# Asymptotically Achieving Centralized Rate on the Decentralized Network MISO
Channel

Antonio Bazco-Nogueras  Antonio Bazco-Nogueras, Paul de Kerret,
David Gesbert, and Nicolas Gresset A. Bazco-Nogueras is with the Mitsubishi
Electric Research and Development Centre Europe, 35708 Rennes, France, and
also with the Communications Systems Department, Eurecom, 06410 Biot, France
(e-mail:bazco@eurecom.fr).P. de Kerret and D. Gesbert are with the
Communications Systems Department, Eurecom, 06410 Biot, France (e-mail:
<EMAIL_ADDRESS>gesbert@eurecom.fr). P. de Kerret and D. Gesbert
also acknowledge the support of the ERC 670896.N. Gresset is with the
Mitsubishi Electric Research and Development Centre Europe, 35708 Rennes,
France (e-mail: n.gresset@fr.merce.mee.com).

###### Abstract

In this paper, we analyze the high-SNR regime of the $M\times K$ Network MISO
channel in which each transmitter has access to a different channel
estimation, possibly with different accuracy. It has been recently shown that,
for some regimes, this setting attains the same Degrees-of-Freedom as the
ideal centralized setting with perfect CSI sharing, in which all the
transmitters are endowed with the best estimate available at any transmitter.
This surprising result is restricted by the limitations of the Degrees-of-
Freedom metric, as it only provides information about the slope of growth of
the capacity as a function of the SNR, without any insight about the possible
performance at a given SNR. In this work, we analyze the affine approximation
of the rate on the high-SNR regime for this decentralized Network MISO
setting, presenting the unexpected result that, for a regime of antenna
configurations, it is possible to asymptotically attain the same achievable
rate as in the ideal centralized scenario. Consequently, it is possible to
achieve the beam-forming gain of the ideal perfect-CSIT-sharing setting even
if only a subset of transmitters is endowed with accurate CSI. This outcome is
a consequence of the synergistic compromise between CSIT accuracy at the
transmitters and consistency between the locally-computed precoders. We
propose a precoding scheme achieving the previous result, which is built on an
uneven structure in which some transmitters reduce the accuracy of their own
precoding vector for sake of using transmission parameters that can be more
easily predicated by the other TXs.

##  I Introduction

###  I-A Cooperative Transmission

Multi-user cooperative networks and the extend of their theoretical
capabilities have been thoroughly analyzed in the literature[1, 2, 3, 4].
Initially, it was shown that cooperation can bring multiplicative gains under
ideal assumptions on the channel knowledge available at the communicating
nodes. As a matter of example, it is known that, under the assumption of
perfect Channel State Information (CSI), the setting in which $M$ transmitters
(TXs) jointly serve $K$ single-antenna users (RXs), the so-called Network MISO
channel, achieves a rate that scales as $\min(N_{T},K)$ times the rate of the
single-antenna point-to-point channel[5], where $N_{T}$ is the total number of
transmit antennas. Conversely, the $K\times K$ interfering channel can attain
a multiplexing gain of $K/2$[6]. This perfect CSI scenario has been profoundly
studied[7, 5, 8, 9]. Unfortunately, the assumption of perfect information is
not practical in most of the current wireless networks. Motivated by the
infeasibility of the previous assumption, the community has investigated
settings in which the information available at the communicating nodes does
not meet the perfect CSI assumption, such as scenarios where the information
available is imperfect [10, 11, 12, 13, 14], or delayed [15, 16, 17, 18].

Yet, even though the aforementioned works assumed an imperfect acquisition or
estimation of the CSI, all the cooperating nodes are assumed to share
_perfectly_ the same _imperfect_ information. However, current and upcoming
wireless networks characteristics make this assumption of _perfect sharing_
also impractical for many applications. This is due to, for example, the
proliferation of heterogeneous networks for which some of the nodes have a
wireless, fluctuating, or limited backhaul[19, 20], or Ultra-Reliable Low-
Latency Communication (URLLC) applications[21, 22], in which the perfect
sharing of the information would result in an intolerable delay. Settings in
which simple devices with low capabilities aim to communicate in a dense
environment, as in IoT applications, also fall into the use cases in which the
sharing of channel information is indispensable but challenging. This
evolution of different use cases boosts the interest of distributed
information settings, in which the information available at the communicating
nodes is not only imperfect but different from one node to another. This type
of settings can be formalized as the so-called Team Decision problems[23], in
which different agents aiming for the same goal attempt to cooperate in the
absence of perfect communication among them.

###  I-B Distributed CSIT Setting

There exists a great number of different distributed settings [24, 25, 26, 27,
28]. In particular, we focus in the Network MISO setting, in which $M$ TXs
jointly serve $K$ different RXs, with Distributed CSI at the TXs (CSIT). In
this scenario, every TX has access to the information symbols of every RX but
does not share the same CSI with the other TXs. This setting arises in
situations in which the data can be buffered of cached[29, 30], but the CSI
needs to be available with very small delay; for example, high mobility
scenarios and IoT (or V2X) networks with fast channel varying but low data
rate[31, 32].

The Network MISO with distributed CSIT has been analyzed in recent works.
Initially, it was shown in [33] that the $2\times 2$ single-antenna scenario
in which one TX has better knowledge of the full channel matrix that the other
TX achieves the same Degrees-of-Freedom (DoF) as the ideal centralized case in
which both TXs are endowed with the best CSI. The DoF metric, which will be
presented in the following section, is defined as the pre-logarithmic factor
of the capacity as function of the SNR[9], and it is also known as
multiplexing gain. The outcome of[33] derives from a master-slave type
precoding, named Active-Passive Zero-Forcing (AP-ZF), in which the TX whose
CSI is less accurate transmits with a fixed precoder. Although this result
could depend on the asymmetric structure of the setting, it has been extended
to more general settings. Indeed, it was shown in[34] that the Generalized
DoF111Denomination of DoF under the assumption that the channel strength
difference between the links does not vanish at high-SNR, see[9] for more
details. of the centralized setting with perfect CSI sharing is attained
whichever TX has the best estimation of every single channel coefficient. By
way of explanation, the pre-logarithmic factor of the centralized $2\times 2$
Network MISO is preserved as long as the estimate of a certain link is
available at one of the TXs. This comprises for example cases in which TXs
have only local CSI or in which each TX knows better the channel towards a
certain RX. The DoF analysis has been also applied to the $K\times K$
scenario, in which two main insights can be outlined: First, the optimal DoF
of the centralized setting with perfect CSIT sharing is also achieved for some
regimes of the $K\times K$ scenario[35]. Second, the quantization of the
information available at a certain TX can be beneficial in distributed
settings, inasmuch as it helps to transform the setting into a hierarchical
configuration in which the structure of the CSI allocation can be used to
increase the DoF[36].

These results provide some understanding on the resilience of cooperating
settings under information mismatches between different nodes. However, the
DoF metric is a limited metric because it only provides information about the
pre-log factor, not offering any information about the achievable rate at any
given SNR. For that reason, it is necessary to take a step beyond and analyze
the affine approximation of the rate at the high-SNR regime.

###  I-C Linear Approximation of Rate at High-SNR

Finding the fundamental limits of communication in distributed settings is a
challenging problem. Indeed, these limits have remained open even for several
important centralized cases. Nevertheless, it is possible to find significant
insights on those complex settings through rate approximations, which helps us
to move towards a better understanding of the behavior of complex wireless
networks.

$\small{{\mathcal{L}}}^{B}_{\infty}$$\small{{\mathcal{L}}}^{A}_{\infty}$zef
$\small\operatorname{DoF}_{\\!A}$zef
$\small\operatorname{DoF}_{\\!B}$$\operatorname{DoF}_{\\!A}=\operatorname{DoF}_{\\!B}$SNR
[dB]Rate [bits/s/Hz] Figure 1: Qualitative illustration of the affine
approximation of two different setting with the same DoF (slope) but different
rate offset
${{\mathcal{R}}}_{\infty}=\operatorname{DoF}{{\mathcal{L}}}_{\infty}$, and
hence different achieved rate.

A very useful metric that has been applied in the literature is the affine
approximation of the rate at high SNR, introduced in [37]. Following this
linear approximation, the rate can be written as

$\displaystyle R=\operatorname{DoF}\log_{2}(P)-{{\mathcal{R}}}_{\infty}+o(1),$
(1)

where $P$ denotes the SNR, $\operatorname{DoF}$ is the pre-logarithmic factor
(or Degrees-of-Freedom), and ${{\mathcal{R}}}_{\infty}$ is the _rate offset_
(or vertical offset). The approximation in (1) can also be written in terms of
the _power offset_ (horizontal offset) ${{\mathcal{L}}}_{\infty}$, where
${{\mathcal{R}}}_{\infty}=\operatorname{DoF}{{\mathcal{L}}}_{\infty}$. An
illustrative representation is shown in Fig. 1. The term
${{\mathcal{L}}}_{\infty}$ represents the zero-order term with respect to a
reference setting with the same slope but whose affine approximation
intersects the origin. These terms are defined as

$\displaystyle\operatorname{DoF}\triangleq\lim_{P\rightarrow\infty}\frac{R(P)}{\log_{2}(P)},\qquad\qquad{{\mathcal{R}}}_{\infty}\triangleq\lim_{P\rightarrow\infty}\operatorname{DoF}\log_{2}(P)-R(P),$
(2)

where $R(P)$ represents the rate as function of the SNR $P$. This measure has
already proved instrumental in several findings. In[3], Lozano _et al._
analyzed the multiple-antenna point-to-point scenario, revealing that some
system features that do not impact the DoF (as antenna correlation, fading…)
do considerably impact the zero-order term, affecting the performance of the
system at any possible SNR. In addition to exposing some limitations of the
DoF metric, [3] also revealed that the affine expansion offers appreciably
tight approximations also at medium-to-low SNR. This characterization has been
also established for the Broadcast Channel (BC) with perfect CSIT using Dirty-
Paper Coding and linear precoding [38], and for the BC with imperfect CSIT
[39]. In [39], the BC setting with quantized feedback was studied under the
assumption of Zero-Forcing (ZF) schemes, showing that the CSI-to-estimation
noise ratio must be proportional to $\operatorname{SNR}^{\alpha}$ in order to
attain a DoF per user of $\operatorname{DoF}_{\operatorname{RX}i}=\alpha$.
Furthermore, having a ratio of $\operatorname{SNR}^{\alpha}$ was shown to be
equivalent to obtaining a quantized feedback of
$\alpha\log_{2}(\operatorname{SNR})$ bits from the RX, which could be attained
if the feedback resources scale proportionally to $\log(\operatorname{SNR})$.

The aforementioned works are yet focused on the centralized scenario. For the
best of our knowledge, this affine approximation was analyzed for first time
for the Distributed CSIT setting in[40]. In that work, it was presented the
surprising result that the $2\times 2$ Network MISO with Distributed CSIT and
quantized feedback achieves the same rate offset as the ideal centralized
scenario of [39] in which the best CSI is shared among the two TXs, which
implies that both settings asymptotically achieve the same rate. Besides that,
it was also shown that the loss of performance at practical SNR values can be
dramatically reduced with the correct design of the transmission schemes.

###  I-D Main Contributions

Motivated by the result of[40] for the $2\times 2$ setting, we aim to
characterize in this work the achievable throughput of ZF precoding techniques
for the general $M\times K$ Network MISO setting with Distributed CSIT by
means of deriving its affine approximation. Our main contributions write as
follows:

  * •

We obtain the affine expansion of the rate achieved with ZF precoding on the
$M\times K$ Network MISO setting with Distributed CSIT. Focusing on the
antenna configurations where the centralized DoF can be achieved by ZF in the
Distributed CSIT setting, we prove that it possible to reach asymptotically
the same rate as in the ideal centralized setting in which the best estimate
available in the network is shared by all the TXs.

  * •

We present an achievable scheme that achieves the previous result by means of
capitalizing on the compromise between precoder accuracy and consistency among
transmitters. Specifically, we demonstrate that decreasing the accuracy of the
precoding at certain transmitters improves the average performance as it helps
to enhance the predictability of the transmission from the other TXs.

In addition to that, the techniques and approaches employed for the design of
the transmission scheme are believed to be worthwhile by themselves for
general decentralized problems, since they deal with the interplay between
local accuracy and global consistency, which is an inherent aspect of
decentralized and Team Decision problems.

##### Notations

${{\mathbb{N}}}_{N}$ stands for
${{\mathbb{N}}}_{N}\triangleq\\{1,2,\dots,N\\}$. We follow the asymptotic
notation presented in[41], based on the prevalent Bachmann–Landau
notation[42]. In consequence, $f(x)=o(g(x))$ implies that
$\lim_{x\rightarrow\infty}\\!\frac{\\!f(x)\\!}{g(x)}=0$,
$f(x)={{\mathcal{O}}}(g(x))$ implies than
$\limsup_{x\rightarrow\infty}\\!\frac{|f(x)|}{g(x)}<\infty$, and
$f(x)=\Theta(g(x))$ implies than
$\lim_{x\rightarrow\infty}\\!\frac{|f(x)|}{g(x)}=c$, $0<c<\infty$. For any
expected value $E$ and event $X$, $E_{|X}$ denotes the conditional expectation
given $X$. $\Pr(X)$ stands for the probability of the event $X$ and
$X^{\mathsf{c}}$ represents the complementary event to $X$. $\mathbf{A}_{i,k}$
or $(\mathbf{A})_{i,k}$ represent the element of the matrix $A$ located in the
$i$-th row and the $k$-th column. $\mathbf{I}_{n}$ writes for the identity
matrix of size $n\times n$. $\mathbf{1}_{M\times N}$ (resp.
$\mathbf{0}_{M\times N}$) represents the all-ones (resp. all-zeros) matrix of
size $M\times N$. $\lVert{\mathbf{A}}\rVert$ denotes the Frobenius norm of the
matrix $\mathbf{A}$.

##  II Problem Formulation

###  II-A Transmission and System Model

We consider the Network MISO setting in which $M$ multi-antenna TXs jointly
serve $K$ single-antenna RXs. We suppose that TX $j$ has $N_{j}$ antennas and
denote the total number of transmit antennas as $N_{T}=\sum_{j=1}^{M}N_{j}$.
The received signal is given by

$\displaystyle\mathbf{y}\triangleq\sqrt{P}\mathbf{H}\mathbf{W}\mathbf{s}+\mathbf{n},$
(3)

where $P$ is transmit power,
$\mathbf{y}\triangleq[\mathrm{y}_{1},\dots,\mathrm{y}_{K}]^{\mathrm{T}}$ is
the received signal vector and $\mathrm{y}_{i}$ is the received signal at RX
$i$. $\mathbf{n}$ stands for the Additive White Gaussian Noise (AWGN)
distributed as $\operatorname{\mathcal{N}_{\mathbb{C}}}(0,1)$, where
$\operatorname{\mathcal{N}_{\mathbb{C}}}(0,\Gamma)$ stands for the circularly-
symmetric complex normal distribution with covariance matrix (or variance)
$\Gamma$. The vector
$\mathbf{s}\triangleq[\mathrm{s}_{1},\dots,\mathrm{s}_{K}]^{\mathrm{T}}$ is
the vector of i.i.d. information symbols, where $s_{i}$ is the message to RX
$i$, that satisfies
$\mathbb{E}\\!\left[{\lVert{\mathbf{s}}\rVert^{2}}\right]=K$. The channel
matrix is given by

$\displaystyle\mathbf{H}\triangleq\begin{bmatrix}\mathbf{h}_{1}\\\ \vdots\\\
\mathbf{h}_{K}\end{bmatrix}\triangleq\begin{bmatrix}\mathbf{h}_{1,1}&\dots&\mathbf{h}_{1,M}\\\
\vdots&\ddots&\vdots\\\
\mathbf{h}_{K,1}&\dots&\mathbf{h}_{K,M}\end{bmatrix}\in{{\mathbb{C}}}^{K\times
N_{T}}.$ (4)

Hence, $\mathbf{h}_{i}\in{{\mathbb{C}}}^{1\times N_{T}}$ denotes the global
channel vector towards RX $i$ and $\mathbf{h}_{i,j}\in{{\mathbb{C}}}^{1\times
N_{j}}$ is the channel vector from TX $j$ to RX $i$. Note that we have defined
the row vectors as $\mathbf{h}_{i}$ and $\mathbf{h}_{i,j}$ in place of the
usual hermitian notation $\mathbf{h}^{{\mathrm{H}}}_{i}$ and
$\mathbf{h}^{{\mathrm{H}}}_{i,j}$. This is done so as to ease the notation for
the remaining of the document. The channel coefficients are assumed to be
i.i.d. as $\operatorname{\mathcal{N}_{\mathbb{C}}}(0,1)$ such that all the
channel sub-matrices are full rank with probability one. The precoding matrix
is given by

$\displaystyle\mathbf{W}\triangleq\begin{bmatrix}\mathbf{T}_{1}\\\ \vdots\\\
\mathbf{T}_{M}\end{bmatrix}\triangleq\mu\begin{bmatrix}\mathbf{w}_{1}&\dots&\mathbf{w}_{K}\end{bmatrix}\triangleq\mu\begin{bmatrix}\mathbf{w}_{1,1}&\dots&\mathbf{w}_{K,1}\\\
\vdots&\ddots&\vdots\\\
\mathbf{w}_{1,M}&\dots&\mathbf{w}_{K,M}\end{bmatrix}\in{{\mathbb{C}}}^{N_{T}\times
K}.$ (5)

Hence, $\mathbf{T}_{j}\in{{\mathbb{C}}}^{N_{j}\times K}$ is the precoding
matrix applied at TX $j$, $\mu\mathbf{w}_{i}\in{{\mathbb{C}}}^{N_{T}\times 1}$
is the global precoding vector for the information symbols of RX $i$
($s_{i}$), and $\mu\mathbf{w}_{i,j}\in{{\mathbb{C}}}^{N_{j}\times 1}$ is the
precoding vector applied at TX $j$ for $s_{i}$. We further define
$\mathbf{w}_{i,j,n}$ as the coefficient at the $n$-th antenna of TX $j$. The
parameter $0<\mu\leq 1$ is a power correction value that will be detailed
afterwards. We define $\mathbf{T}_{j,n}$ as the precoding vector applied at
the $n$-th antenna of TX $j$ for the vector $\bm{s}$, with
$n\in{{\mathbb{N}}}_{N_{j}}$. We assume that the precoder has a per-antenna
instantaneous unit-norm constraint, such that

$\displaystyle\lVert{\mathbf{T}_{j,n}}\rVert\leq 1.$ (6)

The results presented here also hold under the assumption of per-TX
instantaneous constraint ($\lVert{\mathbf{T}_{j}}\rVert\leq 1$). Note that,
even if we set $\lVert{\mathbf{T}_{j}}\rVert=1$, the transmit power varies
over the time as the power of the information symbols $s_{i}$ varies. For sake
of concision, we refer hereinafter to the unit-norm constraint of (6) as
_instantaneous power constraint_ , although strictly speaking it is an
_instantaneous power constraint on the precoding vector_. This is done in
opposition to the less restrictive average power constraint on the precoder
($\mathbb{E}\\!\left[\lVert{\mathbf{T}_{j}}\rVert^{2}\right]\leq 1$) which has
been assumed in other works [34, 35].

###  II-B Network MISO Setting with Distributed CSIT

The Distributed CSIT (D-CSIT) model is characterized by the consideration that
each TX is endowed with a possibly different estimate. The key singularity of
this setting is that, for any channel coefficient, there exist as many
estimates as TXs, each of them _locally available at a single TX_. By way of
example, we can think of a practical scenario in which there are some TXs with
more accurate knowledge for some channel coefficients and some other TXs with
more accurate information about other part of the channel matrix, while some
sort of coarse information could be shared between them.

Although the assumption of Distributed CSIT may seem contradictory with the
assumption of perfect sharing of user data symbols, which is inherent to the
Network MISO setting, both aspects co-exist in many scenarios of interest.
More specifically, this model is inspired by the different timescales of
latency that information data and CSI may experience in a range of emerging
applications. Indeed, CSI sharing is constrained by the channel coherence
time, which can be very short in mobility scenarios, and by the (possibly)
latency-limited backhaul connection; on the other hand, many data applications
have delivery time restrictions which are orders of magnitude weaker, such
that the data can be pre-fetched or cached at the TXs and ready to be
synchronously transmitted. We refer to[35] for a detailed discussion and
motivation on the joint transmission with distributed CSIT and the practical
scenarios in which it arises.

