Robust Streaming Codes based on Deterministic Channel Approximations - PowerPoint PPT Presentation

Robust Streaming Codes based on Deterministic Channel Approximations Ashish Khisti University of Toronto Joint Work with Ahmed Badr (Toronto), Wai-Tian Tan (HP Labs) and John Apostolopoulos (HP Labs) ISIT, 2013 July 9th 2013

Motivation - Delay Sensitive Communication Delay is a central issue in many applications 1 Application Bit-Rate MSDU (B) Delay (ms) Delay (pkts) PLR 10 − 4 Video Conf. 2 Mbps 1500 100 ms 24 Interactive Gaming 1Mbps 512 50 ms 12 10 − 4 10 − 6 SDTV 4Mbps 1500 200 ms 60 Communication Medium: Wireless Channel. 1 IEEE Usage Model Proposal (doc.: IEEE 802.11-03/802r23) ISIT, 2013 July 9th 2013 2/ 17

Motivation - Delay Sensitive Communication Delay is a central issue in many applications 1 Application Bit-Rate MSDU (B) Delay (ms) Delay (pkts) PLR 10 − 4 Video Conf. 2 Mbps 1500 100 ms 24 Interactive Gaming 1Mbps 512 50 ms 12 10 − 4 10 − 6 SDTV 4Mbps 1500 200 ms 60 Communication Medium: Wireless Channel. Prior Work - Real Time Streaming Communication Structural Theorems on Real-Time Encoders (Witsenhausen ’79, Teneketzis ’06) Tree Codes (Schulman ’96, Sahai ’01, Sukhavasi and Hassibi ’11) Real-Time Scheduling (Hou and Kumar ’11, Shakkottai and Srikanth ’11) Low-delay Path Selection (Chen et. al.) 1 IEEE Usage Model Proposal (doc.: IEEE 802.11-03/802r23) ISIT, 2013 July 9th 2013 2/ 17

Real-Time Streaming Model ISIT, 2013 July 9th 2013 3/ 17

Problem Setup Assume s [ t ] ∈ F k q , i.i.d. uniform x [ t ] ∈ F n q . Causal Encoder. Rate: R = k n ISIT, 2013 July 9th 2013 4/ 17

Problem Setup Assume s [ t ] ∈ F k q , i.i.d. uniform x [ t ] ∈ F n q . Causal Encoder. Rate: R = k n Channel C ( N, B, W ) : Any sliding window of length W contains A burst of maximum length B , or, No more than N erasures in arbitrary positions. (N,B,W) = (2,3,6) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 W = 6 N = 2 ISIT, 2013 July 9th 2013 4/ 17

Problem Setup Assume s [ t ] ∈ F k q , i.i.d. uniform x [ t ] ∈ F n q . Causal Encoder. Rate: R = k n Channel C ( N, B, W ) : Any sliding window of length W contains A burst of maximum length B , or, No more than N erasures in arbitrary positions. (N,B,W) = (2,3,6) (N,B,W) = (2,3,6) 0 0 1 1 2 2 3 4 5 3 4 5 6 7 6 7 8 8 9 9 10 10 11 12 11 12 13 13 14 15 14 15 W = 6 W = 6 N = 2 N = 2 ISIT, 2013 July 9th 2013 4/ 17

Problem Setup Assume s [ t ] ∈ F k q , i.i.d. uniform x [ t ] ∈ F n q . Causal Encoder. Rate: R = k n Channel C ( N, B, W ) : Any sliding window of length W contains A burst of maximum length B , or, No more than N erasures in arbitrary positions. (N,B,W) = (2,3,6) (N,B,W) = (2,3,6) (N,B,W) = (2,3,6) 0 0 0 1 1 1 2 2 2 3 4 5 3 4 5 3 4 5 6 7 6 7 6 7 8 8 8 9 9 9 10 10 10 11 12 11 12 11 12 13 13 13 14 15 14 15 14 15 W = 6 W = 6 W = 6 N = 2 N = 2 N = 2 ISIT, 2013 July 9th 2013 4/ 17

Problem Setup Assume s [ t ] ∈ F k q , i.i.d. uniform x [ t ] ∈ F n q . Causal Encoder. Rate: R = k n Channel C ( N, B, W ) : Any sliding window of length W contains A burst of maximum length B , or, No more than N erasures in arbitrary positions. (N,B,W) = (2,3,6) (N,B,W) = (2,3,6) (N,B,W) = (2,3,6) (N,B,W) = (2,3,6) 0 0 0 0 1 1 1 1 2 2 2 2 3 4 5 3 4 5 3 4 5 3 4 5 6 7 6 7 6 7 6 7 8 8 8 8 9 9 9 9 10 10 10 10 11 12 11 12 11 12 11 12 13 13 13 13 14 15 14 15 14 15 14 15 W = 6 W = 6 W = 6 W = 6 N = 2 N = 2 N = 2 B = 3 ISIT, 2013 July 9th 2013 4/ 17

