the discoord algorithms proposed in seek to solve a problem that is described in fig .[ energy_network ] .the system of interest is composed of nodes distributed in space , communication network , and physical energy network .the nodes can generate a certain amount of energy within the node , while it has an initial energy in a certain level , .the nodes also have some desired levels of energy , .the nodes are connected through communication layer as well as through physical energy layer . through the communication layer, they can exchange information about the status of individual node . through the physical layer, the energy flows from a node to another node , .the energy flow is called energy distribution .the main purpose of discoord algorithms is to generate and distribute energies in an attempt to make each node to achieve the desired energy . here ,as the key constraints , there are the lower and upper levels in the generation of energy ; also all the decision is made only through local interactions between neighboring nodes .this paper provides case studies taking account of various scenarios ; while analyzing the simulation results , we will discuss the capability of the discoord algorithms .in discoord algorithms , there are nodes , and each node is able to generate energy within some specified lower and upper boundaries .the nodes are interconnected for energy exchanges and for communications each other . to make the problem clear , the following symbols are provided .[ cols= " < , < , < , < " , ]   the combination of the set of nodes and set of edges is called a graph , i.e. , .thus , the overall topology of the system is defined by the graph .the discoord algorithms seek to solve the following problem .[ prob1 ] given initial and desired energies of each nodes , and , generate energies and exchange energies , , with neighboring nodes such that , by relying upon only local interactions .  in the above problem ,the energy generation should be done with the constraints of   for the energy generation , the node can exchange a certain type of information only with its neighboring nodes .it is supposed that the information is exchanged through communication layer network ( or cyber - layer network ) .the energy exchange means the energy flow between the neighboring nodes ; so the energy exchange is done in physical - layer network .it is also important to notice that there is a relationship between the desired energies and initial and generated energies .that is , the sum of the desired energies should be less than the sum of initial energies and the maximum energy that can be generated by the nodes .it can be constrained as follows :   eventually , the discoord algorithms should generate the energies such as which is called the supply - demand balance .so , given the desired energies of individual nodes , one of the key constraints in the discoord algorithms is to satisfy the supply - demand balance during the generations of .the purpose of energy generation is to generate only using local neighboring interactions under the constraints of ( [ generation_constraint ] ) and ( [ supply_demand_balance ] ) .the algorithm is composed of three steps . in the following algorithm ,the interim parameters and are used .  1.  initial values +  +  2 .find the steady state solutions from the following dynamic equations +  + where is the cardinality of the set .let the steady - state solutions of the above equations denote as and .energy generation of nodes +   note that in the above step , for the update of and , the node needs information of and , which are the values of neighboring interim parameters .so , to compute at the node , it needs to communicate with the neighboring nodes .since and are fixed and can be exchanged by initial local interactions , it is simply assumed that they are available to neighboring nodes .the purpose of the energy distribution is to make the level of energy of each node become , i.e. , by the energy flows . this algorithm is also composed of three steps . in the following algorithm , the interim parameters and are used .  1.  initial values +  + + where is computed from the -rd step in the energy generation algorithm .2 .  find the steady state solutions from the following dynamic equations +  + + where . let the steady - state solution of the above equation denote .energy flow from -th node to the -th node +   note that in the above step , for the update of and , the node needs information of from the neighboring nodes .since are fixed and can be also exchanged by local interactions initially , it is assumed that they are available to neighboring nodes .the convergence of algorithms has been completely proved in .it is assumed that the desired energy of individual node is given as ( [ desired_constraint ] ) .the energy generation algorithm computes ; and then is used for the computation of at the energy distribution algorithm .it is also noticeable that the solutions of the energy generation and energy distribution algorithms are computed through communications among neighboring nodes .so , after obtaining the solutions and , each node generates physical energy and distributes energy to the neighboring nodes simultaneously .in this case , the sum of the desired energies of all nodes is well given as . for the balanced case , we consider the following scenario : + six nodes : , and + seven undirected edges : + initial energies : , and + desired energies : , and + lower boundaries of generation capability : for all + upper boundaries of generation capability : and   clearly , since , , and , the supply - demand balance is satisfied . now using the discoord algorithms given in the previous section ,we obtain + the generated energies : , , , , , + energy flows : , , , , , , + from the above result , we see that the total amount of energy flows is . as shown in fig .[ case11 ] , the desired energies have been well achieved after energy distribution . fig .[ case12 ] shows the overall variation in energy level of each node , and overall energy flows in the graph .  :desired energies(upper figure ) and achieved energies(lower figure ) . ]  : upper figure - initial(blue ) , desired(aqua ) , generated(yellow ) , and initial(red ) ; lower figure - energy flow diagram . ]   in this case , the sum of the desired energy is less than the minimum energy boundary , i.e. , . in the simulation , the same scenario as the case - is considered except the initial energies + initial energies : , and + so , we have . from the discoord algorithms ,we obtain + the generated energies : , , , , , + energy flows : , , , , , , + from the above result , we see that the total amount of energy flows is . after the energy distribution, it is observed that the errors , i.e. , , are .[ case21 ] shows that the desired energies can not be achieved because there are over energies in the network .so , as shown in the lower figure of fig .[ case21 ] , after the energy distribution , each node still has more energy than the desired energy level .it is interesting to observe that the errors of each node are same as ; so it seems that the discoord algorithms evenly distribute the remaining energies to the network . as shown in fig .[ case22 ] , clearly there is no energy generation from the discoord algorithms .thus , we can see that when there are over energy in the network , the algorithms do not generate energy in the whole network .: desired energies(upper figure ) and achieved energies(lower figure ) . ]  : upper figure - initial(blue ) , desired(aqua ) , generated(yellow ) , and initial(red ) ; lower figure - energy flow diagram . ]   in this case , the sum of the desired energy is greater than the maximum energy boundary , i.e. , . for the simulation , the same scenario as the case - considered except the desired energies + desired energies : , and + so , we have . from the discoord algorithms ,we obtain + the generated energies : , , , , , + energy flow : , , , , , , + from the above result , we see that the total amount of energy flows is . after the energy distribution , it is observed that the errors are .