LDPC Communication Project

Transcription

1 Communication Project Implementation and Analysis of codes over BEC Bar-Ilan university, school of engineering Chen Koker and Maytal Toledano

2 Outline Definitions of Channel and Codes. Introduction to. Decoding base on Belief Propagation. Implementation of the decoder. Density Evolution. Results.

3 Project Description Implementation decoder of codes over BEC. Density Evolution algorithm. Asymptotic analytic results versus simulations results.

4 Outline Definitions of Channel and Codes. Introduction to. Decoding base on Belief Propagation. Implementation of the decoder. Density Evolution. Results.

5 Definitions of channel Digital channel:

6 Definitions of channel BEC = Binary Erasure Channel: "E" equal probability for "" and ".

7 Definitions of channel BEC = Binary Erasure Channel: we can assume that the conditional probabilities are

8 Definitions of channel C = Channel Capacity: The amount of information that can be reliably transmitted over a channel. In BEC the Capacity is: Pe = probability of erasure.

9 Definitions of codes Block code:

10 Definitions of codes R = Code Rate: k - number of bits of the effective data. n - number of bits of the effective data + redundancy bits.

11 Definitions of codes G = Generator Matrix: G s T 4 2 c r 2 4 Channel 4

12 Definitions of codes Notice H*G = (H = Parity Matrix) H G

13 Definitions of codes H = Parity Check Matrix: H 2 r 4 4

14 Outline Definitions of Channel and Codes. Introduction to Decoding base on Belief Propagation. Implementation of the decoder. Density Evolution. Results.

15 Introduction to Low density parity check code the parity check matrix H is binary and sparse.

16 Introduction to codes represented by: matrix or by bipartite graph

17 Introduction to Tanner Graph (bipartite graph)

18 Outline Definitions of Channel and Codes. Introduction to. Decoding base on Belief Propagation Implementation of the decoder. Density Evolution. Results.

19 Decoding base on Belief Propagation Decoding is iterative algorithms based on message passing. Messages are passed from check nodes to bit nodes. Messages are passed from bit nodes to check nodes.

20 BEC using Belief Propagation The Iterative Algorithm: Step - All variable nodes send their qij messages. Over BEC for or - for E -

21 BEC using Belief Propagation The Iterative Algorithm. Step 2 - The check nodes calculate their response messages rji,

22 BEC using Belief Propagation The Iterative Algorithm. step 2. For BEC it is Hard-Decision, for or - rji(b) =, for E - rji(b) = /2 In words, if all other bits in the equation are known the message is the correct value of the bit otherwise the message is E.

23 BEC using Belief Propagation The Iterative Algorithm. step 2. The calculate is done by the logic XOR operation. Equation A: X + X2 + X4 = Equation A: + E + = X2 Equation B: X2 + X3 = Equation B: + =

24 BEC using Belief Propagation The Iterative Algorithm. Step 3 - The variable nodes update their response messages to the check nodes.

25 BEC using Belief Propagation The Iterative Algorithm. Step 3 - Over BEC qij(b) =,, /2 If the bit is already revealed then it sends it real value, otherwise it sends E.

26 BEC using Belief Propagation The Iterative Algorithm. Before at step : At step 2 we found: Equation A: X + X2 + X4 = Equation A: + + E = X4 Equation B: X+ X2 + X3 +X4 = Equation B: E =

27 BEC using Belief Propagation The Iterative Algorithm. Now at step 3:

28 BEC using Belief Propagation The Iterative Algorithm. Step 4 - Go to step 2 (The check nodes calculate their response messages rji).

29 Algorithm Performance The algorithm for may be executed for a maximum number of rounds till:. It founds legal codeword not necessarily the right one. 2. It doesn t convergence to solution.

30 Algorithm Performance For BEC :. It fulfils the precise codeword. 2. It reach saturation -no solution. Stopping Set situation.

31 Algorithm Performance The Stooping Set situation. The bits can not be decoded.

32 Outline Definitions of Channel and Codes. Introduction to. Decoding base on Belief Propagation. Implementation of the decoder Density Evolution. Results.

33 Implementation of the Decoder The implementation includes 3 components:. Initialization. 2. Iterative Decoding. 3. Analysis performance.

34 Implementation of the Decoder Iterative Decoding efficient data structure Bit Vector. Value '','', or 'E. 2. Pointer to the equations it take part. 3. Number of equations. Equation Vector. Number of known & unknown bits. 2. Equation value of XOR function. Stack index of unknown bits.

35 Implementation of the Decoder Iterative Decoding The stack is scanned, Is update of unknown bit is possible? Yes: value updates in all the equations it involved. No: skip to the next unknown bit.

36 Implementation of the Decoder Iterative Decoding The process relies on - equation with only one unknown bit. The Stack halt condition. The stack is empty. 2. No bit was updated ( Stopping Set ).

37 Outline Definitions of Channel and Codes. Introduction to. Decoding base on Belief Propagation. Implementation of the decoder. Density Evolution Results.

38 Density Evolution DE = Density Evolution: An asymptotic analysis method for code performance under the Messagepassing decoding.

39 Density Evolution f t = probability of bit to be unknown after t iterations of massage passing algorithm. f t P e ( ( f t ( n ) ) ) ( m ) p e n m initial probability of error. number of bits in each equation. number of equations each bit involved.

40 Outline Definitions of Channel and Codes. Introduction to. Decoding base on Belief Propagation. Implementation of the decoder. Density Evolution. Results

41 Results "waterfall - success per 5 bit.

42 Success rate Results "waterfall - success per bit. Success rate per bit vs probability of error Probability of error

43 Success rate Results Number of iteration to reach a decision: 7 x 4 Number Of Iterations vs probability of error Probability of error

44 Success rate Results Density Evolution: DENSITY EVOLUTION-Success rate per bit vs probability of error Success rate Probability of error

45 Iterations Number Results Density Evolution: Number of iteration to reach a decision. 9 DENSITY EVOLUTION-Number of Iterations vs probability of error Number of Iterations Probability of error

46 Success rate Results Density Evolution versus our decoder performance DENSITY EVOLUTION-Success rate per bit vs probability of error Probability of error

47 Results We expect: P < /2 recover all erased bits. We got: sharp degradation in the success rate around probability P=.43. Why?.Our decoder is not ideal. 2.The code (H matrix) is not optimally.

48 Suggestions for continuing Extending the check of each equation for more than one unknown bit. but. Complexity growing. The decoder would not fit a real time system requirements.

49 Communication Project The End.