Technical Notes on Zak Transforms in 6G Telecom Systems

The Zak transform is a fundamental integral transform that combines aspects of the Fourier transform and the Poisson summation formula. It maps a function on the real line to a function on a two-dimensional torus, effectively representing the function in a joint time-frequency domain. This transform is particularly useful for analyzing systems with periodic structures, such as solid-state physics, where it diagonalizes the Schrödinger equation with a perio dic potential. Furthermore, the Zak transform provides a natural framework for studying Gabor analysis and the theory of coherent states, revealing deep connections between signal processing and quantum mechanics.