11institutetext:  Key Laboratory of Optical Astronomy, National Astronomical
Observatories, Chinese Academy of Sciences, Beijing, 100101, China
22institutetext: University of Chinese Academy of Sciences,Beijing, 100049,
China
33institutetext: Department of Astronomy, Beijing Normal University, Beijing,
100875, China
44institutetext: Key Laboratory of Space Astronomy and Technology, National
Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100101,
China
55institutetext: Yunnan observatories, Chinese Academy of Sciences, Kunming,
650011, China
66institutetext: Nanjing Institute of Astronomical Optics, & Technology,
National Astronomical Observatories, Chinese Academy of Sciences, Nanjing
210042, China
77institutetext: School of Astronomy and Space Science, University of Chinese
Academy of Sciences, Beijing, 100049, China

# The Relative Calibration of Radial Velocity for LAMOST Medium Resolution
Stellar Spectra

Jianping Xiong 1122 Bo Zhang 33 Chao Liu corresponding author:
<EMAIL_ADDRESS>Jiao Li 4455 Yongheng Zhao 112277 Yonghui Hou 6677

###### Abstract

The Large Sky Area Multi-Object Fiber Spectroscopic Telescope (LAMOST) started
median-resolution spectroscopic (MRS, R$\sim$7500) survey since October 2018.
The main scientific goals of MRS, including binary stars, pulsators, and other
variable stars are launched with a time-domain spectroscopic survey. However,
the systematic errors, including the bias induced from wavelength calibration
and the systematic difference between different spectrographs have to be
carefully considered during radial velocity measurement. In this work, we
provide a technique to correct the systematics in the wavelength calibration
based on the relative radial velocity measurements from LAMOST MRS spectra. We
show that, for the stars with multi-epoch spectra, the systematic bias which
is induced from the exposures of different nights can be well corrected for
LAMOST MRS in each spectrograph. And the precision of radial velocity zero-
point of multi-epoch time-domain observations reaches below 0.5 km s-1. As a
by-product, we also give the constant star candidates111The constant stars
here in this work represent for the stars with low radial velocity variations.
They are not necessarily absolute constant stars, but may not show significant
radial velocity variation during the time-domain survey, which can be the
secondary radial-velocity standard star candidates of LAMOST MRS time-domain
surveys.

###### keywords:

methods: data analysis — techniques: radial velocities — stars: statistics —
catalogs — surveys

##  1 Introduction

Over the past years, the large spectroscopic surveys, such as RAVE (Steinmetz
et al., 2006, 2020), SDSS/SEGUE (Yanny et al., 2009), LAMOST (Cui et al.,
2012; Deng et al., 2012; Zhao et al., 2012; Luo et al., 2015), APOGEE
(Majewski et al., 2017), GALAH (De Silva et al., 2015), Gaia (Gilmore et al.,
2012; Katz et al., 2004; Cropper et al., 2018) have yielded a large number of
stellar spectra to understand the galaxy formation and evolution (Gao et al.,
2015; Mesa et al., 2021). As a fundamental property that can be measured
directly from stellar spectra, radial velocity (RV) is the basic ingredient
for many studies, such as the study on the properties of binary (Gao et al.,
2014; Tian et al., 2018; Sana et al., 2012, 2013), Milky Way dynamics (Tian et
al., 2020), asteroseismology (Arentoft et al., 2019), and even searching for
the black holes (Liu et al., 2019).

For large spectroscopic surveys, the radial velocity uncertainty of low-
resolution spectra is larger $3-5$ km s-1for SEGUE in R$\sim$1800 (Yanny et
al., 2009), and around $3-5$ km s-1for LAMOST in R$\sim$1800, (Wu et al.,
2014) . Such precision is very useful in the Milky Way studies, however, it is
not enough in lots of studies of stars and stellar systems, e.g., the internal
dynamics of open clusters (with the radial velocity dispersion of $<1$ km s-1
(Pang et al., 2021a)), the disruption process of open clusters (with the
radial velocity dispersion around $1-3$ km s-1 (Pang et al., 2021b)), the
binary stars with a larger range of period, the low-amplitude variables. The
precision of radial velocity is required down to $<1$ km s-1for these studies.
This is essentially the motivation of the LAMOST medium-resolution
spectroscopic (MRS) with $R\sim 7500$.

Since October 2018, LAMOST has started the 5-year MRS survey (Liu et al.,
2020). A large amount of and medium resolution stellar spectra will be
obtained. It provides an opportunity to study the kinematic and dynamic of
stars more accurately. For LAMOST MRS survey, Wang et al. (2019) derived
radial velocity with cross-correlation method, and the precision achieves to
1.36 km s-1, 1.08 km s-1, 0.91 km s-1 for the spectra at the single-to-noise
ratio (SNR) of 10, 20, 50, respectively, after calibrated with the radial
velocity standard stars (Huang et al., 2018).

However, there are a few issues need to be well addressed. One is that the
wavelength calibration of the spectra of LAMOST demonstrates that it may shift
by a few km s-1 during one night, which is likely due to the variation of the
environments of the spectrographs (Zhang Hao-Tong priv. comm.). The systematic
shift between different exposures may severely affect the orbital parameter
estimates in binary studies. In the mean time, because LAMOST contains 16
spectrographs, one star can be observed by different spectrographs in
different nights, depending on the specific fiber assignment strategy. This
may bring another systematic difference in time-domain spectra, due to the
systematically different wavelength calibration between different
spectrographs. It means that the instability of the wavelength calibration may
induce a large fraction of systematic errors in the uncertainty of radial
velocity.

It is noted that many studies, e.g. the pulsation of stars, the binarities,
etc., do not require absolute radial velocities, but only relative one.
Specifically, at time $t$, a star has a directly measured radial velocity such
as $v_{t}=v_{0}+\Delta v_{t}+\epsilon$, where $v_{0}$ is the systematic radial
velocity of the star, $\Delta v_{t}$ is the term of possible variation if the
star is a pulsator or a companion in a binary system. And $\epsilon$ is the
measurement uncertainty. To understand the pulsational process or the
parameters of the orbits of a binary system using a series time-domain radial
velocities, $\Delta v_{t}$ is sufficient rather than using $v_{t}$.

Therefore, the goal of this work is two-fold: first we correct the systematic
bias in radial velocity; second, we provide relative radial velocity
measurement for the 92,342 time-domain MRS stars. As a by-product, we give the
constant star candidates (The constant stars here in this work represent for
the stars with low radial velocity variations. They are not necessarily
absolute constant stars, but may not show significant radial velocity
variation during the time-domain survey), which do not show significant radial
velocity shift in the MRS time-domain spectra.

This paper is organized as follows. The LAMOST MRS and the data are described
in Section 2. The method of relative calibration is described in Section 3.
The results are indicated in Section 4. The special cases are analyzed in
Section 5. Finally, we summarize in Section 6.

##  2 Data

Guoshoujing Telescope (also known as LAMOST) is a 4-meter class reflective
Schmidt telescope with 5-degree field-of-view. Totally 4000 fibers are
installed at the 1.75 m-diameter large focal plane. 16 spectrographs, each of
which accepts 250 fibers, are used to take spectra simultaneously. Each
spectrograph contains two arms. For medium spectroscopic observation, the blue
arm covers 4950-5350 Å and the red arm covers 6300-6800 Å with $R\sim 7500$.

As mentioned in Liu et al. (2020), the LAMOST MRS take observations in 14
bright and gray nights per month. About 2000 square degree sky area, including
various Galactic latitudes, are designed for the time-domain spectroscopic
survey. It is expected that, after a 5-year MRS survey, there will be around
200 000 stars with $G<14$ mag222$G$ is the band without filter in _Gaia_
survey (Gaia Collaboration et al., 2016). to be observed multiple times.

The seeing, cloud coverage and checking of polluted light are evaluated to
remove the low-quality observations from two-dimensional(2D) frames at first.
Then the 2D medium-resolution frames are processed by LAMOST 2D pipeline,
which includes bias subtraction, fiber tracing, wavelength calibration,
spectral extraction and sky subtraction. For an observation target, the
spectra from the same night will be merged into a coadded spectrum, and the
separate spectrum of each exposure is provided as well.

##  3 Method

The directly measured radial velocity ($\widetilde{v}_{i,t}$) of a star $i$
observed at time $t$ is composed of the intrinsic radial velocity ($v_{i,t}$),
the systematic error ($\Delta u_{t}$) due to the wavelength calibration during
the observation and the measurement uncertainty ($\epsilon_{i,t}$) induced
during measurement, i.e.

$\widetilde{v}_{i,t}=v_{i,t}-\Delta u_{t}+\epsilon_{i,t}.$ (1)

In general, $v_{i,t}$ is independent on $t$. For a constant star, its radial
velocity does not change with observation time (i.e.
$v_{i,t_{1}}=v_{i,t_{2}}$). When star $i$ is in a binary system or is a
variable, $v_{i,t}$ changes periodically with $t$. For the former case, the
relative radial velocities between two exposures should be around zero, while
for the latter case, the differential radial velocities are sufficient for
solving out the velocity curve with higher precision than that derived from
the absolutely calibrated radial velocities. Therefore, in this work, we
concentrate on the relative radial velocity only.

###  3.1 Relative radial velocity with maximum likelihood

We develop a likelihood method to measure the relative radial velocity for a
star from its medium-resolution spectra. First, each spectrum observed by
LAMOST has to be normalized. We normalize each spectrum by dividing by its
pseudo-continuum, the pseudo-continuum is calculated by the follows: each
spectrum is fitted with a smoothing spline function iteratively, then the
pixels with a value greater than the median value by 3 times of standard
deviation are eliminated in each segmented wavelength (e.g. 100Å). And the
iterations are 3 (Zhang et al., 2021).

Then, for the $i$th star with multiple exposures, we select the spectrum,
denoted as $f_{ref,i}(\lambda)$, with highest signal-to-noise ratio as the
reference for the star. For the spectrum $f_{t,i}(\lambda,v)$, which is the
spectrum of $i$th star observed at time $t$, the likelihood distribution of
the relative radial velocity $v$ is

$p(v)=\prod_{\lambda}\frac{\exp[-\frac{(f_{t,i}(\lambda,v)-f_{ref,i}(\lambda))^{2}}{2(\sigma_{t,i}^{2}+\sigma_{ref,i}^{2})}]}{\sqrt{2\pi(\sigma_{t,i}^{2}+\sigma_{ref,i}^{2})}},$
(2)

In which $\sigma_{t,i}$ and $\sigma_{ref,i}$ are the observational errors from
the observed and reference spectra, respectively. In order to avoid the
arithmetic errors caused by minimal values, the logarithmic form as written in
below is actually used:

$\ln
p(v)=\sum_{\lambda}[-\frac{(f_{t,i}(\lambda,v)-f_{ref,i}(\lambda))^{2}}{2(\sigma_{t,i}^{2}+\sigma_{ref,i}^{2})}-0.5\ln(\sigma_{t,i}^{2}+\sigma_{ref,i}^{2})-0.5\ln
2\pi]$ (3)

###  3.2 Calibrate systematic bias in radial velocity

In general, the wavelength calibration is based on the arc lamp, such as Fe-Ar
lamps etc. However, according to the tests, the wavelength calibration may be
suffered from a few km s-1 systematic shift when comparing between different
observations. In addition, the wavelength calibration of the red arm is more
precise than the blue arm for LAMOST. In a few observation fields, radial
velocities of constant stars obtained from high-resolution spectra can be used
to further calibrate the wavelength in absolute sense, since the precision of
these radial velocities can be as high as a few hundreds m s-1 (see Huang et
al. 2018). However, there is no such high-resolution constant stars been
identified in every LAMOST observation field. Therefore, we can not only rely
on the identified constant stars to improve the velocity accuracy but require
an approach in the manner of self-calibration.

