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## Iterative turbo decoder analysis based on density evolution (2001)

Venue: | IEEE J. Select. Areas Commun |

Citations: | 99 - 3 self |

### Citations

1775 | Near Shannon limit errorcorrecting coding and decoding: Turbo codes
- Berrou, Glavieux, et al.
- 1993
(Show Context)
Citation Context ...l-to-noise ratio of the channel. Concatenated coding schemes consist of the combination of two or more simple constituent encoders and interleavers. The parallel concatenation known as a “turbo code” =-=[1]-=- has been shown to yield remarkable coding gains close to the theoretical limits, yet admitting a relatively simple iterative decoding technique. Also, serial concatenation of interleaved codes [2] ma... |

1592 |
Optimal decoding of linear codes for minimizing symbol error rate
- Bahl, Cocke, et al.
- 1974
(Show Context)
Citation Context ...full-degree feedback and feedforward polynomials. If we start with uniform state distributions at the beginning and end of a block for the calculation of α and β in the forward and backward algorithm =-=[6,7]-=-, the distribution of α and β remains uniform. Due to the symmetry of trellis edges for input bits 0 and 1, the output extrinsic information will be zero. If the feedback polynomial is not full-degree... |

638 |
A Recursive Approach to Low Complexity Codes
- Tanner
- 1981
(Show Context)
Citation Context ...); and the rate-1/2 recursive convolutional code, octal (1,1/3). Although convolutional codes are conventionally constructed on a trellis, these 2-state codes have simple Tanner graph representations =-=[13]-=- without loops. This graph representation is shown in Fig. 26. Here the nodes corresponding to the parity bits and the information bits play the role of the variable nodes for LDPC codes, and they are... |

610 | Iterative decoding of binary block and convolutional codes
- Hagenauer, Offer, et al.
- 1996
(Show Context)
Citation Context ...code with feedforward polynomial coefficients {pi} and feedback polynomial coefficients {qi}. Assume without loss of generality that pj = 1 for some j. Then, using the algebra introduced by Hagenauer =-=[17]-=-, we have the following equation relating the extrinsic information {λk} associated with the input bits {xk} and the channel information {λc k } associated with the coded bits {yk}: tanh � � λk−j = 2 ... |

371 | Serial concatenation of interleaved codes: Performance analysis, design and iterative decoding.”
- Benedetto, Divsalar, et al.
- 1998
(Show Context)
Citation Context ...e” [1] has been shown to yield remarkable coding gains close to the theoretical limits, yet admitting a relatively simple iterative decoding technique. Also, serial concatenation of interleaved codes =-=[2]-=- may offer superior performance to parallel concatenation at very low bit-error rates. In both coding schemes, the core of the iterative decoding structure is a soft-input soft-output (SISO) a posteri... |

358 | Codes and decoding on general graphs
- Wiberg
- 1996
(Show Context)
Citation Context ...our article, we apply the density evolution method to analyze the (asymptotic) performance of iterative decoders for turbo codes and turbo-like serially concatenated codes. Wiberg in his dissertation =-=[3]-=- showed that the extrinsic information in iterative decoding can be approximated by a Gaussian density function. This approximation was used by Chung et al. [9] to obtain a threshold on minimum bit si... |

306 | the design of low-density parity-check codes within 0.0045 dB of the Shannon limit
- Chung, Richardson, et al.
- 2001
(Show Context)
Citation Context ...approaches, and also to analysis of the turbo decoder algorithm by Agrawal and Vardy [25] and of turbo decoder thresholds by Richardson and Urbanke [26], Vialle and Boutros [27], and Chung and Forney =-=[28]-=-. Our main contribution is to apply these analyses to gain new insights into designing new turbo-like code structures and to explain the workings of old ones. In this article, we first evaluate iterat... |

244 | Analysis of sumproduct decoding of low-density parity-check codes using a Gaussian approximation,”
- Chung, Richardson, et al.
- 2001
(Show Context)
Citation Context ...nated codes. Wiberg in his dissertation [3] showed that the extrinsic information in iterative decoding can be approximated by a Gaussian density function. This approximation was used by Chung et al. =-=[9]-=- to obtain a threshold on minimum bit signal-to-noise ratio, Eb/N0, for LDPC codes. El Gamal, in his dissertation [4] and with Hammons [5], considered the SISO module in turbo decoders as a signal-to-... |

