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Linear and Nonlinear Precoding Schemes for Centralized Multicell MIMO-OFDM Systems

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Abstract

The aim of this article is to propose and compare linear and nonlinear precoding schemes for multicell multiuser MIMO-OFDM based systems. The considered linear precoder is designed in two phases: first the intercell interference is removed by applying a linear zero-forcing algorithm. Then the system is further optimized by proposing three power allocation algorithms with per base station power constraint and different complexity tradeoffs: one optimal to minimize the average bit-error-rate and two suboptimal. The proposed nonlinear precoding is designed to minimize average bit-error-rate, over the users, conditioned to a channel realization and the transmitted data. In the high SNR regime, this problem simplifies from a constrained quadratic nonlinear optimization to a single quadratic problem with a scaling, allowing to reduce the complexity. The performance of the proposed schemes is evaluated, considering typical pedestrian scenarios based on LTE specifications. Numerical results show that the performance of the nonlinear scheme outperforms the linear ones. The nonlinear algorithm selects and inverts part of the correlation matrix unlike the linear zero-forcing where full inversion is required. This leads to a better performance as the selection allows to get a better conditioned matrix. Also, it is shown that the complexity of the nonlinear scheme is similar to the linear suboptimal closed-form one.

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Acknowledgments

The work presented in this paper was supported by the Portuguese FCT CelCoop (Pest-OE/EEI/LA0008/2011) and CROWN (PTDC/EEA-TEL/115828/2009) projects.

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Correspondence to Daniel Castanheira.

Appendices

Appendix A

In this appendix we prove (15). The KKT conditions of the optimization problem of (13), are given by,

$$\begin{aligned} \left\{ {\begin{array}{l} \frac{\partial L_1 \left( {p_{k,l} ,\mu ,\upsilon _{k,l} } \right)}{\partial p_{k,l} }=-\frac{\eta \sqrt{\chi }e^{\frac{-\chi p_{k,l} }{2\sigma ^{2}}}}{2N_c K\sigma \sqrt{2\pi }\sqrt{p_{k,l} }}+\mu \left[ {\mathbf{W}_l ^{H}\mathbf{W}_l } \right]_{k,k} +\upsilon _{k,l} =0 \\ \frac{\partial L_1 \left( {p_{k,l} ,\mu ,\upsilon _{k,l} } \right)}{\partial \mu }=\sum \limits _{k=1}^K {\sum \limits _{l=1}^{N_c } {p_{k,l} \left[ {\mathbf{W}_l^H \mathbf{W}_l} \right]} } _{k,k} =P_t \\ \\ p_{k,l}.\upsilon _{k,l} =0,\upsilon _{k,l} \ge 0,p_{k,l} \ge 0,\forall k,l \\ \end{array}} \right. \end{aligned}$$
(30)

Let us start by considering that \(\upsilon _{k,l} =0\), then squaring both terms of the first equation of (30) we have,

$$\begin{aligned} \mu ^{2}\left( {\left[ {\mathbf{W}_l ^{H}\mathbf{W}_l } \right]_{k,k} } \right)^{2}=\frac{\eta ^{2}\chi e^{\frac{-\chi p_{k,l} }{\sigma ^{2}}}}{8\pi K^{2}N_c^2 \sigma ^{2}p_{k,l} }=\frac{\eta ^{2}\chi ^{2}}{8\pi K^{2}N_c^2 \sigma ^{4}Xe^{X}} \end{aligned}$$
(31)

where \(X=\frac{\chi p_{k,l} }{\sigma ^{2}}\). Then the problem reduces to solve an exponential equation of type \(Xe^{X}\!=\!a\), with \(a=\frac{\eta ^{2}\chi ^{2} }{ 8\pi \mu ^{2}K^{2}N_c^2 \sigma ^{4}\left( {\left[ {\mathbf{W}_l ^{H}\mathbf{W}_l } \right]_{k,k} } \right)^{2}}\) and the solution can be given by the Lambert function of index 0 [24],

$$\begin{aligned} X=W_0 \left( a \right) \end{aligned}$$
(32)

Finally, replacing \(X\) and \(a\) in (32) we obtain the powers given by (15). If \(\upsilon _{k,l} >0\) then \(p_{k,l} =0,\forall k,l\) and consequently the SNR of the associated user is zero. Therefore, \(\upsilon _{k,l} \) should be set to zero, and in this case the constraints \(p_{k,l} \ge 0,\forall k,l\) are called inactive.

