Channel Hardening in Massive MIMO: Model Parameters and Experimental Assessment
Sara Gunnarsson, José Flordelis, Liesbet Van der Perre, Fredrik Tufvesson
aa r X i v : . [ ee ss . SP ] A p r Channel Hardening in Massive MIMO:Model Parameters and Experimental Assessment
Sara Gunnarsson, Jose Flordelis, Liesbet Van der Perre, Fredrik Tufvesson
Reliability is becoming increasingly important for many applications envisioned for future wireless systems. A technology thatcould improve reliability in these systems is massive MIMO (Multiple-Input Multiple-Output). One reason for this is a phenomenoncalled channel hardening, which means that as the number of antennas in the system increases, the variations of channel gain decreasein both the time- and frequency domain. Our analysis of channel hardening is based on a joint comparison of theory, measurementsand simulations. Data from measurement campaigns including both indoor and outdoor scenarios, as well as cylindrical and planarbase station arrays, are analyzed. The simulation analysis includes a comparison with the COST 2100 channel model with its massiveMIMO extension. The conclusion is that the COST 2100 model is well suited to represent real scenarios, and provides a reasonablematch to actual measurements up to the uncertainty of antenna patterns and user interaction. Also, the channel hardening effectin practical massive MIMO channels is less pronounced than in complex independent and identically distributed (i.i.d.) Gaussianchannels, which are often considered in theoretical work.
Index Terms —Channel hardening, channel model, COST 2100 channel model, massive MIMO, measurements, reliability
I. I
NTRODUCTION M ISSION-CRITICAL applications such as remotesurgery, intelligent transportation systems and industryautomation are envisioned to be based on wireless connectivityin the future. To realize these applications, reliable communi-cation is required. Massive MIMO [1] [2] has in theory showna potential to increase reliability; by deploying a massive num-ber of antennas at the base station side, an unprecedented levelof spatial diversity can be exploited. Additional advantages arethat both spectral and energy efficiency increase. However, theenhanced reliability is the main focus of this paper.Stable channels are essential in order to achieve reliabilityin communication systems. In massive MIMO systems, thechannel gain becomes more concentrated around its meanwhen increasing the number of base station antennas. Thisphenomenon, where a fading channel behaves more determin-istically, is called channel hardening. The decreasing fadingvariations will also render a more stable capacity. The channelhardening effect can be studied from two points of view. Thefirst point is that the fading over frequency is reduced due tothe decrease of experienced delay spread. The second pointis channel hardening in the time domain, where the temporalfading decreases as a result of the coherent combination of thesignals from the many base station antennas.The theory of channel hardening has been discussed in manypapers, see e.g. [1], [3]–[7]. In [3] and [4] a definition ofchannel hardening is given and the authors in [5] introduceand derive closed form results for the coefficient of variationof channel gain, which further relates to the characteristics of
The manuscript was submitted on 2020-02-03. This work has receivedfunding from the strategic research area ELLIIT.All authors are with the Department of Electrical and Information Theory,Lund University, Lund, Sweden. S. Gunnarsson and L. Van der Perre are alsowith the Department of Electical Engineering, KU Leuven, Ghent, Belgium.J. Flordelis is also with Research Center Lund, Sony Corporation, Lund, Swe-den (e-mail: { Sara.Gunnarsson, Jose.Flordelis, Fredrik.Tufvesson } @eit.lth.se,[email protected]).Part of the material was presented at the IEEE International Workshop onSignal Processing Advances in Wireless Communications (SPAWC), 2018,Kalamata, Greece. the channel, which are affecting channel hardening. In [7], aproof of complete convergence of channel hardening in mas-sive MIMO with uncorrelated Rayleigh fading is presented.Although co-located massive MIMO systems are shown toprovide channel hardening, this might not be the case in cell-free massive MIMO [8]. Much research has been carried outon the theoretical aspects of channel hardening, not as manystudies have investigated this phenomenon experimentally [9]–[14].Early theoretical massive MIMO studies and proofs ofchannel hardening have been relying on the assumption thatthe channels experience such rich scattering that they can bemodeled as complex independent and identically distributed(i.i.d.) Gaussian channels. However, it has been shown inmeasurements that this is not the case [15] [16] in practice. Ingeneral, real massive MIMO channels are spatially correlated.This phenomenon has been acknowledged and recently it hasalso been taken into account in theoretical analyses of channelhardening [3] [17]. The other extreme channel is the keyholechannel, which is shown to not provide any channel hardening[4]. Any real massive MIMO system will experience a channelhardening somewhere in between; here we investigate whatactually can be expected in different real scenarios.To develop and assess the system performance of futuremission-critical applications, adequate channel models thatcan capture the relevant channel characteristics are needed.However, the relation between experiments and models hasnot been studied so far. Several channel models could be usedfor this purpose, such as the QuaDRiGa channel model [18][19] with extensions for WINNER-type models [20], the 3GPPchannel model (TR 38.901) [21], the IMT-2020 channel model[22], and the METIS channel model [23]. Here, the COST2100 channel model [24] [25] is chosen as it is a geometry-based stochastic channel model (GSCM) with the importantproperty that it can consistently describe the channel in space,time and frequency. It also realistically captures the behaviorat