2,656 research outputs found
Coding Theorem and Memory Conditions for Abstract Channels with Time Structure
In the first part of this thesis, we generalize a coding theorem and a converse of Kadota and Wyner (1972) to abstract channels with time structure. As a main contribution we prove the coding theorem for a significantly weaker condition on the channel output memory, called total ergodicity for block-i.i.d. inputs. We achieve this result mainly by introducing an alternative characterization of information rate capacity. We show that the Ï-mixing condition (asymptotic output-memorylessness), used by Kadota and Wyner, is quite restrictive, in particular for the important class of Gaussian channels. In fact, we prove that for Gaussian channels the Ï-mixing condition is equivalent to finite output memory. Moreover, we derive a weak converse for all
stationary channels with time structure. Intersymbol interference as well as input constraints are taken into account in a flexible way. Due to the direct use of outer measures and a derivation of an adequate version of Feinsteinâs lemma we are able to avoid the standard extension of the channel input Ï-algebra and obtain a more transparent derivation. We aim at a presentation from an operational perspective and consider an abstract framework, which enables us to treat discrete- and continuous-time channels in a unified way.
In the second part, we systematically analyze infinite output memory conditions for abstract channels with time structure. We exploit the connections to the rich field of strongly mixing random processes to derive a hierarchy for the nonequivalent infinite channel output memory conditions in terms of a sequence of implications. The ergodic-theoretic memory condition used in the proof of the coding theorem and the Ï-mixing condition employed by Kadota and Wyner (1972) are shown to be part of this taxonomy. In addition, we specify conditions for the channel under which memory properties of a random process are invariant when the process is passed through the channel.
In the last part, we investigate cascade and integration channels with regard to mixing conditions as well as properties required in the context of the coding theorem. The results are useful to study many physically relevant channel models and allow a component-based analysis of the overall channel. We consider a number of examples including composed models and deterministic as well as random filter channels. Finally, an application of strong mixing conditions from statistical signal processing involving the Fourier transform of stationary random sequences is discussed and a list of further applications is given.Im ersten Teil der Arbeit wird ein Kodierungstheorem und ein dazugehöriges Umkehrtheorem von Kadota und Wyner (1972) fĂŒr abstrakte KanĂ€le mit Zeitstruktur verallgemeinert. Als wesentlichster Beitrag wird das Kodierungstheorem fĂŒr eine signifikant schwĂ€chere Bedingung an das KanalausgangsgedĂ€chtnis bewiesen, die sogenannte totale ErgodizitĂ€t fĂŒr block-i.i.d. Eingaben. Dieses Ergebnis wird hauptsĂ€chlich durch eine alternative Charakterisierung der InformationsratenkapazitĂ€t erreicht. Es wird gezeigt, dass die von Kadota und Wyner verwendete Ï-Mischungsbedingung (asymptotische GedĂ€chtnislosigkeit am Kanalausgang) recht einschrĂ€nkend ist, insbesondere fĂŒr die wichtige Klasse der GauĂkanĂ€le. In der Tat, fĂŒr GauĂkanĂ€le wird bewiesen, dass die Ï-Mischungsbedingung Ă€quivalent zu endlichem GedĂ€chtnis am Kanalausgang ist. DarĂŒber hinaus wird eine schwache Umkehrung fĂŒr alle stationĂ€ren KanĂ€le mit Zeitstruktur bewiesen. Sowohl Intersymbolinterferenz als auch EingabebeschrĂ€nkungen werden in allgemeiner und flexibler Form berĂŒcksichtigt. Aufgrund der direkten Verwendung von Ă€uĂeren MaĂen und der Herleitung einer angepassten Version von Feinsteins Lemma ist es möglich, auf die Standarderweiterung der Ï-Algebra am Kanaleingang zu verzichten, wodurch die Darstellungen transparenter und einfacher werden. Angestrebt wird eine operationelle Perspektive. Die Verwendung eines abstrakten Modells erlaubt dabei die einheitliche Betrachtung von zeitdiskreten und zeitstetigen KanĂ€len.
FĂŒr abstrakte KanĂ€le mit Zeitstruktur werden im zweiten Teil der Arbeit Bedingungen fĂŒr ein unendliches GedĂ€chtnis am Kanalausgang systematisch analysiert. Unter Ausnutzung der ZusammenhĂ€nge zu dem umfassenden Gebiet der stark mischenden zufĂ€lligen Prozesse wird eine Hierarchie in Form einer Folge von Implikationen zwischen den verschiedenen GedĂ€chtnisvarianten hergeleitet. Die im Beweis des Kodierungstheorems verwendete ergodentheoretische GedĂ€chtniseigenschaft und die Ï-Mischungsbedingung von Kadota und Wyner (1972) sind dabei Bestandteil der hergeleiteten Systematik. Weiterhin werden Bedingungen fĂŒr den Kanal spezifiziert, unter denen Eigenschaften von zufĂ€lligen Prozessen am Kanaleingang bei einer Transformation durch den Kanal erhalten bleiben.
