Page 79 - 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

bBMᅿ gain of a given hierarchical codebook W =
ϵᇎց ᆢ ֍
Channel Domain ℋ beamforming խհմ Bb +QMbB/2`2/ b mMB7Q`KHv /Bbi`B#mi2/
ဈ
վ
rBi?BM ᅺ - r?2`2 ԡ Bb i?2 Mi2MM bT +BM;X .m2 iQ
2

1
{W , W , ⋯ , W } is de ined as the average receive power
The first layer ?(0, 1) i?2 GPa@/QKBM Mi +? MM2H T`QT2`iv BM i?2 +QMbB/2`2/
̄ (W) of the best beam code in the highest layer, i.e.,
b+2M `BQ- i?2 #2 K7Q`KBM; ; BM Q7 ;Bp2M ?B2` `+?B+ H
The second layer ?(1, 1) ?(1, 2) +Q/2#QQF q \q q ੈ q ^ Bb /2}M2/ b i?2 i?2
φ
ϵ
ժ
`2+2Bp2 TQr2` Q7 i?2 #2bi #2 K +Q/2 BM i?2 ?B;?2bi H v2`
2
The third layer ̄ (W) = ∫ max w( , )∈W ‖Hw( , )‖ (H) H, (11)
2 H
?(2, 1) ?(2, 2) ?(2, 3) ?(2, 4) 7Q` i?2 GPa +QKTQM2Mi rBi? bT iB H M;H2 ဈ- BX2X-
H∈ℋ
The last layer
!(3, 1) !(3, 2) !(3, 3) !(3, 4) !(3, 5) !(3, 6) !(3, 7) !(3, 8) Ԗ ဈ K t r୮q Դધ ԃ ဈ rધ URkV
ϵ
յ
յ
֎
ϵ
where (H) is the Probability Density Function (PDF) of
H
ॆ X
h?2
յ
H. The aim of the codebook design is to generate a code‑
օြ
օ կ Կ ਷φ ြ
r?2`2 ԃ ဈ ॅ ੈ Ԕ ੈ Ԕ
֎
յ
+Q/2#QQF /2bB;M T`Q#H2K Bb 2[mBp H2Mi iQ K tBKBxBM;
6B;X j Ĝ h?2 #2 K +Qp2` ;2 Q7 j@H v2` ?B2` `+?B+ H +Q/2#QQFX book W satisfying Criterion 1 meanwhile maximizing the
Fig. 3 – The beam coverage of a 3‑layer hierarchical codebook.
i?2 p2` ;2 #2 K7Q`KBM; ; BM Qp2` ဈ ୮ ᅺ X >2M+2-
+Q/2#QQF Bb irB+2 Q7 i?2 bBx2 Q7 i?2 Ԙ ਷ i? +Q/2#QQFX beamforming gain ̄ (W) over ℋ. Meanwhile, the transmit
then discuss the design of the hierarchical codebooks in power must be limited. Without loss of generality,
i?2 +Q/2#QQF /2bB;M T`Q#H2K + M #2 r`Bii2M b, we as‑
h?2 +Q/2rQ`/b BM i?2 bm#+Q/2#QQF `2 H v2`2/ /m2 iQ
the LOS scenarios and the NLOS scenarios. sume that the ‑th beamforming code word w( , ) in the
i?2B` #2 K +Qp2` ;2 b b?QrM BM 6B;X jX G2i r Ԙ ԙ ମ ‑th layer has unit norm, i.e., w ( , )w( , ) = 1.

ϵᇎ
Ԙମ Ӿ ମ ԙ ମ
3.1 De inition ք਷φ /2MQi2 i?2 ԙi? +Q/2rQ`/ BM i?2 Ԙi? K t q ௷ Ԗ ဈ ԓဈ
H v2`X h?2 #2 K +Qp2` ;2 Q7 HH +Q/2rQ`/b BM i?2 }`bi The codebook
ြ Ј design problem introduced above is formu‑
Before introducing the de inition of a hierarchical lated r Ԙ ԙ r Ԙ ԙ ମ Ԙ ମ Ӿ ମ ԙ ମ ք਷φ
Ӿ਷ H v2`b Kmbi b iBb7v i?2 7QHHQrBM; +`Bi2`BQM, code‑
թ
bXiX on the basis of the codebook design in [25]. How‑
*`Bi2`BQM R, h?2 #2 K +Qp2` ;2 Ӹԋ r Ԙ ԙ Q7 M `@
book, we irst de ine the beam coverage (w ) of a code ever, in [25], mainly the criteria for a hierarchical code‑

