# Asymptotic Analysis of the Downlink in Cooperative Massive MIMO Systems

Itsik Bergel, Siddhartan Govindasamy Itsik Bergel<EMAIL_ADDRESS>is
with the Faculty of Engineering, Bar Ilan University, Ramat-Gan, Israel.
Siddhartan Govindasamy<EMAIL_ADDRESS>is with the Department of
Engineering, Boston College, Chestnut Hill, MA, USA.

(October 2022)

###### Abstract

We consider the downlink of a  cooperative cellular communications system,
where several base-stations around each mobile cooperate and perform zero-
forcing to reduce the received interference  at the mobile. We derive closed-
form expressions for the asymptotic performance of the network as the number
of antennas per base station grows large. These expressions capture the trade
off between various system parameters, and characterize the joint effect of
noise and interference (where either noise or interference is asymptotically
dominant and where both are asymptotically relevant). The asymptotic results
are verified using Monte Carlo simulations, which indicate that they are
useful even when the number of antennas per base station is only moderately
large. Additionally, we show that when the number of antennas per base station
grows large, power allocation can be optimized locally at each base station.
We hence present a power allocation algorithm that achieves near optimal
performance while significantly reducing the coordination overhead between
base stations. The presented analysis is significantly more challenging  than
the uplink analysis, due to the dependence between beamforming vectors of
nearby base stations. This statistical dependence is handled by introducing
novel bounds on marked shot-noise point processes with dependent marks, which
are also useful in other contexts.

##  I Introduction

Multiple-input-multiple-output (MIMO) systems with large numbers of antennas,
also known as massive MIMO, have received tremendous attention in the research
community (e.g., [1, 2, 3, 4, 5, 6, 7]). Early works on massive MIMO systems
(e.g., [1]) used maximal ratio-transmission (MRT) for the downlink or maximal
ratio combining (MRC) for the uplink, to ensure that signals add
constructively. Interference mitigation was accomplished by the approximate
(and asymptotically exact) orthogonality of long channel vectors with random,
independent fading coefficients.

While MRT/MRC are optimal when the number of antennas at the base stations
(BSs) go to infinity, for finite (and even large) numbers of antennas,
significantly higher spectral efficiencies are achievable using other linear
processing techniques. On the uplink, minimum-mean-square-error (MMSE)
processing is the optimal linear processing scheme for receive beamforming [8,
9, 10]. In contrast, in the downlink the precoding decisions of each BS affect
mobiles of all BSs, and optimal processing can only be achieved by numeric
optimization.

Since no optimal processing scheme is known for the downlink, several
heuristic beamforming approaches were considered for interference mitigation.
Most of these are variations of zero-forcing precoding, where transmit
beamformers are designed to produce (near) zero interference at certain
mobiles (e.g., [11, 10, 12, 13, 14, 15]). In [10], regularized zero-forcing
was found to perform as well as MRT, but with an order of magnitude fewer
antennas. Additionally, [13] showed significantly larger spectral efficiencies
with zero forcing compared to MRT, even when the number of antennas is $\sim
200$. Hence, although MRT is asymptotically optimal on the downlink, zero-
forcing precoding is attractive even when the number of antennas per BS is
quite large.

In high-density networks, signals from other base stations may cause
significant interference. Thus, we focus on BS cooperation, which promises
significant performance improvement. In particular, we focus on user-centric
cooperation, where all BSs up to a certain distance from each mobile protect
the mobile from interference. This network structure is strongly related to
cell-free systems (e.g., [8, 16, 17]) which have received a lot of attention
in the literature. Thus our results are also relevant to some cell-free
scenarios. Cell-free systems may differ from our work in 3 main aspects: They
typically use access points with a small number of antennas; They typically
use joint processing of all BSs; and they cooperate both for interference
mitigation and for signal transmission (joint signal transmission is not
considered herein as its contribution is smaller, and it complicates the
analysis).

So far, all analysis of the downlink in  cooperative massive MIMO networks was
done either for specific BS locations, or through simulation (e.g., [10, 8]).
As such, obtaining generalizable conclusions using these approaches is
challenging, and requires extensive numerical simulations. On the other hand,
in the approach taken in this work, we explicitly model how the BSs and
mobiles are positioned in space, which enables us to obtain insights on system
performance such as the impact of BS density and mobile distribution on the
spectral efficiency. Such an approach can enable system designers to trade-off
various parameters such as BS and mobile densities, number of antennas, number
of cooperating base stations and spectral efficiency. These approaches have
been observed to “lead to remarkably precise characterizations” [18]. Further,
analysis of cellular networks that explicitly considers the BS and mobile
distribution in space is quite challenging, even for single antenna systems
[19]. Hence, an analysis of cooperative cellular networks with multiple
antennas, where the spatial distribution of BSs and mobiles is explicitly
considered, is useful and challenging.

Here, we derive closed-form expressions for the asymptotic behavior of the
Signal-to-Interference-plus-Noise Ratio (SINR), which give insights on the
network performance and allow system optimization. The analysis explicitly
addresses the spatial distribution of BSs and mobiles.

We also introduce a novel approach for asymptotic analysis that captures the
effect of both noise and interference. This approach is important because
asymptotic expressions are often used to generate performance approximations
for finite systems. Finite systems are affected by both interference and
noise, while traditional asymptotic analysis either reaches a noise-limited
regime or an interference-limited regime.

Additionally, our results indicate that with large number of antennas, the
effect of each BS on its neighbors can be captured through the network’s
average transmitted power per BS. This finding allows us to present a simple
and efficient power allocation algorithm for large networks which only
involves a minor coupling between the local processing done at each base
station.

To facilitate analysis, this paper considers a relatively simple channel model
with perfect channel knowledge and independent fading. Yet, it is important to
note that practical channels are often characterized both by fading
correlations and estimation errors. E.g., taking advantage of the fading
correlation was shown to reduce the channel estimation load and allow lower
feedback rates in [20, 21].Furthermore, [22] showed that the throughput in
cell-free MIMO is asymptotically unbounded in a correlated fading model that
satisfies specific conditions. Yet, such models are significantly more
complicated and so far, have not yielded closed-form expressions of the
achievable data rates. In contrast, in this work, we provide expressions for
the spectral efficiency that can be evaluated directly, without having to
resort to simulations of a large number of network realizations.

Note that channel estimation for large networks is an active research area
with significant recent progress (e.g., [23, 24] which use Bayesian and
machine-learning approaches). As channel estimation is constantly improving,
it is important to also understand the achievable system performance if
channel estimation errors become negligible.

We emphasize that the downlink scenario is especially challenging for analysis
due to statistical dependence between the actions of the BSs which are coupled
through the locations of nearby mobiles. This form of dependence is different
from the correlation between the fading coefficients as considered in works
such as [25, 16]. Thus, this work is the first to present asymptotic results
for the performance of the downlink in a cooperative network. Note that
several works have presented large network analysis but without closed-form
expressions (e.g.[8, 17]). Such works still require Monte-Carlo simulations to
obtain insight into system performance.

Analysis of spatially distributed systems with statistical dependence  induced
through spatial locations is usually very complicated. Thus, this work was
only enabled through the derivation of novel bounds on the second joint
moments of shot noises driven by marked Poisson Point processes (PPPs), where
the marks are allowed to be statistically dependent111An introduction to shot
noise processes and marked PPPs is given for example in [26] and [27].. These
novel bounds are especially useful for scenarios with weak dependence between
the marks, where the bounds become tight. In this work we use these bounds to
derive tight upper and lower bounds on the joint moments of the interference
and hence to prove the limited effect of the statistical dependence between
the transmitting base stations.

To emphasize the importance of our findings, we also present numerical results
based on Monte-Carlo simulations. These results show agreement with the
theoretical predictions. Moreover, they show that the theoretical predictions
are valuable even when the numbers of antennas per BS are only moderately
large.

In summary, the main contributions of this work are:

  * •

Closed-form expressions for the asymptotic performance in the downlink of
cooperative cellular networks.

  * •

A novel approach for asymptotic analysis of the downlink of cooperative
cellular networks that captures the effect of both noise and interference.

  * •

A proof that the asymptotic effect of each BS on its nearby mobiles depends
only on the network’s average transmitted power per BS. We used this finding
to derive a novel, low-complexity power allocation scheme for large networks.

  * •

Novel bounds on the second joint moments of shot noises driven by marked
Poisson Point processes (PPPs), where the marks are allowed to be
statistically dependent.

##  II Problem Statement

###  II-A Network and channel model

We consider the downlink in a cellular network. The network transmission is
done through BSs which are distributed according to a homogeneous Poisson
Point Process (HPPP), $\Phi_{b}$, with density $\lambda_{\rm b}$ BSs per unit
area. Each BSs has $L$ antennas and serves $M$ active mobiles, which are
located at a distance of at most $R$ from the BS. Hence, the area density of
mobiles is $\lambda=\lambda_{\rm b}M$. Each mobile has one antenna.

Without loss of generality, we analyze the performance of a _typical_ mobile
located at the origin. The typical mobile is labeled as mobile-0, and is
served by BS-0. BSs and mobiles are indexed separately and we denote by
$r_{i,k}$ the distance between mobile $i$ and BS $k$.

Let the BS that serves mobile $i$ be $b_{i}$ (and hence $r_{i,b_{i}}\leq R$).
Also, let $\mathcal{M}_{k}=\left\\{i:b_{i}=k\right\\}$ be the set of mobiles
served by BS $k$ (with $|\mathcal{M}_{k}|=M$). Note that given the existence
of the typical mobile at the origin, its serving BS must be within radius $R$
of the origin (i.e., $r_{0,0}\leq R$). All other BSs still form an HPPP with a
density of $\lambda_{\rm b}$.

