transmission through noisy communication channels is a key problem of information theory .when the quantum nature of information carriers is taken into account , their interaction with the noisy environment is properly described in terms of quantum channels . the classical capacity ,i.e. , the maximal rate of errorless transmission of classical information via a quantum channel is one of the important figures of merit . in general , finding this quantity is a highly non - trivial optimization problem .nevertheless , this problem was recently solved for a wide class of quantum channels with a gaussian noise ( so - called gaussian quantum channels )  which are good models of common physical systems such as free space or fiber optics communications ( see , e.g. ,  ) .the gaussian quantum channels attracted much attention in the last decades  ; however , only a few results on their classical capacity were known  .one of the attempts to approach this problem was to restrict the search for optimal states and encodings by gaussian states and gaussian distributions  .the solution of such reduced optimization problem gives the so - called gaussian classical capacity  which was actively studied during the last years  .although the gaussian capacity is _ a priori _ a lower bound on the actual capacity this quantity is of great interest in practice , because the gaussian states are more accessible in experimental realizations .recently , the progress in finding the gaussian capacity became even more important because it was proved that the classical capacity of bosonic gaussian quantum channels with thermal noise is indeed achieved by gaussian ensembles of ( pure ) gaussian input states  .if input energy is sufficiently high , this result also leads to finding the classical capacity in the case of squeezed thermal noise  .in both cases the calculations were technically performed in terms of the gaussian capacity .however , the equivalence of the gaussian and classical capacity is not yet known for small input energies .the study of gaussian capacity in this region  provides a way to find a lower bound on the classical capacity with the hope that equivalence of this bound and the ( actual ) classical capacity will be eventually proven as well .in addition , the analysis of the properties of the gaussian capacity may give a further insight helping to find such a proof .in this work we focus on the gaussian classical capacity of single - mode gaussian quantum channels and calculate the one - shot ( or single - letter ) gaussian capacity of the fiducial channel  .recently , it was shown that this is equivalent to finding the gaussian capacity of an arbitrary single - mode gaussian channel  .this equivalence is based on the fact that any single - mode gaussian channel can be expressed as the fiducial channel preceded by a rotation and followed by a symplectic transformation ( in some cases only if a proper limit is also taken ) .we also go beyond the known results on the gaussian capacity and present the solution which is valid for all values of input energy .then , we analyze the obtained gaussian capacity as a function of the channel parameters . in particular , we find its rich behavior as a function of the noise squeezing .the present work generalizes previous results obtained for the single - mode lossy ( or attenuation )  and additive noise  gaussian channels .the paper is structured as follows . in section [ sec : chgauss ] , we briefly review gaussian quantum states and gaussian channels . in section [ sec : capgauss ], we introduce the single - mode fiducial gaussian channel and calculate its gaussian capacity . in this calculationwe use a _ _ frequency representation__-representation introduced in  for lossy channel . ] for the covariance matrices to get a better insight about the physical meaning of the solution . in section [ sec : chpara ] , the behavior of the capacity as a function of the channel parameters is analyzed .section [ sec : conclusions ] summarizes the results .quantum mechanics in continuous variables can be introduced as weyl calculus on the system s phase space  ( below we consider only single - mode case , i.e. , 1-d harmonic oscillator ) . in this representation a quantum stateis specified by its wigner function [ is a vector of canonical variables and  , which is a weyl symbol of the density operator acting on the hilbert space and belonging to is = i ] may be calculated straightforwardly : where thus , the von neumann entropy of a thermal state can be expressed as a function of the symplectic eigenvalue or the number of thermal photons .then , any cm of the form can be transformed by a symplectic ( squeezing ) transformation to the cm of a thermal state ( ) with the same number of thermal photons as the initial . noting that this transformation corresponds to a unitary transformation of the corresponding density operator which preserves the von neumann entropy, we recover a well known fact that the von neumann entropy of any single - mode gaussian state is a function of the number of thermal photons given by eq .the one - shot ( single - letter ) _ gaussian capacity _ is the maximal transmission rate of classical information when we are restricted to the use of gaussian input states and gaussian encodings such that different uses of the channel are not correlated .the energy constraint is always imposed on the input state because the unconstrained capacity is trivially infinite . the recently proven additivity conjecture together with the proof of the minimum output entropy conjecture ( see refs . ) assert that the gaussian capacity is the classical capacity for phase - insensitive gaussian channels ( in case of squeezed gaussian noise this holds if the input energy exceeds a certain threshold ) .however , in the absence of the full proof embracing all gaussian channels and the entire input energy regime we distinguish the gaussian capacity from the classical capacity .the former reads  : =2{\bar{n}}+1\}}\chi_g,\\     \chi_g & = g\left({\bar{m}_{\rm out}}\right)-g\left({m_{\rm out}}\right ) .\end{split}\ ] ] where is the cm of the input ( letter ) state and is the cm of the average modulated input state with photons .note that we state in eq .the explicit dependence of the gaussian capacity on .the average ( modulated ) input state with cm is realized by a gaussian distribution of the displacements of pure gaussian states with the same cm ( see , for example , ) .that is why entropies which enter the general formula for the capacity are expressed in terms of functions of thermal photons and corresponding to the _ output _ and _ modulated output states _ with cm and given by , \quad { \bar{{\bm{v}}}_{\rm out}}={\phi}[{\bar{{\bm{v}}}_{\rm in}}].\ ] ] the input energy constraint is expressed in terms of the mean number of photons in the modulated input state .we showed in  that the problem of calculating the gaussian capacity of an arbitrary gaussian channel can be reduced to calculating the capacity of a _ fiducial _ gaussian channel , which is defined in our parametrization by where parameter is a function of , namely , note that and correspond to the perfect transmission channel .otherwise , corresponds to a lossy channel , a classical additive noise channel , an amplification channel , and a phase - conjugating channel .although in the case of the classical additive noise channel the cm of the noise defined in strictly speaking does not correspond to a valid `` state '' of the environment , we will loosely apply the term `` squeezing '' parameter to for all channels .the solution of the optimization problem has already been obtained and intensively discussed for the lossy channel and classical additive noise channel ( see  and  , respectively ) . in the present paperwe generalize those results to the most general single - mode gaussian channel .  in eq .one has to maximize over the two covariance matrices and with the energy constraint stated in eq . , andthe constraint that the input state is pure , i.e. we can straightforwardly use theorem 1 of  ( since the resulting equations are of the same form ) to conclude that the optimal covariance matrices and are diagonal in the same basis as the _ effective noise _ cm .therefore , in the following any covariance matrix will be considered to be diagonal . as a consequence ,the cm of the optimal output state and optimal modulated output state read where now we present the solution to the optimization problem stated in eq . .first , we derive a solution that will only be valid above an input energy threshold , and secondly , we present the solution for input energies below this threshold .we start from an obvious bound on the gaussian capacity , which follows from eq .and the constraints discussed above =2{\bar{n}}+1\ } } g({\bar{m}_{\rm out } } )  - \min_{\{{{\bm{v}}_{\rm in}}:\det{{\bm{v}}_{\rm in}}=1/4\ } } g({m_{\rm out}}).