International Journal of Latest Research in Science and Technology ISSN (Online):2278-5299

Volume 7, Issue 3: Page No.14-18 , May-June 2018

https://www.mnkpublication.com/journal/ijlrst/index.php

THE EUCLIDIAN DISTANCE-BASED DETECTION

METHOD APPLIED ON SCMA CODEWORDS
Sergio Vidal Beltrán1,José Luis López Bonilla1, Fernando Martínez Piñon2
1
Instituto Politécnico Nacional,Escuela Superior de IngenieríaMecánica y Eléctrica- Zacatenco. México
2
Instituto Politécnico Nacional. Centro de Investigación e Innovación Tecnológica- Azcapotzalco, México
svidalb@ipn.mx,fmartinezp@ipn.mx,joseluis.lopezbonilla@gmail.com

Abstract : In this paper, we present the results of IQ parameter detection based on minimum Euclidean distance, when the data is
encoded using Sparse Code Multiple Access and transmitted through a channel affected by Gaussian additive white noise. The results
are presented using different values of signal to noise ratio in the channel.
Keywords – IQ parameters, SCMA, AWGN, Euclidian distance, CodeBooks

I. INTRODUCTION
Fifth generation (5G) mobile communications networks users. These factors benefit the form and the coding gain [6-
will improve the performance of mobile cellular networks by 8].
advancing speed, capacity, latency and connectivity. It is b 11 b 12 SCMA
Codeword 1 SCMA
FEC 1
expected that 5G will achieve transmission rates in the order Encoder Mapping
of Gbps, reliable access with high bandwidth, massive FEC 2 b 21 b 22 SCMA
SCMA Block

connectivity, very low latency (< 5ms) and reliability greater Encoder Mapping

than 99.999% [1-3]. For these reasons, the radio access FEC 3 b 31 b 32 SCMA
U ser 1

U ser 2

network must have efficient administration, control of Encoder Mapping U ser 3

multiple layers and a wide range of air interfaces to allow the FEC 4 b 41 b 42 SCMA U ser 5
U ser 4

Encoder Mapping
aggregation of different networks. In order to comply with U ser 6

Sucarrier 4
the above requirements, we seek to take advantage of the FEC 5 b 51 b 52 SCMA
Sucarrier 1 Sucarrier 2 Sucarrier 3

Encoder Mapping
non-orthogonality of multiple access through SCMA (Sparse
Code Multiple Access), which provides greater spectral FEC 6 b 61 b 62 SCMA
Encoder Mapping Codeword 6 SCMA
efficiency than that achieved in LTE (Long Term Evolution)
and UMTS (Universal Mobile Telecommunications System).
Fig. 1. SCMA Encoder.
A. Sparse Code Multiple Access
SCMA does not use the QAM symbol constellations used II. DESIGN OF SCMA CODE BOOKS
by CDMA methods, instead it directly encodes the user bits SCMA, being low density codes allows to have coding
into multidimensional codewords. Fig. 1 shows the basic gain [9, 10]. For the elaboration of this work the rotation and
diagram of an SCMA encoder with six physical resources interleaving of mother constellations is used to generate the
(subcarriers) and four code words in the SCMA codebook users' code books, considering 6 layers (users), 4 subcarriers,
[4]. Each user or layer assigns the binary data output of the 4 codewords and up to 3 users connected to each radio
FEC (Forward Error Correction) directly to complex code resource, and two dimensions to design the constellations.
words that are assigned to physical resources according to a Next, the process to generate the code books is described.
spreading code defined by the SCMA codebook.
A. Gray Mapping
SCMA is a successor to LDS (Low Density Spreading)
First, a set of Gray mapping vectors is chosen, which are
[5], which uses low-density codebooks to reduce the
the representation of each point of the constellation (mother
complexity of symbol detection. With this method each user
constellation). A subset is defined S1 of Z2, where Z is a set
is assigned a codebook, and the data in it is used to map the
of integers, given by (1):
bit stream directly to a low density vector (sparse vector)
called a codeword. The multidimensional codebooks are used
to complement the QAM modulation and the spreading (1)
depends on this factor. Each contains zero value in the same
two dimensions of all its elements and the position of the M = 4 (code words)
zeros is determined randomly to prevent collisions between A. 2.2 Constellation values for the first dimension

Publication History
Manuscript Received
ISSN:2278-5299 : 30 May 2018 14
Manuscript Accepted : 8 June 2018
Revision Received : 24 June 2018
Manuscript Published : 30 June 2018
 International Journal of Latest Research in Science and Technology.

