On the Achievable Rates of Uplink NOMA with Asynchronized Transmission

Abstract

Non-orthogonal multiple access (NOMA) has been widely recognized as a promising multiple access scheme for realizing next generation wireless communications. Unlike existing NOMA schemes assuming perfectly time synchronized user's signals received at the base station (BS), in this paper, we investigate the achievable rates of uplink NOMA with asyn-chronized transmission. By invoking Szegö's Theorem, we derive both the upper-and lower-bounds of the achievable rates of asynchronized NOMA (aNOMA) systems. In particular, we reveal that the derived lower-bound is essentially the achievable rate for conventional synchronized NOMA systems, which indicates that the asynchronization is not necessarily a foe. More specifically, we show that aNOMA systems are superior to conventional NOMA systems in terms of the achievable rates with non-sinc shaping pulses. Important insights are also unveiled based on the derived bounds. Simulation results confirm the validity of our derived analysis and demonstrate considerable achievable rates gains of aNOMA systems over conventional NOMA systems.

Full text

On the Achievable Rates of Uplink NOMA with
Asynchronized Transmission
Shuangyang Li∗†, Zhiqiang Wei†, Weijie Yuan†, Jinhong Yuan†, Baoming Bai∗, and Derrick Wing Kwan Ng†
∗State Key Lab. of ISN, Xidian University, Xi’an, China
†School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, Australia
Abstract—Non-orthogonal multiple access (NOMA) has been
widely recognized as a promising multiple access scheme for
realizing next generation wireless communications. Unlike ex-
isting NOMA schemes assuming perfectly time synchronized
user’s signals received at the base station (BS), in this paper,
we investigate the achievable rates of uplink NOMA with asyn-
chronized transmission. By invoking Szeg¨
o’s Theorem, we derive
both the upper- and lower-bounds of the achievable rates of
asynchronized NOMA (aNOMA) systems. In particular, we reveal
that the derived lower-bound is essentially the achievable rate for
conventional synchronized NOMA systems, which indicates that
the asynchronization is not necessarily a foe. More specifically, we
show that aNOMA systems are superior to conventional NOMA
systems in terms of the achievable rates with non-sinc shaping
pulses. Important insights are also unveiled based on the derived
bounds. Simulation results confirm the validity of our derived
analysis and demonstrate considerable achievable rates gains of
aNOMA systems over conventional NOMA systems.
I. INTRODUCTION
The massive growth in the number of wireless commu-
nication devices and sensors has fueled the research ad-
vancement in exploring efficient multiple access solutions for
future Internet-of-things (IoT) networks. In particular, non-
orthogonal multiple access (NOMA) has been recognized
as a promising radio access scheme due to its superiority
in handling massive connectivity [1], which is a crucial
requirement for IoT networks. In contrast to conventional
orthogonal multiple access (OMA) schemes, NOMA allows
more than one users to transmit their information via the same
time-frequency resource block, enjoying the improved spectral
efficiency via exploiting their channel disparities [2]–[4].
Although NOMA has been extensively studied in the past
few years, the majority of published works, e.g., [1], in
