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Non-orthogonal Amplify and Forward Relay Selection Systems with Distributed Space–Time Trellis Coded Continuous Phase Modulation

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Abstract

Most of the works on cooperative systems with or without relay selection have focused on linear modulations such as phase shift keying and quadrature amplitude modulation. Whereas, continuous phase modulation (CPM) is a good alternative to these modulations due to its constant envelope property which enables us to use inexpensive and energy-efficient non-linear power amplifiers. In the literature, a few works are based on relaying systems with CPM. To the best of our knowledge, none exists with relay selection. In this paper, we propose single and multiple relay selection CPM schemes based on a new generalized relay selection criterion for non-orthogonal amplify and forward (AF) systems operating with various time sharing protocols. In particular, using the proposed criterion, the best relay or the best two relays over a set of four available relays is/are selected. Error performances of the non-orthogonal AF relay selection systems and their counterparts are compared over frequency non-selective quasi-static Rayleigh fading channels using Viterbi algorithm at the destination terminal. Our numerical results present superiority of the proposed non-orthogonal relay selection systems.

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Author information

Authors and Affiliations

  1. Electronics and Communications Engineering Department, İstanbul Technical University, 34469, İstanbul-Maslak, Turkey

    Ali Serdar Demiroğlu, İbrahim Altunbaş & Mehmet E. Çelebi

  2. Netaş Headquarters, 34912, İstanbul-Kurtköy, Turkey

    Ali Serdar Demiroğlu

Authors
  1. Ali Serdar Demiroğlu
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  2. İbrahim Altunbaş
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  3. Mehmet E. Çelebi
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Corresponding author

Correspondence to Ali Serdar Demiroğlu.

Appendix

Appendix

This section includes simplifications of (3) for M = 3.

1.1 Protocol A

$$ \begin{aligned} {\mathfrak{T}} &=\, \mathop {\hbox{max} }\limits_{{{\mathbb{M}} \subset \left\{ {1, \ldots ,K} \right\}}} \frac{1}{M}\log { \det }\left[ {\left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {\rho_{\text{D}} } & 0 & 0 \\ 0 & {\rho_{\text{D}} } & 0 \\ {{{a}}\tilde{\rho }_{ 1} } & {{{a}}\tilde{\rho }_{ 2} } & {a\rho_{\text{D}} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\rho_{\text{D}}^{*} } & 0 & {{{a}}^{*} \tilde{\rho }_{1}^{*} } \\ 0 & {{\rho_{\text{D}}^{*}}} & {{{a}}^{*} \tilde{\rho }_{2}^{*} } \\ 0 & 0 & {{{a}}^{*} \rho_{\text{D}}^{*} } \\ \end{array} } \right]\times \left( {\left[ {\begin{array}{*{20}c} {N_{0} } & 0 & 0 \\ 0 & {N_{0} } & 0 \\ 0 & 0 & {N_{0} } \\ \end{array} } \right]} \right)^{ - 1} } \right] \\ &=\, \mathop {\hbox{max} }\limits_{{{\mathbb{M}} \subset \left\{ {1, \ldots ,K} \right\}}} \frac{1}{M}\log { \det }\left[ {\begin{array}{*{20}c} {1 + \frac{{\left| {\rho_{\text{D}} } \right|^{2} }}{{N_{0} }}} & 0 & {\frac{{{{a}}^{*} \tilde{\rho }_{1}^{*} \rho_{\text{D}} }}{{N_{0} }}} \\ 0 & {1 + \frac{{\left| {\rho_{\text{D}} } \right|^{2} }}{{N_{0} }}} & {\frac{{{{a}}^{*} \tilde{\rho }_{2}^{*} \rho_{\text{D}} }}{{N_{0} }}} \\ {\frac{{{{a}}\tilde{\rho }_{ 1} \rho_{\text{D}}^{*} }}{{N_{0} }}} & {\frac{{{{a}}\tilde{\rho }_{ 2} \rho_{\text{D}}^{*} }}{{N_{0} }}} & {1 + \frac{{\left| a \right|^{2} \left( {\left| {\tilde{\rho }_{ 1} } \right|^{2} + \left| {\tilde{\rho }_{ 2} } \right|^{2} + \left| {\rho_{\text{D}} } \right|^{2} } \right)}}{{N_{0} }}} \\ \end{array} } \right] \\ &=\, \mathop {\hbox{max} }\limits_{{{\mathbb{M}} \subset \left\{ {1, \ldots ,K} \right\}}} \frac{1}{M}\log \left( {1 + \frac{{2\left| {\rho_{\text{D}} } \right|^{2} }}{{N_{0} }} + \frac{{\left| {\rho_{\text{D}} } \right|^{4} }}{{\left( {N_{0} } \right)^{2} }}} \right. \\ &\quad \left. +\,{ \left| a \right|^{2} \left( {\frac{{\left| {\tilde{\rho }_{ 1} } \right|^{2} + \left| {\tilde{\rho }_{ 2} } \right|^{2} + \left| {\rho_{\text{D}} } \right|^{2} }}{{N_{0} }} + \frac{{\left| {\rho_{\text{D}} } \right|^{2} \left( {\left| {\tilde{\rho }_{ 1} } \right|^{2} + \left| {\tilde{\rho }_{ 2} } \right|^{2} + 2\left| {\rho_{\text{D}} } \right|^{2} } \right)}}{{\left( {N_{0} } \right)^{2} }} + \frac{{\left| {\rho_{\text{D}} } \right|^{6} }}{{\left( {N_{0} } \right)^{3} }}} \right)} \right) \\ \end{aligned} $$
(11)

