# Uplink Performance Analysis of Cell-Free mMIMO Systems under Channel Aging

Ribhu Chopra, , Chandra R. Murthy, , and Anastasios K. Papazafeiropoulos R.
Chopra is with the Department of Electronics and Electrical Engineering,
Indian Institute of Technology Guwahati, Assam, India. (email:
ribhufec@iitg.ac.in). C. R. Murthy is with the Department of Electrical
Communication Engineering, Indian Institute of Science, Bangalore, India.
(email: cmurthy@iisc.ac.in). A. K. Papazafeiropoulos is with the CIS Research
Group, University of Hertfordshire, Hatfield, U. K., and SnT at the University
of Luxembourg, Luxembourg (e-mail: tapapazaf@gmail.com).

###### Abstract

In this paper, we investigate the impact of channel aging on the uplink
performance of a cell-free (CF) massive multiple-input multiple-output (mMIMO)
system with a minimum mean squared error (MMSE) receiver. To this end, we
present a new model for the temporal evolution of the channel, which allows
the channel to age at different rates at different access points (APs). Under
this setting, we derive the deterministic equivalent of the per-user
achievable signal-to-interference-plus-noise ratio (SINR). In addition to
validating the theoretical expressions, our simulation results reveal that, at
low user mobilities, the SINR of CF-mMIMO is nearly $5$ dB higher than its
cellular counterpart with the same number of antennas, and about $8$ dB higher
than that of an equivalent small-cell network with the same number of APs. On
the other hand, at very high user velocities, and when the channel between the
UEs the different APs age at same rate, the relative impact of aging is higher
for CF-mMIMO compared to cellular mMIMO. However, when the channel ages at the
different APs with different rates, the effect of aging on CF-mMIMO is
marginally mitigated, especially for larger frame durations.

###### Index Terms:

Cell Free mMIMO, channel aging, performance analysis.

##  I Introduction

Cell-free (CF) massive multiple-input multiple-output (mMIMO) systems are
considered the natural successor to the cellular mMIMO technology for physical
layer access in next-generation wireless systems [1, 2, 3, 4]. The canonical
CF-mMIMO setup consists of a large number of access points (APs) spread over a
given physical area, and connected to a single central processing unit (CPU).
Since the signals received from all the UEs at multiple APs are processed
jointly at the CPU, a CF system becomes a distributed mMIMO system. In
contrast, a cellular mMIMO system consists of a single base station/AP with a
massive number of antennas serving all the users in the cell. CF-mMIMO systems
offer the advantage of more uniform coverage compared to conventional cellular
systems, while retaining the benefits of mMIMO systems such as high spectral
efficiency [5] and simple linear processing at the APs/CPU [3]. However, CF-
mMIMO also inherits mMIMO’s dependence on accurate channel state information
(CSI) [6, 7]. However, the available CSI may be impaired due to several
factors such as pilot contamination [8], channel aging [9], and hardware
impairments [4].

The phenomenon of channel aging, classically known as time-selectivity, is a
manifestation of the temporal variation in the wireless channel caused due to
user mobility and phase noise [10, 9, 6, 11, 5, 12, 7]. It has been shown that
while the power scaling and the array gains achieved by mMIMO are retained
under channel aging [5], the achievable signal-to-interference-plus-noise
ratio (SINR) gradually decays over time as the channel estimates become
outdated, which, in turn, limits the system dimensions [12, 7]. While there
have been recent efforts to counter the effect of aging via channel prediction
[13], the applicability of these techniques to cellular or cell free mMIMO
systems remains to be seen.

In terms of the effect of aging on the underlying architecture, a key
difference between cellular and CF mMIMO systems is that in the latter system,
the channels between the APs and UEs could potentially evolve over time at
different rates. Together with the CPU based joint processing of the uplink
signals, this makes the analysis (and ultimately the effect) of channel aging
on CF systems different from cellular mMIMO [11]. However, despite its
significance, to the best of our knowledge, the effect of channel aging on CF-
mMIMO systems has not been investigated in the previous literature.

In this paper, we characterize the effect of channel aging on the uplink
achievable SINR of a CF-mMIMO system. First, we develop a model for the
relative speed of the users with respect to the different APs. Then, using
results from random matrix theory [14], we derive an analytical expression for
the deterministic equivalent of the SINR, i.e., the SINR averaged over all
fading channel realizations, as a function of the UE locations and velocities.
Through simulations, we compare the achievable SINR against (a) the
conventional cellular mMIMO setup, and (b) a small cell system [2], to assess
the relative impact of channel aging on the three systems. We conclude that
CF-mMIMO systems generally provide significantly better SINR compared to the
other two systems in the presence of channel aging, but the relative impact of
aging is more severe on CF-mMIMO at very high user velocities and long frame
durations.

##  II System Model and Channel Estimation

We consider a CF-mMIMO system with $M$ APs each having $N$ antennas and
serving a total of $K$ users. The APs are connected to a CPU over a delay-free
unlimited capacity front-haul link. The locations of the APs and the UEs are
considered to be points on the two dimensional plane, with the locations of
the $m$th AP and the $k$th UE represented by using the complex numbers
$\mathtt{d}_{a,m}$ and $\mathtt{d}_{u,k}$, respectively. Also, the path loss
between the $k$th user and the $m$th AP is modeled by using a piecewise linear
multi-slope model as

$\beta_{mk}=\mu_{l}\left({\lvert\mathtt{d}_{a,m}-\mathtt{d}_{u,k}\rvert}\right)^{-\eta_{l}},d_{l-1}<\lvert\mathtt{d}_{a,m}-\mathtt{d}_{u,k}\rvert<d_{l},$
(1)

where $\mu_{l}$ is a normalization constant, $\eta_{l}$ is the $l$th slope,
and $d_{l}$ is the $l$th threshold, $1\leq l\leq L$, with $d_{L}=\infty$ [15].

