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Design of SCMA Codebooks Using Differential Evolution
Kuntal Deka1, Minerva Priyadarsini1 , Sanjeev Sharma2, and Baltasar Beferull-Lozano (Senior Member, IEEE)3
1 Indian Institute of Technology Goa, India, 2 Indian Institute of Technology (BHU) Varanasi, India
3 Department of Information and Communication Technology, University of Agder, Grimstad 4879, Norway
Abstract—Non-orthogonal multiple access (NOMA) is a multi-user detection is usually done by the message passing
promising technology which meets the demands of massive algorithm (MPA) [4].
connectivity in future wireless networks. Sparse code multiple Related work: The performance of an SCMA system is
access (SCMA) is a popular code-domain NOMA technique. The
effectiveness of SCMA comes from: (1) the multi-dimensional mainly determined by the codebooks of the users. The opti-
sparse codebooks offering high shaping gain and (2) sophisticated mization of the codebooks for SCMA system is a convoluted
multi-user detection based on message passing algorithm (MPA). task as multiple users are interfering with multi-dimensional
The codebooks of the users play the main role in determining the complex vectors. In [3, 5], first, the optimum codebook for
performance of SCMA system. This paper presents a framework one user is designed. The remaining codebooks were obtained
to design the codebooks by taking into account the entire system
arXiv:2003.03569v1 [cs.IT] 7 Mar 2020
including the SCMA encoder and the MPA-based detector. The by carrying out user-specific operations on the optimized
symbol-error rate (SER) is considered as the design criterion codebook. In [6] (referred to as “Zhang [6]”), the sum-rate
which needs to be minimized. Differential evolution (DE) is used was considered as one of the design criteria. First, a series of
to carry out the minimization of the SER over the codebooks. one-dimensional complex codewords were designed. Then the
The simulation results are presented for various channel models. angles of these codewords were changed with the objective
of improving the sum-rate. The authors in [7] considered the
Keywords SCMA, codebook design, differential evolution, minimum Euclidean distance and the energy diversity of the
message passing algorithm. constellation points to obtain the optimum codebooks. Yu et
al. (referred to as “Yu [8]”) designed SCMA codebooks based
I. I NTRODUCTION on the star quadrature amplitude modulation (QAM) constel-
lations [8]. In [9], the authors designed codebooks for bit-
The future wireless communication system will comprise interleaved convolutionally-coded SCMA system. This design
of enormous number of interconnected devices. The massive was aided by the analysis based on EXtrinsic Information
connectivity required in such situations cannot be fulfilled Transfer (EXIT) chart. Sharma et al. [10] (referred to as
by the multiple access schemes deployed so far. Earlier, 3G “Sharma [10]”) designed the codebooks by maximizing the
system used code division multiple access (CDMA), 4G used mutual information and shaping gain. Kim et al. proposed a
orthogonal frequency division multiple access (OFDMA) [1]. deep-learning-based method where, the codebook is dynami-
All these technologies are based on orthogonal multiple access cally designed with the objective of minimizing the bit error
(OMA) principle. In OMA, the number of supported users is rate (BER) [11].
limited by the number of available orthogonal resources. The Contributions: The existing codebook-design methods
non-orthogonal multiple access (NOMA) technology provides mainly focus on various geometric properties of the multi-
a gateway for massive connectivity. It supports overloaded dimensional constellations with little emphasis on detection.
systems where the number of users is higher than the number It is difficult to track the MPA-based detection process math-
of orthogonal resources. NOMA techniques can be broadly ematically. The factor graph is finite, not tree-like and it
classified into two groups: power domain and code domain. In contains short cycles. Due to these reasons, the techniques like
power domain NOMA, different levels of powers are allocated density evolution and EXIT chart cannot accurately character-
to different users. In code domain NOMA, different code- ize the MPA-based multi-user detection process. We propose
words or signatures are used for different users. Low-density to consider the symbol error rate (SER) as the cost function
spreading (LDS) is a code-domain NOMA technique where, which is to be minimized. The SER is one such quantity which
a user’s symbol is multiplied with a distinct sparse spreading takes into account every part of the multi-user system be it
signature. This spread-ed sequence is mapped to modulation Euclidean distance profile, product distance profile, MPA etc.
