El estudio de “un sujeto completamente nuevo”

In this section, we propose two methods to optimize beamforming matrix V and equalization matrix U[k] _{while fixing A}[k]_{. Even after fixing A}[k]_{, jointly optimizing}

V and U[k] _{is a non-convex problem, and we will relax this joint optimization problem}

into two separate optimization problems.

For any given beamforming matrix, V, the columns of the optimal equalization matrix U[k] _{is always in the form of MMSE equalizer as in (8.36). On the other}

hand, for a given equalization matrix U[k]_{, we develop two algorithms to update the}

beamforming matrix V. The first algorithm relaxes the problem of choosing V, given A[k] _{and U}[k]_{, to a convex optimization problem. Then uses a convex optimization}

toolbox, like the CVX package [Grant and Boyd, 2014], to solve the relaxed convex problem. The second algorithm optimizes V given U[k] using the idea of channel reciprocity and uplink-downlink duality for integer-forcing [He et al., 2018]. Both algorithms are iterative optimization algorithms in the sense that, we keep updating U[k] and V given A[k] using these two methods. By the end of each iteration, the integer matrix A[k] _{can be updated using the aligned LLL algorithm introduced in}

Section8.3 for fixed V and U[k]_.

We use the CVX package to solve the convex optimization problem for our first algorithm, thus we name the first algorithm CVX-IFIA. The second algorithm is called Dual-IFIA since it borrows the idea of duality for integer-forcing from [He et al., 2018].

8.4.1 CVX-IFIA

Recall that our focus has been shifted towards maximizing the sum of the computation rates rather than the sum of the user rates. This method further relaxes this goal by maximizing the worst computation rate (i.e., largest effective noise power) across all the receivers. Since this computation rate will be mapped to one of the user rates, this corresponds to maximizing the symmetric rate.

The relaxed problem can be written as P1 : min V,U[k]  max k,m σ [k] eff,m
2  s.t kv[`]k2 <sub>≤ 1, ∀`.</sub> (8.63)

As mentioned earlier, the joint optimization problem P1 is a non-convex optimiza- tion problem. However, for a fixed U[k]_{, P1 can be rewritten as}

P2 : min V  max k,m σ [k] eff,m
2  s.t kv[`]k2 ≤ 1, ∀`. (8.64)

which is a convex optimization problem. Since for any fixed beamforming matrix V, (8.36) gives the columns of the optimal U[k]_{, one can iteratively optimize U}[k] _{and V}

using (8.36) and the solution of (8.64). The details of the algorithm is presented in Algorithm 2. To guarantee better performance, the CVX-IFIA algorithm is initialized by the beamforming vectors given by the Max-SINR algorithm described in Section 8.1.

8.4.2 Dual-IFIA

Before giving the details of the algorithm, we introduce the dual channel and dual network for the IFIA. The dual channel for IFIA is a bit trickier than the dual channel for linear receivers (introduced in Section8.1). This is because the number of transmitted codewords (beamforming vectors) from each transmitter and the number of decoded combinations (equalization vectors) at each receiver are not always equal anymore. In the primal network, each receiver k wants to decode M[k] combinations and solve for the desired single codeword s[k] _{sent by the k}th _{transmitter. Overall, we}

have M =PK

k=1M[k]≥ K combinations decoded at all the receivers.

mal receiver becomes the `th _{dual transmitter and the k}th _{primal transmitter becomes}

the kth _{dual receiver (i.e., the dual channel matrix} ←_H−[k,`] <sub>= H</sub>[`,k]†_{). In addition, the}

beamforming (equalization) vectors of the primal network become the equalization (beamforming) vectors of the dual network, respectively. As a result, in the dual net- work, each dual transmitter ` wants to send M[`] _{messages, while each dual receiver}

k wants to decode only one combination of these messages. We have dual beamform- ing matrices ←V−[`] <sub>∈ R</sub>N<sub>Rx</sub>[`]×M[`]

and the dual equalization vectors ←−u[k] _{∈ R}N_Tx[k]_{. Let}

A = [A[1]† _{· · · A}[K]†_]†

∈ ZM ×K _{be the integer matrix of the primal channel, the dual}

integer matrix can be represented as ←−

A = A† _{∈ Z}K×M (8.65)

where K here represents the total number of combinations and M represents the total number of the transmitted messages. Define the channel to the kth dual receiver as

←−

H[k]=h←H−[k,1] · · · ←H−[k,K]i . (8.66)

Following the same steps as in the primal IFIA, the kth _{dual receiver decodes a single}

combination and we can write the power of the effective noise on this combination as ←−_σ[k]

eff

2

, k←u−[k]†k2₊ ←_u−[k]†←_H−[k]←_{V −}− ←−_a[k]†
←−

P ←u−[k]†←H−[k]←V −− ←−a[k]†
†

where←V is the block diagonal matrix of [− ←V−[1] _{· · ·} ←_V−[K]_{] and}←_{P is the diagonal coding}−

power matrix with diagonal elements ←−

Pi,i =

ρ k←v−ik2

, i = 1, ..., M (8.67)

where ←v−i is the ith column of the matrix

←− V.

←−_σ[k] eff at the k th _{dual receiver is} ←−_u[k]† opt = ←− A[k]←P−†←V−†←H−[k]†(I +←H−[k]←V−←P−←V−†←H−[k]†)−1. (8.68)

In order to update V, we use the equalization vectors←−u[k] _{at the k}th_{dual receiver}

(after normalizing) and map it to the beamforming vectors v[k] for the kth primal transmitter. Finally, We can iteratively use the closed form expressions in (8.36) and (8.68) to optimize the beamforming and equalization vectors. The details of the the proposed algorithm is given in Algorithm 6.