[PPT] - On the Fraction of Capacity One Relay can Achieve in Gaussian PowerPoint Presentation

SLIDE 1

On the Fraction of Capacity One Relay can Achieve in Gaussian Half-Duplex Diamond Networks

Authors: Sarthak Jain (Presenter) Soheil Mohajer Martina Cardone

2020 IEEE International Symposium on Information Theory

University of Minnesota, T win Cities

1

(The work of the authors was supported in part by the U.S. National Science Foundation under Grant CCF-1907785.)

SLIDE 2

System Model

Half-Duplex mode of operation

1 2 N S D i

ℓ1 ℓ2 ℓi ℓN

r1 r2 ri rN

2

𝒪 = {(ℓi, ri) ∀i ∈ [1 : N]} Diamond Network

SLIDE 3

C(𝒪) − G1 ≤ ˜ C ≤ C(𝒪) + G2

Capacity and Scheduling

A. S. Avestimehr, et al, IEEE Trans. Inf. Theory, 2011.
A. Ozgur, et al., IEEE Trans. Inf. Theory, 2013.
S. Lim, et al. IEEE Trans. Inf. Theory, 2011.

Shannon Capacity Approximate Capacity

3

1 2 S D 3

ℓ1 ℓ2 ℓ3

r1 r2 r3

SLIDE 4

Capacity and Scheduling

where, G1 + G2 = f(N)

A. S. Avestimehr, et al, IEEE Trans. Inf. Theory, 2011.
A. Ozgur, et al., IEEE Trans. Inf. Theory, 2013.
S. Lim, et al. IEEE Trans. Inf. Theory, 2011.

4

1 2 S D 3

ℓ1

r1 r2

C(𝒪) − G1 ≤ ˜ C ≤ C(𝒪) + G2

ℓ3 ℓ2

r3

SLIDE 5

Capacity and Scheduling

C(𝒪) = max

λ

t 𝗍 . 𝗎 . t ≤∑

𝒯⊆[N]

λ𝒯 ( max

i∈𝒯c∩Ωc ℓi + max i∈𝒯∩Ωri)

∀Ω ⊆ [N] ∑

𝒯⊆[N]

λ𝒯 = 1, λ𝒯 ≥ 0, ∀𝒯 ⊆ [N]

A. S. Avestimehr, et al, IEEE Trans. Inf. Theory, 2011.
A. Ozgur, et al., IEEE Trans. Inf. Theory, 2013.
S. Lim, et al. IEEE Trans. Inf. Theory, 2011.

5

1 2 S D 3

ℓ1

r1 r2

where, G1 + G2 = f(N)

C(𝒪) − G1 ≤ ˜ C ≤ C(𝒪) + G2

ℓ3 ℓ2

r3

SLIDE 6

Capacity and Scheduling

A. S. Avestimehr, et al, IEEE Trans. Inf. Theory, 2011.
A. Ozgur, et al., IEEE Trans. Inf. Theory, 2013.
S. Lim, et al. IEEE Trans. Inf. Theory, 2011.

λ𝒯 : fraction of time in state 𝒯 𝒯 ⊆ [1 : N] i ∈ 𝒯 : Relay i Tx i ∈ 𝒯c : Relay i Rx

6

λ∅, λ{1}, λ{2}, λ{3}, λ{1,2}, λ{1,3}, λ{2,3}, λ{1,2,3} 𝒯 : ∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}

1 2 S D 3

ℓ1

r1 r2

where, G1 + G2 = f(N)

C(𝒪) − G1 ≤ ˜ C ≤ C(𝒪) + G2

C(𝒪) = max

λ

t 𝗍 . 𝗎 . t ≤∑

𝒯⊆[N]

λ𝒯 ( max

i∈𝒯c∩Ωc ℓi + max i∈𝒯∩Ωri)

∀Ω ⊆ [N] ∑

𝒯⊆[N]

λ𝒯 = 1, λ𝒯 ≥ 0, ∀𝒯 ⊆ [N]

ℓ3 ℓ2

r3

SLIDE 7

Capacity and Scheduling

A. S. Avestimehr, et al, IEEE Trans. Inf. Theory, 2011.
A. Ozgur, et al., IEEE Trans. Inf. Theory, 2013.
S. Lim, et al. IEEE Trans. Inf. Theory, 2011.

