Proposed Dynamic Scheduling Strategies

We first propose a less greedy message selection strategy to attack the negative effect

on a new metric with considerations of error correcting performance. Compared with the conventional dynamic schedules, the proposed schedules have not only better error correcting performance but faster convergence speed for short-length codes.

3.2.2.1 Greediness of Dynamic Scheduled Decoding

The residual belief propagation (RBP) scheduling algorithm proposed in [36]

schedules the messages according to how much they change after an update. This strategy improves the convergence speed but may generate new errors which degrade its performance. RBP updates first the C2B message which has the largest residual among all C2B messages. The residual of a message is the difference between the values of the message before and after an update. After the selected C2B message

i j

c v

m _→ from ci to vj is updated, all bit-to-check (B2C) messages

j a

v c

m _→ for ca∈ψ(vj)\ci are updated where ψ(vj)\ca denotes the neighboring CNs of vj excluding ca. Finally, the residuals of C2B messages

a b

c v

m _→ for vb∈ψ(ca)\vj are re-computed where ψ(ca)\vj denotes the neighboring BNs of c_a excluding v_j. Then the next round of message selection and update begins.

RBP uses greedy search that picks the message furthest from convergence so the goal of RBP is to achieve the convergence state with the smallest number of message updates regardless of the decoding performance. In addition, RBP is a greedy approach which only optimizes the decoding process locally for the next message update. Therefore it’s reasonable that RBP converges very fast but cannot correct the errors which static scheduling can after more iterations. RBP may correct trapping-set errors but produce additional greedy errors so a higher error floor than conventional static schedules is

induced.

Node-wise residual belief propagation (NWRBP) [36], which schedules the decoding as a series of CN updates, is less greedy than RBP. The difference from RBP is that when the C2B message m_c→v with the largest residual is selected, the messages

c vj

m_→ for vj∈ψ(c), i.e., all C2B messages from c, will be updated. NWRBP outperforms layered schedule in terms of both error floor performance and convergence speed. Compared with RBP, NWRBP trades the convergence speed for a lower error floor since it schedules a greedy message update followed by several non-greedy message updates. For short-length codes, the problem of additional greedy errors does not solved well by NWRBP. Therefore mixed strategies can be used to decode with layered schedule first and then change to NWRBP when a certain criterion is met. The criterion can be a pre-determined number of iterations or the condition that the number of unsatisfied CNs is smaller than a certain value [36].

In summary, all the above dynamic scheduling strategies are greedy in nature. More greediness leads to faster convergence at the cost of more new greedy errors. We want to use a different way to ease the extreme greediness of RBP, which only involves different criteria to select the message to update next.

3.2.2.2 Less-Greedy Scheduling

In [37], the authors suggest that for combinatorial optimization problems, constructing the solution based on a greedy criterion does not always find the best solution. Instead, choosing one of the best candidates during construction can improve the solution of the

messages must be recognized as selection candidates based on certain criteria and then one of the candidate messages are selected to update. We construct our search strategies from two directions. Let r(m_c→v) denote the residual of the message m_c→v. We first define the subset of interest in two types: 1) cardinality-restricted subset and 2) value-restricted subset.

Type-1 subset is defined as {m_c→v| rx-th ≤ r(m_c→v) ≤ ry-th} where rx-th and ry-th are the x-th and y-th largest residuals of the messages among all residuals. y is always set to 1 in our experiments. Type-2 subset is defined as {m_c→v| rmin ≤ r(m_c→v) ≤ rmax} where rmin and rmax are the lower and upper bound for residuals of the messages in this subset. rmax is always set to infinite in our experiments. Then two methods can be used to select a message from the subset: 1) deterministic and 2) randomized. The deterministic method directly selects the last message in the ordered subset, i.e., the message with the smallest residual in the subset. The randomized method randomly selects one of the messages in the subset. In summary, as shown in Table 3.1, four different strategies S1, S2, S3, and S4 can be used to select the next message to update.

Table 3.1. Message selection strategies.

Subset Type

Selection Method

Cardinality-Restricted Value-Restricted

Deterministic S1: select the message with residual rx-th

S3: select the message with a residual larger than rmin and closest to rmin

Randomized

S2: randomly select a message from messages with residuals larger than rx-th

S4: randomly select a message from messages with residuals larger than rmin

In the following experiments, IEEE 802.16e (576, 288) LDPC code [4] is simulated using BPSK modulation on the AWGN channel with a maximum of 30 iterations. Fig. 3.4 shows the performance of strategies S1, S2, S3, and S4 with various values of rx-th or rmin in terms of block error rate (BLER). We can see that the randomized strategies (S2 and S4) outperform the greedy RBP schedule especially in the error floor region while the deterministic strategies (S1 and S3) have little or no improvement over RBP schedule.

With x ≥ 3, S2 can achieve better BLER performance than RBP schedule. In other words, instead of selecting the message with the largest residual, randomly selecting a message from the three messages with largest residuals can efficiently avoid generating greedy errors. With rmin = 2 or 3, S4 can also achieve better BLER performance than RBP schedule. Hence randomly selecting a message from the messages with residuals larger than 2 or 3 can also efficiently avoid generating greedy errors. S2 requires no additional computation since the ordering of messages by residuals is inherently required by RBP while S4 needs additional computation to find the message with the smallest residual which is greater than or equal to r_min before message selection.