Let $v_{i}$ be the symbol transmitted by BS $b_{i}$ to mobile-$i$ (with
$E[v_{i}]=0$ and $E[|v_{i}|^{2}]=1$) using a normalized beamforming vector
$\mathbf{w}_{i}$ (with $\|\mathbf{w}_{i}\|=1$). The specific form for
$\mathbf{w}_{i}$ is defined in Subsection II-B. For a specific symbol, the
sampled received signal at mobile-$j$ is

$\displaystyle
u_{j}=a_{b_{j}}\varphi_{j,{b_{j}}}\mathbf{h}_{j,{b_{j}}}^{\dagger}\mathbf{w}_{j}v_{j}+\sum_{i\neq
j}a_{b_{i}}\varphi_{i,b_{i}}\mathbf{h}_{j,b_{i}}^{\dagger}\mathbf{w}_{i}v_{i}+n_{j}$

where $\mathbf{h}_{j,k}$ is the (complex conjugate of the) channel between BS
$k$ and mobile $j$, $n_{j}\sim\mathcal{CN}(0,\sigma^{2})$ is the additive
Gaussian noise. $\mathcal{CN}(0,\sigma^{2})$ refers to a circularly symmetric,
zero mean, complex Gaussian random variable with variance $\sigma^{2}$. The
random variables $a_{k}\in\\{0,1\\}$ determine whether or not BS $k$ is
active, as detailed in Subsection II-B. $\varphi^{2}_{i,b_{i}}$ is the power
allocated by BS $b_{i}$ to mobile $i$. We enforce a maximum transmit power
constraint on each BS, such that for each $k$
$\sum_{i\in\mathcal{M}_{k}}\varphi^{2}_{i,k}\leq{P}$. The channel gain vector
is given by: $\mathbf{h}_{i,k}=\mathbf{g}_{i,k}r_{i,k}^{-\frac{\alpha}{2}}$
where $\alpha>2$ is the path loss exponent and $\mathbf{g}_{i,k}$ is the
fading vector between mobile $i$ and BS $k$, which has i.i.d.,
$\mathcal{CN}(0,1)$ entries.

Thus, the SINR for mobile $i$ is:

$\displaystyle\eta_{i}=\frac{\left|a_{b_{i}}\varphi_{i,{b_{i}}}\mathbf{h}_{i,{b_{i}}}^{\dagger}\mathbf{w}_{i}\right|^{2}}{\sum_{j\neq
i}\left|a_{b_{j}}\varphi_{j,b_{j}}\mathbf{h}_{i,b_{j}}^{\dagger}\mathbf{w}_{j}\right|^{2}+\sigma^{2}}.$
(1)

Assuming Gaussian codebooks are used to serve all mobiles, with long enough
blocks and system bandwidth $B$, the achievable rate for mobile $i$, denoted
by $R_{i}$ is

$\displaystyle R_{i}$ $\displaystyle=$ $\displaystyle
B\log_{2}(1+\eta_{i})\,.$ (2)

###  II-B Partial Zero Forcing

The network establishes partial cooperation, where user data is not shared
between BSs, but each BS obtains the channel state information (CSI) for all
neighboring mobiles. This CSI is used by each BS to perform zero-forcing on
all mobiles within a radius $D$ of the BS.222 In some studies, all cooperating
BS also share the user data and jointly transmit it to the user. Such schemes
require a higher level of cooperation between the BSs and hence are not
considered herein. We note that the analysis of such schemes can be done using
the same approach (and has the same complexity) but requires a longer
analysis. Following our comment above, we take the approach of [28] and focus
on characterizing the spectral efficiencies that can be achieved if accurate
channel estimation is possible.

As each BS has $L$ antennas, a proper choice of the precoding weight vector
can null the signal at $L-1$ mobiles. If the number of mobiles in the disk of
radius $D$ around a given BS exceeds $L$, then the BS cannot perform proper
transmission and ZF. Moreover, if the distance to a served mobile is greater
than $D$, the BS is capable of serving it if the number of users in the disk
of radius $D$ does not exceed $L-1$.

In all cases, no BS interferes with a mobile at a distance of less than $D$.
If the BS cannot serve a mobile without creating such interference, it has to
adopt an alternative transmission strategy. As the choice of active users is
beyond the scope of this paper, we consider here a simple scheme, in which a
BS transmits only if it can serve all of its users without violating the
interference constraint. Thus, a BS does not transmit at all if it has more
than $L$ users within a radius $D$, or if it has $L$ users within this radius
and at least one served mobile outside of this radius. Hence

$\displaystyle a_{k}=\begin{cases}1\text{ if
}(\left|\mathcal{D}_{k}\right|<L)\,\mbox{or}\,(\left|\mathcal{D}_{k}\cup\mathcal{M}_{k}\right|\leq
L),\\\ 0\text{ otherwise. }\end{cases}$ (3)

where $\mathcal{D}_{k}=\\{i:r_{i,k}<D\\}$ is the set of mobiles in the radius
$D$ disk around BS $k$. We will ultimately show that the probability of no
transmission ($a_{k}=0$) becomes negligible in the regime of interest, where
$L$ is large.

While we assume that the CSI at BS $k$ includes the channel vectors
$\mathbf{h}_{i,k}$ $\forall i\in\mathcal{D}_{k}\cup\mathcal{M}_{k}$, for
analysis purposes, it is more convenient to use only the fading vectors,
$\mathbf{g}_{i,k}$. Note that the resulting ZF precoder is identical whether
the BSs use $\mathbf{g}_{i,k}$ or $\mathbf{h}_{i,k}$ to construct the
precoders. Thus, for served user $i$ (with $b_{i}=k$) BS $k$ constructs the
interference channel matrix,
$\mathbf{G}_{\bar{i},k}\in\mathbb{C}^{L\times(|\mathcal{D}_{k}\setminus\\{i\\}|)}$
whose columns are the vectors $\mathbf{g}_{\ell,k}$ for all
$\ell\in\mathcal{D}_{k}\setminus\\{i\\}$. Then, the precoding weight vector
used by BS $k$ for user $i$ is given by:

$\displaystyle\mathbf{w}_{i,k}=\frac{\mathbf{Q}_{i,k}\mathbf{g}_{i,k}}{\|\mathbf{Q}_{i,k}\mathbf{g}_{i,k}\|}$
(4)

where
$\mathbf{Q}_{i,k}=\mathbf{I}-\mathbf{G}_{\bar{i},k}(\mathbf{G}_{\bar{i},k}^{H}\mathbf{G}_{\bar{i},k})^{-1}\mathbf{G}_{\bar{i},k}^{H}$
is a projection matrix onto the null space of $\mathbf{G}_{\bar{i},k}$.

##  III Asymptotic behavior of the SINR

In this section, we characterize the asymptotic limit of the typical mobile’s
SINR as the number of antennas increases. The analysis herein is enabled by
the following novel bounds on shot noise functions over marked PPP. The
uniqueness of these bounds is that they hold even if the marks are
statistically dependent. The bounds are presented as a theorem because they
are very general, and can be applied to various scenarios.

######  Theorem 1 (Bounds on the second joint moment of shot noise functions)

Let $\Phi$ be a marked PPP over $\mathbb{R}^{2}$ with density
$\lambda(r,\theta)$, where each point $(r_{i},\phi_{i})$ is associated with
two random marks $p_{i,1}$ and $p_{i,2}$. Also, let $I_{1}$ and $I_{2}$ be
shot noise functions of the form $I_{k}=\sum_{i}f_{k}(r_{i})\cdot p_{i,k}$ for
continuous functions $f_{k}(\cdot)$. If there exists three continuous
functions $q_{0}(r),q_{1}(r),q_{2}(r)$ such that

$\displaystyle
E\left[\left.p_{i,1}p_{j,2}\right|r_{i},r_{j}\right]\leq(\geq)\begin{cases}q_{1}(r_{i})q_{2}(r_{j})&i\neq
j\\\ q_{0}(r_{i})&i=j\end{cases}$ (5)

is satisfied for any two points with indexes $i$,$j$, then the second joint
moment of the shot noises is lower (upper) bounded by:

$\displaystyle E\left[I_{1}I_{2}\right]\leq(\geq)$
$\displaystyle\prod_{k=1}^{2}\left(2\pi\int_{0}^{\infty}f_{k}(r)q_{k}(r)\bar{\lambda}(r)rdr\right)+$
$\displaystyle
2\pi\int_{0}^{\infty}f_{1}(r)f_{2}(r)q_{0}(r)\bar{\lambda}(r)rdr$ (6)

where
$\bar{\lambda}(r)=\frac{1}{2\pi}\int_{0}^{2\pi}\lambda(r,\theta)d\theta$.

Proof: See Appendix A

Before we present our main result, we need to discuss the behavior of the ZF
radius, $D$, and of the noise variance $\sigma^{2}$. As described below the
SINR limit is less interesting if these quantities are constant. Hence we
allow both to be a function of $L$.

For the ZF radius, we set $D=s\cdot L^{\beta}$ (which includes constant $D$ as
a special case). We will show in Appendix B-A that if $\beta>0.5$ or if
$\beta=0.5$ and $s\geq 1/\sqrt{\pi\lambda_{b}M}$, then asymptotically, each BS
will try to zero more users than it has antennas. Thus the network will have a
zero asymptotic rate. Hence, the rest of the manuscript will focus on
$\beta<0.5$ or $\beta=0.5$ and $s^{2}\pi\lambda_{b}M<1$.

Typical network analysis assumes that the noise variance does not depend on
the number of antennas. Yet, such networks are asymptotically (as the number
of antennas grows large) noise limited. Thus, asymptotic analysis with
constant noise variance cannot capture the trade off between noise and
interference. In this work, we allow the noise to change as $\sigma^{2}=\mu
L^{-\zeta}$. Such a dependence allows us to derive asymptotic expressions that
characterize the interference limited regime, the noise limited regime, and
the joint effect of noise and interference. In Section VII we demonstrate that
the resulting asymptotic expressions capture the joint effect of interference
and noise even for finite networks where the number of antennas is moderately
large.

Denote by $\bar{P}$ the average transmit power of the BSs in the network. In
the following, we analyze systems when the BSs apply any power allocation as
long as the correlation of the total BS power on other BSs is decreasing with
distance. Practically, this would apply to any reasonable power allocation
scheme. We mathematically require that for any BS:

$\displaystyle\bar{P}(r_{0,k})\triangleq
E\left[\sum_{i\in{\mathcal{M}}_{k}}\varphi_{i,k}^{2}|a_{k}=1,r_{0,k}\right]=\bar{P}+\xi_{0,k}$
(7)

and for any two BSs, denoting their distance by $\tilde{r}_{k,\ell}$:

$\displaystyle
E\left[\sum_{i\in\mathcal{M}_{k}}\varphi_{i,k}^{2}\sum_{j\in\mathcal{M}_{\ell}}\varphi_{j,\ell}^{2}\Big{|}\Phi_{b},a_{k}=a_{\ell}=1\right]=\bar{P}^{2}+\tilde{\xi}_{k,\ell}$

where $|\xi_{0,k}|\leq\delta r_{0,k}^{-\gamma}$ and
$|\tilde{\xi}_{k,\ell}|\leq\min\\{P^{2},\delta\tilde{r}_{k,\ell}^{-\gamma}\\}$
with $\gamma>0$.

