dune motion , sand encroachment and desertification are based on sand transport by wind .saltation is the basic mechanism of aeolian sand flux  . the simplest and most common devices to measure it in the field are so called sand traps .they consist of cavities into which the sand bearing wind enters , drops the sand inside and then leaves again without the sand . in that way they accumulate inside the sand that would have crossed them during a given time , giving a measure for the flux through their cross section .the difficulty to quantitative gauge them arises in properly estimating this cross section because the trap being an extended fixed object modifies considerably the wind velocity field in its surrounding .the issue boils down to understand how much wind actually enters into the cavity and therefore strongly depends on the geometrical shape of the trap .  among the many different trap designs that have been used inthe past one of the simplest and most popular is a hollow cylinder with two slits , one open and the other covered by a membrane that is impermeable to the grains  see fig .[ fig_1 ] .it has been extensively implemented in field studies along the northern coast of brazil with much success .we will therefore study this particular device in detail by numerically solving the turbulent flow around an object having the corresponding geometry .we will calculate the trajectories of grains released at different positions in the area in front of the trap to assess if they are captured or not and do this for various reynolds and stokes numbers as well as different membrane permeabilities .this paper is organized as follows . in section , we introduce the model and briefly summarize the algorithm used in our simulation . in section we present the qualitative and quantitative results .we conclude in section by discussing the importance of these results in some practical applications .is the diameter and the height .the widths of the vertical slits at the front and back sides are and , respectively .two possible grain trajectories are sketched.,width=377,height=529 ]  as shown in fig .[ fig_2 ] the sand trap that we study consists of a hollow cylinder of diameter and height closed on the top and the bottom , and having two vertical slits , one at the front and another at the back side .for simplicity we will neglect the width of the cylinder walls .a membrane covers the slit of the back side in order to capture particles with diameters larger than .the slits at the front and back sides have widths and , respectively .we chose a height , a diameter and slit widths on the front side and on the back side in agreement with typical dimensions of real devices .the membrane has a thickness of .its permeability can be modified and will be one of the control parameters of our simulation . in order to simulate the conditions of a realistic wind field we consider a box of size with moving boundaries on top , left and right , i.e. in all directions transverse to the wind direction except for the bottom . on the fixed walls , i.e. the trap surface and the bottom we impose non - slip boundary conditions , which means that the velocity is equal to zero at the interfaces between solid and fluid . in order to reduce the size effectsthe simulation box was chosen to be one order of magnitude larger than diameter of the trap .we assume that atmospheric air is an incompressible and newtonian fluid having a viscosity of and a density of .the reynolds number is defined as , where is the average velocity and the linear size of the box in direction .this gives in our case meaning that we can assume to be in a fully developed homogeneous turbulent state .the standard model is therefore an adequate way to solve the corresponding reynolds - averaged navier - stokes equations of motion .their numerical solution can be achieved by discretizing the velocity and pressure fields and using a control volume finite - difference technique  .the convergence criteria of this scheme can be defined in terms of residuals , which measure up to which degree the conservation laws are satisfied . in our simulationwe consider that convergence is achieved when each of the normalized residuals is smaller than .a fully turbulent atmospheric boundary layer over a flat surface shows a logarithmic increase of the velocity with the distance from the surface .therefore , we impose a logarithmic velocity profile on the inlet , i.e. in the plane , of the form where denotes the shear velocity , the so called roughness length and is the von krmn constant  .we model the fabric covering the opening at the backside of the trap as a special type of boundary condition mimicking a porous medium where the velocity / pressure drop characteristics are known .if is the thickness of this membrane then the pressure drop is defined according to darcy s law as : where is the permeability of the medium and is the velocity normal to the membrane . in order to understand the effect of the membrane on the wind velocity field , we perform simulations using different membrane permeabilities , namely , , and .    for simplicity we will only consider spherical particles .we also assume that the density of flying particles is so low that we can neglect collisions between them .for the same reason we will also neglect the momentum loss exerted by the particles on the fluid . consequently the wind velocity and pressure fields can be calculated without knowing the particle positions and one can then obtain the trajectory of each particle by just integrating its equation of motion : where and are the mass and the velocity of a particle of diameter and is the total force acting on this particle . let us assume that drag and gravity are the only relevant forces and that the particles do not interact with each other .then the equation of motion for one particle can be rewritten as where is gravity acceleration and is a typical value for the density of sand particles .the term in eq .( [ eq_4 ] ) represents the drag force per unit particle - mass where here is the particle reynolds number and for the drag coefficient we use an empirical expression taken from ref .the trajectories of the particles are calculated by numerically integrating the equation of motion eq .( [ eq_4 ] ) .we characterize the effect of inertia on the air borne grains in the flow field through the dimensionless _ stokes _ number which is defined by . for , inertia will dominate and the particles will move along straight lines not following the fluid , i.e. they will move in the air ballistically . for particlesbehave as tracers and will perfectly follow the streamlines of the fluid . in our simulation we will vary the stokes number by changing the particle diameter keeping all the other parameters constant .if a particle hits the outer surface of the sand trap it will be reflected elastically .if it hits the inner part of the trap it will stop moving , i.e. is counted as captured .we first present a study of the influence of the permeability of the membrane on the fluid velocity field without the presence of particles . in order to understand essential features of the flow field around and inside the sand trap, we have performed extensive simulations by analyzing the flow for several values of permeability . to see the flow fieldwe focus our attention on horizontal cuts through the trap , at a different distances , and from the ground . and  .the wind velocity profile input comes from eq .( [ eq_1 ] ) .the colors indicate the magnitude of the velocity varying from blue ( low value ) to red ( high value ) .the picture correspond to permeabilities .,width=529 ]  fig .[ fig_3 ] and fig .[ fig_4 ] depict the velocity field around the sand trap for two different values of permeability , and , respectively . herewe focus our discussion on the velocity profile , in the horizontal cuts through the trap at different heights .we will first discuss the case of high permeability . in this casewe can see in fig .[ fig_3 ] that the wind velocity presents a parabolic profile in the channel inside the sand trap and also the existence of stagnation zones beside this channel .view of typical particles trajectories for three different stokes numbers ,  , and  .the membrane permeability is .the particles concentrate in the region of high velocities for all values of .,title="fig:",width=188 ]  view of typical particles trajectories for three different stokes numbers ,  , and  .the membrane permeability is .the particles concentrate in the region of high velocities for all values of .,title="fig:",width=188 ]  view of typical particles trajectories for three different stokes numbers ,  , and  .the membrane permeability is .the particles concentrate in the region of high velocities for all values of .,title="fig:",width=188 ]   surface , for three different stokes numbers ,  and correspond to , and , respectively .the value of the membrane permeability is .,title="fig:",width=188 ]  surface , for three different stokes numbers ,  and correspond to , and , respectively. the value of the membrane permeability is .,title="fig:",width=188 ]  surface , for three different stokes numbers ,  and correspond to , and , respectively. the value of the membrane permeability is .,title="fig:",width=188 ]  for low permeability , see fig .[ fig_4 ] , the velocity decreases along the center line to form a zone of low velocity close to the entrance of the sand trap while it increases around the trap .this increase of flow is due to the obstruction caused by the trap to the incompressible fluid . for larger heights , the zone of low velocity reaches almost the entire region inside of the sand trap . with a decreasing membrane permeability alsothe formation of shadow zones can be observed behind the sand trap .our central issue is the study of sand transport and in particular the effect of the membrane permeability on the sand flux .therefore we calculate particle trajectories by extensive simulations for different stokes numbers and membrane permeabilities . in order to quantify the capture processthe sand trap efficiency is defined as where is the total number of particles released during the interval and is the number of particles among these that have been captured .we will discuss in the following the efficiency as function of membrane permeability and stokes number .   of the sand trap as function of the stokes number .the symbols correspond to different values of permeabilities , namely , ( circles ) , ( squares ) , ( stars ) and ( triangles ) .,width=340 ]   and low stokes number at height .the distance is the maximum distance from the symmetry line for which the particles bend towards the entrance of the sand trap .the stagnation zones are shown in more detail in the inset revealing how the particle trajectories can bend towards the symmetry line of the sand trap.,width=453,height=604 ]   as function of the membrane permeability , for fixed stokes number . the doted line is included as guide to eye and corresponds to . , width=264 ]  initially the grains are placed at , a given height and randomly within the interval , . for a fixed value of and , we release up to particles to determine the efficiency of the sand trap .we show in figs .[ fig_5 ] and [ fig_6 ] the trajectories of particles for two different permeabilities , and , respectively , and different stokes numbers . for both figuresthe stokes numbers are from top to bottom : , and . in the case of high permeability , as shown in fig .