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

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 – 6, 2012, Cambridge, MA, USA Wednesday, June 27, 2012

Most wireless communications are two-way Wednesday, June 27, 2012

Most wireless communications are two-way Video conferencing Wednesday, June 27, 2012

Most wireless communications are two-way Video conferencing Wednesday, June 27, 2012

Information Theory: point-to-point two-way channel [Claude. E. Shannon, 1961] � X 1 ( M 12 ) Y 2 M 12 2 1 Channel � X 2 ( M 21 ) Y 1 M 21 Wednesday, June 27, 2012

Information Theory: point-to-point two-way channel [Claude. E. Shannon, 1961] � X 1 ( M 12 ) Y 2 M 12 2 1 Channel � X 2 ( M 21 ) Y 1 M 21 Capacity in general remains open. Wednesday, June 27, 2012

Why two-way problem is so hard? Wednesday, June 27, 2012

Why two-way problem is so hard? Adaptation! Wednesday, June 27, 2012

Why two-way problem is so hard? Adaptation! x i 1 ( m 12 , y i − 1 y i ) 1 2 2 1 Channel y i x i 2 ( m 21 , y i − 1 ) 2 1 y i − 1 = ( y 2 , 1 , y 2 , 2 , ..., y 2 ,i − 1 ) 2 Wednesday, June 27, 2012

Why two-way problem is so hard? Adaptation! x i 1 ( m 12 , y i − 1 y i ) 1 2 2 1 Channel y i x i 2 ( m 21 , y i − 1 ) 2 1 y i − 1 = ( y 2 , 1 , y 2 , 2 , ..., y 2 ,i − 1 ) 2 Binary Multiplier Channel: Y 1 = Y 2 = X 1 X 2 X 1 , X 2 , Y 1 , Y 2 ∈ { 0 , 1 } Wednesday, June 27, 2012

Why two-way problem is so hard? Adaptation! x i 1 ( m 12 , y i − 1 y i ) 1 2 2 1 Channel y i x i 2 ( m 21 , y i − 1 ) 2 1 y i − 1 = ( y 2 , 1 , y 2 , 2 , ..., y 2 ,i − 1 ) 2 Binary Multiplier Channel: Y 1 = Y 2 = X 1 X 2 X 1 , X 2 , Y 1 , Y 2 ∈ { 0 , 1 } Adaptation is useful and capacity is unknown. Wednesday, June 27, 2012

Two-way deterministic modulo 2 adder channel � X 1 ( M 12 ) Y 2 M 12 2 1 Channel � X 2 ( M 21 ) Y 1 M 21 X 1 , X 2 , Y 1 , Y 2 ∈ { 0 , 1 } Y 1 = Y 2 = X 1 ⊕ X 2 Wednesday, June 27, 2012

Two-way deterministic modulo 2 adder channel � X 1 ( M 12 ) Y 2 M 12 2 1 Channel � X 2 ( M 21 ) Y 1 M 21 X 1 , X 2 , Y 1 , Y 2 ∈ { 0 , 1 } Y 1 = Y 2 = X 1 ⊕ X 2 At each channel use, Y 1 = X 1 ⊕ X 2 Y 2 = X 1 ⊕ X 2 Wednesday, June 27, 2012

Two-way deterministic modulo 2 adder channel � X 1 ( M 12 ) Y 2 M 12 2 1 Channel � X 2 ( M 21 ) Y 1 M 21 X 1 , X 2 , Y 1 , Y 2 ∈ { 0 , 1 } Y 1 = Y 2 = X 1 ⊕ X 2 At each channel use, ⊕ X 1 Y 1 = X 1 ⊕ X 2 Y 2 = X 1 ⊕ X 2 ⊕ X 2 Wednesday, June 27, 2012

Two-way deterministic modulo 2 adder channel � X 1 ( M 12 ) Y 2 M 12 2 1 Channel � X 2 ( M 21 ) Y 1 M 21 X 1 , X 2 , Y 1 , Y 2 ∈ { 0 , 1 } Y 1 = Y 2 = X 1 ⊕ X 2 At each channel use, ⊕ X 1 Y 1 = X 1 ⊕ X 2 Y 2 = X 1 ⊕ X 2 ⊕ X 2 Wednesday, June 27, 2012

Two-way deterministic modulo 2 adder channel � X 1 ( M 12 ) Y 2 M 12 2 1 Channel � X 2 ( M 21 ) Y 1 M 21 X 1 , X 2 , Y 1 , Y 2 ∈ { 0 , 1 } Y 1 = Y 2 = X 1 ⊕ X 2 At each channel use, ⊕ X 1 Y 1 = X 1 ⊕ X 2 = X 2 Y 2 = X 1 ⊕ X 2 ⊕ X 2 = X 1 Wednesday, June 27, 2012