a multipath level. The model has recently been extendedwith features such that it realistically can reflect characteristicsthat are prominent in measured massive MIMO channels [26] [27]. To serve as an adequate channel model for the intendedapplications, all important channel aspects that affect systemperformance should be caught by the model; this leads tothe questions if and how the channel model realistically cancapture the channel hardening effect as well, and what areappropriate model parameters.The contribution of this paper is twofold. First, channelhardening has been validated by experiments; this being akey part in the process of extracting relevant parameters thatcan be used for accurate channel modeling. We extend ouranalysis in [9] with evaluation of more parameters and therebyproviding further insights; the number of analyzed scenarioshas been extended and the analysis is now also includingoutdoor environments as well as different antenna arrays. Wepresent generalized values for what can be expected in terms ofchannel hardening in a co-located massive MIMO system. Thesecond contribution, novel in relation to [9], is an assessmentof the capabilities of the COST 2100 channel model with itsmassive MIMO extension to capture the channel hardeningeffect and an evaluation of its model parameters. This isevaluated by relating simulated results from the model to ourexperimental findings. We highlight key parameters, relatedto propagation and antenna characteristics, that affect thechannel hardening. With this knowledge, appropriate channelmodels and parameter sets can be selected and used to acquirerealistic massive MIMO channels; consequently, these couldbe appropriate for development and assessment of futurewireless applications.The structure of the paper is as follows; Section II describesthe measurement campaigns, including both the scenarios andthe equipment used. Section III presents theory and the defini-tion of channel hardening. The measurement results, includinggeneralized values for channel hardening in different scenarios,are presented in Section IV. Section V describes the COST2100 channel model and the performed simulations. Section VIpresents a joint comparison between theory, measurements andsimulations in order to give a more thorough and completeanalysis of channel hardening in co-located massive MIMOsystems. In Section VII, the paper is summarized and conclu-sions are presented.II. M EASUREMENT SCENARIOS AND EQUIPMENT
The main focus of this work is on indoor propagation, asthis typically is the scenario with the richest scattering, andtherefore the environment where channel hardening is mostpronounced. Only single-antenna user equipment is consid-ered, limiting this study to focus on how channel hardeningis affected when only changing the number of base stationantennas. For this primary scenario, the base station is situatedin the front of an indoor auditorium, and is serving a group ofnine closely-spaced users. We also consider another scenario,which is an outdoor scenario where a base station is located ona roof serving nine closely-spaced users. Two different mea-surement campaigns have been conducted and the measuredchannels have been analyzed in terms of channel hardening.The first measurement campaign considered both scenariosand the measurements were performed with a channel sounder
Cylindrical arrayPlanar array
Fig. 1: The auditorium where the indoor scenario took place.The base station is standing in the front of the room, at differ-ent positions depending on the array used in the measurements,and the users are sitting in the back of the room to the left.Fig. 2: The outdoor scenario for both LOS (position 1) andNLOS (position 2). The base station has a cylindrical arrayand is positioned on the roof. The users were moving withina circle with a diameter of 5 m.deployed with a cylindrical array. This campaign and theperformed measurements therein is further described in [28].The second measurement campaign aimed at repeating andvalidating some of the measurements in the indoor scenariowith a real-time testbed equipped with a planar array. For allmeasurements the user equipment antenna was a verticallypolarized SkyCross SMT-2TO6MB-A.
A. Indoor scenario
The indoor measurement campaign was carried out in anauditorium located at Lund University. As seen in Fig. 1, abase station is standing in the front of the room and is servingusers sitting in the left back corner. With the cylindrical array,simultaneous measurements were done when serving all nineusers. With the planar array, evaluations were performed foruser 1 and 5, which serve as representatives of users in thefront and in the back rows, respectively. The users were mostlystatic during the measurements but were moving the antennasslowly back and forth, tilted approximately 45 degrees, witha speed lower than 0.5 m/s. The measurement campaigns alsoincluded, for both antenna arrays, the case when the rectangle
Fig. 3: The cylindrical base station antenna array as seen fromabove (left) with the numbering of the antenna elements inthe first ring, both vertically and horizontally polarized. Thecylindrical array seen from the side (right) with the numberingper ring.Fig. 4: The planar base station antenna array as seen from thefront (lower). Every other antenna is vertically and horizontallypolarized (upper), where the 100 connected antennas areshown with their corresponding polarizations.in Fig. 1 was almost full with people in the vicinity of theactive users.
B. Outdoor scenario
In the outdoor scenario, the measurement campaign wascarried out in an open area at the campus of Lund University.A base station was placed at the roof on top of the secondfloor and measurements were performed both for a Line-Of-Sight (LOS) scenario (position 1 in Fig. 2) and for a non-LOS (NLOS) scenario (position 2 in Fig. 2). In both cases,nine closely-spaced users were moving around within a circlehaving a diameter of 5 m. The users were holding the antennastilted approximately 45 degrees.