The fast fading channel between the $k$th user and the $m$th AP at the $n$th
instant, denoted by $\mathbf{h}_{mk}[n]\in\mathbb{C}^{N\times 1}$, is assumed
to evolve in time according to the relation [12, 16]

$\mathbf{h}_{mk}[n+\tau]=\rho_{mk}[\tau]\mathbf{h}_{mk}[n]+\bar{\rho}_{mk}[\tau]{\mathbf{z}_{h,mk}[n+\tau;n]},$
(2)

where $\rho_{mk}[\tau]$ is the correlation coefficient of the channel between
the $k$th UE and the $m$th AP at lag $\tau$. The channel state
$\mathbf{h}_{mk}[n]$ and the innovation $\mathbf{z}_{h,mk}[n+\tau;n]$ are
assumed to be distributed as $\mathcal{CN}(0,\mathbf{I}_{N})$, with
$E[\mathbf{h}_{mk}[n]\mathbf{z}^{H}_{h,mk}[n+\tau;n]]=\mathbf{O}_{N}$, where
$\mathbf{I}_{N}$ and $\mathbf{O}_{N}$ are the $N\times N$ identity and all
zero matrices, respectively. We also assume that the innovation process
$\mathbf{z}_{h,mk}[t;n]$ is wide sense stationary, and temporally white over
the time index $t$, for a given anchor index $n$. In (2) and throughout the
paper, for any variable $x\in[-1,1]$, we denote
$\bar{x}\triangleq\sqrt{1-x^{2}}$.

Conventionally, the correlation coefficient $\rho_{mk}[\tau]$ is assumed to
follow the Jakes’ model [16], or the first order autoregressive (AR-1) model
[12]. It is dependent on the user’s speed through the Jakes’ spectrum.
However, this model is developed under the assumption that the scatterers are
localized around the UEs. In case of CF-mMIMO systems, in general, it is not
reasonable to expect that scatterers are localized only around the UE, and
assume that the correlation coefficient is the same across all APs. Moreover,
to the best of our knowledge, measurement campaigns elucidating the variation
of the temporal correlation coefficient across APs are not available in the
literature. Therefore, in this paper, we consider a simple generalization of
the existing Jakes’ spectrum based model, where the correlation coefficient
depends on the relative speed of the UEs with respect to the different APs. To
elaborate, we consider $v_{mk}$ to be a random variable that is i.i.d. across
the APs with mean $v_{k}$ (i.e., depending only on the user index) and support
$[(1-\Delta)v_{k},(1+\Delta)v_{k}]$, where $0\leq\Delta\leq 1$. The
correlation coefficient is then defined as $\rho_{mk}[\tau]=J_{0}(2\pi
f_{d,mk}T_{s}\tau)$, where $J_{0}(\cdot)$ is the Bessel function of zeroth-
order and first kind[16], $f_{d,mk}=v_{mk}f_{c}/c$ is the Doppler frequency of
the $k$th user with respect to the $m$th AP, $f_{c}$ is the carrier frequency,
$c$ is the speed of light, and $T_{s}$ is the sampling interval.

Note that $\Delta=0$ results in the conventional model, where the correlation
coefficient is the same at all APs. Nonetheless, even in this case, the effect
of aging on a CF-mMIMO is different from cellular mMIMO. This is because, in
cellular mMIMO, only the associated base station decodes the signal from a
given UE, and the UE’s signal arriving at other base stations is treated as
interference. When $\Delta>0$, the channel ages at different APs at different
rates, and we will see in the section on numerical results that this has a
marginally beneficial effect on a CF system at high user velocities and large
frame durations.

The uplink frame, consisting of a total of $T$ channel uses, is segmented into
two subframes: training and data transmission. Over the first $P(\leq K)$
channel uses, the UEs transmit uplink pilots to the APs with the $l$th UE
employing a pilot energy $\mathcal{E}_{upl}$. For simplicity, we consider that
the individual pilot signals occupy a single time slot, and the UEs whose
pilots share the $p$th slot are contained in the set $\mathcal{U}_{p}$. These
pilots are used by the respective APs to estimate the channels from these UEs.
In the next phase, consisting of $T-P$ channel uses, the UEs transmit uplink
data. In this paper, we assume the system to have a “level 4” centralization
of data processing according to the classification in [17]. That is, the APs
share all the available information with the CPU, including the channel
estimates and the received data symbols. At the CPU, the received symbols are
processed using an MMSE combiner to obtain estimates of the symbols
transmitted by the users.

