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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 21, NO. 6, JUNE 2022 4549
Performance Analysis and Optimization of
Multihop MIMO Relay Networks in
Short-Packet Communications
Ngo Hoang Tu and Kyungchun Lee ,Senior Member, IEEE
Abstract— This work investigates the multiple-input
multiple-output system in the context of the selective
decode-and-forward multihop relay network under short-
packet communications to facilitate not only ultra-reliability,
but also low-latency communications. For the transmit and
receive diversity techniques, we analyze the transmit antenna
selection (TAS) and maximum-ratio transmission (MRT) schemes
at the transmit side, whereas the selection-combining (SC) and
maximum-ratio combining (MRC) schemes are leveraged at the
receive side. For quasi-static Rayleigh fading channels and the
finite-blocklength regime, we derive the approximate closed-form
expressions of the end-to-end (e2e) block error rate (BLER)
for the TAS/MRC, TAS/SC, and MRT/MRC schemes. The
asymptotic performance in the high signal-to-noise ratio regime
is derived, from which the comparison among diversity schemes
in terms of the diversity order, e2e BLER loss, and SNR gap
is provided. Furthermore, based on the asymptotic results,
we develop power-allocation, relay-location, and joint simplified
optimizations to minimize the asymptotic e2e BLER under the
system constraints. The e2e latency and throughputs are also
analyzed for the considered schemes. The correctness of our
analysis is confirmed via Monte Carlo simulations.
Index Terms—Short-packet communication, ultra-reliable and
low-latency communications, multihop relaying, multiple-input
multiple-output, transmit and receive diversity.
I. INTRODUCTION
FIFTH-GENERATION (5G) networks will reinforce vari-
ous foremost services, including massive Machine-Type
Communications (mMTCs), enhanced Mobile Broadband
(eMBB), and ultra-Reliable and Low-Latency Communica-
tions (uRLLCs) [2]–[4]. In particular, based on 5G ITU-R and
IMT-2020 [2], [3], the 5G networks will fulfill the stringent
Manuscript received March 2, 2021; revised June 7, 2021 and
September 14, 2021; accepted November 22, 2021. Date of publication
December 7, 2021; date of current version June 10, 2022. This work was
supported in part by the Basic Science Research Program through the National
Research Foundation of Korea (NRF) funded by the Ministry of Education
under Grant NRF-2019R1A6A1A03032119 and in part by the NRF funded
by the Korean Government [Ministry of Science and ICT (MSIT)] under
Grant NRF-2019R1F1A1061934. This paper is submitted in part to IEEE
International Conference on Communications (ICC), [1]. The associate editor
coordinating the review of this article and approving it for publication was
M. Wang. (Corresponding author: Kyungchun Lee.)
Ngo Hoang Tu is with the Department of Smart Energy Systems, Seoul
National University of Science and Technology, Seoul 01811, Republic of
Korea (e-mail: ngohoangtu@seoultech.ac.kr).
Kyungchun Lee is with the Department of Electrical and Information
Engineering, Research Center for Electrical and Information Technology,
Seoul National University of Science and Technology, Seoul 01811, Republic
of Korea (e-mail: kclee@seoultech.ac.kr).
Color versions of one or more figures in this article are available at
https://doi.org/10.1109/TWC.2021.3131205.
Digital Object Identifier 10.1109/TWC.2021.3131205
requirements from (i) eMBB with one hundred times the net-
work energy efficiency and three times the spectrum efficiency
of IMT-Advanced, a traffic capacity of 10 −100 Mbps/m2,
user experienced data rates of 100 Mbps, and peak data
rates of 10 −20 Gbps; (ii) mMTCs with the connection
density of 106devices/km2;and(iii) uRLLC with very low
latency of around 1−10 ms [5]. From the standpoint of
uRLLC, the system needs to satisfy not only low latency, but
also ultra-reliability, which refers to packet error rates lower
than 10−5[6]. Motivated by this, in [7], the fundamental short-
packet communication (SPC) conception has been pioneer-
ingly explored to meet the uRLLC requirements in or even
beyond 5G [8] by shortening the frame length to as short
as 100 channel uses (CUs), in contrast to the conventional
systems with the long blocklength (BL). Specifically, in [7],
Polyanskiy et al. have contributed to obtaining the approxima-
tion of the maximum coding rate under the finite-BL (FBL)
regime.
The multiple-input multiple-output (MIMO) system can
provide diversity gains to significantly improve reliability,
whereas the relay network is known to be an efficient solution
to extend coverage, improve reliability for data transmission
through information transmission/reception over short dis-
tances, and reduce transmit power compared with direct trans-
mission between the source and destination [9]. The benefits
of MIMO relaying networks have been well confirmed in the
literature. In particular, [10] revealed the optimal diversity-
multiplexing tradeoff (DMT) for a multihop MIMO decode-
and-forward (DF) relay network with either full-duplex (FD)
or half-duplex (HD) relays. Subsequently, the performance
of multihop MIMO DF relay networks employing transmit
antenna selection (TAS) and selection-combining (SC) tech-
niques was shown to be effective in terms of the outage
probability (OP), ergodic capacity (EC), symbol error rate
(SER), and bit error rate (BER) [11]. Further, the authors
in [12] have studied the TAS scheme for the dual-hop
MIMO system to achieve ultra-reliability (UL). Meanwhile,
in [13], Yilmaz et al. demonstrated the advantages of multihop
MIMO DF relay networks in terms of the OP, EC, and
SER when the TAS, maximum-ratio transmission (MRT), and
orthogonal space-time block coding (OSTBC) techniques were
utilized as transmit diversity schemes, whereas the SC and
maximum-ratio combining (MRC) were employed for the
receive diversity techniques. In [14], the gains of a multihop
amplify-and-forward (AF) relay network utilizing TAS/MRC
were presented, where the OP and SER metrics, as well as
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4550 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 21, NO. 6, JUNE 2022
the simplified optimal power allocation (PA), relay location
(RL), and joint optimizations, were investigated. In [15], the
MRT/MRC-based MIMO with either the dual-hop AF or DF
scheme was analyzed in terms of the secrecy OP (SOP) and
connection OP (COP). Meanwhile, [16] compared the perfor-
mance of the multihop MIMO network for various diversity
schemes, e.g., TAS/MRC, TAS with adaptive receive inter-
ference cancellation (ARIC), and TAS with dominant receive
interference cancellation (DRIC). In addition, the simplified
PA, RL, and joint optimizations were investigated in [16].
In [17], the performance gain of the TAS/MRC-based MIMO
AF relay network was proven in terms of SER.
After the pioneering work in [7], SPC has recently
received much attention, especially in state-of-the-art inte-
gration frameworks between SPC and MIMO. For example,
in [18], Durisi et al. considered a MIMO system for SPC
and revealed that the traditional infinite-BL metrics would
no longer accurately provide estimations for the maximum
coding rate in the short-packet-size scenario, which indicates
the essential need for the FBL regime when considering SPC.
Meanwhile, the investigation in [19] proved that the time-
division multiple access (TDMA) is more affordable to achieve
better performance for UL on the end-to-end (e2e) block
error rate (BLER) than zero-forcing (ZF) beamforming in
MIMO SPC networks. In [20] and [21], massive MIMO links
supporting uRLLC transmissions were successfully leveraged
by SPC, where the received signal-to-noise (SNR) per user and
BLER were evaluated, respectively. Meanwhile, [22] proposed
a status-update process for the MIMO SPC system to facilitate
low-latency (LL), where the age-of-information (AoI) metric,
also known as the freshness of the status information, was
evaluated. In addition, [23] and [24] studied the physical-
layer secrecy throughput (TP) in an uplink massive multi-
user MIMO system using SPC for HD and FD, respectively.