Values are assigned to each point of S1. Considering M = (6)

4, Gray's mapping takes the next form: So the elements for Sl result in (7):
S11 0 0 → -3(1+J) (7)
S12 0 1 → -(1+J)
S13 1 1 → (1+J)
S14 1 0 → 3(1+J) The values of S1,2, S1,1 are generated from the negative
values of S1,4and S1,3. So that:
B. Rotation angle for the second dimension. (8)
Rotation angle is calculated according to (2):
therefore, it is a multidimensional SCMA codebook that is
(2) defined as (9):

Let N be the number of dimensions (N = 2). (9)

, where UN is the Based on the above S2 is defined by:
phase rotation matrix and 1 is an N-dimensional vector of all
S21 0 0 → -0.54 - 1.30j
1's. Fig. 2 shows the rotation of the points of the constellation
S22 0 1 → 1.62 + 3.91j
for 3 dimensions:
S23 1 1 → -1.62 - 3.96j
S24 1 0 → 0.54+1.30j
Fig. 3 shows the mother constellation SCMA (MC), it is
observed that S2 is formed by the rotation of S1 which is a
subset of a QAM constellation.

Fig. 2. Rotation vector of the mother constellation.

The Gray mapping based on the mother constellation (N-
dimensional), takes the form of (3).

(3)

For this case, all dimensions have the same power and the
code words have the same average power.

C. Interleaving and reordering Fig. 3. Mother constellation and rotation of the

constellation
The elements of the even dimensions of the mother
constellation are interleaved, as shown in (4), in order to D. Code books for different users
reduce the average power. The code books are obtained from the mother constellation
(MC), for each user. The code words are rotated, taking care
that the structure and the Euclidean distance remain
(4) unchanged.

Therefore, the mother or base constellation is rewritten as (5): Considering that and that represents the users
that can use a given radio resource, the angle of rotation is
(5) defined by (10):
Whereas M = 4 and N = 2, the angle of rotation it is (10)
calculated according to (2):
Ifl=1, The optimal values of are:

(11)
If l=2,
The values of they are assigned to the non-zero
positions of the graph factor following Latin Squares rules
[9]; considering M = 4 y N =2, the graph factor is given by
With the above, the values for the rotation angle are [11-12]: (12):

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(16)
(12)
Considering that Eb is the bit energy and Esis the energy per
symbol of a signal, so and are the
signal to noise ratio of the bit and the symbol, respectively
Considering that fj be the j-th column of F without zero
[13]. So for an M-ary signaling scheme with
elements, an operator is defined for the j-th user as observed
in (13) bits per symbol, the energy of the signal for each
modulated symbol is given by (17):
(13)
(17)
(14)
Then the signal to noise ratio per symbol turns out to be:
In this example it is considered that J = 6 layers and K = 4
resources, the codebook for each user contains M = 4 (18)
symbols representing the alphabet:
00 01 11 10 A. AWGN Channel model and Euclidian distance
Random noise is added to the signal; noise magnitude
Each symbol in the codebook is a column vector that depends on the signal to noise ratio that it is specified [14].
contains 4 rows; in each row there is a complex number. Two The signal to noise ratio is expressed in dB, in Fig. 5, the
of the rows will be filled and two will contain the complex diagram of a signal to which noise is added is shown.
value 0 + 0j, that is, they are empty. A code book Xj for a
user it is generated by concatenating the code words
corresponding to different symbols transmitted by that user.
The columns represent the code words while the rows
represent the resources (subcarriers). For the first user the
code book is represented by (15). Received
Transmitted
signal signal

