figure table

# Hardware-Impaired Rician-Faded Cell-Free Massive MIMO Systems With Channel
Aging

Venkatesh Tentu, Dheeraj N Amudala, Anish Chattopadhyay and Rohit Budhiraja
The authors are with the Dept. of Electrical Engineering, IIT Kanpur, India.
email: {tentu, dheeraja, anishchat20<EMAIL_ADDRESS>A part of this work
is accepted for presentation in IEEE International Conference on
Communications 2023.

###### Abstract

We study the impact of channel aging on the uplink of a cell-free (CF) massive
multiple-input multiple-output (mMIMO) system by considering i) spatially-
correlated Rician-faded channels; ii) hardware impairments at the access
points and user equipments (UEs); and iii) two-layer large-scale fading
decoding (LSFD). We first derive a closed-form spectral efficiency (SE)
expression for this system, and later propose two novel optimization
techniques to optimize the non-convex SE metric by exploiting the
minorization-maximization (MM) method. The first one requires a numerical
optimization solver, and has a high computation complexity. The second one
with closed-form transmit power updates, has a trivial computation complexity.
We numerically show that i) the two-layer LSFD scheme effectively mitigates
the interference due to channel aging for both low- and high-velocity UEs; and
ii) increasing the number of AP antennas does not mitigate the SE
deterioration due to channel aging. We numerically characterize the optimal
pilot length required to maximize the SE for various UE speeds. We also
numerically show that the proposed closed-form MM optimization yields the same
SE as that of the first technique, which requires numerical solver, and that
too with a much reduced time-complexity.

###### Index Terms:

Cell-free, channel aging, hardware impairments, minorization-maximization
(MM), Rician fading.

##  I Introduction

Cell-free (CF) massive multiple-input multiple output (mMIMO) is being
investigated as a key technology for beyond fifth-generation wireless systems
due to its high spectral efficiency (SE) and improved coverage [1]. A CF mMIMO
system consists of a large number of access points (APs) which are randomly
deployed over a large coverage area, and are connected to a central processing
unit (CPU) via high-speed fronthaul links. The APs cooperate via these
fronthaul links to mitigate the inter-user interference (IUI) caused due to
multiple user equipments (UEs) being served on the same time-frequency
resource [1, 2]. Initial CF mMIMO works commonly considered single layer
decoding (SLD) schemes, wherein APs individually combine their respective
receive signals using the locally-estimated channel information, while the CPU
handle only data detection [1, 2]. Ngo et al. in [1] investigated the SE gains
provided by a CF mMIMO system with SLD, over its conventional small-cell
counterpart. The authors in [2] proposed scalable combiners and precoders for
a spatially-correlated Rayleigh-faded CF mMIMO system with SLD, and showed
that they outperform conventional maximum ratio processing. We know that fifth
generation (5G) cellular networks have recently been deployed. The channel
models considered to evaluate different technologies for 5G systems e.g.,
mMIMO, contain line of sight (LoS) components, along with the non LoS (NLoS)
one[3]. This makes the channel Rician distributed.

Further, the local SLD techniques, despite having a low implementation
complexity, fail to effectively suppress the IUI. This can be improved by
performing the second level of combining at the CPU, referred to as the large-
scale fading decoding (LSFD) [4, 5, 6]. The key difference between LSFD and
SLD receivers is that the former requires additional statistical channel
parameters such as large-scale fading coefficients at the CPU. These large-
scale fading parameters remain constant over multiple coherence intervals, and
thus can be easily made available at the CPU [4, 5, 6]. Ozdogan et al. in [4]
showed the improved SE gains obtained due to LSFD for a CF mMIMO system with
spatially-correlated Rician-faded channels. Demir et al. in [5] maximized the
minimum SE of a wireless-powered CF mMIMO system by considering LSFD and the
spatially-correlated Rician-faded channels. Zhang et al. in [6] proposed local
zero forcing combiners for an uncorrelated CF mMIMO system, and then
integrated these combiners with LSFD, and investigated their SE.

Majority of CF mMIMO works, including the aforementioned ones in [1, 2, 4, 5,
6], considered a block fading channel model, wherein the channel remains
constant over a coherence interval. The 5G cellular systems are designed for a
UE speed of up to $500$ km/h, whose channel then continuously varies with time
[7]. The channel estimated by the APs, consequently, become outdated with time
[8, 9]. This channel aging phenomenon can drastically degrade the gains
accrued by CF mMIMO technology. The authors in [9, 10, 11, 12] and [13, 14,
15] recently investigated the effect of channel aging on cellular and CF mMIMO
systems, respectively. The authors in [9] derived asymptotic power scaling
laws which showed the detrimental effect of channel aging on a single-cell
correlated Rayleigh-faded mMIMO system. The authors in [10, 11] proposed a
machine-learning-based channel predictor for a single-cell mMIMO system with
channel aging, which could be used for both uplink and downlink systems.
Papazafeiropoulos et al. in [12] analyzed the outage probability of a multi-
cell mMIMO system and showed that its preferred to have massive number of
antennas under channel aging conditions. Chopra et. al in [13] showed that the
impact of channel aging is higher on a CF mMIMO system than on a cellular
mMIMO system. This work, however, considered uncorrelated Rayleigh-faded
channels and local SLD scheme at the APs. Zheng et. al in [14] and [15]
analyzed the effect of channel aging in a CF mMIMO system with LSFD, for
uncorrelated and correlated Rayleigh fading channels, respectively. The
authors in [15] also proved that the CF mMIMO system is more robust to channel
aging than a small-cell system.

The above CF mMIMO works [4, 5, 6, 13, 14, 15] assumed high-quality radio
frequency (RF) transceivers and high-resolution analog-to-digital converters
(ADCs)/digital-to-analog converters (DACs) at the APs and UEs. The RF
transceiver chips used to design 5G cellular system, and its evolved CF mMIMO
systems have inherent hardware distortion, which is commonly characterized
using the error vector magnitude (EVM) metric. It is usually specified in the
device data sheet [16]. The effect of hardware distortion caused by the low-
cost hardware can be suppressed by using the calibration schemes and the
compensation algorithms, but the residual impairments still remains [17, 18,
19, 20, 21]. The impact of these residual impairments on CF mMIMO systems
should be further studied. Masoumi et al. in [17] analyzed the effect of
hardware-impairments in a CF mMIMO system with uncorrelated Rayleigh-faded
channels. The authors in [18] derived an approximate SE expression for a
spatially-correlated Rayleigh-faded CF mMIMO system with hardware impairments.
Tentu et. al in [19] investigated the SE of an unmanned aerial vehicle enabled
CF mMIMO system with RF impairments and spatially-correlated Rician channels.
All these works considered only local SLD schemes, and also ignored channel
aging in their analysis. Zhang et al. in [20] analyzed the SE of a hardware-
impaired CF mMIMO system with LSFD, and showed that the hardware distortion
non-trivially impacts the LSFD performance. This work, however, ignored
channel aging. The authors in [21] investigated the joint impact of hardware
impairments and channel aging for a CF mMIMO system, but only for uncorrelated
Rayleigh channels, and that too without LSFD.

Practical CF mMIMO systems, to reduce the energy consumption and
implementation costs, also employ low-resolution ADC/DACs [22, 23, 24]. The
authors in [22, 23, 24] studied the effect of low-resolution ADCs in CF mMIMO
systems. Zhang et. al in [22] analyzed the SE of spatially-uncorrelated
Rayleigh-faded CF mMIMO system with low-resolution ADC/DACs, and showed its
improved energy efficiency over high-resolution ADC/DACs. Zhang et. al in [23]
investigated the uplink and downlink SE of an uncorrelated Rician-faded CF
mMIMO system with low-resolution ADCs at the APs. Hu et. al in [24]
investigated the asymptotic SE of an uncorrelated Rayleigh-faded CF mMIMO
system, and showed that the SE is mainly limited by the UE ADC resolution,
when low-resolution ADCs are used at both APs and UEs. Low resolution ADC/DACs
improve the energy efficiency by reducing the power consumption, but also
cause non-negligible SE loss due to coarse quantization. To alleviate this
problem, the authors in [25, 26] considered a mixed-ADC architecture, wherein
one fraction of AP antennas has a high-resolution ADCs, while the other has
low-resolution. Zhang et. al in [25] and [26] investigated the SE of a CF
mMIMO system with mixed-ADC architecture at the APs by considering spatially-
uncorrelated Rayleigh- and Rician-fading respectively, and showed its
superiority over its low-resolution counterpart.

Recently, the authors in [27, 28] proposed a dynamic ADC architecture and
derived closed-form SE expressions for CF mMIMO with spatially-correlated
Rayleigh channels. The dynamic ADC architecture offers the flexibility to tune
the ADC resolution of each AP antenna. This provides system designers with
extra degrees-of-freedom for system design and optimization. References [22,
23, 24, 25, 26, 27, 28] considered a local SLD scheme and investigated the
effect of low-/mixed-/dynamic-resolution ADCs alone. These works ignored RF
impairments and channel aging. Also, the existing works [23, 19, 26] modeled
the Rician fading channel with a static line-of-sight phase. A small change in
the UE location induces significant phase-shift in the LoS path. The authors
in [4, 5] modeled these phase-shifts as a uniformly distributed random
variable, and derived closed-form SE expression for a CF mMIMO system, but
without considering channel-aging, and hardware impairments. It is crucial to
analyze the joint effect of RF impairments and ADC/DAC quantization in the
presence of channel aging and Rician phase-shifts, which is a crucial gap in
the existing CF mMIMO literature. The current work fills this gap, by
analyzing a CF mMIMO system with two-layer LSFD, channel aging, Rician phase-
shifts, low-cost RF chains and dynamic ADC architecture.

We next summarize in Table I the relevant CF mMIMO literature focusing on
LSFD, channel aging, RF and ADC hardware impairments and spatially-correlated
Rician faded channels with phase-shifts. We infer from Table I that the
existing correlated Rician-faded CF mMIMO literature has not yet investigated
the: i) SE with channel aging and two-stage LSFD; ii) RF impairments and
dynamic ADC architecture; and iii) sum SE optimization. Additionally, for
Rayleigh channels, the CF mMIMO LSFD literature, has not investigated the i)
SE with channel aging and non-ideal hardware and; ii) sum SE optimization.

Table I: Summary of CF mMIMO literature focusing on LSFD and Channel aging. Ref. | LSFD | Channel aging |  Rayleigh/ Rician | Correlation |  RF impairments |  ADC architecture | Optimization
---|---|---|---|---|---|---|---
[4] | ✓ | ✗ | Rician | ✗ | ✗ | ideal | ✗
[13] | ✗ | ✓ | Rayleigh | ✗ | ✗ | ideal | ✗
[15] | ✓ | ✓ | Rayleigh | ✓ | ✗ | ideal | ✗
[20] | ✓ | ✗ | Rayleigh | ✗ | ✓ | ideal | ✗
[21] | ✗ | ✓ | Rayleigh | ✗ | ✓ | ideal | max-min SE
[23] | ✗ | ✗ | Rician | ✗ | ✗ | low-resolution | weighted max-min SE
[26] | ✗ | ✗ | Rician | ✗ | ✗ | mixed-resolution | ✗
[27] | ✗ | ✗ | Rayleigh | ✓ | ✗ | dynamic-resolution | ✗
Pr. | ✓ | ✓ | Rician | ✓ | ✓ | dynamic-resolution | sum-SE

The main contributions of the current work address these gaps as follows:

  * •

We consider a spatially-correlated Rician-faded CF mMIMO system with channels
aging, and investigate the impact of dynamic ADC architecture and low-cost
hardware-impaired RF chains. We also consider the two-layer LSFD, and derive a
closed-form SE expression by addressing the derivation difficulties caused by
the combined modelling of channel aging, RF impairments, dynamic ADC
architecture and spatially-correlated Rician channels.

  * •

We use the derived closed-form SE expression to maximize the non-convex SE
metric by optimizing the UE transmit powers and the LSFD coefficients, which
is the second contribution of this work. We propose two optimal power
allocation schemes using minorization-maximization (MM) technique, which
iteratively maximizes the convex surrogate of the non-convex objective [29].
We propose a novel convex surrogate function, and analytically show that it
satisfies the desirable surrogacy properties [29]. This approach, however, has
a high computation complexity as it requires off-the-shelf optimization
solvers [29]. We next design a low-complexity practically-implementable power
allocation technique by using the Lagrangian dual transform technique [30].
This enables us in designing the closed-form transmit power update, which has
a trivial computational complexity. It is extremely useful for designing
optimal power for practical CF mMIMO systems with channel aging.

  * •

We show that LSFD can effectively mitigate the detrimental effects of i)
channel aging for both low and high UE velocities; and ii) IUI for low-
velocity UEs but not for high-velocity UEs. We also show that the increased
number of AP antennas cannot mitigate the SE degradation due to channel aging.

The rest of the paper is organized as follows. Section II discusses the
channel model, uplink channel estimation and data transmission, and LSFD for
the proposed hardware-impaired CF mMIMO system with channel aging. Section III
derives and analyzes the closed-form SE expression for the aforementioned
system. Section IV proposes two optimal power allocation schemes to optimize
SE. Section V first numerically validates the efficacy of the derived closed-
form SE. It then investigates the effect of channel aging and LSFD on the SE.
It then compares the complexity of the proposed optimization techniques.
Section VI finally concludes the paper.

Notations: Bold-faced lower- and upper-case alphabet denote vectors and
matrices, respectively. Superscripts $(\cdot)^{*}$, $(\cdot)^{T}$,
$(\cdot)^{H}$ and $(\cdot)^{-1}$ denote the conjugate, transpose, conjugate
transpose and inverse, respectively, and $\mathbb{E}\\{\cdot\\}$ is the
expectation operator. Trace and diagonal of a matrix $\mathbf{X}$ are denoted
as $\text{tr}(\mathbf{X})$ and $\text{diag}(\mathbf{X})$, respectively. Also,
$\|\cdot\|$ is the Euclidean 2-norm, $|\cdot|$ is the absolute value and
$\text{real}\\{\cdot\\}$ is the real-part of the argument. The notation
$\mathcal{CN}(\mathbf{0},\mathbf{I})$ denotes a complex circular Gaussian
random vector with zero mean and covariance matrix $\mathbf{I}$.

##  II System Model

We consider the uplink of a CF mMIMO system with $M$ multi-antenna APs and $K$
single-antenna UEs. Each AP has $N$ antennas. We assume, similar to [5, 24],
that the APs are randomly distributed over a large geographical area, and are
connected to a CPU via high-speed fronthaul links. To reduce the system
hardware cost and power consumption, APs and UEs are equipped with low-cost
hardware-impaired RF chains. Further, the APs have a dynamic-resolution ADC
architecture, wherein each AP antenna can be connected to a different
resolution ADC. This is unlike [22, 25, 26], which assume that the ADCs either
have a low or a mixed resolution. The UEs are designed using low-resolution
DACs. In correlated Rician-faded cell-free mMIMO systems, the channel does not
harden. For high speed UEs, the channel will age. To investigate channel aging
effect, we consider, as shown in Fig. 1, a resource block of length $\tau_{c}$
time instants.

Figure 1: Structure of resource block of length $\tau_{c}$ time instants.

The channel remains constant for a time instant, and varies across time
instants in a correlated manner. This temporal correlation is modeled later
using Jake’s model [15]. We also assume that each resource block is divided
into uplink training and data transmission intervals of length $\tau_{p}$ and
$(\tau_{c}-\tau_{p})$ time instants, respectively. We next model the UE-AP
channel.

###  II-A Channel model

The channel from the $k$th UE to the $m$th AP at the $\lambda$th time instant
is denoted as $\mathbf{h}_{mk}[\lambda]$. Due to dense AP deployment, the
channel $\mathbf{h}_{mk}[\lambda]$ contains both LoS and NLoS paths. It,
therefore, has a Rician probability density function (pdf), and is modeled as
follows [4]:

$\displaystyle\\!\\!\\!\mathbf{h}_{mk}[\lambda]=\bar{\mathbf{h}}_{mk}e^{j\phi_{mk}^{\lambda}}\\!+\\!\mathbf{R}_{mk}^{\frac{1}{2}}\tilde{\mathbf{h}}_{mk}[\lambda].$
(1)

Here
$\bar{\mathbf{h}}_{mk}=\sqrt{\frac{\\!K_{mk}\beta_{mk}}{\\!K_{mk}+1}}\breve{\mathbf{h}}_{mk}$,
$\mathbf{R}_{mk}\\!=\frac{\beta_{mk}}{\\!K_{mk}\\!+\\!1}\breve{\mathbf{R}}_{mk}$.
The term $K_{mk}$ is the Rician factor, and $\beta_{mk}$ is the large-scale
fading coefficient. The LoS term $\breve{\mathbf{h}}_{mk}$ is modeled as
$\breve{\mathbf{h}}_{mk}=[1,e^{j\psi_{mk}},\cdots,e^{j(M-1)\psi_{mk}}]^{T}$,
where $\psi_{mk}$ is the angle of arrival from the $k$th UE to the $m$th AP.
The term, $\tilde{\mathbf{h}}_{mk}[\lambda]$, with pdf
$\mathcal{CN}(\mathbf{0},\mathbf{I}_{N})$ models the small-scale NLoS fading.
The term $\breve{\mathbf{R}}_{mk}$ is the spatial correlation matrix. The LoS
phase-shift $\phi_{mk}^{\lambda}$ at the $\lambda$th instant is uniformly
distributed between $[-\pi,\pi]$. The long-term channel statistics i.e.,
$\bar{\mathbf{h}}_{mk}$, $\beta_{mk}$ and $\mathbf{R}_{mk}$, remain constant
over a resource block and, similar to [4], are assumed to be perfectly known
at the AP. Most of the Rician CF works e.g., [31, 32, 19], assume that the LoS
phase-shift $\phi_{mk}^{\lambda}$ is static. A slight change in UEs position,
however, can radically modify the phase. It is, thus, crucial to consider a
dynamic phase-shift $\phi_{mk}^{\lambda}$ in the Rician channel to practically
model it [4]. We accordingly, similar to [4], assume that
$\phi_{mk}^{\lambda}$ varies as frequently as small-scale fading, and that the
AP is unaware of it. As an AP does not have the prior knowledge of
$\phi_{mk}^{\lambda}$, it estimates the channel $\mathbf{h}_{mk}$, without its
knowledge.

As discussed earlier, due to UEs mobility, the AP-UE channel in a resource
block varies across time instants, which causes channel aging [15]. To analyze
its effect, we model the channel $\mathbf{h}_{mk}[n]$ at the $n$th time
instant $1\leq n\leq\tau_{c}$, as a combination of the channel
$\mathbf{h}_{mk}[\lambda]$ at the time instant $\lambda$, and the innovation
component as follows [14, 15]:

$\displaystyle\mathbf{h}_{mk}[n]=\rho_{k}[\lambda-n]\mathbf{h}_{mk}[\lambda]+\sqrt{1-\rho_{k}^{2}[\lambda-n]}\Big{(}\bar{\mathbf{h}}_{mk}e^{j\phi_{mk}^{n}}+\mathbf{f}_{mk}[n]\Big{)}.$
(2)

The temporal correlation $\rho_{k}[\lambda-n]$, based on the Jake’s model
[15], is given as $\rho_{k}[\lambda-n]=J_{0}(2\pi f_{d,i}T_{s}(\lambda-n))$.
Here $J_{0}(\cdot)$ is the zeroth-order Bessel function, and $T_{s}$ is the
sampling time. The term $f_{d,i}=(v_{k}f_{c})/c$ is the Doppler spread, with
$v_{k}$, $f_{c}$ and $c$ being the user velocity, carrier frequency and the
velocity of light, respectively. The innovation component $\mathbf{f}_{mk}[n]$
is independent of the channel at the time instant $\lambda$ i.e.,
${\mathbf{h}}_{mk}[\lambda]$, and has a pdf
$\mathcal{CN}(\mathbf{0},\mathbf{R}_{mk})$ [15].

