Massive MIMO in Small Cell Networks: Wireless Backhaul

Abstract

Dense small cell networks are expected to be deployed in the next generation wireless system to provide better coverage and throughput to meet the ever-increasing requirements of high data rate applications. As the trend toward densification calls for more and more wireless links to forward a massive backhaul traffic into the core network, it is critically important to take into account the presence of a wireless backhaul for the energy-efficient design of small cell networks. In this chapter, we develop a general framework to analyze the energy efficiency of a two-tier small cell network with massive MIMO macro base stations and wireless backhaul. Our analysis reveal that under spatial multiplexing, the energy efficiency of a small cell network is sensitive to the network load, and it should be taken into account when controlling the number of users served by each base station. We also demonstrate that a two-tier small cell network with wireless backhaul can be significantly more energy efficient than a one-tier cellular network. However, this requires the bandwidth division between radio access links and wireless backhaul to be optimally designed according to the load conditions.

Appendix

Proof of Lemma 3.1

The channel matrix between a MBS to its $K_{\mathrm {m}}$ associated UEs can be written as $\hat{\mathbf {H}} = \mathbf {L}^{\frac{1}{2}} \mathbf {H}$, where $\mathbf {L} = \text {diag}\{L_1^{-1}, \ldots , L_{K_\mathrm m}^{-1}\}$, with $L_i = r_i^\alpha / S_i$ being the path loss from the MBS to its ith UE, where $r_i$ is the corresponding distance and $S_i$ denotes the shadowing, $\mathbf {H} = [\mathbf {h}_1,\ldots ,\mathbf {h}_{K_\mathrm m}]^{\text {T}}$ is the $K_\mathrm m \times M_\mathrm m$ small-scale fading matrix, with $\mathbf {h}_i \sim {CN}\left( \mathbf {0}, \mathbf {I}\right)$. The ZF precoder is then given by $\mathbf {W} = \xi \hat{\mathbf {H}}^{*} ( \hat{\mathbf {H}}\hat{\mathbf {H}}^{*})^{-1}$, where $\xi ^2 = 1 / \text {tr}[ (\hat{\mathbf {H}}^{*}\hat{\mathbf {H}})^{-1}]$ normalizes the transmit power [56]. In the following, we use the notation $\varPhi ^{\mathrm {U}}$ as $\varPhi ^{\mathrm {D}}$ to denote the subsets of $\varPhi$ that transmit in uplink and downlink, respectively, we further denote $\mathscr {U}_x$ as the set of UEs that are associated with access point x, and denote $\hat{x}$ as the transmitter that locates closest to the origin. Since the locations of MBSs and SAPs follow a stationary PPP, we can apply the Slivnyark’s theorem [54], which implies that it is sufficient to evaluate the SINR of a typical UE at the origin. As such, by noticing that under dynamic TDD, every wireless link experiences interference from the downlink transmitting MBSs and SAPs, and from the uplink transmitting UEs, the downlink SINR between a typical UE at the origin and its serving MBS can be written as

$$\begin{aligned} \gamma _{\mathrm m}^{\text {DL}} = \dfrac{P_{\mathrm {mt}}\vert \mathbf {h}_{\hat{x}_{\mathrm {m}},o}^{*} \mathbf {w}_{\hat{x}_{\mathrm {m}},o} \vert ^2 L_{\hat{x}_{\mathrm {m}},o}^{-1} }{I_{\mathrm {oc}}^{\mathrm {mu}} + I_\mathrm u + \sigma ^2}, \end{aligned}$$

(3.25)

where $\mathbf {h}_{\hat{x}_{\mathrm {m}},o}$ is the small-scale fading, $\mathbf {w}_{\hat{x}_{\mathrm {m}},o}$ is the ZF precoding vector, $L_{\hat{x}_{\mathrm {m}},o}$ denotes the corresponding path loss, while $I_{\mathrm {oc}}^{\mathrm {mu}}$ is the aggregate interference from other cells to the MBS UE, and $I_\mathrm u$ denotes the interference from UEs, respectively given as follows:

$$\begin{aligned} I_{\mathrm {oc}}^{\mathrm {mu}}&= \sum \limits _{x_\mathrm {m} \in \varPhi _{\text {m}}^{\mathrm {D}} \setminus \hat{x}_{\mathrm {m}}} \dfrac{P_{\mathrm {mt}}g_{x_\mathrm {m},o}}{ K_\mathrm m L_{x_{\mathrm {m}},o}} + \sum \limits _{x_\mathrm {s} \in \varPhi _{\text {s}}^{\mathrm {D}}} \dfrac{P_{\mathrm {st}}g_{x_\mathrm {s},o}}{ K_\mathrm s L_{x_{\mathrm {s}},o} } \end{aligned}$$

