Page 89 - 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

∗

The gradient of Lagrangian with respect to , is given by Next, is set to 2 . By inserting = 2 and Φ , =

(2 −1)2 − into the above equation, the result in (17) can
2 2
̂ ∫ (‖Fa( , )‖ ) be obtained.
∞

0
,
∗
∗
, = , −
2 +1 −1 ( −1) 2
| ∑ | APPENDIX C
,
= ∫ =0 ∗ −
,
2 −1 , Proof of Theorem 3

2 +1 −1
− 2 The objective function in (22) can be reformulated as
= ∫ ∑ ( − ) −
,
2 −1 =0
0 + −1
−1 ( − )(2 +1) ( − )(2 −1) , w ‖ 2
− 2 − 2 ( − ) − ∑ ∑ max w ∈W ‖H 2
= ∑ =1 = 0
=0 ( − )
0 + −1
− = ∑ ∑ max tr((H , ) H , w w )

,
−1 =1 w ∈W
− 2 2 sin(( − ) ) = 0
= ∑ − 0 + −1
− ,
=0 = ∑ ∑ max w ∈W
̂
= − , =1 = 0
,
, ,
(45) 1 ̂ ̂ ̂ ̂

where ̂ , = ∑ −1 2 sin(( − ) ) , which is not dependent − (tr((H , − W )(H , − W ) ))
2
−
=0
1
1
on . Therefore, there exists a Lagrange multiplier that + tr(H ̂ H ̂ ) + tr(W W ),
̂
̂
satis ies the KKT conditions. Hence, F is a local optimum 2 , , 2
for the optimization problem (14) and (42). (49)
̂

where H ̂ , and W are de ined as (H , ) H , and

̂
̂

APPENDIX B w w , respectively. Since tr(H ̂ , H ̂ ) and tr(W W ) are

,

constant, the optimization problem in (22) is equivalent
Proof of Theorem 2 to
0 + −1
̂
min W ∑ ∑ min w ∈W (H ̂ , , W ) (50)

Proof. According to [16], the receive power =1 = 0

2
|a ( , Φ)a ( , Φ )| is given by

,

APPENDIX D
2
sin ( (Φ − Φ))

2
|a ( , Φ)a ( , Φ )| = , (46) Proof of Theorem 4

,

2
sin (Φ , − Φ)
Since (W) is ixed to 1, W can always be decomposed
Inserting Φ , = (2 −1)2 − to the above equation, we ob‑ as

tain: W = w w . (51)

2
2
sin ( (Φ , − Φ)) = sin ( ( (2 −1)2 − − Φ)) Hence, the optimization problem in (26) is equivalent to

2

2
sin ((Φ , − Φ)) sin (Φ , − Φ) (47) max w ∑ w Hw

2
sin ( Φ) H∈ℋ (52)
= . s.t. w w = 1.

2
sin (Φ , − Φ)
The optimal solution for w is given by w , where

Hence, maximizing the receive power w is the eigenvector of ∑ H corresponding to its

|a ( , Φ)a ( , Φ )| 2 over the codebook in the H∈ℋ

,

ℎ layer is equivalent to inding the minimum of largest eigenvalue.
2
sin (Φ , − Φ), which corresponds to minimizing
|Φ , − Φ| with 1 ≤ ≤ 2 −1 . In other words, the beam REFERENCES
coverage of w( , ) is a set of the steering vector with LOS [1] Ian F Akyildiz, Josep Miquel Jornet, and Chong
spatial angle close to Φ . Since the distance between Han. “Terahertz band: Next frontier for wireless
,
Φ , and Φ , −1 is 4 , the beam coverage of w( , ) is communications”. In: Physical Communication 12
2
obtained as (2014), pp. 16–32.
2 2
(w( , )) = [Φ , − 2 , Φ , + 2 ] . (48)
© International Telecommunication Union, 2021 77