###  II-B Uplink training

Recall that the uplink training phase consists of $\tau_{p}$ time instants.
The $k$th UE, similar to [15], transmits its pilot signal
$\sqrt{\tilde{p}_{k}}\phi_{k}[t_{k}]$ at the time instant
$t_{k}\subset\\{1,\dots,\tau_{p}\\}$. Here $|\phi_{k}[t_{k}]|^{2}=1$. We
assume that the uplink training duration $\tau_{p}<K$. The number of UEs
transmitting pilot at a particular time instant is more than one, which causes
pilot contamination [4]. The set of UEs that transmit pilots at the time
instant $t_{k}$ is denoted as $\mathcal{P}_{k}$. The $k$th UE feeds its pilot
signal to the low-resolution DAC, which distorts it. This distortion is
commonly analyzed using Bussgang model [33]. The distorted DAC output, based
on the Bussgang model, is:

$\displaystyle
s_{\text{DAC},k}^{p}[t_{k}]=Q(\sqrt{\tilde{p}_{k}}\phi_{k}[t_{k}])=\alpha_{d,k}\sqrt{\tilde{p}_{k}}\phi_{k}[t_{k}]+\upsilon^{p}_{\text{DAC},k}[t_{k}].$
(3)

Here $\alpha_{d,k}=1\\!-\iota_{d,k}$ with $\iota_{d,k}$ being the DAC
distortion factor. The term $\upsilon^{p}_{\text{DAC},k}[t_{k}]$ is zero-mean
DAC quantization noise, which is uncorrelated with the input pilot signal, and
has a variance
$\alpha_{d,k}(1\\!-\\!\alpha_{d,k})\tilde{p}_{k}\mathbb{E}\big{\\{}|\phi_{k}[t_{k}]|^{2}\big{\\}}$
[4]. The $k$th UE DAC output signal $s_{\text{DAC},k}^{p}[t_{k}]$ is fed to
its hardware- impaired RF chain, which adds a distortion term
$\xi_{\text{RF},k}^{p}[t_{k}]$ [34]. The effective uplink pilot signal is,
therefore, given as
$s_{\text{RF},k}^{p}[t_{k}]=s_{\text{DAC},k}^{p}[t_{k}]+\xi_{\text{RF},k}^{p}[t_{k}]$.
The distortion $\xi_{\text{RF},k}^{p}[t_{k}]$ is independent of the input
signal, and has a pdf
$\mathcal{CN}\big{(}0,\kappa_{t,k}^{2}\big{(}\mathbb{E}\\{s^{p}_{\text{DAC},k}[t_{k}](s_{\text{DAC},k}^{p}[t_{k}])^{H}\\}\big{)}\big{)}$
[34]. The term $\kappa_{t,k}$ in the pdf models the UE transmit EVM [34]. At
the $t_{k}$ time instant, the $m$th AP receives at its antennas, the sum of
pilot signals transmitted by UEs in the set $\mathcal{P}_{k}$ i.e.,

$\displaystyle\mathbf{y}_{m}^{p}[t_{k}]=\sum\limits_{i\in\mathcal{P}_{k}}\mathbf{h}_{mi}[t_{k}]s_{\text{RF},i}^{p}[t_{k}].$
(4)

To reduce the system cost, the APs are designed using hardware-impaired RF
chains and a dynamic-ADC architecture, which enables us to vary the resolution
of each ADC from $1$ to a maximum of $D$ bits. The $m$th AP feeds the above
pilots signal received at its antenna to the hardware-impaired RF chains,
whose distorted output, based on the EVM model, is [34]:

$\displaystyle\mathbf{y}_{\text{RF},m}^{p}[t_{k}]=\mathbf{y}_{m}^{p}[t_{k}]+\boldsymbol{\eta}_{\text{RF},m}^{p}[t_{k}]+\mathbf{z}_{m}^{p}[t_{k}].$
(5)

The term $\boldsymbol{\eta}_{\text{RF},m}^{p}[t_{k}]$, with pdf
$\mathcal{CN}(\boldsymbol{0},\kappa_{r,m}^{2}\mathbf{W}_{m}[t_{k}])$, models
the RF hardware distortion. The term $\kappa_{r,m}^{2}$ is the AP receiver EVM
and
$\mathbf{W}_{m}[t_{k}]=\text{diag}\big{(}\mathbb{E}\big{\\{}\mathbf{y}_{m}^{p}[t_{k}](\mathbf{y}_{m}^{p}[t_{k}])^{H}\big{|}\mathbf{h}_{mk}[t_{k}]\big{\\}}\big{)}$.
Note that the receive hardware impairment is proportional to the power of UEs
transmit signals [34]. The vector $\mathbf{z}_{m}[t_{k}]$, with pdf
$\mathcal{CN}(\boldsymbol{0},\mathbf{I}_{N})$, is the additive white Gaussian
noise (AWGN) at the $m$th AP. The RF chain output is then fed to the dynamic-
resolution ADCs, which distort it by adding quantization noise. The distorted
ADCs output, based on the Bussgang model [33], is given as

$\displaystyle\mathbf{y}_{\text{ADC},m}^{p}[t_{k}]=\mathbf{A}_{m}\mathbf{y}_{\text{RF},m}^{p}[t_{k}]+{\mathbf{n}^{p}_{\text{ADC},m}}[t_{k}]=\mathbf{A}_{m}\big{(}\mathbf{y}_{m}^{p}[t_{k}]+\boldsymbol{\eta}^{p}_{\text{RF},m}[t_{k}]+\mathbf{z}^{p}_{m}[t_{k}]\big{)}+{\mathbf{n}^{p}_{\text{ADC,m}}}[t_{k}].$
(6)

The matrix $\mathbf{A}_{m}=\text{diag}(1-\iota_{m,n},\cdots,1-\iota_{m,n})$,
where $\iota_{m,n}$ is the ADC distortion factor [33]. The zero-mean additive
ADC quantization noise ${\mathbf{n}^{p}_{\text{ADC},m}}[t_{k}]$ is
uncorrelated with $\mathbf{y}_{\text{RF},m}^{p}[t_{k}]$. It has a covariance
of
$\mathbf{B}_{m}\text{diag}\big{(}\mathbb{E}\big{\\{}\mathbf{y}_{\text{RF},m}^{p}[t_{k}](\mathbf{y}_{\text{RF},m}^{p}[t_{k}])^{H}\big{|}\mathbf{h}_{mk}[t_{k}]\big{\\}}\big{)}$
with $\mathbf{B}_{m}=\mathbf{A}_{m}(\mathbf{I}_{N}-\mathbf{A}_{m})$ [33]. The
existing CF mMIMO literature [35, 20, 17, 21, 24, 22, 23], except [27], has
not investigated the dynamic-resolution ADC architecture with different
diagonal elements of $\mathbf{A}_{m}$ i.e., $\iota_{m,p}\neq\iota_{m,q}$ for
$p\neq q$. It significantly complicates the SE analysis and derivation, when
compared with [35, 20, 17, 21, 24, 22, 23], which considers either low- or
mixed-ADC architecture at the APs. Xiong et. al in [27] considered a CF mMIMO
system with dynamic-ADC architecture, but with correlated Rayleigh fading
channels and ideal RF chains at the APs and UEs. The current work, in
contrast, studies the dynamic ADC/DAC architecture for a CF mMIMO system with
i) low-cost RF chains; ii) spatially-correlated Rician channel with phase-
shifts; and iii) channel aging. The proposed architecture is generic, and
reduces to its low-resolution and mixed-resolution counterparts by choosing
$\mathbf{A}_{m}=(1-\iota_{m})\mathbf{I}_{N}$ and
$\mathbf{A}_{m}=\text{blkdiag}\\{(1-\iota_{m})\mathbf{I}_{\gamma},\mathbf{I}_{N-\gamma}\\}$
with $0\leq\gamma\leq N$, respectively.

We next substitute expression of $\mathbf{y}_{m}^{p}[t_{k}]$ from (4) in (6),
and re-express $\mathbf{y}_{\text{ADC},m}^{p}[t_{k}]$ as

$\displaystyle\mathbf{y}_{\text{ADC},m}^{p}[t_{k}]$
$\displaystyle=\\!\sum_{i\in\mathcal{P}_{k}}\\!\mathbf{A}_{m}\big{(}\mathbf{h}_{mi}[t_{k}]\big{(}\\!{\alpha_{d,i}\sqrt{\tilde{p}_{i}}}+\upsilon_{\text{DAC},i}^{p}[t_{k}]\\!+\\!\xi_{\text{RF},i}^{p}[t_{k}]\big{)}\\!+\\!\boldsymbol{\eta}_{\text{RF},m}^{p}[t_{k}]\\!+\mathbf{z}_{m}^{p}[t_{k}]\big{)}\\!+\mathbf{n}_{\text{ADC},m}^{p}[t_{k}].$

Recall that the channel between two different time instants ages, and is
consequently correlated. The received signal
$\mathbf{y}_{\text{ADC},m}^{p}[t_{k}]$ can be exploited while estimating
channel at any other time instant also. The channel estimate quality will,
however, deteriorate with increasing time difference between the pilot
transmission ($1<n<\tau_{p}$) and the considered channel realization
($\tau_{p}+1<n<\tau_{c}$). We, therefore, without loss of generality, estimate
the channel at the time instant $\lambda=\tau_{p}+1$, and use these estimates
to obtain the channels at all other time instants ($n>\lambda$). To estimate
the channel at the $\lambda$th time instant, we express the received pilot
signal $\mathbf{y}_{\text{ADC},m}^{p}[t_{k}]$ in terms of the channel at the
time instant $\lambda$, using (2) as follows

$\displaystyle\mathbf{y}_{\text{ADC},m}^{p}[t_{k}]$
$\displaystyle=\\!\sum_{i\in\mathcal{P}_{k}}\\!\\!\mathbf{A}_{m}\left(\\!\rho_{k}[\lambda\\!-\\!t_{k}]\mathbf{h}_{mk}[\lambda]\\!+\\!\overline{\rho}_{k}[\lambda\\!-\\!t_{k}]\big{(}\bar{\mathbf{h}}_{mk}e^{j\phi_{mk}^{t_{k}}}+\mathbf{f}_{mk}[t_{k}]\big{)}\right)\Big{(}\\!{\alpha_{d,i}\sqrt{\tilde{p}_{i}}}\\!+\\!\upsilon^{p}_{\text{DAC},i}[t_{k}]\\!$
$\displaystyle\qquad+\xi^{p}_{\text{RF},i}[t_{k}]\Big{)}\\!+\mathbf{A}_{m}\boldsymbol{\eta}_{\text{RF},m}^{p}[t_{k}]+\mathbf{A}_{m}\mathbf{z}_{m}^{p}[t_{k}]+\mathbf{n}_{\text{ADC},m}^{p}[t_{k}].$
(7)

Here
$\overline{\rho}_{k}[\lambda\\!-\\!t_{k}]=\sqrt{1-\rho_{k}^{2}[\lambda\\!-\\!t_{k}]}$.
Using $\mathbf{y}_{\text{ADC},m}^{p}[t_{k}]$, the channel
$\mathbf{h}_{mk}[\lambda]$ is estimated in the following Theorem, which is
proved in Appendix A.

######  Theorem 1.

For a hardware-impaired CF mMIMO system with spatially-correlated Rician
fading and phase-shifts, the linear minimum mean square error (LMMSE) estimate
of $\mathbf{h}_{mk}[\lambda]$ is given as

$\displaystyle\hat{\mathbf{h}}_{mk}[\lambda]=\sqrt{\tilde{p}_{k}}\alpha_{d,k}\rho_{k}[\lambda-
t_{k}]\bar{\mathbf{R}}_{mk}\mathbf{A}_{m}\boldsymbol{\Psi}_{mk}\mathbf{y}_{\text{ADC},m}^{p}[t_{k}]\;,\text{where
}\overline{\mathbf{R}}_{mk}=\big{(}\bar{\mathbf{h}}_{mk}\bar{\mathbf{h}}_{mk}^{H}+\mathbf{R}_{mk}\big{)}.$
(8)

The matrix
$\boldsymbol{\Psi}_{mk}=\\!\Big{(}\sum_{i\in\mathcal{P}_{k}}\\!\mathbf{A}_{m}\alpha_{d,i}(1+\kappa_{t,i}^{2})\tilde{p}_{i}\mathbf{\overline{R}}_{mk}\mathbf{A}_{m}^{H}\\!+\\!\left(\mathbf{B}_{a}^{m}+\kappa^{2}_{r,m}\mathbf{A}_{m}\right)\mathbf{J}_{k}+\sigma^{2}\mathbf{A}_{m}\\!\Big{)}^{\\!-1}\\!\\!$
with
$\mathbf{J}_{k}=\sum_{j\in\mathcal{P}_{k}}(1+\kappa_{t,j}^{2})\alpha_{d,j}\tilde{p}_{j}\text{diag}\big{(}\bar{\mathbf{R}}_{mj}\big{)}$.

The LMMSE estimation error
$\tilde{\mathbf{h}}_{mk}[\lambda]\\!=\\!\mathbf{h}_{mk}[\lambda]\\!-\hat{\mathbf{h}}_{mk}[\lambda]$
has a zero mean, and covariance matrix
$\mathbf{C}_{mk}\\!=\\!\bar{\mathbf{R}}_{mk}\\!-\\!\alpha_{d,k}^{2}\tilde{p}_{k}\rho_{k}^{2}[\lambda\\!-\\!t_{k}]\bar{\mathbf{R}}_{mk}\mathbf{A}_{m}\mathbf{A}_{m}^{H}\bar{\mathbf{R}}_{mk}^{H}$.
The estimate $\hat{\mathbf{h}}_{mk}[\lambda]$ is uncorrelated with the error
$\tilde{\mathbf{h}}_{mk}[\lambda]$. The channel estimator, derived in [15], by
assuming Rayleigh fading and ideal RF chains and ADC/DAC, cannot be used
herein. This is because the hardware-impaired RF chains and dynamic/low-
resolution ADC/DAC architecture change the structure and computation of
$\boldsymbol{\Psi}_{mk}$.

######  Corollary 1.

If the $m$-th AP knows the LoS phase-shift $\phi_{mk}^{\lambda}$, the MMSE
estimate of the UE $k$ channel, $\mathbf{h}_{mk}[\lambda]$, is given as

$\displaystyle\hat{\mathbf{h}}_{mk}[\lambda]=\bar{\mathbf{h}}_{mk}e^{j\phi_{mk}^{\lambda}}+\sqrt{\tilde{p}_{k}}\alpha_{d,k}\rho_{k}[\lambda-
t_{k}]\mathbf{R}_{mk}\mathbf{A}_{m}\widetilde{\boldsymbol{\Psi}}_{mk}\big{(}\mathbf{y}_{\text{ADC},m}^{p}[t_{k}]-\overline{\mathbf{y}}_{\text{ADC},m}^{p}[t_{k}]\big{)}\;,\text{where
}$ (9)

$\overline{\mathbf{y}}_{\text{ADC},m}^{p}[t_{k}]=\sum\limits_{i\in\mathcal{P}_{k}}\sqrt{\tilde{p}_{i}}\bar{\mathbf{h}}_{mk}\Big{(}\rho_{k}[\lambda-
t_{k}]e^{j\phi_{mk}^{\lambda}}+\overline{\rho}_{k}[\lambda-
t_{k}]e^{j\phi_{mk}^{t_{k}}}\Big{)}$. The matrix
$\widetilde{\boldsymbol{\Psi}}_{mk}$ can be obtained from
${\boldsymbol{\Psi}}_{mk}$ in (8) by replacing $\bar{\mathbf{R}}_{mk}$ with
$\mathbf{R}_{mk}$. The analytical proof is not included here due to lack of
space. It can, however, be derived on the lines similar to Appendix A.

######  Remark 1.

We note that the LMMSE channel estimation method does not fully utilize the
temporal correlation among different time sample. Its major advantage,
however, is its closed-form solution which crucially helps us in deriving a
closed form SE expression in Section III, which is a function only of the
long-term channel statistics. This closed-form SE expression is then crucially
used to derive a low-complexity solution for its optimization, which we do in
Section IV. In contrast, the iterative Sparse Bayesian Learning (SBL) method
in [10] exploits the channel temporal correlation, but does not provide a
closed-form solution. This will radically complicate the SE closed form
derivation and its optimization. Further, the SBL method proposed in [10]
cannot be trivially extended to our CF mMIMO system with RF impairments,
ADC/DAC quantization noise and Rician phase-shifts. Reference [10] did not
consider these impairments. A closed-form SBL method for the system considered
herein, which can help in deriving closed form SE expression, however, is an
interesting direction of future work.

###  II-C Uplink Transmission

Let $s_{k}[n]$, with $\mathbb{E}\\{|s_{k}[n]|^{2}\\}=1$, be the information
symbol which the $k$th UE wants to transmit at the $n$th time instant. The
symbol $s_{k}[n]$, after scaling with power control coefficient
$\sqrt{p_{k}}$, is fed to the low-resolution DAC. Its distorted output, based
on the Bussgang model [33], is given as
$s_{\text{DAC},k}[n]\\!=\\!\alpha_{d,k}\sqrt{p_{k}}s_{k}[n]\\!+\\!\upsilon_{\text{DAC},k}[n]$.
The term $\upsilon_{\text{DAC},k}[n]$ is the DAC quantization noise [33], and
has a zero mean and variance $\\!(1\\!-\alpha_{d,k})\alpha_{d,k}p_{k}$. It is
uncorrelated with the information signal $s_{k}[n]$. The DAC output
$s_{\text{DAC},k}[n]$ is fed to the low-cost hardware-impaired RF chain, whose
distorted output, based on the EVM model [34], is given as

$\displaystyle
s_{\text{RF},k}[n]=s_{\text{DAC},k}[n]+\xi_{\text{RF},k}[n]=\alpha_{d,k}\sqrt{p_{k}}s_{k}[n]+\upsilon_{\text{DAC},k}[n]+\xi_{\text{RF},k}[n].$
(10)

The distortion term $\xi_{\text{RF},k}[n]$ is the transmit RF hardware
impairment of the $k$th UE. It has a pdf
$\mathcal{CN}(0,\kappa_{t,k}^{2}(\mathbb{E}\\{s_{\text{DAC},k}[n](s_{\text{DAC},k}[n])^{H}\\}))$,
with $\kappa_{t,k}$ being its transmit EVM [4].

The $m$th AP receives the following sum signal at its antennas:
$\mathbf{y}_{m}[n]=\sum\limits_{k=1}^{K}\mathbf{h}_{mk}[n]s_{\text{RF},k}[n]$.
The AP feeds this receive signal to its hardware-impaired RF chains, which
distorts it as[34]:

$\displaystyle\mathbf{y}_{\text{RF},m}[n]=\mathbf{y}_{m}[n]+\boldsymbol{\eta}_{\text{RF},m}[n]+{\mathbf{z}_{m}[n]}.$
(11)

Here $\boldsymbol{\eta}_{\text{RF},m}[n]$, with pdf
$\mathcal{CN}(\mathbf{0},\kappa_{r,m}^{2}\mathbf{W}^{m}[n])$, is the receiver
hardware distortion, and
$\mathbf{W}^{m}[n]=\text{diag}(\mathbb{E}\\{\mathbf{y}_{m}[n]\mathbf{y}_{m}[n]^{H}|\mathbf{h}_{mk}[n]\\})$.
The term $\mathbf{z}_{m}[n]$, with pdf
$\mathcal{CN}(\boldsymbol{0},\mathbf{I}_{N})$, is the AWGN at the $m$th AP.
The RF chain output is then fed to the dynamic-resolution ADCs, whose
quantized and

noisy output, based on the Bussgang model [33], is given as follows:

$\mathbf{y}_{\text{ADC},m}[n]=\mathbf{A}_{m}\mathbf{y}_{\text{RF},m}[n]+\mathbf{n}_{\text{ADC},m}[n]=\mathbf{A}_{m}\Big{(}\sum_{k=1}^{K}\mathbf{h}_{mk}[n]s_{\text{RF},k}[n]+\boldsymbol{\eta}_{\text{RF},m}[n]+\mathbf{z}_{m}[n]\Big{)}+\mathbf{n}_{\text{ADC},m}[n].$

The matrix $\mathbf{A}_{m}=\text{diag}\\{\iota_{m,1},\cdots,\iota_{m,N}\\}$,
with $\iota_{m,i}$ being the ADC distortion factor for the $i$th antenna of
the $m$th AP. The vector $\mathbf{n}_{\text{ADC},m}$ is the quantization
noise, which is uncorrelated with the input signal
$\mathbf{y}_{\text{RF},m}[n]$. It has a zero mean and covariance
$\boldsymbol{\Theta}_{m}=\mathbf{B}_{m}\mathbf{S}^{m}[n]$, where
$\mathbf{B}_{m}=\mathbf{A}_{m}(\mathbf{I}_{N}-\mathbf{A}_{m})$ and
$\mathbf{S}^{m}[n]=\text{diag}(\mathbb{E}\\{{\mathbf{y}_{\text{RF},m}[n]}{\mathbf{y}^{H}_{\text{RF},m}[n]}|\mathbf{h}_{mk}[n]\\})$
[33].
Principles of two-layer decoding: The CF system considered herein, as shown in
Fig. 2, employs two-layer decoding to mitigate IUI. In the first-stage, each
AP combines its received signal by using local channel estimates, which only
partially mitigate the IUI. To mitigate the residual IUI, all APs sends their
locally-combined received signal to the CPU, which performs the second-layer
LSFD. The CPU computes the LSFD weights based on the large-scale fading
coefficients, which remains constant for $100$s of coherence intervals [5,
36]. These coefficients, can thus, be easily computed and stored at the CPU
[5]. We now explain the two-step decoding in detail.

