the oscillations of rotating neutron stars are of great interest in astrophysics .recent efforts to understand the various pulsation modes of compact stars have to a large extent been motivated by gravitational - wave astronomy  .different modes of oscillation depend on different pieces of internal physics and one may hope to use observations to probe , for example , the composition of the high - density region  .work in this area , which has a long history ( for a review see e.g. , * ? ? ?* ) , gained further momentum recently with the relatively successful matching between observed quasiperiodic oscillations in the tails of magnetar flares and calculated torsional oscillations of the neutron star crust  .the excitement following these , the first ever , observations of likely neutron star vibrations is significant .it is clear that our models need to be improved significantly if we expect to carry out a quantitative astero - seismology programme for neutron stars , but the prospects for improvements look good .after all , we already have a reasonable understanding of the dynamics of the crust region ( see * ? ? ?* ; * ? ? ?* ) as well as the superfluid components in the core of the star  .the present work is motivated by the need to consider more realistic models of rapidly rotating stars. our emphasis will be on the role of the internal stratification associated with composition variations .this leads to the presence of the so - called g - modes  , and we want to investigate how these modes are affected by fast rotation ( the analogous problem for thermal ocean g - modes of rapidly rotating neutron stars has already been considered , see ) . in order to determine the rotational effects on the oscillation spectrum we study linear perturbations of axisymmetric stellar configurations , where the coriolis force and the centrifugal flattening of the star are included in the background . in this way, we go beyond the so - called slow rotation approximation , where the rotation itself is treated perturbatively and the mode - frequencies of non - rotating models are determined by a perturbation expansion with respect to the angular velocity of the star .we are particularly interested in the relationship between the g - modes of a stratified stellar model and the inertial modes  , for which the coriolis force is the main restoring agent .the inertial modes are important since they may be driven unstable by the emission of gravitational waves .the strongest instability is associated with the so - called r - modes  .a key issue concerns the amplitude at which an unstable mode saturates .detailed work has shown that an unstable r - mode saturates due to nonlinear couplings to other inertial modes  .however , the mode saturation has only been considered for slowly rotating , non - stratified models and it is important to ask to what extent these results will change if the models are made more realistic . in stratified neutron stars ,the g - modes tend to become rotationally dominated beyond a certain rotation rate  and may then be called inertia - gravity modes  .effects due to the stratification are thus weakened , which indicates that the r - mode saturation estimates may still be reliable , at least for the fastest spinning systems .  improving and extending a numerical code that has been used to study non - stratified stars , we approach the problem via time - evolutions of the equations that govern the linear perturbations of a , potentially rapidly , rotating newtonian star .this strategy has the advantage that one does not have to deal with the many rotationally coupled multipoles , that tend to make a slow - rotation calculation less tractable . on the other hand , a numerical evolution is not expected to have the precision of a frequency domain calculation .neither should one expect it to yield the complete spectrum of modes .after all , the simulation results depend on the chosen initial data . in absence of a clear idea of the nature of the various modesone does not know how to excite specific oscillations .one way around this problem is to follow the `` recycling '' strategy developed by  and .another , more pragmatic , approach is to simply study the modes that are excited by `` generic '' initial data .this is the attitude that we adopt here .we want to investigate the effect that composition variations have on the various oscillation modes of a rapidly rotating neutron star . to do this, we consider an equilibrium configuration representing a rapidly and uniformly rotating star containing three particle species .we assume that matter is composed of degenerate neutrons , protons and electrons . for simplicity, we further assume that neutrons and protons are not superfluid .in essence , our model represents the conditions in the outer core of a very young neutron star that has not yet cooled below the temperature at which the baryons form superfluid condensates ( in practice with a core temperature significantly above  k ) .we also do not consider the various exotic phases of matter that may be present in the deep core of a realistic neutron star model , e.g. hyperons or deconfined quarks ( see * ? ? ?* ; * ? ? ?accounting for the presence of such phases would not be too difficult , but at this stage our focus is on understanding the basic role of composition variations . we are already extending this work to account for the presence of superfluid components and expect to report on results in this direction soon .the matter composition is represented by the three number densities , where is either , or .charge neutrality is provided by the coulomb interaction , that efficiently lock together protons and electrons ( e.g. , * ? ? ?thus we have and , where denotes the velocity of each fluid component . in this model ,the pressure is a function of two variables , e.g. the neutron and proton number densities or alternatively the total baryon number density and the proton fraction . in the unperturbed stellar model, it is assumed that the weak interaction processes : lead to -equilibrium .the proton fraction is then only a function of the baryon number density ( or equivalently the baryon mass density , where for simplicity the proton and neutron masses have been assumed equal ) .then the stellar fluid can be described by a barotropic equation of state . in this paper, we consider a simple polytropic model where is constant and is the adiabatic index .once the equation of state is provided , rotating stellar configurations can be determined by the hachisu self - consistent field method  . for this purposewe use the numerical code described in  .this code generates axisymmetric equilibrium models by specifying the polytropic index and the ratio of the polar to equatorial axes . in table[ tab : back - models ] we give some details relating to the rotation rate , energies and masses of equilibrium configurations used in this paper ..[tab : back - models ] this table displays the main quantities of our rotating equilibrium configurations .the stellar models are described by a polytropic equation of state with adiabatic index . in the first and second columns we show ,respectively , the ratio of polar to equatorial axes and the angular velocity of the star . in the third column , the rotation rate is compared to the kepler velocity that represents the mass shedding limit .the ratio between the rotational kinetic energy and gravitational potential energy and the stellar mass are given in the fourth and fifth columns , respectively .all quantities are given in dimensionless units , where is the gravitational constant , represents the central mass density and is the equatorial radius . [ cols="^,^,^,^,^",options="header " , ]   and . the left panel shows the frequencies of the f- and g - modes , while the right panel shows the corresponding modes for and . in order to emphasize the rotational splitting of the g - modes and the onset of the cfs instability , the frequencies are shown in the reference frame of an inertial observer .[ fig : gf2.05-f - gmodes - polar],title="fig:",height=226 ]  and .the left panel shows the frequencies of the f- and g - modes , while the right panel shows the corresponding modes for and . in order to emphasize the rotational splitting of the g - modes and the onset of the cfs instability ,the frequencies are shown in the reference frame of an inertial observer .[ fig : gf2.05-f - gmodes - polar],title="fig:",height=226 ]  these issues can be directly related to the dynamics of a perturbed fluid element and the balance between the two restoring forces that are acting on it , the coriolis and the buoyancy force . in slowly rotating and stratified neutron stars with large composition gradients ,buoyancy is expected to dominate the coriolis force . as a result , a typical mode should appear as a g - mode . for higher rotation rates , the effect of the coriolis force increases and may dominate the buoyancy force beyond a given stellar spin . in this regime ,the oscillation mode should behave as an inertial mode .the threshold between these two regimes depends on the magnitude of the composition gradient and must obviously be smaller than the mass shedding limit of the star to be of any interest .the order of magnitude of the composition g - mode frequencies can be estimated by the brunt - visl frequency , which is defined by ( see e.g. , * ? ? ?* ) for non - rotating stars this frequency can be written where the equilibrium configuration equations have been used to replace the pressure gradient with the gravitational acceleration .an averaged value of equation ( [ eq : nfreq - nr ] ) can be estimated by approximating the gravitational acceleration as , where is the mass density average , and the density gradients as . as a result ,the averaged and dimensionless brunt - visl frequency is given by : we have tested equation ( [ nn ] ) with the average of equation ( [ eq : nfreq ] ) for the non - rotating and non - barotropic models used in this paper .the results agree to better than .   and in the left panel and in the right panel .the brunt - visl frequency , the lamb frequency and the coriolis term of the dispersion relation ( [ eq : disprel ] ) are given in units .on the horizontal - axis the radial coordinate is normalized with the stellar radius . for simplicity, the and frequencies are presented in the two panels for , while three directions are shown for the coriolis term , respectively and .the two propagation diagrams illustrate the regions of the star where the various restoring forces dominate . increasing the stellar spin , the coriolis term becomes stronger than the brunt - visl frequency in almost all the volume of the star . only in a small region near the equator, buoyancy can dominate even in rapidly rotating stars .