Partially Information Coupled Duo-binary Turbo Codes Xiaowei Wu, Min - - PowerPoint PPT Presentation

SLIDE 1

Partially Information Coupled Duo-binary Turbo Codes

Xiaowei Wu, Min Qiu, and Jinhong Yuan

School of Electrical Engineering and Telecommunications University of New South Wales Sydney, Australia

ISIT 2020

Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 1 / 28

SLIDE 2

Outline

1

Introduction

2

Construction of partially information coupled duo-binary turbo codes (PIC-dTCs)

3

Performance analysis of PIC-dTCs via density evolution

4

Numerical results

5

Conclusion

Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 2 / 28

SLIDE 3

Table of Contents

1

Introduction

2

Construction of partially information coupled duo-binary turbo codes (PIC-dTCs)

3

Performance analysis of PIC-dTCs via density evolution

4

Numerical results

5

Conclusion

Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 3 / 28

SLIDE 4

Introduction

Spatial coupling: Connects a sequence of component codes to form a long codeword chain. Applied to: LDPC codes [1][2], turbo-like codes [3][4]. Spatially coupled codes provide close-to-capacity performance. Spatially coupled codes have threshold saturation phenomenon.

Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 4 / 28 [1]

• A. Jimenez Felstrom and K. S. Zigangirov, “Time-varying periodic convolutional codes with low-density parity-check matrix,” vol. 45,
• no. 6, Sep. 1999, pp. 2181–2191.

[2]

• S. Kudekar, T. J. Richardson, and R. L. Urbanke, “Threshold saturation via spatial coupling: Why convolutional LDPC ensembles perform

so well over the bec,” IEEE Trans. Inf. Theory, vol. 57, no. 2, pp. 803–834, Feb 2011. [3]

• S. Moloudi, M. Lentmaier, and A. Graell i Amat, “Spatially coupled turbo-like codes,” IEEE Trans. Inf. Theory, vol. 63, no. 10, pp.

6199–6215, Oct 2017. [4]

• W. Zhang, M. Lentmaier, K. S. Zigangirov, and D. J. Costello, “Braided convolutional codes: A new class of turbo-like codes,” IEEE
• Trans. Inf. Theory, vol. 56, no. 1, pp. 316–331, Jan 2010.

SLIDE 5

Introduction

In [5][6], we proposed the partially information coupled turbo codes (PIC-TCs). Consecutive component turbo code blocks (CBs) are coupled by sharing a portion of information bits between each other. Achieve a large coupling gain without modifying the component encoder and decoder architecture. However, PIC-TCs have rate loss compared to its component turbo codes, i.e., R < 1

3.

e.g. coupling 1

2 the information bits results in R = 1 5.

Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 5 / 28 [5]

• L. Yang, Y. Xie, X. Wu, J. Yuan, X. Cheng, and L. Wan, “Partially information-coupled turbo codes for LTE systems,” IEEE Trans.

Commun., pp. 1–1, 2018. [6]

• M. Qiu, X. Wu, Y. Xie, and J. Yuan, “Density evolution analysis of partially information coupled turbo codes on the erasure channel,”

in 2019 IEEE Information Theory Workshop (ITW), 2019, pp. 1–5.

SLIDE 6

Introduction

Our contribution in this work: We proposed the partially information coupled duo-binary turbo codes (PIC-dTCs), which can have a consistent code rate R = 1

3 (no rate loss).

We derive the density evolution (DE) equations for the PIC-dTCs, and show that the BP decoding threshold of PIC-dTCs is within a gap of 0.001 to the BEC capacity. Simulation results verify the DE analysis, and show that PIC-dTCs outperform PIC-TCs and the uncoupled turbo codes.

Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 6 / 28

SLIDE 7

Review of PIC-TCs construction

(t 1)-th CB Encoding

1 1

[ , ]

t t

 

u v

1 1

, t t

 

u

1 t

u

1, t t 

u

t-th CB Encoding

[ , ]

t t

u v

, t t

u

t

u

, 1 t t

u

(t+1)-th CB Encoding

1 1

[ , ]

t t

 

u v

1 1

, t t

 

u

1 t

u

, 1 t t

u

• Fig. 1: Block diagram of PIC-TC encoder with

coupling memory m = 1.