The authors of [25] investigated TAS/MRC/SC-based MIMO
systems in SPC along with NOMA support to achieve the
benefits of connectivity, spectral efficiency, UL, and LL.
Furthermore, [26] proposed a blind receiver for the MIMO
network using SPC, which does not need a pilot and can solve
the interblock interference.
From the relaying perspective with SPC, Gu et al. [27]
compared the performance between the FD and HD relaying
under the dual-hop single-input single-output (SISO) SPC
network, where both the UL and LL constraints for uRLLCs
can be satisfied via their proposed scheme. In [28] and [29], the
fundamental works on the dual-hop SISO SPC performance
were investigated, which can be potentially extended to the
MIMO system. The authors of [30] and [31] investigated
the decoding error rate (DER) performance, optimal BL, and
optimal RL for the DF SISO unmanned aerial vehicle (UAV)
relaying SPC network. Li et al. [32] demonstrated that e2e
fountain coding is more suitable for SISO multihop SPCs than
random linear network coding. Meanwhile, Makki et al. [33]
analyzed the e2e latency and TP of SISO multihop SPCs,
where either AF or DF relaying was employed at each
relay node. In [34], a Nelder–Mead simplex method was
proposed to solve the non-convex and nonlinear problems
of the SISO dual-hop SPC network with the reconfigurable
intelligent surface (RIS)-assisted UAV. In addition, in [35],
an incremental relaying strategy was invoked on the dual-hop
SISO SPC network to achieve UL.
Most of the aforementioned works extensively studied
the individual systems, e.g., MIMO relaying, MIMO SPC,
and relaying SPC, as summarized in Table I, whereas
a comprehensive integrated framework of these schemes,
i.e., multihop MIMO SPC, received less attention. Further-
more, some of these systems do not simultaneously sat-
isfy both UR and LL for the uRLLC requirements. This
paper first investigates the multihop MIMO relay network
gains along with SPC to support both UR and LL com-
munications. Our main contributions are summarized as
follows:
•To exploit the full diversity of the MIMO system in
the SPC network, we consider TAS/MRC, TAS/SC, and
MRT/MRC transmission schemes, for which the cumu-
lative density function (CDF) expressions of the output
SNR at the receive side are derived for quasi-static
Rayleigh fading channels. Based on the CDF of the
output SNR, the e2e BLER performance is analyzed
in the FBL regime. The e2e latency and TPs are also
investigated for TAS/MRC, TAS/SC, and MRT/MRC
schemes.
•Based on the asymptotic analysis in the high-SNR region,
the e2e BLERs and diversity orders for TAS/MRC,
TAS/SC, and MRT/MRC schemes are compared, which
reveals that although the diversity orders for TAS/MRC,
TAS/SC, and MRT/MRC are identical, the e2e BLER
for MRT/MRC is the lowest compared to that of the
TAS/MRC and TAS/SC schemes at high SNRs, and the
corresponding performance gap and SNR gap are derived
analytically.
•We solve the optimization problems with the asymptotic
e2e BLER objective function, which is regarded as a
simplified optimization. In particular, to minimize the
asymptotic e2e BLER, we study the PA optimization
problem for a given relay placement and the RL opti-
mization for a fixed transmit power. In addition, the joint
optimization of PA and RL is also investigated.
•Our theoretical analyses of TAS/MRC, TAS/SC, and
MRT/MRC schemes in the SPC network are verified
by Monte Carlo simulations, which show that the e2e
performance obtained via theoretical analyses agree with
that via simulations.
•Our results confirm that the values of the BL and number
of relays should be considered to balance the uRLLC
requirements. Although we consider simplified optimiza-
tion problems with the asymptotic e2e BLER objective
functions in the high-SNR regime, the e2e BLER per-
formance is significantly improved for the TAS/MRC,
TAS/SC, and MRT/MRC schemes over the entire SNR
range.
The remainder of this paper is organized as follows.
In Section II, we introduce the system models. In Section III,
we derive the CDF expressions of the output received SNR for
the TAS/MRC, TAS/SC, and MRT/MRC schemes. Section IV
presents the closed-form expressions of the approximated e2e
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TU AND LEE: PERFORMANCE ANALYSIS AND OPTIMIZATION OF MULTIHOP MIMO RELAY NETWORKS 4551
TAB L E I
COMPARISON OF THE PROPOSED AND EXISTING WORKS
BLERs, latency, and TP for all diversity schemes. Section V
investigates the asymptotic approaches to gain insights on
the characteristics of the considered systems at high SNRs,
from which the simplified optimizations, including PA, RL,
and joint optimizations, are also analyzed. Subsequently,
in Section VI, the Monte Carlo simulation results are shown
to validate the correctness of our analysis. A brief summary
is presented in Section VII.
II. SYSTEM MODEL
We consider a multihop network under the SPC scheme with
Kselective DF (SDF) relays named R1,R
2,...,R
K,which
assist the communication between one source terminal (R0)
and one destination (RK+1 )over Rayleigh fading channels.
We assume that there is no direct link between the source and
destination. The MIMO system under consideration consists
of NTtransmit antennas and NRreceive antennas, which
are installed in each relay node, whereas the source and
destination nodes are equipped with only NTtransmit and NR
receive antennas, respectively. When the ith transmit antenna
of the node Rk−1is chosen for transmission by the TAS
scheme, the received signal at the jth antenna of the node
Rkcan be represented as
y(i,j)
k=Pk−1h(i,j)
kx+n(i,j)
k,(1)
where Pk−1is the transmit power of the node Rk−1,h(i,j)
kis
the fading coefficient from the ith transmit antenna to the jth
receive antenna at the kth hop (k=1,K+1,i= 1,N
T,and
j=1,N
R).1xis a scalar complex baseband transmitted signal
with zero mean and unit variance [36], i.e., E|x|2=1,
and n(i,j)
kis a zero-mean complex Gaussian noise signal with
variance N0.
From (1), the instantaneous received SNR at the jth antenna
of Rkfor the signal transmitted from the ith transmit antenna
of Rk−1can be expressed as γ(i,j)
k=Pk−1
N0h(i,j)
k
2. Under
the assumption that the channel coefficients in each hop are
independent and identically distributed (i.i.d.), the average
SNRs for all links are the same, i.e., ¯γ(i,j)
k=¯γkfor all i
and j,where
¯γ(i,j)
k=Pk−1
N0
Eh(i,j)
k
2.(2)
We note that γ(i,j)
kfollows the i.i.d. exponential distribution
with the mean of ¯γk, thus the CDF of γ(i,j )
kis Fγ(i,j)
k
(γ)=
1−exp −γ
¯γk.
1The value of k= 1,K +1,i= 1,N
T,andj= 1,N
Rwill be used
throughout this paper if we do not specifically mention.
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4552 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 21, NO. 6, JUNE 2022
III. TRANSMIT AND RECEIVE DIVERSITY SCHEMES
In this section, we consider three scenarios, including
TAS/MRC, TAS/SC, and MRT/MRC, to exploit the transmit
and receive diversity of the MIMO system. We note that the
TAS technique at the transmit side of each hop can achieve a
significant reduction in power consumption and hardware cost
because, by selecting only one optimal transmit antenna to
send data, only a single radio-frequency (RF) chain is used.
It is assumed that the receiver at each hop has the perfect
channel-state information (CSI). At the transmit side, based
on feedback information to the transmitter from the receiver,
known as partial CSI feedback [37]–[39], the index of the
best transmit antenna, which maximizes the received SNR,
can be determined. Eventually, the transmitter employs only a
single optimal antenna out of NTantennas to transmit data.