168 |
Convergence of iterative decoding,”
- Brink
- 2001
(Show Context)
Citation Context ... a similar SNR transformer analysis for the soft output Viterbi algorithm (SOVA), and Alexander et al. [21] proposed a similar noise variance trace method to analyze iterative multiuser detection. In =-=[14]-=-, ten Brink developed a method for analyzing the convergence of the decoder based on the evolution of mutual information. Our methods in this article are similar to all of these approaches, and also t... |

91 | Analyzing the turbo decoder using the Gaussian approximation”,
- Gamal, Hammons
- 2001
(Show Context)
Citation Context ...density function. This approximation was used by Chung et al. [9] to obtain a threshold on minimum bit signal-to-noise ratio, Eb/N0, for LDPC codes. El Gamal, in his dissertation [4] and with Hammons =-=[5]-=-, considered the SISO module in turbo decoders as a signal-to-noise ratio (SNR) transformer, and also suggested a method for analyzing the overall turbo decoder. Prior to El Gamal, Hagenauer and Hoehe... |

87 | Analysis of low density codes and improved designs using irregular graphs
- Luby, Mitzenmacher, et al.
- 1998
(Show Context)
Citation Context ...block size corresponding to the girth of the graph representing the overall code, then the effects of cycles on performance can be ignored). 6sThese claims can be justified by a concentration theorem =-=[18,11]-=- similar to that used in [8–11] to prove iterative-decoding decoding thresholds for LDPC codes. The concentration theorem says that the average bit-error probability is concentrated around the ensembl... |

75 | Iterative Detection in CodeDivision Multiple-Access with Error Control Coding
- Alexander, Grant, et al.
- 1998
(Show Context)
Citation Context ...od for analyzing the overall turbo decoder. Prior to El Gamal, Hagenauer and Hoeher [20] proposed a similar SNR transformer analysis for the soft output Viterbi algorithm (SOVA), and Alexander et al. =-=[21]-=- proposed a similar noise variance trace method to analyze iterative multiuser detection. In [14], ten Brink developed a method for analyzing the convergence of the decoder based on the evolution of m... |

71 |
A soft-input soft-output maximum a posteriori module to decode parallel and serially concatenated codes
- Benedetto, Montorsi, et al.
- 1996
(Show Context)
Citation Context ...ce to parallel concatenation at very low bit-error rates. In both coding schemes, the core of the iterative decoding structure is a soft-input soft-output (SISO) a posteriori probability (APP) module =-=[7]-=-. The analysis of iterative decoders for concatenated codes with short blocks is an unsolved problem. However, for very large block sizes, such analysis is possible under certain assumptions. An asymp... |

43 |
The capacity of low density parity check codes under message-passing decoding.
- Richardson, Urbanke
- 2001
(Show Context)
Citation Context ...rried out by the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. 1sevolution proposed by Richardson and Urbanke =-=[8]-=-; see also [10,11]. Their analysis tracks the probability density function of the extrinsic information messages as this density evolves from iteration to iteration. They used this method to compute i... |

35 |
Rate one-half code for approaching the Shannon limit by 0.1dB
- Brink
- 2000
(Show Context)
Citation Context ... codes using doping as a mixing agent. (denoted by “period = 80” in the figure). This technique of sending an occasional uncoded bit amongst many coded bits has been called “code doping” by ten Brink =-=[23]-=-. Without the code doping, the decoding of the inner code would fail to get started, because the rate-1, octal 7/3 code has a nontrivial feedforward component, and so its SNRout versus SNRin character... |

29 | Design of Provably Good LowDensity Parity Check Codes," submitted to - Richardson, Shokrollahi, et al. |

27 |
The turbo decoding algorithm and its phase trajectories. [Online]. Available: http://www.ece.uiuc.edu/ dakshi
- Agrawal, Vardy
(Show Context)
Citation Context ... the decoder based on the evolution of mutual information. Our methods in this article are similar to all of these approaches, and also to analysis of the turbo decoder algorithm by Agrawal and Vardy =-=[25]-=- and of turbo decoder thresholds by Richardson and Urbanke [26], Vialle and Boutros [27], and Chung and Forney [28]. Our main contribution is to apply these analyses to gain new insights into designin... |