Component code: rate- 1

3 turbo code (TC1).

CBt: component code block at time t. ut: information sequence at time t. ut,t: uncoupled info. ut−1,t: info. shared between CBt−1 and CBt. ut,t+1: info. shared between CBt and CBt+1. vt: parity bits. ut−1,t + ut = K. ut−1,t = ut,t+1 = Kc.

Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 7 / 28

SLIDE 8

Review of PIC-TCs construction

t-th CB Encoding

[ , ]

t t

u v

, t t

u

t

u

, 1

[

t t

u

,

]

t t m 

u , ,

,

[

t m t 

u

1, ] t t 

u , ,

• Fig. 2: Block diagram of PIC-TC encoder with

coupling memory m ≥ 1.

ut−i,t: info. shared between CBt−i and CBt, 1≤i≤m. ut,t+i: info. shared between CBt and CBt+i, 1≤i≤m.

• 1≤i≤mut−i,t + ut = K.

ut−i,t = ut,t+i = Kc

m .

Number of CBs: L. Coupling ratio: λ = Kc

K .

Code rate: R = KL−KcL− m

i=1 iKc m

3KL−KcL− m

i=1 iKc m L→∞

= 1−λ 3−λ. ⊲ Rate loss due to coupling.

Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 8 / 28

SLIDE 9

Table of Contents

1

Introduction

2

Construction of partially information coupled duo-binary turbo codes (PIC-dTCs)

3

Performance analysis of PIC-dTCs via density evolution

4

Numerical results

5

Conclusion

Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 9 / 28

SLIDE 10

Construction of PIC-dTCs

Step 1: Construct a rate-2/3 RSC code (RSC2) with a given rate-1/2 RSC code (RSC1). GRSC1 =

• 1 5

7

• , and GRSC2 =
• 1 0 5

7

0 1 3

7

• .

When u′ = 0, the parity sequence from RSC2 is equal to that of RSC1.

D D 𝐯 𝐰 𝐯

(a)

D D 𝐯′ 𝐯 𝐰 𝐯′ 𝐯

(b)

• Fig. 3: Encoder block diagram of (a) RSC1, (b) RSC2.

Step 2: Construct a duo-binary turbo code (TC2) by concatenate RSC2 in parallel.

Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 10 / 28 [7]

• C. Berrou and M. Jezequel, “Non-binary convolutional codes for turbo coding,” Electronics Letters, vol. 35, no. 1, pp. 39–40, Jan 1999.

[8]

• C. Douillard and C. Berrou, “Turbo codes with rate-m/(m+1) constituent convolutional codes,” IEEE Trans. Commun., vol. 53, no. 10,
• pp. 1630–1638, Oct 2005.

SLIDE 11

Construction of PIC-dTCs

(t 1)-th CB Encoding

1

t

 u

1 1

[ , ]

t t

 

u v

1, t t 

u

1, 1 t t  

u

1 t

u

t-th CB Encoding

t

u [ , ]

t t

u v

, 1 t t

u

, t t

u

t

u

(t+1)-th CB Encoding

1

t

 u

1 1

[ , ]

t t

 

u v

1, 2 t t  

u

1, 1 t t  

u

1 t

u

• Fig. 4: Block diagram of PIC-dTC encoder with

coupling memory m = 1.

Step 3: Apply PIC to TC2 (coupling memory m = 1)

ut: the 1st input sequence of TC2 at time t. u′

t: the 2nd input sequence of TC2 at time t.

ut,t: uncoupled info. ut−1,t: info. shared between CBt−1 and CBt. ut,t+1: info. shared between CBt and CBt+1. vt: parity bits. ut = u′

t = K.

ut−1,t = ut,t+1 = Kc.

Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 11 / 28

SLIDE 12

Construction of PIC-dTCs

t-th CB Encoding

t

u [ , ]

t t

u v

, t t

u

t

u

, 1

[

t t

u

,

]

t t m 

u , ,

,

[

t m t 

u

1, ] t t 

u , ,

• Fig. 5: Block diagram of PIC-dTC encoder with

coupling memory m ≥ 1.

Step 3: Apply PIC to TC2 (coupling memory m ≥ 1)

ut−i,t: info. shared between CBt−i and CBt, 1≤i≤m. ut,t+i: info. shared between CBt and CBt+i, 1≤i≤m. ut = u′

t = K.

ut−i,t = ut,t+i = Kc

m .

Number of CBs: L. Coupling ratio: λ = Kc

K .

Code rate: R = KL− m

i=1 iKc m

3KL− m

i=1 iKc m L→∞

= 1 3. ⊲ no rate loss compared with TC1.

Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 12 / 28

SLIDE 13

PIC-TC vs. PIC-dTC

PIC-TC PIC-dTC

Component

TC1 TC2

code

(parallel concatenation of rate- 1

2 RSC)

(parallel concatenation of rate- 2

3 RSC)

Code rate

1−λ 3−λ 1 3

Encoder block diagram

t-th CB Encoding

[ , ]

t t

u v

, t t

u

t

u

, 1

[

t t

u

,

]

t t m 

u , ,

,

[

t m t 

u

1, ] t t 

u , ,

t-th CB Encoding

t

u [ , ]

t t

u v

, t t

u

t

u

, 1

[

t t

u

,

]

t t m 

u , ,

,

[

t m t 

u

1, ] t t 

u , ,

Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 13 / 28

SLIDE 14

Table of Contents

1

Introduction

2

Construction of partially information coupled duo-binary turbo codes (PIC-dTCs)

3

Performance analysis of PIC-dTCs via density evolution

4

Numerical results

5

Conclusion

Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 14 / 28

SLIDE 15

Graph Model Representation

𝐯𝑢 𝐰𝑢

𝑉

𝑔𝑉 𝐰𝑢

𝑀

𝑔𝑀 𝐯𝑢

′

(a) 𝑔𝑉 𝑔𝑉 𝑔𝑉 𝑔𝑀 𝑔𝑀 𝑔𝑀

𝐯t,t 𝟏

𝐰𝑢−1

𝑉

𝐰𝑢

𝑉

𝐰𝑢+1

𝑉

𝐰𝑢−1

𝑀

𝐰𝑢

𝑀

𝐰𝑢+1

𝑀

𝐯t+1,t+1 𝐯t−1,t−1 𝟏

𝟏

𝐯t,t+1 𝐯t−1,t 𝐯t+1,t+2 𝐯t−2,t−1 𝐯𝑢 𝐯𝑢

′

𝐯𝑢+1 𝐯𝑢+1

′

𝐯𝑢−1 𝐯𝑢−1

′

⋮ ⋮

(b)

• Fig. 6: Compact factor graph of (a) uncoupled TC2, and

(b) PIC-dTC with m = 1.

f U: upper decoder. f L: lower decoder. u: first information sequence. u‘: second information sequence. vU, vL: parity sequence which enters upper or lower decoder. : interleaver.

Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 15 / 28 [9]

• B. M. Kurkoski, P. H. Siegel, and J. K. Wolf, “Exact probability of erasure and a decoding algorithm for convolutional codes on the

binary erasure channel,” in IEEE Global Telecommunications Conference, vol. 3, Dec 2003, pp. 1741–1745 vol.3.

