In many audio applications, the signal of interest is corrupted by acoustic background noise, interference, and reverberation. The presence of these contaminations can significantly degrade the quality and intelligibility of the audio signal. This makes it important to develop signal processing methods that can separate the competing sources and extract a source of interest. The estimated signals may then be either directly listened to, transmitted, or further processed, giving rise to a wide range of applications such as hearing aids, noise-cancelling headphones, human-computer interaction, surveillance, and hands-free telephony. Many of the existing approaches to speech separation/extraction relied on beamforming techniques. These techniques approach the problem from a spatial point of view; a microphone array is used to form a spatial filter which can extract a signal from a specific direction and reduce the contamination of signals from other directions. However, when there are fewer microphones than sources (the underdetermined case), perfect attenuation of all interferers becomes impossible and only partial interference attenuation is possible. In this thesis, we present a framework which extends the use of beamforming techniques to underdetermined speech mixtures. We describe frequency domain non-linear mixture of beamformers that can extract a speech source from a known direction. Our approach models the data in each frequency bin via Gaussian mixture distributions, which can be learned using the expectation maximization algorithm. The model learning is performed using the observed mixture signals only, and no prior training is required. The signal estimator comprises of a set of minimum mean square error (MMSE), minimum variance distortionless response (MVDR), or minimum power distortionless response (MPDR) beamformers. In order to estimate the signal, all beamformers are concurrently applied to the observed signal, and the weighted sum of the beamformers’ outputs is used as the signal estimator, where the weights are the estimated posterior probabilities of the Gaussian mixture states. These weights are specific to each timefrequency point. The resulting non-linear beamformers do not need to know or estimate the number of sources, and can be applied to microphone arrays with two or more microphones with arbitrary array configuration. We test and evaluate the described methods on underdetermined speech mixtures. Experimental results for the non-linear beamformers in underdetermined mixtures with room reverberation confirm their capability to successfully extract speech sources.

Due to their ability to provide high data rates, Multiple-Input Multiple-Output (MIMO) wireless communication systems have become increasingly popular. Decoding of these systems with acceptable error performance is computationally very demanding. In the case of large overdetermined MIMO systems, we employ the Sequential Decoder using the Fano Algorithm. A parameter called the bias is varied to attain different performance-complexity trade-offs. Low values of the bias result in excellent performance but at the expense of high complexity and vice versa for higher bias values. We attempt to bound the error by bounding the bias, using the minimum distance of a lattice. Also, a particular trend is observed with increasing SNR: a region of low complexity and high error, followed by a region of high complexity and error falling, and finally a region of low complexity and low error. For lower bias values, the stages of the trend are incurred at lower SNR than for higher bias values. This has the important implication that a low enough bias value, at low to moderate SNR, can result in low error and low complexity even for large MIMO systems. Our work is compared against Lattice Reduction (LR) aided Linear Decoders (LDs). Another impressive observation for low bias values that satisfy the error bound is that the Sequential Decoder's error is seen to fall with increasing system size, while it grows for the LR-aided LDs. For the case of large underdetermined MIMO systems, Sequential Decoding with two preprocessing schemes is proposed – 1) Minimum Mean Square Error Generalized Decision Feedback Equalization (MMSE-GDFE) preprocessing 2) MMSE-GDFE preprocessing, followed by Lattice Reduction and Greedy Ordering. Our work is compared against previous work which employs Sphere Decoding preprocessed using MMSE-GDFE, Lattice Reduction and Greedy Ordering. For the case of large systems, this results in high complexity and difficulty in choosing the sphere radius. Our schemes, particularly 2), perform better in terms of complexity and are able to achieve almost the same error curves, depending on the bias used.

