synchronization is a ubiquitous phenomenon often observed in coupled chaotic and hyperchaotic systems . depending upon the strength and the nature of coupling , various types of synchronization such as phase synchronization ( ps ) , complete synchronization ( cs ) , generalized synchronization , lag and anticipatory synchronizations , inverse phase synchronization ( ips ) and inverse synchronization ( is )have been observed in coupled dynamical systems .all these types of synchronization are achieved in ensembles of dynamical systems through some common coupling schemes , namely linear and nonlinear error feedback couplings , coupling via dissimilar and/or time - delayed variables , inhibitory coupling , coupling via dynamical relaying , adaptive coupling and also systems driven by a common noise , etc .  in many real world systems ,synchronization can occur due to interaction through a common dynamic medium and hence the systems also evolve similar to their environment under the influence of the latter .for example , let us consider the synchronization of self - excited nonlinear oscillators suspended on a movable elastic structure . in this case the oscillators interact with each other through the common elastic beam ( oscillators are excited by the vibrations of the structure ) .it has been shown that for given conditions of the elastic structure initially uncorrelated oscillations of each of the oscillators can synchronize with the frequency of the oscillations of elastic beam ( both inphase and antiphase synchronizations have been identified ) .other examples include synchronization of chemical and genetic oscillators , synchronized behavior with self pulsating periodic and chaotic oscillations produced by an ensemble of cold atoms interacting with a coherent electromagnetic field , synchronization of cells and in coupled circadian oscillators due to common global neurotransmitter oscillation .in all the cases , the coupling function has a dynamics modulated by the system dynamics . in this connection ,resmi et al  have recently shown and verified numerically the existence of various types of synchronization in low - dimensional chaotic systems which are coupled through a dynamic environment without intrinsic or coupling time - delay .time - delay systems are an important class of dynamical systems which occur in many real world systems and synchronizing such systems is having potential applications in diverse areas of science , engineering and technology due to their hyperchaotic nature . during the past few years , researchers are interested synchronizing such time - delay systems and mostly all known synchronizations and their transitions are identified and reported in two coupled time - delay systems using numerical simulations . in contrast , investigations on realizing synchronization in time - delay systems from an experimental point of view is very limited compared to their enormous theoretical results . in particular , despite the possibility of realizing time - delay systems using electronic circuits with some effort , experimental realization of most of the promising theoretical results remain largely unexplored using electronic circuits . experimentally generating and synchronizing such hyperchaotic electronic signals ( with multiple positive lyapunov exponents ) is very important where these signals may be used to hide secret messages in the area of secure communication and cryptography .further experimental realizations of theoretical concepts using electronic circuits with time - delay are being used as test beds before implementing them in photonics systems such as liquid state machines to achieve high speed processing , which are usually very expensive .circuit diagram of a single time delayed feedback oscillator with a nonlinear device ( nd ) unit , a time - delay unit ( delay ) and a low pass first - order filter . ]circuit block diagram of the coupled time delayed feedback oscillator for the mutual coupling configuration ( [ eqn3 ] ) . ]  in this paper , we demonstrate experimentally the occurrence of various types of synchronization and their transitions in indirectly coupled time - delayed electronic circuits with intrinsic time - delay using dynamic environment coupling .we carry out these studies in two different coupling configurations , namely mutual coupling configuration where both the circuits and the environment are mutually sharing their feedback , and subsystem coupling configuration where both circuits are sharing their feedback with the environment , while only one of the circuits is receiving feedback from the environment . it is interesting to note that here synchronization occurs between the independently evolving system and the environmentally affected system .obviously this has important implications in actual physical situations and to our knowledge this problem has not been studied earlier . depending upon the coupling strength and the nature of the feedback ,we observe different types of synchronization transitions in the coupled circuits which include transition from nonsynchronization to cs via ps and from nonsynchronization to is via ips in both coupling configurations .snapshots of time evolution , phase projection and localized sets of the circuits as observed from the oscilloscope confirm the existence of different synchronized states experimentally along with corresponding numerical results .further , the transition to different synchronized states can be numerically quantified from the changes in the largest lyapunov exponents ( les ) , correlation of probability of recurrence ( cpr ) and correlation coefficient ( cc ) of the coupled systems as a function of the coupling strength .we also present a detailed linear stability analysis and obtain synchronization conditions for different synchronized states in the present system .the remaining paper is organized as follows : in sec .[ sec : level2 ] , we will describe briefly the system employed to demonstrate different synchronized states and explain the circuit realization . in sec .[ sec : level3 ] we explain the mutual coupling configuration and present experimental and numerical results to confirm various types of synchronization along with the necessary linear stability analysis . in sec .[ sec : level4 ] , we describe the subsystem coupling configuration and demonstrate the occurrence of different synchronization transitions and , finally , we summarize our results with conclusion in sec . [sec : level5 ] .( color online ) experimental ( vertical scale and horizontal scale ) and numerical observations of phase projections of the coupled systems for mutual coupling configuration : ( a ) experimental ips , ( b ) experimental is , ( c ) numerical ips for and ( d ) numerical is for . ]first we consider a single scalar delay differential equation given as where the nonlinear function , is represented by a piecewise linear function here the system parameters are fixed as , , time - delay , , and throughout the manuscript . for these chosen parameter values the single system ( [ eqn0 ] )exhibits a hyperchaotic attractor with three positive les .( color online ) ( a ) cpr ( continuous line ) , cc ( dotted line ) and ( b ) spectrum of maximal les for ( in mutual coupling configuration as a function of coupling strength . ]   the design of the electronic circuit which describes the dynamics of eq.([eqn0 ] ) along with the threshold piecewise linear function is given in fig .this circuit consists of a diode based nonlinear device ( nd ) with two amplification stages ( and ) , a time - delay unit ( delay ) and a low - pass first order filter . here are engaged as operational amplifiers . and are the constant voltage sources of all active devices ( ) . using thses voltage values and , one can easily adjust the threshold values of the three segment piecewise function ( eq . ( [ eqn2 ] ) ) . by applying the kirchhoff s laws to this circuitthe state equation can be written as  ] is the static characteristic of the nd unit .( color online ) experimental ( vertical scale and horizontal scale ) and numerical phase projection plots of the coupled systems with mutual coupling configuration for ( ) : ( a ) experimental ps , ( b ) experimental cs , ( c ) numerical ps for and ( d ) numerical cs for . ]  to study the circuit equation , we transform it to the dimensionless form ( [ eqn0 ] ) by defining the dimensionless variables and parameters as , , , , and .a nonzero is chosen such that .the circuit parameters are fixed as , , , , ( trimmer - pot ) , , ( trimmer - pot ) , , , ( trimmer - pot ) , , , ( ) , ( ) , . , , and so the time - delay for the chosen circuit parameter values .we now construct a circuit which consists of a system of two identical time - delayed sub - circuits with a threshold piecewise - linear nonlinearity and are coupled indirectly through a common environment . here both the circuits and the environment are mutually sharing their feedback with each other and the state equation for the coupled circuit can be written as   + \nonumber \\ \varepsilon^{\prime}_{1 } \beta^{\prime}_{1}v(t),\\ r_{0}c_{0 } \frac{du_{2}(t)}{dt } & = & -\alpha^{\prime}u_{2}(t)+f[k_{f}u_{2}(t - t_{d})]+ \nonumber \\ \varepsilon^{\prime}_{1 } \beta^{\prime}_{2}v(t),\\ r_{0}c_{0 } \frac{dv(t)}{dt } & = & -k^{\prime}v(t)-\frac{\varepsilon^{\prime}_{2}}{2}[\beta^{\prime}_{1}u_{1}(t)+\beta^{\prime}_{2}u_{2}(t)],\end{aligned}\ ] ]  [ eqn3 ]  where and correspond to the output variables of each circuit and is the output of the environmental equation .the schematic diagram for this coupled circuit is sketched in fig .[ fig2 ] . by defining the normalized variables and parameters as above and , one obtains the equivalent dimensionless equation as follows :  +\varepsilon_{1}\beta_{1}y,\\ \dot{x}_2(t)&=&-\alpha x_2(t)+\beta f[x_{2}(t-\tau)]+\varepsilon_{1}\beta_{2}y,\\ \dot{y } & = & -k y-\frac{\varepsilon_{2}}{2 } ( \beta_{1}x_{1}+\beta_{2}x_{2}),\end{aligned}\ ] ]  [ eqn4 ]    where , , , and . herethe two systems and are not directly coupled to each other , instead they are coupled through a coupling function ( ) which has a dynamics modulated by the system dynamics . is the strength of the feedback to the systems and is the strength of the feedback to the environment ( coupling parameters ) . and are the nature of feedback from and to the environment , respectively . is the damping parameter and we choose it as . in the absence of feedback from the systems to the environment, the strength of the environment decays exponentially fast as .note that in the absence of delay , the system of eqs .( [ eqn4 ] ) reduces to the dynamical system studied by resmi et al  .( color online ) ( a ) cpr ( continuous line ) , cc ( dotted line ) and ( b ) spectrum of maximal les for in mutual coupling configuration as a function of coupling strength . ]we have experimentally observed different types of synchronization in the coupled electronic circuits which are also confirmed using numerical simulations .depending upon the feedback strength and we observe two types of synchronization transitions in the coupled systems .when and are of the same sign , for example ( , we observe a transition from non - synchronization to is via ips and when and are of different signs , ( , then we observe a transition from non - synchronization to cs via ps as a function of the coupling strength .first , we consider the case ( and for simplicity we have chosen the value of the coupling strengths as . in the absence of the coupling ( ) both circuits oscillate independently , and for sufficiently large value of the coupling strength the phase difference between the circuits is exactly , that is the systems exhibit ips . on increasing the coupling strength to further larger values , is occurrs between the circuits .