[ fig_5 ] , the grains concentrate in the regions of large velocities . inside of the trapthe particle trajectories do spread apart more with decreasing stokes number .this effect is more pronounced at lower permeability . for high stokes numbers some particles collide with the trap surfacerevers their direction and leave the box simulation . for intermediate stokes numbersthese particles also collide but are then deviated by the wind flow .this effect is independent of the permeability .the dependence of the sand trap efficiency on the stokes number is shown in fig .[ fig_7 ] for different values of .we performed simulations for membrane permeabilities , , and at many different heights , with an inlet velocity given by eq . [ eq_1 ] .the value of was obtained from the average over different heights , where at each one we released particles .we see that presents two different regimes as function of the permeability .let us first discuss the case of low permeabilities . in the limit of small stokes numbersthe efficiency of the sand trap is small and remains essentially constant close to zero . since the particles can be considered as tracers that nearly exactly follow the streamlines of the flow , avoiding trapping because practically no streamline crosses the interior of the trap as consequence of the low membrane permeability .indeed we observe in these cases stagnation zones inside the sand trap .above the efficiency increases as function of the stokes number . for high values of the efficiency reaches a saturation value close to unity .in this limiting case the particles move ballistically towards the sand trap and those that have been released within the range are captured .  for high permeabilities , the efficiency presents an unexpected behavior in the region of low , as shown in fig . [ fig_7 ] . for large stokes numbers the efficiency remains constant until . below this valuesurprisingly we detect a small minimum followed by an increase to a value that can be above one .we can explain this behavior observing the particle trajectories in the region close to the entrance of the sand trap as shown in fig .[ fig_8 ] .  for small membrane permeabilitythe sand trap behaves like a solid cylinder . in this casea stagnation region appears in front of the trap around the symmetry line .as the permeability increases some air can cross the sand trap and two stagnation zones appear beside the entrance and outside of the trap .each of them has a separation point and all particle trajectories between these two separation points bend towards the center and enter the trap as confirmed in the simulation for low stokes number as shown in fig .[ fig_8 ] . therefore the number of captured particles and the efficiency of the sand trap can increase beyond the ballistic case .  for a fixed value of and different permeabilities ,we search the maximum distance from the symmetry line at which the particles bend towards the entrance of the sand trap . this can be used as a measure for the increase in the sand trap efficiency . in fig .[ fig_9 ] , we show in a semi - log plot the efficiency , which has been calculated from the parameter , against the membrane permeability . clearly , the curves display a strong change starting at the permeability .this is the smallest permeability which still affects the efficiency of the sand trap .in this paper , we have studied numerically the behavior of a sand trap frequently used to measure aeolian sand transport in the field .we solved the turbulent wind velocity field in the presence of the sand trap and investigated the effect of the membrane permeability .we studied quantitatively the particle trajectories carried by the fluid for different membrane permeabilities .we have shown how the efficiency of the sand trap depends on the stokes number and gave some insight about the effect of the membrane permeability on the capturing process .  as previously observedthe membrane permeability strongly influences the capture process at low stokes numbers . the sand trap efficiency exhibits a surprising increase above one , for large permeabilities and small stokes numbers .let us point out that only at higher values , when the trajectories of the particles are completely ballistic , the efficiency becomes independent on the membrane permeability .this is a useful result since for natural sand , for example on dunes , the grain size is around which corresponds to a rather high stokes number .our results therefore confirm that this type of sand trap is adequate to measure aeolian sand flux .the area of the slit at the entrance essentially equals the cross section over which the sand flux is measured independently on the details of the membrane on the back side .we would like to thank andr moreira for helpful discussions .we acknowledge cnpq , capes , finep and funcap / cnpq / ppp and the max planck prize for financial support .procesos costeros y balance sedimentario a lo largo de fortaleza ( ne brazil ) : implicaciones para una gestin adecuada de la zona litoral _ ( phd thesis , faculty of geology , university of barcelona ) ( 1998 ) .

Abstract: sand traps are used to measure aeolian flux . since they modify the surrounding wind velocity field their gauging represents an important challenge . we use numerical simulations under the assumption of homogeneous turbulence based on fluent to systematically study the flow field and trapping efficiency of one of the most common devices based on a hollow cylinder with two slits . in particular , we investigate the dependence on the wind speed , the stokes number , the permeability of the membrane on the slit and the saltation height .


relay - aided transmissions promise significant gains in both coverage extension and performance enhancement .relay - based architecture for cellular network has been considered for next generation wireless systems , such as the lte - a , and is envisioned as part of future heterogeneous networks , along with macro- , pico- , and femtocells .picocells are operator - deployed and help alleviate coverage dead zones and traffic hot zones , while femtocells are usually consumer - deployed in an unplanned manner to increase in - door coverage .both are connected to their own wired backhaul connection , in contrary to relays .the use of low power , low complexity relays in lieu of additional wired connected stations avoids expensive backhaul links .relay nodes are deployed with a more traditional rf planning , with site deployment and maintenance by the operator , thus allowing optimization of their placement within the cell . at the same time , as a result of the tremendous increase of subscribers , jointly with the ever - growing demand for application requirements , energy efficiency became a great challenge for future wireless systems .however , the energy impact of the relay location and of the relay coding scheme remains under - explored in the literature .resource allocation and relaying strategies in cooperative systems have mostly targeted improving the cell capacity or spectral efficiency .a rich literature has investigated relay placement with this perspective .for example , authors of analyze different relaying strategies , amplify - forward , decode - forward and compress - forward , and compare their spectral efficiency .however , the quality of service ( qos ) of value - added mobile applications , such as on - demand video , real - time games or video - conferences , is mostly expressed in terms of a minimum rate to be maintained , along with delay and jitter requirements .this perspective has spurred new analysis and designs for cellular networks .for example , in , relay placement is optimized to enhance capacity while meeting a traffic demand . with another perspective ,references address coverage extension , given a user rate .analysis has also focused on improving cell topology , such as which proposes an interesting novel dual - relay architecture .since user applications mostly run on power - limited devices , reducing the energy consumption , while meeting a rate requirement , has in particular attracted significant attention .energy minimization for relay - aided cellular networks has ignited new techniques such as relay - based handover strategies , sleep mode operations , improved resource management and relay - aided load balancing .nevertheless , little can be found on the specific issue of energy - efficient relay placement and on the performance gains obtained by using advanced coding schemes rather than simpler schemes .moreover , extending cell coverage implies serving far users and consequently , consuming more energy . this trade - offhas neither been investigated .another motivation also brought us to analyze energy - efficient relay placement .although network simulations offer wide possibilities and numerous variable factors for refined investigation , they are often time consuming and may confine the obtained results only to the considered settings .they provide neither generalization to other settings nor performance limits like an analysis based on capacity bounds .thus , a model is needed to give insights of the relay utilization and helps set basis for future investigations and extensive simulations .models for heterogeneous networks has attracted significant attention in recent years and various advanced models have been proposed , such as in for femtocell networks .such models however , are not easy to apply to relay - aided networks , where a cell has to manage both direct and two - hop transmissions and where physical - layer considerations , such as the coding scheme , need to be included .current works using models for relaying are mostly based on simplified patterns composed of inner and outer circular regions for direct and relayed transmissions , such as in .in such a model , the radius of the inner region corresponds to the base - station - to - relay distance , which is neither coverage- nor energy - optimized .  finally ,though relay placement has been addressed in the aforementioned works , none of them allows the characterization of the impact of relay placement based on accurate path - loss models .first , geometrical models for relay positioning are incomplete . in ,the considered model is linear , and the relay is located on a straight line between the source and the destination .this model can not provide accurate solutions for actual relay placement in a network where the user is mobile , while the base station and relay are fixed . in ,two - dimensional models are considered , but the relay height is not taken into account , despite its impact on both path - loss and line - of - sight ( los ) conditions .three - dimensional models are not fully investigated and relay height has been considered only in .second , a majority of results on relay placement , such as , are for awgn channels with path - loss exponent , which may be oversimplified as emphasized in , or for fading channels with the same path - loss model for all links in the network .more accurate path - loss channel models , as in , should be considered .we introduce in this paper the notion of relay efficiency area and propose a geometrical model for energy - efficient relay placement , for both uplink and downlink transmissions , which allows meaningful performance analyses without the need for excessive simulations .the proposed model consists of only a few characteristic distances , but accurately approaches simulations on the probability of relaying within 3% and the energy consumption within 1% .then , we build an analysis framework for the performance of relaying based on time - division full decode - forward and provide comparison with the comprehensive partial decode - forward scheme , which is studied with the perspective of rate maximization in and energy minimization in .in the first part , we consider as a baseline the setting with a fixed number of relays at regular locations in each cell and no inter - cell interference , leaving these factors to future extensions .this baseline model provides a tractable benchmark for performance comparison and analysis , and shows that the classical inner and outer regions as considered in do not reflect the performance gains obtained by relaying .  