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On Power and Rate Allocation for Coded
Uplink NOMA in a Multicarrier System
Jinho Choi
Abstract
In this paper, we propose a simple power and rate allocation scheme for uplink non-orthogonal
multiple access (NOMA) based on a multicarrier system when a practical modulation and coding
scheme is employed at each user. Since information-theoretic approaches with asymptotic settings are not
considered, the proposed power and rate allocation scheme is able to provide a guaranteed performance
with any practical block codes of finite-length in terms of the probability of codeword error. For the
performance prediction in the power and rate allocation with given channel realizations, we employ non-
asymptotic approaches for bounds on the probability of codeword error. Through simulation results, we
can confirm that the probability of codeword error can be lower than or close to the target error rate
using the proposed power and rate allocation.
Index Terms
non-orthogonal multiple access (NOMA); uplink transmissions; power and rate allocation; non-
asymptotic analysis
I. INTRODUCTION
Non-orthogonal multiple access (NOMA) has been extensively studied for 5th generation (5G)
systems [1], [2] due to its various advantages including high spectral efficiency [3] by exploiting
the power difference between users and successive interference cancellation (SIC). For example,
in [4], [5], NOMA is applied to multi-point systems to effectively support cell-edge users with
limited bandwidth. In [6], multiresolution broadcast based on NOMA is proposed for bandwidth-
The author is with School of Electrical Engineering and Computer Science, Gwangju Institute of Science and Technology
(GIST), Korea (Email: jchoi0114@gist.ac.kr). This work was supported by the GIST Research Institute (GRI) in 2018.
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efficient broadcast with multiple transmission qualities. In [7], unicast-multicast based on NOMA
is also considered with various power allocation schemes.
NOMA is a generic multiple access scheme that can be employed with various transmission
schemes such as multiple-input multiple-output (MIMO) transmissions and beamforming. In [8],
NOMA is studied for MIMO systems. In [9], [10], the capacity of MIMO-NOMA is investigated.
Beamforming for NOMA is also extensively studied in [11], [12]. In [13], it is shown that the
spatial correlation in conjunction with user clustering plays a crucial role in NOMA beamforming.
In [14], various multiple-antenna transmission schemes are studied for NOMA.
As NOMA is to exploit the power difference between users, the power allocation becomes
important. In downlink transmissions, the power allocation for NOMA requires the channel state
information (CSI) from users. However, in some cases, the feedback of CSI may not be available
due to high mobility. In this case, statistical knowledge of CSI can be used to perform power
allocation as in [15], while it is also shown that joint power and rate allocation can also be
performed in [16], which can be seen as a generalization of [15].
In most cases, the power allocation for NOMA is based on information-theoretic approaches
[2], which requires capacity achieving codes [17]. In practice, however, it is not easy to achieve
the capacity as coding and modulation are usually separate. Interestingly, in [18], it is shown that
the power allocation can be different from that based on information-theoretic approaches (with
the assumption of Gaussian codebooks to achieve the channel capacity) if practical modulation
schemes are used for NOMA as in [19].
In this paper, we consider uplink NOMA based on a multicarrier system and propose a simple
power and rate allocation when a practical transmission scheme is employed. In particular, we do
not follow information-theoretic approaches to determine the power and rate for users, which is
similar to [18]. However, although a practical approach is considered for the power allocation as
in [18], there are a number of differences. First, we consider uplink transmissions in this paper,
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while downlink transmissions are considered in [18]. Secondly, a specific channel coding scheme
is assumed and the code rate is to be determined in conjunction with users’ transmit powers to
guarantee a given target error rate, which is not considered in [18]. Thus, with a short-length
channel code, a certain performance guarantee can be achieved in terms of the probability of
codeword error without relying on any asymptotic settings. Thirdly, the power and rate allocation
in this paper can be seen as an adaptive transmission scheme that adapts to channel realizations.
In particular, the transmit powers and code rate are determined for given channel realizations.
Related Works: As mentioned earlier, uplink NOMA has been studied in [20]–[23]. However,
most of them are based on information-theoretic approaches and do not consider any practical
channel coding. Note that for coded uplink transmissions, interleave division multiple access
(IDMA) is proposed in [24], while it does not exploit the power difference between users as
NOMA.
Contributions and Limitations: The main contributions of the paper is a simple power and
rate allocation for uplink NOMA with a practical channel coding scheme based on channel
realizations. Since non-asymptotic approaches are used, it is possible to guarantee a certain
performance in terms of the probability of codeword error with any short-length channel codes.
As a result, for short packet transmissions in uplink, the notion of NOMA can be exploited to
achieve a higher spectral efficiency.
There are some limitations of the proposed approach in this paper. While the power and
rate allocation depends on channel realizations and could guarantee a certain performance, the
number of users sharing the same radio resource can be limited due to certain impairment that
results in a poor prediction of the probability of codeword error. Simulation results show that it
is possible to support two users with guaranteed performances. However, if there are 3 users, in
some cases, a guaranteed performance cannot be achieved due to a poor performance prediction.
The rest of the paper is organized as follows. We present the system model in Section II for
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coded uplink NOMA. A simple power and rate allocation approach is proposed in Section III.
In Section IV, a more realistic channel model is considered for the power and rate allocation.
Simulation results are presented in Section V. Finally, we conclude the paper with remarks in
Section VI.
Notation:E[·]and Var(·)denote the statistical expectation and variance, respectively. CN(a,R)
represents the distribution of circularly symmetric complex Gaussian (CSCG) random vectors
with mean vector aand covariance matrix R. The Q-function is given by Q(x) = R∞
x
1
√2πe−t2
2dt.
