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

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