the behavior of granular materials under deformation and flow is often studied by means of numerical methods that solve the equations of motion for every single body and account for the interactions between particles through appropriate models ( see , e.g. , refs . ) . these methods , commonly grouped under the name of discrete element methods ( dem )  , can be extremely useful for simulating discrete media with the purpose of establishing micro - macro relations , and in this perspective they need consistent averaging procedures  .an averaging method computes continuum - like variables ( e.g. , a velocity field ) starting from their discrete , particle - scale counterparts ( e.g. , particle velocities ) .when using such an approach , two issues have to be considered with utmost attention : the representativity of the average and the physical meaning of the obtained estimates .we can say that an average performed at a given point is representative if a sufficiently large domain in space and time can be defined , where particles share the same properties .such a volume , usually called representative volume element , " may be defined with respect to only some variables .an important step in the development of averaging techniques was made by babic  . in that workthe author developed a general framework for weighted space - time averages , applicable to a wide range of conditions , and giving a large set of self - consistent continuum balance equations .in babic s method , the definition of a representative volume is intrinsically that of a zone where affine velocity fields are locally uniform , i.e. , no gradients occur .the assumption that such a volume can be defined is called by babic the continuum assumption ." now , the lack of scale separation typical of granular flows ( the scale of spatial variation of variables has the same order of magnitude of the particle size ) implies that in a three - dimensional ( 3d ) flow we can not in practice define such a volume . as a consequence ,the physical meaning of averages obtained using babic s approach in a 3d flow is generally questionable .at least some average estimates will then probably depend on the size of the averaging volume  .we can see a simple example of this by considering a uniform shear flow ; if all particles follow the mean flow with no frustration , we will tend to say that fluctuations are negligible . but this depends on the definition of fluctuation .if we calculate the fluctuational kinetic energy [ which is related to granular temperature  ; see also eq .( [ eqn : epsilont ] ) ] by considering fluctuations as babic does , we find , where is the averaging scale .thus the fluctuations depend on the averaging scale which casts doubt on such a method .also the kinetic contribution to the stress tensor will suffer the same problem .thus , as we will show below , such an approach may lead to errors that can reach several hundred percentages .  in this work we focus on this problem , with a particular attention on the mechanical energy equation .we discuss both the issue of dependence on the size of the averaging domain and the energy cascade picture associated with the averaging method .it is clear that the dependence of some average variables on the averaging domain size is intrinsic to babic s definition of fluctuations . in the followingwe show how a different fluctuation decomposition taking into account gradients yields a new derivation of continuum balance equations .these balance equations have the merit of clearly separating affine and nonaffine fields and identifying the terms responsible for averaging scale dependence . with this decompositionit is possible to attribute a more sound physical meaning to the terms appearing in the final balance equations .we show how these balance equations may be greatly simplified by a local homogeneity assumption , " which is strongly related to the fluctuation decomposition itself and which yields less strict requirements than babic s continuum assumption ." applying this method to a simple class of flows , we find that two separate paths for fluctuating energy dissipation are implied : while kinetic stress power enters a nonlocal balance , the contact stress power seems to be dissipated locally . this result , as well as the robustness of the method ,is verified with discrete element simulations of a simple shear flow .given a particle property , babic defined the averaged property at point and time as a weighted space - time average : where is a normalized weighting function , the average density , and the mass of a grain .the fluctuation was simply defined by babic with respect to the center of the averaging volume :   this may seem harmless , but in fact as shown in ref . the dependence of some terms of the continuum averaged equations on the size of the averaging domain is strictly related to this choice for the fluctuation decomposition .we propose to define the fluctuation not with respect to the average value at point ( as was done in ref . ) , but with respect to the particle center , in the following we will redevelop continuum - averaged equations in the spirit of babic s approach , but using this new fluctuation decomposition .  in order to simplify the result , we will make a constitutive assumption . we suppose that a scale exists where the gradients of are smooth and that if this scale is the averaging scale intrinsic to the weighting function , we can approximate as a constant near the averaging point .we call this assumption the _ local homogeneity assumption_. together with this assumption we also assume a homogeneous distribution of particle centers near the averaging point and the decorrelation of positions and velocities . these three assumptions are strictly related . +while the absence of scale separation usually prevents identification of a scale where gradients are zero , such a first - order approximation is reasonable and as we will see very fruitful .the local homogeneity assumption , involving gradients instead of variables , is less strict than babic s continuum hypothesis and therefore more likely to be valid under a proper choice of the weighting function .based on this assumption , we can therefore approximate by   following babic s approach , it is possible to develop a generic balance equation for the particle property .details are given in appendix a ; here we show the main findings , with a particular regard to the continuity , momentum , and translational energy equations . when performing the derivation , many new terms appear ( with respect to babic s treatment ) , but most of them vanish due to the local homogeneity hypothesis .the local homogeneity hypothesis leaves the continuity equation unchanged :   where average density and velocity are defined by :      as for the momentum balance , we end up with the classical equation : where the total stress tensor is composed by three contributions : .the first two terms are the same as in babic .the first contribution is due to contact forces ( both long - lasting contacts and collisions ) and is given by where is a weight function related to the fraction of the branch vector joining the two particles and lying within the averaging function .due to the new decomposition , babic s kinetic stress tensor is decomposed in a truly kinetic part , and a new term , which contains all the dependence on averaging domain size through the vector .it is useful to recall that the emergence of a part of the stress tensor depending on velocity gradients has nothing to do with constitutive relations but is the joint product of coarse graining and scale dependence . due to their definition , both and are symmetric . evidently ,in absence of velocity gradients , will vanish ; in that case , the kinetic stress tensor as defined by babic will no longer depend on the coarse - graining length. the components of , related to the characteristic size of the domain along each direction , are defined by the following equation :    where is the _i_-th coordinate of the averaging point .if the local homogeneity hypothesis holds , can be shown to scale with , the size of the averaging domain in the _ i _ direction ( see appendix b ) .the scaling will depend also on the shape of the weighting function .moreover , if the assumption of local homogeneity holds , contains all the dependence on the coarse - graining width of the stress tensor while the kinetic stress tensor defined by means of the present fluctuation decomposition appears to be independent on averaging domain size .the translational kinetic energy balance is only slightly more complicated than the one obtained by babic .once more , most of the new terms are negligible under the local homogeneity assumption .the total translational kinetic energy is found to be the sum of three contributions , , which are respectively the translational kinetic energy of the mean flow , the translational kinetic energy related to the velocity gradients , and the truly fluctuational translational kinetic energy .these three variables are defined as  the translational kinetic energy balance equation is derived as : where and are babic s energy fluxes ( see appendix a ) .the last two terms correspond to the rate of conversion of kinetic energy to other forms of energy ( dissipation due to friction , collisions , plasticity , but also to storage and restitution of elastic energy ) . in the rigid limitthey represent dissipation through friction and inelastic collisions and are therefore positive .the first one contains the contribution of fluctuations : and the second one the rate of conversion of mechanical energy due to affine deformations : where is the mean of and .both terms are independent of the size of the averaging domain .we can see here again the benefit of our approach : on one hand the domain size - dependent terms are clearly identified in the equation ( and ) ; on the other hand , the contribution of affine deformations is separated from the contribution of the fluctuations . in fact , by superficially interpreting babic s approach , one could conclude that all the energy is dissipated by means of fluctuations .this is incorrect , since affine deformations do also dissipate energy ; the reason for this misunderstanding is that affine deformations are considered as fluctuations in babic s method .we can remove the mean flow kinetic energy terms by manipulating the momentum equation in a classical way ( for example , see ref .the result is an equation expressing the balance for the fluctuating energy related to fluctuations and to affine deformations : where stands for transposition .let us do some more manipulation . by looking at the term accounting for the rate of conversion of mechanical energy due to affine deformations , manipulation of eq .( [ gg ] ) , making use of , allows us to rewrite in the following form : where is a new contact stress tensor , differing from the by the weight used . indeed , as mentioned above, is the simple mean of and , whereas is related to the fraction of the branch vector lying within the averaging region .in this section we will apply our method to a simplified flow situation in order to gather some information on the effect of the new terms on force balance and on energy cascade .let us consider a one directional , steady , fully developed flow [ defined by , , .in such a case , the continuity equation is identically zero .the momentum equation becomes : and the fluctuating energy equation according to the definition of , it is easy to see that its only nonzero component is : and it is therefore straightforward to see that that is , the contribution of affine deformations to the stress tensor does not perform any work .moreover , it is obvious that directly affects only the component of the force balance . given that the flow is fully developed , we can safely remove it out from the balances , obtaining :  according to eq .( [ ggg ] ) : it is clear that and are not in principle the same , particularly when the averaging domain is smaller than a particle diameter .however , if the averaging domain spans more than one particle diameter , we can assume that .equation ( [ fluc1d2 ] ) becomes therefore : this means that the work provided by the contact part of the stress tensor is converted _ locally _ by affine deformations , while only the kinetic part of the stress tensor enters into a nonlocal energy balance .this result has important consequences , since it shows that separate paths may exist for the dissipation of the work of kinetic and contact stresses .however , it does not cast doubt on the validity of hydrodynamic theories  since our results do not deal with constitutive relations but suggest that a revision of the balances used for such theories could lead to a simplified description in some cases . concerning the size of the averaging domain, it may seem strange to talk about domains smaller than a particle diameter , because it may seem obvious that such domains are inappropriate : few particles are indeed contained in a snapshot of such a domain .however , we are always dealing with space - time averages , so even if the averaging domain is small , in many cases ( e.g. , slowly evolving or steady - state processes ) the domain indeed contains a lot of particles since many of them traveled through it during their history without affecting local homogeneity .in other cases , periodic boundary conditions may help too , allowing use of small domains along some directions .that is one of the reasons why in the next section we will employ the averaging method to some dem simulation results pertaining to this class of simplified flows .and is submitted to a constant vertical stress . ]  [ cols=">,^,>,^ " , ]   figure [ fig3](a ) displays the profiles of the components of the contact part of the stress tensor ( here mainly due to collisions , since the flow is quite rapid ) .again , a plateau is found at the center ; when approaching the walls all the components decrease in absolute value .  