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On the Capacity of Multi-user Two-way Linear Deterministic Channels Zhiyu Cheng, Natasha Devroye Department of Electrical and Computer Engineering University of Illinois at Chicago IEEE International Symposium on Information Theory, July 1

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SLIDE 1

On the Capacity of Multi-user Two-way Linear Deterministic Channels

Zhiyu Cheng, Natasha Devroye

IEEE International Symposium on Information Theory, July 1 – 6, 2012, Cambridge, MA, USA

Department of Electrical and Computer Engineering University of Illinois at Chicago

Wednesday, June 27, 2012

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SLIDE 2

Most wireless communications are two-way

Wednesday, June 27, 2012

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SLIDE 3

Video conferencing

Most wireless communications are two-way

Wednesday, June 27, 2012

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SLIDE 4

Video conferencing

Most wireless communications are two-way

Wednesday, June 27, 2012

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SLIDE 5

Information Theory: point-to-point two-way channel

[Claude. E. Shannon, 1961]

1

2

Channel

X 1(M12) X2(M21) Y1 Y2

  • M12
  • M21

Wednesday, June 27, 2012

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SLIDE 6

Information Theory: point-to-point two-way channel

Capacity in general remains open.

[Claude. E. Shannon, 1961]

1

2

Channel

X 1(M12) X2(M21) Y1 Y2

  • M12
  • M21

Wednesday, June 27, 2012

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SLIDE 7

Why two-way problem is so hard?

Wednesday, June 27, 2012

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SLIDE 8

Why two-way problem is so hard?

Adaptation!

Wednesday, June 27, 2012

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SLIDE 9

Why two-way problem is so hard?

Adaptation!

xi

1(m12, yi−1 1

) xi

2(m21, yi−1 2

)

yi

1

yi

2

1

2

Channel

yi−1

2

= (y2,1, y2,2, ..., y2,i−1)

Wednesday, June 27, 2012

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SLIDE 10

Why two-way problem is so hard?

Adaptation! Binary Multiplier Channel:

Y1 = Y2 = X1X2

X1, X2, Y1, Y2 ∈ {0, 1}

xi

1(m12, yi−1 1

) xi

2(m21, yi−1 2

)

yi

1

yi

2

1

2

Channel

yi−1

2

= (y2,1, y2,2, ..., y2,i−1)

Wednesday, June 27, 2012

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SLIDE 11

Why two-way problem is so hard?

Adaptation! Binary Multiplier Channel:

Y1 = Y2 = X1X2

X1, X2, Y1, Y2 ∈ {0, 1}

xi

1(m12, yi−1 1

) xi

2(m21, yi−1 2

)

yi

1

yi

2

1

2

Channel

yi−1

2

= (y2,1, y2,2, ..., y2,i−1)

Adaptation is useful and capacity is unknown.

Wednesday, June 27, 2012

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SLIDE 12

Two-way deterministic modulo 2 adder channel

Y1 = Y2 = X1 ⊕ X2

1

2

Channel

X 1(M12) X2(M21) Y1 Y2

  • M12
  • M21

X1, X2, Y1, Y2 ∈ {0, 1}

Wednesday, June 27, 2012

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SLIDE 13

Two-way deterministic modulo 2 adder channel

Y1 = Y2 = X1 ⊕ X2

1

2

Channel

X 1(M12) X2(M21) Y1 Y2

  • M12
  • M21

X1, X2, Y1, Y2 ∈ {0, 1}

Y1 = X1 ⊕ X2 Y2 = X1 ⊕ X2

At each channel use,

Wednesday, June 27, 2012

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SLIDE 14

Two-way deterministic modulo 2 adder channel

Y1 = Y2 = X1 ⊕ X2

1

2

Channel

X 1(M12) X2(M21) Y1 Y2

  • M12
  • M21

X1, X2, Y1, Y2 ∈ {0, 1}

⊕X1

⊕X2 Y1 = X1 ⊕ X2 Y2 = X1 ⊕ X2

At each channel use,

Wednesday, June 27, 2012

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SLIDE 15

Two-way deterministic modulo 2 adder channel

Y1 = Y2 = X1 ⊕ X2

1

2

Channel

X 1(M12) X2(M21) Y1 Y2

  • M12
  • M21

X1, X2, Y1, Y2 ∈ {0, 1}

⊕X1

⊕X2 Y1 = X1 ⊕ X2 Y2 = X1 ⊕ X2

At each channel use,

Wednesday, June 27, 2012

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SLIDE 16

Two-way deterministic modulo 2 adder channel

Y1 = Y2 = X1 ⊕ X2

1

2

Channel

X 1(M12) X2(M21) Y1 Y2

  • M12
  • M21

X1, X2, Y1, Y2 ∈ {0, 1}

⊕X1

⊕X2 = X2 = X1 Y1 = X1 ⊕ X2 Y2 = X1 ⊕ X2

At each channel use,

Wednesday, June 27, 2012

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SLIDE 17

Self-interference

Two-way deterministic modulo 2 adder channel

Y1 = Y2 = X1 ⊕ X2

1

2

Channel

X 1(M12) X2(M21) Y1 Y2

  • M12
  • M21

X1, X2, Y1, Y2 ∈ {0, 1}

⊕X1

⊕X2 = X2 = X1 Y1 = X1 ⊕ X2 Y2 = X1 ⊕ X2

At each channel use,

Wednesday, June 27, 2012

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SLIDE 18

Self-interference

Two-way deterministic modulo 2 adder channel

Y1 = Y2 = X1 ⊕ X2

Capacity region= 1 bit / channel use / user

1

2

Channel

X 1(M12) X2(M21) Y1 Y2

  • M12
  • M21

X1, X2, Y1, Y2 ∈ {0, 1}

⊕X1

⊕X2 = X2 = X1 Y1 = X1 ⊕ X2 Y2 = X1 ⊕ X2

At each channel use,

Wednesday, June 27, 2012

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SLIDE 19

Self-interference

Two-way deterministic modulo 2 adder channel

Y1 = Y2 = X1 ⊕ X2

Capacity region= 1 bit / channel use / user

1

2

Channel

X 1(M12) X2(M21) Y1 Y2

  • M12
  • M21

X1, X2, Y1, Y2 ∈ {0, 1}

⊕X1

⊕X2 = X2 = X1 Y1 = X1 ⊕ X2 Y2 = X1 ⊕ X2

At each channel use, Adaptation is not needed.

