The uncertainty principle for the short-time Fourier transform on finite cyclic groups: Cases of equality

A well-known version of the uncertainty principle on the cyclic group Z_N states that for any couple of functions f,g∈ℓ^2(Z_N)∖{0}, the short-time Fourier transform V_g f has support of cardinality at least N. This result can be regarded as a time-frequency version of the celebrated Donoho-Stark uncertainty principle on Z_N. Unlike the Donoho-Stark principle, however, a complete identification of the extremals is still missing. In this note we provide an answer to this problem by proving that the support of Vgf has cardinality N if and only if it is a coset of a subgroup of order N of Z_N×Z_N. Also, we completely identify the corresponding extremal functions f,g. Besides translations and modulations, the symmetries of the problem are encoded by certain metaplectic operators associated with elements of SL(2,Z_N/a), where a is a divisor of N. Partial generalizations are given to finite Abelian groups.