116 research outputs found
Nonasymptotic noisy lossy source coding
This paper shows new general nonasymptotic achievability and converse bounds
and performs their dispersion analysis for the lossy compression problem in
which the compressor observes the source through a noisy channel. While this
problem is asymptotically equivalent to a noiseless lossy source coding problem
with a modified distortion function, nonasymptotically there is a noticeable
gap in how fast their minimum achievable coding rates approach the common
rate-distortion function, as evidenced both by the refined asymptotic analysis
(dispersion) and the numerical results. The size of the gap between the
dispersions of the noisy problem and the asymptotically equivalent noiseless
problem depends on the stochastic variability of the channel through which the
compressor observes the source.Comment: IEEE Transactions on Information Theory, 201
Universal Sampling Rate Distortion
We examine the coordinated and universal rate-efficient sampling of a subset
of correlated discrete memoryless sources followed by lossy compression of the
sampled sources. The goal is to reconstruct a predesignated subset of sources
within a specified level of distortion. The combined sampling mechanism and
rate distortion code are universal in that they are devised to perform robustly
without exact knowledge of the underlying joint probability distribution of the
sources. In Bayesian as well as nonBayesian settings, single-letter
characterizations are provided for the universal sampling rate distortion
function for fixed-set sampling, independent random sampling and memoryless
random sampling. It is illustrated how these sampling mechanisms are
successively better. Our achievability proofs bring forth new schemes for joint
source distribution-learning and lossy compression
Universal Minimax Discrete Denoising under Channel Uncertainty
The goal of a denoising algorithm is to recover a signal from its
noise-corrupted observations. Perfect recovery is seldom possible and
performance is measured under a given single-letter fidelity criterion. For
discrete signals corrupted by a known discrete memoryless channel, the DUDE was
recently shown to perform this task asymptotically optimally, without knowledge
of the statistical properties of the source. In the present work we address the
scenario where, in addition to the lack of knowledge of the source statistics,
there is also uncertainty in the channel characteristics. We propose a family
of discrete denoisers and establish their asymptotic optimality under a minimax
performance criterion which we argue is appropriate for this setting. As we
show elsewhere, the proposed schemes can also be implemented computationally
efficiently.Comment: Submitted to IEEE Transactions of Information Theor
Coding for Communications and Secrecy
Shannon, in his landmark 1948 paper, developed a framework for characterizing the fundamental limits of information transmission. Among other results, he showed that reliable communication over a channel is possible at any rate below its capacity. In 2008, Arikan discovered polar codes; the only class of explicitly constructed low-complexity codes that achieve the capacity of any binary-input memoryless symmetric-output channel. Arikan's polar transform turns independent copies of a noisy channel into a collection of synthetic almost-noiseless and almost-useless channels. Polar codes are realized by sending data bits over the almost-noiseless channels and recovering them by using a low-complexity successive-cancellation (SC) decoder, at the receiver. In the first part of this thesis, we study polar codes for communications. When the underlying channel is an erasure channel, we show that almost all correlation coefficients between the erasure events of the synthetic channels decay rapidly. Hence, the sum of the erasure probabilities of the information-carrying channels is a tight estimate of the block-error probability of polar codes when used for communication over the erasure channel. We study SC list (SCL) decoding, a method for boosting the performance of short polar codes. We prove that the method has a numerically stable formulation in log-likelihood ratios. In hardware, this formulation increases the decoding throughput by 53% and reduces the decoder's size about 33%. We present empirical results on the trade-off between the length of the CRC and the performance gains in a CRC-aided version of the list decoder. We also make numerical comparisons of the performance of long polar codes under SC decoding with that of short polar codes under SCL decoding. Shannon's framework also quantifies the secrecy of communications. Wyner, in 1975, proposed a model for communications in the presence of an eavesdropper. It was shown that, at rates below the secrecy capacity, there exist reliable communication schemes in which the amount of information leaked to the eavesdropper decays exponentially in the block-length of the code. In the second part of this thesis, we study the rate of this decay. We derive the exact exponential decay rate of the ensemble-average of the information leaked to the eavesdropper in Wyner's model when a randomly constructed code is used for secure communications. For codes sampled from the ensemble of i.i.d. random codes, we show that the previously known lower bound to the exponent is exact. Our ensemble-optimal exponent for random constant-composition codes improves the lower bound extant in the literature. Finally, we show that random linear codes have the same secrecy power as i.i.d. random codes. The key to securing messages against an eavesdropper is to exploit the randomness of her communication channel so that the statistics of her observation resembles that of a pure noise process for any sent message. We study the effect of feedback on this approximation and show that it does not reduce the minimum entropy rate required to approximate a given process. However, we give examples where variable-length schemes achieve much larger exponents in this approximation in the presence of feedback than the exponents in systems without feedback. Upper-bounding the best exponent that block codes attain, we conclude that variable-length coding is necessary for achieving the improved exponents
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