the literature assumed perfectly time-synchronous transmis-
sion among NOMA users. However, this assumption is often
unrealistic in practical uplink transmissions, due to the dif-
ferent clock generators adopted at geographically distributed
uplink users and the inherent heterogeneous signal propagation
delays. As a consequence, the signals from multiple users
cannot be synchronously superimposed at the BS as commonly
assumed in the literature [1], [2], [5], [6]. Inspired by this,
the authors in [7] proposed an iterative interference cancella-
tion scheme for asyncronized NOMA (aNOMA) systems and
analyzed the corresponding bit error rate and capacity perfor-
mances. In specific, they considered an orthogonal frequency-
division multiplexing (OFDM)-based aNOMA system where
the pulse shaping filter is time-limited and the inter-user
interference (IUI) caused by asynchronism only originates
from adjacent symbols. However, in typical narrowband IoT
systems, the shaping pulse is normally bandlimited rather than
time-limited. In this case, the IUI caused by asynchronism
originates from all the transmitted symbols of all users,
resulting in a much more complex system model. Therefore,
the analysis and conclusions from [7] cannot be directly
extended to general IoT systems and therefore the theoretical
achievable rate analysis for narrowband aNOMA systems is
still unknown.
In this paper, we investigate the achievable rate for narrow-
band uplink NOMA with asynchronized transmission, where
each user experiences a random link delay. Since the link delay
for each user may randomly vary for each channel realization,
we investigate the upper- and lower-bounds for the achievable
rates of the aNOMA system by taking advantage of the
bandlimited property of the aNOMA signals. To this end, we
first derive the closed-form expression of the achievable rates
under the successive interference cancellation (SIC) detection
and then invoke Szeg ¨
o’s Theorem [8] to obtain some important
insights on the asynchronized transmission. We note that the
application of Szeg¨
o’s Theorem requires the derivation of
the inverse Fourier series of the IUI coefficients, which is
closely related to the spectrum of the employed shaping pulse
and the corresponding link delay. Therefore, we consider the
upper- and lower-bounds of the inverse Fourier series that are
independent from the link delay of each user. Based on the
derived inverse Fourier series bounds, we further derive the
upper- and lower-bounds of the achievable rates for aNOMA
systems. Specially, we characterize the conditions for achiev-
ing the derived rate bounds and show that the asynchronization
actually leads to lesser multiuser interference with non-sinc
shaping pulses under the SIC detection. Therefore, it can be
noticed that aNOMA systems are superior to conventional
NOMA systems in terms of the achievable rates with non-sinc
shaping pulses. Simulation results match our derived analysis
and demonstrate considerable achievable rate gains of aNOMA
systems over conventional NOMA systems.
Notations: ⊗denotes the convolution operation. I(·;·)de-
notes the mutual information. I(·;·|·)denotes the conditional
mutual information. IN×Ndenotes the identity matrix of size
N×N. The notations (·)T,(·)∗denote the transpose and the
conjugate operations for a matrix, respectively. The blackboard