λ𝒯 : fraction of time in state 𝒯 𝒯 ⊆ [1 : N] i ∈ 𝒯 : Relay i Tx i ∈ 𝒯c : Relay i Rx

7

λ∅, λ{1}, λ{2}, λ{3}, λ{1,2}, λ{1,3}, λ{2,3}, λ{1,2,3} 𝒯 : ∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}

1 2 S D 3

ℓ1

r1 r2

where, G1 + G2 = f(N)

C(𝒪) − G1 ≤ ˜ C ≤ C(𝒪) + G2

C(𝒪) = max

λ

t 𝗍 . 𝗎 . t ≤∑

𝒯⊆[N]

λ𝒯 ( max

i∈𝒯c∩Ωc ℓi + max i∈𝒯∩Ωri)

∀Ω ⊆ [N] ∑

𝒯⊆[N]

λ𝒯 = 1, λ𝒯 ≥ 0, ∀𝒯 ⊆ [N]

ℓ3 ℓ2

r3

SLIDE 8

Capacity and Scheduling

A. S. Avestimehr, et al, IEEE Trans. Inf. Theory, 2011.
A. Ozgur, et al., IEEE Trans. Inf. Theory, 2013.
S. Lim, et al. IEEE Trans. Inf. Theory, 2011.

λ𝒯 : fraction of time in state 𝒯 𝒯 ⊆ [1 : N] i ∈ 𝒯 : Relay i Tx i ∈ 𝒯c : Relay i Rx

8

λ∅, λ{1}, λ{2}, λ{3}, λ{1,2}, λ{1,3}, λ{2,3}, λ{1,2,3} 𝒯 : ∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}

1 2 S D 3

ℓ1

r1 r2

where, G1 + G2 = f(N)

C(𝒪) − G1 ≤ ˜ C ≤ C(𝒪) + G2

C(𝒪) = max

λ

t 𝗍 . 𝗎 . t ≤∑

𝒯⊆[N]

λ𝒯 ( max

i∈𝒯c∩Ωc ℓi + max i∈𝒯∩Ωri)

∀Ω ⊆ [N] ∑

𝒯⊆[N]

λ𝒯 = 1, λ𝒯 ≥ 0, ∀𝒯 ⊆ [N]

ℓ3 ℓ2

r3

SLIDE 9

Capacity and Scheduling

λ∅, λ{1}, λ{2}, λ{3}, λ{1,2}, λ{1,3}, λ{2,3}, λ{1,2,3}

A. S. Avestimehr, et al, IEEE Trans. Inf. Theory, 2011.
A. Ozgur, et al., IEEE Trans. Inf. Theory, 2013.
S. Lim, et al. IEEE Trans. Inf. Theory, 2011.

λ𝒯 : fraction of time in state 𝒯 𝒯 ⊆ [1 : N] i ∈ 𝒯 : Relay i Tx i ∈ 𝒯c : Relay i Rx 𝒯 : ∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} Ω = ∅, {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3}

1 2 S D 3

ℓ1

r1 r2

9

where, G1 + G2 = f(N)

C(𝒪) − G1 ≤ ˜ C ≤ C(𝒪) + G2

C(𝒪) = max

λ

t 𝗍 . 𝗎 . t ≤∑

𝒯⊆[N]

λ𝒯 ( max

i∈𝒯c∩Ωc ℓi + max i∈𝒯∩Ωri)

∀Ω ⊆ [N] ∑

𝒯⊆[N]

λ𝒯 = 1, λ𝒯 ≥ 0, ∀𝒯 ⊆ [N]

ℓ3 ℓ2

r3

SLIDE 10

Capacity and Scheduling

λ∅, λ{1}, λ{2}, λ{3}, λ{1,2}, λ{1,3}, λ{2,3}, λ{1,2,3}

A. S. Avestimehr, et al, IEEE Trans. Inf. Theory, 2011.
A. Ozgur, et al., IEEE Trans. Inf. Theory, 2013.
S. Lim, et al. IEEE Trans. Inf. Theory, 2011.