[ fig : prop - diag],title="fig:",height=226 ]  and in the left panel and in the right panel .the brunt - visl frequency , the lamb frequency and the coriolis term of the dispersion relation ( [ eq : disprel ] ) are given in units .on the horizontal - axis the radial coordinate is normalized with the stellar radius . for simplicity ,the and frequencies are presented in the two panels for , while three directions are shown for the coriolis term , respectively and .the two propagation diagrams illustrate the regions of the star where the various restoring forces dominate . increasing the stellar spin , the coriolis term becomes stronger than the brunt - visl frequency in almost all the volume of the star . only in a small region near the equator, buoyancy can dominate even in rapidly rotating stars .[ fig : prop - diag],title="fig:",height=226 ]  we want to study the effects of composition gradients on the spectrum in detail .we consider non - barotropic models with and frozen composition indices between and in order to investigate stars with low and high composition gradients .it is useful to begin by estimating at what rotation rate the coriolis force will start to dominate the buoyancy . when , equation ( [ nn ] ) suggests that for a model with . for models rotating much faster than this, we would expect the coriolis force to dominate . as a result, one would expect the low - frequency modes of such models to have a predominantly inertial character .as we will see later , the trend suggested by this rough estimate is brought out by the detailed mode - calculations .   for a set of slowly rotating barotropic and non - barotropic models .all these models have axis ratio ( corresponding to ) , but different frozen composition adiabatic index .the solid line corresponds to a barotropic star with .the counter - rotating mode has been determined , from left to right in the figure , for stars with .this figure shows the dependence of the mode frequency on the composition gradients . in the limit of , the counter - rotating mode seems to approach the inertial mode .this behaviour is clearer in fig ( [ fig : g1r - i31 ] ) .[ fig : sp],height=226 ]   mode with the adiabatic index for a sequence of rotating stellar models .the polar - led inertial mode frequencies of the barotropic models with are shown as a circle - solid line .this figure suggests that in rapidly rotating stars the counter rotating mode behaves similarly to the barotropic inertial mode .this is expected when the rotation rate is larger than the brunt - visl frequency , namely , and the coriolis force starts to dominate over the buoyancy force .[ fig : g1r - i31],height=226 ]  the non - barotropic evolution code has been tested against two eigenvalue codes developed by  and  , respectively . solve for linear perturbations of slowly rotating neutron stars in newtonian gravity , whereas  study the pulsation spectra of slowly rotating relativistic stars in the cowling approximation .we have compared the first two g - modes frequencies with the non - rotating and weakly stratified newtonian model of  , which has and .the relevant frequencies , which are given in table [ tab : gmodes - yl ] , show a very good agreement even though we are using the cowling approximation .the second g - mode have an error of four percent , while the others are accurate to within two percent .this difference is due the low frequency of the second g - mode that requires a longer time evolution for extracting a more accurate value with an fft .we have further tested our code on neutron star models with larger composition gradients by using the  code . since the time evolution code used in this paperis newtonian , we have made a comparison between the two codes for a spherically symmetric background model with low compactness . for a polytropic model with and a star with central density , the tolman - oppenheimer - volkoff equations determine a non - rotating star with mass and radius  km , giving a compactness ratio of . in terms of the dimensionless quantity used in our time domain code , this model has which agrees very well ( within one percent ) with the non - rotating model in table [ tab : back - models ] . by using these two approaches , we determined the g - mode frequencies of non - rotating stratified neutron stars in table [ tab : gmodes ] . for non - barotropic stars with frequencies of the first three modes agree to better than eight percent , while for they agree to better than one percent .this level of agreement is satisfactory given that the calculations are carried out within completely different frameworks .thus we are confident that our time - evolutions are able to provide accurate g - mode results .the rotational splitting of the f- and g - modes for stratified stars with is shown , in the frame of an inertial observer , in fig .[ fig : gf2.05-f - gmodes - polar ] .the left panel represents modes with type i parity , while the right panel shows results for type ii parity .the two classes of modes clearly behave in a similar fashion .results for other values of are similar . generally , we observe that the frequencies of the f- and g - modes tend to increase for stars with larger composition gradients , i.e. larger . from the results in fig . [fig : gf2.05-f - gmodes - polar ] it is easy to read off when the low - order g - modes become susceptible to the gravitational - radiation driven cfs instability  .the instability condition is that the mode - frequency changes sign according to an inertial observer .the results in the figure suggest that this happens for .this is in accordance with the typical values expected for the g - mode cfs instability .in fact , we can compare our result to an order of magnitude estimate based on the result of  .he estimates that the rotation rate of the g - mode instability onset corresponds to where is the g - mode frequency for the non - rotating model . from thiswe find that one would expect the instability of the mode for the model to be triggered in a star rotating at .this is close to the ( presumably more accurate ) value determined with our numerical code .however , the g - mode instability is not thought to be very important since the g - modes are not radiating gravitational waves efficiently .viscosity is expected to suppress the instability in more realistic models . in comparison ,the f - mode , which radiates gravitational waves efficiently does not go unstable until the star spins extremely fast . for stellar models with , the f - mode is not expected to become unstable before the mass shedding limit rotational rate is reached  . as is clear from fig .[ fig : gf2.05-f - gmodes - polar ] , the f - mode instability point occurs for faster spins than we are able to study .this accords well with previous results in the literature .let us now study the low frequency band of rotating non - barotropic stars .a local analysis of uniformly rotating stars in the cowling approximation leads to the following dispersion relation for low - frequency waves  : where is the wave vector and is its component orthogonal to the apparent gravity .these modes are generally referred to as inertia - gravity waves by  , as they are restored by both gravity and the coriolis force . in the barotropic case, , equation ( [ eq : disprel ] ) describes the inertial waves , while for a non - rotating model it reduces to the usual gravity waves . in stratified and rotating neutron stars ,it is thus to be expected that the low - frequency oscillation modes have an hybrid character with gravity dominating at low rotation rates and the coriolis force taking over at fast stellar spin .   for rotating stellar models with rotation rate ( left panel ) and ( right panel ) .the yellow surfaces refer to the inertial mode of barotropic models with , while the red surfaces represent the counter - rotating mode of stratified stars with and .this figure illustrates how g- and inertial mode eigenfunctions become similar for increasing rotation rates .[ fig : fphi2d],title="fig:",height=226 ]  for rotating stellar models with rotation rate ( left panel ) and ( right panel ) .the yellow surfaces refer to the inertial mode of barotropic models with , while the red surfaces represent the counter - rotating mode of stratified stars with and .this figure illustrates how g- and inertial mode eigenfunctions become similar for increasing rotation rates .[ fig : fphi2d],title="fig:",height=226 ]   for the rotating stellar model with rotation rate .the left panel shows the contour plot of the inertial mode for the barotropic model with .the right panel displays the contour plot of the counter - rotating mode of a stratified star with and .these plots give a complementary view of the eigenfunctions shown in the left panel of fig .( [ fig : fphi2d ] ) , and confirm the difference between the two modes in a slowly rotating model .[ fig : fphi_contour_slow],title="fig:",height=207 ]  for the rotating stellar model with rotation rate .the left panel shows the contour plot of the inertial mode for the barotropic model with .the right panel displays the contour plot of the counter - rotating mode of a stratified star with and .these plots give a complementary view of the eigenfunctions shown in the left panel of fig .( [ fig : fphi2d ] ) , and confirm the difference between the two modes in a slowly rotating model .[ fig : fphi_contour_slow],title="fig:",height=207 ]   for the rotating stellar model with rotation rate the left panel shows the contour plot of the inertial mode for the barotropic model with .the right panel displays the contour plot of the counter - rotating mode of a stratified star with and .these plots confirm the results of fig .( [ fig : fphi2d ] ) showing that the structure of these two eigenfunctions becomes similar in fast rotating models .[ fig : fphi_contour_fast],title="fig:",height=207 ]  for the rotating stellar model with rotation rate the left panel shows the contour plot of the inertial mode for the barotropic model with .the right panel displays the contour plot of the counter - rotating mode of a stratified star with and .these plots confirm the results of fig .