# How much Training is Needed in Downlink Cell-Free mMIMO under LoS/NLoS
channels?

Sai Manikanta Rishi Rani, Ribhu Chopra, Kumar Appaiah S.M.R. Rani was with the
Department of Electronics and Electrical Engineering, Indian Institute of
Technology Guwahati, Assam, India for the duration of this work, he is now
with the Department of Electrical and Computer Engineering, University of
California San Diego, San Diego, CA 92093 USA. R. Chopra is with the
Department of Electronics and Electrical Engineering, Indian Institute of
Technology Guwahati, Assam, India. K. Appaiah is with the department of
Electrical Engineering, Indian Institute of Technology Bombay, Mumbai,
Maharashtra, India. (emails<EMAIL_ADDRESS>ribhu@outlook.com,akumar@ee.iitb.ac.in).

###### Abstract

The assumption that no LoS channels exist between wireless access points (APs)
and user equipments (UEs) becomes questionable in the context of the recent
developments in the direction of cell free massive multiple input multiple
output MIMO (CF-mMIMO) systems. In CF-mMIMO systems, the access point density
is assumed to be comparable to, or much larger than the the user density,
thereby leading to the possibility of existence of LoS links between the UEs
and the APs, depending on the local propagation conditions. In this paper, we
compare the rates achievable by CF-mMIMO systems under probabilistic LoS/ NLos
channels, with and without acquiring the channel state information (CSI) of
the fast fading components. We show that, under sufficiently large AP
densities, statistical beamforming that does not require the knowledge about
the fast fading components of the channels, performs almost at par with full
beamforming, utilizing the information about the fast fading channel
coefficients, thus potentially avoiding the need for training during every
frame. We validate our results via detailed Monte Carlo simulations, and also
elaborate the conditions under which statistical beamforming can be
successfully employed in massive MIMO systems with LoS/ NLoS channels.

###### Index Terms:

Cell Free massive MIMO, LoS/ NLoS channels, Performance Analysis, Statistical
Beamforming.

##  I Introduction

###  I-A Motivation

The idea of using a large number of co-located antennas to simultaneously
serve a smaller number of user equipments (UEs) over the same time frequency
resource, dubbed massive multiple input multiple output (mMIMO), has been of
much research interest over the past decade [1, 2, 3]. Greater spectral and
energy efficiencies coupled with analytical tractability and simple linear
processing are some of the key benefits of this architecture [4, 5].
Consequently, mMIMO has been widely accepted as a key enabling technology for
5G cellular systems [6], with a large body of literature discussing the
requirement of the availability of accurate channel state information (CSI) at
the BS. However, concerns have been raised on the non-uniform coverage offered
by massive MIMO systems, that is inherently unfair to UEs located at the cell
edges. The focus has therefore shifted to providing a uniform quality of
service to all users in an mMIMO like setup. As a consequence, several
distributed realizations of the mMIMO setup have recently been explored, with
cell free (CF)-mMIMO, emerging as a front-runner [7, 8].

The canonical CF-mMIMO setup employs a large number of access points (APs),
each equipped with a small antenna array or mostly a single antenna,
distributed over a large area and simultaneously serving a large number of
users over the same time frequency resource. Here, the AP antenna density is
assumed to be much larger than the UE density, with all the APs connected to a
central processing unit (CPU) via a high rate, loss free backbone link. It has
been shown that CF-mMIMO systems, in addition to offering a more uniform
coverage to users, also inherit several advantages of conventional cellular
mMIMO systems, such as power scaling and simple linear processing [7]. Another
interesting aspect of CF-mMIMO systems is the distributed architecture that
results in close proximity between the users and the serving APs. This also
increases the probability of having a line of sight (LoS) link between a user
and one or more of the associated APs. Most of the existing literature on CF-
mMIMO systems assumes either Rayleigh fading channels with an LoS link
existing with probability 0 [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], or
Rician fading channels with an LoS link existing with probability 1 [19, 20,
21]. However, due to the uncertainties in the physical distribution of various
blockages (e.g. buildings in urban areas, trees, hills, etc. in rural areas),
it is impossible to accurately determine the presence of an LoS link between
the APs and the UEs apriory. Therefore, in this paper, we discuss the
performance of CF-mMIMO systems under probabilistic LoS components.

###  I-B Prior Work

The idea of CF-mMIMO was first discussed in [8, 9], and it was shown that,
even with local conjugate beamforming, it can outperform both colocated mMIMO
as well as the small cell networks model, in terms of throughput and energy
efficiency. The idea of using only locally available channel state information
(CSI) to serve a large number of users in CF-mMIMO is based on the earlier
idea of cooperative MIMO [12], wherein the basic idea is to group multiple
devices via point-to-point links into a virtual antenna array to emulate MIMO
communications. In [8], the impracticality of forwarding the CSI from all APs
to the CPU was considered, and CPU data detection based on preprocessed
samples from the APs was suggested. Alternatively, the authors in [13]
analyzed the performance of CF-mMIMO systems with centralized MMSE combining
under the assumption of CSI availability at the CPU. The issue of
impracticality of an infinite rate backhaul was raised in [15] and [16], where
the effects of quantization in the backhaul link were derived, and
corresponding impairments to the achievable rate were quantified. The authors
in [14] make a detailed comparison of the backhaul requirements, communication
and computational complexities, and the achievable performances of different
combining schemes in CF-mMIMO systems. It has been demonstrated that the
advantage of channel hardening enjoyed by cellular mMIMO systems is also
inherited by CF-mMIMO systems [8]. However, it was argued in [10] that, due to
the distributed nature of the system, the channels between different APs and
users are independent but non-identically distributed. This leads to CF
massive MIMO inheriting this property only when the number of antennas at
individual APs is large.

In recent years, considerable research efforts have gone into characterizing
the behaviour of MIMO systems under LoS/NLoS channels [22, 23, 24]. We note
that the effect of probabilistic LoS/NLoS channels is more pronounced in dense
cellular networks than sparse networks due to the increased AP and UE
densities. Depending on the degree of accuracy of the modelling for LoS/NLoS
channels, the behaviour of the area spectral efficiency (ASE) of the system
has also been observed to change [22, 23]. In [25] the uplink performance of
CF-mMIMO systems under probabilistic LoS/NLoS channels was analysed for
different combining schemes being used at the CPU, and it was found that, for
sufficiently large AP densities, ubiquitous LoS link coverage can be assumed
for all the UEs. It has recently been argued that, in case LoS links are
present between the BS and the UEs, the downlink beamforming based purely on
the statistical or slow fading parameters of the channel model, dubbed
statistical beamforming, can be used [26, 27, 28]. In [26, 27], the second
order statistics of the channel were exploited to create a two tier precoding
matrix. Since statistical CSI varies slowly as compared to the real-time
instantaneous CSI, it can therefore be obtained accurately by the BS/CPU via
long-term feedback [29]. Moreover, during the transmission, the UEs only need
to estimate and feedback the dimension-reduced effective channel. This greatly
reduces the channel estimation and feedback overhead in mMIMO systems. The
authors in [28] perform LoS based beamforming on a mMIMO downlink system under
a Rician fading channel. The proposed statistical three-dimensional (3D)
beamforming precoding scheme is based only on the statistical LOS information
of each user. The scheme achieves considerably high sum rates while requiring
much less CSI overhead at the BS. This results in an interesting proposition
for CF-mMIMO systems, where the cost of CSI acquisition at the CPU is high,
and the probability of having an LoS channel between a user and one or more
APs has been shown to be large [25].

###  I-C Contributions

In this paper, we examine the need for the availability of fast fading CSI in
a CF-mMIMO system under a probabilistic LoS/NLoS channel. We use the
instantaneous achievable rate per channel use as a metric to quantify the
effect of the availability of the fast fading CSI at the APs. In this
discussion, we restrict ourselves to using conjugate beamforming for downlink
data precoding. This is mainly due to the simplicity of implementation, and
the reduced reliance on the backhaul network due to its distributed
implementation. For simplicity of exposition, we dub schemes with and without
the availability of fast fading CSI as full and statistical beamforming
schemes, respectively. We summarize our main contributions as follows:

  1. 1.

To benchmark the performance of all the other schemes we first derive bounds
on the rate achievable by the system under the availability of accurate CSI at
both the APs and the UEs (see Section III-A.).

  2. 2.

Following this, we analyze the achievable rates of the system, with the
statistical CSI being ubiquitously available, but the fast fading CSI being
obtained via pilot training at both the APs and the UEs. We note that the
estimation errors in the fast fading components lead to the introduction of
self interference impairing the system performance (see Section III-B.).

  3. 3.

Subsequently, we derive theoretical bounds on the system performance under
pure statistical beamforming, and with no information being available about
the fast fading channel components (see Section IV-A.).

  4. 4.

We then obtain bounds on the performance achievable with statistical
beamforming, but aided by downlink pilot training. This is seen to noticeably
reduce the inter-stream interference caused by the fast fading channel
components (see Section IV-B.).

  5. 5.

Finally, via detailed simulations, we validate the derived theory and compare
the performance of the four precoding schemes, and draw conclusions on the
power and performance trade offs involved in LoS reliant CF-mMIMO systems (see
Section V.).

The key takeaway of this work is that, in the case of CF-mMIMO systems, with a
sufficiently large AP density, transmission schemes that are designed to rely
more on the LoS component of the channels perform significantly well; almost
at par with schemes that utilize the full CSI for beamforming. This implies
that having accurate and up to date CSI of the fast fading channel components
becomes unnecessary in CF-mMIMO systems that possess sufficiently large AP
densities, thereby allowing us to skip pilot based CSI estimation in each
coherence interval. We next describe the system model, including the system
setup, the channel model, the LoS link probability model, and the power
control and channel estimation schemes used in this paper.

##  II System Model

We consider a CF-mMIMO system with $M$ APs that operate in the time division
duplex (TDD) mode, each equipped with a uniform linear array of $N$ antennas,
serving a total of $K\ll MN$ UEs. The height of the $m$th AP is assumed to be
$\ell_{m}$, with its antennas separated by a distance $d$. Similarly, the
$k$th UE’s height is assumed to be $\ell_{k}^{\prime}$.

###  II-A Channel Model

We note that the line-of-sight (LoS) path between a UE and an AP may be
obstructed due to the presence of blockages (e.g. buildings in urban areas,
trees/hillocks in rural areas, etc.). However, due to the random nature of the
locations of the UEs, APs, and these blockages, the existence of LoS paths
between these cannot be guaranteed. Therefore, in this work, we characterize
the channel between a UE and an AP as a combination of both LoS and NLoS (non-
line-of-sight) channels. As a result, the channel between the $m$th AP and the
$k$th UE, $\mathbf{h}_{mk}\in\mathbb{C}^{N\times 1}$, is given by

$\mathbf{h}_{mk}=\alpha_{mk}\dot{\mathbf{h}}_{mk}+\sqrt{\beta_{mk}}\mathbf{\bar{h}}_{mk}.$
(1)

Here $\mathbf{\bar{h}}_{mk}$ represents the fast fading component of the NLoS
channel that is assumed to consist of independent and identically distributed
(i.i.d.) zero mean circularly symmetric complex Gaussian (ZMCSCG) entries with
unit variance, i.e.
$\mathbf{\bar{h}}_{mk}\sim\mathcal{C}\mathcal{N}(0,\textbf{I}_{N})$. The
coefficient $\beta_{mk}$ denotes the slow fading path loss component of the
NLoS channel. Also, $\alpha_{mk}$ is a Bernoulli random variable that
indicates the presence or the absence of an LoS component between the $m$th AP
and the $k$th UE, such that $\Pr\\{\alpha_{mk}=1\\}=q_{mk}$ (a detailed
discussion on the modeling of $q_{mk}$ follows in Section II-B). Finally,
$\dot{\mathbf{h}}_{mk}$ denotes the LoS channel, and is given as

$\dot{\mathbf{h}}_{mk}=\mathbf{a}(\theta_{mk})\sqrt{{G_{m}G_{k}}}\left(\frac{\ell_{k}^{{}^{\prime}}\ell_{m}}{4\pi
x_{mk}}\right)e^{\iota 2\pi\frac{x_{mk}}{\lambda_{c}}},$ (2)

with $x_{mk}$ being the three dimensional link distance between the $k$th UE
and the $m$th AP, $\lambda_{c}$ denoting the carrier wavelength,
$\iota\triangleq\sqrt{-1}$, $G_{m}$ and $G_{k}$ being the gains associated
with the antennas of the $m$th AP and $k$th UE respectively, and
$\mathbf{a}(\theta_{mk})$ representing the array response vector for a signal
leaving from the $m$th AP towards the $k$th UE at an angle $\theta_{mk}$,
given as

$\mathbf{a}(\theta_{mk})=\left[1,e^{\iota
2\pi\frac{d}{\lambda_{c}}\sin(\theta_{mk})},e^{\iota
4\pi\frac{d}{\lambda_{c}}\sin(\theta_{mk})},\ldots,e^{\iota
2(N-1)\pi\frac{d}{\lambda_{c}}\sin(\theta_{mk})}\right]^{T}.$ (3)

The downlink frame consists of three sub-frames, viz. uplink training
comprising at most $K$ channel uses for acquisition of CSI at the APs,
downlink training comprising at most $K$ channel uses for acquisition of the
effective downlink channels at the UEs, and actual data transmission. We next
describe the LoS link probability model used in this work.

