first principles calculations based on the density functional theory ( dft ) have been widely used for investigating material properties and phenomena in condensed matter physics .while calculations with hundreds of atoms are currently routine , there is a demand for simulations of larger systems .however , the computational time for conventional dft calculations grows as o( ) and such systems require massive computational effort which is often impossible to realize .linearly scaling methods , or o( ) methods , for dft allow us to study such large systems .in particular , the divide - and - conquer approach to density functional theory ( dc - dft ) has robust convergence properties . in this approach ,the total system is divided into overlapping subsystems often called fragments .the total energy is minimized through an iterative procedure of two steps : ( i ) solving the kohn - sham equation for each fragment independently , and ( ii ) aggregating the electron densities of the fragments to calculate the electronic potential .a difficult problem with the o( ) methods is that the electronic structure calculation for the total system requires a time - consuming post - processing procedure .generally , if we want to obtain the one - electron orbitals and orbital energies for the electronic state spatially extended over the total system , it is necessary to diagonalize the hamiltonian matrix for the total system with one - shot conventional o( ) calculations by using the electron density obtained in the o( ) procedure .these calculations require a large number of basis functions and a time - consuming calculation for the hamiltonian matrix elements .for the former problem , a method is proposed to reduce the atomic - like basis functions to some extent .the latter problem is due to that the matrix elements are calculated by explicitly treating the hamiltonian operator . in dc - dft ,this calculation seems redundant since the kohn - sham equations for overlapping fragments have already been solved in the dc - dft calculation : the effects of kinetic and potential terms should in principle be embedded in the fragment orbitals .the fragment molecular orbital ( fmo ) method is one of the dc approaches specialized for covalent - bonding molecules .there are some post - processing schemes of fmo to evaluate the electronic state of a whole molecule by utilizing information of the fragments . among them , the fmo linear combination of molecular orbitals ( lcmo ) method is a particularly efficient scheme . in fmo - lcmo ,the one - electron hamiltonian matrix is formulated and calculated using the molecular orbitals ( mos ) of each fragment as basis functions , so that the wave function of the whole molecule is represented by a lcmo of the fragments without the redundant recalculation .this scheme can reduce the dimension of the hamiltonian matrix and the computational cost for calculating the matrix elements .nonetheless , this scheme can not be adapted to general materials straightforwardly because fmo crucially relies on the specific property of the orbitals of carbon atoms .  in this work, we present a new post - processing method of dc - dft , which allows us to calculate the eigenstate for general materials utilizing output of the dc - dft calculation . to this end , we use dc - dft with plane - wave basis functions , especially the lean divide - and - conquer ( ldc ) dft , that can be applied to general materials with a systematic procedure . in our method ,a small number of basis functions are constructed by reducing the fragment orbitals derived from dc - dft . the total hamiltonian matrix defined by the localized basis setis derived easily from the fragment orbitals and their orbital energies in the manner of fmo - lcmo .each wave function of the whole system is represented by a linear combination of the fragment orbitals ( lcfo ) .the diagonalization process can be done with little computational cost thanks to the reduced number of the basis functions .thus our procedure , called dc - lcfo hereafter , has advantages of a low cost and versatility .this paper is organized as follows .section [ sec : formalism ] describes dc - dft and our dc - lcfo .section [ sec : results ] presents the computational results .we analyze the parameter dependence for resultant eigenenergies and the feasibility of the matrix - size reduction of the hamiltonian in our scheme .we also compare the eigenenergies and the wave functions with those of the conventional method in p - doped si and ingan / gan superlattice systems . finally , conclusion is given in sec .[ sec : conclusion ] . in appendix[ appendix ] , we compare the formulation of dc - lcfo with that of fmo - lcmo .we first review the fundamental formalism of dc - dft . in dc - dft ,the physical space is represented as a union of non - overlapping core domains , , where .an additive quantity defined for is described as the sum of those for the respective domains .practically , the summands are approximately evaluated by constructing modified domains , or fragments . here , is the buffer region surrounding ( fig .[ fig : dc ] ) .thus the electron density of the whole system is given by where is the electron density in clipped from that of the fragment : for , while otherwise . the orbitals andcorresponding eigenenergies for each fragment are evaluated from the fragment kohn - sham ( ks ) equation , |\phi^{\alpha}_i\rangle    = \varepsilon^{\alpha}_i |\phi^{\alpha}_i\rangle ,    \label{eq : fragmentkseq}\ ] ] with the boundary potential that represents artificial effects of the buffer regions such as termination of the bonds and the boundary condition at . here $ ] is the ks potential with the density of the total system .the fermi energy , or the chemical potential , in the fermi distribution function is determined by the electron number condition imposed on the whole system . here , the chemical potential in eq .is common for all the fragments and it is solved by the bisection or newton - raphson method .   schematic 2d image of the dc algorithm .the whole system is represented as a union of non - overlapping core domains .physical quantities of are evaluated on the fragment , where is a buffer layer whose thickness is b. ]   here we briefly describe the strategy and basic ideas in dc - lcfo .there are several methods for dc - dft with different basis functions and different ways of handling the boundary effects of the fragments : their accuracy depends on the target systems . in the present paper , we adopt ldc - dft as the basis , which utilizes the density template potential as and the periodic boundary conditions at on the fragment ks orbitals .this method is suitable for calculations of condensed matter since it has relatively small overhead and convenient buffer configurations for calculations ( see sec . [ subsec : hamiltonian ] ) .note that we use the step - wise projection in eq .  for simplicity , though continuous weight functions are used in the original ldc - dft paper .our fundamental assumption is that the eigenstates near the fermi level can be well represented by patching a small number of fragment orbitals in the corresponding energy region .thereby we shall first develop an algorithm to generate basis functions from the fragment orbitals within the low - energy region .  as the next step, we shall introduce a method to construct the total hamiltonian matrix using the basis functions and the fragment hamiltonians defined in eq . .the matrix elements of the hamiltonian can be constructed simply by inner products among the fragment orbitals .hence the hamiltonian matrix can be obtained without time - consuming calculations even when the exact exchange potential is taken into account .the resultant basis functions are defined on each core domain and therefore it is not necessary to consider the total overlap matrix of the whole system . moreover ,the hamiltonian matrix has a far smaller dimension than the plane - wave basis case because the new basis functions are made of the fragment orbitals in the low - energy region .notably , a typical number of the basis functions per atom for practical accuracy is roughly 1020 , which is comparable to the atomic - like basis case .however the latter case has drawbacks such as the lack of systematic convergence .there is a similar approach that utilizes the ks orbitals of subsystems as a basis set for evaluating the green s functions , though the method is specialized for a quasi-1d system .in contrast , our scheme is the post - processing method of dc - dft for a direct diagonalization of the hamiltonian matrix .dc - dft can be systematically applied to general 3d systems .the basis functions of the present method are constructed as follows .we introduce a cutoff energy for the energy eigenvalues of the fragment orbitals in order to restrict the number of the fragment orbitals used for constructing the basis set : where is the number of the fragment orbitals satisfying , and is the orbital index .  