at the same time[ fig3](b ) ] , the components of the kinetic part of the stress tensor increase when approaching the walls .this is a well - known direct consequence of the decreasing of solid fraction . in our simulation ,near the walls , kinetic stresses are larger than collisional ones .the sum of the two parts gives the total stress tensor , shown in fig .[ fig3](c ) .as predicted by the stress balance , the ( ) and ( ) components of the total stress tensor are uniform along . concerning the third part of the stress tensor , ,it is clear that according to eq .( [ tgxx ] ) its only nonzero component is the ( ) one . for the reference simulationit was verified that , apart for a small difference in a thin zone ( ) close to the walls ( where slight deviations from local homogeneity appear due to the shape of the wall ) , , confirming the scaling given above .figure [ figu ] collects contact and kinetic flux profiles .all the components of these fluxes differ from zero only when approaching the wall ; as regards the component of the fluxes , fluxes are negative when the fluctuating energy decreases with and positive otherwise .this agrees with the sign convention used for in the derivation , and supports a fourier - like expression for the flux dependence on fluctuating energy gradient . on the other hand ,the horizontal components of the fluxes are not negligible near the walls .this contrasts with a linear transport theory since there is no gradient of fluctuating energy in the direction .figure [ fig4 ] displays the estimates of the stress power related to contact and kinetic contributions and of the dissipative terms related to affine and non affine motions .firstly , it can be seen that far from the walls , the algebraic relations hold therefore justifying the separate dissipation paths for kinetic and contact stress powers . then , near the walls , the energy fluxes seem to influence only the kinetic part , therefore supporting the result obtained in the previous section : only kinetic terms enter in a nonlocal energy balance , while contact stress power is dissipated locally .in the previous sections we developed an averaging procedure for granular materials which was based on a weighted average plus an original fluctuation decomposition . through numerical simulation results we proved the strength of the approach which gives consistent predictions for kinetic terms , and more importantly gives a new picture of the energy cascade in granular flows , at least for simple flows . in this sectionwe discuss the validity of such an approach .it is clear that the assumption of local homogeneity holds only when the solid fraction and the shear rate profiles do not display large gradients with respect to the averaging domain size . for the simulations presented above, two zones were identified near the walls where such gradients exist .the independence of the fluctuating energy profile on the size of the averaging window and the respect of the stress balance are two proofs that the assumption is still acceptable .figure [ fig5 ] allows us to discuss another important point : the physical meaning of the variables obtained by the method , with a particular regard to solid fraction . in the figure ,the solid fraction profile was computed by two other methods : estimating the fraction of ( 1 ) a horizontal line and ( 2 ) of a horizontal slice ( wide ) intersecting the particles .these two methods have a clear geometrical meaning , the second being a moving average of the first .  for the rapid flow simulation which was taken as the reference [ fig .[ fig5](a ) ] , where no layering occurs , the three methods agree very well .figure[fig5](b ) displays the results of the three methods for a denser flow ( , , ) .please note that the quite ordered bumpy wall was explicitly prepared to display layering .in this case the estimates on lines strongly fluctuate near the wall , and the average on slices is a smoother profile which still fluctuates , while the estimate produced by the present averaging method seems to be intermediate between the two . far from the wall results from the methodstherefore , in the presence of layering , the solid fraction calculated by use of the present method may not have the direct geometrical meaning that its name implies .this problem may be reduced by choosing a smoother weighting function ( in this paper we chose on purpose a very abrupt weighting function ) ; we also suspect that in three dimensions this rather pathological issue may be less important .another issue caused by layering is that the strong spatial fluctuation of variables may imply that some of the assumptions of the local homogeneity hypothesis do not hold : the distribution of particles around the averaging point is not in principle homogeneous .fluctuations are evidently associated with strong local gradients of solid fraction and shear rate .therefore , when strong layering occurs , care has to be taken in applying the present method due to the approximations therein .  the inhomogeneous distribution of particles inside the averaging domain yields also a deviation of the virtual center of mass from the center of the domain .this is not , in principle , a problem , because the averaging procedure is related to a volume and not to a point ; average variables may be harmlessly referred to the virtual center of mass instead of the simple geometric center of the averaging domain . in other wordsthe relation which seemed to be a consequence of the local homogeneity assumption can be taken as a definition of , the point where averages are supposed to be related .in addition , results from simulations at and ( which show quite strong layering ) , not shown here for the sake of brevity , show that stress balance is respected and confirm locality of the dissipation of the contact stress power . however , if layering occurs , local homogeneity may be difficult to assume , and therefore the simplified balances obtained in this work have always to be critically analyzed when dealing with regions very close to the walls .in this work , we propose a new derivation of continuum balance equations for granular materials which is inspired by the work of babic  . we introduce a new decomposition of particle scale velocities into mean and fluctuations , taking into account velocity gradients .a very reasonable local homogeneity hypothesis allows us to simplify the balances , ensuring at the same time representativity and physical meaning of the continuum estimates obtained .two important results are obtained .first , the method solves the problem of dependence of some variables ( e.g. , fluctuating energy , kinetic stresses ) on the averaging domain size which is intrinsic to previous methods  by clearly identifying scale - dependent terms .second , the development suggests a new physical picture for the energy cascade in granular flows : kinetic terms enter in a nonlocal energy balance , while contact terms ( containing both collisions and long - lasting contacts ) seem to be dissipated locally .both results are verified against discrete element simulations of granular flows , with respect to which the method is shown at work .this point is particularly important with regards to the hope of obtaining a statistical mechanics for granular flows whose predictions can be faithfully tested in physical experiments and numerical simulations .note also that the question of the size of the averaging window close to a boundary  is problematic since it is bounded by the distance between the center of a grain and that boundary .a last note has to be added on the domain of application of the present method .discrete element methods are sometimes divided into smooth and nonsmooth ones  .the present averaging method ( as babic s one ) applies rigorously only to smooth methods , since the idea of smooth ( differentiable ) particle fields was implicit in the expression of the equation of motion .the adaptation of such a method to non smooth dynamics is a very important point which will be the subject of future works .we are indebted to m. y. louge for fruitful discussions and to s. mcnamara for a critical reading of the manuscript .the numerical simulations were carried out at the ccipl ( centre de calcul intensif des pays de la loire ) under the project mteegd .the derivation follows closely that of babic , except for the choice of the fluctuation decomposition . the evolution equation for a generic particle property is : multiply by , sum on , and integrate on time : the left hand side term can be developed into  :   as regards the right hand side of eq .( [ first ] ) , babic also showed that the interaction term could be developed as : where and are two weighting functions respectively defined as : and now let us see the effect of the fluctuation decomposition on eq .( [ lhs ] ) . decomposing the velocity field with the new decomposition leads to : the divergence term on eq .( [ lhs ] ) can be written as : decomposing also yields for the second term in eq .( [ div ] ) : given that where is the average center of mass of the averaging domain , the integral in the third term of eq .( [ div ] ) becomes   following the homogeneity assumption , we neglect correlations of fluctuations and of particle positions , and we consider that the average center of mass coincides with the averaging point .this means that terms under the divergence operator in eq .( [ div ] ) simplify to developing the component of the third term in eq .( [ div2 ] ) [ using the definition yields : assumption of local homogeneity also implies decorrelation of the components of particle position with respect to the averaging point .this means that : where is related to the scale of the averaging volume in the direction . if is the vector with components , the third term of eq .( [ div2 ] ) is this term renders explicit the scale dependence which was otherwise implicit in babic s method .equation ( [ lhs ] ) is finally reduced to : .\ ] ] as regards to the effect of the fluctuation decomposition on the right hand side of eq .( [ first ] ) , the result depends on the particular property considered .thus in the following we will analyze each property separately .  if mass conservation is considered ( , , ) , the right - hand - side term of eq .( [ first ] ) is zero and it is straightforward to see that the classical continuity equation is obtained [ eq .( [ conti ] ] .if force balances are considered ( , , ) , eq .( [ lhs ] ) becomes .\ ] ]  the last two terms can be considered as related to two components of an effective stress tensor as defined by eqs .( [ tgamma ] ) and ( [ tk ] ) . on the other hand , the right - hand side of eq .( [ first ] ) is not affected by the nature of the fluctuation decomposition .substitution of the and yields to the standard momentum equation , eq .( [ momentum ] ) , with the third component of the effective stress tensor given by eq .( [ tc ] ) .let us derive now the translational kinetic energy equation . in this casethe reference particle property is the average translational kinetic energy is and the fluctuation decomposition allows us to split this quantity into six terms , three of which disappear due to the local homogeneity assumption , yielding : where , with respect to babic s development , is likely to be scale independent , and a new term appears , which is the translational kinetic energy related to affine deformation within the average volume .  the divergence term in eq .( [ lhs ] ) corresponds , for the kinetic energy , to : when developing this term , the fluctuation decomposition gives 27 terms , most of which can be deleted as in the previous development through ( 1 ) the postulate , ( 2 ) the assumption of decorrelation among components of , and ( 3 ) the assumption of decorrelation between components of and velocity fluctuations .it can be shown that the irreducible terms are where   let us treat the terms coming from the right hand side term of eq .( [ first ] ) .first , we must identify the particle scale interaction terms . taking the product of and newton s equation , , the particle interaction term is and the particle source term is concerning the source term , local homogeneity implies :   developing the interaction terms yields : using the fluctuation decomposition , the relation , and the assumption of local homogeneity this becomes : ,\end{aligned}\ ] ] which correspond respectively to : where the first two terms are the conversion rates of translational kinetic energy into other forms of energy respectively due to velocity fluctuations ( ) and to affine deformations ( ) ; can be seen as a translational kinetic energy flux related to contact forces .  the translational kinetic energy balance equation is therefore given by eq .( [ fluct ] ) , which can be further simplified to eq .( [ fluct2 ] ) when subtracting the mean flow energy balance .in the following , a simplified scaling for the vector which enters in the definition of is derived using the local homogeneity hypothesis .let us assume equal - sized spheres , and a step function on space and time as a weighting function .let us consider variables computed at a point in space and time .the average density is therefore : where for particles residing in the averaging volume during the period and zero otherwise .if we call the time average of the number of particles lying in the averaging volume , it is clear that . developing the definition of the vector , we find : where is the component of the particle position .the local homogeneity hypothesis says that and that is uniformly distributed around .this yields for large to : therefore if the local homogeneity holds , is likely to scale on the size of the averaging domain in the direction .the proportionality will then depend on the choice of the weighting function , on polydispersity , on shape , and so on .