Ӹԋ r Ԙ ԙ Ӹԋ r Ԙ ԙ ਷ ૌ
#Bi `v +Q/2rQ`/ r Ԙ ԙ b?QmH/ #2 i?2 mMBQM Q7 i?Qb2 Q7
word w within a codebook W. Let ℋ denote the set of all book are proposed and the objective function of the code‑
ք਷φ

Ӹԋ r Ԙ ԙ ମ Ԙ ମ Ӿ ਷ ମ ԙ ମ
irQ +Q/2rQ`/b BM i?2 M2ti H v2`- BX2X- given indoor prop‑
possible channel snapshots within the book design problem is not explicitly stated. In this work,
URjV
agation scenario, which is called the channel domain in the two criteria for the hierarchical codebook in [25] are
r?2`2 r Ԙ ԙ /2MQi2b i?2 ԙi? +Q/2rQ`/ BM i?2 Ԙi? H v2`X
Ӹԋ r Ԙ ԙ Ӹԋ r Ԙ ԙ ਷ ૌ Ӹԋ r Ԙ ԙ
the following. The beam coverage (w ) of a code word modi ied to a single criterion by adjusting the de inition
h?2 #Qp2 QTiBKBx iBQM T`Q#H2K Bb MQM@+QMp2t T`Q#@
URyV

w in a codebook W is a subset of the channel domain ℋ, of the beam coverage. In addition, the beamforming gain
H2K /m2 iQ i?2 MQM+QMp2t +QMbi` BMibX >2M+2- Bi Bb /B7@
r?2`2 ମ Ԙ ମ Ӿ ਷ X r Ԙ ԙ Bb + HH2/ i?2 T `2Mi Q7 r Ԙ
where the receive power with (w ) is strongest among of the codes in a hierarchical codebook is utilized in this
}+mHi iQ Q#i BM i?2 ;HQ# HHv QTiBKmK bQHmiBQM 7Q` i?2

ԙ਷ M/ r Ԙ ԙ - M/ r Ԙ ԙ਷ r Ԙ ԙ
all code words in W, i.e., work as an objective function, which can be regarded as a
+Q/2#QQF qX >Qr2p2`- i?2 Q#D2+iBp2 7mM+iBQM p Hm2 /2@
`2 + HH2/ i?2 +?BH/`2M Q7 r Ԙ ԙ X
T2M/b QMHv QM i?2 bm#+Q/2#QQF BM i?2 ?B;?2bi H v2`- BX2X-
h?2 #Qp2 +`Bi2`BQM ;m ` Mi22b i?2 +Q/2#QQF Bb i`22@ supplement of the codebook design in [25].
(w ) = {H ∣ ‖Hw ‖ = max ‖Hw ‖ , H ∈ ℋ} .
w ∈W
HBF2- r?B+? + M #2 mb2/ BM i?2 ?B2` `+?B+ H #2 K HB;M@ q <r r ੈ r Դ਷ȯ >X PM2 ;`22/v TT`Q +? 7Q` i?2