constellation points for transmission [2]. Nikoupor et al. came The reliability of the system is precisely reflected by the SER.
up with the technique of sparse code multiple access (SCMA) We adopt differential evolution (DE) for the minimization of
in order to improve upon LDS [3]. In SCMA, the operations of the SER as it is not a simple function of the codebooks.
the spreading and the modulation mapping are merged. Based DE is a flexible and effective evolutionary algorithm which is
on a dedicated codebook, a symbol is directly mapped to a used to solve complex optimization problems with real-valued
sparse multi-dimensional codeword. Thus, SCMA provides parameters [12, 13]. First, the structure of the codebooks is
a better opportunity than LDS to attain high shaping gain. represented with the help of a finite number of constellation
Owing to the inherent sparsity in the code-domain NOMA, the points. Then a DE-based optimization process is invoked to
find the optimum constellation points. The codebooks for minimum distance between any pair (xm , xn ) of codewords
the additive white Gaussian noise (AWGN) and Rayleigh in the entire SCMA system:
fading channels are designed. The SER performance of the
dE,min = min ||xm − xn ||.
proposed codebooks are compared with those of the existing m,n
ones. Moreover, various key parameters of the codebooks are • Euclidean kissing number (τE ): It is defined as the number
computed and analyzed. of distinct codeword pairs with Euclidean distance equal to
Outline: Section II describes the preliminaries such as dE,min .
SCMA system model, important parameters of codebooks and • Minimum product distance (dP,min ): The product distance
DE. The proposed method of codebook design is presented in dm,n between two codewords xm and xn is defined as
P
Section III. The simulation results are presented and analyzed Y
in Section IV. Section V concludes the paper. dm,n
P = |xmj − xnj |
j∈Jm,n
II. P RELIMINARIES
A. System Model where, Jm,n is the set of dimensions for which xmj 6= xnj .
The minimum product distance dP,min is given by:
A J × K SCMA system refers to an overloaded multi-
user scenario with J users and K resource elements. The dP,min = min dm,n
P .
J m,n
overloading factor is given by λ = K . A dedicated code-
book Cj containing M K-dimensional codewords: Cj = • Product Kissing Number (τP ): It is defined as the number of
{xj1 , xj2 , . . . , xjM } is assigned to every j th user. Each code- distinct codeword pairs with product distance equal to dP,min .
word xjm is sparse with N non-zero complex components. The objective is to maximize dE,min , dP,min and minimize
Based on the assigned codebook Cj , log2 (M ) data bits τE , τP . In addition to these parameters, mutual information
are directly mapped to the K-dimensional codeword xj = between the received signal and the sum of the interfering
T T
[xj1 , ..., xjK ] . The received signal y = [y1 , . . . yK ] is: codewords is also a vital parameter. Suppose Y is the signal
J received over one resource element. Let S represent the
sum of the codewords of the df users interfering over that
X
y= diag (hj ) xj + n (1) df
j=1 a total of M
resource. We have distinct sum values for
S as given by s1 , s2 , . . . , sM df . A high value of I(Y ; S)
T
where, hj = [h1 , . . . , hK ] is the channel gain vector for the ensures successful recovery of the individual user’s data from
j th user and n is a complex K ×1 AWGN vector. The locations the noisy sum value. Usually a lower bound IL on I(Y ; S) is
considered1. IL for AWGN channel with noise variance N0 is
given by [6]
1
PM df PM df
1 2
IL = log2 M df − log2 1 +
M df j=1 i=1 exp − 4N 0
|s j − s i | .
i6=j
(3)
Fig. 1: SCMA block diagram.
