Novel Multi-cell Precoding Schemes for TDD Massive MIMO Systems

Abstract

Two novel multi-cell precoding schemes, maximum ratio combining precoding with base station cooperation (MRC-BSC) and zero forcing precoding with BS cooperation (ZF-BSC), are proposed for time division duplex massive multiple-input multiple-output system. By making full use of other cells (not target cell) channel estimate information and changing the matrix structure of classical single-cell maximum ratio combining (MRC) precoding and zero forcing (ZF) precoding, proposed precoding schemes are obtained to reduce inter-cell interference with low complexity. The expressions of lower bound per-user achievable rate of proposed precoding schemes for different uplink pilot reuse factors situations are derived through theoretic analysis. Simulation results demonstrate that the proposed MRC-BSC and ZF-BSC can always achieve higher rates than single-cell MRC precoding and ZF precoding. For each precoding scheme, the downlink achievable rate can be improved greatly when enlarging the uplink pilot reuse factor. In addition, the proposed precoding schemes does not affect (or even improve) other cells’ performance while improving the target cell’s performance.

Appendices

Appendix A

Proof of Theorem 1

First, we present Lemma 1 which will be used in the proof of Theorems 1 and 2.

Lemma 1

If vectors $\text{b} \in {\mathbb{C}}^{M \times 1}$,$\text{c} \in {\mathbb{C}}^{M \times 1}$ , and $\text{b} \sim \mathcal{C}\mathcal{N}\left( {0,\sigma_{b}^{2} {\mathbf{I}}_{M} } \right)$, $\text{c} \sim \mathcal{C}\mathcal{N}\left( {0,\sigma_{c}^{2} {\mathbf{I}}_{M} } \right)$, $\text{b}$ and $\text{c}$ are independent, thus $\frac{{\text{b}^{H} }}{{\| \text{b} \|}}\text{c} \sim \mathcal{C}\mathcal{N}\left( {0,\sigma_{c}^{2} } \right)$.

Proof

Let $d = \frac{{\text{b}^{H} }}{{\left\| \text{b} \right\|}}\text{c}$. we can see that $d \in {\mathbb{C}}^{1 \times 1}$, d is the weighted sum of i.i.d. complex Gaussian random variables with distribution of $\mathcal{C}\mathcal{N}\left( {0,\sigma_{c}^{2} } \right)$, and the powers of the weighted sum up to 1, so d is also a complex Gaussian random variable. Due to the independence between $\text{b}$ and $\text{c}$, ${\mathbb{E}}\left\{ {\frac{{\text{b}^{H} }}{{\left\| \text{b} \right\|}}\text{c}} \right\} = 0$ and $\text{var} \left\{ {\frac{{\text{b}^{H} }}{{\left\| \text{b} \right\|}}\text{c}} \right\} = {\mathbb{E}}\left\{ {\left| {\frac{{\text{b}^{H} }}{{\left\| \text{b} \right\|}}\text{c}} \right|^{2} } \right\} = {\mathbb{E}}\left\{ {\text{c}^{H} \frac{\text{b}}{{\left\| \text{b} \right\|}} \cdot \frac{{\text{b}^{H} }}{{\left\| \text{b} \right\|}}\text{c}} \right\} = \sigma_{c}^{2}$, the end of the proof of Lemma 1.

Here we begin the proof Theorem 1:

From expression (11), we can write the achievable rate lower bound of user k in cell i as

$$R_{ik,dl}^{mrc} = \log_{2} \left( {1 + \frac{{\left| {{\mathbb{E}}\{ \sqrt p h_{iik}^{T} w_{ik} s_{ik} \} } \right|^{2} }}{{{\mathbb{E}}\{ \left| {I_{ik}^{mrc} } \right|^{2} \} }}} \right)$$

(18)

where $I_{ik}^{mrc}$ is given in expression (13), substitute $w_{ik} = \frac{{\hat{h}_{iik}^{*} }}{{\left\| {\hat{h}_{iik} } \right\|}}$ into (13) and we can rewritten $I_{ik}^{mrc}$ into five mutually independent items added together.