Appendix B

The aim of this appendix is to derive the dual of problem of (24). The Lagrangian associated with the aforementioned problem is:

$$\begin{aligned} L_3 (z,\beta ,{\bar{{\mathbf{x}}}}_b ,\bar{{u}}_b ,\bar{{\gamma }},{\bar{{{{\varvec{\lambda }}}}}})=z+\sum _{b=1}^B {\bar{{u}}_b ({\bar{{\mathbf{x}}}}_b ^{H}{\bar{{\mathbf{x}}}}_b -P_{t_b } )} +{\bar{{{{\varvec{\lambda }}}}}}^{T}\left(\beta {\bar{{\mathbf{b}}}}-\sum _{b=1}^B {{\bar{{\mathbf{A}}}}_b {\bar{{\mathbf{x}}}}_b } -z\mathbf{1}\right)-\bar{{\gamma }}\beta \nonumber \\ \end{aligned}$$
(33)

The variables \(\bar{{u}}_b (\bar{{u}}_b \ge 0), \gamma (\gamma \ge 0)\) and the vector \({\bar{{{{\varvec{\lambda }}}}}}({\bar{{{{\varvec{\lambda }}}}}}\ge \mathbf{0})\) are called the dual variables and the variables \(z, \beta \), and the vector \({\bar{{\mathbf{x}}}}_b \) are called the primal variables. The Lagrange dual function can be obtained as the minimum value of the Lagrangian over the primal variables. Since \(L_3 (z,\beta ,{\bar{{\mathbf{x}}}}_b ,\bar{{u}}_b ,\bar{{\gamma }},{\bar{{{{\varvec{\lambda }}}}}})\) is quadratic in \({\bar{{\mathbf{x}}}}_b \) and a linear function of \(z\) and \(\beta \) we can find the minimizing pair \((z,\beta ,{\bar{{\mathbf{x}}}}_b )\) from the corresponding KKT conditions, that are necessary and sufficient:

$$\begin{aligned} \frac{\partial L_3 (z,\beta ,{\bar{{\mathbf{x}}}}_b ,\bar{{u}}_b ,\bar{{\gamma }},{\bar{{{{\varvec{\lambda }}}}}})}{\partial z}&= 1-\mathbf{1}^{T}{\bar{{{{\varvec{\lambda }}}}}}=0 \nonumber \\ \frac{\partial L_3 (z,\beta ,{\bar{{\mathbf{x}}}}_b ,\bar{{u}}_b ,\bar{{\gamma }},{\bar{{{{\varvec{\lambda }}}}}})}{\partial \beta }&= {\bar{{\mathbf{b}}}}^{T}{\bar{{{{\varvec{\lambda }}}}}}-\bar{{\gamma }}=0 \nonumber \\ \frac{\partial L_3 (z,\beta ,{\bar{{\mathbf{x}}}}_b ,\bar{{u}}_b ,\bar{{\gamma }},{\bar{{{{\varvec{\lambda }}}}}})}{\partial {\bar{{\mathbf{x}}}}_b }&= 2\bar{{u}}_b {\bar{{\mathbf{x}}}}_b -{\bar{{\mathbf{A}}}}_b ^{T}{\bar{{{{\varvec{\lambda }}}}}}=0\Leftrightarrow {\bar{{\mathbf{x}}}}_b =\frac{{\bar{{\mathbf{A}}}}_b ^{T}{\bar{{{{\varvec{\lambda }}}}}}}{2\bar{{u}}_b } \end{aligned}$$
(34)