Since maximization of the last term of the logarithmic expression of (11) determines the optimal selection criterion for Protocol A, (8) is obtained.

1.2 Protocol B

$$ \begin{aligned} {\mathfrak{T}} &=\, \mathop {\hbox{max} }\limits_{{{\mathbb{M}} \subset \left\{ {1, \ldots ,K} \right\}}} \frac{1}{M}\log { \det }\left[ {\left[ {\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} } \right] + \left[ {\begin{array}{ccc} {a_{0} \rho_{\text{D}} } & 0 & 0 \\ {a_{1} \tilde{\rho }_{ 1} } & {a_{1} \rho_{\text{D}} } & 0 \\ {a_{2} \tilde{\rho }_{ 1} } & {a_{2} \tilde{\rho }_{ 2} } & {a_{2} \rho_{\text{D}} } \\ \end{array} } \right]\left[ {\begin{array}{ccc} {a_{0}^{*} \rho_{\text{D}}^{*} } & {a_{1}^{*} \tilde{\rho }_{1}^{*} } & {a_{2}^{*} \tilde{\rho }_{1}^{*} } \\ 0 & {a_{1}^{*} \rho_{\text{D}}^{*} } & {a_{2}^{*} \tilde{\rho }_{2}^{*} } \\ 0 & 0 & {a_{2}^{*} \rho_{\text{D}}^{*} } \\ \end{array} } \right] \times \left( {\left[ {\begin{array}{ccc} {N_{0} } & 0 & 0 \\ 0 & {N_{0} } & 0 \\ 0 & 0 & {N_{0} } \\ \end{array} } \right]} \right)^{ - 1} } \right] \\ &=\, \mathop {\hbox{max} }\limits_{{{\mathbb{M}} \subset \left\{ {1, \ldots ,K} \right\}}} \frac{1}{M}\log { \det }\left[ {\begin{array}{ccc} {1 + \frac{{\left| {a_{o} } \right|^{2} \left| {\rho_{\text{D}} } \right|^{2} }}{{N_{0} }}} & {\frac{{a_{0} a_{1}^{*} \rho_{\text{D}} \tilde{\rho }_{1}^{*} }}{{N_{0} }}} & {\frac{{a_{0} a_{2}^{*} \rho_{\text{D}} \tilde{\rho }_{1}^{*} }}{{N_{0} }}} \\ {\frac{{a_{1} a_{0}^{*} \tilde{\rho }_{1} \rho_{\text{D}}^{*} }}{{N_{0} }}} & {1 + \frac{{\left| {a_{1} } \right|^{2} \left( {\left| {\tilde{\rho }_{1} } \right|^{2} + \left| {\rho_{\text{D}} } \right|^{2} } \right)}}{{N_{0} }}} & {\frac{{a_{1} a_{2}^{*} \left( {\left| {\tilde{\rho }_{1} } \right|^{2} + {\rho_{\text{D}} } \tilde{\rho }_{2}^{*} } \right)}}{{N_{0} }}} \\ {\frac{{a_{2} a_{0}^{*} \tilde{\rho }_{1} \rho_{\text{D}}^{*} }}{{N_{0} }}} & {\frac{{a_{2} a_{1}^{*} \left( {\left| {\tilde{\rho }_{1} } \right|^{2} + \tilde{\rho }_{2} \rho_{\text{D}}^{*} } \right)}}{{N_{0} }}} & {1 + \frac{{\left| {a_{2} } \right|^{2} \left( {\left| {\tilde{\rho }_{1} } \right|^{2} + \left| {\tilde{\rho }_{2} } \right|^{2} + \left| {\rho_{\text{D}} } \right|^{2} } \right)}}{{N_{0} }}} \\ \end{array} } \right] \\ &=\, \mathop {\hbox{max} }\limits_{{{\mathbb{M}} \subset \left\{ {1, \ldots ,K} \right\}}} \frac{1}{M}\log \left( {1 + \frac{{\left| {a_{o} } \right|^{2} \left| {\rho_{\text{D}} } \right|^{2} }}{{N_{0} }} + } \right.\frac{{\left| {a_{1} } \right|^{2} \left( {\left| {\tilde{\rho }_{1} } \right|^{2} + \left| {\rho_{\text{D}} } \right|^{2} } \right) + \left| {a_{2} } \right|^{2} \left( {\left| {\tilde{\rho }_{1} } \right|^{2} + \left| {\tilde{\rho }_{2} } \right|^{2} + \left| {\rho_{\text{D}} } \right|^{2} } \right)}}{{N_{0} }} \\ &\quad +\, \frac{{\left| {a_{o} } \right|^{2} \left| {a_{1} } \right|^{2} \left| {\rho_{\text{D}} } \right|^{4} + \left| {a_{o} } \right|^{2} \left| {a_{2} } \right|^{2} \left( {\left| {\tilde{\rho }_{2} } \right|^{2} \left| {{\rho }_{\text{D}} } \right|^{2} + \left| {\rho_{\text{D}} } \right|^{4} } \right)}}{{\left( {N_{0} } \right)^{2} }} \\ &\quad +\, \frac{{\left| {a_{1} } \right|^{2} \left| {a_{2} } \right|^{2} \left( {\left| {\tilde{\rho }_{1} } \right|^{2} \left| {\tilde{\rho }_{2} } \right|^{2} + \left| {\rho_{\text{D}} } \right|^{4} + 2\left| {\tilde{\rho }_{1} } \right|^{2} \left| {\rho_{\text{D}} } \right|^{2} - \left| {\tilde{\rho }_{1} } \right|^{2} \left( {\tilde{\rho }_{2} \rho_{\text{D}}^{*} + \rho_{\text{D}} \tilde{\rho }_{2}^{*} } \right)} \right)}}{{\left( {N_{0} } \right)^{2} }} \\ &\quad\left. {+ \frac{{\left| {a_{0} } \right|^{2} \left| {a_{1} } \right|^{2} \left| {a_{2} } \right|^{2} \left| {\rho_{\text{D}} } \right|^{6} }}{{\left( {N_{0} } \right)^{3} }}} \right) \\ \end{aligned} $$
(12)

