18,845 research outputs found
Coherence Optimization and Best Complex Antipodal Spherical Codes
Vector sets with optimal coherence according to the Welch bound cannot exist
for all pairs of dimension and cardinality. If such an optimal vector set
exists, it is an equiangular tight frame and represents the solution to a
Grassmannian line packing problem. Best Complex Antipodal Spherical Codes
(BCASCs) are the best vector sets with respect to the coherence. By extending
methods used to find best spherical codes in the real-valued Euclidean space,
the proposed approach aims to find BCASCs, and thereby, a complex-valued vector
set with minimal coherence. There are many applications demanding vector sets
with low coherence. Examples are not limited to several techniques in wireless
communication or to the field of compressed sensing. Within this contribution,
existing analytical and numerical approaches for coherence optimization of
complex-valued vector spaces are summarized and compared to the proposed
approach. The numerically obtained coherence values improve previously reported
results. The drawback of increased computational effort is addressed and a
faster approximation is proposed which may be an alternative for time critical
cases
Spectral densities of Wishart-Levy free stable random matrices: Analytical results and Monte Carlo validation
Random matrix theory is used to assess the significance of weak correlations
and is well established for Gaussian statistics. However, many complex systems,
with stock markets as a prominent example, exhibit statistics with power-law
tails, that can be modelled with Levy stable distributions. We review
comprehensively the derivation of an analytical expression for the spectra of
covariance matrices approximated by free Levy stable random variables and
validate it by Monte Carlo simulation.Comment: 10 pages, 1 figure, submitted to Eur. Phys. J.
The Statistical Physics of Regular Low-Density Parity-Check Error-Correcting Codes
A variation of Gallager error-correcting codes is investigated using
statistical mechanics. In codes of this type, a given message is encoded into a
codeword which comprises Boolean sums of message bits selected by two randomly
constructed sparse matrices. The similarity of these codes to Ising spin
systems with random interaction makes it possible to assess their typical
performance by analytical methods developed in the study of disordered systems.
The typical case solutions obtained via the replica method are consistent with
those obtained in simulations using belief propagation (BP) decoding. We
discuss the practical implications of the results obtained and suggest a
computationally efficient construction for one of the more practical
configurations.Comment: 35 pages, 4 figure
Typical performance of low-density parity-check codes over general symmetric channels
Typical performance of low-density parity-check (LDPC) codes over a general
binary-input output-symmetric memoryless channel is investigated using methods
of statistical mechanics. Theoretical framework for dealing with general
symmetric channels is provided, based on which Gallager and MacKay-Neal codes
are studied as examples of LDPC codes. It has been shown that the basic
properties of these codes known for particular channels, including the property
to potentially saturate Shannon's limit, hold for general symmetric channels.
The binary-input additive-white-Gaussian-noise channel and the binary-input
Laplace channel are considered as specific channel noise models.Comment: 10 pages, 4 figures, RevTeX4; an error in reference correcte
Precoding by Pairing Subchannels to Increase MIMO Capacity with Discrete Input Alphabets
We consider Gaussian multiple-input multiple-output (MIMO) channels with
discrete input alphabets. We propose a non-diagonal precoder based on the
X-Codes in \cite{Xcodes_paper} to increase the mutual information. The MIMO
channel is transformed into a set of parallel subchannels using Singular Value
Decomposition (SVD) and X-Codes are then used to pair the subchannels. X-Codes
are fully characterized by the pairings and a real rotation matrix
for each pair (parameterized with a single angle). This precoding structure
enables us to express the total mutual information as a sum of the mutual
information of all the pairs. The problem of finding the optimal precoder with
the above structure, which maximizes the total mutual information, is solved by
{\em i}) optimizing the rotation angle and the power allocation within each
pair and {\em ii}) finding the optimal pairing and power allocation among the
pairs. It is shown that the mutual information achieved with the proposed
pairing scheme is very close to that achieved with the optimal precoder by Cruz
{\em et al.}, and is significantly better than Mercury/waterfilling strategy by
Lozano {\em et al.}. Our approach greatly simplifies both the precoder
optimization and the detection complexity, making it suitable for practical
applications.Comment: submitted to IEEE Transactions on Information Theor
Improved Upper Bounds to the Causal Quadratic Rate-Distortion Function for Gaussian Stationary Sources
We improve the existing achievable rate regions for causal and for zero-delay
source coding of stationary Gaussian sources under an average mean squared
error (MSE) distortion measure. To begin with, we find a closed-form expression
for the information-theoretic causal rate-distortion function (RDF) under such
distortion measure, denoted by , for first-order Gauss-Markov
processes. Rc^{it}(D) is a lower bound to the optimal performance theoretically
attainable (OPTA) by any causal source code, namely Rc^{op}(D). We show that,
for Gaussian sources, the latter can also be upper bounded as Rc^{op}(D)\leq
Rc^{it}(D) + 0.5 log_{2}(2\pi e) bits/sample. In order to analyze
for arbitrary zero-mean Gaussian stationary sources, we
introduce \bar{Rc^{it}}(D), the information-theoretic causal RDF when the
reconstruction error is jointly stationary with the source. Based upon
\bar{Rc^{it}}(D), we derive three closed-form upper bounds to the additive rate
loss defined as \bar{Rc^{it}}(D) - R(D), where R(D) denotes Shannon's RDF. Two
of these bounds are strictly smaller than 0.5 bits/sample at all rates. These
bounds differ from one another in their tightness and ease of evaluation; the
tighter the bound, the more involved its evaluation. We then show that, for any
source spectral density and any positive distortion D\leq \sigma_{x}^{2},
\bar{Rc^{it}}(D) can be realized by an AWGN channel surrounded by a unique set
of causal pre-, post-, and feedback filters. We show that finding such filters
constitutes a convex optimization problem. In order to solve the latter, we
propose an iterative optimization procedure that yields the optimal filters and
is guaranteed to converge to \bar{Rc^{it}}(D). Finally, by establishing a
connection to feedback quantization we design a causal and a zero-delay coding
scheme which, for Gaussian sources, achieves...Comment: 47 pages, revised version submitted to IEEE Trans. Information Theor
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