Cooperative Strategies and Capacity Theorems for Relay Networks - PowerPoint PPT Presentation

Cooperative Strategies and Capacity Theorems for Relay Networks Desmond Lun 22 November 2004 6.454 Graduate Seminar in Area I

What are relay networks? • Relay network: – multi-terminal network; – single pair of terminals wish to communicate (source and destination); – all other terminals assist (relays). • Special case of single relay → relay channel. • Relay channel: – first studied by van der Meulen (1971); – capacity of the relay channel is an open problem. 6.454 Graduate Seminar in Area I 1

What problem is addressed? • We have little hope of finding the capacity of relay networks. • So we focus on achievable rates → design particular schemes and assess the rates they achieve. • Schemes: – decode-and-forward (related to multi-antenna transmission); – compress-and-forward (related to multi-antenna reception); – mixtures of the two. 6.454 Graduate Seminar in Area I 2

Model • T terminals: source is terminal 1, destination is terminal T . • Network is memoryless and time-invariant. . . . Terminal 2 Terminal T − 1 Y n X n Y n X n 2 2 T − 1 T − 1 X n Y n ˆ W W 1 T p Y 2 ··· YT | X 1 ··· XT − 1 Terminal 1 Terminal T 6.454 Graduate Seminar in Area I 3

Upper bound on capacity • Can get upper bound from cut-set bound for multi-terminal networks: C ≤ max S⊂T I ( X 1 X S ; Y S c Y T | X S c ) min p X 1 X 2 ··· XT − 1 • For relay channel: C ≤ max min( I ( X 1 ; Y 2 Y 3 | X 2 ) , I ( X 1 X 2 ; Y 3 )) . p X 1 X 2 6.454 Graduate Seminar in Area I 4

Decode-and-forward • Relays fully decode message and use knowledge of the message to assist the source. • In a wireless setting: achieves the gains related to multi-antenna transmission. • Scheme we discuss is due to Xie and Kumar. 6.454 Graduate Seminar in Area I 5

Decode-and-forward: Single relay • Divide message w into B blocks w 1 , w 2 , . . . , w B of nR bits each, where R < max min( I ( X 1 ; Y 2 | X 2 ) , I ( X 1 X 2 ; Y 3 )) . p X 1 X 2 • Transmission is performed in B + 1 blocks using random codewords of length n . • Rate is B · nR B ( B + 1) n = R B + 1 bits per use . Arbitrarily close to R for B arbitrarily large. 6.454 Graduate Seminar in Area I 6

Decode-and-forward: Single relay • Code construction: – Take joint distribution p X 1 X 2 . – For block b : ∗ Generate 2 nR codewords x n 2 b ( v ) , choosing symbols { x 2 bi ( v ) } independently using p X 2 . ∗ Generate 2 nR codewords x n 1 b ( v, w ) , choosing symbols { x 1 bi ( v, w ) } independently using { p X 1 | X 2 ( ·| x 2 bi ( v )) } . 6.454 Graduate Seminar in Area I 7

Decode-and-forward: Single relay • B = 3 : Block 1 Block 2 Block 3 Block 4 X n x n x n x n x n 1 = 11 (1 , w 1 ) 12 ( w 1 , w 2 ) 13 ( w 2 , w 3 ) 14 ( w 3 , 1) X n x n x n x n x n 2 = 21 (1) 22 ( w 1 ) 23 ( w 2 ) 24 ( w 3 ) • After transmission of block b , – relay decodes w b ; – destination decodes w b − 1 . 6.454 Graduate Seminar in Area I 8

Decode-and-forward: Single relay w (2) • Relay decodes w b reliably if n is large, ˆ b − 1 = w b − 1 , and R < I ( X 1 ; Y 2 | X 2 ) . w (3) • Destination decodes w b reliably if n is large, ˆ b − 1 = w b − 1 , and R < I ( X 1 ; Y 3 | X 2 ) + I ( X 2 ; Y 3 ) = I ( X 1 X 2 ; Y 3 ) . • There exists a distribution p X 1 X 2 that satisfies both conditions by assumption. 6.454 Graduate Seminar in Area I 9