In this work, we assume that a limited cooperation between TXs had place
before the transmission phase, leading to a certain CSI accuracy
configuration. Hence, we assume hereinafter that the average CSIT accuracy at
each TX remains constant for a certain time. The problem of studying the best
strategy of CSI sharing in a limited and constrained communication is a very
interesting research problem per-se, and thus we do not discuss the exact CSI
acquisition mechanism.

Let us denote the estimate of the channel matrix $\mathbf{H}$ available at TX
$j$ as ${{\hat{\mathbf{H}}}}^{(j)}\in{{\mathbb{C}}}^{K\times N_{T}}$. Then, we
model the D-CSIT configuration such that ${{\hat{\mathbf{H}}}}^{(j)}$ is given
by

$\displaystyle{{\hat{\mathbf{H}}}}^{(j)}\triangleq\sqrt{1-Z^{{(j)}}}\
\mathbf{H}+\sqrt{Z^{(j)}}\ \bm{\Delta}^{(j)},$ (7)

where
$\bm{\Delta}^{(j)}\triangleq[\bm{\delta}^{(j)}_{1},\,\dots,\,\bm{\delta}^{(j)}_{K}]^{\mathrm{T}}$
is a random matrix that encloses the additive estimation noise whose elements
are i.i.d. as $\operatorname{\mathcal{N}_{\mathbb{C}}}(0,1)$ and independent
of $\mathbf{H}$, and $Z^{{(j)}}$ is a deterministic value that denotes the
variance scaling of the estimation noise with respect to the SNR. Importantly,
$Z^{{(j)}}$ characterizes the average accuracy of the estimate at TX $j$, that
is assumed to remain constant.

######  Remark 1.

In contrast to the precursory work in [40] for the $2\times 2$ setting, in
which a distributed adaptation of the Random Vector Quantization (RVQ)
feedback model of [39] was supposed, we assume in this work an additive
Gaussian model for the estimation noise. This modification allows us to
enlarge the contribution and verify that the previous result is not dependent
on the feedback model. Indeed, the asymptotic results are expected to hold for
a broad family of estimation models, since the key parameter that
characterizes the asymptotic analysis is the variance scaling of the noise.

The variance scaling $Z^{(j)}$ at a given TX is the same for all the channel
coefficients. Hence, each TX knows with the same accuracy the whole channel
matrix. This model encloses e.g. a scenario in which a main, multi or massive-
antenna base station serves a set of users with the help of some single or
multi-antenna remote radio-head or simple TXs, as depicted in Fig. 2. This
setting is denoted as the _Sorted CSIT setting_ , as the TXs can be sorted by
level of average accuracy. This simplification is not limiting, as we will see
later, because the precoding vectors are computed independently for each RX,
and only the normalization parameter depends on all the user’s vectors.
Importantly, it was shown in [40] that this normalization parameter can be
calculated with a less restricting accuracy without affecting the asymptotic
performance. Hence this assumption does not reduce generalization, as we can
extend the results to a setting where the channel vector of each RX is known
with high accuracy at different TXs. We restrict ourselves to the Sorted CSIT
setting for sake of readability and concision.

Figure 2: Master Base Station with remote radio-heads. The Base Station
obtains an estimate of the whole channel matrix, then it transmits noisy or
compressed CSI to the auxiliary TXs.

It is known that, for the centralized CSIT case in which all the transmit
antennas share the same CSI, the estimation signal-to-noise ratio should scale
as $P^{\alpha}$ in order to obtain a multiplexing gain of $\alpha$ [39, 43].
As our focus is on the high-SNR regime, we consider a high-SNR modeling in
which the estimation error scales such that

$\displaystyle Z^{(j)}={\bar{P}}^{-\alpha^{(j)}},$ (8)

where ${\bar{P}}\triangleq\sqrt{P}$ and $0\leq\alpha^{(j)}\leq 1$.
$\alpha^{(j)}$ is the _accuracy scaling_ parameter that measures the average
quality of estimation of the channel matrix at TX $j$. Hence, we can order the
TXs w.l.o.g. such that

$\displaystyle 1\geq\alpha^{(1)}>\alpha^{(2)}>\dots>\alpha{{}^{(M)}}\geq 0,$
(9)

what implies that TX $1$ is the _best-informed_ TX; in other words, the TX
whose CSIT has the highest average accuracy. We define the set of accuracy
parameters of the Distributed CSIT setting as

$\displaystyle{\bm{\alpha}}_{M}=\\{\alpha^{(j)}\\}_{j\in{{\mathbb{N}}}_{M}}.$
(10)

For further use, we define the estimate for the channel of RX $i$ as

$\displaystyle\hat{\mathbf{h}}^{(j)}_{i}\triangleq\breve{z}^{(j)}\mathbf{h}_{i}+z^{(j)}\
\bm{\delta}^{(j)}_{i},$ (11)

where $z^{(j)}\triangleq{\bar{P}}^{-\alpha^{(j)}}$,
$\breve{z}^{(j)}\triangleq\sqrt{1-(z^{(j)})^{2}}$, and
$\hat{\mathbf{h}}^{(j)}_{i}$, $\bm{\delta}^{(j)}_{i}$, are the $i$-th row of
the matrices ${{\hat{\mathbf{H}}}}^{(j)}$, $\bm{\Delta}^{(j)}$, respectively.
As stated before, the accuracy parameters ${\bm{\alpha}}_{M}$ are assumed to
be long-term coefficients that vary slowly. Based on that, it is assumed that
every TX knows the full set ${\bm{\alpha}}_{M}$, as it only requires a sharing
of few bits over a long period of time.

###  II-C Ideal Centralized Setting

Finding purely distributed upper-bounds is a challenging subject that remains
open, although some first results have been shown in[44]. However, any
decentralized scenario with distributed estimates has an _ideal_ centralized
counterpart in which a genie provides the best estimate of each parameter to
every node. Based on that, we define an ideal centralized scenario as follows.

######  Definition 1 (Ideal Centralized Setting).

Suppose the distributed setting introduced in Section II-B. The _Ideal
Centralized Setting_ is defined as the setting in which all the TXs are
endowed with the estimate of best average accuracy for every channel
coefficient.

Hereinafter, we denote the centralized channel estimate as
${{\hat{\mathbf{H}}}}$ and the estimate for the channel vector of RX $i$ as
$\hat{\mathbf{h}}_{i}$. We further denote the CSIT accuracy for the ideal
centralized case as $\alpha^{\star}$. Note that in the sorted setting, where
$\alpha^{(1)}>\dots>\alpha{{}^{(M)}}$, the ideal centralized setting is
equivalent to a MISO Broadcast Channel with $N_{T}$ transmit antennas, CSIT
${{\hat{\mathbf{H}}}}$ equal to ${{\hat{\mathbf{H}}}}^{(1)}$, and
$\alpha^{\star}=\alpha^{(1)}$.

We will compare the rate achieved in the distributed scenario described in the
previous section with the respective ideal centralized MISO BC counterpart.
This provides us with a benchmark for the performance of ZF schemes on the
Distributed CSIT setting. In this way, we are able to analyze which is the
impact of having distributed information or, in other words, _the cost of not
sharing the best CSI_.

######  Remark 2.

It is important to observe that the ideal centralized scenario is such that
every TX owns the best estimate among the available at any TX, _instead of_
its own estimate which by definition would have less accuracy. This is in
opposition to another genie-aided scenario, also assumed in the literature[34,
35], in which each TX shares its CSIT with any other TX, such that every TX
owns the set of M estimates of the M TXs. The former model, here assumed,
permits to compare the distributed scenario with the centralized counterpart.
The later, although it was shown in[35] that does not attain a greater DoF,
would benefit from the fact that the knowledge of $M$ estimates allows to
reduce the noise variance by a factor proportional to $M$. ∎

###  II-D Figure of Merit

Our figure of merit is the expected sum rate over the fading realizations. Let
us define the expected rate of RX $i$ as
$R_{i}\triangleq\mathbb{E}\\!\left[r_{i}\right]$, where $r_{i}$ is the
instantaneous rate of RX $i$. In our $K$-user setting, $r_{i}$ writes as

$r_{i}\triangleq\log_{2}\bigg{(}1+\frac{P|\mathbf{h}_{i}^{{{\mathrm{H}}}}\mathbf{t}_{i}|^{2}}{1+\sum_{j\neq
i}P|\mathbf{h}_{i}^{{{\mathrm{H}}}}\mathbf{t}_{j}|^{2}}\bigg{)},$ (12)

where $\mathbf{t}_{i}$ denotes the precoder vector for the symbols of RX $i$.
Then, the expected sum rate is given by $R\triangleq\sum_{i=1}^{K}R_{i}$.
Specifically, we study the linear approximation presented in (1) for the rate
of the Distributed CSIT setting. Let us denote the rate achieved in a
Distributed CSIT setting characterized by ${\bm{\alpha}}_{M}$ as
$R({\bm{\alpha}}_{M})$. Hence, we aim to find the values
$\operatorname{DoF}_{d}$, ${{\mathcal{R}}}^{d}_{\infty}$, such that

$\displaystyle
R({\bm{\alpha}})=\operatorname{DoF}_{d}\log_{2}(P)-{{\mathcal{R}}}^{d}_{\infty}+o(1).$
(13)

##  III Zero-Forcing Precoding

###  III-A Centralized Zero-Forcing Schemes (Under ideal CSIT sharing)

We restrict in this work to a general type of ZF precoders that we rigorously
characterize in the following. First, in order to distinguish when the
precoding vectors refer to the ideal centralized CSIT setting or the D-CSIT
one, we denote the centralized precoding coefficients as $\mathbf{v}_{i,k}$;
note that the counterpart vector for the D-CSIT setting in (5) is denoted by
$\mathbf{w}_{i,k}$. In addition, $\mathbf{V}$ and $\mathbf{v}_{i}$ are defined
as the centralized counterpart of $\mathbf{W}$ and $\mathbf{w}_{i}$,
respectively. Hence, the vectors $\mathbf{v}_{i}$ are computed from any ZF
precoding algorithm satisfying

$\displaystyle 1)~{}\qquad$
$\displaystyle\hat{\mathbf{h}}_{i}\mathbf{v}_{\ell}\ =\ 0,\quad\forall\ell\neq
i\qquad\qquad\text{(Zero-Forcing condition)}$ (ZF1) $\displaystyle
2)~{}\qquad$
$\displaystyle{\mathbb{E}}\left[{\lVert{\mathbf{v}_{i,j,n}}\rVert^{-1}}\right]\
=\Theta\left(1\right)\quad\qquad\ $ (ZF2) $\displaystyle 3)~{}\qquad$
$\displaystyle f_{\lVert{\mathbf{v}_{i}}\rVert}\leq
f^{\max}_{\lVert{\mathbf{v}_{i}}\rVert}<\infty$ (ZF3)

Note that (ZF1) is nothing but the condition that defines ZF schemes, (ZF2)
implies that the probability of precoding with a vanishing power is
negligible, and (ZF3) that the precoding vector has a bounded probability
density function (pdf), i.e., that it is neither predetermined nor constant.
Hereinafter, we assume that the centralized precoding scheme satisfies (ZF1),
(ZF2), (ZF3). Furthermore, we assume that the precoding vectors and matrices
can be expressed as a combination of summations, products, and generalized
inverses of the channel estimate. As an example, we can use the typical choice
of the projection of the matched filters onto the null spaces of the
interfered users, i.e.,

$\displaystyle\mathbf{v}_{i}=\lambda_{i}\frac{\mathbf{P}_{\mathbf{h}^{\bot}_{{\bar{i}}}}\hat{\mathbf{h}}^{{\mathrm{H}}}_{i}}{\lVert{\mathbf{P}_{\mathbf{h}^{\bot}_{{\bar{i}}}}\hat{\mathbf{h}}^{{\mathrm{H}}}_{i}}\rVert},\qquad\mathbf{P}_{\mathbf{h}^{\bot}_{{\bar{i}}}}\triangleq\left(\mathbf{I}_{N_{T}}-{{\hat{\mathbf{H}}}}^{{\mathrm{H}}}_{\bar{i}}({{\hat{\mathbf{H}}}}_{\bar{i}}{{\hat{\mathbf{H}}}}^{{\mathrm{H}}}_{\bar{i}})^{-1}{{\hat{\mathbf{H}}}}_{\bar{i}}\right),$
(14)

where the matrix ${{\hat{\mathbf{H}}}}_{\bar{i}}$ stands for the global
channel matrix with the $i$-th row removed, and $\lambda_{i}$ is a parameter
to satisfy the power constraint of (6). Note that the inversion in (14) can be
regularized in order to avoid degenerate cases and increase the performance at
low SNR. However, as conventional regularized schemes converge to their non-
regularized counterpart at high SNR, we omit any reference to regularized
inverses. We further model the precoding scheme as a function of the CSIT,
such that ${{\mathcal{V}}}$ denotes the function applied to the channel
estimate:

$\displaystyle{{\mathcal{V}}}:{{\mathbb{C}}}^{K\times
N_{T}}\rightarrow{{\mathbb{C}}}^{N_{T}\times
K}\qquad\text{and}\qquad\mathbf{V}={{\mathcal{V}}}\big{(}{{\hat{\mathbf{H}}}}\big{)}.$
(15)

###  III-B ZF on Distributed CSIT Settings

It is known that the performance of centralized ZF schemes collapses when they
are naively applied on Distributed CSIT settings[33, 35]. The main reason is
that the interference cancellation achieved through (ZF1) is proportional to
the worst accuracy among the TXs, $\alpha{{}^{(M)}}$ in the sorted case. Thus,
the question is how to prevent the least accurate TXs from harming the
transmission. Intuitively, the TXs should not act naively but take into
account the action of the other TXs. Let us mention two intuitive approaches
and their shortcomings to tackle this problem.

  1. 1.

The first intuitive idea is that the TXs whose CSIT inaccuracy is harming the
transmission should not exploit their instantaneous CSI to precode. Instead,
they could transmit with a fixed or known precoder based on statistical
information. This solution, coined Active-Passive ZF (AP-ZF), surprisingly
achieves the centralized DoF under the less restrictive average power
constraint $\mathbb{E}\\!\left[\lVert{\mathbf{T}_{j}}\rVert^{2}\right]\leq 1$
for the $2\times 2$ setting[34] and for some regimes of the $K\times K$
setting[35]. However, under the instantaneous power constraint
${\lVert{\mathbf{T}_{j,n}}\rVert^{2}}\leq 1$, which is the one that we
consider in this work, the best known performance requires a power back-off
such that the TXs with the best accuracy have enough power to realign the
interference generated by the TXs with fixed precoder. This strategy is flawed
in that the power back-off does not vanish at high SNR.

  2. 2.

Another possible strategy is that the most accurate TXs could estimate the
coefficients of the harming TXs and adjust their own precoder to reduce the
interference. This approach could prove efficient at intermediate SNR. Yet, at
high SNR, it will fail as the error variance of such estimate scales
proportionally to the accuracy of the CSI at the harming TXs.

We can see the two approaches as two extremes among the possible actions at
the least accurate TXs, in which they either disregard their CSIT or fully
trust on it. In this work we bridge the gap between the two strategies, what
will prove instrumental to achieve our main results. We present in the
following several definitions that help to emphasize the two main limitations
of distributed precoding.

######  Definition 2 (Consistency).

Consider two TXs, each endowed with a different CSI. Suppose that the precoder
of TX $1$ depends on the decision at TX $2$ such that we can write
$\mathbf{T}_{1}$ as $\mathbf{T}_{1}=f({{\hat{\mathbf{H}}}}^{(1)},\
g({{\hat{\mathbf{H}}}}^{(2)}))$, where $f,\,g,$ are generic functions. In a
D-CSIT setting, TX $1$ has not access to ${{\hat{\mathbf{H}}}}^{(2)}$. Let us
assume that TX $1$ estimates $g({{\hat{\mathbf{H}}}}^{(2)})$ from
${{\hat{\mathbf{H}}}}^{(1)}$ as $\hat{g}({{\hat{\mathbf{H}}}}^{(1)})$. Then,
the computation is said to be _Consistent_ if and only if
$\hat{g}({{\hat{\mathbf{H}}}}^{(1)})=g({{\hat{\mathbf{H}}}}^{(2)})$.
Otherwise, it is said to be _Inconsistent_.

######  Definition 3 (Power Outage).

Let TX $j$ compute a precoding matrix $\mathbf{T}_{j}$ such that, for a given
function $f$ and value A, the precoding matrix fulfills that
$f(\mathbf{T}_{j})=A$. Then, TX $j$ is said to be in _Power Outage_ if and
only if the precoder $\mathbf{T}_{j}$ obtained from computing
$f(\mathbf{T}_{j})=A$ exceeds the instantaneous power constraint.

In other words, the concept of _consistency_ in Definition 2 stresses the fact
that the TXs should precode in a coherent manner with respect to the actions
of the other TXs. In turn, the concept of _Power Outage_ reflects that even
with a perfect knowledge of the precoder applied at other TXs, a certain TX
may not be able to reduce the interference if the required power overpasses
the instantaneous power constraint. Throughout this document, we will be
interested on the cases in which the precoding is _both_ consistent and
_feasible_ :

######  Definition 4 (Feasible Consistency).

Two TXs apply a _Feasible Consistent_ precoder if the precoding coefficients
are _Consistent_ and there is not _Power Outage_.

One of our contributions is to show that these limitations can be overcome by
encouraging _consistency_ among the different TXs, at the cost of reducing the
accuracy of precoding at some TXs.

##  IV Main Results

Our main contributions rely on a novel ZF-type precoding scheme coined
_Consistent Decentralized ZF_ (CD-ZF), which is presented in detail in Section
IV-B. In short, this scheme is an adaptation to distributed scenarios of the
aforementioned centralized ZF precoding, such that the precoding applied at
each TX is different if the TX is the best informed one or not. Let
$R({\bm{\alpha}}_{M})$ be the expected sum rate for our D-CSIT setting.
Similarly, let $R^{\star}(\alpha^{(1)})$ be the expected sum rate achieved by
a Zero-Forcing scheme on the ideal centralized CSIT setting as described in
Section III-A. Accordingly, the rate gap between those settings is defined as

$\displaystyle{\Delta\mathrm{R}}\triangleq{R^{\star}(\alpha^{(1)})}-{R({\bm{\alpha}}_{M})}.$
(16)

We can now state our main result.

######  Theorem 1.

In the Network MISO setting with Distributed CSIT, with $N_{1}\geq K-1$ and
$\alpha{{}^{(M)}}>0$, the expected sum rate achieved by ZF-type schemes in the
ideal centralized CSIT setting is asymptotically achieved, i.e.,

$\displaystyle\lim_{P\rightarrow\infty}{R^{\star}(\alpha^{(1)})}-{R({\bm{\alpha}}_{M})}=0.$
(17)

###### Proof.

The proof is relegated to Section V, and it builds on the proposed CD-ZF
precoding scheme, which is presented in Section IV-B.∎

######  Corollary 1 (Rate-Offset under Distributed CSIT).

It holds from Theorem 1 that the rate offset ${{\mathcal{R}}}^{d}_{\infty}\\!$
(defined in (1)) of ZF with Distributed CSIT is the same as for the ideal
centralized setting, whose rate offset was shown in [39] to be constant for
the case of $\alpha^{\star}=1$ with respect to Perfect CSIT ZF—and thus with
respect to the capacity-achieving Dirty Paper Coding (DPC).

Remarkably, Theorem 1 implies that it is possible to achieve not only the
multiplexing gain but also the beamforming gain achieved by the ideal
$N_{T}$-antenna MISO BC, even if only $N_{1}$ antennas are endowed with the
maximum accuracy. The constraint $N_{1}\geq K-1$, i.e., that the TX with the
most accurate CSI has a number of antennas at least equal to the number of
interferable RXs, comes from the fact that if $N_{1}<K-1$ the use of only ZF
is not enough to achieve the DoF of the centralized setting[35], and thus
$\lim_{P\rightarrow\infty}{R^{\star}(\alpha^{(1)})}-{R({\bm{\alpha}}_{M})}=\infty$.
Despite that, it was shown in[35] that for certain regimes of the parameters
$\alpha^{(j)}$ it is possible to reach the ideal centralized DoF. This is
achieved by means of an elaborated transmission scheme which comprises
interference quantization and retransmission, superposition coding at the TXs,
and successive decoding at the RXs. Since in this work we focus in a simple ZF
transmission, we restrict to the DoF-achieving regime $N_{1}\geq K-1$.

