  We describe a setup for obtaining uncertainty relations for arbitrary pairs
of observables related by Fourier transform. The physical examples discussed
here are standard position and momentum, number and angle, finite qudit
systems, and strings of qubits for quantum information applications. The
uncertainty relations allow an arbitrary choice of metric for the distance of
outcomes, and the choice of an exponent distinguishing e.g., absolute or root
mean square deviations. The emphasis of the article is on developing a unified
treatment, in which one observable takes values in an arbitrary locally compact
abelian group and the other in the dual group. In all cases the phase space
symmetry implies the equality of measurement uncertainty bounds and preparation
uncertainty bounds, and there is a straightforward method for determining the
optimal bounds.