in order to eliminate a redundant contribution from the buffer region , we project the fragment wave functions onto , to avoid overcompleteness with the projected orbitals , we construct a smaller set of orbitals from them .namely , we define an overlap matrix within each fragment , next , we diagonalize it , where and are the transformation matrix and the eigenvalue of , respectively , and is the number of the linearly independent eigenvectors .practically , we set a sufficiently small cutoff parameter for the eigenvalues in order to control .the new basis functions are defined as , it is notable that these basis functions are orthonormal :    we shall construct the hamiltonian matrix , where is the conventional ks hamiltonian operator of the total system . a straightforward approach to this calculationis real - space integeration , for example , , etc . in the below , however , we show the hamiltonian matrix elements can be evaluated from the fragment orbital energies without the numerically demanding integration .the local part of the hamiltonian has only diagonal blocks ( ) because of the non - overlapping nature of . on the other hand ,the nonzero off - diagonal blocks ( ) come from the kinetic term and the non - local potential term . in the dc scheme, it is assumed that the non - local terms decay within the buffer region .it means that the terms decay within neighbor core domains , because the thickness of is assumed as the same or the half length of each core domain  . following this assumption ,we consider only the diagonal blocks ( ) and the off - diagonal blocks ( ) between the face- , edge- and corner - sharing neighboring core domains .we shall obtain an approximate expression for the hamiltonian matrix , which is represented by the ks orbitals and eigenenergies of a certain fragment overlapping with the core domains and .the apparent absence of the hamiltonian operator in the resulting expression allows us a consistent treatment regardless of whether or not whether numerically demanding non - local operators ( e.g. , exact exchange operator ) are considered in the calculation within each fragment .the pivotal approximation is that the basis function does not spill far out of the core domain through the hamiltonian operation .now we define a projection operator for the hamiltonian decomposition , where is a region to truncate the spillage of the basis functions upon the hamiltonian operation ( fig .[ fig : truncation ] ) . using the projection operator ,we get the following exact expression , where .furthermore , the projected total hamiltonian can be converted to the fragment hamiltonian due to a feature of that vanishes at the core domain .the second term is presumably small if the range of is sufficiently larger than .therefore the hamiltonian acting on the vector can be reasonably approximated as .thus we get to the following approximate form of the hamiltonian matrix element , from the range of the projector , it is obviously nonzero only for whose core domain is overlapping with ( fig .[ fig : beta ] ) .since the fragment hamiltonian can be represented through the fragment orbitals , we obtain , where . here, we reduced the high - energy region ( ) for approximation .  while we introduced as explained above , we mention here that eq .can be generalized as follows : where is an arbitrary fragment satisfying .we have found that the choice of has quantitatively no effect on results .combining eq .  and eq . , we can represent the matrix elements of the total hamiltonian with the fragment orbitals and the eigenenergies , namely the output of dc - dft .note that the controllable parameters in our eigenstate calculation method are , , and the buffer thickness of ( , see fig .[ fig : truncation ] ) .we need some consideration for the optimum thicknesses and . the thickness is dictated by the nearsightedness principle in the conventional dc algorithms .on the other hand , the thickness is determined by the decay range of the non - local term of the ks hamiltonian . although large value of and may improve results of the calculation , there is a possibility that the limit degrade the accuracy because of the artificial boundary conditions at .thus it seems that accuracy requirements for our method demand a large value which can fully contain the nearsightedness range ( conventional ) plus the hamiltonian decay range ( ) .however , following results show this is not the case .in fact , for the ldc - dft - based scheme , the limit leads to the best results in eigenstate calculations ( see sec . [subsec : parameter ] ) . this might be attributed to the fact that the uniform kinetic - energy term is the main factor of the non - local part .moreover , we can deduce that the periodic boundary conditions at the fragment boundaries also improve the accuracy of the eigenstates in condensed matter .if we utilized other boundary conditions ( e.g. insertion of artificial vacuum regions ) , the accuracy for the eigenstates might require an optimization of the value .the exact exchange potential or the hartree - fock exchange potential can be also contained in the total hamiltonian as a non - local term .as is well known , this operator is defined as follows , ({\bf r } )    & = & -\sum_j^{\mathrm{occ } } \psi_j ( { \bf r } ) \int d^3 r '    \frac { \psi^{\ast}_j({\bf r } ' ) \psi_i({\bf r } ' ) } { |{\bf r}-{\bf r}'| } \nonumber \\     & = & - \int d^3 r ' \frac { \rho({\bf r},{\bf r } ' ) \psi_i({\bf r } ' ) } { |{\bf r}-{\bf r}'| } ,   % \nonumber\end{aligned}\ ] ] where are orbitals of the whole system and is the density matrix .the density matrix is exponentially localized with respect to in many cases .therefore we can take into account only the short - range part of and adopt the dc scheme in such cases .it is expected that our method allows high - speed eigenstate calculations within hybrid functionals . in a naive implementation of the exact exchange ,the computational cost is proportional to the fourth power of the system size . on the other hand , in the present method , the computational cost of the hamiltonian construction is negligible and the cost of the diagonalization is proportional to the third power of the small matrix dimension as explained in sec .[ subsec : strategy ] .  in sec .[ subsec : time ] , we demonstrate the validity of our scheme to construct the hamiltonian with the exact exchange potential .namely , after doing the conventional calculation including the exact exchange in each fragment independently , we construct the basis set and the hamiltonian matrix from the orbitals derived by these calculations .we shall see that the energy eigenvalues are accurately reproduced ( see sec .[ subsec : time ] ) .more thorough divide - and - conquer calculations with the exact exchange will be presented elsewhere .we have implemented ldc - dft and our method in the xtapp code .the ldc - dft code is parallelized using the message passing interface ( mpi ) library . to perform our procedure, we utilized the grid points of the fast fourier transformations ( fft ) for the projection and the inner - product operations of the fragment orbitals .the process to generate the basis functions is completely parallelized , while the hamiltonian matrix operation needs mpi communications for calculating the off - diagonal blocks .the matrix elements of the hamiltonian are gathered in the root process , and then the matrix is diagonalized by a serial computation . in the eigenstate calculations , we shifted the energy origin as to suppress a contamination by matrix elements .we can obtain final results through reshifting , , where are eigenenergies of the total hamiltonian .  in the following calculations, we used the plane - wave basis ( for calculations in each fragment ) , the norm - conserving pseudopotentials , and pbe exchange - correlation functional except some calculations with the exact exchange , in which pbe0 hybrid functional was used .all the calculations were carried out in the paramagnetic case .for comparison we also performed the conventional dft calculations , where we sampled the brillouin zone at the point ( point ) .rms error of the occupied eigenenergies as a function of a ratio for sic systems with zinc - blende structure and amorphous structure , where each system contains 512 atoms in a cubic supercell .the other parameters are fixed as ev ( 0.4 hartree ) and . ]rms errors plotted against a number of the basis functions per atom for the same systems as fig .[ fig : b ] , where and ev ( 0.4 hartree ) .these points correspond to , , , , and respectively from left to right . ]   the dependence of the rms errors for the same systems as fig .[ fig : b ] , where the energy origin is fixed at the fermi energy .the other parameters are fixed as and . ]we examine a parameter dependence for the resultant eigenenergies of the present method with respect to , and .trial systems are a sic 512-atom systems with zinc blende structure and an amorphous structure ( supercells ) .we divide the systems into 64 fragments ( with configurations ) , respectively , whose side lengths of each cubic core domain and the buffer thickness are fixed as  ( experimental lattice constant ) . the plane - wave cutoff is 30 ry and the number of the fft mesh points is equal to in each core domain .  