\ ] ] we would obtain the gaussian capacity if we presented and which saturate this bound .this idea was used , for example , in . however , by considering implications of the energy constraint in , which should respect , we will find the limitations of this method .let us start by finding the maximum of the first term of eq .using the fact that the two terms are independent .since is a monotonously increasing function it is sufficient to maximize its argument in order maximize its value .we can express the argument in the first term of in terms of the mean number of photons of the modulated output state , i.e. -1\right ) .\end{split}\ ] ] then , since is fixed for a given number of input photons and channel parameters , , and ( see eq . ) the only free parameter in is the frequency of the modulated output state .its optimal value , which maximizes in eq . , is easily found : which implies that this corresponds to the well - known fact that given a finite energy the thermal state maximizes the von neumann entropy .next we find the minimum of the second term on the right hand side of eq . .in the same way as we maximized the first term by maximizing the argument , it is enough here to minimize the argument .the number of thermal photons at the output can be expressed in terms of the input frequency from definitions and , where is the only free parameter .then it is straightforward to find that the value of is minimized when the squeezing parameter of the input state is equal to the one of the noise in terms of frequencies one may say that the input and environment modes are `` in resonance '' .this also means that the optimal input state should have the same squeezing as the state of the environment .together with the uniform distribution of energy between the quadratures of the modulated output ( thermal ) state this is also a single - mode form of _ quantum water - filling _ ; a type of solution found also in multimode problems , which corresponds to equal variances of all quadratures of all modes , .using eq .we can express which corresponds to this solution in a simple form together with eqs .- this implies furthermore , that the input and the output state are `` in resonance '' as well , i.e.   the cm of the modulated input state can be expressed from the obtained results using the equation which directly follows from eq . .then the number of thermal photons and the frequency are obtained following definitions and from the cm and the input energy constraint in the form after several steps we obtain and   here we should make an important note that the modulated state with cm is obtained by averaging over displacements of the input letter state with cm and therefore , the right hand side of eq.  should be a positive definite matrix .this requirement together with eqs .  andcan be jointly satisfied by some and only if the input energy is above the following threshold : - \frac{1}{2}.\ ] ] hence , only for , the covariance matrices and found above determine the input letter state and modulated average input state which saturate the upper bound on the gaussian capacity , and therefore realize the gaussian capacity .this shows the limitation of the attempt to minimize and maximize terms in eq .  independently .  using the results obtained above we can express in a closed form the gaussian capacity of the fiducial channel as a function of the channel parameters and the input energy constraint : inserting into eq .the optimal values of and where is given by one obtains -1\right)\right)\\     & -g\left(|\tau|\frac{1}{2}+y-\frac{1}{2}\right ) , \ ; { \bar{n}}\ge { \bar{n}_{\rm thr}}.   \end{split}\ ] ] from eqs .and we can easily recover the well - known solution for thermal noise ( ) , which implies and meaning that the optimal encoding corresponds to a displacement of coherent input states for so - called _ thermal _ or _ phase - insensitive _ channels ( see ) .thanks to the recently proven minimum output entropy conjecture eq .provides not only the gaussian capacity but the classical capacity for an arbitrary single - mode gaussian channel as proven in , i.e.    now we derive the solution for the case when the input energy is not enough to satisfy the water - filling solution .up to now a solution for this case is known only for the lossy and additive noise channels .  by definition of condition implies that eqs .  and could only be satisfied if ( at least ) one of the diagonal terms of the difference was negative .however , this would be unphysical because is the cm of a state which is a mixture of states with cm .this means that the formal maximum of eq .lays outside the physically admissible region of optimization parameters and therefore the real ( physically admissible ) maximum lays in fact at the boundary of this region .this boundary is formed by the modulated input cm with ( at least ) one diagonal term being equal to the corresponding diagonal term of the non - modulated input cm .this implies that the optimal encoding only uses one of the two quadratures for information transmission ( modulating neither of them is clearly not optimal ) . without loss of generalitywe assume that the -quadrature of the noise is squeezed and therefore .therefore , we chose the ( more noisy ) -quadrature to be no longer modulated: and argue below why this choice is optimal .  since below the input energy threshold , there is no valid and which saturate independently the minimum and maximum of , we have to maximize the whole expression on the right hand side of eq . .therefor we use in the following the method of lagrange multipliers .  note that for the case the optimization problem can be formulated in terms of the three parameters , and satisfying the input energy constraint  [ where furthermore , only pure input states are considered , see eq . ] .  for the case we can use the new constraint given by eq .  to express in terms of and , which together with eq .leads to together with the input energy constraint the number of independent variables would be reduced to one . however , we find it more instructive to use the energy constraint in the lagrangian with two independent variables. we shall maximize eq .with respect to the pair of variables , .in addition , we will replace the input energy constraint by the output energy constraint which is equivalent to the original one for fixed channel parameters as one can see in eq . .then , our lagrangian is given by - { \bar{n}_{\rm out}}- \frac{1}{2}\right).\nonumber\end{aligned}\ ] ] the choice of the lagrange multiplier will be justified below .an extremum of the lagrangian fulfills where the gradient reads before taking the derivatives , recall the condition , which implies eq .  and therefore , can be expressed using the definition .thus , we find \frac{|\tau|}{4{\omega_{\rm in}}},\label{eq : dldfreqin}\\   \frac{\partial { \mathcal l}}{\partial { \bar{\omega}_{\rm out } } } & = & \left({\bar{m}_{\rm out}}+\frac{1}{2}\right)\left(g'({\bar{m}_{\rm out}}){\bar{\omega}_{\rm out}}^{-1 } - \frac{{\bar{\beta}_{\rm out}}}{\ln 2}\right),\label{eq : dldfreqoutmod}\end{aligned}\ ] ] where the first derivative of is and eq .was simplified using eq . .by equalizing the right hand side of eq .  to zero we recover the bose - einstein statistics for the amount of _ thermal photons _ in the modulated output state which means that the multiplier can be regarded as an _ inverse temperature _ of the state .note that eq .was already derived previously in by several authors , and in particular , for the lossy channel with phase sensitive noise in  .the physical meaning of the lagrange multiplier as an inverse temperature ( intensive parameter ) becomes more meaningful in the case of an `` ensemble '' , which in our case corresponds to a set of parallel single - mode channels with a priori different noises .we use the notion of temperature for the non - modulated output as well and in order to have unified notations we define the function by inverting eq . .in this way we obtain then , using the definition we simplify eq .  and thus , obtain an equation relating the non - modulated input frequency with the modulated and non - modulated output frequencies and `` temperatures '' . this equation not only fully covers the interval , where it provides solution to the optimization problem , but is valid for arbitrary .  indeed , we confirm that for eq.  is satisfied by the solution which we obtained in eqs .  and ,i.e. and the `` resonance '' between the input and non - modulated output state , i.e. .recall that the latter is equivalent to the `` resonance '' between the input and the environment , i.e. .  the solution to eq .gives the functional dependence of the optimal input frequency on the channels parameters and the amount of photons at the input : ( , , , ) .then in its turn determines in .following eq .  