# Dynamic-TDD Interference Tractability Approaches and Performance Analysis in
Macro-Cell and Small-Cell Deployments

Jalal Rachad, Ridha Nasri and Laurent Decreusefond Jalal Rachad and Ridha
Nasri are with Orange Labs, 40-48 Avenue de la République, 92320 Châtillon,
France. Email<EMAIL_ADDRESS><EMAIL_ADDRESS>Rachad and
Laurent Decreusefond are with LTCI Telecom Paris-Tech, Institut Polytechnique
de Paris, Paris, France. Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Meeting the continued growth in data traffic volume, Dynamic Time Division
Duplex (D-TDD) has been introduced as a solution to deal with the uplink (UL)
and downlink (DL) traffic asymmetry, mainly observed for dense heterogeneous
network deployments, since it is based on instantaneous traffic estimation and
provide more flexibility in resource assignment. However, the use of this
feature requires new interference mitigation schemes capable to handle two
additional types of interference between cells in opposite transmission
direction: DL to UL and UL to DL interference. The aim of this work is to
provide a complete analytical approach to model inter-cell interference in
macro-cell and dense small-cell networks. We derive the explicit expressions
of Interference to Signal Ratio (ISR) at each position of the network, in both
DL and UL, to quantify the impact of each type of interference on the system
performance. Also, we provide the explicit expressions of the coverage
probability as functions of different system parameters by covering different
scenarios. Finally, through system level simulations, we analyze the
feasibility of D-TDD implementation in both deployments and we compare its
performance to the static-TDD (S-TDD) configuration.

###### Index Terms:

Dynamic TDD, Interference, Macro-cells, Small-cells, Coverage probability,
SINR, ISR, ASE, FeICIC.

##  I Introduction

###  I-A Context of the study

As the number of mobile users and the volume of data traffic are expected to
continue increasing in the upcoming years, future mobile cellular networks
need to support this proliferation through upgrading their features and key
technologies. The upcoming fifth generation (5G) of cellular network radio
interface, known as New Radio (NR), is being designed by considering
flexibility in features definition in order to satisfy diverse use cases with
different users’ requirements. D-TDD is expected to be one of the major keys
of 5G NR. It has been proposed in order to deal with traffic asymmetry since
it enables the dynamic adjustment of UL and DL resource assignment according
to the instantaneous traffic variations. However, D-TDD system is severely
limited by a strong mutual interference between the UL and DL transmissions
because those two directions share the same frequency band. Hence, two types
of interference appear: DL to UL (impact of DL other cell interference on UL
signal received by the studied cell) and UL to DL (impact of UL mobile users
transmission, located in other cells, on DL signal received by a mobile user
located in the studied cell). Those additional interference, mainly DL to UL,
are usually more difficult to deal with because of the LOS (Line Of Sight)
presence between highly elevated base stations (BSs) transmitting with high
power level and also because the mobiles can move around randomly. Thus, this
duplexing mode can be more convenient with heterogeneous networks (HetNets) as
small-cells are considered well isolated from each others and also from the
macro-cell layers.

Furthermore, in order to mitigate interference in D-TDD system, 3GPP (3rd
Generation Partnership Project) standard advices new approach for enhanced
Interference Mitigation and Traffic Adaptation (eIMTA) in dynamic environment
[1]. Cell clustering scheme is an efficient technique that can be used to deal
with D-TDD interference. Cells that suffer from high DL to UL interference
level between each others can be gathered in the same cluster and use the same
UL-DL configuration. In the same time, transmission directions in different
clusters can be dynamically adapted. This technique can be very efficient for
HetNet when it is combined with enhanced Inter Cell Interference Coordination
(eICIC), introduced in 3GPP Release 10, or Further eICIC (FeICIC), introduced
in 3GPP Release 11. eICIC and FeICIC are based on time domain partitioning:
macro-cell BSs reduce their transmitted power level during some sub-frames
called Almost Blank Sub-frames (ABS) so that small-cells can adjust UL-DL
portions dynamically during those sub-frames according to the traffic
variations.

###  I-B Related works

D-TDD has been widely investigated in the available scientific literature; see
for instance [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. The first study dates back
to 2002 with the work in [2] where performance of a D-TDD fixed cellular
network in UL transmission were investigated. Authors in [2] proposed a time
slot assignment method to improve the UL outage performance. In our recent
work [10], we have proposed an analytical tractability approach to model
interference generated by D-TDD in a macro-cell deployment. We have shown that
D-TDD is only used in favor of DL transmission cycle. However, during the UL
transmission cycle, DL to UL interference may cause a substantial performance
degradation. To reduce the impact of DL transmission of other cells on the UL
received signal, we have proposed a cell clustering scheme that somehow
improves D-TDD system UL performance in a dense small-cells’ network.
Performance of D-TDD system were also investigated in [13] for a particular
small-cells’ architecture, known as phantom cells, in UL and DL transmission
directions. For the analytical approach, authors of [13] used tools from
stochastic geometry to model phantom cells and user locations in order to
derive $SINR$ distributions in DL and UL. Also, an inter-cell interference
coordination scheme has been proposed. Similarly in [14], D-TDD has been
analyzed considering a dense small-cell network. In [15], a two-tier Device to
Device enhanced HetNet operating with D-TDD has been studied. Authors have
proposed an analytical framework to evaluate the coverage probability and
network throughput using stochastic geometry. Likewise in [16], authors have
provided a comparison between static and dynamic TDD in millimeter wave (mm-
wave) cellular network, in terms of $SINR$ distributions and mean rates,
considering synchronized and unsynchronized access-backhaul. Additionally, in
order to make D-TDD feasible, some interference mitigation techniques have
been proposed in literature, such as cell clustering; see for instance [17,
18] and [10]. It was discussed in [18] a soft reconfiguration method based on
cell clustering so as to allow cells in the same cluster to change dynamically
the UL/DL configuration but inter-cluster interference still exists. Also,
several works have discussed radio resource management and optimization
approaches to deal with cross slot interference generated by D-TDD [11].

On the other hand, inter-cell interference are the major issue that obstructs
the achievement of high performance in terms of data rate and spectral
efficiency, especially in dense cellular networks deployments. Interference
tractability is mainly related to the network geometry modeling. It has been
proposed in literature several approaches to tackle this problem in both UL
and DL transmission directions. The most frequently used approaches adopt
random models, in which BSs are distributed in the plane according to a
Poisson Point Process (PPP) [19, 20, 21], and the regular hexagonal model
[22]. Each model has its own advantages and disadvantages. Fro and engineering
point of view, it is always desirable to evaluate inter-cell interference
received at each location taken by a mobile user, which cannot be accomplished
by using random models where only the interference distribution is determined.
However, as one of the major ways leading 5G networks implementation is
network densification and small-cell deployments, it becomes knotty to model
the topology with deterministic models. Therefore, stochastic geometry appears
to be efficient to determine the relevant metrics required to analyze radio
performance in terms of the probabilistic parameters of HetNets.

###  I-C Contribution

The main contribution of this paper is to provide a complete analytical
framework for interference tractability in macro-cell deployment and dense
small-cell network. We model macro-cells by using a regular hexagonal network
with infinite number of sites. We treat, in particular, the explicit
evaluation of $ISR$ at each position taken by a mobile user in the network, in
terms of convergent series, by covering the four types of interference
generated in D-TDD based network. This metric is very useful for link budget
tools in which the expression of the average perceived interference is
required in each position. To model small-cells, we adopt the widely used
spatial PPP and we show how to exploit the mathematical framework based on
stochastic geometry and satisfy in the same time D-TDD assumptions.
Additionally, we derive the explicit expressions of the coverage probability
(SINR distribution) for a typical cell in DL and UL for both macro-cell and
small-cell networks. This metric is related to throughput distribution and it
is useful for cell throughput dimensioning. Finally, we analyze, through
system level simulations, performance of D-TDD based network and its
comparison with Static-TDD (S-TDD) considering different system parameters.