[ case31 ] shows that the desired energies at each node are not achieved due to the energy deficiency .[ case32 ] reveals that the energy at each node is generated in maximum .similarly to the case , the discoord algorithms still attempt to distribute the energies to the network evenly .so , the errors at each node are same .the algorithms have attempted to generate the maximum energy ; after generating the maximum energy , the energies have been distributed evenly to the network to make the same errors at all nodes .  :desired energies(upper figure ) and achieved energies(lower figure ) . ]  : upper figure - initial(blue ) , desired(aqua ) , generated(yellow ) , and initial(red ) ; lower figure - energy flow diagram . ]   in this case , it is supposed that . then , it does not need to generate energy ; only by distributing the initial energy among the nodes , the desired energy of individual node can be achieved . for the simulation , the same scenario as the case - considered except the desired energies + initial energies : , and + so , the sum of initial energies is equal to the sum of the desired energies . from the discoord algorithms ,we obtain + the generated energies are all zero .+ energy flows : , , , , , , + from the above result , we see that the total amount of energy flows is .  :desired energies(upper figure ) and achieved energies(lower figure ) . ]  : upper figure - initial(blue ) , desired(aqua ) , generated(yellow ) , and initial(red ) ; lower figure - energy flow diagram . ]  fig .[ case41 ] shows that the desired energies have been well achieved without generating any energy .so , as expected , the discoord algorithms only distribute the initial energies to the network to make errors of each node zero .[ case42 ] also shows that there is no energy generation ; but only there are energy flows .this paper has presented several cases as applications of discoord algorithms .as shown in the previous examples , the discoord algorithms can deal with various cases even though the supply - demand balance is not ensured any more .the main capability of the algorithms is to decide all the generation and distribution through local interactions , i.e. , by distributed coordination .it is recommended to see ` byeong - yeon kim , `` coordination and control for energy distribution using consensus algorithms in interconnected grid networks '' , ph.d .dissertation , school of information and mechatronics , gwangju institute of science and technology , 2013 ' for applications to various engineering problems of the algorithms developed in this paper .the reader can send email to hyosung.ac.kr to get the matlab source code .kim , `` coordination and control for energy distribution using consensus algorithms in interconnected grid networks , '' in _ph.d dissertation _, school of information and mechatronics , gwangju institute of science and technology , 2013 .

Abstract: this paper presents case studies of the algorithms called `` energy generation and distribution via distributed coordination '' , which was proposed in . for a convenience , we call `` energy generation and distribution via distributed coordination '' as `` discoord algorithms '' . after concisely summarizing the `` discoord algorithms '' in compact forms , we will list several scenarios for numerical tests . then , while analyzing the simulation results , we will discuss the capability of the algorithms in handling the various cases .


longitudinal recording has been the standard method in magnetic recording for decades .recent research has shown that this method seems to reach its physical limits in the near future due to the superparamagnetic effects .on the other hand , the technique which has been known prior to the longitudinal recording perpendicular recording , has recently been the centre of research attention .perpendicular magnetic recording offers promising increased in recording densities , up to 1 terabit per square inch seems feasible  .as the areal density is increased , however , the signal processing aspects of magnetic recording becomes more difficult .sources of distortion including media noise , electronics and head noise , jitter noise , inter - track interference , thermal asperity , partial erasure and dropouts become more apparent and unless appropriate mitigation techniques are present , signals can not be retrieved reliably from the recording media .  since the discovery of turbo codes, soft - decision iterative decoding has been shown to be able to provide significant coding gain over the conventional detection method on magnetic recording .the utilisation of iterative decoding on the concatenation of partial - response ( pr ) channel and powerful error - correcting codes such as low - density parity - check ( ldpc ) and turbo codes has been proposed in many literatures .iterative decoding is a reduced - complexity method to achieve the optimum solution the maximum - likelihood solution and as such , iterative decoding is sub - optimal .  in this paper , we present a method to improve the sub - optimality of the iterative decoding and demonstrate its applications to perpendicular magnetic recording in the presence of media noise .the improved method , which is known as the received - vector - coordinate - modification ( rvcm ) algorithm , follows on the previous works of the authors ,, , , .this method is similar to the works of  and  .this paper also investigates the use of signal - to - noise ratio ( snr ) mismatch  to mitigate the effect of media noise .the rest of the paper is organised as follows .section [ sec : channel ] describes the perpendicular recording channel used .the description of the rvcm algorithm is outlined in section [ sec : rvcm ] and the performance of this algorithm is demonstrated in section [ sec : performance ] .section [ sec : conclusions ] concludes this paper .[ fig : block - diagram ] shows the block diagram of the perpendicular recording system model used in this paper .the user data , denoted as , is a sequence of of input symbols taking values of .some error - protection redundancy is added to the sequence by the error - correcting - codes ( ecc ) encoder forming codeword sequence . to simulate the write current ,the sequence is mapped to according to operation .the scaling factor of is to ensure the transition takes values of .we assume that the read head produces zero voltage in the region of magnetic transitions and some voltage in the region of constant magnetic polarity .we approximate the single - transition step response , denoted as , using the hyperbolic - tangent function  : where is the saturation level or the amplitude from zero to peak ( normalised to unity ) and is the time taken for to go from to .it is assumed that and are normalised to the symbol period , . throughout the paper , it is assumed that .we define the response of two adjacent transitions ( dibit - response ) as : and the readback signal is simply the convolution of and plus some noise : where is the overall noise in the recording system which consists of media , jitter and electronic noise , i.e. .  the media noise , originates from the imperfections of the media and its effect is significant in the magnetic transition regions .typically , media noise is approximately four times the electronic noise at transition regions . in our system model, we consider the media noise as additive - white - gaussian - noise ( awgn ) with mean of and variance of , which exists in the transition region only . asshown in fig .[ fig : block - diagram ] , the media saturation noise depends on the saturation level and it is evaluated as . unlike media noise which is media dependent , the jitter noise is due to timing imperfection only . to model the sampling jitter noise , , the order taylor approximation is used . the jitter estimation block shown in fig .