# The normalized Min-Sum algorithm and other related algorithms

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## Standard Belief Propagation LDPC decoding

In this subsection, the soft iterative decoding of binary LDPC codes is pre-sented. The iterative process, first introduced by Gallager [25] and rediscov-ered by MacKay [41] is more known as the Belief Propagation (BP) algo-rithm. This algorithm propagates probability messages to the nodes through the edges. The probability gives a soft information about the state (0 or 1) of a VN. In the Log-BP algorithm, the probabilities are in the log domain and are called Log Likelihood Ratio (LLR). The LLRs are defined by the following equation: LLR = log P (v = 0) (1.2) P (v = 1).
where P (v = x) is the probability that bit v is egual to x. The order in which the nodes are updated is called the scheduling. The schedule proposed by Gallager is known as the flooding schedule [25]. Flood-ing schedule consists in four steps. First step is the initialization. Second step is the update of all the check nodes. The third step is the update of all the variable nodes. The fourth step consists in going back to step two until a codeword is found or a maximum number of iteration is reached. At the end of the iterative process, a hard decision is made on the VN to output the codeword. The update of a node means that a node reads the incoming messages and then updates the outgoing messages. The initialization, the VN update and the CN update are described hereafter. The test to find the codeword or stopping criteria is described in Section 4.4.3.
Let x denote the transmitted BPSK symbol corresponding to a codeword bit, and let y denote the noisy received symbol, then y = z + y where z is a Gaussian random variable with zero mean. Let us assume that x = +1 when the transmitted bit is 0 and x=-1 when the transmitted bit is 1. Let LLRin = log p(x=+1|y) denote the a-priori LLR for the transmitted bit. The p(x=−1|y) sign of LLRin gives the hard decision on the transmitted bit, whereas the magnitude of LLRin gives an indication on the reliability of the decision: the greater the magnitude is, the higher is the reliability. On a Gaussian channel LLRin is given by: LLRin = 2y/σ2.

### The normalized Min-Sum algorithm and other related algorithms

A lot of researches tend to reduce computational complexity by simplifying the Check Node Processor (CNP). The Min-Sum algorithm is the simplest decoding method which approximates the sum-product algorithm by ignoring other terms except the most minimum incoming message. Therefore the complex computation of function f (x) can be eliminated. The CN Min-Sum processing of Min-Sum algorithm can be expressed as: | Mnew | ≈ min | M ′ →c| (1.8) c→v v′∈vc/v v The Min-Sum algorithm can dramatically reduce the computation com-plexity. However, the resulting approximated magnitude is always overesti-mated. This inaccurate approximation causes significant degradation on the decoding performance. In the Normalized Min-Sum (NMS) algorithm [11], the output of the CN processor is multiplied by a normalization factor α. This compensates the overestimation of Min-Sum approximation.

#### The LDPC code in the DVB-S2, -T2 and -C2 standards

The DVB-S2, -T2, -C2 standards features variable coding and modulation to optimize bandwidth utilization based on the priority of the input data, e.g., SDTV could be delivered using a more robust setting than the correspond-ing HDTV service. These DVB standadards also features adaptive coding and modulation to allow flexibly adapting transmission parameters to the reception conditions of terminals, e.g., switching to a lower code rate during fading.
The DVB-S2, -T2, -C2 standards [20, 22, 21] are characterized by a wide range of code rates (from 1/4 up to 9/10) as shown in table 1.2. Further-more, FEC frame may have either 64800 bits (normal) or 16200 bits (short). Each code rate and frame length corresponds to an LDPC matrix: this is 21 matrices for the DVB-S2 standard, 13 matrices for the DVB-T2 standard and 11 matrices for the DVB-C2 standard. The matrices construction is identical for the three standards. The advantage is that the same LDPC decoder can be used for the 3 standards. Due to the fact that the decoder is identical for the 3 standards, hereafter DVB-X2 refers to DVB-2, -T2, -C2 standards.

