Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes
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1 Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes ISIT 2004 Mostafa El-Khamy, Robert J. McEliece, and Jonathan Harel fmostafa, rjm, California Institute of Technology Pasadena, CA 91125, USA ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 1
2 Introduction Goal: What is the ultimate performance of ASD of RS Codes? ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 2
3 Introduction Goal: What is the ultimate performance of ASD of RS Codes? Find better Multiplicity Assignment algorithms ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 2
4 Introduction Goal: What is the ultimate performance of ASD of RS Codes? Find better Multiplicity Assignment algorithms Existing Multiplicity Assignment Algorithms for ASD: ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 2
5 Introduction Goal: What is the ultimate performance of ASD of RS Codes? Find better Multiplicity Assignment algorithms Existing Multiplicity Assignment Algorithms for ASD: Kötter and Vardy Algorithm [KV03] Gaussian Approximation [PV03] ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 2
6 Outline Introduction Algebraic Soft Decoding Multiplicity Assignment Problem Soft Multiplicity Matrices The Chernoff Bound The Lagrangian Iterative Algorithm Numerical results & Conclusions ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 3
7 Introduction n-dimensional vector over F : u = (u 1 ; : : : ; u n ) ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 4
8 Introduction n-dimensional vector over F : u = (u 1 ; : : : ; u n ) q n arrays: W = (w i (fi)), where i = 1; : : : ; n and fi 2 F. ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 4
9 jw j = 1 2 nx Introduction n-dimensional vector over F : u = (u 1 ; : : : ; u n ) q n arrays: W = (w i (fi)), where i = 1; : : : ; n and fi 2 F. Cost: X w i (fi) (w i (fi) + 1) : i=1 fi2f ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 4
10 jw j = 1 2 nx hu; W i = nx Introduction n-dimensional vector over F : u = (u 1 ; : : : ; u n ) q n arrays: W = (w i (fi)), where i = 1; : : : ; n and fi 2 F. Cost: X w i (fi) (w i (fi) + 1) : Score: i=1 fi2f w i (u i ): i=1 ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 4
11 jw j = 1 2 nx hu; W i = nx Introduction n-dimensional vector over F : u = (u 1 ; : : : ; u n ) q n arrays: W = (w i (fi)), where i = 1; : : : ; n and fi 2 F. Cost: X w i (fi) (w i (fi) + 1) : Score: i=1 fi2f w i (u i ): i=1 c 2 C will be an (n; k; d) Reed-Solomon code over F. ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 4
12 jw j = 1 2 nx hu; W i = nx Introduction n-dimensional vector over F : u = (u 1 ; : : : ; u n ) q n arrays: W = (w i (fi)), where i = 1; : : : ; n and fi 2 F. Cost: X w i (fi) (w i (fi) + 1) : Score: i=1 fi2f w i (u i ): i=1 c 2 C will be an (n; k; d) Reed-Solomon code over F. Channel Output: APP matrix: Π = (ß i (fi)) = (Prfc i = fijr i g) ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 4
13 Algebraic Soft Decoding Π ) Multiplicity Assignment Algorithm; A ) M c! Π A! M m i (fi) is a non-negative integer. ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 5
14 Algebraic Soft Decoding Π ) Multiplicity Assignment Algorithm; A ) M c! Π A! M m i (fi) is a non-negative integer. c is on the GS list if hc; Mi > D k 1 (jmj) c ` M v» v D + p v3=2 2vfl + (fl) p 2fl : 8 2 ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 5
15 Algebraic Soft Decoding Pr fe A g = P Π2AP P Pr fe AjΠg Pr fπg ; E A = fc 0 Mg ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 6
16 Algebraic Soft Decoding fe A g = P Π2AP P Pr fe AjΠg Pr fπg ; Pr Theorem: E A = fc 0 Mg 1 Pr fe A jπg = [c 0 M ] P(c): c2c P P(c) X c2c ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 6
17 Algebraic Soft Decoding fe A g = P Π2AP P Pr fe AjΠg Pr fπg ; Pr Theorem: E A = fc 0 Mg 1 Pr fe A jπg = [c 0 M ] P(c): c2c P P(c) X c2c Find A: Hard Problem! ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 6
