# Encoding information in the mutual coherence of spatially separated light
beams

Alfonso Nardi  Photonics Laboratory, ETH Zürich, Zürich, Switzerland  Shawn
Divitt  Photonics Laboratory, ETH Zürich, Zürich, Switzerland  Massimiliano
Rossi  Photonics Laboratory, ETH Zürich, Zürich, Switzerland  Felix
Tebbenjohanns  Photonics Laboratory, ETH Zürich, Zürich, Switzerland
Currently with the Department of Physics, Humboldt-Universität zu Berlin,
10099 Berlin, Germany  Andrei Militaru  Photonics Laboratory, ETH Zürich,
Zürich, Switzerland  Martin Frimmer  Photonics Laboratory, ETH Zürich, Zürich,
Switzerland  Lukas Novotny  Photonics Laboratory, ETH Zürich, Zürich,
Switzerland  Quantum Center, ETH Zürich, Zürich, Switzerland  Corresponding
author<EMAIL_ADDRESS>
###### Abstract

Coherence has been used as a resource for optical communications since its
earliest days. It is widely used for multiplexing of data, but not for
encoding of data. Here we introduce a coding scheme, which we call mutual
coherence coding, to encode information in the mutual coherence of spatially
separated light beams. We describe its implementation and analyze its
performance by deriving the relevant figures of merit (signal-to-noise ratio,
maximum bit-rate, and spectral efficiency) with respect to the number of
transmitted beams. Mutual coherence coding yields a quadratic scaling of the
number of transmitted signals with the number of employed light beams, which
might have benefits for cryptography and data security.

††journal: ol

Introduction – Optical coherence is a fundamental degree of freedom of the
light field. It accounts for the statistical properties of the electromagnetic
radiation [1]. Not only has the study of optical coherence been crucial to the
foundations and development of modern optics [2], but it has also found
countless applications in optical technology [3]. In particular, a coherence-
based multiplexing technique (known as coherence multiplexing) has been
proposed as the first coherent optical code-division multiple access (CDMA)
technique [4]. Its appeal (in common with other CDMA techniques) lies in the
fact that multiple users share the same optical bandwidth (relaxing the
requirements for wavelength control) and transmit asynchronously to each other
(providing independence of network synchronization). Coherence multiplexing,
in all the proposed configurations [5, 6], makes use of optical coherence to
discriminate one pair of transmitted signal-reference beams from all the other
pairs. Although this CDMA technique uses coherence as a resource for
multiplexing, its data encoding scheme still relies on the relative optical
path difference between two fields, requiring each signal to have a dedicated
reference. The dependence on a reference field can be eliminated by encoding
information directly into the mutual coherence between pairs of transmitted
light beams. This scheme would not only eliminate the need of reference
fields, but would also reduce the number of fields to be multiplexed. In
recent work we have introduced an approach to independently control the mutual
coherence between pairs of spatially separated light beams [7]. Here, we use a
similar approach to implement a reference-less, coherence-based coding scheme,
which we term mutual coherence coding.

By encoding information in the mutual coherence between pairs of spatially
separated transmitted fields, we eliminate the need of a reference in the
decoding step. Furthermore, we gain a quadratic scaling of the number of
transmitted signals with the number of transmitted light beams. After
introducing basic definitions we describe a general implementation that allows
to encode and decode information in field-field correlations. We discuss the
dependence of relevant figures of merit (signal-to-noise ratio, maximum bit-
rate and spectral efficiency) on the number of transmitted beams. Finally, we
discuss the benefits of using the proposed technique in combination with
coherence multiplexing and generalizing the coherence control to other degrees
of freedom.

Principle –

Figure 1: Implementation of mutual coherence coding. A set of broadband light
sources (here: 4) are fed into a linear port $\hat{T}$ to control their mutual
coherences. Afterward, the fields reach the receiver through a transmission
line. The received fields are then split to recover the encoded signals. The
mutual coherence is reconstructed through pair-wise interference experiments,
which can be implemented using a beam splitter and a balanced detector.
Insets: (s1) sketch of four transmission fibers. The mutual coherence is
defined for each pair, leading to a quadratic scaling of the number of
transmitted signals; (s2) detection scheme. Two fields are mixed by a beam
splitter and measured by a differential detector.

The term mutual coherence quantifies the correlation between a pair of
stochastic fields. The mutual coherence can be defined with respect to any
degree of freedom, e.g., space (spatial coherence), time (temporal coherence),
polarization (degree of polarization [8]), and transverse modes. In this work
we consider the special case of spatially separated optical beams. We
represent two random, statistically stationary, narrow-band electric fields at
two different points in space by the complex analytic signals $E_{i}$ and
$E_{j}$. The mutual coherence (also known as equal-time degree of coherence
[1]) $\gamma_{ij}$ between $E_{i}$ and $E_{j}$ is defined as

$\gamma_{ij}=\frac{\overline{E_{i}E_{j}^{\ast}}}{\sqrt{\overline{|E_{i}|^{2}}\leavevmode\nobreak\
\overline{|E_{j}|^{2}}}}\;,$ (1)

where the overline stands for ensemble or infinite time average (equivalent
for ergodic and stationary fields). The mutual coherence is a normalized
quantity, spanning from full incoherence ($|\gamma_{ij}|=0$) to partial
($0<|\gamma_{ij}|<1$) to full coherence ($|\gamma_{ij}|=1$). Let us now
consider a set of $N_{f}$ mutually incoherent input fields
$X_{1},X_{2},\dots,X_{N_{f}}$. We use an optical linear port implementing the
transformation $\hat{T}$ to get the output fields
$Y_{1},Y_{2},\dots,Y_{N_{f}}$. We define an output coherence matrix
$\mathbb{K}$, such that the element $[\mathbb{K}]_{ij}=\gamma_{ij}$ is the
mutual coherence between the fields $Y_{i}$ and $Y_{j}$. The linear port can
be chosen in such a way that we can program the desired output coherence
matrix, as previously shown in Ref. [7]. We note that $\mathbb{K}$ must be
positive semi-definite, which is always fulfilled if $|\gamma_{ij}|\leq
1/(N_{f}-1)$, for all $i\neq j$. This choice, while allowing us to
independently choose the values of all the mutual coherences, restricts the
analysis to a subspace of possible coherence matrices. One could envision a
more general scenario where the value of $\gamma_{ij}$ can exceed the
mentioned bound (see Sec. 1 in Supplement 1).

Thanks to our ability to control the output coherence matrix, we can use the
mutual coherence between pairs of output fields as our signal. The
transmission rate of a bit stream is then limited by the time needed to
generate a desired coherence matrix, e.g., the modulators’ refresh rate in
Ref. [7]. Considering $N_{f}$ output fields, the number of signals equals the
number of field pairs, that is $N_{p}=N_{f}(N_{f}-1)/2$. Thus, for large
$N_{f}$ we achieve a quadratic scaling of the number of transmitted signals
with the number of transmitted beams.

Implementation – A general implementation of mutual coherence coding is
sketched in Fig. 1, for the case of $N_{f}=4$. The incoherent input fields
$X_{1},X_{2},\dots,X_{N_{f}}$ have the same central frequency $\nu_{0}$,
linewidth $\Delta\nu$ and power $P_{\mathrm{in}}$. The light sources employed
can be LEDs or laser diodes, characterized by a large linewidth, but can still
be considered narrow-band, i.e., $\Delta\nu\ll\nu_{0}$. Through a linear port,
each input $X_{j}$ is connected to an output field $Y_{i}$, with
$i\in\\{1,2,\dots,N_{f}\\}$, through a complex coefficient $t_{ij}$, i.e.,
$Y_{i}=\sum_{j=1}^{N_{f}}t_{ij}X_{j}$. The mutual coherence $\gamma_{ij}$
between two output fields $Y_{i}$ and $Y_{j}$ takes the form

$\gamma_{ij}=\frac{\overline{Y_{i}Y_{j}^{*}}}{P_{\mathrm{out}}}=\sum_{n=1}^{N_{f}}t_{in}\sum_{m=1}^{N_{f}}t_{jm}^{*}\frac{\overline{X_{n}X_{m}^{*}}}{P_{\mathrm{out}}}=\frac{P_{\mathrm{in}}}{P_{\mathrm{out}}}\sum_{n=1}^{N_{f}}t_{in}t_{jn}^{*}\;,$
(2)

where we used the condition of input mutual incoherence
$\overline{X_{i}X_{j}^{*}}=P_{\mathrm{in}}\delta_{ij}$, and we assumed the
output fields having the same power
$\overline{|Y_{i}|^{2}}=P_{\mathrm{out}}=\eta P_{\mathrm{in}}$, where
$\eta=P_{\mathrm{out}}/P_{\mathrm{in}}$ is the transmission efficiency. By
tuning the values of the modulation coefficients $t_{ij}$, we can then control
the values of $\gamma_{ij}$ [7]. The linear port realizing the transformation
$\hat{T}$ can be implemented in different ways, e.g., with free-space optics
[7] or photonic integrated circuit [9]. The output fields $Y_{i}$ are
transmitted to the receiver through a transmission line. The transmission line
in the case of signals carried by spatially separated light beams can be a
bundle of single mode fibers, or a multi-core fiber, making use of the
platform developed for spatial division multiplexing [10]. At the receiver
side, we reconstruct the values of mutual coherences to decode the transmitted
data. We divide each received light beam into $N_{f}-1$ copies, and
subsequently perform $N_{f}(N_{f}-1)/2$ pair-wise interference experiments.
The interference experiment is carried out by mixing two fields with a beam
splitter, and measuring the intensities of the two output ports with a
differential detector (see inset s2 of Fig. 1). The two fields $E_{ij}^{+}$
and $E_{ij}^{-}$ at the output ports are

$E_{ij}^{\pm}=\frac{Y_{i}\pm Y_{j}}{\sqrt{2(N_{f}-1)}}\;,$ (3)

while the signal measured by the balanced detector is

$S_{ij}=\langle|E_{ij}^{+}|^{2}\rangle-\langle|E_{ij}^{-}|^{2}\rangle=\frac{2\leavevmode\nobreak\
\mathrm{Re}\big{(}\langle Y_{i}Y_{j}^{*}\rangle\big{)}}{N_{f}-1}\;.$ (4)

The angle brackets stand for the finite-time average performed by the
detector, defined as

$\langle
Y_{i}Y_{j}^{*}\rangle=\frac{1}{T}\int_{-T/2}^{T/2}Y_{i}Y_{j}^{*}dt\;.$ (5)

The detected signal is a random process, which we can characterize with an
expectation value and a variance. From the expectation value, we confirm that
the designed measurement corresponds indeed to an unbiased estimate of the
real value of the mutual coherence

$\overline{S_{ij}}=\frac{2}{T(N_{f}-1)}\int_{-T/2}^{T/2}\leavevmode\nobreak\
\mathrm{Re}\left(\overline{Y_{i}Y_{j}^{*}}\right)dt=\frac{2P_{\mathrm{out}}}{N_{f}-1}\mathrm{Re}(\gamma_{ij})\;,$
(6)

where we used Eq. (2) for the definition of $\gamma_{ij}$. However, the
stochastic nature of the underlying fields leads to a non-zero variance even
in the case where no noise is considered (see Sec. 2 in Supplement 1 for the
derivation)

$\text{Var}(S_{ij})=\overline{S_{ij}^{2}}-\left(\overline{S_{ij}}\right)^{2}=\frac{2P_{\mathrm{out}}^{2}\left[1+\mathrm{Re}\left(\gamma_{ij}^{2}\right)\right]}{(N_{f}-1)^{2}T\Delta\nu}\;.$
(7)

This variance is inversely proportional to the number of statistically
independent realizations that we sample during the measurement time $T$.
Fields separated by a time longer than the inverse of the linewidth are
uncorrelated, hence the number of independent realizations of the random
process is given by $\Delta\nu T$ [11]. Moreover, in the limit of small mutual
coherence $|\gamma_{ij}|\ll 1$, the variance saturates to a non-zero value
determined by the number of collected samples. The variance in Eq. (7) gives a
contribution to the total noise, which is known in the literature as optical
beat noise, and it is regarded as the main limitation for the performance of
traditional coherence multiplexing [12].

Performance –

Figure 2: System performance. (a) $\mathrm{SNR}$, (b) maximum bit-rate
($\mathrm{BR_{max}}$) and (c) spectral efficiency ($\mathrm{SE}$) as a
function of the number of transmitted fields ($N_{f}$) and input power
($P_{\mathrm{in}}$). The dashed lines correspond to (a) Eq. 11, (b) Eq. 12 and
(c) Eq. 13. The system parameters are: $T=1\leavevmode\nobreak\ \mathrm{ns}$,
$\gamma_{\mathrm{max}}=1/(N_{f}-1)$, $\nu_{0}=870\leavevmode\nobreak\
\mathrm{nm}$, $\Delta\nu=1\leavevmode\nobreak\ \mathrm{nm}$ and $\eta=1\%$.

