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Spatially-Consistent Human Body BlockageModeling: A State Generation Procedure
Margarita Gapeyenko, Andrey Samuylov, Mikhail Gerasimenko, Dmitri Moltchanov, Sarabjot Singh,Mustafa Riza Akdeniz, Ehsan Aryafar, Sergey Andreev, Nageen Himayat, and Yevgeni Koucheryavy
Abstract—Spatial correlation has been recognized by 3GPP as one of the key elements in millimeter-wave (mmWave) channelmodeling. Correlated channel behavior is induced by macro objects, such as buildings, as well as by micro objects, includinghumans around the mmWave receivers. The 3GPP’s three-dimensional (3D) spatially consistent channel model designed tocapture these phenomena assumes a-priori knowledge of the correlation distance between the receivers. In this paper, wepropose a novel spatially-consistent human body blockage state generation procedure, which extends the standardized 3Dchannel model by 3GPP to capture the correlation between the line-of-sight (LoS) links and the reflected cluster states affectedby human body blockage. The proposed model is based on analytical expressions for the conditional link state probability, thuspermitting the parametrization of the spatial field of receivers. It also does not require any a-priori information on the correlationdistance as the latter is identified explicitly based on the environmental parameters. We compare the results for the proposedmodel with those obtained with the uncorrelated blockage model and conclude that in many special cases correlation manifestsitself in quantitatively different propagation conditions experienced at the nearby receivers.
Index Terms—5G, mmWave, 3GPP 3D channel model, human body blockage, spatial consistency, correlation.
F
1 INTRODUCTIONMillimeter-wave (mmWave) communication is con-sidered to be the core part of the emerging 5G mobilenetworks, which are capable of supporting the strin-gent requirements of IMT-2020 [1]. Larger availablebandwidths make the extremely high frequency bandsan attractive candidate for serving advanced futureapplications [2]–[5].
Despite a number of benefits delivered bymmWave, there are also several challenges to besolved. For example, due to shorter wavelengths,smaller objects in the channel may produce aconsiderable impact on the mmWave propagation.According to the recent studies, human bodyblockage leads to a significant attenuation of themmWave signal [6]–[8] and should be taken intoaccount in mmWave channel modeling [9], [10].
An example of such models is the 3GPP three-dimensional (3D) stochastic channel model (SCM)proposed in [11] that has further been improved andratified by 3GPP in Release 15 [9]. It is currentlyutilized by both academia and industry to capture themmWave channel properties in system-level simula-tors (SLS) [12], [13].
If mmWave-based receivers are located next to oneanother, they often experience similar propagation
M. Gapeyenko, A. Samuylov, M. Gerasimenko, D. Moltchanov, S. An-dreev, and Y. Koucheryavy are with Tampere University, Tampere, Finland(e-mail: firstname.lastname, [email protected]).S. Singh is with Uhana (e-mail: [email protected])M. R. Akdeniz and N. Himayat are with Intel Corporation, Santa Clara,CA, USA (e-mail: mustafa.akdeniz, [email protected])E. Aryafar is with Portland State University, Portland, OR, USA (e-mail:[email protected])
conditions [14]–[16]. This effect, known as spatial cor-relation, has been recently recognized by 3GPP asan important consideration in the mmWave chan-nel modeling [9]. The correlated state of the chan-nel at the receiver (Rx) may affect implementationand performance of beamsearching and beamtrackingmechanisms, resource allocation strategies, as well asmultiple-input multiple-output (MIMO) system de-sign [9], [17]–[20].
1.1 Background and Related Studies
3GPP has recently extended its 3D SCM channelmodel to capture the correlation of large and smallscale parameters (LSP and SCP) as well as (non)-line-of-sight (n)LoS states. In [9], [21], three such methodshave been proposed. In the first one, named methodof spatially-consistent random variables, the spatial cor-relation of channel clusters is accounted for by intro-ducing the so-called spatial consistency to the channelcluster specific random variables taken from the 3GPP3D SCM model [9].
The second method is known as geometric stochas-tic approach. In this case, the large scale parameters(LSP) are pre-computed for every grid, and each Rxinside it is associated with these LSP. The grid hasa rectangular shape with the side length of correla-tion distance that is provided a-priori. In the thirdalternative, called method of geometrical cluster locations(grid-based GSCM, GGSCM), the cluster, path angles,and delays are defined by the geometrical positions(x, y) of the appropriate scatterers. In all these models,the correlation distance is arbitrarily chosen as an
2
input parameter, which leads to non-uniqueness of theresulting propagation environment.
With the help of a ray-based simulator, the authorsin [22] demonstrated a profound impact of spatialconsistency on the path loss modeling. The work in[23] described a spatially consistent path loss modelfor the street urban scenarios. In [24], the authors pre-sented their simulator wherein the spatially consistentchannel model is integrated.
In addition to macro objects affecting the LoS/nLoSstate, smaller objects (e.g., humans, vehicles, etc.)induce blockage of the mmWave channel. When mod-eling highly crowded realistic urban environments,such as squares and stadiums, the use of ray-tracingapproaches is often difficult due to the associatedcomputational complexity. Recently, a number of an-alytical models for blockage phenomena were pro-posed [25]–[27]. However, these constructions eitherdo not capture the effect of spatial correlation or donot offer a method to optimize the channel model.In [28], the authors measured the channel from dif-ferent base station (BS) locations and demonstratedhow the knowledge of correlation distance may helpfind another best BS in case of blockage. Further,in [29], the authors argued for the importance ofcorrelated blockage consideration. They proposed amodel to establish the probability that a certain targetis blocked, while having more than one transceivercommunicating with that target.
Recently, there have been multiple attempts within3GPP to extend the SCM model by capturing thespatial correlation of blockage caused by micro ob-jects, including the human crowd [9]. Particularly,in [9], a human body has been modeled and twoblockage models, namely, A and B, were introduced.The model A stochastically generates M 2D blockageregions uniformly distributed around each Rx. Theparameter M is a fixed number that may be changedin case the density of blockers varies. The latter modeldoes not account for the height of blockers and as-sumes a fixed distance between Rx and blocker, whichsignificantly reduces the applicability of this model. Inorder to account for the spatial correlation betweenblockers, a certain autocorrelation function is appliedto the centers of blockers. The limitation of this modelis in the correlation distance, which is a parameter thatneeds to be specified in advance.
In model B, a total of M rectangular screens arephysically placed on a map. This allows to accountfor any density, dimensions of blockers, as well asspatial consistency during the simulation time. How-ever, the computational complexity of this methodincreases significantly with the growing numbers ofblockers, thus making it infeasible to apply for mod-eling densely crowded environments.
1.2 Contributions of This Work
In this paper, we complement the existing 3GPP3D channel model for the frequencies of 0.5 to100 GHz [9] by accounting for the spatial correlationcaused by micro objects (particularly, humans bodies)in crowded scenarios, e.g., squares, stadiums, etc.Particularly, we propose a novel spatially consistentblockage state generation procedure, which employsour analytical framework for the conditional link stateprobabilities. Compared to 3GPP blockage models,our approach features the following advantages: (i)correlation distance does not need to be specified inadvance, (ii) spatial correlation is captured across allof the blockers, and (iii) computational complexitydoes not depend on the density of blockers, whichmakes it possible to model crowded environments.
Our main contributions are therefore as follows:• We analyze the spatial correlation of human body
blockage caused by a dense crowd around thereceivers for the mmWave channel by propos-ing a novel mathematical framework. We derivethe conditional channel state probabilities for ev-ery link generated within the scenario, such asTransmitter-Rx links or Reflector-Rx links;
• We integrate the proposed analytical model intothe SLS software [30], with our new block-age state generation procedure taking into ac-count the actual correlation across micro objects.This blockage generation procedure allows tooptimize the channel model by introducing anadditional channel state (blocked/non-blocked)for every link, which captures the spatially-consistent human body blockage;
• We characterize the absolute received power dif-ference when disregarding the correlation due tomicro objects in the mmWave channel by con-ducting extensive simulations. We thus demon-strate that the spatial correlation is a local effect,which leads to significant received power varia-tions at nearby locations when it is not accountedfor.
The rest of this paper is organized as follows. Thesystem model and its assumptions are introducedin Section 2. Our mathematical framework is devel-oped in Section 3. The spatially-consistent blockagestate generation procedure for mmWave propagationmodeling is outlined in Section 4. The key numericalresults are reported in Section 5. Conclusions aredrawn in the last section.