With these assumptions, the next theorem describes the asymptotic limit of
$\eta_{0}$, the SINR of the typical user:

###### Theorem 2

If $\zeta<\beta(\alpha-2)$, then

$\displaystyle\lim_{L\to\infty}L^{-1-\zeta}\eta_{0}=(1-\tilde{s}^{2}\pi\lambda_{b}M)\frac{\varphi_{0,0}^{2}r^{-\alpha}_{0,0}}{\mu}$
(8)

where $\tilde{s}=s$ if $\beta=0.5$ and zero if $\beta<0.5$. If
$\zeta\geq\beta(\alpha-2)$, then

$\displaystyle\lim_{L\to\infty}L^{-1-\beta(\alpha-2)}\eta_{0}=\frac{(1-\tilde{s}^{2}\pi\lambda_{b}M)\varphi_{0,0}^{2}r^{-\alpha}_{0,0}}{\frac{1}{s}\frac{2\pi\bar{P}\lambda_{b}}{\alpha-2}+\tilde{\mu}}$
(9)

where $\tilde{\mu}=\mu$ if $\zeta=\beta(\alpha-2)$ and zero otherwise. Both
limits hold in probability.

Proof: We show the convergence of appropriately scaled interference and signal
powers at the typical mobile, and then combine them to complete the proof.

Define the interference power at the typical mobile as

$\displaystyle I_{0}$
$\displaystyle=\sum_{i=1}^{\infty}\left|a_{b_{i}}\varphi_{i,b_{i}}\mathbf{h}_{0,b_{i}}^{\dagger}\mathbf{w}_{i}\right|^{2}$
(10)

The convergence of $I_{0}$ with appropriate normalization is given by the
following lemma:

###### Lemma 1

For any $s>0$ with $0<\beta<0.5$ or $0<s<1$ with $\beta=0.5$, we have in mean-
square error (MSE):

$\displaystyle\lim_{D\to\infty}D^{\alpha-2}I_{0}$
$\displaystyle=\frac{2\pi\bar{P}\lambda_{b}}{\alpha-2},$ (11)

and $\mbox{var}\left\\{D^{\alpha-2}I_{0}\right\\}=O(D^{4-2\alpha}).$

Proof: See Appendix B-A.

Next, we define the signal power at the typical mobile:

$\displaystyle S_{0}$
$\displaystyle=\left|\varphi_{0,0}\,a_{0}\,r_{0,0}^{-\alpha/2}\,\mathbf{g}^{\dagger}_{0,0}\,\frac{\mathbf{Q}_{0,0}\mathbf{g}_{0,0}}{\|\mathbf{Q}_{0,0}\mathbf{g}_{0,0}\|}\right|^{2}$
(12)

The convergence of $S_{0}$ is given by:

###### Lemma 2

The normalized signal power converges in probability as follows:

$\displaystyle\lim_{L\to\infty}\frac{S_{0}}{L-s^{2}L^{2\beta}\pi\lambda_{b}M}=\varphi_{0,0}^{2}r^{-\alpha}_{0,0}$
(13)

Proof: See Appendix B-B

Combining (11) with the noise scaling, $\sigma^{2}=\mu L^{-\zeta}$, and the ZF
radius scaling, $D=s\cdot L^{\beta}$, we see that if $\zeta<\beta(\alpha-2)$,
the noise plus interference satisfies:

$\displaystyle\lim_{L\to\infty}L^{\zeta}(I_{0}+\sigma^{2})$
$\displaystyle=\mu$ (14)

while if $\zeta\geq\beta(\alpha-2)$ the noise plus interference satisfies:

$\displaystyle\lim_{L\to\infty}L^{\beta(\alpha-2)}(I+\sigma^{2})$
$\displaystyle=\frac{1}{s}\frac{2\pi\bar{P}\lambda_{b}}{\alpha-2}+\tilde{\mu}.$
(15)

in MSE (and hence, in probability) where $\tilde{\mu}=\mu$ if
$\zeta=\beta(\alpha-2)$ and zero otherwise. Combining (14), (15) and (13)
proves convergence of (8) and (9) in probability. $\blacksquare$

The asymptotic expressions of Theorem 2 are very useful and give important
insights on the network behavior. They capture the effect of both noise and
interference, and allow the evaluation of the optimal ZF radius. It is
important to note the different scenarios described by the values of $\zeta$
and $\beta$ in the theorem. Recall that $\zeta$ determines the scaling of the
noise with the number of antennas. In the first scenario, if
$\zeta<\beta(\alpha-2)$ then the noise decreases too slowly and the network is
noise-dominated (this also includes the case of $\zeta=0$). Thus, the
performance in this case (12) are not affected by the interference. In the
second scenario, if $\zeta>\beta(\alpha-2)$ then the noise decreases too fast
and the network is interference-dominated ($\tilde{\mu}=0$ in (13)). Thus, the
most interesting scenario, where we see the effect of both noise and
interference, is when $\zeta=\beta(\alpha-2)$.

As for the scaling of the ZF radius, $\beta$, we already showed that if
$\beta>0.5$ then the ZF radius grows too fast and the network cannot handle
the needed nulls. The theorem shows that if $\beta<0.5$, the ZF radius growth
is slow enough such that it doesn’t burden the network at all ($\tilde{s}=0$).
Thus, optimal performance, which balances the effect of noise and
interference, is achieved when $\beta=0.5$ and $\tilde{s}=s$. Further
discussion on this optimal scenario, and on the achievable performance with a
finite number of antennas are given in the numerical results section, below.
Note that in the most interesting case, where $\beta=0.5$ (later shown to be
optimal) and $\zeta=\alpha/2-1$, the limit simplifies to:

$\displaystyle\lim_{L\to\infty}L^{-\alpha/2}\eta_{0}=\frac{(1-s^{2}\pi\lambda_{b}M)\varphi_{0,0}^{2}r^{-\alpha}_{0,0}}{\frac{1}{s}\frac{2\pi\bar{P}\lambda_{b}}{\alpha-2}+\mu}.$
(16)

This limiting expression for the SINR is particularly interesting as it
captures the effect of both the interference and the noise and is hence useful
for analyzing the SINR of practical systems with moderately large numbers of
antennas (for which interference is still significant). The theorem also shows
that the asymptotic SINR does not depend on the power allocation scheme, but
only on the average power per BS ($\bar{P}$). This insight significantly
simplifies the analysis of networks with different power allocation schemes.
Further, it allows a simple derivation of an asymptotically optimal power
allocation scheme for massive MIMO networks, as shown in Section VI.

##  IV Optimization of the ZF radius

Using Theorem 2 we can find the zero forcing radius, $D$, that maximizes the
throughput, i.e., to find the optimal values of $\beta$ and $s$. If
$\zeta<\beta(\alpha-2)$ the network is asymptotically noise limited. In this
case, the ZF has no effect and optimization trivially results in $D=0$ (or in
terms of (8), $\tilde{s}=0$). Hence, from hereon, we focus only on the case
that $\zeta\geq\beta(\alpha-2)$.

Inspecting (9), the SINR, $\eta_{0}$ scales as $L^{1+\beta(\alpha-2)}$.
Recalling that $\alpha>2$, the SINR is maximized when $\beta$ takes its
maximal value, that is $\beta=0.5$. For $\beta=0.5$, (16) shows that the SINR
scales as $L^{\alpha/2}$. To optimize the SINR, we need to maximize the RHS of
(16) with respect to $s$. Evaluating the derivative of the RHS of (16) with
respect to $s$ and equating to zero, we conclude that the optimal value,
$s^{*}$, must satisfy:

$\displaystyle\frac{\alpha}{\alpha-2}\pi\lambda_{b}s^{*2}+\frac{\mu}{\bar{P}}s^{*\alpha}=\frac{1}{M}.$
(17)

While the optimal expression has no general closed-from solution, it can
easily be solved numerically. For further insight, we can consider the no
noise case ($\mu=0$) which leads to:

$\displaystyle s^{*}=\sqrt{\frac{\alpha-2}{\alpha\pi\lambda_{b}M}}.$ (18)

Thus, for large enough $L$, the best nulling radius is

$\displaystyle D^{*}\approx\sqrt{\frac{\alpha-2}{\alpha\pi\lambda_{b}M}L}.$
(19)

Since for large $L$ and $D$, the average number of mobiles in a radius $D$
disk is approximately $D^{2}\pi\lambda_{b}M$, on average, each BS protects
approximately $(1-2/\alpha)L$ mobiles from interference.

##  V Throughput

The characterization of the asymptotic SINR of each mobile in Theorem 2 is the
main theoretic contribution of this manuscript. To better understand the
implication of these results, it is beneficial to consider the total
throughput of each BS in the network. For simplicity, this section focuses
only on the more interesting case where $\beta=0.5$ and
$\zeta=\beta(\alpha-2)=\alpha/2-1$.

The spectral efficiency of link $i$, $R_{i}$ is given by (2). The following
theorem uses (16) to characterize the asymptotic rate for user $i$ (recall
that mobile 0 is a typical mobile). It is important to note here that while
Theorem 2 shows convergence in probability of the normalized SINR, Corollary 1
shows convergence in MSE of the achievable rate. We need this stronger form of
convergence in order to make assertions on the mean rate in the subsequent
discussion.

###### Corollary 1

$R_{i}$ converges in MSE to:

$\displaystyle\tilde{R}_{i}=\log_{2}\Big{(}1+L^{\alpha/2}\frac{(1-s^{2}\pi\lambda_{b}M)\varphi^{2}_{i,b_{i}}r_{i,b_{i}}^{-\alpha}}{s^{2-\alpha}\frac{2\pi\bar{P}\lambda_{b}}{\alpha-2}+\mu}\Big{)}.$
(20)

That is,
$\lim_{L\to\infty}E\Big{[}\big{|}R_{i}-\tilde{R}_{i}\big{|}^{2}\Big{]}=0$.