and taking into account eq .we obtain where in the second factor of we also took into account the energy constraint in order to express both and as functions of only one variable .finally , the gaussian capacity of a gaussian channel is found as where is expressed according to with and given by and is a solution of eq .  for given channel parameters .observing that eq .is transcendental due to the logarithm in the definition of we conclude that one can not obtain its solution in a closed analytic form below for . in order to solve the equation numericallywe need to use the expressions of , , and as functions of , the channel parameters , and the input energy constraint : we use this solution in order to discuss the dependencies of the gaussian capacity on the channel parameters in detail in the next sections .  before going to this discussionwe argue now why the constraint eq .is optimal for .this statement was already proven in our previous work on the classical additive noise channel ( ) with phase sensitive noise for the whole domain  . since the transcendental equation , up to a change of variables , is identical to the corresponding transcendental equation in  the optimality of eq .is proven by lemma 1 of  .now we are ready to analyze the capacity of the fiducial channel as function of the channel parameters .we first discuss the dependency of the gaussian capacity on the transmissivity / gain , when the other channel parameters parameters , , and are kept constant .we emphasize here that we keep the amount of thermal photons in the environment constant and not , so that the latter becomes a function of and according to .let us consider first the limits . both limiting cases concern only the amplification channel and phase - conjugating channel , hence since [ defined in eq .] depends on and we first study their behavior . in the limit of large , both arguments of , namely , and diverge , which follows straightforwardly from eqs . - .since =1/\ln 2 ] ) and amplification channel ( ) using lemma [ lempipeline ] ( see appendix [ ap : concatenated ] ) . the classical capacity ( as well as the gaussian capacity )fulfills the pipelining property  then , using lemma [ lempipeline ] for ( or  ] ) we obtain where , ( ) , the effective gain parameter is and the effective noise is characterized by .amplification channels satisfy .an amplification channel with a higher gain can always be decomposed into an amplification channel with gain followed by another amplification channel with gain . as we see above , due to the pipelining property the classical capacity ( and the gaussian capacity ) of the channel with gain not have increased .similar relations for the lossy channel , for which the effective transmissivity satisfies ,  ] .this implies that the capacity can not have increased by increasing .the highest ( gaussian ) capacity is given by the case , i.e. the perfect transmission channel , for which . since for increasing not increase the gaussian capacity , we proved that monotonically tends from above to its limiting value stated in eqs .  and .thus , where and are functions of , and according to and the latter is kept constant .equivalently , the lowest capacity is given by the case , i.e. the zero - transmission channel . since for decreasing can not decrease , we proved that monotonically tends from below to its limiting values stated in eqs .  and .we illustrate the limiting behavior of vs. for different values of squeezing and the same in fig .[ fig : capvstau ] .one confirms the monotonic decrease of for and .in addition , as argued above , on observes that indeed the limit is reached from above , whereas the limit is reached from below ( with decreasing ) .( right axis ) and frequencies ( left axis ) of the input and output states , modulated and non - modulated , vs. for different channels given by difference values of , namely , - upper solid ( red ) curve, - upper dashed ( blue ) curve , - lower densely - dashed ( green ) curve, - dashed - dotted ( black ) curve , - dotted ( black ) curve , right axis, - dashed vertical line.the figures from top to bottom correspond to : phase - conjugating channel , , , , , lossy channel , , , , , additive noise channel , , , , and amplifications channel , , , . for the lossy channel ( ) only ( dashed blue ) and ( dotted black )are plotted ., title="fig : " ]  ( right axis ) and frequencies ( left axis ) of the input and output states , modulated and non - modulated , vs. for different channels given by difference values of , namely , - upper solid ( red ) curve, - upper dashed ( blue ) curve , - lower densely - dashed ( green ) curve, - dashed - dotted ( black ) curve , - dotted ( black ) curve , right axis, - dashed vertical line.the figures from top to bottom correspond to : phase - conjugating channel , , , , , lossy channel , , , , , additive noise channel , , , , and amplifications channel , , , . for the lossy channel ( )only ( dashed blue ) and ( dotted black ) are plotted . , title="fig : " ]  ( right axis ) and frequencies ( left axis ) of the input and output states , modulated and non - modulated , vs. for different channels given by difference values of , namely , - upper solid ( red ) curve, - upper dashed ( blue ) curve , - lower densely - dashed ( green ) curve, - dashed - dotted ( black ) curve , - dotted ( black ) curve , right axis, - dashed vertical line.the figures from top to bottom correspond to : phase - conjugating channel , , , , , lossy channel , , , , , additive noise channel , , , , and amplifications channel , , , . for the lossy channel ( )only ( dashed blue ) and ( dotted black ) are plotted . , title="fig : " ]  ( right axis ) and frequencies ( left axis ) of the input and output states , modulated and non - modulated , vs. for different channels given by difference values of , namely , - upper solid ( red ) curve, - upper dashed ( blue ) curve , - lower densely - dashed ( green ) curve, - dashed - dotted ( black ) curve , - dotted ( black ) curve , right axis, - dashed vertical line.the figures from top to bottom correspond to : phase - conjugating channel , , , , , lossy channel , , , , , additive noise channel , , , , and amplifications channel , , , .for the lossy channel ( ) only ( dashed blue ) and ( dotted black ) are plotted . , title="fig : " ]   now we analyze the behavior of the gaussian capacity as a function of the noise squeezing , while keeping the other channel parameters , , and constant .note that in this case both and are constant because does not change .we first investigate in the limit of infinite squeezing .since ( as discussed above ) the only arguments of are and [ see eq . ] we first study their behavior in this limit . from eqs . , , and it follows that   this implies that the difference of -functions can again be replaced by a difference of logarithms , i.e. where at the last step we used eq . .now , as in the case , it is sufficient to maximize the argument of the logarithm in order to obtain the gaussian capacity in this limit .the argument is maximized by input states with squeezing which leads to interestingly , eq .  together with eq .imply that the corresponding optimal modulated input state is a thermal state , i.e. note that the limit stated in eq .was already derived in  for the phase - sensitive lossy channel and here we confirm that it is valid for an arbitrary single - mode gaussian channel as well .  this result can be interpreted as follows : in the limit of infinite squeezing of the noise , , one of the noise quadratures vanishes ( here : the -quadrature ) and the other becomes infinite . then , only one degree of freedom is available for information transmission . at the same time it no longer suffers from the noise induced by the channel but only from the quantum noise induced by the input state itself .we observe that the one - shot gaussian capacity then coincides with the doubled shannon capacity of a classical channel with signal - to - noise ratio . the limiting value stated in eq .is confirmed in fig .[ fig : capvstau ] where the gaussian capacity of the fiducial channels is plotted as a function of for different .let us analyze next how the gaussian capacity depends on the environment frequency in the whole interval .examples of these curves for different types of channels are presented in fig .