$$I_{ik}^{mrc} { = }\sum\limits_{n = 1}^{5} {I_{ik,n}^{mrc} }$$

(19)

where $I_{ik,1}^{mrc} { = }\sqrt p \left( {\frac{{\hat{h}_{iik}^{H} }}{{\left\| {\hat{h}_{iik} } \right\|}}h_{iik} - {\mathbb{E}}\left\{ {\frac{{\hat{h}_{iik}^{H} }}{{\left\| {\hat{h}_{iik} } \right\|}}h_{iik} } \right\}} \right)s_{ik}$, $I_{ik,2}^{mrc} { = }\sum\limits_{j = 1,j \ne k}^{K} {\sqrt p \frac{{\hat{h}_{iij}^{H} }}{{\left\| {\hat{h}_{iij} } \right\|}}h_{iik} s_{ij} }$,$I_{ik,3}^{mrc} { = }\sum\limits_{l = 1,l \ne i}^{L} {\sum\limits_{j = 1,j \ne k}^{K} {\sqrt p \frac{{\hat{h}_{llj}^{H} }}{{\left\| {\hat{h}_{llj} } \right\|}}\left( {h_{lik} - \frac{{diag\{ \hat{h}_{lik} \} }}{{diag\{ \hat{h}_{iik} \} }}h_{iik} } \right)s_{lj} } }$, $I_{ik,4}^{mrc} { = }\sum\limits_{l = 1,l \ne i}^{L} {\sqrt p \frac{{\hat{h}_{llk}^{H} }}{{\left\| {\hat{h}_{llk} } \right\|}}\left( {h_{lik} - \frac{{diag\{ \hat{h}_{lik} \} }}{{diag\{ \hat{h}_{iik} \} }}h_{iik} } \right)s_{lk} }$, $I_{ik,5}^{mrc} = n_{ik}.$

Thus the signal term is

$${\mathbb{E}}\{ \sqrt p h_{iik}^{T} w_{ik} s_{ik} \} = {\mathbb{E}}\ \left\{ \sqrt p \frac{{\hat{h}_{iik}^{H} }}{{\left\| {\hat{h}_{iik} } \right\|}}h_{iik} s_{ik} \right\} = {\mathbb{E}}\left\{ \sqrt p \frac{{\hat{h}_{iik}^{H} }}{{\left\| {\hat{h}_{iik} } \right\|}}(\hat{h}_{iik} + \varepsilon_{iik} )s_{ik} \right\} = \sqrt p E_{M} \sqrt {\sigma_{{\hat{h}_{iik} }}^{2} }$$

(20)

Because $\frac{{\left\| {\hat{h}_{iik} } \right\|^{2} }}{{\sigma_{{\hat{h}_{iik} }}^{2} }}$ is the Chi square distribution with freedom degree 2M, thus $\frac{{\left\| {\hat{h}_{iik} } \right\|}}{{\sqrt {\sigma_{{\hat{h}_{iik} }}^{2} } }}$ is the chi distribution with freedom degree 2M, so ${\mathbb{E}}\left\{ {\left\| {\hat{h}_{iik} } \right\|} \right\} = \frac{{\varGamma (M + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2})}}{\varGamma (M)}\sqrt {\sigma_{{\hat{h}_{iik} }}^{2} } = E_{M} \sqrt {\sigma_{{\hat{h}_{iik} }}^{2} }$. $\text{var} \left\{ {\left\| {\hat{h}_{iik} } \right\|} \right\} = (M - E_{M}^{2} )\sigma_{{\hat{h}_{iik} }}^{2} = Q_{M} \sigma_{{\hat{h}_{iik} }}^{2}$. In the last equation in (20), we have used the property of chi distribution and the independence between $\hat{h}_{iik}$ and $\varepsilon_{iik}$.

Due to the mutual independence between the interference and noise terms in expression (19), we can get

$${\mathbb{E}}\left\{ {\left| {I_{ik}^{mrc} } \right|^{2} } \right\}{ = }\sum\limits_{n = 1}^{5} {{\mathbb{E}}\left\{ {\left| {I_{ik,n}^{mrc} } \right|^{2} } \right\}}$$

(21)

Here we analyze each of the terms in (21), the first term is

$${\mathbb{E}}\left\{ {\left| {I_{ik,1}^{mrc} } \right|^{2} } \right\}{ = }{\mathbb{E}}\left\{ {\left| {\sqrt p \left( {\frac{{\hat{h}_{iik}^{H} }}{{\left\| {\hat{h}_{iik} } \right\|}}h_{iik} - {\mathbb{E}}\left\{ {\frac{{\hat{h}_{iik}^{H} }}{{\left\| {\hat{h}_{iik} } \right\|}}h_{iik} } \right\}} \right)s_{ik} } \right|^{2} } \right\}{ =}\,p\text{var} \left\{ {\frac{{\hat{h}_{iik}^{H} }}{{\left\| {\hat{h}_{iik} } \right\|}}h_{iik} } \right\} = p(Q_{M} \sigma_{{\hat{h}_{iik} }}^{2} + \sigma_{{\varepsilon_{iik} }}^{2} )$$