It is known that the optimal DoF of the centralized CSIT setting with accuracy
$\alpha^{(1)}$ is $1+(K-1)\alpha^{(1)}$[43], which is attained by means of
superposition coding where a common message is broadcast and intended to be
decoded before the zero-forced messages. It is remarkable that, in the regime
of interest, $N_{1}\geq K-1$, the distributed CSIT setting performance still
converges asymptotically to the centralized performance even if superposition
coding is applied. This comes from the fact that the instantaneous power
applied converges to the one used in the centralized setting (as we prove at a
later stage), such that the common symbol broadcast can be sent with the same
rate.

###  IV-A Achievability: Some insights

Theorem 1 evidences that the issues associated which _feasible consistency_
between the TXs (enunciated in Section III-B) can be overtaken. Intuitively,
the strategies mentioned in Section III-B are the extremes cases of
consistency. Particularly, the naive ZF represents the extreme in which
consistency is not considered, whereas the Active-Passive ZF embodies the
extreme with perfect consistency but limited accuracy and possible Power
Outage, such that there may not be _feasible_ consistency. The block diagrams
of naive ZF and Active-Passive ZF are shown in Fig. 3(a) and Fig. 3(b),
respectively. The main question is whether a good compromise can be found.

We can build on the idea first presented in [40] for the simple $2\times 2$
setting, i.e., that discretizing the decision space of the TXs helps to
enforce consistency. Yet, the application of this idea is not straightforward,
as in [40] the only source of inconsistency was a single scalar power
parameter, and no beamforming was possible. Specifically, the strategy in [40]
was to design the ZF precoder such that the TX having worse channel estimate
for a certain user (i.e., smaller $\alpha^{(j)}$) precodes with a real value,
and thus it does not consider the possible phase of the coefficient. Then, the
other TX (the _best-informed_ one) fully controls the phase tuning necessary
to cancel the interference out, such that both TXs need to agree only on a
single value of transmit power—which must yet be agreed with high accuracy. In
the general $M\times K$ setting here considered, however, beamforming and
phase precoding must be applied at every TX equipped with more than one
antenna, what rules out the aforementioned strategy.

Nevertheless, the main insight is still valid: By means of discretizing the
decision space of the TXs that do not have the best CSI, we construct a
probabilistic hierarchical setting in which the best informed TX is able to
estimate correctly the action taken by the other TXs with a certain
probability. Interestingly, this discretization can be applied to either the
available information (the channel matrix) or the output parameters (the
precoding vector). Both cases are illustrated in Fig. 4. This flexibility is
due to the properties of linear systems and the asymptotic nature of our
analysis. It is however clear that the performance at low-to-medium SNR can
importantly differ for each of the cases. We focus in this document on the
scheme that quantizes the channel matrix for sake of a better understanding,
as the proof is less devious, and because the proof for the other case
(quantizing the precoder) follows the same approach and steps.

The key for attaining the surprising result of Theorem 1 is the proposed
precoding scheme, whose rigorous description is presented in the following
section. The proof of Theorem 1 relies on a simple idea: Let $A$ be a set
enclosing the _feasible consistent_ cases in which the precoders transmit
coordinately, and $A^{\mathsf{c}}$ its complementary event. Hence, the rate
gap
${\Delta\mathrm{R}}\triangleq{R^{\star}(\alpha^{(1)})}-{R({\bm{\alpha}}_{M})}$
can be expressed as

$\displaystyle{\Delta\mathrm{R}}=\Delta\mathrm{R}_{A}\Pr(A)+\Delta\mathrm{R}_{A^{\mathsf{c}}}\Pr(A^{\mathsf{c}}).$
(18)

The transmission scheme has to be both feasible consistent
($\Pr(A^{\mathsf{c}})\rightarrow 0$) and, for the consistent cases, it has to
be accurate ($\Delta\mathrm{R}_{A}\rightarrow 0$). Withal, it turns out that
these two conditions are not enough to obtain Theorem 1. In particular, we
need not only that the scheme is feasible consistent
($\Pr(A^{\mathsf{c}})\rightarrow 0$) but also that it attains consistency
_fast enough_ with respect to the CSI scaling, i.e., that
$\Delta\mathrm{R}_{A^{\mathsf{c}}}\Pr(A^{\mathsf{c}})\rightarrow 0$.

As a matter of example, consider that the rate gap for the inconsistent cases
$\Delta\mathrm{R}_{A^{\mathsf{c}}}$ scales proportionally to $\log(P)$ (as it
will proven later on) such that
$\Delta\mathrm{R}_{A^{\mathsf{c}}}=\Theta(\log(P))$. Then, if the probability
of inconsistency $\Pr(A^{\mathsf{c}})$ approached to $0$ at a convergence rate
proportional to $\frac{1}{\log(P)}$
($\Pr(A^{\mathsf{c}})=\Theta(\frac{1}{\log(P)})$), the term
$\Delta\mathrm{R}_{A^{\mathsf{c}}}\Pr(A^{\mathsf{c}})$ would not vanish and
Theorem 1 would not hold. Fortunately, we can design the achievable scheme so
as to reduce the probability of cases without consistency _and_ without power
outage faster than $\Theta(\frac{1}{\log(P)})$, _while providing an accurate
precoding and vanishing gap $\Delta\mathrm{R}_{A}$ in the consistent cases_.
This last condition is critical: As $P$ increases, the accuracy of the
precoder in the consistent cases must increase faster than
${{\mathcal{O}}}(P)$. Otherwise, $\Delta\mathrm{R}_{A}$ would not vanish. The
key to attaining sufficient accuracy for the consistent cases while reducing
at the same time the probability of inconsistent cases is that they depend on
different scaling of $P$ ($\Theta(\log(P))$ versus $\Theta(P)$). This is
rigorously shown in Section V.

TX 2 :${{\hat{\mathbf{H}}}}^{(2)}$ ZF
---
 Precoding
$\mathbf{w}_{1}^{{{\textbf{(2)}}}}$$\mathbf{w}_{2}^{{{\textbf{(2)}}}}$ Power
---
 Control
$\mathbf{T}^{(2)}_{2}$TX 1 :${{\hat{\mathbf{H}}}}^{(1)}$ ZF
---
 Precoding
$\mathbf{w}_{1}^{{{\textbf{(1)}}}}$$\mathbf{w}_{2}^{{{\textbf{(1)}}}}$ Power
---
 Control
$\mathbf{T}^{(1)}_{1}$ (a) Block diagram of conventional ZF applied naively in
the 2x2 Distributed CSIT scenario (No Consistency).

TX 2 :${{\hat{\mathbf{H}}}}^{(2)}$$\alpha^{(1)},\alpha^{(2)}$x  Power
Control
---
 (Constant)
$\mathbf{T}_{2}$TX 1 :${{\hat{\mathbf{H}}}}^{(1)}$ ZF
---
 Precoding
$\mathbf{w}_{1}^{{{\textbf{(1)}}}}$$\mathbf{w}_{2}^{{{\textbf{(1)}}}}$ Power
---
 Control
$\mathbf{T}^{(1)}_{1}$ (b) Block diagram of AP-ZF applied in the 2x2
Distributed CSIT scenario (Full Consistency but possible infeasibility).

Figure 3: Simple strategies for distributed precoding.

TX 2
:${{\hat{\mathbf{H}}}}^{(2)}$${{\mathcal{Q}}}(\cdot)$${{\hat{\mathbf{H}}}}^{(2)}_{q}$
 ZF
---
 Precoding
 Power
---
 Control
$\mathbf{T}^{(2)}_{2}$TX 1
:${{\hat{\mathbf{H}}}}^{(1)}$${{\mathcal{Q}}}(\cdot)$${{\hat{\mathbf{H}}}}^{(1)}_{q}$
 ZF
---
 Precoding
 Power
---
 Control
$\mathbf{T}^{(1)}_{1}$ContinuousDiscrete (a) Quantizing input (CSIT).

TX 2 :${{\hat{\mathbf{H}}}}^{(2)}$  ZF
---
 Precoding
 Power
---
 Control
$\mathbf{T}^{(2)}_{2}$${{\mathcal{Q}}}(\cdot)$$\mathbf{T}^{(2)}_{2q}$TX 1
:${{\hat{\mathbf{H}}}}^{(1)}$  ZF
---
 Precoding
 Power
---
 Control
$\mathbf{T}^{(1)}_{1}$${{\mathcal{Q}}}(\cdot)$$\mathbf{T}^{(1)}_{1q}$ContinuousDiscrete
(b) Quantizing output (precoder).

Figure 4: Two manners of discretizing decision space: At the input
(information) or at the output (action).

###  IV-B Proposed Transmission Scheme: Consistent Distributed ZF (CD-ZF)

We present in the following the Consistent Distributed ZF (CD-ZF) precoding
scheme, where the CSIT of the TXs that do not have access to the most accurate
estimate is quantized so as to improve the consistency of the scheme. Later,
we analyze the feasibility of the proposed precoder. The CD-ZF scheme presents
an uneven structure, such that each TX applies a different strategy depending
on who has higher average accuracy. Furthermore, the proposed scheme computes
independently the precoder for the symbols of different RXs, except for the
final power normalization. We recall that the precoding vector for the message
intended at RX $i$ is defined in (5) as
$\mu\mathbf{w}_{i}\in{{\mathbb{C}}}^{N_{T}\times 1}$, and the segment of
$\mu\mathbf{w}_{i}$ applied at TX $j$ is given by
$\mu\mathbf{w}_{i,j}\in{{\mathbb{C}}}^{N_{j}\times 1}$.

####  IV-B1 Quantizing the CSIT

The block diagram of this precoding scheme is depicted in Fig. 5. Because of
the uneven structure of the precoder, we describe separately the precoding
strategy at the best-informed TX (TX 1) and at any other TX (TX $2$ to TX
$M$). Let us consider first the later. The main limitation of the distributed
precoding is not the error variance at the restricting TXs, but the
impossibility at TX 1 of knowing what the other TXs are going to transmit. In
order to overtake this problem, all the TXs but TX 1 quantize their estimation
matrix with a known quantizer ${{\mathcal{Q}}}$. Hence, for any $j>1$, TX $j$
does not use directly its CSIT ${{\hat{\mathbf{H}}}}^{(j)}$ to precode, but
first pre-processes it. In other words, TX $j$ applies

$\displaystyle{{\hat{\mathbf{H}}}}^{(j)}_{q}={{\mathcal{Q}}}({{\hat{\mathbf{H}}}}^{(j)}).$
(19)

The characteristics of the quantizer ${{\mathcal{Q}}}$ will be detailed later.
Then, TX $j$ naively applies a centralized ZF scheme as described in Section
III-A but in a distributed manner (based on ${{\hat{\mathbf{H}}}}^{(j)}_{q}$).
Since the quantization transforms the continuous variable
${{\hat{\mathbf{H}}}}^{(j)}$ into a discrete one, it facilitates that the
setting becomes a hierarchical setting in which the information available at
other TXs is estimated without explicit communication.

TX 2 :TX 1
:${{\hat{\mathbf{H}}}}^{(1)}$$\text{MAP}({{\hat{\mathbf{H}}}}^{(2)}_{q})$${{\hat{\mathbf{H}}}}^{(2)\leftarrow(1)}_{q}$ZF
Precoding${{\mathbf{T}^{(1)}_{1}}}$${{\hat{\mathbf{H}}}}^{(2)}$${{\mathcal{Q}}}(\cdot)$${{\hat{\mathbf{H}}}}^{(2)}_{q}$ZF
Precoding${{\mathbf{T}^{(2)}_{2}}}$ContinuousDiscrete Figure 5: Block diagram
of Consistent Distributed ZF applied in the 2x2 Distributed CSIT scenario

We focus now on the precoder at the best-informed TX (TX 1), which attempts to
correct the error of the previous TXs. Let TX $1$ estimate
${{\hat{\mathbf{H}}}}^{(j)}_{q}$ based on its own information
${{\hat{\mathbf{H}}}}^{(1)}$, e.g. by computing the Maximum A Posteriori
estimator (MAP) of ${{\hat{\mathbf{H}}}}^{(j)}_{q}$:

$\displaystyle{{\hat{\mathbf{H}}}}^{(j)\leftarrow(1)}_{q}=\operatorname*{argmax}_{{{\hat{\mathbf{H}}}}^{(j)}_{q}\in{{\mathcal{Q}}}({{\mathbb{C}}}^{K\times
N_{T}})}\Pr\left({{\hat{\mathbf{H}}}}^{(j)}_{q}\mid{{\hat{\mathbf{H}}}}^{(1)}\right).$
(20)

It is important to notice that the quantized value
${{\hat{\mathbf{H}}}}^{(j)}_{q}$ is not intended to be transmitted, but it is
aimed at helping TX 1 to estimate the CSIT used at TX $j$, _without any
explicit communication between them_.

####  IV-B2 Canceling the interference

The CD-ZF scheme assumes naively that TX 1 correctly estimates the CSIT at TX
$j$, for any $j$ in ${{\mathbb{N}}}_{M}$, such that

$\displaystyle{{\hat{\mathbf{H}}}}^{(j)\leftarrow(1)}_{q}={{\hat{\mathbf{H}}}}^{(j)}_{q}.$
(21)

The probability that (21) holds true will be evaluated in the following
section. Therefore, the fact that (21) is fulfilled implies that the setting
is _consistent_ —although it does not guarantee feasibility, which will be
ensured by the parameter $\mu$. The goal of TX $1$ is to imitate the
interference cancellation performance that the centralized ZF scheme would
achieve if every other TX also owned ${{\hat{\mathbf{H}}}}^{(1)}$222We could
develop more convoluted schemes in order to increase the achievable rate at
low or medium SNR regimes. As a matter of example, we could design several
layers of quantization such that, for every $j$, TX $(j-1)$ tries to correct
the interference generated by TX $j$, in a similar manner to the algorithm
presented in [36]. Nevertheless, in this work we focus on the asymptotic
regime and thus maintain the scheme its simplest expression.. In order to
provide some insights, we first describe the single-antenna case with 2 TXs
and 2 RXs.

#####  $2\times 2$ case

Let ${\bar{i}}\triangleq i\pmod{2}+1$. Mathematically, the goal is to have
$|\smash{\hat{\mathbf{h}}^{(1)}_{i}\mathbf{w}_{{\bar{i}}}}|=|\smash{\hat{\mathbf{h}}^{(1)}_{i}\mathbf{v}_{{\bar{i}}}}|$,
what can be rewritten as

$\displaystyle|\smash{\hat{\mathbf{h}}^{(1)}_{i,1}\mathbf{w}_{{\bar{i}},1}+\hat{\mathbf{h}}^{(1)}_{i,2}\mathbf{w}_{{\bar{i}},2}}|=|\smash{\hat{\mathbf{h}}^{(1)}_{i,1}\mathbf{v}_{{\bar{i}},1}+\hat{\mathbf{h}}^{(1)}_{i,2}\mathbf{v}_{{\bar{i}},2}}|.$
(22)

We remind that $\mathbf{w}$ stands for the distributed precoder whereas
$\mathbf{v}$ stands for the centralized precoder. Under the assumption that TX
$1$ correctly estimates ${{\hat{\mathbf{H}}}}^{(2)}_{q}$, it knows
$\mathbf{w}_{{\bar{i}},2}$. Then, TX 1 computes its precoder such that

$\displaystyle\mathbf{w}_{{\bar{i}},1}$
$\displaystyle=\mathbf{v}_{{\bar{i}},1}+\underbrace{(\hat{\mathbf{h}}^{(1)}_{i,1})^{\dagger}\hat{\mathbf{h}}^{(1)}_{i,2}(\mathbf{v}_{{\bar{i}},2}-\mathbf{w}_{{\bar{i}},2})}_{\bm{\phi}_{{\bar{i}}}},$
(23)

where $(\mathbf{x})^{\dagger}$ denotes the pseudo-
inverse333$(\mathbf{x})^{\dagger}$ could also represent the regularized
pseudo-inverse. of $\mathbf{x}$, which is known to have minimal Frobenius norm
among all the generalized inverses [45]. The term $\bm{\phi}_{i}$ represents
the correction term that TX 1 has to apply in order to compensate the error
introduced by TX 2; note that (23) satisfies (22).

#####  $M\times K$ case

The generalization from the $2\times 2$ case to the general $M\times K$ case
needs one more step. Let TX 1 have $N_{1}\geq K-1$ antennas. The goal is again
to obtain the same interference cancellation as for the centralized precoder,
such that, for any RX $i\in{{\mathbb{N}}}_{K}$,

$\displaystyle\sum_{\ell\in{{\mathbb{N}}}_{K}\backslash
i}|\smash{\hat{\mathbf{h}}^{(1)}_{i}\mathbf{w}_{\ell}}|^{2}=\sum_{\ell\in{{\mathbb{N}}}_{K}\backslash
i}|\smash{\hat{\mathbf{h}}^{(1)}_{i}\mathbf{v}_{\ell}}|^{2}.$ (24)

The equality in (24) can be attained if
$\hat{\mathbf{h}}^{(1)}_{i}\mathbf{w}_{\ell}=\hat{\mathbf{h}}^{(1)}_{i}\mathbf{v}_{\ell}$,
for any $i,\ell\in{{\mathbb{N}}}_{K}$ such that $i\neq\ell$. Let us split the
precoding and channel vectors in two parts: $\mathbf{v}_{\ell,{1}}$,
$\mathbf{w}_{\ell,{1}}$ and $\hat{\mathbf{h}}^{(1)}_{i,{1}}$ denote the sub-
vector corresponding to the antennas of TX 1, whereas
$\mathbf{v}_{\ell,\bar{1}}$, $\mathbf{w}_{\ell,\bar{1}}$ and
$\hat{\mathbf{h}}^{(1)}_{i,\bar{1}}$ represent the sub-vector corresponding to
the antennas from TX 2 to TX $M$. The sub-matrices
${{\hat{\mathbf{H}}}}_{\bar{\ell},1}$ and
${{\hat{\mathbf{H}}}}_{\bar{\ell},\bar{1}}$ are defined in the same manner,
and they are illustrated in Fig. (6) for the sake of better comprehension.
Note that we have introduced the notation $N_{\bar{1}}\triangleq N_{T}-N_{1}$.
We can expand the condition
$\hat{\mathbf{h}}^{(1)}_{i}\mathbf{w}_{\ell}=\hat{\mathbf{h}}^{(1)}_{i}\mathbf{v}_{\ell}$
as a matrix equation in which $\mathbf{w}_{\ell,1}$ has to satisfy

$\displaystyle{{\hat{\mathbf{H}}}}_{\bar{\ell},1}\mathbf{w}_{\ell,1}={{\hat{\mathbf{H}}}}_{\bar{\ell},1}\mathbf{v}_{\ell,1}+{{\hat{\mathbf{H}}}}_{\bar{\ell},\bar{1}}(\mathbf{v}_{\ell,\bar{1}}-\mathbf{w}_{\ell,\bar{1}}).$
(25)

The precoding vector at TX $1$ is then selected as

$\displaystyle\mathbf{w}_{\ell,1}=\mathbf{v}_{\ell,1}+\underbrace{{{\hat{\mathbf{H}}}}_{\bar{\ell},1}^{\dagger}{{\hat{\mathbf{H}}}}_{\bar{\ell},\bar{1}}(\mathbf{v}_{\ell,\bar{1}}-\mathbf{w}_{\ell,\bar{1}})}_{\bm{\phi}_{\ell}},$
(26)

######  Remark 3.

The dimensionality of the linear system in (25) explains the restriction to
the case $N_{1}\geq K-1$ in Theorem 1.

In (25), it is ensured that the interference received is the same as for the
centralized setting. It is possible also to ensure that the receive signal
$\hat{\mathbf{h}}^{(1)}_{i}\mathbf{w}_{i}$ is equal to the one of the
centralized setting. This would add an extra equation to the linear system
that would require an extra antenna at TX 1. However, it is not necessary as
the received intended signal turns out to be statistically equivalent in both
distributed and centralized settings.

