1. Introduction 2. Fundamentals of Fourier Analysis 3. Fourier Analysis in Systems Theory 4. Fourier Transforms in Probability, Random Variables and Stochastic Processes 5. The Sampling Theory 6. Generalizations of the Sampling Theorem 7. Noise and Error Effects 8. Multidimensional Signal Analysis 9. Time-Frequency Representations 10. Signal Recovery 11. Signal and Image Synthesis: Alternating Projections Onto Convex Sets 12. Mathematical Morphology and Fourier Analysis on Time Sales 13. Applications 14. Appendices 15. Reference

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Handbook of Fourier Analysis & Its Applications

We present an axiomatic framework for seeking distances between power spectral density functions. The axioms require that the sought metric respects the effects of additive and multiplicative noise in reducing our ability to discriminate spectra, as well as they require continuity of statistical quantities with respect to perturbations measured in the metric. We then present a particular metric which abides by these requirements. The metric is based on the Monge-Kantorovich transportation problem and is contrasted with an earlier Riemannian metric based on the minimum-variance prediction geometry of the underlying time-series. It is also being compared with the more traditional Itakura-Saito distance measure, as well as the aforementioned prediction metric, on two representative examples.

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Metrics for Power Spectra: An Axiomatic Approach

Consider the problem of representing a distribution $\pi$ on a large alphabet of size $k$ up to fidelity $\varepsilon$ in Kullback-Leibler (KL) divergence. Heuristically, arguing as for quadratic loss in high dimension, one expects that about $(k/2)\log(1/\varepsilon)$ bits would be required. We show this intuition is correct by proving explicit non-asymptotic bounds for the minimal average distortion when $\pi$ is randomly sampled from a symmetric Dirichlet prior on the simplex. Our method is to reduce the single-sample problem to the traditional setting of iid samples, but for a non-standard rate distortion question with the novel distortion measure $d(x, y)= x\log(x/y)$ , which we call divergence distortion. Practically, our results advocate using a $x\mapsto x^{2/3}$ compander (for small $x$ ) followed by a uniform scalar quantizer for storing large-alphabet distributions.

Quantization of Random Distributions under KL Divergence

We evaluate the rate-distortion function for the i.i.d. gamma sources with respect to the absolute-log distortion measure. The logarithmic transformation reduces this rate-distortion problem to that under the absolute distortion measure. Extending the explicit evaluation of the rate-distortion function for the Gaussian sources, we obtain the parametric form of the rate-distortion function. We show that the optimal distribution of reconstruction consists of a continuous component enclosed by left and right discrete components and the left discrete component vanishes when the allowed distortion is small.

Rate-distortion function for gamma sources under absolute-log distortion measure

http://www.ece.mcgill.ca/~pkabal/papers/1994/De1994.pdf

Auditory distortion measure for speech coder evaluation— discrimination information approach

In many linear predictive coding (LPC) speech compression systems the encoding of the LPC parameters is performed using a product code book scheme. One possible approach consists in organizing the set of LPC reproduction models as the Cartesian product of a vector code book describing the shape of each reproduction LPC model and a scalar codebook describing the gain or energy. We first present a formal development of rate-distortion theoretic results for distortion functions based on quotients of input and reproduction symbols. We also obtain theoretical bounds for the rate-distortion function of the gain term of LPC systems (which, depending on the quantization scheme, can use about 10-25 percent of the transmission rate), as well as asymptotic (small distortions) approximations to the performance of the optimal scalar quantizer for these gain terms, when the overall fidelity criterion is the Itakura-Saito distortion measure, which is a member of the class of quotient distortion measures. The approximations and bounds are compared finally with experimental results.

F. Kuhlmann

Papers

Rate-distortion bounds for quotient-based distortions with application to Itakura- Saito distortion measures