II. SY ST EM MO DE L
Suppose that a multicarrier system consists of Lsubcarriers for uplink transmissions with
multiple users sharing the same radio resource as shown in Fig. 1. In general, a multicarrier sys-
tem has multiple orthogonal subcarriers and orthogonal frequency division multiplexing (OFDM)
is a typical example [25], [26] and adopted into long-term evolution (LTE) standards [27]. In
OFDM, multiple signal symbols can be simultaneously transmitted in the frequency domain and
each signal per subcarrier may experience flat fading as the bandwidth of subcarrier is narrow.
In this paper, we assume that each user transmits a coded block as an OFDM symbol, where
the length of an OFDM symbol is L. The coded signal block from user kis denoted by {sl,k},
where sl,k represents the lth modulated symbol of user ktransmitted through subcarrier l. Denote
by hl,k the (frequency-domain) channel coefficient of the lth subcarrier from user kto a base
station (BS). Then, the received signal through subcarrier lat the BS is given by
yl=
K
X
k=1
hl,ksl,k +nl, l = 0, . . . , L −1,(1)
where Kis the number of users in the same radio resource block and nl∼ CN(0, N0)is the
background additive white Gaussian noise (AWGN) at the lth subcarrier. Throughput the paper,
let σ2
k=E[|hl,k|2], while Ek=E[|sl,k|2]represents the power of the kth user’s signal. In addition,
we assume that E[hl,k]=0and E[sl,k ] = 0. Note that σ2
krepresents the power of the large-scale
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fading term when hl,k is expressed as
hl,k =σkul,k,(2)
where ul,k denotes the normalized channel coefficient with E[|ul,k|2]=1, which is referred to
as the small-scale fading term [28]. In (2), it is assumed that the large-scale fading term is
independent of subcarrier index las it mainly relies on the distance between the user and the
BS [28]. For Rayleigh fading, we assume that ul,k ∼ CN(0,1), which is considered throughout
the paper.
BS
user 1
user 2
Fig. 1. Uplink NOMA where multiple users transmit signals to a BS.
Throughout the paper, we assume that the BS knows the CSI of each user. To this end, each
user needs to send a pilot signal in order to allow the BS to know the CSI. For simplicity, the
CSI at the BS is assumed to be perfect. On the other hand, the users do not know their CSI
as no CSI feedback from the BS to users is assumed. In addition to the assumption that ul,k is
CSCG, we consider two different types of channels as follows.
C1 The channel coefficients, hl,k, are independent.
C2 The channel coefficients, hl,k, are given by
hl,k =1
pNpath
Npath−1
X
p=0
νp,ke−j2π pl
L,(3)
where Npath is the number of multipaths and νp,k ∼ CN(0, σ2
k)is the pth path gain and
independent of each other.
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Clearly, C1 is not a realistic channel model when the number of multipaths is smaller than
the number of subcarriers, i.e., Npath L, while it is an approximation of C2 when Npath is
sufficiently large and has been widely used for performance analysis of OFDM systems. Note
that (2) is true under both the channel models of C1 and C2.
For simplicity, quadrature phase shift keying (QPSK) is assumed for signal modulation with
the natural mapping. Thus, we have sl,k ∈ Sk=±qEk
2±jqEk
2, where Skrepresents the
signal constellation of user k. Since QPSK is assumed, the length of codeword, denoted by nc,
is 2L. Note that if the length of codeword is odd, we have nc= 2L−1. In this case, one bit
of the last subcarrier is fixed to be 0. It is easy to see that the relationship between ncand Lis
given by L=dnc/2efor either even or odd nc.
If the SIC is carried out in increasing order1, the BS is to decode the signal from user 1. Once
{sl,1}is decoded, it is reconstructed for SIC and removed. The SIC is repeated until the signal
from user Kis decoded. The BS needs to allocate the powers and the code rates for the users,
which are fed back to the users, in order to guarantee that each decoding is successful with an
overwhelming probability.
Throughout the paper, the resulting uplink NOMA scheme in a multicarrier system is referred
to as coded multicarrier NOMA (MC-NOMA). A salient feature of the proposed coded MC-
NOMA is that the decoding complexity at the BS is linearly proportional to the number of users,
K, thanks to SIC.
III. PERFORMANCE ANALYSIS AND ALLOCATIO N FO R INDEPENDENT CHANNELS
In this section, we consider the performance analysis and (power and rate) allocation problem
under the assumption of C1, i.e., independent channels. We first focus on the case of two users,
i.e., K= 2, in order to find the coded performance when QPSK and a block code are employed
1Throughout the paper, we assume that user kis the kth nearest user from the BS. Thus, we have σ2
1≥. . . ≥σ2
K.
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at each user. Then, we extend the result to the case of K≥2.
Note that the assumption of C1 is not realistic as the number of paths is finite. However, most
results in this section are also useful for the power and rate allocation under the assumption of
C2, which will be considered in Section IV.
A. Bit Error Probability
Throughout the paper, we confine ourselves to hard-decision inputs to a channel decoder.
Thus, for coded performances, we need to find an estimate of uncoded bit error rate (BER) of
hard-decision. In this subsection, based on the Gaussian approximation, we find the estimate of
BER.
