On the Optimal Delay Amplification Factor of Multi-Hop Relay - - PowerPoint PPT Presentation

On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels

Dennis Ogbe, Chih-Chun Wang, David J. Love

School of Electrical and Computer Engineering Purdue University West Lafayette, Indiana, USA International Symposium on Information Theory Paris, France, July 12, 2019

This work was supported in parts by the National Science Foundation under Grant CCF-1422997, Grant ECCS-1407604, Grant CCF-1618475, and Grant CCF-1816013.

Overview

◮ Part 1: Motivation & Intuition

Motivation A new metric for multi-hop relay channels: The Delay Amplification Factor Preview of main results

◮ Part 2: Main Results

(Necessary) details of problem set-up Main results & proof sketches Work in progress & Conclusion

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Overview

◮ Part 1: Motivation & Intuition

Motivation A new metric for multi-hop relay channels: The Delay Amplification Factor Preview of main results

◮ Part 2: Main Results

(Necessary) details of problem set-up Main results & proof sketches Work in progress & Conclusion

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Motivation

1

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Motivation

Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications

IoT Autonomous driving MTC

1

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Motivation

Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications

IoT Autonomous driving MTC

IMT-2020 URLLC: ≤ 1ms delay

1

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Motivation

Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures

Small cells Integrated access & backhaul IoT Autonomous driving MTC

IMT-2020 URLLC: ≤ 1ms delay

1

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Motivation

Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures For IT:

Small cells Integrated access & backhaul

Lots of relay channels

IoT Autonomous driving MTC

IMT-2020 URLLC: ≤ 1ms delay

1

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Motivation

Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures

◮ Recent technologies ⇒ renewed interest in delay-throughput tradeoff of relay channels

For IT:

Small cells Integrated access & backhaul

Lots of relay channels

IoT Autonomous driving MTC

IMT-2020 URLLC: ≤ 1ms delay

1

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Motivation

Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures

◮ Recent technologies ⇒ renewed interest in delay-throughput tradeoff of relay channels ◮ Interesting from practical perspective: Separated relay channel over multiple hops

For IT:

Small cells Integrated access & backhaul

Lots of relay channels

IoT Autonomous driving MTC

IMT-2020 URLLC: ≤ 1ms delay

s d r · · · s r1 r2 rL

− 1

d general separated separated multi-hop s r d 1

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Motivation

Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures

◮ Recent technologies ⇒ renewed interest in delay-throughput tradeoff of relay channels ◮ Interesting from practical perspective: Separated relay channel over multiple hops ◮ Capacity analysis is clear, delay-throughput analysis is not

For IT:

Small cells Integrated access & backhaul

Lots of relay channels

IoT Autonomous driving MTC

IMT-2020 URLLC: ≤ 1ms delay

s d r · · · s r1 r2 rL

− 1

d general separated separated multi-hop s r d capacity = min(C1, C2) capacity = min(C1, C2, . . . , CL) 1

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Motivation

Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Low latency communications Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures Flexible/dense network architectures

◮ Recent technologies ⇒ renewed interest in delay-throughput tradeoff of relay channels ◮ Interesting from practical perspective: Separated relay channel over multiple hops ◮ Capacity analysis is clear, delay-throughput analysis is not

For IT:

Small cells Integrated access & backhaul

Lots of relay channels

IoT Autonomous driving MTC

IMT-2020 URLLC: ≤ 1ms delay

s d r · · · s r1 r2 rL

− 1

d general separated separated multi-hop s r d capacity = min(C1, C2) capacity = min(C1, C2, . . . , CL)

◮ This work: When R → C, what relaying schemes minimize (relative) delay?

1

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

A new metric: The Delay Amplification Factor

d rL

− 1

r2 r1 s · · ·

2

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

A new metric: The Delay Amplification Factor

C = min

• {Cl}L

l=1

• C1

C2 CL d rL

− 1

r2 r1 s · · ·

2

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

A new metric: The Delay Amplification Factor

C = min

• {Cl}L

l=1

• end-to-end delay Te2e(R, ǫ)

C1 C2 CL d rL

− 1

r2 r1 s · · ·

2

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

A new metric: The Delay Amplification Factor

C = min

• {Cl}L

l=1

• end-to-end delay Te2e(R, ǫ)

bottleneck delay Tbn(R, ǫ)

C1 CL Cl∗ = Cl∗ d rL

− 1

r2 r1 s · · ·

(assume unique Cl∗)

2

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

A new metric: The Delay Amplification Factor

C = min

• {Cl}L

l=1

• end-to-end delay Te2e(R, ǫ)

bottleneck delay Tbn(R, ǫ)

C1 CL DAFΦ lim

RրC lim ǫ→0

Te2e(R, ǫ) Tbn(R, ǫ) Cl∗ = Cl∗ d rL

− 1

r2 r1 s · · ·

(assume unique Cl∗)

2

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

A new metric: The Delay Amplification Factor

C = min

• {Cl}L

l=1

• end-to-end delay Te2e(R, ǫ)

bottleneck delay Tbn(R, ǫ)

C1 CL

Error exponent approx. [Gallager, 1968]

Tbn(R, ǫ) − 1 Erc,l∗(R) log(ǫ) ǫ ≤ exp [−nE(R)] DAFΦ lim

RրC lim ǫ→0

Te2e(R, ǫ) Tbn(R, ǫ) Cl∗ = Cl∗ d rL

− 1

r2 r1 s · · · E(R) lim

n→∞

− log(ǫ) n

(assume unique Cl∗)

R Erc(R)

BSC C ≈ 0.5

• 1R. G. Gallager, Information Theory and Reliable Communication. New York, NY, USA: John Wiley & Sons, Inc., 1968.

