  Using the group theoretic approach based on the set of digits, we first
investigate a finite collection of functions in $\ell^2 ({\mathbb{Z}}^2_N)$
that satisfies some localization properties in a region of the time-frequency
plane. The digits are associated with an invertible (expansive/non-expansive)
matrix having integer entries. Next, we study and characterize an orthonormal
wavelet system for $\ell^2 ({\mathbb{Z}}^2_N)$. In addition, some results
connecting the uncertainty principle with functions that generate the
orthonormal wavelet system having time-frequency localization properties are
obtained.