Suppose that the signal from user 1 is to be decoded first. For convenience, under the
assumption of C1, we omit the subcarrier index lif there is no risk of confusion and consider
the coherent detection of s1. Since the natural mapping is assumed, the bit of in-phase2of s1,
denoted by bI,1∈ {−1,1}, can be detected from the following phase compensated signal:
x=<h∗
l
|hl|y
=rE1
2|h1|bI,1+<(e−j θ1h2s2) + <(e−jθ1n),(4)
where θ1is the phase of h1, i.e., ejθ1=hl
|hl|. Note that h∗
l
|hl|ycan be seen as the output of the
zero-forcing (ZF) equalizer for OFDM systems [25], [26]. Assuming that bI,1= 1, after some
manipulations, the BER is given by
Pb= Pr x−rE1
2|h1|
2>x+rE1
2|h1|
2!
= Pr a+<(e−jθ1n)
2>b+<(e−jθ1n)
2
=Q b2−a2
p2N0(b−a)2!,(5)
2Similarly, the bit of quadrature-phase can be detected from the imaginary part, i.e., =h∗
l
|hl|y, and its BER is identical to
that of the bit of in-phase.
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where a=<(e−jθ1h2s2)and b=√2E1|h1|+<(e−jθ1h2s2). Note that b > a as long as |h1|>0.
Since b2−a2= (b+a)(b−a), we can further show that
Pb=Qb+a
√2N0=Q√2E1|h1|+ 2<(e−jθ1h2s2)
√2N0.
Since s2∈ S2=nqE2
2(±1±j)o, we can show that e−θ1s2h2=e−θ|E2|h2, where θis an
arbitrary phase. Let γk=Ekσ2
k
N0, which is the average signal-to-noise ratio (SNR) of user k. For
Rayleigh fading channels, h2is a CSCG random variable. Noting that E[|h2|2] = σ2
2, we can
show that √2E1|h1|+ 2<(e−jθ1h2s2)
√2N0
=√γ1|u1|+√γ2X, (6)
where Xcan be assumed to be a Gaussian random variable with mean zero and unit variance,
X∼ N(0,1).
For the detection of s1, the signal from user 2, e−θ1s2h2, is an interference and regarded as
a random variable. In this case, noting that h1=σ1u1, the conditional BER for given user 1’s
CSI, denoted by Pb(u1), is given by
Pb(u1) = E[Pb|u1] = E"Q rE1
N0|h1|+√γ2X!#
=E[Q(√γ1|u1|+√γ2X)|u1]
=βl,1,(7)
where the expectation is carried out over X, and
βl,1=Q sγ1|ul,1|2
1 + γ2!,(8)
which is the BER of the first user’s bits transmitted by subcarrier l. Note that the last equality
in (7) is due to [29, Eq. (3.66)].
In fact, the result in (7) is equivalent to the result that can be obtained under the assumption
that the interfering signal from user 2, hl,2sl,2, is Gaussian. In this case, hl,2sl,2+nlis an
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independent zero-mean CSCG with variance σ2
2E2+N0, and the result in (7) can be readily
obtained.
If decoding of user 1’s signal becomes successful, the signal from user 1 can be removed
using SIC. Then, the BER of user 2’s bits transmitted by subcarrier lis given by
βl,2=Q(√γ2|u2,l|).(9)
B. Probability of Codeword Error
In this subsection, we consider the probability of codeword error when each user transmits a
codeword as a coded OFDM symbol.
Under the assumption of C1,hl,k becomes an independent CSCG. Then, from (7), the average
BER across all subcarriers can be given by
¯
β1=E[βl,1] = 1
21−rγ1
γ1+ 2(1 + γ2).(10)
At a decoder, provided that each bit can have an independent error, the channel can be seen as
a virtual binary symmetric channel (BSC) with cross-over probability ¯
β1when hard-decisions
become the input to the decoder. The virtual BSC model can be reasonable if the channel
coefficients are independent (e.g., C1). However, this model may not be valid for dependent
channel coefficients under the assumption of C2. This will be considered in Section IV.
Suppose that the code of user kcan correct up to tkerrors. For a given average uncoded BER,
denoted by ¯p, from [30], the probability that the transmitted codeword of length ncis incorrectly
decoded is bounded as
Pc,1≤
nc
X
i=t1+1 nc
i¯pi(1 −¯p)nc−i=W(¯p, nc, t1),(11)
where W(p, n, t) = Pn
i=t+1 n
ipi(1−p)n−k. For a large n, since it is not easy to find W(p, n, t),
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we can consider the following Poisson approximation [31]:
1−W(p, n, t) =
t
X
i=0 n
ipi(1 −p)n−i
≈
t
X
i=0
e−λλi
i!=Γ(t+ 1, λ)
t!,(12)
where λ=pn and Γ(a, x) = R∞
xya−1e−ydy is the upper incomplete gamma function. Note
that the approximation in (12) is reasonable for a large n. Thus, for user 1, the probability of
codeword error becomes
Pc,1≤W(¯
β1, nc, t1)≈1−Γ(t1+ 1, nc¯
β1)
t1!.(13)
For Rayleigh fading, the average (uncoded) BER of user 2’s signals becomes
¯
β2=1
21−rγ2
2 + γ2.(14)
Provided that there is no error in decoding for user 1, we have the following conditional
probability of codeword error of user 2:
Bc,2≤W(¯
β2, nc, t2)≈1−Γ(t2+ 1, nc¯
β2)
t2!.(15)
Finally, the probability of codeword error of user 2 becomes
Pc,2= 1 −(1 −Pc,1)(1 −Bc,2)
≤1−(1 −W(¯
β1, nc, t1))(1 −W(¯
β2, nc, t2))
≈1−Γ(t1+ 1, nc¯
β1)
t1!