Wednesday, June 27, 2012

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SLIDE 20

Gaussian two-way point-to-point channel

[T. Han, 1984]

Y1 = aX1 + bX2 + N1 Y2 = cX1 + dX2 + N2

1

2

Channel

X 1(M12) X2(M21) Y1 Y2

  • M12
  • M21

a, b, c d

and are channel gains.

N1 N2

N1, N2 ∼ N(0, σ2)

Wednesday, June 27, 2012

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SLIDE 21

Gaussian two-way point-to-point channel

[T. Han, 1984]

Y1 = aX1 + bX2 + N1 Y2 = cX1 + dX2 + N2

1

2

Channel

X 1(M12) X2(M21) Y1 Y2

  • M12
  • M21

a, b, c d

and are channel gains.

N1 N2

N1, N2 ∼ N(0, σ2)

Wednesday, June 27, 2012

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SLIDE 22

Gaussian two-way point-to-point channel

[T. Han, 1984]

Y1 = aX1 + bX2 + N1 Y2 = cX1 + dX2 + N2

1

2

Channel

X 1(M12) X2(M21) Y1 Y2

  • M12
  • M21

a, b, c d

and are channel gains.

N1 N2

Y2 = cX1 + N2 Y1 = bX2 + N1

≡

1

2

Channel

X1(M12) Y2

  • M12

1

2

Channel

X2(M21) Y1

  • M21

N2 N1

N1, N2 ∼ N(0, σ2)

Wednesday, June 27, 2012

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SLIDE 23

Gaussian two-way point-to-point channel

[T. Han, 1984]

Y1 = aX1 + bX2 + N1 Y2 = cX1 + dX2 + N2

1

2

Channel

X 1(M12) X2(M21) Y1 Y2

  • M12
  • M21

a, b, c d

and are channel gains.

N1 N2

Y2 = cX1 + N2 Y1 = bX2 + N1

≡

1

2

Channel

X1(M12) Y2

  • M12

1

2

Channel

X2(M21) Y1

  • M21

N2 N1

N1, N2 ∼ N(0, σ2)

Again, adaptation does not increase capacity.

Wednesday, June 27, 2012

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SLIDE 24

What about two-way networks?

Wednesday, June 27, 2012

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SLIDE 25

3 channel models we consider

Node 1 Node 3 Node 2

(a) Two-way MAC/BC

Wednesday, June 27, 2012

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SLIDE 26

3 channel models we consider

Node 1 Node 3 Node 2

(a) Two-way MAC/BC Node 1 Node 3 Node 2 Node 4

(b) Two-way Z channel

Wednesday, June 27, 2012

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SLIDE 27

3 channel models we consider

Node 1 Node 3 Node 2

(a) Two-way MAC/BC Node 1 Node 3 Node 2 Node 4

(b) Two-way Z channel

Node 1 Node 3 Node 2 Node 4

(c) Two-way interference channel

Wednesday, June 27, 2012

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SLIDE 28

3 channel models we consider

Node 1 Node 3 Node 2

(a) Two-way MAC/BC Node 1 Node 3 Node 2 Node 4

(b) Two-way Z channel

Node 1 Node 3 Node 2 Node 4

(c) Two-way interference channel

Consider linear deterministic model [Avestimehr; Diggavi; Tse; 2007]

Wednesday, June 27, 2012

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SLIDE 29

Two-way Linear Deterministic MAC/BC

2 3 1 M12 M21 M23 M32

  • M32
  • M23
  • M21
  • M12

R12 R21 R23 R32

Y1 = SN−n11X1 + SN−n21X2 Y2 = SN−n12X1 + SN−n22X2 + SN−n32X3 Y3 = SN−n23X2 + SN−n33X3,

N = max(n11, n22, n33, n12, n21, n32, n23)

S shift matrix, inputs binary vectors

Wednesday, June 27, 2012

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SLIDE 30

Two-way Linear Deterministic MAC/BC

2 3 1 M12 M21 M23 M32

  • M32
  • M23
  • M21
  • M12

R12 R21 R23 R32

Y1 = SN−n11X1 + SN−n21X2 Y2 = SN−n12X1 + SN−n22X2 + SN−n32X3 Y3 = SN−n23X2 + SN−n33X3,

N = max(n11, n22, n33, n12, n21, n32, n23)

S shift matrix, inputs binary vectors

1 3 2

Tx Tx Tx

N N N

Wednesday, June 27, 2012

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SLIDE 31

Two-way Linear Deterministic MAC/BC

2 3 1 M12 M21 M23 M32

  • M32
  • M23
  • M21
  • M12

R12 R21 R23 R32

Y1 = SN−n11X1 + SN−n21X2 Y2 = SN−n12X1 + SN−n22X2 + SN−n32X3 Y3 = SN−n23X2 + SN−n33X3,

N = max(n11, n22, n33, n12, n21, n32, n23)

