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Link-Layer Capacity of Downlink NOMA with
Generalized Selection Combining Receivers
Vaibhav Kumar∗, Barry Cardiff∗, Shankar Prakriya†, and Mark F. Flanagan∗
∗School of Electrical and Electronic Engineering, University College Dublin, Belfield, Dublin 4, Ireland
†Department of Electrical Engineering, Indian Institute of Technology – Delhi, New Delhi, India
Email: vaibhav.kumar@ucdconnect.ie, barry.cardiff@ucd.ie, shankar@ee.iitd.ac.in, mark.flanagan@ieee.org
Abstract—Non-orthogonal multiple access (NOMA) has
drawn tremendous attention, being a potential candidate for
the spectrum access technology for the fifth-generation (5G)
and beyond 5G (B5G) wireless communications standards.
Most research related to NOMA focuses on the system
performance from Shannon’s capacity perspective, which,
although a critical system design criterion, fails to quantify
the effect of delay constraints imposed by future wireless
applications. In this paper, we analyze the performance of
a single-input multiple-output (SIMO) two-user downlink
NOMA system, in terms of the link-layer achievable rate,
known as effective capacity (EC), which captures the per-
formance of the system under a delay-limited quality-of-
service (QoS) constraint. For signal combining at the receiver
side, we use generalized selection combining (GSC), which
bridges the performance gap between the two conventional
diversity combining schemes, namely selection combining
(SC) and maximal-ratio combining (MRC). We also derive
two approximate expressions for the EC of NOMA-GSC
which are accurate at low-SNR and at high-SNR, respectively.
The analysis reveals a tradeoff between the number of
implemented receiver radio-frequency (RF) chains and the
achieved performance, and can be used to determine the
appropriate number of paths to combine in a practical
receiver design.
Index Terms—non-orthogonal multiple access, effective
capacity, generalized selection combining.
I. Introduction
Worldwide efforts to enable 5G and B5G wireless
communications are well underway, including new spec-
trum allocation policies, flexible modulation techniques,
multiple-access schemes, standardization, and imple-
mentation. NOMA has received tremendous attention
as a design paradigm for the radio-access techniques of
future communications standards due to its inherent po-
tential for higher spectral efficiency, denser connectivity
and lower latency [1]. In contrast to the traditional orthog-
onal radio-access schemes, multiple users in NOMA are
served in the same resource block (time slot, subcarrier,
frequency band, or spreading code). It was shown in [2]
that when the channels from the source to each of the
multiple users are significantly different, NOMA can
achieve a higher spectral efficiency as compared to its
This publication has emanated from research conducted with the
financial support of Science Foundation Ireland (SFI) and is co-funded
under the European Regional Development Fund under Grant Number
13/RC/2077.
orthogonal multiple access (OMA) counterpart. In order
to further enhance the spectral efficiency, different ad-
vanced signal processing schemes have been suggested,
including cooperative NOMA [3], multi-antenna-assisted
architectures for NOMA [4] and multiple-input multiple-
output NOMA (MIMO-NOMA) [5]. Most existing re-
search related to the performance analysis of NOMA
systems has dealt primarily with the metrics of outage
probability, achievable rate and system throughput.
Although the classical (Shannon) ergodic capacity
has been extremely successful in communication sys-
tem design, it fails to explicitly characterize the delay-
constrained performance, which is one of the most cru-
cial system design parameters for 5G and B5G net-
works. To facilitate the investigation of wireless networks
under statistical QoS limitations (e.g., data rate, delay
and delay-violation probability), a link-layer analysis
tool, known as effective capacity, was first introduced
in [6]. Given a delay-violation probability, the EC defines
the maximum data arrival rate that can be supported
by a radio link. The EC analysis of a multiple-input
single-output (MISO) system in Nakagami-m, Rician and
generalized-Kfading was presented in [7]. Recently, the
EC analysis of a MISO system under the Fisher-Snedecor
Ffading was presented in [8].
