1 Aging Channel Modeling and Transmission Block Size Optimization for Massive MIMO Vehicular

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Aging Channel Modeling and Transmission Block

Size Optimization for Massive MIMO Vehicular

Networks in Non-Isotropic Scattering Environment

Huafu Li, Graduate Student Member, IEEE, Liqin Ding, Member, IEEE,

Yang Wang, and Zhenyong Wang, Senior Member, IEEE

Abstract—We investigate the effect of channel aging on multi-

cell massive multiple-input multiple-output (MIMO) vehicular

networks in a generic non-isotropic scattering environment.

Based on the single cluster scattering assumption and the von

Mises distribution assumptions of the scatterers’ angles, an

aging channel model is established to capture the joint effect of

spatial and temporal correlations resulting from different angular

spread conditions in various application scenarios. Expressions

of the user uplink transmission spectral efﬁciency (SE) are

derived for maximum ratio (MR) and minimum mean square

error (MMSE) combining. Through numerical studies, the area

spectral efﬁciency (ASE) performance of the network is evaluated

in freeway and urban Manhattan road grid scenarios, and easy-

to-use empirical models for the optimal transmission block size

for ASE maximization are obtained for the evaluated scenarios.

Index Terms—Channel aging, non-isotropic scattering, spatial-

temporal correlation, massive MIMO, vehicular network.

I. INTRODUCTION

Massive multiple-input multiple-output (MIMO) is becom-

ing a reality, but there are still many practical issues to be

addressed [1], [2]. One of them is to understand the impact of

channel aging on massive MIMO systems in highly dynamic

scenarios and to ﬁnd corresponding solutions to accommodate

such impact for efﬁcient network operation. Channel aging

refers to the mismatch between the estimated channel coefﬁ-

cients during the channel estimation phase and the actual ones

over which the data is transmitted, as wireless channels change

inevitably with time [3]. Since most of the claimed advantages

of massive MIMO rely heavily on the availability of accurate

channel state information (CSI) at the base stations (BSs), for

precoding in downlink (DL) and combining in uplink (UL)

the signals sent / received through the antenna arrays [4], [5],

massive MIMO is more suspensible to channel aging than

This work was supported in part by the European Union’s Horizon 2020

research and innovation programme under the Marie Skłodowska Curie grant

agreement No 887732 (H2020-MSCA-IF VoiiComm), in part by the Science

and Technology Project of Shenzhen under Grant JCYJ20200109113424990,

and in part by the Marine Economy Development Project of Guangdong

Province under Grant GDNRC [2020]014. (Corresponding author: Yang

Wang.)

Huafu Li and Zhenyong Wang are with the School of Electronics and

Information Engineering, Harbin Institute of Technology, Harbin 150001,

China (e-mail: {fairme, zywang}@hit.edu.cn).

Liqin Ding is with the Department of Electrical Engineering,

Chalmers University of Technology, 412 96 Gothenburg, Sweden (e-mail:

liqind@chalmers.se).

Yang Wang is with the School of Electronics and Information Engineering,

Harbin Institute of Technology Shenzhen, Shenzhen 518055, China (e-mail:

yangw@hit.edu.cn).

conventional MIMO employing small arrays. For example,

it has been shown that a CSI outdated by four milliseconds

could cause up to 50% degradation in data rate for users

with moderate mobility (30 km/h) compared to low mobility

(3km/h) when the BS employ arrays with 32 and 64 antennas

[6]. Therefore, the study of channel aging effects is crucial,

especially for applications with highly dynamic environments

(e.g., urban scenarios [7], [8]) or users with high mobility (e.g.,

ground vehicles [9] and drones [10], [11]).

In the seminal paper [3] by K. T. Truong et al., an aging

channel model that considers both the channel estimation error

and the aging drifts is developed for massive MIMO systems,

and a performance analysis framework that covers both UL

and DL transmissions is established. The temporal autocorre-

lation of the channel is assessed based on isotropic scattering

(i.e., the Jakes-Clarke model) and equal Doppler shift assump-

tion, resulting in an autocorrelation function (ACF) given by

the zeroth-order Bessel function of the ﬁrst kind. Based on this

model, the effects of channel aging are then more thoroughly

studied by A. K. Papazafeiropoulos et al. in a series of

works [12]–[16], considering different precoding / combining

methods and practical issues such as pilot contamination,

phase noise, and hardware impairment. Lately, the study of

aging effects has also been extended to non-central network

architectures, under the name of distributed antenna system or

cell-free massive MIMO network [17]–[21].

