Power allocation algorithms for massive MIMO systems with multi-antenna users

Abstract

Modern 5G wireless cellular networks use massive multiple-input multiple-output (MIMO) technology. This concept entails using an antenna array at a base station to concurrently service many mobile devices that have several antennas on their side. In this field, a significant role is played by the precoding (beamforming) problem. During downlink, an important part of precoding is the power allocation problem that distributes power between transmitted symbols. In this paper, we consider the power allocation problem for a class of precodings that asymptotically work as regularized zero-forcing. Under some realistic assumptions, we simplify the spectral efficiency functional and obtain tractable expressions for it. We prove that equal power allocation provides optimum for the simplified functional with total power constraint (TPC). We propose low-complexity Intersection methods (IM) that improve equal power allocation in the case of per-antenna power constraints (PAPC). On simulations using Quadriga, the proposed IM method in combination with widely-studied water filling (WF) shows a significant gain in spectral efficiency while using a similar computing time as the reference equal power (EP) solution.

Appendix

1.1 Search of MCS-$\beta$ Effective SINR

The values of $\beta$ for Modulation and Coding Scheme (MCS) [52] are taken from Table 4. There are different $\beta$ values for different MCSes [45]. The Table 4 shows $\beta$ values, which corresponds to Tables 5.1.3.1-1 to 5.1.3.1-2 in [53]. The MCS value depends on the radio quality and therefore on $\text {SINR}^{eff}_\beta$.

Thus, $\text {SINR}^{eff}_\beta$ can be found by simple iteration method on the equation (12), initializing $\text {SINR}^{eff}_\beta$ by geometrical average using (33) and then taking $\beta = \beta (\text {MCS})$ from Table 4 and $\text {MCS} = \text {MCS}(\text {SINR}^{eff}_\beta )$ from Table 5.

Also note that low values of $\text {SINR}^{eff}_\beta$ (up to -5 dB) indicate that the user is almost out of service, and high values of $\text {SINR}^{eff}_\beta$ (after 23 dB) do not make much sense.

Table 4 Optimal $\beta$ values for each MCS

Table 5 Optimal SE values for each MCS

1.2 Derivation of the eq. (47)

From the identity (45) ${\mathcal {L}}^{'}_{{p}_l}=0$:

$$\begin{aligned} x_l=(1-\beta _k\ln (X_k))X_k \beta _k\sigma ^2\Vert \varvec{g}_l\Vert ^2 \lambda _i\Vert {{\varvec{w}'}}_{l}\Vert ^2. \end{aligned}$$

(62)

Taking average of (62):

$$\begin{aligned} X_k= & {} \frac{1}{L_k}\sum \limits _{l \in {\mathcal {L}}_k}x_l \Leftrightarrow X_k=(1-\beta _k\ln (X_k))\nonumber \\{} & {} X_k \frac{1}{L_k}\sum \limits _{l \in {\mathcal {L}}_k}\left( \sigma ^2s_l^{-2} \lambda _i\Vert {{\varvec{w}'}}_{l}\Vert ^2 \right) . \end{aligned}$$

(63)

Dividing (62) by (63) we get:

$$\begin{aligned} \frac{x_l}{X_k}=\frac{\sigma ^2s_l^{-2} \lambda _i\Vert {{\varvec{w}'}}_{l}\Vert ^2}{ \frac{1}{L_k}\sum \limits _{v \in {\mathcal {L}}_k}\left( \sigma ^2s_v^{-2} \lambda _i\Vert {{\varvec{w}'}}_v\Vert ^2 \right) }=\frac{s_l^{-2} \Vert {{\varvec{w}'}}_{l}\Vert ^2}{ \frac{1}{L_k}\sum \limits _{v \in {\mathcal {L}}_k}\left( s_v^{-2} \Vert {{\varvec{w}'}}_v\Vert ^2 \right) }. \end{aligned}$$

(64)

From (63):

$$\begin{aligned} X_k=\exp \left( \frac{1}{\beta _k}-\frac{1}{\beta _k \frac{1}{L_k}\sum \limits _{l \in {\mathcal {L}}_k}\left( \sigma ^2s_l^{-2} \lambda _i\Vert {{\varvec{w}'}}_{l}\Vert ^2 \right) }\right) . \end{aligned}$$