###  II-B LoS link probability model

We note that the channel between the $m$-th AP and the $k$-th UE may or may
not contain an LoS component, depending on the existence of blockages between
them, and the heights of these blockages. Therefore, the existence of an LoS
link depends on geographical distribution parameters of the network, such as
the building height distribution, building locations, heights of APs and UEs,
etc. The existing 3GPP model [30], however, does not account for all these
factors, and hence cannot characterize the exact LoS link probabilities. A
more complete model for the LoS link probabilities has been discussed in [31]
using the ITU blockage model [32]. Using this model, the LoS link probability
$q_{mk}$ is given by

$q_{mk}=(1-\omega)^{\sqrt{\eta\,\mu\,d_{mk}}}$ (4)

where
$\omega\triangleq\sqrt{\frac{\pi}{2}}\frac{\rho}{\ell_{m}–\ell^{\prime}_{k}}\left[\text{erf}\left(\frac{\ell_{m}}{\rho\sqrt{2}}\right)-\text{erf}\left(\frac{\ell^{\prime}_{k}}{\rho\sqrt{2}}\right)\right]$,
and $\text{erf}(z)\triangleq\frac{1}{\sqrt{\pi}}\int_{-z}^{z}e^{-t^{2}}dt$,
$\rho$ is the average altitude of blockages, $\eta$ is the fraction of the
built up area, and $\mu$ is the average number of blockages per unit area
[33]. This completes the description of the channel model. We next describe
the uplink channel estimation procedure.

###  II-C Uplink Channel Estimation

For simplicity, we assume the availability of $K$ orthogonal uplink pilot
sequences in each coherence interval, with the pilot symbol sent by the $k$th
UE at the $n$th instant given as $\psi_{k}[n]$, such that
$\sum_{n=1}^{T}\psi_{k}[n]\psi_{l}^{*}[n]=\delta[k-l]$, where $\delta[\cdot]$
denotes the Kronecker delta. We also assume that all the users share the same
average pilot power that we can normalize to unity without the loss of
generality, and the corresponding normalized noise power being given by
$\sigma^{2}_{u}$. Therefore, the pilot signal received by the $m$th AP at the
$n$th instant is given by

$\mathbf{y}_{m}[n]=\sum_{k=1}^{K}\mathbf{h}_{mk}\psi_{k}[n]+\sigma_{u}\mathbf{w}_{k}[n].$
(5)

where $\mathbf{w}_{k}[n]\sim\mathcal{CN}(0,\textbf{I}_{N})$ denotes the
independent additive white Gaussian noise (AWGN). Defining
$\mathbf{y}^{{}^{\prime}}_{mk}=\left(\sum_{n=1}^{T}\mathbf{y}_{m}[n]\psi_{k}^{*}[n]\right)-\alpha_{mk}\mathbf{\dot{h}}_{mk}$,
we obtain

$\mathbf{y}^{{}^{\prime}}_{mk}=\sqrt{\beta_{mk}}\mathbf{\bar{h}}_{mk}+\sigma_{u}\mathbf{w}^{{}^{\prime}}_{k}[n].$
(6)

where $\mathbf{w}^{{}^{\prime}}_{k}[n]$ is defined implicitly. This is done
under the assumption that the APs have an accurate estimate of the LoS CSI
through long-term feedback techniques [28]. We note that it is common to
assume the exact knowledge of the deterministic LoS component of
$\mathbf{h}_{mk}$, and consider only the estimate of $\mathbf{\bar{h}}_{mk}$
[29].

Now,
$\mathbb{E}\left[\mathbf{\bar{h}}_{mk}\mathbf{\bar{h}}^{H}_{mk}\right]=\mathbf{I}_{N},$
and consequently,
$\mathbb{E}\left[\mathbf{y^{\prime}}_{mk}\mathbf{y^{\prime}}^{H}_{mk}\right]=(\beta_{mk}+\sigma^{2}_{u})\mathbf{I}_{N},$
such that,
$\mathbb{E}\left[\mathbf{\bar{h}}_{mk}\mathbf{y^{\prime}}^{H}_{mk}\right]=\sqrt{\beta_{mk}}\mathbf{I}_{N}.$
Based on this, and (6), we can now write the LMMSE estimate
$\mathbf{\hat{h}}_{mk}$ of $\mathbf{h}_{mk}$ as

$\mathbf{\hat{h}}_{mk}=\frac{\sqrt{\beta_{mk}}}{\beta_{mk}+\sigma^{2}_{u}}\mathbf{y}^{{}^{\prime}}_{mk}.$
(7)

We can now express the NLoS component of the channel in terms of its estimate,
and the associated estimation error as,

$\mathbf{\bar{h}}_{mk}=\mathbf{\hat{h}}_{mk}+\mathbf{\tilde{h}}_{mk},$ (8)

with $\mathbf{\tilde{h}}_{mk}$ being a ZMCSCG random vector with
$\mathbb{E}[\mathbf{\tilde{h}}_{mk}\mathbf{\tilde{h}}_{mk}^{T}]=\frac{\sigma^{2}_{u}}{\beta_{mk}+\sigma^{2}_{u}}\mathbf{I}_{N}$,
and $\\\ E[\mathbf{\tilde{h}}_{mk}\mathbf{\hat{h}}^{H}_{mk}]=\mathbf{O}_{N}$,
where $\mathbf{O}_{N}$ corresponds to the all zero, order $N$ square matrix.
At this point, it is also convenient to define the overall channel estimate
available at the APs as
$\mathbf{\check{h}}_{mk}\triangleq\mathbf{\dot{h}}_{mk}+\mathbf{\hat{h}}_{mk}.$
The APs then use the available channel estimates to precode the downlink data,
as described in the next subsection.

###  II-D Downlink Signal Model and Power Control

Letting $s_{k}$ be the symbol being sent to the $k$th user, such that
$\mathbf{s}=[s_{1},s_{2},\ldots,s_{K}]^{T}$, and
$\mathbf{\check{H}}_{m}=\mathbf{\dot{H}}_{m}+\mathbf{\hat{H}}_{m}$ being the
channel estimate available at the $m$th AP, we can write the downlink signal
vector transmitted by the $m$th AP as

$\mathbf{y}_{m}=\mathbf{\check{H}}_{m}^{*}\text{diag}(\mathbf{x}_{m})\mathbf{s}=\textbf{P}_{m}\textbf{s},$

with the $i$th entry, $x_{mi}$ of $\mathbf{x}_{m}$ corresponding to the power
allocated by the $m$th AP to the $i$th user’s channel, and $\mathbf{P}_{m}$
being the effective precoder matrix at the $m$th AP. Also, $\textbf{y}_{m}$
can be seen as consisting of $K$ components corresponding to the $K$ users,
such that $\textbf{y}_{mk}=\textbf{p}_{mk}s_{k}$, with $\textbf{p}_{mk}$ being
the $k$th column of $\textbf{P}_{m}$.

Now, the signal received at the $k$th UE, $r_{k}$, can be written as,

$r_{k}=\sum_{m=1}^{M}\mathbf{h}_{mk}^{T}\mathbf{y}_{m}+\sigma_{o}w_{k}=\left(\sum_{m=1}^{M}\mathbf{h}_{mk}^{T}\mathbf{P}_{m}\right)\mathbf{s}+\sigma_{o}w_{k}$
(9)

where $w_{k}$ is the additive white Gaussian noise with zero mean and unit
variance, and $\sigma_{o}^{2}$ is the normalized noise power. We now define
the composite channel vector from all the APs the to the $k$th user, as
$\boldsymbol{\gamma}_{k}$ whose $i$th element $\gamma_{ki}$ is the effective
channel coefficient between the $k$th UE and the downlink data stream targeted
towards the $i$th user, such that,
$\boldsymbol{\gamma}_{k}=\left(\sum_{m=1}^{M}\mathbf{h}_{mk}^{T}\mathbf{P}_{m}\right)^{T},$
and $\gamma_{ki}=\sum_{m=1}^{M}\mathbf{h}_{mk}^{T}\mathbf{p}_{mi}.$ Also,
since $\mathbf{p}_{mi}=x_{mi}\mathbf{\check{h}}^{*}_{mi}$,
$\gamma_{ki}=\sum_{m=1}^{M}x_{mi}\mathbf{h}_{mk}^{T}\mathbf{\check{h}}^{*}_{mi}$,
where
$\mathbf{\check{h}}^{*}_{mi}=\mathbf{\dot{h}}^{*}_{mi}+\mathbf{\hat{h}}^{*}_{mi}$.

It is important to note that $\gamma_{kk}$ can be expressed as
$\gamma_{kk}=\dot{\gamma}_{kk}+\bar{\gamma}_{kk}$, where

$\dot{\gamma}_{kk}=\sum_{m=1}^{M}x_{mk}\alpha_{mk}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*},$
(10)

and $\bar{\gamma}_{kk}$ is expanded as

$\bar{\gamma}_{kk}=\sum_{m=1}^{M}x_{mk}\left(\alpha_{mk}\left(\mathbf{\dot{h}}_{mk}^{T}\mathbf{\hat{h}}_{mk}^{*}+\mathbf{\bar{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}\right)+\mathbf{\bar{h}}_{mk}^{T}\mathbf{\hat{h}}_{mk}^{*}\right).$
(11)

The term $\dot{\gamma}_{kk}$ is the component of $\gamma_{kk}$ that consists
of only statistical and slow fading terms that can be safely assumed to be
known at the UEs, whereas $\bar{\gamma}_{kk}$ is composed of the fast fading
channel coefficients.

We consider two forms of downlink power control, viz. fixing the power
transmitted by each AP, i.e.
$\mathbb{E}[|\mathbf{y}_{m}\|^{2}]=\mathcal{E}_{\text{a,max}}$, and fixing the
power expended for each UE, i.e.
$\sum_{m=1}^{M}\mathbb{E}[\|y_{mk}\|^{2}]=\mathcal{E}_{\text{u,max}}$.