###### Proof:

(20) holds in probability by applying the continuous mapping theorem, in a
manner similar to the proof of Theorem 2 in [9]. Proving convergence in MSE is
more challenging and is given in Section C. ∎

To further characterize the throughput, we need to limit the discussion and
assume that the mobile locations admit a probability distribution (while so
far we made no assumption on the mobiles except for their maximal distance
from their BS). Additionally, as the network is ergodic, we can characterize
the network throughput by the throughput of the characteristic BS (BS $0$).
Without loss of generality, we assume that BS $0$ serves mobiles
$0,\ldots,M-1$. Thus, the average throughput per BS is:

$\displaystyle
R_{\textrm{av}}=E\left[\sum_{m=0}^{M-1}\log_{2}(1+\eta_{m})\right].$ (21)

To make this derivation applicable also for finite SNRs, we define an
asymptotic approximation of the average throughput per BS:

$\displaystyle
R_{\textrm{as}}=E\left[\sum_{m=0}^{M-1}\log_{2}\left(1+\tilde{R}_{m}\right)\right].$
(22)

We note that (20) guarantees the convergence in MSE, hence
$\lim_{L\to\infty}\left(R-R_{\textrm{as}}\right)=0.$ Thus, for large enough
number of antennas, $L$, we can claim that $R\approx R_{\textrm{as}}$.

Fig 3. demonstrates the accuracy of the prediction in (22) for two different
mobile distributions (mobiles on the cell edge, and uniformly randomly
distributed within a cell). Markers show the simulated results while solid
lines show the theoretical result based on (22)333Closed form expressions for
the throughput in these scenarios can be easily derived, but are omitted here
due to space constraints. The cell radius is $R=0.15$km, $\alpha=3$, and the
transmit power is set such that the SNR for a mobile at the cell edge is 10 dB
from a single antenna.

Figure 1: Spectral efficiency vs. number of antennas, $L$, for cell-edge and
uniformly distributed mobiles.

##  VI Power allocation

In this section, we consider the power allocation to each mobile. We wish to
maximize

$\displaystyle
R_{\textrm{as}}=\lambda_{b}E\left[\sum_{m=0}^{M-1}\log_{2}\left(1+g\varphi^{2}_{0,m}r_{0,m}^{-\alpha}\right)\right],$
(23)

subject to a peak power constraint $\sum_{m=0}^{M-1}\varphi^{2}_{0,m}\leq P$,
where
$g=L^{\alpha/2}\frac{(1-s^{2}\pi\lambda_{b}M)}{s^{2-\alpha}\frac{2\pi\bar{P}\lambda_{b}}{\alpha-2}+\mu}$.

That is, we search for $M$ functions
$\varphi_{0,0}^{2}=f_{0}(\\{r_{0,M}\\}),\ldots,\varphi_{0,M-1}^{2}=f_{M-1}(\\{r_{0,M}\\})$
that maximize (23) subject to the peak power constraint
$0\leq\sum_{m=0}^{M-1}f_{m}(\\{r_{0,M}\\})\leq P$, where $\\{r_{0,M}\\}$ means
$r_{0,0},\ldots,r_{0,M-1}$. Note that $g$ depends on the power allocation
through the average transmission power
$\bar{P}=E\left[\sum_{m=0}^{M-1}\varphi^{2}_{0,m}\right]$.This is a calculus
of variations problem (or an optimal control problem). Theorem 3 shows that it
can be solved using a variation of the well known water filling algorithm.

###### Theorem 3

The maximum rate of (23), subject to the peak power constraint, is given by:

$\displaystyle\max_{0<\bar{P}\leq P}\breve{R}(\bar{P})$ (24)

where $\breve{R}(\bar{P})$ is the rate obtained using the power allocation
functions $f_{m}(\\{r_{0,M}\\})$ evaluated by:

  1. 1.

For a given $\breve{\lambda}$, define

$\displaystyle\breve{f}_{m}(\breve{\lambda},\\{r_{0,M}\\})=\max\left\\{0,\breve{\lambda}-\frac{1}{gr_{0,m}^{-\alpha}}\right\\}$
(25)

and

$\displaystyle\hat{P}(\breve{\lambda})=E\left[\min\left\\{P,\sum_{m=0}^{M-1}\breve{f}_{m}(\breve{\lambda},\\{r_{0,M}\\})\right\\}\right]$
(26)

  2. 2.

Choose $\breve{\lambda}$ such that $\hat{P}(\breve{\lambda})=\bar{P}$.

  3. 3.

If $\sum_{m=0}^{M-1}\breve{f}_{m}(\breve{\lambda},\\{r_{0,M}\\})\leq P$

     * •

Set $f_{m}(\\{r_{0,M}\\})=\breve{f}_{m}(\breve{\lambda},\\{r_{0,M}\\})$

else

     * •

Set

$\displaystyle
f_{m}(\\{r_{0,M}\\})=\max\left\\{0,\breve{\gamma}-\frac{1}{gr_{0,m}^{-\alpha}}\right\\}$
(27)

where $\breve{\gamma}$ is chosen such that
$\sum_{m=0}^{M-1}f_{m}(\\{r_{0,M}\\})=P$.

Furthermore, the optimization in (24) is convex, and the choices of the
constants $\breve{\lambda}$ and $\breve{\gamma}$ is over monotonic non-
decreasing functions. Thus, the optimization problem can be solved easily,
using one dimensional convex optimization over a power allocation scheme with
two water levels: a water level for the average BS power and a different water
level for each BS that is constrained to the maximal peak power. Note that
$\breve{R}(\bar{P})$ is the maximal average BS throughput subject to an
average power constraint $\bar{P}$.

###### Proof:

We start by constraining the optimization in (23) by the average BS power,
$\bar{P}$. Since $g$ depends on $\bar{P}$, the additional constraint
significantly simplifies the problem (as $g$ becomes constant). The resulting
optimization problem for a single BS given $\bar{P}$ is:

$\displaystyle\breve{R}(\bar{P})=\underset{f_{0},\ldots,f_{M-1}}{\max}:$
$\displaystyle\
E\left[\sum_{m=0}^{M-1}\log_{2}\left(1+gf_{m}r_{0,m}^{-\alpha}\right)\right]$
(28) $\displaystyle\text{Subject to}:$ $\displaystyle\
\sum_{m=0}^{M-1}f_{m}\leq P$ (29) $\displaystyle\
E\left[\sum_{m=0}^{M-1}f_{m}\right]=\bar{P}$ (30)

where we have suppressed dependence on $r_{0,0},\cdots,r_{0,M-1}$ to simplify
notation.

This is a convex maximization problem. The relevant variant of the
Karush–Kuhn–Tucker (KKT) conditions (see for example Theorem 9.4 in [29])
states that there exists scalar $\lambda$ and a function
$\gamma=\gamma(r_{0,0},\ldots,r_{0,M-1})\geq 0$ such that the optimal solution
must maximize

$\displaystyle
E\left[\sum_{m=0}^{M-1}\log_{2}\left(1+gf_{m}r_{0,m}^{-\alpha}\right)\right]$
$\displaystyle+E\left[\gamma\sum_{m=0}^{M-1}f_{m}\right]+\lambda
E\left[\sum_{m=0}^{M-1}f_{m}\right]$ (31)

with respect to $f_{m}$ and must satisfy

$\displaystyle E\left[\gamma\left(\sum_{m=0}^{M-1}f_{m}-P\right)\right]=0$
(32)

and $E\left[\sum_{m=0}^{M-1}f_{m}\right]=P.$ As we can choose $f_{m}$ for each
set of distances, the maximization of (VI) can be written as a set of
maximizations:

$\displaystyle\max_{\phi_{0},\ldots,\phi_{M-1}}$
$\displaystyle\sum_{m=0}^{M-1}\log_{2}\left(1+g\phi_{m}r_{0,m}^{-\alpha}\right)$
$\displaystyle+\gamma\left(\sum_{m=0}^{M-1}\phi_{m}-P\right)+\lambda\sum_{m=0}^{M-1}\phi_{m}$
(33)

for each realization of the mobile distances. Note that (32) indicates that
$\gamma\left(\sum_{m=0}^{M}\phi_{m}-P\right)=0$ for any mobile distances (up
to a set with zero probability). As long as we have a solution with
$\sum_{m=0}^{M}\phi_{m}<P$ we set $\gamma=0$ and say that the constraint is
not active. For these realizations we take the derivative of (VI) with respect
to $\phi_{\ell}$ and equate to zero:

$\displaystyle 0$
$\displaystyle=\frac{gr_{0,\ell}^{-\alpha}\log_{2}e}{1+g\phi_{\ell}r_{0,\ell}^{-\alpha}}+\lambda.$
(34)

Simplifying and defining also the water level
$\breve{\lambda}=-\frac{\log_{2}e}{\lambda}$ leads to (25).

When (25) leads to BS power that exceeds $P$, we have $\gamma>0$. In this
case, (32) states the BS transmits at its max power, i.e.,
$\sum_{m=1}^{M}\phi_{m}=P$. This results in the simpler optimization

$\displaystyle\max_{\phi_{0},\ldots,\phi_{M-1}:\sum_{m=1}^{M}\phi_{m}-P}\sum_{m=1}^{M}\log_{2}\left(1+g\phi_{m}r_{0,m}^{-\alpha}\right)$
(35)

which results in:

$\displaystyle\varphi^{2}_{\ell}=-\frac{\log_{2}e}{\gamma}-\frac{1}{gr_{0,m}^{-\alpha}}.$
(36)

The secondary water level (associated with the peak power constraint for a
specific realization) is then defined as
$\breve{\gamma}=-\frac{\log_{2}e}{\lambda}$ , and leads to (27).

The values of the water levels are determined to satisfy the average and peak
power constraint (recalling that we have a scalar water level,
$\breve{\lambda}$, for the average power constraint, and a separate water
level, $\breve{\gamma}=\breve{\gamma}(r_{0,0},\ldots,r_{0,M-1})$, for each BS
in which the peak power constraint is active. The monotonicity of the average
and peak powers on their respective water level is obvious from the problem
structure. ∎

##  VII Numerical results

In this section, we present numerical results that demonstrate our main
assertions on the convergence of the normalized SINR and spectral efficiency,
and show that localized power allocation (Theorem 3) is sufficient even when
the number of antennas per base station is only moderately large. For all
simulations, the base stations and mobiles were placed on a square area with
the edges wrapped around to reduce edge effects.

