## On the construction of some capacity-approaching coding schemes (2000)

Citations: | 57 - 2 self |

### BibTeX

@TECHREPORT{Chung00onthe,

author = {Sae-Young Chung},

title = {On the construction of some capacity-approaching coding schemes},

institution = {},

year = {2000}

}

### Years of Citing Articles

### OpenURL

### Abstract

This thesis proposes two constructive methods of approaching the Shannon limit very closely. Interestingly, these two methods operate in opposite regions, one has a block length of one and the other has a block length approaching infinity. The first approach is based on novel memoryless joint source-channel coding schemes. We first show some examples of sources and channels where no coding is optimal for all values of the signal-to-noise ratio (SNR). When the source bandwidth is greater than the channel bandwidth, joint coding schemes based on space-filling curves and other families of curves are proposed. For uniform sources and modulo channels, our coding scheme based on space-filling curves operates within 1.1 dB of Shannon’s rate-distortion bound. For Gaussian sources and additive white Gaussian noise (AWGN) channels, we can achieve within 0.9 dB of the rate-distortion bound. The second scheme is based on low-density parity-check (LDPC) codes. We first demonstrate that we can translate threshold values of an LDPC code between channels accurately using a simple mapping. We develop some models for density evolution

### Citations

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Citation Context ... However, with such separation, infinite delay and infinite complexity are required to achieve the Shannon limit. Moreover, the separation theorem breaks down in some cases such as multiuser channels =-=[13]-=-. Finally, there are situations in which combined sourcechannel schemes are much simpler, such as the following well-known example: if an iid Gaussian source and an AWGN channel have the same bandwidt... |

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Citation Context ...nd column of the parity check matrix is small compared to the block length. LDPC codes did not get much attention for 29smany decades until recently when highly successful turbo codes were discovered =-=[4]-=-. LDPC codes were then rediscovered by Spielman et al. [62] and MacKay et al. [46]. For many channels and iterative decoders of interest, low-density parity-check (LDPC) codes exhibit a threshold phen... |

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Citation Context ... states (such as a trellis code) as a graph, thus making a full connection between graphical and trellis representation of codes. Recently, Wiberg’s work was further generalized using “factor graphs” =-=[39]-=- to cover non-coding applications such as Markov random fields, belief networks and fast Fourier transforms. Forney [19] constructed “normal graphs” by restricting symbols to be leaf edges connected t... |

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Citation Context ...le and check nodes and the resulting Fourier transform relationship between (p, q) and (p + q, p − q), where p = p(x = 1|y) and q = p(x = −1|y) [19]. From this, fui , we get the following “tanh rule” =-=[35, 3, 34, 54, 19]-=-: tanh u 2 = dc−1 � tanh vj , (5.2) 2 j=1 where vj, i = 1, . . . , dc − 1, are the incoming LLRs from dc − 1 neighbors of a check node, and u is the message sent to the remaining neighbor. The output ... |

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Citation Context ...t algorithm. What would have been the most powerful codes were forgotten, however, because computing power was not enough at that time to demonstrate the full potential of the code. 107sLater, Tanner =-=[63]-=- replaced parity-checks in LDPC codes with general constraints to construct a more general class of codes known as Tanner codes. By allowing hidden nodes in Tanner graphs, Wiberg [72, 73] made it poss... |

367 | Urbanke “The Capacity of Low-Density Parity-Check Codes Under Message-Passing Decoding
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Citation Context ...be the sum-product algorithm together with density evolution in more detail and explain how cycle problems can be avoided for LDPC codes in the next section. 5.2.2 Density Evolution Richardson et al. =-=[54, 53]-=- demonstrated that the average asymptotic behavior of a sum-product decoder for LDPC codes is numerically computable by using an algorithm called density evolution. They also showed that for many inte... |

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Citation Context ...minimization (+) and maximization (−) problems, respectively. This is an infinite-dimensional linear programming. For finite-dimensional linear programming, we can always use the weak duality theorem =-=[5]-=- to easily find solutions to the primal and dual problems since there is no gap between the two spaces of the objective functions. In general, this is not true when the dimension is infinite [2]. Howe... |

