PROBLEMS OF INFORMATION TRANSMISSION
A translation of Problemy Peredachi Informatsii


Volume 48, Number 3, July–September, 2012
Back to contents page

CONTENTS                   Powered by MathJax

 

On the Reliability Function for a Noisy Feedback Gaussian Channel: Zero Rate
M. V. Burnashev and H. Yamamoto
pp. 199–216

Abstract—A discrete-time channel with independent additive Gaussian noise is used for information transmission. There is also a feedback channel with independent additive Gaussian noise, and the transmitter observes all outputs of the forward channel without delay via this feedback channel. Transmission of a nonexponential number of messages is considered (i.e., the transmission rate is zero), and the achievable decoding error exponent for such a combination of channels is investigated. It is shown that for any finite noise in the feedback channel the achievable error exponent is better than the similar error exponent for a no-feedback channel. The transmission/decoding method developed in the paper strengthens the method earlier used by the authors for a BSC. In particular, for small feedback noise, it provides a gain of 23.6% (instead of 14.3% obtained earlier for a BSC).

 

On the Hilbert Transform of Bounded Bandlimited Signals
H. Boche and U. J. Mönich
pp. 217–238

Abstract—In this paper we analyze the Hilbert transform and existence of the analytical signal for the space $\mathcal{B}_\pi^\infty$ of bandlimited signals that are bounded on the real axis. Originally, the theory was developed for signals in $L^2(\mathbb{R})$ and then extended to larger signal spaces. While it is well known that the common integral representation of the Hilbert transform may diverge for some signals in $\mathcal{B}_\pi^\infty$ and that the Hilbert transform is not a bounded operator on $\mathcal{B}_\pi^\infty$, it is nevertheless possible to define the Hilbert transform for the space $\mathcal{B}_\pi^\infty$. We use a definition that is based on the $\mathcal{H}^1\,$–$\,\mathop{\rm BMO}(\mathbb{R})$ duality. This abstract definition, which can be used for general bounded signals, gives no constructive procedure to compute the Hilbert transform. However, for the practically important special case of bounded bandlimited signals, we can provide such an explicit procedure by giving a closed-form expression for the Hilbert transform. Further, it is shown that the Hilbert transform of a signal in $\mathcal{B}_\pi^\infty$ is still bandlimited but not necessarily bounded. With these results we continue the work of [12].

 

Binary Generalized $(L,G)$ Codes That Are Perfect in a Weighted Hamming Metric
S. V. Bezzateev and N. A. Shekhunova
pp. 239–242

Abstract—We propose a class of binary generalized $(L,G)$ codes that are perfect in a weighted Hamming metric.

 

Multiple Access System for a Vector Disjunctive Channel
D. S. Osipov, A. A. Frolov, and V. V. Zyablov
pp. 243–249

Abstract—We address the problem of constructing a multiple access system for a disjunctive vector channel, similar to a multiuser channel without intensity information as described in [1]. To solve the problem, a signal-code construction based on nonbinary codes is proposed. For the resulting multiple access system, a lower bound on the relative group rate is derived. The bound coincides asymptotically with an upper bound.

 

Construction of Self-Orthogonal Codes from Combinatorial Designs
M. Dzhumalieva-Stoeva, I. G. Bouyukliev, and V. Monev
pp. 250–258

Abstract—Self-orthogonal codes are constructed from matrices generated according to parameters of combinatorial designs. An approach towards generating designs and such matrices is considered. Some classification results on self-orthogonal codes are also presented.

 

Stability of Regime-Switching Stochastic Differential Equations
R. Z. Khasminskii
pp. 259–270

Abstract—The main result is reduction of the asymptotic stability problem for a stochastic differential equation (SDE) with sufficiently rapid Markovian switching to the analogous well-studied problem for the “averaged” SDE without switching. Applications to the switching stabilization problem and to ordinary differential equations (ODE) with switching are also considered.

 

Exponential Weighting and Oracle Inequalities for Projection Estimates
G. K. Golubev
pp. 271–282

Abstract—We consider the problem of recovering an unknown vector from noisy data. The vector is estimated using a family of projection estimates, and the goal is finding a sufficiently good convex combination of these estimates based on the observations. We study an aggregation method for constructing estimates related to the so-called exponential weighting and present an upper bound on the mean-square risk of this method.

 

Fine Structure of a One-Dimensional Discrete Point System
V. A. Malyshev
pp. 283–296

Abstract—We consider a system of $N$ points $x_1 < \ldots < x_N$ on a segment of the real line. An ideal system (crystal) is a system where all distances between neighbors are the same. Deviation from idealness is characterized by a system of finite differences $\nabla_i^1=x_{i+1}-x_i$, $\nabla_i^{k+1}=\nabla_{i+1}^k-\nabla_i^k$, for all possible $i$ and $k$. We find asymptotic estimates as $N\to\infty$, $k\to\infty$, for a system of points minimizing the potential energy of a Coulomb system in an external field.