To alleviate the effects of channel aging, channel prediction

techniques are also proposed, such as the Wiener predictor

[3], [12], [13], Kalman predictor [22], [23], and the autore-

gressive moving average (ARMA) predictor [24]. Another key

measure against channel aging is the optimal design of the

channel training frequency, or equivalently, the duration of the

transmission block, to ensure good system-level throughput /

spectral efﬁciency (SE) [25]–[28]. This problem stems directly

from the reasoning that more frequent channel estimation

ensures more accurate CSI and that there will be a sweet

spot where the resulting performance gain most outweighs the

cost. Obviously, such a sweet spot depends on how quickly

the channel ages and how much the performance metric of

interest is affected by the outdated CSI.

Both performance evaluation and system design optimiza-

tion require an aging channel model that correctly captures

the spatial and temporal correlations of channel coefﬁcients.

However, in most existing works, the assumptions are oversim-

pliﬁed. First, most works adopt the Jakes-Clarke model when

arXiv:2210.10250v1 [cs.IT] 19 Oct 2022

assessing the temporal evolution (aging) of the channel, while

in practice, the angular spread of multipath components of the

channel is typically limited [29]–[31]. In other words, non-

isotropic scattering propagation environment is the common

case. Measurement campaigns have demonstrated this for BSs

in a wide range of scenarios [30], [32] and also for mobile

terminals [29], [33] in many scenarios such as the street

canyon environment [34, Section 7.4]. Our preliminary work

[7] shows that the Jakes-Clarke model may lead to overly

pessimistic performance predictions and more than necessary

channel training. Second, the joint impact of spatial correlation

is poorly captured in the existing study. Most works [12]–

[20], [22]–[27] assume that channels are spatially uncorrelated,

and only a few have included spatial correlation in channel

modeling. For example, when computing spatial correlation

matrices, the Laplace distribution and the Gaussian distribu-

tion are assumed for the multipath angles in [3] and [21],

respectively. In our preliminary work [7], spatial correlation is

modeled by assuming a uniform distribution within a limited

range. However, as we shall discuss in detail later in this paper,

the impact of spatial correlation on massive MIMO links is

quite complex and requires careful study.

In this paper, we investigate the effect of channel aging on a

multi-cell massive MIMO system with vehicle users (VUEs) in

a more realistic non-isotropic scattering scenario with spatially

correlated channels. We also address the optimal transmission

block design problem in typical scenarios such that the area

spectral efﬁciency (ASE) is maximized. Our study focuses on

UL transmission and the main contributions are summarized

as follows.

•We derive the spatial-temporal cross-correlation (STCC)

function of the channel based on the assumption of

the von Mises distribution for both angle-of-departure

(AoD) and angle-of-arrival (AoA) to represent a general

nonisotropic scattering condition, and develop an aging

channel model that allows us to study the joint effect of

space-time correlation in various application scenarios.

The channel model captures the impact of the spatial

distribution of BS, VUE, and scatterers with parameters

including the central direction and degree of spread of

AoA / AoD, as well as the orientation of the BS antenna

array and the velocity of the VUE.

•Based on the developed aging channel model, expressions

for the VUE’s SE performance are derived for both

maximum ratio (MR) and minimum mean square error

(MMSE) combining, taking into account the channel

training overhead and pilot contamination effects. The

system-level ASE performance of the massive MIMO

system is then evaluated in freeway and urban Manhattan

road grid scenarios.

•Based on numerical studies, we obtain easy-to-use empir-

ical models for the transmission block size for optimizing

ASE performance, which turns out to be linear equations

of VUE moving speed and square roots of AoD and AoA

spread. The performance gain brought by the optimal

transmission block design is demonstrated.