2 SYSTEM MODEL AND ASSUMPTIONS
2.1 Propagation Environment
The considered scenario is illustrated in Fig. 1. Thetransmitter (Tx) is assumed to be located at the originand has the height of hT . The human body blockers(referred to as blockers further on) are uniformly
3
mmWaveTx
hR
Reflectionpoint
Blocker
R4
2
3
2
3
4
4 - Uncorrelated clusters links
1 - Correlated LoS links
- Uncorrelated LoS links
- Correlated clusters links
R2R1 R3
1
Blocker
H
hT
C1
C2
C3
Fig. 1. The considered scenario of interest.
distributed in the area and modeled as cylinders [31]with the random height of H as well as the constantbase diameter of dm. The blocker heights are assumedto be approximated by a Normal distribution H ∼N(µH , σH) [32]. The centers of the cylinder bases fol-low a 2D Poisson point process (PPP) with the densityof λB . A total of K Rx are uniformly distributed in thearea S with the coordinates xR,k, yR,k, k = 1, . . . ,K.The size of Rx is assumed to be infinitesimally small.The rest of the parameters are summarized in Table 1.
As one may observe in Fig. 1, there are multipleclusters arriving at Rx.
TABLE 1Summary of notation and parameters
Notation DescriptionhT , hR Height of Tx, RxxR,k, yR,k x-, y- coordinates of Rx k, k = 1, . . . ,KxT , yT x-, y- coordinates of TxxC,kn, yC,kn, zC,kn x-, y-, z- coordinates of reflecting point for
cluster n of Rx kφD,kn, φA,kn Angles of departure and arrival of cluster nθD,kn, θA,kn Zenith angles of departure and arrival of clus-
ter nτkn Delay of cluster nH , dm Height and diameter of a blocker, H ∼
N(µH , σH)FH(x) CDF of the blocker heightλB Density of blockers per unit areahC , h1, h2 Heights of points P , O, and Mr0 Two-dimensional distance from O to Pd Two-dimensional distance from O to Mα Angle ∠POMp00, p01 Conditional probabilities to reside in non-
blocked/blocked states at point M (0 and 1)given that there was non-blocked state at O
p10, p11 Conditional probabilities to reside in non-blocked/blocked states at point M (0 and 1)given that there was blocked state at O
Definition 1. A cluster is a set of rays that travel from Txto Rx with a small variation in their angles of arrival anddeparture caused by diffuse reflections on the same surface.
The correlation between the current channel statesof the links is a consequence of the separation angleof clusters. Particularly, two Rx, R1 and R2, in Fig. 1are located nearby, which leads to the correlationbetween their LoS links. In contrast, R1 and R3 arewell-separated, which implies that the presence ofblockage at R1 does not affect LoS link at R3. Thesituation is similar with the clusters. Two clusters, C1
and C2, arriving at Rx R4 are correlated with eachother as they are not well-separated in space, anda single blocker occludes their paths. At the sametime, clusters C2 and C3 are independent. A spatially-consistent model for mmWave channel needs to takethe effects of spatially-consistent human body block-age into account.
2.2 3GPP 3D Channel ModelIt is important to note that the proposed blockagestate generation procedure is compatible with anychannel model. Below, we briefly review the 3GPPmodels with and without spatial correlation as theyare widely acceptable. 3GPP 3D channel model forbands higher than 6 GHz was introduced in [9]. Theproposed considerations are based on a similar logicas those in the LTE specifications [33], with modifi-cations specific to mmWave frequencies. The modelthus allows to generate a set of correlated (with eachother) parameters (angle-of-departure (AoD), angle-of-arrival (AoA), zenith-of-departure (ZoD), zenith-of-arrival (ZoA), powers, delays, etc.) for each clusterbased on the measurements conducted in a specificradio environment.
In Section 7.6.3 of [9], a spatially-consistent exten-sion to the 3GPP 3D channel model is proposed. Themodeling procedure comprises two parts: 1) genera-tion of spatially-consistent large scale and small scaleparameters for a static Rx drop; and 2) correlatedRx mobility modeling. In step one, a regular hor-izontal grid is generated, whose inter-site distanceequals the correlation distance specified in advance.The standard mmWave model is used to specify thepropagation conditions for each node of the grid. Thepropagation parameters of Rx are then interpolatedbased on the nearest nodes, see [34] for details. WhenRx mobility is added, the delay, departure/arrivalangles, and cluster powers are updated according touser direction and speed.
2.3 3GPP Cluster Localization ProcessAccording to the 3GPP 3D channel model, there areN clusters arriving at every Rx in the scenario, whereN is a scenario-specific parameter [9]. The modelprovides the AoA, AoD, ZoD, ZoA, delay, etc. for
4
Tx (xT, yT)
Rx (xR,k, yR,k)
dT,kn dR,kn
(xC,kn, yC,kn, zC,kn)
φD,kn φA,kn
Reflection pointof cluster n
Fig. 2. Illustration of location of a cluster.
each cluster, which become the input parameters forour analytical framework outlined in Section 3. Tointroduce our spatially-consistent blockage state gen-eration procedure described in Section 4 based on thenovel analytical framework introduced in Section 3, itis also required to obtain the coordinates of the clus-ter’s reflection points for Rx k, (xC,kn, yC,kn, zC,kn). Inorder to do that, we follow [21] by assuming that Txand Rx are located at the foci of an ellipse, and thatthe reflection point of a single bounce is located atthe arc of this ellipse as shown in Fig. 2. With thismethod, we can extract the x-, y-, and z-coordinatesof the reflecting point of cluster n for Rx k, which arerequired to parametrize our analytical framework, asfollows:
• as the random generation of AoA and AoD ac-cording to 3GPP [9] does not guarantee that thearrival and departure clusters will intersect in 3Dspace, we randomly choose Tx-side or Rx-side;
• compute the coordinates (xC,kn, yC,kn) as:– choosing Tx-side, we have:
xC,kn = xT + dT,kn cosφD,n,
yC,kn = yT + dT,kn sinφD,n. (1)
– alternatively, choosing Rx-side we have:
xC,kn = xR,k + dR,kn cosφA,kn,
yC,kn = yR,k + dR,kn sinφA,kn, (2)
where dT,kn and dR,kn are the distances be-tween the reflecting point of cluster n and Txor Rx k, respectively, xT , yT , xR,k, and yR,kare the coordinates of Tx and Rx k, φD,kn andφA,kn are the angles of departure and arrivalof cluster n, dkn = dT,kn + dR,kn is the totaltravel distance of cluster n estimated as cτkn,where τkn is the delay and c is the speed oflight.
• since (xC,kn, yC,kn) are the coordinates in case ofa single reflection, the distance may be chosenuniformly between this point and Tx/Rx location.The new distance from the last or first reflection,depending on which side (Tx or Rx) is consid-ered, is denoted as du,kn;
• compute the coordinate zC,kn as:
– in case of Tx-side, we have:
zC,kn = hT + du,kn tan θD,kn, (3)
– alternatively, in case of Rx-side we have:
zC,kn = hR − du,kn tan θA,kn, (4)
where hT and hR are the heights of Tx andRx k, while θD,kn and θA,kn are the zenithangles of departure and arrival of cluster n,respectively.
3 MATHEMATICAL FRAMEWORK
In this section, we develop a novel mathematicalframework for characterizing the conditional linkstate probabilities. The considerations below are acomprehensive extension of our previous modelin [27] that allow to consider different heights ofpoints O and M . In what follows, we first introducethe preliminary details and then proceed by derivingthe unconditional and conditional link state probabil-ities with respect to the channel with known blockagestate. These metrics form the foundation of the pro-posed spatially-consistent human body blockage stategeneration procedure for the mmWave channel modelintroduced in Section 4.
3.1 Important Preliminaries
Consider Fig. 3 illustrating the top-view of thescenario where two clusters are arriving/departingat/from two location points O and M . These clustersare departing/arriving from/at a common point P . Inthe first case, Tx acts as a common entity located atpoint P , which is associated with two Rx located atpoints O and M . In the second case, Rx is consideredas a common entity located at point P , which receivestwo clusters from the reflector points located at O andM . Note that O and M may also be the coordinatesof two Tx communicating with a single Rx located atP or, in general, the coordinates of any other nodes.
