Page 131 - ITU Journal Future and evolving technologies – Volume 2 (2021), Issue 2

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

Algorithm 1 TX Digital Precoding for a Given Analog can‑ ( ) possible realizations for the multi‑tap canceler.

celer Each of those refers to a different C (ℓ) matrix and corre‑
(ℓ)
Input: P , H , , H , , , max , and a realization of C sponds to a speci ic placement of the tap values inside

A
(ℓ)
satisfying constraint (C2). C ; its remaining elements (i.e., − ) need to be

1: Obtain D including the right‑singular vectors of set to zeros. One reasonable C (ℓ) intended for satisfying

(ℓ)
H , +C corresponding to the singular values in de‑ the SI constraint in 2 is to obtain L , L , and L such
1
3
2
scending order. that the resulting analog canceler matrix C (ℓ) has the

2: Set = 0. tap values at the same elements with the largest in am‑
3: for = max , max − 1, … , 2 do plitude elements of H . This C (ℓ) will result in cancelling
,

4: Set F = [D ] . the largest SI signal components. For example, suppose
(∶, − +1∶ )

5: Set G as the optimum precoding for the effective that = 3, = 4, and = 2 and that [H ] and

, 2,1

F
downlink MIMO (or MISO) channel H , [H ] are the two largest in amplitude elements of H .
, 4,2
,
given P . In this case, we may design L , L , and L matrices as:

3
1
(ℓ)
2
]
]
6: if ‖[(H , + C )F G ] ‖ ≤ A L = diag{[[H , 2,1 [H , 4,2 ]}, [L ] 2 = [L ] = 1, and
( ,∶)

1 1,1
2
1 2,2
∀ = 1, 2, … , , then [L ] = [L ] = 1. Other reasonable C ’s include the
(ℓ)

3 4,2
3 2,1
7: Set = + 1. orderly column‑by‑column and row‑by‑row placement of
(ℓ)
8: Store V (ℓ) = F G for the given C . the available tap values starting with the columns and

,

9: end if rows, respectively, of H having the largest Euclidean
,
10: end for norms. For example, suppose that = 3, = 4,
11: Set F = [D ] and G = P 1/2 .

(∶, )
(ℓ) 2 = 3, and that the second RX antenna is the one most af‑
12: if |[(H , +C )F G ] | ≤ ∀ = 1, 2, … , , then fected by SI (i.e., the one affected by the largest SI energy).

A

13: Set = + 1. Then, havingthethreetapvaluesplacedatthesecondrow
(ℓ) (ℓ)
14: Store V , = F G for the given C . of C (ℓ) will focus on reducing the SI received at the second

(ℓ) (ℓ)
15: Output V = V . RX antenna element. Generally, having tap values placed
,1
16: else at the ‑th row results in reducing SI at the ‑th RX an‑
(ℓ)
17: Output that C does not meet the residual SI tenna. In the simulation results with this architecture we

constraint . opt for the latter canceler design, namely the row‑by‑row
A
18: end if placement of the tap values, starting with H , ’s row
having the largest Euclidean norm and continuing with
inating the uplink. Hence, our goal with Algorithm 1 is the rest rows in descending ordering of Euclidean norms,
(ℓ)
to capture this trade‑off and obtain V ∀ solving 2 if there are more taps to be assigned.
,
(ℓ)
for a given C . Running this algorithm for all possible Realizations C (ℓ) for the Multi‑AUX‑TX canceler. To satisfy

canceler realizations inally results in the joint canceler the constraint of AUX TXs, each canceler matrix needs
(ℓ) (ℓ)
and precoder designs C and V , ∀ℓ = 1, 2, … , and to have − all‑zero rows. The nonzero rows spec‑

∗
∀ = 1, 2, … , , which are feasible candidate solutions ify the connection of the DEMUXs and the linear operation
ℓ
for 2. Those pairs will be used in Section 5.2 for ob‑ applied by L . There are in total ( ) ways to connect the
4

taining the joint analog canceler and the TX/RX digital BF output of the AUX TXs to the RX antennas, and each

solution of 1. way corresponds to a speci ic placement of the non‑zero
Algorithm 1 is executed at the FD MIMO node and has as rows inside the canceler matrix. This results in at most
inputs the MIMO channels H , and H , , as well as a real‑ = ( ) possible realizations for the multi‑AUX‑TX can‑

(ℓ)
(ℓ)
ization C . Both H , and H , can be estimated through celer. One reasonable C realization, which we use in

appropriately designed training processes at nodes and our simulation results for this architecture, corresponds
, respectively. The latter matrix estimation can be fed to the case where the AUX TX RF chains are connected
back or not to node depending on whether open‑loop to the antennas that are receiving the largest SI energy.
or closed‑loop MIMO operation, respectively, is adopted. This realization targets H , ’s rows having the largest Eu‑
(ℓ) clidean norms. Connecting the ‑th AUX TX RF chain to the
We next discuss meaningful C realizations for both the
‑th RX antenna corresponds to setting [L ] = 1.
proposed analog SI canceler architectures that provide in‑ 5 ,
sights on the effects of the C (ℓ) choice. Note that one can

also consider reducing the search of canceler realizations 5.2 Joint design of C , V , and U

in 2 to a realization that is a deterministic function of (ℓ) (ℓ)
H , or to a desired subset of possible realizations. Using the candidate designs C and V , ∀ℓ = 1, 2, … ,
∗
ℓ
(ℓ) and ∀ = 1, 2, … , for solving 2 from the approach
Realizations C for the Multi‑Tap canceler. For a given in Section 5.1, we now proceed to the inal joint design of
number of taps there are in total ( ) ways to con‑ the analog canceler and TX/RX digital BF at node max‑

nect the taps from the available TX antennas to the imizing the instantaneous FD rate. In particular, we for‑

available RX antennas. This results in at most = mulatethefollowing optimizationproblem using (12) and

© International Telecommunication Union, 2021 117