Concatenated codes (CH 15)

Transcription

1 Concatenated codes (CH 15) Simple (classical, single-level) concatenation Length of concatenated code is n 1 n 2 Dimension of concatenated code is k 1 k 2 If minimum distances of component codes are d 1 and d 2, respectively, then the concatenated code has minimum distance d 1 d 2 Decoding: Two-stage: Decode (hard-decision) inner code, then outer code Not optimum! Can decode up to approximately ¼ of d min Good for decoding mixture of random and burst errors 1

2 Multiple inner codes Not necessary that all inner codes are identical Justesen codes: n 2 different inner codes Can show that an asymptotically good class of codes can be constructed this way A class {C i } of codes of increasing lengths {n i } is asymptotically good if the normalized dimensions {k i /n i } and the normalized minimum distances {d i /n i } are both bounded away from zero as i approaches infinity A theoretical result; first known class of good codes 2

3 Generalization of the model m may range from 1 to large Permutes the order of outer code symbols 3

4 Example of interleaved serial concatenation 4

5 Example 5

6 Multilevel concatenated codes Multiple outer and inner codes A 1 A 2... A m {0} k i dimension of A i K 1 K 2 N N d(a i ) min. dist. of A i [A i / A i+1 ] coset code: Set of coset representatives; dimension k i - k i+1 q i = 2 k i - k i+1 B i code over GF(q i ) K m N K = K i (k i - k i+1 ) d(c) min{d(b i )d(a i )} 6

7 Multistage decoding Decode stage B 1 [A 1 / A 2 ] first,..., stage B m A m last 1. Decode r = r (1) into a codeword b 1 in B 1 Inner decoding: Find the closest word in [A 1 / A 2 ] Outer decoding: Use inner decoder s results Set i = 2 2. Let r (i) = r (i-1) f i-1 (b i-1 ) Decode r (i) into a codeword b i in B i Set i = i + 1 If (i m), repeat from 2 7

8 Soft decision multistage decoding a) Requires soft decision (and usually trellis based) decoding at each decoding stage b) Decode stage B 1 [A 1 / A 2 ] first,..., stage B m A m last 1. Decode r = r (1) into a codeword b 1 in B 1 Inner decoding: Find the closest word in [A 1 / A 2 ] Outer decoding: Use inner decoder s results Set i = 2 2. Compute modify received vector r (i) : r j,l (i) = r j,l (i-1) (1-2c j,l (i-1) ) Decode r (i) into a codeword b i in B i Set i = i + 1 If (i m), repeat from 2 8

9 Inner and outer decoding a) Inner decoder: Find the word (label) in each coset in A i / A i+1 with largest metric for each symbol of the outer code. This gives N metric tables Pass these N metric tables to the outer decoder b) Outer decoder: Find word with largest metric c) Not MLD because of possible error propagation d) Simpler than known MLD algorithms for such codes e) Can be improved by passing a list of L candidates from one decoding stage to the next; and by selecting as the final decoded word the one with the largest metric at the final stage 9

10 Code decomposition Expressing a code in terms of a multilevel concatenation µ-level decomposable code: Can be expressed as a µ-level concatenated code Some classical code constructions may be expressed in this way. This may facilitate decoding of such codes, and can provide soft decision (sub-optimum) decoding r-th order Reed-Muller code of length 2 m is denoted by RM(r,m) Idea: Decompose trellis into µ trellises, each trellis is significantly less complex than the original trellis 10

11 v 0 = (1...1) of length 2 m Properties of RM(r,m) v i = (0...0, 1...1, 0...0,..., 1...1) (groups of length 2 i-1 ) RM(r,m) is spanned by vectors v 0, v 1, v 2,..., v m, v 1 v 2, v v,..., v 1 3 m- v 1 m,... all products of degree up to r for r > 0 RM(0,m) is spanned by the vector v 0 and RM(-1,m) = {0} k(r,m) = 1 + q(m,1) q(m,r), where q(m,i) = binom(m,i) Minimum distance is 2 m-r RM(r,m) RM(r-1,m)... RM(0,m) RM(-1,m) RM(m-1,m) is the single parity check code RM(m-r-1,m) is the dual code of RM(r,m) 11

12 Example 15.2 RM(3,3) RM(2,3) RM(1,3) RM(0,3) {0} 12

13 RM codes and interleaving a) RM(r,m) = {RM(0,ν) q(r,m-ν), RM(1,ν) q(r-1,m-ν),..., RM(µ,ν) q(r- µ,m-ν) } {RM(r,m-ν), RM(r-1,m-ν),..., RM(r-µ,m-ν)} where µ = ν for r > ν and µ = r for r ν, 1 ν m-1 b) Example: RM(3,6) is a (64,42,8) code. Select µ = ν = 3 c) RM(3,6) = {RM(0,3) q(3,3), RM(1,3) q(2,3), RM(2,3) q(1,3), RM(3,3) q(0,3) } {RM(3,3), RM(2,3), RM(1,3), RM(0,3)} d) = {(8,1), (8,4) 3, (8,7) 3, (8,8)} {(8,8), (8,7), (8,4), (8,1)} = (8,1) [(8,8) / (8,7)] (8,4) 3 [(8,7) / (8,4)] 13 (8,7) 3 [(8,4) / (8,1)] (8,8) [(8,1) / {0}]

14 Example a) RM(4,7) is a (128,99,8) code b) Can show that the trellis has maximum state space dimension of 19 c) Can be decomposed into a 3-level concatenation d) Subtrellises of length 16, and at most 256 states in each trellis 14

15 Another example a) RM(3,7) is a (128,64,16) code b) Can show that the trellis has maximum state space dimension of 26 c) Can be decomposed into a 3-level concatenation d) Subtrellises of length 16, and at most 512 states in each trellis 15

16 Iterative multistage MLD (IMS-MLD) 16

17 Decoding algorithm (m = 2) IMS-MLD algorithm 1. Compute best estimate b (1),1 of first decoding stage and its metric M(b (1),1 ). If coset label sequence L(b (1),1 ) C, then the best codeword is found, so stop, otherwise proceed to 2 2. Perform second stage decoding and obtain L(b (2),1 ) and M(b (2),1 ). Store b (1),1, b (2),1, and M(b (2),1 ), and set i 0 = 1 3. For i > i 0, calculate b (1),i (the i-th best estimate). If M(b (2),i 0) M(b (1),i ), decoding is finished, and b (1),i 0 and b (2),i 0 give the most likely codeword. Otherwise, go to 4 4. If coset label L(b (1),i ) C, then the best codeword is found, so stop, otherwise proceed to 5 5. Generate b (2),i. Update i 0, b (1),i 0, b (2),i 0, and M(b (2),i 0). Go to 3 Can be generalized to m-level concatenated codes 17

18 Example: IMS-MLD RM(3,7) is a (128,64,16) code whose performance is displayed below 18

19 Example: IMS-MLD Decoder complexity comparisons for the (128,64,16) example code 19

20 Convolutional inner codes a) Can of course use convolutional codes as inner codes. This facilitates soft decision decoding b) Example in book 20

21 Concatenation of binary codes a) Also possible with binary outer codes (block or convolutional) b) More difficult to make statements about overall minimum distance c) Interleaver useful for increasing distance d) SISO algorithms are useful for decoding e) Iterative decoding is useful f) Serial concatenation / parallel concatenation 21