(3.26)

and

$$\begin{aligned} I_\mathrm u&= \sum \limits _{x_{\mathrm {u}} \in \varPhi _{\text {u}}^{\mathrm {U}} } \dfrac{P_{\mathrm {ut}}\vert h_{x_{\mathrm {u}},o} \vert ^2}{L_{x_{\mathrm {u}},o} }, \end{aligned}$$

(3.27)

whereas $g_{x_\mathrm {m},o}$ and $g_{x_\mathrm {s},o}$ represent the effective small-scale fading from the interfering MBS $x_\mathrm {m}$ and SAP $x_\mathrm {s}$ to the origin, respectively, given by [57]

$$\begin{aligned} g_{x_\mathrm {m},o} = \sum _{u \in \mathscr {U}_{x_\mathrm {m}}} K_{\mathrm {m}}\vert \mathbf {h}_{x_\mathrm {m},o}^{*} \mathbf {w}_{x_\mathrm {m},u} \vert ^2 \sim \varGamma \left( K_{\mathrm {m}}, 1\right) \end{aligned}$$

(3.28)

and

$$\begin{aligned} g_{x_\mathrm {s},o} = \sum _{u \in \mathscr {U}_{x_\mathrm {s}}} K_{\mathrm {s}}\vert \mathbf {h}_{x_\mathrm {s},o}^{*} \mathbf {w}_{x_\mathrm {s},u} \vert ^2 \sim \varGamma \left( K_{\mathrm {s}}, 1\right) . \end{aligned}$$

(3.29)

By conditioning on the interference, when $K_{\mathrm {m}}, M_{\mathrm {m}}\rightarrow \infty$ with $\beta _{\mathrm m} = K_{\mathrm {m}}/ M_{\mathrm {m}}< 1$, the SINR under ZF precoding converges to [39]

$$\begin{aligned} \gamma _{\mathrm m }^{\text {DL}} \rightarrow \bar{\gamma }_{\mathrm m }^{\text {DL}} = \dfrac{P_{\mathrm {mt}}M_{\mathrm {m}}}{\left( I_{\mathrm {oc}}^{\mathrm {mu}} + I_\mathrm u + \sigma ^2\right) \sum _{j=1}^{K_{\mathrm {m}}} {e}_j^{-1}}, \,\,\, a.s. \end{aligned}$$

(3.30)

where ${e}_i$ is the solution of the fixed point equation

$$\begin{aligned} \dfrac{L_{\hat{x}_{\mathrm {m}},u_i}^{-1}}{{e}_i} = 1 + \dfrac{J}{M_{\mathrm {m}}},~i = 1, 2, \ldots , K_{\mathrm {m}}\end{aligned}$$

(3.31)

with $J = \sum _{j=1}^{K_{\mathrm {m}}} {L_{\hat{x}_{\mathrm {m}},u_j}^{-1}}{{e}_j^{-1}}$. By summing (3.31) over i we obtain

$$\begin{aligned} J = K_{\mathrm {m}}+ \frac{K_{\mathrm {m}}}{M_{\mathrm {m}}}J. \end{aligned}$$

(3.32)

Solving the equation above results in $J=K_{\mathrm {m}}M_{\mathrm {m}}/ (M_{\mathrm {m}}- K_{\mathrm {m}})$, and by substituting the value of J into (3.31) we can have

$$\begin{aligned} \dfrac{1}{\bar{e}_i} = \frac{M_{\mathrm {m}}}{M_{\mathrm {m}}-K_{\mathrm {m}}} \cdot L_{\hat{x}_{\mathrm {m}},u_i}, \end{aligned}$$

(3.33)

which substituted into (3.30) yields

$$\begin{aligned} \bar{\gamma }_{\mathrm m }^{\text {DL}} = \dfrac{ \left( 1 - \beta _{\mathrm m} \right) M_{\mathrm {m}}P_{\mathrm {mt}}}{\left( I_{\mathrm {oc}}^{\mathrm {mu}} + I_\mathrm u + \sigma ^2\right) \sum ^{K_{\mathrm {m}}}_{j=1} L_{\hat{x}_{\mathrm {m}},u_j}}.\qquad \end{aligned}$$

(3.34)