To establish the conditional link state probabilityfor the general case, we operate with the followingterminology: point P with height hC , point O withheight h1, and link state (i.e., blocked or non-blocked)derived with unconditional link state probability, aswell as point M with height h2 and link state derivedwith conditional link state probability. To capturethe spatial correlation with respect to the blockagebetween two links, we are interested in the conditionallink state probabilities pij .
Definition 2. pij , i, j = 0, 1, are the conditional linkstate probabilities (conditional probabilities) that the stateof a node at point M is non-blocked (j = 0) or blocked(j = 1) given that the state of this node at pointO is non-blocked (i = 0) or blocked (i = 1), pij =P [M is in state j|O is in state i].
5
(0,0) D
BC
N
LG
IPF
SR
K
W
O U E
H
N'
dm
r0
r1
dm
dm
M
2 3
1
α
β
d
A
4b 4a
Fig. 3. A top-view illustration of correlation betweenlinks.
These probabilities can be organized into the fol-lowing matrix
P =
(p00 p01p10 p11
)(5)
where states 0 and 1 reflect the non-blocked andblocked states, respectively. The matrix P is a functionof the following variables:• r0 is the two-dimensional distance from O to P ;• d is the two-dimensional distance from O to M ;• α is the angle ∠POM , see Fig. 3;• λB is the density of blockers;• hC , h1, and h2 are the heights of points P , O, andM .
Since the following holds [27]
p00 = 1− p01, p10 = 1− p11, (6)
in order to parametrize (5), it is required to obtain p00and p10.
The geometry of our proposed methodology isdemonstrated in Fig. 3, where two rectangles ABCDand EFGH represent the areas affecting the link POand PM blockage. The width and length of theserectangles are equal to dm (the diameter of a blocker)and r0/r1 (the 2D distances PO and PM ), respec-tively. One may notice that the intersection area oftwo rectangles is the area affecting both links PO andPM . For further analysis, the rectangles are dividedinto multiple zones having a different impact on theconditional probabilities as illustrated in Fig. 3• Zone 1, NN ′LR, is the square area around the
point P . Further derivations are simplified signif-icantly by omitting this zone, while the generalimpact of this zone is considered to be negligibledue to its smaller size.
• Zone 2, ANSKD, is related to the path PO andinfluences the conditional probability in case POpath is blocked.
• Zone 3, IKEH , affects the PM path and doesnot depend on the state at point O.
• Zone 4, SRK, is the common zone affecting bothlinks simultaneously and impacting the depen-dency between the states at points O and M .
Zone 4 can be split further into two smaller zones,4a and 4b, which represent the area on the right andleft sides, respectively, along with PU , which is theline of intersection of two planes as shown in Fig. 3.These zones are used to determine whether a blockerthat is blocking/not blocking the lower plane, is alsoblocking/not blocking the upper one. Depending onthe heights of the involved entities, these zones willcorrespond to different planes, which is reflected insubsequent derivations.
3.2 Unconditional Link State Probability
We begin with characterizing the unconditional linkstate probability by deriving it for point O located atthe distance of r from point P .
Definition 3. Unconditional link state probability, PnB,is the probability that a given link is not occluded by ahuman body, without taking into account the condition ofblocked/non-blocked state of the neighboring links.
We follow [26] to briefly sketch the derivation inquestion. Consider a rectangular blockage zone inFig. 3 with the width corresponding to the diameterof a blocker, dm, and the length of r. Recalling thatthe process associated with the centers of blockers ishomogeneous Poisson, the coordinates of each par-ticular blocker are distributed uniformly over (0, dm)and (0, r), which corresponds to OY and OX coor-dinates of the rectangle sides, respectively. Hence, theblockage probability is the probability that at least oneblocker resides in the area of interest and occludes thelink at hand.
For different values of hC , h1, and the distributionof the blocker height H , a blocker falling into the con-sidered area occludes the link when PH > hm(x),where x ∈ (0, r) and hm(x) is
hm(x) =hC − h1
rx+ h1, x ∈ (0, r). (7)
The non-blocked state probability can now be ob-tained in terms of the void probability for the PPPas
PnB = exp
[λBdm
∫ r
0
(FH(hm(x))− 1) dx
], (8)
where FH(x) is the CDF of the blocker height.The result in (8) is then employed to derive the
conditional link state probabilities. In subsections 3.3,3.4, and 3.5 we establish the conditional probabilitiesfor the general case, where all of the entities arelocated at different heights in relation to each other.
6
Pnbz = fP (0, λB Sz) +
∞∑i=1
fP (i, λB Sz)
[∑Nj=1
xj+1∫xj
yj+1∫yj
FH
(gy−fx−e
h
)dy dx
]iSiz
. (9)
P∗nbz = exp
λB N∑j=1
xj+1∫xj
yj+1∫yj
[FH
(gy − fx− e
h
)− FH
(by − ax− d
c
)− 1
]dydx
. (10)
3.3 Probabilities p10 and p11 for hC > h2
By observing Fig. 3, one may notice that the planes(ABC) and (EFG) intersect at PU . The zones 4aand 4b are located on the right and left sides of PU ,respectively. By using the 3D view of the scenario inFig. 6, where hC > h2, we may conclude that the zone4a of Zone 1, which corresponds to plane (ABC) fromFig. 3, is always higher than the corresponding zoneof Zone 2 depicted as plane (EFG). Alternatively, inFig. 7, when hC < h2, the zone 4a of Zone 1 is alwayslower than the zone 4a of plane Zone 2. Therefore, theconditional probabilities are independent of the heighth1, and it is sufficient to consider two cases, hC > h2and hC < h2.
Let Lz and Lz be the events of having no blockersin zone z that occlude the link paths at O and M ,respectively. The complementary events to Lz and Lzare denoted by Nz and Nz . Note that Lz and Lzoccur when there are no blockers in zone z or allthe blockers are not high enough to occlude the link.When hC > h2, p10 can be written as
p10 =P [M is non-Blocked ∩O is Blocked]
P [O is Blocked]. (11)
Lemma 1. The probability p10 for hC > h2 given in (11)is simplified to the following form
p10 =Pnb3
(Pb2Pnb4bPnb4a + P∗nb4bPnb4a
)1− Pnb2Pnb4aPnb4b
, (12)
where Pnbz and Pbz are the probabilities of the events Lzand Nz , P∗nb4b is the probability of N4b ∩L4b, and Pnbz isthe probability of having no blockers affecting the link inthe corresponding zone for point O.
Proof: See proof in Appendix A, available in theonline supplemental material.
The probabilities Pnbz are derived similarly to theunconditional link state probability in subsection 3.2.They are given by (9), where the auxiliary parametersare provided in Appendix B, available in the onlinesupplemental material, fP (i, λBSz) is the probabilityof having i blockers in the zone z with the density ofblockers λB , Sz is the area of zone z, and FH([gy −fx − e]/h) is the probability that a blocker with thecoordinates x, y is lower than the link PM . Note
that here each zone is actually a polygon formed bythe intersection of the projections of planes. Since, infact, (9) integrates over the area of the correspondingzone, it is easier to represent the entire zone as a setof trapezoids and/or triangles with their lower andupper bounds represented as variables xj , xj+1 andyj , yj+1. The integration limits are provided in ourtechnical report [35].
Using the Maclaurin series expansion of an expo-nential function, (9) can be simplified as
Pnbz = eλB(I−Sz). (13)
Finally, the probability Pnbz is written as
Pnbz =
N∏j=1
eλB
xj+1∫xj
yj+1∫yj
[FH( gy−fx−eh )−1]dydx
. (14)
Consider now Pnbz, which corresponds to the casewhere all of the blockers are lower than the plane(ABC). These probabilities are obtained similarly toPnbz and read as
Pnbz =
N∏j=1
eλB
xj+1∫xj
yj+1∫yj
[FH( by−ax−dc )−1]dydx
. (15)
To simplify (12), observe that the following holds
Pnb2,4 = Pnb2Pnb4aPnb4b,
Pb2 = 1− Pnb2,4Pnb4
, (16)
Pnb4 = Pnb4aPnb4b,
where Pnb2,4 is the probability of having no blockersin the rectangle ADRN for the link PO. It is calculatedby utilizing the generic form (9), which leads to
Pnb2,4 = p0 +
∞∑i=1
pi
xR∫0
yR∫0
FH
[by−ax−d
c
]dydx
||AD||||DR||
i , (17)
where the integration starts at 0, since the point A islocated in the center of coordinates.