( color online ) ( a ) , ( b ) numerically obtained two parameter diagrams in ( ) plane show various types of synchronization states in the case of mutual coupling configuration for ( ) and ( ) , respectively .we mark ps and ips states when cpr .cs and is states are marked when cc and cc , respectively . in both figures , the white color indicates desynchronized state , grey color in fig .[ fig15](a ) and red color ( dark grey ) in fig .[ fig15](b ) represent ips and ps regimes , respectively .is state is represented by pink color ( dark grey in fig . [ fig15](a ) ) and light blue color ( light grey in fig .[ fig15](b ) ) indicate the cs region .the black line indicates the analytically obtained stability condition of the outer regime of the piecewise linear function [ eq .( [ eqn2 ] ) ] and blue ( dark grey ) filled circles indicates the stability condition for the middle region of the piecewise linear function . ]snapshots of the wave forms of the circuits as seen in the oscilloscope are shown in fig .the existence of ips between the circuits is depicted in fig .[ fig3](a ) where the circuits are evolving with a phase difference of but still the amplitudes are uncorrelated .figure [ fig3](b ) shows the realization of is between both systems where both phase and amplitude are correlated and occur exactly opposite to each other . in the corresponding numerical analysis of eq .( [ eqn4 ] ) , we obtain ips for which is depicted in fig .[ fig3](c ) and this synchronization can also be confirmed both experimentally and numerically using the phase projection plots which are shown in figs .[ fig5](a ) and [ fig5](c ) , respectively . on further increase of the coupling strength to ,the systems exhibit is where the maxima of both systems occur exactly opposite to each other for as depicted in fig .[ fig3](d ) .the corresponding experimental and numerical phase projection plots of the systems are given in figs .[ fig5](b ) and [ fig5](d ) , respectively .phase coherence of the systems is further qualitatively visualized both experimentally and numerically using the framework of localized sets .the basic idea of this characterization is that the set of points obtained by sampling the time - series of the system 1 whenever the maximum occurs in system 2 is plotted along with the attractor of system 1 and vice versa .if the coupled systems are said to be phase synchronized then the sets are localized on the attractor , otherwise they spread over the entire attractor implying asynchronization .this provides an easy and efficient way to detect phase synchronization even in non - phase - coherent and high dimensional attractors .the experimental realization of the localized sets can be obtained as follows : the maxima of the systems state variables of the circuits 1 and 2 are taken as our reference points to demonstrate localized sets . using the circuit given in fig . 7.12 in pp .147 of , we generate the impulse whenever the input signal of the system 2 attains maximum .while the attractor of the system 1 is in the channel of the oscilloscope , we feed the impulse signal to the -input .whenever the impulse hits the attractor , one can see the bright spot in the attractor of the system 1 . the same is observed when the impulse generated from the system 1 and superimposed to the attractor of the system 2 .the experimental observation of the localized sets is shown in fig .[ fig6 ] . in the absence of the coupling ( ) ,the sets are distributed over the entire attractor which corresponds to the absence of phase coherence .figures [ fig6](a ) and [ fig6](b ) show that the attractors of the two systems along with the sets for the case of nonphase synchronization .the corresponding numerically obtained figures are plotted in figs .[ fig6](e ) and [ fig6](f ) for . for sufficiently large value of the coupling strength, the sets are localized on their corresponding attractors which confirm a perfect phase locking of the systems [ figs .[ fig6](c ) and [ fig6](d ) ] .the corresponding numerical figures are plotted in figs .[ fig6](g ) and [ fig6](h ) for the value of .circuit block diagram of the coupled time delayed feedback oscillator for subsystem coupling configuration ( [ eqn12 ] ) . ]    next , the transition from nonsynchronization to is via ips can be characterized by changes in the spectrum of maximal les .also the phase coherence is further quantified using the index cpr .complete and inverse synchronizations can be quantified using the cc .these are given by the expressions , where brackets indicate time average .( [ eqn5a ] ) , we calculate the index cpr . here means that the mean value has been subtracted , are the standard deviations of and and is a generalized autocorrelation function based on recurrence properties .if the phases of the systems are perfectly locked , then the probability of recurrence is maximal at the time and , otherwise the maxima do not occur simultaneously and hence one can expect a drift in both the probability of recurrence resulting in low values of cpr . using eq .( [ eqn5b ] ) , we calculate the cc to characterize the cs and is between the systems. if both systems are in cs state then , and for is state will be .    in fig .[ fig8](a ) we have plotted the numerically calculated cpr ( continuous line ) , cc ( dotted line ) and in fig . [ fig8](b )we have plotted the ten largest les of the coupled systems as a function of the coupling strength . in the absence of the coupling ( ) ,the index cpr and cc are near to zero which confirms that the systems are evolving without any synchronization with six positive les ( three for each systems ) .if the coupling strength increases , the index cpr also starts to increase towards unit value and for , cpr is oscillating near unity ( cpr ) whereas the value of cc becomes negative which indeed confirms the onset of ips .additional confirmation comes from the changes in the les , where the zeroth le of the coupled system becomes negative at which indicates the existence of ips .if we increase the coupling strength further , the cc of the coupled systems decreases and reaches the value at . at this value of , except for the three positive les ,all the other positive les of the coupled systems become negative which indeed confirms the existence of is in the coupled time - delay systems .next , we consider the case with ( ( different signs ) .now we observe a transition from nonsynchronization to cs via ps as a function of the coupling strength . in this case , for lower values of coupling strength the individual circuits evolve independently , while for the circuits exhibit ps and for further larger value of ( ) both systems attain a cs state .the experimental snap shots and numerical plots of the time series of the coupled systems are depicted in figs .[ fig9](a ) and [ fig9](c ) , respectively , for indicating that the circuits are evolving with ps .this is also verified ( both experimentally and numerically ) using the phase projection plots [ figs .[ fig11](a ) and [ fig11](c ) ] , respectively .on further increase of the coupling strength to , the circuits exhibit cs with each other which is depicted in figs .[ fig9](b ) ( experimetnal ) and [ fig9](d ) ( numerical ) .the corresponding experimental and numerical phase projection plots of the systems are given in figs .[ fig11](b ) and [ fig11](d ) , respectively .ps is further confirmed by the frame work of localized sets .the experimental observations of localized sets are presented in fig .[ fig12 ] .the sets are distributed over the entire attractors which confirm no synchronization in the absence of the coupling [ figs .[ fig12](a ) and [ fig12](b ) ] .the corresponding numerical figures are shown in figs .[ fig12](e ) and [ fig12](f ) for . for a sufficiently large value of coupling strength, the sets are localized on the attractors which confirm a perfect phase locking between the systems [ figs .[ fig12](c ) and [ fig12](d ) ] .the equivalent numerical figures are given in figs .[ fig12](g ) and [ fig12](h ) for the value of coupling strength .we note here that we can not distinguish ips and ps using the framework of localized sets .this method is used to characterize the phase coherence / incoherence depending on the localization of the sets on the attractor . to differentiate ips from ps we use other plots such as snapshot of the time evolution and phase projection plots .  in fig .[ fig14](a ) we have plotted the index cpr , cc and in fig . [ fig14 ] ( b ) we have plotted ten maximal les of the coupled systems as a function of the coupling strength ( ) . in the absence of the coupling , the index cpr and cc confirming that the systems are independently oscillating without any synchronization .once the coupling strength increases beyond ( ) , the index cpr increases and reaches a value close to unity ( ) and cc becomes , confirming the onset of ps but still the amplitudes are uncorrelated [ fig . [ fig14](a ) ] .ps is also confirmed from the changes in the les where the zeroth le of the coupled system becomes negative at [ fig .[ fig14](b ) ] .further , for , the cc of the system reaches the unit value ( ) which is depicted in fig .[ fig14](a ) and for this value , except for the three positive les , all the other positive les of the coupled system become negative confirming the existence of cs [ fig . [ fig14](b ) ] .  to identify the global picture of occurrence of different synchronization transition regimes ,we have plotted the phase diagrams ( obtained using numerical simulation ) in the two parameter plane of coupling strengths ( ) .we use the index cpr to mark ps and ips regimes when cpr .in addition , cs region is marked when cc and is region is marked when cc .the phase diagrams in the ( ) plane for the coupled time - delay systems are shown in fig .[ fig15](a ) and fig .[ fig15](b ) for ( ) = ( ) and ( ) , respectively . in both figures the white color represents the desynchronized state , grey color in fig .[ fig15](a ) and red color ( dark grey ) in fig .[ fig15](b ) correspond to ips and ps regimes , respectively .is is represented by pink color ( dark grey ) in fig .[ fig15](a ) and light blue color ( light grey ) indicates cs regime in fig .[ fig15](b ) .( color online ) ( a ) cpr ( continuous line ) , cc ( dotted line ) and ( b ) spectrum of maximal les for in subsystem coupling configuration as a function of coupling strength . ]   in this section , we arrive at the stability condition for the synchronized states of the two dynamically coupled time - delay systems .to find the stability of the synchronization , we apply an infinitesimal perturbation to the system ( [ eqn4 ] ) .let and represent the perturbations . then eq . ( [ eqn4 ] )can be rewritten as  .\end{aligned}\ ] ]  [ eqn6 ]  here $ ] is the derivative of eq .( [ eqn1 ] ) and for the two outer regimes of eq .