Abstract: a new averaging method linking discrete to continuum variables of granular materials is developed and used to derive average balance equations . its novelty lies in the choice of the decomposition between mean values and fluctuations of properties which takes into account the effect of gradients . thanks to a local homogeneity hypothesis , whose validity is discussed , simplified balance equations are obtained . this original approach solves the problem of dependence of some variables on the size of the averaging domain obtained in previous approaches which can lead to huge relative errors ( several hundred percentages ) . it also clearly separates affine and nonaffine fields in the balance equations . the resulting energy cascade picture is discussed , with a particular focus on unidirectional steady and fully developed flows for which it appears that the contact terms are dissipated locally unlike the kinetic terms which contribute to a nonlocal balance . application of the method is demonstrated in the determination of the macroscopic properties such as volume fraction , velocity , stress , and energy of a simple shear flow , where the discrete results are generated by means of discrete particle simulation .


mimo , space - time codes , maximum likelihood decoding , computational complexity .multiple - input multiple - output ( mimo ) is a promising technique that can bring significant improvements to the wireless communication systems . in combination with space - time block code ( stbc ), it provides higher spectrum efficiency with better communication reliability  . in the last decades, mimo has been widely employed in the latest wireless communication standards such as ieee 802.11n , 3gpp long term evolution ( lte ) , wimax and digital video broadcasting next generation handheld ( dvb - ngh ) etc .it is also seen as the key technology for the future digital tv terrestrial broadcasting standards  .a so - called space - time - space ( 3d ) mimo code  was proposed for the future tv broadcasting systems in which the services are delivered by the mimo transmission in a single frequency network ( sfn ) .specifically , it is proposed for a distributed mimo broadcasting scenario where tv programs are transmitted by two geographically separated transmission sites , each site equipping two transmit antennas . on the other hand ,each receiver has two receive antennas , forming a mimo transmission .the 3d mimo code has been shown to be robust and efficient in the distributed mimo broadcasting scenarios where there exists strong received signal power imbalances  .hence , it is a promising candidate for the mimo profile of future broadcasting standards .however , the 3d mimo code suffers from a high computational complexity when the maximum - likelihood ( ml ) decoding is adopted .the decoding complexity is as high as when -qam constellation is used .up to now , no study on the decoding complexity reduction for the 3d mimo code has been carried out in the literature .  recently a lot of efforts have been made in the stbc design to obtain both high code rate and low decoding complexity  .the decoding complexity reduction is commonly achieved by exploiting the orthogonality embedded in the stbc codeword .when there exists group - wise orthogonality in the codeword , the joint detection of many information symbols is converted into independent , group - wise detections  , yielding low decoding complexity .for other cases such as djabba code  , biglieri - hong - viterbo ( bhv ) code  , srinath - rajan code  and ismail - fiorina - sari ( ifs ) code  in which the orthogonality only exists in a part of information symbols , some symbols can be detected in a group - wise manner once we condition them with respect to other symbols .the ml solutions can be obtained with a lower complexity compared with the ml detector . in other words ,their decoding complexity is less than where is the number of information symbols in a codeword .such kind of stbcs are referred to as fast decodable stbcs  .however , most of fast - decodable stbcs are not optimized for distributed mimo broadcasting scenarios and they are not robust under the received signal power imbalance conditions  .  a partial interference cancellation( pic ) group decoding scheme has been presented aiming at reducing the decoding complexity of the stbcs containing group - wise orthogonalities in the codewords  .a number of stbcs that are optimized for this decoding scheme have also been proposed  .this scheme actually uses a linear equalization to convert the joint detection of a large number of symbols to several groups of ml decodings for few symbols .however , the overall performance of this decoding scheme can not achieve the ml optimality .some alternatives with reduced decoding complexity have been presented for the distributed mimo broadcasting .described a stbc with less decoding complexity based on orthogonal basis  .however , such a code does not achieve full - diversity or full - rate for mimo transmissions and therefore performs worse than 3d mimo code .a `` punctured version '' of 3d mimo code that is full - rate for mimo transmissions with low decoding complexity has also been proposed  .however , it does not achieve full - diversity and is hence less robust in harsh channel conditions .  in this paper ,we propose a reduced - complexity ml decoder for the 3d mimo code which exploits the embedded orthogonality in the codeword .the main contributions are :  * we propose to modify the original 3d mimo codeword through some permutations of information symbols which leads to an ml decoding algorithm with reduced complexity without affecting all desirable properties of the 3d mimo code .* we prove that the 3d mimo code is fast decodable .moreover , we show that the worst case decoding complexity is for -ary square qam modulations which is the least among all square full - rate stbcs for mimo transmission .* based on the unique properties of the new form of 3d mimo codeword , we propose a novel implementation of the simplified decoder that achieves a lower average complexity in terms of time latency without losing the ml optimality .the proposed implementation is also applicable for other fast decodable stbcs .the remainder of the paper is organized as follows .some fundamentals of the mimo detection are presented in section [ sec : sys_mdl ] . in section [ sec:3dmimo ] ,the 3d mimo code is first recalled .consequently , a modification of the codeword is proposed to facilitate the decoding process .three important properties of the new codeword are also revealed . with this knowledge , in section [ sec : mldec ] ,the ml decoder with a worst case decoding complexity of is derived .then in section [ sec : impmldec ] , a new implementation of the reduced - complexity ml decoder is described .section [ sec : simu ] presents the symbol error and complexity performance of the new decoder .conclusions are drawn in section [ sec : conclusion ] .  _notations : _ vectors and matrices are written in boldface letters .superscript represents transposition of matrix . and denote the real and imaginary parts of a complex number , respectively .the operator performs the complex - real conversion from to : .\ ] ] when operator is applied to a matrix , the operation in ( [ eq : check_function ] ) is performed for all elements s in the matrix , i.e. the  submatrix of is . for a complex vector ^t\in\mathbb{c}^n ] . for a matrix ] . denotes vectorizing matrix followed by the real / imaginary part separation .the inner product of two real - valued vectors and is denoted by .the identity matrix is denoted by .the operator denotes the kronecker product .finally , represents .we consider a mimo transmission with transmit and receive antennas over flat - fading channel .the received signal is : where is the stbc codeword matrix which is transmitted over channel uses ; is a complex - valued additive white gaussian noise ( awgn ) component ; is the channel matrix whose element denotes the channel coefficient of the link between the transmit antenna and the receive antenna .the channel is assumed to be quasi - static .that is , the channel coefficients keep constant over the duration of one stbc codeword , but change from one codeword to another .moreover , s are assumed to be independent from each other .  for linear stbcs, the codeword matrix can be obtained through a linear operation  : where ^t ] that minimizes the euclidian distance between the channel distorted information signal and received signal , namely : where is the set of the constellation symbols .