SLIDE 16

Performance analysis of PIC-dTC with m = 1 via density evolution

𝑔𝑉 𝑔𝑉 𝑔𝑉 𝑔𝑀 𝑔𝑀 𝑔𝑀 𝐰𝑢−1

𝑉

𝐰𝑢

𝑉

𝐰𝑢+1

𝑉

𝐰𝑢−1

𝑀

𝐰𝑢

𝑀

𝐰𝑢+1

𝑀

⋮ ⋮

𝑞1,𝑀,𝑢

(𝑗)

𝑞2,𝑀,𝑢

(𝑗)

𝑞2,𝑀,𝑢

(𝑗)

𝑞1,𝑀,𝑢

(𝑗)

𝑞2,𝑀,𝑢+1

(𝑗)

𝑞1,𝑀,𝑢−1

(𝑗)

𝑞2,𝑉,𝑢+1

(𝑗−1)

𝑞1,𝑉,𝑢−1

(𝑗−1)

• Fig. 7: Compact factor graph of PIC-dTC with m = 1.

At the upper decoder of CBt: the extrinsic information of ut are from: ⊲ ut at the lower decoder of CBt. ⊲ u′

t+1 at the upper and lower decoder of CBt+1.

the extrinsic information of u′

t are from:

⊲ u′

t at the lower decoder of CBt.

⊲ ut−1 at the upper and lower decoder of CBt−1. Average extrinsic erasure probability from ut and u′

t to the upper decoder:

¯ p(i)

1,L,t = ε·p(i) 1,L,t·

• 1−λ+λ·p(i−1)

2,U,t+1·p(i) 2,L,t+1

• .

¯ p(i)

2,L,t = ε·λ·p(i) 2,L,t·p(i−1) 1,U,t−1·p(i) 1,L,t−1.

⊲ ǫ: BEC erasure probability

Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 16 / 28 [9]

• B. M. Kurkoski, P. H. Siegel, and J. K. Wolf, “Exact probability of erasure and a decoding algorithm for convolutional codes on the

binary erasure channel,” in IEEE Global Telecommunications Conference, vol. 3, Dec 2003, pp. 1741–1745 vol.3.

SLIDE 17

Performance analysis of PIC-dTC with m = 1 via density evolution

𝑞1,𝑉,𝑢

(𝑗)

𝑔𝑉 𝑔𝑉 𝑔𝑉 𝑔𝑀 𝑔𝑀 𝑔𝑀 𝐰𝑢−1

𝑉

𝐰𝑢

𝑉

𝐰𝑢+1

𝑉

𝐰𝑢−1

𝑀

𝐰𝑢

𝑀

𝐰𝑢+1

𝑀

⋮ ⋮

𝑞2,𝑉,𝑢

(𝑗)

• Fig. 7: Compact factor graph of PIC-dTC with m = 1.

At the upper decoder of CBt: The evolution of erasure probability inside the upper decoder: ⊲ ut: p(i)

1,U,t = F U 1

• ¯

p(i)

1,L,t, ¯

p(i)

2,L,t, ε

• .

⊲ u′

t: p(i) 2,U,t = F U 2

• ¯

p(i)

1,L,t, ¯

p(i)

2,L,t, ε

• .

⊲ Transfer functions F U

1 and F U 2 are derived

base on [9].

Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 17 / 28 [9]

• B. M. Kurkoski, P. H. Siegel, and J. K. Wolf, “Exact probability of erasure and a decoding algorithm for convolutional codes on the

binary erasure channel,” in IEEE Global Telecommunications Conference, vol. 3, Dec 2003, pp. 1741–1745 vol.3.

SLIDE 18

Performance analysis of PIC-dTC with m ≥ 1 via density evolution

𝐯t,t 𝑔𝑉 𝑔𝑀 𝐰𝑢

𝑉

𝐰𝑢

𝑀

𝟏 𝐯t,t+1

⋮ 𝐯t,t+𝑛

𝐯t−1,t ⋮ 𝐯t−m,t

• Fig. 8: Compact factor graph of PIC-dTC

with m ≥ 1.

At the upper decoder of CBt: the extrinsic information of ut are from: ⊲ ut at the lower decoder of CBt. ⊲ u′

t+j at the upper and lower decoder of CBt+j, 1 ≤ j ≤ m.

the extrinsic information of u′

t are from:

⊲ u′

t at the lower decoder of CBt.