figure [ fig : b ] shows the relation between the thicknesses and the root mean square ( rms ) errors of the occupied eigenenergies with respect to the conventional results . here ,rms error of eigenenergies is defined as follows : where and are the orbital energies for the whole system obtained by our method and the conventional dft method , respectively .this figure indicates that leads to the best results for not only the zinc - blende structure but also the amorphous structure , although there was the concern of boundary effects in the latter case . with these results, we put in all the following calculations .  in order to examine the dependence, we change value from to .figure [ fig : lcut ] is the rms errors plotted against the number of the basis functions ( ) per atom with the same conditions , where ev ( 0.4 hartree ) .the rms errors are saturated at nearly 15 basis functions per atom when .figure [ fig : ecut ] illustrates the dependence of the rms errors with the same conditions , where the energy origin is fixed at the fermi energy (= the valence band maximum in a gapped system ) . in the amorphous structure , the eigenenergies of the occupied states are accurately evaluated with a large value . on the other hand ,the error for the zinc - blende structure indicates weak dependence on probably due to the band gap . in a gapped state, we can deduce that it is sufficient to utilize the occupied fragment orbitals for representing the occupied states of the total system .  in summary ,the accuracy for the eigenstate requires the parameter values for , , and a sufficiently large value compared to the desired energy range .these conditions provide the accuracy comparable with fmo - lcmo .we demonstrate that the present method can represent a defect state with a spatially extended wave functions with satisfactory accuracy . to this end, we perform a calculation for a p - doped si crystal that contains 512 atoms ( one p atom included ) and divide the system into 64 fragments with the configuration , where the side lengths of each core domain and the buffer thickness are fixed to  ( experimental lattice constant ) .the controllable parameters of the eigenstate calculation are set as , , and ev ( hartree ) .the plane - wave cutoff is 30 ry and the number of the fft mesh points is equal to in each core domain .the rms error and the maximum absolute error ( mae ) for the occupied eigenenergies ( ) are 0.013 ev and 0.084 ev , respectively . here, is the eigenenergy of the total system and the highest occupied state is the half filled donor state ( ) .the absolute error for the donor state eigenenergy is 0.004 ev .the rms error and the mae for the unoccupied eigenenergies ( ) are 0.061 ev and 0.133 ev , respectively .the number of the basis functions , or the dimension of the hamiltonian matrix , is .figure [ fig : wf ] shows the wave function of the donor state calculated with our method and the conventional method for comparison . from this resultit can be seen that our scheme can properly represent the wave function extended over fragments .we applied our method to a superlattice system with the polar interface .ingan / gan heterostructures with [ 0001 ] epitaxial alignments have such interface which induces band bending .we show results of the eigenstates calculation for ( in)/(gan) superlattice which contains 768 atoms in a rectangular cell of side lengths , ( in ) .the system is divided into 12 fragments with 1d ( ) configurations .the buffer thickness is equal to the length of each core domain along the direction , while the - plane is not divided so in the - plane .the parameters for the eigenstate calculation are fixed as , , and ev ( hartree ) .the plane - wave cutoff is 50 ry and the number of the fft mesh points is equal to in each core domain .  the rms error and the mae for the occupied eigenenergies ( ) are 0.031 ev and 0.186 ev , respectively , while the rms error and the mae for the unoccupied eigenenergies ( ) are 0.027 ev and 0.084 ev , respectively .the dimension of the total hamiltonian matrix in this calculation is 3301 ( i.e. the number of the basis functions per atom is nearly equal to 4.3 ) .this small number of the basis functions gives sufficient accuracy because of the 1d configuration of the fragments where each fragment has only two boundary regions in contrast to 26 in case of 3d configuration .figure [ fig : ldos ] shows the local density of states ( ldos ) indicating the band - bending structure .    the computational cost of the hamiltonian diagonalization ( per iteration for the conventional cases ) with different supercells , where the exchange - correlation functional is pbe or pbe0 .lcfo indicates our method , while pw corresponds to the conventional plane - wave calculation .the computation is done on a 24-cores 2.5 ghz intel xeon cluster with 144 nodes . ]finally we performed benchmark tests of the dc - lcfo method using si 96-atom ( ) , 216-atom ( ) , and 432-atom ( ) supercells .the systems are divided into 12 , 27 , and 54 fragments along the direction , respectively .we set ( lattice constant ) for each fragment so that it is equal to supercell .the basis set and the hamiltonian matrix are derived from copied wave functions , which are obtained by the conventional calculation in the cell , where we used pbe and pbe0 functionals .we fixed the parameters as , , and (= occupied + 8 orbitals , instead of ) .for the pbe0 calculation , we used the coulomb potential cutoff  for truncating and set a.u . as a simple implementation .figure [ fig : time ] shows the computational cost for the construction and diagonalization of the hamiltonian ( per iteration for the conventional cases ) .this result suggests that our method enables a high - speed computation of the eigenstate . as reference, we describe the rms error of the occupied eigenenergies and the error of the band gap for the pbe0 hybrid functional in cell ; 0.034 ev and 0.029 ev , respectively .we have developed a method , named dc - lcfo , to compute the orbital wave functions and the corresponding orbital energies of general huge systems based on dc - dft .the method utilizes the output derived by dc - dft for constructing the total hamiltonian matrix without redundant computations .furthermore , this method can dramatically reduce the matrix size of the total hamiltonian .thus its computational cost is much lower than the conventional calculations .we have applied the method to p - doped si and ingan / gan superlattice systems to demonstrate that it reproduces a structure of the wave functions spread over the total system with practical accuracy .dc - lcfo is a powerful tool for studying the nature of electronic states , the mechanism of a chemical reaction for example , of large systems . in principle , our scheme can be applied to the hamiltonian including short - range exchange - correlation terms such as the gw self - energy operator .this work was supported in part by jsps kakenhi grant number 26286085 , and mext as a social and scientific priority issue ( creation of new functional devices and high - performance materials to support next - generation industries ; cdmsi ) to be tackled by using post - k computer .s. y. was supported by the advanced leading graduate course for photon science ( alps ) .the authors thank the supercomputer center , the institute for solid state physics , the university of tokyo for the use of the facilities .ls3df - like decomposition of the fragment hamiltonian . ]  in sec .[ subsec : hamiltonian ] , we formulated the approximate expression for the hamiltonian matrix , starting from the consideration of the hamiltonian operation on the basis functions . in fmo - lcmo , on the other hand , the approximate hamiltonian is represented as a decomposed operator which consists of the fragment hamiltonian operators .let us discuss the relation between these two formulations .  in fmo - lcmo ,the total hamiltonian is first decomposed into the `` fragment monomer '' and `` fragment dimer '' ( the union of two monomers ) terms as follows : where we omit the higher terms such as the `` fragment trimer '' ( fmo2 level ) . and are indices of the fragment monomers and denotes the hamiltonian for the fragment monomer . represents the hamiltonian for the fragment dimer .the second term represents the non - local effects across the monomers , while the first term indicates the monomer effects .one can construct a representation analogous to eq .for the hamiltonian operator so that its matrix elements agree with eq . .this representation is formulated by the summation of the fragment hamiltonians in a similar way to the sum formula of electron density in a dc approach called ls3df . for simplicity , we consider only a case of 2d systems . here , each is divided into 4 small fragments and they are assigned new indices . for each fragment , we assign a sign factor depending on the layout of the fragment ( fig . [fig : ls3df ] ) .the approximated hamiltonian operator of the whole system can be expressed as , where is a projected hamiltonian for the small fragment . with the above ,the matrix elements are completely identical to with a proper configuration of the fragment .specifically , we put ( ) if the fragment is located at the north / north - east / east ( south / south - west / west ) of the fragment in 2d systems , while indicates the other fragment overlapping with both and if the fragment is located at the north - west / south - east of the fragment as fig .[ fig : beta - other ] .