Figure 2: Histogram of the normalized SINR for fixed $\mu$. The vertical line
shows the normalized SINR from (38).

###  VII-A Normalized SINR

To demonstrate the convergence of the signal and interference powers, first
consider the normalized SINR of the mobiles in the network, where the SINR is
normalized by $L^{\frac{\alpha}{2}}$, the transmit power and the path loss
between each mobile and its serving base station. Thus, the normalized SINR
for mobile $i$ is defined as

$\displaystyle\underline{\eta}_{i}=\frac{1}{L^{\frac{\alpha}{2}}\phi_{i,b_{i}}r^{-\alpha}_{i,b_{i}}}\eta_{i}.$
(37)

Recall that the asymptotic analysis in the previous sections was made with an
effective noise power that was changing with the number of antennas $L$. This
is in contrast to the practical scenario where the noise power, $\sigma^{2}$,
is fixed. To demonstrate this difference, Fig. 2 presents the convergence of
the interference power for the case of fixed $\mu$, while Fig. 3 presents this
convergence for the case of fixed noise power ($\sigma^{2}$). For the sake of
practicality, all other simulations (i.e., except those depicted in Fig. 2)
used the latter approach with fixed noise power. Strictly speaking, the
asymptotic predictions of the previous section hold only for constant $\mu$
(i.e., when the noise variance changes with the number of antennas). In such
case, from (16):

$\displaystyle\lim_{L\to\infty}\underline{\eta}_{i}=\frac{1-s^{2}\pi\lambda_{b}M}{s^{2-\alpha}\frac{2\pi\bar{P}\lambda_{b}}{\alpha-2}+\mu}.$
(38)

Figure 3: Histogram of the normalized SINR for fixed $\sigma^{2}$. The
vertical lines show the normalized SINR from (39).

For the case of the normalized SINR, we simulated networks of $1000$ base
stations with density 30 BS/$km^{2}$, path loss exponnent of $\alpha=4$, $M=3$
active mobiles per base station and equal power allocated to each mobile. Each
configuration was simulated 10 times to generate sufficient data for the
histograms.

Fig. 2 depicts histograms of the normalized SINRs, (37), for a constant $\mu$
(and hence, noise power that decreases with $L$). The value of $\mu$ is such
that the SNR at the cell edge is 6dB for $L=25$ (i.e.,
$25^{\alpha/2-1}PR^{-\alpha}/\mu=6$dB). The figure shows that even for $L=25$,
almost all the simulated normalized SINRs are within $\pm 6$dB of the
asymptotic prediction given by the RHS of (38), which is indicated by the
vertical line. As the number of antennas increases, the normalized SINRs
concentrate around the predicted value. For $L=200$ antennas per base station,
almost all the simulated normalized SINRs are within $\pm 2$dB of the
predicted value. These results demonstrate that the normalized SINR does
indeed converge to the asymptotic prediction.

To derive theoretic predictions for the case of fixed noise power, we consider
a finite number of antennas, and set $\mu=\sigma^{2}L^{\frac{\alpha}{2}-1}$.
Hence, $\mu$ becomes a function of $L$ instead of being a constant. Yet, our
asymptotic results show that for every $\mu$ there exists a large enough $L$
for which :

$\displaystyle\underline{\eta}_{i}\approx\frac{1-s^{2}\pi\lambda_{b}M}{s^{2-\alpha}\frac{2\pi\bar{P}\lambda_{b}}{\alpha-2}+\sigma^{2}L^{\frac{\alpha}{2}-1}}.$
(39)

Simulations herein show that when $L$ is large enough, this approximation is
very accurate, and gives good predictions even for the case of fixed noise
power.

Fig. 3 depicts the histograms of the normalized SINRs for a fixed noise power,
$\sigma^{2}$. The value of $\sigma^{2}$ is set such that the SNR at the cell
edge was 6dB (i.e., $PR^{-\alpha}/\sigma^{2}=6$dB). The theoretical
predictions from (39) are shown as vertical lines. Note that in these
simulations $\mu$ increases with $L$, and hence the asymptotic normalized SINR
decreases with $L$. The convergence behavior is quite similar to the one
depicted in Fig. 2, with almost all of the simulated SINRs within $\pm 2$ dB
of the theoretical prediction for $L=200$. These simulations thus demonstrate
that the asymptotic predictions are useful when the number of antennas is
large, even when the noise power is constant.

###  VII-B Power Control

As noted in Section VI, the asymptotic results of this paper can be used to
find the power allocation that maximizes the spectral efficiency. Further,
most of the power allocation can be conducted on a per-base-station basis,
resulting in a significant reduction in system complexity.

To test the performance of the power allocation of Theorem 3 we compare its
performance with two other policies, namely constant power allocation, where
the transmit power for all mobiles is equal, and a power allocation policy
based on the weighted-minimum-mean-square-error (WMMSE) sum-spectral
efficiency maximizing algorithm from [30]. This latter algorithm is
independent of the specific beamforming algorithm, and is applied here with
the partial-zero-forcing approach. Note that the complexity of the latter
algorithm is very high for large networks.

Figure 4: Sum spectral efficiency per base station vs. $L$, for $M=3$ and
different power allocation approaches. $\Delta_{o,e}$, and $\Delta_{o,a}$
denote the percentage difference between the optimized and equal power
allocations, and optimized and asymptotic power allocations, respectively.

Fig. 4 shows the sum spectral efficiency per base station vs. the number of
antennas, $L$, for the three power allocation policies considered. We used a
base station density of 60 BS/$km^{2}$, $\alpha=3$, $M=3$ active mobiles per
cell, cell radius of $0.15km$ and a constant noise power such that the cell-
edge SNR ($PR^{-\alpha}/\sigma^{2}$) is $25dB$. Note that the WMMSE approach
has the highest sum spectral efficiency. At $L=10$, the WMMSE power allocation
achieves a throughput that is 29.4% higher than the throughput of the constant
power allocation, but only 7.4% higher compared to the suggested power
allocation (based on Theorem 3). Thus, our novel power allocation is useful
even for quite a small number of antennas, leading to throughput that is just
a few percent below optimal, while requiring much less computations. For
larger numbers of antennas, this difference diminishes. E.g. at $L=40$
antennas per BS, the discrepancy between the W-MMSE and our approach reduces
to 0.8%.

For reference, Fig. 4 also depicts the sum spectral efficiency with non-
cooperative zero forcing, where each BS performs zero forcing only to users
within its cell. In particular, the figure shows the performance with optimal
W-MMSE power control and with equal power allocation. Note that the non-
cooperative zero-forcing schemes perform considerably worse than the
cooperative schemes as the number of antennas grows large. For a smaller
number of antennas, the non-cooperative zero forcing system with W-MMSE power
control is competitive with the cooperative scheme. This shows that optimal
power allocation is especially important in non-cooperating networks. Yet, it
is no match for BS cooperation with a large number of antennas.

These results indicate that the simple power allocation policy of Theorem 3
results in a sum spectral efficiency that approaches that of a much more
computationally intensive power allocation policy, which requires the power
allocations to be optimized across all base stations. While our scheme was
derived using asymptotic limits, it is shown to be efficient even when the
number of antennas per BS is only moderately large (e.g., 10 or more).
Finally, we observe from Figure 4 that the power allocation policy of Theorem
3 does result in a substantial spectral efficiency increase over an equal
power allocation policy for a small number of antennas. This gain comes for
only a small increase in complexity (as most computations can be performed
locally using equations that only depend on system parameters and in-cell
path-losses).

##  VIII Summary and Conclusions

We have developed a method to analyze the SINR and spectral efficiency on the
downlink of cooperative cellular networks with multi-antenna base stations.
The base stations perform partial zero forcing, nulling the interference to
mobiles within a distance $D$ of them. This system is complicated to analyze
due to the dependence between the beam-forming vectors used by nearby base
stations since base stations also null users of other cells. Such dependence
between nodes in spatial point processes is known to be challenging to
analyze.

To handle this complexity, we derived novel bounds on marked Poisson point
processes with dependent marks. Combining these bounds with an asymptotic
analysis, we found a simple closed-form expression for an appropriately
normalized version of the SINR in the limit as the number of antennas per base
station grows large. This result is further used to derive asymptotic
expressions for the spectral efficiency on the downlink.

Further, these simple asymptotic expressions for the spectral efficiency are
used to derive a simple power allocation scheme which is optimal in the sense
of maximizing the mean spectral efficiency of a BS, if the asymptotic
expressions are taken as accurate. This simple power allocation algorithm
mostly requires base stations to perform power allocation locally, requiring
minimal coordination between base stations on power allocation. The findings
of this work are validated by Monte Carlo simulations which show that the
asymptotic expressions are useful even when the number of antennas per base
station is moderately large.

The asymptotic expressions we derived are simple and reveal the dependence of
the spectral efficiency on tangible system parameters, such as numbers of
antennas per base station, base station density, cell size and mobile
distribution. Therefore, the results of this work are useful for system design
and system optimization, and also allow better understanding of the network.