We will now discuss important figures of merit for assessing the performance
of mutual coherence coding, starting with the bit-rate. For the sake of
generality, we choose as a reference the maximum bit-rate $\mathrm{BR_{max}}$
allowed by the channel capacity in the presence of additive white gaussian
noise [13]. The mutual coherence between each field pair is an independent
signal, hence the number of independent communication channels present in the
system is equal to the number of field pairs $N_{p}$, leading to a maximum
bit-rate

$\mathrm{BR_{max}}=N_{p}\leavevmode\nobreak\
\frac{\log_{2}\left(1+\mathrm{SNR}\right)}{2T}\;.$ (8)

The signal-to-noise ratio ($\mathrm{SNR}$) is defined as

$\mathrm{SNR}=\frac{\left(\overline{\tilde{S}_{ij}}\right)^{2}}{\text{Var}(\tilde{S}_{ij})}\;,$
(9)

where $\tilde{S}_{ij}$ is the detected signal resulting from the mutual
coherence of the pair of fields $Y_{i}$ and $Y_{j}$ in the presence of noise.
For multiple, independent noise sources the variance is the sum of the
variances of the different noise contributions (see Sec. 3 in Supplement 1).
Restricting ourselves to optical beat noise and shot noise, the $\mathrm{SNR}$
can be expressed as

$\mathrm{SNR}=\left[\frac{1+\mathrm{Re}(\gamma_{ij}^{2})}{2\mathrm{Re}(\gamma_{ij})^{2}\leavevmode\nobreak\
\Delta\nu T}+\frac{h\nu_{0}}{\eta
P_{\mathrm{in}}\mathrm{Re}(\gamma_{ij})^{2}\leavevmode\nobreak\
T}\right]^{-1}\;.$ (10)

The first term is the optical beat noise highlighted in (7) and the second
term corresponds to shot noise. The derivation of Eq. (8) assumes a value of
$\mathrm{Re}(\gamma_{ij})=\gamma_{\mathrm{max}}$ that is at the extreme of the
allowed range of values. In fact, with this choice the $\mathrm{SNR}$ provides
the maximum number of discretization steps of the signal. We choose
$\gamma_{\mathrm{max}}=1/(N_{f}-1)$ to ensure the positive semi-definiteness
of the output coherence matrix. We then insert $\gamma_{\mathrm{max}}$ into
Eq. (10) for the $\mathrm{SNR}$ and then into Eq. (8) to obtain the maximum
bit-rate. In Figs. 2a and 2b, we show the $\mathrm{SNR}$ and the
$\mathrm{BR_{max}}$, respectively, as a function of the number of transmitted
fields $N_{f}$ for different input powers. The $\mathrm{SNR}$ decays
monotonically with increasing $N_{f}$, whereas the maximum bit-rate first
increases quadratically and then saturates at large $N_{f}$ values. The
quadratic increase derives from the number of field pairs, while the
saturation is due to the low $\mathrm{SNR}$. In the limit of large $N_{f}$ and
sufficient input power (to be limited by optical beat noise) the
$\mathrm{SNR}$ takes the expression

$\mathrm{SNR}\approx\frac{2\Delta\nu T}{(N_{f}-1)^{2}}\;,$ (11)

where the factor $\Delta\nu T$ can be understood as the number of independent
realizations of the light field within the time $T$. When the $\mathrm{SNR}$
becomes smaller than $1$, we can Taylor expand Eq. (8) and the maximum bit-
rate saturates at the value

$\mathrm{BR_{max}}\approx\frac{N_{f}(N_{f}-1)}{4T}\left[\frac{2\Delta\nu
T}{(N_{f}-1)^{2}\ln(2)}\right]\approx\frac{\Delta\nu}{2\ln(2)}\;.$ (12)

Therefore, the ultimate limit of the bit-rate is given by the input linewidth.
The saturation can be well understood considering the statistical nature of
mutual coherence, since the rate of field realizations needed to reconstruct
the mutual coherence is given by the linewidth.

Alongside the maximum bit-rate, an important figure of merit is the spectral
efficiency (SE), which quantifies how well the available bandwidth is used. It
is defined as the bit-rate per communication channel per unit bandwidth
$(\mathrm{bit/s/Hz})$, i.e.,

$\mathrm{SE}=\frac{\mathrm{BR_{max}}}{N_{p}\leavevmode\nobreak\
\Delta\nu}<\frac{1}{N_{p}\leavevmode\nobreak\
2\ln(2)}=\frac{1}{N_{f}(N_{f}-1)\leavevmode\nobreak\ \ln(2)}\;.$ (13)

The dependence of the $\mathrm{SE}$ on the number of transmitted fields
$N_{f}$ is represented in Fig. 2c. With a value much smaller than one, mutual
coherence coding performs worse in terms of spectral efficiency than standard
coding techniques [14]. On the other hand, the bandwidth $\Delta\nu$ of mutual
coherence coding is considerably larger than the bandwidth of other coding
techniques, which are restricted by the bandwidth of optoelectronic
components, such as detectors and modulators ($\sim 50\leavevmode\nobreak\
\mathrm{GHz}$). The large bandwidth associated with mutual coherence coding is
provided by the input sources, and it is used to build the statistical
ensemble required to achieve a quadratic scaling of the number of transmitted
signals on $N_{f}$. Thanks to this quadratic scaling, mutual coherence coding
can achieve bit-rates on the order of $\Delta\nu$, hence very high for input
with large linewidth. Therefore, even if the spectral efficiency of mutual
coherence coding is much lower than in other coding schemes, we can gain
technical advantage in terms of bit-rate.

Discussion – Mutual coherence coding comes with a quadratic scaling of the
bit-rate with the number of transmitted fields, up to the point where it
saturates. The saturation value is given by the linewidth of the light sources
employed. Selecting broadband light sources, such as LEDs or laser diodes, can
therefore lead to remarkable data transmission rates. These light sources have
the added advantage of being inexpensive, thus facilitating the scalability.
Furthermore, the use of broadband light makes our proposal suitable for free-
space communications, where partially coherent sources have been reported to
be more robust to atmospheric turbulence [15]. Another advantage of the
proposed method is the absence of local oscillators at the receiver side,
since in mutual coherence coding the transmitted beams are mutually
referenced, allowing for a simpler implementation. However, the discussed
implementation requires an accurate control on the length difference between
the transmitted fibers. In fact, through the linear port we can control the
equal-time mutual coherence, which is conserved only as long as the optical
path difference of the interfering beams does not exceed the coherence length
of the input source. This issue becomes more severe for large input
linewidths, as the coherence length is reduced accordingly. A possible
solution to this limitation is to combine mutual coherence coding and
coherence multiplexing, obtaining a fully coherence-based communication
system. To do so, one would introduce, at the sender side, a different delay
for each output beam of the linear port. If the individual delays and their
differences are much longer than the coherence time of the light sources, the
fields will not interfere, even when combined in the same single mode fiber.
The receiver then splits the transmitted field and compensates for the input
delays to reconstruct the multiplexed signals, before proceeding with the
pair-wise interference experiments. Besides allowing to perform the
transmission with a single fiber, this approach relaxes the requirement on the
length difference between the fibers. In fact, since the transmission line is
common, any optical path difference between the interfering beams can only
originate from the introduced delays. Interestingly, in this approach the
delay lines are very short (on the order of the coherence length) and embedded
in a well controlled environment. The needed calibration of the delay length
can then make use of the techniques developed for standard coherent detection
schemes [16].

Moreover, having a coherence based communication system comes with security
benefits. Suppose the sender and the receiver have a pre-shared set of delays
(which can be frequently updated). An eavesdropper cannot record the message
encoded in the mutual coherence, because they are not able to record the
fluctuations of the light field in real time. An eavesdropper could measure
the mutual coherences only through interference experiments and only if they
knew the pre-shared delays. This security benefit is similar to that of
traditional coherence multiplexing [17].

As a concluding remark, we highlight that mutual coherence coding is not
limited to spatial coherence. Coherence based on any other degree of freedom
of light (temporal coherence, degree of polarization, correlations between
transverse modes, etc.) can be used to encode information, allowing for
various implementations. Moreover, resorting to mutual coherence between
different degrees of freedom [18] can lead to a great enhancement of the
scaling with the number of transmitted fields. Considering for example $n$
different beams (spatial degree of freedom), each of them characterized by $m$
transverse modes, the number of mutual coherences that we can control is
$N_{p}\approx(n\times m)^{2}/2$. This can be further generalized (e.g.,
including time and polarization) to greatly improve the capacity of current
communication systems.

Conclusions – In this work, we proposed a new method to encode information in
the mutual coherence of separate light beams. Light coherence is not only used
as a resource for multiplexing different signals in the same medium, but also
for encoding data. We derived the $\mathrm{SNR}$, the maximum bit-rate and the
spectral efficiency with respect to the number of transmitted light beams. We
discussed the benefits of employing mutual coherence coding, in particular the
potentially high bit-rate and the added security resulting from the
combination with coherence multiplexing. Finally, we highlighted that mutual
coherence coding can be generalized by involving other degrees of freedom of
light, which could lead to an even larger scaling of number of transmitted
signals per transmitted field.

### Funding.

ETH Zurich Research Grant (ETH-41 19-1).

### Disclosures.

The authors declare that there are no conflicts of interest related to this
article.

### Data availability.

No data were generated or analyzed in the presented research.

### Supplemental document.

See Supplement 1 for supporting content.

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Encoding information in the mutual coherence of spatially separated light
beams:
supplemental document

##  1 Coherence matrix properties

Given a set of light fields, the coherence matrix $\mathbb{K}$ contains all
the information about the coherence in the system. The diagonal elements of
the matrix are the self degrees of coherence $\gamma_{ii}$, which are always
equal to $1$, and the off-diagonal terms are the values of mutual coherence
$[\mathbb{K}_{ij}]=\gamma_{ij}$. The coherence matrix is also known as a
statistical correlation matrix, which is a normalized covariance matrix, and
must be Hermitian and positive semi-definite [1]. From the Hermitian condition
we have $\gamma_{ji}=\gamma_{ij}^{*}$, while the positive semi-definite
condition poses that [2]

$\frac{tr(\mathbb{K})^{2}}{tr(\mathbb{K}^{2})}\geq N_{f}-1\;,$ (S1)

where $tr$ is the matrix trace, and $N_{f}$ is the dimension of $\mathbb{K}$.
Combining the previous properties, we get a condition on the absolute values
of the mutual coherences

$\sum_{i=1}^{N_{f}}\sum_{j=1}^{N_{f}}|\gamma_{ij}|^{2}\leq\frac{N_{f}^{2}}{N_{f}-1}\;.$
(S2)

Choosing all mutual coherences with the same absolute value, i.e.,
$|\gamma_{ij}|=|\gamma|$ $\forall\;i,j$, Eq. (S2) becomes

$|\gamma|\leq\frac{1}{N_{f}-1}\;.$ (S3)

Choosing $|\gamma_{ij}|<1/(N_{f}-1)$ $\forall\;i,j$ would directly satisfy the
condition in Eq. (S3), ensuring that the resulting coherence matrix is
positive semi-definite, regardless of the phase of each mutual coherence
$\gamma_{ij}$. A more sophisticated choice [based on Eq. (S2)] could lead to
better performance of mutual coherence coding.

##  2 Detected signal variance without added noise

In this appendix, we derive the expression of the variance of the detected
signal in mutual coherence coding. We highlight that the variance considered
here results from the stochastic nature of the mutual coherence: no noise is
added to the transmitted fields in this treatment. Let us consider the
interference experiment between a single pair of fields $Y_{i}$ and $Y_{j}$.
The signal $S_{ij}$ measured by the balanced photodetector is (see main text)

$S_{ij}=\frac{2}{T(N_{f}-1)}\int_{-T/2}^{T/2}\leavevmode\nobreak\
\mathrm{Re}\left(Y_{i}Y_{j}^{*}\right)dt\;,$ (S4)

where $T$ is the integration time of the detector, and $N_{f}$ is the number
of transmitted fields. The fields are stochastic variables, hence we
characterize the signal through its expectation value and its variance. The
expectation value, denoted with an overline, is

$\overline{S_{ij}}=\frac{2}{T(N_{f}-1)}\int_{-T/2}^{T/2}\leavevmode\nobreak\
\mathrm{Re}\left(\overline{Y_{i}Y_{j}^{*}}\right)dt=\frac{2P_{\mathrm{out}}}{N_{f}-1}\mathrm{Re}(\gamma_{ij})\;,$
(S5)

where we used the relation
$P_{\mathrm{out}}\gamma_{ij}=\overline{Y_{i}Y_{j}^{*}}$. To derive the
variance
$\mathrm{Var}(S_{ij})=\overline{S_{ij}^{2}}-\left(\overline{S_{ij}}\right)^{2}$,
we compute the second moment

$\displaystyle\overline{S_{ij}^{2}}$
$\displaystyle=\frac{1}{T^{2}({N_{f}}-1)^{2}}\iint_{-T/2}^{T/2}\leavevmode\nobreak\
\overline{\left[Y_{i}(t_{1})Y_{j}^{*}(t_{1})+c.c.\right]\left[Y_{i}(t_{2})Y_{j}^{*}(t_{2})+c.c.\right]}\leavevmode\nobreak\
dt_{1}dt_{2}$ (S6)
$\displaystyle=\frac{2}{T^{2}({N_{f}}-1)^{2}}\iint_{-T/2}^{T/2}\leavevmode\nobreak\
\mathrm{Re}\bigg{\\{}\overbrace{\overline{\left[Y_{i}(t_{1})Y_{j}^{*}(t_{1})+c.c.\right]Y_{i}(t_{2})Y_{j}^{*}(t_{2})}}^{\mathcal{I}}\bigg{\\}}\leavevmode\nobreak\
dt_{1}dt_{2}\;,$

where $c.c.$ stands for complex conjugate. Recalling that the input fields
$X_{n}$, with $n\in\\{1,2,\dots,N_{f}\\}$, are related to the outputs of the
linear port through the relation $Y_{i}=\sum_{n}^{N_{f}}t_{in}X_{n}$, we can
express $\mathcal{I}$ as:

