New criteria for iterative decoding

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New criteria for iterative decoding

Florence Alberge, Ziad Naja, Pierre Duhamel

To cite this version:

Florence Alberge, Ziad Naja, Pierre Duhamel. New criteria for iterative decoding. ICASSP, Apr 2009, Taipei, Taiwan. �hal-01849643�

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Univ. Paris-Sud, UMR 8506, Orsay

CNRS, Orsay, France

New criteria for iterative decoding

Florence Alberge, Ziad Naja, Pierre Duhamel

International Conference on Acoustic Speech and Signal Processing 09

Laboratoire des Signaux et Systèmes

Supelec, Gif-sur-Yvette, France

Context: Turbo-like algorithms with iterative decoding

Goal: Make the link between iterative decoding and classical optimization tech-

niques. Improve the performance of iterative decoding.

System model (Bit Interleaved Coded Modula-

tion)(Zehavi, 1992; Li, 2002)

Transmission model

Convolutional

Encoder Interleaver Symbol

Mapping Channel

b c d s y

AWGN Channel y_k = s_k + n_k n_k N (0, σ_n²) s_k = M apping(d_km+1, d_km+2, ..., d_(k+1)m)

BICM-ID receiver with soft-decision feedback

Demapping Deinterleaver SISO

Decoder Interleaver

p(b;O) p(c;O) p(d;I)

p(d;O) p(c;I)

y

APP (Demapping) p_{AP P}(d_km+i = b) = K_m^′ X

s_k∈Ψⁱ_b

p(y_k | s_k) Y

j

p(d_km+j; I)

Prior × Channel probability → Marginalization

Extrinsic (Demapping) p(d_km+i = b; O) = K_m X

s_k∈Ψⁱ_b

p(y_k | s_k) Y

j6=i

p(d_km+j; I)

APP (Decoder) p_{AP P}(c_l = b) = K_c^′ X

c∈R^l_b

I_C(c) Y

j

p(c_j; I) (I_C indicator f unction of the code)

Prior × Indicator function → Marginalization

Extrinsic (Decoder) p(c_l = b; O) = K_c X

c∈R^l_b

I_C(c) Y

j6=l

p(c_j; I)

Information geometry and iterative decoding

Basic tools and Notations (Walsh, 2006)

(B_i) ∈ {0, 1}^N binary representation of integer i gathered into matrix B = (B₀, B₁, ..., B₂^N₋₁)^T.

PMF η = (Pr[χ = B₀], Pr[χ = B₁], ..., Pr[χ = B₂^N₋₁])^T

Log-coordinates of PMF η (θ_i)_0≤i≤2^N₋₁ = ln(P r[χ = B_i]) − ln(P r[χ = B₀])

Bitwise log-probability ratio (λ_j)_{0≤j≤N −1} = log(^{P r}_{P r}^[χ_[χ^j^=1]

j=0])

For factorisable probability measures, the log-coordinates take the form θ = Bλ.

Link with iterative decoding

• Demapping subblock

p_{AP P}(d_km+i = b) = K_m^′ P

s_k∈Ψⁱ_b p(y_k | s_k) Q

j p(d_km+j; I)



 y



yLog−coordinates

θ_m Bλ₁ p_{AP P}(d_km+i = b) = K_m^′′ p(d_km+i; I) p(d_km+i; O)



 y



yLog−coordinates

Bλ₁ Bλ₂

The demapping sub-block solves, with respect to λ₂, p_B(λ

1+λ2) = p_Bλ₁_+θ_m

• Decoding sub-block

p_{AP P}(c_l = b) = K_c^′ P

c∈R^l_b I_C(c) Q

j p(c_j; I)



 y



yLog−coordinates

θ_c Bλ₂

p_{AP P}(c_l) = K_c^′′ p(c_l; I) p(c_l; O)



 y



yLog−coordinates

Bλ₂ Bλ₁

The decoding sub-block solves, with respect to λ₁, p_B(λ

1+λ2) = p_Bλ₂_+θ_c

• Global criterion Let D_{F D}(p, q) = P

j p_j ln _p

j

q_j

+ P

j(1 − p_j) ln _1−p

j

1−qj

denote the Fermi-Dirac entropy (Kullback- Leibler distance for bit probabilities).

