Encoding the Messages � Table look-up is possible for small k � For large k, table look-up may become extremely difficult = ⇒ ≈ × k k 30 � e.g., 100 2 1 . 26 10 Use of Generator Matrix T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 32 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
Generator Matrix G (size k x n ) n -tuple n -tuple n -tuple a basis set of k linearly independent n -tuples that spans the subspace = m m m L [ ] message m k 1 2 U = mG (size 1 x n ) codeword T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 33 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
34 ] IWCSN' 2007, Guilin, China 6 u 5 u 4 u 3 u (6, 3) Code Example 2 u 1 u U NIVERSITY NIVERSITY [ = OLYTECHNIC U P OLYTECHNIC K ONG ONG ONG K H ONG HE H T HE P T
Parity-Check Matrix H (size ( n - k ) x n ) For each generator matrix G , there exists an ( n - k ) x n matrix H such that rows of G are orthogonal to rows of H . GH = T G ↔ 0 H k x ( n - k ) all-zeros matrix = = T T UH mGH 0 H can be used to test whether a received vector is a valid codeword. T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 35 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
Parity-Check Matrix H (size ( n - k ) x n ) rows are ⎡ ⎤ 1 0 0 1 0 1 orthogonal ⎢ ⎥ = 0 1 0 1 1 0 H ⎢ ⎥ ⎢ ⎥ 0 0 1 0 1 1 ⎣ ⎦ GH = T 0 T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 36 P P OLYTECHNIC OLYTECHNIC U U NIVERSITY NIVERSITY
Parity-Check Matrix H ⎡ ⎤ 1 0 0 ⎢ ⎥ 0 1 0 ⎢ ⎥ ⎢ ⎥ 0 0 1 = ⇒ = T u u u u u u ⎢ ⎥ [ ] UH 0 0 1 2 3 4 5 6 1 1 0 ⎢ ⎥ ⎢ ⎥ 0 1 1 ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 1 0 1 + + = u u u 0 1 4 6 ⇒ + + = u u u 0 2 4 5 + + = u u u 0 3 5 6 T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 37 P P OLYTECHNIC OLYTECHNIC U U NIVERSITY NIVERSITY
⎡ ⎤ 1 0 0 1 0 1 ⎢ ⎥ Bipartite Graph = 0 1 0 1 1 0 H ⎢ ⎥ ⎢ ⎥ ⎣ 0 0 1 0 1 1 ⎦ u 1 = + + = c u u u u ( 0 ) 1 1 4 6 2 u variable 3 = + + = c u u u ( 0 ) nodes 2 2 4 5 u 4 = + + = c u u u ( 0 ) u 3 3 5 6 5 check nodes u 6 T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 38 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
Decoding Additive White received Codeword U Gaussian vector r ’ 0 � +1 volt Noise Channel 1 � − 1 volt T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 39 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
Eight codewords in a 6-tuple space decoded codeword after error correction Hard r i > 0 volt � 0 after making hard Decoding r i < 0 volt � 1 decision on each bit received vector r ’ (AWGN channel) T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 40 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
Soft Decoding U = P ( 101110 | ' ) r maximum a posteriori (MAP) decision rule: Select codeword U that has U = P ( 110100 | ' ) r P the largest ( | ' ) U r r ’ U = U = P P ( 011101 | ' ) ( 101001 | ' ) r r a posteriori probability (APP) T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 41 P P OLYTECHNIC OLYTECHNIC U U NIVERSITY NIVERSITY
Performance of Some Well- known Block Codes (Coherent BPSK over an AWGN channel) t = maximum number of guaranteed correctable errors per codeword T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 42 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
Performance of BCH Codes (Coherent BPSK over an AWGN channel) T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 43 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
Part 3: Low- -Density Density- -Parity Parity- - Part 3: Low Check (LDPC) Codes Check (LDPC) Codes T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 44 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
Parity-Check Matrix H (size ( n - k ) x n ) ⎡ ⎤ 1 0 0 1 0 1 ⎢ ⎥ = 0 1 0 1 1 0 H ⎢ ⎥ ⎢ ⎥ ⎣ 0 0 1 0 1 1 ⎦ GH = T 0 T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 45 P P OLYTECHNIC OLYTECHNIC U U NIVERSITY NIVERSITY
Low-Density-Parity-Check Codes � proposed by Gallager (1960) � parity-check matrix H � sparse (most elements are zeros) � fraction of 1’s ~ O ( n ) � elements of H determine the connections between variable nodes degree of check and check nodes degree of variable node c 3 = 3 node u 6 = 2 T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 46 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
Low-Density-Parity-Check Codes � sparse (low-density) parity-check matrix H implies that all variable nodes and check nodes have very few connections G ↔ H T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 47 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
Error rates achieved by different coding schemes under the binary AWGN channel. Codeword length = 10 6 . Rate =0.5. T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 48 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
Types of LDPC Codes � Regular LDPC � all nodes of the same type (variable node or check node) have the same degree A (3, 6)-regular LDPC code of length 10 and rate one-half. variable check node node degree degree T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 49 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
