Density Evolution

The density evolution is used to find the limits of performance of combined EM-JED algorithm and LDPC decoding. The density evolution with Gaussian approximation in [21] is summarized here for reference. Firstly, the output log-likelihood ratios (LLRs) passed from EM-JED algorithm and RA decoder to LDPC decoder are generated by Monte Carlo simulation. Then the probability density function (pdf) of these output LLRs is modeled as a mixture of symmetric Gaussians based on the EM algorithm. With the assumption of the symmetric Gaussian mixtures of the output LLRs, the evolution of the pdfs can be easily tracked through message-passing algorithm within LDPC de-coder. The pdf of the output extrinsic messages of the variable nodes would become symmetric Gaussian mixtures due to both the irregularity of variable degree and the pdf of the output LLRs of RA decoder. Similarly the pdf of the output extrinsic messages of the check nodes is affected by the irregularity of the check degree. To reduce the complexity of computing the pdf of the output extrinsic messages of the check nodes, we only consider the check-regular degree profile here. With only check-regular degree profile considered, there still occurs some differences in the histogram of variable-regular degree profile and variable-irregular degree profile. The histogram is shown in the sequel.

In [21], the pdf of the output extrinsic messages of the check nodes is assumed Gaussian

DETECTOR

Figure 6.1: Illustration of density evolution of the iterative receiver.

distributed. Nevertheless, it has not been suitable to make Gaussian approximation of the output extrinsic messages of the check nodes with variable-irregular degree profile.

However, the density evolution with Gaussian approximation would become much easier by only tracking the mean of the pdfs within the LDPC decoder.

The procedure of computing the pdfs of the extrinsic LLRs of LDPC decoding is described as follows. The subscript D → L denotes quantities sent from the EM-JED algorithm and RA decoder to LDPC decoder and vice versa, D ← L. Here the subscript D denotes the DETECTOR which contains both the EM-JED algorithm and RA decoder. The subscript b → c denotes quantities sent from variable nodes to check nodes and vice cersa, b ← c. The superscript p denotes pth iteration time of LDPC decoder. And the superscript q denotes the qth iteration time of the overall iterative receiver. The superscript dl,maxdenotes the maximum variable degree and dr,maxdenotes the maximum check degree.

• Initialization:

– Set f_b←c^0,0 (x) = δ(x) and f_D←L⁰ (x) = δ(x).

• Turbo iterations of receiver: (q = 1, 2, . . . Q)

– In the fading channel, the pdf of the extrinsic messages passed from DETEC-TOR to LDPC decoder is modeled as symmetric Gaussian mixtures based on the EM algorithm. The details can be referred to [21]. Hence, the pdfs of the extrinsic messages passed from DETECTOR are computed as

f_D→L^q = XJ

j=1

π_jN (µ_j, 2µ_j), (6.1)

where J is the number of Gaussian component.

– Computing the pdf of the extrinsic messages within LDPC iteration. (p = 1, 2, . . . P )

∗ At a variable node of degree i:

The pdf of the extrinsic messages passed along an edge from variable node of degree i is denoted by f_b→c,i^p,q and it is computed by the convolution of f_D→L^q with (i − 1) convolutions of f_b←c^p−1,q.

f_b→c,i^p,q = f_D→L^q ⊗ f_b←c^p−1,q⊗ f_b←c^p−1,q⊗ . . . ⊗ f_b←c^p−1,q

| {z }

(i-1) convolutions

(6.2)

· If f_b←c^p−1,q is assumed Gaussian distributed, it means that

f_b←c^p−1,q = N (m^p−1,q_b←c , 2m^p−1,q_b←c )

check nodes

variable nodes

Figure 6.2: Illustration of half-iteration message passing at variable node in density evolution.

∗ The pdf of the extrinsic messages passed from the variable nodes to the check nodes along an edge:

f_b→c^p,q =

dl,maxP

i=2

λ_if_b→c,i^p,q (6.4)

(λ_i: fractions of the edges connected to variable nodes of degree i)

∗ At a check node of degree l: The computation of the tanh rule at the check nodes is shown as

tanh(^Lp,qb←c₂^(er)) = Q^l

k=1,k6=r

tanh(^Lp,qb→c₂^(ek)). (6.5)

check nodes

variable nodes

Figure 6.3: Illustration of half-iteration message passing at check node in density evolu-tion.

· If f_b←c^p,q is assumed Gaussian distributed, we can firstly take the ex-pectation of both sides in (6.5) and it becomes

En

where L^p,q_b→c(e_s) and L^p,q_b→c(e_k) are identically distributed and indepen-dent for k 6= s.

Denote ψ(x) , E©

tanh(^L₂)ª

, where L ∼ N (x, 2x).

Taking the definition of ψ function and (6.3), the equation becomes

ψ(m^p,q_b←c,l) =

Finally, f_b←c,l^p,q = N (m^p,q_b←c,l, 2m^p,q_b←c,l).

∗ The pdf of the extrinsic messages passed from the check nodes to the variable nodes:

f_b←c^p,q =

dr,maxP

l=2

ρ_lf_b←c,l^p,q (6.9)

(ρ_l: fractions of the edges connected to check nodes of degree l)

∗ Messages passed back to the DETECTOR at a variable node of degree i:

The pdf of the extrinsic messages passed along an edge from variable node of degree i back to the DETECTOR is denoted by f_D←L,i^q and it is computed by i convolutions of f_b←c^p,q .

f_D←L,i^q = f_b←c^p,q ⊗ f_b←c^p,q ⊗ . . . ⊗ f_b←c^p,q

| {z }

i convolutions

(6.10)

· If f_b←c^p,q is assumed Gaussian distributed, it becomes

f_D←L,i^q = N (im^p,q_b←c, 2im^p,q_b←c). (6.11)

∗ Message passed back to the DETECTOR:

f_D←L^q =

dr,maxX

i=2

eλ_if_D←L,i^q (6.12)

(eλi: fractions of the variable nodes of degree i)

• Finding the minimum SNR:

1. With Gaussian approximation of the pdf of the extrinsic messages passed from check nodes to variable nodes, find the minimum SNR at which m^Q_D←L or m^P,Q_b←c tens to ∞.

2. Find the minimum SNR at which the residual error R₀

−∞f^q(z)dz tends to zero, where f_i^q = f_D←L,i^q ⊗ f_D→L and f^q =P_dr,max

i=2 eλ_if_i^q.

Obviously, the density evolution becomes much easier with the Gaussian approx-imation. Considering the check-regular degree profiles, we show the histograms of LLRs passed from check nodes to variable nodes with variable-regular degree profile in Fig. 6.4 and variable-irregular degree profile in Fig. 6.5. Hence, we find that Gaussian approximation can work well at the output extrinsic messages of check nodes with variable-regular degree profile but might fail with the irregular-variable degree profile.

Figure 6.4: The histogram of output LLRs passed from check nodes to variable nodes with regular (3,6) degree profile.

Figure 6.5: The histogram of output LLRs passed from check nodes to variable nodes with variable-irregular and check-regular degree profile.

Chapter 7