2

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

A new metric: The Delay Amplification Factor

Te2e(R, ǫ) − 1 EΦ(R) log(ǫ)

End-to-end error exponent

C = min

• {Cl}L

l=1

• end-to-end delay Te2e(R, ǫ)

bottleneck delay Tbn(R, ǫ)

C1 CL

Error exponent approx. [Gallager, 1968]

Tbn(R, ǫ) − 1 Erc,l∗(R) log(ǫ) ǫ ≤ exp [−nE(R)] DAFΦ lim

RրC lim ǫ→0

Te2e(R, ǫ) Tbn(R, ǫ) Cl∗ = Cl∗ d rL

− 1

r2 r1 s · · · E(R) lim

n→∞

− log(ǫ) n

(assume unique Cl∗)

R Erc(R)

BSC C ≈ 0.5

• 1R. G. Gallager, Information Theory and Reliable Communication. New York, NY, USA: John Wiley & Sons, Inc., 1968.

2

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

A new metric: The Delay Amplification Factor

Te2e(R, ǫ) − 1 EΦ(R) log(ǫ)

EΦ(R): “Error exponent of relaying scheme Φ”

◮ Coding scheme ◮ Channel transition probabilities ◮ Operation at relays

End-to-end error exponent

C = min

• {Cl}L

l=1

• end-to-end delay Te2e(R, ǫ)

bottleneck delay Tbn(R, ǫ)

C1 CL

Error exponent approx. [Gallager, 1968]

Tbn(R, ǫ) − 1 Erc,l∗(R) log(ǫ) ǫ ≤ exp [−nE(R)] DAFΦ lim

RրC lim ǫ→0

Te2e(R, ǫ) Tbn(R, ǫ)

}

need more definitions

Cl∗ = Cl∗ d rL

− 1

r2 r1 s · · · E(R) lim

n→∞

− log(ǫ) n

(assume unique Cl∗)

R Erc(R)

BSC C ≈ 0.5

• 1R. G. Gallager, Information Theory and Reliable Communication. New York, NY, USA: John Wiley & Sons, Inc., 1968.

2

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

A new metric: The Delay Amplification Factor

Te2e(R, ǫ) − 1 EΦ(R) log(ǫ)

EΦ(R): “Error exponent of relaying scheme Φ”

◮ Coding scheme ◮ Channel transition probabilities ◮ Operation at relays

End-to-end error exponent

C = min

• {Cl}L

l=1

• end-to-end delay Te2e(R, ǫ)

bottleneck delay Tbn(R, ǫ)

C1 CL

Error exponent approx. [Gallager, 1968]

Tbn(R, ǫ) − 1 Erc,l∗(R) log(ǫ) ǫ ≤ exp [−nE(R)] DAFΦ lim

RրC lim ǫ→0

Te2e(R, ǫ) Tbn(R, ǫ) DAFΦ lim

RրC

Erc,l∗(R) EΦ(R)

• r, equivalently

}

need more definitions

Cl∗ = Cl∗ d rL

− 1

r2 r1 s · · · E(R) lim

n→∞

− log(ǫ) n

(assume unique Cl∗)

R Erc(R)

BSC C ≈ 0.5

• 1R. G. Gallager, Information Theory and Reliable Communication. New York, NY, USA: John Wiley & Sons, Inc., 1968.

2

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

A new metric: The Delay Amplification Factor

Te2e(R, ǫ) − 1 EΦ(R) log(ǫ)

EΦ(R): “Error exponent of relaying scheme Φ”

◮ Coding scheme ◮ Channel transition probabilities ◮ Operation at relays

End-to-end error exponent

C = min

• {Cl}L

l=1

• end-to-end delay Te2e(R, ǫ)

bottleneck delay Tbn(R, ǫ)

C1 CL

Error exponent approx. [Gallager, 1968]

Tbn(R, ǫ) − 1 Erc,l∗(R) log(ǫ) ǫ ≤ exp [−nE(R)] DAFΦ lim

RրC lim ǫ→0

Te2e(R, ǫ) Tbn(R, ǫ) DAFΦ lim

RրC

Erc,l∗(R) EΦ(R)

• r, equivalently

}

need more definitions

Cl∗ = Cl∗ d rL

− 1

r2 r1 s · · · E(R) lim

n→∞

− log(ǫ) n

(assume unique Cl∗)

R Erc(R)

BSC C ≈ 0.5

*Small Lemma: DAFΦ ≥ 1 ∀Φ

• 1R. G. Gallager, Information Theory and Reliable Communication. New York, NY, USA: John Wiley & Sons, Inc., 1968.

2

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Preview of main results

3

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Preview of main results

Definition We define the Delay Amplification factor of an L-hop relaying scheme Φ as DAFΦ lim

RրC

Erc,l∗(R) EΦ(R) .