C. Channel sounder
The RUSK LUND MIMO channel sounder was used forthe first measurement campaign with the cylindrical array.This channel sounder is a multiplexed-array channel sounder,meaning that the transfer functions for each transmit-receiveantenna pair is measured at a fast pace one after the other. Thecylindrical array connected to the channel sounder is shown inFig. 3. The array consists of 64 dual-polarized patch antennasspaced half a wavelength apart at the carrier frequency of2.6 GHz, resulting in 128 antenna ports in total. The bandwidthis 40 MHz. The antenna elements are in the measurementdata numbered according to Fig. 3, i.e. starting with the lowerring and finishing with the upper ring as seen to the rightin the figure. To the left in Fig. 3, the lower ring is seen as from above. Important to note is that odd-numbered antennaports are vertically polarized, even-numbered ones have a hor-izontal polarization. For the indoor scenario the measurementsproduced 129 frequency points and 300 snapshots taken over17 seconds. This was also done for the outdoor scenario, withthe exception that here 257 frequency points were measured.For the indoor scenario, the time for sounding is 3.2 µ s for onetransmit-receive antenna pair, leading to a time per snapshot ofapproximately 7.37 ms. For the outdoor scenario, these valuesdoubles due to measuring twice as many frequency points. D. Real-time testbed
The Lund University Massive MIMO testbed (LuMaMi)was used for the second measurement campaign with theplanar array. This testbed is based on software-defined radiotechnology and is operating in real-time at a carrier frequencyof 3.7 GHz with 20 MHz of bandwidth. More informationabout the technical details of the testbed can be found in[29] and [30]. The planar array connected to the testbed isshown in Fig. 4, where the lower part shows the array fromthe front. The upper four rows with 25 antennas at eachrow are connected to one transceiver chain each, resulting in100 antenna elements, spaced half a wavelength apart. Everyother antenna element is vertically polarized while the otherelement has a horizontal polarization. The measurements whenusing the testbed produced 100 frequency points per user andfor 20 seconds, 2000 snapshots were stored .III. C HANNEL HARDENING IN THEORY
Following [4], let h k be the channel vector between thebase station and user k . The channel h k offers hardening ifVar {k h k k } E {k h k k } → , as M → ∞ , (1)where M is the number of base station antennas. In [5], thecoefficient of variation ( CV ) is introduced and further derivedas CV = Var {k H k F } E {k H k F } = E ( A tx , D tx ) E ( A rx , D rx ) E {k c k − k c k } E {k c k } + Var {k c k } E {k c k } , (2)where k H k F is the Frobenius norm of H , E ( A tx , D tx ) and E ( A rx , D rx ) are the second moments of the inner productsbetween the steering vectors of two different physical pathsfor the transmitter and receiver, respectively, and k c k is theaccumulated channel gain from all physical paths. In our case H = h k . The first term in (2) can be considered as a small-scale fading factor. The second term can be considered as alarge-scale fading factor, which for channel coefficients withindependent channel gains with variance σ and mean µ is For completeness, it should be noted that 9 snapshots were lost duringthe measurements with user 1, and 7 snapshots during the measurements withuser 5, possibly due to bad synchronization. These snapshots were thereforeremoved when analysing the data. shown to be equal to P ( σµ ) [5], where P is the number ofphysical paths in the environment. Under the assumptions thatthe channel coefficients for all physical paths are CN (0 , andthat the steering vectors of the physical paths are distributeduniformly over the unit sphere, (2) can be further rewritten as CV = 1 M (1 − /P ) + 1 /P (3)where we here consider M base station antennas and userantenna [5]. The standard deviation of the channel gain as afunction of the number of base station antennas and physicalpaths is shown in Fig. 5, obtained through a simulation basedon (3) and then taking the square-root of the results. Asexpected, more base station antennas increase the channelhardening, although this is only true if there are enoughphysical paths in the environment. The achievable channelhardening in a specific environment will therefore be limitedeither by the number of base station antennas or the numberof physical paths, assuming a fixed number of antennas for theuser equipment. It is worth noting in (3) that if the numberof physical paths P → ∞ , then the coefficient of variation CV → M . If also the number of antennas M → ∞ then CV → , and the channel offers hardening as defined in (1).The data, acquired as previously described for the two mea-surement campaigns, has been analyzed in order to validatethe theoretical channel hardening results. This is done in thesame way as in [9], and similar to the investigation in [11] [12].Specifically, the measured channel transfer functions have beennormalized according to h k ( n, f ) = h k ( n, f ) q NF M P Nn =1 P Ff =1 P Mm =1 | h km ( n, f ) | , (4)where N is the number of snapshots, F is the number offrequency points and M is the number of selected base stationantennas, i.e. those over which the standard deviation is latercomputed in order to obtain a measure of the experiencedchannel hardening. This normalization makes sure that the PSfrag replacements
Number of base station antennas S t a nd a r dd e v i a ti on [ d B ] Number of physical paths