The sum of the last four term of the logarithmic expression of (12) gives (9).

1.3 Protocol C

$$ \begin{aligned} {\mathfrak{T}} &= \, \mathop {\hbox{max} }\limits_{{{\mathbb{M}} \subset \left\{ {1, \ldots ,K} \right\}}} \frac{1}{M}\log { \det }\left[ {\left[ {\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} } \right] + \left[ {\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ {{{a}}\tilde{\rho }_{ 1} } & {{{a}}\tilde{\rho }_{ 2} } & {a\rho_{\text{D}} } \\ \end{array} } \right]\left[ {\begin{array}{ccc} 0 & 0 & {{{a}}^{*} \tilde{\rho }_{1}^{*} } \\ 0 & 0 & {{{a}}^{*} \tilde{\rho }_{2}^{*} } \\ 0 & 0 & {{{a}}^{*} \rho_{\text{D}}^{*} } \\ \end{array} } \right] \times \left( {\left[ {\begin{array}{ccc} {N_{0} } & 0 & 0 \\ 0 & {N_{0} } & 0 \\ 0 & 0 & {N_{0} } \\ \end{array} } \right]} \right)^{ - 1} } \right] \\ &=\, \mathop {\hbox{max} }\limits_{{{\mathbb{M}} \subset \left\{ {1, \ldots ,K} \right\}}} \frac{1}{M}\log { \det }\left[ {\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & {1 + \left| a \right|^{2} \left( {\frac{{\left| {\tilde{\rho }_{ 1} } \right|^{2} + \left| {\tilde{\rho }_{ 2} } \right|^{2} + \left| {\rho_{\text{D}} } \right|^{2} }}{{N_{0} }}} \right)} \\ \end{array} } \right] \\ &=\, \mathop {\hbox{max} }\limits_{{{\mathbb{M}} \subset \left\{ {1, \ldots ,K} \right\}}} \frac{1}{M}\log \left( {1 + \left| a \right|^{2} \left( {\frac{{\left| {\tilde{\rho }_{ 1} } \right|^{2} + \left| {\tilde{\rho }_{ 2} } \right|^{2} + \left| {\rho_{\text{D}} } \right|^{2} }}{{N_{0} }}} \right)} \right) \\ \end{aligned} $$
(13)

The last term of the logarithmic expression of (13) gives (10).

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Demiroğlu, A.S., Altunbaş, İ. & Çelebi, M.E. Non-orthogonal Amplify and Forward Relay Selection Systems with Distributed Space–Time Trellis Coded Continuous Phase Modulation. Wireless Pers Commun 84, 1209–1225 (2015). https://doi.org/10.1007/s11277-015-2684-8

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