( [ fig : fphi2d ] ) showing that the structure of these two eigenfunctions becomes similar in fast rotating models .[ fig : fphi_contour_fast],title="fig:",height=207 ]  however , it is worth noting that the buoyancy and the coriolis force can be locally dominant in different regions of the star .this can be understood from a propagation diagram that illustrates the spectral properties of the star .for instance , we consider two stellar models with frozen adiabatic index and with angular velocity and , respectively . for these models we determine the brunt - visl frequency  ( [ eq : nfreq ] ) , the lamb frequency  and effect of the coriolis force on the dispersion relation ( [ eq : disprel ] ) , i.e. . the role of the lamb frequency is to estimate the propagation region for acoustic modes .for these two stellar models , the propagation diagram for the modes is shown in fig .[ fig : prop - diag ] using the normalized radial coordinate . forthe slowly rotating star with , the and frequencies do not vary much , while the effect of the coriolis force term changes according to its angular dependence , from zero at the equator to the maximum at the pole . in this case, the star is spinning so slowly that the buoyance ( ) is dominant in most of the star . only near the centre , for , is the coriolis force stronger . as a result, we should expect the inertia - gravity modes to behave as g - modes .later , this will be confirmed by a global mode analysis .when the star rotates faster , as in the model with  ( right panel of fig .[ fig : prop - diag ] ) , the coriolis term is more important and larger than the brunt - visl frequency in the most of the star . in order to keep the figure clear , we have shown the and frequencies only for .the change in the other directions is visible , but does not modify the qualitative properties of the propagation diagram .however , even in weakly stratified and rapidly rotating stars there is a region near the equator where buoyancy dominates over the coriolis force .this region , which is due to the dependence of the coriolis term , shrinks for higher rotation rates .models and polar - led inertial modes ( dashed lines ) of the barotropic model .all frequencies are measured in the rotating frame .the dominance of the coriolis force in rapidly rotating stars is evident from this figure .we see that the g - modes are mainly restored by the coriolis force when , and thus become similar to the barotropic inertial modes .[ fig : gf2.05-g - in - comp - polar],height=226 ]  the inertia - gravity modes can be identified in our time domain simulations by studying the fft of the perturbation evolution and extracting the 1d and 2d mode eigenfunctions with the code developed by  and  . in order to study the dependence of the modes on the stellar spin and stratification, we construct a set of equilibrium models by gradually varying the two parameters and .we can then follow the change of the mode frequency and eigenfunction shape and correctly distinguish it among the other peaks in the spectrum .let us consider , for instance , a sequence of slowly rotating models with fixed rotation rate and different .typical fft results are shown in fig .[ fig : sp ] , where we have identified the inertial mode for the barotropic model and the counter - rotating mode for a set of stars with increasing composition gradients .the results suggest that the g - mode frequency approaches the barotropic inertial mode in the limit .this association becomes more evident if we combine these results for a set of rotating models , see fig . [fig : g1r - i31 ] .when the star rotates with , the retrograde mode seems to behave as the barotropic inertial mode .this transition obviously depends on the level of stratification .this property is also noted in the mode eigenfunctions . in figs .[ fig : fphi2d][fig : fphi_contour_fast ] , we show how the eigenfunctions of the mass flux component of the counter - rotating mode approach the shape of the mode when . in this case, the non - barotropic star has and an estimated brunt - visl frequency , see equation ( [ nn ] ) . in fig .[ fig : fphi2d ] we provide a three - dimensional picture of the eigenfunctions , for rotation rate in the left panel , and for in the right panel ( the latter corresponding to a star with an angular velocity slightly larger than the brunt - visl frequency ). clearly , for the slower rotation rate ( left panel ) the the eigenfunction is quite different from the eigenfunction of the barotropic mode . in particular , the nodal curve of the two modes do not coincide near the equator .in contrast for the more rapidly rotating star ( right panel ) the nodal curves of the two modes are very similar . this can be seen particularly clearly in the contour plots of the eigenfunctions given in fig .[ fig : fphi_contour_slow ] ( for ) and fig .[ fig : fphi_contour_fast ] ( for ) .when we extend this approach to the other g and inertial modes , we find further mode associations . combined results are provided in figs .[ fig : gf2.05-g - in - comp - polar ] and [ fig : gf2.05-g - in - comp - axial ] .these figures display the spectrum of barotropic models and stratified stars with .the results show that individual non - barotropic g - modes tend towards specific barotropic inertial modes as the rotation rate of the star is increased .this agrees with the notion that the coriolis force dominates the buoyancy in fast spinning systems . from the results in the two figureswe can identify various g - modes with barotropic inertial - mode counterparts .for type i parity and this leads to : + where the upper index r and p denotes retrograde and prograde g - modes respectively .meanwhile , for type ii parity and we find that : + from fig .[ fig : gf2.05-g - in - comp - axial ] we also see that it is only the r - mode that does not have a g - mode counterpart .as expected , the r - mode is essentially the same in the barotropic and non - barotropic models . in the type ii parityperturbations we could accurately determine only the first g - mode , as the r - mode peak was dominant in the fft and overwhelmed the higher order g - modes .these results are consistent with the study of  , where the authors worked within the slow rotation approximation and in the frequency domain .they studied inertia - gravity modes of polytropic models with weak stratification and slow rotation .in fact , their frozen adiabatic index corresponds to and the fastest rotating model they consider represents , where and are respectively the mass and radius of the non - rotating models . in our unitsthis rotation rate in equivalent to and an axis ratio .in contrast , in the present work we investigate rapidly rotating models up to ( ) and with larger composition gradients .we have tested our results against those of  by numerically evolving perturbations of the two slowest rotating models reported in table [ tab : back - models ] with .the frequencies have an accuracy to better than 2 percent .finally , equations ( [ eq : connrule1 ] ) and ( [ eq : connrule2 ] ) respect the connection rules reported in  .g - modes ( solid lines ) for models and axial - led inertial modes ( dashed line ) of the barotropic models .frequencies are measured in the rotating frame .the counter- and co - rotating modes approach the frequencies of the axial - led inertial modes in the limit of fast rotation .in contrast , the r - mode is essentially unaffected by composition gradients .[ fig : gf2.05-g - in - comp - axial],height=226 ]information concerning the oscillation spectra of realistic neutron star models helps develop our understanding of the physics required to describe these objects .any spectral property can in principle be attributed to particular physical quantities or configurations of the star .astero - seismology studies , using either electromagnetic or gravitational signals ( or indeed both ) may thus help constrain the state of matter at supernuclear densities . of course, in order to facilitate this , we need to improve our theoretical models and clarify the origin of various stellar pulsation features .  in this paper , we have studied the pulsations of stratified and rapidly rotating neutron stars with the aim of understanding the dependence of the composition g - modes on the rotation rate of the star .the stellar pulsations were studied using the linearised euler and conservation equations on an axisymmetric background . in order to simplify the problem, we used the cowling approximation where the gravitational potential perturbations are neglected .this approximation generates only a small error in the g - mode frequencies .the system of perturbation equations was evolved in time by a 2d numerical code based on a standard finite differencing scheme .this code was tested against various results available in literature and demonstrated good accuracy .  since both coriolis and buoyancy forces act on a perturbed fluid element of a rotating and stratified star , the low - frequency modes have a mixed inertia - gravity character .they typically behave as g - modes in the slowly rotating limit , while they assume the properties of the inertial modes when the star rotates rapidly . by comparing the oscillation frequencies and the associated eigenfunctions of barotropic inertial modes and the inertia - gravity modes of non - barotropic stars ,we have demonstrated how the two sets of modes are associated at fast rotation rates .our results show that it would be difficult to deduce the presence of composition gradients from the inertia - gravity mode spectrum of fast rotating stars . however , as the neutron star ages and spins down , the dependence of the inertia - gravity modes on the star s rotation and their deviation from the inertial barotropic modes can , at least in principle , be used to estimate the brunt - visl frequency and the degree of stratification .  as an extension of this work, we are currently developing a numerical code for studying the dynamics of rapidly rotating superfluid neutron stars .this is an important development since all mature neutron stars are expected to have superfluid components in the core .it is also well known that the associated multi - fluid dynamics leaves an imprint on the stellar oscillation spectrum , see for a recent discussion .we hope to be able to report on the initial results of this investigation soon .this work was supported by stfc through grant number pp / e001025/1 .na also acknowledges support from stfc via senior research fellowship no pp / c505791/1 .