(65)

From (64) and (65) we can derive:

$$\begin{aligned} x_l= & {} \frac{s_l^{-2} \Vert {{\varvec{w}'}}_{l}\Vert ^2}{ \frac{1}{L_k}\sum \limits _{v \in {\mathcal {L}}_k}\left( s_v^{-2} \Vert {{\varvec{w}'}}_v\Vert ^2 \right) }\nonumber \\{} & {} \quad \exp \left( \frac{1}{\beta _k}-\frac{1}{\beta _k \frac{1}{L_k}\sum \limits _{l \in {\mathcal {L}}_k}\left( \sigma ^2s_l^{-2} \lambda _i\Vert {{\varvec{w}'}}_{l}\Vert ^2 \right) }\right) . \end{aligned}$$

(66)

Also, we know that $x_l=\exp \left( -\frac{{p}_l}{\beta _l\sigma ^2 s_l^{-2}}\right)$.

So we know $p_l=-\beta _l\sigma ^2 s_l^{-2}\ln \left( x_l\right)$ and can substitute (66) in the $p_l$ expression.

Taking into account $\sum \limits _{l=1}^{L}(\Vert {{\varvec{w}'}}_{l}\Vert ^2{p}_l)=P$ we obtain:

$$\begin{aligned}{} & {} \sum \limits _{l=1}^{L}(\Vert {{\varvec{w}'}}_{l}\Vert ^2{p}_l)\nonumber \\{} & {} =-\sum \limits _{k=1}^{K}\sum \limits _{l \in {\mathcal {L}}_k}\left[ \sigma ^2s_l^{-2}\Vert {{\varvec{w}'}}_{l}\Vert ^2\left( 1-\frac{1}{ \frac{1}{L_{k}}\sum \limits _{v=1}^{L_{k}}\left( \sigma ^2s_v^{-2} \lambda _i\Vert {{\varvec{w}'}}_v\Vert ^2 \right) }\right) \right] -\nonumber \\{} & {} \quad - \sum \limits _{l=1}^{L}\beta _k\sigma ^2s_l^{-2}\Vert {{\varvec{w}'}}_{l}\Vert ^2\ln \left( \frac{s_l^{-2} \Vert {{\varvec{w}'}}_{l}\Vert ^2}{\frac{1}{L_{k}}\sum \limits _{v=1}^{L_{k}}s_v^{-2} \Vert {{\varvec{w}'}}_v\Vert ^2} \right) =\nonumber \\{} & {} =\lambda _i^{-1}L-\sum \limits _{l=1}^{L}\sigma ^2s_l^{-2}\Vert {{\varvec{w}'}}_{l}\Vert ^2\nonumber \\{} & {} \quad - \sum \limits _{l=1}^{L}\beta _k\sigma ^2s_l^{-2}\Vert {{\varvec{w}'}}_{l}\Vert ^2\ln \left( \frac{s_l^{-2} \Vert {{\varvec{w}'}}_{l}\Vert ^2}{\frac{1}{L_{k}}\sum \limits _{v=1}^{L_{k}}s_v^{-2} \Vert {{\varvec{w}'}}_v\Vert ^2} \right) =P, \end{aligned}$$

(67)

$$\begin{aligned} \lambda _i^{-1}= \frac{P}{L}+ \frac{1}{L}\sum \limits _{l=1}^{L}\sigma ^2s_l^{-2}\Vert {{\varvec{w}'}}_{l}\Vert ^2\left( \beta _k\ln \left( \frac{s_l^{-2} \Vert {{\varvec{w}'}}_{l}\Vert ^2}{\frac{1}{L_k}\sum \limits _{v \in {\mathcal {L}}_k}s_v^{-2} \Vert {{\varvec{w}'}}_v\Vert ^2} \right) +1\right) . \end{aligned}$$

(68)

Substituting (68) into (65) and (65) into (64) we get the required expressions for $x_l$ and then for $p_l$.