 Research
 Open Access
Reflection probability in wireless networks with metasurfacecoated environmental objects: an approach based on random spatial processes
 Marco Di Renzo^{1}Email authorView ORCID ID profile and
 Jian Song^{1}
https://doi.org/10.1186/s1363801914037
© The Author(s) 2019
 Received: 4 January 2019
 Accepted: 19 March 2019
 Published: 23 April 2019
Abstract
An emerging and promising vision of wireless networks consists of coating the environmental objects with reconfigurable metasurfaces that are capable of modifying the radio waves impinging upon them according to the generalized law of reflection. By relying on tools from point processes, stochastic geometry, and random spatial processes, we model the environmental objects with a modified random line process of fixed length and with random orientations and locations. Based on the proposed modeling approach, we develop the first analytical framework that provides one with the probability that a randomly distributed object that is coated with a reconfigurable metasurface acts as a reflector for a given pair of transmitter and receiver. In contrast to the conventional network setup where the environmental objects are not coated with reconfigurable metasurfaces, we prove that the probability that the typical random object acts as a reflector is independent of the length of the object itself. The proposed analytical approach is validated against Monte Carlo simulations, and numerical illustrations are given and discussed.
Keywords
 Wireless networks
 Reconfigurable metasurfaces
 Stochastic geometry
 Random spatial processes
 Reflection probability
1 Methods/experimental
The methods used in the paper are based on the mathematical tools of random spatial processes and stochastic geometry. A new analytical framework for performance analysis is introduced. The theoretical framework is validated against Monte Carlo simulations.
2 Introduction
Future wireless networks will be more than allowing people, mobile devices, and objects to communicate (https://www.comsoc.org/publications/ctn/whatwill6gbe). Future wireless networks will be turned into a distributed intelligent wireless communications, sensing, and computing platform, which, besides communications, will be capable of sensing the environment to provide contextawareness capabilities, of locally storing and processing data to enable its time critical and energyefficient delivery, and of accurately localizing people and objects in harsh propagation environments. Future wireless networks will have to fulfill the challenging requirement of interconnecting the physical and digital worlds in a seamless and sustainable manner [1, 2].
To fulfill these challenging requirements, it is apparent that it is not sufficient anymore to rely solely on wireless networks whose logical operation is softwarecontrolled and optimized [3]. The wireless environment itself needs to be turned into a softwarereconfigurable entity [4], whose operation is optimized to enable uninterrupted connectivity. Future wireless networks need a smart radio environment, i.e., a wireless environment that is turned into a reconfigurable space that plays an active role in transferring and processing information.
Different solutions towards realizing this wireless future are currently emerging [5–13]. Among them, the use of reconfigurable metasurfaces constitutes a promising and enabling solution to fulfill the challenging requirements of future wireless networks [14]. Metasurfaces are thin metamaterial layers that are capable of modifying the propagation of the radio waves in fully customizable ways [15], thus owing the potential of making the transfer and processing of information more reliable [16]. Also, they constitute a suitable distributed platform to perform lowenergy and lowcomplexity sensing [17], storage [18], and analog computing [19]. For this reason, they are particularly useful for improving the performance of nonlineofsight transmission, e.g., to appropriately customize the impact of multipath propagation.
In [13], in particular, the authors have put forth a network scenario where every environmental object is coated with reconfigurable metasurfaces, whose response to the radio waves is programmed in software by capitalizing on the enabling technology and hardware platform currently being developed in [20]. Current research efforts towards realizing this vision are, however, limited to implement hardware testbeds, e.g., reflectarrays and metasurfaces, and on realizing pointtopoint experimental tests [5–13]. To the best of the authors’ knowledge, notably, there exists no analytical framework that investigates the performance of largescale wireless networks in the presence of reconfigurable metasurfaces. In the present paper, motivated by these considerations, we develop the first analytical approach that allows one to study the probability that a random object coated with a reconfigurable metasurface acts as a reflector according to the generalized laws of reflection [15]. To this end, we capitalize on the mathematical tool of random spatial processes [21, 22].
Random spatial processes are considered to be the most suitable analytical tool to shed light on the ultimate performance limits of innovative technologies when applied in wireless networks and to guide the design of optimal algorithms and protocols for attaining such ultimate limits. Several recent results on the application of random spatial processes in wireless networks can be found in [23–31]. Despite the many results available, however, fundamental issues remain open [28]. In the current literature, in particular, the environmental objects are modeled as entities that can only attenuate the signals, by making the links either lineofsight or nonlineofsight, e.g., [25–27]. Modeling anything else is acknowledged to be difficult. Just in [32], the authors have recently investigated the impact of reflections, but only based on conventional Snell’s laws. This work highlights the analytical complexity, the relevance, and the nontrivial performance tradeoffs: The authors emphasize that the obtained trends highly depend on the fact that the total distance of the reflected paths is almost always two times larger than the distance of the direct paths. This occurs because the angles of incidence and reflection are the same based on Snell’s law. In the presence of reconfigurable metasurfaces, on the other hand, the random objects can optimize the reflected signals in anomalous directions beyond Snell’s law. The corresponding achievable performance and the associated optimal setups are unknown.
Motivated by these considerations, we develop an analytical framework that allows one to quantify the probability that a random object coated with reconfigurable metasurfaces acts as a reflector for a given pair of transmitter and receiver. Even though reconfigurable metasurfaces can be used to control and customize the refractions from environmental objects, in the present paper, we focus our attention on controlling and customizing only the reflections of signals, since refractions may be subject to severe signal’s attenuation. Our proposed approach, in particular, is based on modeling the environmental objects with a modified random line process of fixed length and with random orientations and locations. In contrast to the conventional network setup where the environmental objects are not coated with reconfigurable metasurfaces, we prove that the probability that the typical random object acts as a reflector is independent of the length of the object itself. The proposed analytical approach is validated against Monte Carlo simulations, and numerical illustrations are given and discussed. In the present paper, we limit ourselves to analyze 2D network scenarios, but our approach can be applied to 3D network topologies as well. This nontrivial generalization is postponed to a future research work.
The remainder of the present paper is organized as follows. In Section 3, the system model is introduced. In Section 4, the problem is formulated in mathematical terms. In Section 5, the analytical framework of the reflection probability is described. In Section 6, numerical results are illustrated, and the proposed approach is validated against Monte Carlo simulations. Finally, Section 7 concludes the paper.
Main symbols and functions used throughout the paper
Symbol/function  Definition 