Two-way deterministic modulo 2 adder channel � X 1 ( M 12 ) Y 2 M 12 2 1 Channel � X 2 ( M 21 ) Y 1 M 21 X 1 , X 2 , Y 1 , Y 2 ∈ { 0 , 1 } Y 1 = Y 2 = X 1 ⊕ X 2 Self-interference At each channel use, ⊕ X 1 Y 1 = X 1 ⊕ X 2 = X 2 Y 2 = X 1 ⊕ X 2 ⊕ X 2 = X 1 Wednesday, June 27, 2012

Two-way deterministic modulo 2 adder channel � X 1 ( M 12 ) Y 2 M 12 2 1 Channel � X 2 ( M 21 ) Y 1 M 21 X 1 , X 2 , Y 1 , Y 2 ∈ { 0 , 1 } Y 1 = Y 2 = X 1 ⊕ X 2 Self-interference At each channel use, ⊕ X 1 Y 1 = X 1 ⊕ X 2 = X 2 Y 2 = X 1 ⊕ X 2 ⊕ X 2 = X 1 Capacity region= 1 bit / channel use / user Wednesday, June 27, 2012

Two-way deterministic modulo 2 adder channel � X 1 ( M 12 ) Y 2 M 12 2 1 Channel � X 2 ( M 21 ) Y 1 M 21 X 1 , X 2 , Y 1 , Y 2 ∈ { 0 , 1 } Y 1 = Y 2 = X 1 ⊕ X 2 Self-interference At each channel use, ⊕ X 1 Y 1 = X 1 ⊕ X 2 = X 2 Y 2 = X 1 ⊕ X 2 ⊕ X 2 = X 1 Capacity region= 1 bit / channel use / user Adaptation is not needed. Wednesday, June 27, 2012

Gaussian two-way point-to-point channel [T. Han, 1984] N 1 � Y 2 X 1 ( M 12 ) M 12 1 2 Channel � X 2 ( M 21 ) Y 1 M 21 N 2 Y 1 = aX 1 + bX 2 + N 1 Y 2 = cX 1 + dX 2 + N 2 N 1 , N 2 ∼ N (0 , σ 2 ) and are channel gains. a, b, c d Wednesday, June 27, 2012

Gaussian two-way point-to-point channel [T. Han, 1984] N 1 � Y 2 X 1 ( M 12 ) M 12 1 2 Channel � X 2 ( M 21 ) Y 1 M 21 N 2 Y 1 = aX 1 + bX 2 + N 1 Y 2 = cX 1 + dX 2 + N 2 N 1 , N 2 ∼ N (0 , σ 2 ) and are channel gains. a, b, c d Wednesday, June 27, 2012

Gaussian two-way point-to-point channel [T. Han, 1984] N 1 � � Y 2 Y 2 X 1 ( M 12 ) M 12 M 12 X 1 ( M 12 ) 1 2 1 2 Channel Channel � X 2 ( M 21 ) Y 1 M 21 N 2 N 2 Y 2 = cX 1 + N 2 Y 1 = aX 1 + bX 2 + N 1 ≡ N 1 Y 2 = cX 1 + dX 2 + N 2 1 2 Channel X 2 ( M 21 ) � Y 1 M 21 N 1 , N 2 ∼ N (0 , σ 2 ) Y 1 = bX 2 + N 1 and are channel gains. a, b, c d Wednesday, June 27, 2012

Gaussian two-way point-to-point channel [T. Han, 1984] N 1 � � Y 2 Y 2 X 1 ( M 12 ) M 12 M 12 X 1 ( M 12 ) 1 2 1 2 Channel Channel � X 2 ( M 21 ) Y 1 M 21 N 2 N 2 Y 2 = cX 1 + N 2 Y 1 = aX 1 + bX 2 + N 1 ≡ N 1 Y 2 = cX 1 + dX 2 + N 2 1 2 Channel X 2 ( M 21 ) � Y 1 M 21 N 1 , N 2 ∼ N (0 , σ 2 ) Y 1 = bX 2 + N 1 and are channel gains. a, b, c d Again, adaptation does not increase capacity. Wednesday, June 27, 2012

What about two-way networks? Wednesday, June 27, 2012

3 channel models we consider Node 1 Node 2 Node 3 (a) Two-way MAC/BC Wednesday, June 27, 2012

3 channel models we consider Node 1 Node 1 Node 2 Node 2 Node 3 Node 4 Node 3 (a) Two-way MAC/BC (b) Two-way Z channel Wednesday, June 27, 2012