Finally, using the Maclaurin series expansion of theexponential function, the probabilities P∗nbz are givenby (10), while the integration limits are provided inour technical report [35]. The complementary proba-bility p11 is 1− p10.
7
11.5
Hei
ght,
h 2[m
]
206
204
202
200
19850
Y-coordinate, [m] X-coordinate, [m]
(a)
4949.55050.551X-coordinate, [m]
0.6
0.8
1
1.2
1.4
1.6
Hei
ght,
h 2[m
]
(b)
49 50 51X-coordinate, [m]
196
198
200
202
204
206
Y-c
oord
inat
e, [
m]
(c)
Fig. 4. Illustration of spatially-consistent zone for hC = 4 m, h1 = 1.5 m, and h2 ∈ (0.5 m− 1.5 m).
Y-coordinate, [m] X-coordinate, [m]
2400
3
400
Hei
ght,
h 2[m
]
200 200
4
00 -200
(a)
-200-1000100200300X-coordinate, [m]
2
2.5
3
3.5
4
Hei
ght,
h 2[m
]
(b)
-200 -100 0 100 200 300X-coordinate, [m]
0
50
100
150
200
250
300
Y-c
oord
inat
e, [
m]
(c)
Fig. 5. Illustration of spatially-consistent zone for hC = 1.5 m, h1 = 4 m, and h2 ∈ (2.1 m− 4.1 m).
3.4 Probabilities p00 and p01 for hC > h2
First, the conditional probability p00 can be written as
P [M is non-blocked|O is non-blocked] =
P [M is non-blocked ∩O is non-blocked]
P [O is non-blocked]. (18)
Note that the event (M is non-blocked) corre-sponds to the event L3 ∩ L4a ∩ L4b, while the event(O is non-blocked) corresponds to the event L2 ∩L4a ∩ L4b. Observe that zone 4a exists in both planes,while 4a of one plane is always higher than that ofanother plane. If a blocker in zone 4a is lower for theplane (FGH), it is lower for the upper plane as well;therefore, the following holds: L4a ∩ L4a = L4a. Thesame applies to L4b ∩ L4b = L4b. Hence, (18) can bewritten as
p00 =P[L3 ∩ L4a ∩ L4b ∩ L2 ∩ L4a ∩ L4b
]P[L2 ∩ L4a ∩ L4b
]=
P [L3 ∩ L4a]
P[L4a
] . (19)
Finally, the conditional probability p00 can be estab-lished as
p00 =Pnb3Pnb4aPnb4a
, (20)
where Pnbz is the probability of having no blockersin zone z, which is produced by (14), while Pnbz is
derived in (15). The complementary probability p01 is1− p00.
3.5 Conditional Probabilities pij for hC < h2
Consider now the case where a common entity height,hC , is lower than the second entity height, hC < h2.The analysis in this case is similar to the case of hC >h2, which has been completed previously, but withone important exception. Here, zones 4a and 4b arelocated differently with respect to the plane havingthe non-blocked link path. Hence, we modify (19) forp00 as
p00 =P [L3 ∩ L4b]
P[L4b
] , (21)
and the conditional probability p00 thus becomes
p00 =Pnb3Pnb4bPnb4b
. (22)
The complementary probability p01 can now beestablished as p01 = 1− p00. Modifying (11) similarly,we arrive at
p10 =Pnb3
(Pb2Pnb4aPnb4b + P∗nb4aPnb4b
)1− Pnb2Pnb4aPnb4b
. (23)
The complementary probability p11 can now beestablished as p01 = 1− p10.
8
3.6 Shape of Spatially-Consistent ZoneTo assess the effect of correlation caused by humanbodies, consider the shape of the spatially-consistent(SpCon) zone.
Definition 4. A spatially consistent zone is the 2D zonearound the target node where for every node located insidethe SpCon zone the fraction of two conditional link stateprobabilities, p10/p00, is lower than a value ∆x (∆x =0.99). Every point at the edge of the SpCon zone coincideswith the distance where two conditional probabilities, p00and p10, converge to unconditional probability, PnB [27].Any node in the SpCon zone has a spatially consistentblockage state with the target node link state.
The spatially-consistent zone for the case of hC =4 m and h1 = 1.5 m as well as the distance betweenpoints P and O of 50 m, is exemplified in Fig. 4,where the height of point M , h2, varies within therange (0.5, 1.5) m. The SpCon zone for the case ofhC = 1.5 m and h1 = 4 m is illustrated in Fig. 5.
As one may observe, the dimensions of the SpConzone for the second case, hC < h2, are considerablylarger that those for the first case where hC > h2. Toexplain this behavior, consider two three-dimensionalillustrations for the two considered cases, hC > h2and hC < h2, as displayed in Fig. 6 and Fig. 7,respectively. Here, point P represents the commonentity with the height of hC . Points O and M are theentities with known and unknown states, respectively.All the blockers are smaller than the plane Ω, whichis the maximum considered height of a blocker.
Analyzing Fig. 6, which reflects the case of hC > h2,we note that the area of the zone inducing the corre-lation between the states is rather small. Indeed, thezone FcGcHcIc (2D plane in Fig. 6) is the commonzone that affects both O and M , while only zoneJcKcHcIc is responsible for the dependence betweenthe states. Most of the common zone is located highenough, where the blockers do not affect the link inquestion. Hence, the dimension of the SpCon zonein its minor axis is very small, since an increase inthe distance between points O and M decreases theintersection area.
Fig. 6. Illustration of the case hC > h2.
Fig. 7. Illustration of the case hC < h2.
In Fig. 7, the zone FcGcHcIc (2D plane in Fig. 7)is the common zone that affects both O and M . Here,only zone FcGcJcKc is responsible for the dependencebetween the states. Hence, an increase in the distancebetween the two entities maintains the correlationfarther than in the first case.
4 SPATIALLY-CONSISTENT BLOCKAGESTATE GENERATION PROCEDURE
After obtaining the unconditional and conditionallink state probabilities, we proceed with specifying aspatially-consistent blockage state generation proce-dure for micro objects (human bodies). Our proposedprocedure can be overlaid on top of the standard3GPP channel model or on top of any 3GPP-likechannel models.
The main goal of the proposed procedure is toassign the spatially consistent blocked/non-blockedstate to every link in a given scenario. Algorithm1 is responsible for generating blocked/non-blockedstates among all the Tx-Rx links, whereas Algorithm2 is responsible for generating blocked/non-blockedstates among all the Reflector-Rx links. Both algo-rithms employ our analytical framework to derive theconditional link state probabilities given in Section 3.The computation complexity of the algorithms growslinearly with the number of Rx nodes i.e., the overallmodeling complexity is O(K), where K is the numberof Rx generated in the scenario. Note that the algo-rithms do not depend on the blocker density, since thisdensity is a parameter of the analytical framework.
The proposed state generation procedure comprisesthree phases. At the first phase, we associate everyRx with the state (blocked/non-blocked) of the LoSpath. First, we generate K Rx with uniformly dis-tributed x- and y-coordinates. Further, we introducetwo sets, NU1 and NU2, containing the coordinates ofRx without and with generated blockage states. Theset NU,s contains blockage states of LoS links for all Rx.Further, we choose the first Rx from the set NU1 andfind the unconditional state of that Rx by using theunconditional link state probability, PnB, from (8). We
9
Algorithm 1: Blocked/non-blocked LoS state gen-eration
Result: Blocked/non-blocked LoS states for all Rx
1 Generate uniformly distributed coordinates(xR,1, yR,1), . . . , (xR,K , yR,K) of K Rx
2 Define the sets NU1 = ∅ and NU2 = ∅ for thecoordinates of Rx w/o and w/ blocked/non-blockedstate, respectively
3 Define the set NU,s = ∅ for blocked/non-blocked statesof every Rx
4 Save the coordinates of every Rx to the setNU1 = (xR,1, yR,1), . . . , (xR,K , yR,K)
5 while NU1 6= ∅ do6 Choose the coordinates of Rx k from the set NU1
7 Find the unconditional state qRu,k = 0 or 1 of Rxk based on the unconditional probability, PnB,from (8)
8 Save the state of Rx k to the set NU,s = NU,s ∪ qRu,k9 Remove the coordinates of Rx k from the set NU1
10 Add the coordinates of Rx k to the setNU2 = NU2 ∪ (xR,k, yR,k)
11 while NU2 6= ∅ do12 Choose the coordinates of Rx l from the set NU213 Calculate the SpCon zone (see Definition 4) for
Rx l using the conditional probabilities, p10from (12), p00 from (20), and theunconditional probability, PnB from (8)
14 Remove the coordinates of Rx l from NU215 Find any Rx from the set NU1 in the SpCone
zone of Rx l16 if Rx m from the set NU1 is in the SpCone zone
then17 Find the conditional state qRc,m = 0 or 1
of Rx m using (12) or (20)18 Add the coordinates of Rx m to the set
NU2 = NU2 ∪ (xR,m, yR,m)19 Save the state of Rx m to the set
NU,s = NU,s ∪ qRc,m20 Remove the coordinates of Rx m from the
set NU1
continue by calculating the SpCon zone for the firstselected Rx by following the Definition 4. All of theRx that fall into the SpCon zone have the correlatedlink states with the link state of a chosen Rx. In casewhere no Rx are in the SpCone zone, the next Rx fromthe set NU1 is selected and the procedure repeats. Thegeneration procedure continues up until all the Rx areassigned with blocked/non-blocked LoS states.