Pr{A}  Probability of Event A 
\(\Pr \left \{ {\overline A} \right \}\)  Probability of complement of Event A 
H(·), \(\bar H\left (\cdot \right)\)  Heaviside function, complementary Heaviside function 
(x_{Tx},y_{Tx})  Location of the transmitter 
(x_{Rx},y_{Rx})  Location of the receiver 
(x_{object},y_{object})  Location of the center of the typical object 
(x_{end1},y_{end1}), (x_{end2},y_{end2})  Coordinates of the end points of the typical object 
L  Length of the typical object 
R _{net}  Radius of the network 
3 System model
We consider a wireless network on a bidimensional plane, where the transmitters and receivers are distributed independently of each other. Without loss of generality, the location of the transmitter and receiver of interest, i.e., the probe transmitter and receiver, are denoted by (x_{Tx},y_{Tx}) and (x_{Rx},y_{Rx}), respectively.

The center points of the objects form a homogeneous Poisson point process.

The orientation of the objects are independent and identically distributed in [0,2π].

The lengths of the objects are fixed and all equal to L.

The random orientation and the center points of the objects are independent of each other.
We consider a generic environmental object, i.e., the typical object, and denote its center by (x_{object},y_{object}), and the coordinates of its two end points by (x_{end1},y_{end1}) and (x_{end2},y_{end2}).
4 Problem formulation

Scenario I: The first scenario corresponds to the case study where the typical object is coated with a reconfigurable metasurface, which can optimize the angle of reflection regardless of the angle of incidence [15].