The available literature on the performance analysis
of NOMA in terms of EC is relatively limited. In [9],
the achievable link-layer rate of a multi-user downlink
NOMA network under a per-user delay QoS requirement
was presented, where the users were divided into mul-
tiple NOMA pairs and OMA was applied for multiple
access among each such NOMA pair. It was shown that
in such a scenario, OMA outperforms NOMA in terms
of the total link-layer rate in the low signal-to-noise-ratio
(SNR) regime, while at high SNR, the NOMA system
achieves higher EC as compared to its OMA-based coun-
terpart. It was shown in [10] that for a two-user downlink
NOMA system, the optimal power control problem to
maximize the sum effective capacity is non-convex and
thus the notion of partial effective capacity was introduced
to derive a sub-optimal power allocation policy. For the
case of half/full-duplex two-user cooperative NOMA, in
order to maximize the minimum effective capacity of
the user-pair, a bisection-based optimal power allocation
scheme was proposed in [11]. Analysis of the delay-
violation probability and EC for a multi-user downlink
NOMA system using stochastic network calculus (SNC)
was presented in [12]. The analysis of effective secrecy rate
for a multi-user downlink NOMA system was recently
presented in [13].
In all of the aforementioned research related to the EC
analysis of NOMA, a single-input single-output (SISO)
system model was considered. In this paper, we consider
a single-input multiple-output (SIMO) downlink NOMA
system consisting of two users1. Although MRC is an
optimal receiver combining technique for a SIMO sys-
tem, it is highly susceptible to channel estimation error
for the paths having lower instantaneous received SNR.
On the other hand, SC uses the best path (in terms
of SNR) and hence fails to exploit the total diversity
offered by the other independent paths. A comparatively
more flexible diversity combining scheme, called GSC,
bridges this gap between SC and MRC, by adaptively
combining a subset of the strongest paths. Therefore,
GSC receivers are more robust to channel estimation
errors and also require fewer active RF chains at the
receiver, thereby reducing the overall cost of the system
implementation. A detailed moment generating function
(MGF)-based performance analysis of GSC in Rayleigh
fading channels, in terms of bit error rate (BER), symbol
error rate (SER) and outage probability, was presented
in [14]. The performance analysis of GSC in a downlink
NOMA system in terms of average achievable sum-rate
and outage probability was presented in [15].
Against this background, the main contributions in
this paper are listed below:
•We analyze the sum effective capacity of a two-
user downlink NOMA system with GSC receivers.
For the case of the strong user, we derive an exact
closed-form expression for the EC. On the other
hand, for the case of the weak user, since it is some-
what tedious to find an exact closed-form expression
for the EC in NOMA-GSC, we derive closed-form
expressions for two special cases, namely NOMA-
SC and NOMA-MRC. We also show that most of the
gain (in terms of sum EC) is achieved by combining
the strongest diversity paths, and that diminishing
returns are obtained as the number of combined
paths increases.
•We derive low-SNR as well as high-SNR approxi-
mations for the sum EC of NOMA-GSC. From the
analysis of the EC for the weak user, it can be
noted that no significant benefit of multiple receive
antennas is observed at the weak user in the high-
SNR regime. Also, the results indicate that the sum
EC grows exponentially with SNR in the low-SNR
1Note that a two-user downlink version of NOMA, called multiuser
superposition transmission (MUST), has been proposed for the Third
Generation Partnership Project Long Term Evolution Advanced (3GPP-
LTE-A) standard.
regime, and linearly with SNR in the high-SNR
regime.
•Using Jensen’s inequality, we derive a fundamental
upper-bound on the sum EC of NOMA-GSC which
is independent of the delay constraint. We show that
this bound is equal to the average achievable sum-
rate of NOMA-GSC and that the difference between
the achievable sum-rate and the sum EC increases
with increasing SNR and with an increase in the
delay exponent.
•For the purpose of comparison, we also derive a
closed-form expression for the sum EC of a two-
user downlink OMA-GSC system, and show that
NOMA-GSC outperforms OMA-GSC in terms of
sum EC for any number of combined paths. We
also show that as the delay constraint becomes more
strict, the performance difference between NOMA-
GSC and OMA-GSC becomes smaller.