Fig. 5: Standard deviation of channel gain as a function ofnumber of base station antennas and the number of physicalpaths. Interpretation based on [5]. average power of each entry in h k , averaged over frequency,time, and base station antennas, is equal to one.For M selected base station antennas, the instantaneouschannel gain for each user is defined as G k ( n, f ) = 1 M M X m =1 | h km ( n, f ) | , (5)such that the average channel gain µ k = 1 N F N X n =1 F X f =1 G k ( n, f ) = 1 (6)is independent of the number of antennas selected at the basestation. This means that the base station can reduce the totaloutput power with a factor of M , i.e. the beamforming gain.The standard deviation of channel gain is computed for eachuser according tostd k = vuut N F N X n =1 F X f =1 | G k ( n, f ) − µ k | , (7)where the instantaneous channel gain for user k , G k ( n, f ) ,is given in (5) and the average channel gain for user k , µ k , is given in (6). The standard deviation in (7) is indeedan estimate, ˆ CV ( M ) , of the (square root of the) coefficientof variation (3), for some M ≥ . In the following, whenquantifying the channel hardening for some subset of antennasof size M , we use the difference ˆ CV ( M ) − ˆ CV (1) (8)of the standard deviation as given in (7). This means thatdepending on which antenna element that is chosen as the ref-erence element, the channel hardening will result in differentvalues. How this reference element is chosen in our analysiswill be elaborated on in the next section.IV. C HANNEL HARDENING IN PRACTICE
This section presents our contribution to the experimentalresearch on channel hardening in real environments, extendingthe initial results in [9] to include more scenarios, arraygeometries and propagation characteristics. In addition, weextend the analysis and provide further insights on aspectsthat impact the experienced channel hardening and hence, theoverall reliability in massive MIMO systems.For the indoor scenario with the cylindrical array, see Fig. 1and Fig. 3, the un-normalized average channel gain for each ofthe 128 base station antennas is shown in Fig. 6, for all nineusers. Since the channels are not normalized, what is seenare the actual measured differences in average channel gainbetween users and antenna elements in the array. As expected,there are for each user large variations over the array whichcan be seen as the four larger peaks and dips, one for eachring of antennas, due to some antennas experiencing a LOScondition while some are not. Fig. 3 is used as reference forthe numbering of the antennas in the rings as well as thenumbering between the two polarizations. The latter explains
PSfrag replacements
Base station antenna A v e r a g ec h a nn e l g a i n [ d B ] Fig. 6: Unnormalized average channel gain for the 128 basestation antennas in the cylindrical array shown for all nineusers. Numbering of the users is row-wise, starting from thetop left corner.
Time [s] Frequency [MHz] N o r m a li ze d c h a nn e l g a i n [ d B ] Fig. 7: Normalized channel gain for user 1 showing the singleantenna with the median average channel gain (lower) andthe total channel gain for all 128 base station antennas in thecylindrical array (upper).the more local variations in Fig. 6, where the average channelgain is alternating between every two consecutive antennas.Both these variations contributes to an imbalance between theantenna elements in the array. The results in Fig. 6 can also beexplained as an effect of the interaction between the cylindricaldual-polarized array and the environment. The variations thatare seen show that the multipath components are not comingin with equal strength from all directions. They are rathercoming dominantly from some distinct angles, in line withother experimental studies [15] [16], and the previous analysisin [28], where also the distribution of channel coefficients canbe found. Detailed cluster analysis can be found in [27] Thisobservation questions the appropriateness of modeling massiveMIMO channels with the complex Gaussian channel modeland strengthens the conclusion that massive MIMO channelsare indeed spatially correlated.
PSfrag replacements
Number of base station antennas S t a nd a r dd e v i a ti on [ d B ] GaussianOriginalStrongest firstWeakest first
Fig. 8: Standard deviation of channel gain as a function ofnumber of base station antennas when selecting antennas indifferent orders, for user 1 and the cylindrical base stationarray, and the complex Gaussian channel as reference.
A. Channel hardening in time and frequency
A clear visualization of the channel hardening effect is seenin Fig. 7, where the normalized channel gain with a singlebase station antenna versus the channel gain when using all128 base station antennas is shown for user 1 in the indoorscenario. For the single antenna case, M = 1 , the antennaelement in the base station array which has the median averagechannel gain is chosen. For this single antenna, there arelarge dips in both time and frequency. The combination ofall 128 antennas results in both array gain, yielding a higheraverage, and smaller variations relative to this average. It isevident that in the time domain, there is a channel hardeningeffect. The channel flattens out when using a large numberof base station antennas; the remaining variations are causedby the movements of the user equipment. The channel hard-ening is even more evident in the frequency domain; for themeasured scenario and the considered bandwidth, the channelresponse becomes almost entirely flat. For comparison, in thisscenario, the difference between the maximum and minimumnormalized channel gain in the time domain, averaged oversubcarriers and for M = 1 and M = 128 respectively, is36 dB and 9 dB. The corresponding values for the frequencydomain are 16 dB and 2 dB. In general in massive MIMOsystems, the result will be similar as it is straight-forward toapply precoding to flatten out the channel in the frequencydomain. However, shadowing effects in the time domain cannot easily be compensated for. B. Channel hardening with different subsets of antennas