$\left(\;\begin{array}[]{cccc}\hat{\mathbf{h}}^{(1)}_{1,1}&\hat{\mathbf{h}}^{(1)}_{1,2}&\cdots&\hat{\mathbf{h}}^{(1)}_{1,M}\\\\[3.87498pt]
\vdots&\vdots&\ddots&\vdots\\\\[3.87498pt]
\hat{\mathbf{h}}^{(1)}_{\ell-1,1}&\hat{\mathbf{h}}^{(1)}_{\ell-1,2}&\cdots&\hat{\mathbf{h}}^{(1)}_{\ell-1,K}\\\\[11.62495pt]
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}\pgfsys<EMAIL_ADDRESS>\hat{\mathbf{h}}^{(1)}_{\ell+1,1}&\hat{\mathbf{h}}^{(1)}_{\ell+1,2}&\cdots&\hat{\mathbf{h}}^{(1)}_{\ell+1,K}\\\\[3.87498pt]
\vdots&\vdots&\ddots&\vdots\\\\[3.87498pt]
\hat{\mathbf{h}}^{(1)}_{K,1}&\hat{\mathbf{h}}^{(1)}_{K,2}&\cdots&\hat{\mathbf{h}}^{(1)}_{K,K}\end{array}\;\right)$
$\,{{\hat{\mathbf{H}}}}_{\bar{\ell},{1}}\in{{\mathbb{C}}}^{K-1\times
N_{1}}$$\,{{\hat{\mathbf{H}}}}_{\bar{\ell},\bar{1}}\in{{\mathbb{C}}}^{K-1\times
N_{\bar{1}}}$ Figure 6: Illustration of the channel sub-matrices
${{\hat{\mathbf{H}}}}_{\bar{\ell},1}$ and
${{\hat{\mathbf{H}}}}_{\bar{\ell},\bar{1}}$.

####  IV-B3 Feasibility and Consistency

In the previous description of the scheme, it has been assumed that TX 1
obtains a feasible consistent precoder, such that it correctly estimates the
CSI at the other TXs, and that the obtained precoding vector can be used for
transmission. In fact, the CD-ZF scheme naively considers that the precoder is
_always_ feasible and consistent.

However, the transmission scheme will suffer from the two main issues
described in Section III-B: Power outage, since it has to satisfy that
$\lVert{\mathbf{T}_{1,n}}\rVert\leq 1$ for any $n\in{{\mathbb{N}}}_{N_{1}}$,
and _Consistency_ , as the quantization of the CSI at TX $j$ allows TX 1 to
obtain the quantized CSI only with a certain probability. Further, as
explained in Section IV-A, we also need to demonstrate that the interference
cancellation is accurate enough in the feasible consistent cases. In the
following we present some properties that will prove instrumental in dealing
with those limitations. For that, we introduce a set of quantizers that are
essential in the proof of Theorem 1. We first tackle the probability of
inconsistency.

######  Definition 5 (Asymptotically Consistent Quantizers).

A quantizer ${{\mathcal{Q}}}$ is said to be _Asymptotically Consistent_ if the
probability of correct estimation of the MAP estimator at TX 1 satisfies

$\displaystyle\Pr\left({{\hat{\mathbf{H}}}}^{(j)\leftarrow(1)}_{q}\neq{{\hat{\mathbf{H}}}}^{(j)}_{q}\right)=o\left(\frac{1}{\log_{2}(P)}\right)\\!\\!\,,\quad\forall
j\in{{\mathbb{N}}}_{M}.$ (P1)

Property (P1) implies that it is possible to induce that the probability of
having _inconsistent_ precoding among TXs vanishes faster than
$1/{\log_{2}(P)}$. This fact implies that the rate impact of inconsistent
precoding events vanishes asymptotically, as mentioned in Section IV-A.
Clearly, it remains to prove that there exists some quantizer
${{\mathcal{Q}}}_{c}$ satisfying (P1) while also ensuring feasibility and
achieving a sufficient accuracy for the consistent cases. Surprisingly, very
simple quantizers as the scalar uniform quantizer satisfy the requirements. To
prove this statement, we first show that this quantizer is an Asymptotically
Consistent Quantizer.

######  Lemma 1.

Let ${{\mathcal{Q}}}_{u}(\mathbf{X})$ be a scalar uniform quantizer with
quantization step $q={\bar{P}}^{-\alpha_{q}}$, where $\alpha_{q}$ is such that
$\alpha^{(j)}>\alpha_{q}>0$, for all $j\in{{\mathbb{N}}}_{M}$. Then,
${{\mathcal{Q}}}_{u}$ is an _Asymptotically Consistent Quantizer_ and

$\displaystyle\Pr\left({{\hat{\mathbf{H}}}}^{(j)\leftarrow(1)}_{q}\neq{{\hat{\mathbf{H}}}}^{(j)}_{q}\right)=o\left(\frac{1}{\log_{2}(P)}\right),\quad\forall
j\in{{\mathbb{N}}}_{M}.$ (27)

###### Proof.

The proof is relegated to Appendix E. ∎

Note that ${{\mathcal{Q}}}_{u}$ is a scalar quantizer. Thus, for a matrix
$\mathbf{A}$, the notation $\mathbf{A}_{q}={{\mathcal{Q}}}_{u}(\mathbf{A})$
denotes with an abuse of notation that $\mathbf{A}_{q}$ is composed of the
independent scalar quantization of the real and imaginary part of each element
in $\mathbf{A}$. Obviously, using vector quantization would improve the
performance. However, as the proof of Theorem 1 is constructive, we are
interested in an intuitive example, and thus we use ${{\mathcal{Q}}}_{u}$ for
the sake of simplicity. The analysis at medium and low SNR, which requires an
optimization on the applied quantizer, is an interesting research topic that
is however out of the scope of this work.

We focus now on the probability of _power outage_. Let us denote the event of
power outage as ${{\mathcal{P}}}_{\\!\\!o}$ and recall that the precoder at
the $n$-th antenna of TX $j$ is expressed by $\mathbf{T}_{j,n}$. In that case,

$\displaystyle{{\mathcal{P}}}_{\\!\\!o}\triangleq\Big{\\{}\bigcup_{\begin{subarray}{c}n\in{{\mathbb{N}}}_{N_{j}}\\\
j\in{{\mathbb{N}}}_{M}\end{subarray}}\lVert{\mathbf{T}_{j,n}}\rVert>1\Big{\\}}.$
(28)

The power outage probability is handled by the parameter $\mu$ of the
precoding vector $\mu\mathbf{w}_{i}\in{{\mathbb{C}}}^{N_{T}\times 1}$. The
value of $\mu$ acts as power back-off that can be tuned so as to achieve the
required scaling, as stated in the following lemma.

######  Lemma 2.

Let $\mu=1-\varepsilon$, where $\varepsilon>0,\
\varepsilon\in\Theta({\bar{P}}^{-\alpha_{\mu}})$, and
$\alpha_{\mu}<\alpha_{q}$. Then,

$\displaystyle\Pr\left({{\mathcal{P}}}_{\\!\\!o}\right)=o\left(\frac{1}{\log_{2}(P)}\right).$
(P2)

###### Proof.

See Appendix C ∎

Similarly to property (P1), property (P2) in Lemma 2 implies that _power
outage_ events are negligible in terms of asymptotic rate. The only TX that
may incur in power outage is TX $1$, as the other TXs apply the naive
centralized precoder and hence they will always satisfy the power constraint.

To this extent, we have shown that the uniform scalar quantizer
${{\mathcal{Q}}}_{u}$ enables us to reach the requirements regarding the
probability of not feasible consistent cases. Hence, it remains to prove that
it also attains high enough accuracy in the feasible consistent cases. This
will be proven in Section V. Hereinafter, we assume that the uniform quantizer
of Lemma 1 is applied in the CD-ZF.

###  IV-C Hierarchical CSIT Setting

Theorem 1 shows that it is possible to attain asymptotically the rate of the
centralized setting. Its performance at low-to-medium SNR is however limited
by the probability of obtaining a _feasible consistent_ precoder. This
probability depends on the quantizer applied, the power back-off considered
and the values of $\alpha^{(j)}$, and hence it is challenging to obtain. As
shown in Section IV-B, the precoder is computed assuming a correct estimation
of the CSI at the other TXs. Consequently, if the probability of _consistency_
is low, the scheme does not perform properly. Moreover, this probability
decreases as the network size increase, since TX 1 needs to estimate correctly
more parameters.

This limitation is inherent to the D-CSIT setting here assumed, in which each
TX only knows its own CSI. However, there exist another practical setting with
distributed CSI but in which there is more structure in the network CSI: The
Hierarchical CSIT setting (H-CSIT). In this setting, each TX is endowed with
its own multi-user CSI ${{\hat{\mathbf{H}}}}^{(j)}$, as in the D-CSIT setting,
but it is also endowed with the CSI of the TXs having less accuracy than
itself. Namely, in the sorted CSI scenario with
$\alpha^{(1)}>\dots>\alpha{{}^{(M)}}$, TX $j$ has access to
$\\{{{\hat{\mathbf{H}}}}^{(j)},{{\hat{\mathbf{H}}}}^{(j+1)},\dots,{{\hat{\mathbf{H}}}}{{}^{(M)}}\\}$.

This scenario, although it may seem less practical, may arise in many
heterogeneous networks. Fig. 2 depicts an example: Suppose that the RXs are
all connected to the same main TX (e.g. TX 1), and the other TXs are remote
radio-heads that receive a coarse version of the CSI by means of a wireless
link from TX 1. In this use case, TX 1 will know the CSI available at each
other TX. If the CSI sharing is done through dedicated links for each TX, each
TX would receive CSI with accuracy proportional to its own link. If the CSI is
broadcast, they may obtain an estimate with different accuracy if layered
encoding [46] or analog feedback [47] is used.

######  Corollary 2.

Theorem 1 also holds in the Hierarchical CSIT setting and hence
$\lim_{P\rightarrow\infty}{R^{\star}(\alpha^{(1)})}-{R({\bm{\alpha}}_{M})}=0$.

###### Proof.

The proof follows directly from the proof of Theorem 1 in Section V. ∎

In this setting, TX 1 already knows ${{\hat{\mathbf{H}}}}^{(j)}$, for any
$j\in{{\mathbb{N}}}_{M}$. Hence, the discretization of the variables at the
other TXs is not needed, and the precoders are _consistent_ with probability
1. In fact, the idea of quantizing the CSI in the D-CSIT setting comes down to
make the CSIT setting asymptotically hierarchical with a probability high
enough. The only effect that may restrain TX 1 to achieve the centralized
performance is the power outage. Therefore, the performance at medium SNR will
improve with respect to the general Distributed CSIT case, and moreover, this
performance is not affected by the size of the network, as we will see in the
numerical examples of Section VI.

##  V Proof of Theorem 1

In order to prove Theorem 1, we need to demonstrate that the user rate gap
$\Delta\mathrm{R}_{i}=R^{\star}_{i}(\alpha^{(1)})-R_{i}({\bm{\alpha}}_{M})$
vanishes, from what it directly follows that
$\Delta\mathrm{R}=\sum_{i\in{{\mathbb{N}}}_{K}}\Delta\mathrm{R}_{i}$ will also
vanish. The proof is divided in several steps: First, armed with Lemma 1 and
Lemma 2, we show that both _power outage_ and _inconsistent precoding_ events
can be made negligible in terms of rate loss. Then, we prove that the rate gap
vanishes also in the feasible consistent cases thanks to the fine accuracy of
the CD-ZF scheme in these cases. We demonstrate this by showing that both the
interference received and the total power received in the distributed setting
converge to their counterparts of the centralized setting.

###  V-A Neglecting Non-consistent Events

The proof of Theorem 1 builds on Lemma 1 and Lemma 2. Indeed, the proposed
scheme will perform poorly in the cases in which the precoder is not feasible
consistent, as the scheme is built on the naive assumption that it is _always_
feasible and consistent. Nevertheless, both Lemma 1 and Lemma 2 illustrate
that those events can be made very unlikely (In particular, the probability of
these events is $o(\frac{1}{\log(P)})$). Let ${\mathcal{H}}_{\neq}$ denote the
set of _inconsistent_ events, i.e.,
${\mathcal{H}}_{\neq}\triangleq\big{\\{}\bigcup_{2\leq j\leq
M}{{\hat{\mathbf{H}}}}^{(j)\leftarrow(1)}_{q}\neq{{\hat{\mathbf{H}}}}^{(j)}_{q}\big{\\}}$.
Hence, the probability of having _feasible consistent_ precoding is
$\Pr\left({{\mathcal{P}}}_{\\!\\!o}^{\mathsf{c}}\cap{\mathcal{H}}_{\neq}^{\mathsf{c}}\right)$.
By means of the law of total expectation, we can split the expected rate gap
for RX $i$ $\Delta\mathrm{R}_{i}$ as444Given a certain feasible event $A$ and
its complementary event $A^{\mathsf{c}}$, the law of total expectation states
that
$\operatorname{{\mathbb{E}}}[X]=\Pr(A)\operatorname{{\mathbb{E}}}[X|A]+\Pr(A^{\mathsf{c}})\operatorname{{\mathbb{E}}}[X|A^{\mathsf{c}}]$.
Furthermore, For any two events $A_{1}$, $A_{2}$, and union event $A=A_{1}\cup
A_{2}$, it follows that $A^{\mathsf{c}}=(A_{1}\cup
A_{2})^{\mathsf{c}}=(A_{1}^{\mathsf{c}}\cap A_{2}^{\mathsf{c}})$.

$\displaystyle\Delta\mathrm{R}_{i}$
$\displaystyle=\Pr\left({{\mathcal{P}}}_{\\!\\!o}\cup{\mathcal{H}}_{\neq}\right)\Delta\mathrm{R}_{i\mid{{\mathcal{P}}}_{\\!\\!o}\cup{\mathcal{H}}_{\neq}}+\Pr\left({{\mathcal{P}}}_{\\!\\!o}^{\mathsf{c}}\cap{\mathcal{H}}_{\neq}^{\mathsf{c}}\right)\Delta\mathrm{R}_{i\mid{{\mathcal{P}}}_{\\!\\!o}^{\mathsf{c}}\cap{\mathcal{H}}_{\neq}^{\mathsf{c}}}.$
(29)

Let us focus on
$\Delta\mathrm{R}_{i\mid{{\mathcal{P}}}_{\\!\\!o}\cup{\mathcal{H}}_{\neq}}$.
The rate gap can be upper-bound by setting the rate of the D-CSIT setting to
0, such that
$\Delta\mathrm{R}_{i\mid{{\mathcal{P}}}_{\\!\\!o}\cup{\mathcal{H}}_{\neq}}\leq
R^{\star}_{i\mid{{\mathcal{P}}}_{\\!\\!o}\cup{\mathcal{H}}_{\neq}}(\alpha^{(1)})$.
By neglecting the received interference, we can write that

$\displaystyle
R^{\star}_{i\mid{{\mathcal{P}}}_{\\!\\!o}\cup{\mathcal{H}}_{\neq}}(\alpha^{(1)})$
$\displaystyle\leq\operatorname{{\mathbb{E}}}_{\mid{{\mathcal{P}}}_{\\!\\!o}\cup{\mathcal{H}}_{\neq}}\left[\log_{2}\left(1+P\left|\mathbf{h}_{i}\mathbf{v}_{i}\right|^{2}\right)\right]$
(30)
$\displaystyle\leq\log_{2}(P)+\operatorname{{\mathbb{E}}}_{\mid{{\mathcal{P}}}_{\\!\\!o}\cup{\mathcal{H}}_{\neq}}\left[\log_{2}\left(1+\left|\mathbf{h}_{i}\mathbf{v}_{i}\right|^{2}\right)\right].$

Moreover, the set ${{\mathcal{P}}}_{\\!\\!o}\cup{\mathcal{H}}_{\neq}$ depends
on the different estimation noise at each TX, which is absent in the
centralized setting. Accordingly, (30) implies that
$R^{\star}_{i\mid{{\mathcal{P}}}_{\\!\\!o}\cup{\mathcal{H}}_{\neq}}(\alpha^{(1)})=\Theta(\log_{2}(P))$.
Hence, it follows from Lemma 1 and Lemma 2 that

$\displaystyle\Pr\left({{\mathcal{P}}}_{\\!\\!o}\cup{\mathcal{H}}_{\neq}\right)\Delta\mathrm{R}_{i\mid{{\mathcal{P}}}_{\\!\\!o}\cup{\mathcal{H}}_{\neq}}$
$\displaystyle=o\left(\frac{1}{\log_{2}(P)}\right)\Theta(\log_{2}(P)),$ (31)

and consequently

$\displaystyle\Delta\mathrm{R}_{i}$
$\displaystyle=\Delta\mathrm{R}_{i\mid{{\mathcal{P}}}_{\\!\\!o}^{\mathsf{c}}\cap{\mathcal{H}}_{\neq}^{\mathsf{c}}}+o(1).$
(32)

Thus, in the remaining of the proof we assume w.l.o.g. that TX 1 knows
${{\hat{\mathbf{H}}}}^{(j)}_{q}$ for any $j\in{{\mathbb{N}}}_{M}$, and that
there is not power outage, as both cases become negligible at high SNR. This
assumption implies that the setting becomes hierarchical, because TX $1$
correctly estimates the quantized CSI of the other TXs. It is important to
remark that this simplification is only possible because of the proposed
scheme, in which we apply a correct power back-off and quantization step.
Indeed, the surprising outcome—and the main purpose of the careful design of
the scheme—is not (32) but the fact that
$\Delta\mathrm{R}_{i\mid{{\mathcal{P}}}_{\\!\\!o}^{\mathsf{c}}\cap{\mathcal{H}}_{\neq}^{\mathsf{c}}}$
converges to the centralized setting rate.

###  V-B Reformulating the Rate Gap

We can rewrite the rate gap for RX $i$ as

$\displaystyle\Delta\mathrm{R}_{i}$
$\displaystyle=\operatorname{{\mathbb{E}}}\left[\log_{2}\left(1+\frac{P\left|\mathbf{h}_{i}\mathbf{v}_{i}\right|^{2}}{1+P\sum_{\ell\neq
i}\left|\mathbf{h}_{i}\mathbf{v}_{\ell}\right|^{2}}\right)\right]-\operatorname{{\mathbb{E}}}\left[\log_{2}\left(1+\frac{P\left|\mu\mathbf{h}_{i}\mathbf{w}_{i}\right|^{2}}{1+P\sum_{\ell\neq
i}\left|\mu\mathbf{h}_{i}\mathbf{w}_{\ell}\right|^{2}}\right)\right]$ (33)
$\displaystyle=\operatorname{{\mathbb{E}}}\bigg{[}\log_{2}\Big{(}\underbrace{\frac{1+P\sum_{\ell\in{{\mathbb{N}}}_{K}\\!}\left|\mathbf{h}_{i}\mathbf{v}_{\ell}\right|^{2}}{1+P\sum_{\ell\in{{\mathbb{N}}}_{K}\\!}\left|\mu\mathbf{h}_{i}\mathbf{w}_{\ell}\right|^{2}}}_{{{\mathcal{F}}}_{{\mathcal{D}}}}\Big{)}\bigg{]}+\operatorname{{\mathbb{E}}}\bigg{[}\log_{2}\Big{(}\underbrace{\frac{1+P\sum_{\ell\neq
i}\left|\mu\mathbf{h}_{i}\mathbf{w}_{\ell}\right|^{2}}{1+P\sum_{\ell\neq
i}\left|\mathbf{h}_{i}\mathbf{v}_{\ell}\right|^{2}}}_{{{\mathcal{F}}}_{{\mathcal{I}}}}\Big{)}\bigg{]}.$

This rewriting of $\Delta\mathrm{R}_{i}$ allows us to separate the ratio of
received interference power (${{\mathcal{F}}}_{{\mathcal{I}}}$) and the ratio
of total received power (${{\mathcal{F}}}_{{\mathcal{D}}}$). In the following,
we will prove that $\lim_{P\rightarrow\infty}\Delta\mathrm{R}_{i}=0$ by
showing that
$\lim_{P\rightarrow\infty}\mathbb{E}\\!\left[\log_{2}({{\mathcal{F}}}_{i})\right]=0$
for both ${{\mathcal{F}}}_{{\mathcal{D}}}$ and
${{\mathcal{F}}}_{{\mathcal{I}}}$. We start with
${{\mathcal{F}}}_{{\mathcal{I}}}$ for simplicity, and later we apply a similar
argument to ${{\mathcal{F}}}_{{\mathcal{D}}}$.