Γ(t2+ 1, nc¯
β2)
t2!.(16)
We can extend the above approach to find the probability of codeword error when K > 2. For
Rayleigh fading, we assume that the sum of interfering terms is Gaussian. Then, the average BER
or cross-over probability of the virtual BSC for the decoding of the kth user’s signal becomes
¯
βk=1
2
1−sγk
γk+ 2(1 + PK
m=k+1 γm)
.(17)
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Let Bc,k denote the probability of codeword error of user kprovided that the coded OFDM
symbols of users 1 to k−1are all correctly decoded and removed by SIC. Then, we have
Bc,k ≤¯
Bc,k =W(¯
βk, nc, tk),(18)
and the probability of codeword error of user kbecomes
Pc,k = 1 −
k−1
Y
m=1
(1 −Bc,m)≤1−
k−1
Y
m=1
(1 −¯
Bc,m),(19)
where Bc,1=Pc,1. From (19), we can show that
Pc,1≤. . . ≤Pc,K ,(20)
i.e., the probability of codeword error increases with k(due to the error propagation). As in
(16), using the Poisson approximation, it follows
Pc,k /Pk= 1 −
k
Y
m=1
Γ(tm+ 1, nc¯
βm)
tm!,(21)
where /represents the approximate inequality. It can readily show that the approximate upper-
bounds also increase with kas follows: P1≤. . . ≤PK.
C. Power and Rate Allocation
As in (21), we have a closed-form expression for the probability of codeword error of each
user in coded MC-NOMA. In this subsection, we consider a simple power and rate allocation
scheme based on the closed-form expression that allows to predict the performance in terms of
the probability of codeword error for given CSI.
For a simple power and rate allocation, suppose that the following equality holds for all k:
¯
Bc,k =or Γ(tk+ 1, nc¯
βk)
tk!= 1 −, k = 1, . . . , K, (22)
where > 0is sufficiently small, which is the target error rate. Then, we have
Pk= 1 −(1 −)k≤k, (23)
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which means that the highest probability of codeword error, i.e., the Kuser’s probability of
codeword error, is at most K. Furthermore, we assume that all the users have the same code
and their code rates are the same, which results in the same transmission rate for all users. That
is, tk=tfor all k. In this case, with given for (23), to maximize the transmission rate (or
code rate), we can minimize tsubject to a maximum transmit power constraint. Consequently,
we have the following optimization problem:
t∗= mint∈T t
subject to (22) and maxkEk(t)≤Emax ,(24)
where Emax represents the maximum transmit power of each user and Trepresents the set of
the error-correction capability, t, for a given code of length nc.
To solve (24), an exhaustive search can be considered with all the possible values of twhen
the length of codewords is short. In this case, for a given t∈ T , we need to find Ekthat satisfies
(22) and the maximum transmit power constraint.
Lemma 1: For given t,βthat satisfies the following equality is unique:
¯
Bc,k =or Γ(t+ 1, ncβ)
t!= 1 −. (25)
In addition, β(t;)is an increasing function of t, where β(t;)represents the solution of β, and
for given {γk+1, . . . , γK}, the transmit power of user k,Ek, satisfying (23) is given by
Ek(tk;) = AkN0
σ2
k
(1 −2β(tk;))2
1−(1 −2β(tk;))2,(26)
where Ak= 2 1 + PK
m=k+1 γm. Here, Ek(tk;)is a decreasing function of t.
Proof: See Appendix A.
While we can find a closed-form expression for the transmit power satisfying (23) for given
tfrom Lemma 1, the key to solve (24) is based on the following conditions: a) Bc,k depends
only on {Ek, . . . , EK}and t;b) Bc,k is a decreasing function of Ekfor given {Ek+1, . . . , EK}
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and t. If conditions a) and b) hold3, the following exhaustive4search (in terms of t) can find
the solution of (24), where the transmit powers can be found recursively from user Kto user 1
(the proof is omitted since it is straightforward).
*) Input: ,{σ2
k},T,lmax, and Emax
0) Initialize: t=t(1) and l= 1.
1) while (l < lmax)
2) obtain β(t;)by solving (25)
3) for k=Kto k= 1: obtain Ek(t;)from (26); end for
4) if max Ek(t;)> Emax:l⇐l+ 1; else break; end if
5) end while
6) Output: t(l)and {Ek(t(l);)}.
Here, t(m)is the mth smallest tin Tand lmax (>1) is the maximum number of iterations.
The above exhaustive search provides the smallest tthat satisfies the maximum transmit power
constraint in (24). The complexity of the above exhaustive search is dependent on |T |. For
example, if binary Bose-Chaudhuri-Hocquenghem (BCH) codes are considered, for a codeword
length nc= 127,|T | is 18 [32]. Thus, for short-length channel codes, the complexity of the
proposed power and rate allocation is low.
D. Discussions
The BS finds the transmit powers for all the users and t∗by solving (24) as shown above.
Then, it can inform the users of their transmit powers and tso that each user can encode its signal
accordingly and the BS can decode the signals with guaranteed decoding error probabilities. The
salient feature of the resulting power and rate allocation in above is that the amount of feedback
3In Appendix A, we show that b) holds, while a) is true since ¯
βkis a function of Ek,...,EK.
4In general, the number of possible values for tkis small when ncis small. In particular, when nc= 511, there are 57
possible values for tk(or |T | = 57). Thus, the exhaustive search does not require a high computational complexity.
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is limited since one transmit power for all subcarriers per user is assumed, while the code rate, t∗,
can be broadcast to all the users as the same code rate is assumed. Furthermore, as the proposed
power and rate allocation is based on {¯
βk}that depend on the large-scale fading coefficients
or {σ2
k}, we can see that the power and rate allocation is independent of small-scale fading
coefficients and remains unchanged as long as large-scale fading coefficients are not varying
under the assumption of C1, which implies that the rate of feedback for the power and rate
allocation to users can be low. However, as will be shown in Section IV, this is not true under
the assumption of C2.