B. Differential Evolution
of the non-zero elements of the codewords for the users can be
represented with the help of a factor matrix as shown in (2). In DE, an initial set of SP random candidate solutions
The 1s present in the jth column specify the locations of the or vectors {pi : i = 1, 2, . . . , SP } is generated. The length
non-zero components of the codewords for the jth user. The of each vector is the same and it is denoted by D. New
matrix F can be alternatively represented by a factor graph as generations are created iteratively using mutation, cross over
shown in Fig. 1. The degree of a resource node is denoted by and selection process. During mutation, each vector pG i , where
df . This specifies that df users interfere with each others over G denotes the generation index, is selected as a primary parent.
one resource element. As the factor graph is sparse, MPA is For each parent, a mutant vector is generated as follows:
used for multi-user detection. uG
i = pr1 + α(pr2 − pr3 ) (4)
1 0 1 0 1 0
0 1 1 0 0 1 where, r1 , r2 , r3 are randomly selected distinct numbers dif-
F= 1 0 0 1 0 1
(2) ferent from i and α is the scaling factor. uG i is the secondary
0 1 0 1 1 0 parent generated from mutation. Cross-over is applied to
primary and secondary parent to obtain the offspring vector
The performance of an SCMA system is highly sensitive to viG . Each component vik G
of viG = [vi1
G G
, . . . , vik G
, . . . , viD ] is
the codebooks. In literature [14, 15], various key performance inherited from either the primary or the secondary parent as
indicators (KPIs) have been considered for the design of multi-
dimensional constellations. A few of them are highlighted:
• Minimum Euclidean Distance (dE,min): It is defined as the 1 I(Y ; S) or IL is related to the sum-rate [6].
per the following rule: where, ai ∈ C, i = 1, . . . , 6.
( In (6), 12 distinct complex numbers are created from 6 com-
G uG
ik , if hk ≤ Cr plex numbers and their negative counterparts. It is preferable
vik = G
pik , otherwise to design the SCMA codebooks with the minimum possible
where, hk is a random number uniformly distributed over number of constellation points. This reduces the hardware
[0,1], Cr is the cross-over rate. The offspring is made to inherit requirement in implementation.
The one-dimensional codebooks can be assigned to all the
at least one component from the secondary parent to ensure
resource nodes through a structure matrix satisfying Latin
that offspring is different from primary parent. The offspring
vector viG has to compete with the parent vector pG property [16]. This way of generating the entire set of the
k to get a
codebooks was considered in [6, 17]. The structure matrix for
place in the next G + 1 generation. If vkG gives a lower cost
function than pG the 6 × 4 SCMA system is shown below:
k then the former replaces the later, else the o
C1 0 C2o 0 C3o 0
primary parent vector exists in the next generation too, i.e.
( 0 C2o C3o 0 0 C1o
viG , if f (viG ) ≤ f (pG
i )
FL = o
(7)
pi
(G+1)
= C2 0 0 C1 0 C3o
o
G
pi , otherwise o
0 C1 0 C3 C2 0 o o
where f (·) is the objective function that needs to be min- The Latin property in (7) ensures that distinct one-
imized. The new generation performs at least as good as dimensional codebooks are assigned to the users overlapping
the best candidate of the previous generation. These steps over any particular resource. Using FL , the set of all code-
are continued until some stopping criteria are satisfied. The books can be generated as follows:
best vector from the current population is considered as the o o
C1 0 C2,p
optimum vector. 0 C2o C3o
C1 = C2o C2 = 0 C3 = 0
III. P ROPOSED C ODEBOOK D ESIGN BASED ON
D IFFERENTIAL E VOLUTION 0 Co 0
1o (8)
Let C = {C1 , . . . , CJ } denote the set or the collection of 0 C3 0
0 0 C1o
the codebooks of all the users. The task of SCMA codebook C4 = C1o C5 = 0 C6 = C3o
design is formulated as an optimization problem as follows: o
C3o C2,p 0
Eb
Copt = arg min f C; ,F where, 0 is the all-zero row vector of length M = 4.