###  I-D Paper organization

The rest of the paper is organized as follows: In Section 2, we describe the
D-TDD model, the macro-cells’ network, the small-cells’ deployment and the
propagation model. In Section 3, we provide the analytical analysis regarding
the explicit derivation of $ISR$ and the coverage probability in macro-cells’
deployment. The theoretical analysis of small-cells’ network performance is
given in Section 4. It is important to note that Section 3 and 4 are
independent. Simulation results are provided in Section 5. Section 6 concludes
the paper.

##  II System models and notations

###  II-A Dynamic TDD model

To model the D-TDD system, we assume that all cells initially operate
synchronously in DL or UL. This setup can be considered as a baseline scenario
characterizing performance of existing synchronous TDD systems i.e., S-TDD.
After a period of time, it is assumed that all cells select randomly UL or DL
frame portions based on traffic conditions. Four types of interference
henceforth appear depending on the transmission direction: i) when the serving
cell transmits to a given mobile location, DL and UL interference effect on DL
useful transmission appears; ii) when the serving cell receives signals from
mobiles, UL and DL interference impact on UL transmission rises. It is
considered hereafter that the scheduler does not allocate the same spectral
resources to different mobile users in one cell at the same time (e.g., TD-LTE
scheduling). So, intra-cell interference is not considered. Therefore in a
given cell, we consider that during a sub-frame of interest (i.e., when D-TDD
is activated), there is one active transmission whether on DL or UL with full-
buffer traffic model. To characterize the transmission directions of cells,
denoted by $\boldsymbol{s}$, we consider two Bernoulli Random variables (RVs)
$\chi_{d}(\boldsymbol{s})$ and $\chi_{u}(\boldsymbol{s})$ such that
$\mathbb{P}(\chi_{d}(\boldsymbol{s})=1)=\alpha_{d}$ and
$\mathbb{P}(\chi_{u}(\boldsymbol{s})=1)=\alpha_{u}$.
$\chi_{d}(\boldsymbol{s})$ refers to the DL transmission cycle of a cell
$\boldsymbol{s}$ and $\chi_{u}(\boldsymbol{s})$ refers to its UL transmission
cycle during a D-TDD sub-frame. It is important to mention that a cell
$\boldsymbol{s}$ cannot operate in DL and UL during the same TTI. Hence, to
avoid this case, we add the following condition
$\chi_{u}(\boldsymbol{s})=1-\chi_{d}(\boldsymbol{s})$. This means that
$\alpha_{d}=1-\alpha_{u}$.

###  II-B Network models

Interference in cellular networks are the major issue that obstructs the
achievement of high performance in terms of data rate and spectral efficiency.
Telecommunication actors continuously attempt to minimize it during all the
phases of a technology conception, since it is related to network performance.
In radio engineering, interference margin, known also as noise rise, is used
to perform link budget tools. However, this notion does not describe the real
perceived interference and does not take into account the geometry of the
studied area that impacts in one way or another performance. Thus, the
analytical tractability of interference is of prime importance. Having a
tractable mathematical model can always give better results and avoid recourse
to extensive simulations.

In effect, interference is related to the network geometry and the spatial
distribution of users. Most considered models that can be found in literature
are the deterministic models such as the regular hexagonal network and random
models based on spatial point processes. Hexagonal network is the basic model
for network design in radio engineering. It is considered effective for
network having fixed cell radius such as macro-cell deployments. Nevertheless,
this model is not useful to describe heterogeneous networks topology. Small-
cells usually occupy unplanned random positions which makes stochastic point
processes practical to model their random distribution in dense urban
environments. Homogeneous PPP (HPPP) is a very popular model in cellular
networks in which BSs and mobile users spatial distribution are modeled
according to independent PPPs. However, despite the popularity of HPPP and its
tractability, this model can not fit with the geometry of real cellular
networks because of the repulsive behavior of transmitting nodes. Also, with
this model, one cannot evaluate interference at each arbitrary user location
and only its distribution that can be determined [22] and [23]. In the
remainder of this paper, we model the macro-cell deployment according to a
regular hexagonal network with an infinite number of cells while small-cells
and their associated users are modeled according to dependent PPPs.

####  II-B1 Macro-cells deployment

We consider a hexagonal cellular network denoted by $\Lambda$ with an infinite
number of macro-cells having an intersite distance between them denoted by
$\delta$. The hexagonal model means that for each node $\boldsymbol{s}$ $\in$
$\Lambda$, there exists a unique $(m,n)$ $\in$ $\mathbb{Z}^{2}$ such that
$\boldsymbol{s}=\delta(m+ne^{i\frac{\pi}{3}})$. We denote by
$\boldsymbol{s}_{0}$ the serving cell located at the origin of
$\mathbb{R}^{2}$ ($\mathbb{R}^{2}$ is isomorphic to $\mathbb{C}$). Antenna in
each site is assumed to have an omni-directional radiation pattern and covers
a geographical area named Voronoi cell, having a cell radius denoted by $R$.
Furthermore, the location of a mobile served by $\boldsymbol{s}_{0}$ is
denoted by $z_{0}$ such that $z_{0}=re^{i\theta}$ where (r, $\theta$) are the
polar coordinates in the complex plane. We denote also by $z$ the geographical
location of a mobile served by a cell $\boldsymbol{s}$ $\in$ $\Lambda^{*}$ in
the plane, where $\Lambda^{*}$ is the lattice $\Lambda$ without the serving
cell $\boldsymbol{s}_{0}$. Location $z$ is written in the complex plane by
$z=\boldsymbol{s}+\rho e^{i\phi}$, where $\rho$ and $\phi$ represents
respectively the distance and the angle between $z$ and $\boldsymbol{s}$.
Moreover, it is assumed that the locations of mobile $z$ in the plane are
uniformly distributed.

####  II-B2 Small-cells deployment

As we have mentioned previously, stochastic point processes are practical to
model the random distribution of small-cells in a dense urban environment.
Most common approaches model DL cellular networks considering that BSs are
distributed according to a spatial PPP and users distributed uniformly in BS
Voronoi cells [19]. For the UL transmission, it has been proposed in [20] an
interesting approach considering a spatial PPP distribution of users in the
plane with a uniform distribution of each BS in the Voronoi cell of the
associated mobile. However, with D-TDD, for each transmission direction there
is two sources of interference: BSs and mobiles. In order to exploit the
mathematical framework based on stochastic geometry and satisfy in the same
time the D-TDD assumptions, we model the set of active mobiles $z$ served by
the small-cells, denoted hereafter by $\tilde{\boldsymbol{s}}$, by a PPP
$\Phi$ of intensity $\lambda$. This implies that $z$ are uniformly distributed
in the studied area.

Given that each small-cell $\tilde{\boldsymbol{s}}$ has one active mobile
transmitting whether in DL, with probability $\alpha_{d}$, or in UL with
probability $\alpha_{u}$, we assume that each mobile is associated with the
nearest small-cell. Furthermore, the position of each small-cell can be
expressed in the complex plane by
$\tilde{\boldsymbol{s}}=z+\rho_{z}e^{i\phi_{z}}$, with $(\rho_{z},\phi_{z})$
are the polar coordinates of $\tilde{\boldsymbol{s}}$ relatively to $z$. This
means that also the set of small-cells forms a PPP obtained by a displacement
of $\Phi$ (Displacement theorem [24]), i.e., the process of mobiles and small-
cells are two dependent PPPs.