Scenario II: The second scenario corresponds to the case study where the typical object is not coated with a reconfigurable metasurface. This is the stateoftheart scenario, where the angle of reflection needs to be equal to the angle of incidence according to Snell’s law of reflection [15].
4.1 Scenario I: Reflections in the presence of reconfigurable metasurfaces
In the presence of reconfigurable metasurfaces, an arbitrary angle of reflection can be obtained for any angle of incidence. This implies that the typical object acts as a reflector for a transmitter and receiver if they are both located on the same side of the infinite line passing through the end points (x_{end1},y_{end1}) and (x_{end2},y_{end2}) of the typical object.
For ease of exposition, we introduce the following event.
Event 1 The probe transmitter, Tx, and receiver, Rx, are located on the same side of the infinite line passing through the end points (x_{end1},y_{end1}) and (x_{end2},y_{end2}) of the typical object.
Therefore, the typical object acts as a reflector if Event 1 holds true. Our objective is to formulate the probability of Event 1, i.e., to compute Pr{Event 1}. This latter probability can be formulated in two different but equivalent ways.
4.1.1 Approach 1
4.1.2 Approach 2
4.2 Scenario II: Reflections in the absence of reconfigurable metasurfaces
In the absence of reconfigurable metasurfaces, the typical object acts as a reflector, for a given transmitter and receiver, only if the angles of reflection and incidence are the same. This is agreement with Snell’s law of reflection, and imposes some geometric constraints among the locations of the typical object, the transmitter, and the receiver. In order to compute the corresponding probability of occurrence, we introduce the following event.
Event 2 The midperpendicular of the line segment that connects the transmitter and receiver intersects the line segment that represents the typical object.
Let (x_{∗},y_{∗}) denote the intersection between the midperpendicular of the line segment that connects the transmitter and receiver, and the line segment that represents the typical object. According to Snell’s law of reflection, for some given locations of the transmitter and receiver, the typical object acts as a reflector if the midperpendicular of the line segment that connects the transmitter and receiver intersects the line segment that represents the typical object (i.e., Event 2), and, at the same time, the transmitted and receiver are located on the same side of the infinite line passing through the end points of the typical object (i.e., Event 1).
Based on Snell’s law of reflection, therefore, the typical object acts a reflector if the following event holds true.
Event 3 The transmitter and receiver are located on the same side of the infinite line passing through the end points of the typical object, and the midperpendicular of the line segment that connects the transmitter and receiver intersects the line segment that represents the typical object.
5 Analytical formulation of the reflection probability
In this section, we introduce analytical expressions of the probability of occurrence of the three events introduced in the previous sections, and, therefore, characterize the probability that the typical object acts as a reflector in the presence and in the absence of reconfigurable metasurfaces. First, we begin with some preliminary results.
5.1 Preliminary results
Lemma 1
where \(m = \frac {{{y_{{\text {Tx}}}}  {y_{{\text {Rx}}}}}}{{{x_{{\text {Tx}}}}  {x_{{\text {Rx}}}}}}\), and z=y_{Rx}−mx_{Rx}.
Proof
It follows by definition of line passing through two points. □
Lemma 2
where α∈[0,2π].
Proof
The proof follows by noting that the centers of the line segments (the objects) are distributed according to a Poisson point process with random orientations, which implies \(p = {R_{{\text {net}}}}\sqrt u \) and α∈[0,2π]. The rest follows from geometric considerations. □
Lemma 3
where \({m_{p}} =  \frac {1}{m}\), and \({z_{p}} = \frac {1}{{2m}}\left ({{x_{{\text {Tx}}}} + {x_{{\text {Rx}}}}} \right) + \frac {1}{2}\left ({{y_{{\text {Tx}}}} + {y_{{\text {Rx}}}}} \right)\).
Proof
See Appendix 1. □
Lemma 4
Proof
Equation 10 follows by solving the system of equations in (6) and (7). Equation 11 follows by solving the system of equations in (7) and (9). □
Lemma 5

The point (x_{1},y_{1}) is above the line if ax_{1}+by_{1}+c>0 and b>0, or if ax_{1}+by_{1}+c<0 and b<0.

The point (x_{1},y_{1}) is below the line if ax_{1}+by_{1}+c<0 and b>0, or if ax_{1}+by_{1}+c>0 and b<0.
Proof
See Appendix 2. □
5.2 Scenario I: Reflection probability in the presence of reconfigurable metasurfaces
Theorems 1 and 2 provide one with analytical expressions of the probability that the typical object acts as a reflector if it is coated with reconfigurable metasurfaces. Theorem 1 is computed based on Approach 1, and Theorem 2 based on the Approach 2.
Theorem 1
Auxiliary functions used in Theorem 1
Function definition 

