Learning Guide and Examples: Information Theory and Coding
Prerequisite courses: Continuous Mathematics, Probability, Discrete Mathematics
Overview and Historical Origins: Foundations and Uncertainty.
Why the movements and
transformations of information, just like those of a fluid, are law-governed. How concepts of
randomness, redundancy, compressibility, noise, bandwidth, and uncertainty are intricately
connected to information. Origins of these ideas and the various forms that they take.
Mathematical Foundations; Probability Rules; Bayes’ Theorem.
The meanings of proba-
bility. Ensembles, random variables, marginal and conditional probabilities. How the formal
concepts of information are grounded in the principles and rules of probability.
Entropies Defined, and Why They Are Measures of Information.
Marginal entropy, joint
entropy, conditional entropy, and the Chain Rule for entropy.
Mutual information between
ensembles of random variables. Why entropy is a fundamental measure of information content.
Source Coding Theorem; Prefix, Variable-, & Fixed-Length Codes.
Symbol codes. Binary
symmetric channel. Capacity of a noiseless discrete channel. Error correcting codes.
Channel Types, Properties, Noise, and Channel Capacity.
Perfect communication through
a noisy channel. Capacity of a discrete channel as the maximum of its mutual information over
all possible input distributions.
Continuous Information; Density; Noisy Channel Coding Theorem.
Extensions of the dis-
crete entropies and measures to the continuous case.
Signal-to-noise ratio; power spectral
density.
Gaussian channels.
Relative significance of bandwidth and noise limitations.
The
Shannon rate limit and efficiency for noisy continuous channels.
Fourier Series, Convergence, Orthogonal Representation.
Generalized signal expansions in
vector spaces. Independence. Representation of continuous or discrete data by complex expo-
nentials. The Fourier basis. Fourier series for periodic functions. Examples.
Useful Fourier Theorems; Transform Pairs. Sampling; Aliasing.
The Fourier transform for
non-periodic functions. Properties of the transform, and examples. Nyquist’s Sampling Theo-
rem derived, and the cause (and removal) of aliasing.
Discrete Fourier Transform. Fast Fourier Transform Algorithms.
Efficient algorithms for
computing Fourier transforms of discrete data.
Computational complexity.
Filters, correla-
tion, modulation, demodulation, coherence.
The Quantized Degrees-of-Freedom in a Continuous Signal.
Why a continuous signal of fi-
nite bandwidth and duration has a fixed number of degrees-of-freedom. Diverse illustrations of
the principle that information, even in such a signal, comes in quantized, countable, packets.
Gabor-Heisenberg-Weyl Uncertainty Relation. Optimal “Logons.”
Unification of the time-
domain and the frequency-domain as endpoints of a continuous deformation. The Uncertainty
Principle and its optimal solution by Gabor’s expansion basis of “logons.” Multi-resolution
wavelet codes. Extension to images, for analysis and compression.