Additionally, to reduce strong macro-cell interference impact on users served
by small-cells, especially in small-cells range expansion, we assume that
FeICIC is implemented. With this feature, macro-cells reduce their transmitted
power level during some specific sub-frames called Almost Blank Sub-frames
(ABS) so that small-cells use those sub-frames to configure dynamically the
UL-DL frame portions. Moreover, we assume that all macro-cells are well
synchronized and adopt the same frame configurations i.e., S-TDD . This means
that all the ABSs are assigned at the same time portion to all the small-cells
to activate D-TDD and also macro-cell BSs interference can be neglected.

###  II-C Propagation model

To model the wireless channel, we consider the standard power-law path loss
model based on the distance between a mobile $z$ and a BS $\boldsymbol{s}$
such that the path loss $L(\boldsymbol{s},z)$ is given by

$L(\boldsymbol{s},z)=a|\boldsymbol{s}-z|^{2b},$ (1)

with $2b$ is the path loss exponent and $a$ is a propagation factor that
depends on the type of the environment (indoor, outdoor…).

Actually, characterizing the propagation in wireless channels is often
performed through measurements and statistics from field experiments in
different environments and under different conditions. Based on that, several
mathematical formulations are obtained. Nevertheless, those models don’t
describe the real behavior of propagation in wireless channels. The path loss
exponent is an important parameter that can have a tremendous effect on system
performance. It refers to the rate of decay of power with respect to distance
between a transmitting and receiving nodes. Path loss exponent values depend
on the environment of propagation (outdoor, indoor…), the visibility between
the transmitter and the receiver (Line of Sight, Non Line of Sight) and also
the links between nodes (BS to BS, mobile to mobile, BS to mobile… ). In
effect, the propagation environment between a BS and a mobile is not the same
as the one between two mobiles. This latter is more dynamic and undergoes
multiple reflexions and diffractions [turkka2008path]. Small values of the
path loss exponent refers to favorable conditions for electromagnetic waves
propagation (e.g., free space path loss exponent is always taken 2) while big
values of path loss exponent refers to harsh propagation conditions. In this
paper, we choose to fix the same value of the path loss exponent for all the
propagation directions between transmitting and receiving nodes in order to
alleviate notations (since we are dealing with equations having a lot of
parameters). Meanwhile, the model can easily be adapted by taking a convenient
path loss exponent for each propagation link.

In addition to the path loss, the received power by a mobile depends on the
random channel effects, especially shadowing and fast fading. Shadowing refers
to the attenuation of the received signal power caused by obstacles
obstructing the propagation between the transmitter and receiver. The common
approach to model shadowing effect in in cellular networks is to consider a
sequence of independent log-Normal random variables multiplied by the
expression of the path loss provided previously. In this work, shadowing
effect is not considered to keep the tractability of our models when
calculating the coverage probability. Meanwhile, we show in the following
section how log-Normal shadowing can be included in the average $ISR$
derivation.

Likewise, fast fading random model is not considered for macro-cell deployment
analysis in order to simplify calculations. Actually, fading effect can be
compensated through link level performing that maps the $SINR$ to the
throughput ($Th$). Also, for an AWGN (Additive Gaussian Noise Channel),
Shannon’s formula provides the relation between $SINR$ and $Th$. Hence, the
fast fading effect can be compensated by using a modified Shannon’s formula to
have $Th=K_{1}log_{2}(1+K_{2}SINR)$, with $K_{1}$ and $K_{2}$ are constants
calibrated from practical systems [25]. However, for the heterogeneous system
analysis, multi-path Rayleigh fading effect is considered. We denote by
$H_{i}$ and $G_{i}$, with $i=\tilde{\boldsymbol{s}}$ or $i=z$, the fading
coefficients between, a transmitting node $i$ and a typical receiving mobile,
and between $i$ and a typical receiving BS. We assume also that the RVs
$H_{i}$ and $G_{i}$ are independent and identically distributed (i.i.d) for
each propagation link and follow an exponential distribution of mean 1.

For the UL transmission, power control is applied to the Physical Uplink
Shared Channel (PUSCH) in order to set the required mobile transmitted power.
In this paper, it is modeled by the fractional power control model (FPC),
i.e., the path loss is partially compensated by the power control [26]. The
transmitted power by the mobile location $z$ to its serving cell
$\boldsymbol{s}$ is then written

$P(z,\boldsymbol{s})=P^{*}(\boldsymbol{s})\left(\left|z-\boldsymbol{s}\right|^{2b}\right)^{k},$
(2)

where $P^{*}$($\boldsymbol{s}$) is the cell specific target power and $k$
$\in$ [0,1] is the power control compensation factor. When $k=1$ the power
control scheme totally indemnifies the path loss in order to reach the target
power $P^{*}(\boldsymbol{s})$. For the case $0<k<1$ the path loss is partially
compensated and mobile users in cell edge create less interference because
their transmitted power is reduced.

Without loss of generality, we consider that $P^{*}(\boldsymbol{s})$ is the
same for all the cells. Power values $P$ and $P^{*}$ are supposed to include
the path loss constants and antenna gains of BSs and user equipments. It is
important to note that section 3 and section 4 are independent. So, to avoid
confusion, we denote by $\tilde{P}$ and $\tilde{P^{*}}$ respectively the
transmitting power and the cell specific target power of small-cell BSs.

##  III Dynamic TDD interference derivation in a macro-cell deployment

We define the Interference to Signal Ratio $ISR$ in DL as the received power
from an interfering source (interfering mobile or BS) divided by the useful
power received by $z_{0}$ from the serving cell. The average $ISR$ experienced
in DL transmission by a mobile location $z_{0}$ connected to
$\boldsymbol{s}_{0}$ is

$D(z_{0})=\alpha_{d}\leavevmode\nobreak\
D_{\downarrow}(z_{0})+\alpha_{u}\leavevmode\nobreak\ D_{\uparrow}(z_{0})$ (3)

with $D_{\downarrow}$ and $D_{\uparrow}$ are respectively DL to DL and UL to
DL average $ISRs$ experienced during the DL cycle by $z_{0}$.

Likewise, the average $ISR$ experienced by a cell $\boldsymbol{s}_{0}$ in UL
transmission cycle is defined by

$U(z_{0})=\alpha_{u}\leavevmode\nobreak\
U_{\uparrow}(z_{0})+\alpha_{d}\leavevmode\nobreak\ U_{\downarrow}(z_{0})$ (4)

where $U_{\uparrow}$ and $U_{\downarrow}$ are respectively UL to UL and DL to
UL interference to signal ratios experienced during the UL cycle of
$\boldsymbol{s}_{0}$.

###  III-A Downlink $ISR$ derivation $D(z_{0})$

####  III-A1 Expression of DL to DL $ISR$ $D_{\downarrow}(z_{0})$

In [22], it has been shown that the DL $ISR$ function of a location
$z_{0}=re^{i\theta}$ in a hexagonal cellular network with infinite number of
cells admits a series expansion on $r$ and $\theta$ and is a very slowly
varying function on $\theta$. Taking $x=\frac{r}{\delta}$ such that $x<1$ (for
hexagonal networks, we always have $x<\frac{1}{\sqrt{3}}$), the expression of
$D_{\downarrow}$ is recalled from [22]

$D_{\downarrow}(z_{0})=\frac{6x^{2b}}{\Gamma(b)^{2}}\sum_{h=0}^{+\infty}\frac{\Gamma(b+h)^{2}}{\Gamma(h+1)^{2}}\omega(b+h)x^{2h}$
(5)

where $\Gamma(.)$ is the Euler Gamma function and

$\omega(z)=3^{-z}\zeta(z)\left(\zeta(z,\frac{1}{3})-\zeta(z,\frac{2}{3})\right),$
(6)

with $\zeta(.)$ and $\zeta(.,.)$ are respectively the Riemann Zeta and Hurwitz
Riemann Zeta functions [27].