[ fig : block - diagram ] is done with order taylor series expansion of .the jitter probability density function is assumed to be uniform , limited by a maximum value .the electronic noise , is awgn with mean of and variance of .the recording system in fig .[ fig : block - diagram ] caters for many different simulation cases with varying degree of electronics , media and sampling jitter noise .we define the channel snr as :   the noisy readback signal is equalised to pr target which is only optimal for electronic noise at the considered  .it serves for comparison purposes only .the maximum - a - posteriori ( map ) decoder of the pr channel exchanges extrinsic with the ecc decoder to deliver solution which is used for performance evaluation .it has been shown that the rvcm algorithm provides considerable coding gain for parallel and serial concatenated turbo codes , and ldpc codes  .the algorithm can be applied directly to perpendicular recording and is described briefly below .let denote an vector at the output of the map decoder , that is the a - posteriori probability ( app ) of the map decoder .let denote the reliability sequence of , where .assume that is an integer where and is a vector of length .* step 1 . * : :  store the vector , let the integer   be initialised to .* step 2a . * : :  set and   .restart the iterative  decoder , store the decoded vector  ( ) . ** : :  set .restart the iterative  decoder , store the decoded vector  ( ) and restore   , i.e.  . * step 3 . * : :  if then set and  continue to * step 2*. otherwise , stop the algorithm and from the list  of all decoded vectors   ,   , choose a  decoded vector that has the minimum euclidean distance .it is assumed  that the iterative decoder always outputs a codeword .  from the steps above ,it is clear that the complexity of the algorithm depends on .one of the major obstacles concerning the rvcm algorithm is the difficulty in finding the symbol(s ) that , if modified , can converge the iterative decoder to the maximum - likelihood solution , .these symbols are referred as the critical symbols and their distribution is uniform with no sign of vulnerable or favourite symbol positions .on the other hand , due to their uniform distribution , it is likely that we can find one of the critical symbols if we confine our search to a small group , i.e. keeping the value of low . in this way, we can reduce the computational complexity for the price of sub - optimum performance . as we will show later that, the gain obtained by confining to a small value is still significant compared to the performance of the standard iterative decoder .there are various methods for selecting the critical symbols , see  for details . in this paper, we restrict the selection to one method only , that is the reliability of the app at the output of the map decoder .we evaluate the performance of the rvcm algorithm on some short - block length turbo and ldpc codes . fig .[ fig : turbo - sm0 ] and [ fig : turbo - sm4 ] show the error rate performance of the turbo code under the standard iterative and rvcm decoders .the turbo code considered is the tail - biting turbo code , where , and with .significant improvement is noticed and the increase in performance gets better as snr increases .it is worth noting that the results in fig .[ fig : turbo - sm4 ] were obtained using snr mismatch technique . with this technique ,the improvement in performance over standard iterative decoder is even greater as snr increases . in the presence of media noise ,snr mismatch methods do not provide the same performance as observed with electronics noise only  , however better targets for media noise are being investigated by the authors .error performance of rvcm decoder on the turbo code in the presence of electronic noise only , width=220 ]  error performance of rvcm decoder on the turbo code in the presence of electronic and media noise , width=220 ]  error performance of rvcm decoder on the cyclic ldpc code in the presence of electronic and media noise , width=220 ]  performance of the cyclic and codes in the presence of electronic and media noise , width=220 ]  similar performance improvement is observed for the ldpc codes , see fig .[ fig : gf2 - 127 - 84 ] .the cyclic ldpc code , which has minimum - distance of , was constructed using a method described in  . as mentioned earlier , the rvcm algorithm allows one to trade off the performance against the computational complexity . from fig .[ fig : gf2 - 127 - 84 ] , despite the performance obtained by setting is approximately inferior to that by setting , the coding gain from the standard iterative decoding is significant . for the case of , we select the critical symbols based on the reliability measure at the output of the map decoder . from the vector , we construct a vector of length such that . in fig .[ fig : gf2 - 255 - 175 ] , we compare the performance of the cyclic code and that of the quasi - cyclic code .we can see that the rvcm algorithm provides significant gain , within one order of magnitude improvement , over the bp algorithm . at approximately ber ,the performance of the cyclic code with rvcm is within away from the longer code under bp decoding .we have shown that the application of the rvcm algorithm to perpendicular magnetic recording shows promising results .simulation results show that improvement of within one order of magnitude is possible .short block length offers an attractive error - correction scheme in which rvcm algorithm can be fully exploited by setting .a bank consisting of parallel rvcm decoders can be built on chips and the decoding of short - block length data has low latency .the performance of longer block - length codes , up to a certain error - rate , can be outperformed by the application of rvcm algorithm to shorter codes .the exact point , at which the longer codes start to perform better , depends on the code structure .we also extended our investigations on using some non binary cyclic ldpc codes  and we observe similar improvement as in the binary cases .further investigations in identifying the critical symbols will allow the application of rvcm algorithm to long powerful codes .the authors would like to thank prof .barry k. middleton of university of manchester for the channel noise discussion .e. papagiannis , m. a. ambroze , and m. tomlinson , `` analysis of non convergence blocks at low and moderate snr in scc turbo schemes , '' _ spsc 2003 8^th^ international worksop on signal processing for space communications , catania , italy _ , pp . 121128 , sep .e. papagiannis , m. a. ambroze , and m. tomlinson , `` approaching the ml performance with iterative decoding , '' _ international zurich seminar on communications , zurich , switzerland _ , pp . 220223 , feb .c. j. tjhai , e. papagiannis , m. tomlinson , m. a. ambroze , and m. z. ahmed , `` improved iterative decoder for ldpc codes with performance approximating to a maximum likelihood decoder . ''uk patent application 0409306.8 , apr .2004 .n. varnica and m. fossorier , `` belief propagation with information correction : imporved near maximum - likelihood decoding of low - density parity - check codes , '' _ proc . of ieee intl .inform . theory ( isit ) ,chichago , usa _ , p. 343, july 2004 .

Abstract: an algorithm of improving the performance of iterative decoding on perpendicular magnetic recording is presented . this algorithm follows on the authors previous works on the parallel and serial concatenated turbo codes and low - density parity - check codes . the application of this algorithm with signal - to - noise ratio mismatch technique shows promising results in the presence of media noise . we also show that , compare to the standard iterative decoding algorithm , an improvement of within one order of magnitude can be achieved .