Before we start the calibration process, we assume that the relative radial
velocities of constant stars should be zero. The calibration process contains
two steps in each iteration.

As the first step, we consider each star $i$. We calculate its relative radial
velocity variation over all exposures such as

$\sigma_{i}^{2}=\frac{1}{m}\sum_{t=1}^{m}{(v_{i,t}-\bar{v_{i}})^{2}},$ (4)

where $v_{i,t}$ is the relative radial velocity of star $i$ in the $t$th
observation, $\bar{v_{i}}$ the mean relative velocity over all exposures, and
$m$ the number of exposures. For all stars observed by a spectrograph, the
mean radial velocity variation is $\bar{\sigma}=\sum_{i}{\sigma_{i}}/n$. The
stars with $\sigma_{i}<\bar{\sigma}$ are selected as the candidate constant
stars.

In the second step, we consider the velocities of stars in each exposure. We
derive the mean relative velocities, denoted as $\bar{v_{t}}$, of the
candidate constant stars selected from the first step in the $t$th exposure.
Because the ground true relative radial velocity of constant stars should be
zero, the non-zero $\bar{v_{t}}$ is mainly contributed by the systematic bias
in wavelength calibration. Therefore, $\bar{v_{t}}$ is subtracted from
$v_{i,t}$ for each star, including constant and non-constant stars. After
subtraction, the candidate constant stars should be more concentrated around
zero.

Note that some contaminators may be induced in the first step, we go back to
step one and select the new candidate constant stars again with corrected
relative radial velocities.

After a few iterations with the above two steps, when $\bar{v_{t}}$ tends to
be very close to zero, and the variation of $\bar{v_{t}}$ for each exposure is
lower than a predefined small threshold (we adopt any of
$|\bar{v_{t}}-\bar{v}_{t-1}|<0.01$ km s-1 ), the iteration is stopped.

###  3.3 Validation with mock stars

We use mock stars to quantitatively assess the performance of the calibration
approach. We simulated the relative radial velocity of the mock stars as the
observation from one of the spectrographs. We produce 30 exposures in 7 nights
for these mock stars, including 200 constant stars with the ground true
velocity within $\pm$ 0.1km s-1 with the measurement uncertainty in 0$\sim$2
km s-1, 10 non-constant stars with the radial velocity changes of 2$\sim$10 km
s-1, and 40 binary stars with the amplitude of 3$\sim$15 km s-1 in the period
of 0.5$\sim$3 days.

Then we add the systematic shift to these mock stars as the radial velocity
that measured directly from the observation. The systematic shift of each
night is randomly generated in the range of -7$\sim$7km s-1. In the mean time,
each exposure of the mock stars are added a random error with a Gaussian error
of 0.5 km s-1. After systematically shifting the radial velocities of the mock
stars from zero, the constant stars’ RV distribution becomes larger, and the
difference between the observation and the ground true velocity is shown in
figure 1 (a). The residual values between the corrected radial velocities and
the ground true velocities are shown in panel (b). It is seen that the mean
value of the residual velocities is essentially between $\pm$0.50 km s-1,
except non-constant stars.

Figure 1: The result of mock stars. Panel (a) shows the distribution of the
difference between the directly measured radial velocities and the ground true
velocities for mock constant stars. Panel (b) shows the residual value between
the corrected velocities and the ground true velocities for mock constant
stars.

In Figure 2, we show the velocity curves of three mock binary stars. The top
panels in figure 2 show the velocity curves in black-solid lines, the mock
observed radial velocities are shown in red circles, and the corrected
velocities are shown in blue stars. The residual value is shown in the bottom
panels of figure 2, including the residual values between the mock
observational radial velocities and the true velocity curves in red circles,
and the residual value between corrected velocities with the true velocity
curves in blue stars. It is seen that the correction method can well
reconstruct the radial velocity curves of the binary stars. After correction,
the residual value are smaller than before correction. The non-costant offeset
showing in the residual panel of the figure is due to that we have added a
random RV measurement error to the simulation data (ranging from 0 to 2 km
s-1). After correction of the systematic offset in each observation, the
random error of RV added to the mock data still exists, leading to the larger
deviations than $\sim$ 0.5 km s-1 in the residuals. The detailed orbital
parameters of the three mock binary stars are list in Table 1.

Figure 2: The velocity correction for three mock periodic variable stars with different velocity amplitudes. The black-solid line indicate the true radial velocity curve. The red circles show the directly measured radial velocities with shifts. The blue stars display the corrected radial velocities. The bottom panels display the residual values with ground true velocity of non-corrected and corrected velocities with red circles and blue stars, respectively.  Table 1: The detailed parameters of mock binaries shown in figure 2 . panel | Period | velocity amplitude | inclination | eccentricity |
---|---|---|---|---|---
| ($days$) | (km s-1) | (°) |  |
(a) | 0.6894 | 13.23 | 88.65 | 0 |
(b) | 0.6373 | 13.01 | 88.15 | 0 |
(c) | 2.9098 | 5.91 | 81.79 | 0 |

##  4 Result

We apply the approach of calibration based on relative radial velocities (RV)
to the LAMOST-MRS spectra with the SNR larger than 5 that released in DR7
time-domain plate, approximately 2,215,918 spectra, including 1,170,445 of the
red arm and 1,045,473 of the blue arm. Then we determined the relative radial
velocity in the red and blue arms separately. Each MRS spectrum is composed of
two parts, the red covering from 6300 Å to 6800 Å and the blue covering from
4950 Å to 5350 Å. And for measuring the relative radial velocity, the
wavelength from 6400 Å to 6700 Å of the red arm is used, and the wavelength
from 5000 Å to 5300 Å from the blue arm is used. Then, the wavelength of each
part of the spectrum is calibrated separately based on. The precision of
wavelength calibration is better in the red than in blue, since the number of
lamp spectra lines in the red part is more than that in the blue part.
Meanwhile, the SNR of red spectra is in principle better than the blue arm due
to larger throughput in red part. Therefore, we measure the relative radial
velocity only using the red part spectra in the first step. Then we correct
the relative radial velocities measured from the red part spectra. In the mean
time, the candidate constant stars in each spectrograph are identified during
the correction of radial velocities measured from red part spectra. Then, we
directly use these candidate constant stars to correct the systematic shift
induced in the wavelength calibration by the blue part spectra.

Figure 3 (a) and (b) show the distribution of the directly measured relative
radial velocity from the spectra of constant star candidates from one sample
field of LAMOST MRS in 16 spectrographs of red and blue arms, respectively.
The violin-plot shows the dispersion of the relative radial velocities of
constant star candidates in each spectrograph, and the white spots are the
mean relative radial velocities of these constant star candidates in each
spectrograph. It illustrates that the radial velocity zero-points are slightly
different between all spectrographs. And figure 3 (c) and (d) show the
relative radial velocity distribution of constant star candidates for a
spectrograph in 38 different exposures. The violin-plot shows the dispersion
of the relative radial velocities of constant star candidates in each
exposure, and the black dots are the mean relative radial velocities of these
constant star candidates. It can be seen that the mean velocity of the
candidate constant stars can move to as much as 3 km s-1 from zero in some
observations, especially in different nights. And according to panels (c) and
(d), the variation of the mean radial velocities of red arm is more stable
than those in blue arm. To sum up, we perform calibration for each
spectrograph separately, and apply the calibration on red arm at first.

Figure 3: Panels (a) and (b) show the distribution of the directly measured
relative radial velocity from the spectra of constant star candidates from one
sample field of LAMOST MRS in 16 spectrographs of red and blue arms. The
circles are the mean relative radial velocity of candidate constant stars.
Panels (c) and (d) show the distribution of the directly measured relative
radial velocity from the spectra observed by LAMOST MRS of constant star
candidates for a spectrograph in red and blue arms, respectively, but in
different exposures. The black dots indicate the initial mean radial
velocities in different exposures.

Figure 4 (a) and (b) show the relation between signal-to-noise ratio(SNR) with
the measurement uncertainty of radial velocity from a same sample field same
as figure 3 of LAMOST MRS in red and blue arm, respectively, and the
measurement uncertainty of radial velocity is calculated as the standard
deviation of the best-fit Gaussian distribution corresponding to the
likelihood function distribution, and donated as $\epsilon_{RV}$. Panels (a)
and (b) both show the decreasing uncertainty of measurement with the
increasing SNR. The non-physical strips are found in fig 4, it is caused by
the multi-star system such as binaries. For these stars, the regular RV
measurement method can not identify the radial velocity of their components
correctly, resulting in a large measurement uncertainty for some spectra in
high SNR. And it can be seen that the uncertainty of measuring relative radial
velocity of red arm is relatively larger than the blue arm due to the fact
that $H_{\alpha}$ is usually the only prominent feature in the red arm,
although it is more stable than blue arm. So we calibrate the systematic shift
for red arm at first, and the constant star candidates picked out after the
red arm calibration are directly used in the blue arm. Using these constant
star candidates, the systematic shift of different exposures for blue arm is
then be corrected.

Figure 4: Panels (a) and (b) show the relation between measurement uncertainty
($\epsilon_{RV}$) with SNR that selected from the same sample field with
figure 3 of LAMOST MRS in the red and blue arm, respectively.

Figure 5 shows the distribution of the non-calibrated and calibrated radial
velocity variation in for candidate constant stars from the same sample field
as figure 3. The radial velocity variation is measured by the standard
deviation of the radial velocities of an constant candidates over different
exposures, donated as $Var_{rv}$. In the mean time, the relations between the
radial velocity variation of the two arms and SNR are also shown in figure 5.
Panel (a) shows the non-calibrated constant candidates’ $Var_{rv}$ as red
solid histogram, which are directly measured from observational spectra in the
red arm, represents the systematic shift of different exposures, the mean
value of the non-calibrated variation is around 1.30 km s-1. And the
calibrated $Var_{rv}$ of constant stars candidates is shown as blue dash-
dotted histogram, which tightly concentrates to around zero. Panel (b) shows
the relation of $Var_{rv}$ distribution and SNR of constant candidates, with
the top panel showing the $Var_{rv}$ distribution for direct measurement from
the observational spectra as red circles and the bottom show the $Var_{rv}$
distribution of constant candidates after calibration as blue triangles. After
calibrating the systematic shift, the $Var_{rv}$ of constant stars are almost
below 0.5 km s-1. As panel (b) shows, the calibration precision depends on the
SNR, and the relatively lower precision results in high SNR is because of the
relatively large variation of SNR for the same target in time-domain
observation. Panels (c) and (d) show similar distributions for blue-arm
calibration result with panels (a) and (b), respectively. In the blue arm, the
initial constant candidates are adopted from red-arm correction result. Panel
(c) shows the non-calibrated $Var_{rv}$ of blue-arm as red histogram with the
mean of 1.19 km s-1, and the calibrated $Var_{rv}$ is shown in blue dash-
dotted histogram, which has mean $Var_{rv}$ of 0.06 km s-1. Panel(d) also
shows the decreasing $Var_{rv}$ of constant stars with increasing SNR, and
after calibrating, the $Var_{rv}$ is almost below to 0.5 km s-1.