[ fig : freqvsfreq ] .one can see that the gaussian capacity , the optimal modulation and therefore , the output frequencies can be non - monotonous functions of with maxima and/or minima . generalizing several results that were previously obtained for the lossy channel in  we will study the extremal points of the capacity curve at a finite noise squeezing begin with the input energy domain . given and wefirst obtain the _ threshold noise frequency _ from [ stated in eq . ] .although the equation has two roots , there is always only one positive which is a decreasing function of taking value for so that for all we have . for the solution for the capacity stated in eqholds , from which we conclude that where equality is realized only if , and furthermore , where also equality is only realized .hence , for , the gaussian capacity is a monotonously decreasing and convex function with respect to attaining its minimum at the boundary of this interval .this means that although squeezing adds more energy to the noise the thermal noise is worst for the capacity in this interval of .this observation was made first for the lossy channel in  and for the additive noise channel in  .finally , we conclude that the maximum of the gaussian capacity as a function of lays in the interval  ] is changing with .we have need to take into account this functional dependence when taking the derivative of with respect to . as a result , in addition to the two equations stated in eq ., we have the third equation which can be simplified with the help of eqs . , : the trivial solution of the latter is given when both sides of the equation vanish .this is the case of thermal noise , i.e. , for which is optimal .note that corresponds to a ( local ) minimum of the gaussian capacity , and not a maximum [ see fig .[ fig : capsvswenv ] ] .  a joint solution of , , andgives us both the optimal noise and optimal input frequency , and , respectively , corresponding to the extrema of the lagrangian . in order to find the solution we express from eq .  and insert the result into eq .thus obtaining injecting into this equation the definition for , stated in eq . , we simplify its right hand side to then using the expression for , given by eq ., on the left hand side of and equalizing it with the right hand side of eq .we obtain interestingly , this solution is equal to the optimal input frequency obtained for the limit of infinite squeezing , stated in eq . .note that the latter again implies that the optimal modulated input state is a thermal stated , i.e. [ see eq . ] .now , in order to obtain the optimal noise frequency , we insert solutions and into eq .  which leads to which states that the difference between the variances of the modulated output state is exactly matched by the difference between the noise variances .in order to find numerically the solution of eq .we express in the form , in the form taking into account , where we inject .also we use similar expressions for and : where we have taken into account eq . .if the solution of the transcendental eq .  together with eqs .  and exists , it provides the extremal points of corresponding to the local maxima or minima of , which is equal to [ see eq . ] evaluated at these points .   vs. environment frequency ( from bottom to top ) : ( a ) phase - conjugating ch .: , ( b ) : lossy ch . : , ( c ) : additive noise ch .: ( d ) : amplification ch .: .for all plots we chose and for plots ( a ) , ( b ) and ( d ) . for  ] the curve corresponding to has exactly a saddle point at finite squeezing , two extrema can be found for and one maximum is found for . for the classical additive noise channel ( ) and the amplifier ( ) the curves corresponding to are monotonically decreasing with whereas for maximum appears .the dashed straight line separates the solutions `` below '' ( left ) and `` above '' ( right ) the threshold .,title="fig : " ]  vs. environment frequency ( from bottom to top ) : ( a ) phase - conjugating ch . : , ( b ) : lossy ch . : , ( c ) : additive noise ch .: ( d ) : amplification ch .: . for all plots we chose and for plots ( a ) , ( b ) and ( d ) . for ] the curve corresponding to has exactly a saddle point at finite squeezing , two extrema can be found for and one maximum is found for . for the classical additive noise channel ( ) and the amplifier ( ) the curves corresponding to are monotonically decreasing with whereas for maximum appears .the dashed straight line separates the solutions `` below '' ( left ) and `` above '' ( right ) the threshold .,title="fig : " ]   in the following we classify the gaussian capacity with respect to the noise frequency .our numerical analysis ( see some typical graphs in fig . [fig : capsvswenv ] ) revealed that all curves have no more than two inflection points and only the following scenarios in the open interval exist :  1 .[ itm : noext ] there are no extremal points and the capacity is a monotonous function of , 2 .[ itm : onemax ] there is only one extremal point corresponding to a maximum , 3 .[ itm : saddle ] there is only one extremal point which is a saddle point , 4 .[ itm : maxmin ] there are two extrema : one minimum and one maximum .all four different scenarios are observed only for the lossy channel ( see also  ) , whereas for the other channels either no extrema or only one maximum were found .scenarios [ itm : noext ] and [ itm : onemax ] were observed and studied for the classical additive noise channel in  .although we do not have an analytic proof of these observations we take the statements above for granted in order to deduce a classification of the dependency of on .we studied a wide range of parameters and did not find any curve , which violates our classification , and that is why we conjecture our hypothesis to be true .first , we exploit the topological argument based on the observations above and the behavior of the capacity in the limits and .the last limit was discussed in the previous subsection with the conclusion that for the capacity is a monotonically decreasing and convex function of .a simple topological argument allows us to conclude that if at the function is  * _ increasing _ : scenario [ itm : onemax ] can be realized ( one maximum ) with one inflection point if is concave at and with two inflection points if it is convex .all other scenarios can not be realized since they would require more than two inflection points , which was never observed and as we conjecture never happens . * _ decreasing and concave _ : only scenario [ itm : noext ] can be realized , i.e. is monotonously decreasing in the whole interval of with one inflection point .any other scenario would require more than two inflection points .* _ decreasing and convex _ : all possible scenarios except [ itm : onemax ] can be realized .let us find now the domains of the channel parameters where the first and the second derivatives of over have definite signs at .we will study the coefficients of the expansion of given by eq .  in powers of around the point . where we used our result on the optimal encoding at [ see eq . ] .note that the neighborhood of lays in the domain which corresponds to the solution for .since no explicit expression for exists in this domain we obtain the coefficients on the right hand side of eq .  using an the expansion of in powers of : the unknown coefficients , , and are found by expanding the transcendental eq .  in powers of with the help of .in this way we obtain a set of equations , each corresponding to a particular power of .the equations corresponding to the first and second power provide us with the solutions : ,\nonumber\\  k_j & = &  \frac{(-1)^j}{\ln 2 } \frac{1-{\omega_{\infty}}^{2}}{\left({\omega_{\infty}}|\tau|y\right)^j}\nonumber\end{aligned}\ ] ] where the new notations correspond , as we will show below , to the critical parameters determining the domains we are looking for .then the coefficients of the expansion are found in the form : , \\\label{eq : coefc }   c & = & c ' + k_3 \left(2-\frac{15}{2}\tilde{\tau}_c^2\right)\left(y^2-y_c^2\right)\\   & \times & \left[\left(1+\frac{15}{2}\tilde{\tau}_c^2\right)\left(y^2-y_c^2\right)^2   -y_c^2\left(\tau^2- \tilde{\tau}_c^2\right)\right]\\ \label{eq : coefcp }   c ' & = & k_3 \left[\frac{4}{3}\left(y^2-y_c^2\right)^3       \left(\left(1-\frac{15}{2}\tilde{\tau}_c^2\right)^2+\frac{15}{2}\tilde{\tau}_c^2\right)\right .  \\& - & \frac{15}{2}\left(y^2-y_c^2\right)y_c^2\tilde{\tau}_c^2     \left(\tau^2- 2\tilde{\tau}_c^2+\frac{4}{15}\left(1-\frac{2}{15}\tilde{\tau}_c^{-2}\right)\right)\nonumber\\    & - & \left.\frac{1}{48}\left(\left(\tau^2-\tilde{\tau}_c^2\right)^2       -\frac{1}{21}\left(\left(\tilde{\tau}_c^2+\frac{4}{3}\right)^2-\frac{32}{15}\right)\right)\right].\nonumber\end{aligned}\ ] ]  this expansion shows that in the limit the partial derivatives of with respect to up to the third order coincide with the full derivatives for moreover , if the latter is satisfied , the dominant linear term in the expansion is canceled because the coefficient vanishes . as we discussed above , this would imply that is an extremal point of for and therefore , the value of alone decides how the capacity approaches the limiting value , i.e. , from below or above . for find that in the vicinity of .then from the topological arguments discussed above we conclude that scenario [ itm : onemax ] is realized and has one local maximum laying in the interval .thus , the line clearly separates the domain of channel parameters where only scenario [ itm : onemax ] is realized from the domain where only other scenarios are possible ( see fig .