Causal coding of stationary sources and individual sequences with high resolution

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 a...

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Bibliographic Details
Published in:IEEE transactions on information theory 2006-02, Vol.52 (2), p.662-680
Main Authors: Linder, T., Zamir, R.
Format: Article
Language:English
Subjects:
Applied sciences
Byproducts
Causal source codes
Codes
Coding
Coding, codes
Counters
Data compression
Delay
differential entropy
Distortion
Entropy
Entropy coding
Exact sciences and technology
finite-memory codes
High resolution
individual sequences
Information, signal and communications theory
Lempel-Ziv complexity
Mathematical analysis
Optimization
Performance gain
Performance loss
Rate-distortion
Signal and communications theory
Source coding
stationary sources
Systems, networks and services of telecommunications
Telecommunications
Telecommunications and information theory
Time sharing computer systems
Transmission and modulation (techniques and equipments)
uniform quantizer
Vector quantization
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Summary: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.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2005.862075