Abstract: we present an efficient post - processing method for calculating the electronic structure of nanosystems based on the divide - and - conquer approach to density functional theory ( dc - dft ) , in which a system is divided into subsystems whose electronic structure is solved separately . in this post process , the kohn - sham hamiltonian of the total system is easily derived from the orbitals and orbital energies of subsystems obtained by dc - dft without time - consuming and redundant computation . the resultant orbitals spatially extended over the total system are described as linear combinations of the orbitals of the subsystems . the size of the hamiltonian matrix can be much reduced from that for conventional calculation , so that our method is fast and applicable to general huge systems for investigating the nature of electronic states .


random matrices and free probability are areas of applied probability with increasing importance in the area of multiantenna wireless systems , see for example . one key problem in the stochastic analysis of these systems has been the study of their asymptotic performance with respect to the number of antennas . the first answer tothis question is the groundbreaking work by telatar , who , describing the system as a random matrix with statistically independent entries , showed that the capacity of this system is infinite .since this independence condition might be restrictive , several further proposals have been made over the past decade . as a result, a few models have emerged to take into account some instances of correlation in the system , , .operator - valued free probability theory has proved to be a powerful tool to study block random matrices .this has made possible to analyze certain systems exhibiting some block - based dynamics . with recent developments in operator - valued free probability theory ,simple matricial iterative algorithms now allow us to find the asymptotic spectrum of sums and products of free operator - valued random variables .the purpose of the present paper is show the significance of these new tools by studying a particular application in wireless communications .in particular , we study an operator - valued kronecker correlation model based on an arbitrarily correlated finite dimensional multiantenna channel . from a blockmatrix dynamics and a parameter related to the angle diversity of the system , an operator - valued equivalent is derived and then a method to calculate the asymptotic isotropic mutual information is developed using tools from operator - valued free probability .the model allows using information related to the asymptotic antenna patterns of the system . to our best knowledge , this the first time that a model with these characteristics is analyzed .more precisely , a multiantenna system is an electronic communication setup in which both the transmitter and the receiver use several antennas .the input and the output of the system can be thought of as complex vectors and , where is the number of transmitting antennas and is the number of receiving antennas .the system response is characterized by the linear model where is an complex random matrix that models the propagation coefficients from the transmitting to the receiving antennas and is a circularly symmetric gaussian random vector with independent identically distributed unital power entries .  in a correlated multiantenna system ,there is correlation between the propagation coefficients .namely , the random matrix is such that the random variables are not necessarily independent .it is customary to take the random variables composing with circularly symmetric gaussian random law telatar1999 . in this context, the joint distribution of the entries of depends only on the covariance function for and .  for a fixed rate , it is known that the capacity of a multiantenna system grows linearly with the number of antennas of the system as long as the matrix has independent entries .this shows the well - behaved scalability properties of multiantenna systems .however , correlation may have a negative effect on the performance of the system .therefore , it is necessary to estimate quantitatively the effect that correlation may have .  throughout this paperwe will assume that the transmitter uses an isotropical scheme , i.e. , where is the transmitter power .in this case , a canonical way to quantify the effect of correlation is by means of the asymptotic isotropic mutual information per antenna . specifically , suppose that and for each the random matrix describes the channel behavior when there are transmitting antennas and receiving antennas .moreover , suppose that both and are increasing sequences and converges to a positive real number .then , the asymptotic isotropic mutual information per antenna is as long as the limit exists .a common phenomena in random matrix theory is that the sequence of arguments in the expected value above converges almost surely to a constant , and under mild conditions also in mean .therefore , the asymptotic isotropic mutual information per antenna is given , essentially , by the a.s .limit of the aforementioned sequence .therefore , in order to find , it is necessary to derive a model for the sequence of random matrices that approximates the channel behavior in the finite size regime and then compute the asymptotic quantity .in this paper we use an alternative method described in four steps :  1 .  assign an operator - valued matrix to the matrix ; 2 .  compute the operator - valued cauchy transform of ; 3 .  via the stieltjes inversion formula ,recover the distribution of , call it ; 4 .  compute as   the operator - valued matrix can be thought of as the asymptotic operator - valued equivalent of the channel  . in this sense ,the common approach consists of giving a model for the finite size regime , computing the mutual information , and taking the limit . on the other hand, the alternative approach takes _ limits in the model _ , replacing matrices by operator - valued matrices , and then calculates the mutual information .of course , these approaches are intimately related .actually , in the traditional case , they provide the same results converges in distribution almost surely to the marchenko pastur distribution .this is equivalent to saying that converges in distribution to a noncommutative random variable whose analytical distribution is the corresponding marchenko pastur distribution , which gives the asymptotic mutual information ( 1 ) . ] , but we prefer the latter approach since it is conceptually easier to understand and carry out , providing a powerful tool for modelling .we will see that this way of thinking goes well with channels exhibiting a block - based behavior .in particular , the operator - valued matrix assigned in step 1 carries the block structure of the channel and some other features of the system . in the example analyzed here , these features include the effect of the asymptotic antenna patterns and the inclusion of the starting finite dimensional channel correlation . to illustrate the kind of tools that may be useful in the assigning process at step 1 , in the next section we retrieve a block - based kronecker model from an angular - based model andderive the operator - valued equivalent .  in section [ section : model ]we derive the proposed operator - valued kronecker correlation model . in section [ section : asymptoticcapacity ]we discuss the asymptotic isotropic mutual information of our model using tools from operator - valued free probability . in section [ section : diagonalcase ]we consider a particular example where the implementation is simple but at the same time flexible enough to be applied in several interesting cases , like some symmetric channels . in section[ section : numericalcomparison ] we compare , through the example of a finite dimensional system , the mutual information predicted by the usual kronecker correlation model against the results from the proposed operator - valued alternative . in appendix a we summarize the notation , the background , and the prerequisites from operator - valued free probability theory . in appendix bwe prove theorem thm : extremalcases on two extreme behaviors of the model regime . in appendix cwe compute some of the operator - valued cauchy transforms required in this paper .the proposed model to approximate the channel behavior in the finite size regime is derived as follows .suppose that for a fixed , each antenna of the original system is replaced by new antennas located around the position of the original one .thus , the new system has transmitting and receiving antennas .figure [ fig : antennas ] shows the original system for and together with the corresponding virtual one for .   