the problem of inter - cell interference in cellular networks has been the major impediment in delivering a consistent level of quality of service to geographically distributed users with an efficiently high spectral efficiency in terms of bits / sec / hz / user . in a recent attempt to break the interference barrier ,the information theoretic concept of interference alignment ( ia ) was used .this concept was initially introduced by maddah - ali _ et .al _ to solve the mimo -channel . in , authors applied the concept of ia to -user interference channel ( ic ) and obtained the theoretical upper bound of multiplexing gain or degrees of freedom ( dof ) .however , the practical development of ia in cellular networks is highly complex , e.g. , an iterative process has been suggested in to design the transmit and receive beamformers . in this paper , we modify the abstract distributed ia , by introducing a level of structured coordination amongst the transmitters , i.e. , base - stations ( bss ) , in the cellular network , to improve the achievable spectral efficiency at the cost of a limited overhead of backhaul processing . in our proposed coordination model , each user receives its desired signal from its dedicated bs , as the primary transmitter , and a coordinating bs , as the secondary transmitter . in this chain of coordinated transmissionsthe first bs in the network coordinates with the second bs , the second bs coordinates with the third bs and so on , and eventually , the last bs coordinates with the first bs to complete the coordination cycle . in a sense , this transmission model is similar to wyner circular model , as each user is supported by two bss , but slightly departs from this model as each user is modeled to receive interference from all the bss in the network .the coordination link between two bss is uni - directional , e.g. , the first bs shares its channel state information ( csi ) with the second bs and serves as the secondary transmitter to the user of the second bs . as a result , each bs transmits to two users , i.e. , to its own user as the primary transmitter and to the user of the coordinated bs as the secondary transmitter . + through feasibility analysis , we compare the proposed coordination scheme with the generic ia .we establish some sufficient conditions for obtaining a proper system that can attain a feasible ia solution according to .we show that , operating at half of the total available dof , i.e. , the information theoretic upper bound on dof , the generic ia is proper for up to 3-user system , whereas the proposed partially coordinated scheme is proper for up to 5-user system for the same configuration , i.e. , with the same number of transmit / receive antennas and the same number of dof .then , exploiting the rank deficiency of the interference covariance matrix of the reciprocal uplink channel , we propose an algorithm that implements the ia solution in the proposed coordination model in one shot .we establish a dof upper bound for obtaining the one shot , i.e. , non - iterative , solution and analyze the system configurations operating below the one shot upper bound .furthermore , we analyze the backhaul overhead requirement of the proposed coordination model and show that the backhaul overhead in our model increases linearly with increase in the number of participating bss .simulation results confirm that our proposed scheme improves the sum rate performance compared to the iterative solution in , for limited usage of the backhaul .+ this paper is organized as follows . in sectionii , we introduce and formulate the proposed coordination scheme . in section iii ,we analyze and compare the feasibility of our proposed scheme with the generic ia and introduce a one shot ia solution for the proposed coordination scheme .moreover , we compare the backhaul overhead in our proposed partially coordinated system with systems having fully coordinating bss in terms of backhaul data rate requirement . in section iv, we support our analysis with simulation results and discussions .finally , section v concludes this paper .+ the following notations are used throughout this paper .bold lowercase and uppercase letters , e.g. , and , represent vectors and matrices , respectively . the dimension of is shown by and indicates the element at row and column , i.e. , element , in .similarly , after partitioning a matrix into submatrices , we use calligraphic letter to define the submatrix at element of .the notation and respectively represent the the remainder of the fraction and the modulo - arithmetic operation over , where , and are all integers .we use the operators , , , , and to compute the floor , ceil , rank , null space , trace and dot product operations of their respective arguments .the operator generates a block - diagonal matrix using the elements of its argument .the superscripts , and represent the transpose , complex conjugate and conjugate transpose operations , respectively .we consider a partially coordinated downlink transmission in a mimo cellular system with bss . in our proposed coordination , bssare connected in pairs and in succession via backhaul links , as shown in fig .[ fig : sysmodel ] .we assume that the backhaul network is based on a line or a ring topology with the bs , , as the node in the line or the ring .the bs has transmit antennas and its corresponding user terminal has receiving antennas .we also assume time - division duplex ( tdd ) communication with synchronized time slots to ensure channel reciprocity and to use uplink csi in the downlink .the successive pair - wise coordination among bss is such that the bs establishes a coordination link with bs , where , in two steps as follows .first , bs shares its csi with bs .next , bs exploits this csi to compute its transmit beamformer , where and is the dof of user .then , is used to transmit independent data streams towards user through the transmit antennas of both bss and . in this coordination setup ,the bss and serve as the primary and the secondary transmitters for user , respectively .we assume flat fading channel that can be achieved by employing mimo orthogonal frequency division multiplexing ( ofdm ) as the baseband modulation .moreover , we assume no intracell interference within a cell , as this can be obtained by using orthogonal frequency division multiple access ( ofdma ) .if the coordination among bss is ignored then this system reduces to the standard -user ic with mimo links . in the proposed schemethe overall received signal ^t ] and ^t ] , where  ] for with and given as , ^t , \quad i=1,2,\ldots , k.\ ] ] in , is identity matrix and , for and , represents the matrix with all zero elements .we define an equivalent interference channel matrix which can be partitioned into submatrices , with , and , where , ,   &    j = 1  \\\left[\mathcal{h}_{i , j-1 } \\mathcal{h}_{i , j } \right ] ,  &    j = 2 , \ldots k ,   \end{array } \right.\ ] ] represents the submatrix at the location in .next , we define the block - diagonal matrix , where its element is given as , ^t ,  &    j = 1  \\ \big[\mathbf{\tilde{v}}^t_j \     \mathbf{v}^t_j           \big]^t , &   j = 2 , \ldots k. \end{array } \right.\ ] ] using , it can be easily verified that .therefore , , as the permutation matrix satisfies . using this relation ,we transform our partially coordinated multicell model into a -user mimo ic problem , where is the transmit beamformer for user . thus , and can be rewritten as , and respectively .similarly , according to the ia principles , the design constraints in become this section , first , we analyze the sufficient conditions for obtaining a proper uncoordinated generic interference channel system in terms of maximum number of users supported at the information theoretic upper bound on dof and compare it with our proposed partial coordination scheme , in .   