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Rajai Nasser - One of the best experts on this subject based on the ideXlab platform.

  • an Ergodic Theory of binary operations part ii applications to polarization
    2017
    Co-Authors: Rajai Nasser
    Abstract:

    An open problem in polarization Theory is to determine the binary operations that always lead to polarization (in the general multilevel sense) when they are used in Arikan style constructions. This paper, which is presented in two parts, solves this problem by providing a necessary and sufficient condition for a binary operation to be polarizing. This (second) part provides a foundation of polarization Theory based on the Ergodic Theory of binary operations which we developed in the first part. We show that a binary operation is polarizing if and only if it is uniformity preserving and its right-inverse is strongly Ergodic. The rate of polarization of single user channels is studied. It is shown that the exponent of any polarizing operation cannot exceed $\frac {1}{2}$ , which is the exponent of quasi-group operations. We also study the polarization of multiple access channels (MAC). In particular, we show that a sequence of binary operations is MAC-polarizing if and only if each binary operation in the sequence is polarizing. It is shown that the exponent of any MAC-polarizing sequence cannot exceed $\frac {1}{2}$ , which is the exponent of sequences of quasi-group operations.

  • Ergodic Theory Meets Polarization. II: A Foundation of Polarization Theory
    2017
    Co-Authors: Rajai Nasser
    Abstract:

    An open problem in polarization Theory is to determine the binary operations that always lead to polarization (in the general multilevel sense) when they are used in Arikan style constructions. This paper, which is presented in two parts, solves this problem by providing a necessary and sufficient condition for a binary operation to be polarizing. This (second) part provides a foundation of polarization Theory based on the Ergodic Theory of binary operations which we developed in the first part. We show that a binary operation is polarizing if and only if it is uniformity preserving and its right-inverse is strongly Ergodic. The rate of polarization of single user channels is studied. It is shown that the exponent of any polarizing operation cannot exceed $\frac{1}{2}$, which is the exponent of quasigroup operations. We also study the polarization of multiple access channels (MAC). In particular, we show that a sequence of binary operations is MAC-polarizing if and only if each binary operation in the sequence is polarizing. It is shown that the exponent of any MAC-polarizing sequence cannot exceed $\frac{1}{2}$, which is the exponent of sequences of quasigroup operations.

  • an Ergodic Theory of binary operations part i key properties
    2016
    Co-Authors: Rajai Nasser
    Abstract:

    An open problem in polarization Theory is to determine the binary operations that always lead to polarization (in the general multilevel sense) when they are used in Arikan style constructions. This paper, which is presented in two parts, solves this problem by providing a necessary and sufficient condition for a binary operation to be polarizing. This (first) part of this paper introduces the mathematical framework that we will use in the second part to characterize the polarizing operations. We define uniformity preserving, irreducible, Ergodic, and strongly Ergodic operations, and we study their properties. The concepts of a stable partition and the residue of a stable partition are introduced. We show that an Ergodic operation is strongly Ergodic if and only if all its stable partitions are their own residues. We also study the products of binary operations and the structure of their stable partitions. We show that the product of a sequence of binary operations is strongly Ergodic if and only if all the operations in the sequence are strongly Ergodic. In the second part of this paper, we provide a foundation of polarization Theory based on the Ergodic Theory of binary operations that we develop in this part.

  • Ergodic Theory meets polarization i an Ergodic Theory for binary operations
    2014
    Co-Authors: Rajai Nasser
    Abstract:

    An open problem in polarization Theory is to determine the binary operations that always lead to polarization (in the general multilevel sense) when they are used in Arikan style constructions. This paper, which is presented in two parts, solves this problem by providing a necessary and sufficient condition for a binary operation to be polarizing. This (first) part of the paper introduces the mathematical framework that we will use in the second part to characterize the polarizing operations. We define uniformity preserving, irreducible, Ergodic and strongly Ergodic operations and we study their properties. The concepts of a stable partition and the residue of a stable partition are introduced. We show that an Ergodic operation is strongly Ergodic if and only if all its stable partitions are their own residues. We also study the products of binary operations and the structure of their stable partitions. We show that the product of a sequence of binary operations is strongly Ergodic if and only if all the operations in the sequence are strongly Ergodic. In the second part of the paper, we provide a foundation of polarization Theory based on the Ergodic Theory of binary operations that we develop in this part.