II. System Model
Consider the two-user downlink NOMA system con-
sisting of a source Sequipped with a single transmit
antenna, and two users Us(the user with strong av-
erage channel conditions) and Uw(the user with weak
average channel conditions) equipped with Ns≥1and
Nw≥1receive antennas, respectively. We assume a
half-duplex communication protocol. The channel coeffi-
cients between Sand Usare assumed to be independent
and identically distributed (i.i.d.) according to CN(0,Ωs)
with mean-square value Ωs, while those between Sand
Uware assumed to be i.i.d. according to CN(0,Ωw)
with mean-square value equal to Ωw. We assume that
Ωw<Ωs, and that for each link perfect CSI is available
at the receiver side, whereas only the knowledge of Ωs
and Ωwis available at the transmitter side2.
The source transmits a power-domain multiplexed
symbol √asExs+√awExwto both users, where xi, i ∈
{s, w}denotes the information-bearing constellation
symbol intended for user Ui(here we assume that
E{|xi|2}= 1), Eis the energy budget at the source for
each (superposed) symbol, aiis the power allocation
coefficient with as< awand Piai= 1. Upon signal
reception, user Us(resp. Uw) adaptively selects a subset
of ns,1≤ns≤Ns(resp. nw,1≤nw≤Nw) antennas, out
of the total Ns(resp. Nw) antennas which provide the
strongest links (in terms of the received SNR), and then
applies MRC to combine the received signals. Therefore,
the signals received at user Ui(after applying GSC) is
given by
yi=hH
ihhipasExs+pawExw+zii,
2This is different from the two-user NOMA system presented in [9],
where (perfect) instantaneous CSI was assumed to be available at the
transmitter and therefore the users were ordered instantaneously (in
terms of strong and weak users). In our system, the users are ordered
depending on the statistical CSI. This reduces the signaling overhead
at the transmitter and the overall system implementation cost.
where hi= [hi,1hi,2. . . , hi,ni]T∈Cni×1contains the
nilargest-magnitude channel coefficients between Sand
Uiin decreasing order, i.e., |hi,1| ≥ |hi,2| ≥ ··· ≥ |hi,ni|,
and zi= [zi,1zi,2. . . , zi,ni]T∈Cni×1. Each element in zi
represents additive white Gaussian noise (AWGN) with
zero mean and variance σ2.
Both Usand Uwfirst decode xwconsidering the inter-
ference from xsas additional noise. User Usthen applies
successive interference cancellation (SIC) to decode the
intended symbol xs. Since the symbol xwneeds to be
decoded correctly at both users (at Uwas the intended
symbol and at Usfor SIC), while xsneeds to be correctly
decoded only at user Us, the received instantaneous
signal-to-interference-plus-noise ratio (SINR) and SNR
for the correct decoding of xwand xsare, respectively,
given by
γw=awgminρ
asgminρ+ 1 , γs=asgsρ,
where gi,hH
ihi,gmin ,min{gs, gw}and ρ,E/σ2.
Using a transformation of random variables and [14,
eqn. (16)], the probability density function (PDF) of gi
can be given by (1), shown at the top of this page.
III. Performance Analysis
In this section, we present a comprehensive perfor-
mance analysis of the two-user NOMA-GSC system in
terms of effective capacity. We also derive closed-form
approximations to the sum EC of NOMA-GSC, which are
valid in the low-SNR and high-SNR regimes respectively,
as well as an upper-bound on the sum EC. For a fair
comparison, we also present the sum EC analysis of an
OMA-GSC system.
A. Effective capacity of NOMA-GSC
The expression for EC of user Uiin NOMA-GSC can
be given by (c.f. [9])
Ei=−1
θT B ln [E{exp (−θT BRi)}],(2)
where θis the delay QoS exponent (denoting the asymp-
totic decay-rate of the buffer occupancy at the source,
defined as θ,−limx→∞ ln Pr{L>x}/x,Lbeing the
equilibrium queue-length of the buffer at the source),
Tis the length of each fading-block (this is assumed
to be same for all the wireless links and an integer
multiple of the symbol duration3), Bis the total available
bandwidth, and Riis the instantaneous achievable rate
for user Ui(defined as Ri= log2{1 + γi}). It is important
to note that θ→ ∞ represents a system with very
stringent delay constraint, while θ→0corresponds to
a system with no delay constraint. For the case when
θ→0, the effective capacity becomes equal to the
average achievable rate. Substituting the expression for
3We also assume that the symbol duration for both xsand xware
the same.