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Citation Context ...codes did not get much attention for 29smany decades until recently when highly successful turbo codes were discovered [4]. LDPC codes were then rediscovered by Spielman et al. [62] and MacKay et al. =-=[46]-=-. For many channels and iterative decoders of interest, low-density parity-check (LDPC) codes exhibit a threshold phenomenon [54]: as the block length tends to infinity, an arbitrarily small bit error... |

280 | Expander codes
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(Show Context)
Citation Context ...the block length. LDPC codes did not get much attention for 29smany decades until recently when highly successful turbo codes were discovered [4]. LDPC codes were then rediscovered by Spielman et al. =-=[62]-=- and MacKay et al. [46]. For many channels and iterative decoders of interest, low-density parity-check (LDPC) codes exhibit a threshold phenomenon [54]: as the block length tends to infinity, an arbi... |

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Citation Context ...l cube I n = [0, 1] n , where n > 1. 4 In other words, this curve passes through every point in the n-dimensional cube. 4 A bijective mapping from I to I n , where n > 1, is necessarily discontinuous =-=[58]-=-. A Jordan curve is a continuous injective mapping from I to E n , where E n is an n-dimensional Euclidean space. Jordan curves in E n with positive n-dimensional Lebesque measure are called Osgood cu... |

265 | Codes and Decoding on General Graphs
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Citation Context ...07sLater, Tanner [63] replaced parity-checks in LDPC codes with general constraints to construct a more general class of codes known as Tanner codes. By allowing hidden nodes in Tanner graphs, Wiberg =-=[72, 73]-=- made it possible to represent a code with states (such as a trellis code) as a graph, thus making a full connection between graphical and trellis representation of codes. Recently, Wiberg’s work was ... |

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Citation Context ...o generalize this result to a graph with more than one cycle. Another approach to understand the sum-product algorithm on a graph with many cycles is to assume that all variables are jointly Gaussian =-=[23, 71, 57]-=-. In this case, the analysis of the sum-product algorithm can be simplified since a Gaussian distribution is characterized by its mean and variance. A common result in [23, 71, 57] is that the sequenc... |

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Citation Context ...utationally intensive task for most channels other than BECs. In BECs, density evolution becomes one-dimensional, and it is possible to do more analysis and even to construct capacity-achieving codes =-=[45]-=-. For more interesting channels, including AWGN channels, however, density evolution is too complicated to be analyzed. In this thesis, we propose four novel approximation methods to estimate the thre... |

210 | On the design of lowdensity parity-check codes within 0.0045 dB of the Shannon limit
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Citation Context ...age densities, enabling the determination of thresholds. Using this result, they constructed LDPC codes that clearly beat the powerful turbo codes [4] on AWGN channels. Recently, this was improved in =-=[8]-=-, suggesting that LDPC codes might approach the channel capacity of the AWGN channel asymptotically. Calculating thresholds and optimizing degree distributions using density evolution 30sis a computat... |

177 | Correctness of local probability propagation in graphical models with loops
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Citation Context ...cess of turbo codes and low-density parity-check codes has ignited further research in this area. The behavior of the sum-product algorithm on graphs with a single cycle is relatively well understood =-=[1, 20, 70]-=-. In this case, the sum-product algorithm converges to a unique stationary point. If all variables are binary-valued, then the componentwise maximum likelihood estimates produced by running the sum-pr... |

163 | Analysis of sum-product decoding of low-density parity-check codes using a Gaussian approximation
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Citation Context ...al in Chapter 3 was presented in part at [11]. The material in Chapters 2 and 3 is in preparation for publication [10]. The material in Section 7.2 was presented in part at [12], and was submitted to =-=[9]-=-. The material in Sections 5.2.4 and 8.1 was submitted in part to [8]. The material in Chapters 4 and 6 and Sections 5.2.3, 7.1, 7.3, and 7.4 is in preparation for publication [7]. 34sChapter 2 Matche... |

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Principles of Communication Engineering
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Citation Context ... noise does not affect the system performance much until the noise reaches a critical level. A more detailed analysis of bandwidth expansion via n-dimensional curves was done by Wozencraft and Jacobs =-=[74]-=-. In modern digital transmission systems, however, the greater concern is the efficient use of bandwidth. In this thesis, we use space-filling curves for bandwidth reduction for several types of conti... |