Decode-and-forward: Multiple relays • Consider two relays. • Divide message w into B blocks w 1 , w 2 , . . . , w B of nR bits each, where R < max min( I ( X 1 ; Y 2 | X 2 X 3 ) , I ( X 1 X 2 ; Y 3 | X 3 ) , I ( X 1 X 2 X 3 ; Y 4 )) . p X 1 X 2 X 3 • Transmission is performed in B + 2 blocks using random codewords of length n . • Rate is B · nR B ( B + 2) n = R B + 2 bits per use . 6.454 Graduate Seminar in Area I 10

Decode-and-forward: Multiple relays • B = 4 : Block 1 Block 2 Block 3 X n x n x n x n 1 = 11 (1 , 1 , w 1 ) 12 (1 , w 1 , w 2 ) 13 ( w 1 , w 2 , w 3 ) X n x n x n x n 2 = 21 (1 , 1) 22 (1 , w 1 ) 23 ( w 1 , w 2 ) X n x n x n x n 3 = 31 (1) 32 (1) 33 ( w 1 ) • After transmission of block b , terminal 2 decodes w b , terminal 3 decodes w b − 1 , destination decodes w b − 2 . 6.454 Graduate Seminar in Area I 11

Decode-and-forward: Multiple relays • B = 4 : Block 4 Block 5 Block 6 X n x n x n x n 1 = 14 ( w 2 , w 3 , w 4 ) 15 ( w 3 , w 4 , 1) 16 ( w 4 , 1 , 1) X n x n x n x n 2 = 24 ( w 2 , w 3 ) 25 ( w 3 , w 4 ) 26 ( w 4 , 1) X n x n x n x n 3 = 34 ( w 2 ) 35 ( w 3 ) 36 ( w 4 ) • After transmission of block b , terminal 2 decodes w b , terminal 3 decodes w b − 1 , destination decodes w b − 2 . 6.454 Graduate Seminar in Area I 12

Decode-and-forward: Multiple relays w (2) w (2) • Terminal 2 decodes w b reliably if n is large, ˆ b − 2 = w b − 2 and ˆ b − 1 = w b − 1 , and R < I ( X 1 ; Y 2 | X 2 X 3 ) . w (3) w (3) • Terminal 3 decodes w b reliably if n is large, ˆ b − 2 = w b − 2 and ˆ b − 1 = w b − 1 , and R < I ( X 1 X 2 ; Y 3 | X 3 ) . w (4) w (4) • Destination decodes w b reliably if n is large, ˆ b − 2 = w b − 2 and ˆ b − 1 = w b − 1 , and R < I ( X 1 X 2 X 3 ; Y 4 ) . • There exists a distribution p X 1 X 2 X 3 that satisfies all three conditions by assumption. 6.454 Graduate Seminar in Area I 13

Decode-and-forward: Multiple relays • Straightforward to generalize scheme to T -terminal relay networks. • Theorem 1. Decode-and-forward achieves any rate up to R DF = max max 1 ≤ t ≤ T − 1 I ( X π (1: t ) ; Y π ( t +1) | X π ( t +1: T − 1) ) . min p X 1 X 2 ··· XT − 1 π • π is a permutation on T with π (1) := 1 and π ( T ) := T . 6.454 Graduate Seminar in Area I 14

Decode-and-forward: Sub-optimality • Requiring the relays to decode can be a severe constraint. • Consider 2 Links are independent with unit capacity. 1 3 • Capacity is clearly 2 bits per use, but decode-and-forward only achieves 1 bit per use. 6.454 Graduate Seminar in Area I 15

Compress-and-forward • Relays do not decode message and, rather, forward compressed versions of their observations. • In a wireless setting: achieves the gains related to multi-antenna reception. • Scheme we discuss is due to Cover and El Gamal (1979) for the single relay network and Kramer et al. for the multiple relay network. 6.454 Graduate Seminar in Area I 16

