Orthonormal Wavelet System in $\ell^2 ({\mathbb{Z}}^2_N)$

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.