( [ eq : ml_detection ] ) indicates that the ml solution is found by _jointly _ determining independent information symbols .in other words , when the modulation of these symbols is -qam , the ml decoding should exhaustively check all combinations .the search complexity grows dramatically with higher modulation order or larger number of information symbols in one codeword .hence , the ml decoding is computationally demanding .more efficient stbc decoding is achieved with the help of orthogonal - triangular ( qr ) decomposition  .the qr decomposition of the equivalent channel matrix yields , where is a unitary matrix , and is an upper triangular matrix .the definitions of the `` classical gram - schmidt algorithm '' based qr decomposition can be found in the appendices .note that other numerically stable qr decomposition algorithms can also be used without affecting the properties of the 3d mimo code as well as the resulting low - complexity decoding methods . instead of solving ( [ eq : ml_detection ] ), the ml solution can be alternatively found by : where is a linear transformation of received signal ; is a hypersphere centered on the received signal .only the codewords inside the hypersphere are checked during the search in order to reduce the search complexity .the size of the hypersphere is represented by its radius .the decoding process is turned into a bounded search over a -level tree with complex - valued nodes .hence , the worst case decoding complexity is .moreover , according to the property of qr decomposition , some information symbols can be decoded independently from the others if some elements of are equal to zero .it suggests that the joint search in a high dimension is converted into a bunch of parallel , independent searches in low dimensions .this results in a significant reduction of the worst - case decoding complexity  .the 3d mimo code  possesses a better robustness against receive signal power imbalances in the distributed mimo broadcasting scenarios but suffers from high decoding complexity  . in this section , we propose a new 3d mimo codeword that enables low sphere decoding complexity via exchanging the positions of information symbols in the original 3d mimo codeword .the basic idea behind this modification comes from the facts that the orthogonality embedded in the information symbols essentially enables independent detections and the sphere decoding complexity is mainly determined by the orthogonality among the first several symbols .hence , exploiting the underlying orthogonality in the codeword and carefully choosing the sequence of information symbols can bring benefits in terms of decoding complexity .the initially proposed codeword matrix of the 3d mimo code is explicitly written as : = \!\frac{1}{\sqrt{5}}\!\left [ \begin{array}{*{20}c }     \alpha ( s_1+\theta s_2 ) & \alpha ( s_3+\theta s_4 ) & -\alpha^{\ast } ( s_5^{\ast}+\theta s_6^{\ast } ) & -\alpha^{\ast } ( s_7^{\ast}+\theta s_8^{\ast})\\     i\bar{\alpha } ( s_3+\bar{\theta } s_4 ) & \bar{\alpha } ( s_1+\bar{\theta } s_2 )  & i\bar{\alpha}^{\ast } ( s_7^{\ast}+\bar{\theta } s_8^{\ast } ) & -\bar{\alpha}^{\ast } ( s_5^{\ast}+\bar{\theta } s_6^{\ast } ) \\     \alpha ( s_5+\theta s_6 ) & \alpha ( s_7+\theta s_8 ) & \alpha^{\ast } ( s_1^{\ast}+\theta s_2^{\ast } ) & \alpha^{\ast } ( s_3^{\ast}+\theta s_4^{\ast})\\     i\bar{\alpha } ( s_7+\bar{\theta } s_8 ) & \bar{\alpha } ( s_5+\bar{\theta } s_6 )  & -i\bar{\alpha}^{\ast } ( s_3^{\ast}+\bar{\theta } s_4^{\ast } ) & \bar{\alpha}^{\ast } ( s_1^{\ast}+\bar{\theta } s_2^{\ast } ) \\ \end{array }     \right],\ ] ] where , , and . it is constructed in a hierarchical manner : eight information symbols ( ) are first encoded to two golden codewords  , i.e. and , which are consequently arranged in an alamouti manner  over four channel uses ( ) .this results in a code rate of which is _ full - rate _ for the mimo transmission .previous study shows that the 3d mimo code achieves efficient and robust performance .however , since eight information symbols are stacked in one codeword , the ml decoding complexity is up to .it was shown that it is possible to achieve lower sphere decoding complexity through permuting the sequence of information symbols  .we propose to slightly modify the codeword by exchanging the positions of information symbols and , yielding a new form of codeword : .\ ] ] since we only change the sequence of the information symbols in the codeword ( the third and fourth information symbols become the fifth and sixth , and vice versa ) and the information symbols are independent from each other , the new codeword preserves all the good attributes of the original 3d mimo code in distributed mimo scenarios illustrated in .more importantly , this modification is based on the embedded orthogonalities in the 3d mimo codeword and  an interesting codeword structure which will be exploited to achieve lower decoding complexity .the advantages brought by the new codeword structure will be highlighted in the following sections .due to the underlying alamouti and golden structures , the 3d mimo code has some unique properties which lead to simplified decoding algorithms .  for the modified 3d mimo code ( [ eq:3dnew ] ) over a mimo channel , the matrix in ( [ eq : sd_detection_metric ] ) is a real - valued matrix .rewrite in a block - wise form : ,\ ] ] where s are submatrices containing s with and .  based on the new codeword in ( [ eq:3dnew ] ) and taking into account ( [ eq : heq ] ) , ( [ eq : st_coding_by_generator_matrix ] ) and ( [ eq : generator_matrix ] ) , we can obtain a few interesting properties of that can be made use of to achieve a low decoding complexity .[ theo : r11 ] is an upper triangular matrix with .[ theo : r12 ] is a null matrix when the channel is quasi - static , i.e. , and .[ cor : r22 ] is an upper triangular matrix with a similar structure as , i.e. .   matrix of the new 3d mimo codeword ( [ eq:3dnew ] ) in quasi - static channel.,width=259 ]  the proofs of theorem [ theo : r11 ] , theorem [ theo : r12 ] and corollary [ cor : r22 ] are presented in appendices .the above properties are visualized in figure [ fig_r_mat_swt ] .* remark 1 * : theorem [ theo : r11 ] and corollary [ cor : r22 ] actually suggest the independency between real and imaginary parts of the information symbols .for instance , means that the real parts of the first and second received symbols , namely and , do not contain any contribution from and .similarly , means that their imaginary parts , namely and , do not contain any contribution from and , either . as we will show later ,this real / imaginary independency leads to independent and parallel detections for real part and imaginary part , respectively .the real / imaginary part independency comes from the underlying golden and alamouti structures .it has been revealed that the complex - valued matrix of the golden code has a real upper left submatrix  , which coincides with the structure as presented in theorem [ theo : r11 ] .it shows the real / imaginary part independency of the golden code in its codeword matrix .the alamouti - like arrangement of the two golden codewords , on the other hand , helps creating this independency in the codeword matrix of the 3d mimo code .* remark 2 * : theorem [ theo : r12 ] indicates that the information symbols and are uncorrelated with and in the received symbols .it means that these two symbol groups can be determined independently .we only have to jointly determine six information symbols to obtain the ml solutions .in other words , the ml decoding complexity is expected to be instead of .therefore , _the 3d mimo code is fast decodable_. details on the simplified ml decoding will be discussed in section [ sec : mldec ] .it should be noted that theorem [ theo : r12 ] is partially enabled by the embedded alamouti structure in the codeword .the channel coefficients should be constant within the duration of one codeword to validate the orthogonalities in alamouti structure .hence theorem [ theo : r12 ] is only valid in the quasi - static channels .matrix of the original 3d mimo codeword given in ( [ eq:3d ] ) in quasi - static channel.,width=259 ]  figure [ fig_r_mat_swt ] illustrates the matrix of the new 3d mimo codeword .compared with the original one as shown in figure [ fig_r_mat ] , the new structure is actually more favorable for the mimo decoding . in the new codeword , the contributions of information symbol groups and are totally uncorrelated in the received signal , which means that the ml detection of eight information symbols can be achieved by two independent and less complex detections of six information symbols .moreover , the structure of and enables the independent detections of real and imaginary parts of and , which leads to further complexity reduction . yet , this real / imaginary parts separation is not straightforward in the original codeword .it should be emphasized that the new codeword only changes the sequence of the information symbols in the codeword to facilitate the decoding process .it does not affect all the good properties of the 3d mimo code .in this section , a low - complexity ml decoding algorithm exploiting the unique properties highlighted in previous section is proposed for the 3d mimo code . generally speaking ,the complexity reduction is achieved in two steps .based on theorem [ theo : r12 ] the joint detection of eight information symbols is converted into two partially independent detections of six information symbols .this step reduces the worst - case decoding complexity from to .consequently , using theorem [ theo : r11 ] and corollary [ cor : r22 ] , the detections of complex information symbols are converted into independent detections of real and imaginary parts , which further reduces the worst - case complexity to .we divide the information symbols and received symbols into four groups , i.e. ^t} ] , ^t} ] , ^t} ] , ^t} ] . taking into account the structure of and theorem [ theo : r12 ] , the decoding metric in ( [ eq : sd_detection_metric ] ) can be rewritten as : from ( [ eq : ml_metric_original1 ] ) and ( [ eq : ml_metric_original2 ] ) , it can be seen that the contributions from the information symbol groups and are uncorrelated in the received symbol .for instance , does not contain any information from , and is irrelevant to , either .this enables us to use group - wise conditional detections to retrieve the ml solutions  .  in particular , the ml solution ^t ] : \!&=\!\arg\!\!\min_{[\mathbf{c},\mathbf{d}]\in\boldsymbol\theta^4}\!