Problem Setup Assume s [ t ] ∈ F k q , i.i.d. uniform x [ t ] ∈ F n q . Causal Encoder. Rate: R = k n Channel C ( N, B, W ) : Any sliding window of length W contains A burst of maximum length B , or, No more than N erasures in arbitrary positions. (N,B,W) = (2,3,6) (N,B,W) = (2,3,6) (N,B,W) = (2,3,6) (N,B,W) = (2,3,6) (N,B,W) = (2,3,6) 0 0 0 0 0 1 1 1 1 1 2 2 2 2 2 3 4 5 3 4 5 3 4 5 3 4 5 3 4 5 6 7 6 7 6 7 6 7 6 7 8 8 8 8 8 9 9 9 9 9 10 10 10 10 10 11 12 11 12 11 12 11 12 11 12 13 13 13 13 13 14 15 14 15 14 15 14 15 14 15 W = 6 W = 6 W = 6 W = 6 W = 6 N = 2 N = 2 N = 2 B = 3 B = 3 Capacity R ( N, B, W, T ) ISIT, 2013 July 9th 2013 4/ 17

Main Result Theorem Consider the C ( N, B, W ) channel, with W ≥ B + 1 , and let the delay be T . Upper-Bound (Badr et al. INFOCOM’13) For any rate R code, we have: � � R B + N ≤ min( W, T + 1) 1 − R ISIT, 2013 July 9th 2013 5/ 17

Main Result Theorem Consider the C ( N, B, W ) channel, with W ≥ B + 1 , and let the delay be T . Upper-Bound (Badr et al. INFOCOM’13) For any rate R code, we have: � � R B + N ≤ min( W, T + 1) 1 − R Lower-Bound: There exists a rate R code that satisfies: � � R B + N ≥ min( W, T + 1) − 1 . 1 − R The gap between the upper and lower bound is 1 unit of delay. ISIT, 2013 July 9th 2013 5/ 17

Error Correction: Baseline Techniques s 0 s 1 s 2 s 3 s 4 s 5 s 6 s 7 k x i n p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 H i ∈ F k × n − k p i = s i · H 0 + s i − 1 · H 1 + . . . + s i − M · H M , q Erasure Codes: Random Linear Codes Strongly-MDS Codes (Gabidulin’88, Gluesing-Luerssen’06 ) ISIT, 2013 July 9th 2013 6/ 17

Error Correction: Baseline Techniques s 0 s 0 s 1 s 1 s 2 s 2 s 3 s 3 s 4 s 4 s 5 s 5 s 6 s 6 s 7 s 7 k k x i x i n n p 0 p 0 p 1 p 1 p 2 p 2 p 3 p 3 p 4 p 4 p 5 p 5 p 6 p 6 p 7 p 7 H i ∈ F k × n − k p i = s i · H 0 + s i − 1 · H 1 + . . . + s i − M · H M , q Erasure Codes: Random Linear Codes Strongly-MDS Codes (Gabidulin’88, Gluesing-Luerssen’06 ) ISIT, 2013 July 9th 2013 6/ 17

Error Correction: Baseline Techniques s 0 s 0 s 0 s 1 s 1 s 1 s 2 s 2 s 2 s 3 s 3 s 3 s 4 s 4 s 4 s 5 s 5 s 5 s 6 s 6 s 6 s 7 s 7 s 7 k k k x i x i x i n n n p 0 p 0 p 0 p 1 p 1 p 1 p 2 p 2 p 2 p 3 p 3 p 3 p 4 p 4 p 4 p 5 p 5 p 5 p 6 p 6 p 6 p 7 p 7 p 7 H i ∈ F k × n − k p i = s i · H 0 + s i − 1 · H 1 + . . . + s i − M · H M , q Erasure Codes: Random Linear Codes Strongly-MDS Codes (Gabidulin’88, Gluesing-Luerssen’06 ) ISIT, 2013 July 9th 2013 6/ 17

Error Correction: Baseline Techniques s 0 s 0 s 0 s 0 s 1 s 1 s 1 s 1 s 2 s 2 s 2 s 2 s 3 s 3 s 3 s 3 s 4 s 4 s 4 s 4 s 5 s 5 s 5 s 5 s 6 s 6 s 6 s 6 s 7 s 7 s 7 s 7 k k k k x i x i x i x i n n n n p 0 p 0 p 0 p 0 p 1 p 1 p 1 p 1 p 2 p 2 p 2 p 2 p 3 p 3 p 3 p 3 p 4 p 4 p 4 p 4 p 5 p 5 p 5 p 5 p 6 p 6 p 6 p 6 p 7 p 7 p 7 p 7 H i ∈ F k × n − k p i = s i · H 0 + s i − 1 · H 1 + . . . + s i − M · H M , q Erasure Codes: Random Linear Codes Strongly-MDS Codes (Gabidulin’88, Gluesing-Luerssen’06 ) ISIT, 2013 July 9th 2013 6/ 17