3

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Preview of main results

Definition We define the Delay Amplification factor of an L-hop relaying scheme Φ as DAFΦ lim

RրC

Erc,l∗(R) EΦ(R) . Lemmas

• 1. Regardless of the relaying scheme, we always have DAF ≥ 1
• 2. For decode-&-forward schemes, we have DAFDF = O(L)

3

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Preview of main results

Definition We define the Delay Amplification factor of an L-hop relaying scheme Φ as DAFΦ lim

RրC

Erc,l∗(R) EΦ(R) . Lemmas

• 1. Regardless of the relaying scheme, we always have DAF ≥ 1
• 2. For decode-&-forward schemes, we have DAFDF = O(L)

Main results

• 1. A relaying scheme where DAFΦ = 1 if l∗ = L
• 2. A 1-bit stop feedback relaying scheme where DAFΦ′ = 1 regardless of

the position of l∗

3

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Overview

◮ Part 1: Motivation & Intuition

Motivation A new metric for multi-hop relay channels: The Delay Amplification Factor Preview of main results

◮ Part 2: Main Results

(Necessary) details of problem set-up Main results & proof sketches Work in progress & Conclusion

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Starting times, relay encoding & joint decoding

A relaying scheme is defined by

◮ Message M ∈ M {1, . . . , |M|} ◮ L deterministic starting times τ1 = 0 ≤ τ2 ≤ · · · ≤ τL ◮ Transmission duration Tdur

4

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Starting times, relay encoding & joint decoding

A relaying scheme is defined by

◮ Message M ∈ M {1, . . . , |M|} ◮ L deterministic starting times τ1 = 0 ≤ τ2 ≤ · · · ≤ τL ◮ Transmission duration Tdur message encode message decode message encode

Tdur = 9 τ2 = 2 τ3 = 3 τ4 = 6 Hop 1 Hop 2 Hop 3 Hop 4

4

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Starting times, relay encoding & joint decoding

A relaying scheme is defined by

◮ Message M ∈ M {1, . . . , |M|} ◮ L deterministic starting times τ1 = 0 ≤ τ2 ≤ · · · ≤ τL ◮ Transmission duration Tdur message encode message decode message encode

Tdur = 9 τ2 = 2 τ3 = 3 τ4 = 6 Hop 1 Hop 2 Hop 3 Hop 4

◮ Coding scheme

4

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Starting times, relay encoding & joint decoding

· · · X1 Y1 X2 Y2 XL YL s r1 r2 rL

− 1

d

A relaying scheme is defined by

◮ Message M ∈ M {1, . . . , |M|} ◮ L deterministic starting times τ1 = 0 ≤ τ2 ≤ · · · ≤ τL ◮ Transmission duration Tdur message encode message decode message encode

Tdur = 9 τ2 = 2 τ3 = 3 τ4 = 6 Hop 1 Hop 2 Hop 3 Hop 4

◮ Coding scheme

4

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Starting times, relay encoding & joint decoding

◮ Block encoding @ source

X1(t) = f [1]

t (M)

· · · X1 Y1 X2 Y2 XL YL s r1 r2 rL

− 1

d

A relaying scheme is defined by

◮ Message M ∈ M {1, . . . , |M|} ◮ L deterministic starting times τ1 = 0 ≤ τ2 ≤ · · · ≤ τL ◮ Transmission duration Tdur message encode message decode message encode

Tdur = 9 τ2 = 2 τ3 = 3 τ4 = 6 Hop 1 Hop 2 Hop 3 Hop 4

◮ Coding scheme

4

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Starting times, relay encoding & joint decoding

◮ Block encoding @ source ◮ Causal-sequential coding @ relays

Yl(t) = DMCl

• Xl(t)
• X1(t) = f [1]

t (M)

Xl(t) = f [l]

t ([Yl−1]t−1 ∗

)

· · · X1 Y1 X2 Y2 XL YL s r1 r2 rL

− 1

d

A relaying scheme is defined by

◮ Message M ∈ M {1, . . . , |M|} ◮ L deterministic starting times τ1 = 0 ≤ τ2 ≤ · · · ≤ τL ◮ Transmission duration Tdur message encode message decode message encode

Tdur = 9 τ2 = 2 τ3 = 3 τ4 = 6 Hop 1 Hop 2 Hop 3 Hop 4

◮ Coding scheme

4

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Starting times, relay encoding & joint decoding

◮ Block encoding @ source ◮ Causal-sequential coding @ relays ◮ Block decoding @ destination

Yl(t) = DMCl

• Xl(t)
• X1(t) = f [1]

t (M)

Xl(t) = f [l]

t ([Yl−1]t−1 ∗

)

• M = g([YL]τL+Tdur

∗

)

· · · X1 Y1 X2 Y2 XL YL s r1 r2 rL

− 1

d

A relaying scheme is defined by

◮ Message M ∈ M {1, . . . , |M|} ◮ L deterministic starting times τ1 = 0 ≤ τ2 ≤ · · · ≤ τL ◮ Transmission duration Tdur message encode message decode message encode

Tdur = 9 τ2 = 2 τ3 = 3 τ4 = 6 Hop 1 Hop 2 Hop 3 Hop 4

◮ Coding scheme

4

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Starting times, relay encoding & joint decoding

