Non-Uniform Windowed Decoding For Multi-Dimensional - - PowerPoint PPT Presentation

▶
non uniform windowed decoding for multi dimensional
SMART_READER_LITE
LIVE PREVIEW

Non-Uniform Windowed Decoding For Multi-Dimensional - - PowerPoint PPT Presentation

Jun 15, 2023 119 likes •730 views

Non-Uniform Windowed Decoding For Multi-Dimensional Spatially-Coupled LDPC Codes Lev Tauz, Homa Esfahanizadeh, and Lara Dolecek Electrical and Computer Engineering University of California, Los Angeles ISIT 2020 Tauz, Esfahanizadeh, Dolecek

slide-1
SLIDE 1

Non-Uniform Windowed Decoding For Multi-Dimensional Spatially-Coupled LDPC Codes

Lev Tauz, Homa Esfahanizadeh, and Lara Dolecek

Electrical and Computer Engineering University of California, Los Angeles

ISIT 2020

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 1 / 34

slide-2
SLIDE 2

Outline

1

Background and Previous Work Spatially-Coupled LDPC Previous Work: Multi-Dimensional SC-LDPC

2

Non-Uniform Windowed Decoding Motivation and Contributions Ensemble Definition Decoder Design

3

Simulations

4

Ongoing Work

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 2 / 34

slide-3
SLIDE 3

Outline

1

Background and Previous Work Spatially-Coupled LDPC Previous Work: Multi-Dimensional SC-LDPC

2

Non-Uniform Windowed Decoding Motivation and Contributions Ensemble Definition Decoder Design

3

Simulations

4

Ongoing Work

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 3 / 34

slide-4
SLIDE 4

LDPC

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 4 / 34

slide-5
SLIDE 5

Spatially-Coupled LDPC Codes

Spatially-coupled (SC) LDPC codes are constructed by concatenating multiple LDPC codes into a single chain. SC-LDPC codes have capacity approaching performance and allow for low latency windowed decoding. [Iyengar ’12]

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 5 / 34

slide-6
SLIDE 6

Outline

1

Background and Previous Work Spatially-Coupled LDPC Previous Work: Multi-Dimensional SC-LDPC

2

Non-Uniform Windowed Decoding Motivation and Contributions Ensemble Definition Decoder Design

3

Simulations

4

Ongoing Work

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 6 / 34

slide-7
SLIDE 7

Multi-Dimensional SC-LDPC Codes

Multi-dimensional (MD) SC codes are a subclass where copies of SC codes are coupled together, to create a code with outstanding finite and asymptotic performance.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 7 / 34

slide-8
SLIDE 8

Benefits of MD-SC LDPC Codes

Robustness to burst erasures [Ohashi ’13] Improved decoding thresholds [Liu ’15, Truhachev ’19] Improved reliability over parallel channels [Schmalen ’14] Continuous transmission of multiple codewords with good finite-length performance [Olmos ’17] Flexible finite-length design for removing detrimental objects to improve iterative decoding [Esfahanizadeh ’20, Hareedy ’20] New Result: Non-uniform windowed decoding for flexible decoder complexity

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 8 / 34

slide-9
SLIDE 9

Previous Work: Low-Latency Multi-Dimensional Windowed Decoding

Best known previous work on windowed decoding of MD-SC codes: All constituent SC codes have a window that move in unison along the MD-SC chain. [Esfahanizadeh ’20] Traditional uniform decoders allow a large latency reduction with almost the same performance; however, not taking full advantage of the MD-SC code structure maximize decoding flexibility.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 9 / 34

slide-10
SLIDE 10

Outline

1

Background and Previous Work Spatially-Coupled LDPC Previous Work: Multi-Dimensional SC-LDPC

2

Non-Uniform Windowed Decoding Motivation and Contributions Ensemble Definition Decoder Design

3

Simulations

4

Ongoing Work

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 10 / 34

slide-11
SLIDE 11

Outline

1

Background and Previous Work Spatially-Coupled LDPC Previous Work: Multi-Dimensional SC-LDPC

2

Non-Uniform Windowed Decoding Motivation and Contributions Ensemble Definition Decoder Design