Types of LDPC Codes � Irregular LDPC: the degrees of each set of nodes are chosen according to some distribution T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 50 P P OLYTECHNIC OLYTECHNIC U U NIVERSITY NIVERSITY
Degree Distribution of Nodes � Degree distribution of variable nodes d maximum variable node degree v ∑ − λ = λ k x k x 1 ( ) = k 2 fraction of edges connected to the variable nodes with degree k � Degree distribution of check nodes d maximum check node degree c ∑ − ρ = ρ k x k x 1 ( ) = k 2 fraction of edges connected to the check nodes with degree k T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 51 P P OLYTECHNIC OLYTECHNIC U U NIVERSITY NIVERSITY
Code Rate 1 ∫ ρ x x ( ) d = − R 0 1 1 ∫ λ x x ( ) d 0 T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 52 P P OLYTECHNIC OLYTECHNIC U U NIVERSITY NIVERSITY
Question Given specific d d v ∑ c ∑ − λ = λ − k ρ = ρ k x k x 1 x k x 1 ( ) and . ( ) = = k k 2 2 How would the LDPC code perform? T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 53 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
Answer Dependent on the actual design (connections), the channel type, e.g. AWGN, binary symmetric channel (BSC), binary erasure channel (BEC) and the decoding algorithm. T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 54 P P OLYTECHNIC OLYTECHNIC U U NIVERSITY NIVERSITY
More Question Any idea on the optimal performance of practical LDPC decoders? T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 55 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
Belief Propagation (BP) Decoding Part 4: Belief Propagation (BP) Decoding Part 4: Algorithm Algorithm T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 56 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
Belief Propagation (BP) Decoding Algorithm � A kind of message-passing decoding algorithm � Applicable to both regular and irregular LDPC codes � Produces very good error performance T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 57 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
Belief Propagation (BP) Decoding Algorithm variable � Define Log Likelihood nodes Ratio (LLR): ⎡ ⎤ = P ( bit 0 | info) log P ⎢ ⎥ = ⎣ ⎦ ( bit 1 | info) check nodes T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 58 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
BP Decoding 0 r LLR ( ) 1 r Algorithm 1 � Compute initial Log Likelihood Ratio (LLR) for each variable node iteration number based on the received signal vector r (real number elements) ⎡ ⎤ = P r ( bit 0 | ) r i log ⎢ ⎥ i = P r ⎣ ( bit 1 | ) ⎦ i T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 59 P P OLYTECHNIC OLYTECHNIC U U NIVERSITY NIVERSITY
BP Decoding 0 r LLR ( ) 1 Algorithm � Set iteration number k = 1 � Pass the LLR messages from variable nodes to the connected check nodes T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 60 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
BP Decoding Algorithm � Check nodes received the LLR messages from the connected variable nodes T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 61 P P OLYTECHNIC OLYTECHNIC U U NIVERSITY NIVERSITY
BP Decoding Algorithm � Each check node computes a message for each of its connected variable nodes, based on the messages from all other variables nodes T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 62 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
BP Decoding Algorithm � Each check node computes a message for each of its connected variable nodes, based on the messages from all other variables nodes T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 63 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
BP Decoding Algorithm � Each check node computes a message for each of its connected variable nodes, based on the messages from all other variables nodes T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 64 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
BP Decoding Algorithm � Each check node computes a message for each of its connected variable nodes, based on the messages from all other variables nodes T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 65 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
BP Decoding Algorithm � Each check node computes a message for each of its connected variable nodes, based on the messages from all other variables nodes T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 66 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
BP Decoding Algorithm � Each check node computes a message for each of its connected variable nodes, based on the messages from all other variables nodes T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 67 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
r → r BP Decoding LLR ( ) LLR ( ) 0 1 1 1 Algorithm � Each variable node update its LLR based on the messages passed from the check nodes and the initial LLR � Based on the updated LLR, estimate the codeword T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 68 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
r → r BP Decoding LLR ( ) LLR ( ) 0 1 1 1 Algorithm � Estimate the codeword as [ ] = c c c ˆ ˆ ˆ L ˆ c n 1 2 > ⎧ ⎫ r 0 if LLR ( ) 0 = k where 1 c ⎨ ⎬ ˆ i < r ⎩ 1 if LLR ( ) 0 ⎭ k 1 = � If T ˆ ˆ , is the c c H 0 decoded codeword. T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 69 P P OLYTECHNIC OLYTECHNIC U U NIVERSITY NIVERSITY