Abstract: we use time - evolutions of the linear perturbation equations to study the oscillations of rapidly rotating neutrons stars . our models account for the buoyancy due to composition gradients and we study , for the first time , the nature of the resultant g - modes in a fast spinning star . we provide detailed comparisons of non - stratified and stratified models . this leads to an improved understanding of the relationship between the inertial modes of a non - stratified star and the g - modes of a stratified system . in particular , we demonstrate that each g - mode becomes rotation - dominated , i.e. approaches a particular inertial mode , as the rotation rate of the star is increased . we also discuss issues relating to the gravitational - wave driven instability of the various classes of oscillation modes .   [ firstpage ]  methods : numerical stars : neutron stars : oscillation star : rotation .


humanoid robots locomoting and performing manipulation tasks on uneven ground are required to actively apply forces on objects and the floor in order to achieve their task successfully .a direct effect of contact forces is a change of momentum in the robot , which on the one hand is necessary to move the center of mass , but on the other hand restricts the type of limb motion that the robot can perform .the nonlinear nature of the angular momentum dynamics makes preview - based control computationally hard in general and even with an admissible angular momentum trajectory open control parameters like the joint motion and feedback control remain problematic for use in a whole body control framework .successful applications of simplified momentum models have been shown on robots walking quite robustly over flat ground .a common approach is to use the linear inverted pendulum model ( lipm ) , which has been exploited for preview control since it was introduced by kajita et .al  . in  ,a model predictive control ( mpc ) approach is formulated that finds foot steps on a flat ground together with a compatible com location on a horizontal plane .com profiles can then be realized on the full body together with other limb motion ( e.g. swing leg ) controllers  ,   ,  .the assumption of a horizontal com motion and flat ground can be relaxed to a pre - designed com height profile  ,  .although , less restrictive than the original lipm , these approaches are either built for point feet or leave parts of the dynamics uncontrolled . depending on the terrain , pre - defining the com along a fixed direction may result in suboptimal or infeasible reaction forces .approaches that leave the regime of linear models have been proposed , for instance for long jumps  leading to more complex task behavior .however , they often require task specific models that for example take into account the swing leg dynamics . going even further ,we have seen work that optimizes over the whole joint trajectories and the full momentum together  .the impressive near - to physical motions , however , come at a high computational cost .time - local controllers , have shown that balancing performance of robots is improved , when overall angular momentum is damped out directly  .nevertheless , it has remained unclear how angular momentum profiles should be chosen in non - static configurations .  in this paper , we consider the full momentum dynamics of the robot for generation of com and momentum trajectories and the according reaction force profiles at each contact .we formulate the problem as a continuous - time optimal control problem in a sequential form .a mode schedule is predefined together with end effector trajectories .a desired angular momentum trajectory , which is required for the limb motion , is generated from a simple inverse kinematics forward integration and realized with admissible contact forces in a least squares optimal sense .since the kinematics information is used before the optimization over the dynamics , our optimization procedure is relatively fast ( for example compared to ) . in our previous work we have proposed a way to compute control gains for the momentum task using a lqr design approach .we showed that it was able to significantly improve performance for balancing tasks on a real humanoid .however , this approach was only used for stabilization . in this paper , we extend the approach to receding horizon tracking control .we use the planned trajectories to generate feedback gains using a lqr design where we linearize the non - linear momentum dynamics around the optimized trajectory . in a simulated stepping task ,our humanoid traverses a terrain with changing height and angled stepping stones successfully .a whole - body controller computes joint torques that realize feedback loops on momentum as well as the swing leg motion and respect additional balancing and hardware constraints and generates physically realizable contact forces .a good tracking of overall momentum is achieved using our receding horizon lqr design . + this paperis organized as follows . in sec[ sec : momentum_trajectory ] we formulate an optimal control problem to obtain momentum trajectories of the robot .resulting trajectories are controlled with a lqr feedback design as discussed in sec [ sec : momentum_lqr ] which is then incorporated into a whole - body control approach as explained in sec [ sec : whole_body ] . in sec[ sec : results ] we demonstrate our control framework on a simulated stepping task on rough terrain .we discuss our results in sec [ sec : discussion ] and finish with a conclusion in sec [ sec : conclusion ] .in this section , we describe how the robot momentum is planned together with admissible contact forces .we will first discuss the reduced dynamics model which will then be used to phrase an optimal control problem to generate com and momentum profiles . at the end of the sectionwe describe how a desired angular momentum is chosen for optimization .    of the system and contacts with the environment .states and controls are color - labeled blue and orange respectively .flat contacts that consist of a force , center of pressure and normal torque ( here ) can be modeled ( e.g. a hand touching a wall ) as well as point contacts like .there is no restriction on the pose of contacts nor the com .momenta generated around the com are not necessarily zero . ]we reduce the state of a robot to its center of mass ( com ) and overall momentum with linear and angular momenta .the state of the system is controlled by forces and torques acting at points on the mechanical structure .the momentum dynamics as illustrated in fig [ fig : mass_and_contacts ] can be written where is the mass of the robot .the contact points can for instance be point foot locations that are touching the ground or they can represent the position of a handle that the robot holds on to .flat feet can be modeled by several contact points on the foot surface .equivalently , we can represent the effective force at the center of pressure ( cop ) . depending on the type of contact, it will be necessary to express constraints on the forces . in the case of point feet , no torques can be generated .torques that are generated at the cop of a flat foot are required to be normal to the foot surface .further , if we assume stationary contact points , we need to restrict the wrench to remain in a friction cone .   in the following we use the momentum dynamics of the robot to plan for admissible contact forces that generate desired ( linear and angular ) momentum trajectories .our goal is to compute force and contact point trajectories that satisfy the momentum dynamics in eqs ( [ eq : momentum_dynamics_1]-[eq : momentum_dynamics_3 ] ) and contact constraints at all time .these trajectories are also required to steer the momentum through desired states over a time horizon . given the initial state of the robot com and momentum , we can integrate eqs ( [ eq : momentum_dynamics_1]-[eq : momentum_dynamics_3 ] ) to obtain note that the states are expressed as ( nested ) functions of the reaction forces .they are constructed by integrating , summing and applying the cross - product on . given a naive idea of what the desired com and momentum should be ( e.g. coming from an initial kinematic plan ) ,we want to find contact forces that minimize the error    where we compute the errors at  ] ; in practice however , we express them at a finite number of time steps . given the limited flexibility of polynomials , we get minor penetration of those constraints in intervals .note that all our constraints are linear , whereas the objective function has quadratic terms as well as higher order ( non - convex ) terms due to the costs on angular momentum .since the dynamics are not further simplified , we have the benefit that all states and controls are included in the cost and none of them are left uncontrolled .given that the problem in eq  is non - convex , we can not expect to find a global optimum in general but need to find a local optimum starting from an initial guess . in our approachwe use a receding horizon technique .we start out by solving our problem for a horizon and obtain an optimal solution }^* ] , i.e. the robot is standing still and applying gravity compensation is already optimal .then , we formulate our problem for the interval  ] turns a cross - product into a matrix multiplication .the resolution of momentum rate to contact wrenches is then left to our whole - body controller as described in the next section . in this lqr designa quadratic performance cost is set once and then optimal gains are computed at each time step for the corresponding contact configuration . as we discussed in our previous work , in order to achieve compatible results with diagonal pd gain matrices , we had to design gains for different contact configurations , whereas the lqr design requires one performance cost and generated suitable gains automatically .in contrast to our previous work we linearize around desired trajectories , whereas in our robot experiments we used only two key configurations ; this may become limiting in more versatile tasks such as walking over a rough terrain .the trajectories generated with the model in eqs ( [ eq : momentum_dynamics_1]-[eq : momentum_dynamics_3 ] ) define the com , momentum and end effector forces of our humanoid . in order to track these trajectories on the full robot ,we need to generate joint torques accordingly and at the same time control the limb motion and guarantee that other constraints are obeyed , e.g. joint limits . for this time - local control problem ,we use inverse dynamics in qp cascades , which we applied successfully on the real robot in previous work  .it allows us to phrase feedback controllers and constraints as functions of joint accelerations , external generalized forces and joint torques .for instance , we can express a cartesian controller on the swing foot as an affine function of joint accelerations or a momentum controller as a function of reaction forces .the latter , for example , would be  {\times } & \mathbf{i}_{3\times 3 } & \ldots  \end{bmatrix }                             \boldsymbol{\lambda } +                              \begin{bmatrix }                              m\mathbf{g }                              \\\mathbf{0 }                             \end{bmatrix }  = \dot{\mathbf{h}}_{ref}.