according to ,a generic -user ic with mimo links , denoted as a , is a proper system if the number of variables involved in each and every subset of equations is greater than or equal to the number of equations in that subset . based on the intuitive discussion in ( see sec .v and vi ) , authors argue that a proper ic system is likely to attain a feasible ia solution . in this context ,a symmetric ic system , with , and , , described as , is proper if the total number of equations is less than or equal to the total number of variables , i.e. , .moreover , in , the various groups of ic systems are classified such that within a group , where is a constant , and , . based on the classification of a group of systems ,the authors in establish a rule of thumb which states that if any system within a group is proper then almost all the systems of that group are also proper and vice versa .+ following the definitions in , our equivalent ic system in can be modeled as .furthermore , the information theoretic upper bound for dof , introduced in , can be tightly approximated as for the general -user ic and , for the symmetric -user case , where is 0 or 1 for an even or odd values of , respectively .note that is not necessarily an integer multiple of .this requires to privilege a number of users by allowing them to operate at a higher integer number of dof , i.e. , , than the others , i.e. , operating at .however , this may not be the best policy of allocating the dof resources amongst the users due to the fairness considerations as well as the fact that different users experience different fading conditions in the channel . clearly , assigning a larger dof to a user facing an aggressive fading condition in the channelis not an efficient policy , as it may require a higher transmit power to deliver a prescribed level of data rate per stream . in the following ,we propose a time - sharing strategy over a number of time - slots so that the available dof resources are equally distributed , when averaged over all time slots , amongst all users .  _( a uniformly time - shared distribution of dof in the generic ia)_[theorem1 ] a system of -user asymmetric interference channel described by , where user is operating at an integer number of of dof so that is the information theoretic upper bound , can be described as an equivalent symmetric system of defined under a time sharing schedule over a span of time slots , where .let be the total number of allocated time slots , be the remaining number of dof to be shared uniformly among users , and . in the equivalent symmetric system each user operates at a dof of for the duration of time slots and at a dof of for the remaining time slots .the distribution of dof is attained such that in each of the time slots the total number of independent data streams in the interference channel system is . over the span of time slots ,each user attains a total of dof .effectively , the dof per time slot for each user , i.e. , , is given as thus , over the span of time slots the system can be modeled as system ._ example 1 : _ consider an asymmetric 5-user interference channel with mimo links where each user operates at its information theoretic upper bound of dof , i.e. , .this system can be described as . applying theorem [ theorem1 ], one can easily verify that after assigning one dof per user the remaining number of dof to be allocated is , the total number of required time slots is and .hence , over the duration of 10 time slots , each user operates at dof for 4 time slots and at dof for the remaining 6 time slots , i.e. , a total number of 14 dof .therefore over a span of 10 time slots , the equivalent symmetric system can be modeled as , where each user attains dof per time slot . + in the sequel , we calculate the upper bounds on the number of users that can be served at the information theoretic limit of dof in the generic and the proposed partially coordinated proper systems .  _( an upper bound on for generic ia)_[corollary1 ] a system of -user interference channel described by that operates at the information theoretic upper bound , where , , and is either 0 or 1 for even or odd values of , respectively , is proper if   an asymmetric system can be modeled as an equivalent symmetric system , where , using the time sharing concept of dof , developed in theorem [ theorem1 ] . to obtain a proper symmetric system , we require the total number of equations and the total number of variables satisfy , i.e., is a sufficient condition for obtaining a proper system ._ remark 1 : _ it can be easily verified from that the system is proper if . the equality of holds only for and for , we have .the exceptional case of with applies to the group of symmetric systems , , , systems that are given as examples 1 , 3 and 5 in .furthermore , generic ic systems operating at their information theoretic upper bound of dof with , i.e. , with an exception of for , are not proper and , hence , are not feasible to attain the ia solution .this fact is confirmed in fig .4 of , where the dof upper bound can not be achieved for four users with 4 and 5 antenna cases .we will also confirm this through simulation results presented in section [ sec_sim ] .+ next , we establish an upper bound on for a proper partially coordinated ia system .  _( an upper bound on for ia with partial coordination)_[corollary2 ] a system of -user partially coordinated channel described by the equivalent interference channel representation , that operates at the information theoretic upper bound , where , and is either 0 or 1 for even or odd values of , respectively , is proper if   an asymmetric system can be modeled as an equivalent symmetric system , where , using the time sharing concept of dof , developed in theorem [ theorem1 ] .one can easily verify that , here , the total number of equations and the total number of variables . to obtain a proper system , we require , i.e., is a sufficient condition for obtaining a proper system of ia with partial coordination .  _remark 2 : _ it can be easily verified from that the equivalent ic system is proper if , when , , and its information theoretic upper bounds on dof for , and are given as , and , respectively .+ in the sequel , we introduce a one - shot algorithm that achieves dof upper limits within by ia with partial coordination and discuss , in more details , the achievability of various dof configurations for , and .   here, we describe a one shot ia solution to find the receive and the transmit beamformers , i.e. , , , and , , respectively , for the proposed partial coordination scheme in two steps .first , we design , , for any receive beamformer such that the signal power received by each user in the original network is maximized. then , using these receive beamformers in the reciprocal network , where roles of the transmitters and the receivers are reversed , we design , , such that the interference leakage at the receiving ends of the reciprocal network is minimized to zero .these steps are described as follows .note that in the following analysis , we use the notation defined in to demonstrate the variables in the reciprocal network , i.e. , in the reverse link , with a left arrow on top . : in the original network , the desired signal power at receiver due to the partially coordinated transmission from bss and is given by where is the signal covariance matrix at user and is the power of message signal transmitted through bss and , i.e. , the transmitter in the equivalent ic model .first of all , the transmitter computes the singular value decomposition of , , as , where }_{1 } \ldots \mathbf{f}^{[i]}_{m_k}] ] and }_{1 } \ldots \mathbf{m}^{[i]}_{m_k}] ] , i.e. , the first columns of .then , the receiver solves the optimization problem to find the receive beamformers that maximizes in .it can be easily verified that is maximized when }_{1 } \ldots \mathbf{f}^{[i]}_{d_i}] ] and . since , therefore , .finally , by expanding in , we arrive at , , , which satisfies the first condition of the interference alignment solution in .therefore , in this way , bss and can jointly transmit the message signal to user with dof without causing interference to the other users .