During the $p$th uplink training slot, the signal received by the $m$th AP is
given by

$\displaystyle\mathbf{y}_{m}[p]=\sum_{l\in\mathcal{U}_{p}}\sqrt{\beta_{ml}\mathcal{E}_{upl}}\mathbf{h}_{ml}[p]+\sqrt{N_{0}}\mathbf{w}_{l}[p]$
$\displaystyle=\sqrt{\beta_{mk}\mathcal{E}_{upk}}\mathbf{h}_{mk}[p]+\sum_{\begin{subarray}{c}l\in\mathcal{U}_{p}\\\
l\neq
k\end{subarray}}\sqrt{\beta_{ml}\mathcal{E}_{upl}}\mathbf{h}_{ml}[p]+\sqrt{N_{0}}\mathbf{w}_{l}[p].$

The $m$th AP uses the pilot signal received at time $p$, $\mathbf{y}_{m}[p]$,
to obtain the MMSE estimate of the $k$th UEs channel at time $P$, i.e.,
$\mathbf{h}_{mk}[P]$. We denote the MMSE estimate of $\mathbf{h}_{mk}[P]$ by
$\mathbf{\hat{h}}_{mk}$. The estimate $\hat{\mathbf{h}}_{mk}$ is used by the
CPU to decode the $k$th UE’s signal over the entire frame.

Using the well-known time-reversal property [18] of (2), we can write

$\displaystyle\mathbf{y}_{m}[p]$
$\displaystyle=\rho_{mk}[P-p]\sqrt{\beta_{mk}\mathcal{E}_{upk}}\mathbf{{h}}_{mk}[P]$
$\displaystyle+\bar{\rho}_{mk}[P-p]\sqrt{\beta_{mk}\mathcal{E}_{upk}}\mathbf{{z}}_{h,mk}[p;P]$
$\displaystyle+\sum_{\begin{subarray}{c}l\in\mathcal{U}_{p}\\\ l\neq
k\end{subarray}}\sqrt{\beta_{ml}\mathcal{E}_{upl}}\mathbf{h}_{ml}[p]+\sqrt{N_{0}}\mathbf{w}_{l}[p].$
(3)

Consequently, we have
$\mathbf{\hat{h}}_{mk}=\frac{\rho_{mk}[{P-p}]\sqrt{\beta_{mk}\mathcal{E}_{upk}}}{\sum_{l\in\mathcal{U}_{p}}\beta_{ml}\mathcal{E}_{upl}+N_{0}}\mathbf{y}_{m}[p],$
and

$\mathbf{h}_{mk}[P]=a_{mk}\hat{\mathbf{h}}_{mk}+\bar{a}_{mk}\tilde{\mathbf{h}}_{mk},$
(4)

with $E[\hat{\mathbf{h}}_{mk}\tilde{\mathbf{h}}^{H}_{mk}]=\mathbf{O}_{N}$ and
$a_{mk}=\sqrt{\frac{\rho^{2}_{mk}[P-p]\beta_{mk}\mathcal{E}_{upk}}{\sum_{l\in\mathcal{U}_{p}}\beta_{ml}\mathcal{E}_{upl}+N_{0}}}$.

Therefore, for $P+1\leq n\leq T$, we have

$\displaystyle\mathbf{h}_{mk}[n]=$
$\displaystyle\rho_{mk}[n-P]a_{mk}\hat{\mathbf{h}}_{mk}+\rho_{mk}[n-P]\bar{a}_{mk}\tilde{\mathbf{h}}_{mk}$
$\displaystyle+$ $\displaystyle\bar{\rho}_{mk}[n-P]\mathbf{z}_{h,mk}[n;P].$
(5)

With the system model and channel estimates in hand, we can now proceed with
the uplink SINR analysis.

##  III Uplink SINR Analysis

During the data phase, i.e., over the next $T-P$ channel uses, the UEs
simultaneously transmit the data to all the APs. If the $k$th UE transmits the
symbol $s_{k}[n]$ at the $n$th instant (satisfying $E[\lvert
s_{k}[n]\rvert^{2}]=1$) with power $\mathcal{E}_{usk}$, then the signal
received at the $m$th AP is

$\mathbf{y}_{m}[n]=\sum_{k=1}^{K}\sqrt{\beta_{mk}\mathcal{E}_{usk}}\mathbf{h}_{mk}[n]s_{k}[n]+\sqrt{N_{0}}\mathbf{w}_{m}[n].$
(6)

Letting
$\mathbf{y}[n]\triangleq[\mathbf{y}^{T}_{1}[n],\mathbf{y}^{T}_{2}[n],\ldots,\mathbf{y}^{T}_{M}[n]]^{T}$,
$\mathbf{h}_{k}[n]\triangleq[\mathbf{h}^{T}_{1k}[n],\mathbf{h}^{T}_{2k}[n],\ldots,\mathbf{h}^{T}_{Mk}[n]]^{T}$,
$\mathbf{H}[n]\triangleq[\mathbf{h}_{1}[n],\mathbf{h}_{2}[n],\ldots,\mathbf{h}_{K}[n]]$,
$\boldsymbol{\beta}_{k}\triangleq[\beta_{1k},\beta_{2k},\ldots,\beta_{Mk}]^{T}$,
${\mathtt{{B}}}\triangleq[\boldsymbol{\beta}_{1},\boldsymbol{\beta}_{2},\ldots,\boldsymbol{\beta}_{K}]$,
$\boldsymbol{\rho}_{k}[n]\triangleq[\rho_{1k}[n],\rho_{2k}[n],\ldots,\rho_{Mk}[n]]^{T}$,
${\mathtt{{R}}}[n]\triangleq[\boldsymbol{\rho}_{1}[n],\boldsymbol{\rho}_{2}[n],\ldots,\boldsymbol{\rho}_{K}[n]]$,
$\mathbf{a}_{k}\triangleq[a_{1k},a_{2k},\ldots,a_{Mk}]^{T}$,
${\mathbf{{A}}}\triangleq[\mathbf{a}_{1},\mathbf{a}_{2},\ldots,\mathbf{a}_{K}]$,
$\mathbf{s}[n]\triangleq[s_{1}[n],\ldots,s_{K}[n]]^{T}$, we can write the
concatenated signal received by the CPU at the $n$th instant as