The paper organization is as follows. The aging channel

BS VUE

scatterer i

Fig. 1. Propagation geometry of a joint space-time correlated channel with

one cluster of scatterers.

model is developed in Section II. Expressions for the UL SE

are derived in Section III. The numerical studies in the two

scenarios are conducted in Section IV, where empirical models

of the optimal transmission block size are also obtained and

evaluated. Finally, Section V concludes this work.

Notation: We use italic lowercase letters, boldface low-

ercase letters, boldface uppercase letters, and calligraphic

letters to represent scalars, column vectors, matrices, and

sets, respectively. The expectation operator, the absolute value,

the Euclidean norm and the trace operator are denoted by

E{·},|·|,k·k, and tr (·), respectively. The conjugate, conjugate

transpose, and pseudoinverse operations are denoted by (·)∗,

(·)H, and (·)†, respectively. Imstands for the m×midentity

matrix, j=√−1,cdenotes the speed of light, and a deﬁnition

is denoted by ,. Finally, NC(0,R)stands for the multi-variate

circularly symmetric complex Gaussian distribution with zero

mean and covariance matrix R.

II. AGING CHANNEL MODELING

We consider the uplink transmission over one subcarrier in

a massive MIMO system. Time is divided into transmission

blocks. The pilot sequence is transmitted at the beginning of

the transmission block, followed by user data in the rest of the

block. We are interested in the impact of channel aging on the

performance of the transmission in one block. In this section,

we model the narrow-band channel between a single-antenna

VUE and its serving BS, where a uniform linear array (ULA)

of Mantennas is deployed.

A. Space-Time Cross Correlation Matrix

As shown in Fig. 1, the orientation of the ULA is speciﬁed

by an angle αfrom the arbitrarily selected reference direction

X, while the direction of movement of the VUE (of speed v) is

given by the angle γ. The channel between them is composed

of single-bounce multipath elements, caused by a cluster of

Sstatic scatterers. The angle of the ith scatterer seen from

the VUE / BS is denoted by φi/θi. Since we consider uplink

transmission in this paper, φiand θiwill be referred to as AoD

and AoA, respectively. The scatterers are assumed to be distant

enough from the BS such that the latter experiences planar

wavefronts. Denote the ULA antenna spacing by dand the

carrier frequency by fc, and let λc=c/fcbe the wavelength.

The M×1complex channel vector at the time tis therefore

given by

h(t) =

i=1

ξiexp j2πfD

it+j2π

λc

βi+jψi,(1)

where ξiis the path gain, fD

i=fcvcos(φi−γ)/c is

the Doppler frequency, βiis the vector of propagation

distance difference, whose p-th element is given by (p−

1)dcos (α−θi),p= 1, . . . , M, and ψi∈[−π, π)is the ran-

dom phase shift associated with the ith multipath component.

We assume that the AoDs {φi}, AoAs {θi}, and phase shifts

{ψi}are all independent random variables. The M×MSTCC

matrix of h(t)with delay τis deﬁned by

R(τ),Eh(t)hH(t+τ)

pE{kh(t)k2}pE{kh(t+τ)k2},(2)

and its (p, q)-th element can be derived as follows:

[R(τ)]p,q =E(S

i=1

exp ja cos (φi−γ) + jb cos (α−θi)),

(3)

where

a=−2πτfcv/c, b = 2π(p−q)d/λc.(4)

Note that the gains of the multipath components are assumed

to be ﬁxed over the time interval of interest.

In this work, we consider that the number of scatterers

approaches inﬁnity (S→ ∞) and that AoD and AoA follow

two independent continuous distributions. (In this case, the

channel gain of each multipath component is inﬁnitesimal.) In

particular, we adopt the empirically veriﬁed von Mises distri-

bution for both angles to represent a non-isotropic scattering

condition [35, Section 2.1.2]. The PDFs of AoD φ∈[−π, π)

and AoA θ∈[−π, π)are given as follows:

p(φ) = exp [κTcos (φ−φc)]

2πI0(κT),(5)

p(θ) = exp [κRcos (θ−θc)]

2πI0(κR),(6)

where I0(z),1

2πRπ

−πexp(zsin x)dx is the modiﬁed Bessel

function of the ﬁrst kind and zero order [36, Eq. (9.6.16)].