###  V-C Analysis of the Interference Ratio
(${{\mathcal{F}}}_{{\mathcal{I}}}$)

We prove the convergence by upper and lower bounding
${{\mathcal{F}}}_{{\mathcal{I}}}$, and then showing that both bounds converge
to 0. We recall that we assume that TX $1$ is able to transmit the desired
precoding vector of (23) since the opposite case only yields an $o(1)$ rate
contribution.

####  V-C1 Upper-bounding
$\operatorname{{\mathbb{E}}}\left[\log_{2}\left({{\mathcal{F}}}_{{\mathcal{I}}}\right)\right]$

Note that, since $\mu\leq 1$,

$\displaystyle\operatorname{{\mathbb{E}}}\left[\log_{2}\left(\frac{1+P\sum_{\ell\neq
i}\left|\mu\mathbf{h}_{i}\mathbf{w}_{\ell}\right|^{2}}{1+P\sum_{\ell\neq
i}\left|\mathbf{h}_{i}\mathbf{v}_{\ell}\right|^{2}}\right)\right]$
$\displaystyle\leq\operatorname{{\mathbb{E}}}\bigg{[}\log_{2}\Big{(}\underbrace{\frac{1+P\sum_{\ell\neq
i}\left|\mathbf{h}_{i}\mathbf{w}_{\ell}\right|^{2}}{1+P\sum_{\ell\neq
i}\left|\mathbf{h}_{i}\mathbf{v}_{\ell}\right|^{2}}}_{{{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}}\Big{)}\bigg{]},$
(34)

where we have introduced the notation
${{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}$ for the sake of readability. Let
$\eta$ be a scalar satisfying $0\leq\eta\leq 1$. We can split the expectation
under the condition that the term ${{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}$
is smaller than $1+\eta$ or not. Therefore,

$\displaystyle\operatorname{{\mathbb{E}}}\left[\log_{2}\left({{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\right)\right]$
$\displaystyle=\Pr\left({{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}<1+\eta\right)\operatorname{{\mathbb{E}}}_{{{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}<1+\eta}\left[\log_{2}\left({{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\right)\right]\
\ +\ \ \Pr\left({{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\geq
1+\eta\right)\operatorname{{\mathbb{E}}}_{{{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\geq
1+\eta}\left[\log_{2}\left({{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\right)\right].$
(35)

Now we present a useful lemma.

######  Lemma 3.

Let $\eta={\bar{P}}^{-\varepsilon}$, with $\alpha_{q}>\varepsilon>0$ and
$\varepsilon$ arbitrarily small. Then,

$\displaystyle\Pr\left({{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\geq
1+\eta\right)=o\left(\frac{1}{\log_{2}(P)}\right)\qquad\text{and}\qquad\Pr\left(\frac{1}{{{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}}\geq
1+\eta\right)=o\left(\frac{1}{\log_{2}(P)}\right).$ (36)

###### Proof.

The proof is relegated to Appendix B-A. ∎

Let $\eta=P^{-\varepsilon}$, with $\alpha_{q}>\varepsilon>0$ and $\varepsilon$
arbitrarily small. Then, (35) becomes

$\displaystyle\operatorname{{\mathbb{E}}}\left[\log_{2}\left({{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\right)\right]$
$\displaystyle\leq\operatorname{{\mathbb{E}}}_{{{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}<1+\eta}\left[\log_{2}\left({{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\right)\right]\
\ +\ \
o\left(\frac{1}{\log_{2}(P)}\right)\operatorname{{\mathbb{E}}}_{{{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\geq
1+\eta}\left[\log_{2}\left({{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\right)\right]$
(37) $\displaystyle\leq\log_{2}(1+\eta)\ +\ o\left(1\right)$

since
$\operatorname{{\mathbb{E}}}_{{{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\geq
1+\eta}\left[\log_{2}\left({{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\right)\right]={{\mathcal{O}}}(\log_{2}(P))$.
In order to prove that
$\operatorname{{\mathbb{E}}}_{{{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\geq
1+\eta}\left[\log_{2}\left({{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\right)\right]={{\mathcal{O}}}(\log_{2}(P))$,
note that

$\displaystyle\operatorname{{\mathbb{E}}}_{{{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\geq
1+\eta}\left[\log_{2}\left({{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\right)\right]$
$\displaystyle=\frac{1}{\Pr\left({{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\geq
1+\eta\right)}\Big{(}\operatorname{{\mathbb{E}}}\left[\log_{2}\left({{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\right)\right]-\Pr\left({{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}<1+\eta\right)\operatorname{{\mathbb{E}}}_{{{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}<1+\eta}\left[\log_{2}\left({{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\right)\right]\Big{)}$
(38)
$\displaystyle\leq\frac{1}{\Pr\left({{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\geq
1+\eta\right)}\operatorname{{\mathbb{E}}}\left[\log_{2}\left({{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\right)\right].$

Furthermore,

$\displaystyle\operatorname{{\mathbb{E}}}\left[\log_{2}\left({{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\right)\right]$
$\displaystyle\leq\operatorname{{\mathbb{E}}}\bigg{[}\log_{2}\Big{(}{1+P\sum_{\ell\neq
i}\left|\mathbf{h}_{i}\mathbf{w}_{\ell}\right|^{2}}\Big{)}\bigg{]}$ (39)
$\displaystyle\leq\log_{2}(P)+\operatorname{{\mathbb{E}}}\bigg{[}\log_{2}\Big{(}{1+\sum_{\ell\neq
i}\left|\mathbf{h}_{i}\mathbf{w}_{\ell}\right|^{2}}\Big{)}\bigg{]}.$

From (38), (39), and the fact that
$\Pr\left({{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\geq
1+\eta\right)=\Theta(1)$, we obtain that
$\operatorname{{\mathbb{E}}}_{{{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\geq
1+\eta}\left[\log_{2}\left({{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\right)\right]={{\mathcal{O}}}(\log_{2}(P))$.

####  V-C2 Lower-bounding
$\operatorname{{\mathbb{E}}}\left[\log_{2}\left({{\mathcal{F}}}_{{\mathcal{I}}}\right)\right]$

Let us now lower-bound the expectation. Note that

$\displaystyle\operatorname{{\mathbb{E}}}\left[\log_{2}\left({{\mathcal{F}}}_{{\mathcal{I}}}\right)\right]$
$\displaystyle\geq\log_{2}(\mu^{2})+\operatorname{{\mathbb{E}}}\big{[}\log_{2}\big{(}{{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\big{)}\big{]}.$
(40)

Furthermore, lower-bounding (40) is equivalent to upper-bounding
$\operatorname{{\mathbb{E}}}\Big{[}\log_{2}\Big{(}{\frac{1}{{{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}}}\Big{)}\Big{]}$.
By applying Lemma 3 and similar as in (37), we obtain that
$\operatorname{{\mathbb{E}}}\left[\log_{2}\left(\tfrac{1}{{{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}}\right)\right]\leq\log_{2}(1+\eta)\
+\ o\left(1\right)$ and hence

$\displaystyle\operatorname{{\mathbb{E}}}\left[\log_{2}\left({{\mathcal{F}}}_{{\mathcal{I}}}\right)\right]$
$\displaystyle\geq\log_{2}(\mu^{2})-\log_{2}(1+\eta)\ +\ o\left(1\right).$
(41)

Consequently, the term
$\operatorname{{\mathbb{E}}}\left[\log_{2}\left({{\mathcal{F}}}_{{\mathcal{I}}}\right)\right]$
can be bounded as

$\displaystyle\log_{2}(\mu^{2})-\log_{2}(1+\eta)\ +\
o\left(1\right)\leq\operatorname{{\mathbb{E}}}\left[\log_{2}\left({{\mathcal{F}}}_{{\mathcal{I}}}\right)\right]\leq\log_{2}(1+\eta)\
+\ o\left(1\right).$ (42)

Since $\lim_{P\rightarrow\infty}\mu=1$ and $\lim_{P\rightarrow\infty}\eta=0$,
it follows that

$\displaystyle\lim_{P\rightarrow\infty}\operatorname{{\mathbb{E}}}\left[\log_{2}\left({{\mathcal{F}}}_{{\mathcal{I}}}\right)\right]=0.$
(43)

###  V-D Analysis of the Received Signal Ratio
(${{\mathcal{F}}}_{{\mathcal{D}}}$)

It remains to prove that the first expectation in (33) also converges to zero.
As for ${{\mathcal{F}}}_{{\mathcal{I}}}$, we can write

$\displaystyle\mathbb{E}\\!\left[\log_{2}({{\mathcal{F}}}_{{\mathcal{D}}})\right]$
$\displaystyle\leq\log_{2}\left(\frac{1}{\mu^{2}}\right)+\operatorname{{\mathbb{E}}}\bigg{[}\log_{2}\Big{(}\underbrace{\frac{1+P\sum_{\ell\in{{\mathbb{N}}}_{K}\\!}\left|\mathbf{h}_{i}\mathbf{v}_{\ell}\right|^{2}}{1+P\sum_{\ell\in{{\mathbb{N}}}_{K}\\!}\left|\mathbf{h}_{i}\mathbf{w}_{\ell}\right|^{2}}}_{{{\mathcal{F}}}^{\prime}_{{{\mathcal{D}}}}}\Big{)}\bigg{]}.$
(44)

Moreover, the equivalent to Lemma 3 also holds for
${{\mathcal{F}}}^{\prime}_{{\mathcal{D}}}$.

######  Lemma 4.

Let $\eta={\bar{P}}^{-\varepsilon}$, with $\alpha_{q}>\varepsilon>0$ and
$\varepsilon$ arbitrarily small. Then,

$\displaystyle\Pr\left({{\mathcal{F}}}^{\prime}_{{\mathcal{D}}}\geq
1+\eta\right)=o\left(\frac{1}{\log_{2}(P)}\right)\qquad\text{and}\qquad\Pr\left(\frac{1}{{{\mathcal{F}}}^{\prime}_{{\mathcal{D}}}}\geq
1+\eta\right)=o\left(\frac{1}{\log_{2}(P)}\right).$ (45)

###### Proof.

The proof is relegated to Appendix B-B. ∎

Thus, applying the same step as in (37) yields

$\displaystyle\operatorname{{\mathbb{E}}}\big{[}\log_{2}\big{(}{{\mathcal{F}}}_{{{\mathcal{D}}}^{\prime}}\big{)}\big{]}$
$\displaystyle\leq\log_{2}(1+\eta)+o(1).$ (46)

We can lower-bound
$\mathbb{E}\\!\left[\log_{2}({{\mathcal{F}}}_{{\mathcal{D}}})\right]$
similarly to obtain that

$\displaystyle-\log_{2}(1+\eta)\ +\
o\left(1\right)\leq\mathbb{E}\\!\left[\log_{2}({{\mathcal{F}}}_{{\mathcal{D}}})\right]\leq\log_{2}\left(\nicefrac{{1}}{{\mu^{2}}}\right)+\log_{2}(1+\eta)\
+\ o\left(1\right).$ (47)

The fact that $\lim_{P\rightarrow\infty}\mu=1$ and
$\lim_{P\rightarrow\infty}\eta=0$ leads to

$\displaystyle\lim_{P\rightarrow\infty}\mathbb{E}\\!\left[\log_{2}({{\mathcal{F}}}_{{\mathcal{D}}})\right]=0.$
(48)

###  V-E Merging Previous Sections

Given that
$\lim_{P\rightarrow\infty}\Delta\mathrm{R}=\lim_{P\rightarrow\infty}\sum_{i=1}^{K}\Delta\mathrm{R}_{i}$,
we obtain that

$\displaystyle\lim_{P\rightarrow\infty}\Delta\mathrm{R}$
$\displaystyle=\lim_{P\rightarrow\infty}K(\mathbb{E}\\!\left[\log_{2}({{\mathcal{F}}}_{{\mathcal{D}}})\right]+\mathbb{E}\\!\left[\log_{2}({{\mathcal{F}}}_{{\mathcal{I}}})\right])$
(49) $\displaystyle=0,$

what concludes the proof of Theorem 1.

##  VI Numerical Results

In this section we provide some performance analysis for the previous
asymptotic results. We consider a scenario in which the most-informed TX has a
CSI accuracy scaling parameter $\alpha^{(1)}=1$ for the whole channel matrix,
and the rest of TXs have a CSI accuracy scaling parameter $\alpha^{(j)}=0.6$,
for any $j>1$. Intuitively, this configuration can model a setting in which a
main TX receives a quantized CSI feedback from all the RXs, and then it shares
a compressed version of the CSI to the other auxiliary transmit antennas. We
present the performance of several schemes:

  * •

The ideal centralized CSIT setting, in which all the TXs are endowed with the
CSI of TX 1.

  * •

The CD-ZF scheme with Hierarchical CSIT (TX 1 knows the other TXs’ CSI).

  * •

The CD-ZF, AP-ZF, and Naive ZF schemes when the CSIT is non hierarchical
(general D-CSIT setting).

  * •

The performance of transmitting only from TX 1 and turning off the other TXs.

In Fig. 7, we show the rate performance for a setting with 2 single-antenna
TXs and 2 RXs under the assumption of instantaneous power constraint for the
precoder. Several insights emerge from the figure.

First, we observe how the proposed CD-ZF scheme performs almost as good as the
ideal centralized CSIT setting for the Hierarchical CSIT configuration. This
fact holds for any setting configuration and size, yet considering that
$N_{1}\geq K-1$. Besides this, the CD-ZF scheme is shown to tend towards the
centralized rate also for the general D-CSIT setting, where the CSI at TX 2 is
not available at TX 1. However, we can see how the convergence is slow, and at
low SNR the CD-ZF scheme outperforms the single-TX transmission or the Naive
ZF only by a slight gap. This is an aftermath of the scheme definition, as it
is tailored for the asymptotic high-SNR regime. Indeed, the CD-ZF scheme
performs in an almost optimal manner if TX 1 correctly estimate the CSI at the
other TXs; however, the probability of correct estimation increases slowly.
Thus, the performance at medium SNR is limited.

It is important to note that the CD-ZF scheme here presented is not optimized,
as our objective was to show the asymptotic behavior. For example, we assume a
scalar quantizer that independently quantizes every real and imaginary part of
each channel coefficient. Considerably higher probabilities of consistency
would be obtained if the quantization phase is optimized, e.g. by using vector
quantization, or by considering more complex schemes as the ones proposed
in[36, 48]. Nevertheless, the aforementioned points show how important is to
provide the CSI with structure (or hierarchy), as it has been proven
indispensable to boost the performance. Moreover, this CSI structure is
sometimes given by the network configuration, such that it does not imply an
extra aspect to develop.

Another point to be considered is that CD-ZF allows to obtain centralized
performance with one informed antenna less than the single-TX transmission.
This consideration can be seen in Fig. 7, as the single-TX transmission does
not achieve even the centralized DoF.

$0$$10$$20$$30$$40$$50$$60$$70$$80$$0$$5$$10$$15$$20$$25$P [dB]Rate
[bits/s/Hz]Centralized CSITCD-ZF H-CSITCD-ZF D-CSITAP-ZFNaive ZFSingle TX (TX
1) Figure 7: Performance of different precoding schemes for a setting with 2
single-antenna TXs and 2 RXs with instantaneous power constraint and
$\alpha^{(1)}=1$, $\alpha^{(2)}=0.6$.

##  VII Conclusions

We have analyzed the rate achievable through linear precoding in a Network
MISO setting with Distributed CSIT. We have developed an achievable scheme
that asymptotically attains the achievable rate in an ideal centralized
setting where every TX is endowed with the best estimate among all the TXs.
For the case in which the CSIT presents a hierarchical structure among TXs,
the previous asymptotic insight is shown to be valid also at medium-SNR
regime, where the performance obtained at both distributed and centralized
settings are alike. This result implies that we are not only able to reach the
DoF performance of the ideal centralized setting but also its _beamforming
gain_ while obtaining accurate CSIT only at a subset of the transmit antennas.
Thus, the performance degradation on account of the CSI mismatches between TXs
can be overcome by a properly designed precoding scheme which is aware of the
distributed nature of the setting. Besides this, it has been shown that the
quantization of the information available at certain nodes is helpful as it
facilitates the consistency of the decision at all the transmitters. This last
result could be applied to a broad set of distributed problems, in which the
trade-off between global consistency and local accuracy has not been deeply
analyzed yet.

##  Appendix A A useful Lemma

We present in the following a lemma that is key for the next proofs. The
impact of the channel quantization on the precoder design can then be
asymptotically computed as shown below.

######  Lemma 5.

Let TX $j$, $2\leq j\leq M$, quantize its CSIT with a scalar uniform quantizer
with quantization step $q={\bar{P}}^{-\alpha_{q}}$,
$\alpha{{}^{(M)}}>\alpha_{q}>0$. The naive precoder of Section IV-B at TX j
satisfies that

$\displaystyle\operatorname{{\mathbb{E}}}\left[\lVert{\mathbf{v}_{i,j}-\mathbf{w}_{i,j}}\rVert\
\,\right]$ $\displaystyle={{\mathcal{O}}}\left({\bar{P}}^{-\alpha_{q}}\right)$
(50)
$\displaystyle\operatorname{{\mathbb{E}}}\left[\lVert{\mathbf{v}_{i,j}-\mathbf{w}_{i,j}}\rVert^{2}\right]$
$\displaystyle={{\mathcal{O}}}\left(P^{-\alpha_{q}}\right).$ (51)

###### Proof.

The proof is provided in Appendix F. ∎

Lemma 5 is based on error propagation properties of linear systems. Thus, it
is expected to hold for a broad set of noisy estimation models whose error
variance scales as $P^{-a}$ for any $a>0$. For example, it holds for the
quantized feedback model of[39], in which random vector quantization is
assumed and the number of quantization bits scales with $P$, as shown in [40].
Furthermore, Lemma (5) leads to the following corollary.

