One of the differentiating features of PARAFAC decompositions is that, under certain conditions, unique solutions are possible. The search for uniqueness conditions for the PARAFAC Decomposition has a limited past, spanning only three decades. The complex structure of the problem and the need for tensor algebras or other similarly abstract characterizations provided a roadblock to the development of uniqueness conditions. Theoretically, the PARAFAC decomposition surpasses its bilinear counterparts in that it is possible to obtain solutions that do not suffer from the rotational problem. However, not all PARAFAC solutions will be constrained sufficiently so that the resulting decomposition is unique. The work of Kruskal, 1977, provides the most in depth investigation into the conditions for uniqueness, so much so that many have assumed, without formal proof, that his sufficient conditions were also necessary. Aided by the introduction of Khatri-Rao products to represent the PARAFAC decomposition, ten Berge and Sidiropoulos (2002) used the column spaces of Khatri-Rao products to provide the first evidence for countering the claim of necessity, identifying PARAFAC decompositions that were unique when Kruskals condition was not met. Moreover, ten Berge and Sidiropoulos conjectured that, with additional k-rank restrictions, a class of decompositions could be formed where Kruskals condition would be necessary and sufficient. Unfortunately, the column space argument of ten Berge and Sidiropoulos was limited in its application and failed to provide an explanation of why uniqueness occurred. On the other hand, the use of orthogonal complement spaces provided an alternative approach to evaluate uniqueness that would provide a much richer return than the use of column spaces for the investigation of uniqueness. The Orthogonal Complement Space Approach (OCSA), adopted here, would provide: (1) the answers to lingering questions about the occurrence of uniqueness, (2) evidence that necessity would require more than a restriction on k-rank, and (3) an approach that could be extended to cases beyond those investigated by ten Berge and Sidiropoulos.

Today the multisensor array signal processing of noisy measurements has received much attention. The classical problem in array signal processing is determining the location of an energy-radiating source relative to the location of the array, in other words, direction-of-arrival (DOA) estimation. One is considering the signal estimation problem when together with the signal(s) of interest some noise and interfering signals are present. In this report a direction-of-arrival estimation system is described based on an antenna array for detecting arrival angles in azimuth plane of signals pitched by the antenna array. For this, the Multiple Signal Classication (MUSIC) algorithmis first of all considered. Studies show that in spite of its good reputation and popularity among researches, it has a certain limit of its performance. In this subspace-based method for DOA estimation of signal wavefronts, the term corresponding to additive noise is initially assumed spatially white. In our paper, we address the problem of DOA estimation of multiple target signals in a particular noise situation - in correlated spherically isotropic noise, which, in many practical cases, models a more real context than under the white noise assumption. The purpose of this work is to analyze the behaviour of the MUSIC algorithm and compare its performance with some other algorithms (such as the Capon and the Classical algorithms) and, uppermost, to explore the quality of the detected angles in terms of precision depending on different parameters, e.g. number of samples, noise variance, number of incoming signals. Some modifications of the algorithms are also done is order to increase their performance. Program MATLAB is used to conduct the studies. The simulation results on the considered antenna array system indicate that in complex conditions the algorithms in question (and first of all, the MUSIC algorithm) are unable to automatically detect and localize the DOA signals with high accuracy. Other algorithms andways for simplification the problem (for example, procedure of denoising) exist and may provide more precision but require more computation time.

Motivated by results of linear algebra over fields, rings and tropical semirings, we present a systematic way to understand the behaviour of matrices with entries in an arbitrary semiring. We focus on three closely related problems concerning the row and column spaces of matrices. This allows us to isolate and extract common properties that hold for different reasons over different semirings, yet also lets us identify which features of linear algebra are specific to particular types of semiring. For instance, the row and column spaces of a matrix over a field are isomorphic to each others' duals, as well as to each other, but over a tropical semiring only the first of these properties holds in general (this in itself is a surprising fact). Instead of being isomorphic, the row space and column space of a tropical matrix are anti-isomorphic in a certain order-theoretic and algebraic sense. The first problem is to describe the kernels of the row and column spaces of a given matrix. These equivalence relations generalise the orthogonal complement of a set of vectors, and the nature of their equivalence classes is entirely dependent upon the kind of semiring in question. The second, Hahn-Banach type, problem is to decide which linear functionals on row and column spaces of matrices have a linear extension. If they all do, the underlying semiring is called exact, and in this case the row and column spaces of any matrix are isomorphic to each others' duals. The final problem is to explain the connection between the row space and column space of each matrix. Our notion of a conjugation on a semiring accounts for the different possibilities in a unified manner, as it guarantees the existence of bijections between row and column spaces and lets us focus on the peculiarities of those bijections. Our main original contribution is the systematic approach described above, but along the way we establish several new results about exactness of semirings. We give sufficient conditions for a subsemiring of an exact semiring to inherit exactness, and we apply these conditions to show that exactness transfers to finite group semirings. We also show that every Boolean ring is exact. This result is interesting because it allows us to construct a ring which is exact (also known as FP-injective) but not self-injective. Finally, we consider exactness for residuated lattices, showing that every involutive residuated lattice is exact. We end by showing that the residuated lattice of subsets of a finite monoid is exact if and only if the monoid is a group.