####  III-A2 Expression of UL to DL $ISR$ $D_{\uparrow}(z_{0})$

The UL to DL interference is generated from mobile users located at other
cells, mainly from those located at the border of cells adjacent to the
serving cell $\boldsymbol{s}_{0}$. Since there is only one mobile user
transmitting at the same time in UL for each cell, the total UL to DL $ISR$
can be evaluated by averaging over locations $z\in\boldsymbol{s}$ and then
summing over $\boldsymbol{s}\in$ $\Lambda^{*}$. So, if we assume that location
$z$ is uniformly distributed in $\boldsymbol{s}$, $D_{\uparrow}(z_{0})$ is
mathematically written as

$D_{\uparrow}(z_{0})=\frac{1}{\pi
R^{2}}\int\limits_{0}^{R}\int\limits_{0}^{2\pi}\sum_{\boldsymbol{s}\in\Lambda^{*}}\frac{P^{*}\leavevmode\nobreak\
\rho^{2bk}\leavevmode\nobreak\ r^{2b}}{P\left|\boldsymbol{s}+\rho
e^{i\phi}-re^{i\theta}\right|^{2b}}\rho d\rho d\varphi$ (7)

To evaluate equation (7), we can proceed analogously to the proof of $ISR$
formulas in hexagonal omni-directional networks provided in [22]. We start by
taking $z^{{}^{\prime}}=re^{i\theta}-\rho e^{i\phi}$. It is obvious that
$\left|z^{{}^{\prime}}\right|<\left|\boldsymbol{s}\right|$. It follows from
[22] that the sum over $\boldsymbol{s}$ inside the double integral admits a
series expansion on $\left|z^{{}^{\prime}}\right|/\delta$ as in (5). Using
formula (5) and writing $\left|z^{{}^{\prime}}\right|$ in terms of $r$,
$\theta$, $\rho$ and $\phi$, (7) becomes

$\displaystyle D_{\uparrow}(z_{0})$ $\displaystyle=\frac{6P^{*}x^{2b}}{P\pi
R^{2}\leavevmode\nobreak\
\Gamma(b)^{2}}\int\limits_{0}^{R}\int\limits_{0}^{2\pi}\sum_{h=0}^{+\infty}\frac{\Gamma(b+h)^{2}\omega(b+h)}{\Gamma(1+h)^{2}\delta^{2h}}\times$
$\displaystyle(r^{2}+\rho^{2})^{h}(1-\frac{2r\rho}{r^{2}+\rho^{2}}\cos(\phi))^{h}\rho^{2bk+1}d\rho
d\phi$ (8)

The sum and integrals of (8) can be switched and the inside integral can be
evaluated by expanding $(1-\frac{2r\rho}{r^{2}+\rho^{2}}\cos(\phi))^{h}$ as a
binomial sum. After few derivations of known special integrals and
simplifications, the UL to DL $ISR$ $D_{\uparrow}(z_{0})$ can be evaluated by
the following convergent series on $x=r/\delta$

$\displaystyle D_{\uparrow}(z_{0})$
$\displaystyle=\frac{6P^{*}x^{2b}R^{2bk}}{P\leavevmode\nobreak\
\Gamma(b)^{2}}\sum_{h=0}^{+\infty}\sum_{n=0}^{\lfloor\frac{h}{2}\rfloor}\sum_{i=0}^{h-2n}\frac{\Gamma(b+h)^{2}\omega(b+h)}{\Gamma(n+1)^{2}\Gamma(h+1)}\times$
$\displaystyle\frac{(\frac{R}{\delta})^{2n+2i}\leavevmode\nobreak\
x^{2h-2n-2i}}{\Gamma(i+1)\Gamma(h-2n-i+1)(n+i+bk+1)}$ (9)

Since $x<1/\sqrt{3}$ for hexagonal model, it is obvious that the first
elements of this series are sufficient to numerically evaluate $D_{\uparrow}$.
Furthermore, after few simplifications, (9) can be written as an entire series
on $x^{2}$ as follows

$D_{\uparrow}(z_{0})=\frac{6P^{*}x^{2b}R^{2bk}}{P}\sum_{h=0}^{+\infty}\beta_{h}x^{2h}$
(10)

with

$\displaystyle\beta_{h}$
$\displaystyle=\sum_{n=0}^{h}\sum_{i=0}^{+\infty}\frac{\Gamma^{2}(b+h+n+i)\omega(b+h+n+i)}{\Gamma(b)^{2}\Gamma^{2}(n+1)\Gamma(i+1)}\times$
$\displaystyle\frac{(\frac{R}{\delta})^{2n+2i}}{\Gamma(h-n+1)\Gamma(h+n+i+1)(n+i+bk+1)}$
(11)

####  III-A3 How to include shadowing in calculations

As we have mentioned previously, shadowing refers to the attenuation of the
received signal power caused by obstacles obstructing the propagation between
the transmitter and receiver. In general, shadowing between a transmitting
node $t$ and a receiving node $r$ ($t$ and $r$ can be BSs or mobiles) is
modeled by a log-Normal random variable $X_{t}(r)=10^{\frac{Y_{t}(r)}{10}}$
with $Y_{t}(r)$ is a Normal random variable with mean $\mathbb{E}(Y_{t}(r))=0$
and variance $\sigma^{2}$.

To model the shadowing effect, we consider two independent and identically
distributed sequence of log-Normal random variables
$X_{\boldsymbol{s}}(z_{0})$. The expression of the average DL to DL $ISR$
becomes

$D_{\downarrow}(z_{0})=\frac{6x^{2b}}{\Gamma(b)^{2}}\mathbb{E}[10^{\frac{\tilde{Y}_{\boldsymbol{s}}(z_{0})}{10}}]\sum_{h=0}^{+\infty}\frac{\Gamma(b+h)^{2}}{\Gamma(h+1)^{2}}\omega(b+h)x^{2h}$
(12)

where
$10^{\frac{\tilde{Y}_{\boldsymbol{s}}(z_{0})}{10}}=\tilde{X}_{\boldsymbol{s}}(z_{0})$
is a log-normal random variable representing the ratio of the shadowing effect
from interfering cells and the shadowing effect from the serving cell (the
ratio of two log-Normal random variables is a log-Normal random variable),
with $\tilde{Y}_{\boldsymbol{s}}$ is a Normal RV with mean 0 and variance
$\tilde{\sigma}^{2}$. The same reasoning can be followed to include shadowing
when calculating the average UL to DL $ISR$ and for the UL analysis provided
in the next section.

###  III-B Uplink $ISR$ derivation $U(z_{0})$

In this part, we derive the analytical expression of the UL interference to
signal ratio. The UL signal received from location $z_{0}$ at cell
$\boldsymbol{s}_{0}$ experiences interference coming from cells transmitting
in DL and also from mobiles in adjacent cells which are in UL transmission
cycle. The following results may be proved in much the same way as
$D_{\downarrow}$ and $D_{\uparrow}$ in the previous section.

####  III-B1 UL to UL $ISR$ $U_{\uparrow}(z_{0})$

The UL interference is generated by mobiles in neighboring cells which are
randomly distributed in the network as opposed to the DL direction where
cells’ positions are fixed. Thus recalling the fact that mobile location $z$
is uniformly distributed in cell $\boldsymbol{s}$ and taking into account the
definition of the transmitted power with fractional power control model given
by equation (2), $U_{\uparrow}(z_{0})$ can be expressed as

$\displaystyle U_{\uparrow}(z_{0})=$ $\displaystyle\frac{1}{\pi
R^{2}}\int\limits_{0}^{R}\int\limits_{0}^{2\pi}\sum_{\boldsymbol{s}\in\Lambda^{*}}\frac{\rho^{2bk}\leavevmode\nobreak\
\left|\boldsymbol{s}+\rho\leavevmode\nobreak\
e^{i\phi}\right|^{-2b}}{r^{2b(k-1)}}\rho d\rho d\phi$ $\displaystyle=$
$\displaystyle A_{1}(b)\leavevmode\nobreak\ x^{2b(1-k)}$ (13)

where

$A_{1}(b)=\frac{6(R/\delta)^{2bk}}{\Gamma(b)^{2}}\sum_{h=0}^{+\infty}\frac{\Gamma(b+h)^{2}\leavevmode\nobreak\
\omega(b+h)}{\Gamma(h+1)^{2}\leavevmode\nobreak\ (bk+h+1)}(R/\delta)^{2h}$

It is interesting to note that when a mobile is located at the same position
as the serving BS, the UL to DL interference expression becomes similar to the
expression of the UL to UL interference. From equation(11) we have

$\beta_{0}=\sum_{i=0}^{+\infty}\frac{\Gamma(b+i)^{2}\leavevmode\nobreak\
\omega(b+i)}{\Gamma(i+1)^{2}\leavevmode\nobreak\ (bk+i+1)}(R/\delta)^{2i}$
(14)

which is similar to the expression of
$\frac{1}{6}(\frac{\delta}{R})^{2bk}A_{1}(b)$.