Im letzten Teil der Arbeit werden sowohl IntegrationskanĂ€le als auch Hintereinanderschaltungen von KanĂ€len in Bezug auf Mischungsbedingungen sowie weitere fĂŒr das Kodierungstheorem relevante Kanaleigenschaften analysiert. Die erzielten Ergebnisse sind nĂŒtzlich bei der Untersuchung vieler physikalisch relevanter Kanalmodelle und erlauben eine komponentenbasierte Betrachtung zusammengesetzter KanĂ€le. Es wird eine Reihe von Beispielen untersucht, einschlieĂlich deterministischer KanĂ€le, zufĂ€lliger Filter und daraus zusammengesetzter Modelle. AbschlieĂend werden Anwendungen aus weiteren Gebieten, beispielsweise der statistischen Signalverarbeitung, diskutiert. Insbesondere die Fourier-Transformation stationĂ€rer zufĂ€lliger Prozesse wird im Zusammenhang mit starken Mischungsbedingungen betrachtet
Ergodic Classical-Quantum Channels: Structure and Coding Theorems
We consider ergodic causal classical-quantum channels (cq-channels) which
additionally have a decaying input memory. In the first part we develop some
structural properties of ergodic cq-channels and provide equivalent conditions
for ergodicity. In the second part we prove the coding theorem with weak
converse for causal ergodic cq-channels with decaying input memory. Our proof
is based on the possibility to introduce joint input-output state for the
cq-channels and an application of the Shannon-McMillan theorem for ergodic
quantum states. In the last part of the paper it is shown how this result
implies coding theorem for the classical capacity of a class of causal ergodic
quantum channels.Comment: 19 pages, no figures. Final versio
Quantum Channels with Memory
We present a general model for quantum channels with memory, and show that it
is sufficiently general to encompass all causal automata: any quantum process
in which outputs up to some time t do not depend on inputs at times t' > t can
be decomposed into a concatenated memory channel. We then examine and present
different physical setups in which channels with memory may be operated for the
transfer of (private) classical and quantum information. These include setups
in which either the receiver or a malicious third party have control of the
initializing memory. We introduce classical and quantum channel capacities for
these settings, and give several examples to show that they may or may not
coincide. Entropic upper bounds on the various channel capacities are given.
For forgetful quantum channels, in which the effect of the initializing memory
dies out as time increases, coding theorems are presented to show that these
bounds may be saturated. Forgetful quantum channels are shown to be open and
dense in the set of quantum memory channels.Comment: 21 pages with 5 EPS figures. V2: Presentation clarified, references
adde
Short Packets over Block-Memoryless Fading Channels: Pilot-Assisted or Noncoherent Transmission?
We present nonasymptotic upper and lower bounds on the maximum coding rate
achievable when transmitting short packets over a Rician memoryless
block-fading channel for a given requirement on the packet error probability.
We focus on the practically relevant scenario in which there is no \emph{a
priori} channel state information available at the transmitter and at the
receiver. An upper bound built upon the min-max converse is compared to two
lower bounds: the first one relies on a noncoherent transmission strategy in
which the fading channel is not estimated explicitly at the receiver; the
second one employs pilot-assisted transmission (PAT) followed by
maximum-likelihood channel estimation and scaled mismatched nearest-neighbor
decoding at the receiver. Our bounds are tight enough to unveil the optimum
number of diversity branches that a packet should span so that the energy per
bit required to achieve a target packet error probability is minimized, for a
given constraint on the code rate and the packet size. Furthermore, the bounds
reveal that noncoherent transmission is more energy efficient than PAT, even
when the number of pilot symbols and their power is optimized. For example, for
the case when a coded packet of symbols is transmitted using a channel
code of rate bits/channel use, over a block-fading channel with block
size equal to symbols, PAT requires an additional dB of energy per
information bit to achieve a packet error probability of compared to
a suitably designed noncoherent transmission scheme. Finally, we devise a PAT
scheme based on punctured tail-biting quasi-cyclic codes and ordered statistics
decoding, whose performance are close ( dB gap at packet error
probability) to the ones predicted by our PAT lower bound. This shows that the
PAT lower bound provides useful guidelines on the design of actual PAT schemes.Comment: 30 pages, 5 figures, journa
On Capacity Regions of Discrete Asynchronous Multiple Access Channels
A general formalization is given for asynchronous multiple access channels
which admits different assumptions on delays. This general framework allows the
analysis of so far unexplored models leading to new interesting capacity
regions. In particular, a single letter characterization is given for the
capacity region in case of 3 senders, 2 synchronous with each other and the
third not synchronous with them.Comment: It has been presented in part at ISIT 2011, Saint Petersburg. This
extended version is accepted for publication in Kybernetik
- âŠ