Page 129 - ITU Journal Future and evolving technologies – Volume 2 (2021), Issue 2

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

whereas, for the multi‑AUX‑TX canceler architecture, (C2) 5. AN EXAMPLE FD MIMO DESIGN
can be expressed using the description of Section 3.2 as:

Capitalizing on the general optimization framework for

the joint design of C , V , and U at the FD MIMO node

C = L L with (9). (C2b) described in Section 4, we hereinafter present an exam‑
5 4

ple joint design of analog cancellation and digital BF. We
assume that there is no inter‑node interference between
In addition, constraint (C3) including the general vector the half duplex multi‑antenna nodes and due to, for
function 1 ∶ ℂ ×1 → ℝ ×1 sets the threshold val‑ example, appropriate node scheduling [9, 10] for the FD
+
ues inside the vector A ∈ ℝ ×1 on functions of the operation of node . Extensions considering this interfer‑
+
ence for the cases where it is known at either the receiv‑
instantaneous residual SI appearing at the RX an‑

tenna elements after analog cancellation and before the ing node and/or the transmitting node , or unknown

RX RF chains. Two examples of the function are: i) to both, are left for future works. The latter assumption
1
the element‑wise instantaneous powers of the residual translates to setting the channel matrix between the in‑
= 0
volved nodes as H
SI signals; and ii) their summation. For the former 1 , × . For this case, the model
example, (C3) results to |[(H , + C )V s ] | 2 ≤ [ ] given by (1) for the received signal at node reduces to:

A
with = 1, 2, … , , whereas for the latter example y = H , (10)
V s + n .

2
((H , + C )V s ) = ‖(H , + C )V s ‖ and conse‑

1
quently ≡ ∈ ℝ . Finally, constraint (C4) with the We rewrite the signal model (3) that describes the estima‑
+
A
A
̂
×1 ×1 tion for s at the RX of node as:
general vector function ∶ ℂ → ℝ + imposes the
2
values included in the vector D ∈ ℝ ×1 on functions ̂ s = U (H , ̃ , (11)
V s + n ) ,
V s + H
+

of the instantaneous residual SI signals obtained af‑

ter applying analog cancellation and RX digital combining. where ̃ H , ∈ ℂ × denotes the effective SI channel
Similar to , instances of the function are the individ‑ after performing analog cancellation, which is de ined as:
2
1
ual instantaneous powers of the latter signals as well ̃ , ≜ H , + C .
H

as their summation. An important performance objective function (⋅) for the
considered system is the FD rate de ined as the sum rate
ThemainnovelcomponentsoftheproposedFDMIMOop‑ of the downlink and uplink communications. We there‑
timization framework in can be summarized as fol‑ fore focus on designing C , V , and U via the solution of

lows. First, the digital TX and RX BF design takes into the following optimization problem:
explicit account the available number of analog taps ,
or number of AUX TXs , of the analog SI cancellation 1 ∶ max ℛ DL (V ) + ℛ UL (C , V , U )

block. Although some available BF solutions [4, 20, 22] C ,V ,U
for FD MIMO systems consider the presence of an ana‑ s.t. (C1), (C2),
log SI canceler, the details of its hardware limitations are ‖[H , ( ,∶) ‖ ≤ ∀ = 1, 2, … , ,
̃
2
V ]
A

excluded from the BF design. Second, the proposed FD 2

( ,∶)
MIMO framework is the only one that explicitly consid‑ ‖[U ] ‖ = 1 ∀ = 1, 2, … , .
ers the case where < min{ , }, i.e., the avail‑ In the latter problem, the achievable downlink rate ℛ is

able number of analog taps, or AUX TX RF chains, may be DL
a function of only the digital precoding matrix V and is

smaller than both the numbers of TX and RX RF chains.
given by:
This is an important feature for practical FD MIMO de‑
H
V V H
ployments, since current analog SI cancellation solutions ℛ DL (V ) = log (det (I + −2 H , H )) . (12)

,
2
require either very large numbers of taps, of the order of

for the architecture proposed in [6], or very large Note that we have assumed capacity‑achieving combin‑

number of AUX TXs, of the order of for the architec‑ ing at node in (12), like the non‑linear Minimum Mean

ture presented in [16]. Third, our framework has the ad‑ Squared Error (MMSE) successive interference canceler
vantage of a more improved utilization of the spatial DoF [27, Chap. 2]. The uplink rate ℛ UL in 1 is a function of

offered by the available multiple antennas at the FD MIMO V , the analog canceler matrix C , and the digital combin‑

node . For example, if the analog canceler consists of ing matrix U , and is derived as:
only = 1 tap, or = 1 AUX TX, then its cancellation ℛ (C , V , U ) = log ( det (I + −2 U H
capabilities are very limited, and more spatial DoF need UL 2 , (13)
to be devoted from the TX and RX BF blocks for meeting × V V H H U Q )),
H
−1
H
the thresholds and in (C3) and (C4). At the other ,
D
A
extreme, if can be afforded to be large, the digital BF where Q ∈ ℂ ×

design may exploit the fact that a signi icant part of SI mit‑ denotes the covariance matrix of
the interference‑plus‑noise after combining at node that
igation is handled by the analog canceler, and thus make can be expressed as:
use of more of the available spatial DoF for improving the
̃
H
2
H
quality of the incoming and outgoing signals of interest. Q = U H , H ̃ H , U + U U . (14)
V V H

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