[ fig : ytaudivision ] ) .let us analyze the conditions on the other channel parameters which allow to attain the critical value .we notice first , that the condition can only be fulfilled when lays in a particular interval .indeed , for pure noise ( ) we obtain from that which provides a lower bound on the physical region of ( see fig .[ fig : ytaudivision ] ) .in particular , for the lower bound is which is a valid lower bound as for the classical additive noise channel as well as for limiting cases of the lossy and amplification channels .then , for an arbitrary gaussian noise with parameter given by eq . ,equation has two solutions : both solutions belong to the interval bounded by the values of for pure noise ( ) : if lays outside this interval then the parameter can never be attained by any and therefore , scenario [ itm : onemax ] ( one maximum ) can never be realized . in fig .[ fig : ytaudivision ] this corresponds to the physical region above the dark gray triangle .we further observe that in the region defined by the first derivative of at both edges of the interval is negative .it implies that although scenario [ itm : onemax ] is excluded , all three other scenarios are possible .namely , the function may be either monotonously decreasing or have one maximum and one minimum [ both located in the interval ] or have a saddle point , see fig .[ fig : capsvswenv ] .  in order to discriminate the number of extremafurther we need to study the second derivative determining the factor of the next term of taylor expansion , which is quadratic in .note that contrary to , factor [ see eq . ]is not always zero for the critical value . a joint solution of is given by the `` super critical '' point , which satisfies and depends on the amount of photons in the input due to .the range  ] ( see also  ) .the parameters and are the solutions of the equations the domain in terms of corresponds to + , see figs . [fig : ytaudivision ] and [ fig : ytildenmaxeta ] .the zones of channel parameters corresponding to the different number of extrema are summarized in table [ table : wenvbehavior ] and plotted in figs .[ fig : ytaudivision ] and [ fig : ytildenmaxeta ] . for the lossy and additive noise channel thiswas previously observed in  .   and , containing the scenarios [ itm : noext ] - [ itm : maxmin ] of the gaussian capacity with respect to . - north - east lines ( ) : zone where is non - physical .- gray triangular region ( ) : the capacity exhibits one maximum . - light gray region [ : capacity may exhibit one minimum and one maximum , one saddle point or is monotonously decreasing with ( see fig .[ fig : ytildenmaxeta ] for details ) .- white region ( , ) : capacity is a monotonously decreasing function of . ].extremality properties of with respect to . [ cols="^,^,^,^,^",options="header " , ]   summarizing our results , we found the following behavior of the gaussian capacity for the different channels .* for the phase - conjugating channel the linear term in the expansion is always non - negative which means that the limiting value is always approached from below for , and furthermore the second derivative of the gaussian capacity is never zero because we always have .hence , there is never a saddle point at . indeed ,our numerical plots show that the capacity of the phase - conjugating channel is always monotonically decreasing with ( see fig .[ fig : capsvswenv ] ) . *the critical behavior can be observed for the amplification channel and the additive noise channel [ .however , the condition for a saddle point at can not be fulfilled for those channels , since . in this casewe observe numerically only two different behaviors : for a maximum appears at a finite and for the capacity is always monotonically increasing . *only the lossy channel )$ ] exhibits all four possible scenarios : in addition to one maximum for and a monotonous increasing for , a saddle point appears when eqs .are satisfied and one maximum followed by one minimum is present when inequality is satisfied .the above results are depicted in figs .[ fig : ytaudivision ] and [ fig : ytildenmaxeta ] .we studied the gaussian capacity of the fiducial gaussian channel in a wide range of channel parameters including limiting cases .as shown previously our solution provides a way of calculating the gaussian capacity of any gaussian channel for its given canonical decomposition  .two types of the solution are separated by an input energy threshold. if the input energy is above the threshold a thermal state at the output can be realized , as well as a squeezed input state that exactly matches the squeezing of the environment state .this type of solution was known before . below this thresholdwe find new solution given by a transcendental equation .this type of solution was found first for the lossy and additive noise channels .here we extended it to all types of gaussian channels .we provides to a physical interpretation of the solution choosing a particular representation for the covariance matrices so that the optimal modulation is found by jointly solving a planck - like equation and a `` resonance '' equation .the `` resonance '' is achieved exactly at the input energy threshold and holds above it .it provides the optimal input frequency matching the noise frequency , which implies that the modulated output state has a frequency equal to 1 .we investigated the capacity with respect to the noise squeezing for given transmissivity / gain and noise determinant and distinguished between four different behaviors for increasing noise squeezing ( or decreasing noise frequency ) : monotonically increasing , one maximum , a saddle point or one maximum followed by a minimum .we classified the precise boundaries for this behavior and concluded that only the lossy channel can exhibit all four behaviors , whereas the capacity of the additive noise channel and amplifier can develop a maximum and the capacity of the phase - conjugating channel is always monotonically increasing with the noise squeezing .  , and . ]furthermore , we found that the lagrange multiplier can be interpreted as the inverse temperature that enters in the planck - like equation . from previous studies  where the solution of a system of single - mode mode channels with joint energy constraint was discussed we know that this multiplier is common for all channels of such a system .this hints to the interpretation of a joint solution for channels as corresponding to a kind of _ thermal equilibrium _ ,similar to the case of connected heat baths with different temperatures which thrive to a joint temperature .in order to study the monotonicity of the classical capacity and gaussian capacity on the parameter the following lemma will be useful .     in the following proof we use the parametrization the total output covariance matrix of the concatenated channel stated on the left hand side of eq .  reads where and has an additional minus sign in front of its off - diagonal elements .it is straight forward to show that for all three conditions stated in the lemma the second line of eq .can be simplified to which completes the proof .v. giovannetti , s. guha , s. lloyd , l. maccone , j. h. shapiro , and h. p. yuen , `` classical capacity of the lossy bosonic channel : the exact solution , '' _ phys ._ , vol . 92 , pp . 027902 - 1027902 - 4 , 2004 .    j. schfer , e. karpov , r. garca - patrn , o. v. pilyavets , and n. j. cerf , `` equivalence relations for the classical capacity of single - mode gaussian quantum channels , '' _ phys . rev . lett ._ , vol . 111 , pp .030503 - 1030503 - 5 , 2013 .v. giovannetti , a. s. holevo , s. lloyd , and l. maccone , `` generalized minimal output entropy conjecture for one - mode gaussian channels : definitions and some exact results , '' _ j. phys .a _ , vol . 43 , pp . 415305 - 1415305 - 10 , 2010 .r. garca - patrn , c. navarrete - benlloch , s. lloyd , j. h. shapiro , and n. j. cerf , `` majorization theory approach to the gaussian channel minimum entropy conjecture , '' _ phys ._ , vol . 108 , pp .110505 - 5 , 2012 .h. p. yuen and j. h. shapiro , `` optical communication with two - photon coherent states part iii : quantum measurements realizable with photoemissive detectors , '' _ ieee trans .inf . theory _it-26 , no . 1 ,pp . 7892 , jan .1980 .b. r. bardhan , r. garca - patrn , m. m. wilde , and a. winter , `` strong converse for the classical capacity of optical quantum communication channels , '' _ ieee trans .inf . theory _ ,61 , no . 4 , pp. 18421850 , apr .