system . on the right the virtual system corresponding to .,scaledwidth=70.0% ]  for any given , the channel matrix for this system will have the form where is the matrix whose entries are the coefficients between the new antennas that come from the original receiving and transmitting antennas .we now derive a model for that takes into account the statistics of the channel matrix and the block structure exhibited above .first , fix a block , and for notational simplicity denote it by .this matrix should reflect the behavior of a scalar channel between two antennas of the original system when these are replaced by antennas each .  in a regime of a very high density of antennas per unit of space ,any two pairs of antennas close enough are likely to experience very similar fading .since as the new antennas are closer to each other , then the propagation coefficients between them are prone to be correlated . as an extreme case , we suppose that all the propagation coefficients between the antennas involved in have the same norm , and without loss of generality we set this to be one .this means that these coefficients produce the same power losses and the differences between them come from the variation that they induce in the signal s phases . with this in mind, we will suppose that for , where , is a real random variable and is a physical parameter that reflects the statistical variation of the phases of the incoming signals . in some geometrical models ,this statistical variation of the phases has been used , along with the angle of arrival and the angle spread , to study the capacity of multiantenna channels .some of the physical factors that have the most impact on the correlation of an antenna array are related to either the physical parameters of the antennas or to local scatterers .since these factors are different for each end of the communication link then , borrowing the intuition from the usual kronecker model , it is natural to take the matrix as a separable or kronecker correlated matrix , that is , where and are the square roots of suitable correlation matrices and is a random matrix with independent entries having the standard gaussian distribution .it is important to point out that is not kronecker correlated .   from a modelling point of view, the case represents the situation in which the environment is rich enough to ensure a high diversity in the angles of the propagation coefficients . on the other hand ,the case represents a system in which the propagation coefficients in the given block are almost the same .intuitively , the first case is better in terms of , since we should be able to recover the multiantenna diversity via the angle diversity ; while in the second case we almost lose the diversity advantage of a multiantenna system over a single antenna system .  in these limiting casesthe following holds .we denote by the ordered eigenvalues of an hermitian matrix .[ thm : extremalcases ] assume that and are full rank . for , as ,  is a matrix with i.i.d .entries with uniform distribution on the unit circle .suppose that is a sequence of positive real numbers such that as .then , almost surely , as where is the asymptotic eigenvalue distribution of .  see appendix b.  observe that in the second part of the previous theorem both and depend on as they are matrices .this means that the entries of the matrix become uncorrelated as , and , by universality , the spectrum of must behave similar to the spectrum of a standard gaussian matrix of the same size .observe that in this limiting case , we arrive at the well known case of i.i.d . entries , i.e. , the canonical model of a multiantenna system telatar1999 .as was mentioned before , in this situation the environment has a high diversity in the angles of the propagation coefficients , and thus it is natural that the system behaves as in the i.i.d. case .  on the other hand ,when , the bulk of is close to that of .this suggests approximating .note that this limiting case leads to the well known kronecker correlation model . in the spirit of a worst case analysis , we will use in what follows .  in the proof of theorem [ thm : extremalcases ] we only used the fact that hasan asymptotic eigenvalue distribution with compact support and converges a.s . as .therefore , under these mild conditions , the same analysis yields to the approximation for any model .in terms of the asymptotic behavior of the spectrum and invoking ideas from random matrix theory and free probability , let be a noncommutative probability space where the algebra has unit ( see appendix a ) .we can model the matrix by means of a noncommutative random variable in such that where and are in such that and are the limits in distribution of and respectively , and is a circular operator with a given variance . since the matrices and depend on separate sides of the communication link , and in some contexts , such as mobile communications , the transmitter and receiver are not in any particular orientation with respect to each other , we can assume that the eigenmodes of this matrices are in standard position .in particular , this means that the distributional properties of and are invariant under random rotations , i.e. , where is a haar distributed random matrix independent from and .the latter implies that and are free , and by voiculescu s theorem they both are free from .  if we use the noncommutative random variable representation , as we did with , for every block with and , then where , and are the corresponding correlation and circular random variables for the block . by the same argument as before, we assume that the families and are free . in this way , for any  where   let be the linear map determined by where isthe -unit matrix in and is any matrix .it is clear that . for every , belongs to the -valued probability space is in fact an algebra of random matrices over the complex numbers .this is the only time we use this abuse of notation . ] and to the -valued probability space where .moreover , we will restrict ourselves to working with asymptotic eigenvalue distributions with compact support , which allows us to work within the framework of a -probability space . in this context ,convergence in distribution implies weak convergence of the corresponding analytic distributions , see also appendix a.  in the derivation of this model , we can observe that all the depend on the new antennas around the original receiving antenna , and thus it is reasonable to take all them equal to some random variable . proceeding with this reasoning at the transmitter side ,we conclude that where and are the operators associated to the correlation structure of the antennas at each side , and .thus , we can think of this model as the operator - valued version of the kronecker correlation model for multiantenna systems .moreover , let be the correlation matrix .] of , i.e. , . in terms of the model, must reflect the correlation structure of the channel matrix and the parameter of the system . a reasonable way to dothis is by setting . in the regime , the latter implies that the mutual information decreases proportionally to .since we can incorporate the constant into the correlation operator - valued matrices and , for notational simplicity we will set in our discussion , thus we will take .nonetheless , remember that the model derivation was made in the regime .observe that each depends on the new antennas around the original receiving antenna .thus the distribution of will depend strongly on the specific geometric distribution of the new antennas .for example , if all the new antennas are located in exactly the same place as the original antenna , we would obtain that the distribution of must be zero . in the casewhere the antennas are collinear and equally spaced , we can use some class of toeplitz operators as shown in .a similar argument can be used for the transmission operators .  