129 |
A new multilevel coding method using error-correcting codes
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Citation Context ...Simulation results for the dl = 100 and dl = 200 codes of Tables 8.2 and 8.3, using a block length of 10 7 . 203s8.2 Bandwidth-Limited Case Wachsmann and Huber [68] used the idea of multilevel coding =-=[37]-=- to achieve a very good coding scheme by using turbo codes, as component codes as shown in Figure 8-3. They proved that 2 m -ary multilevel coding with multistage decoding can be equivalently decompos... |

122 |
Codes on graphs: normal realizations
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Citation Context ...of codes. Recently, Wiberg’s work was further generalized using “factor graphs” [39] to cover non-coding applications such as Markov random fields, belief networks and fast Fourier transforms. Forney =-=[19]-=- constructed “normal graphs” by restricting symbols to be leaf edges connected to a constraint node and states to be ordinary edges between two constraint nodes. This form of restriction is without lo... |

96 |
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Citation Context ...LF-L. The meander type (c) can be generated by an L-system with an axiom L and a set of rules L → LF-RFR-FL+F+LFLFL+FRFR- and R → RF+LFL+FR-F-RF RFR-FLFL+. Another famous example is the Hilbert curve =-=[36]-=-, which can be defined by an Lsystem with an axiom L and a set of rules L → +RF-LFL-FR+ and R → -LF+RFR+FLas illustrated in Figure 3-8 (a), where the Hilbert curve is the limit of the sequence. The Hi... |

96 |
Codes and iterative decoding on general graphs
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Citation Context ...07sLater, Tanner [63] replaced parity-checks in LDPC codes with general constraints to construct a more general class of codes known as Tanner codes. By allowing hidden nodes in Tanner graphs, Wiberg =-=[72, 73]-=- made it possible to represent a code with states (such as a trellis code) as a graph, thus making a full connection between graphical and trellis representation of codes. Recently, Wiberg’s work was ... |

89 |
Optimal quantizer design for noisy channels: an approach to combined source-channel coding
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Citation Context ...ource and channel codebooks, where two codebooks are either jointly or separately optimized. 28sSeveral good algorithms exist to numerically design a vector quantizer in the presence of channel noise =-=[18, 65]-=-. This was generalized to the case when the channel input is power constrained in [25, 24]. These algorithms can also be used when the source and channel bandwidths are different. However, due to the ... |

85 |
Sur un courbe, qui remplit toute un aire plane
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Citation Context ...+F--F+F+F+F--F+F--F+ . . . +F--F+F--F+F+F+F--F+F) Figure 3-6: Generation of the Koch island 68sCurves with this property are called Peano’s space-filling curves or simply spacefilling curves. 5 Peano =-=[49]-=- constructed the first such curve, which can be generated by an Lsystem with an axiom L and a set of rules L → LFRFL-F-RFLFR+F+LFRFL and R → RFLFR+F+LFRFL-F-RFLFR as shown in Figure 3-7 (a). The limit... |

76 | Analysis of low density codes and improved designs using irregular graphs
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(Show Context)
Citation Context ... generalized this idea to randomly constructed irregular LDPC codes, showed that irregular codes perform better than regular ones, and also showed that the threshold phenomenon occurs for these codes =-=[44]-=-. In [54], this observation was further generalized by Richardson and Urbanke to a large range of binary-input channels, including binary erasure, binary symmetric, Laplace, and AWGN channels, and to ... |

66 |
Theoretical limitations on the transmission of data from analog sources
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Citation Context ... an AWGN channel have the same bandwidth, then a simple uncoded system combined with a linear least-square error (LLSE) receiver can achieve Shannon’s rate-distortion bound for any given value of SNR =-=[30]-=-. In this thesis we will construct another interesting example of source and channel in which an uncoded scheme achieves optimality for all values of SNR. It is generally impossible to approach optima... |

63 | Analyzing the Turbo decoder using the Gaussian approximation
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- 2001
(Show Context)
Citation Context ...one-dimensional approximation for densities was first used by ten Brink [64]. However, he concentrated only on parallel concatenated codes and did not optimize codes. A similar idea was later used in =-=[16]-=- for turbo decoding including serial concatenated codes, which appeared at the same time as [12]. Since there is no simple formula for updating message densities for turbo decoding, Monte Carlo simula... |