##  Appendix A Proof of Theorem 1

Consider a finite (large) radius $R$, and define
$I_{k}(R)=\sum_{i}f_{k}(r_{i})p_{i,k}1_{[r_{i}<R]}$. Also, consider a division
of the area to rings of width $\Delta=R/N$ and define an indicator function
$u_{n}(r)=1_{[(n-1)\Delta\leq r<n\Delta}$. Let $q_{k}[n]=\sup_{(n-1)\Delta\leq
r<n\Delta}q_{k}(r)$ (or $q_{k}[n]=\inf_{(n-1)\Delta\leq r<n\Delta}q_{k}(r)$
for an upper bound), and similarly, $f_{k}[n]=\sup_{(n-1)\Delta\leq
r<n\Delta}f_{k}(r)$ ($f_{k}[n]=\inf_{(n-1)\Delta\leq r<n\Delta}f_{k}(r)$). We
thus have:

$\displaystyle E\left[I_{1}(R)I_{2}(R)\right]$ (40)
$\displaystyle=\sum_{n_{1}=1}^{N}\sum_{n_{2}=1}^{N}E\Big{[}\sum_{i,j}f_{1}(r_{i})p_{i,1}u_{n_{1}}(r_{i})f_{2}(r_{j})p_{j,2}u_{n_{2}}(r_{j})\Big{]}$

Next, we apply the bounds in (5) which give:

$\displaystyle\sum_{i,j}E\Big{[}f_{1}(r_{i})p_{i,1}u_{n_{1}}(r_{i})f_{2}(r_{j})p_{j,2}u_{n_{2}}(r_{j})|r_{i},r_{j}\Big{]}$
(41)
$\displaystyle\leq(\geq)\sum_{i}E\Big{[}\sum_{i,j}f_{1}(r_{i})f_{2}(r_{i})q_{0}(r_{i})u_{n_{1}}(r_{i})|r_{i},r_{j}\Big{]}$
$\displaystyle+\\!\sum_{i\neq
j}\\!E\Big{[}f_{1}(r_{i})f_{2}(r_{j})u_{n_{1}}(r_{i})u_{n_{2}}(r_{j})q_{1}\\!(r_{i})q_{2}(r_{j})|r_{i},\\!r_{j}\Big{]}$

Applying the chain rule for expectations and using the quantized bounds
$q_{k}(r)\leq(\geq)q_{k}[n]$ and $f_{k}(r)\leq(\geq)f_{k}[n]$ for
$(n-1)\Delta\leq r<n\Delta$ results in:

$\displaystyle E\left[I_{1}(R)I_{2}(R)\right]$ $\displaystyle\leq(\geq)$
$\displaystyle\sum_{n_{1}=1}^{N}f_{1}[n_{1}]f_{2}[n_{1}]q_{0}[n_{1}]E\Big{[}\sum_{i}u_{n_{1}}(r_{i})\Big{]}$
$\displaystyle+\sum_{n_{1}=1}^{N}\sum_{n_{2}=1}^{N}f_{1}[n_{1}]f_{2}[n_{2}]q_{1}[n_{1}]q_{2}[n_{2}]$
$\displaystyle\cdot E\Big{[}\sum_{i\neq
j}u_{n_{1}}(r_{i})u_{n_{2}}(r_{j})\Big{]}$ (42)

Since $\sum_{i}u_{n}(r_{i})$, is a Poisson random variable with parameter
$\lambda_{n}^{N}=\int_{(n-1)\Delta}^{n\Delta}\int_{0}^{2\pi}r\lambda(r,\theta)d\theta
dr$, $E\big{[}\sum_{i}u_{n_{1}}(r_{i})\big{]}=\lambda_{n_{1}}^{N}$ and
$E\big{[}\sum_{i\neq
j}u_{n_{1}}(r_{i})u_{n_{2}}(r_{j})\big{]}=E\big{[}\sum_{i,j}u_{n_{1}}(r_{i})u_{n_{2}}(r_{j})\big{]}-E\big{[}\sum_{i}u_{n_{1}}(r_{i})u_{n_{2}}(r_{j})\big{]}=\lambda_{n_{1}}^{N}\lambda_{n_{2}}^{N}$.
Substituting in (A) we have:

$\displaystyle E\left[I_{1}(R)I_{2}(R)\right]\leq(\geq)$
$\displaystyle=\sum_{n_{1}=1}^{N}\sum_{n_{2}=1}^{N}f_{1}[n_{1}]q_{1}[n_{1}]f_{2}[n_{2}]q_{2}[n_{2}]\lambda_{n_{1}}^{N}\lambda_{n_{2}}^{N}$
$\displaystyle+\sum_{n_{1}=1}^{N}f_{1}[n_{1}]f_{2}[n_{1}]q_{0}[n_{1}]\lambda_{n_{1}}^{N}$
(43)

To conclude, we note that for any continuous function, $g(r)$, and any family
of sample sequences $r_{n,N}$ satisfying $(n-1)\frac{R}{N}\leq
r_{n,N}<n\frac{R}{N}$, we have

$\displaystyle\lim_{N\to\infty}\sum_{n=1}^{N}g(r_{n,N})\lambda_{n}^{N}$
$\displaystyle=\int_{0}^{R}\int_{0}^{2\pi}g(r)r\lambda(r,\theta)d\theta dr$
$\displaystyle=2\pi\int_{0}^{R}g(r)r\bar{\lambda}(r)dr.$ (44)

Thus,

$\displaystyle E\left[I_{1}(R)I_{2}(R)\right]$
$\displaystyle\leq(\geq)\prod_{k=1}^{2}\Big{(}2\pi\int_{0}^{R}f_{k}(r)q_{k}(r)r\bar{\lambda}(r)dr\Big{)}$
$\displaystyle+2\pi\int_{0}^{R}f_{1}(r)f_{2}(r)q_{0}(r)r\bar{\lambda}(r)dr.$

Taking the limit as $R\to\infty$ completes the proof.

##  Appendix B Proofs of Lemmas

###  B-A Proof of Lemma 1

Since for large enough $L$, $D>R$, without loss of generality, we only need to
consider the $D>R$ case. The interference power at the typical mobile is given
by

$\displaystyle I_{0}$
$\displaystyle=\sum_{k=1}^{\infty}a_{k}\sum_{i\in\mathcal{M}_{k}}\left|\varphi_{i,k}\mathbf{h}_{0,k}^{\dagger}\mathbf{w}_{i}\right|^{2}=\sum_{k=1}^{\infty}1_{\left\\{r_{0,k}>D\right\\}}r_{0,k}^{-\alpha}\tilde{a}_{k}.$

where

$\displaystyle\tilde{a}_{k}$
$\displaystyle=a_{k}\sum_{i\in\mathcal{M}_{k}}\left|\varphi_{i,k}r_{0,k}^{\alpha}\mathbf{h}_{0,k}^{\dagger}\mathbf{w}_{i}\right|^{2}=a_{k}\sum_{i\in\mathcal{M}_{k}}\left|\varphi_{i,k}\zeta_{i,k}\right|^{2}.$

is the effective transmitted power of BS $k$ to the typical mobile, and
$\zeta_{i,k}=\mathbf{g}_{0,k}^{\dagger}\mathbf{w}_{i}$. Since BSs that are
further than $D$ from the origin do not consider the typical mobile in their
beamforming design, their precoding vectors, $\mathbf{w}_{i}$
$\mathbf{g}_{0,k}$ are independent. Recalling also that
$\mathbf{g}_{0,k}\sim\mathcal{CN}(0,\mathbf{I})$ and $\|\mathbf{w}_{i}\|=1$ we
have that $\zeta_{i,k}\sim\mathcal{CN}(0,1)$ and are i.i.d. random variables
for all BSs outside of a disk of radius $D$ from the origin. For $r_{0,k}\leq
D$, $\tilde{a}_{k}$ is always multiplied by zero. Thus, we can assume that
$\zeta_{i,k}\sim\mathcal{CN}(0,1)$ and i.i.d. for any $k>0$.

To show convergence of the normalized interference in the mean-square sense,
we first find the limit of the expectation and then show that its variance
goes to zero.

####  B-A1 Expected Interference

From Prop. 2.4 in [31]:

$\displaystyle E[I_{0}]$
$\displaystyle=2\pi\int_{D}^{\infty}r^{1-\alpha}E[\tilde{a}_{k}|r_{0,k}=r]\lambda_{b}dr.$
(45)

To bound the expectation, we need upper and lower bounds (for $k>0$ and
$r_{0,k}>D$) on

$\displaystyle
E[\tilde{a}_{k}|r_{0,k}=r]=\Pr(a_{k}=1|r_{0,k}=r)\cdot\bar{P}(r)$ (46)

where we used the fact that $a_{k}\in\\{0,1\\}$ and $\bar{P}(r)$ was defined
in (7). We also define:

$\displaystyle\bar{a}(r)=\Pr(a_{k}=1|r_{0,k}=r)=E\left[a_{k}|r_{0,k}=r\right].$
(47)

Recall that $a_{k}=1$ if the number of mobiles in a distance less than $D$
from BS $k$ is at most $L$. As we have $M$ mobiles around each BS, $a_{k}=1$
if (but not only if) there are less than or equal to $L/M$ base stations
closer than distance $D+R$ to BS $k$. Since we condition on an active mobile
being at the origin, there must be one base station at a distance at most $R$
from the origin. This BS should be treated separately from the others. As a
worst case assumption, we will derive a bound by assuming that BS $0$ is at a
distance of less than $D+R$ from BS $k$. All other BSs are from a HPPP with
intensity $\lambda$. Thus, the number of BSs at a distance of at most $D+R$
from BS $k$ is a Poisson random variable with parameter
$\pi\lambda_{b}(D+R)^{2}$, and the probability that it is below $L/M-1$ is

$\displaystyle Q(\lfloor L/M\rfloor,\pi\lambda_{b}(D+R)^{2})$
$\displaystyle\quad\quad=e^{-\pi\lambda_{b}(D+R)^{2}}\sum_{i=0}^{\lfloor
L/M\rfloor-1}\frac{(\pi\lambda_{b}(D+R)^{2})^{i}}{i!}$ (48)

where $Q(a,z)=\Gamma(a,z)/\Gamma(a)$. Thus

$\displaystyle\bar{a}(r)\geq
Q\left(\left\lfloor{L}/{M}\right\rfloor,\pi\lambda_{b}(D+R)^{2}\right)$ (49)

Similarly, $a_{k}=1$ only if (but not necessarily if) there are less than
$L/M$ base stations closer than $D-R$ to BS $k$. For this bound we assume that
BS $0$ is far from BS $k$. Using again the upper regularized gamma function to
bound the probability that the number of BSs within a distance $D-R$ from BS
$k$ is smaller than $L/M$ we get:

$\displaystyle\bar{a}(r)\leq
Q\left(\left\lfloor\frac{L}{M}\right\rfloor+1,\pi\lambda_{b}(D-R)^{2}\right).$
(50)

As we assume that $\bar{P}(r)=\bar{P}+\xi(r)$ where $|\xi(r)|\leq\delta
r^{-\gamma}$ with $\gamma>0$, using (45), we have

$\displaystyle E[I]\geq$ $\displaystyle
2\pi\int_{D}^{\infty}r^{1-\alpha}Q\left(\left\lfloor\frac{L}{M}\right\rfloor,\pi\lambda_{b}(D+R)^{2}\right)$
$\displaystyle\cdot(\bar{P}+\xi(r))\lambda_{b}dr$ $\displaystyle\geq$
$\displaystyle
2\pi\lambda_{b}Q\left(\left\lfloor\frac{L}{M}\right\rfloor,\pi\lambda_{b}(D+R)^{2}\right)\frac{D^{2-\alpha}}{\alpha-2}$
$\displaystyle\cdot\left(\bar{P}-\delta
D^{-\gamma}\frac{\alpha-2}{\alpha+\gamma-2}\right)$ (51)

and on the other hand, using (50) we get:

$\displaystyle E[I]\leq$ $\displaystyle
2\pi\int_{D}^{\infty}r^{1-\alpha}Q\left(\left\lfloor\frac{L}{M}\right\rfloor+1,\pi\lambda_{b}(D+R)^{2}\right)$
$\displaystyle\cdot(\bar{P}+\xi(r))\lambda_{b}dr$ $\displaystyle\leq$
$\displaystyle
2\pi\lambda_{b}Q\left(\left\lfloor\frac{L}{M}\right\rfloor+1,\pi\lambda_{b}(D+R)^{2}\right)\frac{D^{2-\alpha}}{\alpha-2}$
$\displaystyle\cdot\left(\bar{P}+\delta
D^{-\gamma}\frac{\alpha-2}{\alpha+\gamma-2}\right).$ (52)

Note that both in (B-A1) and (B-A1) the second term in the parentheses $\to 0$
as $D\to\infty$.