18 Multiplicity Assignment Problem KV Simplificaton: x! Π A! M ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 7
19 n P(x) = Independence i=1 Assumption: i(x Q ) ß i Multiplicity Assignment Problem KV Simplificaton: x! Π A! M X P(Π; M ) = x2f n [x 0 M ] P(x) ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 7
20 n P(x) = Independence i=1 Assumption: i(x Q ) ß i (Π; fl) = min P P(Π; M ) jmj»fl (Π; fl) = arg M min M P(Π; M ) jmj»fl Multiplicity Assignment Problem KV Simplificaton: x! Π A! M X ASD Decoder: P(Π; M ) = x2f n [x 0 M ] P(x) P (Π; 1) lim = P (Π; fl) fl!1 ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 7
21 n P(x) = Independence i=1 Assumption: i(x Q ) ß i (Π; fl) = min P P(Π; M ) jmj»fl (Π; fl) = arg M min M P(Π; M ) jmj»fl Multiplicity Assignment Problem KV Simplificaton: x! Π A! M X ASD Decoder: Hard Problem! P(Π; M ) = x2f n [x 0 M ] P(x) P (Π; 1) lim = P (Π; fl) fl!1 ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 7
22 Soft Multiplicity Matrices Relax Integer Constraint: Q = (q i (fi)) soft matrix. ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 8
23 fl!1 P Λ (Π; fl): Soft Multiplicity Matrices Relax Integer Constraint: Q = (q i (fi)) soft matrix. Relaxed problem: X P(Π; Q) = x2f n [x 0 Q] P(x) P Λ (Π; fl) min = P(Π; Q) jqj»fl Q Λ (Π; fl) arg min = P(Π; Q) jqj»fl P Λ (Π; 1) lim = ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 8
24 Soft Multiplicity Matrices Theorem : P Λ (Π; 1) = P (Π; 1) ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 9
25 Soft Multiplicity Matrices Theorem : P Λ (Π; 1) = P (Π; 1) Q Λ =? Still a Hard Problem? ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 9
26 Soft Multiplicity Matrices Theorem : P Λ (Π; 1) = P (Π; 1) Q Λ =? Still a Hard Problem? Minimize the Chernoff Bound on the Error Probability! ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 9
27 n Q) = E SQ e sp n i=1 S i o Π; Φ(s; Φ S Q» ffi Ψ» min Pr s 0 n o Φ( s; Π; Q) : sffi e The Chernoff Bound-Finite Cost S i are independent r.v.; hx; Qi = S Q = S S n : X i (s; ß i ; q i ) = E Si ne ss i o ffi ß i (fi)e sq i(fi) : = fi2f = ny ffi i (s; ß i ; q i ): Chernoff Bound: i=1 ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 10
28 2 = 2fl + nq L ; D0 = D v (fl) + n 2 : 4 The Chernoff Bound-Finite Cost Transformation [PV03] X i (fi) = q i (fi) + 1=2; ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 11
29 2 = 2fl + nq L ; D0 = D v (fl) + n 2 : 4 n o Φ( s; Π; X ) sd0 e n o Φ( s; Π; X ) sd0 e The Chernoff Bound-Finite Cost Transformation [PV03] Transformed Problem: X i (fi) = q i (fi) + 1=2; P Λ (Π; fl)» : min 2 =L 2 min s 0 kxk The (sub)optimum matrix : X Λ = arg X min 2 =L 2 min s 0 kxk ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 11
30 Chernoff Bound-Infinite Cost Theorem :(v = k 1) X P (Π; 1) = min 2 =1 krk x2f n hx; Ri» p v Λ P(x) ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 12
31 Chernoff Bound-Infinite Cost Theorem :(v = k 1) X P (Π; 1) = Chernoff Bound: min 2 =1 krk x2f n hx; Ri» p v Λ P(x) nφ( s; Π; R)e sp v o P (Π; 1)» (Sub)Optimum Matrix: min min =1 s 0 2 krk nφ( s; Π; R)e sp v o R χ (Π) = arg R min min =1 s 0 2 krk ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 12
32 s 0 sd 0 + nx The Lagrangian Constrained Optimization Problem:! ψ subject to i=1 min ln ffi i ( s; ß i ; X i ) kx k 2 = L 2 = 2fl nq: ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 13
33 L(s; X ; ) = sd 0 + s 0 nx sd 0 + nx ln ffi i ( s; ß i ; X i ) + 2 kx k 2 L 2 : The Lagrangian Constrained Optimization Problem:! ψ subject to i=1 min ln ffi i ( s; ß i ; X i ) kx k 2 = L 2 = 2fl nq: The Lagrangian: i=1 ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 13
34 The fi = Λ = 0 ) kx k 2 = L ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 14
35 ψpfi2f X i(fi)ß i (fi)e sx i(fi)!fi fifififi nx The fi = Λ = 0 ) kx k 2 = L D 0 i=1 ffi i ( s; ß i ; X i ) s=s Λ = 0 ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 14
