Energy efficiency maximization in FDD massive MIMO systems with channel aging

Abstract

In this paper, energy efficiency (EE) of the frequency division duplexing (FDD) massive multiple-input multiple-output (MIMO) systems in the downlink transmission is investigated by considering the channel aging effect. First, we present a model for channel aging in the FDD massive MIMO systems and study its effect on the EE of the system. We assume that the channel aging exists in the entire transmission frame of the FDD systems which includes both pilot and data transmission phases. Then, we propose methods for compensating for this effect and increasing the EE. These methods include channel prediction method in the pilot transmission phase, system’s parameters optimization for maximizing the EE and a hybrid method. By numerical simulations, we compare the performance of the proposed methods and their abilities to combat the channel aging effect in the massive MIMO systems and EE improvement.

Appendices

Appendix A

Proof of the Theorem 1

In [9], an optimal pilot matrix has been obtained for a spatially correlated mode. We calculate the optimal pilot matrix, considering that the channel is changing from one symbol to another. Optimal pilot vectors that minimize MSE are obtained as

$$\begin{aligned}&\text {arg min }\mathbf {MSE}\nonumber \\&{\text{subject to }}{\varvec{\varphi }}^{H}_{i} {\varvec{\varphi }}_{i}=1;~~i=1,\ldots ,\tau \nonumber \\&\qquad\qquad\quad {\varvec{\varphi }}^{H}_{i} {\varvec{\varphi }}_{k}=0;~~k\ne l\nonumber \\ =&~ \text {arg min }\text {tr}\left( \mathbf {R}-\varvec{\Psi } \right) \nonumber \\&{\text{subject to }}{\varvec{\varphi }}^{H}_{i} {\varvec{\varphi }}_{i}=1;~~i=1,\ldots ,\tau \nonumber \\&\qquad\qquad\;\quad {\varvec{\varphi }}^{H}_{i} {\varvec{\varphi }}_{k}=0;~~k\ne l\nonumber \\ =&~ \text {arg max } \text {tr}\left( \varvec{\Psi } \right) \nonumber \\&{\text{subject to }}{\varvec{\varphi }}^{H}_{i} {\varvec{\varphi }}_{i}=1;~~i=1,\ldots ,\tau \nonumber \\&\qquad\qquad\;\quad {\varvec{\varphi }}^{H}_{i} {\varvec{\varphi }}_{k}=0;~~k\ne l \end{aligned}$$

(54)

By substituting (8) in the above equation and choosing the vectors $\varvec{\varphi }_n$ appropriately, the value of the $\varvec{\varphi } _n^H \alpha ^k \mathbf {R} {{ \varvec{\varphi } }_{n-k}} =0;~k\ne 0$, can be reduced to zero. Therefore, we have

$$\begin{aligned}&\text {arg min }\mathbf {MSE}\nonumber \\&\quad = \text {arg max } \sum \limits _{l = 1}^\tau {\alpha ^{2 l }}{\varepsilon _p} \varvec{ \varphi }_l^H{\mathbf {R}^2}{\varvec{ \varphi }_l} {\left( {{\varepsilon _p} \varvec{\varphi } _l^H \mathbf {R} {{ \varvec{\varphi } }_l} + {\sigma }_{P}^2} \right) ^{- 1}} \end{aligned}$$

(55)

By substituting (2) in (55)

and considering $\tilde{\varvec{\varphi } }_l = {\mathbf {U}^H}\varvec{\varphi }_l$, we have

$$\begin{aligned}&\mathop {\text {arg max }\sum \limits _{l = 1}^\tau {\alpha ^{2l}}{\varepsilon _p} \tilde{\varvec{\varphi }}_l^H \varvec{\Lambda ^2}\tilde{\varvec{\varphi } }_l \left( {\tilde{\varvec{\varphi } }_l}^H \left( {\varepsilon _p} \varvec{\Lambda } +{\sigma }_{P}^2 \mathbf {I}_{M} \right) \tilde{\varvec{\varphi } }_l \right) ^{- 1}}\nonumber \\&{\text{subject to }}{\varvec{\varphi }}^{H}_{i} {\varvec{\varphi }}_{i}=1;~~i=1,\ldots ,\tau \nonumber \\&\qquad\qquad\quad {\varvec{\varphi }}^{H}_{i} {\varvec{\varphi }}_{k}=0;~~k\ne l \end{aligned}$$