C N0 (5)
In order to obtain more diversity and higher shaping gain,
s.t. ||c||2 = 1 ∀c ∈ C
the authors in [6, 17] have proposed to carry out the di-
Eb mensional permutation switching algorithm (DPSA). As per
where, f (·) is the SER at SNR N 0
and F is the factor
graph matrix. Every K-dimensional codeword c present in DPSA, the permuted version of C2o denoted by C2,p o
is used
o
the SCMA codebook system C is constrained to have unit in the codebooks C3 and C5 in place of C2 . We consider
o
Euclidean norm. C2,p = [−a4 , a3 , −a3 , a4 ]. With this dimensional permu-
Note that an already-designed factor graph matrix F is fed tation, using (6) in (8), the codebooks can be written as
to the optimization process in (5). Suppose the factor graph is shown in TABLE I. Observe that the codebooks for the 6 × 4
regular with the resource node degree of df . Then, df users TABLE I: Structure S of the codebooks C = {C1 , . . . , C6 }
overlap on a particular resource node. One needs to carefully
assign constellation points to these users. The task is to assign
a1 a2 −a2 −a1
0 0 0 0
0 0 0 0 a3 a4 −a4 −a3
df distinct one-dimensional constellation2 codebooks to the
C1 = a3 a4 −a4 −a3 C 2 = 0 0 0 0
df overlapping users. These codebooks may be represented
0 0 0 0 a1 a2 −a2 −a1
by C1o , C2o , . . . , Cdof , where the superscript ‘o’ signifies that
−a4 a3 −a3 a4
0 0 0 0
these are one-dimensional. The size of the constellation is M . a5 a6 −a6 −a5 C4 = 0 0 0 0
C3 = 0 0 0 0 a1 a2 −a2 −a1
For example, consider the case of 6×4 SCMA system with
0 0 0 0 a5 a6 −a6 −a5
the factor graph shown in Fig. 1 and the matrix in (2). Here,
a5 a6 −a6 −a5 0 0 0 0
df = 3 and M = 4. The three one-dimensional codebooks
0 0 0 0 C6 = a1 a2 −a2 −a1
can be represented as follows: C5 = 0 0 0 0 a5 a6 −a6 −a5
−a4 a3 −a3 a4 0 0 0 0
C1o = [a1 a2 − a2 − a1 ]
C2o = [a3 a4 − a4 − a3 ] (6) SCMA system can be represented with the help of 6 complex
C3o = [a5 a6 − a6 − a5 ] numbers ai , i = 1, . . . , 6. These complex numbers can be
compactly represented by a = [a1 , . . . , a6 ]. Any particular
2 One-dimensional codebook is assigned as only one resource node is complex vector ai ,ai ∈ C6 refers to a particular set of
considered at a time. codebooks Ci = C1i , . . . , C6i . With these notations, the
SCMA codebook design problem in (5) can now be written mum vector a which contains 6 complex numbers3. However,
as DE is a real-valued optimization technique. Therefore, in this
Eb case, the number of real-valued variables to be optimized is
aopt = arg min f a; (9)
a N0 D = 12. The detailed steps for the DE-based codebook design
Eb
are presented in Algorithm 1. The inputs to the algorithm
where, f a; N is the SER of the SCMA system defined
0 are the structure S for the codebooks as shown in TABLE I,
Eb
by a at SNR N0. In the rest of the paper, for notational SNR N Eb
and the DE parameters: α, Cr ,SP and Imax .
0
Eb The candidate vectors are stored in a population matrix P
brevity, f a; N 0
will be represented as f (a) with the
understanding that the SNR is fixed at a particular value during in row-wise manner. The total number of candidate vectors
the optimization process. In (9), aopt is that vector a which is SP . The size of P is SP × D. The elements of P
yields the optimum codebook Copt as per the structure given in are initialized to random numbers uniformly distributed over
TABLE I. Moreover, we have the constraint that the Euclidean [−1, 1]. Every row ps , s = 1, . . . , SP , of P corresponds to an
norm of every codeword is 1 although it is not explicitly SCMA codebook system specified by the 6 complex numbers
mentioned in (9). as = [as,1 , . . . , as,6 ]. Suppose as,t = ars,t +acs,t i, t = 1, . . . , 6,
where ars,t and acs,t are real uniform numbers in [−1, 1]. The
sth row ps of P is given by
Algorithm 1: Codebook design using differential evolution
ps = ars,1 , acs,1 , ars,2 , acs,2 , . . . , ars,6 , acs,6 .
