Anytime Reliability of Systematic LDPC Motivation Convolutional - PowerPoint PPT Presentation

Anytime Reliability of Systematic... L. D¨ ossel et al Anytime Reliability of Systematic LDPC Motivation Convolutional Codes LDPC Convolutional Codes Anytime LDPC Convolutional Codes Asymptotic Analysis L. D¨ ossel, L. K. Rasmussen , R. Thobaben and M. Skoglund Communication Theory Laboratory Numerical Examples School of Electrical Engineering Summary and KTH Royal Institute of Technology Concluding Remarks ACCESS Linnaeus Center LCCC Workshop: Information and Control in Networks October 2012, Lund, Sweden 1 / 30

Overview Anytime Reliability of Systematic... L. D¨ ossel et al 1 Motivation Motivation LDPC Convolutional Codes 2 LDPC Convolutional Codes Anytime LDPC Convolutional Codes Asymptotic Analysis 3 Anytime LDPC Convolutional Codes Numerical Examples Summary and 4 Asymptotic Analysis Concluding Remarks 5 Numerical Examples 6 Summary and Concluding Remarks 2 / 30

Overview Anytime Reliability of Systematic... L. D¨ ossel et al 1 Motivation Motivation LDPC Convolutional 2 LDPC Convolutional Codes Codes Anytime LDPC Convolutional Codes 3 Anytime LDPC Convolutional Codes Asymptotic Analysis Numerical Examples 4 Asymptotic Analysis Summary and Concluding Remarks 5 Numerical Examples 6 Summary and Concluding Remarks 3 / 30

Automatic Control over Noisy Channels Anytime Reliability of Systematic... L. D¨ ossel et al Motivation LDPC Convolutional Codes Anytime LDPC Convolutional Codes Asymptotic Analysis Numerical Examples Summary and Concluding Remarks A. Sahai and S. Mitter, “The necessity and sufficiency of anytime capacity for W stabilization of a linear system over a noisy communication link - Part I: Scalar O ; C systems,” IEEE Trans. Inf. Theory , vol. 52, no. 8, pp. 3369–3395, Aug. 2006. 4 / 30

Model for Anytime Communications Anytime Reliability of Systematic... � � ���� �9������� &� � #�� �����" L. D¨ ossel et al �� ) ������� . ��#�� � � � Motivation ���� � � ���� � � ������� LDPC Convolutional �� �� #� �� #� Codes � � Anytime LDPC Convolutional Codes � � � Asymptotic Analysis � Numerical Examples ) ������� ��""�� 8��#�"���� � ,���� � Summary and � ������� � � Concluding Remarks � 7 � � � � � ���� � ������� � $� � #�� $� � #�� &���"���� ���� �9������� $� � #�� �����" ,����� B"B0 $�� ���������� ������� � ���� � ���� ������ A. Sahai, “Anytime information theory,” Ph.D. dissertation, MIT, 2001. 5 / 30 B=

Model for Anytime Channel-Coded Transmission Anytime Reliability of Systematic... u 1 , ..., u t L. D¨ ossel et al Encoder v t Source E ( u 1 , ..., u t ) Motivation Binary LDPC Convolutional erasure Codes channel Anytime LDPC Convolutional Codes u 1 , ..., ˆ ˆ u t Decoder v t ˜ Asymptotic Analysis D (˜ v 1 , ..., ˜ v t ) Numerical Examples Summary and Concluding Remarks Encoding and Decoding u [1 , t ] = [ u 1 , u 2 ..., u t ] v t = E ( u 1 , ..., u t ) u [1 , t ] = [ˆ ˆ u 1 , ..., ˆ u t ] = D (˜ v 1 , ..., ˜ v t ) 6 / 30

Anytime Reliability Anytime Reliability of Systematic... L. D¨ ossel et al Anytime Reliability Motivation LDPC Convolutional Codes • The receiver can decide to start decoding at anytime Anytime LDPC Convolutional Codes . . . . . . . . . 1 2 3 4 j - 1 j j +1 t - 1 t Asymptotic Analysis d ( t , j ) Numerical Examples Summary and • Anytime reliability can formally be defined as Concluding Remarks u j � = u j | u [1 , t ] was transmitted) ≤ β 2 − α d ( t , j ) P (ˆ (1) • For a particular code at rate R , the largest α such that (1) is fulfilled is referred to as the anytime exponent of the code 7 / 30