\(f\left ({\alpha,\xi } \right) = \frac {1}{{{R_{{\text {net}}}}}}\left ({\left [ {m\sin \alpha + \cos \alpha } \right ]\xi + z\sin \alpha } \right)\) 
\(g\left ({\alpha,\omega } \right) = \frac {1}{{{R_{{\text {net}}}}}}\left ({\frac {{\left [ {m\sin \alpha + \cos \alpha } \right ]\left [ {\omega  z} \right ]}}{m} + z\sin \alpha } \right)\) 
\(\Theta \! \!\left (\! \!{\alpha \!\left \! {\begin {array}{*{20}{c}} {{\mu _{1}}}&{{\mu _{2}}} \\ {{\mu _{3}}}&{{\mu _{4}}} \end {array}} \right.}\! \!\!\right)\! \!= \!\left [\! {{{\left ({\min \left \{ {{\mu _{1}},{\mu _{2}},1} \right \}} \right)}^{2}}\! \! {{\left ({\max \left \{ {{\mu _{3}},{\mu _{4}},0} \right \}} \right)}^{2}}} \right ]H\!\left ({\min \left \{ {{\mu _{1}},{\mu _{2}},1} \right \} \! \!\max \!\left \{ {{\mu _{3}},{\mu _{4}},0} \right \}}\! \right)\) 
\({\theta _{1}}\left ({\alpha,{x_{{\text {Tx}}}},{x_{{\text {Rx}}}},{y_{{\text {Tx}}}},{y_{{\text {Rx}}}}} \right) = \Theta \left ({\alpha \left  {\begin {array}{*{20}{c}} {f\left ({\alpha,\max \left ({{x_{{\text {Tx}}}},{x_{{\text {Rx}}}}} \right)} \right)}&{g\left ({\alpha,\max \left ({{y_{{\text {Tx}}}},{y_{{\text {Rx}}}}} \right)} \right)} \\ {f\left ({\alpha,\min \left ({{x_{{\text {Tx}}}},{x_{{\text {Rx}}}}} \right)} \right)}&{g\left ({\alpha,\min \left ({{y_{{\text {Tx}}}},{y_{{\text {Rx}}}}} \right)} \right)} \end {array}} \right.} \right) \times H\left (m \right)\) 
\({\theta _{2}}\left ({\alpha,{x_{{\text {Tx}}}},{x_{{\text {Rx}}}},{y_{{\text {Tx}}}},{y_{{\text {Rx}}}}} \right) = \Theta \left ({\alpha \left  {\begin {array}{*{20}{c}} {f\left ({\alpha,\max \left ({{x_{{\text {Tx}}}},{x_{{\text {Rx}}}}} \right)} \right)}&{g\left ({\alpha,\min \left ({{y_{{\text {Tx}}}},{y_{{\text {Rx}}}}} \right)} \right)} \\ {f\left ({\alpha,\min \left ({{x_{{\text {Tx}}}},{x_{{\text {Rx}}}}} \right)} \right)}&{g\left ({\alpha,\max \left ({{y_{{\text {Tx}}}},{y_{{\text {Rx}}}}} \right)} \right)} \end {array}} \right.} \right) \times \bar H\left (m \right)\) 
\({\theta _{3}}\left ({\alpha,{x_{{\text {Tx}}}},{x_{{\text {Rx}}}},{y_{{\text {Tx}}}},{y_{{\text {Rx}}}}} \right) = \Theta \left ({\alpha \left  {\begin {array}{*{20}{c}} {f\left ({\alpha,\min \left ({{x_{{\text {Tx}}}},{x_{{\text {Rx}}}}} \right)} \right)}&{g\left ({\alpha,\min \left ({{y_{{\text {Tx}}}},{y_{{\text {Rx}}}}} \right)} \right)} \\ {f\left ({\alpha,\max \left ({{x_{{\text {Tx}}}},{x_{{\text {Rx}}}}} \right)} \right)}&{g\left ({\alpha,\max \left ({{y_{{\text {Tx}}}},{y_{{\text {Rx}}}}} \right)} \right)} \end {array}} \right.} \right) \times H\left (m \right)\) 
\({\theta _{4}}\left ({\alpha,{x_{{\text {Tx}}}},{x_{{\text {Rx}}}},{y_{{\text {Tx}}}},{y_{{\text {Rx}}}}} \right) = \Theta \left ({\alpha \left  {\begin {array}{*{20}{c}} {f\left ({\alpha,\min \left ({{x_{{\text {Tx}}}},{x_{{\text {Rx}}}}} \right)} \right)}&{g\left ({\alpha,\max \left ({{y_{{\text {Tx}}}},{y_{{\text {Rx}}}}} \right)} \right)} \\ {f\left ({\alpha,\max \left ({{x_{{\text {Tx}}}},{x_{{\text {Rx}}}}} \right)} \right)}&{g\left ({\alpha,\min \left ({{y_{{\text {Tx}}}},{y_{{\text {Rx}}}}} \right)} \right)} \end {array}} \right.} \right) \times \bar H\left (m \right)\) 
Proof
See Appendix 3. □
Theorem 2
Proof
See Appendix 4. □
Remark 1
Theorems 1 and 2 are two analytical formulations of the same event. In the sequel, we show that they coincide. □
5.3 Scenario II: Reflection probability in the absence of reconfigurable metasurfaces
The probability of occurrence of Event 3 is not easy to compute. The reason is that Event 3 is formulated in terms of the intersection of Events 1 and 2, which are not independent. In order to avoid the analytical complexity that originates from the correlation between Events 1 and 2, we propose a upperbound to compute the probability of occurrence of Event 3. Before stating the main result, we introduce the following proposition that provides one with the probability of occurrence of Event 2.
Proposition 1
Auxiliary functions used in Proposition 1
Function definition 
\(F\left ({\alpha,t} \right) = \frac {1}{{{R_{{\text {net}}}}}}\left ({t + \frac {{{z_{p}}\sin \alpha }}{{{m_{p}}\sin \alpha + \cos \alpha }}} \right){\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }}  \cos \alpha } \right)^{ 1}}\) 
\(G\left ({\alpha,v} \right) = \frac {1}{{{R_{{\text {net}}}}}}\left ({v + \frac {{{m_{p}}{z_{p}}\sin \alpha }}{{{m_{p}}\sin \alpha + \cos \alpha }}  {z_{p}}} \right){\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }}  \sin \alpha } \right)^{ 1}}\) 
\(\Gamma _{1}^{a}\left (\alpha \right) = \Theta \left ({\alpha \left  {\begin {array}{*{20}{c}} {F\left ({\alpha,  \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,  \frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }}  \cos \alpha } \right)H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }}  \sin \alpha } \right)\) 
\(\Gamma _{1}^{b}\left (\alpha \right) = \Theta \left ({\alpha \left  {\begin {array}{*{20}{c}} {F\left ({\alpha,  \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,  \frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }}  \cos \alpha } \right)\bar H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }}  \sin \alpha } \right)\) 
\(\Gamma _{1}^{c}\left (\alpha \right) = \Theta \left ({\alpha \left  {\begin {array}{*{20}{c}} {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha,  \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,  \frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)\bar H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }}  \cos \alpha } \right)H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }}  \sin \alpha } \right)\) 
\(\Gamma _{1}^{d}\left (\alpha \right) = \Theta \left ({\alpha \left  {\begin {array}{*{20}{c}} {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,  \frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha,  \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)\bar H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }}  \cos \alpha } \right)\bar H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }}  \sin \alpha } \right)\) 
\(\Gamma _{2}^{a}\left (\alpha \right) = \Theta \left ({\alpha \left  {\begin {array}{*{20}{c}} {F\left ({\alpha,  \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,  \frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }}  \cos \alpha } \right)H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }}  \sin \alpha } \right)\) 
\(\Gamma _{2}^{b}\left (\alpha \right) = \Theta \left ({\alpha \left  {\begin {array}{*{20}{c}} {F\left ({\alpha,  \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,  \frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }}  \cos \alpha } \right)\bar H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }}  \sin \alpha } \right)\) 
\(\Gamma _{2}^{c}\left (\alpha \right) = \Theta \left ({\alpha \left  {\begin {array}{*{20}{c}} {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,  \frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha,  \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)\bar H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }}  \cos \alpha } \right)H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }}  \sin \alpha } \right)\) 