###  II-E Downlink LMMSE estimation of effective channel vector

We again assume the availability of $K$ orthogonal downlink pilot sequences in
each coherence interval, with the pilot symbol being sent to the $k$th UE at
the $n$th instant given as $\xi_{k}[n]$, such that
$\sum_{n=1}^{T}\xi_{k}[n]\xi_{l}^{*}[n]=\delta[k-l]$. We also assume
$\sigma^{2}_{d}$ to be the noise power normalized by the downlink pilot power.
Therefore, the pilot signal received by the $k$th AP at the $n$th instant is
given by

$y_{k}[n]=\sum_{i=1}^{K}\gamma_{ki}\xi_{i}[n]+\sigma_{d}w_{k}[n].$ (12)

where $w_{k}[n]\sim\mathcal{CN}(0,1)$ denotes the independent additive white
Gaussian noise (AWGN). Defining
$y^{{}^{\prime}}_{kl}=\sum_{n=1}^{T}y_{k}[n]\xi_{l}^{*}[n]$, we obtain

$y^{{}^{\prime}}_{kk}=\gamma_{kk}+\sigma_{d}w^{{}^{\prime}}_{k}.$ (13)

Now letting

$\bar{y}_{kk}=y^{{}^{\prime}}_{kk}-\dot{\gamma}_{kk}=\bar{\gamma}_{kk}+\sigma_{d}w^{{}^{\prime}}_{k},$
(14)

we obtain
$\mathbb{E}\left[\bar{y}_{kk}\bar{y}_{kk}^{*}\right]=\mathbb{E}\left[\lvert\bar{\gamma}_{kk}\rvert^{2}\right]+\sigma^{2}_{d},$
$\mathbb{E}\left[\bar{\gamma}_{kk}\bar{y}_{kk}^{*}\right]=\mathbb{E}\left[\lvert\bar{\gamma}_{kk}\rvert^{2}\right],$
and based on this, we can write the LMMSE estimate $\hat{\gamma}_{kk}$ of
$\bar{\gamma}_{kk}$ as,

$\hat{\gamma}_{kk}=\mathbb{E}\left[\bar{\gamma}_{kk}\bar{y}_{kk}^{*}\right]\mathbb{E}\left[\bar{y}_{kk}\bar{y}_{kk}^{*}\right]^{-1}\bar{y}_{kk}=\frac{\mathbb{E}\left[\lvert\bar{\gamma}_{kk}\rvert^{2}\right]\bar{y}_{kk}}{\mathbb{E}\left[\lvert\bar{\gamma}_{kk}\rvert^{2}\right]+\sigma^{2}_{d}}$
(15)

Therefore,

$\bar{\gamma}_{kk}=\hat{\gamma}_{kk}+\tilde{\gamma}_{kk}.$ (16)

with $\mathbb{E}[\hat{\gamma}_{kk}\tilde{\gamma}_{kk}^{*}]=0,$ and

$\mathbb{E}\left[\lvert\hat{\gamma}_{kk}\rvert^{2}\right]=\frac{\left(\mathbb{E}\left[\lvert\bar{\gamma}_{kk}\rvert^{2}\right]\right)^{2}}{\mathbb{E}\left[\lvert\bar{\gamma}_{kk}\rvert^{2}\right]+\sigma^{2}_{d}}.$
(17)

Furthermore,
$\mathbb{E}\left[\lvert\bar{\gamma}_{kk}\rvert^{2}\right]=\mathbb{E}\left[\lvert\hat{\gamma}_{kk}\rvert^{2}\right]+\mathbb{E}\left[\lvert\tilde{\gamma}_{kk}\rvert^{2}\right]$,
and similarly,
$\mathbb{E}\left[\lvert\gamma_{kk}\rvert^{2}\right]=\mathbb{E}\left[\lvert\dot{\gamma}_{kk}\rvert^{2}\right]+\mathbb{E}\left[\lvert\hat{\gamma}_{kk}\rvert^{2}\right]+\mathbb{E}\left[\lvert\tilde{\gamma}_{kk}\rvert^{2}\right].$

We next evaluate the performance of the system described above, under various
full and statistical beamforming schemes.

##  III Performance Under Full Beamforming

In this section, we discuss the case where the APs use full beamforming, that
is, the knowledge of both the LoS and the fast fading components to beamform
the data to the different UEs. We first consider the availability of accurate
CSI at the APs and the UEs to obtain an upper bound on the system performance.

###  III-A Accurate CSI at APs and UEs

In case accurate CSI is available at all the APs, the precoder matrix becomes,
$\mathbf{P}_{m}=\mathbf{H}_{m}^{*}\text{diag}(\mathbf{x}_{m})$, and the signal
received at the $k$th UE, $r_{k}$, can be written as,

$r_{k}=\left(\sum_{m=1}^{M}\mathbf{h}_{mk}^{T}\mathbf{P}_{m}\right)\mathbf{s}+\sigma_{o}w_{k}=\boldsymbol{\gamma}_{k}^{T}\mathbf{s}+\sigma_{o}w_{k}.$
(18)

Equivalently,

$r_{k}=\gamma_{kk}s_{k}+\sum^{K}_{\begin{subarray}{c}i=1\\\ i\neq
k\end{subarray}}\gamma_{ki}s_{i}+\sigma_{o}w_{k},$ (19)

where the first term indicates the signal of interest, the second indicates
the inter-stream interference, and the last term denotes the AWGN. Now based
on this, the achievable data rate for the $k$th user can be upper bounded
according to Theorem 1.

######  Theorem 1.

An upper bound on the rate achievable by the $k$th UE can be written as,

$R_{k}\leq\log\left(1+\frac{\mathbb{E}\left[\lvert\gamma_{kk}\rvert^{2}\right]}{\sum^{K}_{\begin{subarray}{c}i=1\\\
i\neq
k\end{subarray}}\mathbb{E}\left[\lvert\gamma_{ki}\rvert^{2}\right]+\sigma^{2}_{o}}\right),$
(20)

where,

$\displaystyle\mathbb{E}\left[\lvert\gamma_{kk}\rvert^{2}\right]=\sum_{m=1}^{M}|x_{mk}|^{2}\left(q_{mk}\zeta_{mkk}^{2}+N(N+1)\beta_{mk}^{2}+2(N+1)q_{mk}\beta_{mk}\zeta_{mkk}\right)$
$\displaystyle+\sum_{m=1}^{M}\sum_{\begin{subarray}{c}m^{\prime}=1\\\
m^{\prime}\neq
m\end{subarray}}^{M}x_{mk}x_{m^{\prime}k}\left(q_{mk}\zeta_{mkk}+\beta_{mk}N\right)\left(q_{m^{\prime}k}\zeta_{m^{\prime}kk}+\beta_{m^{\prime}k}N\right)^{*},$
(21)

and

$\displaystyle\mathbb{E}\left[\lvert\gamma_{ki}\rvert^{2}\right]=\sum_{m=1}^{M}|x_{mi}|^{2}\left(q_{mk}q_{mi}\lvert\zeta_{mki}\rvert^{2}+\beta_{mk}\beta_{mi}N+q_{mk}\beta_{mi}\zeta_{mkk}+\beta_{mk}q_{mi}\zeta_{mii}\right)$
$\displaystyle+\sum_{m=1}^{M}\sum_{\begin{subarray}{c}m^{\prime}=1\\\
m^{\prime}\neq
m\end{subarray}}^{M}x_{mi}x_{m^{\prime}i}q_{mk}q_{mi}q_{m^{\prime}k}q_{m^{\prime}i}\zeta_{mki}\zeta_{m^{\prime}ki}^{*},$
(22)

with $q_{mk}\triangleq\Pr\\{\alpha_{mk}=1\\}$, and
$\zeta_{mki}\triangleq\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mi}^{*}$.

###### Proof.

Treating the interference in (19) as noise [34], and under the assumption that
$\mathbb{E}\left[s_{i}s_{j}^{*}\right]=\delta\left[i-j\right]$, that is, the
symbols sent to the different users are independent of each other, the rate
achievable by the $k$th user, $R_{k}$, can be written as,

$R_{k}=\mathbb{E}\left[\log\left(1+\frac{\lvert\gamma_{kk}\rvert^{2}}{\sum^{K}_{\begin{subarray}{c}i=1\\\
i\neq
k\end{subarray}}\mathbb{E}\left[\lvert\gamma_{ki}\rvert^{2}\right]+\sigma^{2}_{o}}\right)\right].$
(23)

Invoking Jensen’s inequality, we can upper bound (23) as,

$R_{k}\leq\log\left(1+\frac{\mathbb{E}\left[\lvert\gamma_{kk}\rvert^{2}\right]}{\sum^{K}_{\begin{subarray}{c}i=1\\\
i\neq
k\end{subarray}}\mathbb{E}\left[\lvert\gamma_{ki}\rvert^{2}\right]+\sigma^{2}_{o}.}\right).$
(24)

The statistics for the effective direct and interfering downlink channels have
been derived in Appendix -A. ∎

###  III-B Estimated CSI at APs and UEs

In this case, the precoding matrix consists of the estimated uplink channel
coefficients, as discussed in Section II-C, and therefore,
$\mathbf{P}_{m}=\left(\mathbf{\dot{H}}^{*}_{m}diag(\boldsymbol{\alpha_{m}})+\mathbf{\hat{H}}^{*}_{m}\right)\mathbf{X}_{m}=\mathbf{\check{H}}^{*}_{m}\mathbf{X}_{m}$.

Based on this, we can write the effective downlink channel for the $k$th
user’s data stream as, $\gamma_{kk}=\dot{\gamma}_{kk}+\bar{\gamma}_{kk}$, with
the slow fading component $\dot{\gamma}_{kk}$ accurately known and the UEs
and, $\bar{\gamma}_{kk}$ being estimated as $\hat{\gamma}_{kk}$, as described
in Section II-E.

Therefore, the received signal at the $k$th user, $r_{k}$, can be expressed
as,

$r_{k}=\left(\dot{\gamma}_{kk}+\hat{\gamma}_{kk}\right)s_{k}+\tilde{\gamma}_{kk}s_{k}+\sum^{K}_{\begin{subarray}{c}i=1\\\
i\neq k\end{subarray}}\gamma_{ki}s_{i}+\sigma_{o}w_{k},$ (25)

where the first term denotes the desired signal, the second term denotes the
self interference due to channel estimation error, the third term denotes the
inter-stream interference and the last term is the AWGN. Now based on this,
the achievable data rate for the $k$th user can be upper bounded according to
Theorem 2.

######  Theorem 2.

An upper bound on the rate achievable by the $k$th UE can be written as,

$R_{k}\leq\log\left(1+\frac{\mathbb{E}\left[\lvert\dot{\gamma}_{kk}\rvert^{2}\right]+\mathbb{E}\left[\lvert\hat{\gamma}_{kk}\rvert^{2}\right]}{\mathbb{E}\left[\lvert\tilde{\gamma}_{kk}\rvert^{2}\right]+\sum^{K}_{\begin{subarray}{c}i=1\\\
i\neq
k\end{subarray}}\mathbb{E}\left[\lvert\gamma_{ki}\rvert^{2}\right]+\sigma^{2}_{o}}\right)$
(26)

where,

$\mathbb{E}\left[\lvert\gamma_{ki}\rvert^{2}\right]=\sum_{m=1}^{M}|x_{mi}|^{2}\Bigg{(}q_{mk}q_{mi}\lvert\zeta_{mki}\rvert^{2}+N\beta_{mk}\beta_{mi}\frac{\beta_{mi}}{\beta_{mi}+\sigma_{u}^{2}}+q_{mk}\beta_{mi}\frac{\beta_{mi}}{\beta_{mi}+\sigma_{u}^{2}}\zeta_{mkk}\\\
+q_{mi}\beta_{mk}\zeta_{mii}\Bigg{)}+\sum_{m=1}^{M}\sum_{\begin{subarray}{c}m^{\prime}=1\\\
m^{\prime}\neq
m\end{subarray}}^{M}x_{mi}x_{m^{\prime}i}q_{mk}q_{mi}q_{m^{\prime}k}q_{m^{\prime}i}\zeta_{mki}\zeta_{m^{\prime}ki}^{*},$
(27)

$\mathbb{E}\left[\lvert\dot{\gamma}_{kk}\rvert^{2}\right]=\sum_{m=1}^{M}\lvert
x_{mk}\rvert^{2}q_{mk}\zeta_{mkk}^{2}+\sum_{m=1}^{M}\sum_{\begin{subarray}{c}m^{\prime}=1\\\
m^{\prime}\neq
m\end{subarray}}^{M}x_{mi}x_{m^{\prime}i}q_{mk}q_{m^{\prime}k}\zeta_{mkk}\zeta_{m^{\prime}kk}^{*},$
(28)