observe that in this way we have incorporated the finite dimensional statistics in our operator - valued equivalent . moreover, the correlation matrix does not need to be separable , i.e. , with a kronecker structure .this shows that the operator - valued kronecker model is slightly more flexible than the classical version : it allows an arbitrary correlation resulting from the channel , and it also allows different correlations for different regions of the transmitter and receiver antenna arrays , which in our notation is encoded in the matrices and .in this section we derive a method to calculate the asymptotic isotropic mutual information ( [ asymptoticcapacity ] ) of our model using the tools of operator - valued free probability . for simplicity of exposition , in what followswe will take .note that if , for example , , then we can proceed by just taking and by taking equal to for .let , and as in ( [ hmodel ] ). the goal is to find the distribution of in ( [ asymptoticcapacity ] ) by means of the -valued cauchy transform of ( see appendix a ) .  using the symmetrization technique , we define as notice that the distribution of is the same as the distribution of , and that is selfadjoint .since all the odd moments of are 0 , the distribution of is symmetric .we can then obtain the -valued cauchy transform ( [ ctoper ] ) of from the corresponding transform of using the formula nica2006 where and is the identity matrix in . since the spectrum of is the same as the spectrum of the -valued cauchy transform of well known ( see ) .thus , we just need to find the -valued cauchy transform of in order to be able to apply the operator - valued subordination theory ; see ( [ ctoperprod ] ) in theorem [ thm : subbordination ] of the appendix a.  \a ) the above mentioned operator valued subordination theory allows us to compute , via iterative algorithms over matrices , the distribution of sums and products of operator valued random variables free over some algebra ( theorem [ thm : subbordination ] in appendix a ) . for a rigorous exposition of the concept of freeness over an algebra ,we refer the reader to far2008,speicher2012 and the references therein .observe that this relation is similar to the usual freeness in free probability .\b ) the cauchy transform of is not given explicitly , instead , it is given as a solution of a fixed point equation .in general , the -valued cauchy transform has to be computed for any matrix .however , in some cases it is enough to compute it for diagonal matrices , which simplifies the practical implementation ( see section 4 ) .if the correlation matrices associated to the correlation operators are either constant or exhibit a distribution invariant under random rotations , as we supposed for and in the previous section , then these correlation operators will be free among themselves . in some applications ,these correlation operators come from constant matrices since they model the antenna array architecture which in principle is fixed .suppose that this is the case , and that also the are free among themselves. in some cases this hypothesis will be unnecessary ( see section [ section : diagonalcase ] ) . observe that by the assumed freeness relations between the random variables , we have that the coefficients of the operator - valued matrices in the previous sums are free , and thus the operator - valued matrices are free over .so we just have to compute the -valued cauchy transform of each operator - valued matrix in the above sum , and then apply the results from the free additive subordination theory ( ( ctopersum ) of theorem [ thm : subbordination ] in appendix a ) .[ thm : cauchytrk2 ] let be a noncommutative random variable , a fixed integer and . for , {k , k}^{-2}\left ( g_{r}([b^{-1}]_{k , k}^{-1})-[b^{-1}]_{k , k}\right ) b^{-1}e_{k , k}b^{-1}.\end{aligned}\ ] ]  see appendix c.  with the previous theorem , we can compute the -valued cauchy transforms of  . with these transforms , we have all the elements to compute the -valued cauchy transform of , and in consequence the scalar cauchy transform of the spectrum of is obtained from ( ctopsca ) : using the stieltjes inversion formula ,one then obtains and this gives the asymptotic isotropic mutual information ( [ asymptoticcapacity ] ) .from section [ section : model ] , we have that is an operator - valued matrix composed of circular random variables with correlation observe that in this case , where the ( ) are free circular random variables. thus there exist complex matrices for such that i.e. , can be written as the sum of free circular random variables multiplied by some complex matrices . in this way ,the summands in ( [ eq : sumfreeoverm ] ) are free over .observe that the previous procedure is exactly the same as writing a matrix of complex gaussian random variables as a sum of independent complex gaussian random variables multiplied by some complex matrices .  for ,define so .recall that the operator - valued matrices are free over . as an alternative to the technique given in to compute the operator - valued cauchy transform of , we can use the subordination theory by computing the individual operator - valued cauchy transforms for all and then using equation ( [ ctopersum ] ) .this technique is particularly neat in the following setup .suppose that for all the operator - valued cauchy transforms send diagonal matrices to diagonal matrices . from this assumption and equations ( [ ctopersum ] ) and ( ctoperprod ), it follows that this property is also shared by .moreover , the following theorem shows that this is also true for the operator - valued cauchy transform of .[ thm : cauchytq ] let be a diagonal matrix in . then   see appendix c.  since this diagonal invariance property is also satisfied by , again from equations ( [ ctopersum ] ) and ( [ ctoperprod ] ) , we conclude that satisfies this property .therefore , all the computations involved in this case are within the framework of diagonal matrices .also , in this diagonal case , any assumption of freeness between the noncommutative random variables in and is unnecessary since they do not interact when evaluating the cauchy transform of in diagonal matrices .intuitively , the structure of behaves well enough to destroy the effect that any possible dependency between the correlation operators may have in the spectrum of .it is easy to prove that the condition that sends diagonal matrices to diagonal matrices is equivalent to requiring that is diagonal for any diagonal matrix .this last condition can be shown to be equivalent to requiring that for all , we have that where is a diagonal matrix in and is a permutation matrix .if in a concrete application the correlation matrix can be suitably decomposed , or approximated , in such a way that this latter condition holds , then the method of this example can be applied .[ thm : cauchytxkl ] let .suppose that is a circular random variable , a diagonal matrix in , and a permutation matrix of the same size .let and .then , for with and diagonal matrices in , {\pi(1)}|d_{1}|^{-2}g_{xx^{\ast } } ( [ j_{1}]_{1}[j_{2}]_{\pi(1)}|d_{1}|^{-2}),\ldots  \notag \label{eq : cauchytxkl } \\ & \quad\quad\ldots,[j_{1}]_{\pi^{-1}(n)}|d_{\pi^{-1}(n)}|^{-2}g_{x^{\ast}x}([j_{1}]_{\pi^{-1}(n)}[j_{2}]_{n}|d_{\pi^{-1}(n)}|^{-2})).\end{aligned}\ ] ]  see appendix c.  it is important to remark that and have a marchenko pastur distribution , of which the scalar cauchy transform is given by   taking and , we obtain the -valued cauchy transform of explicitly .given the scalar cauchy transforms of the variables , the corresponding operator - valued transform of is also explicit , as given in equation ( eq : cauchytq ) .nonetheless , the operator - valued cauchy transform of and are not given explicitly , and need to be computed by means of equations ( [ ctopersum ] ) and ( [ ctoperprod ] ) , respectively .suppose that we have an operator - valued equivalent given by which corresponds to a channel with symmetric behavior .let and be defined as follows in the notation of ( [ qc ] ) , .moreover , using the same notation as above , , and . by equation ( [ eq : cauchytxkl ] ) , the -valuedcauchy transforms of and are given , for , by to denote that is the scalar cauchy transform in equation ( [ eq : cauchytmp ] ) .] respectively .figure [ fig : histogram ] shows the asymptotic spectrum of against the corresponding matrix of size when the correlations are assumed to obey the uniform distribution on  ] : the entry of the matrix ;   the transpose of the matrix , and , its conjugate transpose ;   : the -unit matrix in ;   : the identity matrix in ;   : valued with respect to a classical probability space ;   in what follows , will denote a noncommutative unital -algebra with unit , and is a unit - preserving positive linear functional , i.e. , and for any .the pair is called a noncommutative probability space and the elements of are called noncommutative random variables . unless otherwise stated , we use greek letters to denote scalar numbers , lower case letters for noncommutative random variables in , upper case letters for matrices or random matrices in , and upper case bold letters for matrices in .the latter are called operator - valued matrices and is a noncommutative probability space .  