$\displaystyle\mathbf{y}[n]=\sum_{k=1}^{K}\sqrt{\mathcal{E}_{usk}(\mathbf{1}_{N}\otimes\boldsymbol{\beta}_{k})}\odot\mathbf{h}_{k}[n]s_{k}[n]+\sqrt{N_{0}}\mathbf{w}[n]$
$\displaystyle=(\mathbf{1}_{N}\otimes(\mathbf{A}\odot\mathtt{R}[n-P]\odot\sqrt{\mathtt{B}}))\odot\mathbf{\hat{H}}\text{diag}(\sqrt{\boldsymbol{\mathcal{E}}_{us}})\mathbf{s}[n]$
$\displaystyle+(\mathbf{1}_{N}\otimes(\mathbf{\bar{A}}\odot\mathtt{R}[n-P]\odot\sqrt{\mathtt{B}}))\odot\mathbf{\tilde{H}}\text{diag}(\sqrt{\boldsymbol{\mathcal{E}}_{us}})\mathbf{s}[n]$
$\displaystyle+(\mathbf{1}_{N}\otimes(\mathtt{\bar{R}}[n-P]\odot\sqrt{\mathtt{B}}))\odot\mathbf{Z}_{h}[n;P]\text{diag}(\sqrt{\boldsymbol{\mathcal{E}}_{us}})\mathbf{s}[n]$
$\displaystyle+\sqrt{N_{0}}\mathbf{w}[n],$ (7)

where $\odot$ and $\otimes$ represent the Hadamard product and the Kronecker
product of two matrices, respectively, and $\mathbf{1}_{N}$ is the $N\times 1$
all ones vector. The square-root operation above (and elsewhere in this paper)
represents the element-wise square root; note that all the entries of these
matrices are nonnegative. To obtain the last equation above, we use (5), and
anchor time at the $P$th instant. The first term in (7) corresponds to the
signal $s[n]$ received over the known channel $\mathbf{\hat{H}}$, the second
term corresponds to the interference caused due to the channel estimation
error at time $P$, the third term corresponds to the interference due to the
evolution of the channel between time $P$ and time $n$, and the last term
corresponds to AWGN.

Now, the received vector is first pre-processed using an MMSE combiner. We
will use the expression for the combined signal to find the achievable rate of
the system via a deterministic equivalent analysis. The MMSE combiner requires
the covariance matrix of the received signal, conditioned on the available
estimate $\hat{\mathbf{H}}$, which can be expressed as

$\displaystyle\mathbf{R}_{yy|\mathbf{\hat{H}}}[n]=\left(\mathbf{1}_{N}\otimes(\mathtt{R}[n-P]\odot\mathbf{A}\odot\sqrt{\mathtt{B}})\odot\hat{\mathbf{H}}\right)\text{diag}(\boldsymbol{\mathcal{E}})$
$\displaystyle\times\left(\mathbf{1}_{N}\otimes(\mathtt{R}[n-P]\odot\mathbf{A}\odot\sqrt{\mathtt{B}})\odot\hat{\mathbf{H}}\right)^{H}+(\mathbf{1}_{N}\mathbf{1}_{N}^{T})$
$\displaystyle\otimes\Bigg{(}\\!\left(\mathtt{R}[n-P]\odot\bar{\mathbf{A}}\\!\odot\\!\sqrt{\mathtt{B}}\\!\right)\\!\text{diag}(\boldsymbol{\mathcal{E}})\\!\left(\mathtt{R}[n-P]\\!\odot\\!\bar{\mathbf{A}}\\!\odot\\!\sqrt{\mathtt{B}}\right)^{T}$
$\displaystyle+\left(\\!\mathtt{\bar{R}}[n-P]\\!\odot\\!\sqrt{\mathtt{B}}\\!\right)\\!\text{diag}(\boldsymbol{\mathcal{E}})\left(\mathtt{\bar{R}}[n-P]\\!\odot\\!\sqrt{\mathtt{B}}\right)^{T}\\!\Bigg{)}\\!+\\!N_{0}\mathbf{I}_{MN}.$
$\displaystyle=\mathbf{\hat{G}}\mathbf{\hat{G}}^{H}+\boldsymbol{\Psi}\otimes\mathbf{I}_{N}.$
(8)