Notice that $\{L_{\hat{x}_{\mathrm {m}},u_j}\}_{j=1}^{K_{\mathrm {m}}}$ is an independent i.i.d. sequence with finite first moment, given by

$$\begin{aligned} \mathbb {E} \left[ L_{\hat{x}_{\mathrm {m}},u_j} \right] = \varGamma \left( 1 + \frac{1}{\delta }\right) G_\mathrm {m}^{-1} < \infty , \end{aligned}$$

by applying the strong law of large numbers (SLLN) to (3.34), we have

$$\begin{aligned} \bar{\gamma }_{\mathrm m }^{\text {DL}} \rightarrow \frac{ \left( 1 - \beta _{\mathrm m} \right) G_\mathrm {m}^{1/ \delta } P_{\mathrm {mt}}}{\beta _{\mathrm m} \varGamma \left( 1 + \frac{1}{\delta }\right) \left( I_{\mathrm {oc}}^{\mathrm {mu}} + I_\mathrm u + \sigma ^2\right) }, \quad a.s. \end{aligned}$$

(3.35)

As such, using the continuous mapping theorem and the lemma in [58], we can compute the ergodic rate as

$$\begin{aligned} \mathbb {E}\left[ \log _2\left( 1+\bar{\gamma }_{\mathrm m }^{\text {DL}}\right) \right]&= \frac{1}{\ln 2}\mathbb {E}\left[ \ln \left( 1+\frac{\nu _{\mathrm {m}}^{\mathrm {D}}}{I_{\mathrm {oc}}^{\mathrm {mu}} + I_\mathrm u + \sigma ^2}\right) \right] \nonumber \\&= \int _0^\infty \frac{e^{-\sigma ^2 z}}{z \ln 2} \left( 1 - e^{-\nu _\mathrm {m}^{\mathrm {D}} z} \right) \mathbb {E}\left[ e^{-z I_{\mathrm {u}}}\right] \mathbb {E}\left[ e^{-z I_{\mathrm {oc}}^{\mathrm {mu}}}\right] dz. \end{aligned}$$

(3.36)

Due to the composition of independent PPPs and the displacement theorem [59], the interference $I_{\text {u}}$ follows a homogeneous PPP with spatial density $\tilde{\lambda }_{\text {u}} = \left( 1-\tau _{\mathrm {m}}\right) \lambda _{\mathrm {m}} K_{\mathrm {m}}+ \left( 1-\tau _{\mathrm {s}}\right) \lambda _{\mathrm {s}}K_{\mathrm {s}}$, and the corresponding Laplace transform is given as [54]

$$\begin{aligned} \mathbb {E} \left[ e^{-z I_{\text {u}}} \right] = \exp \left( - \dfrac{2 \pi ^2 \tilde{\lambda }_{\text {u}} \mathbb {E}[S_{\mathrm {D}}^{\frac{2}{\alpha }}] P_{\mathrm {ut}}^{\frac{2}{\alpha }} z^{\frac{2}{\alpha }}}{\alpha \sin \left( \frac{2 \pi }{\alpha } \right) } \right) . \end{aligned}$$

(3.37)

As for the Laplace transform of $I_{\mathrm {oc}}^{\mathrm {mu}}$, the conditional Laplace transform on $L_{\hat{x}_{\mathrm {m}},o}$ can be computed as

$$\begin{aligned}&\mathbb {E}\left[ e^{-z I_{\mathrm {oc}}^{\mathrm {mu}}} | L_{\hat{x}_{\mathrm {m}},o} = t\right] \nonumber \\&= \exp \! \left( - \tau _{\mathrm {m}}a_\mathrm {m} \mathscr {C}_{\alpha , K_{\mathrm {m}}} \! \left( z P_{\mathrm {mt}}, t \right) \! \left( \frac{z P_{\mathrm {mt}}}{K_{\mathrm {m}}} \right) ^{\delta } \!\!- \tau _{\mathrm {s}}a_\mathrm {s} \mathscr {C}_{\alpha , K_{\mathrm {s}}} \! \left( z P_{\mathrm {mt}}, t \right) \! \left( \frac{z P_{\mathrm {st}}}{K_{\mathrm {s}}} \right) ^{\delta } \right) . \end{aligned}$$

(3.38)

Notice that $L_{\hat{x}_{\mathrm {m}},o}$ has its distribution given by (3.13), and the rate $R_{\mathrm {m}}^{\mathrm {DL}}$ given as

$$\begin{aligned} R_{\mathrm {m}}^{\mathrm {DL}} = \left( 1 - \zeta _{\mathrm {b}} \right) \mathbb {E}\left[ \log _2\left( 1+\bar{\gamma }_{\mathrm m }^{\text {DL}}\right) \right] , \end{aligned}$$

(3.39)

substituting (3.37) and (3.38) into (3.36), and decondition $L_{\hat{x}_{\mathrm {m}},o}$ with respect to (3.13) we have the corresponding result.