2
2
ժ
յ
φ
ϵ
(8)
ϵ
K2MiX AM i?2 >" H;Q`Bi?K- r?B+? rBHH #2 BMi`Q/m+2/ +Q/2#QQF /2bB;M T`Q#H2K Bb iQ +?QQb2 i?2 bm#+Q/2#QQF
BM i?2 ?B;?2bi H v2` q 7Q` K tBKBx iBQM Q7 i?2 Q#@
Note that for each channel realization H in ℋ, there is al‑ 3.2 LoS codebook ժ design
BM a2+iBQM 9- i?2 H;Q`Bi?K +QMp2`;2b iQ #2 K +Q/2 BM
ways a code word in W aligned to H. In other words, the D2+iBp2 7mM+iBQM p Hm2X h?2M- i?2 ?B;?2bi bm#+Q/2#QQF
i?2 ?B;?2bi H v2` rBi? ?B;? T`Q# #BHBivX h?mb- i?2 p2`@
union of the beam coverages of all code words composes In ժ this subsection, a LoS propagation scenario is consid‑
q Bb 2ti2M/2/ iQ ?B2` `+?B+ H +Q/2#QQFX h?2 /2bB;M
;2 #2 K7Q`KBM; ; BM Q7 ;Bp2M ?B2` `+?B+ H +Q/2#QQF
T`Q#H2K 7Q` i?2 ?B;?2bi H v2` bm#+Q/2#QQF q Bb ;Bp2M
the channel φ domain ℋ, i.e., ered, where the LoS component is always obtained. Based
q \q q ੈ q ^ Bb /2}M2/ b i?2 p2` ;2 `2@
ժ
ϵ
ժ
#v,
+2Bp2 TQr2` ۽ԡ q Q7 i?2 #2bi #2 K +Q/2 BM i?2 ?B;?2bi on the measurement results in [20], the power of the LOS
(w ) = ℋ.
∪
(9)
H v2`- BX2X- w ∈W component is nearly 20 dB larger than the power of the
NLOS components in the THz band in a typical propaga‑
௷K t
K t
ϵ
A hierarchical codebook is a family of sub‑codebooks tionscenarioduetothehighre lectionandscatteringloss.
յ
r୮q Դધ ԃ ᅿ rધ ԓᅿ
q Դ
UR9V
K t
ϵ
յ
ϵ

ϵ >
۽ԡ q ௷
r ժӴօ ୮q Դધ>r Ӿ ԙ ધ ԕ > ԓ>
W , 1 ≤ ≤ . In this work, we only consider the bi‑ Therefore, the channel in THz LOS scenario is in principle
ᇓ
bXiX
>୮ป
թ
ժ਷φ
URRV
nary hierarchical codebook, which means the size of the LOS‑dominant. In this case, the beam should be focused

r Ӿ ԙ r Ӿ ԙ ମ ԙ ମ
th codebook is twice of the size of the ( −1)th codebook. on the LOS component.
r?2`2 ԕ > Bb i?2 T`Q# #BHBiv /2MbBiv 7mM+iBQM US.6V
h?Bb QTiBKBx iBQM T`Q#H2K Bb MQM@+QMp2t bBM+2 i?2 Q#@
>
Q7 >X h?2 BK Q7 i?2 +Q/2#QQF /2bB;M Bb iQ ;2M2` i2
The code words in the sub‑codebook are layered due to D2+iBp2 7mM+iBQM Bb MQM@+QM+ p2 7mM+iBQMX h?2Q`2K R
+Q/2#QQF q b iBb7vBM; *`Bi2`BQM R K2 Mr?BH2 K tBKBx@
their beam coverage as shown in Fig. 3. Let w( , ), 1 ≤ In this paper, the receiver position is assumed to be
b?Qrb i? i i?2 .6h +Q/2#QQF rBHH #2 QM2 Q7 i?2 HQ+ HHv
BM; i?2 #2 K7Q`KBM; ; BM ۽ԡ q Qp2` ෧X
≤ , 1 ≤ ≤ 2 −1 denote the th code word in the th uniformly distributed within the considered indoor en‑
QTiBKmK bQHmiBQMb 7Q` i?2 #Qp2 QTiBKBx iBQM T`Q#H2K
layer. The beam coverage of all code words in the irst vironment in Fig. 2. Thus, the LOS spatial angle
B7 i?2 ?B;?2bi H v2` bm#+Q/2#QQF bBx2 Bb }t2/ iQ ԃ X
2
GQa *Q/2#QQF .2bB;M
jXk
− 1 layers must satisfy the following criterion: Φ = sin is considered as uniformly distributed
յ
h?2Q`2K R, h?2 mMBi `v .6h K i`Bt 6 Bb QM2 HQ+ HHv
Criterion 1: The beam coverage (w( , )) of an arbitary within (0, 2 ), where is the antenna spacing. Due to the
QTiBKmK bQHmiBQMb 7Q` i?2 #Qp2 QTiBKBx iBQM T`Q#H2K
AM i?Bb bm#b2+iBQM- GQa T`QT ; iBQM b+2M `BQ Bb +QMbB/@
code word w( , ) should be the union of those of two code LOS‑dominant channel property in the considered sce‑
2`2/- r?2`2 i?2 GQa +QKTQM2Mi Bb Hr vb T`2bmBiX " b2/
B7 i?2 +Q/2#QQF bBx2 Bb }t2/ iQ ԃ
յ
words in the next layer, i.e., nario, the beamforming gain of a given hierarchical code‑
QM i?2 K2 bm`2K2Mi `2bmHib BM (kR)- i?2 TQr2` Q7 i?2 GPa book W = {W , W , ⋯ , W } is de ined as the the receive

