Encryption Based Strategy to Overcome the Problem of Pilot Contamination Within Multi-cellular Massive MIMO Systems

Abstract

The problem of inter-cell interferences keeps its site as the most challenging constraint that faces the massive multi-input multi-output (M-MIMO) technology. This constraint, known as the problem of pilot contamination (PC), is a direct result of reusing the same set of orthogonal pilot sequences (OPSs) across several cells, due to the scarcity of available pilot resources, compared to the number of the user equipments (UEs) that must be served, the reuse of the same OPSs within different cells is unavoidable. Accordingly, assigning the available OPSs to the UEs should be managed to address the PC problem. To reach that goal, a new decontaminating strategy is proposed herein, which is referred to as the ENCryption-based decontaminating strategy (ENC). The ENC strategy is based on injecting some recognition pilot symbols within the pilot sequence of the UEs; hence, the base stations become able to distinguish between the desired pilot signals from the undesired ones, therefore, the quality of servicecan be enhanced, furthermore, the proposed ENC strategy is of low computational complexity compared to the existing strategies.

Appendix

1.1 The Proof of (10)

A RPS is injected between the PS of each UE, where the same pilot symbol is reused for the UEs of the same cell, while orthogonal ones are used for the adjacent cells. Thus, the set of the PSs used by the UEs of the jth cell can be expressed as follows:

$$\begin{aligned} \varPhi _j= \begin{pmatrix} \psi _{1}^j \\ \psi _{2}^j \\ \vdots \\ \psi _{K}^j \end{pmatrix} = \begin{pmatrix} \phi _{j}^{rec}&{} \phi _{1}^2 &{}\phi _{1}^3&{} \ldots &{} \phi _{1}^{\tau +1} \\ \phi _{2}^1 &{}\phi _{j}^{rec} &{}\phi _{2}^3&{} \ldots &{} \phi _{2}^{\tau +1} \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{}\vdots \\ \phi _{K}^1 &{}\phi _{K}^2&{}\ldots &{} \phi _{K}^\tau &{}\phi _{j}^{rec} \end{pmatrix} \end{aligned}$$

(22)

It is well seen that the same RPS (i.e. $\phi _{j}^{rec}$ for the UEs of the jth cell) is employed by the UEs of the same cell in a non-overlapped time slots i.e. the first row of (22) corresponds to the UE$-1$ of cell$-j$ and it uses $\phi _{j}^{rec}$ in its first time slot, while in the second row the same RPS is used in the second time slot and so on for the other UEs. By considering the orthogonality condition, we have the following:

• 1. $\phi _{j}^{rec} \phi _{k}^i=0$ $\forall$ {$i=1,\ldots ,\tau$, $k=1,\ldots ,K$ } and $\forall j=1,\ldots ,L$: no intra-cell interferences.

• 2. $\phi _{j}^{rec} \phi _{l}^{rec}=\delta _{j,l}$: no inter-RPS interferences.

• 3. $\phi _{k}^p \phi _{k'}^{n}=\delta _{k,k'}\delta _{p,n}$: no intra-cell interferences.

Therefore, the orthogonality condition can be expressed as:

$$\begin{aligned} \varPhi _l \varPhi _j^H&= \begin{pmatrix} \phi _{j}^{rec}&{} \phi _{1}^2 &{}\phi _{1}^3&{} \ldots &{} \phi _{1}^{\tau +1} \\ \phi _{2}^1 &{}\phi _{j}^{rec} &{}\phi _{2}^3&{}\ldots &{} \phi _{2}^{\tau +1} \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{}\vdots \\ \phi _{K}^1 &{}\phi _{K}^2&{}\ldots &{} \phi _{K}^\tau &{}\phi _{j}^{rec} \end{pmatrix} \begin{pmatrix} (\phi _{j}^{rec})^H &{} (\phi _{2}^1)^H &{} \ldots &{}(\phi _{K}^1)^H \\ (\phi _{1}^2)^H &{}(\phi _{j}^{rec})^H &{}\ldots &{} (\phi _{K}^2)^H \\ \vdots &{} \vdots &{} \vdots &{}\vdots \\ (\phi _{1}^{\tau +1})^H&{}(\phi _{2}^{\tau +1})^H&{}\ldots &{}(\phi _{j}^{rec})^H \end{pmatrix} \\&=\begin{pmatrix} \tau +\delta _{j,l}, &{} 0 &{} \ldots &{} \ldots &{}0&{}0 \\ 0,&{} \tau +\delta _{j,l},&{} &{} &{}&{} \\ 0&{} 0&{} \tau +\delta _{j,l}, &{}0 &{} \ldots &{} \vdots \\ \vdots &{} \vdots &{}0 &{} \ldots &{}&{} 0 \\ 0,&{} , &{} 0&{} \ldots &{}0&{} \tau +\delta _{j,l} \end{pmatrix} \\&=(\tau +\delta _{j,l})\mathcal {I}_{\mathcal{K}} \end{aligned}$$