Such conditions imply that Lagrange dual function, \(g(u,y,{{\varvec{\lambda }}})\)is given by:

$$\begin{aligned} g(\bar{{u}}_b ,\bar{{\gamma }},{\bar{{{{\varvec{\lambda }}}}}})=\left\{ {{\begin{array}{ll} {-\sum \limits _{b=1}^B {\left( {\frac{{\bar{{{{\varvec{\lambda }}}}}}^{T}{\bar{{\mathbf{A}}}}_b {\bar{{\mathbf{A}}}}_b^T {\bar{{{{\varvec{\lambda }}} }}}}{4\bar{{u}}_b }+P_{t_b } \bar{{u}}_b } \right)} ,}&{1-\mathbf{1}^{T}{\bar{{{{\varvec{\lambda }}}}}}=0;\;{\bar{{\mathbf{b}}}}^{T}{\bar{{{{\varvec{\lambda }}}}}}-\bar{{\gamma }}=0} \\ {-\infty ,}&{\text{ otherwise}} \\ \end{array} }} \right. \end{aligned}$$
(35)

Consequently, the Lagrange dual problem of (24) is to maximize \(g(\bar{{u}}_b ,\bar{{\gamma }},{\bar{{{{\varvec{\lambda }}}}}})\) subject to the positivity of the dual variables:

$$\begin{aligned}&\min _{\bar{{u}}_b ,\bar{{\gamma }},{\bar{{{{\varvec{\lambda }}}}}}} \sum _{b=1}^B {\left( {\frac{{\bar{{{{\varvec{\lambda }}}}}}^{T}{\bar{{\mathbf{A}}}}_b {\bar{{\mathbf{A}}}}_b^T {\bar{{{{\varvec{\lambda }}} }}}}{4\bar{{u}}_b }+P_{t_b } \bar{{u}}_b } \right)} \nonumber \\&s.t.\mathbf{1}^{T}{\bar{{{{\varvec{\lambda }}}}}}=1,{\bar{{\mathbf{b}}}}^{T}{\bar{{{{\varvec{\lambda }}}}}}=\bar{{\gamma }},{\bar{{{{\varvec{\lambda } }}}}}\ge \mathbf{0},\bar{{u}}_b \ge 0,\bar{{\gamma }}\ge 0 \end{aligned}$$
(36)

However the dual problem can be further simplified since the value of \(\bar{{u}}_b \)can be obtained analytically\(\left( {2\bar{{u}}_b =\sqrt{{{\varvec{\lambda }}}^{T}{\bar{{\mathbf{A}}}}_b {\bar{{\mathbf{A}}}}_b ^{T}{{\varvec{\lambda }}}/P_{t_b } }} \right)\), consequently, the dual problem can be show to be equal to:

$$\begin{aligned} \widehat{{{\varvec{\lambda }}}}=arg\min _{\bar{{\gamma }},{\bar{{{{\varvec{\lambda }}}}}}} \sum _{b=1}^B {\sqrt{{\bar{{{{\varvec{\lambda }}}}}}^{T}{\bar{{\mathbf{A}}}}_b {\bar{{\mathbf{A}}}}_b^T {\bar{{{{\varvec{\lambda }} }}}}}} \nonumber \\ \text{ s.t.}\mathbf{1}^{T}{\bar{{{{\varvec{\lambda }}}}}}=1,{\bar{{\mathbf{b}}}}^{T}{\bar{{{{\varvec{\lambda }}}}}}=\bar{{\gamma }},{\bar{{{{\varvec{\lambda }}}}}}\ge \mathbf{0},\bar{{\gamma }}\ge 0 \end{aligned}$$
(37)

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Castanheira, D., Silva, A. & Gameiro, A. Linear and Nonlinear Precoding Schemes for Centralized Multicell MIMO-OFDM Systems. Wireless Pers Commun 72, 759–777 (2013). https://doi.org/10.1007/s11277-013-1041-z

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