At the second phase, we generate the cluster chan-nel parameters (AoA, AoD, ZoA, ZoD, and delay)for every Rx by following the 3GPP channel modelgeneration procedure [21]. Finally, at the last phase,the state of each cluster for every Rx is generated. Theset NC contains the aforementioned cluster channelparameters of all clusters for all the Rx. The sets NCk1and NCk2 contain cluster coordinates without and withblocked/non-blocked states. The set NCk,s contains thestates of every cluster for each Rx in the scenario. Wechoose the first cluster of Rx k = (1, . . . ,K) and findthe state by using the unconditional probability, PnB,
Algorithm 2: Blocked/non-blocked link state gen-eration for every cluster of each Rx
Result: Blocked/non-blocked states for all clusters
1 Generate the cluster channel parameters(φA,kn, φD,kn, θA,kn, θD,kn, τkn) for all K Rx [9]
2 Define the set NC = ∅ for the parameters of clustersfor every Rx w/o state
3 Define the set NCk,s = ∅ for the blocked/non-blockedstates of every cluster for each Rx
4 Add all the parameters of clusters for each Rx to theset NC = (φA,kn, φD,kn, θA,kn, θD,kn, τkn)
5 while NC 6= ∅ do6 Choose the parameters of clusters for Rx k from
the set NC7 Find xC,kn, yC,kn, and zC,kn coordinates of clusters
n = 1, . . . , N , see Section 2.38 Remove the parameters of clusters for Rx k from
the set NC9 Define the sets NCk1 = ∅ and NCk2 = ∅ for the
coordinates of clusters for Rx k, w/o and w/state respectively
10 Add the coordinates of clusters for Rx k to the setNCk1 = (xC,k1, yC,k1, zC,k1), . . . , (xC,kN , yC,kN ,zC,kN )
11 while NCk1 6= ∅ do12 Choose the coordinates of cluster l from the set
NCk1
13 Find the unconditional state qCu,l = 0 or 1 ofcluster l based on the unconditionalprobability, PnB, from (8)
14 Save the state of cluster l to the setNCk,s = NCk,s ∪ qCu,l
15 Add the coordinates of cluster l to the setNCk2 = NCk2 ∪ (xC,kl, yC,kl, zC,kl)
16 Remove the coordinates of cluster l from theset NCk1
17 while NCk2 6= ∅ do18 Choose the coordinates of cluster e from
the set NCk219 Find the conditional probabilities p10 and
p00 for the link state of cluster m from theset NCk1
20 Remove the coordinates of cluster e fromthe set NCk2
21 if p10/p00 < 0.99 then22 Find the conditional state
qCc,m = 0 or 1 of the cluster m basedon the conditional probabilities p10 orp00
23 Save the state of cluster m to the setNCk,s = NCk,s ∪ qCc,m
24 Add the coordinates of cluster m to theset NCk2
25 Remove the coordinates of cluster mfrom the set NCk1
(8). Due to different heights of the arrived clusters,there is no common 2D SpCon zone for a given clusterreflection point. Therefore, the rest of the clusterswithout the link states are compared individuallywith the chosen cluster to identify any correlated linksamong the clusters. Any chosen cluster is considered
10
0 10 20 30 40 50 60 70Tx-Rx distance, [m]
0
0.2
0.4
0.6
0.8
1P
roba
bili
ty o
f no
n-bl
ocke
d st
ate
Rx is mostly blocked
Rx is mostly non-blocked
No dominant state
Fig. 8. Unconditional probability, PnB, of non-blockedstate, λB = 1.
to have a correlated link state with the link state ofcluster 1 if the fraction of two conditional link stateprobabilities, p10/p00, is lower than a certain value ∆x(∆x = 0.99). The conditional state probabilities p10and p00 are calculated based on (12)/(23) and (20)/(21)by taking into account the heights of the consideredclusters and the Rx. The generation procedure contin-ues up until all the clusters of all the Rx are assignedwith blocked/non-blocked states.
The pseudo code for the above phases is providedin Algorithms 1 and 2, where Algorithm 1 covers thefirst phase, while Algorithm 2 implements the secondand third phases.
5 KEY NUMERICAL RESULTS
In this section, we illustrate the performance of thegeneration procedure as well as study the effects ofcorrelation caused by micro objects (human bodies)in the mmWave channel, which we compare with thecase of no correlation among the links. We consider acrowded environment, where the density of blockersin the modeled area is λB = 1 blockers per squaremeter. The height of Tx is 4 m and the height ofeach Rx is 1.5 m. The Tx is located at the origin(0, 0), while K Rx are distributed uniformly withinthe area of interest. The received power of every Rxis established by using the 3GPP urban micro (UMi)street canyon path loss model and the cluster basedchannel model [9]. In case the link is blocked, weassume 20 dB loss [36]. The link blockage state isderived by employing our spatially-consistent humanbody blockage state generation algorithm as well asindependent generation of the link blockage states.The remaining parameters are collected in Table 2.
Let the coordinates of K Rx be uniformly dis-tributed in three different areas of 10 by 10 squaremeters. The left bottom edges of these areas arelocated at (xL,B , yL,B) coordinates of (1, 1), (15, 15),and (40, 40) referring to the different 2D separationdistances between the Tx and the closest Rx. These
coordinates where chosen to study the impact of themost probable link state in three different cases: (i) Rxare mostly in non-blocked state (probability of non-blocked state is about 0.8 in Fig. 8); (ii) there is nodominant state (probability of non-blocked state isabout 0.5 in Fig. 8), and (iii) Rx are mostly in blockedstate (probability of non-blocked state is about 0.2 inFig. 8). Below, we formulate our key findings in termsof important statements.
We first compare the output of our algorithms withthe data of our simulator [30], where blockers weremodeled explicitly. We model blockers as cylinderswith the density of λB = 1 blockers per square meterfor the scenario described above. In Fig. 9(b), it may beobserved that the simulation results match closely theones derived with the proposed algorithm. A slightmismatch between the data sets is explained by theassumption of the analytical model, where the blockeroccludes the link if its height is lower than the LoSlink at the point, where the blocker’s center is located.In the simulator, the blocker might still block the pathwith its edge in some cases.
Observation 1: Spatial consistency across theblockage states of the links does not impact themean received power averaged over large area ofinterest but affects the received power of the neigh-boring links. Fig. 9 demonstrates the CDFs of thereceived power, PR, for three different assumptionsregarding the blockage state generation: (i) no block-ers (all the Rx are in LoS); (ii) uncorrelated block-age states; (iii) correlated blockage states for threedifferent positions of the left bottom edge of the Rxdistribution area of 10 m×10 m: (1, 1), (15, 15), and(40, 40). Here, the number of Rx, K, was set to 100.As one may observe in Fig. 9, the CDFs of the receivedpower coincide for the scenarios with correlated anduncorrelated blockage states. The reason is that the re-ceived power averaged across all of the Rx distributedin the area larger than the SpCon zone of one Rxremains the same on average. However, if we considerthe area that is much smaller than the SpCon zone ofone Rx, e.g., 1 m×1 m, the difference in the CDFsof the received power for correlated and uncorrelatedblockage states becomes noticeable as confirmed in
TABLE 2Baseline system parameters
Parameter ValueHeight of Tx, hT 4 mHeight of Rx, hR 1.5 mHeight of a blocker, N(µH , σH) N(1.7 m, 0.1 m)Diameter of a blocker, dm 0.5 mCoordinates of the left bottom edge of the areawith distributed K Rx, (xL,B , yL,B)
(1, 1), (15, 15),and (40, 40)
Density of blockers, λB 1 bl/m2
Carrier frequency 28 GHzTransmit power 35 dBmBlockage penalty 20 dBNumber of clusters, N 5
11
-80 -60 -40 -20Received Power, P
R[dBm]
0
0.2
0.4
0.6
0.8
1
CD
F
No blockersUncor. blockageCor. blockage
(a) (xL,B , yL,B) = (1, 1)
-90 -80 -70 -60 -50 -40Received Power, P
R[dBm]
0
0.2
0.4
0.6
0.8
1
CD
F
No blockersUncor. blockageCor. blockageSimulator
(b) (xL,B , yL,B) = (15, 15)
-90 -80 -70 -60Received Power, P
R[dBm]
0
0.2
0.4
0.6
0.8
1
CD
F
No blockersUncor. blockageCor. blockage
(c) (xL,B , yL,B) = (40, 40)
Fig. 9. Received power CDF: 100 Rx distributed in 10 m×10 m area.
