Adaptive Coding for Two-Way Lossy Source-Channel Communication Jian-Jia Weng, Fady Alajaji, and Tam´ as Linder Department of Mathematics and Statistics Queen’s University, Kingston, Canada IEEE International Symposium on Information Theory, June 2020
Two-Way Communication Channel [Shannon’61] • Shannon’s two-way channel (TWC) provides in-band full-duplex data transfer between two terminals T 1 T 2 TWC • Discrete memoryless two-way channel (DM-TWC): • Channel inputs and outputs: X j ∈ X j and Y j ∈ Y j , j = 1 , 2 • Channel transition probability: P Y 1 ,Y 2 | X 1 ,X 2 2 / 20
The Capacity Region C of DM-TWCs • The region C can be fully characterized using limiting expressions [Shannon, 1961], [Kramer, 1998], but these are often incomputable • In general, it is only known that C I ⊆ C ⊆ C O : R 2 Outer Bound C O Inner Bound C I R 1 3 / 20
Some Results on the Capacity Region of DM-TWCs • Inner bounds C I : • Shannon (1961): general TWCs • Schalkwijk (1982, 1983): binary multiplying TWC • Han (1984): general TWCs • Sabag and Permuter (2018): common-output TWCs • Outer bounds C O : • Shannon (1961): general TWCs • Zhang, Berger, and Schalkwijk (1986): general TWCs • Hekstra and Willems (1989): common-output TWCs • Tightness conditions for Shannon’s inner bound: [Shannon, 1961], [Hekstra et al. , 1989], [Varshney, 2013], [Chaaban et al. , 2017], [Weng et al. , 2019] 4 / 20
Lossy Transmission of Correlated Sources over DM-TWCs S K X N X N S K 1 1 2 2 DM-TWC T1 T2 P Y 1 ,Y 2 | X 1 ,X 2 ˆ Y N Y N ˆ S K S K 2 1 2 1 • Correlated sources: • K : block length of source messages • { ( S 1 ,k , S 2 ,k ) } is a memoryless stationary process with S j,k ∈ S j for finite source alphabets S j , j = 1 , 2 • Reconstruction and average distortion: • ˆ S j,k ∈ S j : the reconstruction of S j,k • D j � K − 1 � K k =1 E [ d j ( S j,k , ˆ S j,k )], where d j is a single-letter distortion measure • Overall rate: K N (source symbol/channel use), where N is the total number of channel uses for the overall transmission 5 / 20
Joint Source-Channel Codes • Adaptive encoding: for j = 1 , 2 and 1 ≤ n ≤ N , • f j = ( f j, 1 , f j, 2 , . . . , f j,N ) • X j,n = f j,n ( S K j , Y n − 1 ), where S K j = ( S j, 1 , S j, 2 , . . . , S j,K ) j • Decoding with side-information: for j, j ′ = 1 , 2 with j � = j ′ and 1 ≤ k ≤ K , • g j = ( g j, 1 , g j, 2 , . . . , g j,K ) • ˆ S j ′ ,k = g j,k ( S K j , Y N j ) S K S K f 1 f 2 1 2 DM-TWC ˆ ˆ S K g 1 g 2 S K 2 1 6 / 20
Our Research Problem For a pair of correlated sources and given DM-TWC, we seek forward achiev- ability joint source-channel coding (JSCC) theorem for the transmissibility under fidelity constraints 1 , ˆ E [ d 1 ( S K S K 1 )] ≤ D 1 X N X N 2 ˆ 1 S K S K 1 1 T 1 T 2 DM-TWC ˆ S K S K 2 Y N Y N 2 1 2 2 , ˆ E [ d 2 ( S K S K 2 )] ≤ D 2 7 / 20