S shift matrix, inputs binary vectors

1 3 2

Tx Tx Tx

N N N

Rx

⊕ ⊕ ⊕

n11 n21

⊕

Rx Rx

n23 n33 n12 n22 n32

Wednesday, June 27, 2012

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SLIDE 32

Self-interference

Two-way Linear Deterministic MAC/BC

2 3 1 M12 M21 M23 M32

  • M32
  • M23
  • M21
  • M12

R12 R21 R23 R32

Y1 = SN−n11X1 + SN−n21X2 Y2 = SN−n12X1 + SN−n22X2 + SN−n32X3 Y3 = SN−n23X2 + SN−n33X3,

N = max(n11, n22, n33, n12, n21, n32, n23)

S shift matrix, inputs binary vectors

1 3 2

Tx Tx Tx

N N N

Rx

⊕ ⊕ ⊕

n11 n21

⊕

Rx Rx

n23 n33 n12 n22 n32

Wednesday, June 27, 2012

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SLIDE 33

Two-way Linear Deterministic MAC/BC

2 3 1 M12 M21 M23 M32

  • M32
  • M23
  • M21
  • M12

R12 R21 R23 R32

Y1 = SN−n11X1 + SN−n21X2 Y2 = SN−n12X1 + SN−n22X2 + SN−n32X3 Y3 = SN−n23X2 + SN−n33X3,

N = max(n11, n22, n33, n12, n21, n32, n23)

S shift matrix, inputs binary vectors

Theorem 1: The capacity region of the two-way linear deterministic MAC/BC is the set of non-negative rate tuples (R12, R32, R21, R23) such that MAC →

  • R12 ≤ n12, R32 ≤ n32,

R12 + R32 ≤ max(n12, n32) (1) BC ←

  • R21 ≤ n21, R23 ≤ n23

R21 + R23 ≤ max(n21, n23). (2)

Wednesday, June 27, 2012

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SLIDE 34

Two-way Linear Deterministic MAC/BC

2 3 1 M12 M21 M23 M32

  • M32
  • M23
  • M21
  • M12

R12 R21 R23 R32

Y1 = SN−n11X1 + SN−n21X2 Y2 = SN−n12X1 + SN−n22X2 + SN−n32X3 Y3 = SN−n23X2 + SN−n33X3,

N = max(n11, n22, n33, n12, n21, n32, n23)

Adaptation useless (does not increase capacity region)!

S shift matrix, inputs binary vectors

Theorem 1: The capacity region of the two-way linear deterministic MAC/BC is the set of non-negative rate tuples (R12, R32, R21, R23) such that MAC →

  • R12 ≤ n12, R32 ≤ n32,

R12 + R32 ≤ max(n12, n32) (1) BC ←

  • R21 ≤ n21, R23 ≤ n23

R21 + R23 ≤ max(n21, n23). (2)

Wednesday, June 27, 2012

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SLIDE 35

Self-interference

Two-way Linear Deterministic MAC/BC

2 3 1 M12 M21 M23 M32

  • M32
  • M23
  • M21
  • M12

R12 R21 R23 R32

Y1 = SN−n11X1 + SN−n21X2 Y2 = SN−n12X1 + SN−n22X2 + SN−n32X3 Y3 = SN−n23X2 + SN−n33X3,

N = max(n11, n22, n33, n12, n21, n32, n23)

S shift matrix, inputs binary vectors

Achievability: cancel ``self-interference’’ and use non-adaptive one-way schemes Converse: cut-set or alternative Adaptation useless (does not increase capacity region)!

Wednesday, June 27, 2012

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SLIDE 36

Two-way Linear Deterministic Z Channel

2 3 1 M12 M21

  • M21
  • M12
  • M34
  • M43

M34 M43 R12 R21 R34 R43 4 M23

  • M32

M32

  • M23

R23 R32

Y1 = SN−n11X1 + SN−n21X2 Y2 = SN−n12X1 + SN−n22X2 + SN−n32X3 Y3 = SN−n23X2 + SN−n33X3 + SN−n43X4 Y4 = SN−n34X3 + SN−n44X4

N = max(n11, n22, n33, n44, n12, n21, n32, n23, n34, n43)

Theorem 2: The capacity region of the two-way linear deterministic Z chan- nel is the set of all rate-tuples (R12, R21, R23, R32, R34, R43) which satisfy the following: Z →        R12 ≤ n12, R32 ≤ n32, R34 ≤ n34 R12 + R32 ≤ max(n12, n32) R32 + R34 ≤ max(n32, n34) R12 + R32 + R34 ≤ max(n12, n32) + [n34 − n32]+ Z ←        R43 ≤ n43, R23 ≤ n23, R21 ≤ n21 R43 + R23 ≤ max(n43, n23) R23 + R21 ≤ max(n23, n21) R43 + R23 + R21 ≤ max(n43, n23) + [n21 − n23]+.

Wednesday, June 27, 2012

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SLIDE 37

Two-way Linear Deterministic Z Channel

2 3 1 M12 M21

  • M21
  • M12
  • M34
  • M43

M34 M43 R12 R21 R34 R43 4 M23

  • M32

M32

  • M23

R23 R32

Y1 = SN−n11X1 + SN−n21X2 Y2 = SN−n12X1 + SN−n22X2 + SN−n32X3 Y3 = SN−n23X2 + SN−n33X3 + SN−n43X4 Y4 = SN−n34X3 + SN−n44X4

N = max(n11, n22, n33, n44, n12, n21, n32, n23, n34, n43)

Theorem 2: The capacity region of the two-way linear deterministic Z chan- nel is the set of all rate-tuples (R12, R21, R23, R32, R34, R43) which satisfy the following: Z →        R12 ≤ n12, R32 ≤ n32, R34 ≤ n34 R12 + R32 ≤ max(n12, n32) R32 + R34 ≤ max(n32, n34) R12 + R32 + R34 ≤ max(n12, n32) + [n34 − n32]+ Z ←        R43 ≤ n43, R23 ≤ n23, R21 ≤ n21 R43 + R23 ≤ max(n43, n23) R23 + R21 ≤ max(n23, n21) R43 + R23 + R21 ≤ max(n43, n23) + [n21 − n23]+.