However, there is a tradeoff among the reliable performance,
complexity, power consumption, and hardware cost when
comparing TAS to MRT. The MRT scheme sends the linearly
precoded transmit signals to achieve spatial diversity on the
transmit side, leading to the maximum output SNR [40].
Meanwhile, the MRC and SC are considered at the receive
side to achieve receive diversity at each hop. The detailed
analysis and comparison of these two schemes in conjunction
with the TAS approach and the well-known optimal diversity
scheme, i.e., MRT/MRC, will be shown in this section.
A. TAS/MRC
To achieve the spatial diversity gain at the receive side,
the MRC scheme can be used [37], [38], where the received
signals from each channel are coherently combined. Therefore,
the output SNR of the MRC combiner increases with the
number of diversity branches. When the ith transmit antenna is
chosen for transmission at the kth hop, the instantaneous SNR
of the MRC output after combining all the received signals can
be expressed as γMRC(i)
k=
NR
j=1
γ(i,j)
k.
According to the aforementioned TAS principle, when we
apply TAS and MRC simultaneously for each hop, the instan-
taneous SNR at the kth hop can be written as
γTAS /MRC
k=max
i=1,2,...,NT
γMRC(i)
k=max
i=1,2,...,NT
NR
j=1
γ(i,j)
k.(3)
Because γ(i,j)
kfollows an i.i.d exponential distribution,
NR
j=1
γ(i,j)
kfollows the Chi-square distribution with a mean
of NR¯γkand variance of 2NR¯γk[41]. Then, the
CDF of γMRC(i)
kcan be expressed as FγMRC(i)
k
(γ)=
1−exp −γ
¯γkNR−1
n=0
1
n!γ
¯γkn. Eventually, the CDF of
γTAS /MRC
kbecomes
FγTAS /MRC
k
(γ)=
NT
i=1
FγMRC(i)
k
(γ)
=1−exp −γ
¯γkNR−1
n=0
1
n!γ
¯γkn
NT
.(4)
B. TAS/SC
When the SC scheme is employed at the receive side of
each hop, the link with the highest received SNR is selected to
perform the detection process [42], [43]. Once this approach
is leveraged along with the TAS scheme, a single transmit
antenna out of NTantennas and a single receive antenna out
of NRantennas are jointly chosen. Therefore, we not only
achieve transmit/receive diversity gains, but also further reduce
the number of required RF chains, which leads to lower power
consumption.
In this scheme, the output SNR can be expressed as
γTAS /SC
k=max
i=1,2,...,NT
j=1,2,...,NR
γ(i,j)
k.TheCDFofγTAS /SC
kcan be
obtained easily as
FγTAS /SC
k
(γ)=
NT
i=1
NR
j=1
Fγ(i,j)
k
(γ)
=1−exp −γ
¯γkNTNR
.(5)
C. MRT/MRC
Similar to MRC, the MRT scheme is applied on the transmit
side to achieve the optimal transmit diversity gain. When MRT
and MRC are utilized simultaneosly, the maximum output
SNR at the kth hop is determined as
γMRT/MRC
k=
NR
j=1
NT
i=1
γ(i,j)
k.(6)
Apparently, γMRT/MRC
kfollows the Chi-square distribution
with a mean of NTNR¯γkand variance of 2NTNR¯γk[41].
Hence, the CDF for γMRT/MRC
kis calculated as
FγMRT/MRC
k
(γ)=1−exp −γ
¯γkNTNR−1
n=0
1
n!γ
¯γkn
.(7)
IV. PERFORMANCE ANALYSIS
To evaluate the system performance of SPC, we derive the
closed-form expression for the e2e BLER. The source encodes
Tinformation bits into a BL of βCUs, where the signals are
transmitted from R0to RK+1 with the help of R1,R
2,...R
K
via quasi-static fading channels [44]. For quasi-static fading
channels, we assume that the channel fading coefficients are
random, but remain constant during each transmission block
and change independently at other blocks. The coding rate of
the considered system for each time slot is given by R=T/β.
In contrast, in the context of the FBL scheme with the SPC,
the BL employed should be minimized, but it still needs to
be longer than 100 CUs [45]. In this scenario, the maximum
coding rate can be approximately expressed as [7]
R≈CγX
k−VγX
k/βQ−1εX
k,(8)
where X∈{TAS/MRC,TAS /SC,MRT/MRC}denotes
one type of the diversity transmission schemes, εX
k
represents the instantaneous BLER at the kth hop,
CγX
kΔ
=log
21+γX
kis the channel capacity,
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TU AND LEE: PERFORMANCE ANALYSIS AND OPTIMIZATION OF MULTIHOP MIMO RELAY NETWORKS 4553
VγX
kΔ
=1−1
(1+γX
k)2(log2e)2denotes the channel
dispersion measuring the stochastic variability of the channel
relative to a deterministic channel for the same capacity [7],
and Q−1(·)is the inverse function of the Q-function with
Q(z)= 1
√2π∞
zexp −u2
2du.
From (8), the average BLER at each hop can be written as
¯εX
k≈E⎧
⎨
⎩
Q⎛
⎝CγX
k−R
VγX
k/β ⎞
⎠⎫
⎬
⎭
=
∞
0
Q⎛
⎝CγX
k−R
VγX
k/β ⎞
⎠fγX
k(γ)dγ, (9)
where fγX
k(·)denotes the probability density function of γX
k.
Because of the complicated form of Q C(γX
k)−R
V(γX
k)/β !in (9),
the closed-form expression for the average BLER at each
hop cannot be calculated directly. Fortunately, motivated by
the work in [45, eq. (14)], an approximation approach of
the Q-function can be leveraged to solve the problem in (9).
In particular, we use the approximation Q C(γX
k)−R
V(γX
k)/β !≈
ΨγX
k, with ΨγX
krepresented as
ΨγX
k≈⎧
⎪
⎨
⎪
⎩
1,γ
X
k≤ϕL,
0.5−ξβγX
k−τ,ϕ
L<γ
X
k<ϕ
H,
0,γ
X
k≥ϕH,
(10)
where ξ=#2π22R−1$−1/2,τ=2
R−1,ϕH=τ+
1%2ξ√β,andϕL=τ−1%2ξ√β.
With this feasible approximation, the average BLER at each
hop can be written as
¯εX
k≈
∞
0
ΨγX
kfγX
kγX
kdγX
k
(a)
=ξβ
ϕH
ϕL
FγX
k(γ)dγ, (11)
where step (a)is conducted by the partial integration method.
By inserting (4) into (11), the average BLER at each hop
for the TAS/MRC scheme can be written as
¯εTAS /MRC
k
(b)
=1+ξβ
ϕH
ϕL
NT
m=1 NT
m(−1)mexp −mγ
¯γk·J
1dγ
(12)
where step (b)is achieved via Newton’s binomial theorem and
J1=&NR−1
n=0
1
n!γ
¯γkn'm
. To calculate J1in (12), we use [46,
eq. (16)]; as a result, (12) becomes
¯εTAS /MRC
k≈1+ξβ
NT
m=1 NT
m(−1)m
·
NR−1
j1=0
NR−1
j2=0
...