Eb
input : Codebook structure S (TABLE I), N0
, DE parameters
(α, Cr , SP and Imax ) The elements of P are normalized so that every codeword
output: Codebooks Copt = {C1 , . . . , C6 }.
in a codebook system has unit norm. Against each row
Initialize the population matrix P to a matrix of size SP × D ps , s = 1, . . . , SP , a trial vector u is generated with the
having random numbers uniformly distributed over [−1, 1]
where D = 12, Normalize these entries so that the every
given values of the DE parameters: crossover rate (Cr ) and
codeword of the codebook has unit norm; scaling factor (α). Algorithm 1 shows the detailed steps of the
while termination criteria not fulfilled do generation of the trial vector u. The decision regarding the
for i ← 1 to SP do replacement of the current population vector ps by the trial
Select three distinct vectors (rows) pr0 ,pr1 and pr2 vector u is made by computing the SER values for the two
uniformly at random from P such that they are also
different from pi ; SCMA systems through Monte Carlo simulations. If the SER
Generate an integer jrand uniformly at random from value f (u) of the SCMA system defined by u is less than
{1, 2, . . . , D}; the SER f (pi ) of the system defined by pi , then the current
/* Generation of trial vector u */ population vector pi is set to u. In this way, every candidate
for j ← 1 to D do vector in P is examined and updated with the trial vectors
if rand[0,1] ≤ Cr or j = jrand then
uj,i = pj,r0 + α × (pj,r1 − pj,r2 ) if necessary. From the updated population P, the best vector
// Crossover and Mutation popt with the minimum SER value is identified. The above-
else mentioned process is repeated unless f (popt ) is not changing
uj,i = pj,i significantly during two consecutive iterations or the maximum
Normalize the entries of u so that every codeword has number of iterations are exhausted. The real numbers in popt
unit norm. converted to the corresponding complex numbers to obtain
/* Evaluation and Selection */ aopt . The complex numbers in aopt = [a1 , . . . , a6 ] are plugged
Suppose Cu and Cpi are the two sets of the into S in TABLE I to obtain the optimum codebooks.
codebooks for all users as per u and pi respectively.
Run Monte Carlo simulation for the SCMA system Example 1. Consider the case of J = 6, K = 4 SCMA system
Eb
with the codebooks Cu and Cpi at SNR N 0
. with constellation size M = 4. The structure given in TABLE I
Suppose, f (u) and f (pi ) are the respective SER
is used. The number of variables is D = 12. We can consider
values ;
if f (u) < f (pi ) then SP = 20, Cr = 0.95 and α = 0.6. The population matrix P
pi = u // Replace the ith row of P by u; may be initialized to the following 20 × 12 matrix:
From the updated population matrix P, find the vector
0.46 −0.53 −0.90 −0.22 −0.68 0.16 −0.34 −0.08 0.22 0.91 0.49 0.48
(row) popt which yields the minimum value of SER; −0.02 0.90 0.44 0.08 −0.44 0.01 0.32 −0.84 0.10 0.24 0.82 0.12
If f (popt ) is not changing significantly from the previous
−0.65 −0.62 0.80 0.11 −0.41 −0.15 −0.55 0.22 0.56 0.22 0.41 0.15
P= .
· · · · · · · · · · · ·
iteration or the maximum number of iterations Imax are
· · · · · · · · · · · ·
exhausted, then break from loop; · · · · · · · · · · · ·
Using pmin , form the optimum constellation vector Due to space constraint, only 3 rows out of 20 are shown. It
aopt = [a1 , . . . , a6 ]. Then inserting a in S, the optimized set
of the codebooks of all users Copt = {C1 , . . . , CJ } are
may be verified that the norm of every codeword in P is 1.
obtained. ; 3 It is noteworthy that the proposed method of codebook design is universal
in the sense that it can be applied to an SCMA system with any values of
J and K. The structure as shown in TABLE I must be fed to the DE-based
We solve (9) with the help DE. The job is to find the opti- optimizer.For every row of P, a trial vector is generated by carrying
out the mutation and the crossover operations as specified in
Algorithm 1. If the trial vector yields less SER than the current
row vector, then the current row is overwritten by the trial
vector. In this way every row of P is examined and updated if
needed. Suppose, finally, the population matrix P is as given
below where the first row is the best row pmin .