⊲ ut−j at the upper and lower decoder of CBt−j, 1 ≤ j ≤ m. Average extrinsic erasure probability from ut and u′

t to

upper decoder: ¯ p(i)

1,L,t = ε·p(i) 1,L,t·

• 1 − λ + λ

m

m

j=1 p(i−1) 2,U,t+j·p(i) 2,L,t+j

• .

¯ p(i)

2,L,t = ε·p(i) 2,L,t· λ m

m

j=1 p(i−1) 1,U,t−j·p(i) 1,L,t−j.

Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 18 / 28 [9]

• B. M. Kurkoski, P. H. Siegel, and J. K. Wolf, “Exact probability of erasure and a decoding algorithm for convolutional codes on the

binary erasure channel,” in IEEE Global Telecommunications Conference, vol. 3, Dec 2003, pp. 1741–1745 vol.3.

SLIDE 19

Table of Contents

1

Introduction

2

Construction of partially information coupled duo-binary turbo codes (PIC-dTCs)

3

Performance analysis of PIC-dTCs via density evolution

4

Numerical results

5

Conclusion

Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 19 / 28

SLIDE 20

Density evolution results

0.25 0.5 0.75 1

Coupling ratio

0.645 0.65 0.655 0.66

Decoding threshold

0.6428 0.6578 0.6594 0.6547 0.6590

• Fig. 9: BP decoding thresholds of rate- 1

3 PIC-dTCs with m = 1.

When λ = 0, parity sequence of PIC-dTC is equal to the parity sequence of TC1. BP decoding threshold of TC1: 0.6428. PIC-dTC with m = 1 approaches the BEC capacity with a gap of 0.0073.

Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 20 / 28

SLIDE 21

Density evolution results

10 20 30 40 50

Coupling memory

0.65 0.652 0.654 0.656 0.658 0.66 0.662 0.664 0.666

Decoding threshold

λ=1/8 λ=1/4 λ=3/8 λ=1/2 λ=3/4 λ=1

0.6577 0.6527 0.6625 0.6607 0.6646 0.6656

• Fig. 10: BP decoding thresholds of rate- 1

3 PIC-dTCs over coupling memory m.

BP decoding threshold of TC1: 0.6428. PIC-dTC approaches the BEC capacity with a gap of 0.0011 when m = 50 and λ = 1.

Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 21 / 28

SLIDE 22

Density evolution results

Benchmark codes [3]: Spatially coupled parallel concatenated convolutional codes (SC-PCCs). Spatially coupled serially concatenated convolutional codes (SC-SCCs). Braided convolutional codes (BCCs).

Ensemble εBP , m = 1 εBP , m = 5 εBP , m = 50 PIC-TC1, λ = 1

2

0.6566 0.6625 0.6639 PIC-dTC, λ = 1 0.6594 0.6644 0.6656 SC-PCC 0.6553 0.6553 0.6553 SC-SCC 0.6437 0.6654 0.6654 BCC Type-I 0.6609 0.6650 0.6653 BCC Type-II 0.6651 0.6653 0.6653

Table 1: BP Decoding thresholds εBP of rate- 1

3 spatially coupled codes. Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 22 / 28

1Parity sequences of PIC-TCs are randomly punctured to increase the code rate to 1 3 .

[3]

• S. Moloudi, M. Lentmaier, and A. Graell i Amat, “Spatially coupled turbo-like codes,” IEEE Trans. Inf. Theory, vol. 63, no. 10, pp.

6199–6215, Oct 2017.

SLIDE 23

Error performance simulation results

0.644 0.646 0.648 0.65 0.652 0.654 0.656 0.658 0.66

BEC erasure prob.

10-6 10-5 10-4 10-3 10-2 10-1 100

Bit erasure rate

sim, λ=1/8

BP, λ=1/8

sim, λ=1/4

BP, λ=1/4

sim, λ=1/2

BP, λ=1/2

sim, λ=3/4

BP, λ=3/4

• Fig. 11: Error performance of rate- 1

3 PIC-dTCs.

• Info. length: K = 105.

Coupling memory: m = 1. Number of CBs: L = 100. All the PIC-dTCs are within 0.002 to the BP decoding threshold at a BER of 10−5.

Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 23 / 28

SLIDE 24

Error performance simulation results

0.64 0.642 0.644 0.646 0.648 0.65 0.652 0.654 0.656 0.658

BEC erasure prob.

10-6 10-5 10-4 10-3 10-2 10-1 100

Bit erasure rate

TC1 PIC-dTC, λ=1/8 PIC-dTC, λ=1/4 PIC-dTC, λ=1/2 PIC-dTC, λ=3/4 PIC-TC, λ=1/8 PIC-TC, λ=1/4 PIC-TC, λ=1/2

• Fig. 12: Error performance of rate- 1

3 PIC-dTCs and PIC-TCs.

For TC1:

• Info. length: K = 106.

For PIC-dTCs and PIC-TCs:

• Info. length: K = 104.

Coupling memory: m = 1. Number of CBs: L = 100. Parity sequences of PIC-TCs are randomly punctured to increase the code rate to 1

3.

PIC-dTCs outperform TC1. PIC-dTCs outperform PIC-TCs when λ ≥ 1

4.

Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 24 / 28

SLIDE 25

Table of Contents

1

Introduction

2

Construction of partially information coupled duo-binary turbo codes (PIC-dTCs)

3

Performance analysis of PIC-dTCs via density evolution

4

Numerical results

5

Conclusion

Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 25 / 28

SLIDE 26

Conclusion Our contributions:

We construct the partially information coupled duo-binary turbo codes (PIC-dTCs), which solve the rate loss problem of the PIC-TCs. The PIC-dTCs have close-to-capacity performance. The PIC-dTCs outperform some existing spatially coupled turbo-like codes.

Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 26 / 28

SLIDE 27

References

[1] A. Jimenez Felstrom and K. S. Zigangirov, “Time-varying periodic convolutional codes with low-density parity-check matrix,” vol. 45, no. 6, Sep. 1999, pp. 2181–2191. [2] S. Kudekar, T. J. Richardson, and R. L. Urbanke, “Threshold saturation via spatial coupling: Why convolutional LDPC ensembles perform so well over the bec,” IEEE Trans. Inf. Theory, vol. 57, no. 2, pp. 803–834, Feb 2011. [3] S. Moloudi, M. Lentmaier, and A. Graell i Amat, “Spatially coupled turbo-like codes,” IEEE Trans. Inf. Theory,

• vol. 63, no. 10, pp. 6199–6215, Oct 2017.

[4] W. Zhang, M. Lentmaier, K. S. Zigangirov, and D. J. Costello, “Braided convolutional codes: A new class of turbo-like codes,” IEEE Trans. Inf. Theory, vol. 56, no. 1, pp. 316–331, Jan 2010. [5] L. Yang, Y. Xie, X. Wu, J. Yuan, X. Cheng, and L. Wan, “Partially information-coupled turbo codes for LTE systems,” IEEE Trans. Commun., pp. 1–1, 2018. [6] M. Qiu, X. Wu, Y. Xie, and J. Yuan, “Density evolution analysis of partially information coupled turbo codes on the erasure channel,” in 2019 IEEE Information Theory Workshop (ITW), 2019, pp. 1–5. [7] C. Berrou and M. Jezequel, “Non-binary convolutional codes for turbo coding,” Electronics Letters, vol. 35, no. 1, pp. 39–40, Jan 1999. [8] C. Douillard and C. Berrou, “Turbo codes with rate-m/(m+1) constituent convolutional codes,” IEEE Trans. Commun., vol. 53, no. 10, pp. 1630–1638, Oct 2005. [9] B. M. Kurkoski, P. H. Siegel, and J. K. Wolf, “Exact probability of erasure and a decoding algorithm for convolutional codes on the binary erasure channel,” in IEEE Global Telecommunications Conference, vol. 3, Dec 2003, pp. 1741–1745 vol.3.

Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 27 / 28

SLIDE 28

Thank You!

Xiaowei Wu (UNSW) PIC-dTC ISIT 2020 28 / 28