+ due to the update of , , in step 2 , the received signal power at the user of the original network deviates from the single user performance , i.e. , , in , as follows .starting with and substituting for with updated values calculated in step 2 and , we can write } ( \mathbf{w}_j^{[i]})^h \right)\nonumber\end{aligned}\ ] ] but , it can be easily verified that , therefore })^h\tilde{\mathbf{m}}_i\tilde{\mathbf{\sigma}}^2_i \tilde{\mathbf{m}}^h_i \mathbf{w}_j^{[i ] } \right)}\nonumber\\ & = \frac{p_i}{d_i } \sum_{j=1}^{d_i}{(\sigma^{[i]}_{j})^2 \sum_{l=1}^{d_i}{|\langle \mathbf{m}^{[i]}_j , \mathbf{w}^{[i]}_l \rangle|^2 } } , \label{eqn : s_i2}\end{aligned}\ ] ] where }_1 \ldots \sigma^{[i]}_{d_i}) ] .a comparison of and shows that each term }_{j})^2 ] , where }_j , \mathbf{w}^{[i]}_l \rangle ] , , on }_j ] . however , despite this degradation , the monte - carlo simulation results , described in section [ sec_sim ] , show that the one shot partially coordinated ia offers higher sum rate than the distributed ia in similar scenarios .   in a symmetricdistributed ia system , with and , , the achievable dof information theoretic upper bound satisfies for , whereas in partially coordinated ia , it is limited by , for , as established in _ remark 2_. furthermore , for a symmetric distributed ia , or equivalently , for , because has a rank of and contains only columns .this implies that can not be designed in one shot in the null - space of and an unpredictable number of iterations is needed to reduce the rank of .these iterations continue until the required number of dimensions of is reached for the null - space of .for instance , the distributed ia in a symmetric interference channel system with , , , i.e. , , , requires iterative adjustment of transmit beamformers until is reached at each receiver .whereas , for a symmetric ia system with partial coordination a dof of that satisfies can be achieved in one shot because , i.e , the null space of has enough dimensions to contain dof , or independent streams per user , from the starting point of iterations .let the columns of form a set of orthonormal basis for the null - space of and the columns of be selected from the columns of .then , for each user , we have the flexibility of choosing a set of columns from possible selections of sets for .clearly , the optimum set would be a set of columns that maximizes the received signal power at user according to , i.e. , }_1 \ldots \mathbf{w}^{[k]}_{d_k } ] } s_k} ] , construct  ] .compute the geometric snr coefficient }_{j}|} ] , , in fig .[ fig : sysmodel ] , within .whereas , in the line model , the inter - bs data rate of is required due to the broken link between the first and the coordinating bss and the fact that the csi of the bs , i.e. , , should travel through all other links to reach the first bs .therefore , the data rate requirement for csi exchange in the backhaul grows linearly with in the proposed coordination structure in fig .[ fig : sysmodel ] . whereas , in a system with fully coordinating bss , e.g. , - , it can be easily verified in a similar way that the data rate requirement for overall csi exchange among all bss within the same allowed latency interval of requires a data rate of and in the line and the ring based backhaul models , respectively .in this section , we report our simulation results on the achievable sum rate of the proposed partial coordination scheme and compare them with those of generic ic model , employing distributed ia algorithm , and full coordination , i.e. , block - diagonalization ( bd ) based zero - forcing ( zf ) scheme . transmitting bss use the same transmit power and each bs equally distributes its power across its independent data streams[ fig : k3_2x2]-[fig : k5_3x3 ] compare the achievable sum rate of different schemes for , and , where and mimo links are established between a bs and a single user in each cell .results confirm that the bd based zf in a full coordinated system , where separate base - stations form a giant single bs , achieves the highest sum rates amongst the other schemes .these results also confirm that the proposed one shot ia outperforms the distributed ia in terms of sum rate performance .it should be noted that degradation of performance of the distributed ia is because of the lack of coordination among the precoders .furthermore , we compare the flexible and rigid systems for , discussed as _ example 2 _ in section [ subsec : gains ] . in fig .[ fig : k3_2x2 ] , i.e. , with mimo links , the flexible and rigid systems operate at and , respectively .similarly , in fig .[ fig : k3_3x3 ] , i.e. , with mimo links , we have two flexible systems , with and , and the rigid one operates at .results confirm that systems with higher offer higher sum rate at high snr as the slope of their sum rate curves grow faster at high snr , as shown in figs .[ fig : k3_2x2 ] and [ fig : k3_3x3 ] .however , we observe that at lower snr the flexible system , with lower dof , outperforms the rigid one .this is due to the fact that in the flexible system the geometric mean of the individual snrs is maximized through the selection of the transmit beamformers , whereas , in the rigid one the transmit beamformers are fixed . in fig .[ fig : k3_2x2 ] , the rigid system , i.e. , , offers higher sum rate compared to the flexible one , with , at snrs higher than 15.5 db .similarly , in fig . [ fig : k3_3x3 ] , the rigid system , i.e. , , outperforms the flexible systems of and at snrs higher than 6 db and 13.5 db , respectively . in figs .[ fig : k4_2x2]-[fig : k5_3x3 ] , the sum rate curves of distributed ia in and with no coordination at the precoders saturate at higher snrs due to leakage of interference into the interference free dimensions carrying the information streams .this confirms the result from as these improper systems at and , defined by _corollary [ corollary1 ] _ , are unable to attain a feasible ia solution . in figs .[ fig : k4_2x2 ] and [ fig : k4_3x3 ] , i.e. , system with and mimo links , the sum rate curves of the proposed one shot partially coordinated ia operating the information theoretic dof upper bound , i.e. , at and , respectively , grows linearly with increasing snr .this confirms our discussion in _ example 3 _ as the proposed partially coordinated scheme solves at the information theoretic dof upper bound in one shot .following the discussion from _ example 4 _ , in fig .[ fig : k5_2x2 ] , i.e. , with mimo links , the distributed ia achieves the information theoretic dof upper bound at , i.e. , , , in the partially coordinated system whereas , our one shot solution achieves , i.e. , , , as described in theorem [ theorem1 ] .similarly , in fig .[ fig : k5_3x3 ] , i.e. , system with mimo links , the distributed ia solves the dof upper limit at , while the proposed one shot solution achieves . in figs .