Abstract: the full solution of the optimization problem giving the gaussian capacity of the single - mode fiducial gaussian quantum channel is provided . since it was shown that the gaussian capacity of an arbitrary ( phase - sensitive or insensitive ) single - mode gaussian quantum channel is equal to the gaussian capacity of this fiducial channel , the solution presented in this work can be regarded as universal . the analytical study of this solution , below and above the energy threshold , shows that the dependence of the gaussian capacity on the environment noise squeezing is not monotonic . in particular , the capacity may have a saddle point , one or two extrema at finite squeezing , or be a monotonically increasing function of the squeezing parameter . the exact dependence is defined by the determinant of the noise covariance matrix and by the transmissivity ( or gain ) of the fiducial gaussian channel .   schfer : classical capacity of phase - sensitive gaussian quantum channels  classical capacity of quantum channels , gaussian quantum channels , quantum information .


in the seminal work of marzetta , a massive mimo ( multiple - input multiple - output ) system refers to a multi - user mimo communication system where a base station comprises a very large number of antennas , much larger than the number of served users . in this under - determined multi - user system , the extra spatial degrees of freedom ( dof ) are exploited to make the multi - antenna multi - user mimo channel asymptotically orthogonal .in addition , relying on the knowledge of the channel , a proper processing at the base station ( bs ) averages out the fading at the receivers in the downlink direction and at the bs in the uplink direction . based on those features , enabled by the extra dof ,massive mimo is recognized as a promising technology for very high system throughput and energy efficiency .  in massive mimo systems ,the spatial dofs available in the multi - antenna multi - user channel play a central role . with an independent and identically distributed ( i.i.d . )modelling of fading , the number of dof is simply defined and limited by the number of antennas .this simple modelling provides an inappropriate account of the dof limitation though . for a given propagation environment , the array physical characteristics , its physical size and geometry , define the number of dof as it defines the angular resolvabilitythis can be easily understood in a line - of - sight environment , where users can be separated if they are further apart than the resolution unit . in a scattering environment , the array dimension defines the degree of resolvability of the scattering clusters and hence the dof ( see e.g. ) . in this paper ,_ array aperture _ refers to the dimension of the arrays .we have two types of array , a linear array and a square array : the array aperture is the length of a linear array and the length of the side of a square array .  increasingthe number of antennas within a fixed array aperture is useful to grab all the available dof up to the limit imposed by the aperture .after this limit is exceeded , increasing the number of antennas does not bring improvement .there is another important point that advocates for very large aperture arrays : as pointed out in  an increased aperture implies an extended vision range of the environment .the more the array can capture of his environment , the more diversity it can capture , implying more dof .the importance of the aperture to achieve the theoretical performance of massive mimo systems is the motivation for a series of channel measurements performed at aalborg university where the primary focus is on the impact of the array aperture .the measurements presented in this paper targets a deployment of massive arrays that differs from the current mainstream of a cellular deployment where the massive bs is placed outdoors . in general , we address a deployment in large venues , possibly indoors , where massive arrays with a very large aperture ( vla ) are designed as an integral part of a new large infrastructure . referring to the 5 g scenarios defined within the eu project metis , vla arrays can be deployed along walls or ceiling of a shopping mall or airport , around the structure of a football stadium or along the facade of a building .such scenarios benefit from line - of - sight propagation with the potential of very high rank point - to - point mimo channels and acute discrimination between users , especially in crowd scenarios .they also benefit from rich scattering as the arrays see a wide range of diverse scatterers .our vision is that the theoretical benefits described in can be achieved in those large infrastructure deployment , while cellular deployment with bs located in high towers would benefit from the capability of massive mimo for sharp beams .our measurement campaign provides a first positive echo to this vision .  to this date ,still very few measurement campaigns with published results exist .there are 5 of them   .all target outdoor scenarios at frequency 2.6 ghz .following the first publications on massive mimo , the major stress of those measurements is set on the impact of number of antennas , not relating it directly to array aperture . only in  we find a comparison between two different aperture arrays where the larger aperture array was found to have better performance . in  ,the orthogonality between the channels of 2 users ( the normalized scalar product ) has been measured to increase with the number of bs antennas .all the existing measurement campaigns confirm the promises held by the theoretical studies .however , as pointed out in  , a saturation is observed creating a gap with the performance of i.i.d . channels , likely coming from the limitation of the dof . all those campaignsface the problems induced by the large number of antennas and the impossibility to achieve simultaneous measurements with the current technology .the solution adopted is to create a virtual massive array and/or virtual multi - user set - up .measurements are performed with a set of small number of antennas and users , which are moved for a subsequent measurement .this methodology creates heavy constraints of the measurement protocols imposing the environment to remain static within a whole measurement interval .our measurement campaign involves 64 antennas that are rearranged in 3 different geometrical forms ( see fig . [ fig_allarrays ] ) . in the first array ,the antennas , separated by half the wavelength , are placed within a square .we refer to this array as `` compact 2d '' .it corresponds to a common view on the design of massive arrays , where the massive array is as compact as possible . in the `` large aperture array '', the 64 antennas are spread along a line of 2 meters . in the `` very large aperture array '', the antennas are spread along a line of 6 meters .note that one advantage of the compact 2d array is to offer beamforming capabilities in 2 dimensions : this capability will be tested against the one - dimensional beamforming ( in los ) offered by the linear arrays .the distinctive features of our measurements can be described as follows : first , we have a multi - user set - up where 8 users transmit simultaneously to the massive bs .second , we test mimo capabilities within the same device with and without user proximity effects : the 8 users hold devices with 2 antennas .last , we perform quasi - simultaneous measurements : the set - up includes 16 fully parallel transmitters ( 8 mobile units with 2 antennas ) and 8 fully parallel receive units . at the bs ,the received signals from 8 antenna elements are measured simultaneously . using fast switching ,the parallel system is extended to allow measurements with 64 element bs arrays .our general conclusion is that performance is better as the array aperture increases .more specifically , our main findings are summarized as follows :  * inter - user link orthogonality : the impact of the aperture is mainly visible when the users are closely grouped . as the number of users increases , the very large array is able to hold performance that is closed to the i.i.d . channel . * intra - user link orthogonality ( mimo capability ) : we see a good channel resolvability that is better for the very large array .furthermore , our measurements confirm the strong effect of the user hand grip .* power variations across the array : the largest variations are seen for the very large aperture array .however , even for the compact 2d array , we can see power variations larger than 10db .the measurements campaign was performed in one of the canteens at aalborg university .this location can be considered as a large indoor venue and has a similar structure as a shopping mall : a big open space with high ceiling , stairs ( to go to an upper level ) , and some small areas on the side with a lower ceiling ( see fig .[ fig_scenarios ] ) .we use a total of 64 monopole elements . for practical reasons ,the 64 elements are grouped in sets of 8 elements . the distance between elements in the same set is at 5.8 ghz .each set has 2 dummy elements at the ends to provide balanced properties among the active elements ( load , coupling , correlation ) .those sets are arranged in 3 different array shapes , pictured in fig .