3 channel models we consider Node 1 Node 1 Node 2 Node 2 Node 3 Node 4 Node 3 (a) Two-way MAC/BC (b) Two-way Z channel Node 1 Node 2 Node 4 Node 3 (c) Two-way interference channel Wednesday, June 27, 2012

3 channel models we consider Node 1 Node 1 Node 2 Node 2 Node 3 Node 4 Node 3 (a) Two-way MAC/BC (b) Two-way Z channel Node 1 Node 2 Consider linear deterministic model [Avestimehr; Diggavi; Tse; 2007] Node 4 Node 3 (c) Two-way interference channel Wednesday, June 27, 2012

Two-way Linear Deterministic MAC/BC M 12 N = max( n 11 , n 22 , n 33 , n 12 , n 21 , n 32 , n 23 ) 1 � R 12 M 21 Y 1 = S N − n 11 X 1 + S N − n 21 X 2 R 21 M 21 M 23 2 Y 2 = S N − n 12 X 1 + S N − n 22 X 2 + S N − n 32 X 3 � � R 23 M 12 M 32 Y 3 = S N − n 23 X 2 + S N − n 33 X 3 , M 32 R 32 3 � S shift matrix, inputs binary vectors M 23 Wednesday, June 27, 2012

Two-way Linear Deterministic MAC/BC M 12 N = max( n 11 , n 22 , n 33 , n 12 , n 21 , n 32 , n 23 ) 1 � R 12 M 21 Y 1 = S N − n 11 X 1 + S N − n 21 X 2 R 21 M 21 M 23 2 Y 2 = S N − n 12 X 1 + S N − n 22 X 2 + S N − n 32 X 3 � � R 23 M 12 M 32 Y 3 = S N − n 23 X 2 + S N − n 33 X 3 , M 32 R 32 3 � S shift matrix, inputs binary vectors M 23 Tx 1 N Tx N 2 Tx N 3 Wednesday, June 27, 2012

Two-way Linear Deterministic MAC/BC M 12 N = max( n 11 , n 22 , n 33 , n 12 , n 21 , n 32 , n 23 ) 1 � R 12 M 21 Y 1 = S N − n 11 X 1 + S N − n 21 X 2 R 21 M 21 M 23 2 Y 2 = S N − n 12 X 1 + S N − n 22 X 2 + S N − n 32 X 3 � � R 23 M 12 M 32 Y 3 = S N − n 23 X 2 + S N − n 33 X 3 , M 32 R 32 3 � S shift matrix, inputs binary vectors M 23 Tx Rx ⊕ 1 n 21 N n 11 Rx Tx ⊕ ⊕ N 2 Tx Rx n 12 n 22 n 32 n 33 N ⊕ n 23 3 Wednesday, June 27, 2012

Two-way Linear Deterministic MAC/BC M 12 N = max( n 11 , n 22 , n 33 , n 12 , n 21 , n 32 , n 23 ) 1 � R 12 M 21 Y 1 = S N − n 11 X 1 + S N − n 21 X 2 R 21 M 21 M 23 2 Y 2 = S N − n 12 X 1 + S N − n 22 X 2 + S N − n 32 X 3 � � R 23 M 12 M 32 Y 3 = S N − n 23 X 2 + S N − n 33 X 3 , M 32 R 32 3 � S shift matrix, inputs binary vectors M 23 Self-interference Tx Rx ⊕ 1 n 21 N n 11 Rx Tx ⊕ ⊕ N 2 Tx Rx n 12 n 22 n 32 n 33 N ⊕ n 23 3 Wednesday, June 27, 2012

Two-way Linear Deterministic MAC/BC M 12 N = max( n 11 , n 22 , n 33 , n 12 , n 21 , n 32 , n 23 ) 1 � R 12 M 21 Y 1 = S N − n 11 X 1 + S N − n 21 X 2 R 21 M 21 M 23 2 Y 2 = S N − n 12 X 1 + S N − n 22 X 2 + S N − n 32 X 3 � � R 23 M 12 M 32 Y 3 = S N − n 23 X 2 + S N − n 33 X 3 , M 32 R 32 3 � S shift matrix, inputs binary vectors M 23 Theorem 1 : The capacity region of the two-way linear deterministic MAC/BC is the set of non-negative rate tuples ( R 12 , R 32 , R 21 , R 23 ) such that � R 12 ≤ n 12 , R 32 ≤ n 32 , MAC → (1) R 12 + R 32 ≤ max( n 12 , n 32 ) � R 21 ≤ n 21 , R 23 ≤ n 23 BC ← (2) R 21 + R 23 ≤ max( n 21 , n 23 ) . Wednesday, June 27, 2012