Figure 5: The non-calibration and calibrated radial velocity variation
($Var_{rv}$) from the same sample field with figure 3 of LAMOST MRS for
selected candidate constant stars, and their relation with SNR for two arms.
In panels (a) and (c) the red solid, blue dash-dotted histograms show the
$Var_{rv}$ distributions for constant stars with non-calibration and
calibrated in the red and blue arm, respectively. In panel (b) and (d), the
red circles, blue triangles show the distributions of $Var_{rv}$ for constant
stars with non-calibration and calibrated in red and blue arm, respectively.

In total, 2,215,918 spectra from the 59 time-domain plates of LAMOST MRS of
DR7 were calibrated in this work, including 1,170,445 of the red arm and
1,045,473 of the blue arm. The catalog of calibrated relative radial velocity
for these time-domain observations is listed in Table 2. which including ra,
dec, hjd, SNR, directly measured relative radial velocity (DRV_r and DRV_b),
the uncertainty of measurement (DRV_r_err and DRV_b_err), and the calibrated
result (DRV_r_corr and DRV_b_corr), and the “r” and “b” label the red and blue
arm, respectively. The whole catalog is available in its entirety in machine-
readable form.

After self-calibration in red and blue arms, we obtain 65,123 constant star
candidates. And the catalog of these constant star candidates is also
available in machine-readable form. As the observations increasing, these
constant samples will be evolved and become more accurate. This is a long-term
process that can be used for LAMOST time-domain medium resolution survey.

Table 2: The catalog of relative radial velocity for LAMOST MRS time-domain plates. ra | dec | hjd | SNR_r | DRV_r | DRV_r_err | DRV_r_corr | SNR_b | DRV_b | DRV_b_err | DRV_b_corr
---|---|---|---|---|---|---|---|---|---|---
|  | 2458411.145602 | 24.34 | -4.25 | 1.05 | 0.04 | 15.32 | -4.90 | 0.72 | 0.02
|  | 2458416.177853 | 5.84 | -4.83 | 2.88 | 0.00 | 5.02 | -6.12 | 1.66 | 0.00
16.601013 | 3.819259 | 2458420.162413 | 15.43 | 0.51 | 1.56 | 0.03 | 9.21 | -0.26 | 1.06 | 0.00
|  | 2458449.087430 | 6.01 | -8.11 | 3.83 | 0.14 | 5.0 | -3.14 | 2.01 | 0.05
|  | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$
|  | 2458411.145607 | 7.95 | -4.40 | 3.21 | 0.05 | 6.06 | -7.05 | 1.83 | -0.83
|  | 2458439.058429 | 8.14 | 4.29 | 3.7 | 0.02 | 6.99 | -0.51 | 1.73 | 0.01
16.602702 | 4.362617 | 2458451.078222 | 11.4 | -1.95 | 2.86 | -0.03 | 10.9 | 0.06 | 1.24 | 0.01
|  | 2458453.050072 | 13.61 | 0.73 | 2.33 | 0.02 | 10.35 | 0.27 | 1.29 | -0.02
|  | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$
|  | 2458411.129353 | 42.14 | -3.37 | 0.51 | -0.02 | 21.26 | -3.77 | 0.37 | -0.01
|  | 2458420.112249 | 15.56 | 0.35 | 1.20 | -0.05 | 7.45 | -1.22 | 0.81 | -0.03
16.604269 | 5.121394 | 2458439.058448 | 22.54 | 0.95 | 0.83 | -0.03 | 11.16 | 0.83 | 0.58 | -0.01
|  | 2458451.061743 | 32.21 | -0.03 | 0.62 | 0.00 | 18.48 | -0.06 | 0.41 | -0.01
|  | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$
|  | 2458411.145590 | 16.02 | -6.16 | 1.57 | 0.05 | 5.82 | -11.03 | 1.28 | -0.83
16.612777 | 2.792158 | 2458439.058398 | 12.91 | -0.19 | 1.83 | 0.00 | 5.48 | -3.09 | 1.28 | 0.00
|  | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$
$\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$

Figure 6 and figure 7 show the typical RV variations for each fiber after
calibration in red and blue arms, respectively. The vertical red and blue
lines indicate the range of the RV variations of constant stars for each
fiber. It indicates that the mean value of RV variations due to the fiber-to-
fiber is around 0 with a dispersion of 0.1 km s-1 , which is corresponding to
the limit of the calibration.

Figure 6: The RV bias due to fiber-to-fiber variations after calibration in
red arm. Figure 7: The RV bias due to fiber-to-fiber variations after
calibration in blue arm.

##  5 Discussion

The precision of the calibration is highly correlated with the precision of
the radial velocity measurement method, and the measurement precision is
limited by the spectra type and SNR. So, the precision of this relative radial
velocity measurement method is discussed in different types of spectra, as
well as the pulsating variable stars and binary system.

We conducted the test for single line spectra at first, we selected 7 spectra
with different types with high SNR (SNR$>$100) in LAMOST MRS. For these 7
selected spectra, we simulated 1,000 spectra at different SNR for each type.
For each SNR, we randomly selected a spectrum as the template to measure the
relative radial velocity of them. And the ground true radial velocity of these
spectra should be zero. For the different types of spectra in the varying SNR,
we obtained the standard deviation ($\sigma_{RV}$) from Monte Carlo simulation
results as the measurement precision. Figure 8 (a) and (b) show the precision
improves with the increasing SNR in the red and blue arm, respectively. We
infer from figure 8 that there is a relatively low precision of type O, B, and
A when SNR less than 40, especially in the blue arm (see panel (b)). For the
type O, B, and A, there are fewer lines to measure radial velocity in the blue
arm and the signal is mixed up with noise, so there is a relatively low
precision of type O, B, and A when SNR is less than 40 in the blue arm. This
is improved a lot in the red arm, since $H_{\alpha}$ line is dominated in red
arm. For type F, G, K, and M, we can derive the more accurate radial velocity
in relatively high precision even though they are in a low SNR, because of
their rich populations of spetral lines.

Figure 8: Panels (a) and (b) show the precision ($\sigma_{rV}$) of this
relative radial velocity measurement method for different types spectra in
different SNR for red and blue arms, respectively.

Then we test on pulsating variable stars. As a pulsating variable star, its
effective temperature, spectra feature, and radial velocity may change in a
short period, such as RR-Lyrae. Because the periodic expansion and contraction
of these stars cause a temperature change, and the luminosity also
periodically increases and decreases, it appears that its brightness
periodically brightened and darkened. It indicates that the spectra of the
pulsating variable star may significantly change in a short period with its
large changes in temperature.

So, we mimic pulsating variable stars’ spectra by using the standard template
with different temperatures of 6000 K, 6500 K, 7000 K, with fixed metallicity
of -1 dex and logg of 2.5 dex. We assumed that the spectra in 3 temperatures
are the observational spectra in the time-domain survey of an RR-Lyrae star.
And we produced 1000 mock spectra by the different SNR for the temperatures in
6500 K and 7000 K, and measured their radial velocity by comparing with the
spectrum with temperature in 6000 K. Then we obtained the standard deviation
from Monte Carlo simulation results as the measurement precision. Figure 9
shows $\sigma_{RV}$ in different temperatures with changing SNR in the red and
blue arms. In figure 9, the red solid line is the result of the temperature of
7000 K and the blue dash-dotted line is for the spectrum with 6500 K. It is
seen that the precision is also increasing with the increasing SNR and is
relatively higher for the smaller temperature change. Figure 9 shows that even
the pulsators change their temperature by 1000K, the relative radial
velocities are still reliable.

Figure 9: Panels (a) and (b) show the relative radial velocity measurement
precision($\sigma_{rv}$) for the stars with large temperature change in
different SNR for red and blue arm, respectively.

For the binary systems, we can derive the differential radial velocity for
single-line spectroscopic binary. But for double-line spectroscopic binary, we
can derive the radial velocity with this method only if that we can obtain
both of the spectra at the time of eclipse and non-eclipse. Because the basic
idea for this radial velocity measurement method is to calculate the
similarity for two spectra of the same target in different exposures. If the
double-line spectroscopic binary are always observed at the time of non-
eclipse, i.e. shows double-line in all exposures, the relative radial velocity
with assumption of single star is valid anymore. So, for double-line
spectroscopic binary, the differential radial velocity can not be measured
accurately with this likelihood if the observational spectra are always in
double-line. Their velocities should be dealt with in a different way but we
can pick them out as the variable stars. Therefore, in this paper, the double-
line binary stars are only marked and will be treated with other approach.

##  6 Conclusion

In this work, we mainly develop an approach in the manner of self-calibration
by using differential radial velocity for LAMOST time-domain medium resolution
search, and it is practicable to correct systematic shift in different
exposures by using the differential radial velocities. The precision of radial
velocity zero-point for different exposures after correcting systematic shift
can reach below 0.5 km s-1.

In total, 2,215,918 spectra from the 59 time-domain plates of LAMOST MRS of
DR7 were calibrated in this work, including 1,170,445 of the red arm and
1,045,473 of the blue arm. And the whole catalog of calibrated relative radial
velocities of the spectra for the LAMOST time-domain medium resolution survey
is provided in this paper. And after self-calibration in red and blue arms, we
obtain the 65,123 constant stars candidates, whose radial velocity variation
reach around zero after calibration. This self-calibration method can be used
for the long-term observation of LAMOST time-domain medium resolution survey
and can be used to select constant stars in a relatively high precision. These
initial constant star candidates can provide a library of standard star
candidates of radial velocity for the LAMOST time-domain medium resolution
search. As the observation time increases, the library of the constant stars
becomes more accurate. For the scientific significance, the variable stars
with a relatively small amplitude can be detected and their radial velocities
are calibrated effectively by this self-calibration method.

###### Acknowledgements.

This work is supported by National Key R&D Program of China No.
2019YFA0405500. C.L. Thanks the National Natural Science Foundation of China
(NSFC) with grant No. 11835057, the State Natural Sciences Foundation
Monumental Projects with Nos. 12090040. Z.W.H, 12090041. H.W. Guoshoujing
Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope
LAMOST) is a National Major Scientific Project built by the Chinese Academy of
Sciences. Funding for the project has been provided by the National
Development and Reform Commission. LAMOST is operated and managed by the
National Astronomical Observatories, Chinese Academy of Sciences.

