Flujo de Caja de Gonzáles S.A.C Despachadora de Aduanas

◃ Proposing a new technique to combine the time synchronization metrics in a cen-

tralized MIMO system, we showed that our approach achieves a more robust syn- chronization in low SNR doubly-dispersive channels than the conventional MIMO synchronization techniques. In addition, since the CP based systems are limited to preamble-based synchronization, we showed that the CFO estimation in UW based systems has smaller MSE bounds than the primary CFO estimation (which is solely based on preamble.).

◃ In Sec. 4.4, we derived the LMMSE based CE for basic and Circ.-UW sequences,

and we also derived the adaptive Wiener-Hopf filters for a more robust channel estimation. There, we observed that if the channel is stationary, the Wiener fil- tered Circ.-UW sequences achieves the most robust channel estimation with small- est MSE, which is due to the strong temporal correlation of UW sequences, and also smaller condition number of its observation matrix. On the other hand, in an extreme doubly-dispersive channel, we observed that the CE of basic UW sequences has smaller error floor, because their observation matrix considers a shorter time with respect to the channel coherence time.

◃ Unlike the conventional UW-based equalization techniques that considers the FFT

size over Payloud-UW slot, we proposed the UW-free equalization that takes the FFT only over the payload block. Deriving the CWCU-LMMSE for joint channel- equalization-and-demodulation with imperfect channel knowledge, we showed that the UW-based MIMO transmission with a non-orthogonal multi-carrier over a doubly-dispersive channel has nearly two orders of magnitude smaller coded FER than the conventional pilot-aided CP-OFDM. This performance improvement is jus- tified by higher quality channel estimation of UW-based system than the pilot-aided CP based system in time-varying conditions. Furthermore, we have also observed that if the channel remains stationary, the Wiener filtered Circ.-UW GFDM achieves the smallest gap with respect to its Genie-aided receiver, which is due to the fact that i) Circ-UW achieves higher CE quality than basic UW, and ii) B number of channel estimations are being averaged via the Wiener-hopf filter.

5.2

Open Challenges

Below, we outline further open topics and aspects that can extend what has been covered in this thesis.

◃ The algorithm developments in this thesis were mostly relying on the assumptions

of centralized MIMO systems. In distributed MIMO systems, the scenario becomes more challenging as it influences every module from synchronization to channel estimation and to equalization. In distributed MIMO systems, synchronization must

5 Conclusions and Perspectives

find the starting position of the frame that has been transmitted from each Tx antenna independently (likewise the CFO estimation). Channel estimation not only needs to consider different PDP between each Tx-Rx antenna pair, but also the timing and frequency misalignment of the frames. Similarly, the equalization must consider the timing and frequency mismatches.

◃ The iterative LMMSE-PIC approach of joint channel-estimation-and-equalization

that has been derived in Sec. 3.4 was using the remodulated equalized signal con- stellations as feedback. This iterative approach can be further improved by taking into account the bit LLRs after the channel decoder for interference-cancellation of the channel estimation unit. Making a trade-off between complexity and estimation- detection quality, it would make more room for employing higher MCS values.

◃ In synchronization algorithms of Chapter 4, we assumed to have integer STO and

we also further assumed that CFO remains constant over multiple blocks. The syn- chronization can be further extended to the case where fractional STO terms exist and they can be handled by means of early-late synchronizer. Moreover, if the CFO varies slightly during multiple blocks, an adaptive filter can be designed to better track the CFO in UW-based systems.

◃ The UW-free equalization approach presented in Sec. 4.5 employed a non-iterative

detection approach based on joint channel-equalization-and-demodulation. The ap- proach can be further extended to iterative detection based on LMMSE-PIC, par- ticularly, in an extreme doubly-dispersive channel scenario, where a time-varying channel needs to be equalized. By employing the LMMSE-PIC equalizer, the spec- tral efficiency of higher MCS e.g. 64-QAM can be exploited in such extreme channel conditions.

◃ In our UW-based MIMO frame design, we have considered skipping the first IBI

limited L samples of the UW sequence. By doing so, the peak of the timing metric of synchronization unit in AWGN condition depends on the length of UW sequence. One may also consider employing a CP for the UW sequence to improve the channel estimation quality. However, if the UW block length remains identical, the peak of the timing metric may reduce with respect to SNR. Thus, the trade-off between CP-aided UW and CP-less UW for synchronization and channel estimation can be further studied.

Appendix A

Complementary Materials

A.1

Linear Algebra Identities

We have the following well-known identities from linear algebra (available, e.g., from [Gen07],[RC91]):

• Given the quadratic form ⃗XHA⃗X, the following identity holds:

⃗

XHA⃗X = Tr(⃗XHA⃗X). (A.1)

• Given the matrices A, B, C such that their dimensions are compatible, the following identity holds:

Tr(ABC) = Tr(CAB). (A.2)

• Given a deterministic matrix A and a random vector ⃗X, the following identity holds: E[Tr(A⃗X⃗XH_{)] = Tr(AE[⃗X⃗X}H_{]). (A.3)}

• Given the matrices A, B, C such that their dimensions are compatible, the following identity holds:

vec(ABC) = (CT ⊗ A)vec(B). (A.4)

• Given the matrices A, B, C such that their dimensions are compatible, the following identity holds:

AB(I + CHAB)−1 = (A−1+ BCH)−1B. (A.5)

• Given the matrix A ∈CN ×N _{and the vector ⃗X ∈C}N_{, the following identity holds:}

A Complementary Materials

Proof: Through element-wise computation, one obtains:

[ diag(⃗X)Adiag(⃗X)H] i,j = ⃗XiAi,j ⃗ X∗_j = ⃗Xi⃗X∗jAi,j = (⃗X⃗XH)i,jAi,j = [⃗X⃗XH ◦ A] i,j.