62 | New sequences of linear time erasure codes approaching the channel capacity
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(Show Context)
Citation Context ...ge to zero if it is initially small enough, and if λ ′ (0)ρ ′ (1) > S(N) −1 , then the error probability is bounded away from zero [53, 55], which is a generalization of Shokrollahi’s result for BECs =-=[61]-=-. S(N) is the stability function 3 of the channel and is given by the following [53]: � S(N) = where p(u) is the initial density of the channel. R e −u/2 p(u) du, (5.5) Using the compact initial densi... |

58 |
Space-filling curves: Their generation and their application to bandwidth reduction
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(Show Context)
Citation Context ...rocessing [67, 51, 31, 40] or image 8 After scaling and rotation. 72sFigure 3-9: Generation of the hexagonal space-filling curve data compression [43]. A less known application is bandwidth reduction =-=[6]-=-, where a multi-dimensional analog source is mapped into a one-dimensional interval using a space-filling curve. Since space-filling curves are continuous, they are good candidates for compressing a u... |

47 |
Linear programming in infinite-dimensional spaces. Wiley-interscience series in discrete mathematics and optimization, Theory and Applications, A Wiley-Interscience Publication
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Citation Context ...eorem [5] to easily find solutions to the primal and dual problems since there is no gap between the two spaces of the objective functions. In general, this is not true when the dimension is infinite =-=[2]-=-. However, as we will see later, there is no “duality gap” in our problem and feasible solutions to the primal and dual problems of equal cost are indeed optimal solutions to both problems. The dual p... |

45 |
Probabilistic Information Theory
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(Show Context)
Citation Context ...submitted in part to [8]. The material in Chapters 4 and 6 and Sections 5.2.3, 7.1, 7.3, and 7.4 is in preparation for publication [7]. 34sChapter 2 Matched Sources and Channels Pilc [52] and Jelinek =-=[38]-=- defined that a source and channel pair is matched if the optimal distortion is achieved independent of the block length or, in other words, without any coding. In this chapter, we define three types ... |

43 | Iterative decoding on graphs with a single cycle
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(Show Context)
Citation Context ...chemes that show graceful degradation when the channel is mismatched. 58sWe require such curves be continuous due to the following reason. Suppose a discontinuous curve is used to map an interval I = =-=[0, 1]-=- to a square I 2 as in the following Cantor-type example by Shannon [60]: 1 x = .a1a2a3 . . . y = .b1b2b3 . . . ⎫ ⎬ ⎭ ⇔ z = .a1b1a2b2a3b3 . . . , where x, y, and z are represented in binary numbers an... |

40 |
An optimum symbol--by--symbol decoding rule for linear codes
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(Show Context)
Citation Context ...le and check nodes and the resulting Fourier transform relationship between (p, q) and (p + q, p − q), where p = p(x = 1|y) and q = p(x = −1|y) [19]. From this, fui , we get the following “tanh rule” =-=[35, 3, 34, 54, 19]-=-: tanh u 2 = dc−1 � tanh vj , (5.2) 2 j=1 where vj, i = 1, . . . , dc − 1, are the incoming LLRs from dc − 1 neighbors of a check node, and u is the message sent to the remaining neighbor. The output ... |

37 | On the Metric Properties of Discrete Space-Filling Curves
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(Show Context)
Citation Context ...noise. Surprisingly, there has been little understanding in this direction. To date, several attempts have been made to define measures of locality and in some cases to design space-filling curves in =-=[48, 67, 51, 32]-=-. Some authors [67, 32] conclude that the Hilbert curve is good compared to some other curves under their locality measures. However, their locality measures such as the average ratio of the distance ... |

35 |
Sphere-bound-achieving coset codes and multilevel coset codes
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(Show Context)
Citation Context ...annel lattice is different from the source lattice in the previous section. MAWN channels, or mod-Λ channels were used in [17, 22]. In this section, we summarize the mod-Λ channel capacity results in =-=[21, 22]-=-. The input, output, and noise are all n-dimensional vectors. Let us denote the channel input by X, the output by Y, and the noise by Z. Then, Y = (X + Z) mod Λ. The output Y has finite support R(Λ). ... |

35 |
Optimal linear coding for vector channels
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(Show Context)
Citation Context ...in F,G � E[|| ˆW − W|| 2 ] = tr(GF − I)Λw(F ′ G ′ − I) + tr GΛzG ′ � subject to E[||F W|| 2 ] = tr F ΛwF ′ ≤ P. (2.1) 39 §¨£ §©¥ ��£ � ��� � ��� �sThe matrix equation (2.1) was solved analytically in =-=[41, 42]-=-. 2 The optimal encoder is composed of three stages. It first uncorrelates the source into n independent components by diagonalizing Λw. It also diagonalizes the noise covariance matrix Λz at the last... |