$\displaystyle\mathcal{I}$
$\displaystyle=\overline{\left[Y_{i}(t_{1})Y_{j}^{*}(t_{1})+Y_{i}^{*}(t_{1})Y_{j}(t_{1})\right]Y_{i}(t_{2})Y_{j}^{*}(t_{2})}$
(S7)
$\displaystyle=\sum_{n,m,p,q}^{N_{f}}\left(t_{jn}^{*}t_{im}+t_{in}^{*}t_{jm}\right)t_{jp}^{*}t_{iq}\overline{X^{*}_{n}(t_{1})X_{m}(t_{1})X^{*}_{p}(t_{2})X_{q}(t_{2})}\;.$

To simplify this equation, we need to derive the expression for the second-
order correlation function $\Gamma$ of the input fields, i.e.,

$\Gamma_{nmpq}(t_{1}-t_{2})=\overline{X^{*}_{n}(t_{1})X_{m}(t_{1})X^{*}_{p}(t_{2})X_{q}(t_{2})}\;.$
(S8)

The input fields are independent and have a null expectation value, hence
$\Gamma_{nmpq}(t_{1}-t_{2})$ is non-zero only if the values of the indexes are
all equal, or equal in pairs. Therefore, only for the following four, mutually
exclusive cases, the expression does not vanish. Firstly, if $n=m\neq p=q$, we
have

$\Gamma_{nnpp}(t_{1}-t_{2})=\overline{|X_{n}(t_{1})|^{2}}\;\overline{|X_{p}(t_{2})|^{2}}=P_{\mathrm{in}}^{2}.$
(S9)

Secondly, if $n=q\neq m=p$, we have

$\Gamma_{nmmn}(t_{1}-t_{2})=\overline{X^{*}_{n}(t_{1})X_{n}(t_{2})}\;\overline{X_{m}(t_{1})X^{*}_{m}(t_{2})}\;.$
(S10)

To solve this equation, we need to introduce the degree of first-order
coherence $g^{(1)}$ [3]:

$g^{(1)}(t_{1}-t_{2})=\frac{\overline{X^{*}_{n}(t_{1})X_{n}(t_{2})}}{P_{\mathrm{in}}}\;.$
(S11)

We can consider the process stationary, hence not dependent on the particular
value of $t_{1}$ and $t_{2}$, but only on their difference $\tau=t_{1}-t_{2}$.
Moreover, for our input sources (which we consider chaotic) the expression of
$g^{(1)}(\tau)$ is [3]

$g^{(1)}(\tau)=\exp\Big{(}-i2\pi\nu_{0}\tau-|\tau|\Delta\nu\Big{)}\;,$ (S12)

where $\nu_{0}$ and $\Delta\nu$ are the central frequency and the linewidth of
the input light fields, respectively. Thus, $\Gamma_{nmmn}$ takes the form

$\Gamma_{nmmn}(\tau)=P_{\mathrm{in}}^{2}\left|g^{(1)}(\tau)\right|^{2}\;.$
(S13)

Thirdly, if $n=p\neq m=q$, we have

$\Gamma_{npnp}(t_{1}-t_{2})=\overline{X^{*}_{n}(t_{1})X^{*}_{n}(t_{2})}\;\overline{X_{p}(t_{1})X_{p}(t_{2})}=0\;.$
(S14)

Finally, we have for $n=m=p=q$

$\Gamma_{nnnn}(t_{1}-t_{2})=\overline{X^{*}_{n}(t_{1})X_{n}(t_{1})X_{n}^{*}(t_{2})X_{n}(t_{2})}=P_{\mathrm{in}}^{2}g^{(2)}(\tau),$
(S15)

where we introduced the degree of second-order coherence $g^{(2)}(\tau)$. With
classical chaotic light sources we have [3]

$g^{(2)}(\tau)=1+|g^{(1)}(\tau)|^{2}\;,$ (S16)

which allows us to join the four cases into a single expression:

$\Gamma_{nmpq}(\tau)=P_{\mathrm{in}}^{2}\left[\delta_{nm}\delta_{pq}+\delta_{nq}\delta_{mp}|g^{(1)}(\tau)|^{2}\right]\;.$
(S17)

Going back to the integral $\mathcal{I}$ of Eq. (S7) we get

$\mathcal{I}=P_{\mathrm{in}}^{2}\sum_{n,p}^{N_{f}}\left[\left(t_{jn}^{*}t_{in}+t_{in}^{*}t_{jn}\right)t_{jp}^{*}t_{ip}+\left(t_{jn}^{*}t_{ip}+t_{in}^{*}t_{jp}\right)t_{jp}^{*}t_{in}|g^{(1)}(t_{1}-t_{2})|^{2}\right]\;.$
(S18)

From our choice of the linear port, we have
$P_{\mathrm{out}}\gamma_{ij}=P_{\mathrm{in}}\sum_{n}^{N_{f}}t_{in}t_{jn}^{*}$
[see Eq. (2) in the main text], leading to

$\mathcal{I}=P_{\mathrm{out}}^{2}\left[\left(\gamma_{ij}+\gamma_{ij}^{*}\right)\gamma_{ij}+\left(\gamma_{ij}^{2}+\gamma_{ii}\gamma_{jj}\right)|g^{(1)}(\tau)|^{2}\right]\;.$
(S19)

Including the derived expression of $\mathcal{I}$ in the second moment of the
signal and recalling that $\gamma_{ii}=1$ we get

$\overline{S_{ij}^{2}}=\frac{4P_{\mathrm{out}}^{2}\mathrm{Re}(\gamma_{ij})^{2}}{(N_{f}-1)^{2}}+\frac{2P_{\mathrm{out}}^{2}\left[1+\mathrm{Re}(\gamma_{ij}^{2})\right]}{T(N_{f}-1)^{2}}\int_{-T/2}^{T/2}\leavevmode\nobreak\
|g^{(1)}(\tau)|^{2}\leavevmode\nobreak\ d\tau\;.$ (S20)

The integral can be analytically solved from the definition of $g^{(1)}(\tau)$
given in Eq. (S12):

$\int_{-T/2}^{T/2}\leavevmode\nobreak\ |g^{(1)}(\tau)|^{2}\leavevmode\nobreak\
d\tau=\frac{1-\exp(-T\Delta\nu)}{\Delta\nu}\to\frac{1}{\Delta\nu}$ (S21)

where we considered the limit of integration time $T$ much longer than the
coherence time $\tau_{c}=1/\Delta\nu$ of the light sources. Finally, the
variance of the signal is

$\text{Var}(S_{ij})=\frac{2P_{\mathrm{out}}^{2}\left[1+\mathrm{Re}\left(\gamma_{ij}^{2}\right)\right]}{(N_{f}-1)^{2}T\Delta\nu}\;.$
(S22)

##  3 Signal-to-noise ratio (SNR) derivation

In this section, we derive the expression of the SNR, defined as

$\mathrm{SNR}=\frac{\overline{\tilde{S}_{ij}}^{2}}{\mathrm{Var}(\tilde{S}_{ij})}\;,$
(S23)

where $i\neq j$ and $\tilde{S}_{ij}$ is the signal measured by the balanced
detector. Differently than Sec. 2, we consider here the presence of noise
$N_{i}$ (hence the different symbol $\tilde{S}_{ij}$), which is added to the
received signal $\tilde{Y}_{i}=Y_{i}+N_{i}$. Noise related to different
signals are uncorrelated, i.e., $\overline{N_{i}N_{j}^{*}}=0$, with $i\neq j$,
they are stationary and all characterized by zero mean value and an
autocorrelation
$\mathcal{R}_{N}(\tau)=\overline{N_{i}(t)^{*}N_{i}(t+\tau)}=P_{N}\exp(-\Delta\nu_{N}|\tau|)$,
where $\Delta\nu_{N}$ is the cut-off frequency of the transmission channel,
and $P_{N}$ is the noise power.

The expectation value of the detected signal in the presence of noise is

$\overline{\tilde{S}_{ij}}=\int_{-T/2}^{T/2}\frac{2\leavevmode\nobreak\
\mathrm{Re}\left(\overline{\tilde{Y}_{i}\tilde{Y}_{j}^{*}}\right)}{T({N_{f}}-1)}dt=\int_{-T/2}^{T/2}\frac{2\leavevmode\nobreak\
\mathrm{Re}\left(\overline{Y_{i}Y_{j}^{*}}+\overline{Y_{i}N_{j}^{*}}+\overline{N_{i}Y_{j}^{*}}+\overline{N_{i}N_{j}^{*}}\right)}{T({N_{f}}-1)}dt\;.$
(S24)

Since the noise and the signals are uncorrelated
($\overline{Y_{i}N_{j}^{*}}=\overline{N_{i}Y_{j}^{*}}=0$) and we know that
$\overline{N_{i}N_{j}^{*}}=0$, the expression simplifies to

$\overline{\tilde{S}_{ij}}=\frac{2P_{\mathrm{out}}\mathrm{Re}(\gamma_{ij})}{T({N_{f}}-1)}\;,$
(S25)

which is the same expression obtained without the noise [see Eq. (S5)].

As for the second moment of the detected signal, we have

$\overline{\tilde{S}_{ij}^{2}}=\frac{2}{T^{2}({N_{f}}-1)^{2}}\int_{T^{2}}\leavevmode\nobreak\
\mathrm{Re}\Big{\\{}\overbrace{\overline{\left[\tilde{Y}_{i}^{*}(t_{1})\tilde{Y}_{j}(t_{1})+c.c.\right]\tilde{Y}_{i}^{*}(t_{2})\tilde{Y}_{j}(t_{2})}}^{\tilde{\mathcal{I}}}\Big{\\}}dt_{1}dt_{2}\;.$
(S26)

We can analyze the two terms of $\mathcal{I}$ separately:

$\tilde{\mathcal{I}}=\overbrace{\overline{\tilde{Y}_{i}^{*}(t_{1})\tilde{Y}_{j}(t_{1})\tilde{Y}_{i}^{*}(t_{2})\tilde{Y}_{j}(t_{2})}}^{\eta_{1}}+\overbrace{\overline{\tilde{Y}_{i}(t_{1})\tilde{Y}_{j}^{*}(t_{1})\tilde{Y}_{i}^{*}(t_{2})\tilde{Y}_{j}(t_{2})}}^{\eta_{2}}\;.$
(S27)

Of the $16$ terms obtained expanding the product in $\eta_{1}$, the only non-
zero term is

$\eta_{1}=\overline{Y_{i}^{*}(t_{1})Y_{j}(t_{1})Y_{i}^{*}(t_{2})Y_{j}(t_{2})}=\sum_{n,m,p,q}^{N}t_{in}^{*}t_{jm}t_{ip}^{*}t_{jq}\Gamma_{nmpq}(t_{1}-t_{2})\;,$
(S28)

where we used the definition of the linear port
$Y_{i}=\sum_{n=1}^{N_{f}}t_{in}X_{n}$ and the definition of
$\Gamma_{nmpq}(t_{1}-t_{2})$ given in Eq. (S8). Using the expression derived
in Eq. (S17) we get

$\eta_{1}=P_{\mathrm{out}}^{2}\gamma_{ij}^{2}\left(1+|g^{(1)}(\tau)|^{2}\right)\;.$
(S29)

For $\eta_{2}$, instead, more terms are non-zero:

$\displaystyle\eta_{2}$
$\displaystyle=\sum_{n,m,p,q}^{N_{f}}t_{jn}^{*}t_{im}t_{ip}^{*}t_{jq}\Gamma_{nmpq}(t_{1}-t_{2})+|\mathcal{R}_{N}(\tau)|^{2}+$
(S30)
$\displaystyle\qquad+\sum_{n,m}^{N_{f}}t_{in}t_{im}^{*}\overline{X_{n}(t_{1})X_{m}^{*}(t_{2})}\mathcal{R}(-\tau)+\sum_{n,m}^{N_{f}}t_{jn}^{*}t_{jm}\overline{X_{n}^{*}(t_{1})X_{m}(t_{2})}\mathcal{R}(\tau)\;,$

where $\tau=t_{1}-t_{2}$. Using again Eq. (S17) and considering that
$\gamma_{ii}=\gamma_{jj}=1$ we obtain

$\eta_{2}=P_{\mathrm{out}}^{2}\left[|\gamma_{ij}|^{2}+|g^{(1)}(\tau)|^{2}\right]+|\mathcal{R}_{N}(\tau)|^{2}+P_{\mathrm{out}}\mathrm{Re}\left[g^{(1)}(\tau)\mathcal{R}_{N}(-\tau)\right]\;.$
(S31)

To solve the integral we need the following relations:

$\frac{1}{T}\int_{-T/2}^{T/2}\leavevmode\nobreak\
|g^{(1)}(\tau)|^{2}d\tau=\frac{1-\exp(-\Delta\nu T)}{\Delta\nu
T}\approx\frac{1}{\Delta\nu T}\;,$ (S32)
$\frac{1}{T}\int_{-T/2}^{T/2}\leavevmode\nobreak\
|\mathcal{R}_{N}(\tau)|^{2}d\tau=\frac{P_{N}^{2}\left[1-\exp(-\Delta\nu_{N}T)\right]}{\Delta\nu_{N}T}\approx\frac{P_{N}^{2}}{\Delta\nu_{N}T}\;,$
(S33) $\frac{1}{T}\int_{-T/2}^{T/2}\leavevmode\nobreak\
\mathrm{Re}\left\\{g^{(1)}(\tau)\mathcal{R}_{N}(-\tau)\right\\}d\tau\approx\frac{P_{N}(\Delta\nu+\Delta\nu_{N})}{T\left[(\Delta\nu+\Delta\nu_{N})^{2}+4\pi^{2}\nu_{0}^{2}\right]}\to
0\;.$ (S34)