3

Simulations

4

Ongoing Work

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 11 / 34

slide-12
SLIDE 12

Non-Uniform Windowed Decoder Motivation

In MD-SC codes, not all constituent codes are directly connected. Intuitively, codes that are not directly connected, may provide less information to decode the target VNs. Example: let (1, 0) be the target VN section and all sections in the first column are already decoded.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 12 / 34

slide-13
SLIDE 13

Non-Uniform Windowed Decoder Motivation

In MD-SC codes, not all constituent codes are directly connected. Intuitively, codes that are not directly connected, may provide less information to decode the target VNs. Example: let (1, 0) be the target VN section and all sections in the first column are already decoded. If the graph is strongly connected, the uniform windowed decoder is generally a good solution.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 12 / 34

slide-14
SLIDE 14

Non-Uniform Windowed Decoder Motivation

In MD-SC codes, not all constituent codes are directly connected. Intuitively, codes that are not directly connected, may provide less information to decode the target VNs. Example: let (1, 0) be the target VN section and all sections in the first column are already decoded. If the sections are sparsely connected, select sections into the decoding window that are closer to the target VNs.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 12 / 34

slide-15
SLIDE 15

Non-Uniform Windowed Decoder Motivation

In MD-SC codes, not all constituent codes are directly connected. Intuitively, codes that are not directly connected, may provide less information to decode the target VNs. Example: let (1, 0) be the target VN section and all sections in the first column are already decoded. Distance may not be a good metric to select sections in the window. For the target VNs, connections from CNs to VNs can be more important.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 12 / 34

slide-16
SLIDE 16

Non-Uniform Windowed Decoder Motivation

In MD-SC codes, not all constituent codes are directly connected. Intuitively, codes that are not directly connected, may provide less information to decode the target VNs. Example: let (1, 0) be the target VN section and all sections in the first column are already decoded. Uniform windowed decoders do not fully exploit all these features. Thus, motivating the need for a non-uniform decoder.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 12 / 34

slide-17
SLIDE 17

Key Contributions

A new, more general MD-SC ensemble to allow for varying the coupling along different dimensions. A novel non-uniform shaped windowed decoder and its key design parameters. Optimization of design parameters using density evolution for the BEC channel and insight into decoder design.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 13 / 34

slide-18
SLIDE 18

Outline

1

Background and Previous Work Spatially-Coupled LDPC Previous Work: Multi-Dimensional SC-LDPC

2

Non-Uniform Windowed Decoding Motivation and Contributions Ensemble Definition Decoder Design

3

Simulations

4

Ongoing Work

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 14 / 34

slide-19
SLIDE 19

Quick Notation

(n)B = n mod B. [A] {0, 1, . . . , A − 1} [(L1, L2)] {(j, k) : j ∈ [L1], k ∈ [L2]}.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 15 / 34

slide-20
SLIDE 20

Multi-Dimensional SC-LDPC ensemble: Parameters

Code Parameters dl, dr: degrees of VNs and CNs.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 16 / 34

slide-21
SLIDE 21

Multi-Dimensional SC-LDPC ensemble: Parameters

Code Parameters dl, dr: degrees of VNs and CNs. L1: 1D coupling length, length of individual codes.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 16 / 34

slide-22
SLIDE 22

Multi-Dimensional SC-LDPC ensemble: Parameters

Code Parameters dl, dr: degrees of VNs and CNs. L1: 1D coupling length, length of individual codes. L2: MD coupling length, number of codes coupled.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 16 / 34

slide-23
SLIDE 23

Multi-Dimensional SC-LDPC ensemble: Parameters

Code Parameters dl, dr: degrees of VNs and CNs. L1: 1D coupling length, length of individual codes. L2: MD coupling length, number of codes coupled. γ1 < L1: 1D coupling depth, maximum coupling distance along a constituent code.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 16 / 34

slide-24
SLIDE 24

Multi-Dimensional SC-LDPC ensemble: Parameters

Code Parameters dl, dr: degrees of VNs and CNs. L1: 1D coupling length, length of individual codes. L2: MD coupling length, number of codes coupled. γ1 < L1: 1D coupling depth, maximum coupling distance along a constituent code. γ2 < L2: MD coupling depth, maximum coupling distance among constituent codes.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 16 / 34

slide-25
SLIDE 25

Multi-Dimensional SC-LDPC ensemble: Parameters

Code Parameters dl, dr: degrees of VNs and CNs. L1: 1D coupling length, length of individual codes. L2: MD coupling length, number of codes coupled. γ1 < L1: 1D coupling depth, maximum coupling distance along a constituent code. γ2 < L2: MD coupling depth, maximum coupling distance among constituent codes. 0 ≤ T ≤ 1: edge density among constituent codes. Higher T means more edges among codes and vice versa.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 16 / 34