◮ Block encoding @ source ◮ Causal-sequential coding @ relays ◮ Block decoding @ destination

Yl(t) = DMCl

• Xl(t)
• X1(t) = f [1]

t (M)

Xl(t) = f [l]

t ([Yl−1]t−1 ∗

)

• M = g([YL]τL+Tdur

∗

)

· · · X1 Y1 X2 Y2 XL YL s r1 r2 rL

− 1

d

A relaying scheme is defined by

◮ Message M ∈ M {1, . . . , |M|} ◮ L deterministic starting times τ1 = 0 ≤ τ2 ≤ · · · ≤ τL ◮ Transmission duration Tdur message encode message decode message encode

Tdur = 9 τ2 = 2 τ3 = 3 τ4 = 6 Hop 1 Hop 2 Hop 3 Hop 4

◮ Coding scheme

Definition: A (T, R, ǫ)-tuple is achievable by relaying scheme Φ if (i) T ≥ τL + Tdur (ii) R ≤ log(|M|) Tdur (iii) ǫ ≥ P(M = M)

4

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Starting times, relay encoding & joint decoding

◮ Block encoding @ source ◮ Causal-sequential coding @ relays ◮ Block decoding @ destination

Yl(t) = DMCl

• Xl(t)
• X1(t) = f [1]

t (M)

Xl(t) = f [l]

t ([Yl−1]t−1 ∗

)

• M = g([YL]τL+Tdur

∗

)

· · · X1 Y1 X2 Y2 XL YL s r1 r2 rL

− 1

d

A relaying scheme is defined by

◮ Message M ∈ M {1, . . . , |M|} ◮ L deterministic starting times τ1 = 0 ≤ τ2 ≤ · · · ≤ τL ◮ Transmission duration Tdur

pipelining!

Pipelining: Source sends next message before previous message is received by destination

message encode message decode message encode

Tdur = 9 τ2 = 2 τ3 = 3 τ4 = 6 Hop 1 Hop 2 Hop 3 Hop 4

◮ Coding scheme

Definition: A (T, R, ǫ)-tuple is achievable by relaying scheme Φ if (i) T ≥ τL + Tdur (ii) R ≤ log(|M|) Tdur (iii) ǫ ≥ P(M = M)

4

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Error exponent of relaying scheme Φ

Definition: A (T, R, ǫ)-tuple is achievable by relaying scheme Φ if (i) T ≥ τL + Tdur (ii) R ≤ log(|M|) Tdur (iii) ǫ ≥ P(M = M)

5

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Error exponent of relaying scheme Φ

Definition: A (T, R, ǫ)-tuple is achievable by relaying scheme Φ if (i) T ≥ τL + Tdur (ii) R ≤ log(|M|) Tdur (iii) ǫ ≥ P(M = M)

◮ Let Aφ

• all (T, R, ǫ) achievable by Φ
• 5

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Error exponent of relaying scheme Φ

Definition: A (T, R, ǫ)-tuple is achievable by relaying scheme Φ if (i) T ≥ τL + Tdur (ii) R ≤ log(|M|) Tdur (iii) ǫ ≥ P(M = M)

◮ Let Aφ

• all (T, R, ǫ) achievable by Φ
• ◮ Then define the error exponent of an L-hop relaying scheme Φ as

EΦ(R) lim sup

T →∞

sup

ǫ:(T,R,ǫ)∈AΦ

− log(ǫ) T

5

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Aside: decode-&-forward, amplify-&-forward

Definitions & analysis framework captures many possible relaying schemes, examples:

6

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Aside: decode-&-forward, amplify-&-forward

message decode

Tdur = 6 τ2 = Tdur τ3 = 2Tdur Hop 1 Hop 2 Hop 3

message encode message encode

Simplified example: C1 = C2 = · · · = CL

decode-&-forward

◮ Random coding over every hop ◮ Tdur is the maximum of the block lengths

Definitions & analysis framework captures many possible relaying schemes, examples:

6

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Aside: decode-&-forward, amplify-&-forward

message decode

Tdur = 6 τ2 = Tdur τ3 = 2Tdur Hop 1 Hop 2 Hop 3

message encode message encode

Simplified example: C1 = C2 = · · · = CL

decode-&-forward amplify-&-forward

◮ Random coding over every hop ◮ Tdur is the maximum of the block lengths ◮ Random coding between source and

destination

◮ Relays forward received symbols

Definitions & analysis framework captures many possible relaying schemes, examples:

message decode

Hop 1 Hop 2 Hop 3

message encode message encode τ2 = 1 τ3 = 2 Tdur = 6 6

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Main Result 1: Feedback-free setting

7

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Main Result 1: Feedback-free setting

· · · s r1 r2 rL

− 1

d

l∗ = L In the feedback-free setting, if the bottleneck hop is the last hop (i.e. l∗ = L), then transcoding achieves DAFTC = 1 ≤ DAFΦ, ∀Φ.