\(\Gamma _{2}^{d}\left (\alpha \right) = \Theta \left ({\alpha \left  {\begin {array}{*{20}{c}} {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha,  \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,  \frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)\bar H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }}  \cos \alpha } \right)\bar H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }}  \sin \alpha } \right)\) 
\(\Gamma _{3}^{a}\left (\alpha \right) = \Theta \left ({\alpha \left  {\begin {array}{*{20}{c}} {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha,  \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,  \frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }}  \cos \alpha } \right)H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }}  \sin \alpha } \right)\) 
\(\Gamma _{3}^{b}\left (\alpha \right) = \Theta \left ({\alpha \left  {\begin {array}{*{20}{c}} {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,  \frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha,  \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }}  \cos \alpha } \right)\bar H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }}  \sin \alpha } \right)\) 
\(\Gamma _{3}^{c}\left (\alpha \right) = \Theta \left ({\alpha \left  {\begin {array}{*{20}{c}} {F\left ({\alpha,  \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,  \frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)\bar H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }}  \cos \alpha } \right)H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }}  \sin \alpha } \right)\) 
\(\Gamma _{3}^{d}\left (\alpha \right) = \Theta \left ({\alpha \left  {\begin {array}{*{20}{c}} {F\left ({\alpha,  \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,  \frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)\bar H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }}  \cos \alpha } \right)\bar H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }}  \sin \alpha } \right)\) 
\(\Gamma _{4}^{a}\left (\alpha \right) = \Theta \left ({\alpha \left  {\begin {array}{*{20}{c}} {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,  \frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha,  \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }}  \cos \alpha } \right)H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }}  \sin \alpha } \right)\) 
\(\Gamma _{4}^{b}\left (\alpha \right) = \Theta \left ({\alpha \left  {\begin {array}{*{20}{c}} {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha,  \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,  \frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }}  \cos \alpha } \right)\bar H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }}  \sin \alpha } \right)\) 
\(\Gamma _{4}^{c}\left (\alpha \right) = \Theta \left ({\alpha \left  {\begin {array}{*{20}{c}} {F\left ({\alpha,  \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,  \frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)\bar H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }}  \cos \alpha } \right)H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }}  \sin \alpha } \right)\) 
\(\Gamma _{4}^{d}\left (\alpha \right) = \Theta \left ({\alpha \left  {\begin {array}{*{20}{c}} {F\left ({\alpha,  \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,  \frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)\bar H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }}  \cos \alpha } \right)\bar H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }}  \sin \alpha } \right)\) 
Proof
See Appendix 5. □
Theorem 3
where Pr{Event 1} is formulated in Theorem 1 or Theorem 2, and Pr{Event 2} is given in Proposition 2.
Proof
:The proof follows by applying the Frechet inequality [33]. □
Remark 2
By comparing Theorems 1 and 2 against Theorem 3, we observe that the probability of being a reflector highly depends on the length of the typical object if is it not coated with a reconfigurable metasurfaces, while it is independent of it if it is coated with a reconfigurable metasurface. This is a major benefit of using reconfigurable metasurfaces in wireless networks. This outcome is determined by the assumption that the metasurfaces can modify the angle of reflection regardless of their length. The analysis of the impact of the constraints imposed by the size of the metasurface on its capability of obtaining a given set of angles of reflection as a function of the angle of incidence is an open but very important research issue, which is left to future research. □
6 Numerical results and discussion: validation against Monte Carlo simulations
The aim of this section is to validate the analytical frameworks developed in the previous sections against Monte Carlo simulations and to study the potential of using reconfigurable metasurfaces in wireless networks. The results are illustrated either as a function of the length, L, of the typical object or as a function of the locations of the transmitter and receiver. The simulation setup is detailed in the caption of each figure.
7 Conclusion and discussion
In this paper, we have proposed the first analytical approach that provides one with the probability that a random object coated with reconfigurable metasurfaces acts as a reflector, and have compared it against the conventional setup in which the object is not coated with reconfigurable metasurfaces. This result has been obtained by modeling the environmental objects with a modified random line process with fixed length, and random orientations and locations. Our proposed analytical approach allows us to prove that the probability that an object is a reflector does not depend on the length of the object if it is coated with metasurfaces, while it strongly depends on it if the Snell’s law of reflection needs to be applied. The reason of this major difference in system performance lies in the fact that the angles of incidence and reflection need to be the same according to the Snell’s law of reflection.
In spite of the novelty and contribution of the present paper, it constitutes only a first attempt to quantify the potential of reconfigurable metasurfaces in largescale wireless networks, and to develop a general analytical approach for understanding the ultimate performance limits, and to identify design guidelines for system optimization. For example, the performance trends are based on the assumption that, for any angle of incidence, an arbitrary angle of reflection can be synthetized. Due to practical constraints on implementing metasurfaces, only a finite subset of angles may be allowed, which needs to account for the concept of fieldofview of the metasurfaces. Also, the analytical models and the simulation results have been obtained by using ray tracing assumptions, and ignore, e.g., the radiation pattern of the metasurfaces, and nearfield effects. A major step is needed to obtain tractable analytical expressions of relevant performance metrics that are suitable to unveil scaling laws, are amenable for optimization, and account for different functions applied by the metasurfaces (not just reflections).
8 Appendix 1: Proof of Lemma 3
where their slopes m and m_{p} are assumed to be nonzero.
which implies that the following identity need to be fulfilled mm_{p}=−1.
This concludes the proof.
9 Appendix 2: Proof of Lemma 5