\! \big ( \|\mathbf{v}_{12}-\mathbf{r}_{11}{\mathbf{a}}^{(\ast)}(\mathbf{c } , \mathbf{d } ) \|^2 + \|\mathbf{v}_{34 } - \mathbf{r}_{22}{\mathbf{b}}^{(\ast)}(\mathbf{c } , \mathbf{d } ) \|^2 + \|\mathbf{z}_{56}-\mathbf{r}_{33}\mathbf{c}-\mathbf{r}_{34}\mathbf{d } \|^2+\|\mathbf{z}_{78}-\mathbf{r}_{44}\mathbf{d } \|^2 \big),\end{aligned}\ ] ] and two independent `` inner '' searches for and , respectively : where with , .the outer search is carried out over the combinations of  ] , the search of and the search of are performed in parallel .the concatenation of outer and inner searches ( either or ) results in a joint search of six information symbols .therefore , the worst - case decoding complexity is reduced from to .we note that this complexity reduction does not rely on the constellation that is adopted by the information symbols . in other words ,the 3d mimo code requires a worst decoding complexity of for arbitrary modulation .if square shape qam modulations are considered , the decoding complexity can be further improved .the square -qam symbol can be separated into two independent -pam symbols on the real and imaginary axes , respectively .using theorem [ theo : r11 ] and corollary [ cor : r22 ] , the real and imaginary parts can be decoded separately .take the detection of as an example .denote its real and imaginary parts as ^t ] , respectively. given  ] , ^t ] given  ] given  ] , the search of two complex symbols  ] is realized by a complex sphere decoder with schnorr - euchner enumeration , which is visualized by the search over a 4-level tree as shown in figure [ fig_2step_dec ] .the nodes of the same level represent all the solutions of a complex information symbol .each path from the root to a leaf node represents a possible combination of  ] and  ] is found .consequently , the tree search process is suspended and the new  ] ,  ] and ] ,  ] and  ] and the resulting distance are returned to the tree search phase .the tree search process is resumed .the overall distance is compared with the current radius ( line [ algo_radius_cmp ] in algorithm [ algo_simp_tree ] ) to determine whether the current solution is a better one .if a better solution is found , the radius is updated accordingly ( line [ algo_radius_new ] in algorithm [ algo_simp_tree ] ) .the tree search process is moved on to the next unchecked node .   in the proposed algorithm, the search of eight symbols is divided into a tree search for four symbols and parallel detections for the other four symbols . due to the symmetric structure of the codeword matrix ( [ eq:3dnew ] ), some parts of the codewords can be exchanged without changing the properties of the 3d mimo code .for instance , we have the same properties as illustrated in section [ sec:3dmimo ] after exchanging the positions of  ] .similarly , if we exchange  ] and exchange  ] simultaneously , the structure of matrix maintains , as well . that is to say , besides the original symbol sequence ,the proposed low - complexity decoding algorithm is also valid with other three permuted symbol sequences , i.e.  ] and  ] = ,  ] [algo_radius_new ] ;   ,    ]   compute s ( ) [algo_cmp_2half ]  ] ( )  ] permute and according to permutation results , .if we write  ] , where columns s are orthogonal , and ,\ ] ] where , , , .based on , after some straightforward computation , it yields . according to the definition of qr decomposition , .hence , .in addition , , . taking into account that , it yields .  moreover , and . therefore , .this completes the proof of theorem [ theo : r11 ] .based on , after some straightforward computation , it yields , and .  using and which have been proven in the proof of theorem [ theo : r11 ] , it yields , and .  using which has been proven in the proof of theorem [ theo : r11 ], it yields , .  similarly , since , it yields , .this completes the proof of theorem [ theo : r12 ] .   using the similar method as in the proof of theorem [ theo : r11 ], it can be computed from the definition of that .in addition , using theorem [ theo : r12 ] , it can be obtained that .hence , .  using and theorem [ theo : r12 ] , it yields .hence , .  finally , using and theorem [ theo : r12 ] , it yields and .therefore , .this completes the proof of corollary [ cor : r22 ] .this work is supported by french anr `` mobile multi - media ( m3 ) '' project and `` ple images & rseaux '' .illustration of the matrix of the new 3d mimo codeword given in ( [ eq:3dnew ] ) in quasi - static channel .illustration of the matrix of the original 3d mimo codeword given in ( [ eq:3d ] ) in quasi - static channel .illustration of the two - stage sphere decoding .parallel decisions of  ] ,  ] .symbol error rate obtained with guo - nilson s sphere decoder with s - e for 3d mimo code , ml decoder and proposed simplified ml decoders in quasi - static rayleigh channel with qpsk and 16qam .computational complexity in terms of visited nodes required by guo - nilson s sphere decoder with s - e for 3d mimo code and proposed simplified ml decoders , in quasi - static rayleigh channel with qpsk modulation .computational complexity in terms of visited nodes required by guo - nilson s sphere decoder with s - e for 3d mimo code and proposed simplified ml decoders , in quasi - static rayleigh channel with 16-qam modulation .computational complexity in terms of multiplications required by guo - nilson s sphere decoder with s - e for 3d mimo code and proposed simplified ml decoders , in quasi - static rayleigh channel with qpsk modulation .computational complexity in terms of multiplications required by guo - nilson s sphere decoder with s - e for 3d mimo code and proposed simplified ml decoders , in quasi - static rayleigh channel with 16-qam modulation .computational complexity in terms of divisions required by guo - nilson s sphere decoder with s - e for 3d mimo code and proposed simplified ml decoders , in quasi - static rayleigh channel with qpsk modulation .computational complexity in terms of divisions required by guo - nilson s sphere decoder with s - e for 3d mimo code and proposed simplified ml decoders , in quasi - static rayleigh channel with 16-qam modulation .the table presents the ml decoding complexities of several stbcs that are suitable for mimo transmission .

Abstract: the 3d mimo code is a robust and efficient space - time block code ( stbc ) for the distributed mimo broadcasting but suffers from high maximum - likelihood ( ml ) decoding complexity . in this paper , we first analyze some properties of the 3d mimo code to show that the 3d mimo code is fast - decodable . it is proved that the ml decoding performance can be achieved with a complexity of instead of in quasi static channel with -ary square qam modulations . consequently , we propose a simplified ml decoder exploiting the unique properties of 3d mimo code . simulation results show that the proposed simplified ml decoder can achieve much lower processing time latency compared to the classical sphere decoder with schnorr - euchner enumeration .   [ 1995/12/01 ]


software testing aims at understanding the risks of software implementation and providing proper quality of the system under test .one of fundamental problems is that testing under all possible combinations of inputs is not feasible .this holds for functional testing , while non - functional aspects ( performance , compatibility , etc . )are left apart . for business critical applicationsblack - box testing is a widely - used approach .it examines the functionality of a system under test without dealing with its implementation .it is important to notice that static testing involves verification , while dynamic testing involves validation .one of the biggest challenges in black - box dynamic testing is choosing the proper approach to enumerate all possible cases .domain testing helps quality assurance engineer to define classes of input values , that are crucial to test .the naive way is just to check for possible combinations of such parameters , and it immediately leads to exponential complexity . even in the case of 6 boolean parameters an expert has to check 64 combinations .a possible solution is to generate test - cases in manual way .a software test engineer considers all possible cases ordered in a certain way .the main risk is to lose some information behind the cases . actually , it could be rather exhausting to cover all possibilities for a large number of parameters and not to skip some scenarios .a popular alternative way is pairwise testing  , or its generalization , n - wise testing .we define parameters and domains , and pass them like a model into a black - box algorithm  , which gives us a set of test - cases satisfying a certain condition ( for each pair of input parameters all possible discrete combinations of those parameters are tested ) . in general case it can produce different cases in different runs , while it can be fixed by passing a random seed to it .the main advantage of the approach is its insensitivity to parameter values .however , it can be a rather computationally demanding task .  