7

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Main Result 1: Feedback-free setting

· · · s r1 r2 rL

− 1

d

l∗ = L In the feedback-free setting, if the bottleneck hop is the last hop (i.e. l∗ = L), then transcoding achieves DAFTC = 1 ≤ DAFΦ, ∀Φ. Proof includes:

◮ Description of coding scheme (source, relays, destination) ◮ Derivation of error probability & delay ⇒ Error exponent ⇒ DAF

7

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Feedback-free setting — proof outline I

To reduce delay: Break codewords into K micro-blocks, each of length ∆ symbols

◮ Each protected by random coding with rate CL ◮ Decode-&-forward over the first L − 1 hops

LN L + (K − 1) K

N

8

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Feedback-free setting — proof outline I

To reduce delay: Break codewords into K micro-blocks, each of length ∆ symbols

◮ Each protected by random coding with rate CL ◮ Decode-&-forward over the first L − 1 hops

LN L + (K − 1) K

N

Problem: Error explodes!

e−NErc,2(R) e−N/KErc,2(R)

8

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Feedback-free setting — proof outline I

(Example: L = 3, K = 3)

Hop 1 Hop 2 Hop 3

∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆

DF DF DF DF DF DF

τ3 = 2∆ τ2 = ∆

{

{

Cl > CL CL

To reduce delay: Break codewords into K micro-blocks, each of length ∆ symbols

◮ Each protected by random coding with rate CL ◮ Decode-&-forward over the first L − 1 hops

Message M

Encoder Decoder

• M

LN L + (K − 1) K

N

Problem: Error explodes!

e−NErc,2(R) e−N/KErc,2(R)

8

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Feedback-free setting — proof outline I

(Example: L = 3, K = 3)

Hop 1 Hop 2 Hop 3

∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆

DF DF DF DF DF DF

τ3 = 2∆ τ2 = ∆

Low error rate: Capacity Cl > coding rate CL

{

{

Cl > CL CL

To reduce delay: Break codewords into K micro-blocks, each of length ∆ symbols

◮ Each protected by random coding with rate CL ◮ Decode-&-forward over the first L − 1 hops

Message M

Encoder Decoder

• M

LN L + (K − 1) K

N

Problem: Error explodes!

e−NErc,2(R) e−N/KErc,2(R)

8

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Feedback-free setting — proof outline I

(Example: L = 3, K = 3)

Hop 1 Hop 2 Hop 3

∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆

DF DF DF DF DF DF

τ3 = 2∆ τ2 = ∆

High error rate! (i) Capacity CL = coding rate CL (ii) reduced length!

Low error rate: Capacity Cl > coding rate CL

{

{

Cl > CL CL

To reduce delay: Break codewords into K micro-blocks, each of length ∆ symbols

◮ Each protected by random coding with rate CL ◮ Decode-&-forward over the first L − 1 hops

Message M

Encoder Decoder

• M

LN L + (K − 1) K

N

Problem: Error explodes!

e−NErc,2(R) e−N/KErc,2(R)

8

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Feedback-free setting — proof outline II

(Example: L = 3, K = 3)

Hop 1 Hop 2 Hop 3

∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆

DF DF DF DF DF DF

τ3 = 2∆ τ2 = ∆

High error rate! (i) Capacity CL = coding rate CL (ii) reduced length!

Low error rate: Capacity Cl > coding rate CL

Cyclically shifted outer code (CSOC) Inner encoder Inner/Outer decoder Inner encoder Inner encoder

{

{

Cl > CL CL

Protect message over last hop using additional coding

◮ Inspired by concatenated coding ◮ CSOC outer code + random codes for micro-blocks ◮ Maximum-likelihood decoding at destination

Cyclically shifted outer code (CSOC):

• (c[1]

i1 , · · · , c[K] iK ) : K

• k=1

ik mod eK∆(RI−R) = 0

• Number of messages: eK∆RI /eK∆(RI−R) = eK∆R

Message M

• M

9

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Feedback-free setting — proof outline II

(Example: L = 3, K = 3)

Hop 1 Hop 2 Hop 3

∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆

DF DF DF DF DF DF

τ3 = 2∆ τ2 = ∆

Low error rate: Capacity Cl > coding rate CL

Cyclically shifted outer code (CSOC) Inner encoder Inner/Outer decoder Inner encoder Inner encoder

{

{

Cl > CL CL Low error rate: Protected by outer code

Protect message over last hop using additional coding

◮ Inspired by concatenated coding ◮ CSOC outer code + random codes for micro-blocks ◮ Maximum-likelihood decoding at destination

Cyclically shifted outer code (CSOC):

• (c[1]

i1 , · · · , c[K] iK ) : K

• k=1

ik mod eK∆(RI−R) = 0

• Number of messages: eK∆RI /eK∆(RI−R) = eK∆R

Message M

• M

9

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Feedback-free setting — proof outline III

(Example: L = 3, K = 3)

Hop 1 Hop 2 Hop 3

∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆

DF DF DF DF DF DF

τ3 = 2∆ τ2 = ∆

Low error rate: Capacity Cl > coding rate CL

Cyclically shifted outer code (CSOC) Inner encoder Inner/Outer decoder Inner encoder Inner encoder

{

{

Cl > CL CL Low error rate: Protected by outer code

Message M

• M

◮ Choose K such that

(i) K · Erc,L(R) < min

l∈[1,L−1] Erc,l(CL)

(ii) K · (CL − R) ≤ CL for any R < CL.