in the second part , using the proposed geometrical model ,we highlight relay configurations which are beneficial for coverage extension and for energy efficiency . given that a network designer is constrained by the urban topography and by administrative by - laws for station placement , flexibility in the relay locationis also a relevant performance criteria .we thus consider performance both at and nearby the optimal relay location .furthermore , we show that the optimal relay placements for coverage and energy efficiency are different .we thus analyze the trade - off between energy gain and coverage extension and highlight the deployment cost required to reduce the energy consumption .we finally propose a new definition for optimal relay placement regarding the average energy consumed per unit area .we define our 6-sector urban cell framework and channel path - loss model , in section [ sec : system_model ] , and the considered coding schemes in section [ sec : schemes ] .section [ sec : model ] introduces the characteristic distances that constitute the proposed geometrical model .these distances are derived for both full decode - forward and partial decode - forward in section [ sec : analysis_model ] .we then deduce the system performance in section [ sec : p_rtx_energy ] .simulation results are described in section [ sec : simulation ] .finally , section [ sec : conclusion ] concludes this paper ._ notation : _ stands for the base station , for the relay and for the user .calligraphic refers to the user rate .upper - case letters denote constant distances , such as , or , while lower - case letters stand for variable distances or angles , such as or .subscripts , and respectively refer to the link from user to base station ( direct link ) , user to relay station and relay to base station . is used for probabilities and for expectation .finally , dtx refers to direct transmission , while rtx stands for relay - aided transmission , in the considered coding schemes .in this section , we present the cell and path - loss models . in particular ,we analyze two different path - loss models , depending on the relay being below or above the rooftop .     as presented in and depicted in figure [ fig : cell_model](a ), we consider a 6-sector hexagonal cell .note that , however , the analysis presented in this paper can be adapted to other cell models , such as circular shape or varying number of sectors . refers to the maximum radius , which includes the coverage extension provided by the relay station .the base station is located at the center of the cell and each sector is provided with a relay station , located at a distance from the base station .the user is positioned at a distance from the base station , with an angle , as illustrated in figure [ fig : cell_model](b ) . refers to the distance from the user to relay .we consider a three - dimensional geometrical model .we analyze the impact of the relay height on the system performance .we denote , and the heights of the base station , the relay station and the user respectively .the multiple access strategy allows orthogonality between users , such that only one user is served for a given time and frequency resource .we assume that the relay operates in the same frequency resource as the user it serves ( _ in - band _ relaying ) .we consider half - duplex relaying performed in time division .nevertheless , the following analysis and results are valid with other multiplexing schemes . without loss of generality, we describe the channel model for uplink transmissions . here , a transmission of unitary length is carried out in two phases of equal duration .we denote the transmitted codewords and for the user at each phase respectively , and for the relay . and respectively are the received signals at the relay and destination at the end of phase 1 , is the received signal at the destination at the end of phase 2 .thus , the half - duplex relay channel is written as follows : where , and are independent additive white gaussian noises ( awgn ) with equal variance .we respectively denote , and the channel from user to base station ( direct link ) , from user to relay and from relay to base station . for the channel coefficients , we use the path - loss models as proposed in the winner ii model : = & a_{i } \log_{10}(d ) + b_{i } + c_{i } \log_{10 } \left(\frac{f_c}{5 } \right )  \\ &  + d_{i } \log_{10}\left((\mathsf{h}_\text{tx}-1)(\mathsf{h}_\text{rx}-1)\right )  \end{split } \label{eq : pathloss}\ ] ] where and are the respective heights ( in meters ) of transmitter and receiver , is the distance ( in meters ) between them and the carrier frequency ( in ghz ) . the parameters , , and are constants dependent on the global location of the transmitter and receiver ( street level , rooftop ... ) . for convenience, we will use the following notation   since the relay height is allowed to vary from below to above rooftop , the propagation environment for and changes notably .however , the winner ii project does not provide a continuous path - loss model as a function of the relay height .we will therefore consider the two following situations : 1/ _ vicinity relay : _ the relay station is located below rooftop , implying that the user - to - relay link is generally strong but at the cost of a weaker relay - to - base station link ; 2/ _ base - station - like relay : _ the relay station is located above the rooftop , like a base station , the link is very strong and can be assumes as free space ( fs ) .nevertheless , the user experiences the same path - loss model to access the relay as the base station .we refer the reader to appendix [ app : pathloss ] for further details on the path - loss model .note that this paper only considers path - loss , and not fading .the analysis and the proposed model can thus be understood as the mean coverage and relaying area over a sufficient period of time such that fading is averaged out .  in this channel model , we assume reasonably low inter - cell interference , as a starting point to create a baseline model .indeed , techniques such as fractional frequency reuse ( ffr ) , scheduling based on inter - base - station coordination and directional antennas between the relay and the base station , have been spurred for next cellular ofdma systems to suppress or avoid interference from neighbouring cells .we present the three reference coding schemes for half - duplex time - division relaying which are used for performance analysis . once again , and without loss of generality , we describe the schemes for uplink transmissions .we consider gaussian signaling rather than practical modulations and coding .gaussian signaling is not only suitable for theoretical investigation ( by providing upper bound on practical system performance ) , but is also appropriated for analysis of practical systems . indeed , practical systems such as lte are ofdm - based and data is transmitted over numerous parallel frequency channels such that the output is well approximated by a complex gaussian distribution , as developed in .  for each coding scheme , the power allocation minimizes the energy consumption while maintaining a given user rate .the user and the relay have individual power constraints over the two transmission phases within the same bandwidth , respectively denoted as and .the power constraint of the base station is set to . at each phase, each node allocates to each transmitted codeword a portion of its available power .we consider _ in - band _ relaying , where the relay and the user operate in the same frequency band .   as the first reference, we consider direct transmissions from user to base station . for fair comparison, direct transmission scheme is performed over the same duration as the relaying scheme , and not the half duration as generally done in the literature . by doing this ,relaying does not create excess delay , nor uses more resource ( time and/or frequency ) than direct transmissions .the user rate is feasible as long as it is below the channel capacity , i.e. the minimum required energy is equal to .outage occurs when the user power constraint is not satisfied .   for relay - aided uplink transmissions ,two schemes are considered : full - decode forward and partial decode - forward . for each, we use the power allocation that minimizes total energy consumption , i.e. consumption of the user and the relay station together .we furthermore assume that the user chooses to use either direct transmission ( dtx ) or relayed transmission ( rtx ) depending on which one consumes less energy .   during phase 1, the user sends its message to the base station with rate and power .then , the relay decodes the message , re - encodes it with rate and forwards it to the destination during phase 2 with power .  in the literature ,two main decoding techniques are generally considered i ) _ two - hop relaying _ , as used in practical system , in which the base station only considers the signal received from the relay and thus the relay is merely a repeater ; ii ) _ repetition - coded full decode - forward _ , as defined in , in which the user repeats its message in phase 2 .then , the base station uses mrc and combines the signals received from both the user and relay .rate bounds and energy consumption of both schemes can be similarly written using a parameter , with for two - hop relaying and for repetition - coded full decode - forward , as follows : where the two lines describe the decoding constraints at the relay and the base station respectively . the power allocation set satisfies the following constraints : the allocation set which minimizes the energy consumption is   however , we can show that , in an urban cellular network considering path - loss only , combining the signals received from both the user and relay station as done in repetition - coded full decode - forward only brings little energy gain compared to two - hop relaying .this small gain is present only when the direct link is strong but , in this case , direct transmission is most of the time more energy - efficient than relaying .subsequently , we will consider two - hop relaying and repetition - coded full decode - forward as two variants of the same reference scheme , denoted as full decode - forward ( full - df ) .   as the second relay - aided scheme , we consider the partial decode - forward scheme optimized for energy .this scheme was proposed in our previous work , where it is referred as g - ee . in this paper , however , we will denote this scheme as eo - pdf , for energy - optimized partial decode - forward , for the consistency of notation . in the eo - pdf scheme ,only part of the initial message is relayed , the rest being sent via the direct link .the scheme is based on rate splitting and superposition coding at transmitter , and joint decoding at receivers .it uses beamforming between the relay and the user in the second transmission phase .this scheme arguably requires complex implementation and fine synchronization , however , from the aspect of information theory , it is both energy- and rate - optimal as discussed in , and thus , gives the theoretical upper - bound of the performance that can be achieved with decode - forward based relaying .the optimal rate and power allocation of this coding scheme depends on the desired user rate and is composed of three sub - schemes as follows : + 1 ) if the desired user rate is below a certain threshold , + in phase 1 , the user sends with power and the relay decodes as .