(22)

In the last equation in (22), we have used $h_{iik} = \hat{h}_{iik} + \varepsilon_{iik}$, the independence between $\hat{h}_{iik}$ and $\varepsilon_{iik}$, and the Lemma 1. For $I_{ik,2}^{mrc}$, due to the independence between $\hat{h}_{iij}$ and $h_{iik}$($j \ne k$), according to Lemma 1, we can get $\frac{{\hat{h}_{iij}^{H} }}{{\left\| {\hat{h}_{iij} } \right\|}}h_{iik} \sim \mathcal{C}\mathcal{N}\left( {0,\beta_{iik} } \right)$ for all $j \ne k$. Thus

$${\mathbb{E}}\left\{ {\left| {I_{ik,2}^{mrc} } \right|^{2} } \right\}{ = }{\mathbb{E}}\left\{ {\left| {\sum\limits_{j = 1,j \ne k}^{K} {\sqrt p \frac{{\hat{h}_{iij}^{H} }}{{\left\| {\hat{h}_{iij} } \right\|}}h_{iik} s_{ij} } } \right|^{2} } \right\} = p(K - 1)\beta_{iik}$$

(23)

For $I_{ik,3}^{mrc}$, we found that ${\mathbb{E}}\left\{ {\left( {h_{lik} - \frac{{diag\{ \hat{h}_{lik} \} }}{{diag\{ \hat{h}_{iik} \} }}h_{iik} } \right)^{H} \left( {h_{lik} - \frac{{diag\{ \hat{h}_{lik} \} }}{{diag\{ \hat{h}_{iik} \} }}h_{iik} } \right)} \right\}$ $= {\mathbb{E}}\left\{ {h_{lik}^{H} h_{lik} - h_{lik}^{H} \frac{{diag\{ \hat{h}_{lik} \} }}{{diag\{ \hat{h}_{iik} \} }}h_{iik} - h_{iik}^{H} \frac{{diag\{ \hat{h}^{*}_{lik} \} }}{{diag\{ \hat{h}^{*}_{iik} \} }}h_{lik} + h_{iik}^{H} \frac{{diag\{ \hat{h}^{*}_{lik} \} }}{{diag\{ \hat{h}^{*}_{iik} \} }}\frac{{diag\{ \hat{h}_{lik} \} }}{{diag\{ \hat{h}_{iik} \} }}h_{iik} } \right\} = \beta_{lik} - \sigma_{{\hat{h}_{lik} }}^{2} - \sigma_{{\hat{h}_{lik} }}^{2} + \sigma_{{\hat{h}_{lik} }}^{2} = \sigma_{{\varepsilon_{lik} }}^{2}$. The second last equation is obtained by polynomial expansion. So

$${\mathbb{E}}\left\{ {\left| {I_{ik,3}^{mrc} } \right|^{2} } \right\} = {\mathbb{E}}\left\{ {\left| {\sum\limits_{l = 1,l \ne i}^{L} {\sum\limits_{j = 1,j \ne k}^{K} {\sqrt p \frac{{\hat{h}_{llj}^{H} }}{{\left\| {\hat{h}_{llj} } \right\|}}\left( {h_{lik} - \frac{{diag\{ \hat{h}_{lik} \} }}{{diag\{ \hat{h}_{iik} \} }}h_{iik} } \right)s_{lj} } } } \right|^{2} } \right\} = \sum\limits_{l = 1,l \ne i}^{L} {p(K - 1)} \sigma_{{\varepsilon_{lik} }}^{2}$$

(24)

