Signal to Interference Ratio Based Antenna Selection for Spatial Multiplexing

Abstract

To achieve reliable high throughput wireless links spatial multiplexing and diversity modes of multiple-input multiple-output are used in combination. Antenna selection (AS) has minimal complexity among other spatial diversity methods. Bell Labs Layered Space Time (BLAST) is a low complexity spatial multiplexing technique, especially when minimum mean squared error (MMSE) receivers are considered. However, optimal AS is known to be computationally intensive when used along with spatial multiplexing, since an orthogonal subset channel matrix is required to be found. Known suboptimal algorithms are still relatively complex and incur performance penalties. In this paper we propose a low complexity AS algorithm for BLAST. It uses an approximate signal to interference ratio metric as a heuristic measure to select a given number of antennas. It produces the selection choice after a single iteration only. A structure to reduce hardware complexity by reusing the MMSE equalizer block is also proposed. Such reuse can be applied to several AS algorithms. We compare the performance of the proposed algorithm against others using mean spectral efficiency (SE), 10 % outage SE, symbol error rate performance and implementation complexity. The impact of approximate expressions used in the proposed algorithm is also analyzed.

Appendix

Note: $\varXi$ represents complexity of the abbreviated algorithm mentioned in the superscript in terms of the number of operations of the specific type (RM/RA etc.) mentioned in the subscript. The algorithm MaxCap is abbreviated as ‘MXCP’. $\varPsi$ and $\varPsi _{A}$ represent the multiplication and addition complexity (respectively) of the operation $\hat{\mathbf {L}}= \hat{\mathbf {H}}^{\mathrm{H}}\hat{\mathbf {H}}$ (see Table 1).

$$\begin{aligned} \Psi (p,q)&= \frac{pq}{6}(3q+1). \end{aligned}$$

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$$\begin{aligned} {\Psi _{A}}(p,q)&= \frac{1}{4}(7q^{2}p - 3pq - 2q^{2}). \end{aligned}$$

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Part 1: Complexity Expressions Without Reuse

$$\begin{aligned} \varXi ^{MXCP}_{RM}&= 3\varXi ^{MXCP}_{CM} \end{aligned},$$

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where

$$\begin{aligned} \varXi ^{MXCP}_{CM}&= \frac{2}{3}{\mathrm {{M}}}_{\mathrm{T}}\mathrm {{M}}_{\mathrm{R}}+ \frac{4}{3}+2{\mathrm {{M}}}_{\mathrm{T}}\nonumber \\&\quad+ \max ({\mathrm {{M}}}_{{\mathrm{R}}_{\mathrm{s}}}-2,0)({\mathrm {{M}}}_{\mathrm{T}}^2 + \frac{2}{3}{\mathrm {{M}}}_{\mathrm{T}}+ \frac{1}{3}) \nonumber \\&\quad+ ({\mathrm {{M}}}_{{\mathrm{R}}_{\mathrm{s}}}-1)\Psi (1,{\mathrm {{M}}}_{\mathrm{T}}) + (\frac{2}{3}{\mathrm {{M}}}_{\mathrm{T}}+ 1)({\mathrm {{M}}}_{\mathrm{R}}- 1) \nonumber \\&\quad+({\mathrm {{M}}}_{\mathrm{R}}+ \frac{2}{3})\max ({\mathrm {{M}}}_{{\mathrm{R}}_{{\mathrm{s}}}}-2,0)({\mathrm {{M}}}_{\mathrm{R}}- \frac{\mathrm {{M}}_{{\mathrm{R}}_{\mathrm{s}}}+1}{2}).\end{aligned}$$

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$$\begin{aligned} \varXi ^{MXCP}_{RA}&= 5\varXi ^{MXCP}_{CM} + 2\varXi ^{MXCP}_{RA1},\end{aligned}$$

where

$$\begin{aligned} \varXi ^{MXCP}_{RA1}&= 1 + \max ({\mathrm {{M}}}_{{\mathrm{R}}_{\mathrm{s}}}- 2,0)(1 + 2{\mathrm {{M}}}_{\mathrm{T}}({\mathrm {{M}}}_{\mathrm{T}}-1)) \nonumber \\&\quad+ ({\mathrm {{M}}}_{\mathrm{R}}-1)(\mathrm {{M}}_{\mathrm{T}}+1) \nonumber \\&\quad+ {\mathrm {{M}}}_{\mathrm{T}}\max ({\mathrm {{M}}}_{{\mathrm{R}}_{\mathrm{s}}}-2,0)(2{\mathrm {{M}}}_{\mathrm{R}}-{\mathrm {{M}}}_{{\mathrm{R}}_{\mathrm{s}}}-1).\end{aligned}$$

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$$\begin{aligned} \varXi ^{SBS}_{RM}&= 3\varXi ^{SBS}_{CM}, \end{aligned}$$

where

$$\begin{aligned} \varXi ^{SBS}_{CM}&= \Psi ({\mathrm {{M}}}_{\mathrm{R}},\mathrm {{M}}_{\mathrm{T}}) + \frac{2}{3}{\mathrm {{M}}}_{\mathrm{T}}^2. \end{aligned}$$

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$$\begin{aligned} \varXi ^{SBS}_{RA}&= {\Psi _{A}}({\mathrm {{M}}}_{\mathrm{R}},{\mathrm {{M}}}_{\mathrm{T}}) + {\mathrm {{M}}}_{\mathrm{T}}({\mathrm {{M}}}_{\mathrm{T}}-2). \end{aligned}$$

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$$\begin{aligned} \varXi ^{HB}_{RM}&= 3\varXi ^{HB}_{CM} \hbox {, where } \end{aligned}$$