We next consider the behavior of these expectations at the limit when
$L\to\infty$ and $D=s\cdot L^{\beta}$. Lemma 2 of [32], shows that
$\lim_{L\to\infty}Q(L,qL)$ converges to $1$ if $0\leq q<1$ and converges to
$0$ if $q\geq 1$. Additionally, since $Q(L,x)$ monotonically decreases with
$x$, this is easily extended to show that $\lim_{L\to\infty}Q(L,qL^{2\beta})$
converges to $1$ if $\beta<0.5$ and converges to $0$ if $\beta>0.5$. Hence,
for any $s$ and $B$ combination satisfying the assumptions of the lemma the
bounds in (49) and (50) become

$\displaystyle 1\leq\lim_{L\rightarrow\infty}\bar{a}(r)\leq 1.$ (53)

substituting in (B-A1) and (B-A1), we conclude that

$\displaystyle\lim_{L\to\infty}E[D^{\alpha-2}I]$
$\displaystyle=\frac{2\pi\bar{P}\lambda_{b}}{\alpha-2}.$ (54)

####  B-A2 Second Moment of the Interference

Next, to evaluate the variance of the interference, $I_{0}$, we bound the
second moment of $I_{0}$ using Theorem 1. We need to bound
$E[\tilde{a}_{k}\tilde{a}_{\ell}|r_{0,k},r_{0,\ell}]$ for $k>0$ and $\ell>0$.
Recalling that $a_{k}\in\\{0,1\\}$ we start with:

$\displaystyle E[\tilde{a}_{k}\tilde{a}_{\ell}|r_{0,k},r_{0,\ell}]\leq
E[\tilde{a}_{k}\tilde{a}_{\ell}|r_{0,k},r_{0,\ell},a_{k}=a_{\ell}=1]$

For the case that $k\neq\ell$, we use the fact that $\zeta_{i,k}$ are i.i.d.
$\mathcal{CN}(0,1)$ random variables. Thus

$\displaystyle E[\tilde{a}_{k}\tilde{a}_{\ell}|r_{0,k},r_{0,\ell}]$ (55)
$\displaystyle\quad\leq
E\left[\left.\sum_{i\in\mathcal{M}_{k}}\varphi_{i,k}^{2}\sum_{i\in\mathcal{M}_{\ell}}\varphi_{i,\ell}^{2}\right|r_{0,k},r_{0,\ell},a_{k}=a_{\ell}=1\right].$

This expectation is bounded by the following lemma.

###### Lemma 3

The correlation between the transmitted power of two base stations, given
their distances from the origin is upper bounded by

$\displaystyle
E\left[\sum_{i\in\mathcal{M}_{k}}\varphi_{i,k}^{2}\sum_{j\in\mathcal{M}_{\ell}}\varphi_{j,\ell}^{2}\Big{|}r_{0,k},r_{0,\ell},a\quad_{k}=a_{\ell}=1\right]$
$\displaystyle\quad\leq(\bar{P}+cr_{0,k}^{-\tilde{\gamma}/4})(\bar{P}+cr_{0,\ell}^{-\tilde{\gamma}/4})$
(56)

where $\tilde{\gamma}=\min\\{\gamma,0.5\\}$ and
$c^{2}=\delta+(1-\frac{\pi^{2}}{12})^{-1/2}P^{2}/\pi$.

###### Proof:

See Appendix B-C ∎

In the case that $k=\ell$,

$\displaystyle E[\tilde{a}_{k}^{2}|r_{0,k}]\leq
E\Bigg{[}\Big{(}\sum_{i\in\mathcal{M}_{k}}\varphi_{i,k}^{2}\left|\zeta_{i,k}\right|^{2}\Big{)}^{2}\Big{|}r_{0,k}\Bigg{]}$
(57)
$\displaystyle\quad=E\Big{[}\sum_{i\in\mathcal{M}_{k}}\varphi_{i,k}^{4}\left|\zeta_{i,k}\right|^{4}$
$\displaystyle\quad\
+\sum_{i\in\mathcal{M}_{k}}\sum_{j\in\mathcal{M}_{k},j\neq
i}\varphi_{i,k}^{2}\varphi_{j,k}^{2}\big{|}\zeta_{i,k}\big{|}^{2}\left|\zeta_{j,k}\right|^{2}\Big{|}r_{0,k}\Big{]}$
$\displaystyle\quad=E\left[\left.\sum_{i\in\mathcal{M}_{k}}\varphi_{i,k}^{4}+\sum_{i\in\mathcal{M}_{k}}\sum_{j\in\mathcal{M}_{k}}\varphi_{i,k}^{2}\varphi_{j,k}^{2}\right|r_{0,k}\right]$

The power constraint, $\sum_{i\in\mathcal{M}_{k}}\varphi_{i,k}^{2}\leq P$,
also induces $\sum_{i\in\mathcal{M}_{k}}\varphi_{i,k}^{4}\leq P^{2}$, and
hence we have:

$\displaystyle E[\tilde{a}_{k}^{2}|r_{0,k}]\leq 2{P}^{2}.$ (58)

Using Theorem 1 with $f_{1}(r)=f_{2}(r)=r^{-\alpha}\cdot 1_{\\{r>D\\}}$,
$p_{1,k}=p_{2,k}=\tilde{a}_{k}$, $q_{0}(r)=2{P}^{2}\cdot$ and
$q_{1}(r)=q_{2}(r)=(\bar{P}+cr^{-\tilde{\gamma}/4})$, yields

$\displaystyle E\left[I_{0}^{2}\right]\leq$
$\displaystyle\left(2\pi\lambda_{b}\int_{D}^{\infty}(\bar{P}+cr^{-\tilde{\gamma}/4})r^{-\alpha+1}dr\right)^{2}$
$\displaystyle+4\pi{P}^{2}\lambda_{b}\int_{D}^{\infty}r^{-2\alpha+1}dr$ (59)

Solving the integral and using also (B-A1) we have

$\displaystyle\mbox{var}\left\\{I_{0}\right\\}=E[I_{0}^{2}]-E^{2}[I_{0}]$ (60)
$\displaystyle\leq\left(\frac{2\pi\bar{P}\lambda_{b}D^{2-\alpha}}{\alpha-2}\right)^{2}\left(1+\frac{c(\alpha-2)D^{-\tilde{\gamma}/4}}{\bar{P}(\alpha+\tilde{\gamma}/4-2)}\right)^{2}$
$\displaystyle+\frac{4\pi{P}^{2}\lambda_{b}D^{2-2\alpha}}{2\alpha-2}-Q^{2}\left(\lfloor\frac{L}{M}\rfloor+1,\pi\lambda_{b}(D+R)^{2}\right)$
$\displaystyle\cdot\left(\frac{2\pi\lambda_{b}D^{2-\alpha}}{\alpha-2}\right)^{2}\left(\bar{P}-\delta
D^{-\gamma}\frac{\alpha-2}{\alpha+\gamma-2}\right)^{2}.$

With the nulling radius $D$ such that $s>0$ with $0<\beta<0.5$ or $0<s<1$ with
$\beta=0.5$, the limits in (53) and the fact that $\alpha>2$, we get for any
$\gamma>0$ that $\mbox{var}\left\\{D^{\alpha-2}I_{0}\right\\}$ scales as
$O(D^{4-2\alpha})$. So
$\lim_{D\to\infty}\mbox{var}\left\\{D^{\alpha-2}I_{0}\right\\}=0$ which
combined with (54) completes the proof.

###  B-B Proof of Lemma 2

$\mathbf{Q}_{0,0}$ is a projection matrix and thus is idempotent, and is also
Hermitian. Hence,
$\|\mathbf{Q}_{0,0}\mathbf{g}_{0,0}\|=\sqrt{\mathbf{g}_{0,0}^{\dagger}\mathbf{Q}_{0,0}\mathbf{g}_{0,0}}$.
Substituting in (12) and using also $a_{0}\in\\{0,1\\}$ gives

$\displaystyle
S_{0}=\varphi_{0,0}^{2}a_{0}r^{-\alpha}_{0,0}\mathbf{g}^{\dagger}_{0,0}\,\mathbf{Q}_{0,0}\mathbf{g}_{0,0}$
(61)

Note that given the number of zero-forced mobiles, $\sum_{j}1_{r_{j,0}<D}$,
the quantity
$2\cdot\mathbf{g}^{\dagger}_{0,0}\mathbf{Q}_{0,0}\mathbf{g}_{0,0}$ is a
$\chi^{2}$ random variable with $2(L-\sum_{j}1_{r_{j,0}<D})$ degrees of
freedom. Hence, $\mathbf{g}^{\dagger}_{0,0}\mathbf{Q}_{0,0}\mathbf{g}_{0,0}$
is the sum of $(L-\sum_{j}1_{r_{j,0}<D})$, unit mean, exponential random
variables. Recalling that $D=sL^{\beta}$, with $0<\beta\leq 0.5$, from the
strong law of large numbers we have with probability 1 that:

$\displaystyle\lim_{L\to\infty}\frac{\mathbf{g}^{\dagger}_{0,0}\mathbf{Q}_{0,0}\mathbf{g}_{0,0}}{L-\sum_{j}1_{r_{j,0}<D}}=1.$
(62)

As the BSs form a HPPP, and there are $M$ mobiles per base station, with
probability 1,

$\displaystyle\lim_{D\to\infty}\frac{1}{D^{2}}\sum_{j}1_{r_{j,0}<D}=\pi\lambda_{b}M$
(63)

Combining with (62):

$\displaystyle\lim_{L\to\infty}\frac{\mathbf{g}^{\dagger}_{0,0}\mathbf{Q}_{0,0}\mathbf{g}_{0,0}}{L-s^{2}L^{2\beta}\pi\lambda_{b}M}=1$
(64)

Substituting into (61), and recalling that $a_{0}\to 1$ as $L\to\infty$ in MSE
completes the proof.