( [ eqn2 ] ) , it can be written as and for the middle region .it is also to be noted that in the recent papers , the time average of the derivative of the nonlinear function is approximated as a constant .this makes the entire stability analysis to be generic for all forms of nonlinearity when the systems are coupled through the environmental coupling scheme ( [ eqn4 ] ) .however , the derivative of in our analysis naturally becomes different constants in different regions because of the piecewise linear nature of and hence we assume , for generality . here in the two outer regions and in the middle region .equation ( [ eqn6 ] ) is difficult to solve in the present form and so we consider the special case of complete synchronization state ( ) . now( [ eqn6 ] ) can be simplified by defining so eq . ( [ eqn6 ] ) becomes    [ eqn8 ]  note that eqs .( [ eqn8]a ) and ( [ eqn8]b ) are coupled while eq .( [ eqn8]c ) is uncoupled from the other two and corresponds to the perturbation of the uncoupled time - delay system ( [ eqn0 ] ) confirming the underlying dynamics is chaotic / hyperchaotic for chosen parameter values .we note here that for ( )=(1,-1 ) , is a perturbation to the complete synchronization manifold and for ( )=(1,1 ) corresponds to a perturbation to the inverse synchronization manifold .the synchronization state determined by is stable if the real part of all the eigenvalues of eqs .( [ eqn8]a ) and ( [ eqn8]b ) are negative .  for ( )=(1,1 ) or ( 1,-1 ) , ( and eliminating from eq .( [ eqn8 ] ) , we get consider a solution of the form which on substitution in ( [ eqn9.1 ] ) and after simple algebraic manipulation yields the relation we have considered the middle region of the piecewise linear function where most of the system dynamics is confined . for ( )=(1,-1 )the stable synchronization is achieved at the threshold value and for the two outer regimes ( ) we obtain the condition from eqs .( [ eqn10 ] ) and ( [ eqn10.1 ] ) one can obtain the threshold value of the coupling strength for the synchronized state . for the above chosen values and the feedback ( we obtain a transition from asynchronization to anis state as shown in fig .[ fig15 ] .the blue ( dark grey ) filled circles in fig .[ fig15](a ) corresponds to the stability condition for the middle regime and the black line to the stability condition of the two outer regimes of the piecewise linear function .this shows that the numerically obtained is regime exactly fits with the analytically obtained stability condition of the middle regime of the piecewise linear function where most of the dynamics is confined for the asymptotic synchronized state .similar transition curves are also observed for transition from asynchronization to cs in the case of [ fig .[ fig15](b ) ] .further , we have also considered a second form of coupling called subsystem coupling configuration , where both circuits share their feedback with the environment , while only one of the circuits is receiving a feedback from the environment . the schematic circuit block diagram for this coupling configurationis sketched in fig .[ fig16 ] and the state equations can be given as follows :  ,\\ r_{0}c_{0 } \frac{du_{2}(t)}{dt } & = & -\alpha^{\prime}u_{2}(t)+f[k_{f}u_{2}(t - t_{d})]+ \nonumber\\ \varepsilon^{\prime}_{1 } \beta^{\prime}_{2}v(t),\\ r_{0}c_{0 } \frac{dv(t)}{dt } & = & -k^{\prime}v(t)-\frac{\varepsilon^{\prime}_{2}}{2}[\beta^{\prime}_{1}u_{1}(t)+\beta^{\prime}_{2}u_{2}(t)],\end{aligned}\ ] ]  [ eqn11 ]  and the corresponding dimensionless equations for this configuration can be written as  ,\\ \dot{x}_2(t)&=&-\alpha x_2(t)+\beta f[x_{2}(t-\tau)]+\varepsilon_{1}\beta_{2}y,\\ \dot{y } & = & -ky-\frac{\varepsilon_{2}}{2 } ( \beta_{1}x_{1}+\beta_{2}x_{2}).\end{aligned}\ ] ]  [ eqn12 ]    here the circuits of and ( and ) are sharing their feedback with the environment but only the circuit of is receiving a feedback from the environment .the parameters are fixed as in the previous case .first we consider the case ( and obtain the same kind of transition from ips ( for ) to is ( for ) .the experimental and numerical snap shots of the time evolution of the coupled systems show ips for the coupling strength which is evident from figs . [ fig17](a ) and[ fig17](c ) , respectively .if we increase the coupling to larger values , both circuits attain is at as depicted in figs . [ fig17](b ) ( experimental ) and [ fig17](d ) ( numerical ) .the localized sets plots are again presented to confirm the existence of ips .the experimentally obtained attractors along with the sets are shown in figs .[ fig20](a ) and [ fig20](b ) where the sets are distributed over the entire attractors in the absence of any coupling and the corresponding numerical figures are depicted in figs .[ fig20](e ) and [ fig20](f ) for . for sufficiently large values of the coupling strength, one can observe that the sets are localized on the attractors confirming the perfect locking of the phases of the systems [ figs .[ fig20](c ) and [ fig20](d ) ] .the equivalent numerical figures are plotted in figs .[ fig20](g ) and [ fig20](h ) for .the transition from ips to is can again be confirmed by plotting the index cpr , cc and the changes in the maximal les as a function of the coupling strength . in the absence of the coupling, the systems are evolving without any synchronization ( with six positive les ) and so cpr and cc .for , the value of index cpr increases towards unity and reaches the value cpr , whereas cc becomes negative which confirms the onset of ips state [ fig .[ fig22](a ) ] .this transition is also confirmed from the changes in the les where the zeroth le of the coupled system becomes negative for this value of indicating the existence of ips which is evident from fig .[ fig22](b ) . on further increase of the coupling strength to the systems exhibit is transition where the cc becomes [ fig .[ fig22](a ) ] and from fig .[ fig22](b ) , we notice that except for the three positive les , all the other positive les become negative confirming the existence of is in the coupled time - delay system .finally , we consider the case ( . we obtain a transition from no synchronization to cs via ps in the coupled time - delayed electronic circuit .the experimental snap shots of the time evolution of both circuits are displayed in fig .[ fig23](a ) indicating that the systems exhibit ps and the numerically obtained figure is depicted in fig .[ fig23](c ) for .cs is observed between the circuits for larger value of the coupling strength and the experimental wave forms for suitable is shown in fig .[ fig23](b ) exhibiting cs . figure [ fig23](d ) shows the numerically obtained time traces of the two systems displaying cs for .ps between the circuits can be once again visualized by plotting the localized sets .figure [ fig26 ] demonstrates the experimentally observed attractors along with the sets . in figs . [fig26](a ) and [ fig26](b ) the sets are distributed over the entire attractor for lower values of coupling strength due to the absence of ps .the corresponding numerically obtained figures are plotted in figs .[ fig26](e ) and [ fig26](f ) for .if we increase the coupling strength to a sufficiently large value , the sets are localized on the attractor as depicted in figs . [ fig26](c ) and [ fig26](d ) confirming the phase locking of both systems .the corresponding numerically obtained figures are plotted in figs .[ fig26](g ) and [ fig26](h ) for .( color online ) ( a ) cpr ( continuous line ) , cc ( dotted line ) and ( b ) spectrum of maximal les for subsystem coupling configuration with in subsystem coupling configuration as a function of coupling strength . ]  in the absence of the coupling the systems are evolving freely without any synchronization with six positive les ( three for each systems ) and so cpr and cc . beyond ( ) ,cpr becomes which confirms the existence of ps which is depicted in fig ., [ fig28](a ) . at this value of , the zeroth le of the coupled system becomes negative which is shown in fig .[ fig28](b ) .if we increase the coupling strength further , the cc of the systems increases and reaches the unit value at which is evident from fig .[ fig28](a ) and for this value of coupling strength all the positive les ( except three positive les ) become negative confirming the existence of cs in the coupled time - delay systems [ fig .[ fig28](b ) ] .( color online ) ( a ) , ( b ) numerically obtained two parameter diagrams in ( ) plane shows various types of synchronization states in subsystem coupling configuration for ( ) and ( ) , respectively .color codes are similar to the fig .[ fig15 ] . ]the numerically obtained phase diagrams in the parameter plane of coupling strengths are plotted to identify the global picture of the regimes of different types of synchronization states of the coupled time - delay systems ( eq . [ eqn12 ] ) in fig .[ fig29](a ) and [ fig29](b ) for ( ) and ( ) , respectively . in these figures, we mark ps and ips regimes when cpr . also cs region is marked when cc and is region is marked when cc . in both figures the white color represents the desynchronized state , grey color in fig .[ fig29](a ) and red color ( dark grey ) in fig .[ fig29](b ) correspond to ips and ps regimes , respectively .is is represented by the pink color ( light grey ) in fig .[ fig29](a ) and the light blue ( light grey ) color indicate cs region in fig .[ fig29](b ) .we have also carried out a linear stability analysis for the above coupling configuration as in sec .[ sec : level3b ] and it gives a condition for the stability of the synchronized states as . from this relationone can obtain the threshold value of the coupling strength for different synchronized states .for example , we choose values as in sec . [sec : level3 ] and the feedback ( , we yield a transition from no synchronization to an is state which is plotted along with the numerically obtained synchronized region .the black points in fig . [ fig29](a ) correspond to the outer regimes of the piecewise linear function ( [ eqn2 ] ) and the blue ( dark grey ) filled circles indicate the stability state of the middle regime .similar to fig.