Fig. 5. AWGN channel

(15) Considering Fig. 5, given a given signal-to-noise ratio, it is
carriers 2 and 4 are occupied, while subcarriers 1 and 3 desired to incorporate Gaussian additive white noise.
contain zeros (they are empty). The user data is transmitted Considering the signal to noise ratio , refers to
using two subcarriers; the digital value 0 0 is represented in when using binary modulations (BPSK), for
its complex values by in subcarrier 2 and by multilevel modulations (QPSK, MQAM) is considered
in the subcarrier 4. Fig. 4 shows the .
representation of the code book in the complex plane (left The amount of noise added by the AWGN channel is
figure), and the representation of the digital values in its controlled by the signal-to-noise ratio . Considering the
complex representation (code words). complex IQ plane for all digital modulations, the variance of
the required noise (noise power) to generate random noise of
the Gaussian type is given by (19).
(19)

To generate a noise vector n with a normal distribution

with zero mean and standard deviation given by (19), we
have the following [14].

(20)
Fig. 4. Code books and code words for user 1
where
In this work, only the codebook for the first user is n = it is the noise vector
presented; to generate all the books of the other users, the s = transmitted signal
process of rotation of the matrix is followed. aleat = random values
III. SIGNAL TO NOISE RATIO Finally, the received signal that passes through an AWGN
channel is:
Assuming a cannel with bandwidth B[13], with a reception
power Pr, and a spectral density of noise power N0/2 the (21)
signal-to-noise ratio is given by (16).

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B. IQ detector based on minimum Euclidean distance From Fig. (6), we can observe the influence of the AWGN
For coherent detection, the transmitter and the receiver channel, the received data are far from the points of the
agree on the same reference constellation to modulate and reference constellation, then the Euclidean distance between
demodulate user information. The first step in IQ detection is each received data and the reference constellation is
to calculate the Euclidean distance between two given vectors calculated. The minimum distance is selected to associate the
(the reference array and the symbols received with noise). received data with the closest point of the reference. Fig. 7
Each symbol in the received vector must be compared with shows the association of the received data to its closest
each symbol of the reference array, and then calculate the reference point using the minimum Euclidean distance.
minimum Euclidean distance.
Considering that ,
, be two vectors in the p-plane. The
Euclidean distance is given by (22).

(22)

IV. Simulation and Results

Considering the following user data: [1 0] [0 0] [1 0] [1 1]
[1 0] [0 0] [0 1][0 1][1 0][1 1], its representation in code
words using (15), is given by:
[0 0]  [ 0.202-0.754j ]
[0 1]  [ 0.113-0.422j ]
[0 0]  [ 0.202-0.754j ]
[1 0]  [ -0.202+0.754j]
[0 0]  [ 0.202-0.754j ]
[1 0]  [ -0.202+0.754j]
[1 1]  [ -0.113+0.422j] Fig. 7. association of the received data to its closest
[0 1]  [ 0.113-0.422j ] reference point
[1 1]  [ -0.113+0.422j] Based on the minimum Euclidean distance, the received
[0 1]  [ 0.113-0.422j ]
data is decoded as:
Considering a SNR = -3dB, the transmitted signals are
affected by the AWGN channel (Fig.5), so the values that are 0.483-0.589j  [0 0]
recovered in the receiver are: 0.128-0.153j  [0 1]
0.483-0.589j 0.154-0.532j  [0 1]
0.128-0.153j -0.308+1.093j  [1 0]
0.154-0.532j 0.070-0.58j  [0 1]
-0.308+1.093j -0.063+1.053j  [1 0]
0.070-0.58j 0.096+0.779j  [1 0]
-0.063+1.053j 0.012-0.586j  [0 1]
0.096+0.779j 0.006-0.071j  [0 1]
0.012-0.586j 0.087-0.113j  [0 1]
0.006-0.071j
0.087-0.113j For the third transmitted data [0 0], of (15) it is known that
The received data is shown in the IQ plane in Fig. 6. the code word corresponds to [0.202-0.754j], however, the
received value was [0.154-0.532j], which is closest to the
point [0.113-0.422j], which causes that when calculating the
minimum Euclidean distance, the IQ detector erroneously
decides that the data received is [0 1] instead of [0 0], it is
observed that for this example 4 of the 10 transmitted data are
misinterpreted, this is due because the SNR proposed is -3dB,
for higher SNRs, errors in detection decrease considerably; in