NR−1
jm=0 m
t=1
1
jt!!1
¯γkS·J
2,
(13)
where S=
m
t=1
jtand J2=
ϕH
ϕL
exp −mγ
¯γkγSdγ. By [47,
eq. (3.351.1)], the integration J2in (13) is given by
J2=m
¯γk−S−1ΥS+1,mϕH
¯γk−ΥS+1,mϕL
¯γk,
(14)
where Υ(α, z)=
z
0
e−ttα−1dt denotes the lower incomplete
gamma function. Subsequently, by substituting (14) into (13),
the closed-form expression of the average BLER at each hop
for the TAS/MRC scheme is given by
¯εTAS /MRC
k≈1+ξβ
NT
m=1 NT
m(−1)m
·
NR−1
j1=0
NR−1
j2=0
...
NR−1
jm=0 m
t=1
1
jt!!m−S−1¯γk
·ΥS+1,mϕH
¯γk−ΥS+1,mϕL
¯γk.
(15)
By inserting (5) into (11), the average BLER at each hop
for the TAS/SC scheme can be written as
¯εTAS /SC
k
(b)
≈1+ξβ
NTNR
m=1 NTNR
m(−1)m+1 ¯γk
m
·exp −m
¯γk
ϕH−exp −m
¯γk
ϕL,(16)
where step (b)is based on the same method in (12).
By substituting (7) into (11), the average BLER at each hop
for MRT/MRC is given by
¯εMRT/MRC
k
(c)
≈1−ξβ
NTNR−1
n=0
¯γk
n!
·Υn+1,ϕH
¯γk−Υn+1,ϕL
¯γk,(17)
where we utilize [47, eq. (3.351.1)] in step (c).
For the given average BLERs at each hop of all diversity
schemes in (15)–(17), we propose the following proposition
to calculate the e2e BLERs via the SDF principle.
Proposition 1: Based on the SDF mechanism [48], only
when a relay decodes the received data correctly, it forwards
the data to the next hop. If the incorrect decoding occurs
at a relay node, the BLERs for all later nodes become one,
i.e., ¯εX
n=1(∀n>k),wherekis the index of the relay that
performs erroneous decoding. Therefore, the e2e BLER for the
multihop SDF network is given by
¯εX
e2e ≈¯εX
1+1−¯εX
1¯εX
2+···+1−¯εX
1...1−¯εX
K¯εX
K+1
=¯εX
1+
K+1
k=2 ¯εX
k·
k
m=2 1−¯εX
m−1!.(18)
Proposition 2 (e2e Latency and TP): Let Fbe the feed-
back delay in each hop, measured in CUs, and D(β)denote
the delay for decoding a packet with the BL β[49]. In prac-
tice, D(β)depends on the decoding scheme, the number of
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4554 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 21, NO. 6, JUNE 2022
iterations in the decoder, etc. [49]. We consider the e2e latency
in the retransmission scheme, where the retransmission contin-
ues until the packet is decoded correctly or the system reaches
the maximum number of retransmission times, denoted by L.
As a result, the e2e packet transmission latency (measured in
CUs) is expressed as
LX
e2e =1−¯εX
e2e
(1−¯εX
e2eL+1)*L
r=0 ¯εX
e2errT X
F+TS+,(19)
where TSand TFrepresent the average latency caused by
e2e decoding success and failure, respectively, given by TS=
(K+1)(β+D(β)) and
TX
F=(β+D(β)+F) ¯εX
1+
K+1
k=2
k¯εX
k·
k
m=2 1−¯εX
m−1!.
(20)
Furthermore, the e2e TP is defined as the ratio of the num-
ber of information bits successfully received at the destination
to the average duration time, which is measured in bits per
CU (BPCU). As a result, the e2e TP for the considered system
is given by
δX
e2e =T(1−¯εX
e2eL+1)
LX
e2e (1−¯εX
e2eL+1
)+(L+1)TX
F¯εX
e2eL+1 .(21)
V. A SYMPTOTIC ANALYSIS AND SIMPLIFIED
OPTIMIZATION
A. Asymptotic Analysis
To obtain more insights into the influences of system
parameters on system performance, we present the asymp-
totic analysis in this subsection. In particular, the high-SNR
regime is considered to analyze the e2e BLER for TAS/MRC,
TAS/SC, and MRT/MRC schemes, from which qualitative
conclusions about the diversity orders, e2e BLER loss, and
SNR gap are made.
We define ¯γ=PS/N0as an average SNR and assume that
we allocate power for the source and relay nodes equally, i.e.,
Pk−1=PS/(K+1),wherePSis the total transmit power
of the system. It is noted that ¯γk=Pk−1
N0
Eh(i,j)
k
2=ck¯γ,
where ck=Eh(i,j)
k
2/(K+1). Therefore, in the high-
SNR regime with ¯γ→∞,wehave¯γk→∞.
In the high-SNR regime, we use [47, eq. (8.352.6)] and
[50, eq. (14)] for (11). The asymptotic BLER at each hop for
TAS/MRC can be given by
˜ε∞TAS /MRC
k≈ξβ
ϕH
ϕL
1
(NR!)NTγ
¯γkNTNR
dγ
=
ξ√β(ϕNTNR+1
H−ϕNTNR+1
L)
(NR!)NT¯γNTNR
k(NTNR+1).(22)
For TAS/SC, when ¯γk→∞, we utilize 1−exp −γ
¯γk∼γ
¯γk
for (5). The asymptotic BLER at each hop for the TAS/SC
scheme is expressed as
˜ε∞TAS /SC
k≈ξβ
ϕH
ϕLγ
¯γkNTNR
dγ
=
ξ√β(ϕNTNR+1
H−ϕNTNR+1
L)
¯γNTNR
k(NTNR+1) .(23)
Similar to that of TAS/MRC, the asymptotic BLER at each
hop for MRT/MRC is obtained as
˜ε∞MRT/MRC
k≈ξβ
ϕH
ϕL
1
(NTNR)!γ
¯γkNTNR
dγ
=
ξ√β(ϕNTNR+1
H−ϕNTNR+1
L)
(NTNR)!¯γNTNR
k(NTNR+1).(24)
Proposition 3: In the same manner as Proposition 1,
we invoke (18) for the average BLER at each hop in the high-
SNR regime such that given in (22)–(24).Thevalueof˜ε∞X
k
is very small for ¯γk→∞,i.e.,˜ε∞X
k1. Therefore, the
asymptotic e2e BLER can be expressed approximately as
˜ε∞X
e2e =˜ε∞X
1+1−˜ε∞X
1˜ε∞X
2+···
+1−˜ε∞X
1...1−˜ε∞X
K˜ε∞X
K+1 ≈
K+1
k=1
˜ε∞X
k.(25)
Proposition 4 (e2e Diversity Order, BLER Loss, and SNR
Gap): For the diversity order in the form of [10]
DX=−lim
¯γ→∞
log ˜ε∞X
e2e
log (¯γ),(26)
the maximum diversity orders for all TAS/MRC, TAS/SC, and
MRT/MRC schemes can be achieved, i.e., DX=NTNR.Fo r
the BLER loss, we consider the ratios among their asymptotic
e2e BLER. In particular,
˜ε∞MRT/MRC
e2e
˜ε∞TAS /MRC
e2e
=(NR!)NT
(NTNR)!,
˜ε∞MRT/MRC
e2e
˜ε∞TAS /SC
e2e
=1
(NTNR)!,
˜ε∞TAS /MRC
e2e
˜ε∞TAS /SC
e2e
=1
(NR!)NT.(27)
To gain further insight, we can rewrite the asymptotic
e2e BLER of Xas ˜ε∞X
e2e ≈(GX¯γ)−DX,where GX=
YX
K+1
k=1
1
cNTNR
k−1
DX
is the array gain, which represents
the SNR gain in the e2e BLER curve with respect to (w.r.t.)