[fig15 ] , the numerically obtained is regime exactly fits with the analytically obtained middle regime of the piecewise linear function .figure [ fig29](b ) shows the transition curves for the case of which indicates the transition from no synchronization to cs in the subsystem coupling configuration .in this paper , we have experimentally demonstrated the occurrence of various types of synchronization in coupled piece - wise linear time - delayed electronic circuits with threshold nonlinear function where the circuits are coupled indirectly through a common dynamic environment .we have carried out these studies in two different coupling configurations , namely mutual and subsystem coupling configurations . in both configurations , depending uponthe strength of the coupling and the nature of the feedback , we observe different types of synchronization transitions such as transition from ips to is and from ps to cs in hyperchaotic regimes .snapshots of the time evolution , phase projection and localized sets plots of the circuits observed from the oscilloscope confirm the various synchronization phenomenon experimentally .the corresponding numerical simulations are also presented in detail .further , the transition to different synchronization states can be verified from the changes in the maximal les , index cpr and cc of the coupled systems as a function of the coupling strength .also we have presented a detailed linear stability analysis to obtain synchronization conditions for different synchronized states . from our investigation , it is also clear that the concept of localized sets and recurrence quantification measures can be potentially used to analyze other experimental data and to identify the underlying synchronization transitions .it is also to be noted that the study carried out in this paper can also be extended to other environmentally coupled systems with and without delay because the coupling scheme does not alter the system dynamics in phase - space ( the structure of the systems is the same for both synchronized and desynchronized states ) . in particular, the dynamic coupling occurs in many biological systems ( where delay is inherent in general ) where the coupling occurs through many steps and direct coupling is often considered as a simplified concept to interpret the dynamics .an important example of such systems is the population of cells in which oscillatory reactions are taking place , which communicate via chemicals that diffuse in the surrounding medium .the work of r.s . , k.s . and m.l .has been supported by the department of science and technology ( dst ) , government of india sponsored irhpa research project .m.l . has also been supported by a dst ramanna project and a dae raja ramanna fellowship .d.v.s . and j.k .acknowledge the support from eu under project no .240763 phocus(fp7-ict-2009-c ) .  9 appeltant , l. , soriano , m. c. , van der .sande , g. , danckaert , j. , masser , s. , dambre , j. , schrauwen , s. , mirasso , c. r. &  fischer , i. 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Abstract: we experimentally demonstrate the effect of dynamic environment coupling in a system of coupled piecewise linear time - delay electronic circuits with mutual and subsystem coupling configurations . time - delay systems are essentially infinite - dimensional systems with complex phase - space properties . dynamic environmental coupling with mutual coupling configuration has been recently theoretically shown to induce complete ( cs ) and inverse synchronizations ( is )  in low - dimensional dynamical systems described by ordinary differential equations ( odes ) , for which no experimental confirmation exists . in this paper , we investigate the effect of dynamic environment for the first time in mutual as well as subsystem coupling configurations in coupled time - delay differential equations theoretically and experimentally . depending upon the coupling strength and the nature of feedback , we observe a transition from asynchronization to cs via phase synchronization and from asynchronization to is via inverse - phase synchronization in both coupling configurations . the results are corroborated by snapshots of the time evolution , phase projection plots and localized sets as observed from the oscilloscope . further , the synchronization are also confirmed numerically from the largest lyapunov exponents , correlation of probability of recurrence and correlation coefficient of the coupled time - delay system . we also present a linear stability analysis and obtain conditions for different synchronized states .


the rapid advances in communications technology and mobile network market have urged the research community to direct its efforts towards novel communications paradigms that can offer a better system performance .already resources as time , frequency and space have been exhaustively explored , leading to the suggestion of systems with high degrees of freedom including the random access ( ra ) , the multiple - input - multiple - output ( mimo ) and the orthogonal - frequency - division - multiple - access ( ofdma ) channel .one of the current tendencies , which extends the above , is to understand wireless networks as cooperative entities .conceptually , the idea of cooperative communications is that nodes should communicate with each other and try to adapt their transmission behavior in such a way , that their own as well as the entire system performance is improved .cooperation can be understood either as ( a ) coordination of node actions after certain exchange of messages between neighbours , or ( b ) as full cooperation by concurrent transmission of the same data towards a receiver node .the latter type of cooperation is possible , when full data and/or channel state information is exchanged between the cooperating elements .  for cellular architectures ,a recent suggestion in 3 g and 4 g systems has been the cooperative multipoint transmission ( comp ) , where neighbouring base stations ( bss ) exchange data and/or channel information over backhaul links . with use of appropriate precoding and by exploiting the extra information , interference within this - so called - bs cluster can be cancelled out for the users in service . the concept is also known in the literature as network mimo . theoretically , the idea originates from the multiple access channel ( mac ) with conferencing , whose capacity region has been derived by willems in for the discrete memoryless case .its gaussian extension is provided by bross et al . in .grouping of bss into cooperative clusters normally requires a centralized controller , where information over the channel quality network - wide is available .furthermore , the full benefits of cooperation in throughput and coverage are known to be achieved when both data and channel state information are fully known to all the transmitters of each cluster . however , such information exchange requires a considerable amount of feedback .this is often unrealistic because of the system resources reserved .the research community aims nowadays at resolving this major issue , .approaches to solve this investigate cases of partial , delayed , quantized and limited channel state information exchange between the cooperating entities , as more realistic towards a practical system implementation .there has been a large amount of both theoretical and practical investigations of the aforementioned concepts in the literature .considering mac with conferencing , starting from the work by willems and its gaussian variant by bross et al .numerous extensions of the basic model have been considered . in ,wiese et al . have extended the discrete memoryless region to the case of compound channels with an arbitrary number of channel states .maric et al . in have derived the region for the case of 2-transmitter mac with conferencing and 2-receivers .further extensions for general -transmitters and -receivers are provided in , while wigger and kramer have derived the region for the 3-user mimo gaussian mac with conferencing in . a work which departs from the standard model , but considers channels of 2-transmitters and 2-receivers with cooperation in either of the two or both ends of the channel , is found in by jindal et al . there , the transmitters are equipped with multiple antennas and jointly encode both messages using dirty paper coding .  related to comp ,the authors in investigate a 2-transmitter 2-receiver model and derive from willems capacity region an achievable rate region , when each bs sends a private and a common message to both receivers .further works treating models are , , , and .practical schemes to implement cooperation , combining private / common message and dirty paper coding are found in .optimal methods to form cooperation clusters between bss are investigated in using tools from integer programming and in based on the concept of sparse precoding . in hardjawanainvestigate transmitter structures for a system with cooperating bss  users . finally , many tutorial papers on implementation issues of comp deal with the lingering issue of channel state information feedback , that is necessary to achieve the promised cooperation benefits , , , .the existing research , as shown above , has been limited to models with a small number ( usually two ) of cooperating transmitters and one or more receivers .the aim is to derive performance benefits when cooperation is incorporated and to suggest optimal or practical transmission and reception schemes within the cluster of communicating nodes . on the other hand , the influence of cooperation and the inter - cluster interference effects on a network levelis not yet thoroughly investigated .we aim here at analyzing the performance of cooperation in terms of coverage , in a network with an infinite number of nodes .a first approach to this problem ( done in parallel to the authors work ) is found by akoum and heath in , where bss and users are positioned on the euclidean plane as independent poisson point processes .the concept of random clustering is used to define cooperation clusters , whose bss transmit using intercell interference nulling .although this work aims at analyzing the network in its generality , it does neither consider the issue of the optimal cluster choice ( because of the random way cooperation clusters are formed ) nor questions of power consumption .furthermore , the model does not deal with the novel interference field which arises when cooperation is applied network - wide .our work here primarily aims at providing a _ general framework _ to treat problems of cooperation between pairs of cellular nodes at a network level .the framework utilizes stochastic geometry to derive exact closed form or integral expressions and evaluate network performance when its nodes are assumed randomly scattered on the infinite plane .we consider scenarios of cooperation with data exchange between the optimal pairs of nodes .the present paper is focused on the case where only limited information over the channel is available such as phase shift and second neighbour interference .the reason for such approach is due to the fact that full information exchange , although may allow perfect adaptation of the transmitters to the channel state , it requires a considerable - often unrealistic - amount of inter - cell feedback exchange .the current analysis can however easily be extended to the case full channel knowledge and beamformer adaptation to the instantaneous channel state , but this is left for future investigation and comparison with the current work . more specifically , our basic model * assumptions * in this work are as follows :  * the bss are distributed on the infinite two - dimensional ( 2d ) plane as the realization of a poisson point process ( p.p.p . ) with intensity . *each bs is the center of a cell , defined as the locus of planar points closest to than any other atom of the point process .this is called a 1-voronoi cell .* one user is randomly positioned within each cell of the network .* at most two bss may cooperate for the service of a single user .these are the first and second closest geographic neighbours of each user , who then lies within their 2-voronoi cell . in this sense ,the bs cooperation pair is chosen by each user and not by a central controller entity .practical issues on knowledge acquisition of the bs - user distance are resolved by averaging over multiple instantaneous channel realizations to cancel out the random effect of fast - fading . *the cooperation scenario is similar to the concept of willems coding with conferencing .this means here that the original message is split into a private and a common part sent by the cooperating pair to the user .the common part is exchanged over the backhaul . *the power consumed per user is fixed to . no power constraint per bsis considered . *fast - fading from each bs to each user is modeled by a complex number , whose absolute value is an independent random variable ( r.v . ) with rayleigh distribution ( consequently its power follows the exponential distribution ) and whose angle is an independent r.v . with uniform distribution in .