Fig. 8 the representation in the IQ map of 500 data

transmitted and SNR = 0dB is shown.
In Fig 9, for each point the calculation of the Euclidean
distance is made to associate the received values to its nearest
reference point.
Fig. 6. Received data and its reference points

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[3] ITU. Framework and overall objectives of the future development of

IMT for 2020 and beyond. 2015
[4] Dai, L., Wang, B., Yuan, Y., Han, S., y Wang, Z. “Non-orthogonal
multiple access for 5G: solutions, challenges, opportunities, and future
research trends”. IEEE Communications Magazine, 53(9):74–81, Sept
2015.
[5] Van de Beek, J. y Popovic, B. M. “Multiple access with low-density
signatures”. In GLOBECOM 2009 - 2009 IEEE Global
Telecommunications Conference, pages 1–6, Nov 2009.
[6] Bayesteh, A., Nikopour, H., Taherzadeh, M., Baligh, H., y Ma, J.
“Low complexity techniques for scma detection”. 2015 IEEE
Globecom Workshops (GC Wkshps), pages 1–6, Dec 2015.
[7] Nikopour, H. y Baligh, H. “Sparse code multiple access”. 2013 IEEE
24th Annual International Symposium on Personal, Indoor, and
Mobile Radio Communications (PIMRC), pages 332–336, Sept 2013.
[8] Yu, L., Lei, X., Fan, P., y Chen, D. “An optimized design of SCMA
codebook based on star-QAM signaling constellations”. In 2015
International Conference on Wireless Communications Signal
Processing (WCSP), pages 1–5, Oct 2015.
Fig. 8. Received data and its reference points [9] Minnick, R. C., Elspas, B., Short, R. A. “Symmetric Latin Squares”.
IEEE Transactions on Electronic Computers. 1963
[10] Yang, H., Fang, X., Liu, Y., Li, X., Luo, Y., y Chen, D. “Impact of
overloading on link-level performance for sparse code multiple
access”. 2016 25th Wireless and Optical Communication Conference
(WOCC), pages 1–4, May 2016.
[11] Bao, J., Ma, Z., Ding, Z., Karagiannidis, G. K. y Zhu, Z. “On the
design of multiuser codebooks for uplink SCMA systems”. IEEE
Communications Letters, 20(10):1920–1923, Oct 2016.
[12] Bao, J., Ma, Z., Mahamadu, M. A., Zhu, Z., y Chen, D. “Spherical
codes for SCMA codebook”. 2016 IEEE 83rd Vehicular Technology
Conference (VTC Spring), pages 1–5, May 2016.
[13] Proakis J.G, Digital Communications, fifth edition, New York,
McGraw–Hill, 2008
[14] Andrea Goldsmith, Wireless Communications, Cambridge University
Press, first edition, August 8, 2005

Fig. 9. association of the received data to its closest

reference point
From Fig. 9, it can be seen that the received data are very
close to the reference constellation whereby the error rate
decreases considerably.
V. CONCLUSIONS
This work presents a detailed description of the method of
phase rotation and interleaving of mother constellations to
construct the code words in an SCMA system, additionally
the Euclidian distance-based detection method is presented as
an easy-to-implement alternative to demodulate SCMA data
even in the presence of low signal-to-noise ratio values.
ACKNOWLEDGMENT
The authors thank the InstitutoPolitécnico Nacional for the
support received in carrying out this work.

REFERENCES
[1] METIS. Mobile and wireless communications Enablers for the
Twenty-twenty Information Society. Novel radio link concepts and
state of the art analysis. 2013
[2] Gupta A. y Jha R. K. “A survey of 5g network: Architecture and
emerging technologies”. IEEE Access, 3:1206–1232, 2015.

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