+ in phase 2 , the relay sends with power and the user sends with power .+ the allocation minimizes the total energy consumption of the user and the relay .+ 2 ) if the desired user rate is above , the source splits its message into two parts and , of rates and respectively , with .+ in phase 1 , the source sends with power .the relay decodes as .+ in phase 2 , the relay sends with power and the source sends with power , using superposition coding .the base station jointly decodes and . + the power and rate allocation  minimizes the total energy consumption , with .+ 3 ) if this second sub - scheme is in outage due to very high user rate , the same coding scheme is used , i.e. rate splitting , superposition coding and joint decoding , but with other power allocation . here, minimizes the energy consumption of the relay only .we refer the reader to for the detailed power and rate allocation , as well as for the expression of the threshold .in this section , we define the relaying efficiency , both in terms of energy consumption and coverage extension .to do so , we introduce the notion of relay efficiency area ( rea ) , and build our geometrical model based on this rea for both uplink and downlink transmissions .note that we will consider cartesian coordinates and polar coordinates for the user position .as a first insight of the potential energy gain brought by relaying , we plot in figure [ fig : energy_gain ] the energy gain in percentage ( i.e. ) that is obtained by using either full decode - forward ( full - df ) or the energy - optimized partial decode - forward scheme ( eo - pdf ) over direct uplink transmission , as a function of the user location in the 6-sector cell . the gain is set to 100% when direct transmission is not feasible , i.e. when relaying enhances the cell coverage . as shown in the figure , relaying can significantly reduce the energy consumption .considering the cell - edge of conventional systems using direct transmission only , an average of 46% of energy gain is obtained with full decode - forward and 70% with eo - pdf .in addition to energy savings , relaying also increases the radius of the corresponding hexagonal cell by 33% for full decode - forward and by 56% for eo - pdf .figure [ fig : energy_gain ] shows that conventional relaying schemes , such as two - hop relaying or repetition - coded full decode - forward , can provide notable performance enhancement , especially in terms of coverage extension .nevertheless , those schemes are clearly suboptimal compared to the potential gains of decode - forward , upper - bounded by the eo - pdf scheme .it is then necessary to estimate the gains that implementing advanced coding schemes can bring over simpler schemes , both in terms of energy savings and coverage extension .furthermore , the efficiency of the relay obviously depends on its location in the cell , as well as its propagation environment .estimating the network performance can not be reasonably done by simulating all possible relay network configurations and user locations .thus , a model is needed to give insight of relay utilization and energy consumption .a relay - aided cell is characterized by the probability for a user to be served by the relay station and by the energy saved by using the relay .those two performance criteria determine if a relay location is beneficial , and thus , characterize relay efficiency . to investigate relay efficiency ,we introduce here the concept of _ relay efficiency area ( rea ) _ for a cellular network served by relay stations .it is defined as the set of all user locations for which relaying is more energy - efficient than dtx or for which dtx is not feasible .therefore , the rea covers both performance improvement and coverage extension .the simulated rea s of full - df and the eo - pdf scheme are illustrated in figure [ fig : energy_gain ] , and its related model in figure [ fig : rea_model ] . in figure[ fig : energy_gain ] , the rea corresponds to the cell area where the energy gain is strictly positive ( non - white area ) . when the user is close to the base station , the link is very strong and dtx consumes less energy than relaying .otherwise , relaying is more energy - efficient or allows successful transmission while dtx is in outage .we propose to characterize the rea of each coding scheme using four characteristic distances , as depicted in figure [ fig : rea_circle_model ] : and for the inner bound , and for the outer bound .the inner bound is modelled by combining the strait line of equation and a portion of the circle of radius and centred at the base station , which refers to the minimum cell radius over which direct transmission is infeasible given outage requirement .the outer bound is characterized by and .it is modelled by a portion of the circle of radius and centred at . the relay efficiency area is thus modelled as follows : where is the user location . stands for the portion of the disk centred at with radius and delimited by the sector bounds . stands for the exterior of this disk .note that this model for rea is independent of the user distribution and of the cell shape .these factors are only considered for the computation of the performance metrics , e.g. the coverage radius , the probability of relaying and the average energy consumption .  using this first characterization of rea, we derive the model for hexagonal cells and define the maximum cell radius that guarantees coverage for all users located inside the hexagonal cell without overlap , as depicted in figure [ fig : rea_hexagone_model ] .three characteristic distances are sufficient to define this hexagonal model for rea : , and , where this expression of insures no coverage hole at sector - edge , contrary to the coverage radius defined in . as we will show later , is function of the transmit power .the area of a cell sector can then be expressed as follows .  these characteristic distances , as defined above , allow the direct computation of the overall probability of relaying a communication , given that the user is randomly located in the cell , as well as the average consumed energy to successfully transmit data . in the following sections, we apply this model to compute these characteristic distances for each coding scheme and investigate the probability of relaying in the cell and the total average energy consumption .we now compute the four characteristic distances , , and for full - df and eo - pdf , as defined in section [ sec : schemes ] , and for both uplink and downlink transmissions .these characteristic distances are derived from the channel conditions for transmission feasibility and for energy efficiency .first , we focus on , the maximum radius over which dtx is infeasible .we have where and are given by the path - loss model and defined in eq . and , and where for uplink and for downlink .second , we define for the uplink .it refers to the relaying condition for which relaying can outperform dtx . at the limit , we get where is the distance between the relay and base station .given the cell sector bounds and regarding cartesian coordinates , we can show that the derivative of x with respect to y is small , such that eq .can be approximated by a straight line .thus , in the proposed model , we approximate the relaying condition by , where is defined as the smallest positive solution of in eq . with ,i.e. in particular , when and have the same path - loss exponent , as it is the case when both and are in los , we get : this relaying condition must be satisfied for both full - df and eo - pdf . in practice however , this condition is not limiting for full - df since , for this coding scheme , transmission is over two phases of half - duration and data is thus sent at twice the desired rate .actually , the user decides if relay - aid is necessary based on energy consumption , as we will see in next paragraph . on the contrary ,the relaying condition is relevant for the eo - pdf scheme since only part of the message is relayed , leading to sufficient energy gain to compensate for the shorter transmission duration .  with regardsto downlink , this condition becomes , which is always satisfied since the relay - to - base - station link is very strong .first , we focus on energy efficiency and we derive the conditions for which the energy consumption using dtx ( denoted ) is greater than the consumption using full decode - forward ( denoted ) .this energy - efficient condition can be seen as the dual of the throughput - oriented condition . by using eq . , we get  \\ \text{where } \quad r_s^2  = \mathsf{d}_r^2 & + r^2 - 2 \mathsf{d}_r r \cos(\theta ) .\nonumber\end{aligned}\ ] ]  note that eq .is symmetric in and , meaning that it is valid for both uplink and downlink .we can also show that a solution of eq .for is almost solution for .this implies that the bounds for two - hop relaying can approximate the ones of repetition - coded full df .simulations in section [ sec : simulation ] show that the rea of repetition - coded full df is only increased by less than 1% compared to two - hop relaying .the solution of eq . can be well approximated by the intersection of the half - plane defined by ( inner bound ) with the disk ( outer bound ) , where superscript stands for energy efficiency . and can be computed easily as follows .noting that there exists only one circle with its center on the x - axis and going through any two given points of the plane , using geometrical principles , it is therefore sufficient to find the two user locations for which with and to deduce and .we propose to reject all relay positions for which , i.e. for which is strictly included in , i.e. the area covered by the base station alone .consequently , is only used to determine the acceptance or rejection of a relay configuration , but does not appear in the expression of the rea itself , as defined in eq . .next , we focus on the outage condition . assuming the relay power is sufficient , outage occurs when the transmission between the relay and user is not feasible , i.e. when the channel is too weak , given the power constraint and user rate .note that direct transmission is not feasible either in this case .relayed transmission is in outage for all users positions outside the disk centred at the relay station and of radius : where stands for outage and where for uplink and for downlink .based on the previous analysis , we can deduce the characteristic distances for full decode - forward as follows : where is given by eq ., by eq . and by eq . .these values define the relay efficiency area of full decode - forward , as defined in eq . .note that rea of full - df regarding uplink is strictly included in the rea for downlink . , which indicates whether the user should use dtx or rtx is the same , but the coverage radius is extended for downlink since the relay station disposes of more power than the user .the computation of the characteristic distances for the eo - pdf scheme follows the same basis as for full decode - forward .we recall that the inner - bound is modelled by the combination of the strait line of equation and a portion of the circle centred at the base station and of radius . usually depends on both the relaying condition defined in eq . and on the energy - efficiency condition .  