Figure 2: LSFD architecture for CF mMIMO system.

The $m$th AP first uses channel estimate $\hat{\mathbf{h}}_{mk}[\lambda]$ to
combine the distorted received signal as

$\displaystyle\breve{s}_{km}[n]=\hat{\mathbf{h}}^{H}_{mk}[\lambda]{\mathbf{y}_{\text{ADC},m}[n]}.$
(12)

It sends the combined signal $\breve{s}_{km}[n]$ to the CPU which performs
second-layer LSFD as
$\hat{s}_{k}[n]=\sum_{m=1}^{M}a^{*}_{mk}[n]\breve{s}_{km}[n]$. Here $a_{mk}$
is the complex LSFD coefficient of the $m$th AP and the $k$th UE link. This
reduces the IUI by weighing the received signals from all APs. We note that
$a^{*}_{mk}=1\;\forall m,k$, corresponds to the conventional matched filtering
based SLD considered in [1].

We now re-express the signal $\hat{s}_{k}[n]$ as

$\displaystyle\hat{s}_{k}[n]=\underbrace{\sum_{m=1}^{M}a_{mk}^{*}[n]\alpha_{d,k}\sqrt{p_{k}}\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\mathbf{h}_{mk}[n]s_{k}[n]}_{\text{Desired
Signal, }\widetilde{\text{DS}}_{k,n}}+\sum_{i\neq
k}^{K}\underbrace{\sum_{m=1}^{M}a_{mk}^{*}[n]\alpha_{d,k}\sqrt{p_{i}}\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\mathbf{h}_{mi}[n]s_{i}[n]}_{\text{Inter-
user interference, }{\text{IUI}}_{ki,n}}$
$\displaystyle\;\;+\underbrace{\sum_{m=1}^{M}a_{mk}^{*}[n]\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\Big{(}\sum_{i=1}^{K}\mathbf{h}_{mi}[n]\upsilon_{\text{DAC},i}[n]\Big{)}}_{\text{UE
DAC impairment,
}{\text{DAC}}_{k,n}}+\underbrace{\sum_{m=1}^{M}a_{mk}^{*}[n]\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\Big{(}\sum_{i=1}^{K}\mathbf{h}_{mi}[n]\xi_{\text{RF},i}[n]\Big{)}}_{\text{UE
RF impairment, }{\text{TRF}}_{k,n}}$
$\displaystyle\;\;+\underbrace{\sum_{m=1}^{M}a_{mk}^{*}[n]\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\boldsymbol{\eta_{\text{RF},m}}[n]}_{\text{AP
RF impairment,
}{\text{RRF}}_{k,n}}+\underbrace{\sum_{m=1}^{M}a_{mk}^{*}[n]\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{n}_{\text{ADC},m}[n]}_{\text{AP
ADC impairment,
}{\text{ADC}}_{k,n}}+\underbrace{\sum_{m=1}^{M}a_{mk}^{*}[n]\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}{\mathbf{z}}_{m}[n]}_{\text{AWGN,
}{\text{NS}}_{k,n}}.$

The first term $\widetilde{\text{DS}}_{k,n}$ denotes the desired signal, the
second term $\sum_{i\neq k}^{K}{\text{IUI}}_{ki,n}$ is due to the signal
transmitted by the UEs $i\neq k$ to APs. The third and fourth terms denote the
distortion due to the low-resolution DAC and the RF hardware at the UE,
respectively. The fifth and the sixth terms denote the impairments due to the
dynamic-resolution ADCs and the low-cost RF hardware at the AP. The last term
denotes the AWGN at the AP. We see that, in contrast to [15, Eq. (15)], the
low-cost RF and dynamic-resolution ADC/DAC adds four extra terms
${\text{DAC}}_{k,n}$, ${\text{TRF}}_{k,n}$, ${\text{RRF}}_{k,n}$ and
${\text{ADC}}_{k,n}$. These impairments also non-trivially modify the other
terms in (II-C), and require novel mathematical results to calculate the
closed-form SE expression.

##  III Spectral efficiency Analysis

We now derive a closed-form SE expression for the CF mMIMO system with i)
channel aging; ii) RF and dynamic ADC/DAC architecture; and iii) spatially-
correlated Rician channel with phase-shifts. The closed-form SE expression,
derived using use-and-then-forget (UatF) technique [15, 36], is valid for a
finite number of antennas, and requires only long-term channel statistics. The
UatF technique re-expresses (II-C) by decomposing the desired signal
$\widetilde{\text{DS}}_{k,n}$ therein as follows:

$\displaystyle\hat{s}_{k}[n]=\underbrace{\sum\limits_{m=1}^{M}a_{mk}^{*}[n]\alpha_{d,k}\rho_{k}[n-\lambda]\sqrt{p_{k}}\mathbb{E}\big{\\{}\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\mathbf{h}_{mk}[\lambda]\big{\\}}}_{\text{Desired
signal, }{\text{DS}}_{k,n}}\\!s_{k}[n]+{\text{IUI}}_{ki,n}+{\text{DAC}}_{k,n}$
(14)
$\displaystyle\\!\\!+\underbrace{\sum\limits_{m=1}^{M}a_{mk}^{*}[n]\alpha_{d,k}\rho_{k}[n-\lambda]\sqrt{p_{k}}\Big{(}\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\mathbf{h}_{mk}[\lambda]-\mathbb{E}\big{\\{}\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\mathbf{h}_{mk}[\lambda]\big{\\}}\Big{)}}_{\text{Beamforming
uncertainty, }{\text{BU}}_{k,n}}\\!s_{k}[n]$
$\displaystyle\\!\\!+\underbrace{\sum\limits_{m=1}^{M}\\!a_{mk}^{*}[n]\alpha_{d,k}\overline{\rho}_{k}[n\\!-\\!\lambda]\sqrt{p_{k}}\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\bar{\mathbf{f}}_{mk}[t_{k}]}_{\text{Channel
aging,
}{\text{CA}}_{k,n}}\\!s_{k}[n]+{\text{TRF}}_{k,n}+{\text{RRF}}_{k,n}+{\text{ADC}}_{k,n}+{\text{NS}}_{k,n}.$

We see that the desired signal term $\widetilde{\text{DS}}_{k,n}$ is now
decomposed as
$\widetilde{\text{DS}}_{k,n}={\text{DS}}_{k,n}+{\text{BU}}_{k,n}+{\text{CA}}_{k,n}$.
The modified desired signal term ${\text{DS}}_{k,n}$, which is used for data
detection, requires only long term channel information [15, 36]. The term
${\text{BU}}_{k,n}$ is the beamforming uncertainty [15, 36]. The term
${\text{CA}}_{k,n}$, which denotes channel aging, is because the channel
$\mathbf{h}_{mk}[n]$ at the $n$th time instant is now expressed using (2) as a
combination of the channel $\mathbf{h}_{mk}[\lambda]$ at the time instant
$\lambda$, and its innovation component
$\bar{\mathbf{f}}_{mk}[t_{k}]=\bar{\mathbf{h}}_{mk}e^{j\phi_{mk}^{t_{k}}}+\mathbf{f}_{mk}[t_{k}]$.
In (14), the sum of various terms, except ${\text{DS}}_{k,n}$ can be treated
as effective noise. Using central limit theorem, the sum can be approximated
as a worst-case Gaussian noise [36]. With this assumption, a SE lower bound
for a hardware-impaired spatially-correlated Rician-faded CF mMIMO system with
channel aging for a given LSFD weights is given as:
$\text{SE}_{k,n}=\text{log}_{2}(1+\overline{\text{SINR}}_{k,n})$, where

$\\!\\!\\!\\!\overline{\text{SINR}}_{k,n}=\frac{\overline{\text{DS}}_{k,n}}{\overline{\text{BU}}_{k,n}\\!+\overline{\text{CA}}_{k,n}\\!+\\!\sum\limits_{i\neq
k}^{K}\\!\overline{\text{IUI}}_{ki,n}\\!+\\!\overline{\text{DAC}}_{k,n}\\!+\\!\overline{\text{TRF}}_{k,n}\\!+\overline{\text{RRF}}_{k,n}\\!+\overline{\text{ADC}}_{k,n}\\!+\overline{\text{NS}}_{k,n}}.$
(15)

The term $\overline{\text{DS}}_{k,n}$ is the desired signal power,
$\overline{\text{BU}}_{k,n}$ is the beamforming uncertainty power,
$\sum\limits_{i\neq k}^{K}\overline{\text{IUI}}_{ki,n}$ is the interference
power, $\overline{\text{DAC}}_{k,n}$ is the UE DAC impairment power,
$\overline{\text{TRF}}_{k,n}$ is the UE RF impairment power,
$\overline{\text{RRF}}_{k,n}$ is the AP RF impairment power,
$\overline{\text{ADC}}_{k,n}$ is the AP ADC impairment noise and
$\overline{\text{NS}}_{k,n}$ is the AWGN variance. These terms are
mathematically defined in Table II, where
$\bar{a}_{mk}^{\rho*}[n]=a_{mk}^{*}[n]\alpha_{d,k}\rho_{k}[n_{\lambda}]$ and
$\bar{a}_{mk}^{\bar{\rho}*}[n]=a_{mk}^{*}[n]\alpha_{d,k}\bar{\rho}_{k}[n_{\lambda}]$.

$\overline{\mbox{DS}}_{k,n}\\!=\\!\Big{|}\sum\limits_{m=1}^{M}\bar{a}_{mk}^{\rho*}[n]\sqrt{p_{k}}\mathbb{E}\big{\\{}\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\mathbf{h}_{mk}[\lambda]\big{\\}}\Big{|}^{2}\\!$
$\overline{\mbox{CA}}_{k,n}\\!=\\!\mathbb{E}\Big{\\{}\Big{|}\sum\limits_{m=1}^{M}\bar{a}_{mk}^{\bar{\rho}*}[n]\sqrt{p_{k}}\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\bar{\mathbf{f}}_{mk}[\lambda]\Big{|}^{2}\Big{\\}}$
$\overline{\mbox{IUI}}_{ki,n}\\!=\\!\\!\mathbb{E}\Big{\\{}\Big{|}\sum\limits_{m=1}^{M}a_{mk}^{*}[n]\alpha_{d,i}\sqrt{p_{i}}\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\mathbf{h}_{mi}[n]\Big{|}^{2}\Big{\\}}$
$\overline{\text{RRF}}_{k,n}\\!=\\!\mathbb{E}\Big{\\{}\Big{|}\sum\limits_{m=1}^{M}a_{mk}^{*}[n]\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\boldsymbol{\eta}_{r,m}^{\text{AP}}[n]\Big{|}^{2}\Big{\\}}$
$\overline{\text{TRF}}_{k,n}\\!=\\!\mathbb{E}\Big{\\{}\Big{|}\sum_{m=1}^{M}a_{mk}^{*}[n]\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\mathbf{h}_{mi}[n]\xi_{\text{RF},i}[n]\Big{|}^{2}\Big{\\}}\\!$
$\overline{\text{ADC}}_{k,n}=\mathbb{E}\Big{\\{}\Big{|}\sum\limits_{m=1}^{M}a_{mk}^{*}[n]\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{n}_{\text{ADC},m}[n]\Big{|}^{2}\Big{\\}}$
$\overline{\mbox{DAC}}_{k,n}=\mathbb{E}\Big{\\{}\Big{|}\sum\limits_{m=1}^{M}a_{mk}^{*}[n]\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\mathbf{h}_{mi}[n]\upsilon_{\text{DAC},i}[n]\Big{|}^{2}\Big{\\}}$
$\overline{\text{NS}}_{k,n}\\!\\!=\mathbb{E}\Big{\\{}\Big{|}\sum\limits_{m=1}^{M}a_{mk}^{*}[n]\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\mathbf{z}_{m}[n]\Big{|}^{2}\Big{\\}}$
$\overline{\mbox{BU}}_{k,n}\\!=\\!\mathbb{E}\Big{\\{}\Big{|}\sum\limits_{m=1}^{M}\bar{a}_{mk}^{\rho*}[n]\sqrt{p_{k}}\Big{(}\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\mathbf{h}_{mk}[\lambda]-\mathbb{E}\big{\\{}\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\mathbf{h}_{mk}[\lambda]\big{\\}}\Big{)}\Big{|}^{2}\Big{\\}}\\!$.

Table II: Simulated expressions for the desired signal and interference terms.

The expectations therein need to be computed to derive the closed-form SE
expression. Simplifying these terms requires multiple novel mathematical
results, which are proposed in the following lemma. Its proof is relegated to
Appendix B. We note that these results are a non-trivial extension of [15].

######  Lemma 1.

Consider two correlated random vectors
$\mathbf{h}_{mk}[\tilde{n}]=\rho_{k}[\lambda-\tilde{n}]\mathbf{h}_{mk}[\lambda]+\bar{\rho}_{k}[\lambda-\tilde{n}]\Big{(}\bar{\mathbf{h}}_{mk}e^{j\phi_{mk}^{\tilde{n}}}+\mathbf{f}_{mk}[\tilde{n}]\Big{)}$
where $\tilde{n}\in\\{n,t_{k}\\}$ and
$\mathbf{h}_{mi}[\lambda]=\bar{\mathbf{h}}_{mk}e^{j\phi_{mk}^{\lambda}}+\mathbf{R}_{mk}^{\frac{1}{2}}\tilde{\mathbf{h}}_{mk}[\lambda]$.
The vectors $\mathbf{f}_{mk}[\tilde{n}],\tilde{\mathbf{h}}_{mk}[t_{k}]$ and
$\tilde{\mathbf{h}}_{mk}[n]$ are distributed as
$\mathcal{CN}(\mathbf{0},\mathbf{R}_{mk})$, and with the deterministic
diagonal matrices $\mathbf{A}_{m},\mathbf{P}_{mk}\in\mathbb{R}^{N\times N}$,
then the following results in Table III hold. Here
$\varepsilon(\rho_{i})=\big{(}\rho_{i}^{2}[\lambda-
t_{k}]\bar{\rho}_{i}^{2}[n-\lambda]+1\big{)}$.

i)
$\mathbb{E}\left\\{\mathbf{h}_{mi}^{H}[t_{k}]\mathbf{A}_{a}^{m}\mathbf{P}_{mk}\text{diag}(\mathbf{h}_{mi}[n]\mathbf{h}_{mi}^{H}[n])\mathbf{P}_{mk}^{H}\mathbf{A}_{a}^{m}\mathbf{h}_{mi}[t_{k}]\right\\}=\varepsilon(\rho_{i})\text{Tr}\left(\bar{\mathbf{R}}_{mi}\mathbf{A}_{a}^{m}\mathbf{P}_{mk}\text{diag}\big{(}\bar{\mathbf{R}}_{mi}\big{)}\mathbf{P}_{mk}^{H}\mathbf{A}_{a}^{m}\right)$
$+\rho_{i}^{2}[\lambda\\!-\\!t_{k}]\Big{(}2\text{real}\\!\left\\{\text{Tr}\\!\left(\bar{\mathbf{h}}_{mi}\bar{\mathbf{h}}_{mi}^{H}\mathbf{A}_{m}\mathbf{P}_{\\!mk}\text{diag}\left(\mathbf{P}_{\\!mk}^{H}\mathbf{A}_{m}\mathbf{R}_{mi}\right)\\!\right)\right\\}\\!+\\!\text{Tr}\left(\mathbf{R}_{mi}\mathbf{A}_{m}\mathbf{P}_{\\!mk}\text{diag}\left(\mathbf{P}_{\\!mk}^{H}\mathbf{A}_{m}\mathbf{R}_{mi}\right)\right)\\!\Big{)}$
ii)
$\mathbb{E}\\{\mathbf{h}_{mi}^{H}[\lambda]\mathbf{A}_{m}\mathbf{P}_{mk}\mathbf{h}_{mi}[\lambda]\mathbf{h}_{mi}^{H}[\lambda]\mathbf{P}_{mk}^{H}\mathbf{A}_{m}\mathbf{h}_{mi}[\lambda]\\}$
$=\text{tr}\left(\bar{\mathbf{R}}_{mi}\mathbf{A}_{m}\mathbf{P}_{\\!mk}\bar{\mathbf{R}}_{mi}\mathbf{P}_{\\!mk}^{H}\mathbf{A}_{m}\right)\\!+\left|\text{tr}\left(\mathbf{R}_{mi}\mathbf{A}_{m}\mathbf{P}_{\\!mk}\right)\right|^{2}\\!+2\text{real}\big{\\{}\bar{\mathbf{h}}_{mi}^{H}\mathbf{A}_{m}\mathbf{P}_{\\!mk}{\bar{\mathbf{h}}}_{mi}\text{tr}\left(\mathbf{R}_{mi}\mathbf{P}_{\\!mk}^{H}\mathbf{A}_{m}\right)\\!\big{\\}}$

Table III: Closed-form expressions obtained in Lemma 1.

We next derive the closed-form SE in the following theorem, which depends on
the long-term channel statistics. This theorem extensively uses Lemma 1.

######  Theorem 2.

The closed-form SE expression for a CF mMIMO system with an arbitrary number
of antennas, whose spatially-correlated Rician channel experience aging, has
RF and ADC/DAC impairments both at the APs and UEs, and employs two-stage
LSFD, is given as
$\text{SE}_{sum}=\frac{1}{\tau_{c}}\sum\limits_{n=\lambda}^{\tau_{c}}\sum\limits_{k=1}^{K}\log_{2}(1+\overline{\text{SINR}}_{k,n})$
where

$\displaystyle\\!\\!\\!\\!\overline{\text{SINR}}_{k,n}\\!=\\!\frac{\alpha_{d,k}^{2}p_{k}|\mathbf{a}_{k}^{H}[n]\boldsymbol{\delta}_{k,n}|^{2}}{\begin{Bmatrix}\\!\mathbf{a}_{k}^{H}[n]\Big{(}\mathbf{B}_{k,n}\\!+\boldsymbol{\Lambda}_{k,n}\\!+\sum\limits_{i\neq
k}^{K}\alpha_{d,i}^{2}p_{i}\mathbf{C}_{ki,n}\\!+\sum\limits_{i=1}^{K}\big{(}1\\!-\alpha_{d,i}\\!+\\!\kappa_{t,i}^{2}\big{)}\alpha_{d,i}p_{i}\mathbf{C}_{ki,n}\\\\[-3.0pt]
+\sum\limits_{i=1}^{K}\kappa^{2}_{r,m}{\mathbf{D}}_{ki,n}+\sum\limits_{i=1}^{K}\overline{\mathbf{D}}_{ki,n}+\text{diag}\big{(}\sigma^{2}\text{tr}\big{(}\overline{\boldsymbol{\Gamma}}_{mk}\mathbf{B}_{m}\big{)}\big{)}+\sigma^{2}\mathbf{Q}_{k}\Big{)}\mathbf{a}_{k}[n]\\!\\!\end{Bmatrix}}=\frac{\Delta_{k,n}}{\Omega_{k,n}}.$
(16)

Here
$\boldsymbol{\delta}_{k,n}=[\delta_{k,n}^{1},\ldots,\delta_{k,n}^{M}]^{\text{T}}\;\in\mathbb{C}^{M\times
1}$,
$\mathbf{B}_{k,n}=\text{diag}([b_{k,n}^{1},\ldots,b_{k,n}^{M}])\;\in\mathbb{C}^{M\times
M}$. The terms $\delta_{k,n}^{m}$, $b_{k,n}^{m}$,
$\boldsymbol{\Lambda}_{k,n},\mathbf{C}_{ki,n}$,
$\mathbf{D}_{ki,n},\overline{\mathbf{D}}_{ki,n},\overline{\boldsymbol{\Gamma}}_{mk}$
and $\mathbf{Q}_{k}$ are derived in Appendix C.

