Capacity analysis for LOS millimeter–wave quadrature spatial modulation

Abstract

Capacity analysis for millimeter-wave (mmWave) quadrature spatial modulation (QSM) multiple-input multiple-output (MIMO) system is presented in this paper. QSM is a new MIMO technique proposed to enhance the performance of conventional spatial modulation while retaining almost all its inherent advantages. Furthermore, mmWave utilizes a license-free wide-bandwidth spectrum and is a very promising candidate for future wireless systems. Detailed and novel analysis of the mutual information and the capacity for line of sight (LOS) mmWave-QSM system are presented in this study. The conditions under which theoretical capacity can be achieved are derived and discussed. Also, mmWave channel design is conducted and a novel algorithm is proposed to overcome existing limitation for unbalanced MIMO configurations, i.e., when the number of receive antennas is less than that of the transmit antennas. Monte Carlo simulation results are provided to corroborate derived formulas. It is shown that significant performance enhancements can be achieved under different system and channel configurations.

Appendix: Derivation of \(I(\imath ^{\mathfrak{R}},\imath ^{\mathfrak{I}},s;{\mathbf{y}})\) in (8)

Proof

The entropy of the received signal vector, $H(\mathbf{y})$, is given by

$$\begin{aligned} H\left( \mathbf{y}\right)= & {} -\int _{\mathbf{y}} p_{\mathbf{y}}\left( \mathbf{y}\right) \log _2 p_{\mathbf{y}}\left( \mathbf{y}\right) dy \\= & {} -\mathrm {E}_{\mathbf{y}} \left\{ \log _2 p_{\mathbf{y}}\left( \mathbf{y}\right) \right\} \; \end{aligned}$$

(19)

where $p_y(\cdot )$ is the PDF of the received vector y,

$$\begin{aligned} p_{\mathbf{y}}= & {} \int \limits _{\mathbf{h}_{\imath ^{\mathfrak{R}}_t},\mathbf{h}_{\imath ^{\mathfrak{I}}_t},s_t} \left( p_{\mathbf{h}_{\imath ^{\mathfrak{R}}_t}}\cdot p_{\mathbf{h}_{\imath ^{\mathfrak{I}}_t}}\cdot p_{s_t}\right) p_{\left( \mathbf{y}|\mathbf{h}_{\imath ^{\mathfrak{R}}_t},\mathbf{h}_{\imath ^{\mathfrak{I}}_t},s_t\right) } \\= & {} \frac{ \sum \limits _{\mathbf{h}_{\imath ^{\mathfrak{R}}_t},\mathbf{h}_{\imath ^{\mathfrak{I}}_t},s_t} \exp \frac{-\left\| {\mathbf{y} - \left( {\mathbf{{h}}_{{\imath ^{\mathfrak{R}}}}}{s_t^{\mathfrak{R}}} + {\mathbf{{h}}_{{\imath ^{\mathfrak{I}}}}}{s_t^{\mathfrak{I}}}\right) } \right\| ^2_\mathrm{F}}{\sigma _n^2}}{N_t^2M \times \left( \pi \sigma _n^2\right) ^{N_r}} \\ \end{aligned}$$

(20)

From (19) and (20), the entropy of $\mathbf{y}$ is,

$$\begin{aligned} H\left( \mathbf{y}\right) =&\log _2\left( N_t^2M\right) +N_r\log _2\left( \pi \sigma _n^2\right) \\&- \mathrm {E}_{\mathbf{y}}\left\{ \log _2 \sum \limits _{\mathbf{h}_{\imath ^{\mathfrak{R}}_t},\mathbf{h}_{\imath ^{\mathfrak{I}}_t},s_t} \exp \frac{-\left\| {\mathbf{y} - \left( {\mathbf{{h}}_{{\imath ^{\mathfrak{R}}}}}{s_t^{\mathfrak{R}}} + {\mathbf{{h}}_{{\imath ^{\mathfrak{I}}}}}{s_t^{\mathfrak{I}}}\right) } \right\| ^2_\mathrm{F}}{\sigma _n^2}\right\} \end{aligned}$$

(21)

The conditional entropy of $\mathbf{y}$ on $\mathbf{h}_{\imath ^{\mathfrak{R}}_t},\mathbf{h}_{\imath ^{\mathfrak{I}}_t}$, and $s_t$ is,

$$\begin{aligned} H\left( \mathbf{y}|\mathbf{h}_{\imath ^{\mathfrak{R}}_t},\mathbf{h}_{\imath ^{\mathfrak{I}}_t}, s_t\right)= & {} -\int _{\mathbf{y}}p_{\left( \mathbf{y}| \mathbf{h}_{\imath ^{\mathfrak{R}}_t},\mathbf{h}_{\imath ^{\mathfrak{I}}_t},s_t\right) } \times \log _2p_{\left( \mathbf{y}|\mathbf{h}_{\imath ^{\mathfrak{R}}_t}, \mathbf{h}_{\imath ^{\mathfrak{I}}_t},s_t\right) } \\= & {} -\mathrm {E}_{\mathbf{y}}\left\{ \log _2 p_{\mathbf{y}| \mathbf{h}_{\imath ^{\mathfrak{R}}_t},\mathbf{h}_{\imath ^{\mathfrak{I}}_t},s_t}\right\} \\= & {} -\mathrm {E}_{\mathbf{n}}\left\{ \log _2 p_{\mathbf{n}} \left( {\mathbf{n} + \mathbf{u_y} }\right) \right\} \\= & {} {N_r}\log _2\left( \pi e \sigma _n^2\right) \end{aligned}$$

(22)

where $\mathbf{u_y}= {{\mathbf{h}_{{\imath ^{\mathfrak{R}}}}}{s_t^{\mathfrak{R}}} + {\mathbf{h}_{{\imath ^{\mathfrak{I}}}}}{s_t^{\mathfrak{I}}}}$.

Substituting (21) and (22), in (7) lead to $I(\imath ^{\mathfrak{R}}_t,\imath ^{\mathfrak{I}}_t,s_t;\mathbf{y})$ given in (8). $\square$