The approximate result comes from considering $T\gg 1/\Delta\nu$, $T\gg
1/\Delta\nu_{N}$ and $\nu_{0}\gg\Delta\nu,\Delta\nu_{N}$. Therefore, going
back to the expression of the second moment with the approximate results of
the integrals we get

$\overline{\tilde{S}_{ij}^{2}}=\frac{2P_{\mathrm{out}}^{2}}{(N_{f}-1)^{2}}\left[\mathrm{Re}\left(\gamma_{ij}^{2}\right)+|\gamma_{ij}|^{2}+\frac{1+\mathrm{Re}\left(\gamma_{ij}^{2}\right)}{\Delta\nu
T}+\frac{P_{N}^{2}}{P_{\mathrm{out}}^{2}\Delta\nu_{N}T}\right]\;.$ (S35)

Considering that
$|\gamma_{ij}|^{2}+\mathrm{Re}(\gamma_{ij}^{2})=2\mathrm{Re}(\gamma_{ij})^{2}$,
we finally derive the variance of the detected signal:

$\mathrm{Var}(\tilde{S}_{ij})=\overline{\tilde{S}_{ij}^{2}}-\left(\overline{\tilde{S}_{ij}}\right)^{2}=\frac{2P_{\mathrm{out}}^{2}}{(N_{f}-1)^{2}}\left[\frac{1+\mathrm{Re}\left(\gamma_{ij}^{2}\right)}{\Delta\nu
T}+\frac{P_{N}^{2}}{P_{\mathrm{out}}^{2}\Delta\nu_{N}T}\right]\;.$ (S36)

Finally, the expression of the SNR is the following

$\mathrm{SNR}=\frac{\overline{\tilde{S}_{ij}}^{2}}{\mathrm{Var}(\tilde{S}_{ij})}=\left[\frac{1+\mathrm{Re}(\gamma_{ij}^{2})}{2\mathrm{Re}(\gamma_{ij})^{2}}\frac{1}{\Delta\nu
T}+\frac{P_{N}^{2}}{2P_{\mathrm{out}}^{2}\mathrm{Re}(\gamma_{ij})^{2}}\frac{1}{\Delta\nu_{N}T}\right]^{-1}\;.$
(S37)

In case we choose to consider only shot noise, the power spectral density for
a balanced detector is
$\mathcal{S}_{NN}=P_{N}^{2}/\Delta\nu_{N}=2h\nu_{0}P_{\mathrm{out}}$.
Moreover, considering a transmission efficiency
$P_{\mathrm{out}}/P_{\mathrm{in}}=\eta$, the final expression of the
$\mathrm{SNR}$ is

$\mathrm{SNR}=\left[\frac{1+\mathrm{Re}(\gamma_{ij}^{2})}{2\mathrm{Re}(\gamma_{ij})^{2}\leavevmode\nobreak\
\Delta\nu T}+\frac{h\nu_{0}}{\eta
P_{\mathrm{in}}\mathrm{Re}(\gamma_{ij})^{2}\leavevmode\nobreak\
T}\right]^{-1}\;.$ (S38)

## References

  * [1] L. Devroye, _Non-Uniform Random Variate Generation_ , (Springer New York, New York, NY, 1986).
  * [2] H. Wolkowicz and G. P. Styan, “Bounds for eigenvalues using traces,” Linear Algebr. its Appl. 29, 471–506 (1980).
  * [3] R. Loudon, _The quantum theory of light_ , Oxford science publications (Oxford Univ. Press, Oxford, 2006), 3rd ed.


# Non-Orthogonal Multiple Access Performance for Millimeter Wave in Vehicular
Communications

Baha Eddine Youcef Belmekki  Abdelkrim Hamza  LISIC Laboratory, Electronic and
Computer Faculty, USTHB, Algiers, Algeria,  Benoît Escrig  University of
Toulouse, IRIT Laboratory, School of ENSEEIHT, Institut National Polytechnique
de Toulouse, France, e-mail: $\\{$bahaeddine.belmekki,
<EMAIL_ADDRESS>
###### Abstract

In this paper, we study the performance of millimeter wave (mmWave) vehicular
communications (VCs) using non-orthogonal multiple access scheme (NOMA) at
road intersections, since there areas are more prone to accidents. We study
the case when the intersection involves two perpendicular lanes, we then
extend the study to an intersection with several lanes. The transmission
occurs between a source, and two destinations. The transmission experiences
interference originated from a set of vehicles that are distributed as a
Poisson point process (PPP). Our analysis includes the effects of blockage
from buildings and vehicles at intersections. Closed form outage probability
expressions are obtained. We show that as the nodes reach the intersection,
the outage probability increases. Counter-intuitively, we show that the non
line of sigh (NLOS) scenario has a better performance than the line of sigh
(LOS) scenario. Finally, we compare the performance of mmWave NOMA with OMA,
and we show that NOMA offers a significant improvement over OMA mmWave
vehicular networks. The analysis is verified by Monte-Carlo simulation.

###### Index Terms:

5G, NOMA, mmWave, interference, outage probability, vehicular communications.

##  I Introduction

###  I-A Motivation

Road traffic accidents are a major issue, especially at road intersections
[1]. In that regard, vehicular communications (VCs) offer several applications
for road safety and traffic management. These applications can prevent
accidents or alerting vehicles of accidents happening in their vicinity.
Hence, these applications need high data rate and high spectral efficiency, to
insure high reliability and low latency transmissions. In this context, non-
orthogonal multiple access (NOMA) has been show to increase the data rate and
spectral efficiency [2]. Unlike orthogonal multiple access (OMA), NOMA allows
multiple users to share the same resource with different power allocation
levels. On the other hand, the needs of VCs for the fifth generation (5G) of
wireless networks in terms of resources require a larger bandwidth. Since the
spectral efficiency of sub-6 GHz bands has already reached the theoretical
limits, millimeter wave (mmWave) frequency bands (20-100 GHz and beyond) offer
a very large bandwidth [3].

###  I-B Related Works

####  I-B1 NOMA Works

NOMA is an efficient multiple access technique for spectrum use. It has been
shown that NOMA outperforms OMA (see [4] and reference therein). However, few
works investigate the effect of co-channel interference and their impact on
the performance [5, 6, 7]. The authors in [5] and [6] analyze downlinks of
NOMA networks. The authors in [8] and [7] analyze uplinks of NOMA networks. In
[9], the authors analyze the performance of NOMA transmissions and propose an
interference aware NOMA design that takes into account both intercell and
intracell interference.

####  I-B2 mmWave Works

In mmWave bands, few works studied communications using tools from stochastic
geometry [10, 11, 12, 13]. However, in [10, 11, 12], the effect of small-scale
fading is not taken into consideration. In [13], the authors investigate the
performance of mmWave relaying networks in terms of coverage probability with
best relay selection.

####  I-B3 VCs at Road Intersections Works

Several works studied the effect of the interference at intersections,
considering OMA. The performance in terms of success probability are derivated
considering direct transmission in [14, 15]. The performance of vehicle to
vehicle (V2V) communications are evaluated for multiple intersections schemes
considering direct transmission in [16]. In [17], the authors derive the
outage probability of a V2V communications with power control strategy of a
direct transmission. In [18], the authors investigate the impact of a line of
sight and non line of sight transmissions at intersections considering
Nakagami-$m$ fading channels. The authors in [19] study the effect of mobility
of vehicular communications at road junctions. In [20, 21, 22, 23, 24], the
authors respectively study the impact of non-orthogonal multiple access,
cooperative non-orthogonal multiple access, and maximum ratio combining with
NOMA at intersections. Following this line of research, we study the
performance of VCs at intersections in the presence of interference. However,
at the best of the author’s knowledge, there are no prior works that consider
both an intersection scenario with NOMA and considering mmWave networks. Our
analysis includes the effects of blockage from the building and vehicles at
intersections, and Nakagami-$m$ fading channels between the transmitting nodes
with difference values of $m$ for LOS and NLOS are considered. Unlike other
works that uses approximations, closed form expressions are obtained for
Nakagami-$m$ fading channels.

##  II System Model

Figure 1: NOMA system model for mmWave VCs involving a source and two
destinations. The nodes can be vehicles or as part of the communication
infrastructure.

###  II-A Scenario Model

We consider a mm-Wave vehicular network using a NOMA transmission between a
source $S$, and two destinations $D_{1}$ and $D_{2}$. We consider an
intersection scenario with two perpendicular roads, an horizontal road $X$,
and a vertical road $Y$.

Since both V2V and V2I communications111The Doppler shift and time-varying
effect of V2V and V2I channels are beyond the scope of this paper. are
considered, any node of the triplet $\\{{S,D_{1},D_{2}}\\}$ can be on the road
(e.g., vehicle), or outside the roads (e.g., infrastructure). We denote by
$D_{i}$ the receiving node, and by $d_{i}$ the distance between the node
$D_{i}$ and the intersection, where $i\in\\{1,2\\}$, as shown in Fig.1. The
angle $\theta_{D_{i}}$ is the angle between the node ${D_{i}}$ and the $X$
road (see Fig.1).

The transmission is subject to interference that are originated from vehicles
located on the roads. The set of interfering vehicles located on the $X$ road
that are in a LOS with $\\{{S,D_{1},D_{2}}\\}$, denoted by
$\Phi^{\textrm{LOS}}_{X}$ (resp. on the $Y$ road, denoted by
$\Phi^{\textrm{LOS}}_{Y}$) are modeled as a one-dimensional homogeneous
Poisson point process (1D-HPPP), that is,
$\Phi^{\textrm{LOS}}_{X}\sim\textrm{1D-HPPP}(\lambda^{\textrm{LOS}}_{X},x)$
(resp.$\Phi^{\textrm{LOS}}_{Y}$
$\sim\textrm{1D-HPPP}(\lambda^{\textrm{LOS}}_{Y},y)$, where $x$ and
$\lambda^{\textrm{LOS}}_{X}$ (resp. $y$ and $\lambda^{\textrm{LOS}}_{Y}$) are
the position of the LOS interfering vehicles and their intensity on the $X$
road (resp. $Y$ road).

Similarly, the set of interfering vehicles located on the $X$ road that are in
a NLOS with $\\{{S,D_{1},D_{2}}\\}$, denoted by $\Phi^{\textrm{NLOS}}_{X}$
(resp. on the $Y$ road, denoted by $\Phi^{\textrm{NLOS}}_{Y}$) are modeled as
a 1D-HPPP, that is,
$\Phi^{\textrm{NLOS}}_{X}\sim\textrm{1D-HPPP}(\lambda^{\textrm{NLOS}}_{X},x)$
(resp.$\Phi^{\textrm{NLOS}}_{Y}$
$\sim\textrm{1D-HPPP}(\lambda^{\textrm{NLOS}}_{Y},y)$, where $x$ and
$\lambda^{\textrm{NLOS}}_{X}$ (resp. $y$ and $\lambda^{\textrm{NLOS}}_{Y}$)
are the position of the NLOS interfering vehicles and their intensity on the
$X$ road (resp. $Y$ road). The notation $x$ and $y$ denotes both the
interfering vehicles and their locations.

###  II-B Blockage Model

At the intersection, the mmWave signals cannot penetrate the obstacles (e.g.,
building, vehicles), which causes the link to be in LOS, or in NLOS. The event
of a link between a node $a$ and $b$ is in a LOS and NLOS, are respectively
defined as $\textrm{LOS}_{ab}$, and $\textrm{NLOS}_{ab}$. The LOS probability
function $\mathbb{P}(\textrm{LOS}_{ab})$ is used, where the link between $a$
and $b$ has a LOS probability $\mathbb{P}(\textrm{LOS}_{ab})=\exp(-\beta
r_{ab})$ and NLOS probability
$\mathbb{P}(\textrm{NLOS}_{ab})=1-\mathbb{P}(\textrm{LOS}_{ab})$, where the
constant rate $\beta$ depends on the building size, shape and density [25].

###  II-C Transmission Model

The transmission between the nodes $a$ and $b$ experiences a path loss,
denoted $r_{ab}^{-\alpha}$ , where $r_{ab}=\|a-b\|$, and $\alpha$ is the path
loss exponent. The path exponent
$\alpha\in\\{\alpha_{\textrm{LOS}},\alpha_{\textrm{NLOS}}\\}$, where
$\alpha=\alpha_{\textrm{LOS}}$, when the transmission is in LOS, whereas
$\alpha=\alpha_{\textrm{NLOS}}$, when transmission is in NLOS.

###  II-D Medium Access Control (MAC) Protocol

The medium access protocols used in VCs are mainly based on carrier sense
multiple access (CSMA) schemes (e.g., IEEE 802.11 p) [26]. However, [27, 28]
showed that the performance of CSMA tends to the performance of ALOHA in dense
networks. Hence, we assume that vehicles use slotted Aloha MAC protocol with
parameter $p$, i.e., every node can access the medium with a probability $p$.

###  II-E NOMA Model

Several works in NOMA order the receiving nodes by their channel states [29].
However, we consider that the receiving nodes are ordered according to their
quality of service (QoS) priorities, since it has been show that it is more
realistic assumption [30, 31]. We consider a scenario in which $D_{1}$ needs a
low data rate but has to be served immediately, whereas $D_{2}$ needs a higher
data rate but can be served later. This can be the case when $D_{1}$ is a
vehicle that needs to receive safety data information about an accident,
whereas $D_{2}$ can be a user that accesses the internet connection.