Wednesday, June 27, 2012

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SLIDE 38

Two-way Linear Deterministic Z Channel

2 3 1 M12 M21

  • M21
  • M12
  • M34
  • M43

M34 M43 R12 R21 R34 R43 4 M23

  • M32

M32

  • M23

R23 R32

Y1 = SN−n11X1 + SN−n21X2 Y2 = SN−n12X1 + SN−n22X2 + SN−n32X3 Y3 = SN−n23X2 + SN−n33X3 + SN−n43X4 Y4 = SN−n34X3 + SN−n44X4

N = max(n11, n22, n33, n44, n12, n21, n32, n23, n34, n43)

Theorem 2: The capacity region of the two-way linear deterministic Z chan- nel is the set of all rate-tuples (R12, R21, R23, R32, R34, R43) which satisfy the following: Z →        R12 ≤ n12, R32 ≤ n32, R34 ≤ n34 R12 + R32 ≤ max(n12, n32) R32 + R34 ≤ max(n32, n34) R12 + R32 + R34 ≤ max(n12, n32) + [n34 − n32]+ Z ←        R43 ≤ n43, R23 ≤ n23, R21 ≤ n21 R43 + R23 ≤ max(n43, n23) R23 + R21 ≤ max(n23, n21) R43 + R23 + R21 ≤ max(n43, n23) + [n21 − n23]+.

Wednesday, June 27, 2012

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SLIDE 39

Not cut-set

Two-way Linear Deterministic Z Channel

2 3 1 M12 M21

  • M21
  • M12
  • M34
  • M43

M34 M43 R12 R21 R34 R43 4 M23

  • M32

M32

  • M23

R23 R32

Y1 = SN−n11X1 + SN−n21X2 Y2 = SN−n12X1 + SN−n22X2 + SN−n32X3 Y3 = SN−n23X2 + SN−n33X3 + SN−n43X4 Y4 = SN−n34X3 + SN−n44X4

N = max(n11, n22, n33, n44, n12, n21, n32, n23, n34, n43)

Theorem 2: The capacity region of the two-way linear deterministic Z chan- nel is the set of all rate-tuples (R12, R21, R23, R32, R34, R43) which satisfy the following: Z →        R12 ≤ n12, R32 ≤ n32, R34 ≤ n34 R12 + R32 ≤ max(n12, n32) R32 + R34 ≤ max(n32, n34) R12 + R32 + R34 ≤ max(n12, n32) + [n34 − n32]+ Z ←        R43 ≤ n43, R23 ≤ n23, R21 ≤ n21 R43 + R23 ≤ max(n43, n23) R23 + R21 ≤ max(n23, n21) R43 + R23 + R21 ≤ max(n43, n23) + [n21 − n23]+.

Wednesday, June 27, 2012

slide-40
SLIDE 40

Fano as in deterministic Z channel

[Cadambe, Jafar, Vishwanath 2009]

n(R12 + R32 + R34 − ǫ) ≤ I(M12; Y n

2 |M21, M23, M43) + I(M32, M34; Y n 2 , Y n 4 |M43, M12, M21, M23)

≤ H(Y n

2 |M21, M23, M43) + H(Y n 4 |M43, M12, M21, M23, Y n 2 )

≤

n

  • i=1

[H(Y2,i|Y i−1

2

, M21, M23, Xi

2) + H(Y4,i|M12, M21, M23, M43, Y i−1 4

, Xi

4, Y n 2 , Xn 2 )] (a)

≤

n

  • i=1

[H(SN−n12X1,i + SN−n32X3,i) + H(SN−n34X3,i|M12, M21, M23, M43, Y i−1

4

, Xi

4, SN−n12X1,i + SN−n22X2,i + SN−n32X3,i, Xn 2 , Xn 1 )]

≤

n

  • i=1

[H(SN−n12X1,i + SN−n32X3,i) + H(SN−n34X3,i|SN−n32X3,i)] ≤ n(max(n12, n32) + [n34 − n32]+). In (a), Xn

1 in the second entropy term follows since given, M12 and Xn 2 , we may

construct Xn

1 .

Wednesday, June 27, 2012

slide-41
SLIDE 41

Fano as in deterministic Z channel

[Cadambe, Jafar, Vishwanath 2009]

n(R12 + R32 + R34 − ǫ) ≤ I(M12; Y n

2 |M21, M23, M43) + I(M32, M34; Y n 2 , Y n 4 |M43, M12, M21, M23)

≤ H(Y n

2 |M21, M23, M43) + H(Y n 4 |M43, M12, M21, M23, Y n 2 )

≤

n

  • i=1

[H(Y2,i|Y i−1

2

, M21, M23, Xi

2) + H(Y4,i|M12, M21, M23, M43, Y i−1 4

, Xi

4, Y n 2 , Xn 2 )] (a)

≤

n

  • i=1

[H(SN−n12X1,i + SN−n32X3,i) + H(SN−n34X3,i|M12, M21, M23, M43, Y i−1

4

, Xi

4, SN−n12X1,i + SN−n22X2,i + SN−n32X3,i, Xn 2 , Xn 1 )]

≤

n

  • i=1

[H(SN−n12X1,i + SN−n32X3,i) + H(SN−n34X3,i|SN−n32X3,i)] ≤ n(max(n12, n32) + [n34 − n32]+). In (a), Xn

1 in the second entropy term follows since given, M12 and Xn 2 , we may

construct Xn

1 .