The demapping sub-block solves the minimization problem min_λ₂ D_{F D}(p_Bλ₁_+θ_m, p_B_(λ₁_+λ₂₎). The decoding sub-block solves the minimization problem min_λ₁ D_{F D}(p_Bλ₂_+θ_c, p_B(λ₁_+λ₂₎).

New criteria

An hybrid proximal point algorithm

Goal: Link the two (independent) criteria using proximal point technique (Luque, 1984).

Demapping min

λ2

J_θ_m(λ₁, λ₂) = min

λ2

D_{F D}(p_Bλ₁_+θ_m, p_B(λ₁_+λ₂₎) + µ_mD_{F D}(p

B(λ^(k)₁ +λ^(k)₂ ), p_B(λ₁_+λ₂₎) Decoding min

λ₁ J_θ_c(λ₁, λ₂) = min

λ₁ D_{F D}(p_Bλ₂_+θ_c, p_B(λ

1+λ2)) + µ_cD_{F D}(p

B(λ^(k)₁ +λ^(k+1)₂ ), p_B(λ

1+λ2)) Optimal choice of the step-size :

Choose µ_m such that J_θ_m(λ^(k)₁ , λ^(k+1)₂ ) ≤ J_θ_c(λ^(k)₁ , λ^(k)₂ ) Choose µ_c such that J_θ_c(λ^(k+1)₁ , λ^(k+1)₂ ) ≤ J_θ_m(λ^(k)₁ , λ^(k+1)₂ )

Both criteria decrease with the iterations. Convergence towards the same stationary point than the classical iterative decoding.

An hybrid minimum entropy algorithm

Goal: Improve the performance of the iterative decoding.

Rationale: The entropy of the APP gives a measure of the reliability of the decision : E_B(λ

1+λ₂) → 0 means that the iterative decoding is confident about its decisions (Kocarev, 2006).

Demapping min

λ₂ D_{F D}(p_Bλ₁_+θ_m, p_B_(λ

1+λ2)) + η_mE_B_(λ

1+λ₂)

Decoding min

λ₁ D_{F D}(p_Bλ₂_+θ_c, p_B_(λ

1+λ2)) + η_cE_B_(λ

1+λ₂)

Simulation

The generator polynomial of the encoder is g = [111; 001; 100]. The bits are mapped using subset partitioning to a 8-PSK modulation. The length of the coded bit sequence is L_c = 6000. The

step-sizes η_m and η_c in the HMEA are both chosen equal to 0.05.

4 4.5 5 5.5 6 6.5 7 7.5 8 8.5

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

EbN0

BER

Classical iterative decoding Hybrid Proximal Point Hybrid Min Entropy

4 4.5 5 5.5 6 6.5 7 7.5 8 8.5

5 10 15 20 25 30

EbN0

iterations

Classical iterative decoding Hybrid Proximal Point Hybrid Min Entropy

Left: BER versus EbN0 – Right: Iteration number versus EbN0

References

E. Zehavi. 8-PSK trellis codes for a Rayleigh fading channel. IEEE. Trans on Commun., 40:873–883, May 1992.

X. Li, A. Chindapol and J.A. Ritcey. Bit interleaved coded modulation with iterative decoding and 8-PSK signaling. IEEE. Trans on Commun., 50:1250–1257, Aug. 2002.

J.M. Walsh, P.A. Regalia and C.R. Johnson. Turbo decoding as Iterative Constrained Maximum Likelihood sequence Decoding. IEEE Trans on Inform Theory, 52:5426–5437, Dec. 2006.

F.J. Luque. Asymptotic convergence analysis of the proximal point algorithm. SIAM Journal on Control and Optimization, 22(2), pp. 277–293, 1984.

L. Kocarev, F. Lehmann, G.M. Maggio, B. Scanavino, Z. Tatsev and A. Vardy. Nonlinear dynamics of iterative decoding systems: Analysis and Applications. IEEE Trans. on Inform. Theory, 52(4):

1366–1384, 2006.