-90 -80 -70 -60 -50 -40Received Power, P
R[dBm]
0
0.2
0.4
0.6
0.8
1
CD
F
No blockersUncor. blockageCor. blockage
Fig. 10. Received power CDF: 100 Rx, 1 m×1 m area,(xL,B , yL,B) = (15, 15).
Fig. 10. It is explained by the fact that most of the Rxare falling into the SpCon zone of one Rx, so that allthe links are correlated with each other. Therefore, thecorrelation across the blockage states highly affects theperformance of the neighboring links.
However, note that these results do not imply that theknowledge of correlated states of receivers cannot improvethe system performance. Even though there is no difference
40 45 50X coordinate, [m]
40
42
44
46
48
50
Y c
oord
inat
e, [
m]
1
2
3
4
Pow
er d
iffe
renc
e, [
dBm
]
45% of all receivers LoS links are correlated
Fig. 11. Number of correlated receivers among 20 Rxin 10 m×10 m area, (xL,B , yL,B) = (40, 40).
in the CDFs, one can schedule for service those Rx, whichcurrently reside in non-blocked conditions by benefitingfrom knowing the correlated states of Rx.
Observation 2: Increased density of Rx leads toan increased number of correlated links. The effectof Rx density on the proportion of correlated links isdemonstrated in Fig. 11 and Fig. 12, where the totalnumbers of Rx per 10 m×10 m area are 20 and 100,respectively. The plots illustrate the difference in thereceived power calculated for the same set of Rx butwith different blockage state generation procedure:(i) correlated blockage states and (ii) uncorrelatedblockage states. The figures reflect only those Rx,which see a difference in the received power. Thepositions of the small squares in the plots representthe aforementioned Rx locations and the color of thesquares demonstrates the absolute power difference.It is observed that for one particular realization of thefield of Rx displayed in the plots the percentage of re-ceivers with correlated link states increases from 45%among 20 Rx to 97% Rx among 100 Rx in 10 m×10 marea.
However, the number of Rx with different receivedpower is significantly smaller referring to the 10%among 20 Rx and 16% among 100 Rx. The reason
42 44 46 48X coordinate, [m]
42
44
46
48
50
Y c
oord
inat
e, [
m]
0
5
10
15
Pow
er d
iffe
renc
e, [
dBm
]
97% of all receiversLoS links are correlated
Fig. 12. Number of correlated receivers among 100 Rxin 10 m×10 m area, (xL,B , yL,B) = (40, 40).
12
2 4 6 8 10X coordinate, [m]
2
4
6
8
10
Y c
oord
inat
e, [
m]
0
5
10
15
Pow
er d
iffe
renc
e, [
dBm
]
(a) Random state generator for first Rx
4 6 8 10X coordinate, [m]
2
4
6
8
10
Y c
oord
inat
e, [
m]
0
5
10
15
Pow
er d
iffe
renc
e, [
dBm
]
(b) Forcing first Rx to be in non-blocked state
2 4 6 8 10X coordinate, [m]
4
6
8
10
Y c
oord
inat
e, [
m]
0
5
10
15
Pow
er d
iffe
renc
e, [
dBm
]
(c) Forcing first Rx to be in blocked state
Fig. 13. Absolute received power difference between correlated and uncorrelated blockage state generation for50 Rx, (xL,B , yL,B) = (1, 1).
16 18 20 22 24X coordinate, [m]
16
18
20
22
24
Y c
oord
inat
e, [
m]
5
10
15
Pow
er d
iffe
renc
e, [
dBm
]
(a) Random state generator for first Rx
16 18 20 22 24X coordinate, [m]
16
18
20
22
24
Y c
oord
inat
e, [
m]
5
10
15
Pow
er d
iffe
renc
e, [
dBm
]
(b) Forcing first Rx to be in non-blocked state
16 18 20 22 24X coordinate, [m]
16
18
20
22
Y c
oord
inat
e, [
m]
0
5
10
15
Pow
er d
iffe
renc
e, [
dBm
]
(c) Forcing first Rx to be in blocked state
Fig. 14. Absolute received power difference between correlated and uncorrelated blockage state generation for50 Rx, (xL,B , yL,B) = (15, 15).
is that there is a number of Rx, which are assigneda blocked/non-blocked state based on the uncondi-tional probability in the spatially consistent blockagestate generation procedure. For high unconditionalblockage probability, these states will most probablybe blocked states. The receivers that are located in theSpCon zone of the Rx with unconditional blockagestates having a high probability will follow the linkstate of the latter Rx. Therefore, the number of Rx withdifferent received power is low, since most of the Rxwill be assigned the most probable link state.
Observation 3: The difference between receivedpower calculated with the help of the correlatedand uncorrelated state generation procedures in asingle realization increases when the first chosenRx is assigned a less probable state (e.g., the uncon-ditional non-blockage probability is 0.8, but withthe probability 0.2 the Rx could be assigned theblocked state, which is a less probable state). It wasnoted in the previous plots that the number of Rx withdifferent received power for two different blockagestate generation cases (conditional and unconditional)is rather small, since the Rx is assigned the mostprobable link states. With Fig. 13 and Fig. 14, westudy the effect of the link state that deviates fromthe most probable case. For the three subplots inFig. 13, a set of 50 Rx was generated once in the10 m×10 m area with the left bottom coordinates(xL,B , yL,B) = (1, 1). The small squares refer to the
positions of Rx with different received power for thecorrelated and uncorrelated blockage state generationcases. Also, the position of every Rx is the same foreach subplot of Fig. 13. The first Rx chosen by the spa-tially consistent blockage state generation procedureis assigned a blockage state following three differentstrategies: (i) the first chosen Rx is assigned a blockagestate based on the unconditional probability; (ii) thefirst chosen Rx is assigned a non-blocked state; (iii)the first chosen Rx is assigned a blocked state. Thelatter strategies were chosen to study the effect of asingle link state.
It was observed that the number of Rx with dif-ferent received power for the two scenarios (uncondi-tional and conditional blockage state generation cases)increases when the first chosen Rx is assigned a lessprobable state. For Fig. 13, this state is the blockedstate, since the unconditional non-blockage probabil-ity is about 0.8. To further demonstrate the effect of themost probable state of the Rx, we generate 50 Rx in thearea of 10 m×10 m area with (xL,B , yL,B) = (15, 15);the results are depicted in Fig. 14. These coordinateswere chosen, since there is no most probable state ofthe Rx (unconditional blockage probability is around0.5). The results indicate no major difference betweenthe number of Rx with different received power whenassigning the first Rx with a blocked or non-blockedstate.
13
6 CONCLUSIONS
In this paper, we proposed a new 3GPP-compatiblespatially-consistent human body blockage state gen-eration procedure for dense urban mmWave deploy-ments. The contributed model is built on top of aglobally accepted 3GPP model and extends it to thecase of correlated signal behavior caused by a hu-man crowd around the mmWave receivers. We alsoinvestigated in detail the effects of correlation createdby human bodies to demonstrate that it manifestsitself in a local fluctuation of the received signalstrength as well as heavily depends on the densityof the receivers. The main application area for theproposed state generation procedure is in system-level simulations of the emerging cellular mmWavetechnology. In this setting, our model can be usedas an extension to the 3GPP’s model with correlatedmacro objects, thus inducing additional dependencyin the received signal strength field caused by thehuman crowd. The computational complexity of ourmodel does not depend on the blocker density, whichmakes it suitable for characterizing densely crowdedenvironments.