Related Work • Correlation-preserving coding scheme for almost lossless transmission of correlated sources [G¨ und¨ uz et al ., 2009] • Two-way lossy transmission of correlated sources [Weng et al ., 2017, 2019] • separate source-channel coding (SSCC) scheme that combines Wyner-Ziv (WZ) source coding and Shannon’s channel coding • two-way hybrid analog/digital coding scheme • Two-way interactive lossy transmission of correlated sources: • noiseless TWCs [Kaspi, 1985] • orthogonal one-way noisy channels [Maor and Merhav, 2008] 8 / 20
Contributions • We propose an adaptive coding scheme, which proves a forward JSCC theorem • We show that the proposed scheme strictly generalizes prior results • Our scheme also yields a simple SSCC scheme that combines Wyner-Ziv (WZ) source coding and Han’s adaptive channel coding 9 / 20
Main Idea We couple the two terminals’ encoding and transmission processes through a stationary Markov chain Two-Way Coded Channel ( S 1 , U 1 , ˜ S 1 , ˜ U 1 , ˜ ( S 2 , U 2 , ˜ S 2 , ˜ U 2 , ˜ W 1 ) X 1 X 2 W 2 ) F 1 F 2 DM-TWC Y 1 Y 2 Markov Transmission Process 10 / 20
Auxiliary Coded Two-Way Channels • Function F j transforms the inputs of the coded channel into physical inputs • S j and U j : current source message and its coded data • ˜ S j and ˜ U j : some prior source message and its coded data • ˜ W j : some prior channel inputs and outputs • Joint input distribution of the coded channel: P S 1 ,S 2 ,U 1 ,U 2 , ˜ W 2 = P S 1 ,S 2 P U 1 | S 1 P U 2 | S 2 P ˜ S 1 , ˜ S 2 , ˜ U 1 , ˜ U 2 , ˜ W 1 , ˜ S 1 , ˜ S 2 , ˜ U 1 , ˜ U 2 , ˜ W 1 , ˜ W 1 ( S 1 , U 1 , ˜ S 1 , ˜ U 1 , ˜ ( S 2 , U 2 , ˜ S 2 , ˜ U 2 , ˜ X 1 X 2 W 1 ) W 2 ) F 1 F 2 DM-TWC Y 1 Y 2 11 / 20
Markov Transmission Process - State Space • A time-homogeneous Markov chain Z ( t ) is constructed with state space: S 1 × S 2 × U 1 × U 2 × ˜ S 1 × ˜ S 2 × ˜ U 1 × ˜ U 2 × ˜ W 1 × ˜ W 2 × X 1 × X 2 × Y 1 × Y 2 • For all t , ( S ( t ) 1 , S ( t ) 2 , U ( t ) 1 , U ( t ) S ( t ) S ( t ) U ( t ) U ( t ) W ( t ) W ( t ) 2 ) is independent of ( ˜ 1 , ˜ 2 , ˜ 1 , ˜ 2 , ˜ 1 , ˜ 2 ) • For t ≥ 2 and j = 1 , 2, we set S ( t ) ˜ = S ( t − 1) , ˜ U ( t ) = U ( t − 1) , and ˜ W ( t ) = ( X ( t − 1) , Y ( t − 1) ) j j j j j j j Two-Way Coded Channel ( S 1 , U 1 , ˜ S 1 , ˜ U 1 , ˜ ( S 2 , U 2 , ˜ S 2 , ˜ U 2 , ˜ W 1 ) X 1 X 2 W 2 ) F 1 F 2 DM-TWC Y 2 Y 1 Markov Transmission Process 12 / 20
Markov Transmission Process - Transition Kernel • For t ≥ 2, the transition kernel of { Z ( t ) } is given by w 2 , x 1 , x 2 , y 1 , y 2 | s ′ 1 , s ′ 2 , u ′ 1 , u ′ P Z ( t ) | Z ( t − 1) ( s 1 , s 2 , u 1 , u 2 , ˜ s 1 , ˜ s 2 , ˜ u 1 , ˜ u 2 , ˜ w 1 , ˜ 2 , s ′ s ′ u ′ u ′ w ′ w ′ 2 , x ′ 1 , x ′ 2 , y ′ 1 , y ′ ˜ 1 , ˜ 2 , ˜ 1 , ˜ 2 , ˜ 1 , ˜ 2 ) = P S 1 ,S 2 ( s 1 , s 2 ) P U 1 | S 1 ( u 1 | s 1 ) P U 2 | S 2 ( u 2 | s 2 ) s 1 = s ′ s 2 = s ′ u 1 = u ′ u 2 = u ′ w 1 = ( x ′ 1 , y ′ w 2 = ( x ′ 2 , y ′ · ✶ { ˜ 1 } ✶ { ˜ 2 } ✶ { ˜ 1 } ✶ { ˜ 2 } ✶ { ˜ 1 ) } ✶ { ˜ 2 ) } · ✶ { x 1 = F 1 ( s 1 , u 1 , ˜ s 1 , ˜ u 1 , ˜ w 1 ) } ✶ { x 2 = F 2 ( s 2 , u 2 , ˜ s 2 , ˜ u 2 , ˜ w 2 ) } · P Y 1 ,Y 2 | X 1 ,X 2 ( y 1 , y 2 | x 1 , x 2 ) where ✶ {·} denotes the indicator function • Parameters: F 1 , F 2 , P U 1 | S 1 , P U 2 | S 2 , P ˜ 2 , and P ˜ S (1) S (1) U (1) S (1) W (1) W (1) S (1) S (1) U (1) U (1) , ˜ , ˜ , ˜ , ˜ | ˜ , ˜ , ˜ , ˜ 1 2 1 1 2 1 2 1 2 13 / 20