$\displaystyle\mathbb{E}\left[\lvert\hat{\gamma}_{kk}\rvert^{2}\right]=\sum_{m=1}^{M}|x_{mk}|^{2}\left(N(N+1)\beta_{mk}^{2}\left(\frac{\beta_{mk}}{\beta_{mk}+\sigma_{u}^{2}}\right)^{2}+N\beta_{mk}^{2}\left(\frac{\sqrt{\beta_{mk}}\sigma_{u}}{\beta_{mk}+\sigma_{u}^{2}}\right)^{2}+\right.$
(29)
$\displaystyle\left.q_{mk}\beta_{mk}\left(1+\frac{\beta_{mk}}{\beta_{mk}+\sigma_{u}^{2}}\right)\zeta_{mkk}+2Nq_{mk}\beta_{mk}\frac{\beta_{mk}}{\beta_{mk}+\sigma_{u}^{2}}\zeta_{mkk}\right)$
$\displaystyle+\sum_{m=1}^{M}\sum_{\begin{subarray}{c}m^{\prime}=1\\\
m^{\prime}\neq
m\end{subarray}}^{M}x_{mk}x_{m^{\prime}k}\left(q_{mk}\zeta_{mkk}N\beta_{m^{\prime}k}\frac{\beta_{m^{\prime}k}}{\beta_{m^{\prime}k}+\sigma_{u}^{2}}\right.$
$\displaystyle\left.+q_{m^{\prime}k}\zeta_{m^{\prime}kk}N\beta_{mk}\frac{\beta_{mk}}{\beta_{mk}+\sigma_{u}^{2}}+N^{2}\beta_{mk}\beta_{m^{\prime}k}\frac{\beta_{mk}}{\beta_{mk}+\sigma_{u}^{2}}\frac{\beta_{m^{\prime}k}}{\beta_{m^{\prime}k}+\sigma_{u}^{2}}\right),$

$\mathbb{E}\left[\lvert\hat{\gamma}_{kk}\rvert^{2}\right]=\frac{\left(\mathbb{E}\left[\lvert\bar{\gamma}_{kk}\rvert^{2}\right]\right)^{2}}{\mathbb{E}\left[\lvert\bar{\gamma}_{kk}\rvert^{2}\right]+\sigma^{2}_{d}},$
and
$\mathbb{E}\left[\lvert\tilde{\gamma}_{kk}\rvert^{2}\right]=\frac{\mathbb{E}\left[\lvert\bar{\gamma}_{kk}\rvert^{2}\right]\sigma^{2}_{d}}{\mathbb{E}\left[\lvert\bar{\gamma}_{kk}\rvert^{2}\right]+\sigma^{2}_{d}},$
with $q_{mk}\triangleq\Pr\\{\alpha_{mk}=1\\}$, and
$\zeta_{mki}\triangleq\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mi}^{*}$.

###### Proof.

Treating the interference in (25) as noise, assuming
$\mathbb{E}\left[s_{i}s_{j}^{*}\right]=\delta\left[i-j\right]$, the rate
achievable by the $k$th user, $R_{k}$, can be written as,

$R_{k}=\mathbb{E}\left[\log\left(1+\frac{\lvert\dot{\gamma}_{kk}+\hat{\gamma}_{kk}\rvert^{2}}{\mathbb{E}\left[\lvert\tilde{\gamma}_{kk}\rvert^{2}\right]+\sum^{K}_{\begin{subarray}{c}i=1\\\
i\neq
k\end{subarray}}\mathbb{E}\left[\lvert\gamma_{ki}\rvert^{2}\right]+\sigma^{2}_{o}}\right)\right].$
(30)

Invoking Jensen’s inequality we can upper bound (30) as,

$R_{k}\leq\log\left(1+\frac{\mathbb{E}\left[\lvert\dot{\gamma}_{kk}\rvert^{2}\right]+\mathbb{E}\left[\lvert\hat{\gamma}_{kk}\rvert^{2}\right]}{\mathbb{E}\left[\lvert\tilde{\gamma}_{kk}\rvert^{2}\right]+\sum^{K}_{\begin{subarray}{c}i=1\\\
i\neq
k\end{subarray}}\mathbb{E}\left[\lvert\gamma_{ki}\rvert^{2}\right]+\sigma^{2}_{o}}\right).$
(31)

The statistics for the effective direct and interfering downlink channels,
have been derived in Appendix -B. ∎

Having looked at the performance of the CF-mMIMO system under the availability
of accurate/ estimated fast fading components, we next consider the case where
no information about the fast fading components is available at the CPU/ APs.
As a consequence, the APs use only the information about the LoS components to
beamform the data to the respective UEs.

##  IV Performance Analysis with Statistical Beamforming

We now consider the case where the BS employs statistical beamforming in the
downlink [28]. It is important to note that, even in the absence of any fast
fading CSI at the UEs, the APs can still perform limited downlink training to
allow the UEs to estimate the effective downlink channels, and use those to
detect the received data symbols. As a consequence, we consider two variants
of the statistical downlink beamforming scheme, viz. with and without downlink
pilots. However, in either case, the precoding matrix is composed of only the
LoS components of the channel matrix and is given as
$\mathbf{P}_{m}=\mathbf{\dot{H}}^{*}_{m}\text{diag}(\boldsymbol{\alpha}_{m})\mathbf{X}_{m}$.
Consequently,

$\gamma_{ki}=\sum_{m=1}^{M}x_{mi}\alpha_{mi}\mathbf{h}_{mk}^{T}\mathbf{\dot{h}}_{mi}^{*},$
(32)

with $\gamma_{kk}=\dot{\gamma}_{kk}+\bar{\gamma}_{kk}$ such that,
$\dot{\gamma}_{kk}=\sum_{m=1}^{M}x_{mk}\alpha_{mk}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*},$
and
$\bar{\gamma}_{kk}=\sum_{m=1}^{M}x_{mk}\alpha_{mk}\sqrt{\beta_{mk}}\mathbf{\bar{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}.$
We first consider the case where no downlink pilots are transmitted by the
APs.

###  IV-A No Downlink Pilots

We note that,

$r_{k}=\dot{\gamma}_{kk}s_{k}+\bar{\gamma}_{kk}s_{k}+\sum^{K}_{\begin{subarray}{c}i=1\\\
i\neq k\end{subarray}}\gamma_{ki}s_{i}+\sigma_{o}w_{k},$ (33)

with the first term denoting the desired signal, the second term denoting the
self interference due to the fast fading component of the channel, the third
term denoting the inter-stream interference and the last term being the AWGN.
Now based on this, the achievable data rate for the $k$th user can be upper
bounded according to Theorem 3.

######  Theorem 3.

An upper bound on the rate achievable by the $k$th UE can be written as,

$R_{k}\leq\log\left(1+\frac{\mathbb{E}\left[\lvert\dot{\gamma}_{kk}\rvert^{2}\right]}{\mathbb{E}\left[\lvert\bar{\gamma}_{kk}\rvert^{2}\right]+\sum^{K}_{\begin{subarray}{c}i=1\\\
i\neq
k\end{subarray}}\mathbb{E}\left[\lvert\gamma_{ki}\rvert^{2}\right]+\sigma^{2}_{o}}\right)$
(34)

where,

$\begin{split}\mathbb{E}\left[\lvert\gamma_{ki}\rvert^{2}\right]=\sum_{m=1}^{M}\lvert
x_{mi}\rvert^{2}\left(q_{mk}q_{mi}\lvert\zeta_{mki}\rvert^{2}+\beta_{mk}q_{mi}\zeta_{mii}\right)\\\
+\sum_{m=1}^{M}\sum_{\begin{subarray}{c}m^{\prime}=1\\\ m^{\prime}\neq
m\end{subarray}}^{M}x_{mi}x_{m^{\prime}i}q_{mk}q_{mi}q_{m^{\prime}k}q_{m^{\prime}i}\zeta_{mki}\zeta_{m^{\prime}ki}^{*},\end{split}$
(35)

$\mathbb{E}\left[\lvert\dot{\gamma}_{kk}\rvert^{2}\right]=\sum_{m=1}^{M}\lvert
x_{mk}\rvert^{2}q_{mk}\zeta_{mkk}^{2}+\sum_{m=1}^{M}\sum_{\begin{subarray}{c}m^{\prime}=1\\\
m^{\prime}\neq
m\end{subarray}}^{M}x_{mi}x_{m^{\prime}i}q_{mk}q_{m^{\prime}k}\zeta_{mkk}\zeta_{m^{\prime}kk}^{*},$
(36)

and

$\mathbb{E}\left[\lvert\bar{\gamma}_{kk}\rvert^{2}\right]=\sum_{m=1}^{M}\lvert
x_{mk}\rvert^{2}\beta_{mk}q_{mk}\zeta_{mkk},$ (37)

with $q_{mk}\triangleq\Pr\\{\alpha_{mk}=1\\}$, and
$\zeta_{mki}\triangleq\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mi}^{*}$.

###### Proof.

Treating the interference in (33) as noise, assuming
$\mathbb{E}\left[s_{i}s_{j}^{*}\right]=\delta\left[i-j\right]$, we can write,

$R_{k}=\mathbb{E}\left[\log\left(1+\frac{\lvert\dot{\gamma}_{kk}\rvert^{2}}{\mathbb{E}\left[\lvert\bar{\gamma}_{kk}\rvert^{2}\right]+\sum^{K}_{\begin{subarray}{c}i=1\\\
i\neq
k\end{subarray}}\mathbb{E}\left[\lvert\gamma_{ki}\rvert^{2}\right]+\sigma^{2}_{o}}\right)\right].$
(38)

Invoking Jensen’s inequality, we can upper bound (38) as

$R_{k}\leq\log\left(1+\frac{\mathbb{E}\left[\lvert\dot{\gamma}_{kk}\rvert^{2}\right]}{\mathbb{E}\left[\lvert\bar{\gamma}_{kk}\rvert^{2}\right]+\sum^{K}_{\begin{subarray}{c}i=1\\\
i\neq
k\end{subarray}}\mathbb{E}\left[\lvert\gamma_{ki}\rvert^{2}\right]+\sigma^{2}_{o}}\right).$
(39)

The statistics for the effective direct and interfering downlink channels,
have been derived in Appendix -C.

∎

###  IV-B With Limited Downlink Training

Using limited downlink pilots, we can estimate the fast fading component of
the effective downlink channel to the $k$th user, $\bar{\gamma}_{kk}$ as
$\hat{\gamma}_{kk}$. We note that the received signal can be expressed as

$r_{k}=\left(\dot{\gamma}_{kk}+\hat{\gamma}_{kk}\right)s_{k}+\tilde{\gamma}_{kk}s_{k}+\sum^{K}_{\begin{subarray}{c}i=1\\\
i\neq k\end{subarray}}\gamma_{ki}s_{i}+\sigma_{o}w_{k},$ (40)

with the first term denoting the desired signal, the second term denoting the
self interference due to the error in estimating the fast fading component of
the channel, the third term denoting the inter-stream interference and the
last term being the AWGN. Based on this, the achievable data rate for the
$k$th user can be upper bounded according to Theorem 4.

