Analytical Approximation for Capacity in Massive MIMO Systems

Abstract

In most existing research on massive multiple-input multiple-output (MIMO) systems, theoretical analysis relies on the assumption that the number of antennas at the base station is infinite. Under this assumption, channel vectors for different users will be asymptotically orthogonal; therefore, the calculation of channel capacity can be greatly simplified. However, in practical systems, the number of antennas is always finite, and the channel vectors for different users cannot be completely orthogonal. In this paper, we propose an analytical approximation for the channel capacity of massive MIMO systems, with a finite number of antennas. Numerical results show that the derived closed-form expression is more accurate than the one assuming that the channel vectors are asymptotically orthogonal.

Appendices

Appendix 1

Substituting (7) into the term of ${\mathrm {E_H}}\left( \mathbf{{\Delta }} \right)$ in Eq. (10), we obtain

$$\begin{aligned} {\mathrm {E}}\left\{ {{{[{\mathbf {\Delta }}]}_{(i,j)}}} \right\} \nonumber &={\mathrm {E}}\left[ {\frac{1}{M}{{\mathbf {h}}_{{i}}}^\dag {{\mathbf {h}}_{{j}}} - \delta (i - j)} \right] \nonumber \\ &=\frac{1}{M}{\mathrm {E}}\left( {{{\mathbf {h}}_{{i}}}^\dag {{\mathbf {h}}_{{j}}}} \right) - \delta (i - j), \end{aligned}$$

(14)

where ${\mathbf{{h}}_{{i}}}$ and ${\mathbf{{h}}_{{j}}}$ represent the channel responses of the i-th and j-th users.

(14) can be rewritten as

$$\begin{aligned} \mathrm {E}\left\{ {{{\left[ \mathbf{{\Delta }} \right] }_{\left( {i,j} \right) }}} \right\} \nonumber&= \frac{1}{M}{\mathrm {E}}\left( {\sum \limits _{m = 1}^M {h_{im}^*{h_{jm}}} } \right) - \delta (i - j)\\ \nonumber&= \frac{1}{M}\sum \limits _{m = 1}^M {{\mathrm {E}}(h_{im}^*){\mathrm {E}}({h_{jm}})} - \delta (i - j)\\ \nonumber&= \frac{1}{M}\left[ {\mathrm{{M}}\delta (i - j)} \right] - \delta (i - j)\\&= 0. \end{aligned}$$

(15)

This completes the proof of Eq. (11).

Appendix 2

From (7), the expressions of ${\left[ {\mathbf {\Delta }} \right] _{\left( {i,p} \right) }}$ and ${\left[ {\mathbf {\Delta }} \right] _{\left( {p,j} \right) }}$ can be given by

$$\begin{aligned} {\left[ {\mathbf {\Delta }} \right] _{\left( {i,p} \right) }} = \frac{1}{M}{\mathbf{{h}}_{{i}}}^\dag {\mathbf{{h}}_{{p}}} - \delta (i - p). \end{aligned}$$

(16)

$$\begin{aligned} {\left[ {\mathbf {\Delta }} \right] _{\left( {p,j} \right) }} = \frac{1}{M}{\mathbf{{h}}_{{p}}}^\dag {\mathbf{{h}}_{{j}}} - \delta (p-j). \end{aligned}$$

(17)

Substituting (16) and (17) into ${\mathrm {E_H}}\left( {{\mathbf{{\Delta }}^2}} \right)$ in (12), we have

$$\begin{aligned}&{\mathrm {E_\mathbf {H}}}\left\{ {{{\left[ \mathbf{{\Delta }}^2 \right] }_{\left( {i,j} \right) }}} \right\} \nonumber \\&\quad = {\mathrm {E_\mathbf {H}}}\left\{ {\sum \limits _{p = 1}^K {{{\left[ \mathbf{{\Delta }} \right] }_{\left( {i,p} \right) }}{{\left[ \mathbf{{\Delta }} \right] }_{\left( {p,j} \right) }}} } \right\} \nonumber \\&\quad = {E_\mathbf {H}}\left\{ {\sum \limits _{p = 1}^K {\left[ {\frac{1}{{{M^2}}}{} \mathbf{{h}}_i^\dag {\mathbf{{h}}_p}{} \mathbf{{h}}_p^\dag {\mathbf{{h}}_j} - \frac{1}{M}\delta (i - p)\mathbf{{h}}_p^\dag {\mathbf{{h}}_j}} \right. } } \right. \nonumber \\&\qquad \left. {\left. { - \frac{1}{M}\delta (p - j)\mathbf{{h}}_i^\dag {\mathbf{{h}}_p} + \delta (i - p)\delta (p - j)} \right] } \right\} \end{aligned}$$

(18)

Using the relation between moment and cumulant [7], the first term in (18) is obtained as

$$\begin{aligned}&\frac{1}{{{M^2}}}{\mathrm {E_\mathbf {H}}}\left( {\mathbf{{h}}_{{i}}^{\dag }{\mathbf{{h}}_{{p}}}{} \mathbf{{h}}_{{p}}^{\dag }{\mathbf{{h}}_{{j}}}} \right) \nonumber \\&\quad = \frac{1}{{{M^2}}}\sum \limits _{{m_1} = 1}^M {\sum \limits _{{m_2} = 1}^M {{\mathrm {E_\mathbf {H}}}} } \left( {h_{i{m_1}}^*{h_{p{m_1}}}h_{p{m_2}}^*{h_{j{m_2}}}} \right) \nonumber \\&\quad = \frac{1}{{{M^2}}}\sum \limits _{{m_1} = 1}^M {\sum \limits _{{m_2} = 1}^M {\left\{ {{\mathrm {E_\mathbf {H}}}\left( {\left. {h_{i{m_1}}^*{h_{p{m_1}}}} \right) } \right. } \right. } } + {\mathrm {E_\mathbf {H}}}\left( {\left. {h_{p{m_2}}^*{h_{j{m_2}}}} \right) } \right. \nonumber \\&\qquad + {\mathrm {E_\mathbf {H}}}\left( {\left. {h_{i{m_1}}^*{h_{j{m_2}}}} \right) } \right. + \left. {{\mathrm {E_\mathbf {H}}}\left( {\left. {{h_{p{m_1}}}h_{p{m_2}}^*} \right) } \right. } \right\} \nonumber \\&\quad = \delta (i - p)\delta (p - j) + \frac{1}{M}\delta (i - j). \end{aligned}$$

(19)

Similarly,

$$\begin{aligned} \frac{1}{M}{\mathrm {E_\mathbf {H}}}\left[ {\delta (i - p)\mathbf{{h}}_{{p}}^{\dag }{\mathbf{{h}}_{{j}}}} \right]&= \delta (i - p)\delta (p - j), \end{aligned}$$

(20)

$$\begin{aligned} \frac{1}{M}{\mathrm {E_\mathbf {H}}}\left[ {\delta (p - j)\mathbf{{h}}_{{i}}^{\dag }{\mathbf{{h}}_{{p}}}} \right]&= \delta (i - p)\delta (p - j). \end{aligned}$$

(21)

Substituting (19) through (22) into (18), we have

$$\begin{aligned} \mathrm {Tr}\left\{ {{\mathrm {E_\mathbf {H}}}\left[ {{{\left( {{{\mathbf {\Delta }}^2}} \right) }_{\left( {i,j} \right) }}} \right] } \right\} = \frac{{{K^2}}}{M}. \end{aligned}$$

(22)