Using the similar method as $I_{ik,3}^{mrc}$ and due to the independence between $\hat{h}_{llk}$ and $\varepsilon_{lik}$, we can obtain

$${\mathbb{E}}\left\{ {\left| {I_{ik,4}^{mrc} } \right|^{2} } \right\}\,{=}\,{\mathbb{E}}\left\{ {\left| {\sum\limits_{l = 1,l \ne i}^{L} {\sqrt p \frac{{\hat{h}_{llk}^{H} }}{{\left\| {\hat{h}_{llk} } \right\|}}\left( {h_{lik} - \frac{{diag\{ \hat{h}_{lik} \} }}{{diag\{ \hat{h}_{iik} \} }}h_{iik} } \right)s_{lk} } } \right|^{2} } \right\} = \sum\limits_{l = 1,l \ne i}^{L} p \sigma_{{\varepsilon_{lik} }}^{2}$$

(25)

In addition,

$${\mathbb{E}}\left\{ {\left| {I_{ik,5}^{mrc} } \right|^{2} } \right\}{ = }{\mathbb{E}}\left\{ {\left| {n_{ik} } \right|^{2} } \right\} = 1$$

(26)

Combining expressions (20)–(26), we can get Theorem 1.

Appendix B

Proof of Theorem 2

Using the similar analysis method as in theorem 1, we can write achievable rate lower bound of user k in cell i for ZF-BSC is

$$R_{ik,dl}^{zf} = \log_{2} \left( {1 + \frac{{\left| {{\mathbb{E}}\{ \sqrt p h_{iik}^{T} w_{ik} s_{ik} \} } \right|^{2} }}{{{\mathbb{E}}\{ \left| {I_{ik}^{zf} } \right|^{2} \} }}} \right)$$

(27)

where $w_{ik} = \frac{{a_{ik}^{*} }}{{\left\| {a_{ik} } \right\|}}$, $a_{ik}$ is the kth column of matrix ${\mathbf{A}}_{i}$, and ${\mathbf{A}}_{i} = {\hat{\mathbf{H}}}_{ii} ({\hat{\mathbf{H}}}_{ii}^{H} {\hat{\mathbf{H}}}_{ii} )^{ - 1}$. $I_{ik}^{zf} { = }\sum\limits_{n = 1}^{5} {I_{ik,n}^{zf} }$,${\mathbb{E}}\left\{ {\left| {I_{ik}^{zf} } \right|^{2} } \right\}{ = }\sum\limits_{n = 1}^{5} {{\mathbb{E}}\left\{ {\left| {I_{ik,n}^{zf} } \right|^{2} } \right\}}$.

The signal term is

$${\mathbb{E}}\left\{ \sqrt p h_{iik}^{T} w_{ik} s_{ik} \right\} = {\mathbb{E}}\left\{ \sqrt p \frac{{a_{ik}^{H} }}{{\left\| {a_{ik} } \right\|}}h_{iik} s_{ik} \right\} = {\mathbb{E}}\left\{ \sqrt p \frac{{a_{ik}^{H} }}{{\left\| {a_{ik} } \right\|}}(\hat{h}_{iik} + \varepsilon_{iik} )s_{ik} \right\} = \sqrt p {{\mathbb{E}}}\left\{ \frac{1}{{\left\| {a_{ik} } \right\|}}\right\} = \sqrt p E_{M - K + 1} \sqrt {\sigma_{{\hat{h}_{iik} }}^{2} }$$

(28)

We have used the $a_{ik}^{H} \hat{h}_{iij} = \delta_{jk}$ (the property of ZF precoding) in expression (28). The interference and noise terms are (29)–(33),

$${\mathbb{E}}\left\{ {\left| {I_{ik,1}^{zf} } \right|^{2} } \right\} = {\mathbb{E}}\left\{ {\left| {\sqrt p \left( {\frac{{a_{ik}^{H} }}{{\left\| {a_{ik} } \right\|}}h_{iik} - {\mathbb{E}}\left\{ {\frac{{a_{ik}^{H} }}{{\left\| {a_{ik} } \right\|}}h_{iik} } \right\}} \right)s_{ik} } \right|^{2} } \right\} = p\text{var} \left\{ {\frac{{a_{ik}^{H} }}{{\left\| {a_{ik} } \right\|}}h_{iik} } \right\} = p\left( {\text{var} \left\{ {\frac{1}{{\left\| {a_{ik} } \right\|}}} \right\} + \sigma_{{\varepsilon_{iik} }}^{2} } \right) = p(Q_{M - K + 1} \sigma_{{\hat{h}_{iik} }}^{2} + \sigma_{{\varepsilon_{iik} }}^{2} )$$

(29)