The resulting decoding approach can be seen as a multistage or multilayer decoding approach
for multilevel coding (MLC) [33], where layer kcorresponds to the signal from user k. The
code rate according to the above power and rate allocation becomes R=kc
nc, where kcis the
number of message bits corresponding to t∗. Thus, the resulting coded MC-NOMA system can
support a transmission rate of R=kc
ncfor each user with a decoding error probability of order
.
Note that since it is assumed that tk=tfor all k, according to the proposed rate and power
allocation, from (22), it is expected to have ¯
βk=¯
βfor all k. Thus, the cross-over probabilities
of the Kvirtual BSCs are the same and the resulting MC-NOMA can be seen as a layered
system and each layer can be characterized by a BSC of cross-over probability ¯
β.
IV. PERFORMANCE ANA LYSI S AN D ALL OC ATION PROBLEM FOR DEPENDENT CHANNELS
In this section, we consider the performance analysis and (power and rate) allocation problem
under the assumption of C2, i.e., dependent channels. When the channel coefficients, hl,k, are
not independent, we are not able to assume that the BER of each bit is independent. Therefore,
it is not easy to obtain the probability of codeword error as in Subsection III-B. In particular,
since the channel coefficients are dependent, the cross-over probability of virtual BSC cannot
15
be independent. As a result, (11) cannot be used to predict the probability of codeword error
in the power and rate allocation. Thus, we consider non-asymptotic approaches for bounds on
the probability of codeword error with given channel realizations to perform the power and rate
allocation in this section.
A. Bounds via Instantaneous BER Estimates
With known channel coefficients of all the users, i.e., {hl,k}, the BS can have the instantaneous
signal-to-interference-plus-noise ratio (SINR) as follows:
γl,k =Ek|hl,k|2
N0+PK
m=k+1 Em|hl,m|2(27)
under the assumption that SIC becomes successful to remove the signals in layers 1, . . . , k −1.
Then, the instantaneous (uncoded) BER of the bits transmitted by subcarrier lcan be estimated
as
βl,k =Q√γl,k∈[0,1
2).(28)
Note that the estimate of the instantaneous BER in (28) is based on the Gaussian assumption
that the received signal from the kth user at subcarrier lis assumed to be Gaussian with mean
zero and variance Ek|hl,k|2, which is an approximation as QPSK (not Gaussian signal) is used.
Another important fact is that βl,k depends on the channel realizations (i.e., the actual values of
hl,k).
For convenience, denote by XI
l,k and XQ
l,k ∈ {0,1}the actual error events of the in-phase and
quadrature bits, respectively, transmitted by subcarrier l, where each of XI
l,k and XQ
l,k becomes
0 if there is no bit error and 1 otherwise. For given {hl,k}, we have
E[XI
l,k |{hl,k}] = E[XQ
l,k |{hl,k}] = βl,k.
Let Sk=PL−1
l=0 XI
l,k +XQ
l,k. Since the bit error events for given CSI, {hl,k}, are independent
as they only depend on the background noise terms, XI
l,k and XQ
l,k are independent, but have
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different means (or distributions). Thus, an upper-bound on the following conditional probability
of codeword error of user k, which is the tail probability of Sk,Bc,k = Pr(Sk≥t+ 1), can be
found using Hoeffding’s inequality [34] or Bennett’s inequality [35].
Noting that XI
l,k, XQ
l,k ∈ {0,1}and E[Sk] = 2 PL−1
l=0 βl,k, from [34], the following upper-bound
can be obtained:
Bc,k ≤exp
−t+ 1 −2PL−1
l=0 βl,k2
L
.(29)
Since XI
l,k and XQ
l,k are binary random variables, we can easily show that Var(XI
l,k) =
Var(XQ
l,k) = βl,k(1 −βl,k ), which results in
Var(Sk)=2
L−1
X
l=0
βl,k(1 −βl,k).(30)
For convenience, let v= Var(Sk). Then, from [35], the following upper-bound can be obtained:
Bc,k ≤exp −vh t+ 1 −2PL−1
l=0 βl,k
v!!,(31)
where h(u) = (1 + u) ln(1 + u)−u. Although it is not shown in this paper, we can observe that
the bound obtained by Bennett’s inequality is tighter than that by Hoeffding’s inequality. Thus,
we will consider the bound in (31) for simulations in Section V.
B. Power and Rate Allocation
Using the bound in (29), we can use the exhaustive search method in Subsection III-C.
In particular, we assume that the upper-bound in (29) is to be for all k. For given tand
{Ek+1, . . . , EK}, we can see that βl,k in (28) is a decreasing function5of Ekas
βl,k =Q pEks|hl,k|2
N0+PK
m=k+1 Em|hl,m|2!
5Thus, condition b) holds, while condition a) is readily satisfied as Bc,k is a function of {Ek,...,EK}as shown in (27),
(28), and (31).
17
from (27) and (28). Thus, there is a unique Ekthat satisfies
t+ 1 −2
L−1
X
l=0
βl,k =c > 0
for given c, if the solution exists. From (29), the relationship between and cis given by
e−c2
L=. Thus, for given t, we can decide the transmit powers successively from EKto E1.
Among all the values of tthat allows the existence of {E1, . . . , EK}with the maximum transmit
power constraint, we can choose the smallest one to solve (24).