for quality assurance engineersit is sufficient to have knowledge about possible behavior of the program .usually , there are dependencies between input parameters .a natural form to express such dependencies in mathematical terms is implication , a statement of the form : ` if ... , then ... ' .the ` if'-part is called premise , and the ` then ' is called conclusion .consideration of parameter s interdependence can decrease the complexity of result test - cases in terms of their quantity , by excluding some of possible parameter combinations .proposed approach to test - case generation is focused on implications .we use an approach based on formal concept analysis ( fca) .fca provides software engineers with a tool for exploring the domain of interest in semi - automatic way .the algorithm outputs sound and complete description of the problem , but it is still highly dependent on the accuracy of expert s answers. surveys on fca techniques and applications can be found in  .fca - based approaches have already been applied in software engineering , e.g. , for inference of class hierarchies  , class design  , refactoring  .another application of fca technique is analysis of object - oriented approach for a given system  .the questions of mapping lines of source code to the functionality from requirements is of crucial importance for big systems  .the rest of the paper is organized as follows : in section 2 we introduce basic notions of formal concept analysis . in section 3we focus on the procedure of attribute exploration .section 4 provides example of attribute exploration in the field of positive integers .we make conclusions in section 5 .formal concept analysis  ( fca ) provides mathematical technique for deriving applied ontologies from data .fca relies on lattice and order theories  numerous applications are found in the field of machine learning , data mining , text mining and biology , see  .a formal context is a triple , where g denotes a set of objects , m is a set of attributes , and is a binary relation between and .  it can be interpreted in the following way : for objects in there exists a description in terms of attributes in , and relation reflects that an object has an attribute : object possesses  .an example of a formal context is provided below :    ( 10,10 ) ( 0,0)(1,0)10 ( 0,0)(1,2)5 ( 10,0)(-1,2)5  & & & & +  ( 10,10 ) ( 0,0)(1,0)10 ( 0,0)(0,1)10 ( 0,10)(1,-1)10  & & & & +  ( 20,10 ) ( 0,0)(1,0)20 ( 0,0)(0,1)10 ( 0,10)(1,0)20 ( 20,0)(0,1)10  & & & & +  ( 10,10 ) ( 0,0)(1,0)10 ( 0,0)(0,1)10 ( 0,10)(1,0)10 ( 10,0)(0,1)10  & & & & +  objects : attributes :  p4cmp4 cm  * equilateral triangle * right triange , * rectangle , * square ,  &  * 3 vertices , * 4 vertices , * has a right angle , * all sides are equal  for a given context two following mappings are considered :  * *   for all ,   1 . 2 . 3 .  mappings and , satisfying properties 1 - 3 above , define a galois connection between and , which means :   traditionally , notation is used instead of and . stands both for and ( depending on the argument ) . for arbitrary ,   _ ( formal ) concept _ is a pair : , , ,   in the example with geometric figures a pair is a formal concept . for a formal context , - set of objects , - set of attributes , the following statements hold for operation ( ) :  1 . 2 . 3 . 4 . and 5 . 6 .  _ closure operator _ on set is a mapping , which maps every to _ closure _ , under the following conditions :  1 . ( _ idempotence _ ) 2 . ( _ extensivity _ ) 3 . ( _ monotonicity _ )  _ implication _ , where , takes place if , in other words if each object having also has all attributes from .implications comply with armstrong axioms :  of advantages of building lattices of formal concepts is to get effective navigation from more general concepts to more specific .for example , the line diagram for context with figures from previous section is shown in fig . 1 .  [ figures_lat ]   that property could be beneficial in two ways : regression testing , system description .first of all , for the context of regression testing it implements the algorithm to determine the classes of tests that fail .it could be considered in the following way .let us define set of attributes for all tests that are being run .[ cols="^,^,^,^,^",options="header " , ]   implication list :  1 .  * if * iqv * then * iqp 2 .  * if * isv * then * isp 3 .  * if * iiv * then * iip 4 .  *if * = * then * ibs , not < , not > 5 .  *if * < * then * ibs , not = , not > 6 .  *if * ibs * then * not > 7 .  *if * not iqp * then * not iqv 8 .* if * not isp * then * not isv 9 .  *if * not iip * then * not iiv 10 . *if * not iiv , not isv , not iqv * then * not = , not < , not > , not ibs 11 . *if * not < , not = * then * not ibs 12 . * if * not > , iqv , iqp * then * ibs 13 . * if * not > , isv , isp * then * ibs 14 . *if * not > , iiv , iip * then * ibs 15 . * if * not > , not isv , not iqv , ibs * then * iiv 16 . *if * not > , not isv , not iiv , ibs * then * iqv 17 .* if * not > , not iiv , not iqv , ibs * then * isv 18 . *if * not > , not = , ibs * then * < 19 . * if * not > , not < , ibs * then * = 20 .* if * not ibs * then * not = , not < 21 . * if * not ibs , not < , not = , iqv , iqp * then * > 22 . * if * not ibs , not < , not = , isv , isp * then * > 23 . *if * not ibs , not < , not = , iiv , iip**then * * > 24 . *if * not ibs , not < , not = , > , not iqv , not isv**then * * iiv , iip 25 . *if * not ibs , not < , not = , > , not iiv , not iqv**then * * isv , isp 26 . *if * not ibs , not < , not = , > not iiv , not isv**then * * iqv , iqp 27 . *if * not ibs , not > , not = , not < * * then * * not iqv , not isv , not iivthe most important advantage of the proposed approach is the plug - in design .it can be easily incorporated in existing process of test - case generation . for manual test - case designone can develop test - cases in proposed system .moreover , the step of implication extraction could be postponed up to the review of obtained cases .it is important to notice that attribute exploration is a good technique to extract dependencies that could be formulated as requirements for the system under test . for the case of automatic test case generation ,extraction of dependencies could eliminate the step of test debug and replace it with review of obtained implications and the model of the system under test .also the proposed technique is applicable to verify the correctness of defining type of model , precisely wiseness of it . since big values of in n - wise modeling impose more test cases and it results in the growing time of test run execution .formal concept analysis provides us with useful framework for software testing tasks .it is especially beneficial in regression testing meta report construction , feature navigation , test case analysis and derivation .it unites best practices of manual development and automatic generation of test scenarios .it provides sound and complete description of the investigated domain , based on expert knowledge .the output of the system consists of two main parts : the description of typical objects in the domain , and interdependence between parameters in terms of implications .an important advantage of proposed technique is extensibility .if we add a new attribute , we can just copy the previous examples into new formal context , assuming that new attribute is absent for all objects and proceed with the procedure of attribute exploration .it holds even for the very start of procedure .we can start with non - empty set of objects and implications simultaneously .the described algorithm could be used as a standalone solution for the test case design , as well as , tool to get exiting dependencies in the domain .the obtained implications could be valuable in pairwise testing to adjust the model description .however , we should admit that the current approach is limited in terms of attribute description .for now , it is highly dependent on the boolean nature of attributes .one of the main directions of future work is to work with descriptions of general form by means of patterns structures  , an extension of fca .ganter , b. and kuznetsov , s.o . ,pattern structures and their projections . in : proc .g. stumme and h. delugach , eds . , 9th international conference on conceptual structures ( iccs 2001 ) , lecture notes in artificial intelligence ( springer ) , vol . 2120 , pp .129 - 142 , 2001 .kuznetsov , s.o . ,fitting pattern structures for knowledge discovery in big data . in : cellier , peggy ; distel , felix ; ganter , bernhard , eds . , proc . 11th international conference on formal concept analysis ( icfca 2013 ) , lecture notes in artificial intelligence ( springer ) , vol .7880 , pp .254 266 , 2013 .poelmans j. , kuznetsov s. o. , ignatov d. i. , dedenea g. , formal concept analysis in knowledge processing : a survey on models and techniques . in :expert systems with applications , vol .16 , pp . 6601 - 6623 , 2013 .moha , n. , hacene , a. m. r. , valtchev , p. , gueheneuc , y. g. ( 2008 ) .refactorings of design defects using relational concept analysis . in formal concept analysis ( pp . 289 - 304 ) .springer berlin heidelberg .tilley , t. , cole , r. , becker , p. , eklund , p. ( 2005 ) .a survey of formal concept analysis support for software engineering activities . in formal concept analysis ( pp .250 - 271 ) .springer berlin heidelberg .poshyvanyk , d. , marcus , a. ( 2007 , june ) .combining formal concept analysis with information retrieval for concept location in source code . in program comprehension , 2007 .15th ieee international conference on ( pp .37 - 48 ) .

Abstract: software testing uses wide range of different tools to enhance the complicated process of defining quality of the system under test . formal concept analysis ( fca ) provides us with algorithms of deriving formal ontology from a set of objects and their attributes . with the use of fca we can considerably improve the efficiency of test case derivation . moreover , an fca - based machine learning system supports the analysis of regression testing results .