####  III-B2 DL to UL $ISR$ $U_{\downarrow}(z_{0})$

The signal coming from neighboring cells is often very strong with respect to
mobile transmit power, especially if neighboring cells’ antennas are in LOS
condition or inter-site distance is lower (path loss is low). Contrary to the
UL to UL interference, here the interfering signals come from cells, which
have fixed positions. Hence, under the same system model assumptions,
$U_{\downarrow}$ is given by

$U_{\downarrow}(z_{0})=\sum_{\boldsymbol{s}\in\Lambda^{*}}\frac{P\left|\boldsymbol{s}\right|^{-2b}}{P^{*}\leavevmode\nobreak\
r^{2b(k-1)}}=A_{2}(b)x^{2b(1-k)}$ (15)

where $A_{2}(b)=\frac{P\leavevmode\nobreak\
\omega(b)}{P^{*}\leavevmode\nobreak\ \delta^{2bk}}$.

Figure 1: DL ISR: Static TDD vs Dynamic TDD. Figure 2: UL ISR: Static TDD vs
Dynamic TDD.

Fig. 1 shows the developed $ISR$ in DL transmission direction for different
values of path loss exponent (2b=2.4, 2b=3.5). The first obvious observation
is that the DL interference level decreases in the studied cell when other
cells use more frequently the UL transmission cycle. This means that the
impact of DL interference coming from other cells is relatively higher than
the impact of interference coming from mobiles. Consequently, one can conclude
that DL interference level in DL cycle for D-TDD should be lower than Static
TDD.

The system behavior during the UL cycle is completely different. As shown in
Fig. 2, interference level significantly increases when 25% or 50% of cells
switched to the opposite direction, i.e., DL transmission. The UL performance
degradation is mainly related to the higher DL transmit power of other cells,
especially when they are in LOS conditions. This makes D-TDD system very
limited by DL to UL interference. These conclusions are in agreement with the
results of [9], which showed that there is an improvement of 10dB in the DL
$SINR$ of the serving cell when 50% of other cells switch from DL to UL
transmission cycle; whereas the UL $SINR$ of the same serving cell degrades by
20dB. This UL performance loss is expected to be more significant in macro-
cell deployment. Therefore, DL to UL interference can seriously deteriorate
system performance if no action is taken to mitigate it.

###  III-C Coverage probability

The coverage probability (CCDF of $SINR$) is the probability that a mobile
user is able to achieve a threshold $SINR$, denoted by $\gamma$, in UL and DL
transmissions.

$\Theta(\gamma)=P(SINR>\gamma)$ (16)

For any scenario of user location distributions, the coverage probability is
given by

$\Theta(\gamma)=\int_{\boldsymbol{s}_{0}}\mathds{1}(SINR>\gamma)dt(z)$ (17)

such that $\int_{\boldsymbol{s}_{0}}dt(z)=1$ (e.g.,
$dt(z)=\frac{rdrd\theta}{\pi R^{2}}$ for uniform user locations distribution).

Based on the expressions of the DL and UL $ISR$ derived previously, we define
the DL and UL $SINR$, denoted respectively by $\Pi_{DL}$ and $\Pi_{UL}$, as
follows

$\displaystyle\Pi_{DL}(x)=\frac{1}{\eta\leavevmode\nobreak\
D(x)+y_{0}x^{2b}}=\frac{1}{d(x)}$ (18)
$\displaystyle\Pi_{UL}(x)=\frac{1}{\eta\leavevmode\nobreak\
U(x)+y^{{}^{\prime}}_{0}x^{2b(1-k)}}=\frac{1}{u(x)}$ (19)

where $y_{0}=\frac{P_{N}\delta^{2b}}{P}$,
$y^{{}^{\prime}}_{0}=\frac{P_{N}\delta^{2b(1-k)}}{P^{*}}$, $P_{N}$ is the
thermal noise power and $\eta$ is the average load over the interfering cells.

Under a uniform user locations distribution, the expression of the coverage
probability is given by

$\Theta(\gamma)=\frac{2}{R^{2}}\int_{0}^{R}\mathds{1}(r<\delta
g^{-1}(\frac{1}{\gamma}))\leavevmode\nobreak\
rdr=\frac{\varTheta^{2}(\gamma)}{R^{2}}$ (20)

where

$\varTheta(\gamma)=\min\left(\delta\times g^{-1}(\frac{1}{\gamma}),R\right)$

with $g=d$ for the DL coverage probability and $g=u$ for the UL coverage
probability.

The explicit formulas of the coverage probability require the inverse
functions of $d$ and $u$. This can be calculated by using series reversion
methods; see for instance [22]. The inverse of $u$ is easy to derive, and it
is given by

$u^{-1}(y)=(\frac{y}{\eta\alpha_{u}A_{1}(b)+\eta\alpha_{d}A_{2}(b)+y^{{}^{\prime}}_{0}})^{\frac{1}{2b(1-k)}}$
(21)

Now, to derive the inverse function of $d$, we shall follow the same approach
as in [22]. To do so, let $y=d(x)$, using the series expansion of
$D_{\downarrow}(m)$ in (5) and the simplified expression of $D_{\uparrow}(m)$
given in (10), it is clear that $d$ admits an analytic expansion on
$x=r/\delta$ and can be expressed as follows

$y=x^{2b}f(b)\left(1+\sum_{h=1}^{+\infty}c_{h}x^{2h}\right)$ (22)

where

$f(b)=6\eta\alpha_{d}\omega(b)+\frac{6\eta
P^{*}R^{2bk}\beta_{0}}{P\Gamma^{2}(b)}+y_{0}$ (23)

and

$c_{h}=\frac{6\eta\alpha_{d}\Gamma(b+h)^{2}\omega(b+h)\beta_{h}}{f(b)\Gamma(b)^{2}\Gamma(h+1)^{2}}$
(24)

Equation (22) can be transformed to

$\displaystyle\left(\frac{y}{f(b)}\right)^{\frac{1}{b}}$
$\displaystyle=x^{2}\left(1+\sum_{h=1}^{+\infty}c_{h}x^{2h}\right)^{\frac{1}{b}}$
$\displaystyle=x^{2}+\frac{c_{1}}{b}x^{4}+O(x^{6})$ (25)

Eliminating the error terms in equation (25) gives a second-order equation in
$x^{2}$ that admits two solutions:

$\displaystyle
x_{\pm}^{2}=\frac{V(y,b)^{2}}{\frac{1}{2}\pm\sqrt{\frac{1}{4}+\frac{c_{1}}{b}V(y,b)^{2}}}$
(26)

where

$V(y,b)=(\frac{y}{f(b)})^{\frac{1}{2b}}$ (27)

Since $x=\frac{r}{\delta}$ is a positive real number, only the positive
solution is valid. Hence we obtain the following approximation of
$x=d^{-1}(y)$