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TU AND LEE: PERFORMANCE ANALYSIS AND OPTIMIZATION OF MULTIHOP MIMO RELAY NETWORKS 4555
¯γ−DX.Here,YXis defined as
YX=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
ξ√βϕNTNR
+1
H−ϕNTNR+1
L
(NR!)NT(NTNR+1) ,
for X=TAS/MRC,
ξ√βϕNTNR
+1
H−ϕNTNR+1
L
(NTNR+1) ,
for X=TAS/SC,
ξ√β(ϕNTNR
+1
H−ϕNTNR+1
L)
(NTNR)! (NTNR+1) ,
for X=MRT/MRC.
(28)
Accordingly, the SNR gaps among the diversity schemes are
represented by the ratio of their array gains [51, eq. (16)],
given by
GMRT/MRC
GTAS /MRC
=&YTAS /MRC
YMRT/MRC '1
NRNT=(NTNR!) 1
NTNR
(NR!) 1
NR
,
(29)
GMRT/MRC
GTAS /SC
=&YTAS /SC
YMRT/MRC '1
NRNT=(NTNR!) 1
NTNR,
(30)
GTAS /MRC
GTAS /SC
=&YTAS /SC
YTAS /MRC '1
NRNT=(NR!) 1
NR.(31)
Remark 1: Although TAS/MRC, TAS/SC, and MRT/MRC
offer the same full diversity order, they have different array
gains, which leads to the e2e BLER loss and SNR gap,
as described in Proposition 4. In particular, the MRT/MRC
scheme provides the highest array gain, whereas the TAS/SC
provides the lowest. As a result, the e2e BLER of MRT/MRC
is reduced by factors of (NTNR)!
(NR!)NTand (NTNR)! compared
to that of the TAS/MRC and TAS/SC schemes, respectively;
meanwhile, the e2e BLER of TAS/MRC is reduced by a factor
of (NR!)NTcompared to that of TAS/SC. Accordingly, the SNR
gaps among these diversity schemes are given by (29)–(31).
B. Simplified Optimization Problems
In this subsection, we investigate the problems of PA
optimization with a fixed RL configuration, RL optimization
with a fixed PA configuration, and joint PA/RL optimization
via the Lagrangian multiplier method in order to minimize the
e2e BLER under the uRLLC consideration. For simplicity,
the asymptotic e2e BLERs are considered as the objective
functions [14], [16]. We note that our simplified consideration
is valid for the uRLLC requirements when the e2e BLER falls
below 10−5, which requires a high SNR. In this regime,we can
invoke the concept of Proposition 3 to obtain solutions to the
optimization problems. By inserting (22)–(24) into (25), the
asymptotic e2e BLER for the scheme Xis expressed as
˜εX
e2e ≈
K+1
k=1 YXN0dη
k
Pk−1NTNR
,(32)
where the simplified path-loss model [41] with the average
channel power gain Eh(i,j)
k
2=d−η
kis employed, dkis
the distance between two adjacent nodes, and ηdenotes the
path-loss exponent.
1) Optimal PA: In this part, we present how to optimally
allocate the transmit power to the source and relays to mini-
mize the e2e BLER objective function given in (32) subject to
(s.t.) the constraint of a total transmit power under the defined
RL configuration. The simplified optimization problem in this
context is expressed as
min
P˜εX
e2e (P)≈min
P
K+1
k=1 YXN0dη
k
Pk−1NTNR
s.t.
K+1
k=1
Pk−1=PS.(33)
Lemma 1: The optimization problem (33) is a convex prob-
lem. In such a scenario, we can find a single global optimal
solution P∗using the Lagrangian multiplier method.
Proof: See Appendix A.
Based on Lemma 1, we invoke the Lagrangian multiplier
method to determine P∗. The Lagrangian function of (33) is
given by
Φ1(P,ς
1)=˜εX
e2e (P)−ς1 K+1
k=1
Pk−1−PS!,(34)
where ς1=0is a Lagrangian multiplier constant. By con-
sidering the partial derivatives of Φ1(P,ς
1)w.r.t. Pk−1and
ς1equal to zero, we get
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
∂Φ1(P,ς
1)
∂Pk−1
=−YXNTNR(dη
kN0)NTNR
PNTNR+1
k−1−ς1=0,
∂Φ1(P,ς
1)
∂ς1
=
K+1
k=1
Pk−1−PS=0.
(35)
By solving (35), the optimal value of the transmit PA is
eventually obtained as
P∗
k−1=PS
1+
K+1
u=1
u=kdu
dk
ηNTNR
NTNR+1
,(36)
which is proved in Appendix B.
2) Optimal RL: In this part, we aim to optimize the relay
placement to minimize the e2e BLER under the constraint
of the normalized transmission distance, which is denoted
by D, and the given PA assumption. We consider that all
system nodes are located in a one-dimensional linear topology,
yielding K+1
k=1 dk=D. Then, the RL simplified optimization
problem can be formulated as
min
d˜εX
e2e (d)≈min
d
K+1
k=1 YXN0dη
k
Pk−1NTNR
s.t.
K+1
k=1
dk=D,
(37)
where dis a set of multiple variables of dk, i.e., d=
(d1,d
2,...,d
K+1). By the same proof in Lemma 1,the
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4556 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 21, NO. 6, JUNE 2022
problem (37) is determined as a convex optimization problem.
Accordingly, we use the Lagrangian multiplier method to
determine d∗. The Lagrangian cost function of (37) is given by
Φ2(d,ς
2)=˜εX
e2e (d)−ς2 K+1
k=1
dk−D!,(38)
where ς2=0is a Lagrangian multiplier constant.
Subsequently, by solving the set of equations
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
∂Φ2(d,ς
2)
∂dk
=YXN0
Pk−1NTNR
ηNTNRdηNTNR−1
k−ς2=0,
∂Φ2(d,ς
2)
∂ς2
=
K+1
k=1
dk−D=0,
(39)
we obtain the optimal values of the relay distances as
d∗
k=D
1+
K+1
u=1
u=kPu−1
Pk−1
NTNR
ηNTNR−1
.(40)
The proof of (40) follows the same footsteps as in Appendix B.
It is worth noting that, because of the linear topology
assumption, we can easily determine the optimal relay coor-
dination based on the derived optimal distances given in (40).
3) Joint Optimization: We consider a joint optimization
problem for PA and RL in this part. The problem under
consideration is the optimal configuration of both the PA
and RL strategies to minimize the e2e BLER s.t. the two
constraints of K+1
k=1 Pk−1=PSand K+1
k=1 dk=D.
In particular, the simplified optimization can be formulated
as
min
P,d˜εX
e2e (P,d)≈min
P,d
K+1
k=1 YXN0dη
k
Pk−1NTNR
s.t.
K+1
k=1
Pk−1=PSand
K+1
k=1
dk=D. (41)
The optimization problem (41) is convex, which is proved
in Appendix C. The Lagrangian function of (41), which
is employed to determine a single global optimal solution
(P∗,d∗),isgivenby
Φ3(P,d,ς
3,ς
4)=˜εX
e2e (P,d)−ς3 K+1
k=1
Pk−1−PS!
−ς4 K+1
k=1
dk−D!,(42)
where ς3=0 and ς4=0 are the Lagrangian multiplier con-
stants. A set of equations expressed as
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
∂
∂Pk−1
Φ3(P,d,ς
3,ς
4)=0,
∂
∂dk
Φ3(P,d,ς
3,ς
4)=0,
∂
∂ς3
Φ3(P,d,ς
3,ς
4)=0,
∂
∂ς4
Φ3(P,d,ς
3,ς
4)=0,
(43)
Fig. 1. e2e BLER versus the number of relays K,whereNT=NR=2,
T= 1024 bits, and β=128 CUs.
yields the solution in the form of
P∗
k−1=PS/(K+1).
d∗
k=D/ (K+1).(44)
The proof of (44) is provided in Appendix D.