given a selfadjoint element ,its algebraic distribution is the collection of its moments , i.e. , .let and for be noncommutative probability spaces .if and for are selfadjoint elements , we say that converges in distribution to as if the corresponding moments converge , i.e. , for all . if there is a probability measure in with compact support such that for all  we call the analytical distribution of .a family of noncommutative random variables is said to be _ free _ if \cdots\lbrack p_{k}(a_{i_{k}})-\varphi\left ( p_{k}(a_{i_{k}})\right ) ] \right ) = 0\ ] ] for all , polynomials and such that for .let and be random matrices in for every .if there exists such that and are free and converge in distribution to , i.e. , for all , we say that and are asymptotically free .  given a probability measure in ,its ( scalar ) cauchy transform is defined as the stieltjes inversion formula states that if has density then for all , where denotes the imaginary part and the real part .let denote the set of matrices such that is positive definite , and define .for an operator - valued matrix we define its -valued cauchy transform by where the last power series converges in a neighborhood of infinity .the scalar cauchy transform of is given by   the freeness relation over is defined similarly to the usual freeness , but taking instead of and non - commutative polynomials over instead of complex polynomials .the main tools that we use from the subordination theory are the following formulas to compute the -valued cauchy transforms of sums and products of free elements in ; see .if is an operator - valued matrix in , we define the and transforms , for , by   [ thm : subbordination ] let be selfadjoint elements free over .\i ) for all , we have that where for any and   \ii ) in addition , if is positive definite , and invertible , and we define for all with the function for all , then there exists a function such that for all , and   the functions above are defined in .whenever we evaluate any of these functions in we have to do so by means of the relation .it is a well known result that the eigenvalues are continuous functions of the entries of a selfadjoint matrix .if the entries of a matrix lie in the unit circle , then its frobenius norm is bounded and so its operator norm . in particular , is a bounded and continuous function of the entries of .therefore , if we prove that the entries of converge in distribution to the entries of , i.e. , then as required .the entries of and lie in the unit circle , so we are dealing with compact support distributions .thus , it is enough to show the convergence of the joint moments of the entries of to those of to ensure the multivariate convergence in distribution , and so the claimed convergence in the first part of theorem [ thm : extremalcases ] .let be fixed , for  since and are full rank , a linear algebra argument shows that the previous exponents are all zero if and only if are all zero .therefore , the joint moments of the entries of vanish as except when for all and .it is easy to show that these limiting moments are indeed the joint moments of the entries of .this conclude the proof of the first part .the following lemma and two theorems are from appendix a in   [ lem : normhadamardprod ] let .then where denotes the pointwise or hadamard product of and .[ thm : perturbationinq ] let .then where and are the singular values of .[ thm : inqeigdistrank ] let and be two complex matrices .then , for any hermitian complex matrices and we have that   in this rest of this subsection , will denote the empirical distribution of the _ singular values _ of . since the classical convergence theorems in random matrices hold almost surely , it is enough to deal with the case of non - random matrices .[ lem : inqsingularvaluessqrttrace ] let . then   an straightforward application of theorem [ thm : perturbationinq ] and the generalized means .we define the entrywise exponential function by for all .[ prop : entrywiseexpdist ] let for and .let , then   using the power series for the exponential function we obtain that where .define and . by lemma [ lem : normhadamardprod ] andthe fact that , by lemma [ lem : inqsingularvaluessqrttrace ] we have that and in particular applying theorem [ thm : inqeigdistrank ] to the matrices and we obtain and are equal , i.e. for all . ] which implies that for all .this implies that for  and equivalently therefore and consequently using the same argument that in equation ( [ eq : interleaving ] ) we have that and thus and in particular by the triangle inequality we conclude that as claimed .observe that the previous analysis exclude the biggest singular value of . in the following propositionwe study the behavior of this singular value .  in the notation of the previous proposition ,  this shows that is roughly , while the bulk of is essentially the same as .  by inequality ( [ eq : estimatexz ] ) in the first part of the previous proof using lemma [ lem : inqsingularvaluessqrttrace ] for and  a straightforward computation shows that , so by the triangle inequality as claimed .  finally , with the previous quantitative results we prove the following qualitative result .let such that converge as and .define .if is a sequence of positive real numbers such that as , then as .recall that if and only if for all bounded lipschitz function .let be any bounded lipschitz function , by the previous propositions where is the lipschitz constant of . since is bounded and converge as , by proposition [ prop : entrywiseexpdist ] the previous expression converges to 0 as . finally , since as we have that and by the triangle inequality the result follows .the second part of theorem [ thm : extremalcases ] is an straightforward application of the previous theorem ._ proof of theorem [ thm : cauchytrk2 ] . _ the identities and lead to {k , k}^{n-1}e_{k , k}b^{-1 } \\ & = b^{-1}+[b^{-1}]_{k , k}^{-2}\left ( \sum_{n\geq0}\varphi\left ( r_{k}^{2n}\right ) [ b^{-1}]_{k , k}^{n+1}-[b^{-1}]_{k , k}\right ) b^{-1}e_{k , k}b^{-1 } \\ & = b^{-1}+[b^{-1}]_{k , k}^{-2}\left ( g_{r_{k}^{2}}([b^{-1}]_{k , k}^{-1})-[b^{-1}]_{k , k}\right ) b^{-1}e_{k , k}b^{-1}.\end{aligned}\ ] ] of course , the previous equations do not hold for every matrix , in particular , the power series expansion is valid only in a neighborhood of infinity .however , the previous computation can be carried out at the level of formal power series , and then extended via analytical continuation to a suitable domain ._ proof of theorem [ thm : cauchytq ] ._ a straightforward computation shows that   _ proof of theorem [ thm : cauchytxkl ] . _observe that thus , since and are diagonal for any diagonal matrix , and diagonal matrices commute , we have for that recalling that the odd moments of are zero , the previous equation implies finally , let be the permutation associated to , then {k}=[d^{\prime}]_{\pi(k)}$ ] for any diagonal matrix and any .therefore denote the entry of the diagonal matrix . ]{1}[j_{2}]_{\pi ( 1)}|d_{1}|^{-2}),\ldots , g_{x^{\ast}x}([j_{1}]_{\pi^{-1}(n)}[j_{2}]_{n}|d_{\pi^{-1}(n)}|^{-2})\right )  \notag \\ & = \mathrm{diag}([j_{2}]_{\pi(1)}|d_{1}|^{-2}g_{xx^{\ast}}([j_{1}]_{1}[j_{2}]_{\pi(1)}|d_{1}|^{-2}),\ldots  \notag \\ & \quad\quad\ldots,[j_{1}]_{\pi^{-1}(n)}|d_{\pi^{-1}(n)}|^{-2}g_{x^{\ast}x}([j_{1}]_{\pi^{-1}(n)}[j_{2}]_{n}|d_{\pi^{-1}(n)}|^{-2 } ) ) .\notag\end{aligned}\ ] ]mario diaz was supported in part by the centro de investigacin en matemticas a.c . , mxico and the government of ontario , canada .

Abstract: a model to estimate the asymptotic isotropic mutual information of a multiantenna channel is considered . using a block - based dynamics and the angle diversity of the system , we derived what may be thought of as the operator - valued version of the kronecker correlation model . this model turns out to be more flexible than the classical version , as it incorporates both an arbitrary channel correlation and the correlation produced by the asymptotic antenna patterns . a method to calculate the asymptotic isotropic mutual information of the system is established using operator - valued free probability tools . a particular case is considered in which we start with explicit cauchy transforms and all the computations are done with diagonal matrices , which make the implementation simpler and more efficient .


