12,567 research outputs found
Complex Random Vectors and ICA Models: Identifiability, Uniqueness and Separability
In this paper the conditions for identifiability, separability and uniqueness
of linear complex valued independent component analysis (ICA) models are
established. These results extend the well-known conditions for solving
real-valued ICA problems to complex-valued models. Relevant properties of
complex random vectors are described in order to extend the Darmois-Skitovich
theorem for complex-valued models. This theorem is used to construct a proof of
a theorem for each of the above ICA model concepts. Both circular and
noncircular complex random vectors are covered. Examples clarifying the above
concepts are presented.Comment: To appear in IEEE TR-IT March 200
The geometry of proper quaternion random variables
Second order circularity, also called properness, for complex random
variables is a well known and studied concept. In the case of quaternion random
variables, some extensions have been proposed, leading to applications in
quaternion signal processing (detection, filtering, estimation). Just like in
the complex case, circularity for a quaternion-valued random variable is
related to the symmetries of its probability density function. As a
consequence, properness of quaternion random variables should be defined with
respect to the most general isometries in , i.e. rotations from .
Based on this idea, we propose a new definition of properness, namely the
-properness, for quaternion random variables using invariance
property under the action of the rotation group . This new definition
generalizes previously introduced properness concepts for quaternion random
variables. A second order study is conducted and symmetry properties of the
covariance matrix of -proper quaternion random variables are
presented. Comparisons with previous definitions are given and simulations
illustrate in a geometric manner the newly introduced concept.Comment: 14 pages, 3 figure
Complex-Valued Random Vectors and Channels: Entropy, Divergence, and Capacity
Recent research has demonstrated significant achievable performance gains by
exploiting circularity/non-circularity or propeness/improperness of
complex-valued signals. In this paper, we investigate the influence of these
properties on important information theoretic quantities such as entropy,
divergence, and capacity. We prove two maximum entropy theorems that strengthen
previously known results. The proof of the former theorem is based on the
so-called circular analog of a given complex-valued random vector. Its
introduction is supported by a characterization theorem that employs a minimum
Kullback-Leibler divergence criterion. In the proof of latter theorem, on the
other hand, results about the second-order structure of complex-valued random
vectors are exploited. Furthermore, we address the capacity of multiple-input
multiple-output (MIMO) channels. Regardless of the specific distribution of the
channel parameters (noise vector and channel matrix, if modeled as random), we
show that the capacity-achieving input vector is circular for a broad range of
MIMO channels (including coherent and noncoherent scenarios). Finally, we
investigate the situation of an improper and Gaussian distributed noise vector.
We compute both capacity and capacity-achieving input vector and show that
improperness increases capacity, provided that the complementary covariance
matrix is exploited. Otherwise, a capacity loss occurs, for which we derive an
explicit expression.Comment: 33 pages, 1 figure, slightly modified version of first paper revision
submitted to IEEE Trans. Inf. Theory on October 31, 201
Simultaneous diagonalisation of the covariance and complementary covariance matrices in quaternion widely linear signal processing
Recent developments in quaternion-valued widely linear processing have
established that the exploitation of complete second-order statistics requires
consideration of both the standard covariance and the three complementary
covariance matrices. Although such matrices have a tremendous amount of
structure and their decomposition is a powerful tool in a variety of
applications, the non-commutative nature of the quaternion product has been
prohibitive to the development of quaternion uncorrelating transforms. To this
end, we introduce novel techniques for a simultaneous decomposition of the
covariance and complementary covariance matrices in the quaternion domain,
whereby the quaternion version of the Takagi factorisation is explored to
diagonalise symmetric quaternion-valued matrices. This gives new insights into
the quaternion uncorrelating transform (QUT) and forms a basis for the proposed
quaternion approximate uncorrelating transform (QAUT) which simultaneously
diagonalises all four covariance matrices associated with improper quaternion
signals. The effectiveness of the proposed uncorrelating transforms is
validated by simulations on both synthetic and real-world quaternion-valued
signals.Comment: 41 pages, single column, 10 figure
Internal DLA and the Gaussian free field
In previous works, we showed that the internal DLA cluster on \Z^d with t
particles is a.s. spherical up to a maximal error of O(\log t) if d=2 and
O(\sqrt{\log t}) if d > 2. This paper addresses "average error": in a certain
sense, the average deviation of internal DLA from its mean shape is of constant
order when d=2 and of order r^{1-d/2} (for a radius r cluster) in general.
Appropriately normalized, the fluctuations (taken over time and space) scale to
a variant of the Gaussian free field.Comment: 29 pages, minor revisio
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