\end{aligned}\ ] ]  a reference momentum rate can be chosen from a lqr design as demonstrated in our previous work  and extended in sec [ sec : momentum_lqr ] . given a set of controllers and tasks, we find torques that satisfy the dynamics equations of the full robot and at the same time generate the desired task feedback as good as possible .tasks , however , may conflict , for instance moving the com may require moving the swing leg and vice versa . in these cases ,qp cascade allow for two types of trade - offs ; we can either weigh tasks against each other or we can prioritize them strictly . especially when it comes to trade - offs between constraints , tasks of interest and redundancy resolution , prioritization can facilitate successful control design .this section describes our simulation results of the proposed control framework .we use a model of our sarcos humanoid in the sl simulation environment .contacts are simulated with a penalty method and stiff springs .all experiments are performed on a 2.7 ghz intel i7 processor with 16 gb ram .a task is generated where the robot is to walk on stepping stones that increase in height from one step to the other as visualized in fig [ fig : task_snapshots ] and shown in the attached video ] .the z - axis of the inertial frame points up and the robot walks along the y - axis to the front .two out of the four steps are tilted by . both the change in com height as well as the angled supports break the assumptions made in lipm models and require the consideration of two separate contact wrenches .we generate swing leg trajectories using cubic splines to parameterize the pose of the foot .the humanoid starts out in double support at rest .after the first 3 seconds the left foot breaks contact and moves to an angled support surface located to the front left - hand side of the robot at an increased height ( as shown in fig [ fig : task_snapshots ] ). then a contact switch occurs at every second , changing from single support to double support or vice versa .the second step is again a support surface angled inwards to the robot and located to the front right - hand side .finally , the robot takes two steps onto a horizontal plateau located slightly below knee height .+ the planner is initialized with a naive idea of the robot motion where we simply keep the base at a certain height above the feet . after integrating the desired swing foot positions together with the base height using inverse kinematics ( as described in sec [ seq : optimization_procedure ] ), we obtain resulting trajectories that take into account the angular momentum required to swing the legs from one stepping stone to the other . the inverse kinematics solution , which is physically not consistent , is then adjusted in the trajectory optimization step to be admissible with respect to constrained forces and cops .none of the force constraints were notably violated . as can be seen in fig [ fig : momentum_plan ]the inverse kinematics - generated com trajectory is modified significantly in the lateral direction and as a consequence the angular momentum in the y - direction is modified as well . after a second iteration of inverse kinematics integration and trajectory optimization , the results have sufficiently converged and we stop .polynomials of order 3 are chosen and initialized with zero .the planning process took 4 min and converged after only two iterations .this allows us to generate a complex motion rather quickly compared to motion planners that are based on more extensive models of the robot .the numerical optimization problem in eq [ eq : optimization_problem ] is solved with snopt  , a sequential quadratic programming method .+    next , we construct a hierarchy of feedback controllers and constraints in order to realize the momentum profile on the full humanoid . at the highest prioritywe express the physical model of the full robot to obtain physically - consistent torques .this is followed by force and joint limits together with contact constraints . in the third prioritywe control the swing leg motion and the momentum and add a posture pd control with a relatively - low weight . in the priorities below , we regularize end effector forces and stabilize the base orientation .a summary of the task setup is given in tab [ tab : task_hierarchy ] .p.1p.27x * rank * & * nr . of eq / ineq constraints* & * constraint / task * + 1 & eq & newton euler equation + 2 & eq & contact constraints + & ineq & center of presure + & ineq & friction cone + & ineq & joint acceleration limits + 3 & eq & lqr momentum control + & eq & cartesian swing foot control + & eq & pd control on posture + 4 & eq & contact forces control + 5 & eq & base link orientation +  the momentum is controlled with feedback gains designed as described in sec [ sec : momentum_lqr ] , where our performance cost is fixed for the whole run .we penalize state errors by , forces by and torques by .gains are generated over a 2sec horizon with a granularity of 200 time steps .a sequence of gains is recomputed every 10ms .+ the robot was able to traverse the terrain successfully .the momentum trajectories , which were both dynamically consistent as well as compatible with the robots limb motions , could be tracked well as shown in figs ( [ fig : com_tracking]- [ fig : mom_tracking ] ) .torques were generated that are in the bounds of the physical robot s torque limits . in the beginning of the taskthe planned com height forces the knees to stretch , which prevents the robot from using its knees to lift the com further , but instead it lifts the arms up rapidly .this can be avoided by adding a box constraint on the com in the optimization problem in order to consider kinematic limitations .+ the lqr gain matrices have non - zero off - diagonal values as expected ( cf .fig [ fig : momentum_gain ] ) .for instance , we can see that angular momentum is generated in order to correct for errors in com and linear momentum , which would not be possible with diagonal pd gains .in fact , we tried to track the planned motion using diagonal pd gains .we started with values similar to the diagonal of the lqr gain , but we could not find parameters that were stable throughout several contact situations as was the case for the lqr gains .   and indicate contact switches .gain profiles change significantly over time and contact configurations .the gains and momentum dynamics are discontinuous at contact switches .nevertheless , discontinuities in joint torques were negligible . ]increasing the terrain difficulty was mainly problematic due to kinematic limits because our naive swing leg trajectories required to keep the heel on the ground during the whole support phase , thus limiting the stepping height .further , we noted problems with the simulator s contact model , which caused sporadic spikes in the force profile when we applied strong forces on the ground .+ overall , a complex task could be planned quickly including a non - trivial angular momentum profile that respected the end effector motion .the proposed feedback control law on the momentum showed good performance when it was embedded in an inverse dynamics task hierarchy .the momentum model in eqs ( [ eq : momentum_dynamics_1]-[eq : momentum_dynamics_3 ] ) is often simplified further in order to obtain linear dynamics , which then leads to computationally more efficient algorithms . however , turning multiplicative terms between variables in eq  into linear terms requires potentially restrictive assumptions .e.g. in the lipm the com height is assumed to be constant , there is one effective , which lies in one horizontal plane , and the force is required to act along . +since a constant com height may be limiting for tasks that require vertical movement , the authors in  allow a predefined ( not constant ) . substituting this assumption into eq , and assuming constant , turns the horizontal angular momentum into a linear function of forces and torques .assuming in addition that the nonlinear vertical angular momentum can be neglected , the authors end up again with linear dynamics .if we restrict our dynamics model further into a lipm and use discretized dynamics ( instead of polynomial trajectories ) we can recover the approach of . however , this can be restrictive if we want to step on various slopes or when a non - zero angular momentum is required , e.g. to consider limb motion .   in our current work , we focused on the capabilities of the multi - contact dynamics model to generate dynamic motions on uneven terrain and we showed that these behaviors can be controlled on a full humanoid robot .however , the goal of separating the control process into a predictive control generation on a lower dimensional model and time - local control on the full dynamics has the potential for a fast implementation in a mpc fashion . in our experiments we saw potential drawbacks in our numerical optimization in eq .increasing the order of polynomials leads to slower convergence rates , whereas reducing the polynomial dimensions too far may lead to poor flexibility of the trajectory representation .this may be explained by the discrepancy of basis functions evaluated close to 0 and close to t leading to poorly conditioned problem .piecewise constant trajectories may overcome this problem .another point for improvement of the numerical procedure is the formulation of the optimal control problem . in the current formulation , the objective function in eq  has terms up to 4th order ( squared norm of cross products ) .our numerical solver is based on approximations up to second order , which may limit the region in which approximations are valid .it is possible ( but out of the scope of this paper ) to rewrite the problem into a ( non - convex ) quadratically constrained quadratic program where the objective function as well as the constraints are quadratic , leading to a better approximation in second order methods and potentially improving convergence .pushing the implementation towards an online control algorithm is part of our future work .we presented an approach to control contact forces and momentum for humanoid robots .com and momentum profiles were obtained in an optimal control framework together with admissible contact forces .feedback gains are generated from a lqr design , which generates time and contact - configuration dependent gains from a single performance cost .the resulting controller is embedded in an inverse dynamics - based whole - body controller together with other limb controllers and constraints .we demonstrated the control framework on a simulation of the sarcos humanoid traversing rough terrain .physically admissible momentum and force trajectories could be found relatively quickly and were tracked well during the task execution . in future work , we will implement the discussed speed - up and push the optimal controller to run online .