the instantaneous achievable rate into (2), the expression
for the effective capacity of user Uiin NOMA-GSC can
be given by
Ei=−1
νlog2hEγin(1 + γi)−νoi,(3)
where ν,θT B / ln 2. Therefore, for Us, we have
Es=−1
νlog2Z∞
0
(1 + x)−νfγs(x)dx.(4)
Substituting the expression for fgs(x)from (1) into (4), a
closed-form expression for Escan be given by (5), shown
on the next page, where G·,·
·,·(·)denotes Meijer’s G-
function, Γ(·,·)represents the upper-incomplete Gamma
function and Φi,l ,(1 + l
ni)/Ωi. The integration in I1
is solved using [16, eqns. (7), (10), (11), (21) and (22)],
the integration in I2(l)is solved using [17, eqn. (3.382-
4), p. 347], and the integration in I3(m)is solved similarly
to I1.
On the other hand, using (3), the EC of Uwcan be
given by
Ew=−1
νlog2"Z∞
01 + awρx
asρx + 1−ν
fgmin (x)dx#.(6)
The PDF of gmin can be given by fgmin(x) = fgw(x)[1 −
Fgs(x)] + fgs(x)[1 −Fgw(x)]. It can be noted that the
expression for the PDF of gmin is very complicated and
hence deriving an exact closed-form expression for Ew
is somewhat tedious. Therefore, we discuss two special
cases for the EC of Uw.
Case I (ns=nw=n= 1): In this case,
the NOMA-GSC system reduces to the NOMA-
SC system where the signal from a single path
(which has the highest instantaneous channel
gain) is selected at both Usand Uw. Defining
gmin,SC ,min{|hs,1|2,|hw,1|2}, using a transformation
of random variables, it can be shown that fgmin,SC(x) =
PNs
k=1 PNw
j=1(−1)k+jNs
kNw
jχk,j exp(−χk,j x), where
χk,j ,k
Ωs+j
Ωw. Therefore, replacing fgmin (x)in (6)
by fgmin,SC(x), a closed-form expression for the EC
of Uwin a NOMA-GSC system with n= 1 can be
given by (7), shown on the next page, where G·,·:·,·:·,·
·,·:·,·:·,·(·)
denotes the extended generalized bivariate Meijer’s G-
function (EGBMGF) and the integral is solved using [16,
eqns. (10), (11)] and [18, eqn. (9)].
Case II (ns=Ns,nw=Nw): In this case, the
NOMA-GSC system reduces to the NOMA-MRC system
where signals from all the diversity paths are combined.
Defining gmin,MRC ,mini∈{s,w}nPNi
n=1 |hi,n|2o, using
a transformation of random variables it can be shown
that fgmin,MRC (x) = xNs−1
Γ(Ns)ΩNs
s
exp(−χ1,1x)PNw−1
j=0 xj
j!Ωj
w+
xNw−1
Γ(Nw)ΩNw
w
exp(−χ1,1x)PNs−1
k=0 xk
k!Ωk
s. Therefore, replacing
fgmin (x)in (6) by fgmin,MRC(x), a closed-form expression
for the EC of Uwin a NOMA-GSC system with ns=Ns
fgi(x) = Ni
ni
xni−1exp −x
Ωi
Ωni
i(ni−1)! +1
Ωi
Ni−ni
X
l=1
(−1)ni+l−1Ns−ns
lni
lni−1exp −x
Ωi
×(exp −lx
niΩi−
ni−2
X
m=0
1
m!−lx
niΩim)#.(1)
Es=−1
νlog2"Ns
ns(I1
Ωns
s(ns−1)! +1
Ωs
Ns−ns
X
l=1
(−1)ns+l−1Ns−ns
lns
lns
−1
(I2(l)−
ns−2
X
m=0
I3(m)
m!−l
nsΩsm))#,
I1,Z∞
0
(1 + asρx)−νxns−1exp −x
Ωsdx=1
Γ(ν)(asρ)nsG2,1
1,21
Ωsasρ
1−ns
0, ν−ns,
I2(l),Z∞
0
(1 + asρx)−νexp −1
Ωs1 + l
nsxdx=Φν−1
s,l
(asρ)νexp Φs,l
asρΓ1−ν, Φs,l
asρ,
I3(m),Z∞
0
(1 + asρx)−νexp −x
Ωsxmdx=1
Γ(ν)(asρ)m+1 G2,1
1,21
Ωsasρ
−m
0, ν−m−1.