######  Theorem 4.

An upper bound on the rate achievable by the $k$th UE can be written as,

$R_{k}\leq\log\left(1+\frac{\mathbb{E}\left[\lvert\dot{\gamma}_{kk}\rvert^{2}\right]+\mathbb{E}\left[\lvert\hat{\gamma}_{kk}\rvert^{2}\right]}{\mathbb{E}\left[\lvert\tilde{\gamma}_{kk}\rvert^{2}\right]+\sum^{K}_{\begin{subarray}{c}i=1\\\
i\neq
k\end{subarray}}\mathbb{E}\left[\lvert\gamma_{ki}\rvert^{2}\right]+\sigma^{2}_{o}}\right)$
(41)

where,

$\begin{split}\mathbb{E}\left[\lvert\gamma_{ki}\rvert^{2}\right]=\sum_{m=1}^{M}\lvert
x_{mi}\rvert^{2}\left(q_{mk}q_{mi}\lvert\zeta_{mki}\rvert^{2}+\beta_{mk}q_{mi}\zeta_{mii}\right)\\\
+\sum_{m=1}^{M}\sum_{\begin{subarray}{c}m^{\prime}=1\\\ m^{\prime}\neq
m\end{subarray}}^{M}x_{mi}x_{m^{\prime}i}q_{mk}q_{mi}q_{m^{\prime}k}q_{m^{\prime}i}\zeta_{mki}\zeta_{m^{\prime}ki}^{*},\end{split}$
(42)

$\mathbb{E}\left[\lvert\dot{\gamma}_{kk}\rvert^{2}\right]=\sum_{m=1}^{M}\lvert
x_{mk}\rvert^{2}q_{mk}\zeta_{mkk}^{2}+\sum_{m=1}^{M}\sum_{\begin{subarray}{c}m^{\prime}=1\\\
m^{\prime}\neq
m\end{subarray}}^{M}x_{mi}x_{m^{\prime}i}q_{mk}q_{m^{\prime}k}\zeta_{mkk}\zeta_{m^{\prime}kk}^{*},$
(43)

$\mathbb{E}\left[\lvert\bar{\gamma}_{kk}\rvert^{2}\right]=\sum_{m=1}^{M}\lvert
x_{mk}\rvert^{2}\beta_{mk}q_{mk}\zeta_{mkk},$ (44)

$\mathbb{E}\left[\lvert\hat{\gamma}_{kk}\rvert^{2}\right]=\frac{\left(\mathbb{E}\left[\lvert\bar{\gamma}_{kk}\rvert^{2}\right]\right)^{2}}{\mathbb{E}\left[\lvert\bar{\gamma}_{kk}\rvert^{2}\right]+\sigma^{2}_{d}},$

and

$\mathbb{E}\left[\lvert\tilde{\gamma}_{kk}\rvert^{2}\right]=\frac{\mathbb{E}\left[\lvert\bar{\gamma}_{kk}\rvert^{2}\right]\sigma^{2}_{d}}{\mathbb{E}\left[\lvert\bar{\gamma}_{kk}\rvert^{2}\right]+\sigma^{2}_{d}},$

with $q_{mk}\triangleq\Pr\\{\alpha_{mk}=1\\}$, and
$\zeta_{mki}\triangleq\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mi}^{*}$.

###### Proof.

Treating the interference in (40) as noise, and under the assumption that
$\mathbb{E}\left[s_{i}s_{j}^{*}\right]=\delta\left[i-j\right]$,we can write,

$R_{k}=\mathbb{E}\left[\log\left(1+\frac{\lvert\dot{\gamma}_{kk}+\hat{\gamma}_{kk}\rvert^{2}}{\mathbb{E}\left[\lvert\tilde{\gamma}_{kk}\rvert^{2}\right]+\sum^{K}_{\begin{subarray}{c}i=1\\\
i\neq
k\end{subarray}}\mathbb{E}\left[\lvert\gamma_{ki}\rvert^{2}\right]+\sigma^{2}_{o}}\right)\right].$
(45)

Invoking Jensen’s inequality, we can upper bound (45) as,

$R_{k}\leq\log\left(1+\frac{\mathbb{E}\left[\lvert\dot{\gamma}_{kk}\rvert^{2}\right]+\mathbb{E}\left[\lvert\hat{\gamma}_{kk}\rvert^{2}\right]}{\mathbb{E}\left[\lvert\tilde{\gamma}_{kk}\rvert^{2}\right]+\sum^{K}_{\begin{subarray}{c}i=1\\\
i\neq
k\end{subarray}}\mathbb{E}\left[\lvert\gamma_{ki}\rvert^{2}\right]+\sigma^{2}_{o}}\right).$
(46)

The statistics for the effective direct and interfering downlink channels have
been derived in Appendix -C. ∎

We next validate our derived results via numerical simulations.

##  V Numerical Results

In this section we present simulation and numerical results to validate the
derived theoretical bounds, and to analyze the downlink performance of the CF-
mMIMO system under LoS/ NLoS channels. Unless stated otherwise, the simulation
parameters used for these experiments are listed in Table I.

Table I: Simulation Parameters Parameter | Value | Parameter | Value
---|---|---|---
Reference distance ($d_{0}$) | 1 $\mathrm{m}$ | Network Area |  $1$ ${\mathrm{km}}^{2}$=$10^{6}$ ${\mathrm{m}}^{2}$
Number of Users ($K$) | 64 | Number of APs (M) | 1024
UE antenna height | 1.5 $\mathrm{m}$ | AP Height | 10 $\mathrm{m}$
Carrier frequency ($f_{c}$) | 3.5 GHz | $\eta$ | 0.5
$\mu$ |  $300/$${\mathrm{km}}^{2}$ | Average building height ($\rho$) | 20 $\mathrm{m}$
$N$ | 1 | Normalised Signal Noise Power ($\sigma_{o}$) | 20dB
Figure 1: PMF of number of LoS channels per UE for varying AP densities

In Fig. 1, we plot the probability mass function (PMF) for the number of LoS
channels for an arbitrarily placed UE. We observe that increasing the number
of APs from 128 to 1024 dramatically reduces the probability of the absence of
an LoS channel from 60% to 2%. This, coupled with the deterministic and high
gain nature of LoS channels motivates us to deploy more single antenna APs
instead of a smaller number of APs with multiple antennas. Therefore, for AP
densities greater than $10^{3}$ APs /${\mathrm{km}}^{2}$, transmission schemes
that rely heavily on LoS communication can be deployed effectively without
significantly affecting the 5th percentile performance.

Figure 2: Average per user achievable rate as a function of the data SNR for
different beamforming schemes with a fixed amount of power being transmitted
by each AP. Figure 3: Average per user achievable rate as a function of the
data SNR for different beamforming schemes with a fixed amount of power being
transmitted to each UE.

In Fig. 2, we plot the average achievable rate per user against the normalized
transmitted data power while constraining the power transmitted by each AP. We
note that the performance of statistical beamforming with downlink training
matches closely with the performance achieved under the availability of
accurate CSI. This is due to the fact that, under statistical beamforming, the
APs allocate no power to the NLoS channels between themseleves and the UEs,
reducing the inter-stream interference, and improving the SINR. It is
important to note that, since there is a 76% probability that a certain AP has
no LoS channels to UEs, it will remain silent, thus making statistical
beamforming more power efficient. We keep the maximum power of the system
constant across power allocation schemes, and the respective maximum power
constants are set as,
$\frac{\mathcal{E}_{u,max}}{\mathcal{E}_{a,max}}=\frac{M}{K}$.

The average user throughput as a function of the normalized transmitted data
SNR while constraining the power transmitted to each UE is plotted in Fig. 3.
We observe that, in this case, statistical beamforming, both with and without
downlink training, outperforms full beamforming. This is due to the fact that
in the former case, the power is allocated only to the LoS channels rather
than all the possible channels to a user, most of which are NLoS. We also
observe that all the schemes other than statistical beamforming with downlink
training seem to saturate at high SNRs. In case of full beamforming, this is
explained by high levels of inter-stream interference due to all $1024\times
64$ channels being in use. On the other hand, pure statistical beamforming
suffers from strong self-interference since the scheme neglects the fast-
fading channel components.

Figure 4: Simulated CDF of the per user spectral efficiency for different
beamforming schemes and a fixed amount of power being transmitted by each AP.
Figure 5: Simulated CDF of the per user spectral efficiency for different
beamforming schemes and a fixed amount of power being transmitted to each UE.

In addition to the average throughputs, it is also necessary to analyse the
distribution functions of the user throughputs for all the schemes, as done in
in Figs. 4 and 5. We observe that, as expected, the statistical beamforming
schemes have an abrupt jump of about 2% at rate 0. This corresponds to the
fraction of users that do not have an available LoS channel as observed in
Fig. 1. It interesting to note that for both the power control schemes the
performance of users from the 5th percentile to about 50th percentile is
identical.

In the case where the power being transmitted by each AP remains fixed, the
performance of the high throughput users is commensurate with the observations
from the achievable rate vs. SNR plots (Figs. 2 and 3). It is interesting to
note how closely the CDFs of the statistical schemes track the accurate CSI
case. Apart from the fairly low performance loss at high throughput, the only
major limitation is the abrupt 2% jump observed at 0 due to the unavoidable
outage. This outage is because of the scheme’s reliance on high AP densities.

In case we limit the power transmitted to each UE, the statistical schemes
work at full power capacity, unlike in the case where we limit the power
transmitted by each AP. Constraining the power transmitted to each UE allows
the CPU to divert the bulk of its power to APs located near active users. This
fact, coupled with the reliance of statistical schemes on LoS transmissions
makes it a superior choice for high throughput users. We also note that the
self interference experienced by the pure statistical transmission at high
SNRs make this scheme interference limited and saturate its performance.

Figure 6: Average per user achievable rate as a function of the AP density for
different beamforming schemes with a fixed amount of power being transmitted
by each AP. Figure 7: Average per user achievable rate as a function of the AP
density for different beamforming schemes with a fixed amount of power being
transmitted to each UE.

In Figs. 6 and 7 we plot the average throughput of each precoding scheme with
AP density varying from 128 AP/${\mathrm{km}}^{2}$ to 2048
AP/${\mathrm{km}}^{2}$ for both the power control schemes. At first, it is
apparent that the performance gains from increasing the AP density start
saturating at about $10^{3}$ AP/${\mathrm{km}}^{2}$. This can be attributed to
the increased inter-stream interference. In Fig. 6, where the case of
constraining the power transmitted by each AP is considered, the observations
are mostly in line with Fig. 2. Here, accurate CSI precoding performs the
best, followed by statistical beamforming with downlink training, pure
statistical beamforming and finally, estimated CSI precoding. The only
exception is at $M=128$, where pure statistical beamforming performs worse
than estimated CSI precoding. This is to be expected, since only about 40% of
UEs have an LoS channel. We again note that statistical beamforming is
operating at only 24% of the total power available, as opposed to the full
beamforming schemes that use all the available power. For the case where power
transmitted to each UE is constrained (refer Fig. 7), since the statistical
schemes utilize nearly all the power available, their performance far exceeds
that obtained with full beamforming, especially for low AP densities. It must
be noted that, since a relatively high proportion of UEs have no LoS channels
at low AP densities, we expect a trade-off in their CDFs. This overwhelming
gain in throughput at low AP densities, as mentioned earlier, is because all
the energy of the system is being diverted to APs with LoS channels near
active users. However, this phenomenon causes a reduction in the throughput of
the statistical beamforming schemes at an AP density of $M=2048$.

It must be noted that the statistical beamforming scheme without downlink
training has the unique advantage of not allocating any resources for uplink/
downlink training, and using the entire frame for data transmission.

##  VI Conclusion

In this paper, we have discussed the downlink performance of CF-mMIMO systems
under probabilistic LoS/ NLoS channels. We have evaluated the performance of
these systems under full, as well as statistical beamforming schemes. At
sufficiently high AP densities, it was observed that statistical beamforming
schemes provide superior throughput with little or no loss in coverage and at
much higher energy efficiency. Furthermore, for an AP density of 1024
AP/${\mathrm{km}}^{2}$ the achievable rates of users between the fifth and the
fiftieth percentile is nearly identical, with only 2% of the users in
operating in the statistical schemes experiencing zero throughput, and most of
the variation being among users who are among the top half in terms of
achievable rates. We also show that the performance and coverage gains
obtained from increasing the AP density saturates at about 1000
AP/${\mathrm{km}}^{2}$. This shows that LoS reliant statistical beamforming
schemes can lead to performance that is commensurate or even superior to full
beamforming. Further research would focus on optimal power control schemes for
CF-mMIMO systems under probabilistic LoS/NLoS channel conditions as well as on
precoding schemes that are intended to mitigate inter-stream interference such
as regularised zero forcing.