bold letter E[·],A, and Cdenote the expectation operator, an
energy normalized complex constellation set, and the complex
number field, respectively.
II. SYSTEM MODEL
We consider a narrowband uplink single cell multiple access
system with Kusers, where each user transmits Nsymbols,
i.e., the transmitted symbol vector of the k-th user is given by
xk=[xk[0] ,x
k[1] ,...,x
k[N−1]]Tand the entries in xk
are uniformly taken from the normalized constellation set A.
The transmitted signal sk(t)of the k-th user is of the linear
form
sk(t)=Es[k]
N−1

n=0
xk[n]p(t−nT ),(1)
where Es[k]is the average symbol energy of the k-th user,
Tis the Nyquist symbol time, and p(t)is a bandlimited
real-valued T-orthogonal transmitter shaping pulse with a
normalized energy, i.e., ∞
−∞ |p(t)|2dt=1. Without loss of
generality, we consider the root raised cosine (RRC) pulse with
a roll-off factor βas the transmitter shaping pulse. Assuming
that the wireless channel between each user and the base
station (BS) is slow fading with a specific link delay, the signal
received at the BS is given by
r(t)=
K

k=1
hksk(t−τ[k]) + w(t)
=
K

k=1
N−1

n=0
hkEs[k]xk[n]
×p(t−nT −τ[k]) + w(t),(2)
where τ[k]≥0and hk∈Care the link delay and channel
coefficient for the k-th user, respectively, and w(t)∈Cis the
additive white Gaussian noise (AWGN) at the BS with zero
mean and one-sided power spectral density (PSD) N0. For the
ease of derivation, we assume τ[1]=0and that the delays
corresponding to all users are sorted in an ascending order,
i.e., τ[1] ≤τ[2] ≤...≤τ[K]. At the BS, the received signal
r(t)is matched-filtered based on the transmitter shaping pulse
p(t)and is sampled according to the symbol time in order to
obtain a set of sufficient statistics for symbol detection. In
particular, the sampling procedure after the matched-filtering
for aNOMA systems is illustrated in the upper part of Fig. 1,
where K=3users are considered and their link delays are 0,
2
5T, and 6
5T, respectively. We refer to the moment of sampling
as the sampling index (the vertical dashed lines in Fig. 1) and
the time difference between adjacent sampling indices as the
sampling spacing, respectively. For the notational simplicity,
the lower part of Fig. 1 shows the corresponding sampling
for aNOMA systems by separating the overlapped signals
according to each user. In specific, each user’s signal is
sampled according to the Nyquist symbol time with respect
to their link delays that are assumed to be known at the
receiver. After matched-filtering and sampling, we obtain the
n-th element of the received symbol vector corresponding to
the k-th user yk=[yk[0] ,y
k[1] ,...,y
k[N−1]]Tas
yk[n]=
∞

−∞
r(t)p∗(t−nT −τ[k])dt
=
K

l=1
N−1

m=0
hlEs[l]xl[m]
×g[m−n, τ [l]−τ[k]] + ηk[n].(3)
In (3), the term g[m−n, τ [l]−τ[k]] represents the IUI
between different users, which is given by
g[k, Δτ]Δ
=
∞

−∞
p(t)p∗(t+kT +Δτ)dt
=
∞

−∞
|Hp(f)|2exp (j2πf (kT +Δτ))df, (4)
where Hp(f)is the Fourier transform of p(t)and the second
equality in (4) is due to the Parseval’s Theorem. In particular,
we have g[k, 0] = 0 for 1−N≤k<0and 0<k≤
N−1and g[0,0] = 1, owing to the T-orthogonal property
of the shaping pulse. On the other hand, the term ηk[n]in
(3) denotes the corresponding colored noise sample, where
E{ηk[n]η∗
l[m]}=N0g[m−n, τ [l]−τ[k]].
For conventional synchronized NOMA transmission, the
impact of link delay is assumed to be perfectly eliminated
at the BS [2], in which case the received symbol in (3) is
simplified as
yk[n]=
K

l=1
hlEs[l]xl[n]+η[n],(5)
where the noise samples η[n]are white due to the T-
orthogonal property of the shaping pulse. Therefore, conven-
tional synchronized NOMA transmission can be regarded as
a special case of aNOMA, where the link delays of different
users are set to zero. Consequently, the sampling indices of
each user are perfectly aligned with each other. Comparing
the sampling procedure of aNOMA and conventional synchro-
nized NOMA systems, we can see that the aNOMA system
has KN received samples (received symbols) at the BS,
while conventional synchronized NOMA only has Nreceived
samples. On the other hand, the aNOMA system occupies a
slightly more time resources to be received at the BS compared
to that of the conventional synchronized NOMA due to the
different link delays. However, this extra expense of time
resource is usually negligible compared to the time duration of
the user signals in practical system settings [2]. For the ease
of presentation, (3) can be equivalently expressed in a matrix
form,
yk=
K

l=1
hlEs[l]Gl,kxl+ηk,(6)
where Gl,k is the IUI channel matrix characterizing the IUI
from the l-th user to the k-th user and is given at the top of