The standard deviation of channel gain as the number ofbase station antennas M increases from 1 to 128 is shownin Fig. 8 for user 1. The channels are normalized accordingto (4) and the instantaneous channel gain for every subset iscomputed for all frequencies and snapshots as in (5) beforethe standard deviation of channel gain for each subset is computed using (7). The blue solid line shows the result forthe complex Gaussian channel, which for 128 base stationantennas has a channel hardening, measured as the decreasein standard deviation of channel gain when going from 1 to128 base station antennas as in (8), of around . dB. Thisis close to the theoretical value of
10 log ( √ dB. Thestandard deviation of channel gain is also shown in Fig. 8when selecting the antennas in different orders as the numberof base station antennas increases. All the curves for themeasured responses end up at the same point but they evolvedifferently as the number of antennas increases and havedifferent reference elements and therefore different startingpoints.The case ’Original’ is where the antennas are chosen inthe order shown in Fig. 3. For the indoor scenario, the firstfew selected antennas are in NLOS and then some antennasin LOS get included in the selected subset resulting in atemporary increase of the standard deviation, as seen for theblack dashed line. The case ’Strongest first’ means choosingthe antennas with the highest average channel gain first, ascomputed for all frequency points and snapshots. This seemsto be the most reasonable choice and leads to quite a steadydecrease of the standard deviation in logarithmic scale. Forcomparison, the case ’Weakest first’ is also shown, which issimply the reverse of the ’Strongest first’ label. This results ina increase of the standard deviation when the antennas withhigher channel gain, relative to the already included antennas,also are included in the subset. An observation from Fig. 8is that whether the subset of selected antennas is in LOS,NLOS or a combination therefore, affects the behavior of thestandard deviation curve, both in terms of starting point andregarding the course of the curve. For further analysis see [9].An additional remark is that even though the order ’Weakestfirst’ has the least variations around its average for M inthe range from 5 to 50 antennas, and thus offers the mostchannel hardening, one should note that this still may not bethe option to go for since the power level is normalized forthe selected antennas. This makes them comparable in termsof variations relative to their respective means but that does notguarantee that choosing the weakest antenna first results in agood enough channel in terms of absolute channel gain. As anexample, for M = 10 , the un-normalized total channel gain,as summed for the first ten antennas and the whole bandwidth,is -30 dB for ’Strongest first’ and -42 dB for ’Weakest first’,averaged over snapshots. Therefore, only the ’Strongest first’order will be used in the analysis from now on, in order toavoid any confusion.As previously indicated, the polarization state is causingvariations in the channel gain over the array. Computing thestandard deviation for M = 1 , . . . , when using one of thepolarizations or a combination of both yields the results inFig. 9. For ’Both polarizations’, the set of antennas are selectedas one polarization per antenna, starting with vertical polariza-tion, and then alternating between the two polarizations as ittraverses through the rings. This means that the three optionshave the same aperture. The experienced channel hardeningdepends on the polarization set selected, and thereby alsoon the selected reference element. Fig. 9 shows the decrease PSfrag replacements
20 0-2-4-6-8-10 1010 1
Number of base station antennas S t a nd a r dd e v i a ti on [ d B ] GaussianBoth polarizationsVertical polarizationHorizontal polarization
Fig. 9: Standard deviation of channel gain as a functionof number of base station antennas when using one of thepolarizations or a combination of both, for user 1 and thecylindrical base station array, and the complex Gaussianchannel as reference.of standard deviation when going from 1 to 64 antennas,using the definition in (8). When using the 64 horizontallypolarized antennas, the channel hardening is 2.7 dB, while thecase with the 64 vertically polarized antennas gives a channelhardening of 2.8 dB. It is worth noting that even though thechannel hardening is almost the same for these two curves,the specific value of the standard deviation of channel gainat each point is smaller when using the vertical polarizationsince these antennas already have smaller variations relative toits average compared to the horizontally polarized antennas.An interpretation of this is that the channel already can beconsidered as ’pre-hardend’, which could be a result of theLOS component being mainly vertically polarized. This issupported by the fact that when computing the mean of the un-normalized average channel gain, this results in -64 dB for thevertically polarized antennas and -68 dB for the horizontallypolarized antennas. The most channel hardening, and also thelowest standard deviation of channel gain for M = 128 , isachieved when using a combination of both polarizations; thenthe channel hardening is 4.2 dB. This is partly explained bythe ability to exploit polarization diversity in the environment,i.e. an increased number of physical paths, but is also an effectof having different reference elements. C. Analysis of different scenarios