We see from (16) that the $\overline{\text{SINR}}_{k,n}$ expression is a
generalized Rayleigh quotient with respect to $\mathbf{a}_{k}[n]$. We now
state a Lemma from [36], which is then used to optimize the LSFD coefficients.

######  Lemma 2.

For a fixed vector $\mathbf{a}\in\mathbb{C}^{N\times 1}$ and positive definite
matrix $\mathbf{B}\in\mathbb{C}^{N\times N}$, it holds that
$\underset{\mathbf{v}}{\text{Max}}\;\frac{|\mathbf{v}^{H}\mathbf{a}|^{2}}{\mathbf{v}^{H}\mathbf{B}\mathbf{v}}=\mathbf{a}^{H}\mathbf{B}^{-1}\mathbf{a}$,
where the maximum is attained at $\mathbf{v}=\mathbf{B}^{-1}\mathbf{a}$.

The optimal LSFD coefficients, using Lemma 2, are given as

$\displaystyle\mathbf{a}_{k}^{*}[n]=$
$\displaystyle\Big{(}\mathbf{B}_{k,n}+\boldsymbol{\Lambda}_{k,n}+\sum\limits_{i\neq
k}^{K}\alpha_{d,i}^{2}p_{i}\mathbf{C}_{ki,n}+\sum\limits_{i=1}^{K}\big{(}1\\!-\alpha_{d,i}\\!+\\!\kappa_{t,i}^{2}\big{)}\alpha_{d,i}p_{i}\mathbf{C}_{ki,n}$
$\displaystyle\;\;+\sum\limits_{i=1}^{K}\kappa^{2}_{r,m}{\mathbf{D}}_{ki,n}+\sum\limits_{i=1}^{K}\overline{\mathbf{D}}_{ki,n}+\text{diag}\big{(}\sigma^{2}\text{tr}\big{(}\overline{\boldsymbol{\Gamma}}_{mk}\mathbf{B}_{m}\big{)}\big{)}+\sigma^{2}\mathbf{Q}_{k}\Big{)}^{-1}\boldsymbol{\delta}_{k,n}.$
(17)

We next provide intuitive insights using the closed-form SE expression in
(16).

######  Corollary 2.

We first simplify the SE expression in (16) for ideal hardware, and show that
it matches with the existing ones in [1, 15]. This not only validates our
results, but also shows that the current work subsumes the analysis in [1,
15]. By setting i) Rician factor $K_{mk}=0$; ii)
$\mathbf{R}_{mk}=\beta_{mk}\mathbf{I}_{N}$; and iii) ideal RF hardware and
high ADC/DAC resolution, we have

$\displaystyle\\!\\!\overline{\text{SINR}}_{k,n}=\frac{p_{k}\rho_{k}^{2}[n-\lambda]|\sum\limits_{m=1}^{M}a^{*}_{mk}[n]N\overline{\gamma}_{mk}|^{2}}{\begin{Bmatrix}\sum\limits_{i=1}^{K}\sum\limits_{m=1}^{M}\\!\big{|}a_{mk}^{*}[n]\big{|}^{2}p_{i}N\overline{\gamma}_{mk}\beta_{mi}+N^{2}\sum\limits_{i\in\mathcal{P}_{k}}p_{i}\rho_{i}^{2}[n-\lambda]\Big{|}\sum\limits_{m=1}^{M}\\!a_{mk}^{*}[n]\sqrt{\overline{\gamma}_{mk}\overline{\gamma}_{mi}}\Big{|}^{2}\\\
+\sigma^{2}\sum\limits_{m=1}^{M}|a_{mk}^{*}[n]|^{2}N\overline{\gamma}_{mk}\end{Bmatrix}}.$
(18)

Here
$\overline{\gamma}_{mk}\\!=\text{tr}(\overline{\boldsymbol{\Gamma}}_{mk})$.
The above expression matches with [15, Eq. (21)]. Further, by assuming
$\rho_{k}[n\\!-\\!\lambda]\\!=1$, $N\\!=1$ and $\mathbf{a}_{mk}[n]=1/L$, the
expression in (18) matches with [1, Eq. $(27)$].

######  Corollary 3.

We now investigate the effect of channel aging on the desired signal,
interference, RF and ADC/DAC distortion power terms in (16). We first consider
the term $\overline{\mbox{IUI}}_{ki,n}$, which is simplified in (51) in
Appendix C. For $i\in\mathcal{P}_{k}$, the term $\overline{\mbox{IUI}}_{ki,n}$
can be re-written using (51) as

$\displaystyle\overline{\mbox{IUI}}_{ki,n}=$
$\displaystyle\;\alpha_{d,i}^{2}p_{i}\bigg{(}\sum\limits_{m=1}^{M}\Big{(}|a_{mk}^{*}[n]|^{2}\Big{(}\rho_{i}^{2}[n\\!-\\!\lambda]\big{(}\varsigma_{kim,n}^{(1)}\\!+\varsigma_{kim,n}^{(2)}\\!+\varsigma_{kim,n}^{(3)}\big{)}$
$\displaystyle\quad+\overline{\rho}_{i}^{2}[n\\!-\\!\lambda]\text{tr}\left(\boldsymbol{\overline{\Gamma}}_{mk}\mathbf{A}_{m}\bar{\mathbf{R}}_{mi}\mathbf{A}_{m}\right)\\!\Big{)}+\sum\limits_{m{{}^{\prime}}\neq
m}^{M}a_{mk}^{*}[n]a_{m{{}^{\prime}}k}^{*}[n]c_{kin,mm{{}^{\prime}}}^{(\in)}\Big{)}\bigg{)}.$
(19)

We reproduce $\varsigma^{(1)}_{kin,m}$ from (45) from Appendix C for the sake
of drawing insights:

$\displaystyle\varsigma^{(1)}_{kin,m}=$
$\displaystyle\sum_{j\in\mathcal{P}_{k}}\alpha_{d,j}^{2}\tilde{p}_{j}\text{tr}\left(\bar{\mathbf{R}}_{mj}\mathbf{A}_{m}\mathbf{P}_{mk}\bar{\mathbf{R}}_{mi}\mathbf{P}_{mk}^{H}\mathbf{A}_{m}\right)+\alpha_{d,i}^{2}\tilde{p}_{i}\rho_{i}^{2}[\lambda-
t_{k}]\Big{(}\left|\text{tr}\left(\mathbf{R}_{mi}\mathbf{A}_{m}\mathbf{P}_{mk}\right)\right|^{2}$
$\displaystyle+2\text{real}\big{(}\overline{\mathbf{h}}_{mi}^{H}\mathbf{A}_{m}\mathbf{P}_{mk}{{\overline{\mathbf{h}}}_{mi}}\text{tr}(\mathbf{R}_{mi}\mathbf{P}_{mk}^{H}\mathbf{A}_{m})\big{)}\Big{)}.$
(20)

We see from (3) that $\varsigma^{(1)}_{kin,m}$ consists of
$\rho_{i}^{2}[\lambda-t_{k}]$, which does not vary with the time instant $n$.
Similarly, we see from (46) and (47) that the terms $\varsigma_{kin,m}^{(2)}$
and $\varsigma_{kin,m}^{(3)}$ are also independent of the time instant $n$.
Further, the term $c_{kin,mm^{{}^{\prime}}}^{(\in)}$ in the second summation
of (19), contains $\rho_{i}^{2}[n-\lambda]$. As a result, the interference
term $\overline{\mbox{IUI}}_{ki,n}$ can be expressed as
$e_{1}\rho_{i}^{2}[n-\lambda]+e_{2}(1-\rho_{i}^{2}[n-\lambda])+e_{3}\rho_{i}^{2}[n-\lambda]\triangleq
e_{2}+e_{4}\rho_{i}^{2}[n-\lambda]$, where $e_{1},\dots,e_{4}$ are positive
constants. The term $\overline{\text{IUI}}_{ki,n}$, due to channel aging,
decreases with time instant $n$ as
$\mathcal{O}\big{(}e_{2}+e_{4}\rho^{2}_{i}[n\\!-\\!\lambda]\big{)}$.

Similarly, the desired signal power $\overline{\mbox{DS}}_{k,n}$ and
beamforming uncertainty power $\overline{\mbox{BU}}_{k,n}$ in (16) reduce as
$\mathcal{O}(\rho_{k}^{2}[n-\lambda])$ and
$\mathcal{O}\big{(}\rho^{2}_{k}[n\\!-\\!\lambda](e_{1}+\rho^{2}_{k}[\lambda\\!-\\!t_{k}])\big{)}$,
respectively. The UE transmit and AP receiver RF impairments and ADC/DAC
quantization noise power reduce as
$\mathcal{O}(\epsilon+\rho_{k}^{2}[n-\lambda])$. We see that as $n>\lambda$
increases, the channel aging factor $\rho_{k}[n-\lambda]$ severely reduces
$\overline{\mbox{DS}}_{k,n}$ and $\overline{\mbox{BU}}_{k,n}$. This is also
validated later in Section V, where we analyze the power of desired signal, RF
impairments, and ADC/DAC quantization noise for different UE velocities.

######  Remark 2.

In CF mMIMO systems, each AP may have a small number of two to four antennas.
The system designer, may not connect each AP antenna to a different resolution
ADC, as it will increase the system complexity. The current framework,
however, also allows a designer to analytically evaluate the SE of such CF
mMIMO system by varying the ADC resolution across different APs, and by
keeping them same within an AP. This design is practical as each AP is
designed as a separate subsystem. For CF mMIMO systems, if AP has greater than
four antennas, the dynamic ADC architecture could be implemented within each
AP itself.

##  IV SE Optimization of Hardware-Impaired CF System With Channel Aging

We now optimize the SE for a given time instant $n$. The SE optimization
problem, by using the SE expression in Theorem 2, can be cast as follows:

$\displaystyle\textbf{P1:}\;\underset{\mathbf{p}}{\text{Max}}\;R_{n}(\mathbf{p})=\sum_{k=1}^{K}\log_{2}\bigg{(}1+\frac{{\Delta}_{k,n}(\mathbf{p})}{{\Omega}_{k,n}(\mathbf{p})}\bigg{)},\;\text{subject
to }0\leq p_{k}\leq P_{max}\;\forall k.$ (21)

The term $P_{max}$ is the maximum allowed UE transmit power, and the vector
$\mathbf{p}=[p_{1},\cdots,p_{K}]^{T}$. For a given time-instant $n$, the
objective of P1 is not a concave function. To optimize P1, we now use MM
technique, which first finds a surrogate function that locally approximates
the objective function with their difference minimized at the current point,
and then iteratively maximizes it. The surrogate functions should be designed
to have the following desirable features: i) convexity and smoothness; and ii)
existence of a closed-form minimizer [29]. The first property helps in easily
optimizing the surrogate problem, which can be easily shown to converge to a
stationary point of the original non-convex problem [29]. The second property
helps in designing a low-complexity optimization solution. We use MM framework
to propose two solutions. The surrogate function in the first case only has
the first property but does not have the second one. The first solution thus
has a high complexity and requires a numerical optimization solver e.g., CVX
[37]. The second solution, builds upon the first one, and combines the MM
approach with Lagrangian dual transform from [30] to design another surrogate
function, which satisfies the second property also. This leads to its highly
reduced complexity.

We now briefly explain the MM technique from [29]. Consider the following non-
concave maximization problem with a non-concave objective $f(\mathbf{x})$:

$\displaystyle{\text{Maximize}}\;\;f(\mathbf{x})\text{ subject to
}\mathbf{x}\in\mathcal{X},\text{ with $\mathcal{X}$ being a convex set}.$

The MM technique solves it by first constructing a convex surrogate
$g(\mathbf{x}|\hat{\mathbf{x}}^{(t)})$ of $f(\mathbf{x})$ at a feasible point
$\hat{\mathbf{x}}^{(t)}$ [29]. It then iteratively generates a sequence of
feasible points $\hat{\mathbf{x}}^{(t+1)}$ by maximizing
$g(\cdot|\hat{\mathbf{x}}^{(t)})$. The surrogate function
$g(\mathbf{x}|\hat{\mathbf{x}}^{(t)})$, along with two aforementioned
desirable properties, should be continuous in
$(\mathbf{x},\hat{\mathbf{x}}^{(t)})$. It should also lower bound the
objective function $f(\mathbf{x})$. The surrogate function has to additionally
satisfy the following two technical conditions [29]:

$\displaystyle\text{C1}:g(\hat{\mathbf{x}}^{(t)}|\hat{\mathbf{x}}^{(t)})=f(\hat{\mathbf{x}}^{(t)}),\text{
and
}\text{C2}:\nabla_{\mathbf{x}}g(\mathbf{x}|\hat{\mathbf{x}}^{(t)})|_{\mathbf{x}=\hat{\mathbf{x}}^{(t)}}=\nabla_{\mathbf{x}}f(\mathbf{x})|_{\mathbf{x}=\hat{\mathbf{x}}^{(t)}}.$
(22)

Each MM iteration generates a feasible point $\hat{\mathbf{x}}^{(t)}$. The
function value at each iteration $f(\hat{\mathbf{x}}^{(t)})$ increases, and
then finally converges to a stationary point of the original problem [29].

###  IV-A Minorization-maximization approach

We begin by constructing a novel surrogate function for the non-concave
objective $R_{n}(\mathbf{p})$. We achieve this aim by first proposing a lower
bound on $R_{n}(\mathbf{p})$, and then show that it satisfies C1 and C2. This
will make it a valid surrogate function. We now state a Lemma, which is then
used to construct a lower bound on $R_{n}(\mathbf{p})$.

######  Lemma 3.

For $N_{k}(\mathbf{x}):\mathbb{R}^{n}\rightarrow\mathbb{R}_{+}$,
$D_{k}(\mathbf{x}):\mathbb{R}^{n}\rightarrow\mathbb{R}_{++}$ and an increasing
function $f(\cdot)$, we have

$\displaystyle\sum_{k=1}^{2K}f\Big{(}\frac{N_{k}(\mathbf{x})^{2}}{D_{k}(\mathbf{x})}\Big{)}\geq\sum_{k=1}^{2K}f(2N_{k}(\mathbf{x})-D_{k}(\mathbf{x})).$
(23)

By substituting $N_{k}(\mathbf{x})=y_{k,n}\sqrt{{\Delta}_{k,n}(\mathbf{p})}$,
$D_{k}(\mathbf{x})=y_{k,n}^{2}{\Omega}_{k,n}(\mathbf{p})$, and
$f(\cdot)=\text{log}_{2}(1+\cdot)$ in Lemma 3, the objective function
$R_{n}(\mathbf{p})$ is lower-bounded as follows:

$\displaystyle
R_{n}(\mathbf{p})\geq\sum_{k=1}^{2K}\text{log}_{2}\Big{(}1+2y_{k,n}\sqrt{{\Delta}_{k,n}(\mathbf{p})}-y_{k,n}^{2}{\Omega}_{k,n}(\mathbf{p})\Big{)}=\widetilde{R}_{n}(\mathbf{p},\mathbf{y}_{n}).$
(24)

Here $y_{k,n}$ is a function of the feasible point $\mathbf{p}^{(t)}$ as
$y_{k,n}=\sqrt{{\Delta}_{k,n}(\mathbf{p}^{(t)})}/{\Omega}_{k,n}(\mathbf{p}^{(t)})$,
and $\mathbf{y}_{n}=[y_{1,n},\cdots,y_{K,n}]^{T}$. We now show that
$\widetilde{R}_{n}(\mathbf{p},\mathbf{y}_{n})$ in (24) satisfies conditions C1
and C2, and is a valid surrogate function.

  * •

Condition C1 can be proved by substituting the variable update of $y_{k,n}$
and $\mathbf{p}=\mathbf{p}^{(t)}$ in $R_{n}(\mathbf{p},\mathbf{y}_{n})$.

  * •

Condition C2 can be verified by differentiating both
$\widetilde{R}_{n}(\mathbf{p},\mathbf{y}_{n})$ and $R_{n}(\mathbf{p})$ at
$\mathbf{p}=\mathbf{p}^{(t)}$.

By replacing $R_{n}(\mathbf{p})$ with the surrogate function
$\widetilde{R}_{n}(\mathbf{p},\mathbf{y}_{n})$, problem P1 is recast as
follows:

P2:
$\displaystyle\underset{\mathbf{p}}{\text{Maximize}}\;\;\;\;\sum_{k=1}^{2K}\text{log}_{2}\Big{(}1+2y_{k,n}\sqrt{{\Delta}_{k,n}(\mathbf{p})}-y_{k,n}^{2}{\Omega}_{k,n}(\mathbf{p})\Big{)}$
(25a) $\displaystyle\text{subject to }0\leq p_{k}\leq P_{max}\;\forall k.$
(25b)

We note that the ${\Delta}_{k,n}(\mathbf{p})$ and ${\Omega}_{k,n}(\mathbf{p})$
in the objective of P2 are concave and convex in $\mathbf{p}$, respectively.
Problem P2 now becomes concave which can be solved using CVX [37]. The
procedure to calculate optimal power coefficients begins by first constructing
the surrogate function $R_{n}(\mathbf{p},\mathbf{y}_{n})$, with
$y_{k,n}=\sqrt{{\Delta}_{k,n}(\mathbf{p}^{(t)})}/{\Omega}_{k,n}(\mathbf{p}^{(t)})$,
and then by solving P2 using CVX.

###  IV-B Closed-form MM approach

The MM-based SE optimization has a high complexity. We now propose a
practically-implementable SE optimization algorithm by combining MM approach
with the Lagrangian dual transform [30], which provides an iterative closed-
form solution for the UE transmit powers $\mathbf{p}$. To use this approach,
problem P1 is equivalently expressed as

$\displaystyle\textbf{P3:}\;\underset{\mathbf{p},\gamma_{k,n}}{\text{Maximize}}\;\;\sum_{k=1}^{K}\log_{2}\big{(}1+\gamma_{k,n}\big{)}\;\text{
subject to
}\gamma_{k,n}\leq\frac{{\Delta}_{k,n}(\mathbf{p})}{{\Omega}_{k,n}(\mathbf{p})},\;0\leq
p_{k}\leq P_{max}\;\forall k.$ (26)

The epigraph variable $\gamma_{k,n}$ moves the ratio out of the logarithm
[37]. Problem P3 can be decomposed into outer and inner optimizations over
$p_{k}$ and $\gamma_{k,n}$, respectively. The inner optimization in
$\gamma_{k,n}$ is convex. The strong, duality therefore, holds. Its equivalent
Lagrangian function is [37]:

$\displaystyle\mathcal{L}(\gamma_{k,n},\lambda_{k,n})$
$\displaystyle=\sum_{k=1}^{K}\log_{2}\big{(}1+\gamma_{k,n}\big{)}-\sum_{k=1}^{K}\lambda_{k,n}\left(\gamma_{k,n}-\frac{{\Delta}_{k,n}(\mathbf{p})}{{\Omega}_{k,n}(\mathbf{p})}\right).$
(27)

Here $\lambda_{k,n}$ is the Lagrangian dual variable. Let
$(\gamma_{k,n}^{\ast},\lambda_{k,n}^{\ast})$ be the saddle point which
satisfies the first-order condition i.e.,
${\partial\mathcal{L}(\gamma_{k,n},\lambda_{k,n})}/{\partial\gamma_{k,n}}\big{|}_{\gamma_{k,n}^{*}}=0$,
then

$\displaystyle\lambda_{k,n}^{*}=\frac{1}{(1+\gamma_{k,n}^{*})\ln{2}}\stackrel{{\scriptstyle\mathclap{(a)}}}{{=}}\frac{\Omega_{k,n}(\mathbf{p})}{(\Delta_{k,n}(\mathbf{p})+\Omega_{k,n}(\mathbf{p}))\ln{2}}.$
(28)

Equality $(a)$ is because
$\gamma_{k,n}^{\ast}\\!=\\!{\Delta_{k,n}(\mathbf{p})}/{\Omega_{k,n}(\mathbf{p})}$,
for a fixed $\mathbf{p}$. Substituting (28) in (27), the problem P3 is
rewritten as:

$\displaystyle\textbf{P4}:\;$
$\displaystyle\underset{\mathbf{p},\gamma_{k,n}}{\text{Maximize}}\quad\mathcal{L}(\gamma_{k,n},\lambda_{k,n}^{*})=\sum_{k=1}^{K}\left[\\!\log_{2}\left(1+\gamma_{k,n}\right)-\frac{1}{\ln{2}}\left(\gamma_{k,n}-{\frac{\Delta_{k,n}(\mathbf{p})(1+\gamma_{k,n})}{\Delta_{k,n}(\mathbf{p})+\Omega_{k,n}(\mathbf{p})}}\right)\\!\right]$
(29) $\displaystyle\text{subject to }\quad 0\leq p_{k}\leq P_{max}.$