## References

  * Arentoft et al. (2019) Arentoft, T., Grundahl, F., White, T. R., et al. 2019, A&A, 622, A190
  * Cropper et al. (2018) Cropper, M., Katz, D., Sartoretti, P., et al. 2018, A&A, 616, A5
  * Cui et al. (2012) Cui, X.-Q., Zhao, Y.-H., Chu, Y.-Q., et al. 2012, Research in Astronomy and Astrophysics, 12, 1197
  * De Silva et al. (2015) De Silva, G. M., Freeman, K. C., Bland-Hawthorn, J., et al. 2015, MNRAS, 449, 2604
  * Deng et al. (2012) Deng, L.-C., Newberg, H. J., Liu, C., et al. 2012, Research in Astronomy and Astrophysics, 12, 735
  * Gaia Collaboration et al. (2016) Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016, A&A, 595, A1
  * Gao et al. (2015) Gao, H., Zhang, H.-W., Xiang, M.-S., et al. 2015, Research in Astronomy and Astrophysics, 15, 2204
  * Gao et al. (2014) Gao, S., Liu, C., Zhang, X., et al. 2014, ApJ, 788, L37
  * Gilmore et al. (2012) Gilmore, G., Randich, S., Asplund, M., et al. 2012, The Messenger, 147, 25
  * Huang et al. (2018) Huang, Y., Liu, X.-W., Chen, B.-Q., et al. 2018, AJ, 156, 90
  * Katz et al. (2004) Katz, D., Munari, U., Cropper, M., et al. 2004, MNRAS, 354, 1223
  * Liu et al. (2020) Liu, C., Fu, J., Shi, J., et al. 2020, arXiv e-prints, arXiv:2005.07210
  * Liu et al. (2019) Liu, J., Zhang, H., Howard, A. W., et al. 2019, Nature, 575, 618
  * Luo et al. (2015) Luo, A. L., Zhao, Y.-H., Zhao, G., et al. 2015, Research in Astronomy and Astrophysics, 15, 1095
  * Majewski et al. (2017) Majewski, S. R., Schiavon, R. P., Frinchaboy, P. M., et al. 2017, AJ, 154, 94
  * Mesa et al. (2021) Mesa, V., Alonso, S., Coldwell, G., Lambas, D. G., & Nilo Castellon, J. L. 2021, MNRAS, 501, 1046
  * Pang et al. (2021a) Pang, X., Li, Y., Yu, Z., et al. 2021a, ApJ, 912, 162
  * Pang et al. (2021b) Pang, X., Yu, Z., Tang, S.-Y., et al. 2021b, arXiv e-prints, arXiv:2106.07658
  * Sana et al. (2012) Sana, H., de Mink, S. E., de Koter, A., et al. 2012, Science, 337, 444
  * Sana et al. (2013) Sana, H., de Koter, A., de Mink, S. E., et al. 2013, A&A, 550, A107
  * Steinmetz et al. (2006) Steinmetz, M., Zwitter, T., Siebert, A., et al. 2006, AJ, 132, 1645
  * Steinmetz et al. (2020) Steinmetz, M., Guiglion, G., McMillan, P. J., et al. 2020, AJ, 160, 83
  * Tian et al. (2020) Tian, H., Liu, C., Wang, Y., et al. 2020, ApJ, 899, 110
  * Tian et al. (2018) Tian, Z.-J., Liu, X.-W., Yuan, H.-B., et al. 2018, Research in Astronomy and Astrophysics, 18, 052
  * Wang et al. (2019) Wang, R., Luo, A. L., Chen, J. J., et al. 2019, arXiv e-prints, arXiv:1908.04773
  * Wu et al. (2014) Wu, Y., Du, B., Luo, A., Zhao, Y., & Yuan, H. 2014, in Statistical Challenges in 21st Century Cosmology, ed. A. Heavens, J.-L. Starck, & A. Krone-Martins, Vol. 306, 340
  * Yanny et al. (2009) Yanny, B., Rockosi, C., Newberg, H. J., et al. 2009, AJ, 137, 4377
  * Zhang et al. (2021) Zhang, B., Li, J., Yang, F., et al. 2021, arXiv e-prints, arXiv:2105.11624
  * Zhao et al. (2012) Zhao, G., Zhao, Y.-H., Chu, Y.-Q., Jing, Y.-P., & Deng, L.-C. 2012, Research in Astronomy and Astrophysics, 12, 723


# On the Outage Performance of SWIPT-NOMA-CRS with imperfect SIC and CSI

Ferdi KARA1
1Wireless Communication Technologies Laboratory (WCTLab<EMAIL_ADDRESS>Department of Electrical and Electronics  Faculty of Engineering
Zonguldak Bulent Ecevit University  Zonguldak  Turkey
ORCID iD: https://orcid.org/0000-0001-8038-2747

.202 .202 ..202

###### Abstract

In this paper, a non-orthogonal multiple access based cooperative relaying
system (NOMA-CRS) is considered to increase spectral efficiency. Besides, the
simultaneous wireless information and power transfer (SWIPT) is proposed for
the relay in NOMA-CRS. In SWIPT-NOMA-CRS, three different energy harvesting
(EH) protocols, power sharing (PS), time sharing (TS) and ideal protocols are
implemented. The outage performances of the SWIPT-NOMA-CRS are studied for all
three EH protocols. In the analysis, to represent practical/reasonable
scenarios, imperfect successive interference canceler (SIC) and imperfect
channel state information (CSI) are taken into consideration. The derived
outage probability (OP) expressions are validated via computer simulations.
Besides, the OP for the benchmark scheme, NOMA-CRS without EH, is also derived
under imperfect SIC and CSI. Based on extensive simulations, it is revealed
that the SWIPT-NOMA-CRS outperforms NOMA-CRS without EH. Finally, the effects
of all parameters on the outage performance of the SWIPT-NOMA-CRS are
discussed and for the given scenarios, the optimum PS factor, TS factor and
power allocation coefficients are demonstrated.

###### doi:

10.3906/elk-2005-155

###### keywords:

simultaneous wireless information and power transfer, non-orthogonal multiple
access, cooperative relaying systems, outage analysis, imperfect SIC, channel
estimation errors

##  1 Introduction

The new era of the wireless communications is beyond the personal
communication and it has now many different applications such as Internet of
Things (IoT) networks, vehicular communication etc. Hence, the future wireless
networks (5G and beyond) are to meet challenging requirements such as very
high spectral efficiency, ultra wide coverage and low energy consumption [1].
To this end, the future networks will have the interplay between physical
layer techniques such as non-orthogonal multiple access (NOMA) [2],
cooperative relaying system (CRS) [3] and wireless power transfer [4].

The main idea of the NOMA is to allow multiple users sharing the same resource
block. NOMA has generally divided into two groups as code-domain (CD)-NOMA and
power-domain (PD)-NOMA. In CD-NOMA, the users share the same resource block
with sparse codes, so it is mostly called sparse code multiple access (SCMA)
whereas in PD-NOMA, the users are multiplexed with different power allocation
(PA) coefficients. Compared to orthogonal multiple access schemes, the main
advantage of the NOMA schemes is the spectral efficiency since all users are
allocated in the same resource block whereas the main disadvantage is the
error performance decay due to the inter-user-interference (IUI) [2]. Thanks
to its potential, NOMA111This paper deals with PD-NOMA, thus NOMA is used for
PD-NOMA after this point. has been implemented in other physical layer
techniques such as cooperative communication [5], visible light communication
[6] and index modulation [7].

CRS schemes have been studied for two decades because they provide a
remarkable performance gain in device-to-device communications. However, due
to the cooperative phase, the spectral efficiency of the CRS schemes
decreases. To alleviate this performance loss, NOMA-based CRS is proposed in
[8] where the source implements NOMA in the first phase to increase spectral
efficiency and it is proved that NOMA-CRS is superior to conventional CRS.
Therefore, NOMA-CRS has attracted recent attention from both academia and
industry [8, 9, 10, 11, 12, 13, 14, 15, 16]. In those works, NOMA-CRS has been
investigated in terms of capacity, outage probability (OP) and error
probability. Ergodic capacity of the NOMA-CRS is analyzed over Rayleigh [8]
and Rician [9] fading channels. Then, two different NOMA-CRS schemes have been
considered in [10] and OP expression is derived over Rayleigh fading channels.
The error performance of the NOMA-CRS is investigated in [11] and a machine
learning aided power optimization is proposed to minimize bit error rate.
Then, two more NOMA-CRS schemes are proposed in [12, 13], and capacity/outage
performances are investigated. Moreover, NOMA-CRS schemes have been considered
when multiple relays are available. NOMA-based diamond relaying, as a subset
of the NOMA-CRS with two relays, is analyzed in [14] and [15] in terms of
capacity and error performances, respectively. Contrary to [8, 9, 10, 11, 12,
13, 14, 15], NOMA-CRS is investigated in [16] when an amplify-forward relay is
used rather than a decode-forward relay. However, the aforementioned studies
assume that the relay node has its own independent (infinite) energy source to
help the device-to-device communication. It is neither feasible nor fair for
the relay which consumes its energy/battery for a communication where its own
symbol is not transmitted. Furthermore, those studies mostly assume perfect
successive interference canceler (SIC) at the relay (except for [11, 15]) and
perfect channel state information (CSI) at all nodes (except for [10]). These
are also not practical assumptions and should be relaxed.

On the other hand, simultaneous wireless information and power transfer
(SWIPT) is very promising to increase energy efficiency [4]. SWIPT is proved
to be practical and it can support nodes with limited energy [17]. To this
end, SWIPT has attracted tremendous attention [18, 19]. Besides, since it is
easily applicable, SWIPT integration into all other physical layer techniques
has also taken remarkable consideration [20, 21, 22, 23, 24, 25, 26, 27, 28,
29]. Specifically, the relay node can harvest its energy to re-transmit the
symbols of the source, thus SWIPT-cooperative NOMA schemes have been analyzed
in terms of capacity, outage and error performances [20, 21, 22, 23]. SWIPT
usage with NOMA has also been investigated in many studies. However, to the
best of the author’s knowledge, the SWIPT integration into NOMA-CRS has not
been studied well although other NOMA-involved systems have been analyzed with
SWIPT [24, 25, 26, 27]. Only two studies in the literature consider NOMA-CRS
with SWIPT. In [28], the authors consider a CRS where the source transmits two
symbols in two time slots and in the second phase, an energy harvesting relay
re-transmits the symbol of the first phase. The considered model in [28] is
different from the conventional NOMA-CRS schemes [8, 9, 10, 11, 12, 13, 14,
15, 16] since a power allocation is not implemented. Nevertheless, it is
called as SWIPT-CRS deploying NOMA since an SIC should be performed at the
destination due to the second phase. The authors provide outage probability
analysis for the considered model. Then, in [29], the author’s preliminary
work, a NOMA-CRS with SWIPT is considered and only the achievable rate is
analyzed. However, these two studies assume perfect SIC at the
relay/destination and perfect CSI at all nodes. Thus, these assumptions should
be relaxed and the analysis should be further extended for practical
scenarios.

