# Interference Alignment Based on Rank Constraint in MIMO Cognitive Radio Networks

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## Abstract

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## 1. Introduction

## 2. System Model

## 3. Proposed IA Schemes

#### 3.1. Interference Rank Minimization (IRM)

**Proof:**

- Set $c=1$ and ${f}_{IRM}(0)={10}^{10}$, start with arbitrary decoding matrices ${\mathbf{U}}_{k},\phantom{\rule{4pt}{0ex}}{\mathbf{U}}_{k}^{H}{\mathbf{U}}_{k}={\mathbf{I}}_{{d}_{p}},\phantom{\rule{4pt}{0ex}}k=1,\cdots \phantom{\rule{4pt}{0ex}},{K}_{p}$;
- Calculate ${\mathbf{S}}_{k}^{p},\phantom{\rule{4pt}{0ex}}k=1,\cdots \phantom{\rule{4pt}{0ex}},{K}_{p}$, according to Equation (3);
- If c is odd, solve,$$\begin{array}{c}\underset{{\mathbf{V}}_{k},k=1,\cdots ,{K}_{p}}{min}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\sum _{k=1}^{{K}_{p}}({\u2225{\mathbf{S}}_{k}^{p}\u2225}_{*}+\nu \sum _{j=1,j\ne k}^{{K}_{p}}{\u2225{\mathbf{U}}_{k}^{H}{\mathbf{H}}_{kj}{\mathbf{V}}_{j}\u2225}_{F})\hfill \\ s.t.:\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mathbf{U}}_{k}^{H}{\mathbf{H}}_{kk}^{}{\mathbf{V}}_{k}\succ {\mathbf{0}}_{{d}_{p}\times {d}_{p}}\hfill \\ \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\lambda}_{min}({\mathbf{U}}_{k}^{H}{\mathbf{H}}_{kk}^{}{\mathbf{V}}_{k})\ge \tilde{\alpha}\phantom{\rule{4pt}{0ex}},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}k=1,\cdots ,{K}_{p}\hfill \end{array}$$If c is even, solve,$$\begin{array}{c}\underset{{\mathbf{U}}_{k},k=1,\cdots ,{K}_{p}}{min}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\sum _{k=1}^{{K}_{p}}({\u2225{\mathbf{S}}_{k}^{p}\u2225}_{*}+\nu \sum _{j=1,j\ne k}^{{K}_{p}}{\u2225{\mathbf{U}}_{k}^{H}{\mathbf{H}}_{kj}{\mathbf{V}}_{j}\u2225}_{F})\hfill \\ \phantom{\rule{4pt}{0ex}}s.t.:\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mathbf{U}}_{k}^{H}{\mathbf{H}}_{kk}^{}{\mathbf{V}}_{k}\succ {\mathbf{0}}_{{d}_{p}\times {d}_{p}}\hfill \\ \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\lambda}_{min}({\mathbf{U}}_{k}^{H}{\mathbf{H}}_{kk}^{}{\mathbf{V}}_{k})\ge \tilde{\alpha}\phantom{\rule{4pt}{0ex}},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}k=1,\cdots ,{K}_{p}\hfill \end{array}$$
- If ${f}_{IRM}(c-1)-{f}_{IRM}(c)\le \epsilon $, go to Step 5. If not, set $c=c+1$, go to Step 2;
- Orthogonalize ${\mathbf{V}}_{k},\phantom{\rule{4pt}{0ex}}{\mathbf{U}}_{k},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}k=1,\cdots ,{K}_{p}$.