λ𝒯 : fraction of time in state 𝒯 𝒯 ⊆ [1 : N] i ∈ 𝒯 : Relay i Tx i ∈ 𝒯c : Relay i Rx 𝒯 : ∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} Ω = ∅, {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3}

Ω = {2,3}

1 2 S D 3

ℓ1

r1 r2

10

where, G1 + G2 = f(N)

C(𝒪) − G1 ≤ ˜ C ≤ C(𝒪) + G2

C(𝒪) = max

λ

t 𝗍 . 𝗎 . t ≤∑

𝒯⊆[N]

λ𝒯 ( max

i∈𝒯c∩Ωc ℓi + max i∈𝒯∩Ωri)

∀Ω ⊆ [N] ∑

𝒯⊆[N]

λ𝒯 = 1, λ𝒯 ≥ 0, ∀𝒯 ⊆ [N]

ℓ3 ℓ2

r3

SLIDE 11

Approximate Capacity of One Relay Network

1 S D

ℓ1

r1

11

C(𝒪) = max

λ

t 𝗍 . 𝗎 . t ≤∑

𝒯⊆[N]

λ𝒯 ( max

i∈𝒯c∩Ωc ℓi + max i∈𝒯∩Ωri)

∀Ω ⊆ [N] ∑

𝒯⊆[N]

λ𝒯 = 1, λ𝒯 ≥ 0, ∀𝒯 ⊆ [N]

SLIDE 12

Approximate Capacity of One Relay Network

t ≤ λ∅ . ℓ1 + λ{1}.0 Ω = ∅

Ω = ∅ 1 S D

ℓ1

r1

12

C(𝒪) = max

λ

t 𝗍 . 𝗎 . t ≤∑

𝒯⊆[N]

λ𝒯 ( max

i∈𝒯c∩Ωc ℓi + max i∈𝒯∩Ωri)

∀Ω ⊆ [N] ∑

𝒯⊆[N]

λ𝒯 = 1, λ𝒯 ≥ 0, ∀𝒯 ⊆ [N]

SLIDE 13

Approximate Capacity of One Relay Network

t ≤ λ∅ . ℓ1 + λ{1}.0 t ≤ λ∅.0 + λ{1} . r1 Ω = ∅ Ω = {1}

Ω = ∅ 1 S D

ℓ1

r1

Ω = {1}

13

C(𝒪) = max

λ

t 𝗍 . 𝗎 . t ≤∑

𝒯⊆[N]

λ𝒯 ( max

i∈𝒯c∩Ωc ℓi + max i∈𝒯∩Ωri)

∀Ω ⊆ [N] ∑

𝒯⊆[N]

λ𝒯 = 1, λ𝒯 ≥ 0, ∀𝒯 ⊆ [N]

SLIDE 14

Approximate Capacity of One Relay Network

t ≤ λ∅ . ℓ1 + λ{1}.0 𝖣 = max

λ

t t ≤ λ∅.0 + λ{1} . r1 Ω = ∅ Ω = {1} λ∅ + λ{1} = 1, λ∅, λ{1} ≥ 0

Ω = ∅ 1 S D

ℓ1

r1

Ω = {1}

14

C(𝒪) = max

λ

t 𝗍 . 𝗎 . t ≤∑

𝒯⊆[N]

λ𝒯 ( max

i∈𝒯c∩Ωc ℓi + max i∈𝒯∩Ωri)

∀Ω ⊆ [N] ∑

𝒯⊆[N]

λ𝒯 = 1, λ𝒯 ≥ 0, ∀𝒯 ⊆ [N]

SLIDE 15

Approximate Capacity of One Relay Network

t ≤ λ∅ . ℓ1 + λ{1}.0 𝖣 = max

λ

t t ≤ λ∅.0 + λ{1} . r1 Ω = ∅ Ω = {1} λ∅ + λ{1} = 1, λ∅, λ{1} ≥ 0

C = ℓ1r1 ℓ1 + r1

Ω = ∅ 1 S D

ℓ1

r1

Ω = {1}

15

C(𝒪) = max

λ

t 𝗍 . 𝗎 . t ≤∑

𝒯⊆[N]

λ𝒯 ( max

i∈𝒯c∩Ωc ℓi + max i∈𝒯∩Ωri)

∀Ω ⊆ [N] ∑

𝒯⊆[N]

λ𝒯 = 1, λ𝒯 ≥ 0, ∀𝒯 ⊆ [N]

SLIDE 16

1 2 N S D i

ℓ1 ℓ2 ℓi ℓN

r1 r2 ri rN

𝒪 = {(ℓj, rj), j ∈ [1 : N]}

Approximate Capacity = C(𝒪)

