Mathematics of Signal Design for Communication Systems and Szemerédi’s and Green-Tao’s Theorems

Explore the mathematics behind signal design for communication systems in this lecture by Holger Boche from TU München. Delve into the challenges of orthogonal transmission schemes, focusing on their high dynamical behavior and its impact on system performance and efficiency. Discover the connections between this problem and various mathematical fields, including functional analysis, additive combinatorics, and harmonic analysis. Examine the Peak-to-Average power ratio (PAPR) reduction problem and its relationship to Szemerédi's Theorem on arithmetic progressions. Investigate necessary and sufficient conditions for solving the PAPR problem in OFDM and CDMA systems, and learn about asymptotic tightenings and perfect Walsh sums. Gain insights into the interdisciplinary nature of signal processing and its applications in modern communication standards.

Intro
Motivation - Orthogonal Transmission Scheme
Orthogonal Transmission Scheme - Sketch
Motivation - Dynamics of Orthogonal Transmission Scheme
Drawback of OFDM - The Effect of Clipping
Motivation - High Dynamics of Orthogonal Transmission Scheme
Motivation - Tone Reservation method
PAPR Reduction Problem - Remarks
Necessary and Sufficient Conditions - Essential Subspaces
Necessary Condition - Sketch of Proof
Outline
Solvability of PAPR Problem - OFDM
Szemerédi Theorem on Arithmetic Progressions
Szemerédi Theorem - Historical Remarks
Szemerédi Theorem - Asymptotic Case and Probabilistic Case
Solvability of PAPR reduction problem & Arithmetic Progressions
Asymptotic Tightenings of Thm. 3.7
Walsh functions
Perfect Walsh Sum
PAPR reduction problem for CDMA Case - PWS
Asymptotic results for PWS
Summary and Conclusions
PAPR Reduction Problem - Formulation