Here, for convenience, we have defined,
$\hat{\mathbf{g}}_{k}[n]=(\mathbf{1}_{N}\otimes(\boldsymbol{\rho}_{k}[n-P]\odot\mathbf{a}_{k}\odot\sqrt{\mathcal{E}_{usk}\boldsymbol{\beta}_{k}})\odot\hat{\mathbf{h}}_{k})$,
$\mathbf{\hat{G}}=[\hat{\mathbf{g}}_{1}[n],\ldots,\hat{\mathbf{g}}_{K}[n]]^{T}$,
and $\boldsymbol{\Psi}=\text{diag}(\psi_{m})$, as a diagonal matrix with
$\psi_{m}$ corresponding to the noise and interference power at the $m$th AP.
Also defining
$\tilde{\mathbf{g}}_{k}[n]=(\mathbf{1}_{N}\otimes(\boldsymbol{\rho}_{k}[n-P]\odot\bar{\mathbf{a}}_{k}\odot\sqrt{\mathcal{E}_{usk}\boldsymbol{\beta}_{k}})\odot\tilde{\mathbf{h}}_{k})$,
and
$\boldsymbol{\xi}_{h,l}[n;P]=(\mathbf{1}_{N}\otimes(\boldsymbol{\bar{\rho}}_{l}[n-P]\odot\sqrt{\mathcal{E}_{usl}\boldsymbol{\beta}_{l}})\odot{\mathbf{z}}_{h,l}[n;P])$.
Then, the MMSE combining vector for the $k$th user’s signal is
$\mathbf{R}_{yy|\hat{\mathbf{H}}}^{-1}[n]\hat{\mathbf{g}}_{k}[n]$, and the
decoded signal for $k$th user is given as

$\displaystyle{r}_{k}[n]=\hat{\mathbf{g}}_{k}^{H}[n]\mathbf{R}^{-1}_{yy|\mathbf{\hat{H}}}[n]\hat{\mathbf{g}}_{k}[n]s_{k}[n]+\sum_{\begin{subarray}{c}l=1\\\
l\neq
k\end{subarray}}^{K}\hat{\mathbf{g}}_{k}^{H}[n]\mathbf{R}^{-1}_{yy|\mathbf{\hat{H}}}[n]$
$\displaystyle\times\hat{\mathbf{g}}_{l}[n]s_{l}[n]+\sum_{l=1}^{K}\hat{\mathbf{g}}_{k}^{H}[n]\mathbf{R}^{-1}_{yy|\mathbf{\hat{H}}}[n]\tilde{\mathbf{g}}_{l}[n]s_{l}[n]+\sum_{l=1}^{K}\hat{\mathbf{g}}_{k}^{H}[n]$
$\displaystyle\times\mathbf{R}^{-1}_{yy|\mathbf{\hat{H}}}[n]\boldsymbol{\xi}_{h,l}[n;P]s_{l}[n]\\!+\\!\sqrt{N_{0}}\hat{\mathbf{g}}_{k}^{H}[n]\mathbf{R}^{-1}_{yy|\mathbf{\hat{H}}}[n]\mathbf{w}[n].$
(9)

Since the combining vector is based on the MMSE channel estimate, we can use
the worst-case noise theorem [19], and treat interference as noise to get our
main result, namely, the deterministic equivalent of the uplink achievable
SINR.

######  Theorem 1.

The deterministic equivalent of the uplink achievable SINR for the $k$th user
at the $n$th instant, conditioned on the spatial locations of the users, is
given by

$\gamma_{k}[n]-\frac{\eta_{s,k}[n]}{\eta_{1,k}[n]+\eta_{2,k}[n]+\eta_{3,k}[n]+\eta_{w,k}}\xrightarrow{a.s.}0,$
(10)

where $\eta_{s,k}[n]$ is the desired signal power, $\eta_{1,k}[n]$ corresponds
to the residual inter-user interference after MMSE combining, $\eta_{2,k}[n]$
to the interference due to channel estimation errors, $\eta_{3,k}[n]$ the
interference due to channel aging, and $\eta_{w,k}$ is due to AWGN. These can
be expressed as

$\eta_{s,k}[n]=N\mathcal{E}^{2}_{usk}\left(\sum_{m=1}^{M}\zeta_{mk}[n-P]\varphi_{mk}[n]\right)^{2}$
(11)

with $\zeta_{mk}[n-P]=\beta_{mk}a^{2}_{mk}\rho^{2}_{mk}[n-P]$,

$\varphi_{mk}[n]\triangleq\left({\sum_{\begin{subarray}{c}l=1\\\ l\neq
k\end{subarray}}^{K}\mathcal{E}_{usl}\frac{{\zeta_{ml}[n-P]}}{(1+e_{k,l}[n])}+\psi_{m}}\right)^{-1},$
(12)

and $e_{k,l}[n]$ is iteratively computed as
$e_{k,l}^{(t)}[n]=N\sum_{m=1}^{M}\frac{\zeta_{ml}[n-P]\mathcal{E}_{usl}}{\sum_{i=1;i\neq
k}^{K}\frac{\zeta_{mi}[n-P]\mathcal{E}_{usi}}{1+e^{(t-1)}_{k,i}[n]}+\psi_{m}}$
with the initialization $e_{k,i}^{(0)}[n]=\frac{1}{N_{0}}$.