######  Corollary 3.

Let TX $j$, $2\leq j\leq M$, quantize its CSIT with a scalar uniform quantizer
with quantization step $q={\bar{P}}^{-\alpha_{q}}$,
$\alpha{{}^{(M)}}>\alpha_{q}>0$. The global precoder of Section IV-B satisfies
that

$\displaystyle\operatorname{{\mathbb{E}}}\left[\lVert{\mathbf{v}_{i}-\mathbf{w}_{i}}\rVert\
\,\right]$ $\displaystyle={{\mathcal{O}}}\left({\bar{P}}^{-\alpha_{q}}\right)$
(52)
$\displaystyle\operatorname{{\mathbb{E}}}\left[\lVert{\mathbf{v}_{i}-\mathbf{w}_{i}}\rVert^{2}\right]$
$\displaystyle={{\mathcal{O}}}\left(P^{-\alpha_{q}}\right).$ (53)

###### Proof.

In order to prove that
$\operatorname{{\mathbb{E}}}\left[\lVert{\mathbf{v}_{i}-\mathbf{w}_{i}}\rVert^{2}\right]={{\mathcal{O}}}\left(P^{-\alpha_{q}}\right)$,
let us recall that the vector $\mathbf{w}_{{i}}-\mathbf{v}_{{i}}$ can be
written as

$\displaystyle\mathbf{w}_{{i}}-\mathbf{v}_{{i}}=\begin{bmatrix}\bm{\phi}_{i}\\\
\mathbf{w}_{{i,\bar{1}}}-\mathbf{v}_{{i,\bar{1}}}\end{bmatrix}$ (54)

where $\bm{\phi}_{i}$ (defined in (26) as
$\bm{\phi}_{i}={{\hat{\mathbf{H}}}}_{{\bar{i}},1}^{\dagger}{{\hat{\mathbf{H}}}}_{{\bar{i}},\bar{1}}(\mathbf{v}_{i,\bar{1}}-\mathbf{w}_{i,\bar{1}})$)
is the difference at TX $1$, and
$\mathbf{w}_{{i,\bar{1}}}-\mathbf{v}_{{i,\bar{1}}}$ denotes the difference for
the coefficients of all the TXs but TX $1$, i.e.,

$\displaystyle\mathbf{w}_{{i,\bar{1}}}-\mathbf{v}_{{i,\bar{1}}}=[(\mathbf{w}_{{i,2}}-\mathbf{v}_{{i,2}})^{\mathrm{T}},\
\dots,\ (\mathbf{w}_{{i,K}}-\mathbf{v}_{{i,K}})^{\mathrm{T}}]^{\mathrm{T}}.$
(55)

Recall that we have defined $N_{\bar{1}}$ as $N_{\bar{1}}\triangleq
N_{T}-N_{1}$. Let us further define $\mathbf{G}_{\mathbf{I}}$ as
$\mathbf{G}_{\mathbf{I}}\triangleq\begin{bmatrix}{{\hat{\mathbf{H}}}}_{{\bar{i}},1}^{\dagger}{{\hat{\mathbf{H}}}}_{{\bar{i}},\bar{1}}\\\
\mathbf{I}_{N_{\bar{1}}}\end{bmatrix}$. From the definition of
$\bm{\phi}_{i}$, we can rewrite (54) as

$\displaystyle\mathbf{w}_{{i}}-\mathbf{v}_{{i}}=\mathbf{G}_{\mathbf{I}}(\mathbf{w}_{{i,\bar{1}}}-\mathbf{v}_{{i,\bar{1}}}).$
(56)

From Cauchy-Schwarz inequality, this implies that

$\displaystyle\mathbb{E}\\!\left[\smash{{\lVert{\mathbf{w}_{{i}}-\mathbf{v}_{{i}}}\rVert}}^{2}\right]$
$\displaystyle\leq\sqrt{\mathbb{E}\\!\left[\smash{\lVert{\mathbf{G}_{\mathbf{I}}}\rVert}^{4}\right]\mathbb{E}\\!\left[\smash{\lVert{\mathbf{w}_{{i,\bar{1}}}-\mathbf{v}_{{i,\bar{1}}}}\rVert}^{4}\right]}.$
(57)

Let $g_{s}$ be
$g_{s}\triangleq\sqrt{\mathbb{E}\\!\left[\smash{\lVert{\mathbf{G}_{\mathbf{I}}}\rVert}^{4}\right]}$,
which is $\Theta(1)$. Moreover, the instantaneous power constraint for the
precoder ensures that $\lVert{\mathbf{v}_{i,j}-\mathbf{w}_{i,j}}\rVert^{2}\leq
4N^{2}_{j}$. Hence,

$\displaystyle\mathbb{E}\\!\left[\smash{\lVert{\mathbf{w}_{{i,\bar{1}}}-\mathbf{v}_{{i,\bar{1}}}}\rVert}^{4}\right]\leq\sum_{j=2}^{M}\mathbb{E}\\!\left[\smash{4N^{2}_{j}\lVert{\mathbf{w}_{{i,j}}-\mathbf{v}_{{i,j}}}\rVert}^{2}\right],$
(58)

what, together with Lemma 5, leads to

$\displaystyle\mathbb{E}\\!\left[\smash{{\lVert{\mathbf{w}_{{i}}-\mathbf{v}_{{i}}}\rVert}}^{2}\right]$
$\displaystyle\leq
4N^{2}_{j}g_{s}\sum_{j=2}^{M}\operatorname{{\mathbb{E}}}\big{[}\lVert{\mathbf{v}_{i,j}-\mathbf{w}_{i,j}}\rVert^{2}\big{]}$
(59) $\displaystyle={{\mathcal{O}}}({\bar{P}}^{-\alpha_{q}}),$

what concludes the proof. ∎

##  Appendix B Proof of Lemma 3 and Lemma 4

In this section we prove Lemma 3 and Lemma 4, which are instrumental for the
proof of Theorem 1.

###  B-A Proof of Lemma 3

We aim to prove that, for any $\eta={\bar{P}}^{-\varepsilon}$, with
$\alpha_{q}>\varepsilon>0$ and $\varepsilon$ arbitrarily small, it holds that

$\displaystyle\Pr\left({{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\geq
1+\eta\right)=o\left(\frac{1}{\log_{2}(P)}\right)\qquad\text{and}\qquad\Pr\left(\frac{1}{{{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}}\geq
1+\eta\right)=o\left(\frac{1}{\log_{2}(P)}\right).$ (60)

We start by noting that $\mathbf{h}_{i}$ can be written as
$\mathbf{h}_{i}=\frac{1}{\breve{z}^{(j)}}(\hat{\mathbf{h}}^{(j)}_{i}-z^{(j)}\
\bm{\delta}^{(j)}_{i})$ from the definition of the estimate in (11). Let us
introduce the notations
$\breve{z}^{(j)}_{\operatorname{inv}}=\frac{1}{\breve{z}^{(j)}}$ and
$z^{(j)}_{n}=\frac{z^{(j)}}{\breve{z}^{(j)}}$. Hence, for any $i\neq\ell$, it
follows

$\displaystyle\left|\mathbf{h}_{i}\mathbf{w}_{\ell}\right|$
$\displaystyle\overset{\mathclap{(a)}}{=}\left|\smash{\breve{z}^{(1)}_{\operatorname{inv}}\hat{\mathbf{h}}^{(1)}_{i}\mathbf{w}_{\ell}-z^{(1)}_{n}\bm{\delta}^{(1)}_{i}\mathbf{w}_{\ell}}\right|$
(61)
$\displaystyle\overset{\mathclap{(b)}}{=}\left|\smash{\breve{z}^{(1)}_{\operatorname{inv}}\hat{\mathbf{h}}^{(1)}_{i}\mathbf{v}_{\ell}-z^{(1)}_{n}\bm{\delta}^{(1)}_{i}\mathbf{w}_{\ell}+z^{(1)}_{n}\bm{\delta}^{(1)}_{i}\mathbf{v}_{\ell}-z^{(1)}_{n}\bm{\delta}^{(1)}_{i}\mathbf{v}_{\ell}}\right|$
$\displaystyle\overset{\mathclap{(c)}}{=}\left|\smash{\mathbf{h}_{i}\mathbf{v}_{\ell}-z^{(1)}_{n}\bm{\delta}^{(1)}_{i}(\mathbf{w}_{\ell}-\mathbf{v}_{\ell})}\right|,$

where $(a)$ and $(c)$ come from the D-CSIT model of (11) and $(b)$ from the
precoder definition in (22) since
$\hat{\mathbf{h}}^{(1)}_{i}\mathbf{v}_{\ell}=\hat{\mathbf{h}}^{(1)}_{i}\mathbf{w}_{\ell}$.
Hence, from the triangular inequality it follows that

$\displaystyle\frac{1+P\sum_{\ell\neq
i}\left|\mathbf{h}_{i}\mathbf{w}_{\ell}\right|^{2}}{1+P\sum_{\ell\neq
i}\left|\mathbf{h}_{i}\mathbf{v}_{\ell}\right|^{2}}$ $\displaystyle\leq
1+\frac{P\sum_{\ell\neq
i}\big{(}|z^{(1)}_{n}\bm{\delta}^{(1)}_{i}(\mathbf{w}_{\ell}-\mathbf{v}_{\ell})|^{2}+2P|\mathbf{h}_{i}\mathbf{v}_{\ell}||z^{(1)}_{n}\bm{\delta}^{(1)}_{i}(\mathbf{w}_{\ell}-\mathbf{v}_{\ell})|\big{)}}{1+P\sum_{\ell\neq
i}\left|\mathbf{h}_{i}\mathbf{v}_{\ell}\right|^{2}}.$ (62)

Let us recall that

$\displaystyle\Pr\Big{(}\sum_{k=1}^{K}A_{k}\geq
c\Big{)}\leq\sum_{k=1}^{K}\Pr\Big{(}A_{k}\geq\frac{c}{K}\Big{)}.$ (63)

From (62) and (63), it follows that

$\displaystyle\Pr\left({{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\geq
1+\eta\right)\leq\sum_{\ell\neq
i}\Pr\left(\frac{P\big{(}|z^{(1)}_{n}\bm{\delta}^{(1)}_{i}(\mathbf{w}_{\ell}-\mathbf{v}_{\ell})|^{2}+2P|\mathbf{h}_{i}\mathbf{v}_{\ell}||z^{(1)}_{n}\bm{\delta}^{(1)}_{i}(\mathbf{w}_{\ell}-\mathbf{v}_{\ell})|\big{)}}{1+P\sum_{\ell\neq
i}\left|\mathbf{h}_{i}\mathbf{v}_{\ell}\right|^{2}}\geq\frac{\eta}{K}\right)$
(64) $\displaystyle\qquad\quad\leq\sum_{\ell\neq
i}\Pr\left(\frac{|z^{(1)}_{n}\bm{\delta}^{(1)}_{i}(\mathbf{w}_{\ell}-\mathbf{v}_{\ell})|^{2}+2|\mathbf{h}_{i}\mathbf{v}_{\ell}||z^{(1)}_{n}\bm{\delta}^{(1)}_{i}(\mathbf{w}_{\ell}-\mathbf{v}_{\ell})|}{\left|\mathbf{h}_{i}\mathbf{v}_{\ell}\right|^{2}}\geq\frac{\eta}{K}\right)$
$\displaystyle\qquad\quad\overset{(a)}{=}(K-1)\Pr\left(\frac{|z^{(1)}_{n}\bm{\delta}^{(1)}_{i}(\mathbf{w}_{\ell}-\mathbf{v}_{\ell})|^{2}+2|\mathbf{h}_{i}\mathbf{v}_{\ell}||z^{(1)}_{n}\bm{\delta}^{(1)}_{i}(\mathbf{w}_{\ell}-\mathbf{v}_{\ell})|}{\left|\mathbf{h}_{i}\mathbf{v}_{\ell}\right|^{2}}\geq\frac{\eta}{K}\right)$
$\displaystyle\qquad\quad\overset{(b)}{\leq}(K-1)\Bigg{(}\Pr\bigg{(}\frac{|z^{(1)}_{n}\bm{\delta}^{(1)}_{i}(\mathbf{w}_{\ell}-\mathbf{v}_{\ell})|^{2}}{\left|\mathbf{h}_{i}\mathbf{v}_{\ell}\right|^{2}}\geq\frac{\eta}{2K}\bigg{)}+\Pr\bigg{(}\frac{2|\mathbf{h}_{i}\mathbf{v}_{\ell}||z^{(1)}_{n}\bm{\delta}^{(1)}_{i}(\mathbf{w}_{\ell}-\mathbf{v}_{\ell})|}{\left|\mathbf{h}_{i}\mathbf{v}_{\ell}\right|^{2}}\geq\frac{\eta}{2K}\bigg{)}\Bigg{)}$
$\displaystyle\qquad\quad\overset{(c)}{\leq}2(K-1)\Pr\bigg{(}\frac{|z^{(1)}_{n}\bm{\delta}^{(1)}_{i}(\mathbf{w}_{\ell}-\mathbf{v}_{\ell})|}{\left|\mathbf{h}_{i}\mathbf{v}_{\ell}\right|}\geq\frac{\eta}{4K}\bigg{)}$

where $(a)$ comes from symmetry, $(b)$ from (63), and $(c)$ because $\eta<1$.
Let us introduce now a parameter $\gamma\in{{\mathbb{R}}}$. We can continue as

$\displaystyle\Pr\bigg{(}\frac{|z^{(1)}_{n}\bm{\delta}^{(1)}_{i}(\mathbf{w}_{\ell}-\mathbf{v}_{\ell})|}{\left|\mathbf{h}_{i}\mathbf{v}_{\ell}\right|}\geq\frac{\eta}{4K}\bigg{)}$
$\displaystyle=\Pr\left({|\bm{\delta}^{(1)}_{i}(\mathbf{w}_{\ell}-\mathbf{v}_{\ell})|}\geq\frac{\eta}{4K}{|\bm{\delta}^{(1)}_{i}\mathbf{v}_{\ell}|}\right)$
(65)
$\displaystyle\leq\Pr\left(|\bm{\delta}^{(1)}_{i}\mathbf{v}_{\ell}|<{\bar{P}}^{-\gamma}\right)+\\!\int_{|\bm{\delta}^{(1)}_{i}\mathbf{v}_{\ell}|\geq{\bar{P}}^{-\gamma}}\\!\\!\\!\\!\frac{\mathbb{E}\\!\left[{|\bm{\delta}^{(1)}_{i}(\mathbf{w}_{\ell}-\mathbf{v}_{\ell})|}\right]}{\frac{\eta}{4K}{y}}f_{|\bm{\delta}^{(1)}_{i}\mathbf{v}_{\ell}|}(y)\mathop{}\\!\mathrm{d}y$

where the first equality comes from the fact that
${|\mathbf{h}_{i}\mathbf{v}_{\ell}|}=z^{(1)}_{n}{|\bm{\delta}^{(1)}_{i}\mathbf{v}_{\ell}|}$,
and the last inequality from the Law of Total Probability and Markov’s
Inequality. $f_{|\bm{\delta}^{(1)}_{i}\mathbf{v}_{\ell}|}$ stands for the
probability density function of $|\bm{\delta}^{(1)}_{i}\mathbf{v}_{\ell}|$.
Let us focus first on the first term of (65),
$\Pr\left(|\bm{\delta}^{(1)}_{i}\mathbf{v}_{\ell}|<{\bar{P}}^{-\gamma}\right)$,
which satisfies the following proposition.

######  Proposition 1.

Let $\gamma>0$. Then,

$\displaystyle\Pr\left(|\bm{\delta}^{(1)}_{i}\mathbf{v}_{\ell}|<{\bar{P}}^{-\gamma}\right)=o\left(\frac{1}{\log_{2}(P)}\right).$
(66)

###### Proof.

The proof is relegated to Appendix D. ∎

On the other hand, the integral term of (65) can be rewritten as

$\displaystyle\int_{{|\bm{\delta}^{(1)}_{i}\mathbf{v}_{\ell}|\geq{\bar{P}}^{-\gamma}}}\mathbb{E}\\!\left[{|\bm{\delta}^{(1)}_{i}(\mathbf{w}_{\ell}-\mathbf{v}_{\ell})|}\right]\frac{f_{|\bm{\delta}^{(1)}_{i}\mathbf{v}_{\ell}|}(y)}{\frac{\eta}{4K}{y}}\mathop{}\\!\mathrm{d}y$
$\displaystyle=\frac{4K}{\eta}\mathbb{E}\\!\left[{|\bm{\delta}^{(1)}_{i}(\mathbf{w}_{\ell}-\mathbf{v}_{\ell})|}\right]\mathbb{E}_{\mid|\bm{\delta}^{(1)}_{i}\mathbf{v}_{\ell}|\geq{\bar{P}}^{-\gamma}}\\!\left[\frac{1}{|\bm{\delta}^{(1)}_{i}\mathbf{v}_{\ell}|}\right]$
(67)
$\displaystyle\leq\frac{4K}{\eta}\mathbb{E}\\!\left[{|\bm{\delta}^{(1)}_{i}(\mathbf{w}_{\ell}-\mathbf{v}_{\ell})|}\right]{\bar{P}}^{\gamma}.$

Now, we introduce another useful proposition, whose proof is also relegated to
Appendix D.

######  Proposition 2.

It holds that

$\displaystyle\mathbb{E}\\!\left[{|\bm{\delta}^{(1)}_{i}(\mathbf{w}_{\ell}-\mathbf{v}_{\ell})|}\right]$
$\displaystyle={{\mathcal{O}}}({\bar{P}}^{-\alpha_{q}}).$ (68)

By applying Proposition 1 and Proposition 2 in (65), it is straight to see
that

$\displaystyle\Pr\left({{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\geq
1+\eta\right)$ $\displaystyle\leq
o\left(\frac{1}{\log_{2}(P)}\right)+\frac{8K(K-1)}{\eta}{{\mathcal{O}}}({\bar{P}}^{-\alpha_{q}}){\bar{P}}^{\gamma}.$
(69)

Since $\eta={\bar{P}}^{-\varepsilon}$, with $\alpha_{q}>\varepsilon>0$,

$\displaystyle\Pr\left({{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\geq
1+\eta\right)$ $\displaystyle\leq
o\left(\frac{1}{\log_{2}(P)}\right)+{\bar{P}}^{\varepsilon}{{\mathcal{O}}}({\bar{P}}^{-\alpha^{(2)}}){\bar{P}}^{\gamma}.$
(70)

Let us select $\gamma$ such that $\gamma>0$ and
$\varepsilon+\gamma-\alpha_{q}<0$. Then, it follows that

$\displaystyle\Pr\left({{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}\geq
1+\eta\right)$ $\displaystyle=o\left(\frac{1}{\log_{2}(P)}\right),$ (71)

what concludes the proof of the first statement of Lemma 3. We prove in the
following the second statement, i.e.,

$\displaystyle\Pr\left(\frac{1}{{{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}}\geq
1+\eta\right)=o\left(\frac{1}{\log_{2}(P)}\right).$ (72)

This is obtained by switching the vectors $\mathbf{v}_{\ell}$ and
$\mathbf{w}_{\ell}$ and applying the same steps as in the proof of the first
statement. To begin with, by following the steps in (64) we can easily obtain
that

$\displaystyle\Pr\left(\frac{1}{{{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}}\geq
1+\eta\right)$ $\displaystyle=\Pr\left(\frac{1+P\sum_{\ell\neq
i}\left|\mathbf{h}_{i}\mathbf{v}_{\ell}\right|^{2}}{1+P\sum_{\ell\neq
i}\left|\mathbf{h}_{i}\mathbf{w}_{\ell}\right|^{2}}\geq 1+\eta\right)$ (73)
$\displaystyle\leq
2(K-1)\Pr\bigg{(}\frac{|z^{(1)}_{n}\bm{\delta}^{(1)}_{i}(\mathbf{w}_{\ell}-\mathbf{v}_{\ell})|}{\left|\mathbf{h}_{i}\mathbf{w}_{\ell}\right|}\geq\frac{\eta}{4K}\bigg{)}.$

Furthermore, the final expression in (73) is equal to the one in (64) except
from the fact that the denominator is
$\left|\mathbf{h}_{i}\mathbf{w}_{\ell}\right|$ instead of
$\left|\mathbf{h}_{i}\mathbf{v}_{\ell}\right|$. Hence, continuing as in
(65)-(70), we obtain that

$\displaystyle\Pr\left(\frac{1}{{{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}}\geq
1+\eta\right)$ $\displaystyle=o\left(\frac{1}{\log_{2}(P)}\right),$ (74)

what concludes the proof of Lemma 3.∎

###  B-B Proof of Lemma 4

We aim to prove that, for any $\eta={\bar{P}}^{-\varepsilon}$, with
$\alpha_{q}>\varepsilon>0$ and $\varepsilon$ arbitrarily small, it holds that

$\displaystyle\Pr\left({{\mathcal{F}}}^{\prime}_{{\mathcal{D}}}\geq
1+\eta\right)=o\left(\frac{1}{\log_{2}(P)}\right)\qquad\text{and}\qquad\Pr\left(\frac{1}{{{\mathcal{F}}}^{\prime}_{{\mathcal{D}}}}\geq
1+\eta\right)=o\left(\frac{1}{\log_{2}(P)}\right).$ (75)

Firstly, we focus on the first statement. Note that, applying similar steps as
in (61), it holds that

$\displaystyle\left|\mathbf{h}_{i}\mathbf{v}_{\ell}\right|^{2}$
$\displaystyle\leq|\mathbf{h}_{i}\mathbf{w}_{\ell}|^{2}+|z^{(1)}_{n}\bm{\delta}^{(1)}_{i}(\mathbf{w}_{\ell}-\mathbf{v}_{\ell})|^{2}+2|\mathbf{h}_{i}\mathbf{w}_{\ell}||z^{(1)}_{n}\bm{\delta}^{(1)}_{i}(\mathbf{w}_{\ell}-\mathbf{v}_{\ell})|,$
(76) $\displaystyle\left|\mathbf{h}_{i}\mathbf{v}_{{i}}\right|^{2}$
$\displaystyle\leq\left|\mathbf{h}_{i}\mathbf{w}_{{i}}\right|^{2}+|\mathbf{h}_{i}(\mathbf{w}_{{i}}-\mathbf{v}_{{i}})|^{2}+2\left|\mathbf{h}_{i}\mathbf{w}_{{i}}\right|\left|\mathbf{h}_{i}(\mathbf{w}_{{i}}-\mathbf{v}_{{i}})\right|.$
(77)