Markov Transmission Process - Stationary Configuration • To obtain time-invariant achievability conditions, we only consider stationary chain, in which P ˜ = P S 1 ,S 2 P U 1 | S 1 P U 2 | S 2 for all t S ( t ) 1 , ˜ S ( t ) 2 , ˜ U ( t ) 1 , ˜ S ( t ) 2 • For given F j and P U j | S j , j = 1 , 2, we find appropriate P ˜ W (1) , ˜ W (1) | ˜ S (1) , ˜ S (1) , ˜ U (1) , ˜ U (1) 1 2 1 2 1 2 • For source reconstruction, we also need the consider decoding functions G j : ˜ U j ′ × S j × U j × ˜ S j × ˜ U j × ˜ W j × Y j → ˆ S j ′ • Stationary configuration: { P U 1 | S 1 , P U 2 | S 2 , P U 1 | S 1 , P ˜ U 2 P ˜ U 2 , F 1 , F 2 , G 1 , G 2 } S 1 , ˜ S 2 , ˜ U 1 , ˜ W 1 , ˜ W 2 | ˜ S 1 , ˜ S 2 , ˜ U 1 , ˜ S j , ˆ • Π Z ( D 1 , D 2 ): the set of all stationary configurations with E [ d j ( ˜ ˜ S j )] ≤ D j , j = 1 , 2 14 / 20
Adaptive Joint Source-Channel Coding Theorem A distortion pair ( D 1 , D 2 ) is achievable for the rate-one lossy transmission of correlated sources over a DM-TWC if there exists a stationary configuration in Π Z ( D 1 , D 2 ) such that I ( ˜ S 1 ; ˜ U 1 ) < I ( ˜ U 1 ; S 2 , U 2 , ˜ S 2 , ˜ U 2 , ˜ W 2 , X 2 , Y 2 ) , I ( ˜ S 2 ; ˜ U 2 ) < I ( ˜ U 2 ; S 1 , U 1 , ˜ S 1 , ˜ U 1 , ˜ W 1 , X 1 , Y 1 ) . 15 / 20
Special Cases • Uncoded transmission scheme • Correlation-preserving coding scheme • A SSCC scheme based on WZ source coding and Shannon’s channel coding • Two-way hybrid analog/digital coding • A SSCC scheme based on WZ source coding and Han’s adaptive channel coding 16 / 20
Coding Scheme used in the Proof • Block-wise encoder structure (including three coding components): S (1) S ( 2) S ( 3) S ( B ) j j j j U (1) U ( 2) U ( 3) U ( B ) j j j j … X ( 3) X ( 4) X ( B +1) X (1) X ( 2) X ( B ) j j j j j j Y (1) Y ( 2) Y ( 3) Y ( B − 1) Y ( B ) Y ( B +1) j j j j j j Hybrid Analog/Digital Coding Superposition Coding Adaptive Channel Coding • Rate: B B +1 , which approaches 1 as B → ∞ • Sliding-window decoder with window size two blocks 17 / 20
A Simple Example • Independent binary uniform sources S 1 and S 2 • Dueck’s DM-TWC model: X j = { 0 , 1 } 2 , Y j = { 0 , 1 } 3 , and j = 1 , 2 • Channel input: X j = ( X j 1 , X j 2 ) • Channel output: Y 1 = ( X 1 , 1 · X 2 , 1 , X 2 , 2 ⊕ N 1 , N 2 ) and Y 2 = ( X 1 , 1 · X 2 , 1 , X 1 , 2 ⊕ N 2 , N 1 ) where ⊕ is binary addition • N 1 and N 2 are correlated with joint distribution P N 1 ,N 2 (0 , 0) = 0 and P N 1 ,N 2 ( n 1 , n 2 ) = 1 / 3 for ( n 1 , n 2 ) � = (0 , 0); they are also independent of S j ’s and X j ’s • Hamming distortion measure • Let K = 1 and choose B large enough 18 / 20
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