arXiv subjects

Rajai Nasser

Publications and source records attributed to Rajai Nasser.

29 records · Page 2Linked to original sources

arXiv2017

Continuity of Channel Parameters and Operations under Various DMC Topologies

We study the continuity of many channel parameters and operations under various topologies on the space of equivalent discrete memoryless channels (DMC). We show that mutual information, channel capacity, Bhattacharyya parameter, probability of error of a fixed code, and optimal probability of error for a given code rate and blocklength, are continuous under various DMC topologies. We also show that channel operations such as sums, products, interpolations, and Arıkan-style transformations are continuous.

cs.IT↗

arXiv2017

Topological Structures on DMC spaces

Two channels are said to be equivalent if they are degraded from each other. The space of equivalent channels with input alphabet $X$ and output alphabet $Y$ can be naturally endowed with the quotient of the Euclidean topology by the equivalence relation. A topology on the space of equivalent channels with fixed input alphabet $X$ and arbitrary but finite output alphabet is said to be natural if and only if it induces the quotient topology on the subspaces of equivalent channels sharing the same output alphabet. We show that every natural topology is $σ$-compact, separable and path-connected. On the other hand, if $|X|\geq 2$, a Hausdorff natural topology is not Baire and it is not locally compact anywhere. This implies that no natural topology can be completely metrized if $|X|\geq 2$. The finest natural topology, which we call the strong topology, is shown to be compactly generated, sequential and $T_4$. On the other hand, the strong topology is not first-countable anywhere, hence it is not metrizable. We show that in the strong topology, a subspace is compact if and only if it is rank-bounded and strongly-closed. We introduce a metric distance on the space of equivalent channels which compares the noise levels between channels. The induced metric topology, which we call the noisiness topology, is shown to be natural. We also study topologies that are inherited from the space of meta-probability measures by identifying channels with their Blackwell measures. We show that the weak-* topology is exactly the same as the noisiness topology and hence it is natural. We prove that if $|X|\geq 2$, the total variation topology is not natural nor Baire, hence it is not completely metrizable. Moreover, it is not locally compact anywhere. Finally, we show that the Borel $σ$-algebra is the same for all Hausdorff natural topologies.

math.GN↗

arXiv2017

On the Input-Degradedness and Input-Equivalence Between Channels

A channel $W$ is said to be input-degraded from another channel $W'$ if $W$ can be simulated from $W'$ by randomization at the input. We provide a necessary and sufficient condition for a channel to be input-degraded from another one. We show that any decoder that is good for $W'$ is also good for $W$. We provide two characterizations for input-degradedness, one of which is similar to the Blackwell-Sherman-Stein theorem. We say that two channels are input-equivalent if they are input-degraded from each other. We study the topologies that can be constructed on the space of input-equivalent channels, and we investigate their properties. Moreover, we study the continuity of several channel parameters and operations under these topologies.

cs.IT↗

arXiv2017

Polar Codes for Arbitrary Classical-Quantum Channels and Arbitrary cq-MACs

We prove polarization theorems for arbitrary classical-quantum (cq) channels. The input alphabet is endowed with an arbitrary Abelian group operation and an Arıkan-style transformation is applied using this operation. It is shown that as the number of polarization steps becomes large, the synthetic cq-channels polarize to deterministic homomorphism channels which project their input to a quotient group of the input alphabet. This result is used to construct polar codes for arbitrary cq-channels and arbitrary classical-quantum multiple access channels (cq-MAC). The encoder can be implemented in $O(N\log N)$ operations, where $N$ is the blocklength of the code. A quantum successive cancellation decoder for the constructed codes is proposed. It is shown that the probability of error of this decoder decays faster than $2^{-N^β}$ for any $β<\frac{1}{2}$.

cs.IT↗

arXiv2016

Ergodic Theory Meets Polarization. I: An Ergodic Theory for Binary Operations

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 Arıkan 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.