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$$\begin{aligned} \varXi ^{HB}_{CM}&= \Psi ({\mathrm {{M}}}_{\mathrm{R}},{\mathrm {{M}}}_{\mathrm{T}}) + \frac{2}{3}{\mathrm {{M}}}_{\mathrm{T}}({\mathrm {{M}}}_{\mathrm{T}}-1). \end{aligned}$$

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$$\begin{aligned} \varXi ^{HB}_{RA}&= {\Psi _{A}}({\mathrm {{M}}}_{\mathrm{R}},{\mathrm {{M}}}_{\mathrm{T}}) + ({\mathrm {{M}}}_{\mathrm{T}}-1)(\mathrm {{M}}_{\mathrm{T}}-2). \end{aligned}$$

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$$\begin{aligned} \varXi ^{XBS}_{RM}&= 3\Psi ({\mathrm {{M}}}_{\mathrm{R}},{\mathrm {{M}}}_{\mathrm{T}}) + {\mathrm {{M}}}_{\mathrm{T}}({\mathrm {{M}}}_{\mathrm{T}}-1). \end{aligned}$$

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$$\begin{aligned} \varXi ^{XBS}_{RA}&= 2{\Psi _{A}}({\mathrm {{M}}}_{\mathrm{R}},{\mathrm {{M}}}_{\mathrm{T}}) + \frac{\mathrm {{M}}_{\mathrm{T}}}{2}({\mathrm {{M}}}_{\mathrm{T}}-1). \end{aligned}$$

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$$\begin{aligned} \varXi ^{NBS}_{RM}&= 2{\mathrm {{M}}}_{\mathrm{T}}{\mathrm {{M}}}_{\mathrm{R}}. \end{aligned}$$

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$$\begin{aligned} \varXi ^{NBS}_{RA}&= {\mathrm {{M}}}_{\mathrm{T}}(2{\mathrm {{M}}}_{\mathrm{R}}-1). \end{aligned}$$

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$$\begin{aligned} \varOmega (p,q)&= \frac{1}{6}[(p(p+1)(2p+1))-(q(q+1)(2q+1))]. \end{aligned}$$

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$$\begin{aligned} \varXi ^{SBS,iter}_{RM}&= (2 + \frac{3{\mathrm {{M}}}_{\mathrm{R}}}{2})\varOmega ({\mathrm {{M}}}_{\mathrm{T}},{\mathrm {M}}_{\mathrm{T}_\mathrm{s}}) \nonumber \\&\quad+\frac{\mathrm {{M}}_{\mathrm{R}}}{4}({\mathrm {{M}}}_{\mathrm{T}}- {\mathrm {M}}_{\mathrm{T}_{\mathrm{s}}})(\mathrm {{M}}_{\mathrm{T}}+ {\mathrm {M}}_{\mathrm{T}_{\mathrm{s}}}+ 1).\end{aligned}$$

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$$\begin{aligned} \varXi ^{SBS,iter}_{RA}&= {\mathrm {{M}}}_{\mathrm{R}}\varOmega (\mathrm {{M}}_{\mathrm{T}},{\mathrm {M}}_{\mathrm{T}_{\mathrm{s}}}) \nonumber \\&\quad +(\mathrm {{M}}_{\mathrm{T}}- {\mathrm {M}}_{\mathrm{T}_{\mathrm{s}}})(\mathrm {{M}}_{\mathrm{T}}+ {\mathrm {M}}_{\mathrm{T}_\mathrm{s}}+ 1). \end{aligned}$$

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Part 2: Complexity Expressions With Reuse

Note: the underlined variables indicate complexity with reuse.

$$\begin{aligned} \underline{\varXi ^{SBS}_{RM}}&= 2{\mathrm {{M}}}_{\mathrm{T}}^{2}. \end{aligned}$$

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$$\begin{aligned} \underline{\varXi ^{SBS}_{RA}}&= {\mathrm {{M}}}_{\mathrm{T}}(\mathrm {{M}}_{\mathrm{T}}-1). \end{aligned}$$

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$$\begin{aligned} \underline{\varXi ^{HB}_{RM}}&= 2{\mathrm {{M}}}_{\mathrm{T}}(\mathrm {{M}}_{\mathrm{T}}-1). \end{aligned}$$

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$$\begin{aligned} \underline{\varXi ^{HB}_{RA}}&= (\mathrm {{M}}_{\mathrm{T}}-1)(\mathrm {{M}}_{\mathrm{T}}-2). \end{aligned}$$

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$$\begin{aligned} \underline{\varXi ^{XBS}_{RM}}&= {\mathrm {{M}}}_{\mathrm{T}}(\mathrm {{M}}_{\mathrm{T}}-1). \end{aligned}$$

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$$\begin{aligned} \underline{\varXi ^{XBS}_{RA}}&= \frac{\mathrm {{M}}_{\mathrm{T}}}{2}(\mathrm {{M}}_{\mathrm{T}}-1). \end{aligned}$$

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$$\begin{aligned} \underline{\varXi ^{MXCP}_{RM}}&= 3(\varXi ^{MXCP}_{CM} - (\mathrm {{M}}_{{\mathrm{R}}_{\mathrm{s}}}-1)\Psi (1,{\mathrm {{M}}}_{\mathrm{T}}) - \frac{2}{3}{\mathrm {{M}}}_{\mathrm{T}}\mathrm {{M}}_{\mathrm{R}}). \nonumber \\ \underline{\varXi ^{MXCP}_{RA}}&= \frac{5}{3}\underline{\varXi ^{MXCP}_{RM}} + 2\varXi ^{MXCP}_{RA1}. \end{aligned}$$

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