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Fig. 1. An example of the sampling for the aNOMA system, where 3users are considered and their link delays are 0,2
5T, and 6
5T, respectively. The vertical
dashed lines represent the corresponding sampling indices. The upper part of the figure indicates the sampling of the received signal after matched-filtering.
By separating the overlapped signals, this sampling procedure for the signal of each user can be represented as shown in the lower part of the figure.
Gl,k =⎡
⎢
⎢
⎢
⎣
g[0,τ[l]−τ[k]] g[1,τ[l]−τ[k]] ··· g[N−1,τ[l]−τ[k]]
g[−1,τ[l]−τ[k]] g[0,τ[l]−τ[k]] ··· g[N−2,τ[l]−τ[k]]
.
.
.....
.
.
g[1 −N,τ [l]−τ[k]] g[2 −N,τ [l]−τ[k]] ··· g[0,τ[l]−τ[k]]
⎤
⎥
⎥
⎥
⎦
.(7)
this page. It can be shown that Gl,k is a Toeplitz matrix and
we have Gk,k =IN×Nfor 1≤k≤K. Meanwhile, the noise
vector ηkis given by ηk=[ηk[0] ,η
k[1] , ..., ηk[N−1]]T,
and it can be shown that E{ηk[m]η∗
k[n]}=N0for 1≤k≤
Kand 0≤n, m ≤N−1.
III. ACHIEVABLE RATE ANA LYS IS
In this section, we focus on the asymptotic instantaneous
achievable rate of each user. For the ease of derivation, we
assume that the elements in the transmitted symbol vector xk
are independent and identically distributed Gaussian variables
with average symbol energy Es[k], for 1≤k≤K. In order
to derive the achievable rate, we will first provide several
lemmas and theorems serving as important building blocks
for the analysis.
Without loss of generality, let us assume that the channel
coefficients are sorted in the descending order, i.e., |h1|2≥
|h2|2≥... ≥|hK|2. Conventionally, SIC detection is usu-
ally applied at the BS for general NOMA systems. For the
data detection of the k-th user, the IUI introduced by users
1,2,...,k−1is cancelled in the aNOMA system. Therefore,
the asymptotic instantaneous achievable rate for the k-th user
under SIC detection is defined as
RSIC
k
Δ
= lim
N→∞
1
NIh,τ(yk;xk|x1,...,xk−1)
bits per channel use.(8)
The closed-form expression of Ih,τ(yk;xk|x1,...,xk−1)is
given in the following lemma.
Lemma 1 (Conditional Mutual Information for SIC Detec-
tion): For SIC detection, the conditional mutual information
Ih,τ(yk;xk|x1,...,xk−1)is given by
Ih,τ(yk;xk|x1,...,xk−1)
=1
2log2det IN×N+|hk|2Es[k]
N0
Pl,k,(9)
where
Pl,k
Δ
=⎛
⎜
⎜
⎜
⎝
IN×N+
K

l=k+1
|hl|2Es[l]Gl,kGT
l,k
N0
⎞
⎟
⎟
⎟
⎠
−1
.(10)
Proof sketch: The lemma follows the translation invariance
of the differential entropy [9]. Due to the space limitation the
detailed proofs are omitted here and we refer to the interested
readers to our journal paper. 
It can be observed from Lemma 1 that due
to the link delay of each user, the corresponding
interference term in the aNOMA systems for the
k-th user is K
l=k+1 |hl|2Es[l]Gl,kGT
l,k instead of
K
l=k+1 |hl|2Es[l]IN×Nin the conventional synchronized
NOMA systems. With the help of Lemma 1, the asymptotic
instantaneous achievable rate RSIC
kcan be obtained by
invoking Szeg¨
o’s Theorem [8], [10]. To apply Szeg¨
o’s
Theorem, we first need to verify that Pl,k is a positive
definite Toeplitz matrix in the asymptotic regime, i.e.,
N→∞, for any 1≤l, k ≤K. Due to the space limitation,

we omit the proof here but this argument can be proved based
on Gram’s criterion [11] and Riemann-Lebesgue Lemma [12].
Next, we apply Szeg¨
o’s Theorem to (8). The application of
Szeg¨
o’s Theorem requires the derivation of the corresponding
inverse Fourier series. It can be shown that in the asymptotic
regime, the coefficients {tl,k [n]}of the asymptotical Toeplitz
matrix Tl,k
Δ
=Gl,kGT
l,k are given by
tl,k [n]=
∞

