Page 81 - ITU Journal Future and evolving technologies Volume 2 (2021), Issue 7 – Terahertz communications

ITU Journal on Future and Evolving Technologies, Volume 2 (2021), Issue 7

According to Criterion 1 of the hierarchical codebook, the where H , is the NLOS channel frequency response of

beam coverage of a beam code w( , ), = 1, 2, ⋯ , − 1 is t h realization at frequency . The corresponding

determined by modi ied codebook design problem reads
2 ( − 1)2 − 2 2 −
(w( , )) = ( , ) . (17) max W (W)

s.t. w ( , )w( , ) = 1,
In the LOS scenario, a code word w( , ) is identical to the = 1, 2, ⋯ , , = 1, 2, ⋯ , 2 −1
steering vector located in the center of its corresponding
beam coverage (w( , )), which is given by (w( , )) =
w( , ) = a( , Φ ), (18) (w( + 1, 2 − 1)) ∪ (w( + 1, 2 )).

,
(21)
−
where Φ , = (2 −1)2 . Theorem 1: The beam coverage The above optimization problem cannot be solved di‑

(w( , )) of the ‑th code word in the ‑th layer can be rectly since the second hierarchical structure constraint
expressed as is non‑convex. Hence, before solving the overall hierar‑
chical codebook design problem, a single layer codebook
2 (2 − 2) 4
(w( , )) = {Φ ∣ Φ ∈ [ , ]} . design problem is solved since the objective function de‑
2 2 pends only on the highest layer sub‑codebook. For a sin‑
gle layer codebook W = [w , w , ⋯ , w ], where is the

2
1
Proof. See Appendix B.
codebook size, the optimization problem is given by
3.3 NLOS codebook design
max W ∑ max w ∈W ‖H , w ‖ 2
2

In an indoor scenario, the LOS component might be (22)
=1
blocked by obstacles such as persons, furniture, or many s.t. w w = 1, = 1, 2, ⋯ ,

other diverse objects. In this case, utilizing the power
from NLOS components will be essential for the beam‑
forming performance. Consider a hierarchical codebook Theorem 3: Let H ̂ and denote (H , ) H , . The

,

1
2
W = {W , W , ⋯ , W } with layers. The codebook de‑ distance (X, Y) between two × matrices X and Y
sign problem can be written as
is de ined as
max W ∫max w( , )∈W ‖w( , )H ( , )‖ 2 2

(X, Y) = tr((X − Y)(X − Y) ). (23)
( )d

s.t. w ( , )w( , ) = 1,
The optimization problem in (22) is equivalent to a dis‑
= 1, 2, ⋯ , , = 1, 2, ⋯ , 2 −1 tance minimization problem, i.e.,
(w( , ))
= (w( + 1, 2 − 1)) ∪ (w( + 1, 2 )) 0 + −1
(19) min ∑ ∑ min (H ̂ , w w )

where ( ) is the PDF of the geometry information. How‑ W =1 = 0 w ∈W , (24)

ever, modeling the NLOS components’ behavior in the

THz band is still an open problem. It is dif icult to model s.t. w w = 1, = 1, 2, ⋯ ,
the NLOS channel analytically including the re lection and
scattering behavior. Faced with this challenge in the
NLOS scenario, it is generally intricate to ind an analyt‑ Proof. See Appendix C.
ical method to generate the codebook in the NLOS sce‑
nario. Therefore, a data‑driven codebook design such as
One well‑known algorithm to solve the distance mini‑
the one proposed in [17] is of great bene it for the con‑
mization problem is k‑means clustering, which is an un‑
sidered indoor THz NLOS scenario. In the data‑driven ap‑ ̂
proach, the integral over the indoor scenario is approxi‑ supervised machine learning algorithm. Here W , =
̂

mated by the average over realizations of the indoor 1, 2, ⋯ , and H , , = 1, 2, ⋯ , are regarded as the

channel. In particular, the objective function is de ined as clustering centers in k‑means clustering and the training
data set, respectively. The overall training data set is de‑
noted as ℋ
. The algorithm proceeds by alternating
2
(W) = ∑ max w( , )∈W ‖H , w( , )‖ , (20) between assignment step and update step as described in

2
=1 the following.
© International Telecommunication Union, 2021 69