Remark 2: From (36) and (40), it is revealed that the PA or
RL configuration that minimizes the e2e BLER is determined
by P∗
k−1or d∗
k, respectively. Meanwhile, joint optimization
of both the PA and RL configurations is achieved by equal
allocation for both PA and RL, as shown in (44).
VI. SIMULATION RESULTS AND DISCUSSION
We have performed Monte Carlo simulations to validate our
theoretical results and evaluate the e2e performance of the
TAS/MRT/MRC/SC-based MIMO multihop system in the SPC
network for various system parameters, e.g., the number of
relays, transmit power, number of antennas, and BL.
A. Performance Evaluation
As stated in Remark 2, we employ equal allocation for both
the PA and RL configurations, e.g., dk=D/ (K+1) and
Pk−1=PS/(K+1), to achieve the minimum e2e BLER in
this subsection. The normalized transmission distance D=1
and path-loss exponent η=3are assumed.
Fig. 1 presents the influence of the number of relays K
on the system performance with NT=NR=2,T=
1024 bits, and β= 128 CUs. The average SNR is set to 5 and
15 dB. From Fig. 1, we make various observations, including
(i) TAS/MRC, TAS/SC, and MRT/MRC performance compar-
ison, (ii) the influence of the number of relays K,and(iii)the
effect of the average SNR. First, it is clear that the MRT/MRC
scheme achieves the highest performance compared to that of
TAS/MRC and TAS/SC over the entire Kand SNR ranges.
However, it is worth noting that TAS/SC requires fewer
RF chains than TAS/MRC and MRT/MRC, which leads to
lowest power consumption and hardware complexity. From
the second perspective, the gain of multihop employment in
terms of e2e performance is clearly shown. The e2e BLER
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TU AND LEE: PERFORMANCE ANALYSIS AND OPTIMIZATION OF MULTIHOP MIMO RELAY NETWORKS 4557
Fig. 2. The comparison for various MIMO configurations, where K=3,
T=1024 bits, and β=128CUs.
dramatically drops when the number of relays Kis increased.
This property can be explained by the fact that when K
increases, dkdecreases, which reduces the BLER at each hop,
resulting in a reduction in e2e BLER. In Fig. 1, we also
observe the e2e BLER performance versus ¯γ. It is seen that
as ¯γincreases, the performance is improved.
In Fig. 2, we investigate the system performance versus the
average SNR for various MIMO configurations. The system
parameters of K=3,T= 1024 bits, and β= 128 CUs are
assumed. Fig. 2 shows that, as expected, better performance
is achieved by employing more antennas, owing to higher
diversity gains. It is also observed that for the two scenarios
of NT=3,N
R=4and NT=4,N
R=3,TAS/SCand
MRT/MRC have identical performance, whereas the perfor-
mance of the TAS/MRC scheme varies. This is because, from
asetofNT×NRtransmit/receive antenna combinations,
the TAS/SC scheme chooses only one link out of them and
MRT/MRC combines all signals from them; and the size of
this set is the same for both scenarios, which leads to the
same e2e BLER performance. In contrast, for the TAS/MRC
scheme, the MRC is employed at the receiver, and hence the
employment of more receive antennas is more beneficial than
that of more transmit antennas, which results in better perfor-
mance for (NT=3,N
R=4)than for (NT=4,N
R=3).
In Figs. 1 and 2, it is also worth noting that the simulation
results are in good agreement with all the analysis results,
which verifies our analytical correctness. Additionally, the
analysis and simulation results converge to the asymptotic
results at high SNRs in Fig. 2. It is noted that as the
SNR increases, the e2e BLER losses among the diversity
schemes (e.g., MRT/MRC–TAS/MRC, MRT/MRC–TAS/SC,
and TAS/MRC–TAS/SC) converge to (27); and the SNR
gaps among them converge to (29)–(31), as presented in
Proposition 4. For example, the e2e BLER losses at SNR =
26 dB and NT=NR=2for MRT/MRC–TAS/MRC,
MRT/MRC–TAS/SC, and TAS/MRC–TAS/SC are 5.87, 22.86,
and 3.89, respectively, which are close to their asymp-
totic values of (NTNR)!
(NR!)NT=6,(NTNR)! = 24,and
(NR!)NT=4, respectively. Similarly, the SNR gaps at
Fig. 3. The effect of NTand NRon the e2e BLER, where K=3,
¯γ=10 dB, T= 1024 bits, and β=128 CUs.
Fig. 4. e2e BLER versus β,whereK=3,T= 1024 bits, and ¯γ=0 dB.
e2e BLER =10
−5and NT=NR=2for MRT/MRC–
TAS/MRC, MRT/MRC–TAS/SC, and TAS/MRC–TAS/SC are
2.0, 3.5, and 1.5 dB, respectively, which are close to
10 log 4
√4!
√2! dB, 10 log 4
√4!dB, and 10 log √2!dB.
Fig. 3 illustrates the effect of the numbers of transmit and
receive antennas on the e2e BLER, which is presented in three-
dimensional space, where K=3,¯γ=10dB, T= 1024 bits,
and β= 128 CUs are assumed. In Fig. 3, it is clear that
the MRT/MRC scheme corresponding to the lowest layer
substantially outperforms the TAS/MRC and TAS/SC schemes
corresponding to the upper layers. Furthermore, it is observed
that the simulation results, which are presented by dots, match
the analysis planes, which verifies our theoretical analysis in
Section IV.
To observe the impact of the BL βon the system perfor-
mance, we plot the e2e BLER as a function of BL βfor
K=3,T= 1024 bits, and ¯γ=0dB in Fig. 4. Fig. 4
reveals that our analytical results match the simulation results
for the entire region of β. It also shows that the increase
in βleads to better system performance. We note that the
main purpose of SPC is to maintain the BL as short as
possible but it still employs more than 100 CUs to achieve
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4558 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 21, NO. 6, JUNE 2022
Fig. 5. The e2e latency and TP versus the decoding delay factor α,where
NT=NR=3,K=2,¯γ=10dB, T=1024bits, L=20,F=40CUs,
and β=128CUs.
both low latency and reliable communications. Considering
this tradeoff, the value of βis chosen conservatively to satisfy
both the minimum-latency constraint and BLER performance.
For example, let us assume NT=2,N
R=3,anaverage
SNR of 0dB, and quality of service with a BLER of 10−5.
In this scenario, βcan be chosen as β≈370,β≈460,
and β≈650 for the MRT/MRC, TAS/MRC, and TAS/SC
schemes, respectively, which implies that our analysis results
for e2e BLER, presented in Section IV, can be employed to
optimize the SPC system.
To evaluate the e2e latency and TP given in Proposition 2,
we consider a linear decoding delay profile D(β)=αβ,
where αis the constant decoding delay factor [49], which
depends on different decoding schemes. We assume NT=
NR=3,K=2,L=20,F=40CUs, and T= 1024 bits
for the evaluations in Figs. 5−7. In Fig. 5(a) and Fig. 5(b),
for various decoding delay factors, the e2e performance is
compared in terms of both the e2e latency and TP, where the
parameters of ¯γ=10dB, β= 128 CUs, and K=2are
set. In Fig. 5, it is observed that MRT/MRC achieves the best
performance in terms of both e2e latency and TP compared
to TAS/MRC and TAS/SC.