with regards to uplink ,simulations show that there can exist user locations for which direct transmissions is more energy - efficient than the eo - pdf scheme .indeed , in dtx , messages are sent once over the two phases of transmission , with rate , whereas in the eo - pdf scheme , messages are sent twice , over each phase , with rate .for some specific user locations , the energy gain obtained by eo - pdf does not compensate the loss due to transmitting at a rate of during each phase instead of during both phases , and thus , direct transmission becomes more energy - efficient .nevertheless , simulations show that these user locations are negligible for relay - aided cellular networks .first , they occur only for uncommon relay configurations , specifically when the relay is very close to base station and with very low user rates .moreover , the area of concern remains small compared to the cell size and the energy gain obtained by dtx over eo - pdf is admittedly positive but almost zero .for these reasons , we can consider that reduces to the relaying condition , as described in eq ., and is not limited by the energy - efficiency constraint as it is the case for full decode - forward .  considering downlink transmission , and as said earlier , the relaying condition is always satisfied and for downlink is large compared to the cell radius .the inner - bound thus only depends on , given by the energy - efficiency condition . is computed numerically as the user - to - base - station distance for which with .the outer bound for the eo - pdf scheme is modelled by a portion of circle centred at and of radius .both are determined by the outage condition and are computed similarly to and for full decode - forward in section [ sec : ee_full_df ] , using the coordinates of two user locations at the outage limit . however , generally , and .based on the previous analysis , we can deduce the characteristic distances for the eo - pdf scheme as follows : where is given by eq . .these values define the relay efficiency area of the eo - pdf scheme , as defined in eq . .note that for both full - df and eo - pdf , we have .this shows that models based on inner and outer regions as in do not reflect the performance gains obtained by relaying and that the cell area for energy - efficient relaying is much wider than the classical outer region .so far , we have introduced the relay efficiency area and derived characteristic distances for both full - df and eo - pdf . based on this model ,we now derive the estimated probability of relaying and average energy consumption of the cell .   here, we analyze the probability of relaying a communication .we assume that the user is randomly located in the cell and that the location distribution is uniform .such distribution allows us to derive a simple expression of the probability of relaying .the probability of relaying , denoted , is obtained by the ratio of relaying area vs. total area and is expressed as follows for hexagonal cells . where we assume . and are as in eq . at the top of the page , with   conditions on and defined based on geometrical principles of a cell sector and depend on whether or , as defined in , which is the limiting condition for the rea .the complete proof can be found in appendix b. note that , for circular shape cells , we have and the constraints regarding are relaxed .we now focus on the energy consumption , averaged over all user locations .the expressions for downlink and uplink are similar , and we thus only focus on the uplink case . as before , we assume that the user is randomly located in the cell and that its distribution is uniform .we respectively denote  ] and  ] , can thus be computed as follows . =  \left(1-\mathbb{p}_{\text{rtx}}\right ) \mathbb{e } \left[e_{\text{dtx}}\right ] + \mathbb{p}_{\text{rtx } } \left ( \mathbb{e } \left[e_{u}\right ] + \mathbb{e } \left[e_{r}\right]\right ) .\label{eq : energy}\end{aligned}\ ] ] this expression reflects both the energy consumed by the user only when dtx is used , and the energy consumed by both the user and the relay station when rtx is employed .  we first obtain  ] for the user consumption and  ] in eq . , we have \nonumber  \\ = & \underset{(x , y ) \in \text{rea}}{\iint } \left(2^{2\mathcal{r}}-1\right ) \frac{n}{2 \vert h_s \vert^2 } \mathbb{p}\left((x , y ) \in \text{rea}\right )  dx dy  \\ % & = \frac{\left(2^{2\mathcal{r}}-1\right)n k_s } { \mathcal{a}_{\text{sector } } \mathbb{p}_{\text{rtx } } } \left [ \int_0^{\varphi } \int_{\frac { d_{\min}}{\cos(\theta ) } } ^{r_{\max}\left ( \theta \right ) }  r_s^{a_s/10 } r dr d\theta  % \nonumber \\ &  \nonumber \right .  \\& \quad  \left .\nonumber  + \int_\varphi^{\phi } \int_{\mathsf{r}_\text{dtx}}^{r_{\max}\left ( \theta \right ) } r_s^{a_s/10 } r dr d\theta % \quad  + % \int_{\phi}^{\frac{\pi}{6 } } \int_0^{r_{\max}\left ( \theta \right ) }  r_s^{a_s/10 } r dr d\theta \right ] \nonumber \\ & \text{where } \quad \left\lbrace \begin{array}{ll } r_s & = \sqrt{\mathsf{d}_r^2 + r^2 - r\mathsf{d}_r \cos ( \theta ) } \\ r_{\max}\left ( \theta \right ) & = \frac{\sqrt{3}\mathsf{r}_{\text{cov}}}{\sin(\theta)+\sqrt{3}\cos(\theta ) } \end{array } \right .\nonumber\end{aligned}\ ] ] in this , describes the cell bound . and are as defined in eq . .note that , in most of cases , we have , for which the integral becomes a polynomial and is equal to , which can be computed easily .we compute the average relay energy consumption  ] .however , for a vicinity relay , this range is extended by more than 200 m , which offers much more flexibility for a designer to locate a relay station adequately . in figure[ fig : relay_height_illustration ] , the maximal coverage extension and the corresponding optimal are respectively plotted with a green line and a green diamond .note that the vicinity relay outperforms the base - station - like relay when optimally positioned and allows to increase the maximal coverage by 300 m .the system performance is strongly impacted by the effective relay height using a vicinity relay compared to a base - station - like relay .  even if a vicinity relay still outperforms a base - station - like relay when being in a low position , decreasing the relay height from 20 m ( high relay position ) to 10 m ( low position ) causes a loss of more than 400 m in the coverage and a 5db energy loss , considering los condition ( green triangles ) , as shown in figure [ fig : perf_fulldf ] .thus , with respect to both coverage and energy efficiency , the relay should be placed at the highest location possible as long as it remains below rooftop , which allows higher quality and also more probable los condition for the user - to - relay link than a base - station - like relay .the energy - optimal distance is different from the coverage - optimal distance .regarding coverage , it is beneficial to locate the relay far from the base station . on the contrary , regarding energy consumption , the relay should be placed around the middle of the cell , so as to decrease the value for and consequently , to increase the probability of energy - efficient relaying .for example , the coverage - optimal is 1180 m for a vicinity relay with los conditions ( green triangles ) while the energy - optimal is only 520 m .the loss of los of mostly affects the range for beneficial distances , and limitedly impacts the maximum cell coverage , especially when relay is positioned in a high location but still below rooftop .this is illustrated in figure [ fig : los_condition_illustration ] , considering a vicinity relay .indeed , with los for the direct link , a user located in the middle of the cell can use dtx , allowing the relay station to be far from the base station . with nlos , middle - cell users then require relaying to perform the transmission at same rate , meaning that the relay can not be away from the cell center .when both and are nlos , performance are significantly degraded , but same comments as above can be made , i.e. vicinity relays outperform base - station - like relays but are more sensitive to the effective relay height .so far , we have analyzed the system performance for the practical full - df scheme .we now compare the results with the upper - bound performance obtained by using the energy - optimal scheme eo - pdf .we plot in figure [ fig : with_eo_pdf ] the maximum coverage radius and the average total energy consumed by each scheme as functions of the relay - to - base - station distance .we consider various environment conditions and several user rates .the energy consumption is computed with .also note that it consists of two parts due to the hexagonal cell constraint since the coverage radius is written as a minimum of two terms , as shown in eq . .the first part of the curve corresponds to being the minimum .  with regards to both coverage and energy ,the eo - pdf performs significantly better than full - df for high rates and under weaker channel conditions , i.e. when the direct link is nlos ( plotted with squares ) or when the relay is above rooftop ( diamonds ) .given good channel conditions ( plotted with triangles ) , i.e. vicinity relay and los conditions , full - df performs close to the eo - pdf scheme for low rates ( / s / hz ) , with regards to both coverage and energy .however , performances of full - df are rapidly degraded when the user rate increases .as illustrated in figure [ fig : relay_height_illustration ] for / s / hz , the maximum cell coverage obtained with full - df remains close to the coverage given by the eo - pdf , but the range of beneficial relay locations for which is far smaller .also note that high rates ( / s / hz ) are only achievable with base - station - like relay ( i.e. above rooftop ) by using the eo - pdf , and with vicinity relays , the coverage using full - df is only 65% of the one obtained using eo - pdf .same comments can be made with regards to energy efficiency .for example , when m with / s / hz , the energy loss of full - df is around 1.2db compared to eo - pdf .this loss is relatively low compared to the increase in complexity necessary to implement eo - pdf .however , the performance of full - df is significantly degraded when the relay is not optimally positioned .if the relay is closer to the base station ( m ) , between 2 and 4db energy loss is observed . for / s / hz , this loss attains 6db .so far , we have analyzed the performance of relaying for uplink transmissions , we know investigate downlink . due to the symmetry of the scheme, full - df has the same performance for both .we thus focus only on the eo - pdf scheme . in figure[ fig : downlink ] , we plot the average total energy consumed by the eo - pdf scheme given the cell coverage obtain for uplink ( ) .vicinity relays still outperforms base - station - like relays .interestingly , eo - pdf is more energy - efficient for uplink transmissions than for downlink .this is due to equal time division which is suboptimal , especially when the relay is located above rooftop . in this case , the link from base station to relay station is very strong .thus , the eo - pdf scheme tends to relay the whole message , using limited rate splitting ( ) , and consumes most of energy in the second phase to take advantage of the beamforming gain between the base station and the relay .on the contrary , locating the relay station below rooftop allows more balance between and .this implies better use of rate splitting which is more suitable for equal time division .   