To investigate differences and similarities between scenar-ios, the channel hardening is also analysed for measurementsfrom the outdoor scenario, both in LOS and NLOS, as depictedin Fig. 2. For all three scenarios, the users with the most andleast channel hardening are shown in Fig. 10a and Fig. 10b,respectively. Note that these are not necessarily the users withthe smallest and largest standard deviation for a specific subsetof antennas, such as for M = 128 . In Fig. 10a it can beseen that the user with the most channel hardening is inthe outdoor NLOS scenario. However, the channel from the indoor scenario could already be interpreted as being ’pre-hardened’, as it for each subset of antennas already has asmaller standard deviation of channel gain relative to its mean.The users with the least channel hardening, and also the largeststandard deviation of channel gain for M = 128 , are in generalfound in the outdoor LOS scenario. Recalling that the channelsfor the different users are normalized, it should be noted thateven though users in the outdoor LOS scenario have the leastchannel hardening it does not necessarily imply that it is abad channel since the un-normalized channel could still havea considerable high received channel gain and that a strongLOS in general gives good communication performance. Acomparison of the mean and maximum un-normalized channelgain in the three scenarios is found in Table I. It can beobserved that it is indeed the case that both the mean andthe maximum channel gain, as per antenna, are as expectedhigher in the outdoor LOS scenario than in the NLOS scenario,although lower than in the indoor scenario. However, thedistances are different and thereby also the pathloss.Fig. 11 shows the empirical CDFs of their channel gainswhen M = 128 for all scenarios and users, providing furtherinsights about the behaviour of the curves in the differentscenarios. The marked users experience the least and mostchannel hardening, also shown in Fig. 10a and 10b, respec-tively. Evidently, the steeper the CDF is, the more channelhardening. Another observation is that if there are strongoutliers, the standard deviation level is increased; this is thecase for the outdoor scenarios. As a reference, the exponentialdistribution with λ = 1 is also provided, corresponding to thechannel gain, i.e. | h | , with a mean of for the one-antennacase where the channel is complex Gaussian distributed; thismeans that there is no channel hardening. Given this, theobservation is that although the outdoor LOS scenario is to alarge extent similar to the reference curve for higher channelScenario Mean MaximumFig. 10a Indoor -71 -66Outdoor LOS -100 -93Outdoor NLOS -107 -104Fig. 10b Indoor -72 -66Outdoor LOS -88 -81Outdoor NLOS -105 -99TABLE I: Comparison of the antennas with the mean andmaximum average un-normalized channel gain in the differentscenarios. Scenario Mean ratio Std ratioFig. 10a Indoor -0.7 8.0Outdoor LOS -0.2 7.8Outdoor NLOS -0.3 7.8Fig. 10b Indoor 1.5 7.9Outdoor LOS 1.4 7.9Outdoor NLOS 1.1 7.9TABLE II: Mean and standard deviation (std) of the ratiobetween the channel gains of the vertical and horizontalpolarization for the users in Fig. 10. gains, the risk of lower gains, affecting the probability ofoutage, is reduced when combining the many antennas. Asa final remark on the comparison between different scenarios,a quantification of the average channel hardening results in . dB for the indoor scenario, . dB for the outdoor LOSscenario and . dB for the outdoor NLOS scenario. D. Influence of polarization
After exploring the influence of LOS and NLOS on theexperienced channel hardening, further investigations includeda revisit to take a closer look into the polarization aspect. InTable II, the mean and standard deviation of the ratio betweenthe channel gains of the vertical and horizontal polarization,as per snapshot, frequency point and antenna element, areshown for the users in Fig. 10. Although, they all seem tohave similar standard deviation, the mean is in general closerto zero for the users with the most channel hardening; this islikely contributing to a more even distribution of the channelgain.
E. Interaction between environment and antenna array
The last aspect we want to further investigate is the in-teraction between the environment and the array. Therefore,measurements were also performed in the indoor scenario,as previously has been described (see Fig. 1), with the pla-nar array shown in Fig. 4 for user 1 and 5. The resultingchannel hardening curves are shown in Fig. 12. For thesemeasurements the users were surrounded with a crowd, suchthat the square in Fig. 1 was almost filled with people. Theresults in Fig. 12 show that significantly smaller standarddeviations of channel gain, i.e. down to -6.8 dB, are achievedfor both users when using the planar array in comparison tousing the cylindrical array. This demonstrates that for a typicaldeployment geometry, the planar array can better exploit thediversity in the environment as more of the antennas areeffective and the distribution of channel gain over the array ismore even; the latter can be seen in Fig. 13 where the slopesof the empirical CDFs of the normalized channel gain for thescenario with the planar array are steeper.V. C
HANNEL HARDENING IN THE
COST 2100
CHANNELMODEL
This section provides a short description of the COST 2100channel model, the simulations performed with the model,which are later used for comparison, and an example resultfrom the simulations. Initial answers to the questions if andhow well the COST 2100 channel model with its modelparameters in general can capture the channel hardening effectare also given. This is further elaborated on in the comparisonin Section VI.The COST 2100 channel model [24], including the massiveMIMO extension described in [27], is a GSCM that canstochastically describe massive MIMO channels over time,frequency and space in a spatially consistent manner. The goalwith the extension is to capture channel characteristics thatbecome more prominent in massive MIMO channels. Theseextensions are:
PSfrag replacements
Number of base station antennas S t a nd a r dd e v i a ti on [ d B ] GaussianIndoorOutdoor LOSOutdoor NLOS (a) The users with the most channel hardening
PSfrag replacements
Number of base station antennas S t a nd a r dd e v i a ti on [ d B ] GaussianIndoorOutdoor LOSOutdoor NLOS (b) The users with the least channel hardening
Fig. 10: Standard deviation of channel gain as a function of number of base station antennas for the three different scenarioswith the cylindrical array, and the complex Gaussian channel as reference.