(23)

1.2 The Proof of (12)

In order to estimate the channels, the UEs uplink their PSs, therefore, the BS of jth cell correlates its received pilot signal (11) with its local PSs $\varPhi _j=[(\psi _{1}^j)^T, (\psi _{2}^j)T,\ldots ,(\psi _{K}^j)T]^T \in \mathcal {C}^{K\times (\tau +1)}$. Thus, the estimated CSI can be expressed as:

$$\begin{aligned} \hat{H}_{j,j}&=\frac{Y_j \varPhi _j^H}{\sqrt{\rho _p}} \\&=\sum _{l=1}^{L} H_{j,l} \begin{pmatrix} \phi _{l}^{rec}&{} \phi _{1}^2 &{}\phi _{1}^3&{}\ldots &{} \phi _{1}^{\tau +1} \\ \phi _{2}^1 &{}\phi _{l}^{rec} &{}\phi _{2}^3&{}\ldots &{} \phi _{2}^{\tau +1} \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{}\vdots \\ \phi _{K}^1 &{}\phi _{K}^2&{}... &{} \phi _{K}^\tau &{}\phi _{l}^{rec} \end{pmatrix} \begin{pmatrix} (\phi _{j}^{rec})^H &{} (\phi _{2}^1)^H &{} \ldots &{}(\phi _{K}^1)^H \\ (\phi _{1}^2)^H &{}(\phi _{j}^{rec})^H &{}\ldots &{} (\phi _{K}^2)^H \\ \vdots &{} \vdots &{} \vdots &{}\vdots \\ (\phi _{1}^{\tau +1})^H&{}(\phi _{2}^{\tau +1})^H&{}\ldots &{}(\phi _{j}^{rec})^H \end{pmatrix} +\frac{N_j \varPhi _j^H}{\sqrt{\rho _p}} \end{aligned}$$

(24)

$$\begin{aligned} \\&= H_{j,j} \begin{pmatrix} \alpha _{j,j}+\sum _{i=1}^{\tau } \phi _{1,j}^i(\phi _{1,j}^i)^H,&{} \sum _{i=1}^{\tau } \phi _{1,j}^i(\phi _{2,j}^i)^H+\alpha _{j,j}, &{} &{},\ldots ,&{}&{} \sum _{i=1}^{\tau } \phi _{1,j}^2(\phi _{K,j}^i)^H+\alpha _{j,j} \\ \sum _{i=1}^{\tau } \phi _{2,j}^i(\phi _{1,j}^i)^H+\alpha _{j,j},&{} \sum _{i=1}^{\tau } \phi _{2,j}^i(\phi _{2,j}^i)^H+\alpha _{j,j}, &{} &{},\ldots ,&{}&{} \sum _{i=1}^{\tau } \phi _{2,j}^2(\phi _{K,j}^i)^H+\alpha _{j,j} \\ \vdots &{} \vdots &{} &{} &{}&{} \vdots \\ \vdots &{} \vdots &{} &{} &{}&{} \vdots \\ \sum _{i=1}^{\tau } \phi _{K,j}^i(\phi _{1,j}^i)^H+\alpha _{j,j},&{} \sum _{i=1}^{\tau } \phi _{K,j}^i(\phi _{2,j}^i)^H+\alpha _{j,j}, &{} &{},\ldots ,&{}&{} \sum _{i=1}^{\tau } \phi _{K,j}^2(\phi _{K,j}^i)^H+\alpha _{j,j} \end{pmatrix} \end{aligned}$$