In this thesis, for the under-determined system Fx = b with the matrix F ∈m¡Ñn (m ≤ n), new error bounds involving the traditional condition number and the effective condition number are established. Such error bounds are simple than those of over-determined system. The errors results implies that for stability, the condition number and the effective condition numbers are important if the perturbation of matrix F and vector b are dominant, respectively. This thesis is also devoted to the application of Neumann problems, where the consistent condition holds to guarantee the existence of multiple solutions. For the traditional Neumann conditions, the discrete consistent condition has to be satisfied to guarantee the existence of numerical solutions. Such a discrete consistent condition can be removed, to greatly simplify the numerical algorithms, and to retain the same convergence rates. For Neumann Problems, we may solve its ordinal discrete linear equations, or the underdetermined systems by ignoring some dependent equations, or the fixed variables methods. Moreover, we may choose different equations to be ignored, and different variables to be fixed. The comparisons of these different methods and choices are important in applications. In this thesis, the new comparisons and relations of stability and accuracy are first explored, and some interesting results and new discoveries are found. Numerical examples of Neumann problem in 1D are carried out, to support the analysis made. However, the algorithms and stability analysis can be applied to the complicated Nuemann problems in 2D and 3D, such as the traction problems in linear elastic problems.

The Sherman Morrison iteration method is developed to solve regularized least squares problems. Notions of pivoting and splitting are deliberated on to make the method more robust. The Sherman Morrison iteration method is shown to be effective when dealing with an extremely underdetermined least squares problem. The performance of the Sherman Morrison iteration is compared to classic direct methods, as well as iterative methods, in a number of experiments. Specific Matlab implementation of the Sherman Morrison iteration is discussed, with Matlab codes for the method available in the appendix. / Master of Science

A realistic field monitoring application to evaluate close proximity tunneling effects of a new tunnel on an existing tunnel is presented. A blind source separation (BSS)-based monitoring framework was developed using sensor data collected from the existing tunnel while the new tunnel was excavated. The developed monitoring framework is particularly useful to analyze underdetermined systems due to insufficient sensor data for explicit input force-output deformation relations. The analysis results show that the eigen-parameters obtained from the correlation matrix of raw sensor data can be used as excellent indicators to assess the tunnel structural behaviors during the excavation with powerful visualization capability of tunnel lining deformation. Since the presented methodology is data-driven and not limited to a specific sensor type, it can be employed in various proximity excavation monitoring applications.

Orientador: Prof. Dr. Ricardo Suyama / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Engenharia da Informação, 2016. / O problema de separação cega de fontes no contexto de misturas subparametrizadas tem sido investigado por meio de abordagens que exploram diferentes características dos sinais de interesse, dentre as quais se destaca a esparsidade. Mesmo que os sinais de interesse não sejam originalmente esparsos, é possível, em algumas aplicações, que estes sejam modificados a fim de apresentar um maior grau de esparsidade, e assim facilitar o processo de separação dos sinais. No presente trabalho, comparamos o desempenho de diferentes técnicas de estimação dos sinais de áudio, no contexto de misturas sub-parametrizadas, considerando que os mesmos tenham sido esparsificados antes do processo de mistura. Os resultados obtidos estendem análises preliminares realizadas, e indicam que este préprocessamento traz ganhos efetivos para o processo de estimação dos sinais. Além disso, distintos estudos foram unificados e uma nova proposta estabelecida, obtendo-se um resultado de estimação considerável no âmbito perceptual, segundo análises realizadas. / The blind source separation problem have been investigated through approaches that explore specific characteristics of the signals of interest, among which stands out the sparsity. Even if the original sources aren¿t sparse, some modifications can be done to rebuild signal with greater degree of sparsity, making the separation process easier. In this work we compare the performance of different estimation methods of audio signals, in the underdetermined context, considering that they have been sparsified before the mixing process. The results extend preliminary studies and show that this process may increase the performance of the estimation process. In addition to that, different studies were merged and a new proposal was established, which results are remarkable according to the perceptual analysis.