10

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Feedback-free setting — proof outline III

(Example: L = 3, K = 3)

Hop 1 Hop 2 Hop 3

∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆

DF DF DF DF DF DF

τ3 = 2∆ τ2 = ∆

Low error rate: Capacity Cl > coding rate CL

Cyclically shifted outer code (CSOC) Inner encoder Inner/Outer decoder Inner encoder Inner encoder

{

{

Cl > CL CL Low error rate: Protected by outer code

◮ Then error probability ǫ ≤ (K(L − 1) + 2K − 1)e−K∆Erc,L(R) ◮ Total delay T = (L − 1 + K)∆

Message M

• M

◮ Choose K such that

(i) K · Erc,L(R) < min

l∈[1,L−1] Erc,l(CL)

(ii) K · (CL − R) ≤ CL for any R < CL.

10

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Feedback-free setting — proof outline III

(Example: L = 3, K = 3)

Hop 1 Hop 2 Hop 3

∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆

DF DF DF DF DF DF

τ3 = 2∆ τ2 = ∆

Low error rate: Capacity Cl > coding rate CL

Cyclically shifted outer code (CSOC) Inner encoder Inner/Outer decoder Inner encoder Inner encoder

{

{

Cl > CL CL Low error rate: Protected by outer code

◮ Then error probability ǫ ≤ (K(L − 1) + 2K − 1)e−K∆Erc,L(R) ◮ Total delay T = (L − 1 + K)∆

Message M

• M

◮ Choose K such that

(i) K · Erc,L(R) < min

l∈[1,L−1] Erc,l(CL)

(ii) K · (CL − R) ≤ CL for any R < CL.

◮ Then ETC(R) =

K K+L−1Erc,L(R) 10

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Feedback-free setting — proof outline III

(Example: L = 3, K = 3)

Hop 1 Hop 2 Hop 3

∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆

DF DF DF DF DF DF

τ3 = 2∆ τ2 = ∆

Low error rate: Capacity Cl > coding rate CL

Cyclically shifted outer code (CSOC) Inner encoder Inner/Outer decoder Inner encoder Inner encoder

{

{

Cl > CL CL Low error rate: Protected by outer code

◮ Then error probability ǫ ≤ (K(L − 1) + 2K − 1)e−K∆Erc,L(R) ◮ Total delay T = (L − 1 + K)∆

Message M

• M

◮ Choose K such that

(i) K · Erc,L(R) < min

l∈[1,L−1] Erc,l(CL)

(ii) K · (CL − R) ≤ CL for any R < CL.

◮ Then ETC(R) =

K K+L−1Erc,L(R)

ETC(R) = Erc,L(R)

• Then, when R → CL, K → ∞ and we get

10

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Main Result 2: Stop feedback setting

11

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Main Result 2: Stop feedback setting

Some new definitions:

◮ Allow Tdur to be a stopping time, use E[Tdur] ◮ Use random coding error exponent with stop feedback

[Polyanskiy, Poor, Verdu 2011] Esf,l(R) = (Cl − R)+

• ne-time stop feedback

· · · s r1 r2 rL

− 1

d

l∗ = L

◮ Then

DAFΦ lim

RրCl∗

Esf,l(R) EΦ(R) = lim

RրCl∗

Cl∗ − R EΦ(R)

• 2Y. Polyanskiy, H. V. Poor, and S. Verdu, “Feedback in the non-asymptotic regime,” IEEE Trans. Inform. Theory, vol. 57, no. 8, pp. 4903–4925, Aug. 2011.

11

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Main Result 2: Stop feedback setting

Some new definitions:

◮ Allow Tdur to be a stopping time, use E[Tdur] ◮ Use random coding error exponent with stop feedback

[Polyanskiy, Poor, Verdu 2011] Esf,l(R) = (Cl − R)+

• ne-time stop feedback

· · · s r1 r2 rL

− 1

d

l∗ = L

◮ Then

DAFΦ lim

RրCl∗

Esf,l(R) EΦ(R) = lim

RրCl∗

Cl∗ − R EΦ(R) In the stop feedback setting, regardless of the position of the bottleneck link, transcoding achieves DAFTC = 1

• 2Y. Polyanskiy, H. V. Poor, and S. Verdu, “Feedback in the non-asymptotic regime,” IEEE Trans. Inform. Theory, vol. 57, no. 8, pp. 4903–4925, Aug. 2011.

11

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Stop feedback setting — proof outline

Hop 1 . . . Hop l∗−1 Hop l∗+1 Hop l∗ Hop L . . .

Outer code Inner encoder Inner encoder Inner encoder

Message M DF DF DF DF DF DF ∆ ∆ ∆

{

{

Cl > Cl∗ Cl∗

{

Cl > Cl∗

◮ When l∗ = L, all post-bottleneck hops have high error rate due to error propagation

12

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Stop feedback setting — proof outline

Hop 1 . . . Hop l∗−1 Hop l∗+1 Hop l∗ Hop L . . .

Outer code Inner encoder Inner encoder Inner encoder

Message M DF DF DF DF DF DF ∆ ∆ ∆

{

{

Cl > Cl∗ Cl∗

{

Cl > Cl∗

Low error

◮ When l∗ = L, all post-bottleneck hops have high error rate due to error propagation

12

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Stop feedback setting — proof outline

Hop 1 . . . Hop l∗−1 Hop l∗+1 Hop l∗ Hop L . . .