$\eta_{2,k}[n]=\mathcal{E}_{usk}\sum_{l=1}^{K}\mathcal{E}_{usl}\sum_{m=1}^{M}\varphi^{\prime}_{mk}[n]\check{\zeta}_{ml}[n-P],$
(13)

with $\check{\zeta}_{ml}[n-P]=\beta_{ml}\bar{a}^{2}_{ml}\rho^{2}_{ml}[n-P]$,

$\varphi_{mk}^{\prime}[n]=\varphi_{mk}^{2}[n]\left(\zeta_{mk}[n-P]+\sum_{\begin{subarray}{c}l=1\\\
l\neq
k\end{subarray}}^{K}\mathcal{E}_{usl}\frac{{\zeta_{ml}[n-P]}e^{\prime}_{k,l}[n]}{(1+e_{k,l}[n])}\right),$
(14)

$e^{\prime}_{k,l}[n]=[(\mathbf{I}_{K-1}-\mathbf{J})^{{-1}}\mathbf{u}]_{l}$,
$[\mathbf{J}]_{pq}=\frac{\mathcal{E}_{usp}\mathcal{E}_{usq}\text{Tr}\\{\boldsymbol{\Xi}_{p}\mathbf{T}(\boldsymbol{\Psi})\boldsymbol{\Xi}_{q}\mathbf{T}(\boldsymbol{\Psi})\\}}{1+e_{k,l}[n]}$,
${u}_{p}=\mathcal{E}_{usp}\text{Tr}\\{\boldsymbol{\Xi}_{p}[n]\mathbf{T}(\boldsymbol{\Psi})\boldsymbol{\Xi}_{k}[n]\mathbf{T}(\boldsymbol{\Psi})\\}$,
$\mathbf{T}(\boldsymbol{\Psi})=\left(\text{diag}(\boldsymbol{\varphi}_{k}[n])\right)\otimes\mathbf{I}_{N}$

$\eta_{3,k}[n]=\mathcal{E}_{usk}\sum_{l=1}^{K}\mathcal{E}_{usl}\sum_{m=1}^{M}\varphi^{\prime}_{mk}[n]\dot{\zeta}_{ml}[n-P]$
(15)

with $\dot{\zeta}_{ml}[n-P]=\beta_{ml}\bar{\rho}_{ml}^{2}[n-P]$,

$\eta_{w,k}[n]=N_{0}\mathcal{E}_{usk}\sum_{m=1}^{M}\varphi^{\prime}_{mk}[n],$
(16) $\eta_{1,k}[n]={\mathcal{E}_{usk}}\sum_{\begin{subarray}{c}l=1\\\ l\neq
k\end{subarray}}^{K}\mathcal{E}_{usl}\sum_{m=1}^{M}\zeta_{mk}[n-P]\zeta_{ml}[n-P]{\dot{\epsilon}_{kl}[n]},$
(17)

where

$\displaystyle\dot{\epsilon}_{kl}[n]$
$\displaystyle=\dot{\varphi}^{\prime}_{kl}[n]+\frac{\lvert\sum_{m=1}^{M}\mathcal{E}_{usl}\zeta_{ml}[n-P]\rvert^{2}N^{2}\dot{\varphi}^{2}_{kl}[n]\dot{\varphi}^{\prime}_{kl}[n]}{\lvert
1+\sum_{m=1}^{M}\mathcal{E}_{usl}\zeta_{ml}[n-P]N\dot{\varphi}_{kl}[n]\rvert^{2}}$
$\displaystyle-2\Re\left\\{\frac{\sum_{m=1}^{M}\mathcal{E}_{usl}\zeta_{ml}[n-P]N\dot{\varphi}_{kl}[n]\dot{\varphi}^{\prime}_{kl}[n]}{1+\sum_{m=1}^{M}\mathcal{E}_{usl}\zeta_{ml}[n-P]N\dot{\varphi}_{kl}[n]}\right\\}$
(18)

with $\Re\\{.\\}$ denoting the real part of a complex number, and

$\dot{\varphi}_{mkl}[n]\triangleq\left({\sum_{\begin{subarray}{c}p=1\\\ p\neq
k,l\end{subarray}}^{K}\mathcal{E}_{usp}\frac{{\zeta_{mp}[n-P]}}{(1+e_{k,p}[n])}+\psi_{m}}\right)^{-1},$
(19)

$\dot{e}_{k,lp}[n]$ is iteratively computed as
$\dot{e}_{k,lp}^{(t)}[n]=N\sum_{m=1}^{M}\frac{\zeta_{ml}[n-P]\mathcal{E}_{usl}}{\sum_{i=1;i\neq
k}^{K}\frac{\zeta_{mi}[n-P]\mathcal{E}_{usi}}{1+e^{(t-1)}_{k,i}[n]}+\psi_{m}}$
with the initialization $e_{mk,i}^{(0)}[n]=\frac{1}{N_{0}}$. Also,

$\dot{\varphi}_{mkl}^{\prime}[n]\\!=\\!\dot{\varphi}_{mkl}^{2}[n]\\!\\!\left(\\!\\!\zeta_{mk}[n-P]\\!+\\!\sum_{\begin{subarray}{c}l=1\\\
l\neq
k\end{subarray}}^{K}\frac{\mathcal{E}_{usl}{\zeta_{ml}[n-P]}\dot{e}^{\prime}_{k,l}[n]}{(1+\dot{e}_{k,l}[n])}\right),$
(20)

with
$\dot{e}^{\prime}_{k,l}[n]=[(\mathbf{I}_{K}-\mathbf{\dot{J}})\mathbf{\dot{u}}]_{l}$,
$[\mathbf{\dot{J}}]_{pq}=\frac{\mathcal{E}_{usp}\mathcal{E}_{usq}\text{Tr}\\{\boldsymbol{\Xi}_{p}\mathbf{T}(\boldsymbol{\Psi})\boldsymbol{\Xi}_{q}\mathbf{T}(\boldsymbol{\Psi})\\}}{1+e_{k,l}[n]}$,
$\dot{u}_{p}=\mathcal{E}_{usp}\text{Tr}\\{\boldsymbol{\Xi}_{p}[n]\mathbf{T}(\boldsymbol{\Psi})\boldsymbol{\Xi}_{l}[n]\mathbf{T}(\boldsymbol{\Psi}\\}$.