here , we refine the analysis of the energy consumption vs. the covered area .indeed , increasing the coverage radius implies that far - users are included within the cell .those far - users consuming a lot of energy due to distance , the energy efficiency is thus decreased . however , extending the cell coverage reduces the number of base stations required to cover a given area , which is cost - efficient for the network deployment .consequently , a network designer may opt for better energy efficiency regardless of the coverage , or may prefer to insure a larger coverage even if more energy is consumed in average .this constitutes a fundamental trade - off on relay deployment .we thus propose to simulate performance based on the average energy consumption per unit area .as defined in , this is the ratio of the average energy consumption over the coverage area }{\mathcal{a}_{\text{sector}}} ] and as expressed in eq . and respectively .we plot in figure [ fig : trade_off_energy ] the average energy consumption per unit area considering several coverage radius , expressed as , where refers to the coverage extension in percentage and to the maximum coverage radius as plotted in figure [ fig : with_eo_pdf_c ] . by plotting all curves given by  ]taken in this region , there exists a coverage for which this energy per unit area is optimal with this relay - to - base - station distance . in figure[ fig : trade_off_bs ] , we plot the ratio of the area covered by direct transmission over the extended area .it refers to the normalized number of required base stations , i.e. the deployment cost , and is directly linked to the maximal coverage extension , as plotted in figure [ fig : with_eo_pdf_c ] .for example , at m , using eo - pdf with maximal coverage extension requires only 30% of the base stations needed to cover the same area without cell extension .the lower bound of the shaded area in figure [ fig : trade_off_energy ] gives the minimal energy consumed per unit area as a function of and does not refer to a single coverage extension .we observe once again that the eo - pdf scheme offers better performances than full - df both at the optimal distance m and around this optimum .however , when maximum coverage extension ( ) is considered , full - df outperforms the energy - optimized scheme eo - pdf for some values of .as explained earlier , far - users that consume large amount of energy are included within the cell by using eo - pdf , thus illustrating the need for defining a trade - off between energy and coverage .the minimal average energy consumption per unit area is achieved when the cell coverage is not extended , i.e. when only energy efficiency is considered .this minimum is attained with m for both full - df and eo - pdf . up to this distance ,no coverage extension ( ) is more energy - efficient than extended cell . above this distance, coverage extension has to be considered for better energy efficiency .note that , when the relay is far from the base station ( over 800 m for eo - pdf and 600 m for full - df ) , reducing the coverage extension is no more energy - efficient .the minimal deployment cost , i.e. the minimal required number of base stations , is achieved when the relay is far from the base station , but shows in return severe loss regarding the energy consumed per unit area .  between the energy - optimal relay location ( m ) and the cost - optimal location ( m for eo - pdf and 575 m for full - df ) , the energy loss , compared to the lower - bound of the shaded area , is equal to 5.2db for eo - pdf and 3.3db for full - df .there exists a relay - to - base - station distance for which extending or reducing the cell coverage has minimum impact on the energy consumed per unit area .this impact is illustrated in figure [ fig : trade_off_energy ] by the thickness of the shaded area at a given .for example , at the energy - optimal m , extending the cell coverage from ( extension : 0% ) to ( extension : 100% ) induces an significant energy loss of 7db for eo - pdf .this loss is reduced to 2.5db when m , distance for which the deployment cost is only 31% of the cost of non - extended cells .  to summarize results ,energy - efficient relay placement should be jointly considered with cell coverage extension , and a moderate extension provides better performance in terms of energy . however , the optimal trade - off is , to our mind , neither the energy - optimal nor the cost - optimal relay location , but rather the location for which modifying the cell coverage has minimum impact on the energy consumed per unit area .this is particularly suitable for heterogeneous networks .indeed , the deployment of unplanned nodes , especially at cell edge , implies that users are served by those nodes rather than by the base station , which virtually reduces the coverage radius of the macrocell .relay deployment can not be considered to be energy - efficient only for a given coverage and showing up a 7db loss as soon as a pico- or femtocell is added , or reversely , switched off .this optimal trade - off is also relevant in networks where dynamic resource allocations and sleep mode operations enable the change of cell coverage as a function of the cell load and desired user rates .we have investigated relay placement for noise - limited urban cells and considered both cell coverage and energy efficiency . using an approach complementary to designing algorithms for optimal placement existing in the literature, we analyzed how the propagation environment , user rate and relay coding scheme affect the choice of the relay location and the network performance .to do so , we proposed a geometrical model for evaluating energy efficiency and coverage extension , and highlighted that a trade - off exists between them .two options can be deduced from this work to efficiently deploy relays in a cell . on the one hand , a network designer may choose to keep terminals simple and use two - hop routing .in this case , attention has to be paid on carefully positioning the relay station , so as to provide close to optimal coverage or energy gain . on the other hand ,this designer may be constrained by the cell topology and restricted in potential relay locations .then , more complex coding schemes , such as partial decode - forward , should be considered to fully benefit from the relay station . with regards to heterogeneous networks ,the relay should be located such that increasing the cell coverage has a minimal impact on energy efficiency .in this paper , we propose to model the path - loss depending on the relay height and base our model on the the winner ii project .we consider an urban environment where the user is located outdoors at street level and the base station is clearly above surrounding buildings .both the base station and the relay station are fixed . as recommended by 802.16j / m and 3gpp specifications , the relay station establishes a high quality link with the base station and los is assumed for this channel . since the relay height is allowed to vary from below to above rooftop , the propagation environment for and changes notably .however , the winner ii project does not provide a continuous model as a function of the relay height .we will therefore consider the two situations summarized in table [ sim_pathloss ] .furthermore , we assume that the terminal are sufficiently robust against small - scale environment parameters , such as multipath components ..considered winner ii scenarios [ cols="^,^,^",options="header " , ]the expression of eq . is obtained using geometrical principles . is equal to the surface of the rea divided by the surface of the cell .we can reduce the analysis to one cell sector and rather focus on the surface of , where dtx is more energy - efficient than rtx .we denote the strait line of equation and the circle centred at the base station and of radius .this is illustrated in figure [ fig : proof_prtx ] .first , we focus on the angles and . when , the cell - edge is never limiting and is upper - bound by either or ( left part of the hexagonal cell sector ) .this gives .furthermore , if , the set of all user positions within the sector for which is strictly included in the sphere .therefore , is upper - bounded by only and . otherwise , both and upper - bound and intersect at , with .second , when ( right part of the hexagonal cell sector ) , the cell edge can be limiting .we denote the x - coordinate of the intersection of with the cell edge . if , as before , is upper - bounded by only and . if , is upper - bounded by both and the cell edge , which gives . finally , if , is upper - bounded by , and the cell edge , and angles are computed using geometrical properties .this gives the expressions of and as given in eq . .we can now deduce the surface of . to do so, we decompose this surface into elementary geometrical shapes , whose surface can be easily computed . for  ] , is upper - bounded by .the related surface here is a portion of sphere . finally , when , is upper - bounded by the cell edge for $ ] .this also corresponds to the surface of a triangle .using , this decomposition , we get eq . .fanny parzysz received the m.sc .degree in digital communications and cellular networks from telecom paristech ( enst ) , paris , france , in 2009 .she is currently pursuing the ph.d .degree at the lacime laboratory , cole de technologie suprieure , montreal , canada .is part of the nserc - ultra electronics industrial chair in wireless emergency and tactical communication .her research interest covers technologies and models for next generation relay - aided wireless networks , with a particular focus on information theory , energy - efficient coding and resource allocation .mai vu received a phd degree in electrical engineering from stanford university after having an mse degree in electrical engineering from the university of melbourne and a bachelor degree in computer systems engineering from rmit , australia . between 2006 - 2008 , she worked as a lecturer and researcher at the school of engineering and applied sciences , harvard university . during 2009 - 2012 , she was an assistant professor in electrical and computer engineering at mcgill university . since january 2013, she has been an associate professor in the department of electrical and computer engineering at tufts university .vu conducts research in the general areas of wireless systems , signal processing , and network communications .she has published extensively on cooperative and cognitive communications , relay networks , mimo capacity and precoding .vu has served on the technical program committee of numerous ieee conferences and is currently an editor for the ieee transactions on wireless communications .she is a senior member of the ieee .franois gagnon received the b.eng . anddegrees in electrical engineering from cole polytechnique de montral , montreal , quebec , canada .since 1991 , he has been a professor with the department of electrical engineering , cole de technologie suprieure , montreal , quebec , canada .he chaired the department from 1999 to 2001 , and is now the holder of the nserc ultra electronics chair , wireless emergency and tactical communication , at the same university .his research interest covers wireless high - speed communications , modulation , coding , high - speed dsp implementations , and military point - to - point communications .he has been very involved in the creation of the new generation of high - capacity line of - sight military radios offered by the canadian marconi corporation , which is now ultra electronics tactical communication systems .the company has received , for this product , a `` coin of excellence '' from the u.s .army for performance and reliability .gagnon was awarded the 2008 nserc synergy award for the fruitful and long lasting collaboration with ultra electronics tcs .