[ fig_allarrays ] : 1 ) a square compact 2d array of 25cmx28 cm , where the sets are placed in a square , 2 ) a large aperture linear array of 2 meter long where the sets are placed near to each other along a line , 3 ) a very large aperture linear array of 6 meter long , where the sets are placed further away .the array is placed along a wall of the room , parallel to the stairs .a total of 8 handsets transmit to the massive array ( fig .[ fig_handset ] ) .eight users hold the handsets as if in data mode in one or both hands .the device is located at a few centimeters from the body : see figure [ fig_scenarios ] .the users move randomly in an area of 1 square meter .our original intent in having this small mobility area was to measure small scale fading .however , after analyzing the data , we found that power fluctuations originating from the user movement were larger than expected for the measurements to reflect small scale fading . as a result, we decided to normalize the channel as described in ( [ eq : norm ] ) , where the channel at each measurement is normalized to the same value .the devices are also tested in free space ( no user proximity effect ) and also moved within a 1 square meter area .six scenarios are tested , each one with specific propagation properties , with los ( line of sight ) and nlos ( without los ) and with a specific distribution of the users : see fig .[ fig_scenarios ] and fig .[ fig_users ] .  ** spread los parallel * : the users face the array holding the devices so that the two - antenna array is parallel to the bs array .four users are in a line parallel to the bs array between the stairs and the bs array and four of them in a line behind the stairs .the distance between users is 3 m and the distance from the users to the array varies from 5 m to 12 m . * * spread los perpendicular * : the distribution of users is the same but the users do not face the array but rather look in a direction parallel to the array . the two - antenna array in the handsets andthe bs array are perpendicular . in pure los conditions ,the two - antenna in the array can not be likely resolved .the purpose of this scenario is to test whether this is the case or whether scattering is rich enough to allow discrimination between the two antennas . ** grouped los * : the users are behind the stairs , all in a small area of 1.5mx6 m , shoulders to shoulders and moving away from each other .the distance to the array varies from 10 m to 12 m . ** free space in front the stairs * : the 8 handsets are fastened to a table and the table was moved randomly within a small area .the table is placed between the stairs and the bs array . * * free space behind the stairs * : the same as before but the table is placed behind the stairs . ** spread nlos * : the users are in non los conditions , in the lateral room with a distance between users of 2 m and a distance to the bs array from 5 m to 11 m .[ fig_users ] provides an illustrative floor map of the large room where the measurement were conducted .the stairs are in orange , the bs array is in blue .the 8 users ( with 2 antennas each ) are depicted in los and nlos conditions . to ease the analysis in section [ sec_analysis ] , we associate a user number to its geographical position . in the scenarios grouped los and free space , the relative position of the users ( and their numbers )is the same but the distance between them is reduced .   a correlation based channel sounder is used to make the measurements .the carrier frequency is 5.8 ghz and the bandwidth about 100 mhz .there are 16 fully parallel transmitters ( the 8 devices with the 2 antennas ) , and 8 fully parallel receivers .a fast switching mechanism between the receive sets ( 8 time switch ) allow for a capture of the complete mimo channel in 655 .for each deployment of the bs array and scenario , 1200 realizations of the channel were recorded in 20s , while the users move inside the 1 square meter area .the measurements are calibrated up to the antenna connectors , i.e. the antennas are considered part of the channel .therefore , any mutual coupling and non - ideal characteristics of the arrays are included , as they would be in an actual system .our analysis is based on narrow band channel data obtained via fourier transforms of the wideband measurements .the statistics are taken over the 1200 measurements .in this section we focus on the analysis of the results obtained in the measurement campaign .the notations are as follows : the number of users is , the number of antennas for each user is , the number of base station antennas is , and the number of channel realizations is . in this paper, we report results for the most significant scenarios .the results obtained are compared with the i.i.d .gaussian channel , to provide a comparison between the real measurements and the theoretical results .at the same time the three different arrays are evaluated and compared to each other to show the effect of the aperture on the channel performance .  to analyze the results ,the communication system considered is a single cell mu - mimo ( multi - user mimo ) system with users , having antennas each one , and a base station with antennas serving the users .we denote the matrix corresponding to realization of the channel .later some subsets of this matrix will be considered .notice that .we denote as the channel vector from antenna in the handset of user to the bs array . in matrix ,the two user channel vectors are placed in two consecutive columns .the system has a power gain control so all the users receive the same power regardless their distance to the base station .the power of channel vector is normalized : in this way it is fair to compare users in different location and the results will be determined by the channel conditions and not by the path loss of the user . we denote as the channel matrix made out of the normalized vectors in ( [ eq : norm ] ) .several figures of merit are obtained from the channel matrix giving a basis to understand the channel .   a first step to analyze the behavior of the channelis to look at the correlation between channel vectors , encompassed in the following matrix :   the elements of matrix are pictured in fig .[ fig_sp_gl ] and fig .[ fig_sp_ff ] .the 2x2 blocks along the diagonal represent the correlation between the channels for the same device .the off - diagonal blocks represent the correlation between the channels of different devices .we notice a significant difference in the behavior of the intra - user and inter - user channels .this is the reason why we later detail the performance separately for those two kinds of channels .matrix is shown for two scenarios : `` grouped user in los '' and `` free space '' .both scenarios illustrate a crowd scenario with devices closer to each other in the `` free space '' case .furthermore , by comparing both scenarios , user proximity effects can be assessed . in the figures ,the two antennas associated with each user are denoted and , respectively .  * inter - user properties * : the very large aperture array clearly performs the best and the compact 2d array clearly performs the worse . while the very large aperture and large aperture arrays can satisfactorily discriminate all the users in the grouped los scenario , correlation among users appear for the compact 2d array .the correlations appear not only for users in the same line , but also for users behind each other ( 5,1 or 6,2 ) . as the devices become more packed ( free space ) , performance degrades for all arrays with the apparition of correlated users . for the large aperture array ,strong correlations appear for some pairs of users ( 1,5 and 2,6 and 3,7 and 4,8 ) that are behind each other .so we deduce that the large aperture array loses its resolution in the elevation angle .finally in the compact 2d array the correlation is large even for neighboring users ( specially the ones further away from the base station ) .channels with low correlation can be seen for users that are far from each other ( 1,4 and 1,8 and 4,5 and 5,8 ) .* intra - user properties * : it is expected that the channel correlation properties within a same device are worse than across devices as the antenna are close by . in the free space scenario ,the very large aperture array performs remarkably better than the other arrays . comparing now with the `` grouped los '' scenario, we can notice the large impact of user proximity effects which can equalize the performance among arrays .       herethe inter - user channel properties are considered .the goal is to explore what happens when the number of users in the system increases .we call the number of users in the system and we increase from 1 to .we collect all the combinations of channels from different users ( only one channel per user out of the two channels is selected ) .we form a new channel matrix for realization denoted generically as and take statistics on our metric over the different user combinations and channel realizations .the metric adopted here is the sum of the eigenvalues normalized by largest eigenvalue ( npcg , normalized parallel channel gain ) . denoting as the eigenvalue of the matrix ,then the metric is defined as :  is the largest eigenvalue .this metric is preferred to the condition number of the channel as the latter gives only information on the ratio between the larger and the smaller eigenvalue , whereas metric ( [ eq : se ] ) reflects the behavior of all the eigenvalues .notice that , for a channel that is very well conditioned with equal eigenvalues , and for a channel that is poorly conditioned .  