Abstract: simplified models of the dynamics such as the linear inverted pendulum model ( lipm ) have proven to perform well for biped walking on flat ground . however , for more complex tasks the assumptions of these models can become limiting . for example , the lipm does not allow for the control of contact forces independently , is limited to co - planar contacts and assumes that the angular momentum is zero . in this paper , we propose to use the full momentum equations of a humanoid robot in a trajectory optimization framework to plan its center of mass , linear and angular momentum trajectories . the model also allows for planning desired contact forces for each end - effector in arbitrary contact locations . we extend our previous results on lqr design for momentum control by computing the ( linearized ) optimal momentum feedback law in a receding horizon fashion . the resulting desired momentum and the associated feedback law are then used in a hierarchical whole body control approach . simulation experiments show that the approach is computationally fast and is able to generate plans for locomotion on complex terrains while demonstrating good tracking performance for the full humanoid control .


, the exploitation of multiple - input multiple - output ( mimo ) transmission techniques has become of great interest to support the massive demand for high - speed data services in communication networks .although mimo techniques have been extensively investigated within the wireless ( radio ) scenario , they have been only recently analyzed in the context of power line communications ( plc ) . in practice ,when an electric power distribution network deploys multiple conductors , plc can exploit mimo transmission by injecting and receiving signals among the multiple wires connecting the nodes .this is the case , for instance , of in - home low voltage ( lv ) power distribution networks that deploy three conductors : the phase , the neutral and the protective earth wires .the recent home plug av2 system and the itu g.hn standard have adopted mimo transmission together with a bandwidth extension ( from 2 to 86 mhz ) , precoding , parameters adaptation and a number of advanced techniques at the mac layer that enable high data rates in excess of 1 gbps of peak rate .this impressive performance level stimulates the interest in plc solutions , their evolution and their application from in - home networking to smart grid applications .  in order to develop algorithms and assess performance, it is very important to characterize and model the mimo plc channel .single - input single - output ( siso ) plc channel modeling is a rather more established area of research than mimo plc channel modeling ( * ? ? ?* ch . 2 ) .essentially , two different approaches can be followed : a deterministic approach , which assumes a specific network topology , and a statistical approach , where some variability is introduced to produce random channel realizations .the statistical method is more suitable for a comprehensive model since it statistically emulates the variability encountered within a certain scenario .contrariwise , a deterministic method well represents a specific configuration , but with a poor overall representativeness .both these approaches can then be specialized into a bottom - up or a top - down methodology .the bottom - up methodology has a tight connection with the physical propagation effects , enabling a faithful emulation of an individual channel frequency response ( cfr ) by exploiting the transmission line theory .its application requires a full network topology knowledge in terms of cable characteristics , layout , connection lengths and loads , resulting into a considerable computational effort .an example of a bottom - up deterministic siso plc channel model can be found in . a statistical bottom - up channel modelcan then be obtained by using a random topology model as shown in .contrariwise , the top - down methodology uses an analytic function , whose parameters are chosen to fit an actual channel measurement in an empirical way .the top - down approach facilitates the statistical extension with respect to ( w.r.t . ) the bottom - up approach .one of the first and well - known top - down approaches was discussed in , where the cfr was modeled taking into account the multipath signal propagation and the cable losses .later , it was shown that , by introducing variability in some parameters of the fitting function , the model in can be statistically extended .although much effort has been spent in providing faithful siso models , summarized in ( * ? ? ?2 ) , only few proposals have been made to define a general mimo plc channel model so far .this is also due to the fact that mimo plc channel measurement and characterization has started only recently .the most comprehensive in - home measurement campaign has been conducted by the european telecommunication standards institute ( etsi ) specialist task force 410 ( stf-410 ) in 2012 .a first work that extends the siso bottom - up model in to the mimo case by using multi - conductor transmission line theory based on a large measurement database was described in .very recent improvements have been proposed in .conversely , a first attempt to model the mimo plc channel impulse response via a top - down approach was detailed in , while another frequency domain model was proposed in ( * ? ? ?* ch . 5 ) andrecently improved in .these mimo models start from an analytic fitting function similar to that proposed in and then they render some parameters random similarly to the siso approach in .the main contribution of this paper is to show that a synthetic top - down approach , based on a pure phenomenological philosophy , can be followed to provide a simple model that is statistically representative of a real wideband mimo plc environment . due to the model baseline methodology, it can be potentially applied to other communication scenarios , since it is not connected to any physical assumption or propagation phenomena .furthermore , its underlying concept significantly differs from what has been already proposed in the literature . indeed , differently from the typically more complex transmission line bottom - up methods , e.g. , or the conventional top - down approaches that use a fitting function related to some physical assumptions , e.g. , the proposed statistical model is purely phenomenological and directly synthesizes the mimo cfr and its statistical properties . for this reason ,an initial characterization part is fundamental to first assess the mimo channel properties and then show how to reproduce them by using a reduced set of parameters .this is done by considering all the measurements of a specific database , obtained from the in - home measurement campaign in the band 1.8100 mhz , carried out by the etsi stf-410 .the model is named `` synthetic '' since this terminology encompasses two meanings , namely `` artificial '' , i.e. obtained from synthesis without physical assumptions ( phenomenological ) , and `` compact '' , since the model adopts a small number of parameters .the theoretical formulation resembles the top - down strategy that we presented in ( * ? ? ?3.1 ) for the in - home siso channel , but it is extended to the far more complex mimo scenario . moreover , the proposed model adopts a completely new strategy to take into account both the frequency and the mimo ports ( or modes ) correlation , represented by the mimo statistical covariance matrix .in particular , the proposed synthetic model implementation strategy is able to reconstruct the overall mimo matrix by using a significantly reduced number of parameters .conversely , the method based on the mimo extension of , despite its theoretical simplicity , must have available all the matrix coefficients and adopts a different and more complex strategy for the phase generation .this is why it is referred to as not - fully synthetic in the rest of the paper .the model philosophy is first introduced in section [ sec : modelbasics ] .the experimental data is analyzed in section [ sec : exprevid ] , enabling the derivation of the channel properties , such as the statistical behavior and the exhibited statistical correlation among the frequencies and between the mimo modes .then , the synthetic modeling strategy , which uses a reduced set of parameters to reconstruct the mimo plc channel characteristics , is detailed and justified by the experimental evidence in section [ sec : modeldescript ] .this description guarantees the methodology replicability .the results are validated in section [ sec : results ] by showing that the proposed statistical model is capable to replicate the measured channels in terms of statistical metrics , i.e. average channel gain ( acg ) , root - mean - square delay spread ( rms - ds ) , and coherence bandwidth ( cb ) , as well as in terms of mimo modes correlation measured by the condition number  .the channel capacity distribution ( or maximum achievable rate according to shannon s definition ) is also considered as a further benchmarking tool .all the mentioned quantities have been computed considering the overall measurements database . a comparison with the not - fully synthetic procedure , as well as with the results provided by the synthetic strategy , when considering a reduced set of mimo modes ,is also performed within this section .finally , the conclusions follow in section [ sec : conc ] .most of the proposed top - down channel models , e.g. , are based on an analytic function that represents some physical phenomena , such as the multipath propagation due to the several branches and load mismatches that cause multiple signal reflections in a plc network .similarly , the model presented in is based on the theoretical formulation provided in .therefore , some physical assumptions are made , resulting into a hybrid top - down and bottom - up model .contrariwise , the basic idea underlying the proposed model is fully phenomenological and focuses on a pure top - down procedure .to start , the generic siso cfr at frequency , namely , is a complex number that can be expressed in amplitude and phase , i.e. . in order to simplify the discussion ,the cfr in db scale , namely , is considered , yielding where the real part is the amplitude in db scale , while the imaginary part is still the phase in radians multiplied by the constant value .  