The objective function of problem P4 contains non-convex fractional terms. We
now aim to solve P4 by proposing a surrogate function that lower bounds the
objective. The relevant surrogate function is obtained by substituting
$N_{k}(\mathbf{x})=y_{k,n}\sqrt{\Delta_{k,n}(\mathbf{p})(1+\gamma_{k,n})}$ and
$D_{k}(\mathbf{x})=y_{k,n}^{2}\big{(}\Delta_{k,n}(\mathbf{p})+\Omega_{k,n}(\mathbf{p})\big{)}$
in Lemma 3, and is given as

$\displaystyle{\overline{R}}_{n}(\mathbf{p},\gamma_{k,n},\mathbf{y}_{n})=\sum_{k=1}^{K}\log_{2}(1+\gamma_{k,n})-$
$\displaystyle\frac{1}{\ln{2}}\Big{(}\gamma_{k,n}-\big{(}2y_{k,n}\sqrt{\Delta_{k,n}(\mathbf{p})(1+\gamma_{k,n})}\Big{.}$
$\displaystyle\Big{.}-y_{k,n}^{2}(\Delta_{k,n}(\mathbf{p})+\Omega_{k,n}(\mathbf{p}))\big{)}\Big{)}.$
(30)

Here $\mathbf{y}_{n}=[y_{1,n},\cdots,y_{K,n}]^{T}$ with $y_{k,n}$ being a
function of the feasible point
$\big{(}\mathbf{p}^{(t)},\gamma_{k,n}^{(t)}\big{)}$:
$y_{k,n}^{(t)}=\frac{\sqrt{\Delta_{k,n}(\mathbf{p}^{(t)})(1+\gamma_{k,n}^{(t)})}}{\Delta_{k,n}(\mathbf{p}^{(t)})+\Omega_{k,n}(\mathbf{p}^{(t)})}$.
The surrogate function
${\overline{R}}_{n}(\mathbf{p},\gamma_{k,n},\mathbf{y}_{n})$ can be shown to
follow conditions C1 and C2, on the lines similar to that of
$\widetilde{R}_{n}(\mathbf{p},\mathbf{y}_{n})$. The resultant problem is,
therefore, given as

$\displaystyle\textbf{P5:}\;\;\underset{\mathbf{p},\gamma_{k,n}}{\text{Maximize}}\quad$
$\displaystyle{\overline{R}}_{n}(\mathbf{p},\gamma_{k,n},\mathbf{y}_{n})\quad\text{subject
to}\quad 0\leq p_{k}\leq P_{max}.$ (31)

Problem P5, for a fixed $\mathbf{p}$, is convex in $\gamma_{k,n}$ and its
optimal value is given as
$\gamma_{k,n}^{\ast}=\Delta_{k,n}(\mathbf{p})/\Omega_{k,n}(\mathbf{p})$. We
now provide the closed-form expression to calculate optimal power in the
following lemma.

######  Lemma 4.

For a fixed $\gamma_{k,n}$, problem P5 is concave in $p_{k}$, and the optimal
value of $p_{k}$, obtained using the first-order optimality condition, is
given as

$\displaystyle p_{k}$
$\displaystyle=\min\left(P_{{max}},\frac{y_{k,n}^{2}\left(1+\gamma_{k,n}\right)\alpha_{d,k}^{2}|\mathbf{a}_{k}^{H}[n]\boldsymbol{\delta}_{k,n}|^{2}}{\left(\alpha_{d,k}^{2}y_{k,n}^{2}|\mathbf{a}_{k}^{H}[n]\boldsymbol{\delta}_{k,n}|^{2}+l_{k}^{d}\right)^{2}}\right).$
(32)

Here
$l_{k}^{d}=\sum\limits_{i=1}^{K}\mathbf{a}_{i}^{H}[n]\Big{(}y_{i,n}^{2}\alpha_{d,i}(1+\kappa_{t,i}^{2})\mathbf{C}_{ik,n}+\sum\limits_{i=1}^{K}y_{i,n}^{2}(\kappa^{2}_{r,m}\mathbf{D}_{ik,n}+\overline{\mathbf{D}}_{ik,n})\Big{)}\mathbf{a}_{i}[n]+\mathbf{a}_{k}^{H}[n]y_{k,n}^{2}(\mathbf{B}_{k,n}+\boldsymbol{\Lambda}_{k,n}-\alpha_{d,k}^{2}\mathbf{C}_{kk,n})\mathbf{a}_{k}[n]$.

###### Proof.

Refer to Appendix D. ∎

The procedure to calculate optimal power coefficients is summarized in
Algorithm 1. By iteratively constructing the function
$\overline{R}_{n}(\mathbf{p},\gamma_{k,n},\mathbf{y}_{n})$, and by updating
the transmit powers $p_{k}$ and the epigraph variable $\gamma_{k,n}$, the
algorithm converges to a local optimum of P1 [29].

Input: i) Initialize $\\{p_{k}\\}_{k=1}^{K}$ with equal power allocation, set
maximum number of iteration $I$ and stopping tolerance $\epsilon$.

Output: $\mathbf{p}^{\ast}$

1 for  _$i\leftarrow 1$ to $I$_ do

2 For a given feasible $\mathbf{p}=\mathbf{p}^{(i)}$, compute the updates of
variables $y_{k,n}$ and $\gamma_{k,n}$.

3 Obtain $\mathbf{p}^{(i+1)}$ from (32).

4 [-2pt] Do until convergence
$(\|\mathbf{p}^{(i+1)}-\mathbf{p}^{(i)}\|^{2}\leq\epsilon)$.

5 return _$\mathbf{p}^{\ast}=\mathbf{p}^{(i+1)}$ ._

Algorithm 1 SE optimization using closed-form MM approach

Computational complexity: The complexity of CVX-based approach is dominated by
that of solving P2 in each iteration, which has $K$ optimization variables and
$2K$ linear constraints. This approach has a worst-case complexity of
$\mathcal{O}((3K)^{3/2}K^{2})$ [37]. The update of variable $\mathbf{y}_{n}$
has trivial complexity. The Algorithm 1, which computes optimal transmit power
in a closed-form, has a trivial complexity. Also, the proposed closed-form MM
approach in Algorithm 1 has the same complexity as that of [38] and [39], when
applied for the system models therein.

##  V Simulation Results

We now numerically validate the derived closed-form SE expression in (16), and
investigate the effect of i) channel aging; ii) RF and ADC/DAC impairments;
iii) Rician channels with phase-shifts; and iv) LSFD. For these studies, we
consider a CF mMIMO system, wherein $M$ APs and $K$ UEs are randomly
distributed within a geographical area of $1\times 1\;\text{Km}^{2}$. We,
similar to [1, 15], assume that the coverage area is wrapped around the edges
to avoid the boundary effect. We assume a system bandwidth of $B=20$ MHz, and
a resource block length of $\tau_{c}=100$ instants. Each AP has a uniform
linear antenna array, whose correlation matrix $\mathbf{R}_{mk}$, is modelled
using the Gaussian local scattering model with $\text{ASD}=30^{\circ}$ [36].
The large scale fading coefficients and the Rician factors corresponding to
the UE-AP channels are modeled as follows [3, Table-5.1]:

$\displaystyle\beta_{mk}=-30.9-26\text{log}_{10}(d_{mk})+\varrho_{mk}\;\;\text{and
}K_{mk}=13-0.03d_{mk}[\text{dB}].$ (33)

Here $\varrho_{mk}$ is the correlated shadow fading, with $\sigma_{sh}$ being
the standard deviation. The term $d_{mk}$ is the 2D distance from the $k$th UE
to the $m$th AP. We set the transmit and receiver RF impairment levels as
$\kappa_{r,m}=\kappa_{r}$ and $\kappa_{t,k}=\kappa_{t}$. We assume that each
UE is equipped with a $b$-bit DAC, and the APs have the following dynamic ADC
architecture: $[b_{1},b_{2},b_{3},b_{4}]$ i.e., out of $M$ APs, each of the
$25\%$ of the APs have $b_{1}$, $b_{2}$, $b_{3}$, and $b_{4}$ bit ADC
resolution, respectively. The ADC/DAC distortion factor $\rho$ for $b$ bits is
given in [22]. We set the noise variance $\sigma^{2}_{m}=-94$ dBm, $M=64$ APs,
$N=4$ antennas per AP, pilot power $\tilde{p}_{k}=10$ dBm, the velocity of UEs
$v_{k}=54$ Km/hr. These parameters remain fixed unless explicitly specified.

####  V-1 Validation of closed-form SE

We first validate in Fig. 3(a) the derived closed-form SE expression in (16)
by comparing it with its simulated counterpart in (15), which numerically
computes the expectations. For this study, we consider the following RF
impairment and ADC/DAC combinations: i) ideal RF with
$\kappa_{r}=\kappa_{t}=\kappa=0$, and ideal ADC/DAC with $b=\infty$-bit
resolution; ii) non-ideal RF with $\kappa=0.1$, and ideal ADC/DACs with
$b=\infty$ bit; iii) non-ideal RF with $\kappa=0.1$, and the following dynamic
ADC architecture at the APs: $[b_{1},b_{2},b_{3},b_{4}]=[1,2,4,6]$; and iv)
ideal RF with $\kappa=0$ and $1$-bit ADC/DACs. We see that for all of the
aforementioned RF and ADC/DAC impairments, the derived closed-form SE exactly
matches with its simulated counterpart. This validates the derived analytical
closed-form SE expression, which can thus be used for realistic evaluation of
hardware-impaired CF mMIMO systems with channel aging. The dynamic ADC
architecture with, varied ADC resolution at APs and non-ideal RF with
$\kappa=0.1$, is able to provide $85\%$ (resp. $75\%$) of the SE achieved by
ideal ADCs (resp. ideal RF and ADC) case. The dynamic ADC architecture, with a
careful bit resolution choice, is able to recover the spectral loss due to
low-resolution ADCs and a non-ideal RF.

(a)

(b)

(c)

(d)

Figure 3: a) Validation of closed-form SE; b) SE versus time instant $n$; c)
SE versus the length of resource block $\tau_{c}$; and d) SE versus pilot
length $\tau_{p}$.

####  V-2 Impact of velocity on the length of the resource block $\tau_{c}$

We now investigate the combined effect of UE velocity and the resource block
length $\tau_{c}$ on the sum SE. This will help in deciding the appropriate
$\tau_{c}$ value for different UE velocities. We perform this study by
plotting in Fig. 3(b) the SE versus the time instant $n$ in a resource block.
For this study, we fix $\tau_{c}=500$, and consider three different UE
velocities. We first see that for each UE velocity, the SE reduces with time
instant $n$, and becomes zero. This is due the channel aging. The resource
block length should be, thus, much lesser than the instant at which the SE
becomes zero. This will avoid SE degradation towards the end of resource
block. We also note that a small-length resource block will also increase the
pilot overhead. We next investigate this trade-off in Fig. 3(c) by plotting
the SE versus the resource block length $\tau_{c}$ for different UE
velocities. We see that, for all UE velocities, the SE is relatively low for
very small $\tau_{c}$ values. It then increases with $\tau_{c}$, and then
reduces. This is because for very small $\tau_{c}$ values, the pilot overhead
is high, even though the channel remains fresh. The increased pilot overhead
dominates for such low $\tau_{c}$ values, which leads to low SE values. The
increase in SE with $\tau_{c}$ is due to the reduction in pilot overhead, even
though the channel starts aging now. The reduced overhead dominates the
degradation due to channel aging, which increases the SE. The final decrease
in SE with increase in $\tau_{c}$ is because the channel ages too much for a
long resource block. The degradation due to channel aging now dominates the
reduced pilot overhead, which reduce the SE. We also note that the SE for a
higher UE velocity, peaks for smaller $\tau_{c}$ value. This is due to the
increased channel aging. A system designer can thus, depending on the UE
velocity, decide the $\tau_{c}$ value, where the SE peaks.

####  V-3 Impact of channel estimation on SE

We now jointly investigate in Fig. 3(d) the impact of channel estimation on SE
and resource allocation between channel training and data transmission, by
plotting the sum SE versus the pilot sequence length $\tau_{p}$ for different
antenna values at the AP. We consider $M=100$ APs, $K=60$ UEs and velocity of
UEs $v_{k}=54$ Km/hr. We first observe that with increase in $\tau_{p}$, the
SE first increases till a threshold value of $\tau_{p}=20$, and reduces after
that. This is because the increase in $\tau_{p}$ increases the number of
pilots. This reduces the number of UEs sharing the same pilot, which in turn,
reduces the pilot contamination. This improves the channel estimation quality.
For $\tau_{p}\leq 20$, the increase in SINR due to improved channel estimates
dominates the linear decrease in data transmission duration (resources), which
increases the SE. For $\tau_{p}>20$, the decrease in SE due to the reduced
data transmission duration dominates the increased SE due to improved pilot
channel estimation, which reduces the SE. This shows that $\tau_{p}=20$ is
optimal for these system configurations.

####  V-4 Impact of number of APs and UEs on SE

We numerically investigate in Fig. 4(a) that the SE versus the number of APs
$M$ for LSFD and SLD. We perform this study for $K=20$ and $K=40$ UEs, and
consider a pilot length of $\tau_{p}=K/2$. We see that the SE increases with
increase in $M$. This is due to the increased array gain. We also note that
the percentage of LSFD gain over SLD is higher for $K=40$ UEs than $K=20$ UEs.
Specifically, for $K=40$ UEs and lower (resp. higher) number of APs, the LSFD
offers $93\%$ (resp. $82\%$) SE gain over SLD, whereas these gains are $78\%$
(resp. $56\%$) for $K=20$ case. This is because the increased number of UEs
increases the interference experienced by the desired UE, which the LSFD can
efficiently mitigate at the CPU by performing an extra level of decoding.

####  V-5 Effect of channel aging on the power of desired and interference
terms

We now numerically investigate in Fig. 4(b), the behavior of the desired
signal power $\overline{\text{DS}}_{k,n}$, channel aging power
$\overline{\text{CA}}_{k,n}$, inter-user interference power $\sum_{i\neq
k}^{K}\overline{\text{IUI}}_{ki,n}$, and UE RF and DAC distortion power
$\overline{\text{DAC}}_{k,n}+\overline{\text{TRF}}_{k,n}$ (labelled as
$\overline{\text{DS}}$, $\overline{\text{CA}}$, $\overline{\text{IUI}}$ and
$\overline{\text{DAC}}+\overline{\text{TRF}}$, respectively), for LSFD and SLD
schemes. For this study, we consider $K=20$ UEs. The $K/2$ UEs have a velocity
of $v_{k}=54$ Km/hr, and the remaining $K/2$ UEs have $v_{k}=212$ Km/hr. We
plot in Fig. 4(b), the power level of signal and interference terms of all UEs
with LSFD. We observe the following:

  * •

For high-velocity UEs i.e., $v_{k}=212$ Km/hr, the $\overline{\text{DS}}$ and
$\overline{\text{DAC}}+\overline{\text{TRF}}$ powers reduce with time (see the
bottom subplot in Fig. 4(b)). For example, at $n=100$, they are roughly $20$
dB lesser than at $n=10$. This is because both these powers are functions of
the temporal correlation coefficient $\rho_{k}[n-\lambda]$, whose value, for a
high UE velocity, reduces greatly with time. For low-velocity UEs, as shown in
the top subplot in Fig. 4(b), these powers are almost time-invariant. This is
because for such UEs, $\rho_{k}[n-\lambda]$ varies very slowly with $n$.

  * •

For high-velocity UEs and $n\geq 60$, we see that $\overline{\text{DS}}$
reduces monotonically, while $\overline{\text{DAC}}+\overline{\text{TRF}}$
power first tapers out, and then floors to a constant value. This is because,
as also shown in Corollary 3, the $\overline{\text{DS}}$ power decreases as
$\rho_{k}^{2}[n-\lambda]$, which decreases monotonically with $n$. The
transmit RF + DAC impairments reduce as $\epsilon+\rho_{k}^{2}[n-\lambda]$,
with $\epsilon>1$ being a constant. For $n\geq 60$, the constant $\epsilon$
term dominates the reduction due to $\rho_{k}^{2}[n-\lambda]$, which makes the
$\overline{\text{DAC}}+\overline{\text{TRF}}$ impairment power floor to a
constant value.

  * •

For both UE velocities, $\overline{\text{IUI}}$ power almost remains constant.
This observation is in line with the result in Corollary 3, which showed that
IUI power remains constant for all UE velocities.

(a)

(b)

(c)

(d)

Figure 4: a) Comparison of LSFD and SLD for different UE configuration; Effect
of channel aging on the power of desired and interference terms for b) LSFD;
and c) SLD; and d) SE versus number of antenna per AP for different UE
velocities.

We next compare Fig. 4(b) with Fig. 4(c), which plots the above power values
for SLD. We see that for low-velocity UEs, LSFD has a much lower
$\overline{\text{IUI}}$, $\overline{\text{CA}}$ and
$\overline{\text{DAC}}+\overline{\text{TRF}}$ power than SLD. This is because
the channels of low-velocity UEs do not significantly age, and consequently
their channel estimate quality do not deteriorate. The LSFD can thus better
suppress the IUI. For high-velocity UEs, LSFD yields much lower
$\overline{\text{CA}}$ values than SLD, while both LSFD and SLD yield similar
$\overline{\text{IUI}}$ values. This implies that LSFD can mitigate the effect
of channel aging for high-velocity UEs, but not IUI. This study not only
validates the interference-related insights in Corollary 3, but also shows
that LSFD can mitigate i) the effect of channel aging for low/high UE
velocities; and ii) IUI, but only for low-velocity UEs.

####  V-6 Impact of AP antennas on the UE velocity

We now investigate in Fig. 4(d) the SE versus number of antennas per AP ($N$)
for different UE velocities: $v_{k}=\\{0,54,128\\}$ Km/hr. We see that the SE
increasing with $N$. This is due to the increased array gain. We also infer
that for different $N$ values, the $\%$ of SE loss for $v_{k}=54$ Km/hr (resp.
$v_{k}=128$ Km/hr) over $v_{k}=0$ Km/hr is same i.e., $5\%$ (resp. $19\%$).
This crucially informs that the SE degradation due to channel aging is
independent of the $N$ value, and depends only on the UE speed.

####  V-7 Dynamic ADC architecture across APs/antennas

To investigate the effect of dynamic ADC architecture, we plot in Fig. 5(a)
the SE versus transmit power and compare for the following ADC architectures:
i) ideal ADCs; ii) Dynamic ADC - antennas: $25\%$ of the antennas at each AP
has $1$, $2$, $3$ and $4$ bit ADC resolutions; and iii) Dynamic ADC - APs:
$25\%$ of the APs have $1$, $2$, $3$ and $4$ bit resolution. We see from Fig.
5(a) that Dynamic ADC $-$ antennas architecture yields a similar SE as that of
Dynamic ADC $-$ APs architecture. Also, both dynamic architectures provide
nearly $84\%$ of the ideal ADCs SE. For this study, we assume that each AP has
$4$ antennas.

This shows the flexibility of dynamic ADC architecture, which allows the
system designer to either have different ADC resolution across the antennas of
each AP or across APs.

####  V-8 SE versus UE transmit power

We now plot in Fig. 5(b), the SE obtained using the proposed optimization
algorithms: CVX-based (labelled as MM) and Algorithm 1 (labelled as closed-
form MM) for LSFD and SLD schemes. We also compare them with full power
allocation (FPA) scheme, where each UE transmits at its maximum power. We fix
$M=64$ APs, $K=20$ UEs, $N=4$ antennas, $\kappa=0.1$ and $b=4$ bits at both
APs and UEs. We first see that both CVX-based and closed-form MM provide a
similar SE, but the latter with its closed-form transmit power updates, has a
trivial complexity. It can, therefore, be easily implemented in commercial CF
systems. Also, both proposed optimizations improve the SE by $20\%$ in SLD,
whereas they provide a reduced gain of $6.25\%$ in LSFD. This is because SLD
is unable to effectively mitigate the IUI, which in turn is mitigated by using
the optimal powers designed by the proposed algorithms. The LSFD, in contrast,
mitigates the IUI by performing an extra level of decoding at the CPU, and
optimizing transmit power, therefore, only provides marginal SE gain.