Fig. 6 depicts the e2e latency and TP of the considered
systems for various BLs β. Here, the results are obtained for
the scenario of ¯γ=−10 dB, α=2,andK=3.InFig.6,very
high e2e latency and very low e2e TP were attained for short
BLs. It is due to the fact that, with the short BL, the e2e BLER
is high, which leads to the more frequent restransmissions.
In contrast, when the BL is sufficiently long, the probability of
retransmitting the packet converges to zero. In this scenario,
increasing βleads to an increase in the e2e latency and a
decrease in the e2e TP. Therefore, choosing the optimal value
for βis worth considering. In Fig. 6(a) and Fig. 6(b), it can
be seen that for a given number of information bits T,β=
325,β= 450 CUs and β= 600 CUs are the optimal BL
values for the MRT/MRC, TAS/MRC, and TAS/SC schemes,
respectively, which minimize the e2e latency and maximize
the e2e TP.
Fig. 6. The e2e latency and TP versus the BL β,whereNT=NR=3,
K=3,¯γ=−10 dB, α=2,L=20,F=40CUs, and T= 1024 bits.
Fig. 7. The influence of the number of relays Kon the e2e latency and TP,
where NT=NR=3,¯γ=10dB, T=1024 bits, L=20,F=40CUs,
and β=128CUs.
In multihop relay networks for uRLLCs, the number of relay
nodes plays an important role. Fig. 7 shows the influence of
Kon the e2e latency and TP performance, where ¯γ=15dB,
β= 128 CUs, and α=0.5,2are assumed. In Fig. 7,
it is observed that the MRT/MRC scheme requires fewer K
than TAS/MRC and TAS/SC to achieve the minimum e2e
latency and maximum TP. In addition, increasing Kholds
the tradeoff between the reliability and latency. Intuitively,
a larger number of relays leads to shorter hops, i.e., shorter
distance between two relay nodes to improve the reliability,
but it can suffer from a longer transmission time via relays
and even e2e TP degradation. Nonetheless, under the tradeoff
consideration among e2e BLER, latency, and TP, we can
determine the optimal value of the number of relays. For
example, for the environment shown in Fig. 7, K=1,K=2,
and K=3are the optimal values to achieve the highest
e2e TP and lowest e2e latency for the TAS/MRC, TAS/SC,
and MRT/MRC schemes, respectively. For a CU duration of
3μs[5]andα=2, the e2e latencies for TAS/MRC, TAS/SC,
and MRT/MRC are 1376, 1856, and 898 CUs corresponding
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TU AND LEE: PERFORMANCE ANALYSIS AND OPTIMIZATION OF MULTIHOP MIMO RELAY NETWORKS 4559
Fig. 8. An illustration of (a) linear PA configurations and (b) linear RL
configurations, where K=4is assumed.
to 4.13, 5.57, and 2.69 ms, respectively, which satisfy the low-
latency constraint of uRLLCs (≤10 ms [5]).
B. Evaluation of Optimization Strategies
In this subsection, we compare the performance among
the optimized PA/RL configurations and nonoptimized
schemes. In particular, the optimal PA and RL configurations
of (36) and (40) are compared to various nonoptimal config-
urations, which are formed by the linear functions
Pk−1(k)=A(k−1) + PS
K+1−K
2A(45)
and
dk(k)=B(k−1) + D
K+1−K
2B,(46)
where Aand Bare the slopes of the linear functions. It is noted
that (45) and (46) satisfy the constraints K+1
k=1 Pk−1=PS
and K+1
k=1 dk=D, respectively. Furthermore, when Aand
Bare equal to zero, we obtain the uniform configurations
of Pk−1(k)= PS
K+1 and dk(k)= D
K+1 , respectively.
An illustration of various linear PA and RL configurations,
e.g., A1−A5and B1−B5, is shown in Fig. 8, where K=4is
assumed. In Fig. 8, the values of Pk−1and dkare the fractions
of PSand D, respectively. To evaluate the performance in
this subsection, the system parameters of T= 1024 bits,
β= 128 CUs, NT=NR=3antennas, and η=3are
employed.
First, the comparison between the PA optimization and
nonoptimization is shown in Fig. 9. In this scheme, we assume
that the distance of each hop is set to dk=k·D/
K+1
i=1
i,which
satisfies the constraint of K+1
k=1 dk=D. For the optimal
case, the PA in (36) is employed, whereas for the nonoptimal
case, A1−A
5are assumed. In Fig. 9, it is clear that the PA
optimization strategy achieves significantly better e2e perfor-
mance than the nonoptimal linear PA schemes in all diversity
systems over the entire range of the average SNR. Moreover,
substantial power savings of the PA optimization scheme are
observed. In particular, the optimal PA scheme approximately
Fig. 9. The performance comparison among various PA configurations for
(a) TAS/MRC, (b) TAS/SC, and (c) MRT/MRC schemes.
Fig. 10. The performance comparison among various RL configurations for
(a) TAS/MRC, (b) TAS/SC, and (c) MRT/MRC systems.
achieves SNR gains in the range between 1.5 dB and 9 dB at
an e2e BLER of 10−5w.r.t. cases A1−A
5for TAS/MRC,
TAS/SC, and MRT/MRC systems.
In Fig. 10, the optimal RL configuration of (40) is compared
to B1−B
5cases. The PA configuration is set to Pk−1=
(K+2−k)·PS/
K+1
i=1
i, which ensures that K+1
k=1
Pk−1=
PS. In Fig. 10, it is observed that the RL optimized e2e
performance is significantly higher than that of the B1−B5
configurations for the whole range of the average SNR. For the
e2e BLER of 10−5, the power savings of the RL optimization
for all diversity schemes are approximately 10 dB and 0.5 dB
compared to the worst case (i.e., B1) and best case (i.e., B4),
respectively.
Fig. 11 depicts the e2e BLERs for the combinations of
A1−A
5and B1−B
5configurations, which are illustrated
in three-dimensional space. Fig. 11 shows that the best e2e
performance is achieved when setting (A3,B3) for the PA and
RL configurations. We note that the (A3,B3) configuration is
a result of the joint optimization given in (44).
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4560 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 21, NO. 6, JUNE 2022
Fig. 11. e2e BLER for Auand Bvconfigurations, where ¯γ=15 dB.
VII. CONCLUSION
In this research, we have studied the MIMO SDF multihop
relay network under the SPC scenario, where the multihop
network can be useful in the context of limited transmit power
because the source and destination terminals are too far apart.
Under the consideration of the FBL regime and quasi-static
Rayleigh fading channels, the closed-form expressions for
the e2e BLERs of the TAS/MRC, TAS/SC, and MRT/MRC
schemes are obtained in approximated forms, from which their
asymptotic forms for high SNRs are derived. Based on the
asymptotic analysis, the qualitative conclusions, including the
e2e diversity order, e2e BLER loss, and SNR gap, are made.
Furthermore, the e2e latency and TPs are also analyzed for
the considered systems. Optimization strategies, namely, the
PA, RL, and joint PA/RL optimizations, are investigated in
the uRLLC context, which are referred to as the simplified
optimizations. The simulation results are shown to verify
the analysis results for the e2e performance. Our results
demonstrate the gains from the employment of a MIMO and
multihop relay network for uRLLC requirements using SPC.
They also show that the values of the BL and number of relays
should be addressed to balance the tradeoff between the UL
and LL constraints. In addition, although the individual and
joint optimizations of PA and RL are considered in the high-
SNR regime, the e2e BLER performance of each diversity
scheme is significantly improved over the entire range of the
average SNR.
