Page 66 - ITU Journal Future and evolving technologies – Volume 2 (2021), Issue 2

ITU Journal on Future and Evolving Technologies, Volume 2 (2021), Issue 2

Table 3 – MPCs with power level above min = −60 dBm and cosine becomes a unit vector with the given elevation and
distance‑based clustering results for = 20 . ∘
azimuth angles.
MPC AoD‑Az AoD‑El AoA‑Az AoA‑El Power Delay Cluster

For clustering the MPCs in (11), an iterative cosine

∘
∘
∘
∘
# (i) ( ) ( ) ( ) ( ) (dBm) (ns) # (i)
distance‑based k‑means clustering algorithm is used. At

1 100 0 ‑80 0 ‑37.33 33.85 1
each iteration X is grouped into a number of clusters that
2 100 ‑20 ‑80 0 ‑43.25 33.20 1

3 ‑60 ‑20 ‑80 ‑20 ‑45.35 42.32 2 is equal to the current iteration count, and this process
4 ‑120 0 ‑80 0 ‑50.75 57.94 4 is repeated until the desired spatial distance between the
5 20 0 ‑20 0 ‑51.27 88.54 3 cluster centroids is achieved. The cosine distance be‑

6 120 0 ‑100 0 ‑51.60 37.11 1 tween any two vectors and is de ined as follows:
7 120 0 ‑120 0 ‑52.35 39.06 1
8 180 0 ‑140 0 ‑52.83 115.88 5 ( , ) = 1 − cos( ), (12)
9 20 0 ‑140 0 ‑53.42 102.21 3
10 ‑100 0 ‑80 0 ‑54.08 54.68 4 where is the angle between the vectors in three‑
11 20 20 ‑20 20 ‑54.52 94.40 3 dimensional space. If and point in the same direction
∘
12 180 20 ‑20 20 ‑55.47 107.42 5 ( = 0 ), then ( , ) = 0, or if they point in opposite
∘
13 ‑100 0 ‑80 0 ‑56.60 55.99 4 directions ( = 180 ), then the distance attains its maxi‑
14 120 ‑20 ‑100 0 ‑57.07 36.46 1 mum, and ( , ) = 2.
15 100 0 ‑120 0 ‑57.50 38.41 1
16 40 20 ‑20 20 ‑58.69 93.74 3 Let Γ = { ,1 , … , , } be the set of cluster centroids at
Γ
17 100 0 120 ‑20 ‑59.93 40.36 1 ‑th ( ≤ ) iteration, where , is a 1×3 vector that rep‑
inding a link between the TX and the RX will be higher resents the centroid of ∈ X. It should be noted that, al‑

when the MPCs spread out over the AoD azimuth and though each MPC will be matched to a centroid, there will
AoD elevation space compared to when the MPCs exhibit be unique clusters and hence centroids. Each centroid is
clusters. Therefore, for a reliable assessment of the the coordinate‑wise mean of the points in a cluster, after
multipath richness of a channel, one should also take into normalizing those points to unit Euclidean length. Once
consideration the spatial diversity of the MPCs. thecentroidsarefoundforthecurrentnumberofclusters,
the angular distance between any MPC and the centroid of
3.2 Cosine distance‑based clustering the cluster to which that MPC belongs is calculated. If all
the MPCs are at an angular distance from their centroids
Having identi ied the paths over which a link can be es‑ of less than a beam separation threshold , i.e.,

tablished, the next step is to ind the number of effec‑
tive alternate paths (i.e., beam directions that are sepa‑ , = ∠ < , ∀ ∈ (1, … , ) , (13)
,

rated by at least a user‑de ined angular distance from any then the iterations are terminated, and the current itera‑
other). Here we assume that the beamwidth is smaller tion count is returned as the number of clusters.
compared to the blockage angle , as shown in Fig. 1. To

achieve higher gains, multiple antenna elements with nar‑ The parameter is a design parameter, and it represents

row beams are used at mmWave frequencies. This also the size or the angular width of the blockages that are
limits the number of beam directions. For example, it is likelytoobscurethepossiblelinksbetweentheTXandthe
shown in [14] that the number of beam directions at the RX in a given environment (see Fig. 1). When is higher,

∘
mmWave BS side (with a 8 × 8 array and beamwidth 13 ) it gets more likely that the paths close to each other will be
is only 10 in azimuth when the BS scans a total of 120 de‑ blocked. As a result, there will be fewer paths over which
grees. the signals can be transmitted. We point out that if the
angular resolution of the TX/RX antennas is much lower
Unlike the approaches that aim to parameterize the chan‑ than , then the clustering process can be skipped. In

nel impulse response by clustering the MPCs based on such a case, each resolved path can be treated as a clus‑
AoA,AoD,and delayinformation(e.g.,[15, 16]), wecluster ter, and their powers can be directly plugged into (14) to
the MPCs only using their angular parameters. This way, calculate the EMR.
it is possible to ind how many alternate paths are avail‑
able if some of the paths are blocked. Omitting the pa‑ It should be recalled that initializing the centroids at dif‑
rameters other than the angles in the interested domain ferent locations may lead the k‑means algorithm to return
(in this case, the AoD), each MPC in X in (10) can be rep‑ different clusters. It is also possible that some initializa‑
resented by a vector in three‑dimensional space as tions may result in local optima. Both scenarios intro‑
duce a bias in the EMR values. Therefore, at each itera‑
= { , AoD,Az , AoD,El } , (11) tion, we randomly initialize centroids, run the k‑means

algorithm, and compute the cost function. This process is
for = 1, … , , where is the number of MPCs in X, and repeated times, and the clustering that yields the low‑
is the magnitude of the vector . While may be de ined est cost is picked for the current iteration. For the chan‑

as the individual powers of the MPCs, since here only the nel measurements used in this study, we observed that it
angles are of interest, can be simply set to 1, so is suf icient to set = 20 for the clusters to converge;

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