

In this paper, we propose space-polarization shift keying (SPSK) modulation scheme for multiple-input multiple-output (MIMO) systems employing dual-polarized antenna array, where the concept of space shift keying (SSK) is extended to include both the space and polarization dimensions. In the SPSK, polarization index also been used to relay the information along with the antenna index. As a result, the proposed transmission scheme reduces the signal processing complexity at the receiver exponentially when compared to the SSK scheme, for a given target transmission rate. In this correspondence, we investigate the performance of $4\times 4$ SPSK–MIMO configurations over different wireless channel scenarios for a range of values of the correlation coefficient and the Rician K-factor. We also highlight the effect of random orientation of antennas at the mobile unit on the bit error rate performance of SSK and SPSK transmission scheme. By using analytical results and simulations, we show that the proposed scheme outperforms the SSK scheme under line-of-sight channel conditions and its performance is comparable to SSK under non-line-of-sight channel conditions, for various receive-antenna inclination angle.

Appendix

1.1 Proof of (42)

$$\begin{aligned} {\mathcal {I}}= \int _{0}^{\infty }\frac{Q\left( \sqrt{\bar{\gamma }\nu }\right) \nu ^{\zeta _{N_r}-1}exp\left( -\frac{N_{r}m\nu }{x\bar{\gamma }}\right) }{\left( \frac{x\bar{\gamma }}{N_{r}m}\right) ^{\zeta _{N_r}}\varGamma \left( \zeta _{N_r}\right) }\,d\nu \end{aligned}$$

(46)

where $\varGamma (\cdot )$ represents the Gamma function. By using [1, Eq.6.5.17], the integral ${\mathcal {I}}$ can be written as follows:

$$\begin{aligned} {\mathcal {I}}=\frac{1}{2\sqrt{\pi }} \int _{0}^{\infty }\frac{\varGamma \left( \frac{1}{2},\frac{\nu \bar{\gamma }}{2}\right) \nu ^{\zeta _{N_r}-1}exp\left( -\frac{N_{r}m\nu }{x\bar{\gamma }}\right) }{\left( \frac{x\bar{\gamma }}{N_{r}m}\right) ^{\zeta _{N_r}}\varGamma \left( \zeta _{N_r}\right) }\,d\nu \end{aligned}$$

(47)

where $\varGamma (\cdot ,\cdot )$ denotes incomplete Gamma function.

From [8, Eq.(6.455)], the integral in (47) can be simplified as

$$\begin{aligned} {\mathcal {I}}&=\frac{\varGamma\,\left( \zeta _{N_r}+\frac{1}{2}\right) }{2\sqrt{2\pi }\varGamma \left( \zeta _{N_r}\right) \left( \frac{x\bar{\gamma }}{N_{r}m}\right) ^{\zeta _{N_r}}\zeta _{N_r}\left( \frac{1}{2}+\frac{N_{r}m}{x\bar{\gamma }}\right) ^{\zeta _{N_r}+\frac{1}{2}}}\nonumber \\&\quad {}_2F_1\left( 1,\zeta _{N_r}+\frac{1}{2};\zeta _{N_r}+1;\frac{2N_{r}m}{x\bar{\gamma }+2N_{r}m}\right) \end{aligned}$$

(48)

where $_2F_1(\cdot ,\cdot ;\cdot ;\cdot )$ is the Gaussian hypergeometric function [1, Eq.15.1.1].

After some algebraic manipulations, the integral ${\mathcal {I}}$ can be written as shown in below:

$$\begin{aligned} {\mathcal {I}}&=\frac{\varGamma (2\zeta _{N_r})}{\varGamma (\zeta _{N_r}) \varGamma (\zeta _{N_r}+1)}\left( \frac{N_{r}m}{2(x\bar{\gamma }+2N_rm)}\right) ^{\zeta _{N_r}}\left( \frac{x\bar{\gamma }}{x\bar{\gamma }+2N_rm}\right) ^{1/2}\nonumber \\&\quad {} _2F_1\left( 1,\zeta _{N_r}+\frac{1}{2};\zeta _{N_r}+1;\frac{2N_{r}m}{x\bar{\gamma }+2N_{r}m}\right) \end{aligned}$$

(49)