The study course is intended to deepen their knowledge of signal sampling and approximation, of linear system theory based on entire analytic function theory, as well as of other communication technology theoretical problems. The theory of entire analytic functions is a valuable tool in communication theory and practice because entire analytic functions in the complex plane corresponding to the functions with limited spectra on the real axis. The latter are just functions that describe signals transmitted over the bandlimited communication channels. In this way, it turns out that the well-known sampling (Kotelnikov) theorem is merely the special case of Lagrange’s interpolation formula of entire analytic functions enabling also other sampling possibilities including nonuniform sampling. Similarly, other new possibilities appear in signal approximation and in approximation error evaluation, as well as in signal restoration if only partial information about the signal is known.
The following main topics are covered in this study course: entire analytic functions and their application in signal sampling, approximation, and restoration; properties of Fourier transform; signal multiplexing in multichannel systems, CDMA systems; the negentropy principle of information and its meaning for telecommunications; the influence of quantum effects on signal transmission; quantum communications; quantum cryptography; quantum computers; stochastic resonance.
Course contents:
- Motivation, goal and content of study course. The main concepts.
- Entire analytic functions, their significance in communication theory and mathematical properties.
- The main theorems of Fourier analysis. Widening of the Fourier transform concept. Uncertainty relation.
- Functions with limited spectra and entire analytic functions
- Sampling (Kotelnikov) theorem as a special case of Lagrange’s interpolation formula.
- Uniform and nonuniform sampling of signals.
- Approximation of signals in communication theory and the approximation errors.
- Classification of signal multiplexing methods.Linear and nonlinear multiplexing methods.
- Code division multiplexing of signals (CDMA). CDMA/DS and CDMA/FH systems.
- Correlation receivers in the case of white noise and signal multipaths propagation.
- Orthogonal coding of channel signals. Walsh-Hadamard and Casami coding.
- Statistical multiplexing in analogue and digital communication systems.
- Linear systems in communication engineering. The inverse problem.
- The main integral equation of linear systems and its solution.
- Solution unitarity and correctness. The practical meaning of these mathematical properties.
- Signal restoration by the analytic continuation method.
- Effect of noise on the restoration precision of the input signal.
- Quantum effects and their influence on the communications.
- Stochastic communication channels with thermal noise.
- The negentropy principle of information, its applications in electronic and photonic channels.
- Quantum limits of electronic and photonic communication channels.
- Peculiarities of quantum systems – their stochasticity and nonlocality.
- Quantum bits – qubits. Quantum communication channels. Schumacher theorem.
- Redundancy of quantum signals, their multiplexing and teleportation.
- Quantum cryptography.BB84 and EPR protocols.
- Operation principles and development trends of modern computer technology.
- Fourier optics and holography.
- Analogue quantum computers.
- RSA code and Shor’s algorithm. The operation principles of digital quantum computers.
- Potentialities of digital quantum computers and their practical realization.
- Stochastic resonance and its mathematical description.
- Stochastic resonance in nature and technology.
- Course project (Research and evaluation of nonlinear transmission system).