Eli Barkai - One of the best experts on this subject based on the ideXlab platform.

  • infinite Ergodic Theory meets boltzmann statistics
    2020
    Co-Authors: Erez Aghion, David A Kessler, Eli Barkai
    Abstract:

    Abstract We investigate the overdamped stochastic dynamics of a particle in an asymptotically flat external potential field, in contact with a thermal bath. For an infinite system size, the particles may escape the force field and diffuse freely at large length scales. The partition function diverges and hence the standard canonical ensemble fails. This is replaced with tools stemming from infinite Ergodic Theory. Boltzmann-Gibbs statistics, even though not normalized, still describes integrable observables, like energy and occupation times. The Boltzmann infinite density is derived heuristically using an entropy maximization principle, as well as via a first-principles calculation using an eigenfunction expansion in the continuum of low-energy states. A generalized virial theorem is derived, showing how the virial coefficient describes the delay in the diffusive spreading of the particles, found at large distances. When the process is non-recurrent, e.g. diffusion in three dimensions with a Coulomb-like potential, we use weighted time averages to restore basic canonical relations between time and ensemble averages.

  • infinite Ergodic Theory for heterogeneous diffusion processes
    2019
    Co-Authors: N Leibovich, Eli Barkai
    Abstract:

    We show the relation between processes which are modeled by a Langevin equation with multiplicative noise and infinite Ergodic Theory. We concentrate on a spatially dependent diffusion coefficient that behaves as $D(x)\ensuremath{\sim}|x\ensuremath{-}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{x}{|}^{2\ensuremath{-}2/\ensuremath{\alpha}}$ in the vicinity of a point $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{x}$, where $\ensuremath{\alpha}$ can be either positive or negative. We find that a nonnormalized state, also called an infinite density, describes statistical properties of the system. For processes under investigation, the time averages of a wide class of observables are obtained using an ensemble average with respect to the nonnormalized density. A Langevin equation which involves multiplicative noise may take different interpretation, It\^o, Stratonovich, or H\"anggi-Klimontovich, so the existence of an infinite density and the density's shape are both related to the considered interpretation and the structure of $D(x)$.

  • from non normalizable boltzmann gibbs statistics to infinite Ergodic Theory
    2019
    Co-Authors: Erez Aghion, David A Kessler, Eli Barkai
    Abstract:

    We study a particle immersed in a heat bath, in the presence of an external force which decays at least as rapidly as $1/x$, e.g., a particle interacting with a surface through a Lennard-Jones or a logarithmic potential. As time increases, our system approaches a non-normalizable Boltzmann state. We study observables, such as the energy, which are integrable with respect to this asymptotic thermal state, calculating both time and ensemble averages. We derive a useful canonical-like ensemble which is defined out of equilibrium, using a maximum entropy principle, where the constraints are normalization, finite averaged energy, and a mean-squared displacement which increases linearly with time. Our work merges infinite-Ergodic Theory with Boltzmann-Gibbs statistics, thus extending the scope of the latter while shedding new light on the concept of Ergodicity.

Erez Aghion - One of the best experts on this subject based on the ideXlab platform.

