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postgraduate thesis: Some results for special families of channels with memory
Title  Some results for special families of channels with memory 

Authors  
Issue Date  2015 
Publisher  The University of Hong Kong (Pokfulam, Hong Kong) 
Citation  Li, Y. [李永龙]. (2015). Some results for special families of channels with memory. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5699932 
Abstract  The main concerns of this thesis are some special families of channels with memory, which are of great importance both in theory and in practical applications. Unlike discrete memoryless channels, the capacity of channels with memory generally admits no singleletter characterization and it is difficult to numerically evaluate the channel capacity.
As prominent examples of channels with memory, finitestate channels have attracted much interest. However, the numerical computation of the channel capacity of a finitestate channel is still an open problem. Recently, two algorithms have been proposed to numerically compute the channel capacity for a special family of finitestate channels with Markovian inputs. One sufficient condition for the convergence of the proposed algorithms is the concavity of mutual information rate with respect to some chosen parameters of the input Markov chains. In this thesis, under mild positivity conditions, concavity is proved at the high signaltonoise ratio and counterexamples are given to show that concavity may fail in general.
The inputconstrained discrete erasure channels are examples of channels with memory and the channel capacity is unknown and has been a longstanding open problem in information theory. In this thesis, an \explicit" formula of the mutual information rate for a discrete erasure channel with a Markovian input is derived. Moreover, if the input Markov chain is of firstorder and supported on the (1; ∞) run length limited constraint, the mutual information rate is shown to be strictly concave with respect to the transition probability of the input Markov chain. By comparing the asymptotics in the high SNR regime, feedback capacity is shown to be strictly larger than the nonfeedback capacity, which disproves a statement of Shannon.
The other example of channels with memory is the recently proposed ash memory channel model, which possesses both input and output memory. In this thesis, the ash memory channels are shown to be indecomposable under rather weak conditions. Then the channel capacity is shown to be achievable by a stationary process and consequently, as its order tends to infinity, the Markov capacity converges to the real channel capacity. This result suggests that it is highly possible to apply the ideas and techniques in the computation of the capacity of finitestate channels, which are relatively better explored, to that of the capacity of ash memory channels. 
Degree  Doctor of Philosophy 
Subject  Digital communications  Mathematics 
Dept/Program  Mathematics 
Persistent Identifier  http://hdl.handle.net/10722/223041 
HKU Library Item ID  b5699932 
DC Field  Value  Language 

dc.contributor.author  Li, Yonglong   
dc.contributor.author  李永龙   
dc.date.accessioned  20160217T23:14:38Z   
dc.date.available  20160217T23:14:38Z   
dc.date.issued  2015   
dc.identifier.citation  Li, Y. [李永龙]. (2015). Some results for special families of channels with memory. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5699932   
dc.identifier.uri  http://hdl.handle.net/10722/223041   
dc.description.abstract  The main concerns of this thesis are some special families of channels with memory, which are of great importance both in theory and in practical applications. Unlike discrete memoryless channels, the capacity of channels with memory generally admits no singleletter characterization and it is difficult to numerically evaluate the channel capacity. As prominent examples of channels with memory, finitestate channels have attracted much interest. However, the numerical computation of the channel capacity of a finitestate channel is still an open problem. Recently, two algorithms have been proposed to numerically compute the channel capacity for a special family of finitestate channels with Markovian inputs. One sufficient condition for the convergence of the proposed algorithms is the concavity of mutual information rate with respect to some chosen parameters of the input Markov chains. In this thesis, under mild positivity conditions, concavity is proved at the high signaltonoise ratio and counterexamples are given to show that concavity may fail in general. The inputconstrained discrete erasure channels are examples of channels with memory and the channel capacity is unknown and has been a longstanding open problem in information theory. In this thesis, an \explicit" formula of the mutual information rate for a discrete erasure channel with a Markovian input is derived. Moreover, if the input Markov chain is of firstorder and supported on the (1; ∞) run length limited constraint, the mutual information rate is shown to be strictly concave with respect to the transition probability of the input Markov chain. By comparing the asymptotics in the high SNR regime, feedback capacity is shown to be strictly larger than the nonfeedback capacity, which disproves a statement of Shannon. The other example of channels with memory is the recently proposed ash memory channel model, which possesses both input and output memory. In this thesis, the ash memory channels are shown to be indecomposable under rather weak conditions. Then the channel capacity is shown to be achievable by a stationary process and consequently, as its order tends to infinity, the Markov capacity converges to the real channel capacity. This result suggests that it is highly possible to apply the ideas and techniques in the computation of the capacity of finitestate channels, which are relatively better explored, to that of the capacity of ash memory channels.   
dc.language  eng   
dc.publisher  The University of Hong Kong (Pokfulam, Hong Kong)   
dc.relation.ispartof  HKU Theses Online (HKUTO)   
dc.rights  The author retains all proprietary rights, (such as patent rights) and the right to use in future works.   
dc.rights  Creative Commons: Attribution 3.0 Hong Kong License   
dc.subject.lcsh  Digital communications  Mathematics   
dc.title  Some results for special families of channels with memory   
dc.type  PG_Thesis   
dc.identifier.hkul  b5699932   
dc.description.thesisname  Doctor of Philosophy   
dc.description.thesislevel  Doctoral   
dc.description.thesisdiscipline  Mathematics   
dc.description.nature  published_or_final_version   