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Design and Performance of Rate Compatible-SCCC Alexandre Graell i Amat †‡, Guido Montorsi ‡, Francesca Vatta* † Universitat Pompeu Fabra. Barcelona, Spain ‡ Politecnico di Torino. Torino, Italy * Università di Trieste. Trieste, Italy NEWCOM, Department 1-SPW1 meeting ENSEA, April 28th, 2005

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2 Politecnico di Torino – Universitat Pompeu Fabra Motivations ■Standard SCCC for high-rates: Outer Encoder Inner Encoder

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3 Politecnico di Torino – Universitat Pompeu Fabra Motivations ■Standard SCCC for high-rates: High-rate Encoder Inner Encoder ■ If the interleaver size is fixed different information block sizes for different rates ■ For very high rates, the increasing value of the outer code rate causes an interleaver gain penalty error floor

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4 Politecnico di Torino – Universitat Pompeu Fabra Motivations ■Standard Rate-compatible SCCC: ■ Rate-compatibility restricts puncturing to the inner encoder ■ In general, the rate of the inner encoder is restricted to be R i 1 t he overall code rate is at most R o Outer Encoder Inner Encoder PiPi

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5 Politecnico di Torino – Universitat Pompeu Fabra A new class of SCCC RC-SCCC ■The inner code may be punctured beyond the unitary rate R SCCC may be greater than the outer code rate ■Puncturing is split between systematic and parity bits: s : systematic permeability p : parity permeability Outer Encoder u Inner Encoder PoPo MUXMUX PsiPsi PpiPpi

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6 Politecnico di Torino – Universitat Pompeu Fabra A new class of SCCC ■Performance depend on puncturing patterns P o,P s i,P p i s and p should be properly selected ■We propose design criteria of this new class of SCCC by deriving the upper bounds to the error probability Outer Encoder PoPo Inner Encoder MUXMUX PpiPpi PsiPsi C’ o C’’ o C’ i

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7 Politecnico di Torino – Universitat Pompeu Fabra Upper bounds to the error probability ■We obtain: ■The dominant contribution to the error probability for (asymptotic with N) is the largest exponent of N, M.

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8 Politecnico di Torino – Universitat Pompeu Fabra Upper bounds to the error probability ■For recursive inner encoder: and ■ h( M ): weight associated to the highest exponent of N

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9 Politecnico di Torino – Universitat Pompeu Fabra Upper bounds to the error probability ■We obtain: ■ d o’ f : free distance of C’ o ■ d o’’ (d o’ f ): minimum weight of C’’ o code sequences corresponding to a C’ o code sequence of weight d o’ f ■ d i’ f,eff : effective free distance of C’ i ■ h (3) m : minimum weight of C’ i sequences generated by weight 3 input sequences

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10 Politecnico di Torino – Universitat Pompeu Fabra Upper bounds to the error probability Outer Encoder PoPo Inner Encoder MUXMUX PpiPpi PsiPsi C’ o C’’ o C’ i ■ d o’ f : free distance of C’ o ■ d o’’ (d o’ f ): minimum weight of C’’ o code sequences corresponding to a C’ o code sequence of weight d o’ f ■ d i’ f,eff : effective free distance of C’ i ■ h (3) m : minimum weight of C’ i sequences generated by weight 3 input sequences

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11 Politecnico di Torino – Universitat Pompeu Fabra Upper bounds to the error probability ■We obtain: ■ d o’ f : free distance of C’ o ■ d o’’ (d o’ f ): minimum weight of C’’ o code sequences corresponding to a C’ o code sequence of weight d o’ f ■ d i’ f,eff : effective free distance of C’ i ■ h (3) m : minimum weight of C’ i sequences generated by weight 3 input sequences

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12 Politecnico di Torino – Universitat Pompeu Fabra Upper Bound to the error probability ■Then, P b (e) (asymptotic with respect to N): ■For large E b /N 0 BER performance is given by: d o’ f odd d o’ f even

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13 Politecnico di Torino – Universitat Pompeu Fabra Upper Bound to the error probability ■Design considerations: ■ P o should be chosen to optimize C’ o distance spectrum ■ P s i and P p i should be chosen so that h( m ) and h m are maximized ■ P p i must be optimized to yield the best C’ i IOWEF ■ P s i must be selected to optimize d o’’ (d o’ f ) P s i turns out to be interleaver dependent

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14 Politecnico di Torino – Universitat Pompeu Fabra Rate-compatible SCCC ■We designed well-performing rate-compatible SCCC following the aforementioned considerations ■ P s i to optimize d o’’ (d o’ f ) ■ P p i to optimize C i’ IOWEF ■ We used a searching algorithm that works incrementally, fulfilling the rate-compatible restriction, so that the punctured positions for a given outer rate are also punctured for all higher rates.

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15 Politecnico di Torino – Universitat Pompeu Fabra The SCCC Scheme Rate-1/2 4 state u Rate-1/2 4 state Fix punct. MUXMUX PsiPsi PpiPpi d o’ f =3 d o’ f =4 outer code puncturingconstituent codes

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16 Politecnico di Torino – Universitat Pompeu Fabra Performance Bounds Bounds of Rate-2/3 SCCC for several p N=200. P o,1 p =2/30 p =4/30 p =6/30 p =8/30 p =10/30

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17 Politecnico di Torino – Universitat Pompeu Fabra Performance Bounds Bounds of Rate-2/3 SCCC for several p N=200. P o,2 p =2/30 p =4/30 p =6/30 p =8/30 p =10/30

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18 Politecnico di Torino – Universitat Pompeu Fabra Simulation Results Performance of Rate-2/3 SCCC for several p N=200. P o,1 p =2/30. Simulation p =2/30. Bound p =4/30. Simulation p =4/30. Bound p =8/30. Simulation p =8/30. Bound p =10/30. Simulation p =10/30. Bound

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19 Politecnico di Torino – Universitat Pompeu Fabra Simulation Results p =2/30. Simulation p =2/30. Bound p =4/30. Simulation p =4/30. Bound p =8/30. Simulation p =8/30. Bound UMTS PCCC SCCC (VTC’01) Performance of Rate-2/3 SCCC for several p N=2000. P o,1

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20 Politecnico di Torino – Universitat Pompeu Fabra Simulation Results p =4/222. Simulation p =4/222. Bound p =10/222. Simulation p =10/222. Bound p =16/222. Simulation p =16/222. Bound UMTS PCCC Performance of Rate-9/10 SCCC for several p N=2000. P o,1

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21 Politecnico di Torino – Universitat Pompeu Fabra Simulation Results Performance versus p for several E b /N 0. R=9/10. N=2000. P o,1 22/222 20/222 18/222 16/222 14/222 12/222 10/222 8/222 6/222 4/222 2/222 pp

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22 Politecnico di Torino – Universitat Pompeu Fabra Simulation Results FER Performance comparison. N=428 SCCC (10 it.) PCCC (8 it.) LDPC (50 it.)

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23 Politecnico di Torino – Universitat Pompeu Fabra Conclusions ■Derived lower bound to the error probability of a new class of SCCC ■Derived suitable design guidelines ■Derived optimal Rate-compatible SCCC families ■The proposed scheme offers good performance for low to moderate block lengths in a large range of rates ■ The interleaver gain for low rates is kept also in the case of heavy puncturing ■This code structure has been proposed as a candidate coding scheme for ESA MHOMS

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24 Politecnico di Torino – Universitat Pompeu Fabra Open Problems ■Convergence analysis EXIT charts and Density Evolution Techniques are difficult to apply ■We are open to cooperations with other groups!!!

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