0 r BP Decoding LLR ( ) 1 Algorithm ≠ � If T ˆ , increment the c H 0 iteration number k . � Each variable node computes a message for each of its connected check nodes, based on its initial LLR and the messages from all other connected check nodes T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 70 P P OLYTECHNIC OLYTECHNIC U U NIVERSITY NIVERSITY
0 r BP Decoding LLR ( ) 1 Algorithm � Each variable node computes a message for each of its connected check nodes, based on its initial LLR and the messages from all other connected check nodes T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 71 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
0 r BP Decoding LLR ( ) 1 Algorithm � Each variable node computes a message for each of its connected check nodes, based on its initial LLR and the messages from all other connected check nodes T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 72 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
BP Decoding Algorithm � Check nodes received the LLR messages from the connected variable nodes T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 73 P P OLYTECHNIC OLYTECHNIC U U NIVERSITY NIVERSITY
BP Decoding Algorithm � Each check node computes a message for each of its connected variable nodes, based on the messages from all other variables nodes � Same iterative process repeated …. until convergence to a valid codeword or maximum number of iterations exceeded T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 74 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
Capacity (Threshold) of LDPC codes Given the degree distributions d d c ∑ v ∑ − − λ = λ ρ = ρ k k x k x 1 x k x 1 ( ) ( ) = = k k 2 2 and the channel type (AWGN, BSC or BEC) and the use of BP decoding algorithm. Richard and Urbanke (2001) proposed an effective algorithm – density evolution – to determine the capacity (threshold) of LDPC codes. T. J. Richardson and R. Urbanke, “The capacity of low-density parity-check codes under message- passing decoding,” IEEE Trans. Inform. Theory , vol. 47, pp. 599–618, Feb. 2001. T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 75 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
Density Evolution Algorithm iterations d v ∑ − λ = λ k x k x 1 ( ) = k 2 Density d c ∑ threshold − ρ = ρ k x k x 1 ( ) Evolution σ * value = k 2 Algorithm channel type a higher threshold value indicates a higher achievable performance of the code achievable T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 76 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
Good Degree Distribution Pairs (Rate = 0.5) T. J. Richardson et al., “Design of capacity-approaching irregular low-density parity-check codes,” IEEE Trans. Inform. Theory , vol. 47, pp. 619–637, Feb.2001. T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 77 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
Good Degree Distribution Pairs (Rate = 0.5) T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 78 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
Density Evolution Algorithm � the threshold value , for example indicates the σ * maximum noise power that can be tolerated for error-free communication in AWGN channels � the threshold value can be achieved σ achieved if * � the message-passing process does not contain any cycles � number of iterations tends to infinity � codeword length is infinite � codeword length is infinite T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 79 P P OLYTECHNIC OLYTECHNIC U U NIVERSITY NIVERSITY
Density Evolution Algorithm � Problems: � optimizing the codes based on DE algorithm is not a d d simple task v c ∑ ∑ − − λ = λ ρ = ρ k k x k x 1 x k x 1 ( ) ( ) = = k k 2 2 � codeword length cannot be infinite � optimizing the threshold value may vary to maximize the threshold value give a more complex code � number of connections T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 80 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
Part 5: Review of Complex Networks Part 5: Review of Complex Networks T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 81 P P OLYTECHNIC OLYTECHNIC U U NIVERSITY NIVERSITY
Basic properties � Path length : the distance between two nodes, which is defined as the number of edges along the shortest path connecting them T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 82 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
Basic properties � Betweenness centrality : the fraction of shortest paths going through a given node � Assortative mixing : preference of high-degree nodes attach to other high-degree nodes � Disassortative mixing : preference of high- degree nodes attach to low-degree nodes T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 83 P P OLYTECHNIC OLYTECHNIC U U NIVERSITY NIVERSITY
Examples of Complex Networks � Random Networks � Given a network with N nodes. Each pair of nodes are connected with a probability of p . Poisson distribution = p 0.1 ER μ − μ k e = p k ( ) k ! μ = T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 84 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