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[24] S. Ju and T. S. Rappaport, “Simulating motion - incorporatingspatial consistency into the NYUSIM channel model,” in IEEE88th Vehicular Technology Conference (VTC Fall), August 2018,pp. 1–6.
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[26] M. Gapeyenko, A. Samuylov, M. Gerasimenko, D. Moltchanov,S. Singh, E. Aryafar, S.-p. Yeh, N. Himayat, S. Andreev, andY. Koucheryavy, “Analysis of human-body blockage in urbanmillimeter-wave cellular communications,” in 2016 IEEE Inter-national Conference on Communications (ICC). IEEE, May 2016,pp. 1–7.
[27] A. Samuylov, M. Gapeyenko, D. Moltchanov, M. Gerasimenko,S. Singh, N. Himayat, S. Andreev, and Y. Koucheryavy, “Char-acterizing spatial correlation of blockage statistics in urbanmmWave systems,” in IEEE Global Communications Workshops(GLOBECOM Wkshps), December 2016, pp. 1–7.
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Margarita Gapeyenko is a Ph.D. candidateat the Laboratory of Electronics and Com-munications Engineering at Tampere Univer-sity, Finland. She earned her M.Sc. degreein Telecommunication Engineering from Uni-versity of Vaasa, Finland, in 2014, and B.Sc.degree in Radio-Engineering, Electronics,and Telecommunications from KaragandaState Technical University, Kazakhstan, in2012. Her research interests include math-ematical analysis, performance evaluation,
and optimization methods of future wireless networks, device-to-device communication, and 5G-grade heterogeneous networks.
Andrey Samuylov received the Ms.C. in Ap-plied Mathematics and Cand.Sc. in Physicsand Mathematics from the RUDN Univer-sity, Russia, in 2012 and 2015, respectively.Since 2015 he is working at Tampere Uni-versity as a researcher, focusing on mathe-matical performance analysis of various 5Gwireless networks technologies. His researchinterests include P2P networks performanceanalysis, performance evaluation of wirelessnetworks with enabled D2D communications,
and mmWave band communications.
Mikhail Gerasimenko is a Researcher atTampere University in the Laboratory ofElectronics and Communications Engineer-ing. Mikhail received Specialist degree fromSaint-Petersburg University of Telecommuni-cations in 2011. In 2013, he obtained Masterof Science degree from TUT. In 2018 healso obtained Doctor of Science degree fromTampere University of Technology. Mikhailstarted his academic career in 2011 andsince then he appeared as (co-)author of
multiple scientific journal and conference publications, as well asseveral patents. He also acted as reviewer and participated in edu-cational activities. His main subjects of interest are wireless commu-nications, machine-type communications, and heterogeneous net-works.
Dmitri Moltchanov is a Senior ResearchScientist in the Laboratory of Electron-ics and Communications Engineering, Tam-pere University, Finland. He received hisM.Sc. and Cand.Sc. degrees from Saint-Petersburg State University of Telecommuni-cations, Russia, in 2000 and 2002, respec-tively, and Ph.D. degree from Tampere Uni-versity of Technology in 2006. His researchinterests include performance evaluation andoptimization issues of wired and wireless IP
networks, Internet traffic dynamics, quality of user experience ofreal-time applications, and traffic localization P2P networks. DmitriMoltchanov serves as TPC member in a number of internationalconferences. He authored more than 80 publications.
Sarabjot Singh (’SM 09, M’ 15) is a PrincipalEngineer at Uhana Inc. CA. He received theB. Tech. from IIT, India, and the M.S.E andPh.D. in EE from UT Austin. His past affilia-tions include Intel, Nokia Technologies, BellLabs, and Qualcomm Inc, where he workedon protocol and algorithm design for nextgeneration of cellular and WiFi networks. Dr.Singh is interested in the system and archi-tecture design of wireless networks leverag-ing theoretical and applied tools. He is a co-
author of more than 40 patent applications, and multiple conferenceand journal papers. He was the recipient of the President of IndiaGold Medal in 2010, the ICC Best Paper Award in 2013, andrecognized for being a prolific inventor at Intel Corp.
Mustafa Riza Akdeniz (S’09) received theB.S. degree in electrical and electronics en-gineering from Bogazici University, Istanbul,Turkey, in 2010 and the Ph.D. degree inelectrical and computer engineering at NewYork University Tandon School of Engineer-ing, Brooklyn, NY in 2016. He is working asa research scientist for Intel Labs in SantaClara, CA. His research interests includewireless communications, wireless channelmodeling, and information theory.
15
Ehsan Aryafar is an Assistant Professor ofComputer Science at Portland State Univer-sity. Prior to that and from 2013 to 2017,he was a Research Scientist at Intel Labsin Santa Clara, CA. He received the B.S.degree in Electrical Engineering from SharifUniversity of Technology, Iran, in 2005, andthe M.S. and Ph.D. degrees in Electrical andComputer Engineering from Rice University,Houston, Texas, in 2007 and 2011, respec-tively. From 2011 to 2013, he was a Post-
Doctoral Research Associate in the Department of Electrical En-gineering at Princeton University. His research interests are in theareas of wireless networks and networked systems, and span bothalgorithm design as well as system prototyping. He has more than30 issued and pending patents in the areas of mobile and wirelesssystems.
Sergey Andreev is a Professor in the Lab-oratory of Electronics and CommunicationsEngineering at Tampere University, Finland.He received the Specialist degree (2006)and the Cand.Sc. degree (2009) both fromSt. Petersburg State University of AerospaceInstrumentation, St. Petersburg, Russia, aswell as the Ph.D. degree (2012) from Tam-pere University of Technology. Sergey (co-)authored more than 100 published researchworks on wireless communications, energy
efficiency, heterogeneous networking, cooperative communications,and machine-to-machine applications.
Nageen Himayat is a Principal Engineerwith Intel Labs, where she leads a teamconducting research on several aspects ofnext generation (5G/5G+) of mobile broad-band systems. Her research contributionsspan areas such as multi-radio heteroge-neous networks, mm-wave communication,energy-efficient designs, cross layer radioresource management, multi-antenna, andnon-linear signal processing techniques. Shehas authored over 250 technical publications,
contributing to several IEEE peer-reviewed publications, 3GPP/IEEEstandards, as well as holds numerous patents. Prior to Intel, Dr.Himayat was with Lucent Technologies and General InstrumentCorp, where she developed standards and systems for both wirelessand wire-line broadband access networks. Dr. Himayat obtained herB.S.E.E degree from Rice University, and her Ph.D. degree from theUniversity of Pennsylvania. She also holds an MBA degree from theHaas School of Business at University of California, Berkeley.
Yevgeni Koucheryavy is a Full Professor inthe Laboratory of Electronics and Commu-nications Engineering of Tampere University,Finland. He received his Ph.D. degree (2004)from Tampere University of Technology. Heis the author of numerous publications inthe field of advanced wired and wirelessnetworking and communications. His currentresearch interests include various aspects inheterogeneous wireless communication net-works and systems, the Internet of Things
and its standardization, as well as nanocommunications. He isAssociate Technical Editor of IEEE Communications Magazine andEditor of IEEE Communications Surveys and Tutorials.
1
Spatially-Consistent Human Body BlockageModeling: A State Generation Procedure
Margarita Gapeyenko, Andrey Samuylov, Mikhail Gerasimenko, Dmitri Moltchanov, Sarabjot Singh,Mustafa Riza Akdeniz, Ehsan Aryafar, Sergey Andreev, Nageen Himayat, and Yevgeni Koucheryavy
F
APPENDIX APROOF OF LEMMA 1Here, the proof of Lemma 1 is explained in details.First, note that the denominator of
p10 =P [M is non-Blocked ∩O is Blocked]
P [O is Blocked]. (1)
can be written as
P [O is Blocked] = P[(N2 ∩ L4b ∩ L4a
)∪
∪(L2 ∩ N4b ∩ L4a
)∪ · · · ∪
(N2 ∩ N4b ∩ N4a
)]. (2)
Adding L2 ∩ L4b ∩ L4a to the events listed in theright-hand side part of (2), we arrive at the completeevent space. Hence, (2) can be simplified to
P [O is Blocked] = 1− P[L2 ∩ L4b ∩ L4a
]. (3)
Observe that the event (M is non-Blocked) is equiv-alent to L3 ∩ L4b ∩ L4a. Taking into account the factthat the events L4a and N4a are mutually exclusive,the numerator of (1) becomes
P [M is non-Blocked ∩O is Blocked] =
= P[(N2 ∩ L4b ∩ L4a ∩ L4b ∩ L4a ∩ L3
)∪
∪(L2 ∩ N4b ∩ L4a ∩ L4b ∩ L4a ∩ L3
)∪
∪(N2 ∩ N4b ∩ L4a ∩ L4b ∩ L4a ∩ L3
)]=
= P[L3 ∩
((N2 ∩ L4b ∩ L4a ∩ L4b ∩ L4a
)∪
∪(N4b ∩ L4a ∩ L4b ∩ L4a
))]. (4)
The event N4b ∩ L4b corresponds to having at leastone blocker occluding the link path at O, while allof them are not high enough to block the link at M .