−0.33 0.63 −0.83 0.43 0.71 0 −0.36 0 −0.42 −0.84 0.59 0.35
· · · · · · · · · · · ·
P= .
· · · · · · · · · · · ·
· · · · · · · · · · · ·
Then the solution of (9) is given by the following:
aopt = [−0.33 + 0.63i, −0.83 + 0.43i, 0.71, −0.36, −0.42 − 0.84i, 0.59 + 0.35i] .
Using the above aopt in TABLE I, we can generate the code-
books for the SCMA system. TABLE II shows these codebooks
in Section IV.
Remark 1. Although the proposed scheme appears to be SNR-
dependent, the codebooks optimized at a high SNR are found
to work well at different SNR values. In practice, we fix an
SNR where the SER is around 10−3 . The results for different Fig. 2: aopt for AWGN channel.
SNR values are not shown here due to space constraint.
TABLE II: Codebooks optimized for AWGN channel
Complexity Analysis: Observe from Algorithm 1 that during
−0.3318 + 0.6262i −0.8304 + 0.4252i 0.8304 − 0.4252i 0.3318 − 0.6262i
each cycle of DE, the objective functions (SERs) for the
0 0 0 0
C1 =
0.7055 −0.3601 0.3601 −0.7055
current population and the trial vector are computed. The SER
0 0 0 0
computation is a time-consuming process due to fact that the
0 0 0 0
0.7055 −0.3601 0.3601 −0.7055
complexity of the MPA is O M df . Thus, the complexity
C2 =
0 0 0 0
−0.3318 + 0.6262i −0.8304 + 0.4252i 0.8304 − 0.4252i 0.3318 − 0.6262i
of proposed algorithm is higher than some of the existing
0.3601 0.7055 −0.7055 −0.3601
codebook designing methods. However, note that the codebook
−0.4202 − 0.8350i 0.5933 + 0.3548i −0.5933 − 0.3548i 0.4202 + 0.8350i
C3 =
0 0 0 0
design is a one-time and offline task. It does not add to the real-
0 0 0 0
time complexity of the system. Therefore, the proposed DE-
0 0 0 0
0 0 0 0
based codebook design method is feasible in practical scenario. C4 =
−0.3318 + 0.6262i −0.8304 + 0.4252i 0.8304 − 0.4252i 0.3318 − 0.6262i
−0.4202 − 0.8350i 0.5933 + 0.3548i −0.5933 − 0.3548i 0.4202 + 0.8350i
IV. S IMULATION R ESULTS
−0.4202 − 0.8350i 0.5933 + 0.3548i −0.5933 − 0.3548i 0.4202 + 0.8350i
0 0 0 0
C5 =
The simulations are carried out to evaluate the performance
0
0.3601
0
0.7055
0
−0.7055
0
−0.3601
of the proposed codebooks. For comparison, the following
0
0
0
0
codebooks are considered: • “Zhang [6]” • “Sharma [10]” −0.8304 + 0.4252i 0.8304 − 0.4252i 0.3318 − 0.6262i
−0.3318 + 0.6262i
C6 = −0.4202 − 0.8350i 0.5933 + 0.3548i −0.5933 − 0.3548i 0.4202 + 0.8350i
• “Yu [8]” • “Ma [18]” (Chapter 12 of [18]). As per the
0 0 0 0
thumb rules provided in [19, 20], the DE parameters are set to:
SP = 20, D = 12, Cr = 0.95, α = 0.6 and Imax = 80. The
simulations are done for uncoded transmission over AWGN in Fig. 3. The SER plots for the other codebooks are also
and flat Rayleigh fading channels. presented. Observe that the proposed codebooks outperform
the others by a significant margin. Specifically, there is cod-
A. AWGN channel ing gain of about 1.4 dB at SER=10−5 over the next best
First the results for the AWGN channel are presented. codebooks (“Sharma [10]”). The performance of “Ma [18]”
An SCMA system with J = 6, K = 4 and M = 4 is over AWGN channel is not satisfactory. Therefore, its SER
Eb
considered. Algorithm 1 is run at N 0
= 10 dB to find the plot is not shown in Fig. 3. However, “Ma [18]” yields good
optimum vector aopt . The complex numbers of aopt and their results over Rayleigh fading channel which is presented later
negative versions are shown in Fig. 2. With this value of in this section.