###### Proof.

We will soon post an updated version of this document with a detailed proof. ∎

In the special case where the channel correlation coefficient $\rho[n]$ is
independent of the UE and AP indices, it can be shown that
$\eta_{s,k}[n],\eta_{1,k},$ and $\eta_{2,k}[n]$ are proportional to
$\rho^{4}[n-P]$, while $\eta_{3,k}[n]$ is proportional to
($\rho^{2}[n-P]-\rho^{4}[n-P]$). Since $\rho[n]$ is a decreasing function of
$n$, their overall effect is an increase in interference and a decrease in the
SINR with time.

##  IV Numerical Results

In this section, we perform from Monte-Carlo simulations to corroborate the
analytical results on the SINR of CF-mMIMO systems under channel aging. We
consider a unit area including K=16 UEs served by N=256 single antenna APs.
The signaling bandwidth is 5 MHz and the carrier frequency is 5 GHz. The frame
duration is T=1024 channel uses. For all the experiments, we assume that the
transmit SNR for both data and pilots is $20$ dB. For the multi-slope path
loss model in (1), we assume $L=2$, $d_{0}=0.1$ units and $d_{1}=\sqrt{2}$
units while $\eta_{0}=0$ and $\eta_{1}=3$. To segregate the effects of pilot
contamination and channel aging, we consider orthogonal pilots, transmitted
over a duration of $16$ channel uses.

For the computation of the average SINR at a given UE, the UEs and APs are
dropped uniformly at random locations over a square with unit area. For each
spatial realization of UEs and APs, $100$ independent channel realizations are
used to compute the average uplink SINR achieved by a UE. This SINR is
averaged over $100$ independent spatial realizations of the UE and AP
locations. Thus, the average SINR is computed by averaging over 10,000
independent channel instantiations.

In Fig. 1, we plot the average per user SINR at the CPU as a function of the
time index for different UE velocities with $\Delta=0$, i.e., the channel ages
at the same rate at all APs. We see that the theoretical and simulated curves
match perfectly for CF-mMIMO. In the figure, theoretical and simulated curves
are represented by the lines (no markers) and the markers (no lines),
respectively. Also, we also compare the relative effects of channel aging on
CF-mMIMO against those on cellular mMIMO and small cells. In the case of
cellular mMIMO, we consider a single BS at the cell center equipped with
$N=256$ antennas. For the case of small cells, we consider $M=256$ single-
antenna APs deployed over the area of interest (the same as in the CF case),
with each UE associated with its nearest AP, under the transmit power
assumptions considered in [2]. The theoretical expressions for the SINR
achieved in these two systems can be derived in a similar manner as the
expressions presented in this paper. We omit the details due to lack of space.
CF-mMIMO achieves much higher SINR than both cellular mMIMO and small cells.
Moreover, we observe that the impact of higher mobility on CF is more severe
than that on cellular mMIMO or small cells.

Figure 1: Uplink achievable SINR as a function of time for different UE
velocities and system configurations.

In Fig. 2, we study effect of variation in the relative speed of the UE with
respect to different APs for different numbers of APs. We do this by varying
the parameter $\Delta$ described earlier under the assumption that the UEs are
moving at an average speed of $v_{\max}=300$ km/h, and for different numbers
of APs. We depict the average SINR of the CF-mMIMO system against the time
index. Interestingly, a larger value of $\Delta$ mitigates the effects of
channel aging. In addition, we see that the relative loss in the SINR due to
channel aging remains approximately unchanged as the number of APs is
increased. In other words, contrary to intuition, an increase in the system
dimension does not worsen the effects of aging. In other words, the benefits
of higher dimensions offsets the degradation due to aging.

Figure 2: Uplink achievable SINR as a function of sample index at a UE
velocity of $300$ km/h for different values of $\Delta$.

In Fig. 3, we illustrate the average SINR per user at the $1024$th sample as a
function of the UE velocity for different values of $\Delta$. Also, we compare
the achievable SINR against cellular mMIMO and small cells. We observe that at
low to medium user mobility, CF-mMIMO significantly outperforms both cellular
mMIMO and small cells. However, at high mobilities, the performance of CF-
mMIMO becomes comparable to, if not worse than both cellular mMIMO and small
cells. This strengthens the observation made in [20], that the effect of
channel aging on wireless communication systems is highly dependent on the
system model, and warrants careful analysis.

Figure 3: Uplink achievable SINR at the $1024$th sample, as a function of the
UE velocity, for different values of $\Delta$.

##  V Conclusions and Future Work

In this paper, we studied the effect of channel aging on the uplink of a
general CF-mMIMO system. We elaborated on the effects of different relative
speeds of the UEs with respect to the APs, and derived the deterministic
equivalent SINR under centralized MMSE combining at the CPU. We observed that
CF-mMIMO systems offer much better uplink SINR compared to cellular mMIMO
systems and small cells at low user mobilities. However, at high velocities
and long frame durations, the relative impact of aging is higher on CF-mMIMO
systems. We also note that unlike the Jakes’ model, there is no well-accepted
model for characterizing the effects of mobility on mmWave channels.
Therefore, the study of the effect of user mobility in mmWave massive MIMO
requires well designed measurement campaigns to quantify the temporal
correlation, variation in the angle of arrival/departures, and other channel
characteristics. Hence, the study of channel aging in CF systems operating at
the mmWave bands could be the topic for future work.