(a)

(b)

(c)

(d)

Figure 5: a) Comparison of different ADC architectures; b) SE versus UE
transmit power; c) SE versus time instant $n$; d) CPU and average run time for
different optimization techniques.

####  V-9 Effect of channel aging on SE optimization

We now study in Fig. 5(c), the effect of optimization time instant on the sum
SE for the following UE velocities $v_{k}=\\{54,212\\}$ Km/hr. We optimize the
SE at a specific time-instant, denoted as $n_{\text{opt}}$, and use the
obtained optimal transmit powers to calculate the SE for all time-instants. We
compare in Fig. 5(c), the SE for:

  * •

Case 1: SE is optimized at every data transmission instant i.e.,
$n_{\text{opt}}=n$.

  * •

Case 2: SE is optimized at one of the following time instants
$n_{\text{opt}}=\\{\tau_{p}+1,\frac{\tau_{c}}{4},\frac{\tau_{c}}{2},\frac{3\tau_{c}}{4}\\}$.

We perform this study for two different UE velocities of $v_{k}=54$ Km/hr and
$v_{k}=212$ Km/hr. We see that for a low UE velocity of $v_{k}=54$ Km/hr, Case
2, irrespective of the optimizing time-instant $n_{\text{opt}}$ yields a
similar SE as that of Case 1 for all time instants. For a high UE velocity of
$v_{k}=212$ Km/hr, Case 1 and Case 2 with
$n_{\text{opt}}\leq\frac{3\tau_{c}}{4}$ has similar SE. But, for
$n_{\text{opt}}=\tau_{c}$, Case 2 yields only a minor $10\%$ lower SE than
Case 1, and that too during initial time instants. The SE thus can be
optimized in the beginning of the resource block and its solution could be
reported only once, which makes the proposed optimization practical.

####  V-10 Optimization complexity

We now analyze the time complexity of both the proposed iterative optimization
MM techniques – CVX-based and closed-form – by first plotting in Fig. 5(d)
their total CPU run time required to converge. We see that the later approach
requires much lesser time to converge. Also, both the techniques converge to
the same SE value. We next also investigate the average per-iteration CPU run
time required by both algorithms (shown in the sub-plot). We see that the
closed-form technique has a much lesser run-time per iteration. This is due to
its closed-form power updates with trivial complexity.

##  VI Conclusion

We derived a closed-form SE expression for a hardware-impaired spatially-
correlated Rician-faded CF mMIMO system with channel aging and two-layer LSFD.
We verified this expression for different transmit power and hardware
impairment values. We optimized the non-convex SE metric by proposing two
novel optimization techniques. The first one has a high complexity, while the
second provides a closed-form solution with a trivial complexity. We
numerically showed that the two-layer LSFD effectively mitigates the
interference due to channel aging for both low- and high-velocity UEs. It,
however, mitigates inter-user interference only for the low-velocity UEs, and
not the fast ones. We also showed that the SE loss due to hardware impairments
can be compensated by the proposed dynamic ADC architecture.

## Appendix A

The LMMSE estimate of the channel $\mathbf{h}_{mk}$, based on received pilot
signal in (7) is given by

$\displaystyle\hat{\mathbf{h}}_{mk}[\lambda]=\mathbf{C}_{\mathbf{h}_{mk}[\lambda]\mathbf{y}_{m}[t_{k}]}\mathbf{C}_{\mathbf{y}_{m}[t_{k}]\mathbf{y}_{m}[t_{k}]}^{-1}\mathbf{y}_{m}[t_{k}].$
(34)

Here
$\mathbf{C}_{\mathbf{h}_{mk}[\lambda]\mathbf{y}_{m}^{p}[t_{k}]}=\mathbb{E}\big{\\{}\mathbf{h}_{mk}[\lambda](\mathbf{y}^{p}_{\text{ADC},m}[t_{k}])^{H}\big{\\}}$
and
$\mathbf{C}_{\mathbf{y}_{m}^{p}[t_{k}]\mathbf{y}^{p}_{m}[t_{k}]}=\mathbb{E}\big{\\{}\mathbf{y}_{\text{ADC},m}^{p}[t_{k}](\mathbf{y}^{p}_{\text{ADC},m}[t_{k}])^{H}\big{\\}}$.

We begin by simplifying the term
$\mathbf{C}_{\mathbf{y}_{m}[t_{k}]\mathbf{y}_{m}[t_{k}]}$ by substituting
$\mathbf{y}_{\text{ADC},m}[t_{k}]$ from (7) as

$\displaystyle\mathbf{C}_{\mathbf{y}_{m}[t_{k}]\mathbf{y}_{m}[t_{k}]}$
$\displaystyle\overset{(a)}{=}\sum_{j\in\mathcal{P}_{k}}\mathbf{A}_{m}\alpha_{d,j}^{2}\tilde{p}_{j}\bar{\mathbf{R}}_{mj}\mathbf{A}_{m}^{H}+\sum_{j\in\mathcal{P}_{k}}\mathbf{A}_{m}(1-\alpha_{d,j}+\kappa_{t,j}^{2})\alpha_{d,j}\tilde{p}_{j}\bar{\mathbf{R}}_{mj}\mathbf{A}_{m}^{H}$
$\displaystyle\qquad+\kappa^{2}_{r,m}\mathbf{J}_{k}+\mathbf{B}_{m}\big{(}(1+\kappa_{r,m}^{2})\mathbf{J}[t_{k}]+\sigma^{2}\mathbf{I}_{N}\big{)}+\sigma^{2}\mathbf{A}_{m}\mathbf{A}_{m}^{H}$
$\displaystyle=\sum_{j\in\mathcal{P}_{k}}\mathbf{A}_{m}\alpha_{d,j}(1+\kappa_{t,j}^{2})\tilde{p}_{j}\mathbf{\overline{R}}_{mj}\mathbf{A}_{m}^{H}+\big{(}\mathbf{B}_{a}^{m}+\kappa^{2}_{r,m}\mathbf{A}_{m}\big{)}\mathbf{J}+\sigma^{2}\mathbf{A}_{m}.$
(35)

Here
$\mathbf{J}_{k}=\sum_{j\in\mathcal{P}_{k}}(1+\kappa_{t,j}^{2})\alpha_{d,j}\tilde{p}_{j}\text{diag}\big{(}\bar{\mathbf{R}}_{mj}\big{)}$.
Equality $(a)$ is obtained by noting that the i) distortion terms
$\upsilon_{\text{DAC},k}^{p}[t_{k}],\xi_{\text{RF},k}^{p}[t_{k}],\boldsymbol{\eta}^{p}_{\text{RF},m}[t_{k}]$
and $\mathbf{n}_{\text{ADC},m}^{p}[t_{k}]$ are independent of each other, and
have a zero mean; and ii) variance of distortion/quantization noises are
$\mathbb{E}\\{\xi_{\text{RF},j}^{p}[t_{k}](\xi^{p}_{\text{RF},j}[t_{k}])^{H}\\}\\!=\kappa_{t,j}^{2}\alpha_{d,j}\tilde{p}_{j}$,
$\mathbb{E}\\{\mathbf{n}_{\text{ADC},m}^{p}[t_{k}](\mathbf{n}_{\text{ADC},m}^{p}[t_{k}])^{H}\\}=\mathbf{B}_{m}\mathbf{S}^{m}[t_{k}]$,
$\mathbb{E}\\{\boldsymbol{\eta}_{\text{RF},m}^{p}[t_{k}](\boldsymbol{\eta}^{p}_{\text{RF},m}[t_{k}])^{H}\\}=\kappa^{2}_{r,m}{\mathbf{W}}^{m}[t_{k}]$;
and iii) impairment/quantization noises are independent of channel
$\mathbf{h}_{mk}$. Here the matrices
$\mathbf{W}^{m}[t_{k}]=\sum_{j\in\mathcal{P}_{k}}(1+\kappa_{t,j}^{2})\alpha_{d,j}\tilde{p}_{j}\text{diag}\big{(}\mathbf{h}_{mj}[t_{k}]\mathbf{h}_{mj}^{H}[t_{k}]\big{)}$
and
$\mathbf{S}^{m}[n]=(1+\kappa_{r,m}^{2})\mathbf{W}^{m}[t_{k}]+\sigma^{2}\mathbf{I}_{N}$.
We, similarly, have
$\mathbf{C}_{\mathbf{h}_{mk}[\lambda]\mathbf{y}_{m}^{p}[t_{k}]}$ as
$\mathbf{C}_{\mathbf{h}_{mk}[\lambda]\mathbf{y}_{m}^{p}[t_{k}]}=\alpha_{d,k}\sqrt{\tilde{p}_{k}}\rho_{k}[\lambda\\!-\\!t_{k}]\bar{\mathbf{R}}_{mk}\mathbf{A}_{m}$.

## Appendix B

To obtain the result (a) in Table III, we first express
$\mathbf{h}_{mi}[t_{k}]$ and $\mathbf{h}_{mi}[n]$ in terms of
$\mathbf{h}_{mi}[\lambda]$, $\mathbf{u}_{mi}[t_{k}]$ and $\mathbf{u}_{mi}[n]$
as

$\displaystyle\mathbb{E}\left\\{\mathbf{h}_{mi}^{H}[t_{k}]\mathbf{A}_{a}^{m}\mathbf{P}_{mk}\text{diag}(\mathbf{h}_{mi}[n]\mathbf{h}_{mi}^{H}[n])\mathbf{P}_{mk}^{H}\mathbf{A}_{a}^{m}\mathbf{h}_{mi}[t_{k}]\right\\}$
$\displaystyle\quad\qquad=\rho_{i}^{2}[\lambda-
t_{k}]\mathbb{E}\big{\\{}\mathbf{h}_{mi}^{H}[\lambda]\mathbf{A}_{a}^{m}\mathbf{P}_{mk}\text{diag}(\rho_{i}^{2}[n-\lambda]\mathbf{h}_{mi}[\lambda]\mathbf{h}_{mi}^{H}[\lambda])\mathbf{P}_{mk}^{H}\mathbf{A}_{a}^{m}\mathbf{h}_{mi}[\lambda]\big{\\}}$
$\displaystyle\qquad\qquad+\big{(}\rho_{i}^{2}[\lambda-
t_{k}]\bar{\rho}_{i}^{2}[n-\lambda]+\bar{\rho}_{i}^{2}[\lambda-
t_{k}]\big{)}\text{tr}\left(\bar{\mathbf{R}}_{mi}\mathbf{A}_{a}^{m}\mathbf{P}_{mk}\text{diag}\big{(}\bar{\mathbf{R}}_{mi}\big{)}\mathbf{P}_{mk}^{H}\mathbf{A}_{a}^{m}\right).$
(36)

Equality $(a)$ is obtained by substituting
$\mathbb{E}\\{\mathbf{u}_{mi}[t_{k}]\mathbf{u}^{H}_{mi}[t_{k}]\\}\\!=\\!\mathbb{E}\\{\mathbf{u}_{mi}[n]\mathbf{u}^{H}_{mi}[n]\\}\\!=\\!\mathbf{R}_{mi}$
as [40]. We now calculate the expectation in the first term of (36) by
substituting the value of $\mathbf{h}_{mi}[\lambda]$ as

$\displaystyle\mathbb{E}\big{\\{}\mathbf{h}_{mi}^{H}[\lambda]\mathbf{A}_{a}^{m}\mathbf{P}_{mk}\text{diag}(\mathbf{h}_{mi}[\lambda]\mathbf{h}_{mi}^{H}[\lambda])\mathbf{P}_{mk}^{H}\mathbf{A}_{a}^{m}\mathbf{h}_{mi}[\lambda]\big{\\}}$
$\displaystyle=\bar{\mathbf{h}}_{mi}^{H}\mathbf{A}_{a}^{m}\mathbf{P}_{mk}\text{diag}(\bar{\mathbf{h}}_{mi}\bar{\mathbf{h}}_{mi}^{H}+\mathbf{R}_{mi})\mathbf{P}_{mk}^{H}\mathbf{A}_{a}^{m}\bar{\mathbf{h}}_{mi}+\text{Tr}\left(\mathbf{R}_{mi}\mathbf{A}_{a}^{m}\mathbf{P}_{mk}\text{diag}(\bar{\mathbf{h}}_{mi}\bar{\mathbf{h}}_{mi}^{H})\mathbf{P}_{mk}^{H}\mathbf{A}_{a}^{m}\right)$
$\displaystyle\quad+2\text{real}\Big{\\{}\mathbb{E}\big{\\{}\bar{\mathbf{h}}_{mi}^{H}\mathbf{A}_{a}^{m}\mathbf{P}_{mk}diag(\bar{\mathbf{h}}_{mi}{\tilde{\mathbf{h}}}_{mi}^{H}[\lambda]\mathbf{R}_{mi}^{1/2})\mathbf{P}_{mk}^{H}\mathbf{A}_{a}^{m}\mathbf{R}_{mi}^{1/2}{\tilde{\mathbf{h}}}_{mi}[\lambda]\big{\\}}\Big{\\}}$
$\displaystyle\quad+\mathbb{E}\Big{\\{}{\tilde{\mathbf{h}}}_{mi}^{H}[\lambda]\mathbf{R}_{mi}^{1/2}\mathbf{A}_{a}^{m}\mathbf{P}_{mk}diag(\mathbf{R}_{mi}^{1/2}{\tilde{\mathbf{h}}}_{mi}[\lambda]{\tilde{\mathbf{h}}}_{mi}^{H}[\lambda]\mathbf{R}_{mi}^{1/2})\mathbf{P}_{mk}^{H}\mathbf{A}_{a}^{m}\mathbf{R}_{mi}^{1/2}{\tilde{\mathbf{h}}}_{mi}[\lambda]\Big{\\}}$
$\displaystyle=\bar{\mathbf{h}}_{mi}^{H}\mathbf{A}_{a}^{m}\mathbf{P}_{mk}\text{diag}(\bar{\mathbf{h}}_{mi}\bar{\mathbf{h}}_{mi}^{H}+\mathbf{R}_{mi})\mathbf{P}_{mk}^{H}\mathbf{A}_{a}^{m}\bar{\mathbf{h}}_{mi}+\text{Tr}\left(\mathbf{R}_{mi}\mathbf{A}_{a}^{m}\mathbf{P}_{mk}\text{diag}(\bar{\mathbf{h}}_{mi}\bar{\mathbf{h}}_{mi}^{H})\mathbf{P}_{mk}^{H}\mathbf{A}_{a}^{m}\right)$
$\displaystyle+\\!2\text{real}\Big{\\{}\mathbb{E}\Big{\\{}\\!\sum_{j}\Big{(}\\!\sum_{n_{1}}[\bar{\mathbf{h}}_{mi}]_{n_{1}}^{*}[\bar{\mathbf{h}}_{mi}]_{j}[\mathbf{A}_{m}\mathbf{P}_{\\!mk}]_{n_{1}j}\\!\Big{)}\Big{(}\\!\sum_{n_{2}}[\mathbf{P}_{\\!mk}^{H}\mathbf{A}_{m}]_{jn_{2}}[{\tilde{\mathbf{h}}}_{mi}^{H}[\lambda]\mathbf{R}_{mi}^{1/2}]_{j}[\mathbf{R}_{mi}^{1/2}\tilde{\mathbf{h}}_{mi}[\lambda]]_{n_{2}}\\!\Big{)}\\!\Big{\\}}\Big{\\}}$
$\displaystyle+\Big{(}\sum_{j}\sum_{n_{1}}\sum_{n_{2}}\sum_{n_{3}}\sum_{n_{4}}\mathbb{E}\Big{\\{}[{\tilde{\mathbf{h}}}_{mi}^{*}[\lambda]]_{n_{1}}[{\tilde{\mathbf{h}}}_{mi}[\lambda]]_{n2}[{\tilde{\mathbf{h}}}_{mi}^{*}[\lambda]]_{n3}[{\tilde{\mathbf{h}}}_{mi}[\lambda]]_{n4}\Big{\\}}$
$\displaystyle\qquad\times[\mathbf{R}_{mi}^{1/2}\mathbf{A}_{a}^{m}\mathbf{P}_{mk}]_{n_{1}j}[\mathbf{R}_{mi}^{1/2}]_{jn_{2}}[\mathbf{R}_{mi}^{1/2}]_{n_{3}j}[\mathbf{P}_{mk}^{H}\mathbf{A}_{a}^{m}\mathbf{R}_{mi}^{1/2}]_{jn_{4}}\Big{)}.$
(37)

Here, the terms $[\bar{\mathbf{h}}_{mi}]_{n_{1}}$ and
$[{\tilde{\mathbf{h}}}_{mi}^{*}[\lambda]]_{n_{1}}$ represent the $n_{1}$th
element of the vectors $\bar{\mathbf{h}}_{mi}$ and
${\tilde{\mathbf{h}}}_{mi}^{*}[\lambda]$, respectively. The terms
$[\mathbf{A}_{a}^{m}\mathbf{P}_{mk}]_{n_{1}j}$ and
$[\mathbf{R}_{mi}^{1/2}\mathbf{A}_{a}^{m}\mathbf{P}_{mk}]_{n_{1}j}$ represent
the $(n_{1},j)$th element of the matrices $\mathbf{A}_{a}^{m}\mathbf{P}_{mk}$
and $\mathbf{R}_{mi}^{1/2}\mathbf{A}_{a}^{m}\mathbf{P}_{mk}$, respectively. We
note that, in the last term, the expectation
$\mathbb{E}\big{\\{}[{\tilde{\mathbf{h}}}_{mi}^{*}[\lambda]]_{n_{1}}[{\tilde{\mathbf{h}}}_{mi}[\lambda]]_{n_{2}}[{\tilde{\mathbf{h}}}_{mi}^{*}[\lambda]]_{n3}[{\tilde{\mathbf{h}}}_{mi}[\lambda]]_{n4}\big{\\}}$
is non-zero only when either
$n_{1}\\!=\\!n_{2}\\!=\\!m_{1};n_{3}\\!=\\!n_{4}\\!=\\!m_{2}$ or
$n_{1}\\!=\\!n_{4}\\!=\\!m_{1};n_{2}\\!=n_{3}\\!=m_{2}$. We can, therefore,
calculate the expectations in (37) as

$\displaystyle\\!\\!\\!\\!\mathbb{E}\Big{\\{}\mathbf{h}_{mi}^{H}[\lambda]\mathbf{A}_{m}\mathbf{P}_{mk}\text{diag}(\mathbf{h}_{mi}[\lambda]\mathbf{h}_{mi}^{H}[\lambda])\mathbf{P}_{mk}^{H}\mathbf{A}_{m}\mathbf{h}_{mi}[\lambda]\Big{\\}}$
$\displaystyle\\!\\!\\!\\!\\!=\bar{\mathbf{h}}_{mi}^{H}\mathbf{A}_{m}\mathbf{P}_{mk}\text{diag}(\bar{\mathbf{h}}_{mi}\bar{\mathbf{h}}_{mi}^{H}+\mathbf{R}_{mi})\mathbf{P}_{mk}^{H}\mathbf{A}_{a}^{m}\bar{\mathbf{h}}_{mi}+\text{Tr}\left(\mathbf{R}_{mi}\mathbf{A}_{m}\mathbf{P}_{mk}\text{diag}(\bar{\mathbf{h}}_{mi}\bar{\mathbf{h}}_{mi}^{H})\mathbf{P}_{mk}^{H}\mathbf{A}_{m}\right)$
$\displaystyle\\!\\!\\!\\!\\!\quad+2\text{real}\Big{\\{}\sum_{j}\Big{(}\sum_{n_{1}}[\bar{\mathbf{h}}_{mi}\bar{\mathbf{h}}_{mi}^{H}]_{jn_{1}}[\mathbf{A}_{m}\mathbf{P}_{mk}]_{n_{1}j}\Big{)}\Big{(}\sum_{n_{2}}[\mathbf{P}_{mk}^{H}\mathbf{A}_{m}]_{jn_{2}}[\mathbf{R}_{mi}]_{n_{2}j}\Big{)}\Big{\\}}$
$\displaystyle\\!\\!\\!\\!\\!\quad+\sum_{j}\sum_{m_{1}}\sum_{m_{2}}[\mathbf{R}_{mi}^{1/2}\mathbf{A}_{m}\mathbf{P}_{mk}]_{m_{1}j}[\mathbf{R}_{mi}^{1/2}]_{jm_{1}}[\mathbf{R}_{mi}^{1/2}]_{m_{2}j}[\mathbf{P}_{mk}^{H}\mathbf{A}_{m}\mathbf{R}_{mi}^{1/2}]_{jm_{2}}$
$\displaystyle\quad+\sum_{j}\sum_{m_{1}}\sum_{m_{2}}[\mathbf{R}_{mi}^{1/2}\mathbf{A}_{m}\mathbf{P}_{mk}]_{m_{1}j}[\mathbf{R}_{mi}^{1/2}]_{jm_{2}}[\mathbf{R}_{mi}^{1/2}]_{m_{2}j}[\mathbf{P}_{mk}^{H}\mathbf{A}_{m}\mathbf{R}_{mi}^{1/2}]_{jm_{1}}$
(38)
$\displaystyle\\!\\!\\!\\!\\!\overset{(d)}{=}\text{Tr}\\!\left(\left(\bar{\mathbf{h}}_{mi}\bar{\mathbf{h}}_{mi}^{H}+\mathbf{R}_{mi}\right)\mathbf{A}_{m}\mathbf{P}_{mk}\text{diag}\left(\bar{\mathbf{h}}_{mi}\bar{\mathbf{h}}_{mi}^{H}+\mathbf{R}_{mi}\right)\mathbf{P}_{mk}^{H}\mathbf{A}_{a}^{m}\right)$
$\displaystyle\\!\\!\\!\\!\\!\quad+2\text{real}\left\\{\text{Tr}\\!\left(\bar{\mathbf{h}}_{mi}\bar{\mathbf{h}}_{mi}^{H}\mathbf{A}_{m}\mathbf{P}_{\\!mk}\text{diag}\\!\left(\mathbf{P}_{\\!mk}^{H}\mathbf{A}_{m}\mathbf{R}_{mi}\right)\right)\right\\}\\!+\\!\text{Tr}\\!\left(\mathbf{R}_{mi}\mathbf{A}_{m}\mathbf{P}_{\\!mk}\text{diag}\\!\left(\mathbf{P}_{\\!mk}^{H}\mathbf{A}_{m}\mathbf{R}_{mi}\right)\right).$
(39)

Equality $(d)$ is obtained by expressing the summation terms in matrix form
and using algebraic operations. Substituting (39) in (36), we obtain first
result in Table III. Similarly, we can simplify the term
$\mathbb{E}\\{\mathbf{h}_{mi}^{H}[\lambda]\mathbf{A}_{m}\mathbf{P}_{mk}\mathbf{h}_{mi}[\lambda]\mathbf{h}_{mi}^{H}[\lambda]\mathbf{P}_{mk}^{H}\mathbf{A}_{m}\mathbf{h}_{mi}[\lambda]\\}$
to obtain the second result in Table III.