Error Correction: Baseline Techniques s 0 s 0 s 0 s 0 s 0 s 1 s 1 s 1 s 1 s 1 s 2 s 2 s 2 s 2 s 2 s 3 s 3 s 3 s 3 s 3 s 4 s 4 s 4 s 4 s 4 s 5 s 5 s 5 s 5 s 5 s 6 s 6 s 6 s 6 s 6 s 7 s 7 s 7 s 7 s 7 k k k k k x i x i x i x i x i n n n n n p 0 p 0 p 0 p 0 p 0 p 1 p 1 p 1 p 1 p 1 p 2 p 2 p 2 p 2 p 2 p 3 p 3 p 3 p 3 p 3 p 4 p 4 p 4 p 4 p 4 p 5 p 5 p 5 p 5 p 5 p 6 p 6 p 6 p 6 p 6 p 7 p 7 p 7 p 7 p 7 H i ∈ F k × n − k p i = s i · H 0 + s i − 1 · H 1 + . . . + s i − M · H M , q Erasure Codes: Random Linear Codes Strongly-MDS Codes (Gabidulin’88, Gluesing-Luerssen’06 ) ISIT, 2013 July 9th 2013 6/ 17

Error Correction: Baseline Techniques s 0 s 0 s 0 s 0 s 0 s 0 s 1 s 1 s 1 s 1 s 1 s 1 s 2 s 2 s 2 s 2 s 2 s 2 s 3 s 3 s 3 s 3 s 3 s 3 s 4 s 4 s 4 s 4 s 4 s 4 s 5 s 5 s 5 s 5 s 5 s 5 s 6 s 6 s 6 s 6 s 6 s 6 s 7 s 7 s 7 s 7 s 7 s 7 k k k k k k x i x i x i x i x i x i n n n n n n p 0 p 0 p 0 p 0 p 0 p 0 p 1 p 1 p 1 p 1 p 1 p 1 p 2 p 2 p 2 p 2 p 2 p 2 p 3 p 3 p 3 p 3 p 3 p 3 p 4 p 4 p 4 p 4 p 4 p 4 p 5 p 5 p 5 p 5 p 5 p 5 p 6 p 6 p 6 p 6 p 6 p 6 p 7 p 7 p 7 p 7 p 7 p 7 H i ∈ F k × n − k p i = s i · H 0 + s i − 1 · H 1 + . . . + s i − M · H M , q Erasure Codes: Random Linear Codes Strongly-MDS Codes (Gabidulin’88, Gluesing-Luerssen’06 ) ISIT, 2013 July 9th 2013 6/ 17

Error Correction: Baseline Techniques s 0 s 0 s 0 s 0 s 0 s 0 s 0 s 1 s 1 s 1 s 1 s 1 s 1 s 1 s 2 s 2 s 2 s 2 s 2 s 2 s 2 s 3 s 3 s 3 s 3 s 3 s 3 s 3 s 4 s 4 s 4 s 4 s 4 s 4 s 4 s 5 s 5 s 5 s 5 s 5 s 5 s 5 s 6 s 6 s 6 s 6 s 6 s 6 s 6 s 7 s 7 s 7 s 7 s 7 s 7 s 7 k k k k k k k x i x i x i x i x i x i x i n n n n n n n p 0 p 0 p 0 p 0 p 0 p 0 p 0 p 1 p 1 p 1 p 1 p 1 p 1 p 1 p 2 p 2 p 2 p 2 p 2 p 2 p 2 p 3 p 3 p 3 p 3 p 3 p 3 p 3 p 4 p 4 p 4 p 4 p 4 p 4 p 4 p 5 p 5 p 5 p 5 p 5 p 5 p 5 p 6 p 6 p 6 p 6 p 6 p 6 p 6 p 7 p 7 p 7 p 7 p 7 p 7 p 7 Recover s 0 , s 1 , s 2 , s 3 H i ∈ F k × n − k p i = s i · H 0 + s i − 1 · H 1 + . . . + s i − M · H M , q Erasure Codes: Random Linear Codes Strongly-MDS Codes (Gabidulin’88, Gluesing-Luerssen’06 ) ISIT, 2013 July 9th 2013 6/ 17

Robust Streaming Codes based on Deterministic Channel Approximations - PowerPoint PPT Presentation

Robust Streaming Codes based on Deterministic Channel Approximations Ashish Khisti University of Toronto Joint Work with Ahmed Badr (Toronto), Wai-Tian Tan (HP Labs) and John Apostolopoulos (HP Labs) ISIT, 2013 July 9th 2013 Motivation -

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