#### 3.2. Interference Subspace Distance Minimization (ISDM)

**Lemma**

**1.**

- Set $c=1$ and ${f}_{ISDM}(0)={10}^{10}$, start with arbitrary ${\mathbf{V}}_{k},\phantom{\rule{4pt}{0ex}}{\mathbf{V}}_{k}^{H}{\mathbf{V}}_{k}={\mathbf{I}}_{{d}_{p}},\phantom{\rule{4pt}{0ex}}k=1,\cdots \phantom{\rule{4pt}{0ex}},{K}_{p}$;
- Calculate ${({\mathbf{A}}_{k}^{p})}_{}^{\perp},\phantom{\rule{4pt}{0ex}}k=1,\cdots \phantom{\rule{4pt}{0ex}},{K}_{p}$ according to Equation (19);
- At the j-th primary transmitter, $j=1,\cdots ,{K}_{p}$Solve:$$\begin{array}{c}\underset{{\mathbf{V}}_{j}^{}}{min}\phantom{\rule{0.166667em}{0ex}}\sum _{k=1,k\ne j}^{{K}_{p}}{\u2225{({\mathbf{A}}_{k}^{p})}_{}^{\perp}{[{({\mathbf{A}}_{k}^{p})}_{}^{\perp}]}^{H}{\mathbf{H}}_{kj}^{}{\mathbf{V}}_{j}^{}\u2225}_{F}^{}\hfill \\ \phantom{\rule{4pt}{0ex}}s.t.:\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{[{({\mathbf{A}}_{j}^{p})}_{}^{\perp}]}^{H}{\mathbf{H}}_{jj}^{}{\mathbf{V}}_{j}\succ {\mathbf{0}}_{{d}_{p}\times {d}_{p}}\hfill \\ \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\lambda}_{min}({[{({\mathbf{A}}_{j}^{p})}_{}^{\perp}]}^{H}{\mathbf{H}}_{jj}^{}{\mathbf{V}}_{j})\phantom{\rule{4pt}{0ex}}\ge \tilde{\alpha}\hfill \end{array}$$
- If ${f}_{ISDM}(c-1)-{f}_{ISDM}(c)\le \epsilon $, go to Step 5. If not, set $c=c+1$, go to Step 2;
- Output: ${\mathbf{V}}_{k},\phantom{\rule{4pt}{0ex}}{({\mathbf{A}}_{k}^{p})}_{}^{\perp},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}k=1,\cdots ,{K}_{p}$.

## 4. Interference Management in the Secondary Network

#### 4.1. Rank Constraint Scheme

#### 4.2. IRM Based on the Rank Constraint (IRM-RC)

- Set $c=1$ and ${f}_{IRM-RC}(0)={10}^{10}$, start with arbitrary ${\mathbf{W}}_{l}\phantom{\rule{4pt}{0ex}},\phantom{\rule{4pt}{0ex}}{\mathbf{W}}_{l}^{H}{\mathbf{W}}_{l}={\mathbf{I}}_{{d}_{l}^{s}},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}l={K}_{p}+1,\cdots \phantom{\rule{4pt}{0ex}},K$;
- Calculate ${\mathbf{S}}_{l}^{s},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}l={K}_{p}+1,\cdots \phantom{\rule{4pt}{0ex}},K$ according to Equation (31);
- If c is odd, take ${\mathbf{F}}_{l},\phantom{\rule{4pt}{0ex}}l={K}_{p}+1,\cdots \phantom{\rule{4pt}{0ex}},K$ as the variable to solve the optimization in Equation (33);
- If c is even, take ${\mathbf{W}}_{l},\phantom{\rule{4pt}{0ex}}l={K}_{p}+1,\cdots \phantom{\rule{4pt}{0ex}},K$ as the variable to solve the optimization in Equation (33);
- If ${f}_{IRM-RC}(c-1)-{f}_{IRM-RC}(c)\le \epsilon $, go to Step 6. If not $c=c+1$, go to Step 2;
- Orthogonalize ${\mathbf{F}}_{l},\phantom{\rule{4pt}{0ex}}{\mathbf{W}}_{k},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}l={K}_{p}+1,\cdots ,K$.