PSfrag replacements F ( x ) Normalized channel gain [dB] (a) Indoor
PSfrag replacements F ( x ) Normalized channel gain [dB] (b) Outdoor LOS
PSfrag replacements F ( x ) Normalized channel gain [dB] (c) Outdoor NLOS
Fig. 11: Empirical CDFs of the normalized channel gain when using all 128 antennas in the cylindrical array, for all scenariosand users. The users marked in (a) black dashed, (b) orange dotted and (c) green dash-dotted lines have the least and the mostchannel hardening, and correspond to the users shown in Fig. 10. To serve as reference, the exponential distribution ( λ = 1 )is in blue full line. •
3D propagation, • the possibility for different parts of a physically largearray to experience different clusters, and • a gain-function regulating the gain for individual mul-tipath components, which is of particular importance toclosely-spaced users.For the evaluation in this paper we use the indoor sce-nario with the cylindrical array and closely-spaced users at2.6 GHz. This scenario has been parameterized based on themeasurement data used in previous experimental analysis andthe complete model can be found at [25]. More specifically,the chosen settings when running the simulations are outlinedin Table III.In the simulations, 300 snapshots were generated at a rate of50 snapshots per second with a user velocity of 0.25 m/s; thismeans that samples are taken over a distance of 1.5 m in total.129 points in frequency are computed. The simulated propaga-tion environment is combined with a synthetic antenna patternof a cylindrical array with 128 antennas at the base station side. At the user side, the multipath components obtained from thesimulated environment are combined with either an antennapattern with user effect or an omni-directional antenna, forcomparison. A description of the antenna pattern with usereffect can be found in [31].Each user antenna starts with a random initial orientationuniformly generated between [ − π π ) , and during the sim-ulation each user antenna undergoes a randomly generatedrotation between [ − π π ) in total in the azimuth plane. Forthe obtained transfer function, the channels are normalized asSettingNetwork ’Indoor CloselySpacedUser 2 6GHz’Link ’Multiple’Antenna ’MIMO Cyl patch’Band ’Wideband’TABLE III: Settings chosen for the simulations with the COST2100 channel model. PSfrag replacements
Number of base station antennas S t a nd a r dd e v i a ti on [ d B ] GaussianCylindrical user 1Cylindrical user 5Planar user 1Planar user 5
Fig. 12: Standard deviation of channel gain as a function ofnumber of base station antennas for user 1 and 5, and thecomplex Gaussian channel as reference. The scenario is theindoor scenario where the users are surrounded by a crowdwith the cylindrical and the planar array, respectively.
PSfrag replacements F ( x ) Normalized channel gain [dB]
ExponentialCylindrical user 1Cylindrical user 5Planar user 1Planar user 5
Fig. 13: Empirical CDFs of the normalized channel gain whenusing all 100 or 128 antennas for the two arrays and users inFig. 12. and the exponential distribution ( λ = 1 ) as reference.in (4) and the standard deviation of channel gain is computedas in (7), in the same manner as for the measurement data.Running the simulation ten times, and averaging the resultingstandard deviation of channel gain over these ten simulations,gives the result in Fig. 14 for a typical user.As an example result from the simulations, Fig. 14 showsthe resulting standard deviation both when using the omni-directional and measured antenna pattern with user effect, forthe same propagation environment. The complex Gaussianchannel is also shown as a reference. The difference of thestarting point of the two curves with different antenna patternsis almost 0.9 dB while at the end point, the difference isabout 3.4 dB. The result here is only for one specific userantenna pattern, and changing this pattern would affect thechannel hardening. However, it shows that there is both alimitation on the channel hardening that is imposed by thesimulated environment, in line with the presented theory inSection III, but also that the antenna pattern of the user can PSfrag replacements
Number of base station antennas S t a nd a r dd e v i a ti on [ d B ] GaussianCOST omniCOST measMeas bestMeas worst
Fig. 14: Standard deviation of channel gain as a functionof number of base station antennas with an omni-directionalantenna (COST omni) or an antenna pattern with user effect(COST meas) in the COST 2100 channel model for a typicaluser, and the complex Gaussian channel as a reference.reduce the experienced channel hardening as much as to halfof the channel hardening experienced with an omni-directionalantenna, on a linear scale. In general, any user equipment witha non omni-directional antenna pattern will likely degrade theexperienced channel hardening as not all scatterers will beeffective.VI. C
OMPARISON BETWEEN THEORY , MEASUREMENTSAND SIMULATIONS
The last part of this study is a joint comparison of the threelevels at which channel hardening has been analyzed: theory,measurements and simulations. This provides a more thoroughanalysis of channel hardening and synthesizes the three pointsof view. We discuss aspects which are relevant when designingreliable massive MIMO systems and highlight channel andantenna characteristics important for channel hardening.In Fig. 15, a comparison between theory, measurementsand simulation results from the COST 2100 channel model isshown. The users shown from the measurements and simula-tions are the users with the most channel hardening (Fig. 15a)and the least channel hardening, (Fig. 15b) among the nineusers when using all 128 base station antennas. As reference,the complex Gaussian channel is included.Quantifying the general behaviour of the curves from thesimulations, Fig. 16 shows the decrease ( ∆ ) of standarddeviation when increasing the number of antennas, averagedover all users. Here, it can be seen that on average, the slopesresulting from the measurements and the COST simulationswith measured antenna pattern have very similar behaviorsand that the model in general shows good correspondence.Meanwhile, the COST simulations with an omni-directionalantenna have in general a larger slope, although not as highas in the complex Gaussian case due to the limitations in theenvironment. Considering the average channel hardening inthe three cases, the measurements give a channel hardening of3.9 dB, while the COST simulations on average give, for the PSfrag replacements
Number of base station antennas S t a nd a r dd e v i a ti on [ d B ] GaussianMeasurementCOST omniCOST meas P = 13 P = 29 P = 5 P = 22 (a) The users with the most channel hardening PSfrag replacements
Number of base station antennas S t a nd a r dd e v i a ti on [ d B ] GaussianMeasurementCOST omniCOST meas P = 4 P = 11 P = 3 P = 22 (b) The users with the least channel hardening Fig. 15: Standard deviation of channel gain as a function of number of base station antennas for one user from (i) themeasurements, (ii) the simulation with an omni-directional antenna and (iii) the simulation with an antenna pattern with usereffect in the indoor scenario with the cylindrical array, and the complex Gaussian channel as reference.