$d^{-1}(y)\approx\frac{V(y,b)^{2}}{\sqrt{\frac{1}{2}+\sqrt{\frac{1}{4}+\frac{c_{1}}{b}V(y,b)^{2}}}}.$
(28)

##  IV Small-cell network performance analysis

Let $\tilde{\boldsymbol{s}}_{0}=z_{0}+Re^{i\theta}$ be the complex location of
a typical small-cell that serves a typical mobile $z_{0}$. Thanks to the
stationarity of the PPP $\Phi$, we can evaluate interference in the mobile
location $z_{0}=0$ having a random distance $R$ to its closest serving BS. To
simplify calculations, we denote hereafter by $R_{\boldsymbol{s}}$, the
distance between an interfering small-cell $\tilde{\boldsymbol{s}}$ and the
typical mobile location $z_{0}$, by $R_{z}$ the distance between an
interfering mobile $z$ and $z_{0}$, by $D_{\boldsymbol{s}}$ the distance
between an interfering small-cell $\tilde{\boldsymbol{s}}$ and the typical
cell $\tilde{\boldsymbol{s}}_{0}$ and by $D_{z}$ the distance between an
interfering mobile $z$ and $\tilde{\boldsymbol{s}}_{0}$; see Fig. 3.

Figure 3: Parameters recapitulation.

Based on the heterogeneous system model provided previously, The DL
interference $\mathcal{I}_{DL}$, perceived by a mobile operating in DL, is
expressed by

$\displaystyle\mathcal{I}_{DL}=\mathcal{I}_{DL->DL}+\mathcal{I}_{UL->DL}$
$\displaystyle=\sum_{z\in\Phi\setminus\\{z_{0}\\}}\big{(}R^{-2b}_{\boldsymbol{s}}H_{\boldsymbol{s}}\tilde{P}\chi_{d}(z)+R^{-2b}_{z}H_{z}\rho^{2bk}_{z}\tilde{P}^{*}(1-\chi_{d}(z))\big{)}.$
(29)

Similarly, the UL interference $\mathcal{I}_{UL}$ experienced by a mobile
$z_{0}$, operating in UL transmission cycle and received at the serving small-
cell BSs position, is given by

$\displaystyle\mathcal{I}_{UL}=\mathcal{I}_{UL->UL}+\mathcal{I}_{DL->UL}$
$\displaystyle=\sum_{z\in\Phi\setminus\\{z_{0}\\}}\big{(}D^{-2b}_{z}G_{z}\rho^{2bk}_{z}\tilde{P}^{*}\chi_{u}(z)+D^{-2b}_{\boldsymbol{s}}G_{\boldsymbol{s}}\tilde{P}(1-\chi_{u}(z))\big{)}.$
(30)

Therefore, the DL and UL $SINR$ can be defined respectively by

$\displaystyle\Pi_{DL}=\frac{\tilde{P}HR^{-2b}}{\mathcal{I}_{DL}+P_{N}}$ (31)
$\displaystyle\Pi_{UL}=\frac{\tilde{P}^{*}GR^{-2b(1-k)}}{\mathcal{I}_{UL}+P_{N}}.$
(32)

with $H$ and $G$ are the fading coefficients between the typical mobile and
its serving small-cell.

Once again, the DL (UL) coverage probability is defined as the CCDF of the DL
(UL) $SINR$. It gives the percentage of locations in which $\Pi_{DL}$
($\Pi_{UL}$) is greater than a threshold value $\gamma$. It can be expressed
for the DL and UL transmission directions as

$\displaystyle\Theta_{DL}=\mathbb{E}_{\\{R\\}}\big{[}\mathbb{P}(\Pi_{DL}>\gamma)|R=r\big{]}$
(33)
$\displaystyle\Theta_{UL}=\mathbb{E}_{\\{R\\}}\big{[}\mathbb{P}(\Pi_{UL}>\gamma)|R=r\big{]}.$
(34)

###  IV-A DL coverage probability derivation

Starting from the definition of the DL coverage probability and the DL $SINR$,
we have

$\Theta_{DL}(\gamma)=\int_{0}^{+\infty}\mathbb{P}\big{[}H>\frac{\gamma(\mathcal{I}_{DL}+P_{N})}{\tilde{P}R^{-2b}}|R=r\big{]}f_{R}(r)dr$
(35)

with $f_{R}(r)$ is the distribution of $R$ which is the distance between
$z_{0}$ and the closest small-cell. Using the null probability of a PPP, it
has been shown that this distance is Rayleigh distributed [19] and its
probability density function is given by

$f_{R}(r)=2\pi\lambda re^{-\lambda\pi r^{2}}$ (36)

Using the fact that $H$ follows an exponential distribution of mean 1 and the
definition of the Laplace transform, it follows that

$\Theta_{DL}(\gamma)=2\pi\lambda\int_{0}^{+\infty}e^{-\lambda\pi
r^{2}}e^{-\gamma\tilde{P}^{-1}P_{N}r^{2b}}\mathcal{L}_{\mathcal{I}_{DL}}(\frac{\gamma
r^{2b}}{\tilde{P}})rdr$ (37)

with $\mathcal{L}_{\mathcal{I}_{DL}}(v)$ is the Laplace transform of
$\mathcal{I}_{DL}$ conditionally on $R$. It is defined by

$\displaystyle\mathcal{L}_{\mathcal{I}_{DL}}(v)=\mathbb{E}\big{[}e^{-v{\mathcal{I}_{DL}}}\big{]}$
$\displaystyle=\mathbb{E}_{\\{R_{\boldsymbol{s}},R_{z},\rho_{z},H_{\boldsymbol{s}},H_{z},\chi_{d}(z)\\}}\bigg{[}\exp\big{(}-v\times$
$\displaystyle\sum_{z\in\Phi\setminus\\{z_{0}\\}}R^{-2b}_{\boldsymbol{s}}H_{\boldsymbol{s}}\tilde{P}\chi_{d}(z)+R^{-2b}_{z}H_{z}\rho^{2bk}_{z}\tilde{P}^{*}(1-\chi_{d}(z))\big{)}\bigg{]}$
(38)

From the complex geometry, the distance $R_{\boldsymbol{s}}$ between a small-
cell $\tilde{\boldsymbol{s}}$ and the typical mobile $z_{0}=0$ can be written
in terms of $R_{z}$ and $\rho_{z}$ as

$R^{2}_{\boldsymbol{s}}=R^{2}_{z}+\rho^{2}_{z}+2R_{z}\rho_{z}\cos(\arg(z)-\phi_{z}).$
(39)

with $\arg(z)$ is the complex argument of $z$ relatively to the origin of the
plane $z_{0}=0$.

Thus, replacing $R_{\boldsymbol{s}}$ by its expression and using the fact that
$H_{\boldsymbol{s}}$ and $H_{z}$ are i.i.d RVs and follow an exponential
distribution of mean 1, (38) can be simplified to

$\displaystyle\mathcal{L}_{\mathcal{I}_{DL}}(v)=\mathbb{E}_{\\{R_{z},\rho_{z},\chi_{d}(z)\\}}\bigg{[}\prod_{z\in\Phi\setminus\\{z_{0}\\}}\mathbb{E}_{\\{H_{\boldsymbol{s}},H_{z}\\}}\big{[}$
$\displaystyle\exp\big{(}-v(R^{2}_{z}+\rho^{2}_{z}+2R_{z}\rho_{z}\cos(\arg(z)-\phi_{z}))^{-b}H_{\boldsymbol{s}}\tilde{P}\chi_{d}(z)\big{)}\times$
$\displaystyle\exp\big{(}-vR^{-2b}_{z}H_{z}\rho^{2bk}_{z}\tilde{P}^{*}(1-\chi_{d}(z))\big{)}\big{]}\bigg{]}$
$\displaystyle=\mathbb{E}_{\\{R_{z}\\}}\bigg{[}\prod_{z\in\Phi\setminus\\{z_{0}\\}}\mathbb{E}_{\\{\chi_{d}(z),\rho_{z}\\}}\big{[}$
$\displaystyle\frac{1}{1+v(R^{2}_{z}+\rho^{2}_{z}+2R_{z}\rho_{z}\cos(\arg(z)-\phi_{z}))^{-b}\tilde{P}\chi_{d}(z)}\times$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
\frac{1}{1+vR^{-2b}_{z}\rho^{2bk}_{z}\tilde{P}^{*}(1-\chi_{d}(z))}\big{]}\bigg{]}$
(40)