APPENDIX A
PROOF OF LEMMA 1
Based on [52, Theorem A.2], we show that the objective
function of (33) is convex. In particular, we consider the
Hessian matrix
H(P)=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
∂2˜εX
e2e (P)
∂P2
0··· ∂2˜εX
e2e (P)
∂P0PK
.
.
.....
.
.
∂2˜εX
e2e (P)
∂PKP0··· ∂2˜εX
e2e (P)
∂P2
K
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,(A.1)
where ∂2˜εX
e2e(P)
∂P2
k−1=YX(dη
kN0)NTNRNTNR(NTNR+1)
PNTNR+2
k−1
>0and
∂2˜εX
e2e(P)
∂Pi−1Pj−1=0,∀i, j = 1,K +1,i =j. It is clear that the
matrix H(P)is a diagonal matrix with positive diagonal
elements, which yields that H(P)is positive definite, leading
to a strictly convex function of ˜εX
e2e (P). Furthermore, because
the constraint K+1
k=1
Pk−1=PSis affine, it forms a convex set.
Therefore, the optimization problem (33) is a convex problem,
which completes the proof.
APPENDIX B
PROOF OF (36)
The equations in (35) can be rewritten as
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
−YXNTNR(dη
kN0)NTNR
PNTNR+1
k−1−ς1=0,(Zk),
K+1
k=1
Pk−1−PS=0,(Z0).
(B.1)
By subtracting (Zk)from (Zu),whereu= 1,K +1 and
u=k, we obtain
−YXNTNR(dη
uN0)NTNR
PNTNR+1
u−1−ς1
− −YXNTNR(dη
kN0)NTNR
PNTNR+1
k−1−ς1!=0,(B.2)
which leads to
Pu−1=Pk−1du
dk
ηNTNR
NTNR+1
.(B.3)
By inserting (B.3) into (Z0), we obtain the desired result (36),
which completes the proof. In particular, we have
Pk−1+
K+1
u=1
u=k
Pk−1du
dk
ηNTNR
NTNR+1
−PS=0
⇔P∗
k−1=PS
1+
K+1
u=1
u=kdu
dk
ηNTNR
NTNR+1
.(B.4)
APPENDIX C
PROOF OF CONVEX OPTIMIZATION PROBLEM (41)
In the joint optimization context, the proof can be extended
from Lemma 1. Specifically, we have
∂2˜εX
e2e (P,d)
∂Pi−1Pj−1
=∂2˜εX
e2e (P,d)
∂didj
=0,
∀i, j =1,K+1,i=j,
(C.1)
∂2˜εX
e2e (P,d)
∂P2
k−1
=YX(dη
kN0)NTNRNTNR(NTNR+1)
PNTNR+2
k−1
>0,
(C.2)
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TU AND LEE: PERFORMANCE ANALYSIS AND OPTIMIZATION OF MULTIHOP MIMO RELAY NETWORKS 4561
and
∂2˜εX
e2e (P,d)
∂d2
k
=YXN0
Pk−1NTNR
ηNTNR(ηNTNR−1)
×dηNTNR−2
k>0.(C.3)
Therefore, the Hessian matrix of ˜εX
e2e (P,d)becomes a diag-
onal matrix with all positive elements, which forms
H(P,d)=blgdiag(E,F),(C.4)
where blgdiag (E,F)denotes a block-diagonal matrix with
matrices Eand Fon the diagonal, i.e., blgdiag (E,F)=
E0
K+1
0K+1 F,where0K+1 is the (K+1)×(K+1) zero
matrix,
E=diag∂2˜εX
e2e (P,d)
∂P2
0
,...,∂2˜εX
e2e (P,d)
∂P2
K,(C.5)
F=diag∂2˜εX
e2e (P,d)
∂d2
1
,...,∂2˜εX
e2e (P,d)
∂d2
K+1 .(C.6)
As a result, H(P,d)is positive-definite, which confirms the
strict convexity of ˜εX
e2e (P,d).Furthermore,K+1
k=1 Pk−1=
PSand K+1
k=1 dk=Dare convex sets; hence, (41) becomes
a convex optimization problem.
APPENDIX D
PROOF OF (44)
The equations in (43) are expressed as
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
−YXNTNR(dη
kN0)NTNR
PNTNR+1
k−1−ς3=0,(Zk),
YXN0
Pk−1NTNR
ηNTNRdηNTNR−1
k−ς4=0,(Fk),
K+1
k=1
Pk−1−PS=0,(Z0),
K+1
k=1
dk−D=0,(F0).
(D.1)
By the same analysis as in Appendix B, we consider
(Zu)−(Zk),whereu= 1,K+1 and u=k, and obtain
Pk−1=PS
1+
K+1
u=1
u=kdu
dk
ηNTNR
NTNR+1
.(D.2)
Subsequently, by considering (Fu)−(Fk),whereu=
1,K+1 and u=k, we obtain
dηNTNR−1
u
PNTNR
u−1
=dηNTNR−1
k
PNTNR
k−1
.(D.3)
Then, by inserting Pu−1=Pk−1du
dk
ηNTNR−1
NTNR,whichis
obtained by solving (D.3), into (Z0),wefind
Pk−1=PS
1+
K+1
u=1
u=kdu
dk
ηNTNR−1
NTNR
.(D.4)
From (D.2) and (D.4), we have
K+1
u=1
u=kdu
dk
ηNTNR
NTNR+1
=
K+1
u=1
u=kdu
dk
ηNTNR−1
NTNR.(D.5)
It is noted that for η=1+ 1
NTNR,wehave ηNTNR
NTNR+1 =
ηNTNR−1
NTNR. Fortunately, we have NTNR>1and 2≤η≤6,
which imply η=1+ 1
NTNR. Therefore, the solution to (D.5) is
given by du=dk, which indicates the equal RL configuration,
i.e., d∗
k=D/ (K+1). The proof of P∗
k−1=PS/(K+1)
follows the same footsteps as the proof of d∗
k=D/ (K+1).
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NgoHoangTureceived the B.S. degree in computer
networking and data communications from the Ho
Chi Minh City University of Transport (UT-HCMC),
Vietnam, in 2020. He is currently pursuing the
M.S. degree with the Department of Smart Energy
Systems, Seoul National University of Science and
Technology, South Korea. From January 2019 to
January 2020, he worked as an Assistant Researcher
with the Wireless Communication Laboratory, Posts
and Telecommunications Institute of Technology
(PTIT), Vietnam. From February 2020 to August
2020, he was a Lecturer with the Department of Computer Engineering, UT-
HCMC. His research interests include the cognitive radio, relay networks,
MIMO systems, short-packet communications, and 6G infrastructures.
Kyungchun Lee (Senior Member, IEEE) received
the B.S., M.S., and Ph.D. degrees in electrical
engineering from the Korea Advanced Institute of
Science and Technology (KAIST), Daejeon, in 2000,
2002, and 2007, respectively. From April 2007 to
June 2008, he was a Postdoctoral Researcher with
the University of Southampton, U.K. From July
2008 to August 2010, he was with Samsung Elec-
tronics, Suwon, South Korea. In 2017, he was a
Visiting Assistant Professor with North Carolina
State University, Raleigh, NC, USA. Since Septem-
ber 2010, he has been with the Seoul National University of Science and
Technology, South Korea. His research interests include wireless communi-
cations and applied machine learning. He received the Best Paper Awards
at the IEEE International Conference on Communications (ICC) and IEEE
Wireless Communications and Networking Conference (WCNC) in 2009 and
2020, respectively.
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