m=−∞
g[m, τ [l]−τ[k]] g[m−n, τ [l]−τ[k]].
(11)
We note that the derivation of inverse Fourier series with
respect to (11) depends on the difference between link de-
lays of each user. However, the link delays’ difference may
randomly vary with respect to various channel conditions,
which is generally intractable. As an alternative, we apply
bounding techniques for the inverse Fourier series to facilitate
the achievable rate analysis. In particular, the bounds of inverse
Fourier series depend on the corresponding spectrum of the
shaping pulse. In comparison to the spectrum of the RRC
pulse with a roll-off factor β, let us consider the spectrum
of the sinc pulse |Hsinc (f)|2, i.e., |Hp(f)|2with β=0, and
a function ¯
Hp(f), which is given by
¯
Hp(f)=⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
|Hp(f)|2,f∈−1−β
2T,1−β
2T,
|Hp(f)|2−Hpf−1
T
2,f∈1−β
2T,1+β
2T,
|Hp(f)|2−Hpf+1
T
2,f∈−1+β
2T,−1−β
2T.
(12)
For example, we plot the function ¯
Hp(f)with β=0.5in
Fig. 2, where the shape of function ¯
Hp(f)is limited within the
frequency interval f∈−1+β
2T,1+β
2T. Particularly, it can be
shown that ¯
Hp(f)=|Hp(f)|2=|Hsinc (f)|2, when β=0.
In the following lemma, we show that inverse Fourier series
corresponding to |Hp(f)|2can be properly bounded according
to |Hsinc (f)|2and ¯
Hp(f).
Lemma 2 (Folded-spectrum and Bounds): Given an RRC
pulse with spectrum |Hp(f)|2and the frequency inter-
val f∈−1
2T,1
2T, the infinite series of the form
∞

k=−∞ Hpf−k
T
2e−j2παk can be upper-bounded and
lower-bounded as
¯
Hp(f)≤
∞

k=−∞ 
Hp f−k
T!
2
e−j2παk ≤|Hsinc (f)|2,
(13)
where αis an arbitrary constant number. Furthermore, the
lower-bound becomes exact if p(t)is strictly bandlimited in
the frequency interval f∈−1
2T,1
2T, i.e., the sinc pulse,
while the upper-bound becomes exact in the case of sinc pulse
or α=0.
Proof : The proof is given in Appendix A.
As a remark to Lemma 2, we point out that the series of the
form
∞

k=−∞ Hpf−k
T
2is commonly known as the folded-
spectrum in the literature of faster-than-Nyquist signaling
Fig. 2. Shape of the function ¯
Hp(f).
[13]–[17]. The folded-spectrum indicates that the frequency
components outside the interval −1
2T,1
2Tare “folded-in” the
interval to form an equivalent frequency domain representation
of the transmitted signal [18]. With Lemma 2, the inverse
Fourier series of with respect to (11) can be upper- and lower-
bounded for our analysis as shown in Theorem 1.
Theorem 1 (Inverse Fourier Series of IUI Coefficients):
Given the set of IUI coefficients tl,k [n]of the form (11) and
an arbitrary link delay difference Δτ=τ[l]−τ[k], the inverse
Fourier series
˜
tl,k (2πfT )=
∞

n=−∞
∞

m=−∞
g[n, Δτ]g[n−m, Δτ]e−j2πmTf ,
(14)
are upper- and lower-bounded by
1
T2¯
Hp(f)
2≤˜
tl,k (2πfT )≤1
T2|Hsinc (f)|4.(15)
Proof sketch: The theorem can be derived based on the
Poisson summation formula and interchanging the order of
the integral and the summation. 
Then, the application of Szeg¨
o’s Theorem to (8) is straight-
forward thanks to Theorem 1. The corresponding results are
summarized in the next theorem.
Theorem 2 (Bounds on the Achievable Rate): The asymp-
totic instantaneous achievable rate for the k-th user RSIC
kunder
SIC detection is upper-bounded by
RSIC
k
≤T
21
2T
−1
2T
log2⎛
⎜
⎜
⎜
⎝
1+ |hk|2Es[k]
N0+
K

