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Power allocation and subchannel pairing for BER minimization in MIMO-OFDM AF relay systems

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Abstract

The present study investigates the problem of bit error rate minimization in multi-input-multi-output orthogonal frequency-division multiplexing amplify-and-forward relay systems. It is supposed that channels are frequency selective fading and under a total power constraint. In the multi-antenna transmitter, an independent symbol sequence is transmitted from each antenna. Zero forcing beamforming is performed in the reception and transmission of the relay to cancel out the inter-antenna interference. A power allocation and subchannel pairing scheme is considered as the optimization method. The goal of the optimization problem is to achieve the best bit error rate performance by applying a joint power allocation and subchannel pairing approach for all antennas. To deal with subchannel pairing, at first, we formulate it as a linear assignment problem and then Jonker-Volgenant algorithm is used to solve this problem. The joint scheme is compared with separate power allocation and subchannel pairing for each antenna as the main target. It is found that the joint scheme is superior to the separate method. Simulation results show that the proposed inter-antenna power allocation and subchannel pairing scheme can improve bit error rate and mean square error dramatically compared with separate subchannel pairing and power allocation at the cost of a little increase in complexity.

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Correspondence to Yasser Attarizi.

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Appendix A

Appendix A

$$ {\left(\underset{n(x)}{\overset{m(x)}{\int }}h(t)\ \mathrm{dt}\right)}^{\prime }={m}^{\prime }(x).h\left(m(x)\right)-{n}^{\prime }(x).h\left(n(x)\right) $$
(A1)
$$ {\left(\sqrt{u}\right)}^{\prime }=\frac{u^{\prime }}{2\sqrt{u}} $$
(A2)
$$ Q(x)=\frac{1}{\sqrt{2\pi }}\ \underset{x}{\overset{\infty }{\int }}{e}^{-\frac{u^2}{2}}\kern0.5em \mathrm{du} $$
(A3)

The function of the optimization problem (36) can be expressed as follows:

$$ L=\sum \limits_{i=1}^{\mathrm{KN}}\sum \limits_{j=1}^{\mathrm{KN}}\frac{\rho_{i,j}u}{\sqrt{2\pi }}\underset{\left(\sqrt{v{\gamma}_{i,j}^{\mathrm{eq}}\frac{q_{i,j}}{\rho_{i,j}}}\right)}{\overset{\infty }{\int }}{e}^{-\frac{u^2}{2}}\kern1em \mathrm{du}\kern0.5em +\lambda \left(\sum \limits_{i=1}^{\mathrm{KN}}\sum \limits_{j=1}^{\mathrm{KN}}{q}_{i,j}-{P}_T\right) $$
(A4)

If it is applied, \( \frac{\partial L}{\partial {q}_{i,j}}=0 \):

$$ \left(0-\frac{\rho_{i,j}u}{2\sqrt{2\pi }}\kern0.5em \frac{v{\gamma}_{i,j}^{\mathrm{eq}}\frac{1}{\rho_{i,j}}}{\sqrt{v{\gamma}_{i,j}^{\mathrm{eq}}\frac{q_{i,j}}{\rho_{i,j}}}}\kern1.25em {e}^{-\frac{v{\gamma}_{i,j}^{\mathrm{eq}}\frac{q_{i,j}}{\rho_{i,j}}}{2}}\kern0.5em \right)+\lambda =0 $$
(A5)

The inverse function of f (x) = xex [34] and z = f−1(zez) = W(zez), where W(.) is the Lambert function. Thus, after some simple mathematical arrangements, it can be expressed as follows:

$$ \left(\ \frac{\rho_{i,j}u}{2\sqrt{2\pi }}\kern0.5em \frac{\sqrt{v{\gamma}_{i,j}^{\mathrm{eq}}\frac{1}{\rho_{i,j}}}}{\sqrt{q_{i,j}}}\kern1.25em {e}^{-\frac{v{\gamma}_{i,j}^{\mathrm{eq}}\frac{q_{i,j}}{\rho_{i,j}}}{2}}\kern0.5em \right)=\lambda $$
(A6)
$$ \left(\ {\left(\frac{\rho_{i,j}u}{2\sqrt{2\pi}\kern0.5em \lambda}\right)}^2\kern0.75em \frac{v{\gamma}_{i,j}^{\mathrm{eq}}}{\rho_{i,j}}\kern0.5em {e}^{-v{\gamma}_{i,j}^{\mathrm{eq}}\frac{q_{i,j}}{\rho_{i,j}}}\kern0.5em \right)={q}_{i,j} $$
(A7)

Next, multiplied the both sides at \( \left(\frac{v{\gamma}_{i,j}^{\mathrm{eq}}}{\rho_{i,j}}\right) \) as:

$$ {\left(\frac{\rho_{i,j}u}{2\sqrt{2\pi}\kern0.5em \lambda}\right)}^2\ {\left(\frac{v{\gamma}_{i,j}^{\mathrm{eq}}}{\rho_{i,j}}\right)}^2={q}_{i,j}\ {e}^{v{\gamma}_{i,j,}^{\mathrm{eq}}\frac{q_{i,j}}{\rho_{i,j}}}\ \left(\frac{v{\gamma}_{i,j}^{\mathrm{eq}}}{\rho_{i,j}}\right) $$
(A8)
(A9)

Finally, qi, j is given by:

$$ {q}_{i,j}=\left(\frac{\rho_{i,j}}{v{\gamma}_{i,j}^{eq}}\right)\ W\left[{\left(\frac{v{\gamma}_{i,j}^{eq}u}{2\sqrt{2\pi}\kern0.5em \lambda}\right)}^2\right] $$
(A10)

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Shamsesalehi, M., Attarizi, Y. & Rajabi, R. Power allocation and subchannel pairing for BER minimization in MIMO-OFDM AF relay systems. Ann. Telecommun. (2020). https://doi.org/10.1007/s12243-019-00737-3

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Keywords

  • MIMO-OFDM
  • AF relay
  • Subchannel pairing
  • Power allocation
  • Linear assignment problem