Now, using the Probability Generating Functional (PGFL) of PPP $\Phi$ with
respect to the function inside the product, (40) becomes

$\displaystyle\mathcal{L}_{\mathcal{I}_{DL}}(v)=\exp\bigg{(}-\lambda\int_{r}^{+\infty}\int_{0}^{2\pi}\big{(}1-\mathbb{E}_{\\{\chi_{d}(z),\rho_{z}\\}}\big{[}$
$\displaystyle\frac{1}{1+v(x^{2}+\rho^{2}_{z}+2x\rho_{z}\cos(\theta))^{-b}\tilde{P}\chi_{d}(z)}\times$
$\displaystyle\frac{1}{1+vx^{-2b}\rho^{2bk}_{z}\tilde{P}^{*}(1-\chi_{d}(z))}\big{]}\big{)}xdxd\theta\bigg{)}$
$\displaystyle=\exp\bigg{(}-\lambda\int_{r}^{+\infty}\int_{0}^{2\pi}\big{(}1-\mathbb{E}_{\\{\rho_{z}\\}}\big{[}$
$\displaystyle\frac{\alpha_{d}}{1+v(x^{2}+\rho^{2}_{z}+2x\rho_{z}\cos(\theta))^{-b}\tilde{P}}+$
$\displaystyle\frac{\alpha_{u}}{1+vx^{-2b}\rho^{2bk}_{z}\tilde{P}^{*}}\big{]}\big{)}xdxd\theta\bigg{)}$
(41)

We have made the assumption that each user is associated with the nearest BS.
Hence, following the same analysis provided in [20], we can approximate the
distribution of $\rho_{z}$ by the same distribution as $R$ i.e., Rayleigh. It
follows that

$f_{\rho_{z}}(\rho)=2\pi\lambda\rho
e^{-\lambda\pi\rho^{2}},\leavevmode\nobreak\ \leavevmode\nobreak\ \rho\geq 0.$
(42)

Finally, by using (42), it follows that

$\displaystyle\mathcal{L}_{\mathcal{I}_{DL}}(v)=\exp\bigg{(}-\lambda\int_{r}^{+\infty}\int_{0}^{2\pi}\big{(}1-$
$\displaystyle\int_{0}^{+\infty}2\pi\lambda
e^{-\lambda\pi\rho^{2}}\big{[}\frac{\alpha_{d}}{1+v(x^{2}+\rho^{2}+2x\rho\cos(\theta))^{-b}\tilde{P}}+$
$\displaystyle\frac{\alpha_{u}}{1+vx^{-2b}\rho^{2bk}\tilde{P}^{*}}\big{]}\rho
d\rho\big{)}xdxd\theta\bigg{)}.$ (43)

###  IV-B UL coverage probability derivation

The derivation of the UL coverage probability is quite similar to the DL one.
The typical cell $\tilde{\boldsymbol{s}}_{0}$ is operating in UL cycle and
perceives interference coming from the other small-cells operating in DL and
also from the mobiles transmitting in UL. Interference in this case is
received at the location of $\tilde{\boldsymbol{s}}_{0}$. Since the small-
cells are distributed also as a PPP, denoted hereafter by
$\Phi_{\boldsymbol{s}}$ and constructed by the displacement of $\Phi$, we can
perform the analysis at the location $\tilde{\boldsymbol{s}}_{0}=0$ (The
stationarity of the PPP). Therefore, using the definition of the UL coverage
probability given by (3.33), we get

$\displaystyle\Theta_{UL}(\gamma)$
$\displaystyle=\int_{0}^{+\infty}\mathbb{P}\big{[}G>\frac{\gamma(\mathcal{I}_{UL}+P_{N})}{\tilde{P}^{*}R^{-2b(1-k)}}|R=r\big{]}f_{R}(r)dr$
$\displaystyle=2\pi\lambda\int_{0}^{+\infty}e^{-\lambda\pi
r^{2}}e^{-\gamma\tilde{P}^{*-1}P_{N}r^{2b(1-k)}}\times$ $\displaystyle\ \ \ \
\ \ \ \ \ \ \ \ \ \mathcal{L}_{\mathcal{I}_{UL}}(\frac{\gamma
r^{2b(1-k)}}{\tilde{P}^{*}})rdr,$ (44)

with $\mathcal{L}_{\mathcal{I}_{UL}}(v)$ is the Laplace transform of
$\mathcal{I}_{UL}$ calculated conditionally on $R$.

Once again, the distance $D_{\boldsymbol{s}}$ represents the distance between
an interfering small-cell $\tilde{\boldsymbol{s}}$ and the typical one taken
at the origin ($\tilde{\boldsymbol{s}}_{0}=0$). Hence using the complex
notations, we get

$D^{2}_{z}=D^{2}_{\boldsymbol{s}}+\rho^{2}_{z}+2D_{\boldsymbol{s}}\rho_{z}\cos(\arg(\boldsymbol{s})-\phi_{z}).$
(45)

Thus, by following the same steps as we did in the derivation of
$\mathcal{L}_{\mathcal{I}_{DL}}(v)$, the Laplace transform of
$\mathcal{I}_{UL}$ is given by

$\displaystyle\mathcal{L}_{\mathcal{I}_{UL}}(v)=\exp\bigg{(}-\lambda\int_{r}^{+\infty}\int_{0}^{2\pi}\big{(}1-$
$\displaystyle\int_{0}^{+\infty}2\pi\lambda
e^{-\lambda\pi\rho^{2}}\big{[}\frac{\alpha_{u}}{1+v\rho^{2bk}(x^{2}+\rho^{2}-2x\rho\cos(\theta))^{-b}\tilde{P}^{*}}+$
$\displaystyle\frac{\alpha_{d}}{1+vx^{-2b}\tilde{P}}\big{]}\rho
d\rho\big{)}xdxd\theta\bigg{)}.$ (46)

###  IV-C Average spectral efficiency

The instantaneous spectral efficiency (SE) is defined as the maximum
information rate that can be transmitted in a given bandwidth. Using the
upper-bound of the well known Shannon’s formula, the instantaneous spectral
efficiency is expressed by

$SE_{s}=\log_{2}(1+\Pi_{s})$ (47)

with $s=DL$ when the serving small-cell is operating in DL and $s=UL$ when the
serving small-cell is operating in UL.

The average spectral efficiency (ASE) is obtained by averaging over (47). It
follows that

$\displaystyle ASE_{s}$ $\displaystyle=\mathbb{E}[\log_{2}(1+\Pi_{s})]$
$\displaystyle=\int_{0}^{+\infty}\mathbb{P}(\log_{2}(1+\Pi_{s})>t)dt$
$\displaystyle\stackrel{{\scriptstyle\mathclap{\mbox{(a)}}}}{{=}}\frac{1}{\ln(2)}\int_{0}^{+\infty}\frac{\Theta_{s}(\gamma)}{(\gamma+1)}d\gamma$
(48)

with $(a)$ comes from the change of variable $\gamma=e^{t}-1$.

##  V Simulation results

We simulate in MATLAB the proposed Macro-cell and small-cell models, for both
DL and UL transmission directions, using the parameters summarized in Table.
I.

