Bases Teóricas Científicas

In conclusion, we can see that the above formalism based on the rank of the channel covariance matrix gives a good understanding of the applica-

bility and behaviour of the parametric channel estimator in comparison to an LMMSE estimator. It is clear that the prior knowledge available to the LMMSE estimator through the covariance matrix is of crucial importance to its performance.

Clearly we have assumed that the parametric channel estimator can per- fectly estimateΣlocal. That is not the case in practice and so the above consid- erations are to be understood as the best possible performance one can obtain with a parametric channel estimator. From the simulation results of Paper A it seems that the parametric channel estimator is able to obtain a reasonably accurate estimate ofΣlocalthough.

The above discussion is based on a numerical investigation of the co- variance matrices which are generated from two different channel models. Transposition of the drawn conclusions to real channels thus stands on the assumption that these models accurately reflect the behaviour of the rank of the covariance matrix rank of these real channels. That is not necessarily the case because the channel models have not been devised with this purpose in mind. It would be interesting to investigate the channel covariance matrix and its rank directly based on a set of measurements, but that falls outside the scope of this work.

Chapter 4

Contributions & Outlook

4.1

Contributions

The overall theme of this thesis is the design of algorithms for sparsity-based LSE. We pay particular attention to the design of algorithms that are prac- tically applicable. This for example means that the algorithms we design inherently can estimate nuisance parameters such as the noise variance. It also means that there is a strong focus on reducing the computational com- plexity of the algorithms. We further demonstrate how sparsity-based LSE algorithms can be incorporated into larger inference frameworks by consider- ing their use for parametric channel estimation in an iterative receiver design. In this section we give a short summary of each of the papers in the thesis and the main contributions and findings of the papers are discussed.

Paper A: In view of the discussion in Chapter 3, it is clear that the parametric approach provides a way to obtain channel estimators with high estimation accuracy. Another very powerful approach to increasing channel estimation accuracy is via joint channel estimation and decoding schemes. In this paper the simultaneous application of these two ideas for wireless receiver design is investigated. Special attention is paid to rigorous methodologies for the de- sign of such receiver algorithms. The Bayesian method for sparsity-based LSE is used to obtain the parametric channel estimator. It is shown that a mean- field implementation of this channel estimator can naturally be merged with belief propagation decoding in a variational Bayesian inference framework.

The proposed receiver algorithm is assessed using an extensive numerical evaluation. It is shown that the proposed algorithm indeed increases the channel estimation accuracy in comparison to a selection of state-of-the-art joint channel estimation and decoding algorithms. It is also demonstrated that the proposed receiver design can have lower bit-error rate (BER) than the

baseline algorithms in scenarios with high SNR and large modulation orders. Further, the proposed design may allow for the number of pilot signals to be lowered (compared to the baseline algorithms) without incurring an increase in BER.

Paper B: In the application of LSE for parametric channel estimation in a receiver (Paper A) it is observed that the high computational complexity of the algorithm precludes it from implementation in an actual wireless com- munication system. Motivated by this limitation we study the computational complexity of LSE algorithms in a general setting in Paper B. The Bayesian method for sparsity-based LSE is investigated and it is shown that an algo- rithm with low computational complexity can be obtained by appropriately formulating an inference scheme using the Bernoulli-Gaussian prior model. At the core of the algorithm lies a so-called superfast method (specifically the generalized Schur algorithm) for decomposing the inverse of a Hermitian Toeplitz covariance matrix along with a non-uniform fast Fourier transform. The asymptotic per-iteration computational complexity of the obtained al- gorithm scales asO(N log2N)in the problem size N. That is a significant im- provement over theO(N3)scaling of most of the state-of-the-art algorithms. The proposed algorithm is also shown to outperform the selected baseline algorithms in terms successful recovery of the line spectral frequencies. Paper C: In this paper we consider the numerical solution of the convex optimization problem that appears in atomic norm soft thresholding (AST). Not surprisingly, this optimization problem has some of the same Toeplitz structure that is exploited in the superfast LSE algorithm of Paper B. We investigate how this structure can be exploited to obtain a fast algorithm for solving AST.

Since AST is a convex optimization problem this work takes quite a dif- ferent approach to algorithm design than Paper B. Specifically we propose a novel primal-dual interior point method (IPM) to solve this problem. AST is usually formulated as a semidefinite programming problem, but that form precludes the derivation of a fast primal-dual IPM due to the presence of O(N2)dual variables. We show that by reformulating AST as a non-symme- tric conic programming problem only O(N)dual variables are needed. The problem structure in the conic formulation depends crucially on a non-self- dual convex cone. The derivation of the primal-dual IPM involves an analysis of the dual of this cone. Specifically we show how membership of the dual cone can approximately be determined in a fast manner.

A numerical investigation shows that the proposed primal-dual IPM ac- curately solves AST and that it does so much faster than any other known method.