Wednesday, June 27, 2012

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SLIDE 42

Two-way linear deterministic IC

2 3 1 M12 M21

  • M21
  • M12
  • M34
  • M43

M34 M43 R12 R21 R34 R43 4

Y1 = SN−n11X1 + SN−n21X2 + SN−n41X4 Y2 = SN−n12X1 + SN−n22X2 + SN−n32X3 Y3 = SN−n23X2 + SN−n33X3 + SN−n43X4 Y4 = SN−n14X1 + SN−n34X3 + SN−n44X4

N = max(n11, n22, n33, n44, n12, n21, n32, n23, n14, n41, n34, n43)

Wednesday, June 27, 2012

slide-43
SLIDE 43

Two-way linear deterministic IC

2 3 1 M12 M21

  • M21
  • M12
  • M34
  • M43

M34 M43 R12 R21 R34 R43 4

Y1 = SN−n11X1 + SN−n21X2 + SN−n41X4 Y2 = SN−n12X1 + SN−n22X2 + SN−n32X3 Y3 = SN−n23X2 + SN−n33X3 + SN−n43X4 Y4 = SN−n14X1 + SN−n34X3 + SN−n44X4

N = max(n11, n22, n33, n44, n12, n21, n32, n23, n14, n41, n34, n43)

X1,i = f1(M12), X2,i = f2(M21, Y i−1

2

) X3,i = f3(M34), X4,i = f4(M43, Y i−1

4

)

If partial adaptation:

Wednesday, June 27, 2012

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SLIDE 44

Two-way linear deterministic IC

2 3 1 M12 M21

  • M21
  • M12
  • M34
  • M43

M34 M43 R12 R21 R34 R43 4

Y1 = SN−n11X1 + SN−n21X2 + SN−n41X4 Y2 = SN−n12X1 + SN−n22X2 + SN−n32X3 Y3 = SN−n23X2 + SN−n33X3 + SN−n43X4 Y4 = SN−n14X1 + SN−n34X3 + SN−n44X4

N = max(n11, n22, n33, n44, n12, n21, n32, n23, n14, n41, n34, n43)

Then capacity:

R12 ≤ n12, R34 ≤ n34, R12 + R34 ≤ max(n12, n32) + [n34 − n32]+ R12 + R34 ≤ max(n34, n14) + [n12 − n14]+ R12 + R34 ≤ max([n12 − n14]+, n32) + max([n34 − n32]+, n14) 2R12 + R34 ≤ max(n12, n32) + [n12 − n14]+ + max([n34 − n32]+, n14) R12 + 2R34 ≤ max(n34, n14) + [n34 − n32]+ + max([n12 − n14]+, n32) R21 ≤ n21, R43 ≤ n43 R21 + R43 ≤ max(n21, n41) + [n43 − n41]+ R21 + R43 ≤ max(n43, n23) + [n21 − n23]+ R21 + R43 ≤ max([n21 − n23]+, n41) + max([n43 − n41]+, n23) 2R21 + R43 ≤ max(n21, n41) + [n21 − n23]+ + max([n43 − n41]+, n23) R21 + 2R43 ≤ max(n43, n23) + [n43 − n41]+ + max([n21 − n23]+, n41)

(A) IC in → direction (B) IC in ← direction

X1,i = f1(M12), X2,i = f2(M21, Y i−1

2

) X3,i = f3(M34), X4,i = f4(M43, Y i−1

4

)

If partial adaptation:

Wednesday, June 27, 2012

slide-45
SLIDE 45

Two-way linear deterministic IC

2 3 1 M12 M21

  • M21
  • M12
  • M34
  • M43

M34 M43 R12 R21 R34 R43 4

Y1 = SN−n11X1 + SN−n21X2 + SN−n41X4 Y2 = SN−n12X1 + SN−n22X2 + SN−n32X3 Y3 = SN−n23X2 + SN−n33X3 + SN−n43X4 Y4 = SN−n14X1 + SN−n34X3 + SN−n44X4

N = max(n11, n22, n33, n44, n12, n21, n32, n23, n14, n41, n34, n43)

Then capacity:

R12 ≤ n12, R34 ≤ n34, R12 + R34 ≤ max(n12, n32) + [n34 − n32]+ R12 + R34 ≤ max(n34, n14) + [n12 − n14]+ R12 + R34 ≤ max([n12 − n14]+, n32) + max([n34 − n32]+, n14) 2R12 + R34 ≤ max(n12, n32) + [n12 − n14]+ + max([n34 − n32]+, n14) R12 + 2R34 ≤ max(n34, n14) + [n34 − n32]+ + max([n12 − n14]+, n32) R21 ≤ n21, R43 ≤ n43 R21 + R43 ≤ max(n21, n41) + [n43 − n41]+ R21 + R43 ≤ max(n43, n23) + [n21 − n23]+ R21 + R43 ≤ max([n21 − n23]+, n41) + max([n43 − n41]+, n23) 2R21 + R43 ≤ max(n21, n41) + [n21 − n23]+ + max([n43 − n41]+, n23) R21 + 2R43 ≤ max(n43, n23) + [n43 − n41]+ + max([n21 − n23]+, n41)

(A) IC in → direction (B) IC in ← direction

X1,i = f1(M12), X2,i = f2(M21, Y i−1

2

) X3,i = f3(M34), X4,i = f4(M43, Y i−1

4

)

If partial adaptation:

Wednesday, June 27, 2012

slide-46
SLIDE 46

Two-way linear deterministic IC

2 3 1 M12 M21

  • M21
  • M12
  • M34
  • M43

M34 M43 R12 R21 R34 R43 4

Y1 = SN−n11X1 + SN−n21X2 + SN−n41X4 Y2 = SN−n12X1 + SN−n22X2 + SN−n32X3 Y3 = SN−n23X2 + SN−n33X3 + SN−n43X4 Y4 = SN−n14X1 + SN−n34X3 + SN−n44X4