Macro-cells power $P$ | 60dBm
---|---
Small cells power $\tilde{P}$ | 26dBm
Target power cell specific $P^{*}$ | 20dBm
Noise power $P_{N}$ | -93dBm
Number of rings (Macro-cells) | 4 (60 interfering BSs)
Inter-site distance $\delta$ | 1km
Antennas gain | 16dBi
User distributions | uniform
small-cells density | 10 cells.$km^{-2}$
Propagation factor $a$ | Outdoor: 130dB, Indoor: 160dB
System bandwidth | Macro:20Mhz, small:10Mhz
Path loss exponent $2b$ | 2.5, 3.5
Table I: Simulation parameters.

We plot, respectively in Fig. 4 and Fig. 5, the DL and UL empirical coverage
probability curves obtained by using Monte Carlo simulations for 20000 mobile
locations $z_{0}$ and for two path loss exponent different values ($2b=2.5$
and $2b=3.5$). As we can see in Fig. 4, starting from a static TDD
configuration where all the macro-cells are transmitting in DL, the coverage
probability increases when D-TDD is activated with $\alpha_{d}=75\%$ and
$\alpha_{d}=50\%$. This behavior is expected since the macro-cells BSs
transmit with high power level and generate strong interference compared to
interfering mobiles $z$ transmitting in UL. However, the system behavior is
completely different during the UL cycle of the serving cell. As it is shown
in Fig. 5, activating D-TDD with a mean number of neighboring UL macro-cells
$\alpha_{u}=75\%$ and $\alpha_{u}=50\%$ deteriorates completely the coverage
probability. For instance, with a threshold SINR of $-20dB$, the coverage
probability undergoes a huge degradation of $80\%$. Hence, one can conclude
that D-TDD has a tremendous effect on performance during UL transmission
especially for macro-cells deployment. Those results are in agreement with
simulation results provided in [9] and quite similar to the behavior of
theoretical $ISR$ curves of Fig. 1 and Fig. 2.

Figure 4: Macro-cells network DL coverage probability. Figure 5: Macro-cells
network UL coverage probability.

In Fig. 6, we plot the DL and UL coverage probability curves, with different
fractional power control factor values ($k=0$, $k=0.4$, $k=0.8$ and $k=1$). We
consider the scenario when D-TDD is activated and the number of DL and UL
interfering cells is quite proportional (i.e., $\alpha_{d}=50\%$ and
$\alpha_{u}=50\%$). When the serving cell is operating in DL, we notice that
changing the power control factor has no impact on the coverage probability.
This is mainly due to the fact that the principal interference impact comes
from the DL BS signals where no power control mechanisms are considered.
During the UL transmission cycle of the serving cell, one can notice that the
coverage probability is decreasing when the fractional power control is
increasing. Actually, FPC aims at providing the required $SINR$ to UL users
while controlling at the same time their interference. When FPC factor $k=1$
the path loss is completely compensated and the target cell-specific power
$P^{*}$ is reached. Thus, the interference coming from mobiles $z$ in UL is
higher especially if a mobile is located in the edge of a neighboring cell.
When FPC factor $0<k<1$, the scheme indemnifies partially the path loss. The
higher is the path loss the lower is the received signal. This means that
there is a compromise between the path loss and the $SINR$ requirements.
Therefore, interference are likely to be controlled, which explain the
enhancement of the coverage probability. This enhancement is more obvious when
$k=0$. Which means that there is no compensation and the signal coming from
the mobiles is weak.

Figure 6: Macro-cells network DL and UL coverage probability with different
power control factors. Figure 7: Small-cells DL coverage probability :
$2b=3.5$, $k=0.4$ and $\lambda=10$ small-cell$/km^{2}$. Figure 8: Small-cells
UL coverage probability : $2b=3.5$, $k=0.4$ and $\lambda=10$ small-
cell$/km^{2}$.

Fig. 7 shows the DL coverage probability obtained from the simulation of a
heterogeneous network. First, small-cells operate with a static TDD
configuration i.e., $\alpha_{d}=1$, then the whole network switch to a D-TDD
configuration during the ABS sub-frames. We plot the coverage probability
curves for an outdoor environment with a propagation parameter $a=130dB$ and a
deep indoor environment with $a=160dB$. For the outdoor environment, the
comparison between the S-TDD and D-TDD shows that the behavior is quite
similar to macro-cells deployment results. For instance, when $50\%$ of small-
cells switch from the DL to UL, there is an enhancement of the coverage
probability by $15\%$ for a threshold $SINR$ of $-10dB$. However, for a deep
indoor environment, one can notice that the coverage probability remains
unchanged even when $50\%$ of the interfering small-cells switch to the
opposite direction. In fact, signal propagation in a deep indoor environment
suffers from high attenuation and delay factors because of the presence of
obstacles and building penetration. During the DL transmission, the major
interference comes from DL small-cells signal. In a deep indoor environment,
not only the signal received from the serving cell is attenuated but also
interference signal is subject to high attenuation. Similarly, Fig. 8
represents the comportment of the system during the UL transmission cycle. As
expected, there is a degradation of the coverage probability when $50\%$ of
small-cells switch from DL to UL for an outdoor environment. This degradation
is not severe like the case of macro-cells because small-cells transmit with
low power and not highly elevated. Also, the comportment in a deep indoor
environment is quite similar to the DL scenario for the same reasons stated
previously.

Figure 9:  DL average spectral efficiency : $2b=3.5$, $k=0.4$. Figure 10: UL
average spectral efficiency : $2b=3.5$, $k=0.4$..

Additionally, we plot in Fig. 9 the average spectral efficiency in DL obtained
from the simulation of the HetNet as a function of small-cells’ density. Once
again, the comparison between the static TDD configuration and dynamic TDD, in
an outdoor environment, shows that there is an enhancement of the ASE when
D-TDD is activated with $\alpha_{d}=50\%$. Also, we observe that the ASE is
slightly decreasing when small-cells density increases before it becomes
almost constant. Actually, when small-cells density increases, the mean
distance between the small-cells decreases because the density is inversely
proportional to the mean distance between nodes. Hence, also the size of
small-cells decreases and then both received signal power and interference
increase simultaneously. Once we reach the interference limited scenario, the
ASE becomes almost constant. For deep indoor environment, interference
undergoes high attenuation as we have explained previously. When $\lambda$
increases, the size of small-cells decrease. Thus, the received signal from
the serving cell overcome interference undergoing bad propagation conditions.
Similarly in Fig. 10, we plot the ASE during the UL cycle of a typical serving
small-cell using the same parameters as in DL. Once again, we observe that the
ASE is decreasing when D-TDD is activated with $\alpha_{u}=50\%$ for the
outdoor environment. Also, the ASE decreases as the small-cells density
increases, especially when D-TDD is active. Moreover, for a deep indoor
environment, the system experiences very bad performances in terms of ASE.

##  VI Conclusions

In this paper, we have investigated inter-cell interference in D-TDD based
network. Explicit formulas of $ISR$ covering different scenarios of
interference have been derived. We have provided the explicit expressions of
the coverage probability in macro-cell and small-cell deployments. The
comparison between static-TDD configuration and D-TDD shows that performance
are better with D-TDD during the DL cycle of typical cell. However, the UL
transmission is severely limited by interference coming from other BSs DL
signals. Also, we have compared two types of environment, outdoor and deep
indoor. As expected, the system experiences bad performance in deep indoor
environment for both static and dynamic TDD. Moreover, we have analyzed the
impact of fractional power control mechanisms on the UL transmission. Results
have shown that small FPC factors enhance the coverage probability for both
D-TDD and S-TDD. Further extension of this work could include the analysis of
interference mitigation schemes such as 3D beamforming for macro-cell
deployment in order to minimize the impact of the strong DL to UL interference
which will make D-TDD feasible for macro-cells. Also, extension could include
a dynamic system level analysis by including traffic model.

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