during the development of neural systems , spontaneous and synchronous activities can appear following the outgrowth of neurites and before the availability of external stimulus inputs .these activities are believed to play an important role in the formation and organization of functional neural circuitries .the investigation of these network activities can help to elucidate the cellular and network mechanisms involved in neural development and will lead to a better understanding of the functioning of a brain . among approaches to study the spontaneous activity of developing neural systems , dissociated cultures of cortical or hippocampal neurons on a multi - electrode array ( mea )have been used for decades as experimental models for observing the dynamics of growing networks .usually , spontaneous activities can be observed after about a week in vitro and the activities are later synchronized into episodic network bursts .interesting patterns of these neural bursts have been reported , where the activity level of firing rate in the burst can have repeated peaks of rise - and - falls called reverberations at a time scale of hundreds of milliseconds following the initial spike of activities .these so - called `` super bursts '' can last for seconds and , for their similarity in the time scales , are thought to be important to understand cognitive functions such as working memory on the cellular and network levels .there have been active studies on the initiation of in vitro neural bursts focusing on both the role of hub neurons and topological effects .the development of high - density mea systems has enabled more detailed investigation of the activity propagation in the neural bursts .notably , the collective dynamics of spiking neurons such as center - of - activity trajectory ( cat ) allow the identification of a propagation phase and a reverberation phase in the progression of a burst event .  in the current study , we use a similar high - density mea system to investigate reverberatory bursts observed in the development of dissociated cortical cultures . instead of considering reduced dynamics such as principal components or cat, we use a propagation model to predict the location of each occurring spike .the effectiveness of such prediction allows the classification of the spikes into _ evoked _ and _ spontaneous _ ones , and can be used in reverse for an inference on the spreading speed of the recorded spiking activity .we find a recovering dominance of the evoked spikes over the reverberatory phase of a burst following their reduction after the initial propagating wave .we implement a physiologically realistic model of neuronal systems on a geometrically - constrained , two - dimensional network and identify sets of parameters that can produce reverberatory bursts qualitatively similar to the experimental observations . with all dynamical variables being available in computer simulations ,we clarify the roles played by the neuronal noise as well as the depletion of synaptic resources in the continuation and termination of the reverberatory bursts .we find that the depletion , which is responsible for terminating the burst events , is also important in restoring the synchrony of reverberatory activity during the bursts .cortical neurons were dissociated from wistar rat at embryonic day 17 ( e17 ) .tissues were digested by 0.125% trypsin and plated on the biochip 4096e ( 3brain , switzerland ) previously coated with poly - d - lysine ( 0.1mg / ml ) and laminin ( 0.1mg / ml ) to promote the adhesion of neurons .about 610 neurons were plated , completely covering an active area of 66 mm^2^ , yielding a density of the culture of about 1.710 neurons / mm^2^. cultures were filled with 1 ml culture medium at 30 min after plating and incubated at 37 in the presence of 5% co~2~. half of the medium was refreshed twice a week .electrophysiological activities of neurons were recorded with the original culture medium once every other day since 6 div in 5% co~2~ at room temperature ( 24 ) . before recording ,the culture was kept at room temperature for 10 min for stabilization and placed back to the incubator immediately after the recording for future measurement .the chip 4096e has a recording area of 5.125.12 mm^2^ covered by 6464 electrodes .the area of each electrode is 2121 ^2^ with an inter - electrode separation of 81 m .the network activity was acquired at a sampling rate of 7.7 khz for each electrode .each recording data set includes network activity of 5 min .but , the data sets containing unstable activity patterns , long silent periods , or abnormal activities with , e.g. , strong noise , were excluded for further processing .the qualified data sets for further processing are listed in table [ tab : selected - cultures ] .ccc > m2.2 cm * culture * & * div * & * reverberation * & * * spreading speed ( mm / s ) * * +  & 12 & no & 93.15 +  & 25 & yes & 140.9 +  & 13 & no & 30.78 +  & 25 & yes & 75.33 +  & 33 & yes & 200.9 +  & 40 & yes & 179.8 +  & 68 & no & 554.9 +  & 26 & no & 145.0 +  & 33 & yes & 458.5 +  & 40 & yes & 480.3 +  & 68 & no & 517.6 +  & 24 & yes & 77.76 +  & 38 & yes & 435.0 +  & 41 & no & 398.5 +  & 12 & yes & 116.6 +  & 39 & yes & 403.4 +  & 54 & no & 448.7 +  spontaneous activities can be observed after about 2 weeks in vitro , comprising isolated spikes and short bursts involving many neurons ( electrodes ) , e.g. , the one shown in fig .[ fig : raster - histograms ] .  ]the isolated spikes produced in neurons are detected by the brainwave software through the precise timing spike detection with threshold values that are 8 times of the standard deviation of spike - free signals .the bursts are detected as follows .of the recording while the horizontal are time axes [ fig : illustrated - parameters ] ]  the spike rate at each instance is measured as an average over the time window of size centered at the time . fordetected spike times from each mea recording , typically of a 5-minute duration , the maximum of the spike rate is first determined .a lower threshold is used to decide whether the culture is in an active state as illustrated in fig .[ fig : illustrated - parameters]a .a reverberatory burst typically starts with a strong activity peak for the initiation phase followed by varying activity level or peaks in the reverberation phase as illustrated in fig .[ fig : illustrated - parameters]a .a burst is registered starting at when the culture becomes active at and stays active until the spike rate reaches an upper threshold of .the registered burst ends at when the culture becomes inactive at and stays inactive at least for a duration of .the empirical values for the parameters used in the burst detection of both the experimental and simulated data are : , , , and .  the peaks or reverberations could also be identified using the same method as described above with a different set of empirical parameter values .however , here we use a simpler definition that is time - symmetric : a peak is defined as a significant maximum ( height > > ) in the firing rate of a continuous time interval where the rate is above half of this maximum firing rate as marked in fig . [fig : illustrated - parameters]b . preceding this interval and following the previous peak ,if the firing rate of the culture stays above the lower threshold , the minimum of firing rate is considered the starting time of this peak . otherwise , the starting time is registered as the time when the rate crosses the lower threshold .the state variables of the system representing the internal noise and degree of depletion , which are only available in simulation results , are determined at the start time of a peak to correlate with the characteristics of the peak .the propagation of the spiking activity in a burst can be visually observed from the animated replay of sustaining spikes ( online resource ) . to quantify the wave - like propagation of the initial sweep of activity and the subsequent distributed activation of neurons ,we introduce a simple linear - spread diffusive model that can be used to predict the electrode for the next spike using spikes that have already been recorded .the probability for the next spike occurring at time to be on the electrode at is given by where and are time and location of the previous spike , is the spreading influence range of the spike , is the decay time of the influence , is the normalization factor , and is the spreading speed of the influence .we note that the probability ( [ eq : linear - spread - diffusive ] ) is conditional on a spike occurring at time , and should be multiplied with the spike rate for predicting the occurrence of a spike at .we define the predictability of spikes as the average of over all spikes in a recording comparing to the uniform distribution , which tells us how well the location of a spike can be predicted from previous spikes using the simple model ( [ eq : linear - spread - diffusive ] ) . for each recording , we find the value of that maximizes the predictability relative to a surrogate with randomized spike positions as shown in fig . [fig : predictability - histogram]a and these values are included in table ( [ tab : selected - cultures ] ) for all recordings .  ]  with the optimal value , a spike is considered an evoked spike if its position satisfies where is the average probability for the spike to occur at an electrode out of the electrodes for our mea .the number ratio of evoked spikes to the total spikes within the rate peaks of a burst are shown next to the corresponding peaks in fig .[ fig : predictability - histogram]b .   to gain insight into the dynamics of the reverberatory bursts, we use a neuronal synaptic model similar to that described by volman et al .the model uses morrislecar ( ml ) neurons connected with tsodyksmarkram ( tm ) synapses .the dynamics of neurons are governed by the ml equations ,  [ subsec : morris - lecar ]   where is the current through the membrane ion channels ,    are the voltage dependent dynamic parameters , and the threshold of membrane potential defines the spiking events which result in synchronous releases of neural transmitters at the efferent synapses . additionally , a residual calcium variable driven by the spiking events , where the spike train is with being the time of the spike event , is used to determine the rate , of synapse - dependent asynchronous releases of neural transmitters ( see below ) following an independent poisson process at each efferent synapse .the neural transmitters released by the spike - driven synchronous and calcium - dependent asynchronous events follow a four - state decaying dynamics based on a modification of the tm model ,  [ subsec : tums - dynamics ]   where summing over the asynchronous release events with a poisson rate given by ( [ eq : asynchro - rate ] ) , to include a super - inactive state .multiplying by the synaptic weights , the fractions of neural transmitters in the active state ( [ eq : active - state ] ) determine the contribution of the afferent synapses to the membrane conductance of a post - synaptic neuron through a linear sum over all pre - synaptic neurons of the given post - synaptic neuron .following , the synaptic weights are randomly drawn from a truncated gaussian distribution with a width that is of its mean for the connected neurons .we place the model neurons on a 2d geometrical network with connection probability between two neurons decaying exponentially with the distance between them .most of the model parameters used in our simulations follow the values given in and can be found in table [ tab : model - parameters ] .0.85  +  &  & &  & &  + &  & &  & &  + &  & &  & &  + &  & &  & &  + &  & &  & &  +  0.8  +  &  & &  & & + &  & &  & & +  0.95  +  &  & & & &  + &  & &  & & + &  & &  & & +  the time constants of tm dynamics , background currents for ml neurons , and synaptic weights are adjusted uniformly to reach simulated time - histograms that qualitatively reproduce the experimental results as seen in fig .[ fig : predictability - histogram ] . the raster plots for the simulated burst and the experimentally observed burst in fig .[ fig : predictability - histogram ] are shown in fig .[ fig : raster plots ] .b. * b * raster plot of simulated spikes and corresponding histogram for the bursting event shown in fig .[ fig : predictability - histogram]c[fig : raster plots ] ]  for current study , we focus on the reverberatory bursts with distinct reverberation peaks or sub - bursts in the spike rate histogram .  the same burst and peak detections for the experimental measurements are applied to the simulation results with slightly different empirical parameters .comparing to the experiments , the full dynamics of the simulations is readily available as numerical data and can be further analyzed to clarify the physical mechanisms of the bursting behavior .beside recording the time and neuron of each spike for the calculation of a time - histogram and keeping track of activity propagation , we are interested in the information of neuronal noise and the depletion of synaptic resources .the former is represented by the average concentration of residual calcium that governs the asynchronous release while the later is represented by the average fraction of inactive and super inactive neural transmitters which deplete the available neural transmitters in a bursting cycle .both of the values are retained at the start time of each detected peak in the spike - rate histogram and used to correlate with the properties of each peak .we implemented the computational model in the c++ programming language using the common simulation tools framework .the simulation codes along with the framework are included in the supplementary materials of the paper .the spike data from the mea recordings as well as the computer simulations were processed with the python3 programming language and most of the data plots were produced using the matplotlib library module . a jupyter notebook containing the python3 codes for data processing and plotting is also included in the supplementary materials .after plating , spontaneous activities are observed in about a week in vitro .such activities become synchronized into network bursts around 10 div and show reverberations after 15 div .the number of peaks per burst reaches a maximum around 30 div as shown in fig .