(25)

$$\begin{aligned} \\&\quad + \sum _{l=1,l\ne j}^{L} H_{j,l} \begin{pmatrix} \sum _{i=1}^{\tau } \phi _{1,l}^i(\phi _{1,j}^i)^H+\alpha _{l,j},&{} \sum _{i=1}^{\tau } \phi _{1,l}^i(\phi _{2,j}^i)^H+\alpha _{l,j}, &{} &{},\ldots ,&{} \sum _{i=1}^{\tau } \phi _{1,l}^2(\phi _{K,j}^i)^H+\alpha _{l,j} \\ \sum _{i=1}^{\tau } \phi _{2,l}^i(\phi _{1,j}^i)^H+\alpha _{l,j},&{} \sum _{i=1}^{\tau } \phi _{2,l}^i(\phi _{2,j}^i)^H+\alpha _{l,j}, &{} &{},\ldots ,&{}\sum _{i=1}^{\tau } \phi _{2,l}^2(\phi _{K,j}^i)^H+\alpha _{l,j} \\ \vdots &{} \vdots &{} &{}&{} \vdots \\ \vdots &{} \vdots &{} &{}&{} \vdots \\ \sum _{i=1}^{\tau } \phi _{K,l}^i(\phi _{1,j}^i)^H+\alpha _{l,j},&{} \sum _{i=1}^{\tau } \phi _{K,l}^i(\phi _{2,j}^i)^H+\alpha _{j,j}, &{} &{},\ldots ,&{}\sum _{i=1}^{\tau } \phi _{K,l}^2(\phi _{K,j}^i)^H+\alpha _{l,j} \end{pmatrix} \\&\quad +\frac{N_j \varPhi _j^H}{\sqrt{\rho _p}} \end{aligned}$$

(26)

where $\alpha _{j,j}=\phi _j^{rec}( \phi _j^{rec})^H$ and $\alpha _{j,l}=\phi _j^{rec}(\phi _l^{rec})^H=\phi _l^{rec}(\phi _{k,j}^i)^H=0$ are considered $\forall (j \ne l)=1,\ldots ,L$, $\forall i=1,\ldots ,\tau$, and $\forall k=1,\ldots ,K$.

Therefore, the estimated CSI is then expressed as:

$$\begin{aligned} \hat{H}_{j,j}&= H_{j,j} \begin{pmatrix} \tau +1, &{} 0 &{} \ldots &{} \ldots &{}0&{}0 \\ 0,&{} \tau +1,&{} &{} &{}&{} \\ 0&{} 0&{} \tau +1, &{}0 &{} \ldots &{} \vdots \\ \vdots &{} \vdots &{}0 &{} \ldots &{}&{} 0 \\ 0,&{} , &{} 0&{} \ldots &{}0&{} \tau +1 \end{pmatrix} \\&\quad + \sum _{l=1,l\ne j}^{L} H_{j,l} \begin{pmatrix} \tau +0 , &{} 0 &{} \ldots &{} \ldots &{}0&{}0 \\ 0,&{} \tau +0 ,&{} &{} &{} &{} \\ 0&{} 0&{} \tau +0 , &{}0 &{} \ldots &{} \vdots \\ \vdots &{} \vdots &{}0 &{} \ldots &{} &{} 0 \\ 0,&{} , &{} 0&{} \ldots &{}0&{} \tau +0 \end{pmatrix} +\frac{N_j \varPhi _j^H}{\sqrt{\rho _p}} \\&=H_{j,j}(\tau +1) I_{K,K}+\sum _{l=1, l\ne j}^{L} H_{j,l} \tau I_{K,K}+\frac{N_j \varPhi _j^H}{\sqrt{\rho _p}} \end{aligned}$$

(27)

Therefore, the estimated CSI at the BS of jth cell is:

$$\begin{aligned} \hat{H}_{j,j} = (\tau +1) H_{j,j} +\tau \sum _{l=1, l\ne j}^{L} H_{j,l} +\frac{N_j \varPhi _j^H}{\sqrt{\rho _p}} \end{aligned}$$

(28)