Page 130 - ITU Journal Future and evolving technologies – Volume 2 (2021), Issue 2

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

Different from the downlink rate in (12), in (13) and (14) that , which denotes the number of taps or AUX TXs, is
we include the considered linear combining matrix U decided of line upon hardware design as a function of size
which jointly with V and C we aim to optimally design. constraints, cost per tap and cost per AUX TX RF chain, or

Note that in the formulation of 1 we have relaxed con‑ other hardware constraints. Examples of realizations for
straint (C3) concerning the instantaneous residual SI af‑ the analog canceler are given at the end of this section.
ter analog cancellation that appears in the general to We use the notation V (ℓ) to represent the precoder design

an average power per RX RF chain constraint, where the solving 2 for each speci ic C . The alternating opti‑
(ℓ)

average is taken over all possible transmit symbol vectors. mization approach is repeated for C (ℓ) ∀ℓ in order to ind

This constraint imposes that, at the input of each of the the best pair of canceler and precoder solving 2. The
RX RF chains, the average power of the SI signal for (ℓ) (ℓ)

all transmitted symbols within a coherent channel block solution for V given C is summarized in Algorithm 1.
cannot be larger than the threshold . Notice also that The precoder is iteratively constructed as the cascade ma‑
A
×
in 1 we have not included a constraint similar to (C4) trix F G with F ∈ ℂ and G ∈ ℂ × , where is a

for the residual SI signal after digital combining. Instead positive integer taking the values 1 ≤ ≤ max and holds
we have only incorporated a constraint on the norm of that ≤ min{ , }. In general, max = , however,

the rows of U . The reason for this simpli ication mainly for large transmission powers and strictly small values for

lies on 1’s sum‑rate objective function. We expect that it is advisable to set max = min{ , }. For each

A
the joint design of C , V , and U optimizing the uplink value of we adopt a similar approach to [7] for the pre‑

rate will naturally result in keeping the average power of coding design. Particularly, its F component aims at min‑

̃
the residual SI signal after both A/D processing at an ac‑ imizing the impact of the residual SI MIMO channel H , ,
ceptable level; acceptable level is any level allowing up‑ whereas the goal of the G component is to maximize the

link communications. Furthermore, the unity constraint rate of the effective downlink channel H , × .
F ∈ ℂ
on the norm of each of the rows of U excludes combin‑ Intuitively, parameter represents the effective number

ing solutions that result in undesired ampli ication of the of TX antennas after squeezing SI in the − least domi‑

̃
received signals (i.e., the signals from node , SI, and nant modes of H , via the ef icient use of F . For the cases

AWGN). where H ,
F is a MIMO channel, the precoder G in Step
We propose to tackle 1 with the following two‑step 5 of Algorithm 1 is given by the open‑loop or closed‑loop
approach. First, as described next in Section 5.1, we precoding for this channel derived using [28], depend‑
consider only the downlink which is usually more rate ing on whether H , is unknown or known, respectively,
demanding than the uplink, and obtain the pairs of C at the transmit side of node . In the simulation results
and V designs optimizing the instantaneous downlink shown later in Section 6 we will use open‑loop precoding.

rate while meeting their respective constraints. Then, we When = 1 and ≥ 2, H ,
F is a Multiple Input Single

solve for the best pair of C and V as well as the U de‑ Output (MISO) channel, and if its knowledge is available

sign that jointly maximize the sum‑rate performance, as at node , the optimum precoding is Maximal Ratio Trans‑
will be explained in Section 5.2. mission (MRT). If H ,
F is a Single Input Multiple Output
(SIMO) (i.e., for ≥ 2 and = 1) or a scalar (i.e., for

5.1 Candidate designs for C and V = = 1) channel, G is a scalar set to P 1/2 .

We irst formulate the following downlink rate maximiza‑ (ℓ)
tion problem using expression (12) for the design of the As seen from Step 15 of Algorithm 1, the V solving 2
(ℓ)
(ℓ)
analog cancellation matrix C and the precoding matrix for a speci ic C is given by V ,1 . This notation repre‑

V at the FD node : sents the precoder corresponding to the largest value of

that results in meeting constraint ; recall that deter‑
A
2 ∶ max ℛ DL (V ) mines F and G dimensions. We denote the maximum

C ,V
(ℓ) ∗
s.t. (C1), (C2), value of for the C design as , and also use the no‑
ℓ
(ℓ)
∗
2
ℓ
‖[(H , + C )V ] ‖ ≤ ∀ = 1, 2, … , . tation V , with = 1, 2, … , for the ‑th candidate
A

( ,∶)
(ℓ)
precoder solution for 2 given C . Although, the in‑

To solve the latter problem we adopt an alternating op‑ cluded iterations for solving this problem could be termi‑
timization approach. Speci ically, supposing that a re‑ (ℓ) (ℓ) ∀
alization of the analog canceler satisfying (C2) is given, nated when V ,1 is found, Algorithm 1 computes V ,
A
we seek the TX digital precoder maximizing the down‑ meeting 2’s threshold and optimizing the downlink
(ℓ)
link rate, while meeting (C1) and the threshold . Note rate for a given C . Among those designs, the ones corre‑

A
that each realization of the analog canceler corresponds sponding to lower values of (i.e., those with increasing
to a distinct MUX/DEMUX con iguration. Let us assume index ) naturally result in larger SI mitigation. Although
that for taps (or AUX TXs, depending on the un‑ this behavior is desirable for maximizing the uplink rate,
(ℓ)
derlying canceler architecture) there are in total dis‑ V , ’s with larger (i.e., obtained from lower ) yield
(ℓ) (ℓ)
tinct realizations for the analog canceler, where C with lower downlink rates. On the contrary, V ,1 maximizing
ℓ = 1, 2, … , denotes the ℓ‑th canceler realization. Recall the downlink rate creates the stronger SI signal contam‑
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