M. Gapeyenko, A. Samuylov, M. Gerasimenko, D. Moltchanov, S. An-dreev, and Y. Koucheryavy are with Tampere University, Tampere, Finland(e-mail: firstname.lastname, [email protected]).S. Singh is with Uhana (e-mail: [email protected])M. R. Akdeniz and N. Himayat are with Intel Corporation, Santa Clara,CA, USA (e-mail: mustafa.akdeniz, [email protected])E. Aryafar is with Portland State University, Portland, OR, USA (e-mail:[email protected])
Hence, accounting for (4), the probability that M isnon-blocked given that O is blocked can be obtainedas
p10 =Pnb3
(Pb2Pnb4bPnb4a + P∗nb4bPnb4a
)1− Pnb2Pnb4aPnb4b
, (5)
APPENDIX BDERIVATION OF GEOMETRICAL PARAMETERS
Here, we estimate the distances in the scenario thatare required by the proposed framework.
Aligning the center of the Cartesian coordinatesystem at 0 and OY axis with OP , the coordinatesof points A, B, C, and D are given by
xA = 0, xB = 0, xC = dm, xD = dm,
yA = 0, yB = r0, yC = r0, yD = 0. (6)
The coordinates of E, H , G, and F are thus
xE =dm2
+ d0 sinα−dm2
cosβ,
yE = d0 cosα−dm2
sinβ,
xH = xE + dm cosβ, yH = yE + dm sinβ,
xG = xH − r1 sinβ, yG = yH + r1 cosβ,
xF = xG − dm cosβ, yF = yG − dm sinβ, (7)
where r1 and β are given by
r1 =√r20 + d20 − 2r0d0 cosα,
β = arcsin(d0 sinα
r1
). (8)
The coordinates of K, the intersection of CD withFE, are
xK = dm, yK = (dm − xF )yE − yFxE − xF
+ yF . (9)
The length of KE is
|KE| =√
(xE − xK)2 + (yE − yK)2. (10)
2
The coordinates of the intersection between LD andGH , I , are
xI = dm, yI = (dm − xG)yH − yGxH − xG
+ yG. (11)
The lengths of IK and IJ are
|IK| =√(xK − xI)2 + (yK − yI)2;
|IJ | =√|IK|2 − d2m. (12)
To determine the probability of having no blockersthat affect the LoS path at M in zone 4, two parts oftwo planes (ABC) and (EFG) will be considered withthe following z coordinates: (zB = hC ; zC = hC ; zD =h1) and (EFG) (zF = hC ; zG = hC ; zH = h2). Theequation for (ABC) is
ax− by + cz + (−axC + byC − czC) = 0, (13)
where the coefficients a, b, and c are given by
a = (yB − yC)(zD − zC)− (yD − yC)(zB − zC),b = (xB − xC)(zD − zC)− (xD − xC)(zB − zC),c = (xB − xC)(yD − yC)− (xD − xC)(yB − yC). (14)
The equation for (EFG) reads as
fx− gy + hz + (−fxG + gyG − hzG) = 0, (15)
where the constants f , g, and h are
f = (yF − yG)(zH − zG)− (yH − yG)(zF − zG),g = (xF − xG)(zH − zG)− (xH − xG)(zF − zG),h = (xF − xG)(yH − yG)− (xH − xG)(yF − yG). (16)
Further, we require the equation for a segment PU ,the intersection of two planes ABC and FGH asfollows
x
gc− bh=
y − yQcf − ah
=z − zQbf − ag
, (17)
where yQ and zQ are
yQ =czQ + d
b, zQ =
gd− bebh− gc
. (18)
while d and e are:
d = −axC + byC − czC , e = −fxG + gyG − hzG. (19)
We also find the intersection point W of CD andPU with the coordinates
xW = dm,
yW =dm(cf − ah)(gc− bh)
+ yQ,
zW =
(dm(cf − ah)−r0(gc− bh)
− yQr0
+ 1
)(h1 − hC
)+ hC . (20)
The coordinates of T are given by
xT =(xW − xP )(yPxR − yNxR)
(yR − yN )(xW − xP )− xR(yW − yP )xRxP (yW − yP )
(yR − yN )(xW − xP )− xR(yW − yP ),
yT =xT (yR − yN )
xR+ yN , (21)
where the coordinates of P and N are
xP =dm2, xN = 0,
yP = r0, yN = r0 −dm2. (22)
The lengths of TR and WR are
|TR| =√
(xR − xT )2 + (yR − yT )2;|WR| =
√(xR − xW )2 + (yR − yW )2. (23)
The intersection point W ′ between PU and FE is
xW ′ =(yF − yQ)(gc− bh)(xE − xF )
(cf − ah)(xE − xF )− (yE − yF )(gc− bh)xF (yE − yF )(gc− bh)
(cf − ah)(xE − xF )− (yE − yF )(gc− bh),
yW ′ =xW ′(cf − ah)
gc− bh+ yQ. (24)
The coordinates of T ′ and S are therefore
xT ′ = xW ′ ; yT ′ = r0 −dm2
;
xS =(yS − yF )(xE − xF )
yE − yF+ xF ;
yS = r0 −dm2. (25)
The lengths of SR, W ′T ′, RK, and ST are
|SR| =√(xR − xS)2 + (yR − yS)2;
|W ′T ′| =√(xT ′ − xW ′)2 + (yT ′ − yW ′)2;
|RK| =√(xK − xR)2 + (yK − yR)2;
|ST | =√(xT − xS)2 + (yT − yS)2. (26)
Intersection of EH and CD is I ′.
xI′ = xD,
yI′ =(xD − xE)(yH − yE)
xH − xE+ yE . (27)
Intersection of FE and AD is K ′.
xK′ =(yD − yF )(xE − xF )
yE − yF+ xF ,
yK′ = yD. (28)
Intersection of AD and EH is D′.
xD′ =(yD − yE)(xH − xE)
yH − yE+ xE ,
yD = yD. (29)
3
Intersection of PU and AD is U ′.
xU ′ =(yD − yQ)(gc− bh)
cf − ah,
yU ′ = yD. (30)
Intersection of PU and EH is E′.
xE′ =xE(yH − yE)(gc− bh)
(yH − yE)(gc− bh)− (cf − ah)(xH − xE),
(yQ − yE)(xH − xE)(gc− bh)(yH − yE)(gc− bh)− (cf − ah)(xH − xE)
,
yE′ =xE′(cf − ah)gc− bh
+ yQ. (31)
All below coordinates belong to the projection ofthe point required for calculating the area in question.The coordinates of D′′, W ′′, T ′′, E′′, and H ′ are:
xD′′ = xD′
xW ′′ = xW ′
xT ′′ = xT
xE′′ = xE′
xH′ = xE
,
yD′′ = yR
yW ′′ = yR
yT ′′ = yD
yE′′ = yR
yH′ = yD
. (32)
Our technical report [1] collects all the cases withthe required limits of integration. The expressionsfor some parameters used in our technical report aregiven next.
LEF =(x− xF )(yE − yF )
xE − xF+ yF ;
LHG =(x− xG)(yH − yG)
xH − xG+ yG;
LHE =(x− xE)(yH − yE)
xH − xE+ yE ;
LPU =cf − ahgc− bh
· x+ yQ. (33)
REFERENCES[1] A. Samuylov et al., “Characterizing spatial correlation of block-
age statistics,” Tampere University of Technology, Techni-cal report available at: http://winter-group.net/downloads/mmWave blockage report.pdf, 2016.