2
1
S`QQ7X a22 TT2M/Bt X
+QKTQM2Mi Bb M2 `Hv ky /" H `;2` i? M i?2 TQr2` Q7 i?2
(w( , )) = (w( + 1, 2 − 1)) ∪ (w( + 1, 2 )), power of the best beam code in the highest layer for the
LGPa +QKTQM2Mib BM i?2 h>x # M/ BM ivTB+ H T`QT @ LOS component with spatial angle Φ, i.e.,
6B;m`2 9U V BHHmbi` i2b i?2 #2 K7Q`KBM; ; BM Q7 i?2 .6h
(10)
where 0 ≤ ≤ − 1. w( , ) is called the parent of w( +Q/2#QQF BM i?2 +QMbB/2`2/ GPa BM/QQ` T`QT ; iBQM b+2@
; iBQM b+2M `BQ /m2 iQ i?2 ?B;? `2~2+iBQM M/ b+ ii2`BM; +
HQbbX h?2`27Q`2- i?2 +? MM2H BM h>x GPa b+2M `BQ Bb BM
1, 2 −1) and w( +1, 2 ), and w( +1, 2 −1), w( +1, 2 ) M `BQX *QKT `2/ iQ i?2 #2 K7Q`KBM; ; BM Q7 i?2 K t@
T`BM+BTH2 GPa@/QKBM MiX AM i?Bb + b2- i?2 #2 K b?QmH/
2

‖a ( , Φ)w‖ .
(12)
(Φ) = max
are called the children of w( , ). BKmK ` iBQ i` MbKBbbBQM UJ_hV BM 6B;X 9U#V- r?B+?
2

w∈W

#2 7Q+mb2/ QM i?2 GPa +QKTQM2MiX codebook is tree‑like,
The above criterion guarantees the Bb +QMbB/2`2/ b i?2 mTT2` #QmM/ Q7 i?2 #2 K7Q`K@
AM i?Bb T T2`- i?2 `2+2Bp2` TQbBiBQM Bb bbmK2/ iQ #2
which can be used in the hierarchical beam alignment. BM; ; BM- i?2 #2 K7Q`KBM; ; BM Q7 i?2 .6h +Q/2#QQF
rBi?BM i?2 2MiB`2 BM/QQ` 2MpB`QMK2Mi Bb +HQb2 iQ i? i
mMB7Q`KHv /Bbi`B#mi2/ rBi?BM i?2 +QMbB/2`2/ BM/QQ` 2M@

Φ
In the HBA algorithm, which will be introduced in Sec‑ where a ( , Φ) = [1, ⋯ , , ⋯ , ( −1)Φ ] . The code‑

h?mb- i?2 GPa bT iB H M;H2
Q7 J_hX >2M+2- i?2 ?B2` `+?B+ H .6h +Q/2#QQF 2Mi BHb
pB`QMK2Mi BM };X
kX
tion 4, the algorithm converges to a beam code in the book design problem is equivalent to maximizing the av‑
highest layer with high probability. Thus, the average erage beamforming gain over Φ ∈ (0, 2 ). Hence, the
8
© International Telecommunication Union, 2021 67