The power and rate allocation with the inequality in (31) is similar to that with the inequality
in (29). In particular, if Ekcan be uniquely decided to guarantee that the bound in (31) is equal
to , the same exhaustive search approach can be used for the power and rate allocation with
the inequality in (31).
Lemma 2: For given tand {Ek+1, . . . , EK}, the solution of Ekthat satisfies the following is
unique (if exists):
α(Ek) = vh t+ 1 −2PL−1
l=0 βl,k
v!= ¯c > 0,(32)
where e−¯c=, for Ek∈ Ek={Ek:t+ 1 −2Plβl,k ≥0}.
Proof: See Appendix B.
Using the bound in (29), we can show that the error probability decreases exponentially with
Lor ncunder a certain condition that is related to the notion of capacity of virtual BSC.
Lemma 3: Let ¯
βk(L) = 1
LPL
l=0 βl,k, which is the conditional average BER of virtual BSC
of user kfor given channel realizations under the assumption of C2. Let
Φ({hl,k}) = 1 −hb(¯
βk(L)),(33)
which is referred to as the effective capacity of virtual BSC. Here, hb(p) = −plog2p−(1 −
p) log2(1 −p)is the binary entropy function. If the code rate kc
ncis lower than Φ({hl,k}), the
error probability exponentially decreases with nc.
18
Proof: See Appendix C.
It is noteworthy that the effective capacity in (33) differs from the conditional ergodic or
average capacity (for given channel realizations) that is given by
C({hl,k}) = 1
L
L−1
X
l=0
(1 −hb(βl,k)) .(34)
According to Jensen’s inequality, since hb(p)is concave in p, we have
Φ({hl,k})≤C({hl,k}).
However, if the instantaneous BER is low (e.g., p≤0.1), hb(p)is nearly linear. Thus, we expect
that the gap between Φ({hl,k})and C({hl,k})may not be large for moderate uncoded BERs,
and it is reasonable to use the bound in (29) or (31) for the power and rate allocation.
V. SIMULATION RESULTS
In this section, we present simulation results to see the performance of coded MC-NOMA.
We mainly consider the channel model in C2 as it is more realistic than that in C1. Furthermore,
we mostly focus on the case of two users with the following large-scale fading terms:
σ2
k=D−η
k,
where ηis the path loss exponent and Dkis the normalized distance between the BS and user
k, where D1is normalized to be 1 and Dkincreases with k. That is, the nearest user to the BS
is user 1 and Dkis the ratio of the distance between the BS and user kto that between the BS
and user 1. We also assume that η= 3. The number of subcarriers, L, is set to 256 throughout
this section. For channel coding, we use the BCH code with nc= 511.
Although we mainly present simulation results under the assumption of C2 as mentioned
earlier, we briefly show some simulation results under the assumption of C1 in order to demon-
strate the impact of the channel types on the power and rate allocation. In Fig. 2, we present
simulation results of the proposed power and rate allocation when the BS assumes C1 for coded
19
MC-NOMA with two users and D2= 2 and Emax = 20 dB. Thus, if C1 is valid, as shown
in Fig. 2 (a), the probability of codeword error is close to the target error rate . However, the
probability of codeword error is not close to the target error rate as shown in Fig. 2 (b), if the
actual channels are given as in C2 with Npath = 10, while the BS assumes C1.
Fig. 2 (c) shows that if Npath increases, the probability of codeword error can decrease although
the power and rate allocation is carried out under the assumption of C1 at the BS. However, the
probability of codeword error cannot approach the target error rate unless Npath is sufficiently
large (near L), which might be impractical. Thus, it is important to take into account the type
of channels in the power and rate allocation.
We now only consider the case that the channels are given as in C2 and the power and
rate allocation is carried out for given realizations of channel coefficients as in Section IV. As
mentioned earlier, we consider the upper-bound in (31) as it is tighter than that in (29) for the
power and rate allocation.
Fig. 3 shows the performances of coded MC-NOMA for various values of the target error rate,
, with Emax = 20 dB, Npath = 10,K= 2, and D2= 2. The number of runs for each value of
is set to d200−1e. Since an upper-bound is used for the performance prediction in the power
and rate allocation, the probability of codeword error from simulation results becomes lower
than the target error rate as shown in Fig. 3 (a). The results of the power and rate allocation are
also shown in Fig. 3 (b) and (c), respectively. It is shown that the code rate increases with ,
while the allocated transmit powers are almost invariant.
For comparison with orthogonal multiple access (OMA), we run simulations and obtain the
performance of coded signals when each user has a half of the total subcarriers, i.e., L/2 = 128
subcarriers, without any overlapping. The length of codewords of OMA is 255 for each user
and the power allocation is carried out to meet a given target error rate with the same values
of parameters as those in Fig. 3. According to Fig. 3 (c), at = 10−3, the total number of
20
message bits, which is ‘code rate ×code length ×K’, is 0.4×511 ×2≈409 with NOMA.
If we define the spectral efficiency as the number of message bits per subcarrier, NOMA has
a spectral efficiency of 1.6 at = 10−3, which is also shown in Fig. 4. We also present the
spectral efficiency of OMA in Fig. 4 to compare it with that of NOMA for given target error
rate. Clearly, the spectral efficiency of NOMA is higher than that of OMA, which confirms that
NOMA is superior to OMA as expected in [1], [2], while we see that the spectral efficiency (for
given target error rate, ) increases with in Fig. 4.
We consider the power and rate allocation for various values of the normalized distance
between the BS and the second user, D2, with Emax = 20 dB, K= 2,Npath = 10, and = 10−2
in Fig. 5. As shown in Fig. 5 (a), it is possible to keep the probability of codeword error (of
both users 1 and 2) to be lower than the target error rate through the power and rate allocation
although D2varies, since the power and rate allocation adapts to actual channel realizations.