23 |
Low complexity turbo-like codes
- Divsalar, Dolinar, et al.
- 2000
(Show Context)
Citation Context ...s the mixture-of-Gaussians model, except that it is slightly more optimistic about the speed of decoder convergence on the far side of the iterative decoding tunnel. Figure 20 shows another variation =-=[22]-=- of a mixing scheme for a rate-1/3 code, wherein the output of the repetition-3 outer code is either sent uncoded to the channel or sent through a rate-1 inner recursive convolutional code, octal 7/3.... |

23 |
Thresholds for turbo codes
- Richardson, Urbanke
- 2000
(Show Context)
Citation Context ...methods in this article are similar to all of these approaches, and also to analysis of the turbo decoder algorithm by Agrawal and Vardy [25] and of turbo decoder thresholds by Richardson and Urbanke =-=[26]-=-, Vialle and Boutros [27], and Chung and Forney [28]. Our main contribution is to apply these analyses to gain new insights into designing new turbo-like code structures and to explain the workings of... |

7 |
iterative decoding of double serially concatenated codes with interleavers
- Benedetto, Divsalar, et al.
- 1998
(Show Context)
Citation Context ...ncatenations of 2-State Codes We can also use the 2-state convolutional code, either rate-1 or rate-1/2, as an outer code or more generally as the middle code of a serial concatenation of three codes =-=[15,16]-=-. Again we use a Tanner graph representation for the 2-state convolutional code. In the general case, there are three types of input and output extrinsic messages that must be analyzed, corresponding ... |

6 |
Analysis and Design of Iterative Decoding Systems
- Richardson, Urbanke
(Show Context)
Citation Context ...he Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. 1sevolution proposed by Richardson and Urbanke [8]; see also =-=[10,11]-=-. Their analysis tracks the probability density function of the extrinsic information messages as this density evolves from iteration to iteration. They used this method to compute iterative decoding ... |

5 |
Coding Theorems for ‘Turbo-Like
- Divsalar, Jin, et al.
- 1998
(Show Context)
Citation Context ... 6 an example of how the convergence properties of the overall decoder can be determined from its noise figure. The code in this case is a simple serially concatenated repeat-and-accumulate (RA) code =-=[19]-=-, tested at three values of Eb/N0 just above the iterative decoding threshold. The noise figure in each case rises with iterations from an initial value of 0 to a maximum just below 1, indicating that... |

4 |
Gamal, On the theory and application of space-time and graph based codes
- El
- 1999
(Show Context)
Citation Context ...imated by a Gaussian density function. This approximation was used by Chung et al. [9] to obtain a threshold on minimum bit signal-to-noise ratio, Eb/N0, for LDPC codes. El Gamal, in his dissertation =-=[4]-=- and with Hammons [5], considered the SISO module in turbo decoders as a signal-to-noise ratio (SNR) transformer, and also suggested a method for analyzing the overall turbo decoder. Prior to El Gamal... |

4 |
Performance limits of concatenated codes with iterative decoding
- Vialle, Boutros
- 1997
(Show Context)
Citation Context ...re similar to all of these approaches, and also to analysis of the turbo decoder algorithm by Agrawal and Vardy [25] and of turbo decoder thresholds by Richardson and Urbanke [26], Vialle and Boutros =-=[27]-=-, and Chung and Forney [28]. Our main contribution is to apply these analyses to gain new insights into designing new turbo-like code structures and to explain the workings of old ones. In this articl... |

3 |
On the Serial Concatenation of Rate-One Codes Through Uniform Random Interleavers
- Pflster, Siegel
(Show Context)
Citation Context ...ncatenations of 2-State Codes We can also use the 2-state convolutional code, either rate-1 or rate-1/2, as an outer code or more generally as the middle code of a serial concatenation of three codes =-=[15,16]-=-. Again we use a Tanner graph representation for the 2-state convolutional code. In the general case, there are three types of input and output extrinsic messages that must be analyzed, corresponding ... |

1 |
Iterative Decoding of Nonsystematic Turbo
- Collins, Takeshita, et al.
- 2000
(Show Context)
Citation Context ...nel bits than input bits, i.e., the code rate is less than 1. Examples of rate-1/2 recursive nonsystematic codes that produce a nonzero SNR at the first iteration are those proposed by Collins et al. =-=[24]-=-. Consider a rate-1/2 recursive nonsystematic code ([p1(D)/q(D)], [p2(D)/q(D)]), where p1(D) and p2(D) do not have any common factors with q(D). If p1(D)+p2(D)=D i for some i ≥ 0, then the code produc... |