36 ψpfi2f X i(fi)ß i (fi)e sx i(fi)!fi fifififi nx ffi i ( s; ß i ; X i ) ψpfi2f X i(fi)ß i (fi)e sx i(fi) nx ffi i ( s; ß i ; X i ) fix=x Λ fi The fi = Λ = 0 ) kx k 2 = L D 0 s=s Λ = 0 i=1! i (fi) X 2 L ß i(fi)e sx i(fi) i ( s; ß i ; X i ) ffi fi fi fi i=1 ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 14
37 ψpfi2f X i(fi)ß i (fi)e sx i(fi)!fi fifififi nx ffi i ( s; ß i ; X i ) ψpfi2f X i(fi)ß i (fi)e sx i(fi) nx ffi i ( s; ß i ; X i ) fix=x Λ fi The fi = Λ = 0 ) kx k 2 = L D 0 s=s Λ = 0 i=1! i (fi) X 2 L ß i(fi)e sx i(fi) i ( s; ß i ; X i ) ffi fi fi fi i=1 0 D 2 X i(fi) L i (fi)e sλ X i (fi) ß Pfi2F i(fi)e sλ X i (fi) ß fi fi fi fi = 0 ;s=s Λ Λ X=X fi ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 14
38 Convexity Define L s (s) = L Λ (s; X )j X=const ; L X (X ) = L Λ (s; X )j s=s Λ : ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 15
39 Convexity Define L s (s) = L Λ (s; X )j X=const ; L X (X ) = L Λ (s; X )j s=s Λ : L s (s) is convex in s ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 15
40 L X (X ) is convex in X Convexity Define L s (s) = L Λ (s; X )j X=const ; L X (X ) = L Λ (s; X )j s=s Λ : L s (s) is convex in s ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 15
41 r X (L Λ (s j ; X )) = fl fl sj s j 1 fl fi fi fi fi s=s j Iterative Algorithm X = Initialize L2 0 Π; s o = 0:1 Λ D0 L and 2 j = 0: D Do o j := j + 1 I. Solve for s j, = 0 II. Solve for X j, s (L Λ (s; X j 1 ) (s; X j 1 ) ( Λ j F)fi X j (s fifififi ; ; i = 1; :::; n; fi 2 (fi) X=X j = 0 While s j 1 fl flfl 1» ffl: ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 16
42 m i (fi) = Round fmax f0; X Λ i Iterative Algorithm For finite cost multiplicity matrix M = (m i (fi)): (fi) 0:5gg : ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 17
43 Iterative Algorithm For finite cost multiplicity matrix M = (m i (fi)): i (fi) = Round fmax f0; X Λ i (fi) 0:5gg : m Solve could be replace by a Newton type algorithm. ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 17
44 Iterative Algorithm For finite cost multiplicity matrix M = (m i (fi)): i (fi) = Round fmax f0; X Λ i (fi) 0:5gg : m Solve could be replace by a Newton type algorithm. To reduce computational complexity: Set X i (fi) = 0 if m i (fi) < threshold ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 17
45 Numerical Results & Conclusions 10 0 ASD of (15,11) RS code BPSK modulated over AWGN Channel Codeword Error Rate HD BM KV,γ= Gauss,γ= Chernoff,γ= KV,γ=10 4 Chernoff,γ= E b /N o (db) ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 18
46 Numerical Results & Conclusions 10 0 ASD of (15,11) RS code over AWGN, Infinite Cost HD BM KV Gauss Chernoff ML sim ML TSB 10 3 Codeword Error Rate E /N b o ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 19
47 Numerical Results & Conclusions 10 0 ASD of (15,11) RS Code 16 PSK modulated over AWGN HD BM KV, C= Gauss, C= Chernoff, C= KV, C=1e4 Chernoff, C=1e4 Codeword Error Rate E b /N o ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 20
48 Numerical Results & Conclusions SD of (31,25) RS code BPSK modulated over AWGN Channel HD BM KV Gauss Chernoff 10 2 Codeword Error Rate SNR (db) ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 21
49 Numerical Results & Conclusions SD of (31,25) RS code BPSK modulated over AWGN Channel HD BM KV Gauss Chernoff ML TSB Codeword Error Rate SNR (db) ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 22
50 Future Work Finding less complex Algorithms with high efficiency. Minimize the true Error Probability directly. Precondition Π to have information from other symbols and the channel. Iterative Algebraic Decoding. ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 23
51 Thank You! ISIT 2004-Performance Enhancements for Algebraic Soft Decision Decoding of Reed-Solomon Codes p. 24
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