If the point A is above the line, then it must be y_{1}−y>0. From Eq. (23), we evince that y_{1}−y>0 if \(\frac {{a{x_{1}} + b{y_{1}} + c}}{b} > 0\). This, in turn, corresponds to the following: i) either ax_{1}+by_{1}+c>0 and b>0 or ii) ax_{1}+by_{1}+c<0 and b<0.

If the point A is below the line, then it must be y_{1}−y<0. From Eq. (23), we evince that y_{1}−y<0 if \(\frac {{a{x_{1}} + b{y_{1}} + c}}{b} < 0\). This, in turn, corresponds to the following: i) either ax_{1}+by_{1}+c<0 and b>0 or ii) ax_{1}+by_{1}+c>0 and b<0.
This concludes the proof.
10 Appendix 3: Proof of Theorem 1
Based on the sign of m sinα+ cosα and m, four cases can be identified.
10.1 Case 1
where (a) follows from \(p = {R_{{\text {net}}}}\sqrt u = {R_{{\text {net}}}}\upsilon \) and (b) follows by computing the integral with respect to υ, whose probability density function is f_{υ}(υ)=2υ, since u is uniformly distributed in [0,1].
The integration limits in (b) are determined from the conditions \(m\sin \alpha + \cos \alpha \geqslant 0\), which implies 0≤α≤δ_{1} and δ_{2}≤α≤2π, where \({\delta _{1}} = 2{\tan ^{ 1}}\left ({m + \sqrt {1 + {m^{2}}}} \right)\) and \({\delta _{2}} = 2\pi + 2{\tan ^{ 1}}\left ({m  \sqrt {1 + {m^{2}}}} \right)\).
The other three case studies can be obtained by using the same approach as for Case 1. Thus, the details are omitted and only the final result is reported.
10.2 Case 2
10.3 Case 3
10.4 Case 4
This concludes the proof.
11 Appendix 4: Proof of Theorem 2

Case 1: The location of the transmitter (x_{Tx},y_{Tx}) is above the line x cosα+y sinα−p=0 given sinα>0.

Case 2: The location of the transmitter (x_{Tx},y_{Tx}) is above the line x cosα+y sinα−p=0 given sinα<0.

Case 3: The location of the receiver (x_{Rx},y_{Rx}) is above the line x cosα+y sinα−p=0 given sinα>0.

Case 4: The location of the receiver (x_{Rx},y_{Rx}) is above the line x cosα+y sinα−p=0 given sinα<0.

Case 5: The location of the transmitter (x_{Tx},y_{Tx}) is below the line x cosα+y sinα−p=0 given sinα>0.

Case 6: The location of the transmitter (x_{Tx},y_{Tx}) is below the line x cosα+y sinα−p=0 given sinα<0.