[ fig : k5_2x2 ] and [ fig : k5_3x3 ] , the iterative ia , operating at and , respectively , offers higher sum rate at high snrs . however , we observe that at lower snrs the proposed one shot ia operating at lower dof , i.e. , at and , respectively , outperforms iterative ia .this improvement at lower snrs can be intuitively explained as follows .in one shot ia , the receive beamformers in the original network ( i.e. , _ step 1 _ ) are designed to maximize the received signal power , whereas in distributed ia , the beamformers are chosen to minimize interference leakage in both the original and the reciprocal networks . in fig .[ fig : k5_2x2 ] , the partially coordinated distributed ia with offers higher sum rate compared to one shot solution with at snrs higher than 26 db .similarly , in fig . [fig : k5_3x3 ] , the distributed ia with outperforms the one shot algorithm with at snrs higher than 32 db .note that for , the fully coordinated system offers very high multiplexing gain , i.e. , with mimo links and with mimo links . hence , to obtain a better resolution for the sum rate curves of the partially coordinated scheme ,the sum rate curves for the fully coordinated bd based zf are omitted in figs .[ fig : k5_2x2 ] and [ fig : k5_3x3 ] . in fig .[ fig : backhaul ] , we compare the data rate required for exchanging the csi overhead in the backhaul in the proposed partially coordinated precoding scheme with non - coordinating interference channel model and the full coordination model - , as discussed in section [ sec : backhaul ] .the data rates are expressed in terms of multiples of , where is the data rate required to send the single mimo link csi through the backhaul within the allowed latency interval .the full coordination model requires exponential increase in backhaul data rate , whereas , the backhaul data rate of the partially coordinated ia increases linearly with .in this paper , we introduced a partially coordinated transmit precoding scheme for downlink multicell mimo systems .we transformed the system model of our scheme to an equivalent interference channel problem and compared its feasibility with standard interference channel model .we established a sufficient condition for obtaining a proper system , in terms of the maximum number of users supported at the information theoretic upper bound on dof , in the proposed partially coordinated ia and the generic ia .furthermore , we introduced a one shot ia algorithm with partial coordination and analyzed the maximum achievable degrees of freedom in the system .simulation results confirmed that , in comparison with the generic ia , the proposed partially coordinated ia offers higher throughput at practical snr levels and avoids an uncontrolled number of back and forth iterations between the original and the reciprocal networks .

Abstract: * in this paper , we introduce an efficient interference alignment ( ia ) algorithm exploiting partially coordinated transmit precoding to improve the number of concurrent interference - free transmissions , i.e. , the multiplexing gain , in multicell downlink . the proposed coordination model is such that each base - station simultaneously transmits to two users and each user is served by two base - stations . first , we show in a -user system operating at the information theoretic upper bound of degrees of freedom ( dof ) , the generic ia is proper when , whereas the proposed partially coordinated ia is proper when . then , we derive a non - iterative , i.e. , one shot , ia algorithm for the proposed scheme when . we show that for a given latency , the backhaul data rate requirement of the proposed method grows linearly with . monte - carlo simulation results show that the proposed one - shot algorithm offers higher system throughput than the iterative ia at practical snr levels . *  * interference alignment , coordinated precoding , multiplexing gain , feasibility . *


a wide variety of coordinated tasks performed by teams of mobile robots critically rely on the topology of the underlying communication network and its spectral properties .examples include , cooperative manipulation lynch1996 , mataric1995 , wang2002 , surveillance and coverage cortes2004 , parker2002 , poduri2004 , distributed averaging jadbabaie2003 , saber2007 , ren2005 , formation control , flocking , and multi - robot placement , that all require some form of network connectivity , structure and , oftentimes , spectral properties . in this paperwe address the problem of controlling a network of mobile robots to a topology with a desired eigenvalue spectrum .this effort is a first step towards the design of controllers that allow robots to perform their assigned tasks , while optimizing coordination within the team .the eigenvalue spectra of a network provide valuable information regarding the behavior of many dynamical processes running within the network p08 .for example , the eigenvalue spectra of the laplacian and adjacency matrices of a graph affects the mixing speed of markov chains , the stability of synchronization of a network of nonlinear oscillators pc98,pv05 , the spreading of a virus in a network , as well as the dynamical behavior of many decentralized network algorithms l97 .similarly , the second smallest eigenvalue of the laplacian matrix ( also called spectral gap ) is broadly considered a critical parameter that influences the stability and robustness properties of dynamical systems that are implemented over information networks .optimization of the spectral gap has been studied both in a centralized and decentralized context .  in this paper, we propose a novel framework to control the structure of a network of mobile robots to achieve a desired eigenvalue spectrum .in particular , we focus on the spectrum of the weighted adjacency matrix of the network , with weights that are ( decreasing ) functions of the inter - robot distances .this construction is relevant , for example , in modeling the signal strength in wireless communication networks .although our framework performs well with different distance metrics , in this paper , we focus on the norm ( manhattan distance ) primarily for analytical reasons , since its composition with convex functions preserves their convexity . additionally , this metric has potential applications in indoor navigation where the presence of obstacles forces the signals to propagate in grid - like environments .we employ a novel abstraction of the eigenvalue spectrum in terms of the associated spectral moments , and define artificial potentials that capture the distance between the network s spectral moments and their desired values .these potentials are minimized via a gradient descent algorithm , for which we show convergence to the globally optimal moments . since the eigenvalue spectrum is uniquely determined by the associated spectral moments, our approach provides a way to indirectly control a network s eigenvalues .this work is related to preciado2010 , which addresses a similar problem for static robots in discrete environments .this formulation , however , is more appropriate for robotics applications , where communication depends continuously on the robot motion .the rest of this paper is organized as follows . in section[ sec _ notation ] , we introduce some graph - theoretical notation and useful results . in section [ sec_problem ] , we formulate the control problem under consideration .we introduce an artificial potential and derive the associated motion controllers in section [ sec_control ] and discuss convergence of our approach in section [ sec_convergence ] . finally , in section [ sec_simulations ], we illustrate our approach with several computer simulations .in this section we introduce some nomenclature and results needed in our exposition .let denote a weighted undirected graph , with  ] , where is the weight of edge . in this paperwe are particularly interested in the spectral properties of .since is a real symmetric matrix , it has a set of real eigenvalues .the powers of the adjacency matrix can be related to walks in :  [ lem_biggs ] let be a weighted undirected graph with no self - loops .the -th entry of can be written in terms of walks in as follows: _ { ij}=\sum_{\mathbf{w}\in w_{i , j}^{\left ( k\right ) } } a_{\mathbf{w}},\]]where is the set of all closed walks of length from node to in the complete graph .the above result comes directly from an algebraic expansion of _ { v_{0},v_{k}} ] and  ] denote the stacked column vector of all robot positions , so that is the -th coordinate of the -th robot position .we assume that we can control the position of the robots by the simple kinematic law ^{t}\in \mathbb{r}^{dn } ,  \label{closed - loop system}\]]where is the control input applied to robot , and is the -th coordinate of .  