#### 4.3. ISDM Based on the Rank Constraint

- Set $c=1$ and ${f}_{ISDM-RC}(0)={10}^{10}$, start with arbitrary ${\mathbf{F}}_{l},\phantom{\rule{4pt}{0ex}}{\mathbf{F}}_{l}^{H}{\mathbf{F}}_{l}={\mathbf{I}}_{{d}_{l}^{s}},\phantom{\rule{4pt}{0ex}}l={K}_{p}+1,\cdots \phantom{\rule{4pt}{0ex}},K$;
- Calculate ${({\mathbf{A}}_{l}^{s})}_{}^{\perp},\phantom{\rule{4pt}{0ex}}l={K}_{p}+1,\cdots \phantom{\rule{4pt}{0ex}},K$ according to Equation (36);
- At each secondary transmitter, solve the optimization in Equation (38) to acquire the precoding matrix;
- If ${f}_{ISDM-RC}(c-1)-{f}_{ISDM-RC}(c)\le \epsilon $, go to Step 5. If not, set $c=c+1$, go to Step 2;
- Output: ${\mathbf{F}}_{l},\phantom{\rule{4pt}{0ex}}{\mathbf{A}}_{l}^{s},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}l={K}_{p}+1,\cdots ,K$.

## 5. Simulation

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 2.**Achievable rate of the system versus signal-to-noise ratio (SNR). IRM-CR: IRM-CR: Interference rank minimization in the CR network. ISDM-CR: interference subspace distance minimization in the CR network. IRM: Interference rank minimization. ISDM: Interference subspace distance minimization. IA: interference alignment.

**Figure 8.**Achievable rate of the system versus SNR. IMWLI: improved version of a minimum weighted leakage interference.

**Figure 12.**Convergence of the IRM based on the rank constraint (IRM-RC) and the ISDM based on the rank constraint (ISDM-RC).

Symbol | Definition |
---|---|

${M}_{t,p}$ | The number of PU transmit antennas |

${M}_{t,s}$ | The number of SU transmit antennas |

${M}_{r,p}$ | The number of PU receive antennas |

${M}_{r,s}$ | The number of SU receive antennas |

${\mathbf{V}}_{k}\in {\mathbb{C}}^{{M}_{t,p}\times {d}_{p}}$ | The precoding matrix of dimension $({M}_{t,p}\times {d}_{p})$ at the k-th primary transmitter |

${\mathbf{U}}_{k}\in {\mathbb{C}}^{{M}_{r,p}\times {d}_{p}}$ | The decoding matrix of dimension $({M}_{r,p}\times {d}_{p})$ at the k-th primary receiver |

${\mathbf{F}}_{l}\in {\mathbb{C}}^{{M}_{t,s}\times {d}_{l}^{s}}$ | The precoding matrix of dimension $({M}_{t,s}\times {d}_{l}^{s})$ at the l-th secondary transmitter |

${\mathbf{W}}_{l}\in {\mathbb{C}}^{{M}_{r,s}\times {d}_{l}^{s}}$ | The decoding matrix of dimension $({M}_{r,s}\times {d}_{l}^{s})$ at the l-th secondary receiver |

${\mathbf{H}}_{kk}$ | The channel coefficient between the k-th transmitter and its corresponding receiver |

${\mathbf{H}}_{kj}$ | The channel coefficient between the j-th transmitter and the k-th receiver |

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**MDPI and ACS Style**

Li, Y.; Diao, X.; Dong, Q.; Tang, C.
Interference Alignment Based on Rank Constraint in MIMO Cognitive Radio Networks. *Symmetry* **2017**, *9*, 107.
https://doi.org/10.3390/sym9070107

**AMA Style**

Li Y, Diao X, Dong Q, Tang C.
Interference Alignment Based on Rank Constraint in MIMO Cognitive Radio Networks. *Symmetry*. 2017; 9(7):107.
https://doi.org/10.3390/sym9070107

**Chicago/Turabian Style**

Li, Yibing, Xueying Diao, Qianhui Dong, and Chunrui Tang.
2017. "Interference Alignment Based on Rank Constraint in MIMO Cognitive Radio Networks" *Symmetry* 9, no. 7: 107.
https://doi.org/10.3390/sym9070107