(5)
Ew,SC =−1
νlog2
1
Γ(ν)Γ(−ν)
Ns
X
k=1
Nw
X
j=1
(−1)k+jNs
kNw
jχk,j
ρG1,1:1,1:1,0
1,1:1,1:0,10
ν−1
1+ν
0
−
0as,χk,j
ρ
,(7)
and nw=Nwcan be given by (8), shown on the next
page.
The sum effective capacity for NOMA-GSC is then
obtained as Esum =Es+Ew.
B. Effective capacity of OMA-GSC
For a fair comparison, we also present a closed-form
analysis for the EC of OMA-GSC in this subsection.
Consider a time-division multiplexed system, where S
transmits √Exwand √Exsin the first and second time
slots, respectively. The instantaneous achievable rate for
Uiis given by ˆ
Ri= 0.5 log2(1 + ˆγi), where ˆγi,ρgi.
Therefore, similar to (3), the EC of Uifor OMA-GSC can
be given by
ˆ
Ei=−1
νlog2hEˆγin(1 + ˆγi)−ν/2oi
=−1
νlog2Z∞
0
(1 + ρx)−ν/2fgi(x)dx.(9)
Substituting the expression for fgi(x)from (1) into (9), a
closed-form expression for ˆ
Eiis given by (10), shown
on the next page, where the integrals in ˆ
Ii,1,ˆ
Ii,2(l)
and ˆ
Ii,3(m)are solved similar to I1,I2(l)and I3(m)
respectively. The sum effective capacity for OMA-GSC
is then obtained by ˆ
Esum =ˆ
Es+ˆ
Ew.
C. High-SNR approximation for the EC of NOMA-GSC
In this subsection, we present a high-SNR approxima-
tion for the EC of NOMA-GSC. For ρ1, it can be noted
from (3) that, for Us, we have
Es≈ −1
νlog2Egs(asρgs)−ν
= log2(asρ)−1
νlog2Z∞
0
x−νfgs(x)dx.
A closed-form expression for the high-SNR approxima-
tion of Esis given by (11), shown on the next page,
where the integral is solved using [17, eqn. (3.351-
3), p. 340] and the integral holds good for ν <
min{ns,1, m + 1}= 1. A similar limitation was encoun-
tered in the analysis in [7]. On the other hand, for the
case of weak user, a high-SNR approximation for the EC
of Uwcan be given by
Ew≈log21 + aw
as.(12)
It is interesting to note from (12) that at high SNR, the
effective capacity of Uwis independent of θand Nw. This
means that at high SNR, the EC of Uwbecomes (almost)
equal to the average achievable rate of Uwand no gain is
obtained (in terms of the EC of Uw) by having multiple
antennas at Uw. For ρ1, an approximation for the sum
EC can be obtained by adding (11) and (12).