###  II-F Directional Beamforming Model

We model the directivity similar to in [32], where the directional gain,
denoted $G(\omega)$, within the half power beamwidth ($\phi/2$) is $G_{max}$
and is $G_{min}$ in all other directions. The gain is then expressed as

$G(\omega)=\left\\{\begin{array}[]{ll}G_{max},&\textrm{if}\>\>|\omega|\leq\frac{\phi}{2};\\\
G_{min},&\textrm{otherwise}.\end{array}\right.$ (1)

In this paper, we consider a perfect beam alignment between the nodes, hence
$G_{eq}=G_{max}^{2}$. The impact of beam misalignment is beyond the scope of
this paper, and can be a topic of future works.

###  II-G Channel and Interference Model

We consider an interference limited scenario, that is, the power of noise is
set to zero ($\sigma^{2}=0$). Without loss of generality, we assume that all
nodes transmit with a unit power. The signal transmitted by $S$, denoted
$\chi_{S}$ is a mixture of the message intended to $D_{1}$ and $D_{2}$. This
can be expressed as

$\chi_{S}=\sqrt{a_{1}}\chi_{D_{1}}+\sqrt{a_{2}}\chi_{D_{2}},$

where $a_{i}$ is the power coefficients allocated to $D_{i}$, and $\chi_{Di}$
is the message intended to $D_{i}$, where $i\in\\{1,2\\}$. Since $D_{1}$ has
higher power than $D_{2}$, that is $a_{1}\geq a_{2}$, then $D_{1}$ comes first
in the decoding order. Note that, $a_{1}+a_{2}=1$.

The signal received at $D_{i}$ is expressed as

$\mathcal{Y}_{D_{i}}=h_{SD_{i}}\sqrt{r_{SD_{i}}^{-\alpha}\Upsilon}\>\chi_{S}\mathds{1}(\textrm{LOS}_{SD_{i}})+h_{SD_{i}}\sqrt{r_{SD_{i}}^{-\alpha}\Upsilon}\>\chi_{S}\mathds{1}(\textrm{NLOS}_{SD_{i}})+\sum_{x\in\Phi^{\textrm{LOS}}_{X_{D_{i}}}}h_{D_{i}x}\sqrt{r_{D_{i}x}^{-\alpha_{\textrm{LOS}}}\Upsilon}\>\chi_{x}\\\
+\sum_{y\in\Phi^{\textrm{LOS}}_{Y_{D_{i}}}}h_{D_{i}y}\sqrt{r_{D_{i}y}^{-\alpha_{\textrm{LOS}}}\Upsilon}\>\chi_{y}+\sum_{x\in\Phi^{\textrm{NLOS}}_{X_{D_{i}}}}h_{D_{i}x}\sqrt{r_{D_{i}x}^{-\alpha_{\textrm{NLOS}}}\Upsilon}\>\chi_{x}+\sum_{y\in\Phi^{\textrm{NLOS}}_{Y_{D_{i}}}}h_{D_{i}y}\sqrt{r_{D_{i}y}^{-\alpha_{\textrm{NLOS}}}\Upsilon}\>\chi_{y},$
(2)

where $\mathcal{Y}_{D_{i}}$ is the signal received by $D_{i}$, and $\chi_{S}$
is the message transmitted by $S$. The messages transmitted by the interfere
node $x$ and $y$, are denoted respectively by $\chi_{x}$ and $\chi_{y}$. The
term $\Upsilon=G_{eq}\eta^{2}/(4\pi)^{2}$ models the directional gain, the
reference path loss at one meter, and $\eta$ is the wavelength of the
operating frequency.

The coefficient $h_{SD_{i}}$ denotes the fading of the link $S-D_{i}$. The
fading coefficient $h_{SD_{i}}$ is distributed according to a Nakagami-$m$
distribution with parameter $m$ [13], that is

$f_{h_{SD_{i}}}(x)=2\Big{(}\frac{m}{\mu}\Big{)}^{m}\frac{x^{2m-1}}{\Gamma(m)}e^{-\frac{m}{\mu}x^{2}},$
(3)

where the parameter $m\in\\{m_{\textrm{LOS}},m_{\textrm{NLOS}}\\}$. Note that
$m=m_{\textrm{LOS}}$ when $S-D_{i}$ is in a LOS, whereas
$m=m_{\textrm{NLOS}}$, when $S-D_{i}$ is in a NLOS. The parameter $\mu$ is the
average received power. Hence, the power fading coefficient $|h_{SD_{i}}|^{2}$
is distributed according to a gamma distribution, that is,

$f_{|h_{SD_{i}}|^{2}}(x)=\Big{(}\frac{m}{\mu}\Big{)}^{m}\frac{x^{m-1}}{\Gamma(m)}e^{-\frac{m}{\mu}x}.$
(4)

The fading coefficients $h_{D_{i}x}$ and $h_{D_{i}y}$ denote the fading of the
link $D_{i}-x$, and $D_{i}-y$. The fading coefficients are modeled as Rayleigh
fading [33]. Thus, the power fading coefficients $|h_{D_{i}x}|^{2}$ and
$|h_{D_{i}y}|^{2}$, are distributed according to an exponential distribution
with unit mean.

The aggregate interference from the $X$ road at ${D_{i}}$, denoted
$I_{X_{D_{i}}}$, is expressed as

$\displaystyle I_{X_{D_{i}}}$ $\displaystyle=$ $\displaystyle
I^{\textrm{LOS}}_{X_{D_{i}}}+I^{\textrm{NLOS}}_{X_{D_{i}}}$ (5)
$\displaystyle=$
$\displaystyle\sum_{x\in\Phi^{\textrm{LOS}}_{X_{D_{i}}}}|h_{{D_{i}}x}|^{2}r_{{D_{i}}x}^{-\alpha_{\textrm{LOS}}}\Upsilon+\sum_{y\in\Phi^{\textrm{NLOS}}_{X_{{D_{i}}}}}|h_{{D_{i}}x}|^{2}r_{{D_{i}}x}^{-\alpha_{\textrm{NLOS}}}\Upsilon,$

where $I^{\textrm{LOS}}_{X_{D_{i}}}$ denotes the aggregate interference from
the $X$ road that are in a LOS with ${D_{i}}$, and
$I^{\textrm{NLOS}}_{X_{{D_{i}}}}$ denotes the aggregate interference from the
$X$ road that are in a NLOS with ${D_{i}}$. Similarly,
$\Phi^{\textrm{LOS}}_{X_{D_{i}}}$ and $\Phi^{\textrm{NLOS}}_{X_{D_{i}}}$,
denote respectively, the set of the interferers from the $X$ road at ${D_{i}}$
in a LOS, and in NLOS. In the same way, the aggregate interference from the
$Y$ road at ${D_{i}}$, denoted $I_{Y_{D_{i}}}$, is expressed as

$\displaystyle I_{Y_{D_{i}}}$ $\displaystyle=$ $\displaystyle
I^{\textrm{LOS}}_{Y_{{D_{i}}}}+I^{\textrm{NLOS}}_{Y_{{D_{i}}}}$ (6)
$\displaystyle=$
$\displaystyle\sum_{y\in\Phi^{\textrm{LOS}}_{Y_{{D_{i}}}}}|h_{{D_{i}}y}|^{2}r_{{D_{i}}y}^{-\alpha_{\textrm{LOS}}}\Upsilon+\sum_{y\in\Phi^{\textrm{NLOS}}_{Y_{{D_{i}}}}}|h_{{D_{i}}y}|^{2}r_{{D_{i}}y}^{-\alpha_{\textrm{NLOS}}}\Upsilon,$

where $I^{\textrm{LOS}}_{Y_{{D_{i}}}}$ denotes the aggregate interference from
the $X$ road that are in a LOS with ${D_{i}}$, and
$I^{\textrm{NLOS}}_{Y_{{D_{i}}}}$ denotes the aggregate interference from the
$Y$ road that are in a NLOS with ${D_{i}}$. Similarly,
$\Phi^{\textrm{LOS}}_{Y_{{D_{i}}}}$ and $\Phi^{\textrm{NLOS}}_{Y_{{D_{i}}}}$,
denote respectively, the set of the interferers from the $Y$ road at ${D_{i}}$
in a LOS, and in NLOS.

##  III NOMA Outage Expressions

###  III-A Signal-to-Interference Ratio (SIR) Expressions

The outage probability is defined as the probability that the SIR at the
receiver is below a given threshold. According to successive interference
cancellation (SIC) [34], $D_{1}$ message is decoded first at the receiver
since it has the higher power allocation, and $D_{2}$ message is considered as
interference. The SIR at $D_{1}$ to decode its desired message, denoted
$\textrm{SIR}^{(\alpha)}_{D_{1}}$, is given by

$\textrm{SIR}^{(\alpha)}_{D_{1}}=\frac{|h_{SD_{1}}|^{2}r_{SD_{1}}^{-\alpha}\Upsilon\,a_{1}}{|h_{SD_{1}}|^{2}r_{SD_{1}}^{-\alpha}\Upsilon
a_{2}+I_{X_{D_{1}}}+I_{Y_{D_{1}}}}.$ (7)

In order for $D_{2}$ to decode its desired message, it has to decode $D_{1}$
message. The SIR at $D_{2}$ to decode $D_{1}$ message, denoted
$\textrm{SIR}^{(\alpha)}_{D_{2-1}}$, is expressed as

$\textrm{SIR}^{(\alpha)}_{D_{2-1}}=\frac{|h_{SD_{2}}|^{2}r_{SD_{2}}^{-\alpha}\Upsilon\,a_{1}}{|h_{SD_{2}}|^{2}r_{SD_{2}}^{-\alpha}\Upsilon
a_{2}+I_{X_{D_{2}}}+I_{Y_{D_{2}}}}.$ (8)

The SIR at $D_{2}$ to decode its desired message, denoted
$\textrm{SIR}^{(\alpha)}_{D_{2-2}}$, is expressed as

$\textrm{SIR}^{(\alpha)}_{D_{2-2}}=\frac{|h_{SD_{2}}|^{2}r_{SD_{2}}^{-\alpha}\Upsilon\,a_{2}}{I_{X_{D_{2}}}+I_{Y_{D_{2}}}}.$
(9)

###  III-B Outage Event Expressions

The outage event that $D_{1}$ does not decode its desired message, denoted
$\textit{O}_{D_{1}}$, is defined as

$\mathcal{O}_{D_{1}}\triangleq\bigcup_{{\textrm{Z}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\big{\\{}\textrm{Z}_{SD_{1}}\cap(\textrm{SIR}^{(\alpha_{\textrm{Z}})}_{D_{1}}<\Theta_{1})\big{\\}},$
(10)

where $\Theta_{1}=2^{2\mathcal{R}_{1}}-1$, and $\mathcal{R}_{1}$ is the target
data rate of $D_{1}$.

Also, the outage event that $D_{2}$ does not decode its desired message,
denoted $\textit{O}_{D_{2}}$, is defined as

$\mathcal{O}_{D_{2}}\triangleq\bigcup_{{\textrm{Z}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\bigcup_{i=1}^{2}\big{\\{}\textrm{Z}_{SD_{2}}\cap(\textrm{SIR}^{(\alpha_{\textrm{Z}})}_{D_{2-i}}<\Theta_{i})\big{\\}},$
(11)

where $\Theta_{2}=2^{2\mathcal{R}_{2}}-1$ ($i=2)$, and $\mathcal{R}_{2}$ is
the target data rate of $D_{2}$.

###  III-C Outage Probability Expressions

The outage probability related to $\mathcal{O}_{D_{1}}$, denoted
$\mathbb{P}(\mathcal{O}_{D_{1}})$, is given when
$\Theta_{1}<\frac{a_{1}}{a_{2}}$, by

$\mathbb{P}(\mathcal{O}_{D_{1}})=1-\sum_{\textrm{Z}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\mathbb{P}(\textrm{Z}_{SD_{1}})\\\
\prod_{\textrm{K}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\sum_{k=0}^{m_{\textrm{Z}}-1}\frac{1}{k!}\>\big{(}-\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SD_{1}}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{)}^{k}\sum_{n=0}^{k}\binom{k}{n}\frac{\textrm{d}^{k-n}\mathcal{L}_{I^{\textrm{K}}_{X_{D_{1}}}}\big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SD_{1}}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{)}}{\textrm{d}^{k-n}\big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SD_{1}}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{)}}\frac{\textrm{d}^{n}\mathcal{L}_{I^{\textrm{K}}_{Y_{D_{1}}}}\big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SD_{1}}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{)}}{\textrm{d}^{n}\big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SD_{1}}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{)}},$
(12)

where $\Psi_{1}=\Theta_{1}/(a_{1}-\Theta_{1}a_{2})$.

Figure 2: Outage probability as a function of distance from the intersection.
Figure 3: Outage probability as a function of $\|S-D_{1}\|=\|S-D_{2}\|$.
Figure 4: Outage probability as function of $\lambda$ for LOS transmission,
NLOS, and LOS/NLOS (equation (12) and (13)).