## Appendix C

We now derive Theorem 2 by computing the following expectations.
Computation of $\overline{\text{DS}}_{k,n}$: The power of the desired signal
can be calculated as

$\displaystyle\overline{\text{DS}}_{k,n}=\Big{|}\sum_{m=1}^{M}\mathbf{a}_{mk}^{*}[n]\rho[n-\lambda]\left(\alpha_{d,k}\right)\sqrt{p_{k}}\mathbb{E}\left\\{\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\mathbf{h}_{mk}[\lambda]\right\\}\Big{|}^{2}.$
(40)

We now compute
$\mathbb{E}\big{\\{}\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\mathbf{h}_{mk}[\lambda]\big{\\}}$
by substituting $\hat{\mathbf{h}}_{mk}$ using (8) as

$\displaystyle\mathbb{E}\left\\{\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\mathbf{h}_{mk}[\lambda]\right\\}=\alpha_{d,k}\sqrt{\tilde{p}_{k}}\rho_{k}[\lambda-
t_{k}]\text{tr}\left(\mathbf{A}_{m}\mathbb{E}\left\\{\mathbf{h}_{mk}[\lambda]\mathbf{y}_{\text{ADC},m}^{H}[t_{k}]\right\\}\boldsymbol{\Psi}_{mk}\mathbf{A}_{m}\bar{\mathbf{R}}_{mk}\right)$
$\displaystyle\qquad=\alpha_{d,k}^{2}\tilde{p}_{k}\rho_{k}^{2}[\lambda-
t_{k}]\text{tr}\left(\mathbf{A}_{m}\bar{\mathbf{R}}_{mk}\mathbf{A}_{m}\boldsymbol{\Psi}_{mk}\mathbf{A}_{m}\bar{\mathbf{R}}_{mk}\right)=\text{tr}\left(\mathbf{A}_{m}\boldsymbol{\overline{\Gamma}}_{mk}\right).$
(41)

Substituting (41) in (40), we get
$\overline{\text{DS}}_{k,n}\\!=\\!\big{|}\sum_{m=1}^{M}\\!\mathbf{a}_{mk}^{*}[n]\rho[n\\!-\\!\lambda]\alpha_{d,k}\\!\sqrt{p_{k}}\text{tr}\left(\mathbf{A}_{m}\boldsymbol{\overline{\Gamma}}_{mk}\right)\big{|}^{2}$.
Computation of $\overline{{\mbox{IUI}}}_{ki,n}$: The power of inter-user
interference can be computed as

$\displaystyle\overline{{\mbox{IUI}}}_{ki,n}\\!$
$\displaystyle=\\!\mathbb{E}\bigg{\\{}\\!\Big{|}\sum\limits_{m=1}^{M}\\!a_{mk}^{*}[n]\alpha_{d,i}\sqrt{p_{i}}\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\mathbf{h}_{mi}[n]\Big{|}^{2}\bigg{\\}}\\!=\\!\alpha_{d,i}^{2}p_{i}\sum\limits_{m=1}^{M}\\!\Big{(}|a_{mk}^{*}[n]|^{2}\underbrace{\mathbb{E}\Big{\\{}\big{|}\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\mathbf{h}_{mi}[n]\big{|}^{2}\Big{\\}}}_{c_{kin,m}}$
$\displaystyle\quad+\sum\limits_{m^{{}^{\prime}}=1}^{M}a_{mk}^{*}[n]a_{m^{{}^{\prime}}k}^{*}[n]\underbrace{\mathbb{E}\Big{\\{}\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\mathbf{h}_{mi}[n]\hat{\mathbf{h}}_{m^{{}^{\prime}}k}^{H}[\lambda]\mathbf{A}_{m^{{}^{\prime}}}\mathbf{h}_{m^{{}^{\prime}}i}[n]\Big{\\}}}_{c_{kin,mm^{{}^{\prime}}}}\Big{)}.$
(42)

We first compute the term
$c_{kin,m}=\mathbb{E}\Big{\\{}\Big{|}\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\mathbf{h}_{mi}[n]\Big{|}^{2}\Big{\\}}$
for $i\in\mathcal{P}_{k}$ as

$\displaystyle
c_{kin,m}^{\in}={\rho_{i}}^{2}[n-\lambda]\underbrace{\mathbb{E}\bigg{\\{}\Big{|}\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\mathbf{h}_{mi}[\lambda]\Big{|}^{2}\bigg{\\}}}_{\gamma_{1}}+\overline{\rho_{i}}^{2}[n-\lambda]\underbrace{\mathbb{E}\bigg{\\{}\Big{|}\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\mathbf{u}_{mi}^{{}^{\prime}}[n]\Big{|}^{2}\bigg{\\}}}_{\gamma_{2}}.$
(43)

We now compute the term $\gamma_{1}$ by first substituting
$\hat{\mathbf{h}}_{mk}$ using (8) as

$\displaystyle\gamma_{1}$
$\displaystyle=\mathbb{E}\Big{\\{}\Big{|}\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\mathbf{h}_{mi}[\lambda]\Big{|}^{2}\Big{\\}}=\mathbb{E}\Big{\\{}\big{|}\alpha_{d,k}\sqrt{\tilde{p}_{k}}\rho_{k}[\lambda-
t_{k}]\mathbf{y}_{\text{ADC},m}^{H}[t_{k}]\boldsymbol{\Psi}_{mk}\mathbf{A}_{m}\bar{\mathbf{R}}_{mk}\mathbf{A}_{m}\mathbf{h}_{mi}[\lambda]\big{|}^{2}\Big{\\}}$
$\displaystyle\overset{(b)}{=}{\varsigma^{(1)}_{kin,m}}+{\varsigma^{(2)}_{kin,m}}+{\varsigma^{(3)}_{kin,m}}.$
(44)

Equality $(b)$ is obtained by substituting the received pilot signal
$\mathbf{y}^{p}_{\text{ADC},m}[t_{k}]$ in (7). Here the terms
${\varsigma^{(1)}_{kin,m}}={\mathbb{E}\\{\mathbf{x}_{1}^{H}\mathbf{P}_{mk}\mathbf{h}_{mi}[\lambda]\mathbf{h}_{mi}^{H}[\lambda]\mathbf{P}_{mk}^{H}\mathbf{x}_{1}\\}}$,
${\varsigma^{(2)}_{kin,m}}={\mathbb{E}\\{\mathbf{x}_{2}^{H}\mathbf{P}_{mk}\mathbf{h}_{mi}[\lambda]\mathbf{h}_{mi}^{H}[\lambda]\mathbf{P}_{mk}^{H}\mathbf{x}_{2}\\}}$
and
${\varsigma^{(3)}_{kin,m}}={\mathbb{E}\\{\mathbf{x}_{3}^{H}\mathbf{P}_{\\!mk}\mathbf{h}_{mi}[\lambda]\mathbf{h}_{mi}^{H}[\lambda]\mathbf{P}_{\\!mk}^{H}\mathbf{x}_{3}\\}}$
with
$\mathbf{x}_{1}\\!=\\!\sum\limits_{j\in\mathcal{P}_{k}}\\!\alpha_{d,j}\sqrt{\tilde{p}_{j}}\mathbf{A}_{m}\mathbf{h}_{mj}[t_{k}]$,
$\mathbf{x}_{2}\\!=\\!\sum\limits_{j\in\mathcal{P}_{k}}\\!\mathbf{A}_{m}\mathbf{h}_{mj}[t_{k}](\upsilon^{p}_{\text{DAC},j}[t_{k}]+\xi^{p}_{\text{RF},j}[t_{k}])$,
$\mathbf{x}_{3}\\!=\\!\mathbf{A}_{m}\boldsymbol{\eta}_{\text{RF},m}^{p}[t_{k}]+\mathbf{A}_{m}\mathbf{z}^{p}_{m}[t_{k}]+\mathbf{n}^{p}_{\text{ADC},m}[t_{k}]$
and $\mathbf{P}_{\\!mk}\\!=\alpha_{d,k}\sqrt{\tilde{p}_{k}}\rho_{k}[\lambda-
t_{k}]\boldsymbol{\Psi}_{mk}\mathbf{A}_{m}\bar{\mathbf{R}}_{mk}\mathbf{A}_{m}$.
We now simplify the term
$\varsigma^{(1)}_{kin,m}=\mathbb{E}\\{\mathbf{x}_{1}^{H}\mathbf{P}_{mk}\mathbf{h}_{mi}[\lambda]\mathbf{h}_{mi}^{H}[\lambda]\mathbf{P}_{mk}^{H}\mathbf{x}_{1}\\}$
in (44) as

$\displaystyle\varsigma^{(1)}_{kin,m}\overset{(b)}{=}\alpha_{d,i}^{2}\tilde{p}_{i}\rho_{i}^{2}[\lambda-
t_{k}]\mathbb{E}\big{\\{}\mathbf{h}_{mi}^{H}[\lambda]\mathbf{A}_{m}\mathbf{P}_{mk}\mathbf{h}_{mi}[\lambda]\mathbf{h}_{mi}^{H}[\lambda]\mathbf{P}_{mk}^{H}\mathbf{A}_{m}\mathbf{h}_{mi}[\lambda]\big{\\}}$
$\displaystyle\;\;+\alpha_{d,i}^{2}\tilde{p}_{i}\overline{\rho}_{i}^{2}[\lambda-
t_{k}]\text{tr}\big{(}\bar{\mathbf{R}}_{mi}\mathbf{A}_{m}\mathbf{P}_{mk}\bar{\mathbf{R}}_{mi}\mathbf{P}_{mk}^{H}\mathbf{A}_{m}\big{)}+\sum_{j\in\mathcal{P}_{k}/i}\alpha_{d,j}^{2}\tilde{p}_{j}\text{tr}\big{(}\bar{\mathbf{R}}_{mj}\mathbf{A}_{m}\mathbf{P}_{mk}\bar{\mathbf{R}}_{mi}\mathbf{P}_{mk}^{H}\mathbf{A}_{m}\big{)}.$
$\displaystyle\quad\overset{(c)}{=}\sum_{j\in\mathcal{P}_{k}}\alpha_{d,j}^{2}\tilde{p}_{j}\text{tr}\left(\bar{\mathbf{R}}_{mj}\mathbf{A}_{m}\mathbf{P}_{mk}\bar{\mathbf{R}}_{mi}\mathbf{P}_{mk}^{H}\mathbf{A}_{m}\right)$
(45) $\displaystyle\qquad+\alpha_{d,i}^{2}\tilde{p}_{i}\rho_{i}^{2}[\lambda-
t_{k}]\Big{(}\left|\text{tr}\left(\mathbf{R}_{mi}\mathbf{A}_{m}\mathbf{P}_{mk}\right)\right|^{2}+2\text{real}\big{\\{}\overline{\mathbf{h}}_{mi}^{H}\mathbf{A}_{m}\mathbf{P}_{mk}{\overline{\mathbf{h}}_{mi}}\text{tr}\left(\mathbf{R}_{mi}\mathbf{P}_{mk}^{H}\mathbf{A}_{m}\right)\big{\\}}\Big{)}.$

Equality $(b)$ is obtained by i) expressing $\mathbf{h}_{mi}[t_{k}]$ in terms
of $\mathbf{h}_{mi}[\lambda]$ using (2); and ii) using the fact that
innovation component and channel $\mathbf{h}_{mi}[\lambda]$ are independent,
and have a zero mean. We can obtain equality $(c)$ by applying the second
result in Lemma 1.

We can similarly calculate $\varsigma^{(2)}_{kin,m}$ in (44) as follows:

$\displaystyle\varsigma^{(2)}_{kin,m}=\sum_{j\in\mathcal{P}_{k}}\alpha_{d,j}(1-\alpha_{d,j}+\kappa_{t,j}^{2})\tilde{p}_{j}\text{tr}\left(\bar{\mathbf{R}}_{mj}\mathbf{A}_{m}\mathbf{P}_{mk}\bar{\mathbf{R}}_{mi}\mathbf{P}_{mk}^{H}\mathbf{A}_{m}\right)$
(46)
$\displaystyle+\alpha_{d,i}(1\\!-\\!\alpha_{d,i}+\kappa_{t,i}^{2})\tilde{p}_{i}\rho_{i}^{2}[\lambda\\!-\\!t_{k}]\Big{(}\left|\text{tr}\left(\mathbf{R}_{mi}\mathbf{A}_{m}\mathbf{P}_{mk}\right)\right|^{2}\\!+2\text{real}\big{\\{}\bar{\mathbf{h}}_{mi}^{H}\mathbf{A}_{m}\mathbf{P}_{mk}{\overline{\mathbf{h}}_{mi}}\text{tr}\left(\mathbf{R}_{mi}\mathbf{P}_{mk}^{H}\mathbf{A}_{m}\right)\\!\big{\\}}\Big{)}.$

We now simplify $\varsigma^{(3)}_{kin,m}$ in (44) by substituting
$\mathbf{x}_{3}\\!=\\!\mathbf{A}_{m}\boldsymbol{\eta}^{p}_{\text{RF},m}[t_{k}]+\mathbf{A}_{m}\mathbf{z}^{p}_{m}[t_{k}]+\mathbf{n}^{p}_{\text{ADC},m}[t_{k}]$
as

$\displaystyle\varsigma^{(3)}_{kin,m}\\!=\text{tr}\left(\mathbb{E}\left\\{\left(\mathbf{B}_{m}\\!+\kappa_{r,m}^{2}\mathbf{A}_{m}\right)\\!\mathbf{W}^{m}[t_{k}]\mathbf{P}_{mk}\mathbf{h}_{mi}[\lambda]\mathbf{h}_{mi}^{H}[\lambda]\mathbf{P}_{mk}^{H}\right\\}\right)\\!+\\!\text{tr}\left(\mathbb{E}\left\\{\sigma^{2}\mathbf{A}_{m}\mathbf{P}_{mk}\mathbf{h}_{mi}[\lambda]\mathbf{h}_{mi}^{H}[\lambda]\mathbf{P}_{mk}^{H}\right\\}\right).$
$\displaystyle\overset{(d)}{=}\\!\\!\sum_{j\in\mathcal{P}_{k}}\\!(1\\!+\\!\kappa_{t,j}^{2})\alpha_{d,j}\tilde{p}_{j}\text{tr}\big{(}\\!\\!\left(\mathbf{B}_{m}\\!+\\!\kappa_{r,m}^{2}\mathbf{A}_{m}\right)\text{diag}\left(\bar{\mathbf{R}}_{mj}\\!\right)\mathbf{P}_{\\!mk}\bar{\mathbf{R}}_{mi}\mathbf{P}_{\\!mk}^{H}\big{)}\\!+\\!\text{tr}\left(\sigma^{2}\\!\mathbf{A}_{m}\mathbf{P}_{\\!mk}\bar{\mathbf{R}}_{mi}\mathbf{P}_{\\!mk}^{H}\right)$
$\displaystyle\quad+(1+\kappa_{t,i}^{2})\alpha_{d,i}\tilde{p}_{i}\rho_{i}^{2}[\lambda-
t_{k}]\Big{(}2\text{real}\big{\\{}\text{tr}\big{(}\overline{\mathbf{h}}_{mi}\overline{\mathbf{h}}_{mi}^{H}\mathbf{P}_{mk}^{H}\left(\mathbf{B}_{m}+\kappa_{r,m}^{2}\mathbf{A}_{m}\right)\text{diag}\left(\mathbf{P}_{mk}\mathbf{R}_{mi}\right)\big{)}\big{\\}}$
$\displaystyle\quad+\text{tr}\left(\text{diag}\left(\mathbf{R}_{mi}\mathbf{P}_{mk}^{H}\right)(\mathbf{B}_{m}+\kappa_{r,m}^{2}\mathbf{A}_{m})\text{diag}\left(\mathbf{P}_{mk}\mathbf{R}_{mi}\right)\right)\Big{)}.$
(47)

Here
$\mathbf{B}_{m}\\!=\\!\mathbf{A}_{m}\left(\mathbf{I}_{N}\\!-\\!\mathbf{A}_{m}\right)$.
Equality $(d)$ is obtained by i) substituting $\mathbf{W}^{m}[t_{k}]$ in the
first expectation; and ii) applying the first result from Lemma 1.

$\displaystyle\text{tr}\big{(}\mathbb{E}\left\\{\left(\mathbf{B}_{m}+\kappa_{r,m}^{2}\mathbf{A}_{m}\right)\mathbf{W}^{m}[t_{k}]\mathbf{P}_{mk}\mathbf{h}_{mi}[\lambda]\mathbf{h}_{mi}^{H}[\lambda]\mathbf{P}_{mk}^{H}\right\\}\big{)}$
$\displaystyle=\text{tr}\big{(}\mathbb{E}\big{\\{}\left(\mathbf{B}_{m}+\kappa_{r,m}^{2}\mathbf{A}_{m}\right)\sum_{j\in\mathcal{P}_{k}}(1+\kappa_{t,j}^{2})\alpha_{d,j}\tilde{p}_{j}\text{diag}(\mathbf{h}_{mj}[t_{k}]\mathbf{h}_{mj}^{H}[t_{k}])\mathbf{P}_{mk}\mathbf{h}_{mi}[\lambda]\mathbf{h}_{mi}^{H}[\lambda]\mathbf{P}_{mk}^{H}\big{\\}}\big{)}$

Using (45), (46) and (47) we can write $c_{kin,m}^{(\in)}$ in (43) as

$\displaystyle
c_{kin,m}^{(\in)}={\rho_{i}}^{2}[n-\lambda]\left(\varsigma_{kin,m}^{(1)}+\varsigma_{kin,m}^{(2)}+\varsigma_{kin,m}^{(3)}\right)+\bar{\rho}_{i}^{2}[n-\lambda]\text{tr}\left(\boldsymbol{\overline{\Gamma}}_{mk}\mathbf{A}_{m}\bar{\mathbf{R}}_{mi}\mathbf{A}_{m}\right).$
(48)

For $i\not\in\mathcal{P}_{k}$, the term $c_{kin,m}$ in (43) can be computed as

$\displaystyle
c_{kin,m}^{(\notin)}\\!=\\!{\rho_{i}}^{2}[n\\!-\\!\lambda]\mathbb{E}\big{\\{}\big{|}\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\mathbf{h}_{mi}[\lambda]\big{|}^{2}\big{\\}}\\!+\\!\bar{\rho}_{i}^{2}[n\\!-\\!\lambda]\mathbb{E}\big{\\{}\big{|}\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\mathbf{u}_{mi}^{{}^{\prime}}[n]\big{|}^{2}\big{\\}}\\!\overset{(e)}{=}\\!\text{tr}(\boldsymbol{\overline{\Gamma}}_{\\!mk}\mathbf{A}_{m}\bar{\mathbf{R}}_{mi}\mathbf{A}_{m}).$

Equality (e) follows from the fact that i) the channels
$\hat{\mathbf{h}}_{mk}[\lambda]$ and $\mathbf{h}_{mi}[\lambda]$ are
uncorrelated, ii) the variance of actual and estimated channel is
$\bar{\mathbf{R}}_{mi}$ and $\boldsymbol{\overline{\Gamma}}_{mk}$,
respectively.