###  -A Channel Statistics with accurate CSI

We know that,
$\mathbf{h}_{mk}=\alpha_{mk}\mathbf{\dot{h}}_{mk}+\sqrt{\beta_{mk}}\mathbf{\bar{h}}_{mk}$,
where $\alpha_{mk}$ is a Bernoulli RV with $\Pr\\{\alpha_{mk}=1\\}=q_{mk}$.
Defining $\zeta_{mki}=\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mi}^{*}$, we
can write,

$\mathbb{E}\left[\gamma_{ki}\right]=\sum_{m=1}^{M}x_{mi}q_{mk}q_{mi}\zeta_{mki}+x_{mk}N\beta_{mk}\delta[k-i].$
(47)

Similarly,

$\mathbb{E}\left[\lvert\gamma_{kk}\rvert^{2}\right]=\sum_{m=1}^{M}|x_{mk}|^{2}\mathbb{E}\left[|\mathbf{h}_{mk}^{T}\mathbf{h}_{mk}^{*}|^{2}\right]\\\
+\sum_{m=1}^{M}\sum_{\begin{subarray}{c}m^{\prime}=1\\\ m^{\prime}\neq
m\end{subarray}}^{M}x_{mk}x_{m^{\prime}k}\mathbb{E}\left[\mathbf{h}_{mk}^{T}\mathbf{h}_{mk}^{*}\right]\mathbb{E}\left[\mathbf{h}_{m^{\prime}k}^{T}\mathbf{h}_{m^{\prime}k}^{*}\right]^{*},$
(48)

where
$\mathbb{E}\left[\mathbf{h}_{mk}^{T}\mathbf{h}_{mk}^{*}\right]=q_{mk}\zeta_{mkk}+\beta_{mk}N$
and,

$\displaystyle\mathbb{E}\left[|\mathbf{h}_{mk}^{T}\mathbf{h}_{mk}^{*}|^{2}\right]$
$\displaystyle=\mathbb{E}\left[\lvert\alpha_{mk}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}+\beta_{mk}\mathbf{\bar{h}}_{mk}^{T}\mathbf{\bar{h}}_{mk}^{*}+\alpha_{mk}\sqrt{\beta_{mk}}(\mathbf{\bar{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}+\mathbf{\dot{h}}_{mk}^{T}\mathbf{\bar{h}}_{mk}^{*})\rvert^{2}\right]$
$\displaystyle=\mathbb{E}\left[\alpha_{mk}\lvert\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}\rvert^{2}+\beta_{mk}^{2}\lvert\mathbf{\bar{h}}_{mk}^{T}\mathbf{\bar{h}}_{mk}^{*}\rvert^{2}+2\alpha_{mk}\beta_{mk}\lvert\mathbf{\dot{h}}_{mk}^{T}\mathbf{\bar{h}}_{mk}^{*}\rvert^{2}\right.$
$\displaystyle\left.+2\Re(\alpha_{mk}\beta_{mk}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}\mathbf{\bar{h}}_{mk}^{T}\mathbf{\bar{h}}_{mk}^{*}+\alpha_{mk}\sqrt{\beta_{mk}}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\bar{h}}_{mk}^{*}\right.$
$\displaystyle\left.+\alpha_{mk}\sqrt{\beta_{mk}}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}\mathbf{\dot{h}}_{mk}^{H}\mathbf{\bar{h}}_{mk}+\alpha_{mk}\beta_{mk}^{\frac{3}{2}}\mathbf{\bar{h}}_{mk}^{T}\mathbf{\bar{h}}_{mk}^{*}\mathbf{\bar{h}}_{mk}^{H}\mathbf{\dot{h}}_{mk}\right.$
$\displaystyle\left.+\alpha_{mk}\beta_{mk}^{\frac{3}{2}}\mathbf{\bar{h}}_{mk}^{T}\mathbf{\bar{h}}_{mk}^{*}\mathbf{\bar{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}+\alpha_{mk}\beta_{mk}(\mathbf{\bar{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*})^{2})\right]$
$\displaystyle=q_{mk}\zeta_{mkk}^{2}+N(N+1)\beta_{mk}^{2}+2(N+1)q_{mk}\beta_{mk}\zeta_{mkk}.$

This simplifies to the expression in (1).

Similarly for $\gamma_{ki}$ and $k\neq i$,

$\mathbb{E}\left[\lvert\gamma_{ki}\rvert^{2}\right]=\sum_{m=1}^{M}|x_{mi}|^{2}\mathbb{E}\left[|\mathbf{h}_{mk}^{T}\mathbf{h}_{mi}^{*}|^{2}\right]+\sum_{m=1}^{M}\sum_{\begin{subarray}{c}m^{\prime}=1\\\
m^{\prime}\neq
m\end{subarray}}^{M}x_{mi}x_{m^{\prime}i}\mathbb{E}\left[\mathbf{h}_{mk}^{T}\right]\mathbb{E}\left[\mathbf{h}_{mi}^{*}\right]\mathbb{E}\left[\mathbf{h}_{m^{\prime}k}^{T}\right]^{*}\mathbb{E}\left[\mathbf{h}_{m^{\prime}i}^{*}\right]^{*},$
(49)

where, $\mathbb{E}\left[\mathbf{h}_{mk}\right]=q_{mk}\mathbf{\dot{h}}_{mk}$
and,

$\displaystyle\mathbb{E}\left[|\mathbf{h}_{mk}^{T}\mathbf{h}_{mi}^{*}|^{2}\right]$
$\displaystyle=\mathbb{E}\left[\lvert\alpha_{mk}\alpha_{mi}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mi}^{*}+\sqrt{\beta_{mk}\beta_{mi}}\mathbf{\bar{h}}_{mk}^{T}\mathbf{\bar{h}}_{mi}^{*}+\alpha_{mk}\sqrt{\beta_{mi}}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\bar{h}}_{mi}^{*}\right.$
$\displaystyle\left.+\sqrt{\beta_{mk}}\alpha_{mi}\mathbf{\bar{h}}_{mk}^{T}\mathbf{\dot{h}}_{mi}^{*}\rvert^{2}\right]$
$\displaystyle=\mathbb{E}\left[\alpha_{mk}\alpha_{mi}\lvert\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mi}^{*}\rvert^{2}+\beta_{mk}\beta_{mi}\lvert\mathbf{\bar{h}}_{mk}^{T}\mathbf{\bar{h}}_{mi}^{*}\rvert^{2}+\alpha_{mk}\beta_{mi}\lvert\mathbf{\dot{h}}_{mk}^{T}\mathbf{\bar{h}}_{mi}^{*}\rvert^{2}\right.$
$\displaystyle\left.+\beta_{mk}\alpha_{mi}\lvert\mathbf{\bar{h}}_{mk}^{T}\mathbf{\dot{h}}_{mi}^{*}\rvert^{2}+2\Re\Big{(}\alpha_{mk}\alpha_{mi}\sqrt{\beta_{mk}\beta_{mi}}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mi}^{*}\mathbf{\bar{h}}_{mk}^{H}\mathbf{\bar{h}}_{mi}\right.$
$\displaystyle\left.+\alpha_{mk}\alpha_{mi}\sqrt{\beta_{mi}}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mi}^{*}\mathbf{\dot{h}}_{mk}^{H}\mathbf{\bar{h}}_{mi}+\alpha_{mk}\alpha_{mi}\sqrt{\beta_{mk}}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mi}^{*}\mathbf{\dot{h}}_{mi}^{T}\mathbf{\bar{h}}_{mk}^{*}\right.$
$\displaystyle\left.+\alpha_{mk}\sqrt{\beta_{mk}}\beta_{mi}\mathbf{\bar{h}}_{mk}^{T}\mathbf{\bar{h}}_{mi}^{*}\mathbf{\bar{h}}_{mi}^{T}\mathbf{\dot{h}}_{mk}^{*}+\alpha_{mk}\beta_{mk}\sqrt{\beta_{mi}}\mathbf{\bar{h}}_{mk}^{T}\mathbf{\bar{h}}_{mi}^{*}\mathbf{\bar{h}}_{mk}^{H}\mathbf{\dot{h}}_{mi}\right.$
$\displaystyle\left.+\alpha_{mk}\alpha_{mi}\sqrt{\beta_{mk}\beta_{mi}}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\bar{h}}_{mi}^{*}\mathbf{\bar{h}}_{mk}^{H}\mathbf{\dot{h}}_{mi}\Big{)}\right]$
$\displaystyle=q_{mk}q_{mi}\lvert\zeta_{mki}\rvert^{2}+\beta_{mk}\beta_{mi}N+q_{mk}\beta_{mi}\zeta_{mkk}+\beta_{mk}q_{mi}\zeta_{mii}.$

This simplifies to the expression in (1).

###  -B Channel Statistics with Estimated CSI at APs and UEs

$\displaystyle\mathbb{E}\left[\gamma_{kk}\right]=$
$\displaystyle\sum_{m=1}^{M}x_{mi}\Big{(}\mathbb{E}\left[\alpha_{mk}\right]\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}+\sqrt{\beta_{mk}}\mathbb{E}\left[\alpha_{mk}\right]\mathbb{E}\left[\mathbf{\bar{h}}_{mk}^{T}\right]\mathbf{\dot{h}}_{mk}^{*}+\mathbb{E}[\alpha_{mk}]\sqrt{\beta_{mk}}\mathbf{\dot{h}}_{mk}^{T}\mathbb{E}[\mathbf{\hat{h}}_{mk}^{*}]$
(50)
$\displaystyle+\beta_{mk}\mathbb{E}[\mathbf{\bar{h}}_{mk}^{T}\mathbf{\hat{h}}_{mk}^{*}]\Big{)}=\sum_{m=1}^{M}x_{mk}\Big{(}q_{mk}\zeta_{mkk}+\frac{N\beta_{mk}}{\beta_{mk}+\sigma_{u}^{2}}\Big{)}$
$\displaystyle\mathbb{E}\left[\gamma_{ki}\right]=$
$\displaystyle\sum_{m=1}^{M}x_{mk}\Big{(}\mathbb{E}\left[\alpha_{mk}\alpha_{mi}\right]\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mi}^{*}+\sqrt{\beta_{mk}}\mathbb{E}\left[\alpha_{mi}\right]\mathbb{E}\left[\mathbf{\bar{h}}_{mk}^{T}\right]\mathbf{\dot{h}}_{mi}^{*}+\mathbb{E}[\alpha_{mk}]\sqrt{\beta_{mi}}\mathbf{\dot{h}}_{mk}^{T}\mathbb{E}[\mathbf{\hat{h}}_{mi}^{*}]$
(51)
$\displaystyle+\sqrt{\beta_{mk}\beta}_{mi}\mathbb{E}[\mathbf{\bar{h}}_{mk}^{T}\mathbf{\hat{h}}_{mi}^{*}]\Big{)}=\sum_{m=1}^{M}x_{mi}q_{mk}q_{mi}\zeta_{mki}$
$\mathbb{E}\left[\dot{\gamma}_{kk}\right]=\sum_{m=1}^{M}x_{mk}\mathbb{E}\left[\alpha_{mk}\right]\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}=\sum_{m=1}^{M}x_{mk}q_{mk}\zeta_{mkk}$
(52)
$\mathbb{E}\left[\bar{\gamma}_{kk}\right]=\mathbb{E}\left[\gamma_{kk}\right]-\mathbb{E}\left[\dot{\gamma}_{kk}\right]=\sum_{m=1}^{M}x_{mk}N\frac{\beta_{mk}}{\beta_{mk}+\sigma_{u}^{2}}$
(53)

Similarly,

$\displaystyle\mathbb{E}\left[\lvert\gamma_{kk}\rvert^{2}\right]=\sum_{m=1}^{M}|x_{mk}|^{2}\mathbb{E}\left[|\mathbf{h}_{mk}^{T}\mathbf{\check{h}}^{*}_{mk}|^{2}\right]+\sum_{m=1}^{M}\sum_{\begin{subarray}{c}m^{\prime}=1\\\
m^{\prime}\neq
m\end{subarray}}^{M}x_{mk}x_{m^{\prime}k}\mathbb{E}\left[\mathbf{h}_{mk}^{T}\mathbf{\check{h}}^{*}_{mk}\right]\mathbb{E}\left[\mathbf{h}_{m^{\prime}k}^{T}\mathbf{\check{h}}^{*}_{m^{\prime}k}\right]^{*},$
(54)

where
$\mathbb{E}\left[\mathbf{h}_{mk}^{T}\mathbf{\check{h}}^{*}_{mk}\right]=q_{mk}\zeta_{mkk}+N\beta_{mk}\frac{\beta_{mk}}{\beta_{mk}+\sigma_{u}^{2}}$
and,