l=k+1
|hl|2Es[l]1
T2¯
Hp(f)
2
⎞
⎟
⎟
⎟
⎠
df
bits per channel use,(16)

and lower-bounded by
RSIC
k≥1
2log2⎛
⎜
⎜
⎜
⎝
1+ |hk|2Es[k]
N0+
K

l=k+1
|hl|2Es[l]
⎞
⎟
⎟
⎟
⎠
bits per channel use.(17)
Proof : The proof is given in Appendix B.
Based on Theorem 2, the normalized achievable rate bounds
can be easily calculated as shown in the following corollary.
Corollary 1 (Normalized Achievable Rate Bounds): The
normalized achievable rate RSIC
kis upper-bounded by
RU
k
Δ
=1
2T
−1
2T
log2⎛
⎜
⎜
⎜
⎝
1+ |hk|2Es[k]
N0+
K

l=k+1
|hl|2Es[l]1
T2¯
Hp(f)
2
⎞
⎟
⎟
⎟
⎠
df
bits per second,(18)
and lower-bounded by
RL
k
Δ
=1
Tlog2⎛
⎜
⎜
⎜
⎝
1+ |hk|2Es[k]
N0+
K

l=k+1
|hl|2Es[l]
⎞
⎟
⎟
⎟
⎠
bits per second.
(19)
Proof : The corollary comes naturally from Theorem 2
by considering the symbol rate of each user, which is 1/T
samples per second, and the signaling dimensions, which is 2
for complex Gaussian distributed symbols. 
According to Corollary 1, some interesting observations and
insights can be revealed.
•Recalling Lemma 2, we observe that both the upper-
bound and the lower-bound in Corollary 1 are achievable.
Theoretically, the upper-bound RU
kcan be achieved if the
shaping pulse’s roll-off factor β=0, i.e., the sinc pulse,
while the lower-bound RL
kcan be achieved if all the users
share the same link delay, i.e., the conventional synchro-
nized NOMA system, and the corresponding achievable
rate lower-bound is essentially the achievable rate for
conventional synchronized NOMA systems.
•With practical RRC pulses, i.e., β=0, the asynchroniza-
tion leads to a greater achievable rate region compared
to that of the conventional NOMA systems. Moreover,
the upper-bound RU
kindicates that the potential data rate
improvement of aNOMA systems is due to less multiuser
interference energy, which is the result of different link
delays. An intuitive explanation of this observation is that
the aNOMA system naturally avoids the fully superposi-
tion of transmitted symbols from different users due to the
diverse link delays such that it is unlikely to acquire the
peak interference energy at each sampling index, which
is consistent from the observations in Fig. 1. However,
for the sinc pulse, i.e., β=0, the aNOMA system
does not provide any improvement in terms of achievable
rates compared to the conventional asynchronized NOMA
system, as the upper-bound RU
kbecomes exactly the same
as the lower-bound RL
k. This observation indicates that
compared to conventional synchronized NOMA systems,
the aNOMA system has lesser multiuser interference
caused by the different link delays and offers the potential
of achieving higher achievable rates. On the other hand,
compared with conventional OMA systems, aNOMA
systems allow multiple users to transmit their information
over almost the same time-frequency resources at a cost
of manageable multiuser interference.
•The instantaneous achievable rate region of the aNOMA
system is directly connected to the spectrum of the
shaping pulse. In particular, the region enlarges with the
increases of β, as the corresponding shaping pulse spec-
trum has less energy in the frequency interval −1
2T,1
2T.
Now let us focus on the achievable sum rate RSIC
SUM of the
aNOMA system. Based on Corollary 1, we can obtain the
following corollary.
Corollary 2 (Normalized Achievable Sum Rate Bounds):
The normalized achievable sum rate RSIC
SUM is upper-bounded
by
RU
SUM
Δ
=1
2T
−1
2T
K