Examples of Complex Networks � Regular Coupled Networks 1 � high clustering � large average path 0.8 length 0.6 P(k) 0.4 � Fully-connected 0.2 Networks 0 2 2.5 3 3.5 4 4.5 <k> T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 85 P P OLYTECHNIC OLYTECHNIC U U NIVERSITY NIVERSITY
Examples of Complex Networks = p 0 WS � Small-World Networks � Each edge of a regular coupled network is re-wired with a probability of p � high clustering = p 0.2 WS � small average path length T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 86 P P OLYTECHNIC OLYTECHNIC U U NIVERSITY NIVERSITY
Examples of Complex Networks � Scale-Free Networks T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 87 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
Examples of Complex Networks � Scale-Free (SF) Networks − γ − γ n n f x x Pr( ) ~ ( ) ~ i i T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 88 P P OLYTECHNIC OLYTECHNIC U U NIVERSITY NIVERSITY
Characteristics of Typical Complex Networks Average Distance Degree Clustering Distribution Coefficient Ο N Random Networks Short Poisson Low (log( )) Ο Regular Coupled Long N Uniform High ( ) Network Ο N Fast to disseminate Small World Short (log( )) - High information Networks Scale Free Very Short Power-Law - Ο N (log(log( )))* Networks *The exponent parameter should be valued between 2 and 3. See reference “ Scale-Free Networks Are Ultrasmall ”, PRL, vol. 90, no. 5 T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 89 P P OLYTECHNIC OLYTECHNIC U U NIVERSITY NIVERSITY
Part 6: Scale- -free Networks to free Networks to Part 6: Scale LDPC Codes LDPC Codes T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 90 P P OLYTECHNIC OLYTECHNIC U U NIVERSITY NIVERSITY
Scale-free Networks meet LDPC Codes Can the “very short distance” property of scale-free network helps passing/spreading messages quickly in the decoding of LDPC codes? If so, how? T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 91 P P OLYTECHNIC OLYTECHNIC U U NIVERSITY NIVERSITY
Scale-free Networks meet LDPC Codes ? ? T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 92 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
From Bipartite Graph to Unipartite Graph T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 93 P P OLYTECHNIC OLYTECHNIC U U NIVERSITY NIVERSITY
From Bipartite Graph to Unipartite Graph power-law degree distribution Power-law degree distribution ! T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 94 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
Typical node degree distribution of the corresponding unipartite graph. Codeword length = 10000 and maximum variable node degree = 20. X. Zheng, F.C.M. Lau and C.K. Tse, " Study of LDPC Codes Built on Scale-Free Networks," Proceedings, NOLTA'06, Bologna, Italy, September 2006, pp. 563-566. T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 95 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
Building LDPC Codes From SF Networks � Assume that the variable nodes have the power-law k γ − P k degree distribution and the check nodes ( ) ~ λ obey the Poisson-law degree distribution − μ μ l e = P l ( ) ρ l ! � Use the (Density Evolution) DE to select the γ μ . optimized parameters and T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 96 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
[1] σ Threshold value and average variable node degrees < k > of LDPC codes built * from scale-free networks and the optimized ones reported in [1] for an AWGN channel. Rate equals 0.5. [1] T. J. Richardson et al., “Design of capacity-approaching irregular low-density parity-check codes,” IEEE Trans. Inform. Theory , vol. 47, pp. 619–637, Feb.2001. T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 97 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
Typical node degree distribution of the corresponding unipartite graph. Codeword length = 1000 and maximum variable node degree = 15. X. Zheng, F.C.M. Lau and C.K. Tse, “Error Performance of Short-Block-Length LDPC Code Built on Scale-Free Networks,” Proceedings, The Third Shanghai International Symposium on Nonlinear Sciences and Applications , Shanghai, China, June 2007, pp. 55-57. T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 98 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
Building LDPC Codes From SF Networks σ � Threshold values lower compared with those * reported in the literature � Average variable node degrees < k > lower compared with those reported in the literature Which one is better in practice ? T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 99 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
Threshold Value � The threshold value can be achieved σ * achieved if � the message-passing process does not contain any cycles and � number of iterations tends to infinity and � codeword length is infinite � codeword length is infinite T HE HE H H ONG ONG K K ONG T ONG IWCSN' 2007, Guilin, China 100 P OLYTECHNIC P OLYTECHNIC U U NIVERSITY NIVERSITY
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