Compress-and-forward: Single relay • Divide message w into B blocks w 1 , w 2 , . . . , w B of nR bits each, where I ( X 1 ; ˆ R < max Y 2 Y 3 | X 2 ) p X 1 p X 2 p ˆ Y 2 | X 2 Y 2 subject to the constraint I ( ˆ Y 2 ; Y 2 | X 2 Y 3 ) ≤ I ( X 2 ; Y 3 ) . • Transmission is performed in B + 1 blocks using random codewords of length n . • Rate is again R · B/ ( B +1) . Arbitrarily close to R for B arbitrarily large. 6.454 Graduate Seminar in Area I 17

Compress-and-forward: Single relay • Code construction: – Take distributions p X 1 , p X 2 and p ˆ Y 2 | X 2 Y 2 . – For block b : ∗ Generate 2 nR codewords x n 1 b ( w ) , choosing symbols { x 1 bi ( w ) } independently using p X 1 . ∗ Generate 2 nR codewords x n 2 b ( v ) , choosing symbols { x 2 bi ( v ) } independently using p X 2 . 2 n ( R ′ 2 + R 2 ) ∗ (“Quantization” codebook:) Generate codewords y n ˆ 2 b ( v, t, u ) , choosing symbols { ˆ y 2 bi ( v, t, u ) } independently using { p ˆ Y 2 | X 2 ( ·| x 2 bi ( v )) } . 6.454 Graduate Seminar in Area I 18

Compress-and-forward: Single relay • B = 3 : Block 1 Block 2 Block 3 Block 4 X n x n x n x n x n 1 = 11 ( w 1 ) 12 ( w 2 ) 13 ( w 3 ) 14 ( w 4 ) X n x n x n x n x n 2 = 21 (1) 22 ( v 2 ) 23 ( v 3 ) 24 ( v 4 ) ˆ Y n y n y n y n 2 = ˆ 21 (1 , t 1 , v 2 ) ˆ 22 (1 , t 2 , v 3 ) ˆ 23 ( v 2 , t 3 , v 4 ) • After transmission of block b , – relay encodes to ( t b , v b +1 ) , – destination decodes v b , then t b − 1 , then w b − 1 . 6.454 Graduate Seminar in Area I 19

Compress-and-forward: Single relay • Relay encodes to ( t b , v b +1 ) reliably if n is large and 2 > I ( ˆ R 2 + R ′ Y 2 ; Y 2 | X 2 ) . • Destination decodes ( v b , t b − 1 , w b − 1 ) reliably if n is large, ˆ v b − 1 = v b − 1 , 2 < I ( ˆ R < I ( X 1 ; ˆ R ′ R 2 < I ( X 2 ; Y 3 ) , Y 2 ; Y 3 | X 2 ) , Y 2 Y 3 | X 2 ) . • Can find R 2 and R ′ 2 to satisfy these conditions given that assumption on R is satisfied. 6.454 Graduate Seminar in Area I 20

Compress-and-forward: Multiple relays • Compress-and-forward does not generalize to multiple relays as straightforwardly as decode-and-forward. • Main complication: Relays forward their observations simultaneously → interference at other relays and at destination. • Kramer et al. deal with complication by allowing for partial decoding at relays of each other’s codewords. 6.454 Graduate Seminar in Area I 21

Compress-and-forward: Multiple relays • Theorem 2. Compress-and-forward achieves any rate up to I ( X 1 ; ˆ R CF = max Y T Y T | U T X T ) pX 1 { pUtXtp ˆ } t ∈T Yt | U T XtYt where I ( ˆ Y S ; Y S | U T X T ˆ I ( ˆ X Y S c Y T ) + Y t ; X T \{ t } | U T X t ) t ∈S M X ≤ I ( X S ; Y T | U S X S c ) + I ( U K m ; Y r ( m ) | U K c m X r ( m ) ) m =1 for all S ⊂ T , all partitions {K m } M m =1 of S , and all r ( m ) ∈ { 2 , 3 , . . . , T } such that r ( m ) / ∈ K m . For r ( m ) = T , we set X T := 0 . 6.454 Graduate Seminar in Area I 22