in practice ,when considering a plc scenario , the quantity changes depending on the considered network topology and the specific pair of nodes , e.g. , outlets in a home network .thus , from a statistical perspective , can be modeled as a random variable ( rv ) , while the ensemble of all the available frequency samples can be grouped into a channel vector of rvs .similarly , a general mimo channel can be represented by a three - dimensional ( 3d ) matrix of rvs with dimensions , where , are the number of receiving and transmitting ports , respectively .it follows that a mimo plc channel is completely described by an equivalent 3d matrix of rvs characterized by certain statistics and relationships .therefore , a typical communication scenario can be emulated by generating a set of simulated channels having a statistical behavior and , in particular , a frequency and mimo modes correlation that are equivalent to those derived from experimental data .hence , to provide an effective channel emulator , real data must be analyzed to propose a replicable modeling approach , possibly with a reduced set of parameters .this is the subject of the next sections .[ ssec : thform ] the proposed model considers the marginal distribution of the cfr , i.e. the distribution function of the amplitude and phase , as well as the their statistical correlation between the different mimo modes and among the different frequencies .however , as it will be clarified in section [ sec : exprevid ] , the cfr amplitude and phase can be approximated as independent .thus , their marginal distribution and statistical auto - correlation are sufficient to provide an order two statistics of the channel .  in order to obtain such statistics from real data and to simplify the modeling procedure ,the 3d matrix is reshaped into a vector of size .thus , the first elements concern all the frequency samples of the first receiving and transmitting port .the second elements correspond to the frequencies of the second receiving port and the first transmitting port , and so on for all the receiving ports .afterwards , all the other transmitting modes are considered accordingly .hence , the 3d matrix is reshaped as ^t,\ ] ] where denotes the transpose operator , while is the row vector of rvs associated to the frequency samples vector for the cfr between the -th transmitting and the -th receiving ports .the statistical covariance matrix , of size , of the reshaped vector is given by ,\ ] ] where is the complex conjugate operator and  ] is the corresponding average value .moreover , and are the row and column indexes of that , according to the previously described reshaping method , are function of and , respectively. in particular , the row index is given by , and the same applies to the column index for the values , and .  in the following, the normalized covariance matrix is also considered .it is computed as the covariance matrix in ( [ eq : covmtx ] ) normalized by the product of the related rvs standard deviation , i.e. and .each element of ranges from 0 ( no correlation ) to 1 ( fully correlated ) and thus it is equivalent to the pearson correlation coefficient .hence , each coefficient of the matrix represents the correlation degree among the corresponding frequency samples and between the different transmitting and receiving ports ( mimo modes correlation ) .  by exploiting ( [ eq : cfrdb ] ) and( [ eq : covmtx ] ) , the covariance matrix , of size , of the cfr in db can be obtained as ,\ ] ] where is the covariance matrix between the amplitude ( in db scale ) and the phase , while and are the auto - covariance matrices of the amplitude in db and the phase , respectively .obviously , in this case , the amplitude and the phase are vectors reshaped as in ( [ eq : reshmtx ] ) , equivalently to the cfr vector .as it can be noted , the real part of depends only on the amplitude and phase auto - covariance matrices , while the imaginary part is a function of the mutual covariance matrix , as discussed in .the knowledge of the covariance matrices , , and provide information about the joint distribution function of the cfr amplitude and phase , which gives a second order characterization of the channel in a certain frequency range , as detailed in . hence , according to the above formulation , only the amplitude and phase properties need to be assessed and reproduced to obtain the desired behavior , providing an extremely compact procedure , named synthetic modeling strategy .in this section , a database of mimo plc in - home experimental cfrs is considered to obtain the statistics needed by the model , whose basic elements were described in the previous section .the mimo in - home channel scenario has relevant application value .the measurements are briefly discussed first , assessing the main statistical properties and the relationships required for the generation process .this analysis highlights several properties that enable further simplifications to the general model , as detailed in section [ sec : modeldescript ] .the database of experimental channels collected during the measurement campaign across europe by the etsi stf-410 is herein evaluated .the database consists of 353 mimo cfrs with 1588 samples in the extended 1.8100 mhz frequency range measured via a vector network analyzer .the in - home lv power distribution networks herein considered deploy three wires , two for the power supply , namely the phase ( p ) and the neutral ( n ) , and one for protection in case of an insulation fault , i.e. the protective earth ( e ) .plc modems can instantiate mimo communication as follows . at the transmitter side a differential transmission can be implemented by signaling among a pair of wires . at the receiver side , instead, the signal can be observed between each wire and a reference plane in a star - style configuration . beyond the three standard star - style receiving modes , an additional mode , named common mode ( cm ) , can be extracted . due to kirchhoff s law ,only two out of three transmitting voltages are linearly independent .the third is a linear combination of the other two .thus , only two signals can be injected .the same thing applies at the receiver , i.e. , among the four acquired signals , only three are linearly independent ( see for details ) .hence , the mimo channel that includes the pn and pe transmitting ports and the p , n , and cm star - style receiving ports , is considered in the following .   from the analysis of the measurements in ,the following properties have been found .the amplitude of is log - normally distributed , hence is normally distributed , while the phase is uniformly distributed in , at any frequency and for each individual transmitting - receiving mode combination .this confirms the findings about siso channels in and in . in particular , the mean ( ) and the standard deviation ( ) for the best normal fit of the cfr amplitudes in db are reported in fig .[ fig : musigma ] as a function of frequency and for all the considered transmitter - receiver mode combinations in the 1.8100 mhz band .    .robust regression fit coefficients for the mean and standard deviation parameters of the amplitude in db scale . [ cols="^,^,^ " , ]   mimo case in the 1.8100 mhz band . ]the statistical metrics for the experimental and the simulated channels for both the mimo and siso channels are listed in table [ tab : avgvalsubset ] . a good agreement is observed in terms of and statistical metrics average value , with some differences only in terms of cb .this is mainly due to the approximation adopted for the amplitude covariance matrix anti - diagonal profile discussed in section [ ssec : ampgen ] .in addition , similar values are obtained also for the acg average standard deviation , while the rms - ds and cb experimental and simulated standard deviations are not so close . as discussed in section [ ssec:2x3mimo ] , this discrepancy is mainly related to the adopted simplifications .nevertheless , a good performance replicability is obtained , as confirmed by the capacity ccdf .the experimental and simulated channel capacity ccdf for both the schemes mimo and siso is shown in fig .