Time Varying Convolutional Codes for Punctured Turbocodes Yannick Saouter Claude Berrou 09/04/2019

Turbocodes Turbo code encoding : data frame X=(x1,…,xn) is permuted by  to give X’=(x(1),…,x(n)). The two frames are encoded by a RSC and gives redundancies Y1 and Y2. Turbo code decoding : Iterative MAP or SUB-MAP. Invented by Claude Berrou and Alain Glavieux [BerrouGlavieux93]

Criteria for good performance Low autocorrelations in permutations : Lee distance (S-Random) [DolinarDivsalar95], autocorrelation girth [Saouter10] … Large minimum distance for turbocodes as error-correcting codes. Theoretical result: turbocode with N information symbols, minimum distance D, RSC components with minimum distance d [Breiling04] Good turbocodes requires large d for components. DK d log(N)

Heller’s bound Heller’s bound : Convolutional codes with encoding rate R=1/n, constraint length  and minimum distance d [Heller68]: Example : R=1/2 dmin1k (n/2).2k/(2k-1).(+k-1) Heller’s bound vs. Non periodic convolutional codes Bound unreached for =5, 8, 10, 12, 14

Periodic convolutional codes eventually meet Heller’s bound The period 4 convolutional code ([37,27], [37,35], [33,25], [33,25]) has d=8 for =5 [Lee89]. In most cases, Heller’s bound can be reached

From punctured non systematic convolutional codes … Search for non systematic high rate punctured convolutional codes [Yasuda et al. 84]. Criterion PV : bit error probability with Viterbi decoding algorithm. Let R=n/(n+1) and C optimal 1/2 convolutional code. Every n periods, 2n bits are produced by C. Puncture at each period one of the two redundancies produced. Final rate = n/(n+1). Compute PV for 2n-1 possibilities and select the minimum value. Example : for C=(15,17) and n=3, best puncturing pattern (110,101).

… to punctured recursive systematic codes Previous codes naturally produce PRS when rendered recursive. Mother code C=(R1,R2), puncturing pattern (1b…b,1b…b) of length n, if b=0 (resp. 1) in the first pattern then b=1 (resp. 0) in the second. If pattern index >1, no choice possible. If pattern=(0,1) then PRS=R1/R2. If pattern=(1,0) then PRS=R2/R1. At index pattern 0, choose either R1/R2 or R2/R1. Example : C=(15,17), n=3, pattern=(110,101). PRS=(17/15,17/15,15/17) or (15/17,17/15,15/17).

New criteria for search Decoding of convolutional codes: nowadays Maximum a Posteriori or SUBMAP. MAP minimizes bit error probability. Upper bound: rate R=k/n, minimum distance d, multiplicity ad [Viterbi71] Tail-biting codes : encoding initial state = encoding final state. Maximize d. If equality, minimize ad. Codes are checked for circularity.

Mother codes : (13,15) for = 4 and (23,35) for  =5. Summary of the search Mother codes : (13,15) for = 4 and (23,35) for  =5. Common choices in turbocode domain. Best PRS for mother code (13,15) Best PRS for mother code (23,35)

Application to turbocodes Simulations : Time varying vs. Standard convolutional codes 8 iterations of Max-Log-MAP S-Random Permutation, tail biting 8-state binary, frame length = 720 bits 16-state binary, frame length = 560 bits

Decoding : done iteratively through state lattice Decoding complexity Decoding : done iteratively through state lattice Now, 2 state lattices instead of 1. Small difference : switch of two branch metrics. Can be done easily  small increase of VLSI complexity

Further axes of research Time Varying convolutional codes can improve turbocode performance S-Random permutations are not the best (QPP, ARP, DRP) Design specific permutations for TV turbocodes. Drawback of TV turbocodes : one encoder for each encoding rate. Obtain improvements with punctured full rate TV convolutional code.

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