Based on above discussions, in this paper, the NOMA-CRS with SWIPT is proposed
where the relay harvests its energy from radio frequency (RF) waves to
transmit symbols. In SWIPT integration, three different energy harvesting (EH)
protocols, power sharing (PS), time sharing (TS) and ideal protocols, are
implemented. All non-practical assumptions are relaxed, to this end, the OP
expressions for all EH protocols are derived with imperfect SIC and CSI.
Besides, the OP for the benchmark, NOMA-CRS without EH, is also derived. It is
revealed that the SWIPT-NOMA-CRS outperforms NOMA-CRS without EH. Then, the
effects of EH factors and PA coefficient are discussed for the outage
performance of the SWIPT-NOMA-CRS.

The rest of the paper is organized as follows. In Section 2, the SWIPT-NOMA-
CRS is introduced where all EH protocols and power transfers are defined. The
information transferring in SWIPT-NOMA-CRS and the related signal-to-
interference plus noise (SINR) definitions are also given in this section.
Then, in Section 3, the theoretical analysis of OP is performed. In Section 4,
the analysis is validated via computer simulations and performance comparisons
are presented. Finally, Section 5 discusses the results and concludes the
paper.

##  2 System Model

A NOMA-based relaying system is considered for a device-to-device
communication where a source (S), a decode-forward (DF) relay (R) and a
destination (D) are located. All nodes are assumed to be equipped with single
antenna. The channel fading coefficient between each node follows
$CN(0,\Omega_{k}),k=\\{sr,sd,rd\\}$ where $\Omega_{k}$ denotes the large-scale
path loss component. The imperfect channel state information (CSI) is
considered and the estimated channel at each node is given by
$\hat{h}_{k}=h_{k}+\epsilon$ where $\epsilon=CN(0,\kappa)$ which is an
appropriate model for the practical channel estimation techniques. The DF
relay operates in half duplex mode. Thus, the total communication is completed
in two phases. In order to alleviate inefficiency of the CRS, NOMA is
implemented at the source so that the spectral efficiency is increased.
Besides, it is assumed that the relay do not have independent energy source
and it harvests its energy from the RF signal in the first phase to transmit
symbols in the second phase. Therefore, the source implements a SWIPT so that
the system is called SWIPT-NOMA-CRS. The illustration of the SWIPT-NOMA-CRS is
given in Figure 1.

Figure 1: The illustration of SWIPT-NOMA-CRS

###  2.1 Transmit Power and Energy Harvesting

In the energy-constrained networks, a node can harvest energy from the RF
signals so that it can use it to transmit symbols. In the EH, the RF signals
are converted to Direct Current (DC) power via energy receiver (called
rectenna) [18]. This rectenna can be placed on the same circuit with a
transceiver, hence the SWIPT becomes possible. According to internal
characteristics (due to the diode ) of the rectenna, the EH protocols are
categorized in two groups : 1) linear EH 2) non-linear EH. In this paper, the
linear EH protocols are considered where the received RF waves are converted a
DC power with a energy conversion coefficient.

As explained above, the relay node harvests its energy from the RF signal
transmitted by the source. In this paper, within the linear EH models, three
different EH protocols (i.e., PS, TS and ideal protocols) [23] are
implemented. Time schedules for all three EH protocols are shown in Figure2.
In Figure2, the time schedule of the benchmark is also given where no EH is
implemented and the total energy is shared among the source and relay.

####  2.1.1 Benchmark (No EH)

As considered in the literature, if the relay has no ability to harvest
energy, it has its own energy source, and the total consumed energy/power is
the sum of the energy/power consumed by the source and the relay. As shown in
Figure 2(a), the source and the relay consume their energy within
$\nicefrac{{T}}{{2}}$ seconds. Thus, for fairness, considering the total
energy consumption during $T$ seconds, the transmit powers of both source and
relay are given as

$P_{s}=P_{r}=P_{T}.$ (1)

where $P_{s}$ and and $P_{r}$ are the transmit powers of the source and relay.
$P_{T}$ is the total consumed power during the whole communication ($T$
seconds).

####  2.1.2 Power Sharing (PS) Protocol

As shown in Figure 2(b), in the PS protocol, the communication times from
source-to-relay (S-R) and from relay-to-destination (R-D) are equal and cover
half of the total $T$ duration. Thus, by considering the total consumed energy
during $T$ seconds, the transmit power of the source within
$\nicefrac{{T}}{{2}}$ seconds is equal to

$P_{s}=2P_{T}.$ (2)

The relay harvests its energy during the first $\nicefrac{{T}}{{2}}$ seconds,
hence the harvested energy in PS protocol is given as

$E_{H}=\eta\rho P_{s}\left|\hat{h}_{sr}\right|^{2}(\nicefrac{{T}}{{2}}),$ (3)

where $\eta$ is the energy conversion coefficient and it is given as
$0<\eta<1$. In (3), $\rho$ is the PS factor as represented in Figure 2(b). It
is noteworthy that the energy is harvested according to the estimated channel
$\hat{h}_{sr}$. The harvested energy is consumed by the relay to transmit
symbols between R-D within the remained $\nicefrac{{T}}{{2}}$ seconds.
Therefore, the transmit power of the relay in PS protocol is obtained as

$P_{r}=\eta\rho P_{s}\left|\hat{h}_{sr}\right|^{2}.$ (4)

Figure 2: Time schedules for a) Benchmark (No EH) b) EH with PS protocol c) EH
with TS protocol d) EH with ideal protocol

####  2.1.3 Time Sharing (TS) Protocol

As shown in Figure 2(c), in TS protocol, the source transmits power for EH in
the first $\xi T$ seconds and then, it transmits information (data) in the
next $\nicefrac{{(1-\xi)T}}{{2}}$ second. Thus, the total consumed energy by
the source in TS mode is given by $\nicefrac{{P_{s}(1+\xi)T}}{{2}}$. For
fairness, regarding the total consumed energy in $T$ seconds with total power
$P_{T}$, the source power in TS protocol is obtained as

$P_{s}=\frac{2P_{T}}{(1+\xi)},$ (5)

where $\xi$ is the TS factor as given in Figure 2(c). The harvested energy
from this source power within $\xi T$ seconds is given as

$E_{H}=\eta P_{s}\hat{h}_{sr}(\xi T).$ (6)

Since this harvested energy at the relay is consumed within the remained
$\nicefrac{{(1-\xi)}}{{2}}$ seconds, the transmit power of the relay in TS
protocol is derived as

$P_{r}=2\eta\frac{\xi}{(1-\xi)}P_{s}\hat{h}_{sr}.$ (7)

####  2.1.4 Ideal Protocol

In the ideal protocol, as shown in Figure 2(d), the source transmits power
during $\nicefrac{{T}}{{2}}$ seconds. Thus, again by considering the total
consumed power within $T$ seconds, the transmit power of the source, in ideal
protocol, is given as

$P_{s}=2P_{T}.$ (8)

During the first $\nicefrac{{T}}{{2}}$ seconds, the relay harvests energy from
the RF wave between S-R, thus the harvested energy is given as

$E_{H}=\eta P_{s}\left|\hat{h}_{sr}\right|^{2}(\nicefrac{{T}}{{2}}).$ (9)

It is noteworthy that the energy and data transfers are achieved with the same
power $P_{s}$, in the ideal protocol whereas the transmit power of the source
is allocated by $\rho$ for energy and data transfers in the PS protocol. This
is the main difference between the PS and ideal protocols.

Then, this harvested energy is consumed by the relay to transmit symbols
between R-D link within $\nicefrac{{T}}{{2}}$ seconds. Therefore, the transmit
power of the relay, in ideal protocol, is obtained as

$P_{r}=\eta P_{s}\left|\hat{h}_{sr}\right|^{2}.$ (10)

###  2.2 Information Transfer

As explained above and shown in Figure 1, the information transfer is
completed in two phases. In the first phase, the source implements NOMA for
the consecutive two symbols of the destination to increase spectral efficiency
and broadcasts this superposition coded symbol to both relay and destination.
Thus, the received signals by both nodes are given as

$y_{k}=\sqrt{pP_{s}}\left(\sqrt{\alpha}x_{1}+\sqrt{\left(1-\alpha\right)}x_{2}\right)h_{k}+n_{k},\
k=sr,sd,$ (11)

where $\alpha$ is the PA coefficient (i.e., $\alpha<0.5$) among the symbols.
$n_{k}$ is the additive white Gaussian noise (AWGN) and follows
$CN(0,\sigma^{2})$. It is hereby noted that the $h_{k}$ in (11) is the actual
channel fading coefficient between nodes and it is estimated as
$\hat{h}_{k}=h_{k}+\epsilon$ at the receiving nodes as explained above.
Besides, it is noteworthy that the transmit power of the source changes
according to the used EH protocol as explained in detail in the previous
subsection (see eq. (1), (4), (7) and (10)). Thus, the $p$ coefficient denotes
the ratio how much the source power is allocated for the information transfer.
According to the EH protocol, it is given as

$p=\begin{cases}1-\rho,&\text{in PS protocol},\\\ 1,&\text{in TS, ideal
protocols and in no EH}.\end{cases}$ (12)

In the first phase, according to the received signal $y_{sd}$, $x_{2}$ symbols
are detected with a conventional detector (e.g., maximum likelihood (ML)) at
the destination by pretending $x_{1}$ symbols as a noise. At the same time,
the relay implements a successive interference canceler (SIC) to detect
$x_{1}$ symbols. In the SIC process, the relay firstly detects $x_{2}$ symbols
with a ML detector, then it subtracts these estimated $x_{2}$ symbols from the
received signal $y_{sd}$. Finally, it implements one more ML detector to
detect $x_{1}$ symbols based on the remained signal after subtraction. In the
second phase, the relay re-transmits the detected $x_{1}$ symbols to the
destination. Hence, the received signal in the second phase is given as

$y_{rd}=\sqrt{P_{r}}\hat{x}_{1}h_{rd}+n_{rd},$ (13)

where $\hat{x}_{1}$ is the detected symbol at the relay after SIC. It is worth
noting that $P_{r}$ is the transmit power of the relay and changes according
to the EH protocol since it is harvested from the source-relay link in the
first phase (or it has its own energy in no EH benchmark) (see eq. (1), (4),
(7) and (10)). Besides, the channel fading coefficient $h_{rd}$ is estimated
as $\hat{h}_{rd}$ at the destination. Lastly, the destination detects $x_{1}$
symbols based on $y_{rd}$.

###  2.3 Received Signal-to-Interference plus Noise Ratios (SINRs)

As given in (11), the source transmits superposition-coded NOMA symbols in the
first phase, thus an interference occurs. Both relay and destination detect
$x_{2}$ symbols firstly by pretending $x_{1}$ symbols as noise. Therefore, by
considering imperfect CSI at the nodes, the received SINRs for the $x_{2}$
symbols at the nodes are given as

$\begin{split}&SINR_{x_{2}}^{(sr)}=\frac{(1-\alpha)pP_{s}\hat{\gamma}_{sr}}{\alpha
pP_{s}\hat{\gamma}_{sr}+pP_{s}\kappa+\sigma^{2}},\\\
&SINR_{x_{2}}^{(sd)}=\frac{(1-\alpha)pP_{s}\hat{\gamma}_{sd}}{\alpha
pP_{s}\hat{\gamma}_{sd}+pP_{s}\kappa+\sigma^{2}}.\end{split}$ (14)

where $\hat{\gamma}_{k}\triangleq|\hat{h}_{k}|^{2}$ is defined and it follows
exponential distribution with the parameter
$\hat{\Omega}_{k}=\Omega_{k}-\kappa$. In (14), the first terms in the
denominators define the interference due to the $x_{1}$ symbols (NOMA
signalling) whereas the second and the third terms are the effects of the
imperfect CSI and the AWGN, respectively.