The outage probability related to $\mathcal{O}_{D_{2}}$, denoted by
$\mathbb{P}(\mathcal{O}_{D_{2}})$ is given, when
$\Theta_{1}<\frac{a_{1}}{a_{2}}$, by

$\mathbb{P}(\mathcal{O}_{D_{2}})=1-\sum_{\textrm{Z}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\mathbb{P}(\textrm{Z}_{SD_{2}})\\\
\prod_{\textrm{K}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\sum_{k=0}^{m_{\textrm{Z}}-1}\frac{1}{k!}\>\big{(}-\dfrac{m_{\textrm{Z}}\>\Psi_{\mathrm{max}}}{\mu\>r_{SD_{2}}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{)}^{k}\sum_{n=0}^{k}\binom{k}{n}\frac{\textrm{d}^{k-n}\mathcal{L}_{I^{\textrm{K}}_{X_{D_{2}}}}\big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{\mathrm{max}}}{\mu\>r_{SD_{2}}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{)}}{\textrm{d}^{k-n}\big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{\mathrm{max}}}{\mu\>r_{SD_{2}}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{)}}\frac{\textrm{d}^{n}\mathcal{L}_{I^{\textrm{K}}_{Y_{D_{2}}}}\big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{\mathrm{max}}}{\mu\>r_{SD_{2}}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{)}}{\textrm{d}^{n}\big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{\mathrm{max}}}{\mu\>r_{SD_{2}}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{)}},$
(13)

where $\Psi_{\mathrm{max}}=\mathrm{max}(\Psi_{1},\Psi_{2})$, and
$\Psi_{2}=\Theta_{2}/a_{2}$.
Proof: See Appendix A. $\blacksquare$

##  IV Laplace Transform Expressions

In this section, we present the Laplace transform expressions of the
interference from the $X$ road at $D_{i}$, denoted
$\mathcal{L}_{I^{\textrm{K}}_{X_{D_{i}}}}$, and from the $Y$ road at $D_{i}$,
denoted $\mathcal{L}_{I^{\textrm{K}}_{Y_{D_{i}}}}$.

The Laplace transform of the interference originating from the $X$ road at the
received node denoted $D_{i}$, is expressed as

$\mathcal{L}_{I^{\textrm{K}}_{X_{D_{i}}}}(s)=\exp\Bigg{(}-\emph{p}\lambda^{\textrm{K}}_{X}\int_{\mathbb{R}}\dfrac{1}{1+\|\textit{x}-{D_{i}}\|^{\alpha_{\textrm{K}}}/s}\textrm{d}x\Bigg{)},$
(14)

where

$\|\textit{x}-{D_{i}}\|=\sqrt{\big{[}d_{i}\sin(\theta_{{D_{i}}})\big{]}^{2}+\big{[}x-d_{i}\cos(\theta_{D_{i}})\big{]}^{2}}.$
(15)

The Laplace transform of the interference originating from the $Y$ road at
$D_{i}$ is given by

$\mathcal{L}_{I^{\textrm{K}}_{Y_{D_{i}}}}(s)=\exp\Bigg{(}-\emph{p}\lambda^{\textrm{K}}_{Y}\int_{\mathbb{R}}\dfrac{1}{1+\|\textit{y}-{D_{i}}\|^{\alpha_{\textrm{K}}}/s}\textrm{d}y\Bigg{)},$
(16)

where

$\|\textit{y}-{D_{i}}\|=\sqrt{\big{[}d_{i}\cos(\theta_{{D_{i}}})\big{]}^{2}+\big{[}y-d_{i}\sin(\theta_{D_{i}})\big{]}^{2}},$
(17)

where $\theta_{D_{i}}$ is the angle between the node ${D_{i}}$ and the $X$
road. Proof: See Appendix B. $\blacksquare$

We only present the case when $\alpha_{\textrm{K}}=2$ due to the lack of
space. The Laplace transform expressions of the interference at $D_{i}$ for an
intersection scenario, when $\alpha_{\textrm{K}}=2$ are given by

$\mathcal{L}_{I^{\textrm{K}}_{X_{D_{i}}}}(s)=\exp\Bigg{(}\dfrac{-\emph{p}\lambda^{\textrm{K}}_{X}s\pi}{\sqrt{\big{[}{d_{i}}\sin(\theta_{{D_{i}}})\big{]}^{2}+s}}\Bigg{)},$
(18)

and

$\mathcal{L}_{I^{\textrm{K}}_{Y_{D_{i}}}}(s)=\exp\Bigg{(}\dfrac{-\emph{p}\lambda^{\textrm{K}}_{Y}s\pi}{\sqrt{\big{[}{d_{i}}\cos(\theta_{{D_{i}}})\big{]}^{2}+s}}\Bigg{)}.$
(19)

Proof: See Appendix C. $\blacksquare$

Figure 5: Outage probability as a function of $\lambda$ considering NOMA and
OMA. Figure 6: Outage probability as a function of $\lambda$ for several
values of the number of lanes.

##  V Simulations and Discussions

In this section, we evaluate the performance of mmWave VCs using NOMA at road
intersections. In order to verify the accuracy of the theoretical results,
Monte-Carlo simulations are carried out by averaging over 10,000 realizations
of the PPPs and fading parameters. Monte Carlo simulations are carried out,
and they match perfectly the theoretical results, which validates the
correctness of our analysis. We set, without loss of generality,
$\lambda^{\textrm{LOS}}_{X}=\lambda^{\textrm{LOS}}_{Y}=\lambda^{\textrm{NLOS}}_{X}=\lambda^{\textrm{NLOS}}_{Y}=\lambda$.
$S=(0,0)$, $D_{1}=(100,10)$, and $D_{2}=(100,-10)$, $\beta=9.5\times 10^{3}$
[25], $\mu=1$. We set $\alpha_{\textrm{LOS}}=2$, $\alpha_{\textrm{NLOS}}=4$,
$m_{\textrm{LOS}}=2$, and $m_{\textrm{NLOS}}=1$. Finally, we set $G_{max}=18$
dBi, $\eta=30$ GHz. Unless stated otherwise, we consider mmWave VCs using NOMA
in all the results.

Fig.2 shows the outage probability as a function of the distance of the
triplet $\\{S,D_{1},D_{2}\\}$ from the intersection. We see from Fig.2 that
the outage probability increases as the vehicles drive toward the
intersection. This is because when the vehicles are far from the intersection,
only the interferes in the same road segment contribute to the aggregate
interference. But, as the vehicles approach the intersection, both road
segments contribute to the aggregate interference. We can also see that the
outage probability when $\|S-D_{1}\|=100$ m ($\|S-D_{2}\|=100$ m) is higher
than the outage probability when $\|S-D_{1}\|=110$ m ($\|S-D_{2}\|=110$ m).
This result is counter-intuitive because large distance decreases the path
loss, and thus, it decreases the SIR at the receiver and increases the outage
probability.

To investigate the effect of the distance between the transmitter and the
receiver, Fig.3 depicts the outage probability as a function of the distance
between the source and the destinations. We can see from Fig.3 that the outage
probability increases, as the distance between the source and the destinations
increases, until it reaches its maximum point and then it deceases. This
because, as the distance between the transmitting and the receiving nodes
increases, the LOS probability decreases, and the NLOS probability increases,
which decreases SIR at the receiver, and hence, decreases the outage
probability. We can also see that the peak of the outage probability is
reached for different values of $\lambda$. In fact, we can see that for high
values of $\lambda$, the peak is reached for short distances ($\|S-D_{1}\|=50$
m and $\|S-D_{2}\|=40$ m). However, for low values of $\lambda$, the peak is
reached for high distances ($\|S-D_{1}\|=110$ m and $\|S-D_{2}\|=90$ m).

To show the effect of LOS and NLOS on the performance, Fig.4 plots the outage
probability as function of $\lambda$ for LOS transmission, NLOS transmission,
and LOS/NLOS, that is equation (12) and (13). We can see that LOS scenario has
the highest outage probability. This is because, when the interference are in
direct line of sight with $D_{1}$ and $D_{2}$, the power of aggregate
interference increases, hence reducing the SIR and increasing the outage
probability. On the other hand, the NLOS scenario has the smallest outage
probability, since the interference are in NLOS with the transmitting nodes.
The model for this paper include a blockage model that includes both LOS and
NLOS. Therefore, we wan see that the performance are between the LOS scenario
and NLOS scenario, which are two extreme cases.

Fig.5 compares the outage probability of mmWave VCs using NOMA and OMA. We can
see that NOMA outperforms OMA. We can also see that when $a_{1}=0,9$, the
outage probability of $D_{1}$ decreases, and the outage probability of $D_{2}$
increases. However, when $a_{1}=0,7$, the outage probability of $D_{1}$
increases, and the outage probability of $D_{2}$ decreases. This because for
low values of $a_{1}$, there is more power allocated to $D_{2}$, which
increases its SIR and decreases its outage probability. On the other hand,
high values of $a_{1}$, means more power is allocated to $D_{1}$, which
increases its SIR and its decreases its outage probability.

Now, we investigate a more realistic intersection scenario involving several
lanes. Fig.6 plots the outage probability as a function of $\lambda$ for
several values of the number of lanes. We can see from Fig.6 that the outage
probability increases as the number of lanes increases. This because, when the
number of lanes increases, the number of interfering vehicles increases as
well, which increases the aggregate interference at the receiver, hence,
increasing the outage probability.

##  VI Conclusion

In this paper, we studied performance of mmWave VCs using NOMA at road
intersections. We studied the case when the intersection involves of two
perpendicular lanes, and then extended the study to an intersection with
several lanes. Our analysis included the effects of blockage from buildings
and vehicles at intersections. Closed form outage probability expressions were
obtained. We showed that as the nodes reach the intersection, the outage
probability increases. This makes intersections a critical areas. We also
showed that that the peak of the outage probability is reached for different
values of $\lambda$. We showed that for high density scenarios (high values of
$\lambda$), the peak is reached for short distances, whereas for low density
scenarios, the peak is reached for high distances. Counter-intuitively, we
showed that NLOS scenario has a better performance than LOS scenario. Finally,
we compared the performance of mmWave NOMA with OMA, and we showed that OMA
offers a significant improvement over cooperative OMA.

## Appendix A

To calculate $\mathbb{P}(\mathcal{O}_{D_{1}})$, we express as follows

$\mathbb{P}(\mathcal{O}_{D_{1}})=1-\mathbb{P}(\mathcal{O}^{C}_{D_{1}}).$ (20)

The probability $\mathbb{P}(\mathcal{O}^{C}_{D_{1}})$ is expressed as

$\displaystyle\mathbb{P}(\mathcal{O}^{C}_{D_{1}})$ $\displaystyle=$
$\displaystyle\sum_{{\textrm{Z}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\mathbb{P}\Bigg{\\{}\
\textrm{Z}_{SD_{1}}\cap(\textrm{SIR}^{(\alpha_{\textrm{Z}})}_{D_{1}}\geq\Theta_{1})\Bigg{\\}}\Bigg{]}$
$\displaystyle=$
$\displaystyle\sum_{{\textrm{Z}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\mathbb{P}(\textrm{Z}_{SD_{1}})\>\>\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\mathbb{P}\Bigg{\\{}\
\textrm{SIR}^{(\alpha_{\textrm{Z}})}_{D_{1}}\geq\Theta_{1}\Bigg{\\}}\Bigg{]}$
$\displaystyle=$
$\displaystyle\sum_{{\textrm{Z}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\mathbb{P}(\textrm{Z}_{SD_{1}})\>\>\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\mathbb{P}\Bigg{\\{}\
\frac{|h_{SD_{1}}|^{2}r_{SD_{1}}^{-\alpha_{\textrm{Z}}}\Upsilon
a_{1}}{|h_{SD_{1}}|^{2}r_{SD_{1}}^{-\alpha_{{\textrm{Z}}}}\Upsilon
a_{2}+I_{X_{D_{1}}}+I_{Y_{D_{1}}}}\geq\Theta_{1}\Bigg{\\}}\Bigg{]}$
$\displaystyle=$
$\displaystyle\sum_{{\textrm{Z}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\mathbb{P}(\textrm{Z}_{SD_{1}})\>\>\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\mathbb{P}\Bigg{\\{}|h_{SD_{1}}|^{2}r_{SD_{1}}^{-\alpha_{\textrm{Z}}}\Upsilon(a_{1}-\Theta_{1}a_{2})\geq\Theta_{1}\big{[}I_{X_{D_{1}}}+I_{Y_{D_{1}}}\big{]}\Bigg{\\}}\Bigg{]}.$

We can notice from (A) that, when $\Theta_{1}\geq a_{1}/a_{2}$,
$\mathbb{P}(\mathcal{O}^{C}_{D_{1}})=0$. Then, when $\Theta_{1}<a_{1}/a_{2}$,
and after setting $\Psi_{1}=\Theta_{1}/(a_{1}-\Theta_{1}a_{2})$, we get

$\mathbb{P}(\mathcal{O}^{C}_{D_{1}})=\sum_{{\textrm{Z}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\mathbb{P}(\textrm{Z}_{SD_{1}})\>\>\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\mathbb{P}\Bigg{\\{}|h_{SD_{1}}|^{2}\geq\frac{\Psi_{1}}{r_{SD_{1}}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{[}I_{X_{D_{1}}}+I_{Y_{D_{1}}}\big{]}\Bigg{\\}}\Bigg{]}.$
(22)

Since $|h_{SD_{1}}|^{2}$ follows a gamma distribution, its complementary
cumulative distribution function (CCDF) is given by