###  B-C Proof of Lemma 3

For any two BSs, we assumed that

$\displaystyle C_{k,\ell}(\Phi_{b})$ $\displaystyle\triangleq
E\left[\sum_{i\in\mathcal{M}_{k}}\varphi_{i,k}^{2}\sum_{j\in\mathcal{M}_{\ell}}\varphi_{j,\ell}^{2}\Big{|}\Phi_{b},a_{k}=a_{\ell}=1\right]$
$\displaystyle=\bar{P}^{2}+\xi_{k,\ell}$ (65)

with $|\xi_{k,\ell}|\leq\min\\{P^{2},\delta\tilde{r}_{k,\ell}^{-\gamma}\\}$.
Removing the condition on all BS locations and conditioning only on $r_{0,k}$,
$r_{0,\ell}$:

$\displaystyle E$
$\displaystyle\left[\sum_{i\in\mathcal{M}_{k}}\varphi_{i,k}^{2}\sum_{j\in\mathcal{M}_{\ell}}\varphi_{j,\ell}^{2}\Big{|}r_{0,k},r_{0,\ell},a_{k}=a_{\ell}=1\right]$
$\displaystyle=E\left[E\left[C_{k,\ell}(\Phi_{b})\right]\Big{|}r_{0,k},r_{0,\ell},a_{k}=a_{\ell}=1\right]$
$\displaystyle\leq\bar{P}^{2}+E\left[|\xi_{k,\ell}|\Big{|}r_{0,k},r_{0,\ell}\right].$
(66)

Let $q=\sqrt{r_{0,k}r_{0,\ell}}.$ We have:

$\displaystyle E$
$\displaystyle\left[|\xi_{k,\ell}|\Big{|}r_{0,k},r_{0,\ell}\right]$
$\displaystyle=\Pr(\tilde{r}_{k,\ell}\leq
q)E\left[|\xi_{k,\ell}|\Big{|}r_{0,k},r_{0,\ell},\tilde{r}_{k,\ell}\leq
q\right]$
$\displaystyle+\Pr(\tilde{r}_{k,\ell}>q)E\left[|\xi_{k,\ell}|\Big{|}r_{0,k},r_{0,\ell},\tilde{r}_{k,\ell}>q\right]$
$\displaystyle\leq\Pr(\tilde{r}_{k,\ell}\leq q)P^{2}+\delta q^{-\gamma}.$ (67)

Let $\theta$ be the angle between the BSs as seen by the typical mobile, and
using the law of cosines:

$\displaystyle\tilde{r}_{k,\ell}=r_{0,k}^{2}+r_{0,\ell}^{2}-2r_{0,k}r_{0,\ell}cos\theta.$
(68)

Using (68), the probability that the distance between the two BSs is smaller
than $q$ is

$\displaystyle\Pr(\tilde{r}_{k,\ell}\leq q)$
$\displaystyle=\Pr\left(cos\theta\geq\frac{r_{0,k}^{2}+r_{0,\ell}^{2}-\sqrt{r_{0,k}r_{0,\ell}}}{2r_{0,k}r_{0,\ell}}\right)$
$\displaystyle\leq\Pr\left(1-cos\theta\leq\frac{\sqrt{r_{0,k}r_{0,\ell}}}{2r_{0,k}r_{0,\ell}}\right).$
(69)

Using also the inequality
$1-\cos\theta\geq\frac{\theta^{2}}{2}-\frac{\theta^{4}}{24}\geq(1-\frac{\pi^{2}}{12})\frac{\theta^{2}}{2}$
for $|\theta|\leq\pi$ we have

$\displaystyle\Pr(\tilde{r}_{k,\ell}\leq
q)\leq\Pr\left((1-\frac{\pi^{2}}{12})\frac{\theta^{2}}{2}\leq\frac{\sqrt{r_{0,k}r_{0,\ell}}}{2r_{0,k}r_{0,\ell}}\right)$
(70)
$\displaystyle\;\;\;\;\;\;=\Pr\left(|\theta|\leq(1-\frac{\pi^{2}}{12})^{-1/2}(r_{0,k}r_{0,\ell})^{-1/4}\right).$

Finally, since the network is homogenous, $\theta$ is uniformly distributed
over $[0,2\pi)$. Thus

$\displaystyle\Pr(\tilde{r}_{k,\ell}\leq
q)\leq\frac{1}{\pi}{\left(1-\frac{\pi^{2}}{12}\right)^{-1/2}(r_{0,k}r_{0,\ell})^{-1/4}}.$
(71)

Substituting in (B-C) and then in (B-C), we have:

$\displaystyle
E\left[\sum_{i\in\mathcal{M}_{k}}\varphi_{i,k}^{2}\sum_{j\in\mathcal{M}_{\ell}}\varphi_{j,\ell}^{2}\Big{|}r_{0,k},r_{0,\ell},a_{k}=a_{\ell}=1\right]$
$\displaystyle\quad\quad\leq\bar{P}^{2}+c^{2}\cdot(r_{0,k}r_{0,\ell})^{-\frac{\tilde{\gamma}}{2}}$
$\displaystyle\quad\quad\leq(\bar{P}+cr_{0,k}^{-\frac{\tilde{\gamma}}{4}})(\bar{P}+cr_{0,\ell}^{-\frac{\tilde{\gamma}}{4}}).$

where we used the lemma definitions for $c^{2}$and $\tilde{\gamma}$.

##  Appendix C Proof of Mean-Square Convergence of the Throughput

Let
$\breve{\eta}_{0}={\varphi_{0,0}^{2}r^{-\alpha}_{0,0}\mathbf{g}^{\dagger}_{0,0}\mathbf{g}_{0,0}}/({\mu
L^{1-\alpha/2}})$,
$H~{}=~{}\lim_{L\to\infty}L^{-\alpha/2}\eta_{0}=\frac{(1-\tilde{s}^{2}\pi\lambda_{b}M)\varphi_{0,0}^{2}r^{-\alpha}_{0,0}}{\frac{1}{s}\frac{2\pi\bar{P}\lambda_{b}}{\alpha-2}+\mu}$
and
$\breve{H}=\lim_{L\to\infty}L^{-\alpha/2}\breve{\eta}_{0}={\varphi_{0,0}^{2}r^{-\alpha}_{0,0}}/{\mu}$.
As $\mathbf{g}^{\dagger}_{0,0}\mathbf{g}_{0,0}$ is a $\chi^{2}$ random
variable with $2L$ degrees of freedom, $L^{-\alpha/2}\breve{\eta}_{0}$ has a
bounded third moment. Noting that $0\leq\eta_{0}\leq\breve{\eta}_{0}$, both
$L^{-\alpha/2}\breve{\eta}_{0}$ and $L^{-\alpha/2}\eta_{0}$ are uniformly
integrable random variables [33] and hence
$\lim_{L\to\infty}E\left[\left|L^{-\alpha/2}\eta_{0}-H\right|^{2}\right]=0$.

Let $\mathcal{Y}$ be $1$ if ${\eta_{0}\geq L^{\alpha/2}H/4}$, and $0$
otherwise, and consider
$\mathcal{E}=E[|\log_{2}(1+\eta_{0})-\log_{2}(1+L^{\alpha/2}H)|^{2}]$. We need
to prove that $\lim_{L\to\infty}\mathcal{E}=0$. We have:

$\displaystyle\mathcal{E}$ $\displaystyle=$ $\displaystyle
E[|\log_{2}(1+\eta_{0})-\log_{2}(1+L^{\alpha/2}H)|^{2}\cdot\mathcal{Y}]$
$\displaystyle+E[|\log_{2}(1+\eta_{0})-\log_{2}(1+L^{\alpha/2}H)|^{2}\cdot\bar{\mathcal{Y}}]$
$\displaystyle\leq$ $\displaystyle
E\left[\left|\log_{2}\left(\frac{\eta_{0}}{L^{\alpha/2}H}\right)\right|^{2}\mathcal{Y}\right]+E\left[\log_{2}^{2}(1+L^{\alpha/2}H)\bar{\mathcal{Y}}\right]$

The first term converges to zero because ${\eta_{0}}/{L^{\alpha/2}H}\to 1$ in
MSE and $|\log x|$ is continuous with bounded slope for $x>1/4$. The second
term is bound by
$\log_{2}^{2}(1+L^{\alpha/2}H)\cdot\Pr\left\\{\mathcal{Y}=0\right\\}$ where
the first term scales as $\log_{2}^{2}(L)$ and the second term can be bounded
by

$\displaystyle\Pr\left\\{\mathcal{Y}=0\right\\}\leq\Pr\left\\{a_{0}<1\right\\}$
(73)
$\displaystyle+\Pr\left\\{\frac{\mathbf{g}^{\dagger}_{0,0}\mathbf{Q}_{0,0}\mathbf{g}_{0,0}}{L}<\frac{1-\tilde{s}^{2}\pi\lambda_{b}M}{2}\right\\}$
$\displaystyle+\Pr\left\\{\frac{\sum_{i=1}^{\infty}\left|a_{b_{i}}\varphi_{i,b_{i}}\mathbf{h}_{0,b_{i}}^{\dagger}\mathbf{w}_{i}\right|^{2}+\mu}{2L^{1-\alpha/2}}>\frac{1}{s}\frac{2\pi\bar{P}\lambda_{b}}{\alpha-2}+\mu\right\\}$

Using the results derived above, it can be shown that all three terms in (73)
decay to zero, where the slowest term (the third) decays as $O(L^{2-\alpha})$.
This is still much faster than the $\log_{2}^{2}L$ term. Thus (C) converges to
zero, which completes the proof.

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