slide-26
SLIDE 26

Multi-Dimensional SC-LDPC ensemble: Parameters

Code Parameters dl, dr: degrees of VNs and CNs. L1: 1D coupling length, length of individual codes. L2: MD coupling length, number of codes coupled. γ1 < L1: 1D coupling depth, maximum coupling distance along a constituent code. γ2 < L2: MD coupling depth, maximum coupling distance among constituent codes. 0 ≤ T ≤ 1: edge density among constituent codes. Higher T means more edges among codes and vice versa. M: number of VNs in a section.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 16 / 34

slide-27
SLIDE 27

Multi-Dimensional SC-LDPC Ensemble: Construction

1

Assume we have sections that have M VNs and M(dl/dr) CNs each.

Figure: Example grid of L1 = 4 and L2 = 3.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 17 / 34

slide-28
SLIDE 28

Multi-Dimensional SC-LDPC Ensemble: Construction

1

Assume we have sections that have M VNs and M(dl/dr) CNs each.

2

Construct an L1 by L2 grid of sections and index them by the set [(L1, L2)]. All sections of the form (i, ·) are the ith constituent SC code. All sections of the form (·, j) are the jth segment of all constituent codes.

Figure: Example grid of L1 = 4 and L2 = 3.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 17 / 34

slide-29
SLIDE 29

Multi-Dimensional SC-LDPC Ensemble: Construction

3

Iterate over all sections (i, j) ∈ [(L1, L2)] and all VNS in the chosen section.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 18 / 34

slide-30
SLIDE 30

Multi-Dimensional SC-LDPC Ensemble: Construction

3

Iterate over all sections (i, j) ∈ [(L1, L2)] and all VNS in the chosen section.

4

For each neighboring dl edges of the selected VN, toss an unfair coin and perform

  • ne of the following:

◮ With probability 1 − T , a section is randomly chosen from

{(i + k, j) : k ∈ [γ1]}.

◮ With probability T , a section is randomly chosen from

{(i + k, (j + r)L2) : k ∈ [γ1], r ∈ [γ2] \ {0}}.

Figure: Example of the coupling choices for γ1 = γ2 = 3.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 18 / 34

slide-31
SLIDE 31

Multi-Dimensional SC-LDPC Ensemble: Construction

3

Iterate over all sections (i, j) ∈ [(L1, L2)] and all VNS in the chosen section.

4

For each neighboring dl edges of the selected VN, toss an unfair coin and perform

  • ne of the following:

◮ With probability 1 − T , a section is randomly chosen from

{(i + k, j) : k ∈ [γ1]}.

◮ With probability T , a section is randomly chosen from

{(i + k, (j + r)L2) : k ∈ [γ1], r ∈ [γ2] \ {0}}.

5

After selecting a section, randomly choose a CN from the M(dl/dr) CNs in that section to connect the edge to.

Figure: Example of the coupling choices for γ1 = γ2 = 3.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 18 / 34

slide-32
SLIDE 32

Multi-Dimensional SC-LDPC Ensemble: Construction

Figure: Example of MD coupling for γ1 = γ2 = 2. Blue lines indicate coupling within the same

constituent code and black lines indicate coupling across segments.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 19 / 34

slide-33
SLIDE 33

Density Evolution Equations

As M → ∞, we use density evolution to analyze the asymptotic reliability over the BEC. After l iterations, x(l)

i,j and y(l) i,j denote the

erasure probability of outgoing messages from a VN and CN in section (i, j), respectively.

y(l+1)

(i,j)

= 1 −

  • 1 − 1 − T

γ1

γ1−1

  • k=0

x(l)

(i−k,j) −

T γ1 (γ2 − 1)

γ1−1

  • k=0

γ2−1

  • r=1

x(l)

  • i−k,(j−r)L2
  • dr−1

x(l+1)

(i,j)

= ǫ

  • 1 − T

γ1

γ1−1

  • k=0

y(l+1)

(i+k,j) +

T γ1 (γ2 − 1)

γ1−1

  • k=0

γ2−1

  • r=1

y(l+1)

  • i+k,(j+r)L2
  • dl−1

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 20 / 34

slide-34
SLIDE 34

Density Evolution Equations

As M → ∞, we use density evolution to analyze the asymptotic reliability over the BEC. After l iterations, x(l)

i,j and y(l) i,j denote the

erasure probability of outgoing messages from a VN and CN in section (i, j), respectively.