PSfrag replacements
20 0-0.5-1-1.5-2-10 101010 1
Number of base station antennas ∆ o f s t a nd a r dd e v i a ti on [ d B ] GaussianCOST omniCOST measMeasurementMeas worst
Fig. 16: Decrease ( ∆ ) of standard deviation of channel gain asa function of the increasing number of base station antennasfor the average user in (i) the measurements, (ii) the simulationwith an omni-directional antenna and (iii) the simulation withan antenna pattern with user effect, and the complex Gaussianchannel as reference.measured and omni-directional antenna, a channel hardeningof 3.5 dB and 5.3 dB, respectively. Looking at the averagestarting points for the three cases, the measurements start at0.3 dB while the COST simulations with the measured antennapattern start at 0.6 dB and the omni-directional antenna at-0.8 dB. Below we summarize the results and discussions fromthree important aspects. The complex Gaussian channel model is overoptimistic:
Starting the discussion from a theoretical point of view, thecomplex Gaussian channel, which is only limited by thenumber of antennas, gives a channel hardening of 10.5 dBwhen having 128 base station antennas. This channel hasoften been assumed in theoretical studies of massive MIMOand is shown in this study to be more optimistic than realmeasured massive MIMO channels, which in general are spa- tially correlated. In reality the environment does generally notcreate as rich scattering as assumed in the complex Gaussianchannel. As a result, channel responses are rather spatiallycorrelated, and thus the environment also imposes a limitationon the achievable channel hardening; this fact is capturedby (3). Moreover, the interactions between the array and theenvironment can result in quite different received power levelsat the antenna elements.
The channel gain distribution over the array is essential:
Repeating the assumptions made in [5] in order to get to (3), itwas assumed that the channel coefficients for all physical pathsare CN (0 , and that the steering vectors of the physical pathsare distributed uniformly over the unit sphere. If the mean µ and the variance σ would deviate from this assumption,then the large-scale fading term in (3) would be bounded by P ( σµ ) , see [5]. With a large variance, relative to the mean, thestandard deviation curves would move up. The curves wouldmove down if the variance is small relative to the mean. Whatcan be seen in e.g. Figs 8-10 is indeed that when havinga higher mean, the standard deviation curves in the indoorscenario move down; this is likely due to having a strong LOScomponent. Having a large imbalance in the channel gain overthe array causes variations and can hence move the standarddeviation curves up; causes of these variations can be dueto having some antennas experiencing a strong LOS whileothers are not. This imbalance can also occur between thepolarization modes. These aspects could as well change overtime and thereby much of the remaining variations of channelgain come from the time domain. The user movements and antenna patterns are uncertain:
The large variation of channel gain in the time domain canbe hard to predict as there is always an uncertainty in theuser movements and as a result of this, in the interactionbetween the environment and the antenna patterns of theuser equipment. It is clear that the user antenna pattern andbehaviour can be a limiting factor for the channel hardening.This can be seen in Fig. 15 as the simulated curves with the antenna pattern with user effect are close to the user withthe least channel hardening in the measurements in Fig. 15b,while a user with an omni-directional antenna experiencesmore channel hardening. Assuming a non-beneficial scenariofrom the user aspect, then this type of simulation, where theuser equipment is rotating, could be used while in reality itcould happen that conditions are more favourable and thus asmaller standard deviation could be achieved. For example, theuser from the measurements with the most channel hardeningin Fig. 15a, experiences a similar channel hardening as thesimulations with the omni-directional antenna in Fig. 15b.Overall, the most significant reasons for the differences be-tween the measurements and the simulations in Fig. 15 arethe different antenna patterns and movements of the users;to further align the results, then these parameters should bethe same. Further on, since the COST 2100 channel modelis a stochastic channel model, comparisons with particularmeasurements are difficult. However, on average the channelhardening curves resulting from the model show very similarbehavior as for the measurements, as seen in Fig. 16.VII. C ONCLUSION
From our experiments and analysis, we can conclude thatthere is a clear channel hardening effect in massive MIMO,which flattens out the channel fading in frequency and timeas the number of antennas increases. However, the complexi.i.d. Gaussian channel model, which is commonly assumedin theoretical studies, is shown to be overoptimistic anddoes not provide an accurate model for channel hardening inmassive MIMO. This is due to the fact that real environmentsdo not provide as rich scattering as assumed in the modelbut are rather spatially correlated, as recently acknowledgedin literature. Moreover, interactions between the array andthe environment will cause variations of the received powerlevels between the different antenna elements. Therefore, byinvestigating the mechanisms that build up the distribution ofchannel gain over the array, the channel hardening effect canbe better understood. Power imbalances can be due to havinga set of antennas where some have a stronger LOS and somenot; this imbalance could also be between the polarizationmodes. As these imbalances change over time, there will stillbe variations of the channel gain that are originating from thetime domain. These changes are hard to predict as there isalways an uncertainty of user movements and antenna pattern.We have also shown that the COST 2100 model with itsmassive MIMO extension statistically can capture the channelhardening effect well, given appropriate propagation param-eters and a realistic antenna pattern and user behavior. As afinal remark, in order to maximize the experienced channelhardening, one needs to find strategies to cope with the powerimbalances over the array and design the massive MIMO sys-tem such that it can best exploit the diversity of the consideredenvironment. However, channel hardening is only one part ofthe complex story of achieving reliable communication and infuture work other aspects, such as received power levels andcoverage, should also be considered. R
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