On the other hand, the relay implements SIC to detect $x_{1}$ symbols in the
first phase, thus, by also considering imperfect SIC, the received SINR for
the $x_{1}$ symbols at the relay is given as

$SINR_{x_{1}}^{(sr)}=\frac{\alpha
pP_{s}\hat{\gamma}_{sr}}{(1-\alpha)pP_{s}|g|^{2}+pP_{s}\kappa+\sigma^{2}},$
(15)

where the first term in the denominator denotes the imperfect SIC effect where
$g$ follows $CN(0,\delta\Omega_{sr})$. $0<\delta<1$ is defined where
$\delta=0$ and $\delta=1$ denote perfect SIC and no SIC cases, respectively.
Just like (14), in (15), the second and the third terms of the denominator are
the effects of the imperfect CSI and the AWGN, respectively. In (14) and (15),
the transmit power of the source is given in (1), (2), (5) and (8) according
to the EH protocol.

Lastly, the received SINR in the second phase is given as

$SINR_{\hat{x}_{1}}^{(rd)}=\frac{P_{r}\hat{\gamma}_{rd}}{P_{r}\kappa+\sigma^{2}}.$
(16)

It is important to note that $P_{r}$ in (16) will include the
$\hat{\gamma}_{sr}$ random variable if any of EH protocols is implemented (see
(4), (7) and (10)). It can be given as independent from $\hat{\gamma}_{sr}$
only if no EH is implemented (1).

##  3 Outage Probability (OP) Analysis

The outage event for a communication system is defined as the probability of
the achievable rate being below the target rate (QoS). To this end, the outage
probability for the symbols in SWIPT-NOMA-CRS is given by

$P_{i}(out)=P(R_{i}<\acute{R}_{i}),\ i=1,2,$ (17)

where $R_{i}$ and $\acute{R}_{i}$ denote the achievable rate and the target
rate of $x_{i},\ i=1,2$ symbols. Hence, as firstly, the achievable rates of
the symbols should be defined. Since a cooperative communication is
considered, the achievable rate of $x_{1}$ symbols is limited by the weakest
link [30]. Thus, according to the Shannon Theory [31], the achievable rate of
$x_{1}$ symbols is given as

$R_{1}=\zeta
B\log_{2}\left(1+\min\\{SINR_{x_{1}}^{(sr)},SINR_{\hat{x}_{1}}^{(rd)}\\}\right),$
(18)

where $B=\nicefrac{{1}}{{T}}$ the bandwidth. In (18), $\zeta$ exists since the
total communication is handled in two phases. According to the time schedules
of EH protocols in Figure 2 and Section 2.1, it is defined as

$\zeta=\begin{cases}\frac{1-\xi}{2},&\text{in TS protocol}\\\
\frac{1}{2},&\text{in PS and ideal protocols and in no EH}\end{cases}$ (19)

On the other hand, although cooperative communication is not considered for
the $x_{2}$ symbols, to guarantee SIC operation, the achievable rate at the
relay for $x_{2}$ symbols should not also cause outage. Thus, the achievable
rate of $x_{2}$ symbols is also given as [8]

$R_{2}=\zeta
B\log_{2}\left(1+\min\\{SINR_{x_{2}}^{(sr)},SINR_{x_{2}}^{(sd)}\\}\right).$
(20)

Firstly, to obtain $P_{2}(out)$, by substituting (14) into (20) then into
(17), the OP222In the following OP analysis, $B$ is removed for notation
simplicity since it is equal in all scenarios and does not affect the
analysis. is defined as

$P_{2}(out)=P\left(\min\left\\{\frac{(1-\alpha)pP_{s}\hat{\gamma}_{sr}}{\alpha
pP_{s}\hat{\gamma}_{sr}+pP_{s}\kappa+\sigma^{2}},\frac{(1-\alpha)pP_{s}\hat{\gamma}_{sd}}{\alpha
pP_{s}\hat{\gamma}_{sd}+pP_{s}\kappa+\sigma^{2}}\right\\}<\phi_{2}\right),$
(21)

where $\phi_{i}=2^{\nicefrac{{\acute{R}_{i}}}{{\zeta}}}-1,\ i=1,2$. The
probability of $P(Z<z)$ is represented by $F_{Z}(z)$ which is called the
cumulative distribution function (CDF) of $Z$. If $Z=\min\\{X,Y\\}$ is defined
and in case X and Y are statistically independent, the CDF of the Z is given
by $F_{Z}(z)=F_{X}(z)+F_{Y}(z)-F_{X}(z)F_{Y}(z)$ [32] where $F_{X}()$ and
$F_{Y}()$ are the CDFs of $X$ and $Y$ random variables, respectively.
Recalling $\hat{\gamma}_{sr}$ and $\hat{\gamma}_{sd}$ are statistically
independent, $X\triangleq\frac{(1-\alpha)pP_{s}\hat{\gamma}_{sr}}{\alpha
pP_{s}\hat{\gamma}_{sr}+pP_{s}\kappa+\sigma^{2}}$,
$Y\triangleq\frac{(1-\alpha)pP_{s}\hat{\gamma}_{sd}}{\alpha
pP_{s}\hat{\gamma}_{sd}+pP_{s}\kappa+\sigma^{2}}$ and/or
$\hat{\gamma}_{sr}\triangleq\frac{X\left(pP_{s}\kappa+\sigma^{2}\right)}{\left(1-(1+X)\alpha\right)pP_{s}}$,
$\hat{\gamma}_{sd}\triangleq\frac{Y\left(pP_{s}\kappa+\sigma^{2}\right)}{\left(1-(1+Y)\alpha\right)pP_{s}}$
are defined. Therefore, the OP of $x_{2}$ symbols is given as

$\begin{split}P_{2}(out)=F_{X}(\phi_{2})+F_{Y}(\phi_{2})-F_{X}(\phi_{2})F_{Y}(\phi_{2})=F_{\hat{\gamma}_{sr}}\left(A_{1}\right)+F_{\hat{\gamma}_{sd}}\left(A_{1}\right)-F_{\hat{\gamma}_{sr}}\left(A_{1}\right)F_{\hat{\gamma}_{sd}}\left(A_{1}\right),\end{split}$
(22)

where
$A_{1}\triangleq\frac{\phi_{2}\left(pP_{s}\kappa+\sigma^{2}\right)}{\left(1-(1+\phi_{2})\alpha\right)pP_{s}}$
is defined for representation/notation simplicity. As defined in (14),
$\hat{\gamma}_{k}=|\hat{h}_{k}|^{2}$ follows exponential distribution, hence
the CDF of the $\hat{\gamma}_{k}$ is
$F_{\hat{\gamma}_{k}}(.)=1-\exp\left(-\nicefrac{{\hat{\gamma}_{k}}}{{\hat{\Omega}_{k}}}\right)$
where $\hat{\Omega}_{k}=\Omega_{k}-\kappa$ is defined. Thus, the OP of the
$x_{1}$ symbols is derived as

$P_{2}(out)=\left\\{1-\exp\left(-\frac{A_{1}}{\hat{\Omega}_{sr}}\right)\right\\}+\left\\{1-\exp\left(-\frac{A_{1}}{\hat{\Omega}_{sd}}\right)\right\\}-\left\\{1-\exp\left(-\frac{A_{1}}{\hat{\Omega}_{sr}}\right)\right\\}\left\\{1-\exp\left(-\frac{A_{1}}{\hat{\Omega}_{sd}}\right)\right\\}.$
(23)

Likewise in the analysis of $x_{2}$ symbols, the OP of the $x_{1}$ symbols is
obtained, by substituting (15) and (16) into (18) then into (17), as

$P_{1}(out)=P\left(\min\left\\{\frac{\alpha
pP_{s}\hat{\gamma}_{sr}}{(1-\alpha)pP_{s}|g|^{2}+pP_{s}\kappa+\sigma^{2}},\frac{P_{r}\hat{\gamma}_{rd}}{P_{r}\kappa+\sigma^{2}}\right\\}<\phi_{1}\right).$
(24)

Recalling $P_{r}$ is a function of $P_{s}$ and $\bar{\gamma}_{sr}$ based on
the EH, the OP of the $x_{1}$ symbols is rewritten as

$P_{1}(out)=P\left(\min\left\\{\frac{\alpha
pP_{s}\hat{\gamma}_{sr}}{(1-\alpha)pP_{s}|g|^{2}+pP_{s}\kappa+\sigma^{2}},\frac{\Upsilon
P_{s}\hat{\gamma}_{sr}\hat{\gamma}_{rd}}{\Upsilon
P_{s}\hat{\gamma}_{sr}\kappa+\sigma^{2}}\right\\}<\phi_{1}\right)$ (25)

where $\Upsilon$ is the power transformation coefficient from the power of the
source in that EH protocol and according to section 2.1, it is given as

$\Upsilon=\begin{cases}\eta\rho,&\text{in PS protocol},\\\
2\eta\frac{\xi}{1-\xi},&\text{in TS protocol},\\\ \eta,&\text{in ideal
protocol}.\\\ \end{cases}$ (26)

In (25), since both SINRs include $\hat{\gamma}_{sr}$ they are statistically
correlated. Besides, the second SINR includes the same random variable on both
nominator and denominator, thus the joint CDF is very hard to obtain.
Nevertheless, without loss of generality, to derive a very tight approximate
expression, the rule of independent random variables given in (22) is applied.
The OP of the $x_{1}$ symbols is obtained as

$P_{1}(out)\cong
F_{\hat{\gamma}_{sr}}\left(A_{2}\right)+F_{\hat{\gamma}_{sr},\hat{\gamma}_{rd}}\left(\phi_{1}\right)-F_{\hat{\gamma}_{sr}}\left(A_{2}\right)F_{\hat{\gamma}_{sr},\hat{\gamma}_{rd}}\left(\phi_{1}\right),$
(27)

where
$A_{2}\triangleq\frac{\phi_{1}\left((1-\alpha)pP_{s}\delta\hat{\Omega}_{sr}+pP_{s}\kappa+\sigma^{2}\right)}{\alpha
pP_{s}}$ is defined for notation simplicity.
$F_{\hat{\gamma}_{sr},\hat{\gamma}_{rd}}$ is the joint CDF for the second SINR
term in (25). Since the $\hat{\gamma}_{sr}$ is exponentially distributed, the
first term in (27) can be easily obtained as

$F_{\hat{\gamma}_{sr}}\left(A_{2}\right)=1-\exp\left(-\frac{A_{2}}{\hat{\Omega}_{sr}}\right).$
(28)