$\displaystyle\mathbb{E}\left[|\mathbf{h}_{mk}^{T}\mathbf{\check{h}}^{*}_{mk}|^{2}\right]$
$\displaystyle=\mathbb{E}\left[\lvert\alpha_{mk}\mathbf{\dot{h}}^{T}_{mk}\mathbf{\dot{h}}^{*}_{mk}+\beta_{mk}\mathbf{\bar{h}}^{T}_{mk}\mathbf{\hat{h}}^{*}_{mk}+\alpha_{mk}\sqrt{\beta_{mk}}(\mathbf{\dot{h}}^{T}_{mk}\mathbf{\hat{h}}^{*}_{mk}+\mathbf{\bar{h}}^{T}_{mk}\mathbf{\dot{h}}^{*}_{mk})\rvert^{2}\right]$
$\displaystyle=\mathbb{E}\left[\alpha_{mk}\lvert\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}\rvert^{2}+\beta_{mk}^{2}\lvert\mathbf{\bar{h}}_{mk}^{T}\mathbf{\hat{h}}_{mk}^{*}\rvert^{2}+\alpha_{mk}\beta_{mk}\lvert\mathbf{\dot{h}}_{mk}^{T}\mathbf{\hat{h}}_{mk}^{*}\rvert^{2}\right.$
$\displaystyle\left.+\alpha_{mk}\beta_{mk}\lvert\mathbf{\bar{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}\rvert^{2}+2\Re\Big{(}\alpha_{mk}\beta_{mk}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}\mathbf{\hat{h}}_{mk}^{T}\mathbf{\bar{h}}_{mk}^{*}\right.$
$\displaystyle\left.+\alpha_{mk}\sqrt{\beta_{mk}}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}\mathbf{\dot{h}}_{mk}^{H}\mathbf{\hat{h}}_{mk}+\alpha_{mk}\sqrt{\beta_{mk}}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\bar{h}}_{mk}^{*}\right.$
$\displaystyle\left.+\alpha_{mk}\beta_{mk}^{\frac{3}{2}}\mathbf{\bar{h}}_{mk}^{T}\mathbf{\hat{h}}_{mk}^{*}\mathbf{\dot{h}}_{mk}^{H}\mathbf{\hat{h}}_{mk}+\alpha_{mk}\beta_{mk}^{\frac{3}{2}}\mathbf{\bar{h}}_{mk}^{T}\mathbf{\hat{h}}_{mk}^{*}\mathbf{\bar{h}}_{mk}^{H}\mathbf{\dot{h}}_{mk}\right.$
$\displaystyle\left.+\alpha_{mk}\beta_{mk}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\hat{h}}_{mk}^{*}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\bar{h}}_{mk}^{*}\Big{)}\right]$
$\displaystyle=q_{mk}\zeta_{mkk}^{2}+N(N+1)\beta_{mk}^{2}\left(\frac{\beta_{mk}}{\beta_{mk}+\sigma_{u}^{2}}\right)^{2}$
$\displaystyle+N\beta_{mk}^{2}\left(\frac{\sqrt{\beta_{mk}}\sigma_{u}}{\beta_{mk}+\sigma_{u}^{2}}\right)^{2}+q_{mk}\beta_{mk}\Bigg{(}1+\frac{\beta_{mk}}{\beta_{mk}+\sigma_{u}^{2}}\Bigg{)}\zeta_{mkk}$
$\displaystyle+2Nq_{mk}\beta_{mk}\frac{\beta_{mk}}{\beta_{mk}+\sigma_{u}^{2}}\zeta_{mkk}.$

Therefore,

$\displaystyle\mathbb{E}\left[\lvert\gamma_{kk}\rvert^{2}\right]$
$\displaystyle=\sum_{m=1}^{M}|x_{mi}|^{2}\Bigg{(}q_{mk}\zeta_{mkk}^{2}+N(N+1)\beta_{mk}^{2}\left(\frac{\beta_{mk}}{\beta_{mk}+\sigma_{u}^{2}}\right)^{2}+N\beta_{mk}^{2}\left(\frac{\sqrt{\beta_{mk}}\sigma_{u}}{\beta_{mk}+\sigma_{u}^{2}}\right)^{2}$
(55)
$\displaystyle+q_{mk}\beta_{mk}(1+\frac{\beta_{mk}}{\beta_{mk}+\sigma_{u}^{2}})\zeta_{mkk}+2Nq_{mk}\beta_{mk}\frac{\beta_{mk}}{\beta_{mk}+\sigma_{u}^{2}}\zeta_{mkk}\Bigg{)}$
$\displaystyle+\sum_{m=1}^{M}\sum_{\begin{subarray}{c}m^{\prime}=1\\\
m^{\prime}\neq
m\end{subarray}}^{M}x_{mi}x_{m^{\prime}i}\left(q_{mk}\zeta_{mkk}+N\beta_{mk}\frac{\beta_{mk}}{\beta_{mk}+\sigma_{u}^{2}}\right)$
$\displaystyle\times\left(q_{m^{\prime}k}\zeta_{m^{\prime}kk}+N\beta_{m^{\prime}k}\frac{\beta_{m^{\prime}k}}{\beta_{m^{\prime}k}+\sigma_{u}^{2}}\right)^{*}.$

Similarly for $\mathbb{E}\left[\lvert\gamma_{ki}\rvert^{2}\right]$,

$\mathbb{E}\left[\lvert\gamma_{ki}\rvert^{2}\right]=\sum_{m=1}^{M}|x_{mi}|^{2}\mathbb{E}\left[|\mathbf{h}_{mk}^{T}\mathbf{\check{h}}^{*}_{mi}|^{2}\right]+\sum_{m=1}^{M}\sum_{\begin{subarray}{c}m^{\prime}=1\\\
m^{\prime}\neq
m\end{subarray}}^{M}x_{mi}x_{m^{\prime}i}\mathbb{E}\left[\mathbf{h}_{mk}^{T}\right]\mathbb{E}\left[\mathbf{\check{h}}^{*}_{mi}\right]\mathbb{E}\left[\mathbf{h}_{m^{\prime}k}^{T}\right]^{*}\mathbb{E}\left[\mathbf{\check{h}}_{m^{\prime}i}^{*}\right]^{*},$
(56)

where,
$\mathbb{E}\left[\mathbf{h}_{mk}\right]=\mathbb{E}\left[\mathbf{\check{h}}_{mk}\right]=\mathbb{E}\left[\alpha_{mk}\mathbf{\dot{h}}_{mk}\right]=q_{mk}\mathbf{\dot{h}}_{mk}$
and,

$\displaystyle\mathbb{E}\left[|\mathbf{h}_{mk}^{T}\mathbf{\check{h}}^{*}_{mi}|^{2}\right]$
$\displaystyle=\mathbb{E}\left[\lvert\alpha_{mk}\alpha_{mi}\mathbf{\dot{h}}^{T}_{mk}\mathbf{\dot{h}}^{*}_{mi}+\sqrt{\beta_{mk}\beta_{mi}}\mathbf{\bar{h}}^{T}_{mk}\mathbf{\hat{h}}^{*}_{mi}+\alpha_{mk}\sqrt{\beta_{mi}}\mathbf{\dot{h}}^{T}_{mk}\mathbf{\hat{h}}^{*}_{mi}\right.$
$\displaystyle\left.+\alpha_{mi}\sqrt{\beta_{mk}}\mathbf{\bar{h}}^{T}_{mk}\mathbf{\dot{h}}^{*}_{mi}\rvert^{2}\right]$
$\displaystyle=\mathbb{E}\left[\alpha_{mk}\alpha_{mi}\lvert\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mi}^{*}\rvert^{2}+\beta_{mk}\beta_{mi}\lvert\mathbf{\bar{h}}_{mk}^{T}\mathbf{\hat{h}}_{mi}^{*}\rvert^{2}+\alpha_{mk}\beta_{mi}\lvert\mathbf{\dot{h}}_{mk}^{T}\mathbf{\hat{h}}_{mi}^{*}\rvert^{2}\right.$
$\displaystyle\left.+\alpha_{mi}\beta_{mk}\lvert\mathbf{\bar{h}}_{mk}^{T}\mathbf{\dot{h}}_{mi}^{*}\rvert^{2}+2\Re(\alpha_{mk}\alpha_{mi}\sqrt{\beta_{mk}\beta_{mi}}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mi}^{*}\mathbf{\bar{h}}_{mk}^{H}\mathbf{\hat{h}}_{mi}\right.$
$\displaystyle\left.+\alpha_{mk}\alpha_{mi}\sqrt{\beta_{mi}}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mi}^{*}\mathbf{\dot{h}}_{mk}^{H}\mathbf{\hat{h}}_{mi}+\alpha_{mk}\alpha_{mi}\sqrt{\beta_{mk}}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mi}^{*}\mathbf{\bar{h}}_{mk}^{H}\mathbf{\dot{h}}_{mi}\right.$
$\displaystyle\left.+\alpha_{mk}\sqrt{\beta_{mk}}\beta_{mi}\mathbf{\bar{h}}_{mk}^{T}\mathbf{\hat{h}}_{mi}^{*}\mathbf{\dot{h}}_{mk}^{H}\mathbf{\hat{h}}_{mi}+\alpha_{mk}\beta_{mk}\sqrt{\beta_{mi}}\mathbf{\bar{h}}_{mk}^{T}\mathbf{\hat{h}}_{mi}^{*}\mathbf{\bar{h}}_{mk}^{H}\mathbf{\dot{h}}_{mi}\right.$
$\displaystyle\left.+\alpha_{mk}\alpha_{mi}\sqrt{\beta_{mk}\beta_{mi}}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\hat{h}}_{mi}^{*}\mathbf{\bar{h}}_{mk}^{H}\mathbf{\dot{h}}_{mi})\right]$
$\displaystyle=q_{mk}q_{mi}\lvert\zeta_{mki}\rvert^{2}+N\beta_{mk}\beta_{mi}\frac{\beta_{mi}}{\beta_{mi}+\sigma_{u}^{2}}+q_{mk}\beta_{mi}\frac{\beta_{mi}}{\beta_{mi}+\sigma_{u}^{2}}\zeta_{mkk}$
$\displaystyle+q_{mi}\beta_{mk}\zeta_{mii}.$

This simplifies to the expression in (27).