aopt = [a1 , a2 , a3 , a4 , a5 , a6 ], the optimum codebooks are The SER performance of the various codebooks can be
generated using the structure S shown in TABLE I. These studied and justified by carrying out the mutual information
codebooks are shown in TABLE II. Observe that Euclidean analysis as mentioned in Section II-A. We plot the lower
norm of every codeword is 1. bound IL on mutual information for various codebooks in
The SER performance of the proposed codebooks are eval- Fig. 4. Observe that in the SNR region below 16 dB, “Shamra
uated through Monte Carlo simulations and these are shown [10]” and “Zhang [6]” provide higher values of IL than theB. Fading channel
In this case, each user observes independent Rayleigh fading
channel coefficients over the resource elements. The same
SCMA framework with J = 6, K = 4, M = 4 and the
structure S given in TABLE I is considered. Algorithm 1 is
Eb
executed at N 0
= 17 dB. It yields the optimum constellation
vector aopt which are plotted in Fig. 5. The complex numbers
Fig. 3: SER performance of the SCMA system using various
codebooks for J = 6 and K = 4 in AWGN channel.
Fig. 5: aopt for fading channel.
in aopt = [a1 , . . . , a6 ] are put in S. The resulting codebooks are
shown in TABLE III. Observe that the optimized codebooks
for the fading channel are different from those for the AWGN
channel. This signifies that the codebook design problem
depends on the underlying channel model.
TABLE III: Codebooks optimized for fading channel
−0.3344 − 0.7316i −0.5754 + 0.2224i 0.5754 − 0.2224i 0.3344 + 0.7316i
0 0 0 0
C1 =
0.4153 − 0.4248i 0.4680 + 0.6328i −0.4680 − 0.6328i −0.4153 + 0.4248i
0 0 0 0
0 0 0 0
Fig. 4: Lower bound on the mutual information for various
0.4153 − 0.4248i 0.4680 + 0.6328i −0.4680 − 0.6328i −0.4153 + 0.4248i
C2 =
0 0 0 0
codebooks.
−0.3344 − 0.7316i −0.5754 + 0.2224i 0.5754 − 0.2224i 0.3344 + 0.7316i
−0.4680 − 0.6328i 0.4153 − 0.4248i −0.4153 + 0.4248i 0.4680 + 0.6328i
−0.1492 − 0.5839i 0.7759 − 0.1713i −0.7759 + 0.1713i 0.1492 + 0.5839i
C3 =
proposed codebooks. However, the IL for “Yu [8]” and the
0 0 0 0
0 0 0 0
proposed codebooks reach the maximum value of 6 quicker
0
0
0
0
than the other codebooks. Observe that IL for “Yu [8]” is
0 0 0 0
C4 =
−0.3344 − 0.7316i −0.5754 + 0.2224i 0.5754 − 0.2224i 0.3344 + 0.7316i
slightly higher than that for the proposed codebooks in the
−0.1492 − 0.5839i 0.7759 − 0.1713i −0.7759 + 0.1713i 0.1492 + 0.5839i
range of 15-20 dB. They reach the maximum value almost
−0.1492 −
0
0.5839i
0.7759 −
0
0.1713i
−0.7759 +
0
0.1713i
0.1492 +
0
0.5839i
C5 =
at the same SNR of 24 dB. However, as described later in
0
0
0
0
−0.4680 − 0.6328i 0.4153 − 0.4248i −0.4153 + 0.4248i 0.4680 + 0.6328i
this section, the Euclidean distance and the product distance
0
0
0
0
profiles of “Yu [8]” are poorer than those of the proposed one.