16

Goal: Find the fraction of approximate capacity guaranteed by

a single relay in an N-relay half-duplex diamond network

SLIDE 17

1 2 N S D i

ℓ1

r1

𝒪 = {(ℓj, rj), j ∈ [1 : N]}

C(𝒪i) = ℓiri ℓi + ri

S D i

ℓi

ri

𝒪i = {(ℓi, ri)}

17

Goal: Find the fraction of approximate capacity guaranteed by

a single relay in an N-relay half-duplex diamond network

Approximate Capacity = C(𝒪)

ℓ2 ℓi ℓN

r2 ri rN

SLIDE 18

1 2 N S D i

ℓ1

r1

𝒪 = {(ℓj, rj), j ∈ [1 : N]}

C(𝒪i) = ℓiri ℓi + ri

S D i

ℓi

ri

𝒪i = {(ℓi, ri)}

18

Goal: Find the fraction of approximate capacity guaranteed by

a single relay in an N-relay half-duplex diamond network

Approximate Capacity = C(𝒪)

C(𝒪i) C(𝒪)

ℓ2 ℓi ℓN

r2 ri rN

SLIDE 19

1 2 N S D i

ℓ1

r1

𝒪 = {(ℓj, rj), j ∈ [1 : N]}

C(𝒪i) = ℓiri ℓi + ri

S D i

ℓi

ri

𝒪i = {(ℓi, ri)}

19

Goal: Find the fraction of approximate capacity guaranteed by

a single relay in an N-relay half-duplex diamond network

Approximate Capacity = C(𝒪)

C(𝒪i) C(𝒪) maxi C(𝒪i) C(𝒪)

ℓ2 ℓi ℓN

r2 ri rN

SLIDE 20

1 2 N S D i

ℓ1

r1

𝒪 = {(ℓj, rj), j ∈ [1 : N]}

C(𝒪i) = ℓiri ℓi + ri

S D i

ℓi

ri

𝒪i = {(ℓi, ri)}

20

Goal: Find the fraction of approximate capacity guaranteed by

a single relay in an N-relay half-duplex diamond network

Approximate Capacity = C(𝒪)

C(𝒪i) C(𝒪) maxi C(𝒪i) C(𝒪) min

𝒪

maxi C(𝒪i) C(𝒪)

ℓ2 ℓi ℓN

r2 ri rN

SLIDE 21

Theorem 1: Minimal Ratio: maxi C(𝒪i) C(𝒪)

For any half-duplex diamond N-relay network 𝒪 with approximate capacity C(𝒪), the best relay has an approximate capacity such that

maxi C(𝒪i) C(𝒪) ≥ 1 2 + 2 cos (

2π N + 2)

Moreover, this bound is tight, i.e., for any positive Tightness: integer N, there exist Gaussian half-duplex diamond N − relay networks

𝒪 for which the bound is tight.

21

SLIDE 22

The bound in Theorem 1 surprisingly depends on the

⋆

22

Half-Duplex Full-Duplex

⋆

Nazaroglu et al, TIT-2014

cosine of a function of the number of relays N .

SLIDE 23

Special Cases: N = 2 and N = ∞ :

maxi C(𝒪i) C(𝒪) ≥

1 2

for N = 2

1 4

for N → ∞

⋆ ⋆

23

Y. H. Ezzeldin et. al., TIT-2019

Half-Duplex Full-Duplex

⋆ ⋆ ⋆ ⋆

Nazaroglu et al, TIT-2014

The bound in Theorem 1 surprisingly depends on the cosine of a function of the number of relays N .

SLIDE 24

Sketch of the Proof

24

maxi C(𝒪i) C(𝒪) ≥ 1 2 + 2 cos (

2π N + 2 )

SLIDE 25

Let 𝒪 * be the collection of (optimal) half-duplex diamond N − relay networks with minimum maxi C(𝒪i)

C(𝒪) .

Then, at least one of those networks follows the following properties:

25

Main Idea 1 : Properties of Minimal Ratio Networks

1 2 N S D i

ℓ1 ℓ2 ℓi ℓN

r1 r2 ri rN

SLIDE 26

1 2 N S D i

C(𝒪1) = 1 C(𝒪2) = 1 C(𝒪i) = 1 C(𝒪N) = 1

26

Let 𝒪 * be the collection of (optimal) half-duplex diamond N − relay networks with minimum maxi C(𝒪i)

C(𝒪) .