N = max(n11, n22, n33, n44, n12, n21, n32, n23, n14, n41, n34, n43)

Then capacity:

R12 ≤ n12, R34 ≤ n34, R12 + R34 ≤ max(n12, n32) + [n34 − n32]+ R12 + R34 ≤ max(n34, n14) + [n12 − n14]+ R12 + R34 ≤ max([n12 − n14]+, n32) + max([n34 − n32]+, n14) 2R12 + R34 ≤ max(n12, n32) + [n12 − n14]+ + max([n34 − n32]+, n14) R12 + 2R34 ≤ max(n34, n14) + [n34 − n32]+ + max([n12 − n14]+, n32) R21 ≤ n21, R43 ≤ n43 R21 + R43 ≤ max(n21, n41) + [n43 − n41]+ R21 + R43 ≤ max(n43, n23) + [n21 − n23]+ R21 + R43 ≤ max([n21 − n23]+, n41) + max([n43 − n41]+, n23) 2R21 + R43 ≤ max(n21, n41) + [n21 − n23]+ + max([n43 − n41]+, n23) R21 + 2R43 ≤ max(n43, n23) + [n43 − n41]+ + max([n21 − n23]+, n41)

(A) IC in → direction (B) IC in ← direction

X1,i = f1(M12), X2,i = f2(M21, Y i−1

2

) X3,i = f3(M34), X4,i = f4(M43, Y i−1

4

)

If partial adaptation:

Wednesday, June 27, 2012

slide-47
SLIDE 47

Two-way linear deterministic IC

2 3 1 M12 M21

  • M21
  • M12
  • M34
  • M43

M34 M43 R12 R21 R34 R43 4

Y1 = SN−n11X1 + SN−n21X2 + SN−n41X4 Y2 = SN−n12X1 + SN−n22X2 + SN−n32X3 Y3 = SN−n23X2 + SN−n33X3 + SN−n43X4 Y4 = SN−n14X1 + SN−n34X3 + SN−n44X4

N = max(n11, n22, n33, n44, n12, n21, n32, n23, n14, n41, n34, n43)

If FULL adaptation:

Wednesday, June 27, 2012

slide-48
SLIDE 48

Two-way linear deterministic IC

2 3 1 M12 M21

  • M21
  • M12
  • M34
  • M43

M34 M43 R12 R21 R34 R43 4

Y1 = SN−n11X1 + SN−n21X2 + SN−n41X4 Y2 = SN−n12X1 + SN−n22X2 + SN−n32X3 Y3 = SN−n23X2 + SN−n33X3 + SN−n43X4 Y4 = SN−n14X1 + SN−n34X3 + SN−n44X4

N = max(n11, n22, n33, n44, n12, n21, n32, n23, n14, n41, n34, n43)

If FULL adaptation: Then we still have the outer bounds:

R12 ≤ n12, R34 ≤ n34, R12 + R34 ≤ max(n12, n32) + [n34 − n32]+ R12 + R34 ≤ max(n34, n14) + [n12 − n14]+ R12 + R34 ≤ max([n12 − n14]+, n32) + max([n34 − n32]+, n14) 2R12 + R34 ≤ max(n12, n32) + [n12 − n14]+ + max([n34 − n32]+, n14) R12 + 2R34 ≤ max(n34, n14) + [n34 − n32]+ + max([n12 − n14]+, n32) R21 ≤ n21, R43 ≤ n43 R21 + R43 ≤ max(n21, n41) + [n43 − n41]+ R21 + R43 ≤ max(n43, n23) + [n21 − n23]+ R21 + R43 ≤ max([n21 − n23]+, n41) + max([n43 − n41]+, n23) 2R21 + R43 ≤ max(n21, n41) + [n21 − n23]+ + max([n43 − n41]+, n23) R21 + 2R43 ≤ max(n43, n23) + [n43 − n41]+ + max([n21 − n23]+, n41)

(A) IC in → direction (B) IC in ← direction

Wednesday, June 27, 2012

slide-49
SLIDE 49

Two-way linear deterministic IC

2 3 1 M12 M21

  • M21
  • M12
  • M34
  • M43

M34 M43 R12 R21 R34 R43 4

Y1 = SN−n11X1 + SN−n21X2 + SN−n41X4 Y2 = SN−n12X1 + SN−n22X2 + SN−n32X3 Y3 = SN−n23X2 + SN−n33X3 + SN−n43X4 Y4 = SN−n14X1 + SN−n34X3 + SN−n44X4

N = max(n11, n22, n33, n44, n12, n21, n32, n23, n14, n41, n34, n43)

If FULL adaptation: Then we still have the outer bounds:

R12 ≤ n12, R34 ≤ n34, R12 + R34 ≤ max(n12, n32) + [n34 − n32]+ R12 + R34 ≤ max(n34, n14) + [n12 − n14]+ R12 + R34 ≤ max([n12 − n14]+, n32) + max([n34 − n32]+, n14) 2R12 + R34 ≤ max(n12, n32) + [n12 − n14]+ + max([n34 − n32]+, n14) R12 + 2R34 ≤ max(n34, n14) + [n34 − n32]+ + max([n12 − n14]+, n32) R21 ≤ n21, R43 ≤ n43 R21 + R43 ≤ max(n21, n41) + [n43 − n41]+ R21 + R43 ≤ max(n43, n23) + [n21 − n23]+ R21 + R43 ≤ max([n21 − n23]+, n41) + max([n43 − n41]+, n23) 2R21 + R43 ≤ max(n21, n41) + [n21 − n23]+ + max([n43 − n41]+, n23) R21 + 2R43 ≤ max(n43, n23) + [n43 − n41]+ + max([n21 − n23]+, n41)

