Page 132 - ITU Journal Future and evolving technologies – Volume 2 (2021), Issue 2

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

(ℓ) (ℓ)
(13) for the computation of the best pair of C and V even without the following extensions, our presented de‑
,
together with the optimum U : sign outperforms the SotA solutions, as will be shown in

Section 6 including our performance evaluation results.
3 ∶ max ℛ (V (ℓ) )
∗ DL ,
(ℓ) (ℓ) ℓ Remark 1: The presented solutions of 3 for the ana‑
U ,{C ,{V , } }
=1 ℓ=1
log cancellation and TX/RX digital BF are functions of the
(ℓ)
+ ℛ UL (C , V (ℓ) , U ) MIMO channel matrices H , , H , , and H , . This implies

,
that the update of the BF settings as well as the settings of
2
s.t. ‖[U ] ‖ = 1 ∀ = 1, 2, … , . the canceler (values for the taps or AUX TX RF chains as

( ,∶)
well as MUX/DEMUX con igurations) depend on the co‑
To solve 3 we adopt the following exhaustive search herence time of the involved wireless channels.
∗
approach. For each of the ∑ pairs of analog can‑
ℓ=1 ℓ
celer and TX digital precoder obtained in the previous Remark 2: Solving 2 is feasible when there exists
step as candidate designs for solving 2, we compute U (ℓ) (ℓ)
A
maximizing the uplink rate given by (13), while meeting at least one pair of C and V , meeting the con‑
straint. When such a pair does not exist, uplink communi‑
its respective constraint included in both 1 and 3.
Then, for each computed U and its corresponding C (ℓ) cations are impossible to take place simultaneously with

(ℓ) the downlink one (i.e., FD communications for the given
and V pair we calculate the achievable FD rate. The
, and is infeasible). We note that, for our FD rate re‑
A
joint design maximizing the FD rate provides the solution sults appearing in Section 6.4, we only focus on scenarios
for 3. To solve the uplink rate maximization problem where solving 2 is feasible. For cases where a C (ℓ) and
̃
we assume that H , and H , appearing in (13) and (14) (ℓ)
are available at node through appropriately designed V , pair satisfying does not exist, 1 can be solved
A
training phases. With the availability of this channel via half duplex communications, and there is no need for
(ℓ) (ℓ) a canceler design. In this case, the 1 solution is either

knowledgeandapairofC andV , , itcanbeshownthat
the U maximizing the UL rate is given using [29, Section the precoder maximizing the downlink rate or the com‑

H
4.2] by U = W , where W ∈ ℂ × has as columns biner maximizing the uplink one, depending on which of

the two results in the maximum half duplex rate. If we

the left singular vectors of −1/2 H , correspond‑ relax the SI constraint in 1 and 2 to a subset, in‑
E H

ing to its respective non‑zero singular values. The diago‑ stead of all, RX RF chains (i.e., suppose that the con‑

nal matrix ∈ ℂ × and the matrix E ∈ ℂ × straint becomes ‖[H V ] ‖ ≤ ∀ = 1, 2, … , ′

̃
2

, ( ,∶)
A

are obtained from the eigenvalue decomposition of the with ′ < ),FD communications are more proba‑

interference‑plus‑noise covariance matrix B ∈ ℂ × ble to be feasible for a given N and . This happens

A

at node , which is de ined as follows: because with this relaxation we allow uplink communi‑

′
cations even when there exists at most − RX RF
(ℓ) (ℓ) (ℓ) H (ℓ) H 2
B ≜ (H , + C )V , [V , ] (H , + C ) + I . chains experiencing average residual SI power larger than

(15)
. However, those saturated RX RF chains should not be
A
The eigenvalues of B are included in the main diagonal of considered for reliable reception, hence, they should be

, while the columns of E include their corresponding deactivated for uplink communications via adequate an‑

eigenvectors. The diagonal matrix ∈ ℝ × ensures tenna selection. Under this strategy, the uplink MIMO ma‑
2
′
the constraint ‖[U ] ‖ = 1 ∀ = 1, 2, … , is met. The trix is denoted by H ′ ∈ ℂ × being a submatrix of

( ,∶)

,
H
‑th entry of is equal to 1/‖[W ] ( ,∶) ‖. For the special case H , , where the rows corresponding to the saturated RX
of = 1 [17], which consequently results in = 1, RF chains have been excluded. It is inally noted that the

′
the solution combining vector w ≜ W ∈ ℂ 1× sim‑ value for , and to which speci ic RX RF chains the A

pli ies to the eigenvector corresponding to the maximum constraint is imposed, will impact the achievable uplink
eigenvalue of the matrix A ∈ ℂ × given by [30] rate, and hence, the feasible FD communications.

−1
H
A ≜ P B h , , , (16)
h

6. SIMULATION RESULTS AND DISCUSSION
where we have used the notation h , ≜ H , ∈ ℂ ×1 .
We note that for the practical case of imperfect analog The performance of the wireless communication scenario
cancellation, signi icant gains with the considered RX dig‑ illustrated in Fig. 1 using the FD MIMO design presented
ital combining are feasible only when the following con‑ in Section 5 is evaluated. In Section 6.1 we describe the

dition holds: − ≥ . benchmark approaches with which our solutions will be

compared. The simulation parameters and assumptions

5.3 Remarks are then detailed in Section 6.2, whereas the SI mitigation
capability and achievable rate results for different hard‑
We next provide some subtleties of our example FD MIMO ware complexity levels are presented in Secs. 6.3 and 6.4.
design and possible extensions. We note however that,
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