Main Idea 1 : Properties of Minimal Ratio Networks

ℓ1 ℓ2 ℓi ℓN

r1 r2 ri rN

Then, at least one of those networks follows the following properties:

1.

ℓiri ℓi + ri = 1 ∀i ∈ [1 : N]

SLIDE 27

27

Let 𝒪 * be the collection of (optimal) half-duplex diamond N − relay networks with minimum maxi C(𝒪i)

C(𝒪) .

Main Idea 1 : Properties of Minimal Ratio Networks

Then, at least one of those networks follows the following properties:

1 2 N S D i

C(𝒪1) = 1 C(𝒪2) = 1 C(𝒪i) = 1 C(𝒪N) = 1

ℓ1 ℓ2 ℓi ℓN

r1 r2 ri rN

2. 1 ≤ ℓ1 ≤ ℓ2 ≤ ⋅ ⋅ ⋅ ≤ ℓn−1 ≤ ℓn ≤ ∞

1.

ℓiri ℓi + ri = 1 ∀i ∈ [1 : N]

SLIDE 28

2. 1 ≤ ℓ1 ≤ ℓ2 ≤ ⋅ ⋅ ⋅ ≤ ℓn−1 ≤ ℓn ≤ ∞
3. 1 ≤ rn ≤ rn−1 ≤ ⋅ ⋅ ⋅ ≤ r2 ≤ r1 ≤ ∞

1.

ℓiri ℓi + ri = 1 ∀i ∈ [1 : N]

Let 𝒪 * be the collection of (optimal) half-duplex diamond N − relay networks with minimum maxi C(𝒪i)

C(𝒪) .

Main Idea 1 : Properties of Minimal Ratio Networks

Then, at least one of those networks follows the following properties:

1 2 N S D i

C(𝒪1) = 1 C(𝒪2) = 1 C(𝒪i) = 1 C(𝒪N) = 1

ℓ1 ℓ2 ℓi ℓN

r1 r2 ri rN

28

SLIDE 29

Consequence of Main Idea 1

29

min

𝒪

maxi C(𝒪i) C(𝒪)

max

𝒪 C(𝒪)

C(𝒪i) = 1 ∀i ∈ [1 : N]

s.t.

SLIDE 30

1 2 N S D i

ℓ1 ℓ2 ℓi ℓN

r1 r2 ri rN

1 2 N S D i

1 + z2

1 + z

i

1 + zN 1 + 1 / z

1

1 + 1/z2 1 + 1 / zi 1 + 1/zN

1 + z

1

ℓi = 1 + zi ri = 1 + 1 zi z1 ≤ z2 ≤ z3 ≤ ⋅ ⋅ ⋅ ≤ zN

30

C(𝒪i) = ℓiri ℓi + ri = 1 ∀i ∈ [1 : N]

