Aspectos externos a las empresas que inciden en un proceso de asociatividad.

Since the MF output yMF in (5.2) is a sufficient statistic for the transmit signal vector s0, a variety of optimal and suboptimal equalization-based data detection

the PD architecture, we use (5.2) to form the estimate ˆzMF =diag(G)−1yMF,

where the diagonal matrix diag(G)−1is computed in the centralized equalization

unit; see Fig. 5.1(a). The MF estimate ˆzMF _{can then be used to perform either}

hard- or soft-output data detection. For soft-output data detection, one requires the error variance vector given by

ˆσ2

MF = diag diag(G)−1Gdiag(G)−HN0

+diag(G)−1G₋I_Udiag(G)−1G₋I_UHEs

that contains the post-equalization SINR for each entry of ˆzMF_{. Note that MF-}

based equalization was shown to be optimal (i) for a fixed number of UEs and infinitely many BS antennas [53], which is equivalent to β → 0 in the large- system limit, or (ii) in the low-SNR regime [13]. The estimate of the ZF equalizer for the PD architecture is given by

ˆzZF =G−1yMF,

where the matrix G−1 _{is computed in the centralized equalization unit. The}

associated error variance vector is given by ˆσ2

ZF =diag(G−1)N0.

For L-MMSE equalization, we have

ˆzL-MMSE = (G+ρI_U)−1yMF.

where the matrix(G+ρIU)−1is computed in the centralized equalization unit.

constellations (e.g., QPSK or 16-QAM). The associated error variance vector is given by ˆσ_L-MMSE2 =diag (G+ρIU)−1G(G+ρIU)−HN0 +(G+ρI_U)−1G−IU  (G+ρI_U)−1G−IU H Es
.

We reiterate that the MF, ZF, and L-MMSE equalizers for the PD architecture deliver exactly the same estimates as their centralized counterparts—the only difference is the way the involved quantities are computed.

As shown in [13] and outlined in Section 5.2.5, centralized MF, ZF, and L- MMSE equalizers decouple MIMO systems in the large-system limit and for Rayleigh fading channels; this implies that the entries of the error variance vectors ˆσ2

MF, ˆσZF2 , and ˆσL-MMSE2 converge to the decoupled noise variance σ2of

the MF, ZF, and L-MMSE equalizer, respectively. Since linear equalizers in the PD architecture yield exactly the same estimates as in a centralized architecture, we can directly characterize the associated decoupled noise variance σ2

PDin the

PD architecture using the following result.

Theorem 26 ([13, Thm. 3.1]). Fix the system ratio β =U/B, and assume the large-

system limit and Rayleigh fading channels. Then, the decoupled noise variance σ_PD2 for MF, ZF, and L-MMSE equalization in a centralized or PD architecture, is the solution to the following fixed-point equation

σ_PD2 =N0+βΨ(σ_PD2 ), (5.5)

where the MSE function Ψ(σ2)is given by

Ψ(σ2) = Es, (MF)

Ψ(σ2) = σ2, (ZF)

Ψ(σ2) = Es

Es+σ2σ

for MF, ZF, and L-MMSE equalization, respectively.

We note that the expression for the ZF equalizer only holds for β <1, whereas the expressions for MF and L-MMSE hold for general system ratios β.1 _From

Theorem 26, we obtain closed-form expressions for the post-equalization SINR in (5.4) for MF, ZF, and L-MMSE equalization in the PD architecture.

Corollary 27. Assume that the conditions of Theorem 26 hold. Then, the post- equalization SINR for MF, ZF, and L-MMSE equalization in the PD architecture are given by SINRMF_PD = Es/N0 1+βEs/N0, (MF) SINRZF_PD= Es N0(1−β), for β<1, (ZF) SINRL-MMSE_PD =1 2 s  1₋ Es N0(1−β) 2 +4Es N0 −1− _NEs 0(1−β)  ! . (L-MMSE)

We note that in the massive MU-MIMO regime, which corresponds to β_→0, all post-equalization SINR expressions converge to Es/N0, which confirms the

well-known fact that MF is optimal in this scenario [53]. It can also be shown that SINRL-MMSE_PD bounds SINRMF_PD and SINRZF_PD from above for all system ratios and in all SNR regimes. Hence, L-MMSE equalization is often the preferred choice in realistic massive MU-MIMO systems [6, 22]. We reiterate that the SINR expressions listed in Corollary 27 are also valid for centralized architectures.

1_{The asymptotic SINR performance of ZF equalization via the Moore-Penrose pseudo inverse}

5.3.2 LAMA for the PD Architecture

The LAMA algorithm presented in Algorithm 2 is a nonlinear equalizer is able to achieve individually-optimal performance in the large-system limit given certain conditions on the antenna ratio β and the noise variance N0are satisfied. LAMA

operates directly on the input-output relation in (2.1) and is, hence, designed for centralized processing. We now develop a novel variant of LAMA that directly operates on the complete MF output yMF and the Gram matrix G in (5.2) to enables its use in the PD architecture. Since the antenna configuration in massive MU-MIMO systems typically satisfies U B, the LAMA-PD algorithm operates on a lower dimension which reduces complexity while delivering exactly the same estimates as the original LAMA algorithm. We note that LAMA was derived in the large-system limit and for Rayleigh fading channels [87], but these assumptions are not required in practice. We next summarize the LAMA-PD algorithm; the derivation can be found in Appendix A.3.1.

Algorithm 4 (LAMA-PD). Initialize s` = ES[S]for ` = 1, . . . , U, φ(1) = VarS[S],

and v(1) _{= 0}

B×0. Then, for every iteration t = 1, 2, . . . , Tmax, compute the following

steps: ˆzt =yMF+ (I−G)ˆst+vt (5.6) ˆst+1 _=F(ˆzt_{, N} 0+ ˆτ_t2) ˆτ_t2₊₁ =β_hG(ˆzt, N0+ ˆτt2)i vt+1 ₌ ˆτt2+1 N0+ ˆτ_t2(ˆz t_−st_).

The estimates and error variances of LAMA are ˆzt and σ_t,LAMA2 = N0+ ˆτ_t2, respectively.

ize that the equalization output ˆzt_{is equivalent to that of the original centralized}

LAMA algorithm in Algorithm 2. As discussed in Section 3.3.3, LAMA (and hence LAMA-PD) decouples the MIMO system into parallel AWGN channels.

For t _→ ∞, the SE recursion in Theorem 2 converges to the same fixed- point equation of linear equalizers in (5.5), where the only difference is the MSE function (3.13). As for linear equalizers, we can use the fixed-point equation in (5.5). Correspondingly, we can use SE to analyze the post-equalization SINR performance of LAMA and LAMA-PD. Unfortunately, there are no closed-form expressions known for the decoupled noise variance or the SINR for LAMA and LAMA-PD with discrete constellations, due to the specific form of the MSE function (3.13). Nevertheless, we can numerically compute (3.13) and, hence, analyze the SINR. A corresponding SINR comparison with linear equalizers is given in Section 5.5.