Notes on a posteriori probability (APP) metrics for LDPC

Transcription

1 Notes on a posteriori probability (APP) metrics for LDPC IEEE 802.3an Task Force November, 2004 Raju Hormis, Xiaodong Wang Columbia University, NY raju@ee.columbia.edu

2 Outline Section of draft D1.1 discusses decoding of LDPC code groups. This presentation outlines a method for calculation of exact LDPC metrics, with examples. General definition of APP metrics for iterative decoders. A simple example: PAM-2. APP metric for 1-D lattices. APP metric for 2-D and N-D lattices. Conclusions. 2

3 APP calculation for iterative decoders Iterative decoders for LDPC and turbo-codes require a posteriori probabilities (APP s) as metrics, rather than direct channel observations. Usually expressed as a log ratio, and often referred to as LLR (log likelihood ratio). Let {b 0, b 1, b 2,..., b n } be the set of bits mapped to a symbol a k in constellation A. We will denote the APP for, say bit b 0, as L b0. Consider the AWGN channel model: channel observation, and υ is AWGN. y = a k + υ, where a k is the transmitted symbol, y is the We can write the definition of L b0 as [ ] p(b0 = 0/y) L b0 log e, and applying Bayes rule, we get p(b 0 = 1/y) [ ] p(y/b0 = 0)p(b 0 = 0) p(y) = log e, p(y) p(y/b 0 = 1)p(b 0 = 1) [ ] p(y/b0 = 0) = log e. p(y/b 0 = 1) 3

4 APP computation for PAM-2 in AWGN Consider the case of PAM-2 which takes a 0 = +1 when b 0 = 0, a 1 = 1 when b 0 = 1. We will make use of the conditional Gaussian pdf of a received symbol y in AWGN: 1 p(y/a) = 2πσ 2 e (y a)2 2σ 2, where σ 2 is the noise variance. We can now derive the APP value for b 0 as p(y/b 0 = 0) L b0 = log e, as derived earlier, p(y/b 0 = 1) e (y a 0) 2 /2σ 2 = log e e, (y a 1) 2 /2σ 2 = log e e (y 1)2 /2σ 2 e (y+1)2 /2σ 2, L b0 = 2 σ 2 y. The APP value is a simpe scaled version of channel observation y. Notice that scaling is inversely proportional to noise variance. 4

5 APP for 1-D lattice: PAM-8 constellation 1 To compute the APP s for PAM-8, notice that b 2 b 1 b 0 PAM level p(b 0 = 1/y) = k p(b 0 = 0/y) = j p(a k /y), where a k A, s.t. b 0 = 1 p(a j /y), where a j A, s.t. b 0 = 0 Using the above, we can now define the LLR for b 0 as [ ] p(b0 = 0/y) L b0 log e, p(b 0 = 1/y) [ a = log k A, b 0 =0 p(a ] k/y) e a j A, b 0 =1 p(a. Now, applying Bayes rule we get, j/y) L b0 = log e a k {,+7,+1, 1, 7, } e (y a k ) 2 2σ 2 j)2 a j {,+5,+3, 3, 5, } e (y a 2σ 2. 1 S. Rao, R. Hormis, and E. Krouk, The 4-D PAM-8 proposal for 10G-Base-T, pdf, Nov

6 APP for 1-D lattice: PAM-8 constellation 500 L b0 400 L b APP channel observation y PAM-8: APP for bits b 0 and b 1, noise σ =

7 APP calculation for 2-D and N-D lattices In this case, the received symbol y and the constellation symbols, a k A, can be viewed as vectors. Let us denote the components of vector a k as [a k0 a k1 ]. Similarly, y = [y 0 y 1 ]. We can write the APP for bit b 0 as [ a L b0 = log k A,b 0 =0 p(a ] k/y) e a j A,b 0 =1 p(a, j/y) [ a = log k A, b 0 =0 p(y/a ] k) e a j A, b 0 =1 p(y/a, assuming that p(a n ) constant n. j) Notice that p(y/a k ) = p(y 0 /a k0 ). p(y 1 /a k1 ), since y 0, y 1, are conditionally independent given a k0, a k1, respectively. We can then simplify the APP for b 0 as [ a L b0 = log k A, b 0 =0 p(y ] 0/a k0 ) p(y 1 /a k1 ) e a j A, b 0 =1 p(y 0/a j0 ) p(y 1 /a j1 ) which involves products of 1-D conditional Gaussian pdf s., 7

8 APP for 2D-lattice: 2D-PAM12 Key-eye proposal Points - Cross Constellation uncoded co-set /27/ Weizhuang Xin, Modified 128 point cross constellation mapping, Aug

9 APP for 2D-lattice: 2D-PAM12 Key-eye proposal 2-D APP function of b 0, noise σ =

10 APP for 2D-lattice: 2D-PAM12 Key-eye proposal 200 cross section of LLR(b 0 ), y 1 = APP(b 0 ) horiz. 1 D PAM 12 symbol y cross section of LLR(b 0 ), y 1 = APP(b 0 ) horiz. 1 D PAM 12 symbol y 0 Cross-sections of 2-D APP function of b 0, at y 0 = +11 and y 0 = 0, noise σ =

11 APP for 2D-lattices: 2D-PAM12 Teranetics proposal 3 One Dimensional Map The Mapping of the Uncoded Bits in Two Dimensions 0.5 One Dimensional Map for 12 PAM Coset Labels L L Point Labels L L C C C C R R R R D. Dabiri and J. Tellado, Modifications to LDPC proposal offering lower symbol rate and lower latency, pdf, Mar

12 APP for 2D-lattices: 2D-PAM12 Teranetics proposal 2-D APP function for bit b 0, noise σ =

13 APP for 2D-lattices: 2D-PAM12 Teranetics proposal 600 cross section of LLR(b 0 ), y 1 = APP(b 0 ) horiz. 1 D PAM 12 symbol y cross section of LLR(b 0 ), y 1 = APP(b 0 ) horiz. 1 D PAM 12 symbol y 0 Cross-section of APP function of b 0, y 0 = +11 and y 0 = 0, noise σ =

14 Conclusions From first principles, we derived the generalized APP metric for multi-dimensional constellations. Non-rectangular constellations lead to APP s that are not always separable into 1-D functions... the 1-D functions would be approximations. Recommendations: Use of 2-D (N-D) APP metrics, depending on the dimensionality of the constellation. The APP metrics tend to be approximately piece-wise linear (planar) perhaps easy to map using LUT s or digital gates. The piece-wise linear APP metric surface should be documented. 14