Case 7: The location of the receiver (x_{Rx},y_{Rx}) is below the line x cosα+y sinα−p=0 given sinα>0.

Case 8: The location of the receiver (x_{Rx},y_{Rx}) is below the line x cosα+y sinα−p=0 given sinα<0.
where (a) follows from \(p = {R_{{\text {net}}}}\sqrt u = {R_{{\text {net}}}}\upsilon \) and (b) follows by solving the integral with respect to υ whose probability density function is f_{υ}(υ)=2υ, since u is uniformly distributed in [0,1].
This concludes the proof.
12 Appendix 5: Proof of Proposition 1
In order to compute this probability, we need to examine four cases depending on the relationship between x_{end1} and x_{end2}, as well as y_{end1} and y_{end2}.
12.1 Case 1
Depending on the sign of \(\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }}  \cos \alpha } \right)\) and \(\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }}  \sin \alpha } \right)\), four subcases need to be studied.
12.1.1 Case 1a
where (a) follows from \(p = {R_{{\text {net}}}}\sqrt u = {R_{{\text {net}}}}\upsilon \) and (b) follows by solving the integral with respect to υ whose probability density function is f_{υ}(υ)=2υ, since u is uniformly distributed in [0,1].
By using a similar approach, we can study the remaining three subcases.
12.1.2 Case 1b
12.1.3 Case 1c
12.1.4 Case 1d
By using a similar line of thought, the remaining three cases can be studied. The final result is reported in the following sections.
12.2 Case 2
12.3 Case 3
12.4 Case 4
This concludes the proof.
Declarations
Funding
Not applicable.
Availability of data and materials
Data sharing is not applicable to this article as no datasets were generated or analysed during the current study. The paper is built upon mathematical analysis.
Publishers Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Authors’ contributions
The authors declare that they have equally contributed to the paper. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 P. Hu, P. Zhang, M. Rostami, D. Ganesan. Braidio: an integrated activepassive radio for mobile devices with asymmetric energy budgets (ACM SIGCOMM (FlorianopolisBrazil, 2016).Google Scholar
 C. Liaskos, S. Nie, A. Tsioliaridou, et al., in 2018 IEEE 19th International Symposium on “A World of Wireless, Mobile and Multimedia Networks” (WoWMoM). Realizing wireless communication through softwaredefined hypersurface environments (IEEE, 2018), pp. 14–15.Google Scholar
 5GPPP, Vision on Software Networks and 5G SN WG, (2017).Google Scholar
 C. Liaskos, S. Nie, A. Tsioliaridou, A. Pitsillides, S. Ioannidis, I. F. Akyildiz, A new wireless communication paradigm through softwarecontrolled metasurfaces. IEEE Commun. Mag.56(9), 162–169 (2018).View ArticleGoogle Scholar
 L. Subrt, P. Pechac, in 2012 6th European Conference on Antennas and Propagation (EUCAP). Controlling propagation environments using intelligent walls (IEEE, 2012), pp. 1–5.Google Scholar
 L. Subrt, P. Pechac, Intelligent walls as autonomous parts of smart indoor environments. IET Commun.6(8), 1004–1010 (2012).View ArticleGoogle Scholar
 X. Tan, Z. Sun, J. M. Jornet, et al., in 2016 IEEE International Conference on Communications (ICC). Increasing indoor spectrum sharing capacity using smart reflectarray (IEEE, 2016), pp. 1–6.Google Scholar
 O. Abari, D. Bharadia, A. Duffield, et al., in 14th USENIX Symposium on Networked Systems Design and Implementation (NSDI 17). Enabling highquality untethered virtual reality, (2017), pp. 531–544.Google Scholar
 A. Welkie, L. Shangguan, J. Gummeson, et al., in Proceedings of the 16th ACM Workshop on Hot Topics in Networks. Programmable radio environments for smart spaces (ACM, 2017), pp. 36–42.Google Scholar
 R. Chandra, K. Winstein, in ACM workshop on hot topics in networks. Programmable Radio Environments for Smart Spaces  HotNetsXVI Dialogue (Palo Alto, 2017).Google Scholar
 S. Hu, F. Rusek, O. Edfors, Beyond Massive MIMO: The potential of data transmission with large intelligent surfaces. IEEE Trans. Sig. Process.66(10), 2746–2758 (2018).MathSciNetView ArticleGoogle Scholar
 X. Tan, Z. Sun, D. Koutsonikolas, et al., in IEEE INFOCOM 2018IEEE Conference on Computer Communications. Enabling indoor mobile millimeterwave networks based on smart reflectarrays (IEEE, 2018), pp. 270–278.Google Scholar