(56)

The above equation is the sum of the positive fractions, and it is maximized when the amount of each fraction is maximized. By derivation from (56) respect to $\tilde{\varvec{\varphi }}_l$ and simplifying, $\tilde{\varvec{\varphi }}_l$ is obtained as

$$\begin{aligned} {\tilde{\varvec{\varphi } }_l} = {\left( {{\mathbf {I}_{{M}}}} \right) _{l}}. \end{aligned}$$

(57)

Therefore, optimal pilot vectors are

$$\begin{aligned} { \varvec{\varphi }_l} = \mathbf {u} _{l};l=1,\ldots,\tau . \end{aligned}$$

(58)

Since the pilots are the eigen vectors of the channel covariance matrix and they are orthogonal to each other, hence, the initial considered assumption is correct.

Appendix B

Proof of the Theorem 2

Optimal pilot vectors that minimize MSE are obtained as

$$\begin{aligned}&\text {arg min }\mathbf {MSE}\nonumber \\&{\text{subject to }}{\varvec{\varphi }}^{H}_{i} {\varvec{\varphi }}_{i}=1;~~i=1,\ldots ,\tau \nonumber \\&\qquad\qquad\quad {\varvec{\varphi }}^{H}_{i} {\varvec{\varphi }}_{k}=0;~~k\ne l \end{aligned}$$

The above optimization problem is equivalent with

$$\begin{aligned}&\text {arg max } \text {tr}\left( \varvec{\Theta } \right) \nonumber \\&{\text{subject to }}{\varvec{\varphi }}^{H}_{i} {\varvec{\varphi }}_{i}=1;~~i=1,\ldots ,\tau \nonumber \\&\qquad\qquad\quad {\varvec{\varphi }}^{H}_{i} {\varvec{\varphi }}_{k}=0;~~k\ne l \end{aligned}$$

(59)

where $\text {tr} (\varvec{\Theta })$ is calculated as

$$\begin{aligned} \text {tr} ~\varvec{\Theta } ={\alpha }^2 {\varepsilon _p} \text {tr} \left( {\varvec{\Upsilon }^H} \left( \varvec{\delta } \left( {n,\alpha } \right) \otimes \mathbf {R} \right) ^H \left( \varvec{\delta } \left( {n,\alpha } \right) \otimes \mathbf{R} \right) {\varvec{\Upsilon }}\mathbf {Y}^{-1}\left( n,\alpha \right) \right) \end{aligned}$$

(60)

By choosing the vectors $\varvec{\varphi }_n$ appropriately, the value of the $\varvec{\varphi } _n^H \alpha ^k \mathbf {R} {{ \varvec{\varphi } }_{n-k}} =0;~k\ne 0$, can be reduced to zero. Therefore, we have

$$\begin{aligned}&\text {arg max }{\alpha ^2} \sum \limits _{i = 1}^\tau {\alpha ^{2\left( {\tau - i} \right) }}{\varepsilon _p} \varvec{ \varphi }_i^H{\mathbf {R}^2}{\varvec{ \varphi } _i} {\left( {{\varepsilon _p} \varvec{\varphi } _i^H \mathbf {R} {{ \varvec{\varphi } }_i} + {\sigma }_{P}^2} \right) ^{- 1}}, \end{aligned}$$

(61)

By substituting (2) in the above equation and considering $\tilde{\varvec{\varphi }}_i = {\mathbf {U}^H}\varvec{\varphi } _i$, the above equation is as

$$\begin{aligned}&\text {arg max }{\alpha ^2} \sum \limits _{i = 1}^\tau {\alpha ^{2\left( {\tau - i} \right) }}{\varepsilon _p} \tilde{\varvec{\varphi }}_i^H \varvec{\Lambda ^2}\tilde{\varvec{\varphi }}_i \left( {\tilde{\varvec{\varphi }}_i}^H \left( {\varepsilon _p} \varvec{\Lambda } +{\sigma }_{P}^2\mathbf {I}_{M}\right) \tilde{\varvec{\varphi }}_i \right) ^{- 1}, \end{aligned}$$

(62)

By derivation of the above equation and equating it to zero, the optimal pilot vectors are obtained as

$$\begin{aligned} {\varvec{\varphi } _n} = \mathbf {u} _{\tau - n + 1}. \end{aligned}$$

(63)