Hence, following the steps applied in (62)-(64), we can write that

$\displaystyle\Pr\left(\frac{1+P\sum_{\ell\in{{\mathbb{N}}}_{K}\\!}\left|\mathbf{h}_{i}\mathbf{v}_{\ell}\right|^{2}}{1+P\sum_{\ell\in{{\mathbb{N}}}_{K}\\!}\left|\mathbf{h}_{i}\mathbf{w}_{\ell}\right|^{2}}\geq
1+\eta\right)$
$\displaystyle\leq\Pr\left({{\mathcal{D}}}_{1}+{{\mathcal{D}}}_{2}+{{\mathcal{D}}}_{3}+{{\mathcal{D}}}_{4}\geq{\eta}\right)\qquad$
(78)
$\displaystyle\leq\sum_{i=1}^{4}\Pr\left({{\mathcal{D}}}_{i}\geq\frac{\eta}{4}\right),$

where we have introduced the notations

$\displaystyle{{\mathcal{D}}}_{1}$ $\displaystyle\triangleq\sum_{\ell\neq
i}\frac{|z^{(1)}_{n}\bm{\delta}^{(1)}_{i}(\mathbf{w}_{\ell}-\mathbf{v}_{\ell})|^{2}}{\left|\mathbf{h}_{i}\mathbf{w}_{i}\right|^{2}}$
(79) $\displaystyle{{\mathcal{D}}}_{2}$ $\displaystyle\triangleq\sum_{\ell\neq
i}\frac{2\left|\mathbf{h}_{i}\mathbf{w}_{\ell}\right||z^{(1)}_{n}\bm{\delta}^{(1)}_{i}(\mathbf{w}_{{i}}-\mathbf{v}_{\ell})|}{\left|\mathbf{h}_{i}\mathbf{w}_{i}\right|^{2}}$
(80) $\displaystyle{{\mathcal{D}}}_{3}$
$\displaystyle\triangleq\frac{\left|\mathbf{h}_{i}(\mathbf{w}_{{i}}-\mathbf{v}_{{i}})\right|^{2}}{\left|\mathbf{h}_{i}\mathbf{w}_{i}\right|^{2}}$
(81) $\displaystyle{{\mathcal{D}}}_{4}$
$\displaystyle\triangleq\frac{2\left|\mathbf{h}_{i}\mathbf{w}_{{i}}\right|\left|\mathbf{h}_{i}(\mathbf{w}_{{i}}-\mathbf{v}_{{i}})\right|}{\left|\mathbf{h}_{i}\mathbf{w}_{i}\right|^{2}}.$
(82)

The first inequality in (78) is obtained by applying (76)-(77) and eliminating
the term $1+P\sum_{\ell\neq
i}\left|\mathbf{h}_{i}\mathbf{w}_{\ell}\right|^{2}$ from the denominator. From
the analysis of ${{\mathcal{F}}}_{{\mathcal{I}}}$ in the previous section (see
(65)) it follows easily that, if $\eta={\bar{P}}^{-\varepsilon}$, with
$\alpha_{q}>\varepsilon>0$,

$\displaystyle\Pr\left({{\mathcal{D}}}_{1}\geq\frac{\eta}{4}\right)$
$\displaystyle\leq\sum_{\ell\neq
i}\Pr\left(z^{(1)}_{n}\frac{|\bm{\delta}^{(1)}_{i}(\mathbf{w}_{\ell}-\mathbf{v}_{\ell})|^{2}}{\left|\mathbf{h}_{i}\mathbf{w}_{i}\right|^{2}}\geq\frac{\eta}{4K}\right)$
(83) $\displaystyle=o\bigg{(}\frac{1}{\log_{2}(P)}\bigg{)}.$

Similarly,

$\displaystyle\Pr\left({{\mathcal{D}}}_{2}\geq\frac{\eta}{4}\right)$
$\displaystyle=\sum_{\ell\neq
i}\Pr\left(z^{(1)}_{n}\frac{2\left|\mathbf{h}_{i}\mathbf{w}_{\ell}\right||\bm{\delta}^{(1)}_{i}(\mathbf{w}_{{i}}-\mathbf{v}_{\ell})|}{\left|\mathbf{h}_{i}\mathbf{w}_{i}\right|^{2}}\geq\frac{\eta}{4K}\right)$
(84) $\displaystyle=o\bigg{(}\frac{1}{\log_{2}(P)}\bigg{)}.$

For the two remaining terms, ${{\mathcal{D}}}_{3}$ and ${{\mathcal{D}}}_{4}$,
note that

$\displaystyle\Pr\left({{\mathcal{D}}}_{3}\geq\frac{\eta}{4}\right)+\Pr\left({{\mathcal{D}}}_{4}\geq\frac{\eta}{4}\right)$
$\displaystyle=\Pr\left({\frac{\left|\mathbf{h}_{i}(\mathbf{w}_{{i}}-\mathbf{v}_{{i}})\right|^{2}}{\left|\mathbf{h}_{i}\mathbf{w}_{i}\right|^{2}}}\geq\frac{\eta}{4}\right)+\Pr\left({\frac{2\left|\mathbf{h}_{i}(\mathbf{w}_{{i}}-\mathbf{v}_{{i}})\right|}{\left|\mathbf{h}_{i}\mathbf{w}_{i}\right|}}\geq\frac{\eta}{4}\right)$
(85) $\displaystyle\leq
2\Pr\left({\frac{\left|\mathbf{h}_{i}(\mathbf{w}_{{i}}-\mathbf{v}_{{i}})\right|}{\left|\mathbf{h}_{i}\mathbf{w}_{i}\right|}}\geq\frac{\eta}{16}\right)$
$\displaystyle=2\Pr\left({\frac{|\tilde{\mathbf{h}}_{i}(\mathbf{w}_{{i}}-\mathbf{v}_{{i}})|}{|\tilde{\mathbf{h}}_{i}\mathbf{w}_{i}|}}\geq\frac{\eta}{16}\right).$

where $\tilde{\mathbf{h}}=\frac{\mathbf{h}}{\lVert{\mathbf{h}}\rVert}$ is
unit-norm and it is distributed isotropically on the $N_{T}$-dimensional unit-
sphere[39]. We can continue as in (65) to write

$\displaystyle\Pr\left({\frac{|\tilde{\mathbf{h}}_{i}(\mathbf{w}_{{i}}-\mathbf{v}_{{i}})|}{|\tilde{\mathbf{h}}_{i}\mathbf{w}_{i}|}}\geq\frac{\eta}{16}\right)$
$\displaystyle\leq\Pr\left(|\tilde{\mathbf{h}}_{i}\mathbf{w}_{i}|<{\bar{P}}^{-\gamma}\right)+\int_{|\tilde{\mathbf{h}}_{i}\mathbf{w}_{i}|\geq{\bar{P}}^{-\gamma}}\frac{\mathbb{E}\\!\left[\smash{{|\tilde{\mathbf{h}}_{i}(\mathbf{w}_{{i}}-\mathbf{v}_{{i}})|}}\right]}{\frac{\eta}{16}{y}}f_{|\tilde{\mathbf{h}}_{i}\mathbf{w}_{i}|}(y)\mathop{}\\!\mathrm{d}y$
(86)
$\displaystyle\leq{{\mathcal{O}}}({\bar{P}}^{-\gamma})+16{\bar{P}}^{\varepsilon}\mathbb{E}\\!\left[\smash{{|\tilde{\mathbf{h}}_{i}(\mathbf{w}_{{i}}-\mathbf{v}_{{i}})|}}\right]{\bar{P}}^{\gamma}.$

The fact that $\lVert{\tilde{\mathbf{h}}_{i}}\rVert=1$ implies that
$\mathbb{E}\\!\left[\smash{{|\tilde{\mathbf{h}}_{i}(\mathbf{w}_{{i}}-\mathbf{v}_{{i}})|}}\right]\leq\mathbb{E}\\!\left[\smash{{\lVert{\mathbf{w}_{{i}}-\mathbf{v}_{{i}}}\rVert}}\right]$.
Moreover, Corollary 3 states that
$\mathbb{E}\\!\left[\smash{\lVert{\mathbf{w}_{i}-\mathbf{v}_{i}}\rVert}\right]={{\mathcal{O}}}({\bar{P}}^{-\alpha_{q}})$.
Consequently, by selecting $\gamma$ such that $\gamma>0$ and
$\varepsilon+\gamma-\alpha_{q}<0$, it follows from (86) that

$\displaystyle\Pr\left({\frac{|\tilde{\mathbf{h}}_{i}(\mathbf{w}_{{i}}-\mathbf{v}_{{i}})|}{|\tilde{\mathbf{h}}_{i}\mathbf{w}_{i}|}}\geq\frac{\eta}{16}\right)$
$\displaystyle\leq{{\mathcal{O}}}({\bar{P}}^{-\gamma})+{\bar{P}}^{\varepsilon}{{\mathcal{O}}}({\bar{P}}^{-\alpha_{q}}){\bar{P}}^{\gamma}$
(87) $\displaystyle=o\left(\frac{1}{\log_{2}(P)}\right).$

We can introduce the result of (87) into (85) to obtain from (78) that

$\displaystyle\Pr\left({{\mathcal{F}}}^{\prime}_{{\mathcal{D}}}\geq
1+\eta\right)=o\left(\frac{1}{\log_{2}(P)}\right).$ (88)

It would remain to prove that
$\Pr\left(\tfrac{1}{{{\mathcal{F}}}^{\prime}_{{\mathcal{D}}}}\geq
1+\eta\right)=o\left(\frac{1}{\log_{2}(P)}\right)$. To do so, we just need to
apply the same previous steps, in which $\mathbf{w}$ and $\mathbf{v}$ are
interchanged. Following those steps and following similar argument as in the
proof for
$\Pr\left(\tfrac{1}{{{\mathcal{F}}}^{\prime}_{{\mathcal{I}}}}\right)$, we
directly obtain the result. For this reason, and for sake of concision, we
omit the derivation. ∎

##  Appendix C Proof of Lemma 2 (Probability of Power Outage)

We denote the event of power outage as ${{\mathcal{P}}}_{\\!\\!o}$. Note that

$\displaystyle\Pr\left({{\mathcal{P}}}_{\\!\\!o}\right)\leq
N_{1}\Pr\left(\lVert{\mathbf{T}_{1,1}}\rVert>1\right),$ (89)

and
$\mathbf{T}_{1,1}=\mu[\mathbf{w}_{1,1,1},\mathbf{w}_{2,1,1},\dots,\mathbf{w}_{K,1,1}]$,
where $\mathbf{w}_{i,j,n}$ represents the $n$-th element of the precoding
vector at TX $j$ for the data symbols of RX $i$. Therefore,

$\displaystyle\Pr\left({{\mathcal{P}}}_{\\!\\!o}\right)$ $\displaystyle\leq
N_{1}\Pr\left(\lVert{\mu[\mathbf{w}_{1,1,1},\mathbf{w}_{2,1,1},\dots,\mathbf{w}_{K,1,1}]}\rVert>1\right)$
(90)
$\displaystyle\overset{(a)}{\leq}N_{1}\Pr\left(\bigcup_{i\in{{\mathbb{N}}}_{K}}\lVert{\mu\mathbf{w}_{i,1,1}}\rVert>\lVert{\mathbf{v}_{i,1,1}}\rVert\right)$
$\displaystyle\overset{(b)}{\leq}N_{1}K\Pr\left(\lVert{\mu\mathbf{w}_{1,1,1}}\rVert>\lVert{\mathbf{v}_{1,1,1}}\rVert\right)$
$\displaystyle\overset{(c)}{\leq}N_{1}K\Pr\left(\mu\lVert{\mathbf{v}_{1,1,1}}\rVert+\mu\lVert{\bm{\phi}_{1}}\rVert>\lVert{\mathbf{v}_{1,1,1}}\rVert\right)$
$\displaystyle=N_{1}K\Pr\left(\lVert{\bm{\phi}_{1}}\rVert>\frac{1-\mu}{\mu}\lVert{\mathbf{v}_{1,1,1}}\rVert\right)$

where $(a)$ is obtained from the precoder definition as
$\lVert{[\mathbf{v}_{1,1,1}\ \dots\ \mathbf{v}_{K,1,1}]}\rVert\leq 1$, $(b)$
because $\mathbf{w}_{i,1,1}$ (resp. $\mathbf{v}_{i,1,1}$) is equally
distributed for any $i\in{{\mathbb{N}}}_{K}$, and $(c)$ from (26). Now, we
obtain the probability by conditioning on $\lVert{\mathbf{v}_{1,1}}\rVert$ and
then averaging over the distribution of $\lVert{\mathbf{v}_{1,1,1}}\rVert$.
Let us denote $\mu^{\prime}\triangleq\frac{1-\mu}{\mu}$. Hence,

$\displaystyle\Pr\left({{\mathcal{P}}}_{\\!\\!o}\right)$ $\displaystyle\leq
N_{1}K\int_{-\infty}^{\infty}\Pr\left(\lVert{\bm{\phi}_{1}}\rVert>\mu^{\prime}\nu\right)f_{\lVert{\mathbf{v}_{1,1,1}}\rVert}(\nu)\mathop{}\\!\mathrm{d}\nu.$
(91)

Using Markov’s inequality we obtain that

$\displaystyle\Pr\left({{\mathcal{P}}}_{\\!\\!o}\right)$ $\displaystyle\leq
N_{1}K\int_{-\infty}^{\infty}\frac{\operatorname{{\mathbb{E}}}[{\lVert{\bm{\phi}_{1}}\rVert}]}{\mu^{\prime}\nu}f_{\lVert{\mathbf{v}_{1,1,1}}\rVert}(\nu)\mathop{}\\!\mathrm{d}\nu$
(92)
$\displaystyle=N_{1}K\operatorname{{\mathbb{E}}}[{\lVert{\bm{\phi}_{1}}\rVert}]\frac{1}{\mu^{\prime}}{\mathbb{E}}\big{[}{\lVert{\mathbf{v}_{1,1,1}}\rVert^{-1}}\big{]},$

where ${\mathbb{E}}\big{[}{\lVert{\mathbf{v}_{1,1,1}}\rVert^{-1}}\big{]}$
exists from property (ZF2). Let us focus on the first expectation term of (92)
($\operatorname{{\mathbb{E}}}[{\lVert{\bm{\phi}_{1}}\rVert}]$). Recalling
(26), $\bm{\phi}_{i}$ is defined as

$\displaystyle\bm{\phi}_{i}={{\hat{\mathbf{H}}}}_{{\bar{i}},1}^{\dagger}{{\hat{\mathbf{H}}}}_{{\bar{i}},\bar{1}}(\mathbf{v}_{i,\bar{1}}-\mathbf{w}_{i,\bar{1}}).$
(93)

Then,

$\displaystyle\operatorname{{\mathbb{E}}}\left[\lVert{\bm{\phi}_{i}}\rVert\right]$
$\displaystyle\overset{(a)}{\leq}\operatorname{{\mathbb{E}}}\left[\lVert{{{\hat{\mathbf{H}}}}_{{\bar{i}},1}^{\dagger}{{\hat{\mathbf{H}}}}_{{\bar{i}},\bar{1}}}\rVert\lVert{\mathbf{v}_{i,\bar{1}}-\mathbf{w}_{i,\bar{1}}}\rVert\right]$
(94)
$\displaystyle\overset{(b)}{\leq}\sqrt{\operatorname{{\mathbb{E}}}\left[\lVert{{{\hat{\mathbf{H}}}}_{{\bar{i}},1}^{\dagger}{{\hat{\mathbf{H}}}}_{{\bar{i}},\bar{1}}}\rVert^{2}\right]\operatorname{{\mathbb{E}}}\left[\lVert{\mathbf{v}_{i,\bar{1}}-\mathbf{w}_{i,\bar{1}}}\rVert^{2}\right]},$

where $(a)$ comes from the sub-multiplicative property of Frobenius norm and
$(b)$ from Cauchy–Schwarz inequality. Let us denote
$g_{m}\triangleq\sqrt{{\operatorname{{\mathbb{E}}}\left[\lVert{{{\hat{\mathbf{H}}}}_{{\bar{i}},1}^{\dagger}{{\hat{\mathbf{H}}}}_{{\bar{i}},\bar{1}}}\rVert^{2}\right]}}$,
which is a value that does not depend on $P$ since the channel estimates are
equally distributed for any estimation error variance. Then, we have that

$\displaystyle\operatorname{{\mathbb{E}}}\left[\lVert{\bm{\phi}_{i}}\rVert\right]$
$\displaystyle\leq
g_{m}\sqrt{\operatorname{{\mathbb{E}}}\left[\lVert{\mathbf{v}_{i,\bar{1}}-\mathbf{w}_{i,\bar{1}}}\rVert^{2}\right]}.$
(95)

Lemma 5 and the fact that
$\operatorname{{\mathbb{E}}}\left[\lVert{\mathbf{v}_{i,\bar{1}}-\mathbf{w}_{i,\bar{1}}}\rVert^{2}\right]=\sum_{j=2}^{M}\operatorname{{\mathbb{E}}}\big{[}\lVert{\mathbf{v}_{i,j}-\mathbf{w}_{i,j}}\rVert^{2}\big{]}$
yield

$\displaystyle\operatorname{{\mathbb{E}}}\left[\lVert{\bm{\phi}_{1}}\rVert\right]={{\mathcal{O}}}\Big{(}{\bar{P}}^{-\alpha_{q}}\Big{)}.$
(96)

Since $\mu=1-\varepsilon$, with
$\varepsilon=\Theta({\bar{P}}^{-\alpha_{\mu}})$ and $\varepsilon>0$, the term
$\frac{1}{\mu^{\prime}}=\frac{\mu}{1-\mu}$ satisfies
$\frac{1}{\mu^{\prime}}=\Theta({\bar{P}}^{\alpha_{\mu}})$. From (ZF2),
${\mathbb{E}}\big{[}{\lVert{\mathbf{v}_{1,1,1}}\rVert^{-1}}\big{]}$ is
$\Theta(1)$. Hence, recalling (92),

$\displaystyle\Pr\left({{\mathcal{P}}}_{\\!\\!o}\right)$ $\displaystyle\leq
N_{1}K{\mathbb{E}}\big{[}{\lVert{\mathbf{v}_{1,1,1}}\rVert^{-1}}\big{]}\operatorname{{\mathbb{E}}}[{\lVert{\bm{\phi}_{1}}\rVert}]\frac{1}{\mu^{\prime}}$
(97)
$\displaystyle=\Theta(1){{\mathcal{O}}}\left({\bar{P}}^{-\alpha_{q}}\right)\Theta({\bar{P}}^{\alpha_{\mu}}),$

what implies that
$\Pr\left({{\mathcal{P}}}_{\\!\\!o}\right)={{\mathcal{O}}}\left({\bar{P}}^{\alpha_{\mu}-\alpha_{q}}\right)$.
By selecting $\alpha_{\mu}<\alpha_{q}$, the probability of power outage
vanishes and it holds that

$\displaystyle\Pr\left({{\mathcal{P}}}_{\\!\\!o}\right)=o\left(\frac{1}{\log_{2}(P)}\right),$
(98)

what concludes the proof. ∎

##  Appendix D Proof of Proposition 1 and Proposition 2

###  D-A Proof of Proposition 1

We prove in the following that
$\Pr\big{(}|\bm{\delta}^{(1)}_{i}\mathbf{v}_{\ell}|<{\bar{P}}^{-\gamma}\big{)}=o\left(\frac{1}{\log_{2}(P)}\right)$,
for any $\gamma>0$ and $i,\ell\in{{\mathbb{N}}}_{K}$ such that $\ell\neq i$.
Let us denote the precoder for RX $\ell$ obtained with perfect knowledge of
$\mathbf{H}$ as $\mathbf{u}_{\ell}$. Then,

$\displaystyle\Pr\big{(}|\bm{\delta}^{(1)}_{i}\mathbf{v}_{\ell}|<{\bar{P}}^{-\gamma}\big{)}$
$\displaystyle=\Pr\big{(}|\bm{\delta}^{(1)}_{i}\mathbf{u}_{\ell}+\bm{\delta}^{(1)}_{i}(\mathbf{v}_{\ell}-\mathbf{u}_{\ell})|<{\bar{P}}^{-\gamma}\big{)}$
(99)
$\displaystyle\leq\Pr\big{(}\big{|}{|\bm{\delta}^{(1)}_{i}\mathbf{u}_{\ell}|-|\bm{\delta}^{(1)}_{i}(\mathbf{v}_{\ell}-\mathbf{u}_{\ell})|}\big{|}<{\bar{P}}^{-\gamma}\big{)},$

where we have applied the inverse triangle inequality. In order to prove
Proposition 1 we capitalize the intuition that the term
$|\bm{\delta}^{(1)}_{i}\mathbf{u}_{\ell}|$ is independent of the quality of
the estimate and $P$, but the value of
$|\bm{\delta}^{(1)}_{i}(\mathbf{v}_{\ell}-\mathbf{u}_{\ell})|$ is directly
proportional to the quality $P^{-\alpha^{(1)}}$. Before applying this
intuition to (99), we first analyze the term
$|\bm{\delta}^{(1)}_{i}(\mathbf{v}_{\ell}-\mathbf{u}_{\ell})|$ to obtain which
is the probability
$\Pr\big{(}|\bm{\delta}^{(1)}_{i}(\mathbf{v}_{\ell}-\mathbf{u}_{\ell})|>{\bar{P}}^{-\beta}\big{)}$,
for $\beta<\alpha^{(1)}$. Let us define the scalar $\varepsilon>0$ such that
$\beta<\beta+\varepsilon<\alpha^{(1)}$. By means of the Cauchy-Schwarz
inequality and the law of total probability we obtain