  • infinite Ergodic Theory meets boltzmann statistics
    2020
    Co-Authors: Erez Aghion, David A Kessler, Eli Barkai
    Abstract:

    Abstract We investigate the overdamped stochastic dynamics of a particle in an asymptotically flat external potential field, in contact with a thermal bath. For an infinite system size, the particles may escape the force field and diffuse freely at large length scales. The partition function diverges and hence the standard canonical ensemble fails. This is replaced with tools stemming from infinite Ergodic Theory. Boltzmann-Gibbs statistics, even though not normalized, still describes integrable observables, like energy and occupation times. The Boltzmann infinite density is derived heuristically using an entropy maximization principle, as well as via a first-principles calculation using an eigenfunction expansion in the continuum of low-energy states. A generalized virial theorem is derived, showing how the virial coefficient describes the delay in the diffusive spreading of the particles, found at large distances. When the process is non-recurrent, e.g. diffusion in three dimensions with a Coulomb-like potential, we use weighted time averages to restore basic canonical relations between time and ensemble averages.

  • from non normalizable boltzmann gibbs statistics to infinite Ergodic Theory
    2019
    Co-Authors: Erez Aghion, David A Kessler, Eli Barkai
    Abstract:

    We study a particle immersed in a heat bath, in the presence of an external force which decays at least as rapidly as $1/x$, e.g., a particle interacting with a surface through a Lennard-Jones or a logarithmic potential. As time increases, our system approaches a non-normalizable Boltzmann state. We study observables, such as the energy, which are integrable with respect to this asymptotic thermal state, calculating both time and ensemble averages. We derive a useful canonical-like ensemble which is defined out of equilibrium, using a maximum entropy principle, where the constraints are normalization, finite averaged energy, and a mean-squared displacement which increases linearly with time. Our work merges infinite-Ergodic Theory with Boltzmann-Gibbs statistics, thus extending the scope of the latter while shedding new light on the concept of Ergodicity.

Bosco Leung - One of the best experts on this subject based on the ideXlab platform.

  • quantization noise spectrum of double loop sigma delta converter with sinusoidal input
    1994
    Co-Authors: Sundeep Rangan, Bosco Leung
    Abstract:

    An exact formula for the output noise spectrum of a double-loop sigma-delta modulator, under the no overloading assumption and with a sinusoidal input, is derived without the use of a white-noise model. In the case of a sinusoidal input with irrational input amplitude and digital frequency, the result agrees with the exact formula derived by Ergodic Theory for two-stage modulators. In addition, the present method also provides an exact formula for sinusoidal inputs with rational frequency and amplitude. Furthermore, the period of the output with rational initial conditions and DC input is also calculated. The results are of primary interest to multibit sigma-delta modulators, which do not overload over the entire input amplitude range. The Ergodic Theory method for calculating the exact noise spectrum involves explicitly determining the autocorrelation of the internal quantization error with Ergodic Theory techniques, and then determining the noise spectrum from the correlation function. The present method, however, directly determines the quantization noise spectrum by using an open-loop model for the coder and applying a Fourier series representation of the quantization error function. The result of both of these methods is that the output noise spectrum for a sinusoidal input is composed of discrete spectral lines shaped by a sin /sup 4/(w/2) envelope. >

  • quantization noise spectrum of double loop sigma delta converter with sinusoidal input
    1993
    Co-Authors: Sundeep Rangan, Bosco Leung
    Abstract:

    An exact formula for the output noise spectrum of a double-loop sigma-delta modulator with a sinusoidal input is derived without the use of a white-noise model. In the case of a sinusoidal input with irrational input amplitude and digital frequency, the result agrees with the exact formula derived by Ergodic Theory for two-stage modulators. In addition, the present method also provides an exact formula for a sinusoidal input with rational frequency and amplitude. Furthermore, the period of the output with rational initial conditions and DC input is also calculated. All results assume that the quantizer does not overload, and hence apply only to multi-bit coders. The Ergodic Theory method for calculating the exact noise spectrum involves explicitly determining the autocorrelation of the internal quantization error with Ergodic Theory techniques, and then determining the noise spectrum from the correlation function. The present method, however, directly determines the quantization noise spectrum by using an open-loop model for the coder and applying a Fourier series representation of the quantization error function. The result of both of these methods is that the output noise spectrum for a sinusoidal input is composed of discrete spectral lines shaped by a sin/sup 4/(/spl omega2) envelope. >

Ian Morris - One of the best experts on this subject based on the ideXlab platform.