We now compute the term $c_{kin,mm^{{}^{\prime}}}$, for the case $m\neq
m^{{}^{\prime}}$ and $i\in\mathcal{P}_{k}$, we have

$\displaystyle c_{kin,mm^{{}^{\prime}}}^{(\in)}$
$\displaystyle\overset{(d)}{=}\rho_{i}^{2}[n-\lambda]\text{tr}\big{(}\mathbb{E}\big{\\{}\mathbf{h}_{mi}[\lambda]\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\big{\\}}\big{)}\text{tr}\big{(}\mathbb{E}\big{\\{}\mathbf{h}_{m^{{}^{\prime}}i}[\lambda]\hat{\mathbf{h}}_{m^{{}^{\prime}}k}^{H}[\lambda]\mathbf{A}_{m^{{}^{\prime}}}\big{\\}}\big{)}$
(49)
$\displaystyle\overset{(e)}{=}\\!\rho_{i}^{2}[n\\!-\\!\lambda]\alpha_{d,i}^{4}\rho_{i}^{2}[\lambda\\!-\\!t_{k}]\tilde{p}_{i}^{2}\text{tr}\left(\bar{\mathbf{R}}_{mi}\mathbf{A}_{m}\boldsymbol{\Psi}_{mk}\mathbf{A}_{m}\bar{\mathbf{R}}_{mk}\mathbf{A}_{m}\right)\text{tr}\big{(}\bar{\mathbf{R}}_{m^{{}^{\prime}}i}\mathbf{A}_{m^{{}^{\prime}}}\boldsymbol{\Psi}_{m^{{}^{\prime}}k}\mathbf{A}_{\\!m^{{}^{\prime}}}\bar{\mathbf{R}}_{m^{{}^{\prime}}k}\mathbf{A}_{m^{{}^{\prime}}}\big{)}.$

Equality (d) is due to fact that innovation component and the channel are
uncorrelated at time instant $\lambda$ and has zero mean. Equality $(e)$ is
obtained by substituting $\hat{\mathbf{h}}_{mk}[\lambda]$ using (8).

For the case $i\not\in\mathcal{P}_{k}$, the term
$c_{kin,mm^{{}^{\prime}}}\forall m\neq m^{{}^{\prime}}$ can be calculated as

$\displaystyle
c_{kin,mm^{{}^{\prime}}}^{(\notin)}=\mathbb{E}\big{\\{}(\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\mathbf{h}_{mi}[n])(\hat{\mathbf{h}}_{m^{{}^{\prime}}k}^{H}[\lambda]\mathbf{A}_{m}^{m^{{}^{\prime}}}\mathbf{h}_{m^{{}^{\prime}}i}[n])^{*}\big{\\}}=0.$
(50)

Using (49) and (50), we can write $\overline{\mbox{IUI}}_{kin}$ in (42)

$\displaystyle\overline{\mbox{IUI}}_{ki,n}\\!=\begin{cases}\alpha_{d,i}^{2}p_{i}\Big{(}\sum\limits_{m=1}^{M}\big{(}|a_{mk}^{*}[n]|^{2}c_{kin,m}^{(\in)}+\sum\limits_{m^{{}^{\prime}}\neq
m}^{M}a_{mk}^{*}[n]a_{m^{{}^{\prime}}k}^{*}[n]c_{kin,mm^{{}^{\prime}}}^{(\in)}\big{)}\Big{)},\;\;i\in\mathcal{P}_{k}\\\\[-1.0pt]
\alpha_{d,i}^{2}p_{i}\Big{(}\sum\limits_{m=1}^{M}\big{(}|a_{mk}^{*}[n]|^{2}c_{kin,m}^{(\notin)}+\sum\limits_{m^{{}^{\prime}}\neq
m}^{M}a_{mk}^{*}[n]a_{m^{{}^{\prime}}k}^{*}[n]c_{kin,mm^{{}^{\prime}}}^{(\notin)}\big{)}\Big{)}.\;\;i\not\in\mathcal{P}_{k}.\end{cases}\\!\\!\\!\\!$
(51)

Computation of $\overline{\text{BU}}_{k,n}$: The power of the beamforming
uncertainty is calculated as

$\displaystyle\overline{\text{BU}}_{k,n}$
$\displaystyle=\sum_{m=1}^{M}|\mathbf{a}_{mk}^{*}[n]|^{2}\alpha_{d,k}^{2}p_{k}\big{(}\mathbb{E}\big{\\{}|\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\mathbf{h}_{mk}[\lambda]|^{2}\big{\\}}-\big{|}\mathbb{E}\big{\\{}\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\mathbf{h}_{mk}[\lambda]\big{\\}}\big{|}^{2}\big{)}$
$\displaystyle\overset{(f)}{=}\sum_{m=1}^{M}|\mathbf{a}_{mk}^{*}[n]|^{2}\alpha_{d,k}^{2}p_{k}\big{(}\varsigma_{k,k,n,m}^{(1)}\\!+\varsigma_{k,k,n,m}^{(2)}\\!+\varsigma_{k,k,n,m}^{(3)}-\big{|}\text{tr}(\mathbf{A}_{m}\boldsymbol{\overline{\Gamma}}_{mk})\big{|}^{2}\big{)}=\mathbf{a}_{k}^{H}[n]\mathbf{B}_{k,n}\mathbf{a}_{k}[n].$

Equality $(f)$ is obtained by following the similar steps as in (43). We have
$\mathbb{E}\big{\\{}|\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\mathbf{h}_{mk}[\lambda]|^{2}\big{\\}}$
as
$c_{kkn,m}^{\in}=\varsigma_{kkn,m}^{(1)}+\varsigma_{kkn,m}^{(2)}+\varsigma_{kkn,m}^{(3)}$.
Here
$\mathbf{B}_{k,n}=\text{diag}\big{(}\alpha_{d,k}^{2}p_{k}\big{(}c_{kkn,m}^{\in}-\big{|}\text{tr}\left(\mathbf{A}_{m}\boldsymbol{\overline{\Gamma}}_{mk}\right)\big{|}^{2}\big{)}\big{)}$.
Computation of $\overline{\text{RRF}}_{k,n}$: The power of the receiver RF
impairments is calculated as

$\displaystyle\overline{\text{RRF}}_{k,n}$
$\displaystyle=\sum\limits_{m=1}^{M}|a_{mk}^{*}[n]|^{2}\mathbb{E}\big{\\{}\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\boldsymbol{\eta_{r,m}^{\text{AP}}}[n]\boldsymbol{\eta}_{\text{RF},m}^{H}[n]\mathbf{A}_{m}\hat{\mathbf{h}}_{mk}[\lambda]\big{\\}}$
(52)
$\displaystyle\overset{(g)}{=}\sum\limits_{m=1}^{M}|a_{mk}^{*}[n]|^{2}\sum_{i=1}^{K}\underbrace{\kappa_{r,m}^{2}(1+\kappa_{t,i}^{2})\alpha_{d,i}p_{i}\mathbb{E}\big{\\{}\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{A}_{m}\text{diag}(\mathbf{h}_{mi}[n]\mathbf{h}_{mi}^{H}[n])\mathbf{A}_{m}\hat{\mathbf{h}}_{mk}[\lambda]\big{\\}}}_{d_{ki,n}^{m}}.$

Equality $(g)$ is obtained by i) using the that the receiver AP RF impairment
$\boldsymbol{\eta_{r,m}^{\text{AP}}}[n]$ has pdf
$\mathcal{CN}(0,\kappa_{r,m}^{2}\mathbf{W}^{m}[n])$, and ii) substituting the
expression of matrix $\mathbf{W}^{m}[n]$ from Appendix A.

For $i\notin\mathcal{P}_{k}$, the term $d_{ki,n}^{m}$ is
$d_{ki,n}^{m}=\kappa_{r,m}^{2}(1+\kappa_{t,i}^{2})\alpha_{d,i}p_{i}\text{tr}\left(\boldsymbol{\overline{\Gamma}}_{mk}\mathbf{A}_{m}\text{diag}\left({\bar{\mathbf{R}}_{mi}}\right)\mathbf{A}_{m}\right)$.
For $i\in\mathcal{P}_{k}$, the term $d_{ki,n}^{m}$ can be obtained as below by
i) following similar to steps given for i) $\gamma_{1}$ in (44),
$\varsigma_{kin,m}^{(1)}$, $\varsigma_{kin,m}^{(2)}$ and
$\varsigma_{kin,m}^{(3)}$, respectively; and ii) applying the results from
Lemma 1.

$\displaystyle d_{ki,n}^{m}$
$\displaystyle=(1+\kappa_{t,i}^{2})\alpha_{d,i}p_{i}\Big{(}\sum_{j\in\mathcal{P}_{k}}(1+\kappa_{t,j}^{2})\alpha_{d,j}\tilde{p}_{j}\Big{(}\text{tr}\big{(}\bar{\mathbf{R}}_{mj}\mathbf{A}_{m}\mathbf{P}_{mk}\text{diag}(\bar{\mathbf{R}}_{mi})\mathbf{P}_{mk}^{H}\mathbf{A}_{m}\big{)}$
$\displaystyle\quad+\text{tr}\left(\left(\mathbf{B}_{m}+\kappa_{r,m}^{2}\mathbf{A}_{m}\right)\text{diag}(\bar{\mathbf{R}}_{mj})\mathbf{P}_{mk}\text{diag}(\bar{\mathbf{R}}_{mi})\mathbf{P}_{mk}^{H}\right)\Big{)}+\sigma^{2}\text{tr}\left(\mathbf{A}_{m}\mathbf{P}_{mk}\text{diag}(\bar{\mathbf{R}}_{mi})\mathbf{P}_{mk}^{H}\right)$
$\displaystyle\quad+(1+\kappa_{t,i}^{2})\alpha_{d,i}\tilde{p}_{i}\rho_{i}^{2}[n-\lambda]\rho_{i}^{2}[\lambda-
t_{k}]\Big{(}\text{tr}\left(\left(\mathbf{B}_{m}+\kappa_{r,m}^{2}\mathbf{A}_{m}\right)\text{diag}(\mathbf{R}_{mi}\mathbf{P}_{mk}^{H})\mathbf{P}_{mk}\mathbf{R}_{mi}\right)$
$\displaystyle\quad+2\text{real}\big{\\{}\text{tr}\big{(}\bar{\mathbf{h}}_{mi}\bar{\mathbf{h}}_{mi}^{H}\mathbf{A}_{m}\mathbf{P}_{mk}\text{diag}\left(\mathbf{P}_{mk}^{H}\mathbf{A}_{m}\mathbf{R}_{mi}\right)\big{)}\big{\\}}+\text{tr}\left(\mathbf{R}_{mi}\mathbf{A}_{m}\mathbf{P}_{mk}\text{diag}\left(\mathbf{P}_{mk}^{H}\mathbf{A}_{m}\mathbf{R}_{mi}\right)\right)$
$\displaystyle\quad+2\text{real}\big{\\{}\text{tr}\big{(}(\mathbf{B}_{m}+\kappa_{r,m}^{2}\mathbf{A}_{m})\big{(}(\bar{\mathbf{h}}_{mi}\bar{\mathbf{h}}_{mi}^{H})\odot\mathbf{P}_{mk}\big{)}\big{(}\mathbf{R}_{mi}\odot\mathbf{P}_{mk}^{H}\big{)}\big{)}\big{\\}}\Big{)}\Big{)}.$
(53)

Substituting (53) in (52), we have
$\overline{\text{RRF}}_{k,n}\\!=\kappa_{r,m}^{2}\mathbf{a}_{k}^{H}[n]\sum\limits_{i=1}^{K}\mathbf{D}_{ki,n}\mathbf{a}_{k}[n]$,
where $\mathbf{D}_{ki,n}\\!=\text{diag}(d_{ki,n}^{1},\cdots,d_{ki,n}^{M})$.
Computation of $\overline{\text{ADC}}_{k,n}$: The power of the dynamic-ADC
distortion at the $k$th UE is given as

$\displaystyle\overline{\text{ADC}}_{k,n}=\mathbb{E}\big{\\{}\big{|}\sum\limits_{m=1}^{M}a_{mk}^{*}[n]\hat{\mathbf{h}}_{mk}^{H}[\lambda]\mathbf{n}_{\text{ADC},m}[n]\big{|}^{2}\big{\\}}\overset{(h)}{=}\mathbf{a}_{k}^{H}[n]\Big{(}\sum\limits_{i=1}^{K}\overline{\mathbf{D}}_{ki,n}+\text{diag}\big{(}\sigma^{2}\text{tr}(\overline{\boldsymbol{\Gamma}}_{mk}\mathbf{B}_{m})\big{)}\Big{)}\mathbf{a}_{k}[n].$

Equality $(h)$ can be derived on lines similar to (52). For
$i\not\in\mathcal{P}_{k}$ the matrix $\mathbf{D}_{ki,n}$ is

$\displaystyle\overline{\mathbf{D}}_{ki,n}$
$\displaystyle=\text{diag}\big{(}(1+\kappa_{t,i}^{2})(1+\kappa_{r,m}^{2})\alpha_{d,i}p_{i}\text{tr}\left(\boldsymbol{\overline{\Gamma}}_{mk}\mathbf{B}_{m}\text{diag}\left(\bar{\mathbf{R}}_{mi}\right)\right)+\sigma^{2}\text{tr}\left(\boldsymbol{\overline{\Gamma}}_{mk}\mathbf{B}_{m}\right)\big{)}.$

For $i\in\mathcal{P}_{k}$ the matrix $\overline{\mathbf{D}}_{ki,n}$ can be
given as
$\overline{\mathbf{D}}_{ki,n}=(1\\!+\\!\kappa_{r,m}^{2})\mathbf{D}_{ki,n}$.

Similarly, the closed-form expressions of the transmit RF impairment
$\overline{\mbox{TRF}}_{k,n}$, DAC impairment $\overline{\text{DAC}}_{k,n}$,
channel aging $\overline{\text{CA}}_{k,n}$ and noise term
$\overline{\text{NS}}_{k,n}$ in (15) are given respectively as:
$\overline{\mbox{TRF}}_{k,n}=\mathbf{a}_{k}^{H}[n]\sum_{i=1}^{K}\kappa_{t,i}^{2}\alpha_{d,i}p_{i}\mathbf{C}_{kin}\mathbf{a}_{k}[n]$,
$\overline{\text{DAC}}_{k,n}=\mathbf{a}_{k}^{H}[n]\sum_{i=1}^{K}\rho_{d,i}\alpha_{d,i}p_{i}\mathbf{C}_{kin}\mathbf{a}_{k}[n]$,
$\overline{\text{CA}}_{k,n}=\mathbf{a}_{k}^{H}[n]\boldsymbol{\Lambda}_{k,n}\mathbf{a}_{k}[n]$
with
$\boldsymbol{\Lambda}_{k,n}\\!=\\!\overline{\rho}_{k}^{2}[n-\lambda]\alpha_{d,k}^{2}p_{k}\text{diag}\big{(}\text{tr}\left(\boldsymbol{\overline{\Gamma}}_{1k}\mathbf{A}_{1}\bar{\mathbf{R}}_{1k}\mathbf{A}_{1},\cdots,\boldsymbol{\overline{\Gamma}}_{Mk}\mathbf{A}_{M}\bar{\mathbf{R}}_{Mk}\mathbf{A}_{M}\right)\\!\big{)}$
and
$\overline{\text{NS}}_{k,n}=\sigma^{2}\mathbf{a}_{k}^{H}[n]\mathbf{Q}_{k}\mathbf{a}_{k}[n]$
with
$\mathbf{Q}_{k}\\!=\\!\text{diag}\big{(}\text{tr}(\overline{\boldsymbol{\Gamma}}_{1k}\mathbf{A}_{1}\mathbf{A}_{1}),\cdots,\text{tr}(\overline{\boldsymbol{\Gamma}}_{Mk}\mathbf{A}_{M}\mathbf{A}_{M})\big{)}$.

## Appendix D

We begin by evaluating the first order derivative of P3 objective with respect
to $p_{k}$ as

$\displaystyle\frac{\partial\sum_{k=1}^{K}\log_{2}(1+\gamma_{k,n})-\frac{1}{\ln{2}}\left(\gamma_{k,n}-\left(2y_{k,n}\sqrt{\Delta_{k,n}(\mathbf{p})(1+\gamma_{k,n})}-y_{k,n}^{2}\left(\Delta_{k,n}(\mathbf{p})+\Omega_{k,n}(\mathbf{p})\right)\right)\right)}{\partial
p_{k}}$
$\displaystyle\quad=y_{k,n}\frac{\sqrt{(1+\gamma_{k,n})}}{\sqrt{\Delta_{k,n}(\mathbf{p})}}\frac{\partial\Delta_{k,n}(\mathbf{p})}{\partial
p_{k}}-\bigg{(}y_{k,n}^{2}\frac{\partial\Delta_{k,n}(\mathbf{p})}{\partial
p_{k}}+\sum_{j=1}^{K}y_{j,n}^{2}\frac{\partial\Omega_{j,n}(\mathbf{p})}{\partial
p_{k}}\bigg{)}.$ (54)

The partial derivative of desired signal $\Delta_{k,n}(\mathbf{p})$, using
(16) is given as $\frac{\partial\Delta_{k,n}(\mathbf{p})}{\partial
p_{k}}=\alpha_{d,k}^{2}|\mathbf{a}_{k}^{H}[n]\boldsymbol{\delta}_{k,n}|^{2}$.
The partial derivative of $\Omega_{j,n}(\mathbf{p})\forall j$ is calculated
using (16) as follows:

$\displaystyle\sum_{i=1}^{K}y_{i,n}\frac{\partial\Omega_{i,n}(\mathbf{p})}{\partial
p_{k}}=$
$\displaystyle\sum\limits_{i=1}^{K}\mathbf{a}_{i}^{H}[n]\Big{(}y_{i,n}^{2}\alpha_{d,i}(1+\kappa_{t,i}^{2})\mathbf{C}_{ik,n}+\sum\limits_{i=1}^{K}y_{i,n}^{2}(\kappa^{2}_{r,m}\mathbf{D}_{ik,n}+\overline{\mathbf{D}}_{ik,n})\Big{)}\mathbf{a}_{i}[n]$
$\displaystyle+\mathbf{a}_{k}^{H}[n]y_{k,n}^{2}(\mathbf{B}_{k,n}+\boldsymbol{\Lambda}_{k,n}-\alpha_{d,k}^{2}\mathbf{C}_{kk,n})\mathbf{a}_{k}[n]=l_{k}^{d}.$
(55)

The optimal $p_{k}$ can obtained by substituting
${\partial\Delta_{k,n}(\mathbf{p})}/{\partial p_{k}}$ and (55) in (54), and by
equating it to zero. It is given as
$p_{k}=\bigg{(}P_{max},\frac{y_{k,n}^{2}\left(1+\gamma_{k,n}\right)\alpha_{d,k}^{2}|\mathbf{a}_{k}^{H}[n]\boldsymbol{\delta}_{k,n}|^{2}}{\left(\alpha_{d,k}^{2}y_{k,n}^{2}|\mathbf{a}_{k}^{H}[n]\boldsymbol{\delta}_{k,n}|^{2}+l_{k}^{d}\right)^{2}}\bigg{)}$.

## References

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