Proof of Lemma 3.2

Let us consider a UE transmitting in uplink to a typical MBS located at the origin, which employs a ZF receive filter $\mathbf {r}_{o,\hat{x}_{\mathrm {u}}}^* = \hat{\mathbf {h}}_{o,\hat{x}_{\mathrm {u}}}^* ( \sum _{u \in \mathscr {U}_o } \hat{\mathbf {{h}}}_{o,u} \hat{\mathbf {h}}_{o,u}^{*} )^{-1}$ [56], the SINR is then given by

$$\begin{aligned} \gamma _{\mathrm {m}}^{\text {UL}} = \dfrac{P_{\mathrm {ut}}L_{o,\hat{x}_\mathrm {u}}^{-1} \vert \mathbf {r}_{o,\hat{x}_\mathrm {u}}^{*} \mathbf {h}_{o,\hat{x}_\mathrm {u}} \vert ^2}{ \left( I_{\mathrm {oc}}^{\mathrm {mbs}} + I_\mathrm u + \sigma ^2 \right) \Vert \mathbf {r}_{o,\hat{x}_\mathrm {u}} \Vert ^2 }, \end{aligned}$$

(3.40)

where $I_{\mathrm {oc}}^{\mathrm {mbs}}$ denotes the interference from other cells received at the MBS. By conditioning on the interference, when $K_{\mathrm {m}}, M_{\mathrm {m}}\rightarrow \infty$ with $\beta _\mathrm m = K_{\mathrm {m}}/ M_{\mathrm {m}}< 1$, the SINR above converges to [39]

$$\begin{aligned} \gamma _{\mathrm m}^{\text {UL}} \rightarrow \bar{\gamma }_{\mathrm m}^{\text {UL}} = \frac{P_{\mathrm {ut}}M_{\mathrm {m}}(1-\beta _\mathrm m) L_{o,\hat{x}_{\mathrm {u}}}^{-1} }{I_{\mathrm {oc}}^{\mathrm {mbs}} + I_\mathrm u + \sigma ^2}, \quad a.s. \end{aligned}$$

(3.41)

By using the continuous mapping theorem [58], the uplink ergodic rate can be calculated as

$$\begin{aligned} \mathbb {E}\left[ \log _2\left( 1 + \bar{\gamma }_{\mathrm {m}}^{\mathrm {UL}} \right) \right]&= \frac{1}{\ln 2} \mathbb {E}\left[ \ln \left( 1 + \frac{\nu _{\mathrm {m}}^{\mathrm {U}} L_{o,x_\mathrm {u}}^{-1} }{ I_{\mathrm {oc}}^{\mathrm {mbs}} + I_{\mathrm {u}} + \sigma ^2 } \right) \right] \nonumber \\&= \int _0^\infty \!\!\! \int _0^\infty \frac{e^{-\sigma ^2 z}}{z \ln 2} \left( 1 - e^{-z \nu _{\mathrm {m}}^{\mathrm {U}}/t } \right) \mathbb {E}\left[ e^{- z I_{\mathrm {u}}} \right] \mathbb {E}\left[ e^{-z I_{\mathrm {oc}}^{\mathrm {mbs}} } \right] f_{L_{\mathrm {m}}}(t) dz dt. \end{aligned}$$

(3.42)

The Laplace transform of $I_{\mathrm {oc}}^{\mathrm {mbs}}$ can be computed as

$$\begin{aligned}&\mathbb {E}\left[ e^{-z I_{\mathrm {oc}}^{\mathrm {mbs}} } \right] \nonumber \\&= \exp \! \left( - \frac{ \varGamma \! \left( 1 \!+\! \delta \right) \delta \pi ^2 z^\delta }{\sin (\delta \pi )} \left[ \frac{ \tau _{\mathrm {m}}a_\mathrm {m} P_{\mathrm {mt}}^\delta \prod _{i=1}^{K_{\mathrm {m}}-1} (i\!+\!\delta ) }{\varGamma (K_{\mathrm {m}}) K_{\mathrm {m}}^\delta } \! + \! \frac{ \tau _{\mathrm {s}}a_\mathrm {s} P_{\mathrm {st}}^\delta \prod _{i=1}^{K_{\mathrm {s}}-1} (i\!+\!\delta ) }{\varGamma (K_{\mathrm {s}}) K_{\mathrm {s}}^\delta }\right] \right) . \end{aligned}$$