The second term in (27) is obtained as

$\begin{split}F_{\hat{\gamma}_{sr},\hat{\gamma}_{rd}}\left(\phi_{1}\right)&=\mathop{\hbox
to12.43668pt{\hss $\displaystyle\mathop{\iint}^{\hskip 4.12494pt\phi_{1}}_{0}$
\hss}}\frac{\Upsilon P_{s}\hat{\gamma}_{sr}\hat{\gamma}_{rd}}{\Upsilon
P_{s}\hat{\gamma}_{sr}\kappa+\sigma^{2}}d\hat{\gamma}_{sr}d\hat{\gamma}_{rd},=\int\limits_{0}^{\infty}\frac{1}{\hat{\Omega}_{rd}}\exp\left(-\nicefrac{{\hat{\gamma}_{rd}}}{{\hat{\Omega}_{rd}}}\right)\int\limits_{\frac{\phi_{1}\sigma^{2}}{\left(\hat{\gamma}_{rd}-\phi_{1}\kappa\right)\Upsilon
P_{s}}}^{\infty}\frac{1}{\hat{\Omega}_{sr}}\exp\left(-\nicefrac{{\hat{\gamma}_{sr}}}{{\hat{\Omega}_{sr}}}\right)d\hat{\gamma}_{sr}d\hat{\gamma}_{rd},\\\
&=1-\int\limits_{0}^{\infty}\frac{1}{\hat{\Omega}_{rd}}\exp\left(-\frac{\hat{\gamma}_{rd}}{\hat{\Omega}_{rd}}-\frac{\phi_{1}\sigma^{2}}{\left(\hat{\gamma}_{rd}-\phi_{1}\kappa\right)\Upsilon
P_{s}\hat{\Omega}_{sr}}\right)d\hat{\gamma}_{rd}.\end{split}$ (29)

Since the exponential expression includes a complex polynomial in (29), to the
best of the author’s knowledge, it has no closed-form solution. Nevertheless,
it can be easily computed by numerical tools such as MATLAB, MAPPLE,
MATHEMATICA. By substituting (28) and (29), the OP of $x_{1}$ symbols is
derived.

Since both symbols are transmitted to the same destination in NOMA-CRS, the
system is considered to be in outage in case any of the symbols being in
outage. Therefore, the OP of the SWIPT-NOMA-CRS is equal to the union outage
probabilities of the symbols and it is given by

$P_{SWIPT-NOMA-CRS}(out)=P_{1}(out)\cup
P_{2}(out)=P_{1}(out)+P_{2}(out)-P_{1}(out)P_{2}(out).$ (30)

It is derived by substituting (23) and (27) into (30).

###  3.1 Benchmark Analysis (NOMA-CRS without EH)

As explained in the system model, the first phases in SWIPT-NOMA-CRS and NOMA-
CRS without EH are the same. The only difference is the power of the source.
Therefore, the OP of $x_{2}$ symbols in NOMA-CRS without EH is also equal to
(23).

On the other hand, since no EH is implemented, the relay will have its own
energy thereby the SINR between R-D will be independent from the first phase
(do not include $\hat{\gamma}_{sr}$ anymore). Therefore, by using (24) and
considering that $\hat{\gamma}_{sr}$ and $\hat{\gamma}_{rd}$ are statistically
independent, the OP of $x_{1}$ symbols is obtained, by repeating steps between
(21)-(23), as

$P_{1}(out)=\left\\{1-\exp\left(-\frac{A_{2}}{\hat{\Omega}_{sr}}\right)\right\\}+\left\\{1-\exp\left(-\frac{A_{3}}{\hat{\Omega}_{rd}}\right)\right\\}-\left\\{1-\exp\left(-\frac{A_{2}}{\hat{\Omega}_{sr}}\right)\right\\}\left\\{1-\exp\left(-\frac{A_{3}}{\hat{\Omega}_{rd}}\right)\right\\}$
(31)

where
$A_{3}\triangleq\frac{\phi_{1}\left(P_{r}\kappa+\sigma^{2}\right)}{P_{r}}$.
Lastly, the overall OP of the NOMA-CRS without EH is derived by substituting
(23) and (31) into (30).

##  4 Numerical Results

In this section, the derived OP expressions of the SWIPT-NOMA-CRS are
validated for all EH protocol and without EH (benchmark). In the following
figures, the lines denote the theoretical curves whereas the markers represent
computer simulations. Besides, unless the figures are presented with respect
to one of the parameters, the simulation parameters are given in Table.
Moreover, in the figures, the curves with the same parameters (e.g., $\kappa$,
$\alpha$) are noted with dashed circles.

Table 1: Simulation parameters Parameter | Value | Parameter | Value
---|---|---|---
$\Omega_{k},\ k=sr,sd,rd$ | $[10,2,10]$ | Time sharing factor ($\xi$) | $0.2$
Imperfect CSI factor ($\kappa$) |  $0$ (perfect CSI), $0.01$ | Imperfect SIC factor ($\delta$) |  $0$ (perfect SIC), $0.001$
Power allocation ($\alpha$) |  $0.1$ and $0.2$ | Bandwidth ($B$) | 1 MHz
Energy conversation coefficient ($\eta$) | $0.95$ | Target Rates ($\acute{R}_{i},\ i=1,2$) | $[500Kbps,100Kbps]$
Power sharing factor ($\rho$) | $0.2$ |  |

In Figure 3 and Figure 4, the outage performances of $x_{1}$ symbols are
presented for perfect SIC ($\delta=0$) and imperfect SIC ($\delta=0.001$)
cases, respectively. In both figures, the results are given for two different
PA coefficient and for both perfect CSI and imperfect CSI cases. Firstly, it
is noteworthy that the derived OP expressions match well with simulations for
all EH protocols which proves that the analysis is very tight. Besides, the
derived OP for without EH protocol is perfectly-matched with simulations. In
both figures, the OP of $x_{1}$ symbols have better performances in all SWIPT-
NOMA-CRS schemes (the EH protocols are implemented) rather than NOMA-CRS
without EH (benchmark). For instance, according to the EH protocol, the SWIPT-
NOMA-CRS schemes achieve the same outage performance with $\sim 2.5\ dB$ to
$4\ dB$ less power than NOMA-CRS without EH. This is very promising for energy
efficiency. However, with the increase of SIC uncertainties, the advantage of
the TS protocol can be diminished in high SNR region. This is explained as
follows. The transmit power of the source is higher in SWIPT-NOMA-CRS (with EH
protocols). Therefore, the effect of imperfect SIC becomes greater in the
first phase in SWIPT-NOMA-CRS since the imperfect SIC effect depends on the
source power given in (15). In addition to this, in TS protocol, since the
power transfer and information transfer are achieved by time division duplex,
the information transfer is implemented with a lower duration and this
increases $\phi_{i}$ in TS protocol due to the $\zeta$ (see (19) and below
(21)). However, one can easily see that the probability of the imperfect SIC
becomes very low in SWIPT-NOMA-CRS when an actual modulator/detector is
implemented since, with the increase of the transmit power, the error
probability decreases significantly. Then, in Figure 5, the outage performance
of $x_{2}$ symbols are presented for two different PA and CSI conditions. As
seen from the figures, the derived OP expressions match perfectly with
simulations for all EH and without EH protocols. The SWIPT-NOMA-CRS is again
superior to NOMA-CRS in terms of outage performances of $x_{2}$ symbols in all
scenarios. Indeed, this performance gain is more than the one of $x_{1}$
symbols. The SWIPT-NOMA-CRS can provide $\sim 3\ dB$ to $5\ dB$ energy
efficiency.

Figure 3: Outage Performance of $x_{1}$ symbols with perfect SIC ($\delta=0$)
a) PA $\alpha=0.1$ b) PA $\alpha=0.2$. Figure 4: Outage Performance of $x_{1}$
symbols with imperfect SIC ($\delta=0.001$)a) PA $\alpha=0.1$ b) PA
$\alpha=0.2$. Figure 5: Outage Performance of $x_{2}$ symbols a) PA
$\alpha=0.1$ b) PA $\alpha=0.2$.

Since the validations of the derived OP expressions for both symbols are
presented in Figure 3-Figure 5 and the overall outage probability of the
SWIPT-NOMA-CRS is defined as the union of the OPs of the symbols (30), in the
following figures, only the OP of the SWIPT-NOMA-CRS (union OP of the symbols)
is presented. In order to investigate the effect of the imperfect SIC, in
Figure 6, the OP of the SWIPT-NOMA-CRS is given with respect to imperfect SIC
effect ($\delta$). In Figure 6, the total transmit SNR is assumed to be
$30dB$. As seen in the previous figures, with the increase of the imperfect
SIC effect, the gap between performances becomes lower. Nevertheless, as
expected from the previous results, the PS and ideal protocols in SWIPT-NOMA-
CRS still outperforms NOMA-CRS without EH even a bit whereas the TS protocol
performs worse with higher imperfect SIC effects. Besides, if the imperfect
SIC effect becomes too high (e.g. $\delta\geq-5\ dB$), all considered
scenarios are always in outage. In Figure 6(a) and Figure 6(b), one can see
that the imperfect CSI does not have too much effect on the outage performance
when imperfect SIC is higher, since the performance is dominated by the
imperfect SIC. This is also seen in comparisons of PA coefficients. Since the
imperfect SIC causes worse performance of $x_{1}$ symbols and it pulls down
the overall performance of SWIPT-NOMA-CRS, the better performance is achieved
by increasing the allocated power to $x_{1}$ symbols (e.g., $\alpha=0.2$).

Figure 6: Outage Performance of the SWIPT-NOMA-CRS with respect to imperfect
SIC effect ($\delta$) a) $\kappa=0$ (perfect CSI) b) $\kappa=0.01$ (imperfect
CSI). Figure 7: Outage Performance of the SWIPT-NOMA-CRS with respect to EH
factor and PA coefficient a) in PS protocol vs. PS factor ($\rho$) b) in TS
protocol vs. TS factor ($\xi$) c) in all protocols vs. PA coefficient
($\alpha$)

In order to reveal the effect of EH protocol and PA parameters, in Figure 7,
the OP of the SWIPT-NOMA-CRS is presented with respect to these parameters. In
Figure 7, $\delta=0$, $\kappa=0$ and the total SNR is $30\ dB$. In Figure 7(a)
and Figure 7(b), the PA coefficient is $\alpha=0.2$. In Figure 7(a), OP of the
SWIPT-NOMA-CRS is given in PS protocol with the change of PS factor ($\rho$).
To compare, the performances of the ideal protocol in SWIPT-NOMA-CRS and of
the NOMA-CRS without EH are also presented. One can easily see that the PS
factor ($\rho$) has a dominant effect on the OP and according to chosen
$\rho$. The SWIPT-NOMA-CRS with PS protocol can achieve similar performance to
ideal protocol or it can have worse performance than NOMA-CRS without EH. To
this end, considering the performance gain over NOMA-CRS without EH, the
optimum PS factor ($\rho^{*}$) can be given as $0.25$ for the given
conditions. Likewise, the OP of the SWIPT-NOMA-CRS in TS protocol is given in
Figure 7(b) with respect to TS factor ($\xi$). Based on provided comparisons
in Figure 7(b), the SWIPT-NOMA-CRS can outperform NOMA-CRS without EH when
only $0.05\leq\xi\leq 0.25$ which is a very short range compared to the PS
protocol. Then, the OPs of all EH and without EH protocol are presented with
respect to PA coefficient ($\alpha$) in Figure 7(c). In Figure 7(c), the PS
and TS factors are chosen as $\rho=0.25$ and $\xi=0.15$ according to the
obtained values from Figure 7(a) and Figure 7(b). As can be seen from Figure
7(c), with the use of optimal EH factor values, SWIPT-NOMA-CRS always
outperforms NOMA-CRS without EH regardless of PA coefficient which is very
promising for energy efficiency. With the increase of $\alpha$, the outage
performances for all cases becomes better. Nevertheless, this increase is
floored after $\alpha\sim 0.35$. To this end, for the considered scenario, the
optimum PA coefficient can be given as $\alpha^{*}=0.35$.