$\bar{F}_{|h_{SD_{1}}|^{2}}(X)=\mathbb{P}(|h_{SD_{1}}|^{2}>X)=\frac{\Gamma(m_{\textrm{Z}},\frac{m_{\textrm{Z}}}{\mu}X)}{\Gamma(m_{\textrm{Z}})},$
(23)

hence (22) becomes

$\displaystyle\mathbb{P}(\mathcal{O}^{C}_{D_{1}})$ $\displaystyle=$
$\displaystyle\sum_{{\textrm{Z}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\mathbb{P}(\textrm{Z}_{SD_{1}})$
$\displaystyle\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\frac{\Gamma\Big{(}m_{\textrm{Z}},\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SD_{1}}^{-\alpha_{{\textrm{Z}}}}\Upsilon}(I^{\textrm{LOS}}_{X_{D_{1}}}+I^{\textrm{LOS}}_{Y_{D_{1}}})\Big{)}}{\Gamma(m_{\textrm{Z}})}\Bigg{]}\times\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\frac{\Gamma\Big{(}m_{\textrm{Z}},\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SD_{1}}^{-\alpha_{{\textrm{Z}}}}\Upsilon}(I^{\textrm{NLOS}}_{X_{D_{1}}}+I^{\textrm{NLOS}}_{Y_{D_{1}}})\Big{)}}{\Gamma(m_{\textrm{Z}})}\Bigg{]}.$

Therefore, we obtain

$\mathbb{P}(\mathcal{O}^{C}_{D_{1}})=\sum_{{\textrm{Z}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\mathbb{P}(\textrm{Z}_{SD_{1}})\>\>\prod_{{\textrm{K}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\frac{\Gamma\Big{(}m_{\textrm{Z}},\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SD_{1}}^{-\alpha_{{\textrm{Z}}}}\Upsilon}(I^{\textrm{K}}_{X_{D_{1}}}+I^{\textrm{K}}_{Y_{D_{1}}})\Big{)}}{\Gamma(m_{\textrm{Z}})}\Bigg{]}.$
(24)

The exponential sum function when $m_{\textrm{Z}}$ is an integer is defined as

$e_{(m_{\textrm{Z}})}=\sum_{k=0}^{m_{\textrm{Z}}-1}\frac{(\frac{m_{\textrm{Z}}}{\mu}X)^{k}}{k!}=e^{X}\frac{\Gamma(m_{\textrm{Z}},\frac{m_{\textrm{Z}}}{\mu}X)}{\Gamma(m_{\textrm{Z}})},$
(25)

then

$\frac{\Gamma(m_{\textrm{Z}},\frac{m_{\textrm{Z}}}{\mu}X)}{\Gamma(m_{\textrm{Z}})}=e^{-\frac{m_{\textrm{Z}}}{\mu}X}\sum_{k=0}^{m_{\textrm{Z}}-1}\frac{1}{k!}{\big{(}\frac{m_{\textrm{Z}}\>X}{\mu}\big{)}}^{k}.$
(26)

We denote the expectation in equation (24) by $\mathcal{E}(I_{X},I_{Y})$, then
$\mathcal{E}(I_{X},I_{Y})$ equals

$\displaystyle\mathcal{E}(I_{X},I_{Y})$ $\displaystyle=$
$\displaystyle\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\exp\Big{(}-\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SD_{1}}^{-\alpha_{\textrm{Z}}}\Upsilon}(I^{\textrm{K}}_{X_{D_{1}}}+I^{\textrm{K}}_{Y_{D_{1}}})\Big{)}\
\sum_{k=0}^{m_{\textrm{Z}}-1}\frac{1}{k!}\Big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SD_{1}}^{-\alpha_{\textrm{Z}}}\Upsilon}(I^{\textrm{K}}_{X_{D_{1}}}+I^{\textrm{K}}_{Y_{D_{1}}})\Big{)}^{k}\Bigg{]}$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{m-1}\frac{1}{k!}\>\Big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SD_{1}}^{-\alpha_{\textrm{Z}}}\Upsilon}\Big{)}^{k}\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\exp\Big{(}-\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SD_{1}}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{(}I^{\textrm{K}}_{X_{D_{1}}}+I^{\textrm{K}}_{Y_{D_{1}}}\big{)}\Big{)}\big{(}I^{\textrm{K}}_{X_{D_{1}}}+I^{\textrm{K}}_{Y_{D_{1}}}\big{)}^{k}\Bigg{]}.$

Applying the binomial theorem in (A), we get

$\displaystyle\mathcal{E}(I_{X},I_{Y})$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{m_{\textrm{Z}}-1}\frac{1}{k!}\>\Big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SD_{1}}^{-\alpha_{\textrm{Z}}}\Upsilon}\Big{)}^{k}$
(28)
$\displaystyle\times\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\exp\Big{(}-\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SD_{1}}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{[}I^{\textrm{K}}_{X_{D_{1}}}+I^{\textrm{K}}_{Y_{D_{1}}}\big{]}\Big{)}\sum_{n=0}^{k}\binom{k}{n}{(I^{\textrm{K}}_{X_{D_{1}}})}^{k-n}\>{(I^{\textrm{K}}_{Y_{D_{1}}})}^{n}\Bigg{]}$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{m_{\textrm{Z}}-1}\frac{1}{k!}\>\Omega^{k}\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\exp\Big{(}-\Omega\big{[}I^{\textrm{K}}_{X_{D_{1}}}+I^{\textrm{K}}_{Y_{D_{1}}}\big{]}\Big{)}\times\sum_{n=0}^{k}\binom{k}{n}{(I^{\textrm{K}}_{X_{D_{1}}})}^{k-n}\>({I^{\textrm{K}}_{Y_{D_{1}}}})^{n}\Bigg{]},$

where
$\Omega=\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon}$.
To calculate the expectation in (28), denoted $\mathcal{T}(I_{X},I_{Y})$, we
proceed as follows

$\displaystyle\mathcal{T}(I_{X},I_{Y})$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{k}\binom{k}{n}$ $\displaystyle\times$
$\displaystyle\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}e^{-\Omega\>I^{\textrm{K}}_{X_{D_{1}}}}\>e^{-\Omega\>I^{\textrm{K}}_{Y_{D_{1}}}}{(I^{\textrm{K}}_{X_{D_{1}}})}^{k-n}\>{(I^{\textrm{K}}_{Y_{D_{1}}})}^{n}\Bigg{]}$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{k}\binom{k}{n}\mathbb{E}_{I_{X}}\Bigg{[}e^{-\Omega\>I^{\textrm{K}}_{X_{D_{1}}}}{(I^{\textrm{K}}_{X_{D_{1}}})}^{k-n}\Bigg{]}$
$\displaystyle\times$
$\displaystyle\mathbb{E}_{I^{\textrm{K}}_{Y_{D_{1}}}}\Bigg{[}e^{-\Omega\>I^{\textrm{K}}_{Y_{D_{1}}}}\>{(I^{\textrm{K}}_{Y_{D_{1}}})}^{n}\Bigg{]}$
$\displaystyle\overset{(a)}{=}$
$\displaystyle\sum_{n=0}^{k}\binom{k}{n}(-1)^{k-n}\frac{\textrm{d}^{k-n}\mathcal{L}_{I^{\textrm{K}}_{X_{D_{1}}}}(\Omega)}{\textrm{d}^{k-n}\Omega}(-1)^{n}\frac{\textrm{d}^{n}\mathcal{L}_{I^{\textrm{K}}_{Y_{D_{1}}}}(\Omega)}{\textrm{d}^{n}\Omega}$
$\displaystyle=$
$\displaystyle(-1)^{k}\sum_{n=0}^{k}\binom{k}{n}\frac{\textrm{d}^{k-n}\mathcal{L}_{I^{\textrm{K}}_{X_{D_{1}}}}(\Omega)}{\textrm{d}^{k-n}\Omega}\frac{\textrm{d}^{n}\mathcal{L}_{I^{\textrm{K}}_{Y_{D_{1}}}}(\Omega)}{\textrm{d}^{n}\Omega}.$

where (a) stems form the following property

$\displaystyle\mathbb{E}_{I}\Big{[}e^{-\Omega I}{I}^{N}\Big{]}$
$\displaystyle=$
$\displaystyle(-1)^{N}\frac{\textrm{d}^{N}\mathbb{E}_{I}\Big{[}e^{-\Omega\>I}{I}^{N}\Big{]}}{\textrm{d}^{N}\Omega}$
$\displaystyle=$
$\displaystyle(-1)^{N}\frac{\textrm{d}^{N}\mathcal{L}_{I}(\Omega)}{\textrm{d}^{N}\Omega}.$

Finally, the expectation becomes

$\mathcal{T}(I_{X},I_{Y})=\sum_{k=0}^{m_{\textrm{Z}}-1}\frac{1}{k!}\>\big{(}-\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SD_{1}}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{)}^{k}\sum_{n=0}^{k}\binom{k}{n}\frac{\textrm{d}^{k-n}\mathcal{L}_{I^{\textrm{K}}_{X_{D_{1}}}}\big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SD_{1}}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{)}}{\textrm{d}^{k-n}\big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SD_{1}}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{)}}\frac{\textrm{d}^{n}\mathcal{L}_{I^{\textrm{K}}_{Y_{D_{1}}}}\big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SD_{1}}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{)}}{\textrm{d}^{n}\big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SD_{1}}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{)}}.$
(29)

Then plugging (29) in (24) yields (12). The expression of
$\textrm{d}^{k-n}\mathcal{L}_{I^{\textrm{K}}_{X}}(s)/\textrm{d}^{k-n}(s)$ and
$\textrm{d}^{n}\mathcal{L}_{I^{\textrm{K}}_{Y}}(s)/\textrm{d}^{n}(s)$ are
given by (39) and (40).

In the same way we express $\mathbb{P}(\mathcal{O}^{C}_{D_{2}})$ as

$\mathbb{P}(\mathcal{O}_{D_{2}})=1-\mathbb{P}(\mathcal{O}^{C}_{D_{2}}).$ (30)

To calculate $\mathbb{P}(\mathcal{O}^{C}_{D_{2}})$ we proceed as follows

$\displaystyle\mathbb{P}(\mathcal{O}^{C}_{D_{2}})$ $\displaystyle=$
$\displaystyle\sum_{{\textrm{Z}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\mathbb{P}\Bigg{\\{}\
\bigcap_{i=1}^{2}\big{\\{}\textrm{Z}_{SD_{2}}\cap(\textrm{SIR}^{(\alpha_{\textrm{Z}})}_{D_{2-i}}\geq\Theta_{i})\Bigg{\\}}\Bigg{]}$
(31) $\displaystyle=$
$\displaystyle\sum_{{\textrm{Z}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\mathbb{P}(\textrm{Z}_{SD_{2}})\>\>\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\mathbb{P}\Bigg{\\{}\
\bigcap_{i=1}^{2}\textrm{SIR}^{(\alpha_{\textrm{Z}})}_{D_{2-i}}\geq\Theta_{i}\Bigg{\\}}\Bigg{]}$
$\displaystyle=$
$\displaystyle\sum_{{\textrm{Z}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\mathbb{P}(\textrm{Z}_{SD_{2}})\>\>\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\mathbb{P}\Bigg{\\{}\
\textrm{SIR}^{(\alpha_{\textrm{Z}})}_{D_{2-1}}\geq\Theta_{1}\cap\textrm{SIR}^{(\alpha_{\textrm{Z}})}_{D_{2-2}}\geq\Theta_{2}\Bigg{\\}}\Bigg{]}.$

Following the same steps as for $\mathbb{P}(\mathcal{O}^{C}_{D_{1}})$, we get

$\mathbb{P}(\mathcal{O}^{C}_{D_{2}})=\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\mathbb{P}\Bigg{\\{}\frac{|h_{SD_{2}}|^{2}r_{SD_{2}}^{-\alpha_{\textrm{Z}}}\Upsilon
a_{1}}{|h_{SD_{2}}|^{2}r_{SD_{2}}^{-\alpha_{\textrm{Z}}}\Upsilon
a_{2}+I_{X_{D_{2}}}+I_{Y_{D_{2}}}}\geq\Theta_{1},\frac{|h_{SD_{2}}|^{2}r_{SD_{2}}^{-\alpha_{\textrm{Z}}}\Upsilon
a_{2}}{I_{X_{D_{2}}}+I_{Y_{D_{2}}}}\geq\Theta_{2}\Bigg{\\}}\Bigg{]}.$ (32)

When $\Theta_{1}>a_{1}/a_{2}$, then $\mathbb{P}(\mathcal{O}_{D_{2}})=1$,
otherwise we continue the derivation We set $\Psi_{2}=\Theta_{2}/a_{2}$, then

$\displaystyle\mathbb{P}(\mathcal{O}^{C}_{D_{2}})$ $\displaystyle=$
$\displaystyle\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\mathbb{P}\Bigg{\\{}|h_{SD_{2}}|^{2}\geq\frac{\Psi_{1}}{r_{SD_{2}}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{[}I_{X_{D_{2}}}+I_{Y_{D_{2}}}\big{]},|h_{SD_{2}}|^{2}\geq\frac{\Psi_{2}}{r_{SD_{2}}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{[}I_{X_{D_{2}}}+I_{Y_{D_{2}}}\big{]}\Bigg{\\}}\Bigg{]}$
$\displaystyle=$
$\displaystyle\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\mathbb{P}\Bigg{\\{}|h_{SD_{2}}|^{2}\geq\frac{\mathrm{max}(\Psi_{1},\Psi_{2})}{r_{SD_{2}}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{[}I_{X_{D_{2}}}+I_{Y_{D_{2}}}\big{]}\Bigg{\\}}\Bigg{]}.$

Following the same steps as for $\mathbb{P}(\mathcal{O}^{C}_{D_{1}})$, we
obtain (13).