* limited channel state information is considered .specifically the analysis assumes that the transmitters know and exchange the angular shifts resulting from complex fading , so that the reception is in - phase at the user side .the purpose of such modeling assumption is to investigate benefits from scenarios of cooperation that avoid exhaustive information exchange between transmitters within a cluster . in our case ,the information exchange is limited to just the phase of the channel and not the amplitude , which is a considerable feedback gain .+ furthermore , we analyze the case where interference from the second base station is non - causally known , and the transmission from the pair is adapted to the latter by application of dirty paper coding .this case does not assume further channel knowledge ( i.e. does not refer to the standard combined zero forcing and dirty paper coding scheme in ) and its purpose is to give an optimistic evaluation of our limited feedback scheme .* a subset of users choses no cooperation ( siso channels ) whereas its complementary subset choses full coopeartion ( miso channel ) . * the choice to cooperate or not is dictated by a family of user - optimal policies , which is one of the main novelties of our work .these policies are strictly geometric and a user choses between them ( as binary actions ) based on the relative value of the ratio ( of distances from its first and second bs neighbor ) to the design parameter .the parameter influences the width of some geometric zone of cooperation between the cells of the two neighbouring bss and is left as an optimization variable .the major * results * can be summarized as follows :  * the chosen communication scenario and action policies result in a specific expression .* the coverage probability of a typical planar point is derived , given an threshold .the coverage further depends on the value of geometric parameter .explicit expressions are provided . *the network - wide interference is modeled as a shot - noise field , which explicitly takes into consideration the special influence of cooperation between bss .the interference r.v . is provided together with its laplace transform and expected value .it can be seen that although different cooperation schemes lead to the same expected value for interference , the same does not hold for its higher moments . *the work considers the possibility of interference elimination of the non - beneficial signal coming from the second closest neighbour .this can be achieved when the interference created to the typical user is measured and known to both miso transmitters . in this case dirty paper coding ( dpc ) can be applied , described in . * the coverage expressions for both scenarios ( with and without dpc )are numerically evaluated and the benefits of cooperation with geometric policies are illustrated in plots .simulation results are compared with the theoretical ones to guarantee the validity of the results . * * it is concluded that : * cooperation ( in the sense of conferencing here ) is substantially beneficial for network coverage even with limited channel adaptive schemes and when cooperation is restricted between two neighbouring bss .dpc significantly improves these benefits .* coverage regions change their shape when varying parameter . due tothe geometric policies suggested , advantage is always given to planar areas closer to the cell boundaries .let us stress that beyond these particular results on cooperation with limited channel knowledge , the main contribution of the present paper is the new framework in which these questions are analyzed .the k - voronoi diagram allows one to identify the region where certain k - tuples of antennas constitute the optimal -cluster .poisson stochastic geometry allows one to analyze e.g. the probability of coverage but potentially other performance measures as well .this framework can be extended to optimal cooperation by more than two base stations , to a variety of cooperation schemes and to a variety of assumptions on channel knowledge .our work is organized as follows .section [ sectionii ] presents the general model of cooperation .specifically , it explains the geometric notions , gives the modeling assumptions and describes the conferencing power - split scenario .the last part presents the policies of no / full cooperation used and the resulting cooperation zones at the cell boundaries , which are controlled by parameter .section [ sectioniii ] provides the derivation of the expressions for the coverage probability using stochastic geometry .the laplace transform of the full cooperation signal is derived and a probabilistic comparison with the siso signal is given . the interference is described as a r.v . and its properties and expected value are provided .the section includes the variation of the above expressions when interference from the second closest neighbour is eliminated by use of dpc .it concludes with the derivation of the expressions for coverage .section [ sectioniv ] presents a list of pros and cons for the total coverage , when the proposed policies are applied to the network , compared to the traditional model without cooperation .section [ sectionv ] illustrates the numerical evaluation of the expressions , as well as their comparison with simulation results .finally section [ sectionvi ] contains the conclusions of our work .a large amount of supplementary material , as well as proofs that are omitted from the main text can be found in the appendix .* notation : * we will use capital letters for random variables ( r.v.s ) , boldface small letters for vectors ( including planar points ) and small letters for real / complex quantities . capitalcalligraphic letters are reserved for sets ( except for the notation on signal and interference ) .the set of real numbers will be denoted by and that of complex numbers by .the unit imaginary number is denoted by .the asterisc notation denotes the complex - conjugate of .the distance between two planar points and is denoted by .for the model under study , the considered base stations ( bss ) are equipped with a single antenna and are positioned at the locations of atoms of the realization of a planar p.p.p . with intensity , denoted by .a planar tessellation , called the 1-voronoi diagram , separates the plane into subregions with the properties that : ( a ) exactly one subregion corresponds to each .the subregion with center is denoted by and is called a * 1-voronoi cell*. ( b ) considering euclidean distance , which we denote by , the subregion consists of all planar points closer to than to any other atom in , in other words , .( c ) the union of all subregions is the plane .the intersection of every pair of cells is of lebesgue measure .when the 1-voronoi cells of two atoms share a common edge , one says that they are * delaunay neighbours*. a dual graph of the 1-voronoi tessellation called the delaunay graph can be formed if all delaunay neighbours are connected by an edge .another tessellation of the plane , called the 2-voronoi diagram , separates the plane into subregions with the properties that : ( a ) exactly one subregion corresponds to each unordered pair , , .this subregion is denoted by and is called a * 2-voronoi cell*. ( b ) consists of all planar points with the property that and , . in other words , it consists of all planar points geographically closest to and second - closest to or the other way around .( c ) the union of all subregions is the plane .the intersection of every pair of cells is of lebesgue measure . rather interestingly , the 2-voronoi cell is non - empty if and only if the atoms and are delaunay neighbours . for formal definitions , proofs and further discussions ,the reader is referred to appendix a and the publications and .the 1- and 2-voronoi tessellation in a square with area 1 [ based on the uniform positioning of 10 atoms is shown in fig .[ fig : voronoi1 ] and [ fig : voronoi2 ] .[ fig : voronoiexmp ]  in the current work , we will consider a geometric cooperation scenario based on the following assumptions :  * the bss are connected via a backhaul network , which consists of links of infinite capacity , one for each pair of delaunay neighbours .the infrastructure in the network thus constitutes a delaunay graph . *exactly one user with a single antenna is located randomly at some point within the 1-voronoi cell of its bs and we write . because of this choice , the user positions do not follow a p.p.p . distribution . * each user may be served by either one or two bss . in the latter case , these are the atoms in , which are its * first * and * second closest geographic neighbours*. in the former case , it is just the first closest neighbour .we use the notation and respectively when referring to these neighbours of .as already mentioned , we assume that the two bss can communicate by information exchange over the backhaul network . * from the point of view of a bs located at , we refer to the user in its 1-voronoi cell as the * primary user * and to all other users served by it but located outside the cell as the * secondary users*. these constitute a set , with cardinality that ranges between zero and the number of delaunay neighbours , depending on the users position relative to .the modeling assumption of exactly one user per bs is similar to the bipolar models suggested in and .this choice is also found in the work of venkatesan for the performance of cooperation schemes in cellular networks .it is useful because it does not pose questions about user scheduling decisions .it is further justified by the fact that , in practical communications systems , users are split among orthogonal dimensions of time and frequency ( e.g. by applying ofdma techniques ) and each frequency slot per time is allocated to exactly one user per cell .the distance between user and its first bs neighbour is equal to .this is the maximum radius of an open ball with center , which is empty of atoms in ( see prop .[ circlev1 ] in appendix a ) .the second nearest neighbour can only be one of the atoms in whose associated voronoi polygons are adjacent to .the distance between user and its second bs neighbour ( say ) , .this is the maximum radius such that contains and only in its interior and on its boundary , or is empty and contains both on the boundary ( in the poisson case , the last scenario has probability .we refer also the reader to prop .[ circlev2 ] in appendix a ) .an example of network for the cooperation scenario under study is illustrated in fig .