Selected Prior Work Anytime Reliability of Systematic... L. D¨ ossel et al A. Sahai, “Anytime information theory,” Ph.D. dissertation, MIT, 2001. Motivation A. Sahai and S. Mitter, “The necessity and sufficiency of anytime capacity for LDPC Convolutional Codes stabilization of a linear system over a noisy communication link - Part I: Scalar systems,” IEEE Trans. Inf. Theory , vol. 52, no. 8, pp. 3369–3395, Aug. 2006. Anytime LDPC Convolutional Codes L. J. Schulman, “Coding for interactive communication,” IEEE Trans. Inf. Theory , Asymptotic Analysis vol. 42, no. 6, pp. 1745–1756, Jun. 1996. Numerical Examples Summary and R. Ostrovsky, Y. Rabani, and L. J. Schulman, “Error-correcting codes for automatic Concluding Remarks control,” IEEE Trans. Inf. Theory , vol. 55, no. 7, pp. 2931–2941, Jul. 2009. G. Como, F. Fagnani, and S. Zampieri, “Anytime reliable transmission of real-valued information through digital noisy channels,” SIAM J. Control and Opt. , vol. 48, no. 6, pp. 3903–3924, Mar. 2010. R. T. Sukhavasi and B. Hassibi, “Linear error correcting codes with anytime reliability,” in IEEE Int. Symp. Inf. Theory , St. Petersburg, Rusia, Jun. 2011. 8 / 30

Our Contributions Anytime Reliability of Systematic... L. D¨ ossel et al Motivation Anytime LDPC Convolutional Codes LDPC Convolutional Codes • Modern coding structures have not yet been considered for anytime Anytime LDPC transmission Convolutional Codes • We propose: Asymptotic Analysis • a tractable protograph structure for an LDPC-CC ensemble Numerical Examples • an expanding-window decoding scheme Summary and Concluding Remarks • We show that the ensemble asymptotically exhibits the desired anytime properties • We show through simulation that the ensemble also exhibits some anytime properties for finite-length codes 9 / 30

Overview Anytime Reliability of Systematic... L. D¨ ossel et al 1 Motivation Motivation LDPC Convolutional 2 LDPC Convolutional Codes Codes Anytime LDPC Convolutional Codes 3 Anytime LDPC Convolutional Codes Asymptotic Analysis Numerical Examples 4 Asymptotic Analysis Summary and Concluding Remarks 5 Numerical Examples 6 Summary and Concluding Remarks 10 / 30

LDPC Convolutional Codes Background • Invented in Anytime Reliability of Systematic... A. J. Felstr¨ om and K. Sh. Zigangirov, “Time-varying periodic convolutional codes with low-density parity-check matrix,” IEEE Trans. on Inf. Theory , vol. 45, no. 6, pp. 2181–2191, Sept. 1999. L. D¨ ossel et al • Good performance has been analysed in Motivation M. Lentmaier, A. Sridharan, D. J. Costello, and K. Sh. Zigangirov, “Iterative decoding threshold LDPC Convolutional analysis for LDPC convolutional codes,” IEEE Trans. on Inf. Theory , vol. 56, no. 10, pp. 5274 – Codes 5289, Oct. 2010. Anytime LDPC ⇒ “For a terminated LDPCC code ensemble, the thresholds are better than for Convolutional Codes corresponding regular and irregular LDPC block codes” Asymptotic Analysis Numerical Examples • Capacity achieving property has been proven in Summary and S. Kudekar, T. Richardson, and R. Urbanke, “Threshold saturation via spatial coupling: Why convo- Concluding Remarks lutional LDPC ensembles perform so well over the BEC,” IEEE Trans. on Inf. Theory , vol. 57, no. 2, pp. 803 – 834, Feb. 2011. ⇒ “Spatial coupling of individual codes increases the belief-propagation (BP) threshold of the new ensemble to its maximum possible value, namely the maximum a posteriori (MAP) threshold of the underlying ensemble.” • Implementation aspects A. E. Pusane, A. J. Felstr¨ om, A. Sridharan, M. Lentmaier, K. Sh. Zigangirov, and D. J. Costello, “Implementation aspects of LDPC convolutional codes,” IEEE Trans. on Comm. , vol. 56, no. 7, pp. 1060 – 1069, July 2008. 11 / 30

LDPC Convolutional Codes • A rate R = b / c LDPC convolutional code is defined as a set of sequences v [0 , L − 1] = [ v 0 , . . . , v L − 1 ] that satisfy Anytime Reliability of Systematic... 0 = v [0 , L − 1] H T [0 , L − 1] = . L. D¨ ossel et al  H T H T  0 (0) . . . ms ( m s ) Motivation H T H T 0 (1) ms ( m s + 1)  . . .    LDPC Convolutional v [0 , L − 1]   ... ... Codes     Anytime LDPC H T H T 0 ( L − 1 − m s ) . . . ms ( L − 1) Convolutional Codes � �� � Asymptotic Analysis H T [0 , L − 1] Numerical Examples where Summary and Concluding Remarks • H T [0 , L − 1] ( t ) is the syndrome former matrix (i.e., the transposed parity check matrix H [0 , L − 1] ), • H T i ( t ) is a c × ( c − b ) binary matrix, • H T 0 ( t ) must have full rank ∀ t , • L is the number of positions; length of the code: cL • m s is the syndrome former memory. • For LDPC-CCs the syndrome former matrix is sparse. 12 / 30