## References

  * [1] E. Björnson, L. Sanguinetti, H. Wymeersch, J. Hoydis, and T. L. Marzetta, “Massive MIMO is a reality—what is next?: Five promising research directions for antenna arrays,” _Digital Signal Processing_ , vol. 94, pp. 3 – 20, 2019.
  * [2] H. Q. Ngo, A. Ashikhmin, H. Yang, E. G. Larsson, and T. L. Marzetta, “Cell-free massive MIMO versus small cells,” _IEEE Trans. Wireless Commun._ , vol. 16, no. 3, pp. 1834–1850, Mar. 2017.
  * [3] T. L. Marzetta, E. G. Larsson, H. Yang, and H. Q. Ngo, _Fundamentals of Massive MIMO_. Cambridge University Press, Cambridge, UK, 2016.
  * [4] H. Masoumi and M. J. Emadi, “Performance analysis of cell-free massive MIMO system with limited fronthaul capacity and hardware impairments,” _IEEE Trans. Wireless Commun._ , vol. 19, no. 2, pp. 1038–1053, 2020.
  * [5] C. Kong, C. Zhong, A. K. Papazafeiropoulos, M. Matthaiou, and Z. Zhang, “Sum-rate and power scaling of massive MIMO systems with channel aging,” _IEEE Trans. Commun._ , vol. 63, no. 12, pp. 4879–4893, Dec. 2015.
  * [6] A. K. Papazafeiropoulos and T. Ratnarajah, “Deterministic equivalent performance analysis of time-varying massive MIMO systems,” _IEEE Trans. Wireless Commun._ , vol. 14, no. 10, pp. 5795–5809, Oct. 2015.
  * [7] R. Chopra, C. Murthy, H. Suraweera, and E. Larsson, “Performance analysis of FDD massive MIMO systems under channel aging,” _IEEE Trans. Wireless Commun._ , vol. 17, no. 2, pp. 1094–1108, Feb. 2018.
  * [8] G. Interdonato, H. Q. Ngo, P. Frenger, and E. G. Larsson, “Downlink training in cell-free massive MIMO: A blessing in disguise,” _IEEE Trans. Wireless Commun._ , vol. 18, no. 11, p. 5153–5169, Nov. 2019.
  * [9] K. T. Truong and R. W. Heath, “Effects of channel aging in massive MIMO systems,” _J. Commun. Net._ , vol. 15, no. 4, pp. 338–351, Aug. 2013.
  * [10] W. A. W. M. Mahyiddin, P. A. Martin, and P. J. Smith, “Massive MIMO systems in time-selective channels,” _IEEE Commun. Lett._ , vol. 19, no. 11, pp. 1973–1976, 2015.
  * [11] A. K. Papazafeiropoulos, “Impact of general channel aging conditions on the downlink performance of massive MIMO,” _IEEE Trans. Veh. Technol._ , vol. 66, no. 2, pp. 1428–1444, Feb. 2016.
  * [12] R. Chopra, C. R. Murthy, and H. A. Suraweera, “On the throughput of large MIMO beamforming systems with channel aging,” _IEEE Signal Process. Lett._ , vol. 23, no. 11, pp. 1523–1527, Nov. 2016.
  * [13] H. Guo, B. Makki, D.-T. Phan-Huy, E. Dahlman, M.-S. Alouini, and T. Svensson, “Predictor antenna: A technique to boost the performance of moving relays,” 2020\.
  * [14] R. Couillet and M. Debbah, _Random matrix methods for wireless communications_ , 1st ed. New York, NY, USA: Cambridge University Press, 2011.
  * [15] X. Zhang and J. G. Andrews, “Downlink cellular network analysis with multi-slope path loss models,” _IEEE Trans. Commun._ , vol. 63, no. 5, pp. 1881–1894, May 2015.
  * [16] W. C. Jakes and D. C. Cox, Eds., _Microwave Mobile Communications_ , 2nd ed. IEEE Press, New York: IEEE Press, 1994.
  * [17] E. Björnson and L. Sanguinetti, “Making cell-free massive MIMO competitive with MMSE processing and centralized implementation,” _IEEE Trans. Wireless Commun._ , vol. 19, no. 1, pp. 77–90, 2020.
  * [18] H. Osawa, “Reversibility of first-order autoregressive processes,” _Stochastic Processes and their Applications_ , vol. 28, no. 1, pp. 61 – 69, 1988.
  * [19] B. Hassibi and B. M. Hochwald, “How much training is needed in multiple-antenna wireless links?” _IEEE Trans. Inf. Theory_ , vol. 49, no. 4, pp. 951–963, Apr. 2003.
  * [20] R. Chopra, C. R. Murthy, H. A. Suraweera, and E. G. Larsson, “Analysis of nonorthogonal training in massive MIMO under channel aging with SIC receivers,” _IEEE Signal Process. Lett._ , vol. 26, no. 2, pp. 282–286, Feb. 2019.