[ fig : examplenet ] .the communications scenario in this work uses the following idea .when two bss cooperatively serve a user in the downlink , its signal is split into one * common part * served by both , and * private parts * served by each one of the serving bss .the common part contains information known to both transmitters , that is exchanged between them over a reliable conferencing link . the idea is originally based on the seminal work of willems , where the author derives the capacity region of the multiple access channel ( mac ) with conferencing , in the case that two transmitters cooperate over backhaul links to provide service to a single receiver .the proof of achievability is precisely based on such a splitting of the user s signal .the capacity region for the gaussian noise case has been derived by bross et al . for the model in and maric et al for the model in . in both casesthe splitting idea is used for achievability ( the reader is referred also to appendix c ) .a variation of the idea has been applied in a network - mimo setting by zakhour and gesbert for the model in , where the beamforming vectors take the role of signal weights and define the power - split - ratios .  in this workwe assume that only the information over channel phase shift is available at the transmission pair ( and the interference from the second base station , in the dpc case later on ) , so that full adaptation of the signal to the instantaneous channel conditions is not possible .the purpose is to analyze schemes with limited information exchange . for each primary user located within the 1-voronoi cell of atom , consider a signal to be transmitted .the user signals are independent realizations of some random process with power = p>0 ] , .the signal destined for user in the downlink is divided into two parts :   * a * private * part sent to from its first bs neighbour , denoted by .the second neighbour does not have a private part to send . *a * common * part sent by both and , which is denoted by .this part is communicated between both bss over the backhaul links .the two parts are uncorrelated random variables , in other words =0 ] , which is named the * power - split - ratio*. this ratio has in general a different value for different users .hence , = \left(1 - 2a_i\right)p , & \mathbb{e}\left[\left|s_{i}^{(c1)}\right|^2\right ] = a_ip , & \mathbb{e}\left[\left|s_{i}^{(c2)}\right|^2\right ] = a_ip . \end{aligned}\ ] ] we will denote the infinite vector of these ratios by and the vector of ratios omitting the entry , by .then bs consumes in total for user , while bs consumes for the same user .each bs transmits a total signal .this signal is a superposition of :  * the private and common message for its primary user : . *the common message for all its secondary users : .we write altogether    the bs signals are transmitted over the wireless medium to reach the users and experience real valued path - lost and complex valued fast - fading . for the former , the signal transmitted by bs is received at the location of user , multiplied by the path - loss factor which depends on the distance . the fast - fading is written as a product of its amplitude and the exponential shift , i.e. , where .its power is a realization of a unit - mean exponential random variable ( r.v . ) , whereas its phase shift , as a realization of a unifom r.v . within the interval .consequently , the total power gain from the first and second neighbour and to user is equal to    while the total power gain from the first and second neighbour , of some other user to is    with . regarding the phase - shift due to fast - fading, the signal transmitted by will be multiplied by the gain amplitude and the complex phase shift .similar factors will appear at the signals from , and .altogether , the signal received at user is and is equal to    the noise is a realization of an independent r.v . of distribution . in the above the sum of signals intended to users is the interference received by user .consequently , the for user takes the form    in the above expression , an extra term , which includes the square root of the channel gain product , appears in both the beneficial signal ( [ sinrui2a2th ] ) and the interfering ones ( [ sinrui2a4th ] ) from all s , .its existence is due to the common part transmitted from both bss serving one user .the signals and are scaled versions of each other . as a result, there is correlation at the reception of these signals transmitted by two different bss .the term would not exist in case and were chosen independently .  in the expression above ,the maximum beneficial signal for is equal to .this is achieved when the two angles are equal and as a result the two parts of the common signal from and are received in - phase after fading through the channel .this knowledge over the phases should be available at the transmitters side .considering the interference term and the angular influence on it , control over the phase shift is not possible because transmission is usually done using local information and aims at the benefit of users within the local cell .hence , the angles , are realizations of independent and uniformly distributed random variables . as a result , it makes sense to consider the interference value for in expectation over the reception angles of the signals related to all . in such case , since = 0 ] such that the policies are called `` user - optimal '' and `` geometric '' because the optimal choice to cooperate or not depends on the relative position of each user to its two closest bss .the ratio  ] fully determines this probability .[ lemrho ] the probability of a user on the 2d plane , not to demand for bs cooperation for its service is = \mathbb{p}\left[r_1\leq \rho r_2\right ] = \rho^2.\end{aligned}\ ] ]   in the current subsection we will derive a general expression for the distribution of the signal power when full cooperation is applied . furthermore , certain properties of this distribution will be provided .the proofs of the results in this paragraph are provided in appendix g.  [ lemz ] given the r.v.s , , the laplace transform ( lt ) of the r.v . is equal to where and for .furthermore , its p.d.f . is square integrable and its expectation is equal to & = & \sqrt{\frac{1}{\mu_1\mu_2}}\left(\frac{\pi}{2 } + \frac{\mu_1+\mu_2}{\sqrt{\mu_1\mu_2 } } \right).\end{aligned}\ ] ]  we further provide an interesting property of the r.v . of the fading for , in the special case of ( the expression given in ( [ rvz ] ) ) , with relation to the r.v . , which is the case for no cooperation .we use the notion of _ stochastic ordering _p.411 ) based on which the r.v . stochastically dominates the r.v . and we write , if \leq \mathbb{p}\left[y > t\right] ] lies within the region . from lemma [ lemrho ]this probability is equal to , whereas with probability the user requests for .then we consider that each bs is associated with a * binary * r.v . such that    each bs is related to an independent .then :  * if an independent mark is associated with the bs .the mark is equal to , and this signal is treated as interference at the typical location , where .the signal has to traverse a distance of from the closest neighbour of user , as shown in fig .[ fig : intref ] . *if an independent mark is associated with the bs .this is the case of , where the interfering signal due to user is jointly emitted from its two closest neighbours . here, we make the * far - field approximation * , where the distances of the two cooperating atoms to the typical location are treated as equal .such a heuristic is allowed , since the two closest neighbours of user are expected to lie close to each other and at the same time , outside of the ball of radius . in this sense we can expect that the error is very small . based on this approximation , bss with primary users requiring ,are considered to emit the entire signal , .it is interesting to notice that , the r.v . , related to the mark follows the exponential distribution , or equivalently the distribution , with expected value , whereas the r.v . related to , follows the , with the same expected value . in other words ,the path - loss of the interfering signals is in expectation equal in both cases ( of either or for the interfering user ) . in this far - field approximation ,the interference random variable takes the expression    where the first two terms come from the interference created by the second neighbour lying on the boundary of the ball .finally we derive the lt of the interference variable .[ thlapli ] the lt of the interference random variable for the model under study , with exponential fading channel power ( rayleigh fading ) , is equal to where   it can be shown that the lt of the interference random variable in ( [ ltinterfd ] ) is a non - decreasing function in and and non - increasing function in .a direct consequence of this is that the lt of the interference takes its maximal value for .this is reasonable based on the laplace - stieltjes transform ordering .the argument is that for any ,  .the larger the smaller the interference .similarly , the larger the distance to the second neighbour , the smaller the interference , because a larger empty ball of interferers around the typical location is guaranteed .finally , we give the expression of the expected value for the interference r.v . & = & \frac{p}{\left(\beta-2\right)r_2^{\beta}}\left(\beta-2 + 2\pi\lambda r_2 ^ 2\right).\end{aligned}\ ] ]  the expected value of the interference r.v . is * independent of the parameter *. this means that all scenarios of cooperation regions for any  ] is equal to in this expression , is the lt of given in ( [ ltinterfd ] ) , ( [ eachid ] ) and is the lt of the general fading r.v . given in ( [ ltz ] ) .for the case of dirty paper coding , , should be replaced by given in ( [ ltinterfdcancel ] ) .significantly higher gains are obtained in the case where some further system knowledge is available at the bs side .specifically , the information over second neighbour interference may eliminate the negative effects from ( and obviously ) and result in high gains in coverage , in the entire domain of .this is illustrated in the plot of coverage in fig .[ fig : covgaininterf ] . from the plotwe get a maximum gain in coverage of over 16% between to .the absolute difference percentage is shown in fig .[ fig : covdiffgaininterf ] and the optimal value of for each threshold in fig .[ fig : optthreshinterf ] .[ fig : dpcgain ]the coverage probability expression in ( [ tint1])+([tint2 ] ) can provide intuition on the change in coverage probability when applying different -dependent 2-cell cooperation policies . in the followingwe summarize the pros and cons of cooperation , given the model of the previous sections .we would like to emphasize here , that all benefits result from pairwise cooperation with use of limited feedback exchange between the transmitters ( only channel phase ) .* pros : * the gains of the cooperation scheme result from four reasons .* the specific coding scheme enabled by conferencing , provides an extra term of  for the beneficial signal in ( [ sinrui2a2 ] ) .given that the channel fast fading r.v.s follow exponential distribution , and for the symmetric case the resulting variable is larger than in the stochastic ordering , as shown in lemmas [ stochord1 ] and [ laplaceord ] . from an engineering perspective, the user transmission power is equally divided between the two closest bs neighbours , who transmit the same signal with power . since the signalsadd - up coherently at the receiver at the typical location , this can provide an increase in the received signal power , but depends also on the distances of the two bss from the user .such benefit can be achieved only when the two cooperating bss transmit exactly the same data in a synchronous way to the receiver .the codeword of the second neighbour is linearly dependent to the codeword of the first neighbour .