Signal processing has been playing a key role in providing solutions to key problems encountered in communications, in general, and in wireless communications, in particular. Time-Frequency Signal Processing (TFSP) provides eective tools for analyzing nonstationary signals where the frequency content of signals varies in time as well as for analyzing linear time-varying systems. This research aimed at exploiting the advantages of TFSP, in dealing with nonstationary signals, into the fundamental issues of signal processing, namely the signal estimation and signal separation. In particular, it has investigated the problems of (i) the Instantaneous Frequency (IF) estimation of Linear Frequency-Modulated (LFM) signals corrupted in complex-valued zero-mean Multiplicative Noise (MN), and (ii) the Underdetermined Blind Source Separation (UBSS) of LFM signals, while focusing onto the fast-growing area of Wireless Communications (WCom). A common problem in the issue of signal estimation is the estimation of the frequency of Frequency-Modulated signals which are seen in many engineering and real-life applications. Accurate frequency estimation leads to accurate recovery of the true information. In some applications, the random amplitude modulation shows up when the medium is dispersive and/or when the assumption of point target is not valid; the original signal is considered to be corrupted by an MN process thus seriously aecting the recovery of the information-bearing frequency. The IF estimation of nonstationary signals corrupted by complex-valued zero-mean MN was investigated in this research. We have proposed a Second-Order Statistics approach, rather than a Higher-Order Statistics approach, for IF estimation using Time-Frequency Distributions (TFDs). The main assumption was that the autocorrelation function of the MN is real-valued but not necessarily positive (i.e. the spectrum of the MN is symmetric but does not necessary has the highest peak at zero frequency). The estimation performance was analyzed in terms of bias and variance, and compared between four dierent TFDs: Wigner-Ville Distribution, Spectrogram, Choi-Williams Distribution and Modified B Distribution. To further improve the estimation, we proposed to use the Multiple Signal Classification algorithm and showed its better performance. It was shown that the Modified B Distribution performance was the best for Signal-to-Noise Ratio less than 10dB. In the issue of signal separation, a new research direction called Blind Source Separation (BSS) has emerged over the last decade. BSS is a fundamental technique in array signal processing aiming at recovering unobserved signals or sources from observed mixtures exploiting only the assumption of mutual independence between the signals. The term "blind" indicates that neither the structure of the mixtures nor the source signals are known to the receivers. Applications of BSS are seen in, for example, radar and sonar, communications, speech processing, biomedical signal processing. In the case of nonstationary signals, a TF structure forcing approach was introduced by Belouchrani and Amin by defining the Spatial Time- Frequency Distribution (STFD), which combines both TF diversity and spatial diversity. The benefit of STFD in an environment of nonstationary signals is the direct exploitation of the information brought by the nonstationarity of the signals. A drawback of most BSS algorithms is that they fail to separate sources in situations where there are more sources than sensors, referred to as UBSS. The UBSS of nonstationary signals was investigated in this research. We have presented a new approach for blind separation of nonstationary sources using their TFDs. The separation algorithm is based on a vector clustering procedure that estimates the source TFDs by grouping together the TF points corresponding to "closely spaced" spatial directions. Simulations illustrate the performances of the proposed method for the underdetermined blind separation of FM signals. The method developed in this research represents a new research direction for solving the UBSS problem. The successful results obtained in the research development of the above two problems has led to a conclusion that TFSP is useful for WCom. Future research directions were also proposed.

We often have to face the fact that several signals are mixed together in unknown environment. The signals must be first extracted from the mixture in order to interpret them correctly. This problem is in signal processing society called blind source separation. This dissertation thesis deals with multi-channel separation of audio signals in real environment, when the source signals outnumber the sensors. An introduction to blind source separation is presented in the first part of the thesis. The present state of separation methods is then analyzed. Based on this knowledge, the separation systems implementing fuzzy time-frequency mask are introduced. However these methods are still introducing nonlinear changes in the signal spectra, which can yield in musical noise. In order to reduce musical noise, novel methods combining time-frequency binary masking and beamforming are introduced. The new separation system performs linear spatial filtering even if the source signals outnumber the sensors. Finally, the separation systems are evaluated by objective and subjective tests in the last part of the thesis.