TY - JOUR
T1 - Causal coding of stationary sources and individual sequences with high resolution
AU - Linder, Tamás
AU - Zamir, Ram
N1 - Funding Information:
Manuscript received Sepember 15, 2003; revised August 12, 2005. This work was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada. T. Linder is with the Department of Mathematics and Statistics, Queen’s University, Kingston, ON K7L 3N6, Canada. He is also affiliated with the Computer and Automation Research Institute of the Hungarian Academy of Sciences, Budapest, Hungary (e-mail: linder@mast.queensu.ca). R. Zamir is with the Department of Electrical Engineering—Systems, Tel-Aviv University, Ramat-Aviv 69978, Tel-Aviv, Israel (e-mail: zamir@ eng.tau.ac.il). Communicated by S. A. Savari, Associate Editor for Source Coding. Digital Object Identifier 10.1109/TIT.2005.862075
PY - 2006/2
Y1 - 2006/2
N2 - In a causal source coding system, the reconstruction of the present source sample is restricted to be a function of the present and past source samples, while the code stream itself may be noncausal and have variable rate. Neuhoff and Gilbert showed that for memoryless sources, optimum performance among all causal source codes is achieved by time-sharing at most two memoryless codes (quantizers) followed by entropy coding. In this work, we extend Neuhoff and Gilbert's result in the limit of small distortion (high resolution) to two new settings. First, we show that at high resolution, an optimal causal code for a stationary source with finite differential entropy rate consists of a uniform quantizer followed by a (sequence) entropy coder. This implies that the price of causality at high resolution is approximately 0.254 bit, i.e., the space-filling loss of the uniform quantizer. Then, we consider individual sequences and introduce a deterministic analogue of differential entropy, which we call "Lempel-Ziv differential entropy." We show that for any bounded individual sequence with finite Lempel-Ziv differential entropy, optimum high-resolution performance among all finite-memory variable-rate causal codes is achieved by dithered scalar uniform quantization followed by Lempel-Ziv coding. As a by-product, we also prove an individual-sequence version of the Shannon lower bound.
AB - In a causal source coding system, the reconstruction of the present source sample is restricted to be a function of the present and past source samples, while the code stream itself may be noncausal and have variable rate. Neuhoff and Gilbert showed that for memoryless sources, optimum performance among all causal source codes is achieved by time-sharing at most two memoryless codes (quantizers) followed by entropy coding. In this work, we extend Neuhoff and Gilbert's result in the limit of small distortion (high resolution) to two new settings. First, we show that at high resolution, an optimal causal code for a stationary source with finite differential entropy rate consists of a uniform quantizer followed by a (sequence) entropy coder. This implies that the price of causality at high resolution is approximately 0.254 bit, i.e., the space-filling loss of the uniform quantizer. Then, we consider individual sequences and introduce a deterministic analogue of differential entropy, which we call "Lempel-Ziv differential entropy." We show that for any bounded individual sequence with finite Lempel-Ziv differential entropy, optimum high-resolution performance among all finite-memory variable-rate causal codes is achieved by dithered scalar uniform quantization followed by Lempel-Ziv coding. As a by-product, we also prove an individual-sequence version of the Shannon lower bound.
KW - Causal source codes
KW - Differential entropy
KW - Finite memory codes
KW - Individual sequences
KW - Lempel-Ziv complexity
KW - Stationary sources
KW - Uniform quantizer
UR - http://www.scopus.com/inward/record.url?scp=31744443412&partnerID=8YFLogxK
U2 - 10.1109/TIT.2005.862075
DO - 10.1109/TIT.2005.862075
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AN - SCOPUS:31744443412
SN - 0018-9448
VL - 52
SP - 662
EP - 680
JO - IEEE Transactions on Information Theory
JF - IEEE Transactions on Information Theory
IS - 2
ER -