629 research outputs found

    Channel combining and splitting for cutoff rate improvement

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    The cutoff rate R0(W)R_0(W) of a discrete memoryless channel (DMC) WW is often used as a figure of merit, alongside the channel capacity C(W)C(W). Given a channel WW consisting of two possibly correlated subchannels W1W_1, W2W_2, the capacity function always satisfies C(W1)+C(W2)C(W)C(W_1)+C(W_2) \le C(W), while there are examples for which R0(W1)+R0(W2)>R0(W)R_0(W_1)+R_0(W_2) > R_0(W). This fact that cutoff rate can be ``created'' by channel splitting was noticed by Massey in his study of an optical modulation system modeled as a MM'ary erasure channel. This paper demonstrates that similar gains in cutoff rate can be achieved for general DMC's by methods of channel combining and splitting. Relation of the proposed method to Pinsker's early work on cutoff rate improvement and to Imai-Hirakawa multi-level coding are also discussed.Comment: 5 pages, 7 figures, 2005 IEEE International Symposium on Information Theory, Adelaide, Sept. 4-9, 200

    Error Exponents for Variable-length Block Codes with Feedback and Cost Constraints

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    Variable-length block-coding schemes are investigated for discrete memoryless channels with ideal feedback under cost constraints. Upper and lower bounds are found for the minimum achievable probability of decoding error Pe,minP_{e,\min} as a function of constraints R, \AV, and τˉ\bar \tau on the transmission rate, average cost, and average block length respectively. For given RR and \AV, the lower and upper bounds to the exponent (lnPe,min)/τˉ-(\ln P_{e,\min})/\bar \tau are asymptotically equal as τˉ\bar \tau \to \infty. The resulting reliability function, limτˉ(lnPe,min)/τˉ\lim_{\bar \tau\to \infty} (-\ln P_{e,\min})/\bar \tau, as a function of RR and \AV, is concave in the pair (R, \AV) and generalizes the linear reliability function of Burnashev to include cost constraints. The results are generalized to a class of discrete-time memoryless channels with arbitrary alphabets, including additive Gaussian noise channels with amplitude and power constraints

    Properties and Construction of Polar Codes

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    Recently, Ar{\i}kan introduced the method of channel polarization on which one can construct efficient capacity-achieving codes, called polar codes, for any binary discrete memoryless channel. In the thesis, we show that decoding algorithm of polar codes, called successive cancellation decoding, can be regarded as belief propagation decoding, which has been used for decoding of low-density parity-check codes, on a tree graph. On the basis of the observation, we show an efficient construction method of polar codes using density evolution, which has been used for evaluation of the error probability of belief propagation decoding on a tree graph. We further show that channel polarization phenomenon and polar codes can be generalized to non-binary discrete memoryless channels. Asymptotic performances of non-binary polar codes, which use non-binary matrices called the Reed-Solomon matrices, are better than asymptotic performances of the best explicitly known binary polar code. We also find that the Reed-Solomon matrices are considered to be natural generalization of the original binary channel polarization introduced by Ar{\i}kan.Comment: Master thesis. The supervisor is Toshiyuki Tanaka. 24 pages, 3 figure

    The Reliability Function of Lossy Source-Channel Coding of Variable-Length Codes with Feedback

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    We consider transmission of discrete memoryless sources (DMSes) across discrete memoryless channels (DMCs) using variable-length lossy source-channel codes with feedback. The reliability function (optimum error exponent) is shown to be equal to max{0,B(1R(D)/C)},\max\{0, B(1-R(D)/C)\}, where R(D)R(D) is the rate-distortion function of the source, BB is the maximum relative entropy between output distributions of the DMC, and CC is the Shannon capacity of the channel. We show that, in this setting and in this asymptotic regime, separate source-channel coding is, in fact, optimal.Comment: Accepted to IEEE Transactions on Information Theory in Apr. 201

    How to Achieve the Capacity of Asymmetric Channels

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    We survey coding techniques that enable reliable transmission at rates that approach the capacity of an arbitrary discrete memoryless channel. In particular, we take the point of view of modern coding theory and discuss how recent advances in coding for symmetric channels help provide more efficient solutions for the asymmetric case. We consider, in more detail, three basic coding paradigms. The first one is Gallager's scheme that consists of concatenating a linear code with a non-linear mapping so that the input distribution can be appropriately shaped. We explicitly show that both polar codes and spatially coupled codes can be employed in this scenario. Furthermore, we derive a scaling law between the gap to capacity, the cardinality of the input and output alphabets, and the required size of the mapper. The second one is an integrated scheme in which the code is used both for source coding, in order to create codewords distributed according to the capacity-achieving input distribution, and for channel coding, in order to provide error protection. Such a technique has been recently introduced by Honda and Yamamoto in the context of polar codes, and we show how to apply it also to the design of sparse graph codes. The third paradigm is based on an idea of B\"ocherer and Mathar, and separates the two tasks of source coding and channel coding by a chaining construction that binds together several codewords. We present conditions for the source code and the channel code, and we describe how to combine any source code with any channel code that fulfill those conditions, in order to provide capacity-achieving schemes for asymmetric channels. In particular, we show that polar codes, spatially coupled codes, and homophonic codes are suitable as basic building blocks of the proposed coding strategy.Comment: 32 pages, 4 figures, presented in part at Allerton'14 and published in IEEE Trans. Inform. Theor
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