(3.43)

On the other hand, to consider the uplink interference from UEs, we use the result in [60] where the path loss from MBS UEs and SAP UEs are modeled as two independent inhomogeneous PPP with intensity measure being

$$\begin{aligned} \varLambda _{\mathrm {mu}}^{(\mathrm {m})}(dx)&= \delta a_{\mathrm {m}} x^{\delta -1} \left[ 1 - \exp \left( - G_{\mathrm {m}} x^\delta \right) \right] ,\end{aligned}$$

(3.44)

$$\begin{aligned} \varLambda _{\mathrm {su}}^{(\mathrm {m})}(dx)&= \delta a_{\mathrm {s}} x^{\delta -1} \left[ 1 - \exp \left( - G_{\mathrm {m}} x^\delta \right) \right] . \end{aligned}$$

(3.45)

The Laplace transform of the UE interference can then be calculated as

$$\begin{aligned} \mathbb {E}[e^{-z I_{\mathrm {u}}}]&= \exp \left( - (1-\tau _{\mathrm {m}}) K_{\mathrm {m}}\int _{0}^\infty \frac{\varLambda _{\mathrm {mu}}^{(\mathrm {m})}(dx)}{1 + z^{-1}x/P_{\mathrm {ut}}} - (1-\tau _{\mathrm {s}}) K_{\mathrm {s}}\int _{0}^\infty \frac{\varLambda _{\mathrm {su}}^{(\mathrm {m})}(dx)}{1 + z^{-1} x/P_{\mathrm {ut}}} \right) \nonumber \\&= \exp \left( - \tilde{\lambda }_{\mathrm {u}} \pi \mathbb {E}[S_{\mathrm {D}}^{\delta }] \int _0^\infty \frac{1 - e^{ - G_\mathrm {m} u }}{1 + z^{-1} u^{\frac{1}{\delta }}/P_{\mathrm {ut}}} du \right) . \end{aligned}$$

(3.46)

As such, noticing that

$$\begin{aligned} R_{\mathrm {m}}^{\mathrm {UL}} = \left( 1 - \zeta _{\mathrm {b}} \right) \mathbb {E}\left[ \log _2\left( 1 + \bar{\gamma }_{\mathrm {m}}^{\mathrm {UL}} \right) \right] \end{aligned}$$

(3.47)

the result follows by substituting (3.43) and (3.46) into (3.42).

Proof of Lemma 3.7

The average rate for a typical UE located at the origin is given by

$$\begin{aligned} R = A_\mathrm m R_\mathrm m + A_\mathrm s R_\mathrm s, \end{aligned}$$

(3.48)

where $R_\mathrm m$ and $R_\mathrm s$ are the data rates when the UE associates to a MBS and a SAP, respectively, given by

$$\begin{aligned} R_\mathrm m =\tau _{\mathrm {m}}{R}^{\mathrm {DL}}_\mathrm m + (1 - \tau _{\mathrm {m}}) {R}^{\mathrm {UL}}_\mathrm m \end{aligned}$$

(3.49)

and

$$\begin{aligned} R_\mathrm s = \tau _{\mathrm {s}}\min \left\{ {R}_{\mathrm s}^{\mathrm {DL}} , {R}_{\mathrm b}^{\mathrm {DL}} \right\} + (1 - \tau _{\mathrm {s}}) \min \left\{ {R}_{\mathrm s}^{\mathrm {UL}} , {R}_{\mathrm b}^{\mathrm {UL}} \right\} . \end{aligned}$$

(3.50)

As each MBS and each SAP serve $K_{\mathrm {m}}$ and $K_{\mathrm {s}}$ UEs, respectively, the total density of active UEs is given by $K_\mathrm m \lambda _{\mathrm {m}}+ K_\mathrm s \lambda _{\mathrm {s}}$. Let B be the available bandwidth, the sum rate per area is obtained as $\mathscr {R} = \left( K_\mathrm m \lambda _{\mathrm {m}}+ K_\mathrm s \lambda _{\mathrm {s}}\right) B R$. Lemma 3.7 then follows from Lemmas 3.1 to 3.6 and by the continuous mapping theorem.