21,534 research outputs found
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
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
A Unifying Approach to Quaternion Adaptive Filtering: Addressing the Gradient and Convergence
A novel framework for a unifying treatment of quaternion valued adaptive
filtering algorithms is introduced. This is achieved based on a rigorous
account of quaternion differentiability, the proposed I-gradient, and the use
of augmented quaternion statistics to account for real world data with
noncircular probability distributions. We first provide an elegant solution for
the calculation of the gradient of real functions of quaternion variables
(typical cost function), an issue that has so far prevented systematic
development of quaternion adaptive filters. This makes it possible to unify the
class of existing and proposed quaternion least mean square (QLMS) algorithms,
and to illuminate their structural similarity. Next, in order to cater for both
circular and noncircular data, the class of widely linear QLMS (WL-QLMS)
algorithms is introduced and the subsequent convergence analysis unifies the
treatment of strictly linear and widely linear filters, for both proper and
improper sources. It is also shown that the proposed class of HR gradients
allows us to resolve the uncertainty owing to the noncommutativity of
quaternion products, while the involution gradient (I-gradient) provides
generic extensions of the corresponding real- and complex-valued adaptive
algorithms, at a reduced computational cost. Simulations in both the strictly
linear and widely linear setting support the approach
Transmit Optimization with Improper Gaussian Signaling for Interference Channels
This paper studies the achievable rates of Gaussian interference channels
with additive white Gaussian noise (AWGN), when improper or circularly
asymmetric complex Gaussian signaling is applied. For the Gaussian
multiple-input multiple-output interference channel (MIMO-IC) with the
interference treated as Gaussian noise, we show that the user's achievable rate
can be expressed as a summation of the rate achievable by the conventional
proper or circularly symmetric complex Gaussian signaling in terms of the
users' transmit covariance matrices, and an additional term, which is a
function of both the users' transmit covariance and pseudo-covariance matrices.
The additional degrees of freedom in the pseudo-covariance matrix, which is
conventionally set to be zero for the case of proper Gaussian signaling,
provide an opportunity to further improve the achievable rates of Gaussian
MIMO-ICs by employing improper Gaussian signaling. To this end, this paper
proposes widely linear precoding, which efficiently maps proper
information-bearing signals to improper transmitted signals at each transmitter
for any given pair of transmit covariance and pseudo-covariance matrices. In
particular, for the case of two-user Gaussian single-input single-output
interference channel (SISO-IC), we propose a joint covariance and
pseudo-covariance optimization algorithm with improper Gaussian signaling to
achieve the Pareto-optimal rates. By utilizing the separable structure of the
achievable rate expression, an alternative algorithm with separate covariance
and pseudo-covariance optimization is also proposed, which guarantees the rate
improvement over conventional proper Gaussian signaling.Comment: Accepted by IEEE Transactions on Signal Processin
On the superiority of improper Gaussian signaling in wireless interference MIMO scenarios
©2016 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.Recent results have elucidated the benefits of using improper Gaussian signaling (IGS) as compared to conventional proper Gaussian signaling (PGS) in terms of achievable rate for interference-limited conditions. This paper exploits majorization
theory tools to formally quantify the gains of IGS along with widely linear transceivers for MIMO systems in interferencelimited scenarios. The MIMO point-to-point channel with interference (P2P-I) is analyzed, assuming that received interference can be either proper or improper, and we demonstrate that the
use of the optimal IGS when received interference is improper strictly outperforms (in terms of achievable rate and mean square error) the use of the optimal PGS when interference is proper.
Then, these results are extended to two practical situations. First, the MIMO Z-interference channel (Z-IC) is investigated, where a trade-off arises: with IGS we could increase the achievable rate of the interfered user while gracefully degrading the rate of the non-interfered user. Second, these concepts are applied to a
two-tier heterogeneous cellular network (HCN) where macrocells and smallcells coexist and multiple MIMO Z-IC appear.Peer ReviewedPostprint (author's final draft
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