k=1
log2⎛
⎜
⎜
⎜
⎝
1+ |hk|2Es[k]
N0+
K

l=k+1
|hl|2Es[l]1
T2¯
Hp(f)
2
⎞
⎟
⎟
⎟
⎠
df
bits per second,(20)
and lower-bounded by
RL
SUM
Δ
=1
Tlog2⎛
⎜
⎜
⎜
⎝
1+
K

k=1
|hk|2Es[k]
N0
⎞
⎟
⎟
⎟
⎠
bits per second.
(21)
Proof : The corollary can be straightforwardly derived from
Corollary 1. 
As expected, we observe that the instantaneous sum rate of
aNOMA systems is lower-bounded by that of the conventional
synchronized NOMA system. Similar to the previous corollary,
this is due to the less severe multiuser interference caused by
the different link delays. In the next section, we will verify
our analysis by means of numerical simulations.
IV. NUMERICAL RESULTS
Without loss of generality, we consider the baseband trans-
mission for simulation, where we set the Nyquist symbol time
as T=1second. We provide the simulation results for both
the normalized instantaneous achievable rates and the sum
rate for the aNOMA system and compare the results with

OMA
R
U
k
aNOMA
R
L
k
Fig. 3. The achievable rate region of NOMA and aNOMA with one BS
and two users. Both users have the same unit transmitted symbol energy and
|h1|2Es[1]
N0=|h2|2Es[2]
N0= 10 dB. The transmitter shaping pulse is the
RRC pulse with β=0.3.
the theoretical bounds derived in Corollary 1 and Corollary 2,
where the number of transmitted symbols for each user is set as
N= 500. In particular, with a given channel coefficient hkfor
each user, we adopt the Monte Carlo method to eliminate the
effect of link delays. With a sufficient number of Monte Carlo
realizations, we obtain the average achievable rates according
to (9), where the link delay for each user is randomly generated
with a uniform distribution between the interval [0,2T]in each
Monte Carlo realization.
Fig. 3 depicts the normalized instantaneous achievable rate
region of NOMA and aNOMA systems, where one BS and
two users are considered. In specific, we consider the case
where both users have the same unit transmitted symbol energy
and we set |h1|2Es[1]
N0=|h2|2Es[2]
N0=10dB. The transmitter
shaping pulse is the RRC pulse with β=0.3. We notice
that the achievable rate based on the Monte Carlo simulation
is perfectly bounded by the derived bounds. Furthermore,
as the lower-bound is essentially the achievable rate for the
conventional NOMA system, we can see that the aNOMA
system is superior to the conventional NOMA system, in terms
of the achievable rates. This observation is consistent with our
derivation in Corollary 1.
Fig. 4 depicts the normalized instantaneous achievable sum
rate performance of aNOMA, NOMA, and OMA, where one
BS and K=4users are considered. In specific, we assume
that all the users have the same unit transmitted symbol energy
and channel condition, i.e., |h1|2=|h2|2=··· =|h4|2=1.
Without loss of generality, we define the receiver signal-to-
noise ratio (SNR) as SNR Δ
=
K