$\displaystyle\Pr\big{(}|\bm{\delta}^{(1)}_{i}(\mathbf{v}_{\ell}-\mathbf{u}_{\ell})|>{\bar{P}}^{-\beta}\big{)}\leq\Pr\big{(}\lVert{\bm{\delta}^{(1)}_{i}}\rVert\lVert{\mathbf{v}_{\ell}-\mathbf{u}_{\ell}}\rVert>{\bar{P}}^{-\beta}\big{)}$
(100)
$\displaystyle\qquad\qquad\qquad\leq\Pr\big{(}\lVert{\bm{\delta}^{(1)}_{i}}\rVert\lVert{\mathbf{v}_{\ell}-\mathbf{u}_{\ell}}\rVert>{\bar{P}}^{-\beta}\mid\lVert{\mathbf{v}_{\ell}-\mathbf{u}_{\ell}}\rVert>{\bar{P}}^{-\beta-\varepsilon}\big{)}\Pr\big{(}\lVert{\mathbf{v}_{\ell}-\mathbf{u}_{\ell}}\rVert>{\bar{P}}^{-\beta-\varepsilon}\big{)}$
$\displaystyle\qquad\qquad\qquad\ \ \
+\Pr\big{(}\lVert{\bm{\delta}^{(1)}_{i}}\rVert\lVert{\mathbf{v}_{\ell}-\mathbf{u}_{\ell}}\rVert>{\bar{P}}^{-\beta}\mid\lVert{\mathbf{v}_{\ell}-\mathbf{u}_{\ell}}\rVert\leq{\bar{P}}^{-\beta-\varepsilon}\big{)}\Pr\big{(}\lVert{\mathbf{v}_{\ell}-\mathbf{u}_{\ell}}\rVert\leq{\bar{P}}^{-\beta-\varepsilon}\big{)}$
$\displaystyle\qquad\qquad\qquad\leq\Pr\big{(}\lVert{\mathbf{v}_{\ell}-\mathbf{u}_{\ell}}\rVert>{\bar{P}}^{-\beta-\varepsilon}\big{)}+\Pr\big{(}\lVert{\bm{\delta}^{(1)}_{i}}\rVert>{\bar{P}}^{\varepsilon}\mid\lVert{\mathbf{v}_{\ell}-\mathbf{u}_{\ell}}\rVert\leq{\bar{P}}^{-\beta-\varepsilon}\big{)}.$

The first term
$\Pr\big{(}\lVert{\mathbf{v}_{\ell}-\mathbf{u}_{\ell}}\rVert>{\bar{P}}^{-\beta-\varepsilon}\big{)}$
can be upper-bounded by means of the Markov’s inequality, such that

$\displaystyle\Pr\big{(}\lVert{\mathbf{v}_{\ell}-\mathbf{u}_{\ell}}\rVert>{\bar{P}}^{-\beta-\varepsilon}\big{)}$
$\displaystyle\leq{\bar{P}}^{\beta+\varepsilon}\
{\operatorname{{\mathbb{E}}}[\lVert{\mathbf{v}_{\ell}-\mathbf{u}_{\ell}}\rVert]}$
(101)
$\displaystyle={{\mathcal{O}}}({\bar{P}}^{\beta+\varepsilon-\alpha^{(1)}}),$

where the last step follows directly after applying Lemma 5 to vectors whose
respective input estimates differ by a
${{\mathcal{O}}}({\bar{P}}^{-\alpha^{(1)}})$ additive error term. For the last
term in (100),
$\Pr\big{(}\lVert{\bm{\delta}^{(1)}_{i}}\rVert>{\bar{P}}^{\varepsilon}\mid\lVert{\mathbf{v}_{\ell}-\mathbf{u}_{\ell}}\rVert\leq{\bar{P}}^{-\beta-\varepsilon}\big{)}$,
it follows that

$\displaystyle\Pr\big{(}\lVert{\bm{\delta}^{(1)}_{i}}\rVert>{\bar{P}}^{\varepsilon}\mid\lVert{\mathbf{v}_{\ell}-\mathbf{u}_{\ell}}\rVert\leq{\bar{P}}^{-\beta-\varepsilon}\big{)}\leq\frac{\Pr\big{(}\lVert{\bm{\delta}^{(1)}_{i}}\rVert>{\bar{P}}^{\varepsilon}\big{)}}{\Pr\big{(}\lVert{\mathbf{v}_{\ell}-\mathbf{u}_{\ell}}\rVert\leq{\bar{P}}^{-\beta-\varepsilon}\big{)}}.$
(102)

From (101), it holds that
$\Pr\big{(}\lVert{\mathbf{v}_{\ell}-\mathbf{u}_{\ell}}\rVert\leq{\bar{P}}^{-\beta-\varepsilon}\big{)}=1-{{\mathcal{O}}}({\bar{P}}^{\beta+\varepsilon-\alpha^{(1)}})$.
Besides this,
$\lVert{\bm{\delta}^{(1)}_{i}}\rVert^{2}=\sum_{n=1}^{N_{T}}|\delta^{(1)}_{i,n}|^{2}$,
where $\delta^{(1)}_{i,n}$ are i.i.d. as
$\operatorname{\mathcal{N}_{\mathbb{C}}}(0,1)$. Consequently,
$|\delta^{(1)}_{i,n}|^{2}$ is distributed following a Rayleigh distribution
and

$\displaystyle\lVert{\bm{\delta}^{(1)}_{i}}\rVert^{2}\sim\Gamma_{d}(N_{T},1),$
(103)

where $\Gamma_{d}(N_{T},1)$ denotes the Gamma distribution. Moreover,
$\Gamma_{d}(N_{T},1)$ is also called the Erlang distribution, and it satisfies
that

$\displaystyle\Pr\big{(}X\sim\Gamma_{d}(N_{T},1)<x\big{)}=1-\sum_{n=0}^{N_{T}-1}\frac{1}{n!}e^{-x}x^{n}.$
(104)

Hence,

$\displaystyle\Pr\big{(}\lVert{\bm{\delta}^{(1)}_{i}}\rVert>{\bar{P}}^{\varepsilon}\big{)}$
$\displaystyle=\Pr\big{(}\lVert{\bm{\delta}^{(1)}_{i}}\rVert^{2}>{\bar{P}}^{2\varepsilon}\big{)}$
(105)
$\displaystyle=\sum_{n=0}^{N_{T}-1}\frac{1}{n!}e^{-{\bar{P}}^{2\varepsilon}}{\bar{P}}^{2n\varepsilon}.$

Since $\varepsilon>0$,
$\Pr\big{(}\lVert{\bm{\delta}^{(1)}_{i}}\rVert>{\bar{P}}^{\varepsilon}\big{)}=o({\bar{P}}^{x})$,
for any $x\in{{\mathbb{R}}}$, and hence it is $o(1/\log_{2}(P))$. This implies
that

$\displaystyle\frac{\Pr\big{(}\lVert{\bm{\delta}^{(1)}_{i}}\rVert>{\bar{P}}^{\varepsilon}\big{)}}{\Pr\big{(}\lVert{\mathbf{v}_{\ell}-\mathbf{u}_{\ell}}\rVert\leq{\bar{P}}^{-\beta-\varepsilon}\big{)}}$
$\displaystyle=\frac{1}{1-{{\mathcal{O}}}({\bar{P}}^{\beta+\varepsilon-\alpha^{(1)}})}o\left(\frac{1}{\log_{2}(P)}\right),$
(106)

what together with (101) and (100) leads to

$\displaystyle\Pr\big{(}|\bm{\delta}^{(1)}_{i}(\mathbf{v}_{\ell}-\mathbf{u}_{\ell})|>{\bar{P}}^{-\beta}\big{)}=o\left(\frac{1}{\log_{2}(P)}\right)$
(107)

for any $\beta<\alpha^{(1)}$. Equipped with this result, we can focus back on
(99), which can be expanded by means of the Law of total probability such that

$\displaystyle\Pr\big{(}\big{|}{|\bm{\delta}^{(1)}_{i}\mathbf{u}_{\ell}|-|\bm{\delta}^{(1)}_{i}(\mathbf{v}_{\ell}-\mathbf{u}_{\ell})|}\big{|}<{\bar{P}}^{-\gamma}\big{)}$
(108)
$\displaystyle\qquad=\Pr\big{(}\big{|}{|\bm{\delta}^{(1)}_{i}\mathbf{u}_{\ell}|-|\bm{\delta}^{(1)}_{i}(\mathbf{v}_{\ell}-\mathbf{u}_{\ell})|}\big{|}<{\bar{P}}^{-\gamma}\mid|\bm{\delta}^{(1)}_{i}(\mathbf{v}_{\ell}-\mathbf{u}_{\ell})|\leq{\bar{P}}^{-\beta}\big{)}\Pr\big{(}|\bm{\delta}^{(1)}_{i}(\mathbf{v}_{\ell}-\mathbf{u}_{\ell})|\leq{\bar{P}}^{-\beta}\big{)}$
$\displaystyle\qquad\ \ \ \
+\Pr\big{(}\big{|}{|\bm{\delta}^{(1)}_{i}\mathbf{u}_{\ell}|-|\bm{\delta}^{(1)}_{i}(\mathbf{v}_{\ell}-\mathbf{u}_{\ell})|}\big{|}<{\bar{P}}^{-\gamma}\mid|\bm{\delta}^{(1)}_{i}(\mathbf{v}_{\ell}-\mathbf{u}_{\ell})|>{\bar{P}}^{-\beta}\big{)}\Pr\big{(}|\bm{\delta}^{(1)}_{i}(\mathbf{v}_{\ell}-\mathbf{u}_{\ell})|>{\bar{P}}^{-\beta}\big{)}$
$\displaystyle\qquad=\Pr\big{(}\big{|}{|\bm{\delta}^{(1)}_{i}\mathbf{u}_{\ell}|-|\bm{\delta}^{(1)}_{i}(\mathbf{v}_{\ell}-\mathbf{u}_{\ell})|}\big{|}<{\bar{P}}^{-\gamma}\mid|\bm{\delta}^{(1)}_{i}(\mathbf{v}_{\ell}-\mathbf{u}_{\ell})|\leq{\bar{P}}^{-\beta}\big{)}+o\left(\frac{1}{\log_{2}(P)}\right)$
$\displaystyle\qquad\leq\Pr\big{(}|\bm{\delta}^{(1)}_{i}\mathbf{u}_{\ell}|<{\bar{P}}^{-\gamma}+{\bar{P}}^{-\beta}\big{)}+o\left(\frac{1}{\log_{2}(P)}\right).$

Let us assume w.l.o.g. that $\beta<\gamma$, such that
$\Pr\big{(}|\bm{\delta}^{(1)}_{i}\mathbf{u}_{\ell}|<{\bar{P}}^{-\gamma}+{\bar{P}}^{-\beta}\big{)}\leq\Pr\big{(}|\bm{\delta}^{(1)}_{i}\mathbf{u}_{\ell}|<2{\bar{P}}^{-\beta}\big{)}$.
Therefore, it remains to prove that
$\Pr\big{(}|\bm{\delta}^{(1)}_{i}\mathbf{u}_{\ell}|<2{\bar{P}}^{-\beta}\big{)}=o\left(\frac{1}{\log_{2}(P)}\right)$.
Let $\epsilon_{\beta}$ be a scalar such that $0<\epsilon_{\beta}<\beta$ and
let us define $\psi$ as the angle satisfying

$\displaystyle\cos(\psi)\triangleq\frac{|\bm{\delta}^{(1)}_{i}\mathbf{u}_{\ell}|}{\lVert{\bm{\delta}^{(1)}_{i}}\rVert\lVert{\mathbf{u}_{\ell}}\rVert}.$
(109)

Then, we use again the Law of Total Probability to obtain

$\displaystyle\Pr\left(|\bm{\delta}^{(1)}_{i}\mathbf{u}_{\ell}|<2{\bar{P}}^{-\beta}\right)$
$\displaystyle=\Pr\left(\lVert{\bm{\delta}^{(1)}_{i}}\rVert\lVert{\mathbf{u}_{\ell}}\rVert\cos(\psi)<2{\bar{P}}^{-\beta}\mid\lVert{\mathbf{u}_{\ell}}\rVert\leq{\bar{P}}^{-\epsilon_{\beta}}\right)\Pr\left(\lVert{\mathbf{u}_{\ell}}\rVert\leq{\bar{P}}^{-\epsilon_{\beta}}\right)$
(110)
$\displaystyle\quad+\Pr\left(\lVert{\bm{\delta}^{(1)}_{i}}\rVert\lVert{\mathbf{u}_{\ell}}\rVert\cos(\psi)<2{\bar{P}}^{-\beta}\mid\lVert{\mathbf{u}_{\ell}}\rVert>{\bar{P}}^{-\epsilon_{\beta}}\right)\Pr\left(\lVert{\mathbf{u}_{\ell}}\rVert>{\bar{P}}^{-\epsilon_{\beta}}\right)$
$\displaystyle\leq\Pr\left(\lVert{\mathbf{u}_{\ell}}\rVert\leq{\bar{P}}^{-\epsilon_{\beta}}\right)+\Pr\left(\lVert{\bm{\delta}^{(1)}_{i}}\rVert\lVert{\mathbf{u}_{\ell}}\rVert\cos(\psi)<2{\bar{P}}^{-\beta}\mid\lVert{\mathbf{u}_{\ell}}\rVert>{\bar{P}}^{-\epsilon_{\beta}}\right)$
$\displaystyle\leq\Pr\left(\lVert{\mathbf{u}_{\ell}}\rVert\leq{\bar{P}}^{-\epsilon_{\beta}}\right)+\Pr\left(\lVert{\bm{\delta}^{(1)}_{i}}\rVert\cos(\psi)<2{\bar{P}}^{-\beta}{\bar{P}}^{\epsilon_{\beta}}\mid\lVert{\mathbf{u}_{\ell}}\rVert>{\bar{P}}^{-\epsilon_{\beta}}\right).$

Importantly, $\bm{\delta}^{(1)}_{i}$ is isotropically distributed (i.e., the
normalized value $\bm{\delta}^{(1)}_{i}/\lVert{\bm{\delta}^{(1)}_{i}}\rVert$
is uniformly distributed in the sphere surface). Besides this,
$\mathbf{u}_{\ell}$ is a function of $\mathbf{H}$. Since $\mathbf{H}$ and
$\bm{\delta}^{(1)}_{i}$ are mutually independent, so $\bm{\delta}^{(1)}_{i}$
and $\mathbf{u}_{\ell}$ are. Hence, from isotropy of $\bm{\delta}^{(1)}_{i}$,
$\cos(\psi)$ is independent of $\mathbf{u}_{\ell}$. On this basis, we can
select $\mathbf{u}_{\ell}=[1,\bm{0}_{1\times N_{T}-1}]$ to obtain that

$\displaystyle\Pr\left(\lVert{\bm{\delta}^{(1)}_{i}}\rVert\cos(\psi)<2{\bar{P}}^{\epsilon_{\beta}-\beta}\mid\lVert{\mathbf{u}_{\ell}}\rVert>{\bar{P}}^{-\epsilon_{\beta}}\right)$
$\displaystyle=\Pr\left(|\bm{\delta}^{(1)}_{i,1,1}|<2{\bar{P}}^{\epsilon_{\beta}-\beta}\right),$
(111)

where $\bm{\delta}^{(1)}_{i,1,1}$ denotes the first element of the vector
$\bm{\delta}^{(1)}_{i}$, and it is distributed as
$\operatorname{\mathcal{N}_{\mathbb{C}}}(0,1)$. Then,

$\displaystyle\Pr\left(|\bm{\delta}^{(1)}_{i,1,1}|<2{\bar{P}}^{\epsilon_{\beta}-\beta}\right)$
$\displaystyle=\frac{2}{\sqrt{2\pi}}\int_{0}^{2{\bar{P}}^{\epsilon_{\beta}-\beta}}\\!\\!\\!\\!\\!\\!e^{-x^{2}/2}\mathop{}\\!\mathrm{d}x$
(112)
$\displaystyle\leq\frac{4}{\sqrt{2\pi}}{\bar{P}}^{\epsilon_{\beta}-\beta}.$

On the other hand, the term
$\Pr\left(\lVert{\mathbf{u}_{\ell}}\rVert\leq{\bar{P}}^{-\epsilon_{\beta}}\right)$
can be bounded by

$\displaystyle\Pr\left(\lVert{\mathbf{u}_{\ell}}\rVert\leq{\bar{P}}^{-\epsilon_{\beta}}\right)$
$\displaystyle=\int_{0}^{{\bar{P}}^{-\epsilon_{\beta}}}\\!\\!\\!\\!f_{\lVert{\mathbf{u}_{i}}\rVert}(x)\mathop{}\\!\mathrm{d}x$
(113) $\displaystyle\leq
f^{\max}_{\lVert{\mathbf{u}_{i}}\rVert}{\bar{P}}^{-\epsilon_{\beta}},$

what follows from (ZF3). By introducing (112) and (113) in (110) we obtain
that

$\displaystyle\Pr\left(|\bm{\delta}^{(1)}_{i}\mathbf{u}_{\ell}|<2{\bar{P}}^{-\beta}\right)\leq{{\mathcal{O}}}({\bar{P}}^{\max(-\epsilon_{\beta},\
\epsilon_{\beta}-\beta)}).$ (114)

Note that $\epsilon_{\beta}$ satisfies $0<\epsilon_{\beta}<\beta$. Hence,

$\displaystyle\Pr\big{(}|\bm{\delta}^{(1)}_{i}\mathbf{v}_{\ell}|<{\bar{P}}^{-\gamma}\big{)}$
$\displaystyle\leq\Pr\big{(}\big{|}{|\bm{\delta}^{(1)}_{i}\mathbf{u}_{\ell}|-|\bm{\delta}^{(1)}_{i}(\mathbf{v}_{\ell}-\mathbf{u}_{\ell})|}\big{|}<{\bar{P}}^{-\gamma}\big{)}$
(115)
$\displaystyle\leq\Pr\big{(}|\bm{\delta}^{(1)}_{i}\mathbf{u}_{\ell}|<2{\bar{P}}^{-\beta}\big{)}+o\left(\frac{1}{\log_{2}(P)}\right)$
$\displaystyle=o\left(\frac{1}{\log_{2}(P)}\right),$

what concludes the proof of Proposition 1. ∎

###  D-B Proof of Proposition 2

We prove in the following that
$\mathbb{E}\\!\left[\smash{|\bm{\delta}^{(1)}_{i}(\mathbf{w}_{\ell}-\mathbf{v}_{\ell})|}\right]={{\mathcal{O}}}({\bar{P}}^{-\alpha_{q}})$
for any $i,\ell\in{{\mathbb{N}}}_{K}:\ell\neq i$. It follows that

$\displaystyle{\mathbb{E}}\big{[}|\bm{\delta}^{(1)}_{i}(\mathbf{w}_{\ell}-\mathbf{v}_{\ell})|\big{]}$
$\displaystyle\leq{\mathbb{E}}\big{[}{\lVert{\bm{\delta}^{(1)}_{i}}\rVert\lVert{\mathbf{w}_{\ell}-\mathbf{v}_{\ell}}\rVert}\big{]}$
(116)
$\displaystyle=\operatorname{cov}\left({\lVert{\bm{\delta}^{(1)}_{i}}\rVert,\lVert{\mathbf{w}_{\ell}-\mathbf{v}_{\ell}}\rVert}\right)+\mathbb{E}\\!\left[\lVert{\bm{\delta}^{(1)}_{i}}\rVert\right]\mathbb{E}\\!\left[\lVert{\mathbf{w}_{\ell}-\mathbf{v}_{\ell}}\rVert\right]$