 C. Liaskos, A. Tsioliaridou, A. Pitsillides, S. Ioannidis, I. F. Akyildiz, Using any surface to realize a new paradigm for wireless communications. Commun. ACM. 61(11), 30–33 (2018).View ArticleGoogle Scholar
 C. Liaskos, A. Tsioliaridou, S. Ioannidis, in 2018 IEEE 23rd International Workshop on Computer Aided Modeling and Design of Communication Links and Networks (CAMAD). Towards a Circular Economy via Intelligent Metamaterials (IEEE, 2018), pp. 1–6.Google Scholar
 N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, Z. Gaburro, Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction. Science. 334(6054), 333–337 (2011).View ArticleGoogle Scholar
 C. L. Holloway, E. F. Kuester, J. A. Gordon, J. O’Hara, J. Booth, D. R. Smith, An overview of the theory and applications of metasurfaces: the twodimensional equivalents of metamaterials. IEEE Antennas Propag. Mag.54(2), 10–35 (2012).View ArticleGoogle Scholar
 L. La Spada, Metamaterials for Advanced Sensing Platforms. Res. J. Opt. Photonics. 7:, 14–16 (2017).View ArticleGoogle Scholar
 T. Nakanishi, T. Otani, Y. Tamayama, et al., Storage of electromagnetic waves in a metamaterial that mimics electromagnetically induced transparency. Phys. Rev. B.87(16), 161110–1–1611104 (2013).View ArticleGoogle Scholar
 A. Silva, F. Monticone, G. Castaldi, et al., Performing mathematical operations with metamaterials. Science. 343(6167), 160–163 (2014).MathSciNetView ArticleGoogle Scholar
 H, 2020 VISORSURF project, A hardware platform for softwaredriven functional metasurfaces. http://www.visorsurf.eu/.
 T. Bai, R. Vaze, R. W. Heath Jr, Analysis of blockage effects on urban cellular networks. IEEE Trans. Wirel. Commun.13(9), 5070–5083 (2014).View ArticleGoogle Scholar
 J. Lee, F. Baccelli, in IEEE International Conference on Computer Communications. On the Effect of Shadowing Correlation on Wireless Network Performance (Honolulu, 2018).Google Scholar
 J. G. Andrews, F. Baccelli, R. K. Ganti, A tractable approach to coverage and rate in cellular networks. IEEE Trans. Commun.59(11), 3122–3134 (2011).View ArticleGoogle Scholar
 M. Di Renzo, A. Guidotti, G. E. Corazza, Average rate of downlink heterogeneous cellular networks over generalized fading channels – a stochastic geometry approach. IEEE Trans. Commun.61(7), 3050–3071 (2013).View ArticleGoogle Scholar
 M. Di Renzo, Stochastic geometry modeling analysis of multitier millimeter wave cellular networks. IEEE Trans. Wirel. Commun.14(9), 5038—5057 (2015).View ArticleGoogle Scholar
 W. Lu, M. Di Renzo, in Proceedings of the 18th ACM International Conference on Modeling, Analysis and Simulation of Wireless and Mobile Systems. Stochastic geometry modeling of cellular networks: Analysis, simulation and experimental validation (ACM, 2015), pp. 179–188.Google Scholar
 M. Di Renzo, W. Lu, P. Guan, The intensity matching approach: a tractable stochastic geometry approximation to systemlevel analysis of cellular networks. IEEE Trans. Wirel. Commun.15(9), 5963—5983 (2016).View ArticleGoogle Scholar
 M. Di Renzo, A. Zappone, T. T. Lam, M. Debbah, Systemlevel modeling and optimization of the energy efficiency in cellular networks  a stochastic geometry framework. IEEE Trans. Wirel. Commun.17(4), 2539—2556 (2018).View ArticleGoogle Scholar
 M. Di Renzo, S. Wang, X. Xi, Modeling and analysis of cellular networks by using inhomogeneous poisson point processes. IEEE Trans. Wirel. Commun.17(8), 5162–5182 (2018).View ArticleGoogle Scholar
 M. Di Renzo, T. T. Lam, A. Zappone, M. Debbah, A tractable closedform expression of the coverage probability in poisson cellular networks. IEEE Wirel. Commun. Lett.8(1), 249–252 (2018). https://doi.org/10.1109/LWC.2018.2868753.View ArticleGoogle Scholar
 M. Di Renzo, A. Zappone, T. T. Lam, M. Debbah, 8. Spectralenergy efficiency Pareto front in cellular networks: a stochastic geometry framework, (2018), pp. 424–427. https://doi.org/10.1109/LWC.2018.2874642.
 A. Narayanan, S. V. Sreejith, R. K. Ganti, in GLOBECOM 20172017 IEEE Global Communications Conference. Coverage Analysis in Millimeter Wave Cellular Networks with Reflections (IEEE, 2017), pp. 1–6.Google Scholar
 M. Di Renzo, W. Lu, Systemlevel analysis and optimization of cellular networks with simultaneous wireless information and power transfer: Stochastic geometry modeling. IEEE Trans. Veh. Technol.66(3), 2251–2275 (2017).View ArticleGoogle Scholar