[ fig : peak - of - reverberation ] and falls back to one without reverberation after 40 div .  , width=302 ]  it has been observed that the reverberatory bursts during the intermediate div can be divided into two phases : a propagation phase where the channels are activated sequentially and diffusively and a reverberation phase where the firings of the neurons are seemingly random and more decoupled .such division was confirmed with cat observation . as evident from streches of the cats shown in the insets of fig .[ fig : predictability - histogram]b for a reverberatory burst , the propagation is indeed more prominent for the initiating peak of spike rate ( blue trajectory ) and reduces to a lingering ( green ) trajectory soon after .however , as the network reverberates , the cat gradually regains its propagating sweeps until the end of the burst ( magenta trajectory ) .the factors driving the spiking activity of a neuron during a bursting event include the synaptic action spreading from its presynaptic neurons and the spontaneous activation driven by its own neuronal or synaptic noises .to identify the dominating factor contributing to a spike , we use the simple linear - spread diffusive model ( [ eq : linear - spread - diffusive ] ) parametrized with a spreading speed , which can be determined by a maximum likelihood method for each recording as documented in table [ tab : selected - cultures ] .while a more sophisticated propagation model might produce a better match to the observed behavior , the added complexity is not expected to change our conclusions qualitatively . using the propagation model ( [ eq : linear - spread - diffusive ] ) , we classify spikes into evoked spikes and spontaneous spikes .we then determine their ratio for all rate peaks of a burst .the results of evoked - spike fractions plotted in fig .[ fig : fraction - of - spread]a for 33 div recording of culture c show an increase in the fraction of evoked spikes as the network reverberates .  ]  to characterize how synchronous the spikes within an activity peak are , we normalize each rate peak with its spike count and use the normalized height to quantify the synchrony . in figure[ fig : fraction - of - spread]b , the synchrony of the activity peaks is plotted against the time of the peaks relative to the start of the bursts . while the synchrony data is more disperse , we can see an upward trend following the time course of the bursts .this demonstrates a correlation between the activity spreading and synchrony of the spikes .the result may not be a surprise considering the activity spreading through synaptic action following presynaptic spikes is how neurons can communicate and should help to orchestrate the synchronous activity .  to further clarify the synaptic dynamics contributing to the increasing dominance of the evoked spikes over spontaneous ones during a reverberatory burst, we turn to our simulations that produce qualitatively similar , reverberatory bursts with the increasing height of activity peaks in the spike - rate , time histogram over the bursts as shown in fig .[ fig : predictability - histogram]c . with a simulated system ,the full set of dynamic variables are available for analysis .we identify two factors of relevance in determining the peak height or the synchrony of the reverberation from our simulations : firstly , the residual calcium concentration controls the rate of asynchronous release at synapses in the model and represents the strength of an internal _ noise _ of the neurons .secondly , the inactive and super - inactive states featured in the model take up the neural transmitters as they are activated and represent the _ depletion _ of synaptic resources .we correlate the system average of these two factors with the height of activity peaks in a 3d scatter plot for all peaks of the simulated recording as shown in fig .[ fig : height - noise - depletion ] .  ] ) , and the depletion of synaptic resources , represented by the super inactive state of neural transmitters . *b * front view of the 3d plot .* c * top view of the 3d plot [ fig : height - noise - depletion ] ]  from the projection fig .[ fig : height - noise - depletion]b , we see that depletion , which increases during a burst , correlates positively with an increase of the peak height and thus the synchrony of the spikes . on the other hand , the noise factor represented by residual calcium , as shown in fig .[ fig : height - noise - depletion]c , is initially pumped up by the spiking activity of a burst , reaching a maximum about half way through the burst , and decreases afterward due to the lengthening intervals between the reverberation peaks until the end of the burst .the detailed dynamics of different factors can be further analyzed in a simulation . in fig .[ fig : mean - variables - over - burst ] , we plot the residual calcium concentration , depleted neural transmitter fraction ( ) , and the active neural transmitter fraction over the very burst shown in fig . [fig : predictability - histogram]c .c. the shaded area shows the time - histogram of the burst [ fig : mean - variables - over - burst ] ]  taken from the computational model , the depletion of neural transmitters to the and states is driven by the activated transmitters from the spiking activity .the spiking activity also increases the level of residual calcium , which controls the noisy , asynchronous releases of the neural transmitters leading to the reverberation in a burst .apart from making the role of synchronous releases more important in the activation of neurons , the depletion also leads to longer recovery time of neural transmitters .coupling with the rapid decay of calcium between the reverberation peaks , this leads to the lowing in the mean level of residual calcium towards the end of a burst .in the current study , we use a high - density mea system to investigate the physical mechanisms underlying the morphological richness of reverberatory neural bursts .our simple linear - spread diffusive model allows a classification of the spikes as well as an inference of the propagation speed of synaptic activities .the change of the predictability of spikes allows us to detect the change in the propagation behavior during a burst as shown in fig .[ fig : fraction - of - spread ] .however , the traveling - wave - like sweep of activity , especially for the initiation of a burst , is not diffusive . a more sophisticated model will be required if one would like to have a more faithful capture of such dynamics .nonetheless , the method of inference for the model parameters using individual spikes as demonstrated remains applicable .the method is enabled by our use of high - density mea and does not resort to data reduction before inferring the propagation dynamics .that is , each spike has a direct contribution to the resolution of the spreading speed and the method can potentially be used to resolve more complex dynamics of the system .the finding from our analysis of the simulated system suggests an interesting phenomena , which we call _ depletion - enhanced synchronization _, at play in the cultured network with the reverberatory bursts .in such a burst , the initiation activity is a fast sweeping wave of propagating spikes across the network that is well synchronized .this activity produces a significant amount of residual calcium , promoting noisy asynchronous releases , and prompting the spontaneous firing of the neurons that results in the subsequent reverberation of the burst .initially , the spontaneous spikes are more or less independent and the heterogeneity in the neurons and their connectivity makes the spike - rate peaks broad and less synchronous .however , as the neural and synaptic resources are increasingly depleted by the continuing spiking activity of the burst , it becomes harder for the neurons to fire independently and they thus increasingly rely on the synchronous releases triggered by the firing of their presynaptic neurons to help them cross the firing threshold .such mechanism accounts for the observed increase of evoked spikes and the synchrony in fig .[ fig : fraction - of - spread ] and may be a general mode of operation for other complex systems .  the synchronized network activities observed in our cultures seem to be similar to the switching between up and down states as observed in other neuronal network preparations .however , since our measurements are carried out on mea , records of the membrane potentials are not available to verify these states .it is known that activities similar to what we reported here can also be induced in acute slice when inhibitory interactions are blocked . presumably , there are too many recurrent connections in our cultures which might correspond to the pathological condition during epilepsy .  in the computational model , the active state is initially stabilized by the residual calcium which promotes the asynchronous releases intrinsic to the neurons , and later revitalized by the synaptic couplings of the network . the role of calcium in the reverberationwas implicated by and we chose to implement the model by based the similarity between the firing - rate time histograms it produces and those were seen in our experiments .alternatively , nmda receptors have been proposed to play a role in persisting a burst .it will be interesting to see in future studies what difference in the bursting morphology will result from an nmda receptor based model .  finally , we note that while synchrony is often associated with coherence , it actually reduces the diversity in the possible dynamics of a system . in the reverberatory bursts that we focused on ,the synchrony results from the depletion of synaptic resources and precedes the termination of the burst .this parallels the recent findings in epilepsy that increasing synchrony can be observed towards the end of seizures .our results may suggest a possible mechanism for such phenomena for systems of similar episodic dynamics .this work has been supported by the ministry of science and technology , taiwan , republic of china under grand no .102 - 2112-m-001 -009 -my3 , and physics division , national center for theoretical sciences , taiwan , republic of china .bermudez contreras ej , schjetnan agp , muhammad a , bartho p , mcnaughton bl , kolb b , gruber aj , luczak a ( 2013 ) formation and reverberation of sequential neural activity patterns evoked by sensory stimulation are enhanced during cortical desynchronization .neuron 79(3):555566                  jiruska p , de curtis m , jefferys jgr , schevon ca , schiff sj , schindler k ( 2013 ) synchronization and desynchronization in epilepsy : controversies and hypotheses .the journal of physiology 591(4):787797                   schroeter ms , charlesworth p , kitzbichler mg , paulsen o , bullmore et ( 2015 ) emergence of rich - club topology and coordinated dynamics in development of hippocampal functional networks in vitro .the journal of neuroscience 35(14):54595470      van pelt j , corner ma , wolters ps , rutten wlc , ramakers gja ( 2004 ) longterm stability and developmental changes in spontaneous network burst firing patterns in dissociated rat cerebral cortex cell cultures on multielectrode arrays .neuroscience letters 361(13):8689

Abstract: developing networks of neural systems can exhibit spontaneous , synchronous activities called neural bursts , which can be important in the organization of functional neural circuits . before the network matures , the activity level of a burst can reverberate in repeated rise - and - falls in periods of hundreds of milliseconds following an initial wave - like propagation of spiking activity , while the burst itself lasts for seconds . to investigate the spatiotemporal structure of the reverberatory bursts , we culture dissociated , rat cortical neurons on a high - density multi - electrode array to record the dynamics of neural activity over the growth and maturation of the network . we find the synchrony of the spiking significantly reduced following the initial wave and the activities become broadly distributed spatially . the synchrony recovers as the system reverberates until the end of the burst . using a propagation model we infer the spreading speed of the spiking activity , which increases as the culture ages . we perform computer simulations of the system using a physiological model of spiking networks in two spatial dimensions and find the parameters that reproduce the observed resynchronization of spiking in the bursts . an analysis of the simulated dynamics suggests that the depletion of synaptic resources causes the resynchronization . the spatial propagation dynamics of the simulations match well with observations over the course of a burst and point to an interplay of the synaptic efficacy and the noisy neural self - activation in producing the morphology of the bursts .