Periodic state-space representations of periodic convolutional codes

In this paper we study the representation of periodically time-varying convolutional codes by means of periodic input-state-output models. In particular, we focus on period two and investigate under which conditions a given two-periodic convolutional code (obtained by alternating two time-invariant encoders) can be represented by a periodic input-state-output system. We first show that one cannot expect, in general, to obtain a periodic input-state-output representation of a periodic convolutional code by means of the individual realizations of each of the associated time-invariant codes. We, however, provide sufficient conditions for this to hold in terms of the column degrees of the associated column reduced generator matrices. Moreover, we derive a sufficient condition to obtain a periodic state-space realization that is minimal. Finally, examples to illustrate the results are presented.publishe

ЧИСЛЕННЫЕ РЕШЕНИЯ ДЛЯ СОБСТВЕННЫХ МОД АБСОЛЮТНОЙ ПАРАМЕТРИЧЕСКОЙ РАСПАДНОЙ НЕУСТОЙЧИВОСТИ НЕОДНОРОДНОЙ ПЛАЗМЫ НА НЕЛИНЕЙНОЙ СТАДИИ

A mathematical model for the nonlinear absolute decay parametric instability of inhomogeneous plasma is developed. Decay of the Langmuir pump wave on the Langmuir and ion-acoustic waves is considered in detail. The numerical solutions describing the eigen modes of the instability are obtained. It is shown that at the nonlinear stage of the instability, a depletion of a pump wave and a broadening of an ion-acoustic wave spectrum are observed. The simulation results are in agreement with the experimental ones.Создана математическая модель, описывающая нелинейную абсолютную параметрическую неустойчивость в неоднородной плазме. Подробно рассмотрен распад ленгмюровской волны накачки на ленгмюровскую и ионно- звуковую волны. Получены численные решения, описывающие собственные моды неустойчивости. Показано, чтов нелинейной стадии неустойчивости наблюдается истощение накачки и уширение спектра ионно-звуковой волны. Результаты численных расчетов согласуются с экспериментальными результатами

A state space approach to periodic convolutional codes

In this paper we study periodically time-varying convolutional codes by means of input-state-output representations. Using these representations we investigate under which conditions a given time-invariant convolutional code can be transformed into an equivalent periodic time-varying one. The relation between these two classes of convolutional codes is studied for period 2. We illustrate the ideas presented in this paper by constructing a periodic time-varying convolutional code from a time-invariant one. The resulting periodic code has larger free distance than any time-invariant convolutional code with equivalent parameters

Forward Error Correction for Optical Transponders

Forward error correction is an essential technique required in almost all communication systems to guarantee reliable data transmission close to the theoretical limits. In this chapter, we discuss the state-of-the-art forward error correction (FEC) schemes for fiber-optic communications. Following a historical overview of the evolution of FEC schemes, we first introduce the fundamental theoretical limits of common communication channel models and show how to compute them. These limits provide the reader with guidelines for comparing different FEC codes under various assumptions. We then provide a brief introduction to the general basic concepts of FEC, followed by an in-depth introduction to the main classes of codes for soft decision decoding and hard decision decoding. We include a wide range of performance curves, compare the different schemes, and give the reader guidelines on which FEC scheme to use. We also introduce the main techniques to combine coding and higher-order modulation (coded modulation), including constellation shaping. Finally, we include a guide on how to evaluate the performance of FEC in transmission experiments. We conclude the chapter with an overview of the properties of some state-of-the-art FEC schemes used in optical communications and an outlook