There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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Table Of Integrals Series And Products

Information Theory and Reliable Communication

Digital Coding of Waveforms

Consider a discrete source producing a sequence of message letters from a finite alphabet. A single-letter distortion measure is given by a non-negative matrix (d ij ). The entryd ij measures the ?cost? or ?distortion? if letteriis reproduced at the receiver as letterj. The average distortion of a communications system (source-coder-noisy channel-decoder) is taken to bed= ? i.j P ij d ij whereP ij is the probability ofibeing reproduced asj. It is shown that there is a functionR(d) that measures the ?equivalent rate? of the source for a given level of distortion. For coding purposes where a leveldof distortion can be tolerated, the source acts like one with information rateR(d). Methods are given for calculatingR(d), and various properties discussed. Finally, generalizations to ergodic sources, to continuous sources, and to distortion measures involving blocks of letters are developed. In this paper a study is made of the problem of coding a discrete source of information, given afidelity criterionor ameasure of the distortionof the final recovered message at the receiving point relative to the actual transmitted message. In a particular case there might be a certain tolerable level of distortion as determined by this measure. It is desired to so encode the information that the maximum possible signaling rate is obtained without exceeding the tolerable distortion level. This work is an expansion and detailed elaboration of ideas presented earlier [1], with particular reference to the discrete case. We shall show that for a wide class of distortion measures and discrete sources of information there exists a functionR(d) (depending on the particular distortion measure and source) which measures, in a sense, the equivalent rateRof the source (in bits per letter produced) whendis the allowed distortion level. Methods will be given for evaluatingR(d) explicitly in certain simple cases and for evaluatingR(d) by a limiting process in more complex cases. The basic results are roughly that it is impossible to signal at a rate faster thanC/R(d) (source letters per second) over a memoryless channel of capacityC(bits per second) with a distortion measure less than or equal tod. On the other hand, by sufficiently long block codes it is possible to approach as closely as desired the rateC/R(d) with distortion leveld. Finally, some particular examples, using error probability per letter of message and other simple distortion measures, are worked out in detail.

Coding Theorems for a Discrete Source With a Fidelity CriterionInstitute of Radio Engineers, International Convention Record, vol. 7, 1959.

The rate distortion behavior of sparse memoryless sources is studied. These serve as models of sparse signal representations and facilitate the performance analysis of “sparsifying” transforms like the wavelet transform and nonlinear approximation schemes. For strictly sparse binary sources with Hamming distortion, R(D) is shown to be almost linear. For nonstrictly sparse continuous-valued sources, termed compressible, two measures of compressibility are introduced: incomplete moments and geometric mean. The former lead to low- and high-rate upper bounds on mean squared error D(R), while the latter yields lower and upper bounds on source entropy, thereby characterizing asymptotic R(D) behavior. Thus, the notion of compressibility is quantitatively connected with actual lossy compression. These bounding techniques are applied to two source models: Gaussian mixtures and power laws matching the approximately scale-invariant decay of wavelet coefficients. The former are versatile models for sparse data, which in particular allow to bound high-rate compression performance of a scalar mixture compared to a corresponding unmixed transform coding system. Such a comparison is interesting for transforms with known coefficient decay, but unknown coefficient ordering, e.g., when positions of highest-variance coefficients are unknown. The use of these models and results in distributed coding and compressed sensing scenarios are also discussed.

/pdf/rate-distortion-behavior-of-sparse-sources-14r28c46db.pdf

Rate Distortion Behavior of Sparse Sources

Recent rate-distortion analyses of image transform coders are based on a trade-off between the lossless coding of coefficient positions versus the lossy coding of the coefficient values. We propose spike processes as a tool that allows a more fundamental trade-off, namely between lossy position coding and lossy value coding. We investigate the Hamming distortion case and give analytic results for single and multiple spikes. We then consider upper bounds for a single Gaussian spike with squared error distortion. The obtained results show a rate distortion behavior which switches from linear at low rates to exponential at high rates.

/pdf/rate-distortion-analysis-of-spike-processes-zu15gnrmtk.pdf

Rate-distortion analysis of spike processes

In joint source-channel arithmetic coding (JSCAC) schemes, additional redundancy may be introduced into an arithmetic source code in order to be more robust against transmission errors. The purpose of this work is to provide analytical tools to predict and evaluate the effectiveness of that redundancy. Integer binary arithmetic coding (AC) is modeled by a reduced-state automaton in order to obtain a bit-clock trellis describing the encoding process. Considering AC as a trellis code, distance spectra are then derived. In particular, an algorithm to compute the free distance of an arithmetic code is proposed. The obtained code properties allow to compute upper bounds on both bit error and symbol error probabilities and thus to provide an objective criterion to analyze the behavior of JSCAC schemes when used on noisy channels. This criterion is then exploited to design efficient error-correcting arithmetic codes. Simulation results highlight the validity of the theoretical error bounds and show that for equivalent rate and complexity, a simple optimization yields JSCACs that outperform classical tandem schemes at low to medium SNR.

/pdf/analytical-tools-for-optimizing-the-error-correction-3oz5ezfzvq.pdf

Analytical tools for optimizing the error correction performance of arithmetic codes

Biorthogonal wavelets have been used with great success in most of the recent transform image coders. By using the new balanced multiwavelets, one can now easily design fully orthogonal linear phase FIR transform schemes. The aim of our work is to assess whether the added orthogonality yields a performance gain compared to traditional biorthogonal transforms. As comparison platform we use the well-known SPIHT codec, which is based on the significance tree quantization (STQ) principle. Without any particular fine-tuning the multiwavelet codec performs within 0.5 dB of SPIHT. A closer inspection shows however that it is hard to improve on this, therefore re-establishing the rule of thumb that strict orthogonality is not a key factor in image transform coding. More details can be obtained on the [WEB] at http://lcavwww.epfl.ch/~weidmann/mwcoder.

/pdf/significance-tree-image-coding-using-balanced-multiwavelets-3prnugq39h.pdf

Significance tree image coding using balanced multiwavelets

Data partitioning in H.264 extended profile video coding enables unequal error protection. Its performance can be improved if the decoder tries to also decode packets containing errors. We propose a soft-input sequential decoding algorithm for the prediction residuals encoded in low-priority packets. Information from the decoding process is then used to control additional error concealment. This combined technique provides significant PSNR gains compared to a simple packet-loss scenario.

/pdf/combined-sequential-decoding-and-error-concealment-of-h-264-2ij2t04z66.pdf

Claudio Weidmann

Papers

Rate Distortion Behavior of Sparse Sources

Rate-distortion analysis of spike processes

Analytical tools for optimizing the error correction performance of arithmetic codes

Significance tree image coding using balanced multiwavelets

Combined sequential decoding and error concealment of H.264 video