Following these distributions, the AoDs and AoAs are con-

centrated around the central directions φc∈[−π, π)and

θc∈[−π, π), and the degrees of concentration are determined

by κT(≥0) and κR(≥0), respectively. The larger κT/κR

is, the more concentrated the distribution of AoD / AoA. If

κT/κRis close to 0, the distribution is close to uniform. We

further adopt

σT=1

√κT

, σR=1

√κR

(7)

as the measure of AoD and AoA spread (in radians) [29], since

1/κ is analogous to the variance in the normal distribution1.

Based on the above, the (p, q)-th element of the STCC

matrix R(τ), given by (3), can be further derived as the

1In fact, when κis very large, the von Mises distribution approximates

closely the normal distribution of variance 1/κ.

following closed-form2

[R(τ)]p,q =Zπ

−π

exp{ja cos (φ−γ)}p(φ)dφ·

Zπ

−π

exp{jb cos (α−θ)}p(θ)dθ

=ρ(τ)·s(p, q),(8)

where 









ρ(τ) = I0√−a2+κ2

T+2ajκTcos(γ−φc)

I0(κT),

s(p, q) = I0√−b2+κ2

R+2bjκRcos(α−θc)

I0(κR).

(9)

It can be easily veriﬁed that ρ(τ)is the ACF of any entry of

the time-varying channel h(t), and that s(p, q)is the spatial

correlation function (SCF) between the p-th and q-th entries at

any time instance. We can see from (9) that ρ(τ)(0≤ |ρ(τ)| ≤

1) is affected by the AoD spread, the speed of the VUE and its

moving direction relative to the AoA central direction (i.e., γ−

φc), besides the time delay; and that s(p, q)(0≤ |s(p, q)| ≤ 1)

is affected by the AoA spread, the spacing of the ULA at the

BS and its orientation relative to the AoD central direction

(i.e., α−θc). Together, these parameters determine the STCC

matrix.

This analytically tractable model allows us to investigate the

joint effects of the spatial-temporal characteristics of channels

on massive MIMO networks under a wide range of scattering

conditions by adjusting the above mentioned parameters, and

hopefully, draw some far-reaching conclusions. For example,

in the case of κT= 0,ρ(τ) = J0(2πτfcv/c)is immediately

obtained, where J0(z),1

2πRπ

−πexp(−jz sin x)dx is zeroth-

order Bessel function of the ﬁrst kind [38, Eq. (9.19)], which is

the Jakes-Clarke ACF model based on the isotropic scattering

assumption (p(φ) = 1/2π) around the mobile user. In the

case of κR= 0 (p(θ)=1/2π), the SCF becomes s(p, q) =

J0[2π(p−q)d/λc], from which it can be seen that although

the BS is surrounded by a large number of isotropic scatterers,

there can still be correlation among the antennas. This is due

to the fact that ULAs have better angular resolution for the

boresight direction than for those directions closer to the end-

ﬁre [39].

Finally, we remark that it is reasonable to assume that the

scatterers that contribute to the multipath components of the

channel are unchanged within the time interval of our interest

(i.e.,, the duration of a transmission block), which typically

lasts several milliseconds [40], [41]. With this assumption, the

STCC matrix R(τ)with elements given by (8) is in fact time

independent. We also note that the total gain of the channel,

G(t),kh(t)k2, is also considered to be unchanged in this

short time interval and is below denoted by G.

B. Numerical Examples

In Fig. 2, the absolute value of the STCC for two adjacent

antenna elements (p−q= 1), as a function of antenna spacing

2The relation Rπ

−πexp(xsin z+ycos z)dz = 2πI0(px2+y2)[37, Eq.

(3.338-4)] is adopted in the derivation.

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1AgingChannelModelingandTransmissionBlockSizeOptimizationforMassiveMIMOVehicularNetworksinNon-IsotropicScatteringEnvironmentHuafuLi,GraduateStudentMember,IEEE,LiqinDing,Member,IEEE,YangWang,andZhenyongWang,SeniorMember,IEEEAbstractWeinvestigatetheeffectofchannelagingonmulti-cellmassivemultiple-inpu...

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