35 |
Compression of two-dimensional data
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Citation Context ...ning multi-dimensional data for the purpose of image processing [67, 51, 31, 40] or image 8 After scaling and rotation. 72sFigure 3-9: Generation of the hexagonal space-filling curve data compression =-=[43]-=-. A less known application is bandwidth reduction [6], where a multi-dimensional analog source is mapped into a one-dimensional interval using a space-filling curve. Since space-filling curves are con... |

29 | Power and bandwidth efficient digital communication using turbo codes in multilevel codes
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(Show Context)
Citation Context ...4 0.45 0.5 −6 E /N [dB] b 0 Figure 8-2: Simulation results for the dl = 100 and dl = 200 codes of Tables 8.2 and 8.3, using a block length of 10 7 . 203s8.2 Bandwidth-Limited Case Wachsmann and Huber =-=[68]-=- used the idea of multilevel coding [37] to achieve a very good coding scheme by using turbo codes, as component codes as shown in Figure 8-3. They proved that 2 m -ary multilevel coding with multista... |

28 |
Replication decoding
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(Show Context)
Citation Context ...le and check nodes and the resulting Fourier transform relationship between (p, q) and (p + q, p − q), where p = p(x = 1|y) and q = p(x = −1|y) [19]. From this, fui , we get the following “tanh rule” =-=[35, 3, 34, 54, 19]-=-: tanh u 2 = dc−1 � tanh vj , (5.2) 2 j=1 where vj, i = 1, . . . , dc − 1, are the incoming LLRs from dc − 1 neighbors of a check node, and u is the message sent to the remaining neighbor. The output ... |

27 |
Trellis precoding: combined coding, precoding and shaping for intersymbol interference channels
- Eyuboglu, Forney
- 1992
(Show Context)
Citation Context ...f the modulo mapping is in a fundamental region R(Λ) of Λ. In general, this channel lattice is different from the source lattice in the previous section. MAWN channels, or mod-Λ channels were used in =-=[17, 22]-=-. In this section, we summarize the mod-Λ channel capacity results in [21, 22]. The input, output, and noise are all n-dimensional vectors. Let us denote the channel input by X, the output by Y, and t... |

26 |
Iterative decoding trajectories of parallel concatenated codes
- Brink
- 2000
(Show Context)
Citation Context ...faster, but it is often almost as good as the optimization methods based on density evolution given in [53]. For turbo codes, a one-dimensional approximation for densities was first used by ten Brink =-=[64]-=-. However, he concentrated only on parallel concatenated codes and did not optimize codes. A similar idea was later used in [16] for turbo decoding including serial concatenated codes, which appeared ... |

21 |
Design of Capacity-Approaching Low-Density Parity-Check Codes
- Richardson, Shokrollahi, et al.
- 2001
(Show Context)
Citation Context ...n limit at a bit error rate of 10 −6 when a block length of 10 7 is used. Our 26 Sinksasymptotic result is more than 10 times closer to the Shannon limit than the previous record by Richardson et al. =-=[53]-=-. Our simulation result is more than 3 times closer to the Shannon limit than the previous record in [53]. Interestingly, these two methods operate in opposite extreme regions. Our joint source and ch... |

21 |
The Behavior of Analog Communication Systems
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(Show Context)
Citation Context ...alog source vector to a continuous AWGN channel input can achieve the rate-distortion bound on the meansquare error for all values of SNR if the channel bandwidth is greater than the source bandwidth =-=[76]-=-. To approach the optimality for a range of SNR via separate source-channel coding, we need to construct a parametrized pair of source and channel codes and adjust the codes adaptively as the channel ... |

16 |
Thresholds for turbo codes
- Richardson, Urbanke
- 2000
(Show Context)
Citation Context ...ion. As in the Gaussian-capacity approximation for LDPC codes, this can be interpreted as approximating normalized channels by AWGN channels at each iteration, since density evolution for turbo codes =-=[56]-=- has essentially the same property as density evolution for LDPC codes. Since density evolution is computationally intensive for turbo codes, Monte-Carlo simulations were used in [64] to estimate the ... |