We calculate $\mathbb{E}\left[\lvert\dot{\gamma}_{kk}\rvert^{2}\right]$,

$\displaystyle\mathbb{E}\left[\lvert\dot{\gamma}_{kk}\rvert^{2}\right]$
$\displaystyle=\sum_{m=1}^{M}\lvert
x_{mk}\rvert^{2}\mathbb{E}\left[\lvert\alpha_{mk}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}\rvert^{2}\right]+\sum_{m=1}^{M}\sum_{\begin{subarray}{c}m^{\prime}=1\\\
m^{\prime}\neq
m\end{subarray}}^{M}x_{mk}x_{m^{\prime}k}\mathbb{E}\left[\alpha_{mk}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}\right]$
(57)
$\displaystyle\times\mathbb{E}\left[\alpha_{m^{\prime}k}\mathbf{\dot{h}}_{m^{\prime}k}^{T}\mathbf{\dot{h}}_{m^{\prime}k}^{*}\right]^{*},$

where,
$\mathbb{E}\left[\alpha_{mk}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}\right]=q_{mk}\zeta_{mkk}$
and,

$\begin{split}\mathbb{E}\left[\lvert\alpha_{mk}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}\rvert^{2}\right]=\mathbb{E}\left[\lvert\alpha_{mk}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}*\rvert^{2}\right]=q_{mk}\zeta_{mkk}^{2}.\end{split}$

This expression simplifies to (28), (36) and (43). Finally for
$\bar{\gamma}_{kk}$,

$\mathbb{E}\left[\lvert\bar{\gamma}_{kk}\bar{\gamma}_{kk}^{*}\rvert\right]=\mathbb{E}\left[\rvert\sum_{m=1}^{M}x_{mk}\left(\alpha_{mk}\left(\mathbf{\hat{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}+\mathbf{\dot{h}}_{mk}^{T}\mathbf{\bar{h}}_{mk}^{*}\right)+\mathbf{\hat{h}}_{mk}^{T}\mathbf{\bar{h}}_{mk}^{*}\right)\rvert^{2}\right],$
$\begin{split}\mathbb{E}\left[\lvert\bar{\gamma}_{kk}\rvert^{2}\right]=&\sum_{m=1}^{M}|x_{mk}|^{2}\left(\mathbb{E}\left[|\mathbf{h}_{mk}^{T}\mathbf{\check{h}}^{*}_{mk}|^{2}\right]-\mathbb{E}\left[\lvert\alpha_{mk}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}\rvert^{2}\right]\right)+\sum_{m=1}^{M}\sum_{\begin{subarray}{c}m^{\prime}=1\\\
m^{\prime}\neq m\end{subarray}}^{M}x_{mk}x_{m^{\prime}k}\\\
&\mathbb{E}\left[\mathbf{h}_{mk}^{T}\mathbf{\check{h}}^{*}_{mk}\right]\mathbb{E}\left[\mathbf{h}_{m^{\prime}k}^{H}\mathbf{\check{h}}^{*}_{m^{\prime}k}\right]-\mathbb{E}\left[\alpha_{mk}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}\right]\mathbb{E}\left[\alpha_{m^{\prime}k}\mathbf{\dot{h}}_{m^{\prime}k}^{T}\mathbf{\dot{h}}_{m^{\prime}k}^{*}\right]^{*},\end{split}$
(58)

where each expectation term has been previously defined. This expression
simplifies to (29). It is important to note that the two components,
$\dot{\gamma}_{kk}$ and $\bar{\gamma}_{kk}$ are uncorrelated. Therefore,

$\mathbb{E}\left[\lvert\gamma_{kk}\rvert^{2}\right]=\mathbb{E}\left[\lvert\dot{\gamma}_{kk}\rvert^{2}\right]+\mathbb{E}\left[\lvert\bar{\gamma}_{kk}\rvert^{2}\right].$
(59)

###  -C Channel Statistics for Statistical Beamforming Schemes

$\begin{split}\mathbb{E}\left[\gamma_{kk}\right]&=\sum_{m=1}^{M}x_{mk}(\mathbb{E}\left[\alpha_{mk}\right]\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}+\mathbb{E}\left[\alpha_{mk}\right]\mathbb{E}\left[\mathbf{\bar{h}}_{mk}^{T}\right]\mathbf{\dot{h}}_{mk}^{*})=\sum_{m=1}^{M}x_{mk}q_{mk}\zeta_{mkk}\end{split}$
(60)
$\begin{split}\mathbb{E}\left[\gamma_{ki}\right]&=\sum_{m=1}^{M}x_{mi}(\mathbb{E}\left[\alpha_{mk}\alpha_{mi}\right]\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mi}^{*}+\mathbb{E}\left[\alpha_{mi}\right]\mathbb{E}\left[\mathbf{\bar{h}}_{mk}^{T}\right]\mathbf{\dot{h}}_{mi}^{*})=\sum_{m=1}^{M}x_{mi}q_{mk}q_{mi}\zeta_{mki}\end{split}$
(61)
$\begin{split}\mathbb{E}\left[\dot{\gamma}_{kk}\right]=\sum_{m=1}^{M}x_{mk}\mathbb{E}\left[\alpha_{mk}\right]\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}=\sum_{m=1}^{M}x_{mk}q_{mk}\zeta_{mkk}\end{split}$
(62)
$\mathbb{E}\left[\bar{\gamma}_{kk}\right]=\mathbb{E}\left[\gamma_{kk}\right]-\mathbb{E}\left[\dot{\gamma}_{kk}\right]=0$
(63)

Similarly,

$\displaystyle\mathbb{E}\left[\lvert\gamma_{kk}\rvert^{2}\right]$
$\displaystyle=\sum_{m=1}^{M}\lvert
x_{mk}\rvert^{2}\mathbb{E}\left[\lvert\alpha_{mk}\mathbf{h}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}\rvert^{2}\right]+\sum_{m=1}^{M}\sum_{\begin{subarray}{c}m^{\prime}=1\\\
m^{\prime}\neq
m\end{subarray}}^{M}x_{mk}x_{m^{\prime}k}\mathbb{E}\left[\alpha_{mk}\mathbf{h}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}\right]\mathbb{E}\left[\alpha_{m^{\prime}k}\mathbf{h}_{m^{\prime}k}^{T}\mathbf{\dot{h}}_{m^{\prime}k}^{*}\right]^{*},$
(64)

where,
$\mathbb{E}\left[\alpha_{mk}\mathbf{h}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}\right]=q_{mk}\zeta_{mkk}$
and,

$\displaystyle\mathbb{E}\left[\lvert\alpha_{mk}\mathbf{h}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}\rvert^{2}\right]$
$\displaystyle=\mathbb{E}\left[\lvert\alpha_{mk}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}+\sqrt{\beta_{mk}}\alpha_{mk}\mathbf{\bar{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}\rvert^{2}\right]$
$\displaystyle=\mathbb{E}\left[\alpha_{mk}\lvert\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}\rvert^{2}+\alpha_{mk}\beta_{mk}\lvert\mathbf{\bar{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}\rvert^{2}+\alpha_{mk}\sqrt{\beta_{mk}}2\Re(\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\bar{h}}_{mk}^{*})\right]$
$\displaystyle=q_{mk}\zeta_{mkk}^{2}+\beta_{mk}q_{mk}\zeta_{mkk}.$

Therefore,

$\displaystyle\mathbb{E}\left[\lvert\gamma_{kk}\rvert^{2}\right]$
$\displaystyle=\sum_{m=1}^{M}\lvert
x_{mk}\rvert^{2}\Big{(}q_{mk}\zeta_{mkk}^{2}+\beta_{mk}q_{mk}\zeta_{mkk}\Big{)}+\sum_{m=1}^{M}\sum_{\begin{subarray}{c}m^{\prime}=1\\\
m^{\prime}\neq
m\end{subarray}}^{M}x_{mk}x_{m^{\prime}k}q_{mk}\zeta_{mkk}q_{m^{\prime}k}\zeta_{m^{\prime}kk}.$
(65)

Similarly, for $\mathbb{E}\left[\lvert\gamma_{ki}\rvert^{2}\right]$,

$\displaystyle\mathbb{E}\left[\lvert\gamma_{ki}\rvert^{2}\right]$
$\displaystyle=\sum_{m=1}^{M}\lvert
x_{mi}\rvert^{2}\mathbb{E}\left[\lvert\alpha_{mi}\mathbf{h}_{mk}^{T}\mathbf{\dot{h}}_{mi}^{*}\rvert^{2}\right]$
(66) $\displaystyle+\sum_{m=1}^{M}\sum_{\begin{subarray}{c}m^{\prime}=1\\\
m^{\prime}\neq
m\end{subarray}}^{M}x_{mi}x_{m^{\prime}i}\mathbb{E}\left[\mathbf{h}_{mk}^{T}\right]\mathbb{E}\left[\alpha_{mi}\mathbf{\dot{h}}_{mk}^{*}\right]\mathbb{E}\left[\mathbf{h}_{m^{\prime}k}^{T}\right]^{*}\mathbb{E}\left[\alpha_{m^{\prime}k}\mathbf{\dot{h}}_{m^{\prime}k}^{*}\right]^{*},$

where,
$\mathbb{E}\left[\mathbf{h}_{mk}\right]=\mathbb{E}\left[\alpha_{mk}\mathbf{\dot{h}}_{mk}\right]=q_{mk}\mathbf{\dot{h}}_{mk}$
and,

$\mathbb{E}\left[\lvert\alpha_{mi}\mathbf{h}_{mk}^{T}\mathbf{\dot{h}}_{mi}^{*}\rvert^{2}\right]=\mathbb{E}\left[\lvert\alpha_{mk}\alpha_{mi}\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mi}^{*}+\sqrt{\beta_{mk}}\alpha_{mi}\mathbf{\bar{h}}_{mk}^{T}\mathbf{\dot{h}}_{mi}^{*}\rvert^{2}\right]\\\
=\mathbb{E}\left[\alpha_{mk}\alpha_{mi}\lvert\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mi}^{*}\rvert^{2}+\alpha_{mi}\beta_{mk}\lvert\mathbf{\bar{h}}_{mk}^{T}\mathbf{\dot{h}}_{mi}^{*}\rvert^{2}+\alpha_{mk}\alpha_{mi}\sqrt{\beta_{mk}}2\Re(\mathbf{\dot{h}}_{mk}^{T}\mathbf{\dot{h}}_{mi}^{*}\mathbf{\dot{h}}_{mi}^{T}\mathbf{\bar{h}}_{mk}^{*})\right]\\\
=q_{mk}q_{mi}\lvert\zeta_{mki}\rvert^{2}+\beta_{mk}q_{mi}\zeta_{mii}.$

This expression simplifies to (35) and (42). Since the definition of
$\dot{\gamma}_{kk}$ remains unchanged, its statistics calculated in Appendix B
still apply. Finally,

$\begin{split}\mathbb{E}\left[\lvert\bar{\gamma}_{kk}\rvert^{2}\right]=\mathbb{E}\left[\lvert\gamma_{kk}-\dot{\gamma}_{kk}\rvert^{2}\right]=\mathbb{E}\left[\left\lvert\sum_{m=1}^{M}x_{mk}\alpha_{mk}\mathbf{\bar{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}\right\rvert^{2}\right]\end{split}$
$\displaystyle\mathbb{E}\left[\lvert\bar{\gamma}_{kk}\rvert^{2}\right]$
$\displaystyle=\sum_{m=1}^{M}\lvert
x_{mk}\rvert^{2}\mathbb{E}\left[\lvert\alpha_{mk}\mathbf{\bar{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}\rvert^{2}\right]$
(67) $\displaystyle+\sum_{m=1}^{M}\sum_{\begin{subarray}{c}m^{\prime}=1\\\
m^{\prime}\neq
m\end{subarray}}^{M}x_{mk}x_{m^{\prime}k}\mathbb{E}\left[\alpha_{mk}\mathbf{\bar{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}\right]\mathbb{E}\left[\alpha_{m^{\prime}k}\mathbf{\bar{h}}_{m^{\prime}k}^{T}\mathbf{\dot{h}}_{m^{\prime}k}^{*}\right]^{*}$

where,
$\mathbb{E}\left[\alpha_{mk}\mathbf{\bar{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}\right]=0$
and,
$\mathbb{E}\left[\lvert\alpha_{mk}\mathbf{\bar{h}}_{mk}^{T}\mathbf{\dot{h}}_{mk}^{*}\rvert^{2}\right]=\beta_{mk}q_{mk}\zeta_{mkk}.$
Therefore,

$\mathbb{E}\left[\lvert\bar{\gamma}_{kk}\rvert^{2}\right]=\sum_{m=1}^{M}\lvert
x_{mk}\rvert^{2}\beta_{mk}q_{mk}\zeta_{mkk}$ (68)

It is important to note that the two components $\dot{\gamma}_{kk}$ and
$\bar{\gamma}_{kk}$ are uncorrelated. Therefore,

$\mathbb{E}\left[\lvert\gamma_{kk}\rvert^{2}\right]=\mathbb{E}\left[\lvert\dot{\gamma}_{kk}\rvert^{2}\right]+\mathbb{E}\left[\lvert\bar{\gamma}_{kk}\rvert^{2}\right]$
(69)

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