y(l+1)

(i,j)

= 1 −

  • 1 − 1 − T

γ1

γ1−1

  • k=0

x(l)

(i−k,j) −

T γ1 (γ2 − 1)

γ1−1

  • k=0

γ2−1

  • r=1

x(l)

  • i−k,(j−r)L2
  • dr−1

x(l+1)

(i,j)

= ǫ

  • 1 − T

γ1

γ1−1

  • k=0

y(l+1)

(i+k,j) +

T γ1 (γ2 − 1)

γ1−1

  • k=0

γ2−1

  • r=1

y(l+1)

  • i+k,(j+r)L2
  • dl−1

When T = 0 or y2 = 1, these equations degenerate to the equations for standard SC-LDPC code ensemble.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 20 / 34

slide-35
SLIDE 35

Properties of Ensemble

Interestingly, our ensemble has the same normalized rate and block decoding threshold regardless of the values of T or γ2. When using a block decoder, our ensemble performs exactly the same as a standard SC ensemble. The benefits of this new construction arise from how we utilize the second dimension to allow for Non-Uniform Windowed Decoding.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 21 / 34

slide-36
SLIDE 36

Outline

1

Background and Previous Work Spatially-Coupled LDPC Previous Work: Multi-Dimensional SC-LDPC

2

Non-Uniform Windowed Decoding Motivation and Contributions Ensemble Definition Decoder Design

3

Simulations

4

Ongoing Work

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 22 / 34

slide-37
SLIDE 37

Non-Uniform Windowed Decoder

Let W = [W0, W1, . . . , WL2−1] be the window shape. δ: the target erasure probability ǫδ,W : windowed decoding threshold such that all VNs reach target erasure probability Let section (i, j) be the target VN section. Wk refers to the number of sections used from ((j + k) mod L2)th code inside the window. Belief propagation is performed on the window until the probability of erasure for Target VNs reaches δ. Example: A non-uniform decoder with targeted VNs in section (1, 2) and window shape W = [5, 4, 3, 3, 4].

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 23 / 34

slide-38
SLIDE 38

Non-Uniform Windowed Decoder

Let W = [W0, W1, . . . , WL2−1] be the window shape. δ: the target erasure probability ǫδ,W : windowed decoding threshold such that all VNs reach target erasure probability Let section (i, j) be the target VN section. Wk refers to the number of sections used from ((j + k) mod L2)th code inside the window. Belief propagation is performed on the window until the probability of erasure for Target VNs reaches δ. Example: A non-uniform decoder with targeted VNs in section (1, 2) and window shape W = [5, 4, 3, 3, 4].

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 23 / 34

slide-39
SLIDE 39

Complexity and Latency Analysis

Decoder Complexity

◮ Can be measured by the total number of updates performed for

iterative decoding.

◮ Given a fixed number of iterations, the number of updates must then

be O(number of VNs) = O(

k Wk).

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 24 / 34

slide-40
SLIDE 40

Complexity and Latency Analysis

Decoder Complexity

◮ Can be measured by the total number of updates performed for

iterative decoding.

◮ Given a fixed number of iterations, the number of updates must then

be O(number of VNs) = O(

k Wk).

Decoder Latency

◮ Latency is the time needed to access and decode the VNs. ◮ It is upper-bounded by O(

k Wk)

◮ Latency reduction from a full block-decoder is O(

  • k Wk

L1L2 ).

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 24 / 34

slide-41
SLIDE 41

Complexity and Latency Analysis

Decoder Complexity

◮ Can be measured by the total number of updates performed for

iterative decoding.

◮ Given a fixed number of iterations, the number of updates must then

be O(number of VNs) = O(

k Wk).

Decoder Latency

◮ Latency is the time needed to access and decode the VNs. ◮ It is upper-bounded by O(

k Wk)

◮ Latency reduction from a full block-decoder is O(

  • k Wk

L1L2 ).

Hence, the measure of interest is the total number of VNs used in the window, i.e.

k Wk.

Let C be the window complexity constraint such that

k Wk ≤ C.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 24 / 34

slide-42
SLIDE 42

Key Design Parameters

Design degrees of freedom:

1

What order should the Windows be processed in?

2

What is the best Window Shape given the latency/complexity constraint?

3

How many iterations of belief propagation should each window perform?

