Non-Commutative Harmonic Analysis on Locally Compact Groups and Applications to Wireless Communications in Nigerian Telecommunications

This dissertation develops original results in non-commutative harmonic analysis on locally compact groups and establishes novel connections between the abstract harmonic analysis and the mathematical foundations of orthogonal frequency division multiplexing waveform design, contributing to the rigorous mathematical underpinning of wireless communications technology of direct relevance to the continuing expansion of mobile broadband networks in Nigeria. The proliferation of 4G LTE and 5G NR services by Nigerian mobile operators has created significant demand for mathematically sophisticated spectrum efficiency and waveform design solutions, yet the deep connections between abstract harmonic analysis and wireless communications have not been systematically explored in the Nigerian research literature. The dissertation formulates the Heisenberg-Weyl group framework for time-frequency analysis of communication signals and establishes original Plancherel theorems and inversion formulas for group-theoretic Fourier transforms appropriate to the Gabor and OFDM signal structures used in contemporary wireless standards. New uncertainty principles on the Heisenberg-Weyl group are proved, quantifying the fundamental trade-offs between time localisation and frequency localisation of communication waveforms under orthogonality constraints. The Zak transform is rigorously developed as a unitary operator intertwining the Heisenberg-Weyl group action and a multiplication action, and its utility for characterising orthonormal Gabor bases relevant to interference-robust waveform design is demonstrated. Applications to spectral efficiency analysis for the Nigerian 700 MHz band are provided. Keywords: harmonic analysis, locally compact groups, OFDM waveforms, Heisenberg-Weyl group, wireless communications