math.CO↗

arXiv2016

Ergodic Theory Meets Polarization. II: A Foundation of Polarization Theory

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 Arıkan 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.

cs.IT↗

arXiv2016

Age of Information: The Gamma Awakening

We consider a scenario where a monitor is interested in being up to date with respect to the status of some system which is not directly accessible to this monitor. However, we assume a source node has access to the status and can send status updates as packets to the monitor through a communication system. We also assume that the status updates are generated randomly as a Poisson process. The source node can manage the packet transmission to minimize the age of information at the destination node, which is defined as the time elapsed since the last successfully transmitted update was generated at the source. We use queuing theory to model the source-destination link and we assume that the time to successfully transmit a packet is a gamma distributed service time. We consider two packet management schemes: LCFS (Last Come First Served) with preemption and LCFS without preemption. We compute and analyze the average age and the average peak age of information under these assumptions. Moreover, we extend these results to the case where the service time is deterministic.

cs.IT↗

arXiv2016

Erasure Schemes Using Generalized Polar Codes: Zero-Undetected-Error Capacity and Performance Trade-offs

We study the performance of generalized polar (GP) codes when they are used for coding schemes involving erasure. GP codes are a family of codes which contains, among others, the standard polar codes of Arıkan and Reed-Muller codes. We derive a closed formula for the zero-undetected-error capacity $I_0^{GP}(W)$ of GP codes for a given binary memoryless symmetric (BMS) channel $W$ under the low complexity successive cancellation decoder with erasure. We show that for every $R<I_0^{GP}(W)$, there exists a generalized polar code of blocklength $N$ and of rate at least $R$ where the undetected-error probability is zero and the erasure probability is less than $2^{-N^{\frac{1}{2}-ε}}$. On the other hand, for any GP code of rate $I_0^{GP}(W)<R<I(W)$ and blocklength $N$, the undetected error probability cannot be made less than $2^{-N^{\frac{1}{2}+ε}}$ unless the erasure probability is close to $1$.

cs.IT↗

arXiv2013

Polar Codes for Arbitrary DMCs and Arbitrary MACs

Polar codes are constructed for arbitrary channels by imposing an arbitrary quasigroup structure on the input alphabet. Just as with "usual" polar codes, the block error probability under successive cancellation decoding is $o(2^{-N^{1/2-ε}})$, where $N$ is the block length. Encoding and decoding for these codes can be implemented with a complexity of $O(N\log N)$. It is shown that the same technique can be used to construct polar codes for arbitrary multiple access channels (MAC) by using an appropriate Abelian group structure. Although the symmetric sum capacity is achieved by this coding scheme, some points in the symmetric capacity region may not be achieved. In the case where the channel is a combination of linear channels, we provide a necessary and sufficient condition characterizing the channels whose symmetric capacity region is preserved by the polarization process. We also provide a sufficient condition for having a maximal loss in the dominant face.

cs.IT↗

arXiv2011

Polar codes for the m-user multiple access channels

Polar codes are constructed for m-user multiple access channels (MAC) whose input alphabet size is a prime number. The block error probability under successive cancelation decoding decays exponentially with the square root of the block length. Although the sum capacity is achieved by this coding scheme, some points in the symmetric capacity region may not be achieved. In the case where the channel is a combination of linear channels, we provide a necessary and sufficient condition characterizing the channels whose symmetric capacity region is preserved upon the polarization process. We also provide a sufficient condition for having a total loss in the dominant face.

cs.IT↗

arXiv2011

Oriented paths in n-chromatic digraphs

In this thesis, we try to treat the problem of oriented paths in n-chromatic digraphs. We first treat the case of antidirected paths in 5-chromatic digraphs, where we explain El-Sahili's theorem and provide an elementary and shorter proof of it. We then treat the case of paths with two blocks in n-chromatic digraphs with n greater than 4, where we explain the two different approaches of Addario-Berry et al. and of El-Sahili. We indicate a mistake in Addario-Berry et al.'s proof and provide a correction for it.

math.CO↗