D. Low-SNR approximation for the EC of NOMA-GSC
A low-SNR (ρ→0) approximation for the EC can be
given by (c.f. [7, eqn. (18)])
Ei=ρ˙
Ei+ 0.5ρ2¨
Ei+O(ρ2),(13)
Ew,MRC =−1
νlog2
1
Γ(Ns)ΩNs
s
Nw−1
X
j=0
ρ−(Ns+j)
j!Ωj
wG1,1:1,1:1,0
1,1:1,1:0,11−(Ns+j)
ν−(Ns+j)
1+ν
0
−
0as,χk,j
ρ
+1
Γ(Nw)ΩNw
w
Ns−1
X
k=0
ρ−(Nw+k)
k!Ωk
sG1,1:1,1:1,0
1,1:1,1:0,11−(Nw+k)
ν−(Nw+k)
1+ν
0
−
0as,χk,j
ρ#.(8)
ˆ
Ei=−1
νlog2"Ni
ni(ˆ
Ii,1
Ωni
i(ni−1)! +1
Ωi
Ni−ni
X
l=1
(−1)ni+l−1Ni−ni
lni
lni−1
(ˆ
Ii,2(l)−
ni−2
X
m=0
ˆ
Ii,3(m)
m!−l
niΩim))#,
ˆ
Ii,1,Z∞
0
(1 + ρgi)−ν/2xni−1exp −x
Ωidx=1
Γ(ν/2)ρniG2,1
1,21
Ωiρ
1−ni
0,ν
2−ni,
ˆ
Ii,2(l),Z∞
0
(1 + ρx)−ν/2exp (−Φi,l x) dx=Φ(ν/2)−1
i,l
ρν/2exp Φi,l
ρΓ1−ν
2,Φi,l
ρ,
ˆ
Ii,3(m),Z∞
0
(1 + ρx)−ν/2xmexp −x
Ωidx=1
Γ(ν/2)ρm+1 G2,1
1,21
Ωiρ
−m
0,ν
2−m−1.
(10)
Es≈log2(asρ)−1
νlog2"Ns
ns Γ(ns−ν)
Ων
sΓ(ns)+1
Ωs
Ns−ns
X
l=1
(−1)ns+l−1Ns−ns
lns
lns−1
×(Γ(1 −ν)Φν−1
s,l −
ns−2
X
m=0
Γ(m−ν+ 1)Ωm−ν+1
s
m!−l
nsΩsm)!#.(11)
where ˙
Eiand ¨
Eidenote the first and second order
derivatives of the EC in (3) with respect to ρand evalu-
ated at ρ= 0, and O(·)denotes the Landau symbol. Fol-
lowing a similar line of reasoning as in [19, Appendix I],
˙
Esand ¨
Esare given by
˙
Es= log2(e)asE{gs},
¨
Es= log2(e)a2
sν(E{gs})2−(ν+ 1)E{g2
s}.(14)
Using [17, eqn. (3.351-3), p. 340], closed-form expressions
for E{gs}and E{g2
s}are given by (15) and (16), respec-
tively, shown on the next page. Therefore, a closed-form
expression for the low-SNR approximation of Escan be
obtained using (13) – (16). Similarly, for Uw, we have
˙
Ew= log2(e)awE{gmin},
¨
Ew= log2(e)awhνaw(E{gmin})2
−{(ν+ 1)aw−2as}E{g2
min}
(17)
Since it is somewhat tedious to obtain closed-form ex-
pressions of E{gmin}and E{g2
min}, we present closed-
form analysis for two special cases.
Case I (ns=nw=n= 1): In this case, NOMA-GSC
reduces to NOMA-SC. Therefore, we have
E{gmin}=E{gmin,SC }=
Ns
X
k=1
Nw
X
j=1 Ns
kNw
j(−1)k+j
χk,j
,
E{g2
min}=E{g2
min,SC}=
Ns
X
k=1
Nw
X
j=1 Ns
kNw
j2(−1)k+j
χ2
k,j
.
(18)
Using (13), (17) and (18), we can obtain a closed-form
expression for the low-SNR approximation of Ewfor
ns=nw= 1.
Case II: (ns=Ns,nw=Nw): In this case NOMA-
GSC reduces to NOMA-MRC. The closed-form expres-
sions for E{gmin}and E{g2
min}are given by (19), shown
on the next page. Using (13), (17) and (19), we can obtain
a closed-form expression for the low-SNR approximation
of Ewfor ns=Nsand nw=Nw.
E. Upper-bound on the effective capacity of NOMA-GSC
In this subsection, we derive a fundamental upper-
bound on the EC of NOMA-GSC, using Jensen’s inequal-
ity. Using the fact that −log2(·)is a log-concave function,
from (3) it follows that
Ei≤ − 1
νEγilog2(1 + γi)−ν
=Eγi{log2(1 + γi)},˜
Ei.(20)
It is important to note that ˜
Eiis independent of the delay
exponent θand represents the average achievable rate of
Uiin NOMA-GSC. This bound is also consistent with the
fact that the EC of NOMA-GSC is always less than or
equal to the average achievable rate (the equality holds
for the case when θ→0). A closed-form expression for
an upper-bound on ˜
Esum =˜
Es+˜
Ew, which is very tight
in the mid-to-high SNR range can be obtained using [15,
eqns. (2), (4)].