In Fig. 5 (b), we see that the transmit power of user 2 increases with D2, while the transmit
power of user 1 is almost independent of D2. However, as D2approaches 2, the code rate
rapidly decreases (or t∗rapidly increases, because the increases of the transmit power may not
be sufficient to support the target error rate), which results in the decrease of E1(since the same
target error rate can be obtained with a lower E1if tincreases or the code rate decreases).
The impact of the number of multipaths, Npath, on the performance of the power and rate
allocation is shown in Fig. 6 with Emax = 20 dB, K= 2,D2= 2, and = 10−2. Ideally,
there should not be any differences due to different values of Npath, because the power and
rate allocation is carried out with channel realizations. However, since the upper-bound in (31)
depends on the instantaneous BER estimates in (28), which are obtained based on the Gaussian
assumption, poor BER estimates result in a poor outcome of the power and rate allocation.
According to Fig. 6 (a), it seems that a large number of multipaths is desirable for a reasonable
performance guarantee in terms of the probability of codeword error. We also note that the code
21
rate and transmit powers increase with the number of multipaths.
In coded MC-NOMA, ideally, it is possible to support any number of users. However, in
practice, the number of users might be limited due to various impairment including the reliability
of instantaneous BER estimates although a high maximum transmit power is assumed. In Fig. 7,
we consider K= 3 users to see the performances of the power and rate allocation for different
values of maximum transmit power, Emax, with K= 3,(D2, D3) = (1.5,2),Npath = 20, and
= 10−2. It is expected to support the first user with the probability of codeword error less than
= 10−2. However, when Emax is low, there are some realizations of the channel coefficients
that do not allow any power and rate allocation for a given target error rate of = 10−2. For
example, when Emax = 10 dB, among 8000 runs (for each value of Emax), the percentage of
successful power and rate allocation is 52.68%. The other cases cannot find any power and rate
allocation satisfying a target error rate of = 10−2. With Emax = 12 dB, the percentage of
successful power and rate allocation becomes 97.50%, while the percentage of successful power
and rate allocation is 100% with Emax ≥14 dB. Thus, the results with Emax ∈ {10,12}dB
are a bit biased as only good channel realizations are taken into account. In general, for a high
maximum transmit power, fortunately, we can see that the power and rate allocation can support
users with the probability of codeword error that is sufficiently close to or less than = 10−2.
In summary, we can see that the power and rate allocation for coded MC-NOMA with real-
izations of channel coefficients under the assumption of C2 can support users with a target error
rate. However, the accuracy of guaranteed performance (in terms of the probability of codeword
error) depends on the reliability of estimated instantaneous BERs. Based on simulations, it is
reasonable to support two users (i.e., K= 2) with not too small Npath, while we can have
imprecise power and rate allocation results if K= 3 with a low maximum transmit power.
22
VI. CONCLUDING RE MA RK S
We studied a simple power and rate allocation for coded MC-NOMA with a practical trans-
mission scheme consisting of QPSK modulation and a block channel code (of finite-length). It
was shown that for given channel realizations, the power and rate allocation can be carried out
for a guaranteed performance in terms of the probability of codeword error. We employed upper-
bounds to predict the probability of codeword error in the proposed power and rate allocation.
Through simulation results, we found that the resulting probability of codeword error can be
lower than or close to the target error rate using the proposed power and rate allocation.
There are other issues to be studied in the future. First, we need to consider soft-decision inputs
to channel decoders at the BS for a better performance. In this case, a closed-form expression for
the probability of codeword error in terms of uncoded bits’ SINRs might be derived. Secondly,
user pairing or clustering can be investigated when there are a number of users with multiple
resource blocks. As shown in Section V, the performance guarantee can be well achieved with
two users. Thus, assuming that two users per resource block can be supported, we may consider
user pairing approaches to maximize the code rate for given channel realizations. Thirdly, of
course, for a better prediction with more than two users, a good instantaneous BER estimate is
required in the power and rate allocation. There might also be other allocation schemes to be
investigated in the future.
APPENDIX A
PROOF OF LEMMA 1
For given 1−and t, there is a unique ¯
βkthat satisfies (25), which is β(t;), because Γ(t+1, λ)
is a decreasing function of λ. From (12), we can show that
Γ(t+ 1, λ)
t!<Γ(t0+ 1, λ)
t0!, t < t0,
23
where tand t0are positive integers. Thus, we have β(t;)< β(t0;)since Γ(t+1,λ)
t!is a decreasing
function of λ, which implies β(t;)is an increasing function of t.
From (17), γkcan be expressed as γk=Ak(1−2¯
βk)2
1−(1−2¯
βk)2. From this, we can derive (26) since
γk=Ekσ2
k
N0with ¯
βk=β(tk;)for given t=tk. We can also show that Ek(t;)is a decreasing
function of t, because Ek(t;)is a decreasing function of β(t;)and β(t;)is an increasing
function of t.
APPENDIX B
PROOF OF LEMMA 2
For given tand {Ek+1, . . . , EK}, if α(Ek)increases with Ek, the solution of (32) is unique
(if exists, which depends on ¯cor ). Thus, we need to show that α(Ek)is an increasing function
of Ek.
Let f(x) = xh 1
x. Then, we can show that f(x)is a decreasing function of x, because df(x)
dx =
ln 1 + 1
x−1
x<0,x > 0. Let x=v
t+1−2Plβl,k . Since v= 2 Plβl,k (1 −βl,k)=2Plβl,k −β2
l,k,
we can see that vdecreases with Ek. In addition, since t+ 1 −2Plβl,k increases with Ek,x
decreases with Ekfor Ek∈ Ek. As a result, f(x)increases with Ek.