According to the previous analysis, $a_{ik}$ is the kth column of matrix ${\mathbf{A}}_{i}$, and ${\mathbf{A}}_{i} = {\hat{\mathbf{H}}}_{ii} ({\hat{\mathbf{H}}}_{ii}^{H} {\hat{\mathbf{H}}}_{ii} )^{ - 1}$, $\hat{h}_{iik}$ is the kth column of ${\hat{\mathbf{H}}}_{ii}$, $\hat{h}_{iik} \sim \mathcal{C}\mathcal{N}\left( {0,\sigma_{{\hat{h}_{iik} }}^{2} {\mathbf{I}}_{M} } \right)$. Using the results of lemma 4 in paper [17], we can get ${\mathbb{E}}\left\{ {\frac{1}{{\left\| {a_{ik} } \right\|}}} \right\} = E_{M + K - 1} \sqrt {\sigma_{{\hat{h}_{iik} }}^{2} }$, $\text{var} \left\{ {\frac{1}{{\left\| {a_{ik} } \right\|}}} \right\} = Q_{M + K - 1} \sigma_{{\hat{h}_{iik} }}^{2}$, where $E_{M - K + 1} = \frac{{\varGamma (M - K + 1 + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2})}}{\varGamma (M - K + 1)}$, $Q_{M - K + 1} = M - K + 1 - E_{M - K + 1}^{2}$.

Due to $\frac{{a_{ij}^{H} }}{{\left\| {a_{ij} } \right\|}}h_{iik} = \frac{{a_{ij}^{H} }}{{\left\| {a_{ij} } \right\|}}(\hat{h}_{iik} + \varepsilon_{iik} ) = \frac{{a_{ij}^{H} }}{{\left\| {a_{ij} } \right\|}}\varepsilon_{iik} \sim \mathcal{C}\mathcal{N}\left( {0,\sigma_{{\varepsilon_{iik} }}^{2} } \right)$ for $j \ne k$, thus

$${\mathbb{E}}\left\{ {\left| {I_{ik,2}^{zf} } \right|^{2} } \right\} = {\mathbb{E}}\left\{ {\left| {\sum\limits_{j = 1,j \ne k}^{K} {\sqrt p \frac{{a_{ij}^{H} }}{{\left\| {a_{ij} } \right\|}}h_{iik} s_{ij} } } \right|^{2} } \right\} = p(K - 1)\sigma_{{\varepsilon_{iik} }}^{2}$$

(30)

Using the similar method as $I_{ik,3}^{mrc}$, $I_{ik,4}^{mrc}$ in Theorem 1 and due to the independence between $a_{lk}$ and $\varepsilon_{lik}$, we can obtain

$${\mathbb{E}}\left\{ {\left| {I_{ik,3}^{zf} } \right|^{2} } \right\}\,{=}\,{\mathbb{E}}\left\{ {\left| {\sum\limits_{l = 1,l \ne i}^{L} {\sum\limits_{j = 1,j \ne k}^{K} {\sqrt p \frac{{a_{lj}^{H} }}{{\left\| {a_{lj} } \right\|}}\left( {h_{lik} - \frac{{diag\{ \hat{h}_{lik} \} }}{{diag\{ \hat{h}_{iik} \} }}h_{iik} } \right)s_{lj} } } } \right|^{2} } \right\} = \sum\limits_{l = 1,l \ne i}^{L} {p(K - 1)} \sigma_{{\varepsilon_{lik} }}^{2}$$

(31)

$${\mathbb{E}}\left\{ {\left| {I_{ik,4}^{zf} } \right|^{2} } \right\}\,=\,{\mathbb{E}}\left\{ {\left| {\sum\limits_{l = 1,l \ne i}^{L} {\sqrt p \frac{{a_{lk}^{H} }}{{\left\| {a_{lk} } \right\|}}\left( {h_{lik} - \frac{{diag\{ \hat{h}_{lik} \} }}{{diag\{ \hat{h}_{iik} \} }}h_{iik} } \right)s_{lk} } } \right|^{2} } \right\} = \sum\limits_{l = 1,l \ne i}^{L} p \sigma_{{\varepsilon_{lik} }}^{2}$$

(32)

In addition,

$${\mathbb{E}}\left\{ {\left| {I_{ik,5}^{zf} } \right|^{2} } \right\}{ = }{\mathbb{E}}\left\{ {\left| {n_{ik} } \right|^{2} } \right\} = 1$$

(33)

Combining expressions (27)–(33), we can obtain Theorem 2.