−0.3344 − 0.7316i −0.5754 + 0.2224i 0.5754 − 0.2224i 0.3344 + 0.7316i
C6 = −0.1492 − 0.5839i 0.7759 − 0.1713i −0.7759 + 0.1713i 0.1492 + 0.5839i
This observation reinforces the superior performance of the
0 0 0 0
proposed DE-based codebooks. Also note that the IL for “Ma
[18]” cannot climb up to the maximum value. This justifies The SER performance of the proposed codebooks along
the poor performance of “Ma [18]” over the AWGN channel. with those of the other existing ones are shown in Fig. 6.The progress of Algorithm 1 with increasing iteration is
depicted in Fig. 7. The minimum SER corresponding to the
best vector/row in the population matrix P as updated in
the current iteration is plotted against the iteration number.
Observe from Fig. 7 that the minimum SER tends to settle
down at a value after 20 iterations. We have observed similar
progression of the DE-based algorithm in the case of AWGN
channel. However, due to space constraint, the plot for AWGN
channel is not included in the paper.
The values of the KPIs mentioned in Section II-A are shown
in TABLE IV for various codebooks. “Zhang [6]” has the best
Euclidean distance profile. Its dE,min is the highest and τE
is the lowest. Proposed codebook (AWGN) has the highest
dP,min , however the τP is not the lowest. The Euclidean
distance parameters for “Ma [18]” are the worst since dE,min
is the lowest and τE is the highest. Its poor performance over
AWGN channel can be attributed to this fact. However, its
product distance profile is impressive. Its dP,min is not the
lowest and τP has the lowest value. However, it is difficult
to justify the reported SER performances completely with the
Fig. 6: SER performance of the SCMA system using various
help of the KPIs mentioned in TABLE IV.
codebooks for J = 6 and K = 4 in Rayleigh fading channel.
TABLE IV: Key performance indicators
Proposed Proposed
Zhang [6] Sharma [10] Yu [8] Ma [18]
Observe that the proposed codebooks produce the best results. (AWGN) (Fading)
At SER=10−5 , we experience a coding gain of about 1.5 dB dE,min 0.8966 0.8625 1.0171 0.9976 0.5351 0.3883
over the next best methods: “Zhang [6]” and “Ma [18]”. We τE 4 4 2 2 2 4
dP,min 0.1103 0.0595 0.0810 0.0544 0.0379 0.0448
also evaluate the SER performance of the proposed codebooks τP 4 4 4 4 2 2
designed for the AWGN channel. However, the performance
is not satisfactory and inferior to “Zhang [6]”, “Ma [18]” and
These KPIs only partially characterize the SCMA system.
“Yu [8]”. This observation reiterates the well known fact that
The SCMA system is a complicated mult-user scenario where
the optimum constellation for AWGN may not be optimum
the detection is carried out by the sophisticated MPA. These
for Rayleigh fading channel and vice versa [14].
KPIs fail to take the MPA-based detection process into ac-
count. Thus they are inadequate to facilitate a conclusive
comparative analysis of various codebooks.
The above codebooks can be enlarged to build J = 8, K =
4 SCMA systems with 200% overloading factor. We consider
the following factor matrix:
1 0 1 0 1 0 1 0
0 1 1 0 0 1 1 0
F= 1 0 0 1 0 1 0 1.
0 1 0 1 1 0 0 1
The 3rd and the 4th columns are repeated as the 8th and
7th ones respectively. The constellation points for C3 and C4
are exchanged between the 7th and the 8th users. The SER
performances of these codebooks are shown in Fig. 8 for
Rayleigh fading channel. In this case also, the proposed DE-
based codebooks yield the best result.
V. C ONCLUSIONS
This paper presented a method to design the codebooks for
an SCMA system. The constellation points are designed with
Fig. 7: Minimum SER versus iteration in Rayleigh fading the objective of minimizing the SER. The SER is considered
Eb
channel at N 0
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