[ fig : capccdfsubset ] .it can be noted as there is a good agreement between the performance achieved by the measurements and those obtained by the simulated channels generated according to the synthetic modeling strategy . in particular , an almost perfect matching is noticeable for the siso transmission scheme , while some deviations are found for the mimo case when considering high capacity values .modeling the plc channel is relevant to support the task of testing and developing communication solutions over power lines .a brief overview has been given concerning the main modeling approaches present in the literature , mostly limited to the siso scenario .in this paper , an extremely compact and effective top - down modeling method has been proposed for the mimo plc channel , which has been derived from a pure phenomenological approach .it is based on the synthesis of the second order channel statistics observed in a real data set .the model is compact and requires few parameters to simulate an in - home mimo plc channel in the band 1.8100 mhz , i.e. the appropriate cfr amplitude and phase statistics and their modes and frequency correlation .the results have validated the proposed strategy as a statistically representative simulation tool , despite its simplicity .  a comparison to the not - fully synthetic procedure proposed in ( * ? ? ?3.1.1 ) for the siso case , herein extended to the mimo case , has also been made .it has been shown that , despite the extremely reduced set of parameters required by the synthetic model w.r.t .the not - fully synthetic model , a good matching with the experimental channel statistics and channel capacity distribution is achieved .l. berger , a. schwager , p. pagani , and d. schneider , _ mimo power line communications : narrow and broadband standards , emc , and advanced processing _ , ser .devices , circuits , and systems.1em plus 0.5em minus 0.4emtaylor & francis , 2014 .l. yonge , j. abad , k. afkhamie , l. guerrieri , s. katar , h. lioe , p. pagani , r. riva , d. m. schneider , and a. schwager , `` an overview of the homeplug av2 technology , '' _ journal of electrical and computer engineering _ , vol .2013 , pp . 120 , 2013 .`` unified high - speed wireline - based home networking transceivers system architecture and physical layer specification , '' recommendation itu - t g.9960 , itu , december 2010 .[ online ] .available : https://www.itu.int/rec/t-rec-g.9960/en  l. lampe , a. m. tonello , and t. g. swart , _ power line communications : principles , standards and applications from multimedia to smart grid _ , 2nd ed.1em plus 0.5em minus 0.4emchichester , uk : john wiley & sons , 2015 .s. galli and t. c. banwell , `` a deterministic frequency - domain model for the indoor power line transfer function , '' _ ieee journal on selected areas in communications _ ,24 , no . 7 , pp .13041316 , july 2006 .a. m. tonello and f. versolatto , `` bottom - up statistical plc channel modeling part i : random topology model and efficient transfer function computation , '' _ ieee transactions on power delivery _ , vol .26 , no . 2 ,891898 , april 2011 .     , `` powerline telecommunications ( plt ) ; mimo plt ; part 3 : setup and statistical results of mimo plt channel and noise measurements , '' european telecommunication standardization institute , tech .rep . , 2012 .f. versolatto and a. m. tonello , `` an mtl theory approach for the simulation of mimo power - line communication channels , '' _ ieee transactions on power delivery _ , vol .26 , no . 3 , pp .17101717 , july 2011 .j. a. corchado , j. a. corts , f. j. caete , and l. dez , `` an mtl - based channel model for indoor broadband mimo power line communications , '' _ ieee journal on selected areas in communications _ , vol .34 , no . 7 , pp .20452055 , july 2016 .a. canova , n. benvenuto , and p. bisaglia , `` receivers for mimo - plc channels : throughput comparison , '' in _ proc .of ieee international symposium on power line communications and its applications ( isplc ) _ , march 2010 , pp . 114119 .p. pagani and a. schwager , `` a statistical model of the in - home mimo plc channel based on european field measurements , '' _ ieee journal on selected areas in communications _34 , no . 7 , pp . 20332044 , july 2016 .a. m. tonello , a. pittolo , and m. girotto , `` power line communications : understanding the channel for physical layer evolution based on filter bank modulation , '' _ ieice transactions on communications _ , vol .e97-b , no . 8 , pp . 14941503 , august 2014 .a. m. tonello , f. versolatto , and a. pittolo , `` in - home power line communication channel : statistical characterization , '' _ ieee transactions on communications _ , vol .62 , no . 6 , pp .20962106 , june 2014 .a. papoulis and s. u. pillai , _ probability , random variables , and stochastic processes _ , ser .mcgraw - hill electrical and electronic engineering series.1em plus 0.5em minus 0.4emmcgraw - hill , 2002 . [ online ] .available : https://books.google.ie/books?id=yywqaqaaiaaj  s. krusevac , p. rapajic , and r. a. kennedy , `` channel capacity estimation for mimo systems with correlated noise , '' in _ ieee global telecommunications conference ( globecom ) _ , vol . 5 , dec 2005 , pp . 28102816 .a. pittolo , a. m. tonello , and f. versolatto , `` performance of mimo plc in measured channels affected by correlated noise , '' in _ proc . of 18th ieee international symposium on power line communications and its applications ( isplc )_ , march 2014 , pp . 261265 .alberto pittolo received the bachelor of science degree in electrical engineering ( 2009 ) , the master of science degree ( 2012 ) in electrical and telecommunications engineering , with honors , and the phd ( 2015 ) in industrial and information engineering from the university of udine , udine , italy .his research interests and activities focus on the channel statistical characterization and on its numerical modeling , on the physical layer security investigation , and on the resource optimization in conjunction with the allocation algorithms , for both wireless and power line communications ( plc ) .andrea m. tonello ( m00-sm12 ) received the laurea degree ( summa cum laude ) and the phd in electronics and telecommunications from the university of padova , italy , in 1996 and in 2003 , respectively . from 1997 to 2002 , he was with bell labs - lucent technologies , whippany , nj , usa , first as a member of the technical staff .then , he was promoted to technical manager and appointed to managing director of the bell labs italy division . in 2003, he joined the university of udine , italy , where he became aggregate professor in 2005 and associate professor in 2014 .he also founded witikee , a spin - off company working on telecommunications for smart grids .currently , he is professor and chair of the embedded communication systems group at the university of klagenfurt , austria .tonello received several awards , including eight best paper awards , the bell labs recognition of excellence award ( 1999 ) , the distinguished visiting fellowship from the royal academy of engineering , u.k .( 2010 ) , the ieee vts distinguished lecturer award ( 2011 - 2015 ) , the italian full professor habilitation ( 2013 ) .he is the chair of the ieee communications society technical committee on power line communications .he serves as associate editor of ieee trans . on communications and ieee accesshe was the general chair of ieee isplc 2011 and ieee smartgridcomm 2014 , and tpc co - chair in a number of conferences .

Abstract: this paper proposes a synthetic statistical top - down mimo power line communications channel model based on a pure phenomenological approach . the basic idea consists of directly synthesizing the experimental channel statistical properties to obtain an extremely compact model that requires a small set of parameters . the model is derived from the analysis of the in - home mimo plc channel data set obtained by the etsi specialist task force 410 measurement campaign in the band 1.8100 mhz . the challenge of modeling the channel statistical correlation , exhibited among the frequencies and between the mimo modes , in compact form is tackled and it is shown that a small set of parameters can be used to reconstruct such a correlation behavior . the model is validated and compared to the measured channels , showing a good agreement in terms of average channel gain , root - mean - square delay spread , coherence bandwidth , and channel capacity distribution .   power line communications , plc , channel modeling , mimo , phenomenological approach , top - down methodology , statistical model .