We note that
α(Ek) = t+ 1 −2X
l
βl,k!f(x).
Since both f(x)and t+ 1 −2Plβl,k increase with Ekfor Ek∈ Ek, we can claim that α(Ek)
increases with Ek, which completes the proof.
APPENDIX C
PROOF OF LEMMA 3
Let
δ=t+ 1
2L−¯
βk(L)>0.(35)
24
Then, the bound in (29) becomes e−4δ2L. Thus, the bound decreases exponentially with L, which
implies that the probability of codeword error decreases exponentially with Lor nc. For a binary
code, we have Pt
i=0 n
i≤2n−k. For a large nc, from [36], we have
t
X
i=0 nc
i= 2nc(hb(t
nc)+o(1)).(36)
Thus, to keep the inequality in (35), we need to have
1−kc
nc≥hbt
nc≈hbt+ 1
2L≥hb¯
βk(L),(37)
which implies that
R=kc
nc≤1−hb¯
βk(L)(38)
is a sufficient condition for an exponentially decreasing error probability. That is, for a large
ncor L, if the code rate Ris less than the effective capacity, the bound in (29) decreases
exponentially with Lor nc, which completes the proof.
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27
10−3 10−2 10−1
10−3
10−2
10−1
100Channel C1
Target Error Rate
Prob. of Codeword Error
User 1
User 2
Target Error Rate
10−3 10−2 10−1
10−3
10−2
10−1
100Channel C2
Target Error Rate
Prob. of Codeword Error
User 1
User 2
Target Error Rate
(a) (b)
2 4 6 8 10 12 14 16 18 20
Number of Paths
10-3
10-2
10-1
100
Prob. of Codeword Error
User 1
User 2
Target Error Rate
(c)
Fig. 2. The results of the proposed power and rate allocation for coded MC-NOMA with two users when D2= 2 and Emax = 20
dB: (a) Probability of codeword error versus target error rate, , for the channels of C1; (b) Probability of codeword error versus
target error rate, , for the channels of C2; (c) Probability of codeword error versus the number of multipaths, Npath, for the
channels of C2.
28
10-3 10 -2 10-1
Target Error Rate
10-4
10-3
10-2
10-1
Prob. of Codeword Error
User 1
User 2
Target Error Rate
(a)
10-3 10 -2 10-1
Target Error Rate
18
18.2
18.4
18.6
18.8
19
19.2
19.4
19.6
19.8
20
Transmit Power (dB)
User 1
User 2
10-3 10 -2 10-1
Target Error Rate
0.4
0.41
0.42
0.43
0.44
0.45
0.46
0.47
0.48
0.49
0.5
Code Rate
(b) (c)
Fig. 3. Performances of the power and rate allocation for various values of the target error rate, , with Emax = 20 dB, K= 2,
D2= 2, and Npath = 10: (a) Probability of codeword error; (b) the allocated transmit powers; (c) the code rate.
29
10-3 10 -2 10-1
Target Error Rate
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Spectral Efficiency
NOMA
OMA
Fig. 4. Spectral efficiencies of NOMA and OMA for various values of the target error rate, , with Emax = 20 dB, K= 2,
D2= 2, and Npath = 10.
30
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
Normalized Distance of 2nd user
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
Prob. of Codeword Error
User 1
User 2
Target Error Rate
(a)
1 1.2 1.4 1.6 1.8 2
Normalized Distance of 2nd user
10
11
12
13
14
15
16
17
18
19
20
Transmit Power (dB)
User 1
User 2
1 1.2 1.4 1.6 1.8 2
Normalized Distance of 2nd user
0.44
0.445
0.45
0.455
0.46
0.465
0.47
Code Rate
(b) (c)
Fig. 5. Performances of the power and rate allocation for various values of the normalized distance between the BS and the
second user, D2, with Emax = 20 dB, K= 2,Npath = 10, and = 10−2: (a) Probability of codeword error; (b) the allocated
transmit powers; (c) the code rate.
31
5 10 15 20 25 30
Number of Paths, Npath
10-3
10-2
10-1
Prob. of Codeword Error
User 1
User 2
Target Error Rate
(a)
5 10 15 20 25 30
Number of Paths, Npath
18.4
18.5
18.6
18.7
18.8
18.9
19
19.1
19.2
19.3
19.4
Transmit Power (dB)
User 1
User 2
5 10 15 20 25 30
Number of Paths, Npath
0.436
0.438
0.44
0.442
0.444
0.446
0.448
0.45
Code Rate
(b) (c)
Fig. 6. Performances of the power and rate allocation for various numbers of multipaths, Npath, with Emax = 20 dB, K= 2,
D2= 2, and = 10−2: (a) Probability of codeword error; (b) the allocated transmit powers; (c) the code rate.
32
10 12 14 16 18 20 22 24 26 28 30
Emax (dB)
10-3
10-2
10-1
Prob. of Codeword Error
User 1
User 2
User 3
Target Error Rate
(a)
10 15 20 25 30
Emax (dB)
5
10
15
20
25
30
Transmit Power (dB)
User 1
User 2
User 3
10 15 20 25 30
Emax (dB)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Code Rate
(b) (c)
Fig. 7. Performances of the power and rate allocation for different values of maximum transmit power Emax, with K= 3,
(D2, D3) = (1.5,2),Npath = 20, and = 10−2: (a) Probability of codeword error; (b) the allocated transmit powers; (c) the
code rate.