E{gs}=Ns
ns"nsΩs+1
Ωs
Ns−ns
X
l=1
(−1)ns+l−1Ns−ns
lns
lns−1(Φ−2
s,l −
ns−2
X
m=0
(m+ 1)Ω2
s−l
nsm)#,(15)
E{g2
s}=Ns
ns"Γ(ns+ 2)Ωns+2
s
Ωns
s(ns−1)! +1
Ωs
Ns−ns
X
l=1
(−1)ns+l−1Ns−ns
lns
lns−1
×(2Ωs
Φs,l 3
−
ns−2
X
m=0
1
m!−l
nsΩsm
Γ(m+ 3)Ωm+3
s)#.(16)
E{gmin}=E{gmin,MRC}=
Nw−1
X
j=0
(Ns+j)!χ−(Ns+j+1)
1,1
Γ(Ns)ΩNs
sj!Ωj
w
+
Ns−1
X
k=0
(Nw+k)!χ−(Nw+k+1)
1,1
Γ(Nw)ΩNw
wk!Ωj
sk,
E{g2
min}=E{g2
min,MRC}=
Nw−1
X
j=0
(Ns+j+ 1)!χ−(Ns+j+2)
1,1
Γ(Ns)ΩNs
sj!Ωj
w
+
Ns−1
X
k=0
(Nw+k+ 1)!χ−(Nw+k+2)
1,1
Γ(Nw)ΩNw
wk!Ωj
sk.
(19)
10 20 30 40
4
6
8
10
12
14
10 20 30 40
4
6
8
10
12
14
10 20 30 40
5
10
15
10 20 30 40
5
10
15
Fig. 1. High-SNR approximation for the sum EC of NOMA-GSC.
Clockwise from the top-left: n=1 (SC), 2, 3, 4 (MRC).
IV. Results and Discussion
In this section, we present the numerical and analyt-
ical results for the EC of NOMA-GSC and OMA-GSC.
We consider a system where Ns=Nw=N= 4,
ns=nw=n,Ωs= 1,Ωw= 0.1,T= 0.01 ms, B= 100
kHz. As reported in [15], the optimal power allocation in
a NOMA-GSC system depends on the target data rate of
Ui. Considering the target data rate for both Usand Uw
to be 2 bps/Hz, it follows from [15] that a valid range for
asis given by 0< as<0.25. Note that throughout this
section, results for the sum EC are presented for the case
where the sum EC is optimized using a one-dimensional
search over as∈ {0.1,0.2,...,0.24}. The optimal value of
asthat maximizes the sum EC of NOMA-GSC is found to
be 0.24 (the maximum possible value considered in the
range 0< as<0.25). This occurs because most of the
EC in Esum is obtained by Es(as xsis decoded without
any inter-symbol interference and the links between S
and Usare comparatively stronger on average).
-20 -15 -10 -5
0
0.2
0.4
0.6
-20 -15 -10 -5
0
0.2
0.4
0.6
0.8
-20 -15 -10 -5
0
0.2
0.4
0.6
0.8
1
-20 -15 -10 -5
0
0.2
0.4
0.6
0.8
1
Fig. 2. Low-SNR approximation for the sum EC of NOMA-GSC.
Clockwise from the top-left: n=1 (SC), 2, 3, 4 (MRC).
Figs. 1 and 2 show the high-SNR and low-SNR approx-
imations for the sum EC of NOMA-GSC for θ= 0.5, re-
spectively. An excellent match can be noticed from both
the figures between the exact (numerically evaluated)
values of Esum and the analytically/semi-analytically
evaluated approximations (in semi-analytical evaluation,
Esis evaluated using the closed-form expression and
Ewis evaluated numerically). It can be noted from these
figures that there is a linear growth in the sum EC in
the high-SNR regime and an exponential growth in the
low-SNR regime.