for a given set of robot positions , , we define the weighted adjacency matrix of the network of robots as  ] with .the adjacency matrix of an undirected graph with no self - loops has independent entries ( for example , the upper triangular entries ) .each one of these entries are a function of the vector of positions .hence , applying the chain rule , we have the following expansion for the partial derivative of the cost function with respect to the entries : , a particular entry depends solely on the position and ; therefore , we have that if both and are different than .thus , for a fixed , only the following summands in ( [ eqn_chain_rule ] ) survive we have used that and .we now analyze each one of the partial derivatives in ( eqn_surviving_terms ) .first , from ( [ eqn_cost_function ] ) and ( eqn_spectral_moments ) , we have that _ { ij},\end{aligned}\]]where we have used lemma [ lem_partial_of_trace ] in the last equality .second , from ( [ eqn_adjacency_entries ] ) we have that ( the fact that is not differentiable at does not affect our analysis ) .let us define the antisymmetric matrix  ] .we applied the proposed control law and studied the evolution of the spectral moments of the network s adjacency matrix .[ fig_hex_conv ] shows the evolution of the second , third and fourth moment .as expected , they all converge to the desired values .the asymptotic values of all spectral moments of the network are are very close to the desired sequence of moments , and slightly on the larger side , as predicted by theorem [ thm_convergence ] .the eigenvalues of the weighted adjacency matrix of the final configuration are are also very close to the desired values .notice that our approach fits better those eigenvalues further away from the origin , since they are more heavily weighted in the expression of spectral moments , than those close to zero .an alternative approach to overcome this limitation would be to modify our cost function ( [ eqn_cost_function ] ) to assign more weight to eigenvalues of small magnitude .  as discussed in section[ sec_control ] , we also consider a relaxation of problem [ problem ] , that involves a truncated sequence of the first four moments of the network . in this case, the cost function is given by ( eqn_alternative_cost_function ) and the closed loop system is defined in , for .the asymptotic values of the first four spectral moments of the network are eigenvalues of the final configuration are are remarkably close to the set of desired eigenvalues , especially those far from zero .the main advantages of using a relaxation of problem [ problem ] are ( _ i _ ) that it reduces the computational cost of the controller , since it only requires the terms and , and ( _ ii _ ) that it improves the stability of the numerical behavior of the gradient descent algorithm .[ example_delta ] in this example we consider the problem of matching the eigenvalue spectrum of a particular realization of a random geometric graph on nodes that are uniformly distributed in \times \left [ 0,1\right ] ] . in this case , matching the whole set of spectral moments directly provides us with a very good approximation of the eigenvalues spectrum . as before , we also considered the relaxation of problem [ problem ] involving only a truncated sequence consisting of the first 4 moments . the evolution of the second , third and fourth moments of the network is shown in fig .the asymptotic values of the spectral moments of the network are are remarkably close to the desired ones and , as in the previous example , slightly on the larger side .similarly , the final set of eigenvalues is is a very good fit to the given eigenvalues , especially for those far from zero .in this paper , we proposed a novel control framework to modify the structure of a mobile robot network in order to control the eigenvalue spectrum of its adjacency matrix .we introduced a novel abstraction of the eigenvalue spectrum by means of the spectral moments and derived explicit gradient descent motion controllers for the robots to obtain a network with the desired set of moments .since the eigenvalue spectrum is uniquely determined by the associated spectral moments , our approach provides a way of controlling the eigenvalues of mobile networks .convergence to the desired moments was always guaranteed due to convexity of the proposed cost functions .efficiency of our approach was illustrated in nontrivial computer simulations .the adjacency matrix eigenvalue spectrum is relevant to the performance of many distributed coordination algorithms run over a network .therefore , our approach is particularly useful in providing network structures that are optimal with respect to networked coordination objectives .future work involves theoretical guarantees for the euclidean distance metric ( in ( [ eqn_adjacency_entries ] ) ) , as well as extension of our results to the spectrum of laplacian matrix of the network .moreover , the relaxation of problem [ problem ] to a truncated sequence of moments does not guarantee ( mathematically ) a good fit of the complete distribution of eigenvalues .therefore , a natural question is to characterize the set of graphs most of whose spectral information is contained in a relatively small set of low - order moments .m. mesbahi and f. y. hadaegh , formation flying control of multiple spacecraft via graphs , matrix inequalities and switching , _ aiaa journal of guidance , control and dynamics _ ,369 - 377 , 2001 .

Abstract: the eigenvalue spectrum of the adjacency matrix of a network is closely related to the behavior of many dynamical processes run over the network . in the field of robotics , this spectrum has important implications in many problems that require some form of distributed coordination within a team of robots . in this paper , we propose a continuous - time control scheme that modifies the structure of a position - dependent network of mobile robots so that it achieves a desired set of adjacency eigenvalues . for this , we employ a novel abstraction of the eigenvalue spectrum by means of the adjacency matrix spectral moments . since the eigenvalue spectrum is uniquely determined by its spectral moments , this abstraction provides a way to indirectly control the eigenvalues of the network . our construction is based on artificial potentials that capture the distance of the network s spectral moments to their desired values . minimization of these potentials is via a gradient descent closed - loop system that , under certain convexity assumptions , ensures convergence of the network topology to one with the desired set of moments and , therefore , eigenvalues . we illustrate our approach in nontrivial computer simulations .