(A) IC in → direction (B) IC in ← direction

Wednesday, June 27, 2012

slide-50
SLIDE 50

Two-way linear deterministic IC

2 3 1 M12 M21

  • M21
  • M12
  • M34
  • M43

M34 M43 R12 R21 R34 R43 4

Y1 = SN−n11X1 + SN−n21X2 + SN−n41X4 Y2 = SN−n12X1 + SN−n22X2 + SN−n32X3 Y3 = SN−n23X2 + SN−n33X3 + SN−n43X4 Y4 = SN−n14X1 + SN−n34X3 + SN−n44X4

N = max(n11, n22, n33, n44, n12, n21, n32, n23, n14, n41, n34, n43)

If FULL adaptation: Then we still have the outer bounds:

R12 ≤ n12, R34 ≤ n34, R12 + R34 ≤ max(n12, n32) + [n34 − n32]+ R12 + R34 ≤ max(n34, n14) + [n12 − n14]+ R12 + R34 ≤ max([n12 − n14]+, n32) + max([n34 − n32]+, n14) 2R12 + R34 ≤ max(n12, n32) + [n12 − n14]+ + max([n34 − n32]+, n14) R12 + 2R34 ≤ max(n34, n14) + [n34 − n32]+ + max([n12 − n14]+, n32) R21 ≤ n21, R43 ≤ n43 R21 + R43 ≤ max(n21, n41) + [n43 − n41]+ R21 + R43 ≤ max(n43, n23) + [n21 − n23]+ R21 + R43 ≤ max([n21 − n23]+, n41) + max([n43 − n41]+, n23) 2R21 + R43 ≤ max(n21, n41) + [n21 − n23]+ + max([n43 − n41]+, n23) R21 + 2R43 ≤ max(n43, n23) + [n43 − n41]+ + max([n21 − n23]+, n41)

(A) IC in → direction (B) IC in ← direction

M i s s i n g : w e a k i n t e r f e r e n c e E T W

  • t

y p e b

  • u

n d s !

Wednesday, June 27, 2012

slide-51
SLIDE 51

Partial adaptation key lemma:

X1,i = f1(M12), X2,i = f2(M21, Y i−1

2

) X3,i = f3(M34), X4,i = f4(M43, Y i−1

4

)

Partial adaptation conditions

Central to many of the converses!

Wednesday, June 27, 2012

slide-52
SLIDE 52

Remark on partial adaptation:

2 3 1 M12 M21

  • M21
  • M12
  • M34
  • M43

M34 M43 R12 R21 R34 R43 4

X1,i = f1(M12), X2,i = f2(M21, Y i−1

2

) X3,i = f3(M34), X4,i = f4(M43, Y i−1

4

)

Wednesday, June 27, 2012

slide-53
SLIDE 53

Blocks routing at node 1,3

Remark on partial adaptation:

2 3 1 M12 M21

  • M21
  • M12
  • M34
  • M43

M34 M43 R12 R21 R34 R43 4

X1,i = f1(M12), X2,i = f2(M21, Y i−1

2

) X3,i = f3(M34), X4,i = f4(M43, Y i−1

4

)

Wednesday, June 27, 2012

slide-54
SLIDE 54

Blocks routing at node 1,3

Remark on partial adaptation:

2 3 1 M12 M21

  • M21
  • M12
  • M34
  • M43

M34 M43 R12 R21 R34 R43 4

X1,i = f1(M12), X2,i = f2(M21, Y i−1

2

) X3,i = f3(M34), X4,i = f4(M43, Y i−1

4

)

2 3 1 M12 M21

  • M21
  • M12
  • M34
  • M43

M34 M43 R12 R21 R34 R43 4

Wednesday, June 27, 2012

slide-55
SLIDE 55

Blocks routing at node 1,3

Remark on partial adaptation:

2 3 1 M12 M21

  • M21
  • M12
  • M34
  • M43

M34 M43 R12 R21 R34 R43 4

X1,i = f1(M12), X2,i = f2(M21, Y i−1

2

) X3,i = f3(M34), X4,i = f4(M43, Y i−1

4

)

2 3 1 M12 M21

  • M21
  • M12
  • M34
  • M43

M34 M43 R12 R21 R34 R43 4

Adaptation useful in general! Routing!

Wednesday, June 27, 2012

slide-56
SLIDE 56

Blocks routing at node 1,3

Remark on partial adaptation:

2 3 1 M12 M21

  • M21
  • M12
  • M34
  • M43

M34 M43 R12 R21 R34 R43 4

X1,i = f1(M12), X2,i = f2(M21, Y i−1

2

) X3,i = f3(M34), X4,i = f4(M43, Y i−1

4

)

2 3 1 M12 M21

  • M21
  • M12
  • M34
  • M43

M34 M43 R12 R21 R34 R43 4

Adaptation useful in general! Routing!

However, it is sometimes useless!

Wednesday, June 27, 2012

slide-57
SLIDE 57

Symmetric two-way linear deterministic IC

1 3 2

Tx Tx Tx

N N N

4

Tx

N

Rx

⊕ ⊕ ⊕

n11 nD

⊕

Rx Rx

nI n33 nD n22 nI

⊕ ⊕

nI nD n44

Rx

⊕ ⊕

nI nD

Wednesday, June 27, 2012

slide-58
SLIDE 58

Symmetric two-way linear deterministic IC

1 3 2

Tx Tx Tx

N N N

4

Tx

N

Notice symmetry!

