Rajai Nasser

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.

Polar Codes for Arbitrary DMCs and Arbitrary MACs

Polar codes are constructed for arbitrary channels by imposing an arbitrary quasi-group structure on the input alphabet. Just as with usual polar codes, the block error probability under successive cancellation decoding is o(2-N1/2ε), 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 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.

An Ergodic Theory of Binary Operations—Part I: Key Properties

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 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.An open problem in polarization theory is to determine the bi nary operations that always lead to polarization when they are used in Arıkan-like constructions. This paper , which is presented in two parts, solves this problem by providing a necessary and sufficient condition for a binar y operation to be polarizing. This (first) part of the paper introduces the mathematical framework that we will us e in the second part [1] to characterize the polarizing operations: uniformity-preserving, irreducible, ergodi c and strongly ergodic operations are defined. The concepts of a stable partition and the residue of a stable partition ar e 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 se qu nce 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.

Polarization theorems for arbitrary DMCs

A polarization phenomenon in a special sense is shown for an arbitrary discrete memoryless channel (DMC) by imposing a quasigroup structure on the input alphabet. The same technique is used to derive a polarization theorem for an arbitrary multiple access channel (MAC) by using an appropriate Abelian group structure. These results can be used to construct capacity-achieving polar codes for arbitrary DMCs with a block error probability of o(2-N1/2-ε), and an encoding/decoding complexity of O(N log N), where N is the block length.

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.

Ergodic theory meets polarization II: A foundation of polarization theory for MACs

An open problem in polarization theory is to determine the binary operations that always lead to polarization when they are used in Arıkan style constructions. This paper solves this problem by providing a necessary and sufficient condition for a binary operation to be polarizing. The characterization is given in terms of a new mathematical framework that we introduce. We show that a binary operation is polarizing if and only if its inverse is strongly ergodic.

Continuity of channel parameters and operations under various DMC topologies

We study the continuity of several 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 Arikan-style transformations are continuous.

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 Arikan-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 that 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−Nβ for any β < ½

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. We show that this topology is compact, path-connected and metrizable. 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| ≥ 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| ≥ 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 provide a necessary and sufficient condition for a sequence of channels to converge in the strong topology. 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| ≥ 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.

Fourier analysis of MAC polarization

One problem with multiple access channel (MAC) polar codes that are based on MAC polarization is that they may not achieve the entire capacity region. The reason behind this problem is that MAC polarization sometimes induces a loss in the capacity region. This paper provides a single letter necessary and sufficient condition, which characterizes all the MACs that do not lose any part of their capacity region by polarization.