in the same figure , fig .[ fig_se_1 ] two scenarios are represented , spread los with their two antennas parallel to the bs array and grouped los. comparing both scenarios , it is obvious that the more spread the users , the better .for all the number of users and all the arrays the channel of grouped users has worse conditioning .the limitation of the compact 2d array in the dof available to 8 users is particularly visible in the grouped los scenario .note that the very large aperture array in the spread los scenario gives performance slightly better than the i.i.d .gaussian case .this can happen in los conditions depending on the location of the users .  in the next figure , fig .[ fig_se_2 ] , two scenarios are plotted .one is the spread nlos and the other is the free space behind the stairs ( the scenario with minimal inter - device distance ) .the three arrays have a worse channel conditioning than the previous scenario . for the compact 2d array adding new users can hardly increase the metric showing again a limitation in the dof .two effects can be observed . on one handthe effect of nlos propagation and on the other hand the effect of closely spaced devices . in both casesthe very large aperture array and the large aperture array have a similar behavior .it appears that the scattering is rich enough so that increasing the aperture from large to very large does not bring improvement in the nlos scenario . at last ,we show the results for the free space scenario , where we can see an overall degradation of the performance .           to evaluate the mimo channel properties within a same device , we go back to the conventional condition number as we have only 2 antennas .the condition number indicates how spread the eigenvalues of the channel are .it is defined as where and are the maximum and minimum eigenvalues of the mimo channel matrix .statistics on the condition number are taken over the realizations of the channel .[ fig_cn_1 ] and fig .[ fig_cn_2 ] shows the cumulative distribution function ( cdf ) of the condition number .first , it can be noticed that there is a significant gap between the i.i.d .case and the measured cases , mostly due to user proximity effect and small spacing between the antennas .in general , the very large aperture array still performs the best with slopes that are steeper than the other arrays . as the aperture increases ,the number of dof increases and the distribution of the eigenvalues tend to a deterministic quantity .this is in line with a similar effect in i.i.d .channels when the number of antennas ( and hence dof ) increases .[ fig_cn_1 ] accounts for the spread user scenarios , where the array formed by the antennas at the device is parallel or perpendicular to the bs array .we do not observe a clear tendency in the comparison between both scenarios , while the scenario with perpendicular arrays performs much worse in pure los than the scenario with parallel arrays .this might indicate that scattering is rich enough to enable mimo capabilities in both cases .this observation is actually positive : it makes the access robust to device orientation as the devices will have a random orientation relative to the bs in general .looking at the fig .[ fig_cn_2 ] , where the users are spread nlos and in free space in front of the stairs , we observe that the very large aperture array remains robust towards both scenario conditions .both large and compact arrays also give robust performance in the nlos scenarios brought by a rich scattering environment .poor performance is observed in the los free space scenario .more analysis is needed to fully understand this case .our interpretation is that performance tend to depend on location in los , suggesting a more favorable location relative to the very large aperture array compared to the other arrays .as mentioned in  where measurements involving a large aperture massive array have been performed , channel characteristics become non - stationary across the array , such as the received power or the direction of arrivals . in general ,the environment that is observed is different from one part of the array to another .  in this section , to illustrate this phenomenon , we examine the variations of the received power across the array for one scenario ( i.e. spread los ) and two arrays , the very large and the compact one , shown in fig .[ fig_power ] .the x axis represents each user ( 2 antennas per user ) , and the y axis represents the received power averaged over the 1200 measurements at each of the 64 bs antennas .several observations can be made .first , obviously , the users that are further away from the array have a smaller received power .second , as the aperture increases , so does the impact of path loss variations across the array .this is more visible for users that are closer to the array in front of the stairs .third , the power variations depend on the environment : users 7 and 8 are positioned behind the stairs that have a shadowing effect and make the receive power uniform compared to other users . at last , the power variations are smaller in the compact 2d array case compared with the very large aperture one .however , even for this small aperture , significant variations of 10db order can be observed .the property of non - stationarity across a large aperture array is a new feature that gives a distinctive edge to the type of communication system studied in this paper .it impacts the performance of the system but also the multi - user access methods which could be beneficial in terms of multi - user diversity .the presented investigation describes a measurement campaign involving a massive array with 64 antennas and 8 users with mimo capabilities in a large indoor environment . the main purpose is to investigate on the impact of the massive array aperture .three different shape and aperture of the base station array were tested as well as different propagation conditions ( los and nlos ) , user device distribution ( spread and grouped ) and user proximity effect .the measurements confirm that the aperture is important to create the spatial dof that brings the benefits promised by the theoretical studies on massive mimo .we found that the array with the largest aperture perform the best with performance close to the i.i.d .the channel tends to be better conditioned bringing very good discrimination among users but also between antennas of a same device , where the user proximity effect still has a major effect .building on this experience drawn from this first measurement campaign , we are planning a new campaign involving a much larger number of antennas in a larger venue as well in outdoor conditions .the research presented in this paper was partly supported by the danish council for independent research ( det frie forskningsrd ) dff133500273 .part of this work has been performed in the framework of the fp7 project ict-317669 metis , which is partly funded by the european union .the authors would like to acknowledge the contributions of their colleagues in metis , although the views expressed are those of the authors and do not necessarily represent the project .x. gao , f. tufvesson , o. edfors , and f. rusek , `` measured propagation characteristics for very - large mimo at 2.6 ghz . '' signals , systems and computers ( asilomar ) , 2012 conference record of the forty sixth asilomar conference on .ieee , 2012 .s. payami , and f. tufvesson , `` channel measurements and analysis for very large array systems at 2.6 ghz . ''6th european conference on antennas and propagation , eucap 2012 , prague , czech republic , 2012 - 03 - 26 .

Abstract: massive mimo is a new technique for wireless communications that claims to offer very high system throughput and energy efficiency in multi - user scenarios . the cost is to add a very large number of antennas at the base station . theoretical research has probed these benefits , but very few measurements have showed the potential of massive mimo in practice . we investigate the properties of measured massive mimo channels in a large indoor venue . we describe a measurement campaign using 3 arrays having different shape and aperture , with 64 antennas and 8 users with 2 antennas each . we focus on the impact of the array aperture which is the main limiting factor in the degrees of freedom available in the multiple antenna channel . we find that performance is improved as the aperture increases , with an impact mostly visible in crowded scenarios where the users are closely spaced . we also test mimo capability within a same user device with user proximity effect . we see a good channel resolvability with confirmation of the strong effect of the user hand grip . at last , we highlight that propagation conditions where line - of - sight is dominant can be favourable .