Rx

⊕ ⊕ ⊕

n11 nD

⊕

Rx Rx

nI n33 nD n22 nI

⊕ ⊕

nI nD n44

Rx

⊕ ⊕

nI nD

Wednesday, June 27, 2012

slide-59
SLIDE 59

Symmetric two-way linear deterministic IC

1 3 2

Tx Tx Tx

N N N

4

Tx

N

Notice symmetry!

Rx

⊕ ⊕ ⊕

n11 nD

⊕

Rx Rx

nI n33 nD n22 nI

⊕ ⊕

nI nD n44

Rx

⊕ ⊕

nI nD

Wednesday, June 27, 2012

slide-60
SLIDE 60

Symmetric two-way linear deterministic IC

1 3 2

Tx Tx Tx

N N N

4

Tx

N

Notice symmetry! Define:

Rx

⊕ ⊕ ⊕

n11 nD

⊕

Rx Rx

nI n33 nD n22 nI

⊕ ⊕

nI nD n44

Rx

⊕ ⊕

nI nD

Wednesday, June 27, 2012

slide-61
SLIDE 61

1 3 2 4

One-way IC

2 messages

[Etkin, Tse, Wang 2008] [Bresler, Tse 2008] [El Gamal, Costa 1982]

Two-way interference channel related work

Wednesday, June 27, 2012

slide-62
SLIDE 62

1 3 2 4

One-way IC

2 messages

[Etkin, Tse, Wang 2008] [Bresler, Tse 2008] [El Gamal, Costa 1982]

Two-way interference channel related work

1 3 2 4

One-way IC with FB

2 messages

[Suh, Tse 2011]

Wednesday, June 27, 2012

slide-63
SLIDE 63

1 3 2 4

One-way IC

2 messages

[Etkin, Tse, Wang 2008] [Bresler, Tse 2008]

1 3 2 4

One-way IC with rate-limited FB

R R

2 messages

[Vahid, Suh, Avestimehr 2012] [El Gamal, Costa 1982]

Two-way interference channel related work

1 3 2 4

One-way IC with FB

2 messages

[Suh, Tse 2011]

Wednesday, June 27, 2012

slide-64
SLIDE 64

1 3 2 4

One-way IC

2 messages

[Etkin, Tse, Wang 2008] [Bresler, Tse 2008]

1 3 2 4

One-way IC with rate-limited FB

R R

2 messages

[Vahid, Suh, Avestimehr 2012] [El Gamal, Costa 1982]

Two-way interference channel related work

1 3 2 4

One-way IC with interfering FB

2 messages

1 3 2 4

Time-share forward + reverse

[Suh, Wang, Tse 2012]

(two-way interference channel) 1 3 2 4 One-way IC with FB

2 messages

[Suh, Tse 2011]

Wednesday, June 27, 2012

slide-65
SLIDE 65

1 3 2 4

One-way IC

2 messages

[Etkin, Tse, Wang 2008] [Bresler, Tse 2008]

1 3 2 4

One-way IC with rate-limited FB

R R

2 messages

[Vahid, Suh, Avestimehr 2012] [El Gamal, Costa 1982]

true adaptation

1 3 2 4

Two-way IC

4 messages

[Cheng, Devroye Allerton 2011] [Cheng, Devroye ISIT 2012] [Cheng, Devroye CISS 2011] [Cheng, Devroye

  • sub. Trans IT, 2012]

Two-way interference channel related work

1 3 2 4

One-way IC with interfering FB

2 messages

1 3 2 4

Time-share forward + reverse

[Suh, Wang, Tse 2012]

(two-way interference channel) 1 3 2 4 One-way IC with FB

2 messages

[Suh, Tse 2011]

Wednesday, June 27, 2012

slide-66
SLIDE 66

0.5 1 1.5 2 2.5 3 0.4 0.6 0.8 1 1.2 1.4 1.6

  • Csym

One−way IC with perfect feedback One−way IC = Two−way IC with partial adaptation One−way IC with rate−limited feedback Two−way IC with full adaptation

Symmetric sum-rate capacity comparison:

Wednesday, June 27, 2012

slide-67
SLIDE 67

Two-way capacity ≡ One-way ➔ One-way ➔

(as in two-way binary adder, two-way Gaussian channels)

Wednesday, June 27, 2012

slide-68
SLIDE 68

Preview on Gaussian two-way interference channel

Under submission to IEEE Trans. on Inf. Theory, 2012

Two-way Interference Constant Gaps per user per direction, in bits (to outer bound) Very Strong 0 (partial) Strong 1 (full) INR < 1 1 (full) HK1 is active 1.5 (full) Weak INR ≥ 1 → direction 1 (partial) HK2 is active ← direction SNR ≤ INR3 1 (partial) SNR > INR3 2 (partial)

TABLE I CONSTANT GAPS BETWEEN NON-ADAPTIVE SYMMETRIC HAN AND KOBAYASHI SCHEMES IN EACH DIRECTION AND PARTIALLY OR

FULLY ADAPTIVE OUTER BOUNDS FOR THE TWO-WAY GAUSSIAN IC. Wednesday, June 27, 2012

slide-69
SLIDE 69

Conclusion

Adaptation appears to be useless when:

Node 1 Node 3 Node 2 Node 1 Node 3 Node 2 Node 4 Node 1 Node 3 Node 2 Node 4

(a) Two-way MAC/BC (b) Two-way Z channel (c) Two-way interference channel

(a) Can cancel “self-interference” (b) No coherent gains to be had (c) No “routing’’ possible

In general, when in adaptation useless (does not increase capacity region) in two-way networks?

[Cheng and Devroye, “Two-way Networks: when Adaptation is Useless”, Submitted to IEEE Trans. IT, available on arxiv.org.]

Wednesday, June 27, 2012

slide-70
SLIDE 70

Thank you!

Zhiyu Cheng zcheng3@uic.edu

Wednesday, June 27, 2012