Abstract: the performance of a relay - based cellular network is greatly affected by the relay location within a cell . existing results for optimal relay placement do not reflect how the radio propagation environment and choice of the coding scheme can impact system performance . in this paper , we analyze the impact on relaying performance of node distances , relay height and line - of - sight conditions for both uplink and downlink transmissions , using several relay coding schemes . our first objective is to propose a geometrical model for energy - efficient relay placement that requires only a small number of characteristic distances . our second objective is to estimate the maximum cell coverage of a relay - aided cell given power constraints , and conversely , the averaged energy consumption given a cell radius . we show that the practical full decode - forward scheme performs close to the energy - optimized partial decode - forward scheme when the relay is ideally located . however , away from this optimum relay location , performance rapidly degrades and more advanced coding scheme , such as partial decode - forward , is needed to maintain good performance and allow more freedom in the relay placement . finally , we define a trade - off between cell coverage and energy efficiency , and show that there exists a relay location for which increasing the cell coverage has a minimal impact on the average energy consumed per unit area .


in the last two decades , discoveries of the kuiper belt , as well as planets orbiting stars other than the sun , have supplied the centuries - old quest to understand the formation of the solar system with fresh constraints and insights into physical processes at play . among a multitude of newly proposed formation scenarios ,the nice " model is particularly notable , as it has attained a considerable amount of success in reproducing the various observed features of the solar system . within the context of the scenario envisioned by the nice model ,giant planets start their post - nebular evolution in a compact , multi - resonant configuration , and following a brief period of dynamical instability , scatter onto their current orbits .the first success of the nice model lies in its ability to quantitatively reproduce the observed orbits of the giant planets , as well as their dynamical architecture ( i.e. secular eigenmodes of the system ) .simultaneously , the brief instability , inherent to the model , provides a natural trigger to the late heavy bombardment , as well as a transport mechanism for emplacement of dynamically hot " kuiper belt objects ( kbos ) from inside of  .meanwhile , it has been recently demonstrated that survival of a dynamically cold " primordial population between neptune s current 3:2 and 2:1 exterior mean - motion resonances ( mmrs ) is fully consistent with a nice model - like evolution of the planets , implying an _ in - situ _ formation of the cold classical population of the kuiper belt .finally , the presence of jupiter s and neptune s trojan asteroids has been attributed to chaotic capture of planetesimals during the instability .all successful realizations of the nice model to date have been comprised exclusively of the four currently present giant planets .however , there exists no strong evidence that suggests that additional planets were not present in the solar system , at the epoch of the dispersion of the nebula .in fact , theoretical arguments , presented by point to a possibility of initially forming as many as five ice giants , three of which get subsequently removed via ejections ( see however ) .the dynamical sensibility of such a scenario is further strengthened by the fact that a considerable fraction of standard nice model simulations result in an ejection of an ice - giant after an encounter with at least one of the gas giants .  in this paper, we explore an instability - driven dynamical evolution of a 5-planet system ( 2 gas giants + 3 ice giants ) with an eye towards identifying a pathway towards reproduction of solar system - like dynamical architecture . in principle, the realm of possibility available to this study is enormous .consequently , rather than performing a comprehensive parameter - search , here we limit ourselves to systems that contain an additional uranus - like planet , with the aim of presenting a few proof - of - concept numerical experiments .the plan of the paper is as follows . in section 2, we describe our numerical setup . in section 3 , we show that in our model , the planetary orbits , their secular eigenmodes , as well as various populations of the kuiper belt are approximately reproduced .we conclude and discuss our results in section 4 .the numerical setup of the simulations performed here was qualitatively similar to those presented by and .particularly , the five giant planets were initialized in a compact , multi - resonant initial condition , surrounded by a massive planetesimal disk that extended between its immediate stability boundary and .two of the three ice - giants were taken to have the same mass as uranus and were initially placed next to saturn and as the outermost planet respectively .the middle ice - giant was taken to have neptune s mass . in all simulations ,jupiter and saturn started out in a 3:2 mmr , in accord with the results of hydrodynamical simulations of convergent migration of the planets in the solar nebula .the ice giants were also sequentially assembled into first order mmrs by applying dissipative forces , designed to mimic the presence of the nebula .following resonant locking , each assembled multi - resonant initial condition was evolved in isolation for 10myr , as an immediate test of orbital stability .this procedure yielded a total of 81 stable multi - resonant initial conditions .the search for adequate dynamical evolutions was performed in two steps .first , we evolved 10 permutations of each initial condition , with planetesimal disks composed of planetesimals .disk masses were chosen randomly between and .the density profiles followed a power - law distribution , where the power - law index , , was chosen randomly between and .the planetesimals were initialized on near - circular orbits ( ) . to reduce the already substantial computational cost , self - gravity of the planetesimal swarm was neglected .subsequently , we eliminated all initial conditions that did not yield any final systems that were comprised of 4 planets , reducing the number of viable initial conditions to 25 .then , an additional 30 permutations of these initial conditions were integrated with disks composed of planetesimals ( but otherwise identical to those described above ) .each integration was performed using the _ mercury6 _ integration software package and spanned 50myr .the calculations were performed on caltech s _ pangu _ super - computer .after their completion , simulations that were deemed successful were reintegrated with the use of tracer simulations ( see ) , to address the dynamical evolution of a locally formed population of kbos . in particular , each run was supplemented with an additional disk of mass - less particles that resided in the cold classical region of the kuiper belt ( i.e. between the final exterior 3:2 and 2:1 mmrs of neptune ) .out of the 810 integrations that were initially performed , 214 ( ) cases featured an ejection of a single ice - giant , yielding a system composed of 4 planets .perhaps unsurprisingly , in most cases the ejected planet is the ice - giant that neighbors saturn . of the 750 simulations that were performed following the elimination of initial conditions, 33 evolutions resulted in orbits reminiscent of the solar system .specifically , we searched for solutions where the saturn - jupiter period ratio exceed , while the final semi - major axes of the ice - giants were within of their observed counterparts .no strong requirements were placed on the planetary eccentricities and inclinations .it is noteworthy that evolving the giant planets onto solar system - like orbits is insufficient for a simulation to be deemed successful for indeed , there are additional constraints that must be satisfied .the first orbital constraint is the reproduction of the secular architecture of the system .the secular orbital angular momentum exchange ( i.e. eccentricity evolution ) of a planetary system containing secondaries can be approximately represented as a superposition of eigenmodes , each corresponding to a fundamental frequency of the system ( see ) .physically , the maximum eccentricity that a given planet attains in its secular cycle is equal to the sum of all of its corresponding eigenmode amplitudes . in the context of the traditional nice model, some difficulty has been noted in correctly reproducing the dynamical character of jupiter s and saturn s eccentricity evolution .particularly , it has been shown that smooth passage of jupiter and saturn through the 2:1 mmr has a tendency to under - excite the eccentricity eigenmode as well as their mutual inclinations .this difficulty can be overcome sometimes by invoking a close - encounter between an ice - giant and the gas giants .consequently , we have checked , using fourier analysis ( see for example ) , the relative strength of the and eigenmodes in all simulations whose orbital end - state resembled the solar system .we did not restrict the success criteria of our simulations to include the correct reproduction of the mean eccentricities of the planets and in some cases , the mean final eccentricities ( namely those of the ice - giants ) exceeded their observed counter - parts by as much as a factor of .this is , however , likely an artifact of the coarse representation of the planetesimal disk and the resulting dynamical friction , that we employed in our calculations and should not be viewed as a major drawback .  