Figure 8: Outage Performance of the SWIPT-NOMA-CRS with respect to target
rates ($\acute{R}_{1}$,$\acute{R}_{2}$) a) NOMA-CRS without EH (benchmark) b)
PS protocol ($\rho=0.25$) c) TS protocol ($\xi=0.15$) d) ideal protocol.

Lastly, to investigate the effects of the target rates (QoS requirements) on
the outage performance, in Figure 8, the OPs of the SWIPT-NOMA-CRS are given
with respect to $\acute{R}_{1}$ and $\acute{R}_{2}$. In Figure 8, the total
transmit SNR is $30dB$. The PS factor, TS factor and PA coefficient are set to
$\rho=0.25$, $\xi=0.15$ and $\alpha=0.35$ according to discussion on Figure 7.
As expected, if one of the target rates is increased, all scenarios have worse
outage performance. The best outage performance is achieved when lower rates
are required as QoS. On the other hand, if too strict QoS requirements (too
high target rates) are demanded, SWIPT-NOMA-CRS with TS protocol may be in
always outage. Based on provided comparisons, it is clear that SWIPT-NOMA-CRS
outperforms NOMA-CRS without EH for any QoS requirement.

##  5 Conclusion

In this paper, the SWIPT-NOMA-CRS is proposed where the relay node harvests
its energy from the RF signal between source and relay. In the SWIPT-NOMA-CRS,
three different EH protocol (e.g., PS, TS and ideal protocols) are
implemented. The closed-form OP expressions are derived for all EH protocols
under imperfect SIC and CSI. The derivations are validated via simulations.
The proposed SWIPT-NOMA-CRS outperforms conventional NOMA-CRS without EH
significantly and it can reduce the energy consumption up to $\sim 5\ dB$ for
the same OP target which is very promising for the energy-constraint networks
(e.g., IoT). Based on simulations, as expected, the ideal protocol has the
best performance. On the other hand, the PS protocol is superior to the TS
protocol. Moreover, the effects of the EH and PA parameters on the OP are
discussed and it is revealed that the PS protocol is more flexible than the TS
protocol. The PS protocols outperforms the conventional NOMA-CRS without EH
within a very large PS factor range whereas it is a very small TS factor range
in the TS protocol. To this end, the optimum parameters are represented for
the minimum OP in given scenarios. Finally, as future works, the energy
efficiency of other NOMA schemes can be increased thanks to the SWIPT
integration and the analysis of imperfect SIC and CSI can be extended for
these systems.

## References

  * [1] David K, Elmirghani J, Haas H, You X -H. Defining 6G: Challenges and Opportunities. IEEE Vehicular Technology Magazine 2019; 14 (3): 14–16. doi:10.1109/mvt.2019.2922512
  * [2] Dai L, Wang B, Ding Z, Wang Z, Chen Z, Hanzo L. A Survey of Non-Orthogonal Multiple Access for 5G. IEEE Communications Surveys&Tutorials 2018; 20 (3): 2294-2323. doi: 10.1109/COMST.2018.2835558.
  * [3] Nosratinia A, Hunter T E, Hedayat A. Cooperative communication in wireless networks. IEEE Communications Magazine 2004; 42 (10): 74-80. doi: 10.1109/MCOM.2004.1341264.
  * [4] Krikidis I, Timotheou S, Nikolaou S, Zheng, Ng D W K, Schober R. Simultaneous wireless information and power transfer in modern communication systems. IEEE Communications Magazine 2014; 52 (11): 104-110. doi:10.1109/MCOM.2014.6957150.
  * [5] Kara F, Kaya H. Threshold-Based Selective Cooperative-NOMA. IEEE Communications Letters 2019; 23 (7): 1263-1266. doi:10.1109/LCOMM.2019.2914918
  * [6] Marshoud H, Kapinas V M, Karagiannidis G K, Muhaidat S. Non-orthogonal multiple access for visible light communications. IEEE Photonics Technology Letters 2015; 28 (1): 51–54. doi:10.1109/LPT.2015.2479600
  * [7] Kara F, Kaya H. Performance Analysis of SSK-NOMA. IEEE Transactions on Vehicular Technology 2019; 68 (7): 6231–6242. doi:10.1109/TVT.2019.2914315
  * [8] Kim J -B, Lee I -H. Capacity Analysis of Cooperative Relaying Systems Using Non-Orthogonal Multiple Access. IEEE Communications Letters 2015; 19 (11): 1949–1952. doi:10.1109/LCOMM.2015.2472414
  * [9] Jiao R, Dai L, Zhang J, Mackenzie R, Hao M. On the Performance of NOMA-Based Cooperative Relaying Systems Over Rician Fading Channels. IEEE Transactions on Vehicular Technology 2017; 66 (12): 11409–11413. doi:10.1109/TVT.2017.2728608
  * [10] Zhang Y, Yang Z, Feng Y, Yan S. Performance Analysis of Cooperative Relaying Systems with Power-Domain Non-Orthogonal Multiple Access. IEEE Access. 2018; 6: 39839–39848. doi:10.1109/ACCESS.2018.2854774
  * [11] Kara F, Kaya H. Error performance of NOMA-based Cooperative Relaying Systems and Power Optimization for Minimum BER. under review.
  * [12] Xu M, Ji F, Wen M, Duan W. Novel Receiver Design for the Cooperative Relaying System with Non-Orthogonal Multiple Access. IEEE Communications Letters 2016; 20 (8): 1679–1682. doi:10.1109/LCOMM.2016.2575011
  * [13] Kader M F, Uddin M B, Islam S M R, Shin S Y. Capacity and outage analysis of a dual-hop decode-and-forward relay-aided NOMA scheme. Digital Signal Processing: A Review Journal 2019; 88: 138–148. doi:10.1016/j.dsp.2019.02.014
  * [14] Wan D, Wen M, Ji F, Yu H, Chen F. On the achievable sum-rate of NOMA-based diamond relay networks. IEEE Transactions on Vehicular Technology 2019; 68 (2): 1472–1486. doi:10.1109/TVT.2018.2886845
  * [15] Kara F, Kaya H. Error Probability Analysis of NOMA-based Diamond Relaying Network. IEEE Transactions on Vehicular Technology 2020; 69 (2): 2280-2285. doi:10.1109/TVT.2019.2956199
  * [16] Abbasi O, Ebrahimi A, Mokari N. NOMA inspired cooperative relaying system using an AF relay. IEEE Wireless Communications Letters 2019; 8 (1): 261–264. doi:10.1109/LWC.2018.2869592
  * [17] Zhang R, Ho C K. MIMO Broadcasting for Simultaneous Wireless Information and Power Transfer. IEEE Transactions on Wireless Communications 2019; 12 (5): 1989-2001. doi: 10.1109/TWC.2013.031813.120224.
  * [18] Clerckx B, Zhang R, Schober R, Ng D W K, Kim D I, Poor H V. Fundamentals of wireless information and power transfer: From RF energy harvester models to signal and system designs. IEEE Journal on Selected Areas in Communications 2019; 37 (1): 4–33. doi: 10.1109/JSAC.2018.2872615.
  * [19] Kashyap S, Björnson E, Larsson E G. On the Feasibility of Wireless Energy Transfer Using Massive Antenna Arrays. IEEE Transactions on Wireless Communications 2016; 15 (5): 3466–3480. doi:10.1109/TWC.2016.2521739
  * [20] Mukherjee A, Acharya T, Khandaker M R A. Outage Analysis for SWIPT-Enabled Two-Way Cognitive Cooperative Communications. IEEE Transactions on Vehicular Technology 2018; 67 (9): 9032-9036. doi:10.1109/TVT.2018.2840140.
  * [21] Krikidis I. Relay Selection in Wireless Powered Cooperative Networks With Energy Storage. IEEE Journal on Selected Areas in Communications 2015; 33 (12): 2596-2610. doi: 10.1109/JSAC.2015.2479015.
  * [22] Zeng Y, Zhang R. Full-Duplex Wireless-Powered Relay With Self-Energy Recycling. IEEE Wireless Communications Letters 2015; 4 (2): 201-204. doi: 10.1109/LWC.2015.2396516.
  * [23] Babaei M, Aygolu U, Basar E. BER analysis of dual-hop relaying with energy harvesting in nakagami-m fading channel. IEEE Transactions on Wireless Communications 2018; 17 (7): 4352–4361. doi:10.1109/TWC.2018.2823711
  * [24] Hedayati M, Kim I M. On the Performance of NOMA in the Two-User SWIPT System. IEEE Transactions on Vehicular Technology 2018; 67 (11): 11258–11263. doi:10.1109/TVT.2018.2866612
  * [25] Diamantoulakis P D, Pappi K N, Ding Z, Karagiannidis G K. Wireless-Powered Communications with Non-Orthogonal Multiple Access. IEEE Transactions on Wireless Communications 2016; 15 (12): 8422–8436. doi:10.1109/TWC.2016.2614937
  * [26] Liu Y, Ding Z, Elkashlan M, Poor H V. Cooperative Non-orthogonal Multiple Access With Simultaneous Wireless Information and Power Transfer. IEEE Journal on Selected Areas in Communications 2016; 34 (4): 938-953. doi: 10.1109/JSAC.2016.2549378.
  * [27] Hedayati M, Kim T. CoMP-NOMA in the SWIPT Networks. IEEE Transactions on Wireless Communications 2020; to appear doi:10.1109/TWC.2020.2985038
  * [28] Nguyen H S, Ly T T H, Nguyen T S, Van Huynh V, Nguyen T. L, Voznak M. Outage performance analysis and SWIPT optimization in energy-harvesting wireless sensor network deploying NOMA. Sensors 2019; 19 (3): 1–21. doi:10.3390/s19030613
  * [29] Kara F. Wireless Powered Cooperative Relaying Systems with Non-orthogonal Multiple Access. IEEE Black Sea Conference on Communications and Networking 2020.
  * [30] Bhatnagar M R. On the Capacity of Decode-and-Forward Relaying over Rician Fading Channels. IEEE Communications Letters 2013; 17 (6): 1100-1103. doi: 10.1109/LCOMM.2013.050313.122813.
  * [31] Shannon C. The Zero-Error Capacity of a Noisy Channel. IRE Transactions on Information Theory 1956; 2: 8–19.
  * [32] Ross S. A First Course in Probability. New Jersey, USA: Prentice Hall, 1988.