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 25 / 34

slide-43
SLIDE 43

Key Design Parameters

Design degrees of freedom:

1

What order should the Windows be processed in?

2

What is the best Window Shape given the latency/complexity constraint?

3

How many iterations of belief propagation should each window perform?

Resolving the first two bullets, the iterations per window is chosen to guarantee enough iterations for the density evolution to succeed. For the first two bullets, we provide rationale and simulations to justify our choices.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 25 / 34

slide-44
SLIDE 44

(1) Processing Order

Through simulation and theory, the ensemble has a decoding wave going along the first coupling dimension while quickly converging along the second dimension. This motivates decoding the same section for each constituent code first before moving on to the next section. Example of decoding wave propagation within a constituent code.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 26 / 34

slide-45
SLIDE 45

(1) Processing Order, Cont.

Since we first decode along the second dimension, we decide the order of processing the VNs across constituent

  • codes. Example: Natural order.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 27 / 34

slide-46
SLIDE 46

(1) Processing Order, Cont.

Since we first decode along the second dimension, we decide the order of processing the VNs across constituent

  • codes. Example: Natural order.

The order among codes does affect the decoding threshold as all orderings face a worst-case window configuration.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 27 / 34

slide-47
SLIDE 47

(1) Processing Order, Cont.

Since we first decode along the second dimension, we decide the order of processing the VNs across constituent

  • codes. Example: Natural order.

The order among codes does affect the decoding threshold as all orderings face a worst-case window configuration. The order does matter when considering the speed of decoding convergence.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 27 / 34

slide-48
SLIDE 48

(1) Processing Order, Cont.

Since we first decode along the second dimension, we decide the order of processing the VNs across constituent

  • codes. Example: Natural order.

The order among codes does affect the decoding threshold as all orderings face a worst-case window configuration. The order does matter when considering the speed of decoding convergence. Example: Average number of iterations for different orderings Code Parameters dl = 4, dr = 8, L1 = 30, L2 = 19, γ1 = γ2 = 2, T = 0.1, C = 19, and δ = 10−12.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 27 / 34

slide-49
SLIDE 49

(2) Optimal Window Shape

Given a fixed processing order, we find the optimal W to maximize the windowed decoding threshold while satisfying

k Wk ≤ C.

Theorem

Given the constraint

k Wk ≤ C, there exists a

W ′ = [W ′

0, W ′ 1, . . . , W ′ L2−1] that satisfies k W ′ k = C and has the largest

windowed decoding threshold. Thus, consider only window shapes that satisfy

k Wk = C which

significantly reduces the search space.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 28 / 34

slide-50
SLIDE 50

Outline

1

Background and Previous Work Spatially-Coupled LDPC Previous Work: Multi-Dimensional SC-LDPC

2

Non-Uniform Windowed Decoding Motivation and Contributions Ensemble Definition Decoder Design

3

Simulations

4

Ongoing Work

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 29 / 34

slide-51
SLIDE 51

Thresholds for Different Code Parameters

Given a decoder complexity C, these are the best window shapes that give the largest threshold ǫδ,W for dl = 4, dr = 8, γ1 = 2, and δ = 10−12. To reduce the search space, window sizes are constrained between 2 and 7 for L2 = 7, and between 2 and 5 for L2 = 9. L2 γ2 T C W ǫδ,W 7 2 0.05 28 (5, 5, 4, 2, 3, 4, 5) ≈ 0.4829 7 2 0.1 28 (5, 5, 4, 3, 3, 4, 4) ≈ 0.4722 7 3 0.05 28 (5, 4, 4, 3, 4, 4, 4) ≈ 0.4723 7 3 0.1 28 (4, 4, 4, 4, 4, 4, 4) ≈ 0.4685 9 2 0.05 36 (5, 5, 4, 3, 2, 3, 4, 5, 5) ≈ 0.4872 9 2 0.1 36 (5, 5, 4, 3, 2, 3, 4, 5, 5) ≈ 0.4806 9 3 0.05 36 (5, 5, 5, 2, 3, 3, 4, 4, 5) ≈ 0.4767 9 3 0.1 36 (4, 4, 4, 4, 4, 4, 4, 4, 4) ≈ 0.4685