1.5 2.0 2.5 3.0 10^-4

10^-3 10^-2 10^-1

BLER

E_b/N₀ (dB)

Layered RBP S1 (2) S1 (3) S1 (5) S1 (10) S1 (20)

(a)

1.5 2.0 2.5 3.0 10^-5

10^-4 10^-3 10^-2 10^-1

BLER

Eb/N

0 (dB)

Layered RBP S2 (3) S2 (5) S2 (10) S2 (20) S2 (40)

(b)

1.5 2.0 2.5 3.0 10^-4

10^-3 10^-2 10^-1

BLER

E_b/N₀ (dB)

Layered RBP S3 (3) S3 (5) S3 (10) S3 (20) S3 (30)

(c)

1.5 2.0 2.5 3.0 10^-5

10^-4 10^-3 10^-2 10^-1

BLER

E_b/N₀ (dB)

Layered RBP S4 (2) S4 (3) S4 (5) S4 (10) S4 (20)

(d)

Fig. 3.4. BLER performance of message selection strategies S1, S2, S3, and S4.

3.2.2.3 Farsighted Scheduling

We already figure out that easing the greediness of residual-based dynamic scheduling strategies by increasing the message selection candidates is effective in improving the decoding performance. Next we consider the ordering metric of candidates itself and try to select the next updated message based on a new metric considering decoding performance.

The shortcoming of RBP is its additional complexity for the computation of residuals.

To generate the residual of an C2B message, we need to compute the new C2B message but

approximation to compute residuals for lowering the overall complexity. Instead of trying to reduce the complexity for residual generation, we try to further utilize those new C2B messages in message selection. We know that RBP only make use of the residual, i.e., the difference between the values of a message before and after an update, which requires the computation of the new message. However, since the new message has been computed, more information can be utilized.

The errors in the trapping sets are difficult to correct by static schedules. An (n, m) trapping set [13] is a set of n BNs in the Tanner graph with m odd-degree CNs in their induced sub-graph. In this sub-graph, most CNs have even degree and only a small number of CNs have odd degree. When decoding with the case that the BNs in a trapping set are all in error by static schedules, those BNs corrected by odd-degree CNs will later be flipped back to the wrong hard decisions by even-degree CNs. However, an efficient dynamic schedule can break the trapping set through continuous updating the C2B messages corresponding to odd-degree CNs. Fig. 3.5 shows a dynamic schedule that correct an (4, 1) trapping set through continuous updating the degree-1 CNs. A bold line is an C2B or B2C message which is updated and a dotted line is a message whose residual is re-computed at that moment. RBP is effective in solving trapping-set errors since the message updated first by RBP has the largest residual and thus is very likely to be the one corresponding to odd-degree CNs.

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(c) (d) Fig. 3.5. A dynamic schedule that breaks the trapping-set errors.

Consider the process how a trapping set is break as shown in Fig. 3.5, each scheduled and updated C2B message

i j

c v

m _→ must be powerful enough to flip the hard decision of BN v_j so that v_j and its connected edges can be removed from the trapping set. In addition, after a message

i j

c v

m _→ is updated, an efficient schedule should keep updating another C2B message

a b

c v

m _→ for c_a∈ψ(vj)\c_i and v_b∈ψ(ca)\v_j, whose residual is recently re-computed (the dotted lines in Fig. 3.5). In other words, after the correction of BN vj, it would be better to continue processing another neighboring CN of vj.

Based the above observations, we propose a farsighted metric for message ordering and selection, i.e.,

( ) ^{new old}

w c v c v c v

r m_→ = × ×α β m_→ −m_→ (3.3)

where m_c^old_→v and m_c^new_→v are the messages from CN c to BN v before and after an update.

new old

c v c v

m_→ −m_→ is the residual. α and β are two weighting factors which are further defined in (3.4) and (3.5) respectively.

if its residual is re-computed after the last message update.

1 otherwise.

w_α

α 

=  (3.4)

if will flip the hard decision of BN . 1 otherwise.

new c v

w_β m v

β _{= }^{ →}

 (3.5)

α is obtained after all necessary residual re-computation is done. β is obtained after each time m_c^new_→v is computed. α increases the possibility to schedule the messages with the newest residuals and β increases the possibility to schedule the messages with the capability to flip the hard decisions. Of course we don’t know the flipped bit is correct or not, but more bit flipping gives more chances for the trapping set to be break. Since trapping-set errors are relatively difficult to correct, we believe that these weightings are also helpful for solving non-trapping-set errors.

3.2.2.4 Low-Complexity Farsighted Scheduling

Consider the schedule that directly restricts the possible candidates to only those messages whose ordering metrics are recently re-computed after the last message update.

This schedule is equivalent to the proposed farsighted schedule with extremely large w^α

since the other messages whose ordering metrics are not recently re-computed will not be selected due to their relatively small ordering metrics. Hence while the original dynamic schedule treats all of the C2B messages as possible candidates and orders them in a long list, this particular schedule limits the candidates to a small group of messages. The complexity in message ordering and selection can be lowered. For IEEE 802.16e (576, 288) code, there exists 1824 edges in its Tanner graph, representing 1824 different C2B messages. Each C2B message update is followed by residual re-computation of maximum 28 messages. Hence this Low-complexity farsighted schedule reduces the size of the ordered list for selection from 1824 messages to maximum 28 messages.

在文檔中 Design of Low-Density Parity-Check Coding Systems (頁 68-80)