k=1
|hk|2Es[k]"N0. In order to
have a comprehensive understanding, we set the shaping pulse
as the RRC pulse with β=0and β=0.3, respectively. As
can be observed from the figure, the normalized achievable
aNOMA, R
U
SUM
, = 0.3
aNOMA, R
L
SUM
, = 0.3
aNOMA, R
SIC
SUM
, = 0.3
aNOMA, R
SIC
SUM
, = 0
NOMA, R
SIC
SUM
, = 0.3
OMA, R
SIC
SUM
, = 0.3
Fig. 4. The achievable sum rate region of aNOMA, NOMA, and OMA with
one BS and K=4users, where |h1|2=|h2|2=··· =|h4|2=1.
sum rate performance for the aNOMA system with β=0.3
is again clearly bounded by the derived bounds. Furthermore,
for the aNOMA system with β=0, the sum rate performance
based on the Monte Carlo simulation aligns perfectly with that
of the lower-bound of the aNOMA system. These observations
substantiate our derivation in Corollary 2. On the other hand,
for the sum rate at around 3bits/s, the aNOMA system shows
an around 10.7dB SNR gain compared to that of the OMA
systems. Moreover, for the sum rate at around 6bits/s, the
aNOMA system shows around 0.7dB SNR gain compared
to that of the conventional synchronized NOMA system.
These observations clearly demonstrates the superiority of the
aNOMA system over the conventional schemes.
V. C ONCLUSION
In this paper, we investigated the achievable rate of uplink
aNOMA systems. We derived both the upper-bound and the
lower-bound of the achievable rates by invoking Szeg¨
o’s
Theorem. In particular, we provided important insights for
aNOMA systems based on the derived bounds. Simulation
results agreed with our analysis and demonstrated consider-
able achievable rate gains of aNOMA systems compared to
conventional synchronized NOMA and OMA systems. Our
future work will focus on the outage performance analysis
of aNOMA systems and neural network assisted detection
algorithms [19] for aNOMA systems.
APPENDIX A
PROOF OF LEMMA 2
As we know that the spectrum of RRC pulse is strictly non-
negative, it is obvious that
∞

k=−∞ 
Hp f−k
T!
2
e−j2παk ≤
∞

k=−∞ 
Hp f−k
T!
2
.
(22)

Therefore, according to the property of RRC pulses, we
have
∞

k=−∞ Hpf−k
T
2=|Hsinc (f)|2. Furthermore, it is
obvious that (22) becomes exact if α=0.
On the other hand, we know that |Hp(f)|2is strictly
bandlimited within the frequency interval f∈−1+β
2T,1+β
2T.
Therefore, for f∈−1
2T,1
2T,wehave
∞

k=−∞ 
Hp f−k
T!
2
e−j2παk
≥|Hp(f)|2−
Hp f−1
T!
2
−
Hp f+1
T!
2
,
=¯
Hp(f),(23)
where the equality only holds if Hpf−k
T
2=0if k=
0, i.e., p(t)is the sinc pulse. This completes the proof of
Lemma 2.
APPENDIX B
PROOF OF THEOREM 2
Applying Szeg¨
o’s Theorem to (8) yields,
RSIC
k
Δ
= lim
N→∞
1
NIh,τ(yk;xk|x1,...,xk−1)
=1
4ππ
−π
log2#1+ |hk|2Es[k]
N0
×⎛
⎜
⎜
⎜
⎝
1+
K

l=k+1
|hl|2Es[l]˜
tl,k (ω)
N0
⎞
⎟
⎟
⎟
⎠
−1⎞
⎟
⎟
⎟
⎟
⎠
dω, (24)
where ω=2πTf. Furthermore, by considering Theo-
rem 1, (24) can be lower-bounded by
RSIC
k
≥T
21
2T
−1
2T
log2⎛
⎜
⎜
⎜
⎝
1+ |hk|2Es[k]
N0+
K

l=k+1
|hl|2Es[l]1
T2|Hsinc (f)|4
⎞
⎟
⎟
⎟
⎠
df
=1
2log2⎛
⎜
⎜
⎜
⎝
1+ |hk|2Es[k]
N0+
K

l=k+1
|hl|2Es[l]
⎞
⎟
⎟
⎟
⎠
bits per channel use,
(25)
where the second equality is due to the fact that |Hsinc (f)|2=
Tin the frequency interval f∈−1
2T,1
2T.
On the other hand, (24) can be upper-bounded according to
Theorem 1 by
RSIC
k
≤T
21
2T
−1
2T
log2⎛
⎜
⎜
⎜
⎝
1+ |hk|2Es[k]
N0+
K

l=k+1
|hl|2Es[l]1
T2¯
Hp(f)
2
⎞
⎟
⎟
⎟
⎠
df
bits per channel use.(26)
This completes the proof of Theorem 2.
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