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Remerciements
R´esum´e
Abstract
Contents
Notations
Introduction
1 Background
1.1 Basic concepts
1.1.1 Digital communication
1.1.2 Channel decoders
1.1.3 Linear block codes
1.1.4 LDPC codes
1.1.5 Standard Belief Propagation LDPC decoding
1.2 Sub-optimal algorithms
1.2.1 The normalized Min-Sum algorithm and other related algorithms
1.2.2 Serial implementation of the NMS algorithm
1.3 LDPC Layered decoder
1.3.1 The turbo message passing schedule
1.3.2 Structured matrices
1.3.3 Soft Output (SO) centric decoder
1.3.4 Architecture overview
1.4 The DVB-S2, -T2 and -C2 standards
1.4.1 Digital Terrestrial Television
1.4.2 DVB group
1.4.3 The LDPC code in the DVB-S2, -T2 and -C2 standards
1.4.4 State-of-the-art on DVB-S2 LDPC implementation
1.5 Testing the performance of a decoder
1.5.1 Software and hardware simulation
1.5.2 Test with all-zero codewords
1.5.3 Test of a communication model
1.5.4 Channel emulator
1.5.5 Interpreting results
1.5.6 Standard requirements
2 Memory update conflicts
2.1 Conflicts due to the matrix structure
2.1.1 State-of-the-art
2.2 Conflict resolution by group splitting
2.2.1 Construction of the sub-matrices
2.2.2 DDSM in DVB-X2 and simulation results
2.3 Parity check matrix equivalent
2.3.1 Principle of the split-extend process
2.3.2 Simulation results
2.3.3 Performance improvement
2.4 Conflict Resolution by Layer duplication
2.4.1 Conflict resolution by Write Disabling the memory
2.4.2 Scheduling of the layers
2.4.3 Write disabling in the Mc→v memory
2.4.4 Write disabling the Mc→v memory when a Min-Sum algorithm is used
2.4.5 Simulations and memory size results
2.4.6 The write-disable architecture
2.4.7 Synthesis results on FPGA
2.4.8 Conclusion
2.5 Memory update conflicts due to pipeline
2.5.1 Non pipelined CNP
2.5.2 Pipelined CNP
2.5.3 The problem of memory update conflicts
2.5.4 Conflict reduction by group splitting
2.5.5 Conflict resolution by scheduling
2.5.6 Conclusion
2.6 Combining layers duplication and scheduling
2.7 Conclusion
3 Memory optimization
3.1 Saturation of the stored values
3.1.1 Channel LLR saturation
3.1.2 SO saturation
3.1.3 Saturation of the extrinsic messages
3.1.4 Combining the saturation processes
3.1.5 Saturation optimization conclusion
3.2 Optimizing the size of the extrinsic memory
3.2.1 Extrinsic memory size requirements
3.2.2 Optimization principle
3.2.3 Results of optimization
3.2.4 Case of the sum-product algorithm
3.2.5 Mc→v memory optimization conclusion
3.3 Finite precision architecture of the layered decoder
3.4 Results of memory optimization
3.4.1 Monte-Carlo Simulations results
3.4.2 Synthesis results on FPGA
3.4.3 Memory capacity comparison
3.5 A single port RAM architecture
3.5.1 Single port ram, dual port ram, pseudo dual port ram and dual port RAM
3.5.2 Memories in ASICS and FPGA
3.5.3 Implementation of dual port RAM with single Port
3.5.4 FIFO memory with single port memory modules
3.5.5 Single port memories banks for the SO memories
3.6 Layer scheduling for single port RAM
3.6.1 An example with two memory banks
3.6.2 Generalization for DVB-X2 matrices
3.6.3 Genetic algorithm to solve scheduling problem
3.7 Conclusion
4 Multi Stream LDPC decoder
4.1 Introduction
4.2 The parallelism option
4.2.1 Area saving compared with x decoders and conclusion
4.3 Share resources in a dual stream decoder
4.3.1 Sharing principle
4.4 Use of a buffer
4.4.1 FIFO buffer principle
4.4.2 Preemptive buffer control
4.4.3 Variable iterative decoder
4.4.4 FIFO Buffer size
4.4.6 Implementation issue
4.5 Conclusion
5 Conclusion
5.1 Produced work
5.2 Perspectives
A DVB-S2 matrices construction
A.1 Standard matrices construction
A.2 Matrix permutations for layered structure
B Hardware Discrete Channel Emulator
B.1 Introduction
B.2 Linear Feedback Shift Register
B.3 The alias method algorithm
B.4 The HDCE architecture
B.5 Resulting distribution
C R´esum´e ´etendu
C.1 Introduction
C.2 Pr´e-requis
C.3 Les conflits de mise `a jour de la m´emoire
C.3.1 Conflits dus `a la structure
C.3.2 Conflits dus au pipelining
C.4 Optimisation de la taille m´emoire
C.4.1 Optimisation de la taille des mots
C.4.2 Optimisation des bancs m´emoire des extrins`eques
C.4.3 Utilisation de RAM simple port
C.5 Un d´ecodeur de flux multiple
C.5.1 Parall´elisme
C.5.2 Partage des ressources
C.5.3 Addition d’un buffer `a un d´ecodeur it´eratif variable .
C.6 Conclusion
C.6.1 Applications
C.6.2 Perspectives
List of figues
List of tables
Bibliography

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