+ evaluating the expression in ( [ ezpdf ] ) for and , we get an expected value }{2}=1+\pi/4\approx 1.8>1=\mathbb{e}\left[g\right] ] is strictly larger than =\frac{1}{2\sqrt{\lambda}} ] , can define the shape of the coverage areas . a change of the parameter results in a change of the shape ,so that certain planar points not covered when can be covered for some other value and the other way round . in this way , by adapting the parameter , we can shape the coverage of the cell depending on the service needs . *the parameter can be chosen optimally as , so that the geometric policy maximizes coverage .the optimal value of depends on the set of model parameters .the expression in ( [ tint1])+([tint2 ] ) can not be shown to be concave in , so that existence of local optima can not be excluded .the global optimum can be found by numerical evaluation , as shown in fig .[ fig : covt01full ] for a specific example . fig .[ fig : zona06 ] exhibits the optimal planar cooperation policy for the considered example .[ fig : voronoireg ]  * cons : * the negative effects of the cooperation scheme result from two reasons .* the parameter of the geometric policy may enforce a user at a planar point to ask for cooperation , even when .the reason is that geometric policies do not adapt to the actual fast - fading realization and depend only on the ratio . since is a design parameter , the optimal is expected to adapt the policy to such events .* by comparing the tail probabilities of the random variable and the exponential random variable we can find many examples of the pair , such that there exist values of the signal threshold , above which it holds <\mathbb{p}\left[g > t\right].\nonumber\end{aligned}\ ] ] in other words , above a certain threshold , the contribution of the exponential signal power to the coverage probability , may be more considerable than the signal power .this is very important if we add also the fact that , as shown in subsection [ paraginterference ] , the interference r.v .has an expected value which is independent of the cooperation parameter .a consequence is that , there are values of the threshold , above which the coverage probability of the everywhere scheme is optimal and cooperation turns out to be suboptimal . an example plot over the threshold , with parameter values , and is shown in fig . [ fig : tailprob ] .it is clearly shown how the tail probability of the signal power is higher than the one , for .the coverage probability given in theorem [ covprob ] as a sum of the two integrals in ( [ tint1 ] ) and ( [ tint2 ] ) can be numerically evaluated using matlab .the evaluation uses routines of nested integration for triple and quadraple integrals .  to guarantee the validity of the analysis and of the derived results , we have further developed a simulator , which builds a p.p.p . of intensity for the positions of the bss , within a finite square observation window of surface ( size of the x axis ) .for each realization , a poisson generator produces a random number of atoms with expected value =\lambda_{sim } w = 20\ \mathrm{atoms} ] ) is low , so that the simulations do not consider the interference created by atoms outside the window of size .the lower the path - loss exponent , the stronger the interfering signal for large distances from the emitting base station . in this sense, a small simulation window underestimates the total interference created .when the size of the window - and consequently the expected number of point process atoms - increases it can be verified that the gap diminishes .  in fig .[ fig : simulthetacurves ] the coverage probability curves using the two different models ( [ sinrui2a1th])-([sinrui2a4th ] ) and ( [ sinrui2a1])-([sinrui2a3 ] ) are compared .the curves are produced by the simulator , in order to evaluate the quality of the approximation , when the expectation over is taken at the interference term .it is evident from the figure that the two models produce almost identical results and hence the approximation is valid for the model under study , since it can simplify the analysis , without considerably affecting the results .  in fig .[ fig : simulb24noapproax ] , we further evaluate , for , the difference between numerical coverage curves and simulated without this approximation , in order to get a comparison with the coverage of the original model . it can be observed that although the curve for is the same as the curve with the approximation in fig .[ fig : simulb4 ] ( as expected ) this does not hold for the full and optimal cooperation simulation curves .the coverage probability deteriorates faster than the analytical results for higher thresholds , which suggests that the approximation underestimates the effect of interference .the main reason is the fact that , in the case for some user not at the typical location , it can occur in our model , that * it can be served as secondary user by the bs closest to the typical location ( )*. in such a case , its signal creates the highest possible interference at the typical location , since it stems from the closest possible distance , i.e. . since the first neighbour interference has such a negative effect , it is therefore reasonable to prefer cooperation policies , which diminishes it .this can be achieved by increasing the information at the transmitter side and applying dirty paper coding ( dpc ) techniques .   in the plots of fig .[ fig : simulb4 ] and fig .[ fig : simulb24noapproax ] , the coverage probability is plotted with respect to the threshold value .the threshold varies between  ] of simulated atoms and the given planar area is not enough to sufficiently approximate tha analytical curves , and a larger area should be considered .this gives a further arguement in favor of the use of point processes and analytical / numerical evaluation , compared to simulations , since the latter may provide erroneous results due to edge effects .[ fig : b2p5 ]in this paper , we have proposed a general methodology to treat problems of cooperation in cellular networks , in the case where data exchange is allowed only between pairs of nodes .the framework developed uses tools from stochastic geometry to model the random positions of nodes on the plane and calculate the network performance with reference to a typical location .  in our workthe focus has been on the evaluation of the coverage probability , using a specific cooperation scheme .this scheme is based on the idea of conferencing by willems , where two transmitters encode a common message after exchange of information over the backhaul .we give to the service of each user two possibilities .either to be served by its closest bs or to ask from the two closest ones to cooperate .the latter choice splits the total user power between these two bss , which encode the same common message .the scheme results in an additional term on the beneficial received signal due to the correlated nature of the two transmissions .the treated cooperation scheme assumes only the knowledge of the fast - fading angular shift at the cooperation pair , so that transmission is done in a way that the two signals add up in - phase at the receiver and the maximum signal power benefit is obtained .the purpose of this assumption on limited channel state information exchange was to investigate possible coverage benefits , that can result without the necessity of exhaustive channel state feedback and full - channel beamformer adaptation .the stochastic geometry framework introduce here , can be applied to the evaluation of other types of cooperation , as is the case where the transmitters have full knowledge of the instantaneous channel gains and adapt their functionality to it .treatment of such cases is very important , since it may reveal higher performance benefits , at the cost of additional information at the transmitter side , and is our future research direction .  to investigate cooperation ,we have defined a set of geometric user - defined policies for the user actions .depending on the ratio and some design parameter  ] since the noise and the encoded signal are uncorrelated independent r.v.s , in ( v ) we consider the codeword to be the sum of the * private * and * common * message r.v.s by applying superposition coding ( see also , and ) , in ( vi ) we further expand the product in the expectation and consider that the primary and common messages are all uncorrelated r.v.s , ( vii ) applies the cauchy - schwarz inequality and ( viii ) splits the maximum transmission power , such that is given for the private and for the common message .an important notice is that the inequality in the application of the cauchy - schwarz holds with equality , if and only if and are linearly dependent . to guarantee this , we consider , where .  given ( [ momenty ] ) the entropy of is bounded by using ( * ? ? ?* th.9.6.5 ) , which states that the normal distribution maximizes the entropy for a given variance . applying this inequality to ( [ ig ] ) above, we reach the result for some fixed .   given a reference user and a fixed cooperation pattern for the rest of the network , it is optimal for the two nearest bss either to fully cooperate or not to cooperate at all for the user service , depending on the value \label{ratiol}\end{aligned}\ ] ] of their relative channel power ratio .more specifically , [ regions ]  to prove the first part of the statement it is easily seen that the signal from ( [ sinrui2a2 ] ) is either monotone increasing or decreasing .the first partial derivative of the sinr over gives and the monotonicity depends on the sign of the expression in parenthesis. for positive sign , the sinr is increasing in and the full cooperation is optimal , otherwise no cooperation is optimal . roots of the above quadratic polynomial are and and the polynomial is negative inbetween and positive otherwise on .furthermore , only the positive roots can be accepted since is non - negative . hence results in full cooperation being optimal .optimality of no cooperation is proved for the negative sign of the expression in parenthesis .let us use the above result with a specific expression for the total gain . specifically , we consider no fast - fading , so that , and the exponential expression for the path - loss , so that , .the distances of the first and second neighbour to user are and respectively .it is rather interesting in the above that the results hold irrespective of the cooperation strategy chosen by the other users .the decision to cooperate or not , is dependent on the user position within the related 2-voronoi cell and the fading characteristics .this can be true , only when we assume that a bs has an unbounded power budget to consume .however , in cases where the number of secondary users is large , we can not expect that the bs can use full cooperation modus for all such users , due to high power consumption expenses .it is reasonable then to ask , which is the optimal bs cooperation scheme under specific power constraints per bs .   