Outer code Inner encoder Inner encoder Inner encoder

Message M DF DF DF DF DF DF ∆ ∆ ∆

{

{

Cl > Cl∗ Cl∗

{

Cl > Cl∗

Low error High error

◮ When l∗ = L, all post-bottleneck hops have high error rate due to error propagation

12

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Stop feedback setting — proof outline

Hop 1 . . . Hop l∗−1 Hop l∗+1 Hop l∗ Hop L . . .

Outer code Inner encoder Inner encoder Inner encoder

Message M DF DF DF DF DF DF ∆ ∆ ∆

{

{

Cl > Cl∗ Cl∗

{

Cl > Cl∗

Low error High error High error

◮ When l∗ = L, all post-bottleneck hops have high error rate due to error propagation

12

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Stop feedback setting — proof outline

Hop 1 . . . Hop l∗−1 Hop l∗+1 Hop l∗ Hop L . . .

Outer code Inner encoder Inner encoder Inner encoder

Message M DF DF DF DF DF DF ∆ ∆ ∆

{

{

Cl > Cl∗ Cl∗

{

Cl > Cl∗

Low error High error High error Low error due to outer code

◮ When l∗ = L, all post-bottleneck hops have high error rate due to error propagation ◮ Bottleneck relay (BR) can decode outer code after K micro-blocks

12

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Stop feedback setting — proof outline

Hop 1 . . . Hop l∗−1 Hop l∗+1 Hop l∗ Hop L . . .

Outer code Inner encoder Inner encoder Inner encoder

Message M DF DF DF DF DF DF ∆ ∆ ∆

{

{

Cl > Cl∗ Cl∗

{

Cl > Cl∗

DF Correction Phase Low error High error High error Low error due to outer code

◮ When l∗ = L, all post-bottleneck hops have high error rate due to error propagation ◮ Bottleneck relay (BR) can decode outer code after K micro-blocks ◮ BR sends additional redundancy until destination signals stop feedback

. . .

12

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Stop feedback setting — proof outline

Hop 1 . . . Hop l∗−1 Hop l∗+1 Hop l∗ Hop L . . .

Outer code Inner encoder Inner encoder Inner encoder

Message M DF DF DF DF DF DF ∆ ∆ ∆

{

{

Cl > Cl∗ Cl∗

{

Cl > Cl∗

Sequential Random Permutation Outer Codes (SRPOC)

DF Correction Phase Low error High error High error Low error due to outer code

◮ When l∗ = L, all post-bottleneck hops have high error rate due to error propagation ◮ Bottleneck relay (BR) can decode outer code after K micro-blocks ◮ BR sends additional redundancy until destination signals stop feedback ◮ With careful code construction can show DAFTC = 1

. . .

12

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Work in progress

l Cl

With stop feedback: Solved Feedback-free: Solved

13

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Work in progress

With stop feedback: Solved Feedback-free: Partially solved

l Cl

With stop feedback: Solved Feedback-free: Solved

l Cl

Proposed scheme Proposed scheme DF over groups

13

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Work in progress

l Cl

With stop feedback: Solved Feedback-free: Partially solved

l Cl

With stop feedback: Solved Feedback-free: Solved

l Cl

Proposed scheme Proposed scheme DF over groups With stop feedback: Solved Feedback-free: ? Can we do better than DF?

13

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Work in progress

What is left? — Case l∗ = L without feedback

14

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Work in progress

∆ ∆ δ ∆ δ ∆ δ ∆ ∆ DF DF DF DF DF DF High error Controlled redundancy Hop 1 . . . Hop l∗−1 Hop l∗+1 Hop l∗ Hop L . . .

{

{

Cl > Cl∗ Cl∗

{

Cl > Cl∗

Message M

Inner encoder Inner encoder Inner encoder Outer Code Inner/Outer decoder ◮ BR adds redundancy per micro-block ◮ Conjecture: 1 < DAFTC′ < DAFDF

What is left? — Case l∗ = L without feedback

14

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Conclusion

15

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Conclusion

Multi-hop relay channel

· · ·

· · · s r1 r2 rL

− 1

d

15

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Conclusion

Multi-hop relay channel

· · ·

· · · s r1 r2 rL

− 1

d

When R → C, what relaying schemes minimize relative delay?

15

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Conclusion

Multi-hop relay channel

· · ·

· · · s r1 r2 rL

− 1

d

When R → C, what relaying schemes minimize relative delay?

C1 CL Cl∗ · · · rL

− 1

d r2 r1 s 15

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Conclusion

Multi-hop relay channel

· · ·

· · · s r1 r2 rL

− 1

d

When R → C, what relaying schemes minimize relative delay?

R Erc(R)

C1 CL Cl∗ · · · rL

− 1

d r2 r1 s 15

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Conclusion

Multi-hop relay channel

· · ·

· · · s r1 r2 rL

− 1

d

Delay Amplification Factor DAFΦ lim

RրC

Erc,l∗(R) EΦ(R)

When R → C, what relaying schemes minimize relative delay?