Change of variables

SLIDE 31

2N possible cuts Ω

1 2

4

S D

3

Main Idea 2: Reducing the number of Cut Constraints

1 2

4

S D

3

31

SLIDE 32

2N possible cuts Ω

1 2

4

S D

3

1 2

4

S D

3

̂ Ω0

32

Main Idea 2: Reducing the number of Cut Constraints

SLIDE 33

2N possible cuts Ω

1 2

4

S D

3

1 2

4

S D

3

̂ Ω1

33

Main Idea 2: Reducing the number of Cut Constraints

SLIDE 34

2N possible cuts Ω

1 2

4

S D

3

1 2

4

S D

3

̂ Ω2

34

Main Idea 2: Reducing the number of Cut Constraints

SLIDE 35

2N possible cuts Ω

1 2

4

S D

3

1 2

4

S D

3

̂ Ω3

35

Main Idea 2: Reducing the number of Cut Constraints

SLIDE 36

2N possible cuts Ω

1 2

4

S D

3

1 2

4

S D

3

̂ Ω4

36

Main Idea 2: Reducing the number of Cut Constraints

SLIDE 37

2N cuts : Ω

1 2

4

S D

3

1 2

4

S D

3

N + 1 cuts : ̂ Ωt

̂ Ω2 ̂ Ω1 ̂ Ω3 ̂ Ω0

37

̂ Ω4

Main Idea 2: Reducing the number of Cut Constraints

SLIDE 38

Main Idea 3: From Network States to Relay States

1 2

4

S D

3

38

SLIDE 39

Main Idea 3: From Network States to Relay States

1 2

4

S D

3

39

λ{1,3}

SLIDE 40

Main Idea 3: From Network States to Relay States

1 2

4

S D

3

λ{3}, λ{4}, λ{3,4}, λ{1,4} λ∅, λ{1}, λ{1,3}, λ{1,3,4}

Relay 2 is in receive mode

40

SLIDE 41

1 2

4

S D

3

λ{3}, λ{4}, λ{3,4}, λ{1,4} λ∅, λ{1}, λ{1,3}, λ{1,3,4}

Relay 2 is in receive mode

λ{3} + λ{4} + λ{3,4} + λ{1,4} λ∅ + λ{1} + λ{1,3} + λ{1,3,4}+

=

α2

Similarly, we have: α1, α2, α3, α4

41

Main Idea 3: From Network States to Relay States

SLIDE 42

1 2

4

S D

3

λ{3}, λ{4}, λ{3,4}, λ{1,4} λ∅, λ{1}, λ{1,3}, λ{1,3,4}

Relay 2 is in receive mode

λ{3} + λ{4} + λ{3,4} + λ{1,4} λ∅ + λ{1} + λ{1,3} + λ{1,3,4}+

=

α2

Similarly, we have: α1, α2, α3, α4

N variables : αi

42

2N variables : λ𝒯

(Network States) (Relay States)

Main Idea 3: From Network States to Relay States

SLIDE 43

2 3 5

S D

4

1 + z1 1 + z2

1

6

1 + 1/z5 1 + 1/z6 ̂ Ω3

43

2 3 5

S D

4

1 + z2

1

6

1 + 1/z5 1 + 1/z6 ̂ Ω3 1 + z1

Main Idea 4: Upper-Bounding the Link Capacities

SLIDE 44

2 3 5

S D

4

1 + z1 1 + z2

1

6

1 + 1/z5 1 + 1/z6

44

2 3 5

S D

4

1 + z2 1 + z2

1

6

1 + 1/z5 1 + 1/z6

Main Idea 4: Upper-Bounding the Link Capacities

̂ Ω3 ̂ Ω3

SLIDE 45

2 3 5

S D

4

1 + z1 1 + z2

1

6

1 + 1/z5 1 + 1/z6

45

2 3 5

S D

4

1 + z2 1 + z2

1

6

1 + 1/z5

Main Idea 4: Upper-Bounding the Link Capacities

1 + 1/z5 ̂ Ω3 ̂ Ω3

SLIDE 46

Convex in zi

46

z⋆

i

Objective Function

z1 ≤ z2 ≤ z3 ≤ ⋅ ⋅ ⋅ ≤ zN

Main Idea 5: Grouping Variables

SLIDE 47

Convex in zi

z⋆

i

z⋆

i−1

z⋆

i+1

47

z⋆

i

z⋆

i−1

z⋆

i+1

Objective Function

Main Idea 5: Grouping Variables

max

zi

z1 ≤ z2 ≤ z3 ≤ ⋅ ⋅ ⋅ ≤ zN

SLIDE 48

Tightness of the bound

48

min

𝒪

maxi C(𝒪i) C(𝒪) = 1 2 + 2 cos (

2π N + 2)

SLIDE 49

2 3 5

S D

4

1

6

2 2 2 2 2 + 2 2 + 2 2 2

2 3 5

S D

4

1

1.8019 3.2470 ∞ 1.8019 3.2470 2.2470 2.2470 1.4450 1.4450

maxi C(𝒪i) C(𝒪) = 1 2 + 2 cos (

2π 6 + 2)

= 0.2929 maxi C(𝒪i) C(𝒪) = 1 2 + 2 cos (

2π 5 + 2)

= 0.3080

Example: N=5 and N=6 relay networks

49

2 2 2 + 2 2 + 2

SLIDE 50

Summary

The main takeaways from this work are as follows:

1. The best relay in an N − relay half-duplex Gaussian diamond wireless network has at least

ℱ = 1 2 + 2 cos (

2π N + 2)

fraction of the approximate capacity of the entire network.

2. This bound is tight, i.e. we have shown that there exist networks for which

the best relay's capacity is exactly equal to ℱ fraction of the capacity of the entire network.

50

SLIDE 51

Any Questions?

(Email: jain0122@umn.edu)

51

arXiv:2001.02851