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 ${\mathbb{Z}}_N$ states that for any couple of functions $f,g\in {\mathrm{\ell }}^2({\mathbb{Z}}_N)\setminus \{0\}$, 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 ${\mathbb{Z}}_N$. 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 $V_{g}f$ has cardinality $N$ if and only if it is a coset of a subgroup of order $N$ of ${\mathbb{Z}}_N\times {\mathbb{Z}}_N$. 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 $\mathrm{S}\mathrm{L}(2,{\mathbb{Z}}_{N\mathrm{/}a})$, where $a$ is a divisor of $N$. Partial generalizations are given to finite Abelian groups.