in 1864 , j. c. maxwell had set forth the problem of ether drift by the declaration of electrodynamics and despaired of a solution by experiment . over the past 150 years , a significant number of experimental investigations have been performed in different ways on the earth and on the space stations .these widely used experiments can be divided into two types based on the robertson - mansoury - sexl ( rms ) test theory : ( a ) one - way [ or first order test ; ( b ) two - way [ or second order test , where is the speed of the laboratory and is the speed of light in vacuum .these tests are mostly based on a traditional approach using two - way experiments where one tries to find an upper limit of the second order variation which is very small compared with the speed of light . in principle , a first order effect is times higher than a second order effect [ assuming / s , the orbital speed of the earth , and km / s , the speed of light in vacuum ] .however , one - way speed of light measurement did not receive wider attention because of the debates of the clock synchronization and successes of the sr . +  + due to the present challenge in the formulation of a quantum theory of gravity , the fundamental postulate of sr - the constancy of the speed of light - is now under further scrutiny in many different ways .the methods and techniques of measurement have been extended and improved with the improvements of technology with time and our dependence on their reliability has increased .therefore , researchers repeat tests of the fundamental postulate of any established theory over time to get more reliable result . here, we attempt to indicate a direction for further investigation of the possibility of the one - way speed of light measurement to test the isotropy of the speed of light using the experiments that directly used the one - way speed of light in the recent years . a treatment of the motion of the earth with respect to a preferred frame is considered for present analysis .we admit that motion of the earth does occur .then we immediately recognize that the concept of velocity of the earth plays a key role in our description and understanding of the isotropy of speed of light tests in a terrestrial experiment .+  + different co - ordinate transformations are proposed to describe space - time . among them , the well - known galilean transformations based on a universal time which leads to the anisotropy of the speed of light while lorentz transformation is based on the constancy of the speed of light .rms transformations ( or test theory ) assume that the speed of light is isotropic in the rest frame but needs to be tested in the moving frame . in order to give an idea of rms test theories compared with galilean ( newtonian ) and lorentz transformations , we introduce fig .[ fig : shematic1 ] .let us consider two inertial reference frames where is the hypothetical rest frame [ the cosmic microwave background ( cmb ) is the best candidate with experimental proof ] and [ the sun - centered celestial equatorial frame ( sccef ) ] is moving at a uniform velocity along the hypothetical rest frame .the cmb and the sccef have been elaborated in detail in to explain different tests of the lorentz invariance .the sccef is the frame in which the sun is at the centre , and is inertial relative to the cmb frame to first order . in addition, we introduce a brief idea of a conventional one - way test where two clocks are required ( clock a and clock b ) as shown in fig .[ fig : shematic1 ] .-frame ) with speed km / s relative to the cmb ( -frame ) with an idea of a conventional one - way speed of light test.,width=480 ]      following .,width=480 ]by examining recent experiments that directly use one - way light , we continue our discussion of the possibility of one - way speed of light to test the isotropy of sr .+  + in 2009 , the paper reported a one - way experiment using a time of flight technique .this experiment used a light beam from a he - ne laser .the beam was amplitude modulated and was directed onto a detector .the response was converted to an electrical signal .the phase - difference between the transmitted and the received signals was measured .the paper presents the average value of the one - way speed of light of two measurements which is m / s .+  + in 2008 , another report indicated similar type of a one - way speed of light measurement of the report using two arrangements with identical equipment of stable frequency sources and pulse generators that make simultaneous one - way speed of light measurements from clock a to clock b and in the opposite direction .the comparison of the two opposite responses was proposed .also the report indicates that the one way light speed to an accuracy around 1 part in .+  + in order to test one - way isotropy of the speed of light , we would like to indicate that this experiment should be performed for a period of 24-hours when the earth makes a complete rotation . then , searching of the possibility of the consequences of the speed of earth in terms of one - way speed of light measurement using this experiment can be performed assuming a galilean transformation as follows :  1 .one can plot the responses with time of the day of measurement and look for any diurnal variation as illustrated in fig .[ figure : shapechange2 ] .in addition , fourier transformation of the data can give the idea about any meaningful frequency that is consistent with the earth s rotation as simulated in fig .[ figure : fourier3 ] .2 .  with the existence of any meaningful diurnal variation , one can repeat the measurement in different seasons of the year and look for any meaningful phase change due to the orbital motion of the earth as illustrated in fig . [figure : shapechange2 ] using simulation in different seasons of the year .one can do this measurement simultaneously with the light propagations in opposite directions as indicated in the report .the plot of the outcomes following numbering 1 would be in opposite phase if the speed of the earth can be measured in terms of the speed of light .a normalization approach is derived and described in the report and used in the report .this approach can make the two opposite responses identical for precise comparison to look for any opposite phase in the responses .fizeau s gear - wheel method for measuring the speed of light was adopted and improved in the reports to measure the hypothetical variation of the one - way speed of light and reported a controversial result .this experiment was proposed to re - consider in the reports in response to the reports .detailed descriptions of the experimental apparatus , different challenges and an interpretation of the experiment have been presented elsewhere . recently in 2013 , report presents the outcome of fizeau s gear - wheel method for measuring the one - way speed of light that is consistent with sr .+  + the interpretation derived in suggests that the test using two detectors is insensitive to measure any velocity of the earth toward the direction parallel to the spin - axis of the earth .taking into account this limitation as well as other challenges to use two - detectors , we propose a new approach of the test of the one - way isotropy of the speed of light using one detector in a fizeau - type - coupled - slotted - discs experiment in this section .this will overcome some of the limitations of our present experiment .+  + a block diagram of our proposed future improvement to test the isotropy of the one - way speed of light using a fizeau - type - coupled - slotted - discs is shown in fig . [figure : block3 ] . in this improvement, we consider the response of the detector to only one direction of the light at a time .when the light travelling in a direction is received by the detector , the light in the opposite direction is blocked by the disc and vice versa ..,width=480 ]  the data collected in the course of a 24-hour period will not present any significant change if the speed of light follows sr . however , if sr is not valid then we will observe a significant regularity of variation in the collected data as simulated in fig .[ figure : diur4 ] following in fig .[ figure : shapechange2 ] .if special relativity is valid then there will not be any diurnal variation .the experiment on the turn - table would be the best setup as one can perform this test within a short period of time .the one - way velocity of light depends on the synchronization parameter .report indicates that experiments which test the isotropy in one - way or two - way have observables that depend on test functions but not on the synchronization procedure .therefore , the consideration of clock synchronization is irrelevant for a one - way speed of light test if one expresses the results in terms of physically measurable quantities .it is possible to measure a one - way speed of light .+  + the authors of the reports emphasized that the `` one - way speed of light has so far never been examined in experiments '' .contrary to this quotation , however , it is already noted in the papers that there are experimental investigations of the one - way speed of light .the controversy about the results on the limits of the isotropy of the one - way speed of light from nasa - experiments and the regularity in the variations of the reported results of the isotropy of the one - way speed of light in different time periods of the graal facility of the european synchrotron radiation facility ( esrf ) in grenoble remain unclear and needs further investigations by one - way experiments .+  + the previous one - way experiments noted above are not repeatable in simple forms in different laboratories as one can do with michelson - morley type two - way tests . in order to look for a one - way experiment to test the isotropy of the speed of light which is repeatable in a simple form in any laboratory and also , in order to ensure the results are clearly valid, we are proposing an improvement to our previous approach in using one detector .in addition we indicate the possibility of a one - way speed of light isotropy test in the work reported in the papers .based on above discussions , we would like to emphasize at this point that a general point of view of the fundamental postulate of sr , i.e. , the `` one - way '' constancy of the speed of light is not accepted unambiguously and is still controversial , and needs further investigations from one - way experiments .+  + we would like to emphasize that one - way tests should not be dismissed due to the perceived problem of clock synchronization .it has been shown how the consequences of a one - way speed of light can be used to perform isotropy test assuming the light follows a galilean transformation . a one - way speed of light measurement in terms of physically measurable quantities for a period of 24-hours with graphical representation can indicate the following phenomena if sr is invalid :  1 .  a diurnal variation 2 .spike at the frequency of cycle /hour in fourier spectral analysis .phase shifts for the measurement in different seasons .opposite phase in the responses of the two simultaneous measurements .a magnitude of the one - way velocity of light can be challenging due to the debates of the clock - synchronisation .however , we conclude that as long as one performs a one - way speed of light measurement for a period of 24-hours and present the outcomes in terms of physically measurable quantities , one - way isotropy test of sr can be performed .this work is supported in part by the national science and engineering research council , thoth technology inc . and york university , canada .  99 swenson l. s. , jr ._ the ethereal aether : a history of the michelson - morley - miller aether - 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Abstract: at first blush , what appears to be a purely physical question to measure any velocity : how to measure the velocity on a one - way trip ? however , due to the debates of the clock - synchronisation and the successes of special relativity ( sr ) , searching of the possibility of one - way speed of light measurement did not receive wider attention since the declaration of the constancy of the speed of light in vacuum by maxwell s electrodynamics in 1864 . however , our analysis suggests that the debates of the clock synchronization are insignificant when one uses a one - way experiment to test the isotropy of the speed of light - the fundamental postulate of sr . searching of the possibility of one - way speed of light to test sr is introduced by reviewing recent one - way tests .