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 30 / 34

slide-52
SLIDE 52

Effect of Complexity on Average Number of Iterations

The plot shows the average number of iterations for the best W given C. The uniform decoder is provided for comparison. Note that for a uniform window decoder, the complexity C must always be a multiple of L2. Code Parameters: dl = 4, dr = 8, γ1 = γ2 = 2, L1 = 30, L2 = 9, T = 0.1, and δ = 10−12.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 31 / 34

slide-53
SLIDE 53

Effect of Complexity on Average Number of Iterations

The plot shows the average number of iterations for the best W given C. The uniform decoder is provided for comparison. Note that for a uniform window decoder, the complexity C must always be a multiple of L2. Code Parameters: dl = 4, dr = 8, γ1 = γ2 = 2, L1 = 30, L2 = 9, T = 0.1, and δ = 10−12.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 31 / 34

slide-54
SLIDE 54

Effect of Complexity on Average Number of Iterations

The plot shows the average number of iterations for the best W given C. The uniform decoder is provided for comparison. Note that for a uniform window decoder, the complexity C must always be a multiple of L2. Code Parameters: dl = 4, dr = 8, γ1 = γ2 = 2, L1 = 30, L2 = 9, T = 0.1, and δ = 10−12.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 31 / 34

slide-55
SLIDE 55

Outline

1

Background and Previous Work Spatially-Coupled LDPC Previous Work: Multi-Dimensional SC-LDPC

2

Non-Uniform Windowed Decoding Motivation and Contributions Ensemble Definition Decoder Design

3

Simulations

4

Ongoing Work

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 32 / 34

slide-56
SLIDE 56

Conclusion and Ongoing Work

Summary:

◮ We introduced a new Non-Uniform Windowed Decoder that allows for

more flexibility in the trade-off between performance and decoder complexity.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 33 / 34

slide-57
SLIDE 57

Conclusion and Ongoing Work

Summary:

◮ We introduced a new Non-Uniform Windowed Decoder that allows for

more flexibility in the trade-off between performance and decoder complexity.

Ongoing Work:

◮ Theoretic understanding of how the code parameters affect the

performance of the non-uniform windowed decoder.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 33 / 34

slide-58
SLIDE 58

Conclusion and Ongoing Work

Summary:

◮ We introduced a new Non-Uniform Windowed Decoder that allows for

more flexibility in the trade-off between performance and decoder complexity.

Ongoing Work:

◮ Theoretic understanding of how the code parameters affect the

performance of the non-uniform windowed decoder.

◮ Additionally, we want to leverage finite-length analysis to guarantee

good non-asymptotic performance.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 33 / 34

slide-59
SLIDE 59

Conclusion and Ongoing Work

Summary:

◮ We introduced a new Non-Uniform Windowed Decoder that allows for

more flexibility in the trade-off between performance and decoder complexity.

Ongoing Work:

◮ Theoretic understanding of how the code parameters affect the

performance of the non-uniform windowed decoder.

◮ Additionally, we want to leverage finite-length analysis to guarantee

good non-asymptotic performance.

◮ Designing heterogeneous MD-SC codes for improved flexibility in code

design, i.e. rate and degree distribution, and use in Time-Varying channels.

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 33 / 34

slide-60
SLIDE 60

References

(Iyengar ’12) A. R. Iyengar et al., ”Windowed decoding of protograph-based LDPC convolutional codes over erasure channels”, IEEE TIT, 2012 (Ohashi ’13) R. Ohashi et al., ”Multi-Dimensional Spatially-Coupled Codes”, ISIT, 2013 (Schmalen ’14) L. Schmalen et al., ”Laterally Connected Spatially Coupled Code Chains for Transmission over Unstable Parallel Channels”, ISTC, 2014 (Liu ’15) Y. Liu et al.,”Spatially coupled LDPC codes constructed by parallelly connecting multiple chains”, IEEE Communication Letters, 2015 (Olmos ’17) P. M. Olmos et al., ”Continuous transmission of spatially coupled LDPC code chains”, IEEE TCOM, 2017 (Truhachev ’19) D. Truhachev et al., ”Code design based on connecting spatially coupled graph chains”, IEEE TIT, 2019 (Esfahanizadeh ’20) H. Esfahanizadeh et al., ”Multi-Dimensional Spatially-Coupled Code Design: Enhancing the Cycle Properties”, IEEE TCOM, 2020 (Hareedy ’20) A. Hareedy et al., ”Minimizing the number of detrimental objects in multi-dimensional graph-based codes”, IEEE TCOM, 2020

Tauz, Esfahanizadeh, Dolecek (UCLA) ISIT 2020 34 / 34