This text presents a theory of time-frequency representations over finite and finitely generated abelian groups which can be used to design algorithms for multidimensional applications in imaging, electromagnetics and communication theory. Emphasis is placed on Weyl-Heisenberg systems and expansions. Algorithms are developed within this abstract setting without reference to co-ordinates or dimension. By not concerning itself with co-ordinates and dimensions, algorithmic structures can be derived which should be of importance to multidimensional applications in mathematics and electrical engineering.