R Erc(R)

C1 CL Cl∗ · · · rL

− 1

d r2 r1 s 15

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Conclusion

Multi-hop relay channel

· · ·

· · · s r1 r2 rL

− 1

d

Delay Amplification Factor DAFΦ lim

RրC

Erc,l∗(R) EΦ(R)

Main Takeaways

◮ Relaying schemes with DAF = 1 are possible ◮ Linearly growing delay of decode-&-forward is NOT

a fundamental limit

When R → C, what relaying schemes minimize relative delay?

R Erc(R)

C1 CL Cl∗ · · · rL

− 1

d r2 r1 s 15

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Conclusion

Multi-hop relay channel

· · ·

· · · s r1 r2 rL

− 1

d

Delay Amplification Factor DAFΦ lim

RրC

Erc,l∗(R) EΦ(R)

Main Takeaways

◮ Relaying schemes with DAF = 1 are possible ◮ Linearly growing delay of decode-&-forward is NOT

a fundamental limit Future Work

◮ l∗ = L without feedback ◮ Applications to Gaussian, Rayleigh fading channels

When R → C, what relaying schemes minimize relative delay?

R Erc(R)

C1 CL Cl∗ · · · rL

− 1

d r2 r1 s 15

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Questions.

16

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

References I

[1]

• R. G. Gallager, Information Theory and Reliable Communication. New York, NY, USA: John Wiley & Sons, Inc., 1968.

[2]

• Y. Polyanskiy, H. V. Poor, and S. Verdu, “Feedback in the non-asymptotic regime,” IEEE Trans. Inform. Theory, vol. 57, no. 8,
• pp. 4903–4925, Aug. 2011.

[3]

• T. Cover and A. E. Gamal, “Capacity theorems for the relay channel,” IEEE Trans. Inform. Theory, vol. 25, no. 5, pp. 572–584, Sep.

1979. [4] R.-A. Pitaval, O. Tirkkonen, R. Wichman, K. Pajukoski, E. Lahetkangas, and E. Tiirola, “Full-duplex self-backhauling for small-cell 5g networks,” IEEE Wireless Commun., vol. 22, no. 5, pp. 83–89, 2015. [5]

• H. Ji, S. Park, J. Yeo, Y. Kim, J. Lee, and B. Shim, “Ultra-reliable and low-latency communications in 5g downlink: Physical layer

aspects,” IEEE Wireless Commun., vol. 25, no. 3, pp. 124–130, Jun. 2018. [6]

• C. Shannon, R. Gallager, and E. Berlekamp, “Lower bounds to error probability for coding on discrete memoryless channels. ii,”

Information and Control, vol. 10, no. 5, pp. 522–552, 1967. [7]

• G. D. Forney, Concatenated codes. Cambridge, MA, USA: MIT Press, 1966.

[8]

• M. V. Burnashev, “Data transmission over a discrete channel with feedback. random transmission time,” Problems Inform.

Transmission, vol. 12, no. 4, pp. 10–30, 1976. [9]

• S. L. Fong and V. Y. F. Tan, “Achievable rates for gaussian degraded relay channels with non-vanishing error probabilities,” IEEE
• Trans. Inform. Theory, vol. 63, no. 7, pp. 4183–4201, Jul. 2017.

[10]

• Y. Hu, J. Gross, and A. Schmeink, “On the capacity of relaying with finite blocklength,” IEEE Trans. Veh. Technol., vol. 65, no. 3,
• pp. 1790–1794, Mar. 2016.

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Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

References II

[11]

• V. Y. F. Tan, “On the reliability function of the discrete memoryless relay channel,” IEEE Trans. Inform. Theory, vol. 61, no. 4,
• pp. 1550–1573, Apr. 2015.

[12]

• N. Wen and R. A. Berry, “Reliability constrained packet-sizing for linear multi-hop wireless networks,” in 2008 IEEE International

Symposium on Information Theory, Jul. 2008, pp. 16–20. [13]

• C. Thommesen, “Error-correcting capabilities of concatenated codes with MDS outer codes on memoryless channels with

maximum-likelihood decoding,” IEEE Trans. Inform. Theory, vol. 33, no. 5, pp. 632–640, Sep. 1987. [14] C.-C. Wang, D. J. Love, and D. Ogbe, “Transcoding: A new strategy for relay channels,” in 55th Annual Allerton Conference on Communication, Control, and Computing (Allerton), Oct. 2017. [15]

• Y. Polyanskiy, H. V. Poor, and S. Verdu, “Channel coding rate in the finite blocklength regime,” IEEE Trans. Inform. Theory, vol. 56,
• no. 5, pp. 2307–2359, May 2010.

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Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Backup Slides.

19

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

Aside: DAF of decode-&-forward

message decode

Tdur = 6 τ2 = Tdur τ3 = 2Tdur Hop 1 Hop 2 Hop 3

message encode message encode

Simplified example: C1 = C2 = · · · = CL DAFDF = L

◮ Operate each hop close to capacity ⇒ T ≈

l B Cl for a B-bit message

◮ But Tbn ≈

B Cl∗

◮ Then DAFDF =

T Tbn = l Cl∗ Cl > 1 20

Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019

N.B.: AF does not achieve capacity, DAF does not apply

Prior Work: Transcoding

Throughput Delay

DF AF TC DF AF TC

Delay Throughput

Error exponent analysis Channel dispersion approx.7

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Dennis Ogbe, Chih-Chun Wang, David J. Love On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels International Symposium on Information Theory Paris, France, July 12, 2019