## Appendix B

The Laplace transform of the interference originating from the $X$ road at
${D_{i}}$ is expressed as

$\displaystyle\mathcal{L}_{{I^{\textrm{K}}_{X_{D_{i}}}}}(s)$ $\displaystyle=$
$\displaystyle\mathbb{E}\Bigg{[}{\exp\big{(}-sI^{\textrm{K}}_{X_{D_{i}}}\big{)}}\Bigg{]}$
(33) $\displaystyle=$
$\displaystyle\mathbb{E}\Bigg{[}{\exp\Bigg{(}-\sum_{x\in\Phi^{\textrm{K}}_{X_{D_{i}}}}s|h_{{D_{i}}x}|^{2}r_{{D_{i}}x}^{-\alpha_{\textrm{K}}}\Bigg{)}}\Bigg{]}$
$\displaystyle=$
$\displaystyle\mathbb{E}\Bigg{[}\prod_{x\in\Phi^{\textrm{K}}_{X_{D_{i}}}}\exp\Bigg{(}-s|h_{{D_{i}}x}|^{2}r_{{D_{i}}x}^{-\alpha_{\textrm{K}}}\Bigg{)}\Bigg{]}$
$\displaystyle\overset{(a)}{=}$
$\displaystyle\mathbb{E}\Bigg{[}\prod_{x\in\Phi^{\textrm{K}}_{X_{D_{i}}}}\mathbb{E}_{|h_{{D_{i}}x}|^{2},p}\Bigg{\\{}\exp\Bigg{(}-s|h_{{D_{i}}x}|^{2}r_{D_{i}x}^{-\alpha_{\textrm{K}}}\Bigg{)}\Bigg{\\}}\Bigg{]}$
$\displaystyle\overset{(b)}{=}$
$\displaystyle\mathbb{E}\Bigg{[}\prod_{x\in\Phi^{\textrm{K}}_{X_{D_{i}}}}\dfrac{p}{1+sr_{{D_{i}}x}^{-\alpha_{\textrm{K}}}}+1-p\Bigg{]}$
$\displaystyle\overset{(c)}{=}$
$\displaystyle\exp\Bigg{(}-\lambda^{\textrm{K}}_{X}\displaystyle\int_{\mathbb{R}}\Bigg{[}1-\bigg{(}\dfrac{p}{1+sr_{{D_{i}}x}^{-\alpha_{\textrm{K}}}}+1-p\bigg{)}\Bigg{]}\textrm{d}x\Bigg{)}$
$\displaystyle=$
$\displaystyle\exp\Bigg{(}-p\lambda^{\textrm{K}}_{X}\displaystyle\int_{\mathbb{R}}\dfrac{1}{1+1/sr_{{D_{i}}x}^{-\alpha_{\textrm{K}}}}\textrm{d}x\Bigg{)}$
$\displaystyle=$
$\displaystyle\exp\Bigg{(}-p\lambda^{\textrm{K}}_{X}\displaystyle\int_{\mathbb{R}}\dfrac{1}{1+r_{{D_{i}}x}^{\alpha_{\textrm{K}}}/s}\textrm{d}x\Bigg{)},$
(34)

where (a) follows from the independence of the fading coefficients; (b)
follows from performing the expectation over $|h_{{D_{i}}x}|^{2}$ which
follows an exponential distribution with unit mean, and performing the
expectation over the set of interferes; (c) follows from the probability
generating functional (PGFL) of a PPP. The expression of
$\mathcal{L}_{{I^{\textrm{K}}_{Y_{D_{i}}}}}(s)$ can be acquired by following
the same steps.

## Appendix C

In order to calculate the Laplace transform of interference originated from
the $X$ road at the node $D_{i}$, we have to calculate the integral in (14).
We calculate the integral in (14) for $\alpha_{\textrm{K}}=2$. We set
${d_{i}}_{x}=d_{i}\cos(\theta_{D_{i}})$, and
${d_{i}}_{y}=d_{i}\sin(\theta_{D_{i}}$), then (14) becomes

$\mathcal{L}_{I^{\textrm{K}}_{X_{D_{i}}}}(s)=\exp\Bigg{(}-\emph{p}\lambda^{\textrm{K}}_{X}s\int_{\mathbb{R}}\dfrac{1}{s+{d_{i}}_{y}^{2}+(x-{d_{i}}_{x})^{2}}\textrm{d}x\Bigg{)},$
(35)

and the integral inside the exponential in (35) equals

$\int_{\mathbb{R}}\dfrac{1}{s+{d_{i}}_{y}^{2}+(x-{d_{i}}_{x})^{2}}\textrm{d}x=\dfrac{\pi}{\sqrt{{d_{i}}_{y}^{2}+s}}.$
(36)

Then, plugging (36) into (35), and substituting ${d_{i}}_{y}$ by
$d_{i}\sin(\theta_{{D_{i}}})$ we obtain

$\mathcal{L}_{I^{\textrm{K}}_{X_{D_{i}}}}(s)=\exp\Bigg{(}-\dfrac{\emph{p}\lambda^{\textrm{K}}_{X}s\,\pi}{\sqrt{{d_{i}}^{2}\sin(\theta_{{D_{i}}})^{2}+s}}\Bigg{)}.$
(37)

Following the same steps above, and without details for the derivation with
respect to $s$, we obtain

$\mathcal{L}_{I^{\textrm{K}}_{Y_{D_{i}}}}(s)=\exp\Bigg{(}-\dfrac{\emph{p}\lambda^{\textrm{K}}_{Y}s\,\pi}{\sqrt{{d_{i}}^{2}\cos(\theta_{{D_{i}}})^{2}+s}}\Bigg{)}.$
(38)

Then, when compute the derivative of (37) and (38), we obtain

$\frac{\textrm{d}^{k-n}\mathcal{L}_{I^{\textrm{K}}_{X_{D_{i}}}}\big{(}s\big{)}}{\textrm{d}^{k-n}s}=\Bigg{[}-\frac{\emph{p}\lambda^{\textrm{K}}_{X}\pi}{\sqrt{{d_{i}}^{2}\sin(\theta_{{D_{i}}})^{2}+s}}+\frac{1}{2}\frac{\emph{p}\lambda^{\textrm{K}}_{X}\pi
s}{({d_{i}}^{2}\sin(\theta_{{D_{i}}})^{2}+s)^{3/2}}\Bigg{]}^{k-n}\\\
\times\exp\Bigg{(}-\frac{\emph{p}\lambda^{\textrm{K}}_{X}\pi
s}{\sqrt{{d_{i}}^{2}\sin(\theta_{{D_{i}}})^{2}+s}}\Bigg{)}.$ (39)
$\frac{\textrm{d}^{n}\mathcal{L}_{I^{\textrm{K}}_{Y_{D_{i}}}}\big{(}s\big{)}}{\textrm{d}^{n}s}=\Bigg{[}-\frac{\emph{p}\lambda^{\textrm{K}}_{Y}\pi}{\sqrt{{d_{i}}^{2}\cos(\theta_{{D_{i}}})^{2}+s}}+\frac{1}{2}\frac{\emph{p}\lambda^{\textrm{K}}_{Y}\pi
s}{({d_{i}}^{2}\cos(\theta_{{D_{i}}})^{2}+s)^{3/2}}\Bigg{]}^{n}\\\
\times\exp\Bigg{(}-\frac{\emph{p}\lambda^{\textrm{K}}_{Y}\pi
s}{\sqrt{{d_{i}}^{2}\cos(\theta_{{D_{i}}})^{2}+s}}\Bigg{)}.$ (40)

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# A note on universal graphs for spanning trees

Ervin Győri Alfréd Rényi Institute of Mathematics, Budapest, Hungary. Binlong
Li School of Mathematics and Statistics, Northwestern Polytechnical
University, Xi’an, 710072, China.Xi’an-Budapest Joint Research Center for
Combinatorics, Northwestern Polytechnical University, Xi’an, 710072, China.
Nika Salia King Fahd University of Petroleum and Minerals, Dhahran, Saudi
Arabia. Casey Tompkins11footnotemark: 1

###### Abstract

Chung and Graham considered the problem of minimizing the number of edges in
an $n$-vertex graph containing all $n$-vertex trees as a subgraph. They showed
that such a graph has at least $\frac{1}{2}n\log{n}$ edges. In this note, we
improve this lower estimate to $n\log{n}$.

##  1 Introduction

We say that an $n$-vertex graph $G$ is universal for spanning trees if $G$
contains every $n$-vertex tree as a subgraph. Chung and Graham [1] considered
the problem of how few edges a graph $G$ can have if it is universal for
spanning trees; let us denote this minimum by ${\rm s}(n)$. They also
considered a related parameter $s^{*}(n)$ defined to be the minimum number of
edges of a graph (with no condition on the number of vertices) containing
every $n$-vertex tree as a subgraph. Obviously ${\rm s}^{*}(n)\leq{\rm s}(n)$.
A lower bound of $\frac{1}{2}n\log(n)\leq{{\rm s}}^{*}(n)$ was given in [1],
where $\log$ denotes the natural logarithm (see also [2]). Upper bounds for
${{\rm s}}^{*}(n)$ or ${\rm s}(n)$ were given in [1], [4] and [3] yielding

$\frac{1}{2}n\log{n}<{{\rm s}}^{*}(n)\leq{\rm
s}(n)\leq\frac{5}{\log{4}}n\log{n}+O(n).$

In regard to the correct leading coefficient for ${\rm s}(n)$, Chung and
Graham write in [3]: “Conceivably, the right value might even be $\frac{1}{2}$
although at present, we certainly do not see how to prove this”. We show that
this is, in fact, not the case by improving the lower bound on ${\rm s}(n)$ by
a factor of $2$. The proof resembles the proof of the lower bound for ${{\rm
s}}^{*}(n)$ given in [1] and [3]. However, we keep more careful track of the
edges rather than immediately considering the degree sequence.

######  Theorem 1.

For every positive integer $n$,

$n\log{n}-O(n)\leq{{\rm s}}^{*}(n).$

##  2 Proof of Theorem 1

Let $G$ be a graph which contains all $n$-vertex trees. Observe that since $G$
contains all $n$-vertex trees as a subgraph, it must also contain all forests
with at most $n$-vertices.

For every integer $i$, such that $1\leq i\leq\frac{n}{2}$, let $F_{i}$ be a
star forest with $i$ connected components each isomorphic to a star with
$\left\lfloor{\frac{n}{i}}\right\rfloor$ vertices.

The proof proceeds as follows: we will sequentially consider forests $F_{i}$,
and at each step, we will color $\left\lfloor{\frac{n}{i}}\right\rfloor-1$
edges red that have not been colored previously.

Let $F_{1}^{\prime}$ be a subgraph of $G$ isomorphic to $F_{1}$. Let $v_{1}$
be the central vertex of $F^{\prime}_{1}$. We color all edges of
$F_{1}^{\prime}$ red. Let $F_{2}^{\prime}$ be a subgraph of $G$ isomorphic to
$F_{2}$. Thus $G$ contains two vertex disjoint stars of
$\left\lfloor{\frac{n}{2}}\right\rfloor$ vertices. At least one of these stars
does not contain $v_{1}$ as a vertex, and so none of the edges of this star
are red. Let the center of this star be $v_{2}$, and color all edges of this
star red.

In general, in step $i$ we consider the forest $F_{i}$ and let
$F_{i}^{\prime}$ be a subgraph of $G$ isomorphic to $F_{i}$. The forest
$F_{i}^{\prime}$ consists of $i$ vertex-disjoint stars, each composed of
$\left\lfloor{\frac{n}{i}}\right\rfloor$ vertices. At least one of these stars
does not contain a vertex from the set $\\{v_{1},v_{2},\dots,v_{i-1}\\}$;
therefore, its edges are still not colored. We let the center of one such star
be $v_{i}$ and color all of its edges red.

At the end of this procedure, we have

$\sum_{k=1}^{\left\lfloor{\frac{n}{2}}\right\rfloor}\left(\left\lfloor{\frac{n}{k}}\right\rfloor-1\right)>\sum_{k=1}^{\left\lfloor{\frac{n}{2}}\right\rfloor}\left(\frac{n}{k}-2\right)>\int_{1}^{\left\lfloor{\frac{n}{2}}\right\rfloor}\left(\frac{n}{k}-2\right)\,dk=n\log{n}-O(n)$

red edges. Thus we have ${\rm s}^{*}(n)\geq n\log{n}-O(n)$, completing the
proof.

##  3 Acknowledgements

Győri and Salia were supported by the National Research, Development and
Innovation Office NKFIH, grant K132696. Li was supported by NSFC (12071370)
and Shaanxi Fundamental Science Research Project for Mathematics and Physics
(22JSZ009). Tompkins was supported by the National Research, Development and
Innovation Office, NKFIH, grant K135800.

## References

  * [1] F. Chung and R. Graham.  On graphs which contain all small trees.  Journal of combinatorial theory, Series B, 24(1):14–23, 1978.
  * [2] F. Chung and R. Graham.  On universal graphs.  Ann. New York Acad. Sci, 319:136–140, 1979.
  * [3] F. Chung and R. Graham.  On universal graphs for spanning trees.  Journal of the London Mathematical Society, 2(2):203–211, 1983.
  * [4] F. Chung, R. Graham, and N. Pippenger.  On graphs which contain all small trees, II.  Colloquia Mathematica Societatis János Bolyai, 1976.