given a fixed parameter and locations of and on the 2d plane and respectively , the geometric locus of points that satisfy for is a * disc * , whose circle describing the boundary is given by the equation and has center and radius , where is the euclidean distance between and .the expression can be rewritten by using the expressions on the euclidean distances ( [ distancem ] ) between a point of the 2d plane and the positions of the bss and ,   simply by setting in the last expression , we reach ( [ linecoop ] ) . for we can divide by .we further add at both sides of the equality the term so that expressions of squares of sums are formed and we reach   this is obviously the equation describing a circle with center and radius given in the proposition . to show why the degenerate case for is a line which passes over the 1-voronoi boundary we argue as follows .the 1-voronoi boundary is by def .[ defv1 ] the perpendicular bisector of the line segment , so that a point on the boundary is .take any other point on the boundary .the tangent of the vector defined by these two points , equals the tangent of the line segment rotated by      to derive the p.d.f .of the distances to the first and second bs neighbour of the typical location , we will make use of prop .[ circlev1 ] and prop .[ circlev2 ] from appendix [ appa ] . based on these , to determine the first closest neighbour , we should find the largest ball of radius which is empty .this ball meets the first neighbour on its boundary .this implies by prop .[ circlev1 ] that belongs to its 1-voronoi cell .we further enlarge the ball until the second closest neighbour is met on the boundary of , and the first ( and no other ) is contained in its interior , so that by prop .[ circlev2 ] belongs to the 2-voronoi cell of these two bss . in this way ,the annulus starting from the ball of radius and reaching the ball of radius is the largest empty .  given a poisson p.p .of intensity , the p.d.f . of the distance between the typical location and its first closest neighbour , equals [ pdfn1 ] the expected value of the distance equals & = & \frac{1}{2\sqrt{\lambda}}. \label{expr1}\end{aligned}\ ] ] [ lemr1 ]  we can derive the p.d.f . from the cumulative distribution of , in other words from the expression of the probability  ] .we can easily conclude that the r.v . follows the rayleigh distribution with p.d.f . and expected value =\sqrt{\frac{\pi}{4\mu_i}} ] .we further plot the p.d.f . for values in fig .[ fig : zplot ] .  [ stochord ] given , and the two r.v.s and from ( [ rvz ] ) , the following stochastic ordering inequality holds \leq \mathbb{p}\left[g > t_l\right] t_l\leq t^* \mathbb{p}\left[z_{r , r}>2 t_h\right ] \geq \mathbb{p}\left[g > t_h\right] t_h > t^* ] .using the same arguments as above , since is positive for the , there will be exactly one such that a level crossing occurs .that means for : \leq \mathbb{p}\left[g > t_l\right] ] .an example is plotted in fig .[ fig : so1 ] for .* [ lemma [ laplaceord ] ] * we first establish a simple result .it can be easily shown that for the p.d.f . . taking the laplacetransform and using the above equality for the p.d.f.s we get =\mathbb{e}_z\left[e^{-\frac{s}{2}2t}\right]=\mathcal{l}_z\left(\frac{s}{2}\right) ] and hence the term in the parenthesis is .this concludes the proof .[ [ proofs - and - supplementary - material - for - the - interference - random - variable ] ] proofs and supplementary material for the interference random variable ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~  * [ theorem [ thlapli ] ] * from the definition of the laplace transform , \nonumber\\                           & = & \mathbb{e}_{\phi,\left\{g_n\right\},\left\{g_{n1},g_{n2}\right\},\left\{b_n\right\}}\left[e^{-s\left(r_{2}^{-\beta } g_{2}b_2 + r_2^{-\beta}\frac{g_1+g_2}{2 } \left(1-b_2\right)\right ) } \prod_{\mathbf{z}_n\in\phi\setminus\left\{\mathbf{b}_1,\mathbf{b}_2\right\}}e^{-s\left(d_{n}^{-\beta}g_{n } b_n + d_n^{-\beta}\frac{g_{n1}+g_{n2}}{2 } \left(1-b_n\right)\right)}\right]\nonumber\end{aligned}\ ] ] observe now that the random variables , , are independently distributed and independent from the point process .furthermore , the contribution of each atom outside the ball of radius is \cdot\mathbb{e}_{g_{n1},g_{n2}}\left[e^{-sd_n^{-\beta}\frac{g_{n1}+g_{n2}}{2 } \left(1-b_n\right)}\right]\right]\nonumber\\ & = & \rho^2\mathbb{e}_{g_n}\left[e^{-sd_{n}^{-\beta}g_{n}}\right ] + \left(1-\rho^2\right)\mathbb{e}_{g_{n1},g_{n2}}\left[e^{-sd_n^{-\beta}\frac{g_{n1}+g_{n2}}{2}}\right]\nonumber\\ & = & \rho^2\mathcal{l}_g\left(sd_{n}^{-\beta}\right)+ \left(1-\rho^2\right)\left(\mathcal{l}_g\left(\frac{sd_{n}^{-\beta}}{2}\right)\right)^2\nonumber\\ & \stackrel{(a)}{= } & \rho^2\frac{1}{1+sd_n^{-\beta}p}+\left(1-\rho^2\right)\frac{1}{\left(1+sd_n^{-\beta}\frac{p}{2}\right)^2}\nonumber\end{aligned}\ ] ] where ( a ) comes fromt the lt of the random variable , .similarly for the atom on the boundary of the ball , the contribution is given from the above expression by replacing by .the lt of the interference finally takes the expression \nonumber\\ & \stackrel{(b)}{= } & \mathcal{l}_{\mathcal{j}}\left(s,\rho , r_2\right)\cdot e^{-2\pi\lambda\int_{r_2}^{\infty } \ !\left(1-\mathcal{l}_{\mathcal{j}}\left(s,\rho , r\right)\right)r \ , \mathrm{d } r } \nonumber\end{aligned}\ ] ] where ( b ) comes from applying the laplace functional expression for the p.p.p .using polar coordinates ( * ? ? ?* proposition 1.5 , pp .18 - 19 ) , for radius ranging from to , since it has already been identified that the second closest neighbour lies at the boundary of the ball of radius .hence is increasing in . from the expression in ( [ ltinterfd ] ) and using the above result on it is easily seen that the partial derivative over is also non - negative .the monotonicity comes from the following arguement .variable is found in the expression only as a product of the form , hence is increasing in ( laplace transform ) . in the exponential term , is the lower limit of the integral and does not appear at some other position .hence , the larger the the smaller the value of the integral , since the quantity can be shown to be non - negative . as a result ,the value of the negative exponential term is non - decreasing with increasing and this concludes the proof .the lt of the interference takes its maximal value for , where is applied in the entire voronoi cell of each interferer .this is reasonable based on the laplace - stieltjes transform ordering .the argument is that for any ,  . the larger the the less the interference .similarly , the larger the distance to the second neighbour , the less the interference , because a larger empty ball of interferers around the typical location is guaranteed .    [ ei ] the expected value of the interference random variable , whose lt is given in ( [ ltinterfd ] ) is equal to & = & \frac{p}{\left(\beta-2\right)r_2^{\beta}}\left(\beta-2 + 2\pi\lambda r_2 ^ 2\right),\end{aligned}\ ] ] and is * independent of the cooperation parameter *. for the special case of path - loss exponent , the expression simplifies to & = & \frac{p}{r_2^{4}}\left(1+\pi\lambda r_2 ^ 2\right).\end{aligned}\ ] ]  by taking the derivative of the lt for the interference r.v . in ( [ ltinterfd ] )we get =-\left.\frac{\partial\mathcal{l}_{\mathcal{i}}\left(s,\rho , r_2\right)}{\partial s}\right|_{s=0}$ ] .the expression in ( [ eispecial2 ] ) results by direct substitution in ( [ eigeneral ] ) of or can alternatively be derived by and the lt expression in ( [ ltinocoop ] ) .\right]\nonumber\\             & = &  \int_0^{\infty } \ !\int_{\frac{r_1}{\rho}}^{\infty } \ !\mathbb{p}\left[g_1>r_1^{\beta}t\left(\sigma^2+\mathcal{i}\right)|r_1,r_2\right ] f_{r_1,r_2}\left(r_1,r_2\right ) \ ,\mathrm{d } r_2 \ ,  \mathrm{d } r_1\nonumber\\             & + &  \int_0^{\infty } \ !\int_{r_1}^{\frac{r_1}{\rho } } \ !\mathbb{p}\left[z_{r_1,r_2}>2t\left(\sigma^2+\mathcal{i}\right)|r_1,r_2\right ] f_{r_1,r_2}\left(r_1,r_2\right ) \ ,\mathrm{d } r_2 \ ,  \mathrm{d } r_1\nonumber\\             & : = & q_{c,1}\left(\rho\right ) + q_{c,2}\left(\rho\right),\nonumber\end{aligned}\ ] ]     & \stackrel{(a)}{= } & \mathbb{e}_{\mathcal{i}}\left[\mathbb{p}\left[g_1>r_1^{\beta}t\left(\sigma^2+\mathcal{i}\right)|r_1,r_2,\mathcal{i}\right]\right]\nonumber\\ & \stackrel{(b)}{= } & \mathbb{e}_{\mathcal{i}}\left[e^{-\frac{r_1^{\beta}}{p}t\left(\sigma^2+\mathcal{i}\right)}|r_1,r_2\right ]  \nonumber\\ & \stackrel{(c)}{= } & e^{-\frac{r_1^{\beta}}{p } t\sigma^2}\mathcal{l}_{\mathcal{i}}\left(\frac{r_1^{\beta}}{p } t,\rho , r_2\right).\nonumber\end{aligned}\ ] ]   for the second integral , we will use the proposition in ( * ? ? ? * prop.5.4 ) or alternatively in ( * ? ? ? *prop.2.2 ) for the expression of the coverage probability for general fading . the conditionsto apply the proposition is that :   then we get that & = & \int_{-\infty}^{\infty } \ !e^{-2j\pi\sigma^2 s}\cdot \mathcal{l}_{\mathcal{i}}\left(2j\pi s,\rho , r_2\right)\frac{\mathcal{l}_{z}\left(-j\pi s / t,\frac{r_1^{\beta}}{p},\frac{r_2^{\beta}}{p}\right)-1}{2j\pi s}\ ,  \mathrm{d } s,\nonumber \label{genfade}\end{aligned}\ ] ] where is the lt of the general fading r.v . , whose expression is provided in ( [ ltz ] ) and is the lt of the interference r.v . , provided in ( [ ltinterfd ] ) . in the case of dpc , the lt for the interference r.v .is provided in ( [ ltinterfdcancel ] ) . substituting the probability expressions in the above and respectively and using the joint distribution over from ( [ pdfn2f ] ) in lemma [ lemjointd ], we conclude the proof .observe that for the above expression of in ( [ tint1 ] ) and ( [ tint2 ] ) equals and coincides with the expression in ( * ? ? ?* eq . ( 2 ) ) , given the presence of an interferer at a distance away from the typical location .

Abstract: cooperation in cellular networks has been recently suggested as a promising scheme to improve system performance , especially for cell - edge users . in this work , we use stochastic geometry to analyze cooperation models where the positions of base stations ( bss ) follow a poisson point process distribution and where voronoi cells define the planar areas associated with them . for the service of each user , either one or two bss are involved . if two , these cooperate by exchange of user data and channel related information with conferencing over some backhaul link . our framework generally allows variable levels of channel information at the transmitters . in this paper we investigate the case of limited channel state information for cooperation ( channel phase , second neighbour interference ) , but not the fully adaptive case which would require considerable feedback . the total per - user transmission power is further split between the two transmitters and a common message is encoded . the decision for a user to choose service with or without cooperation is directed by a family of geometric policies depending on its relative position to its two closest base stations . an exact expression of the network coverage probability is derived . numerical evaluation allows one to analyze significant coverage benefits compared to the non - cooperative case . as a conclusion , cooperation schemes can improve system performance without exploitation of extra network resources .   cellular networks , cooperative multipoint transmission ( comp ) , coverage probability , stochastic geometry , poisson point process , power control , geometric policies