in total, we found ten cases ( corresponding to eight different initial conditions ) where their amplitudes are satisfactorily reproduced .specifically , in these ten cases , the amplitude of jupiter s eigenmode exceeds that of the eigenmode , while the amplitudes are roughly the same for saturn. we made no attempt at quantitatively matching the pair s inclination eigenmodes , however their reproduction does not appear to be problematic .we also examined the amplitudes of the secular eigenmodes of uranus and neptune .generally it appears that the dynamical architecture of the ice - giants is set in an essentially random manner , depending on the particular encounter history .consequently , we decided to not use ice - giant secular architecture as a distinctive property in our analysis .successful formation of the kuiper belt is another important constraint of the nice model . shown that the excited populations of the kuiper belt are naturally emplaced from inside of during the instability . given that this aspect of the planetary evolution is not particularly different between the 4-planet and the 5-planet scenarios , there is little reason to speculate that the dynamical pathway for formation of the resonant , scattered , and hot classical populations of the kuiper belt will be inhibited .the same is likely to be true for chaotic capture of trojan asteroids .the cold classical population of the kuiper belt , however , is a different story .a series of observational dissimilarities between the cold classical population and the rest of the kuiper belt ( e.g. uniquely red colors ; strongly enhanced wide binary fraction ) suggest that the cold classicals formed in situ , and maintained dynamical coherence despite neptune s temporary acquisition of high eccentricity and inclination ( a characteristic orbital feature of the cold classical population is inclination that does not exceed  ) . in a recent study, showed that local formation of the cold classicals is fully consistent with an instability - driven evolution of the planets , given favorable conditions during the instability .particularly , required the apsidal precession and nodal recession rates of neptune to be comparatively fast to prevent secular excitation of the cold classical orbits , in addition to a sufficiently small apohelion distance of neptune , to avoid orbital excitation due to close encounters .    in a 5-planet scenario ,the retention of unexcited orbits of the cold population can be jeopardized by the ejecting planet .this possibility served as a premise for recalculation of the dynamical evolution of test particles in the cold classical region , with the aid of tracer simulations .we found that in only three of the ten simulations where the secular eigenmodes were successfully reproduced , a primordial cold classical population of the kuiper belt also retained unexcited orbits .consequently , it appears that the retention of an unexcited cold belt is not the norm of the 5-planet scenario .this is not surprising , since dynamical excitation can only be avoided if ejection and close encounter timescales are sufficiently short . from a quantitative point of view ,our simulations imply that dynamically cold orbits can only be sustained if the ejecting ice - giant spends years or less crossing the classical kuiper belt region . as a result, we can expect that the incorporation of yet additional ice - giants into the primordial solar system ( as suggested by ) will further diminish the chances of reproducing the cold classical kuiper belt , since all ejecting planets would have to do so very rapidly .two of the three successful simulations discussed above are presented in figures ( 1 - 3 ) .specifically , figure ( 1 ) shows the orbital evolutions of the runs , while figures ( 2 ) and ( 3 ) show the eccentricity and inclination distributions of planetesimals in the kuiper belt region respectively .the red and blue dots , shown in figures ( 2 ) and ( 3 ) depict the orbital distribution of emplaced and local planetesimal populations respectively at .the red and blue triangles depict the planetesimals whose orbits remain stable at .the starting multi - resonant initial condition of the simulation presented in panels ( a ) of the figures is one where saturn and the first ice - giant are locked in a 2:1 mmr , while both pairs of ice - giants are locked in 5:4 mmrs .the planetesimal disk in this simulation was comprised of particles and contained a total of 26 . in this evolution ,the local population of test particles suffers numerous short close encounters with the ejecting ice giant , yielding a more excited and depleted cold classical population , compared to that of run ( b ) .note also that at , the ice - giant eccentricities are considerably greater than that of uranus and neptune .these high eccentricities do not get damped away by dynamical friction in the following 500myr of dormant evolution .consequently , in this simulation , the inner edge of the cold belt gets dynamically depleted over the following 500myr .it is furthermore noteworthy that another simulation that originated from the same initial condition reproduced the eigenmodes of the system correctly , although the primordial cold kuiper belt in this integration was entirely destroyed by close encounters .the starting multi - resonant initial condition of the simulation presented in panels ( b ) of the figures is one where saturn and the first ice - giant are locked in a 3:2 mmr , the inner pair of ice - giants is locked in a 2:1 mmr , and the outer pair of the ice - giants is locked in a 4:3 mmr . in this simulation ,the disk consisted of particles and had a cumulative mass of .incidentally , the frequency spectrum of the eccentricity vectors of jupiter and saturn produced in this simulation , matches that of the real jupiter and saturn exceptionally well , signaling a nearly ideal reproduction of the secular eigenmodes .particularly , the simulation yields ( in the notation of ) and where as the solar system is characterized by and .the scaling of the eigen - vectors are also well reproduced : while the solar system has . in this simulation ,the cold kuiper belt suffers only a single short encounter with the escaping ice - giant , allowing for the orbits ( inclinations in particular ) to remain dynamically cold .in this paper , we have presented a successful realization of the nice model which starts out with 5 planets .the numerical experiments presented here explicitly show that such an evolution is plausible since the resulting 4-planet systems can closely resemble the solar system .particularly , in both simulations presented here , the secular architecture of the outer solar system is well reproduced .furthermore the demonstrated survival of a local , primordial cold classical kuiper belt suggests that all constraints that can be matched with a 4-planet model can also be matched with a 5-planet model to an equal degree of satisfaction .it is noteworthy that ejection was not always necessary in our simulations to generate a 4 planet system . in a handful of runs (one of which successfully reproduced the secular eigenmodes , but not the cold kuiper belt ) , one of the ice - giants ended up merging with saturn . in principle , such a scenario may help explain saturn s enhanced metallicity in comparison with jupiter .although , here again the explanation is not unique ( see and the references therein ) .  in a traditional realization of the nice model ,the rate of successful reproduction of the secular eigenmodes is rather low i.e. of the integrations for a favorable initial condition .this is in part because an ice - giant / gas giant encounter often leads to an ejection of the ice - giant , leaving behind only three planets .thus , the need for an ice - giant / gas - giant encounter in the orbital history of the solar system is in itself motivation for a 5-planet model.  the statistics of simulations presented in this work suggest that a 5-planet model is neither more nor less advantageous .recall that the probability of ending up with only 4 planets is .the probability of reproducing the secular eigenmodes of jupiter and saturn is .naively , this yields an overall probability of success of only .however , it is important to keep in mind that the characteristic outcomes are generally dependent on initial conditions and runs that originated from the initial condition presented in panels ( a ) correctly reproduced the secular eigenmodes in 2 out of 30 simulations .this statistic is similar to the 4-planet model .finally , the probability of also retaining an unexcited cold classical kuiper belt puts the 5-planet model and the 4-planet model on equal footing in terms of success rate .that said , it is important to note that this success rate is only characteristic of the particular 5-planet model that we have constructed .in other words , it is likely that if one allows the mass of the ejected planet to also be a variable parameter , tuning of the initial state may in principle lead to a more frequent reproduction of the solar system .the results presented in this work imply that the solar system is one of many possible outcomes of dynamical evolution , and can originate from many possible initial conditions . as a result ,the possibility of having an extra planet initially present in the system , yet its ejection leaving no observable signature erases any hope for construction of a deterministic model for solar system evolution .the forward process - like nature of the nice model is not surprising , given that the solar system exhibits large - scale chaos , characterized by lypunov times that are comparable to orbital timescales , during the instability .moreover , the similarity between the orbital architectures of simulations whose outcomes were deemed unsuccessful in this work and those of extra - solar planetary systems further confirms that planet - planet scattering is likely to be the physical process responsible for shaping the orbital distribution of planets .consequently , we conclude that an instability - driven dynamical history remains a sensible choice as a baseline scenario for solar system s early dynamical evolution . +* acknowledgments * we thank alessandro morbidelli , hal levison , david nesvorny and peter goldreich for useful conversations .we thank naveed near - ansari for operational help with the _ pangu _ supercomputer .k. batygin acknowledges supported from nasa s nessf graduate fellowship .

Abstract: over the last decade , evidence has mounted that the solar system s observed state can be favorably reproduced in the context of an instability - driven dynamical evolution model , such as the nice " model . to date , all successful realizations of instability models have concentrated on evolving the four giant planets onto their current orbits from a more compact configuration . simultaneously , the possibility of forming and ejecting additional planets has been discussed , but never successfully implemented . here we show that a large array of 5-planet ( 2 gas giants + 3 ice giants ) multi - resonant initial states can lead to an adequate formation of the outer solar system , featuring an ejection of an ice giant during a phase of instability . particularly , our simulations demonstrate that the eigenmodes which characterize the outer solar system s secular dynamics can be closely matched with a 5-planet model . furthermore , provided that the ejection timescale of the extra planet is short , orbital excitation of a primordial cold classical kuiper belt can also be avoided in this scenario . thus the solar system is one of many possible outcomes of dynamical relaxation and can originate from a wide variety of initial states . this deems the construction of a unique model of solar system s early dynamical evolution impossible .