Fig. 3 shows a comparison of the sum EC in NOMA-
GSC and OMA-GSC for different values of n. Markers
in the figure denote the numerically evaluated results.
The dashed lines (without markers) denote the closed-
form analytical results for the sum EC of OMA-GSC.
For the case of NOMA-GSC, the solid lines (without
markers) for n= 1 and n= 4 denote the closed-form
analytical results, whereas the solid lines (without mark-
10 15 20 25 30 35 40
2
4
6
8
10
12
14
16
Fig. 3. Comparison of EC for NOMA-GSC and
OMA-GSC with θ= 1.
10 15 20 25 30 35 40
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
= 2
= 1
Fig. 4. Difference between the ECs of NOMA-
GSC and OMA-GSC.
10-1 10 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
= 40 dB
= 5 dB
Fig. 5. Difference between average achievable
sum-rate and sum effective capacity.
ers) for n= 2 and n= 3 represent the semi-analytical
results. An excellent agreement between the numerical
and analytical/semi-analytical results confirms the cor-
rectness of the analysis.
It can be noted from the figure that NOMA-GSC
outperforms OMA-GSC for any value of n. More inter-
estingly, it can be observed that the gain in the sum EC
achieved by moving from nto n+ 1 combined paths
decreases with increasing n. These results are further
elaborated in Fig. 4. In the figure, markers denote the
numerically evaluated results, whereas the solid lines
(without markers) denote the analytical/semi-analytical
results. It can be noted from the figure that the difference
between Esum and ˆ
Esum increases in the low-to-mid SNR
regime, with increasing ρand also with increasing n.
This difference then saturates in the high-SNR range
(which is in line with the results in [9]) and most of the
gain is obtained by using only two strongest diversity
paths. As the value of θincreases, the figure shows a
decrease in the performance difference between NOMA
and OMA, which means that the performance of NOMA
and OMA both degrade severely under a stringent delay
requirement.
Fig. 5 shows the effect of the delay QoS constraint θ
on the difference between the average achievable sum-
rate and sum EC of NOMA-GSC (i.e., ˜
Esum −Esum) for
different values of n. First, it can be noted from the
figure that an increase in the value of θresults in a
very severe system performance degradation as the value
of ˜
Esum −Esum grows very rapidly for large θ. This
means that a system with strict delay constraints will
have a much lower sum EC as compared to the aver-
age achievable sum-rate. Furthermore, it can be noted
from the figure that with an increase in the value of
ρ, the difference between the average achievable rate
and the effective capacity increases. It is also interesting
to note that for small values of ρ, although increasing
the number of combined paths increases both ˜
Esum and
Esum, the difference between these two quantities also
increases. For large values of ρ, by combining more
diversity paths at the receiver, the difference between
˜
Esum and Esum decreases. Therefore, in order to achieve a
desired quality of service in terms of delay QoS exponent
and data rate, a higher transmit power is required than
that predicted by the traditional achievable rate analysis.
V. Conclusion
In this paper, we have presented the effective capac-
ity analysis of a two-user single-input multiple-output
downlink NOMA system with generalized selection
combining receivers. We derived closed-form expres-
sions for the sum EC of NOMA-GSC and OMA-GSC. We
also presented high-SNR and low-SNR approximations
for the sum EC of NOMA-GSC. Our results indicate
that the sum EC grows exponentially in the low-SNR
regime and linearly in the high-SNR regime. The high-
SNR analysis of the sum EC confirms that no benefit
is obtained in terms of EC by having multiple antennas
at the weak user. The results also indicate that most of
the gain in terms of sum EC is obtained by combining
the signals received from the strongest diversity paths,
while diminishing returns are obtained by increasing the
number of combined paths. By quantifying the difference
between average achievable sum-rate and sum effective
capacity, we showed that for a system with stringent
delay requirements, the achievable link-layer rate is sig-
nificantly smaller than the ergodic rate, and this differ-
ence increases with an increase in SNR. The presented
analysis serves as a practical system design tool which
can be efficiently applied to any configuration in order
to determine the appropriate number of diversity paths
to be combined to achieve a delay-constrained target
quality-of-service.
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