This paper continues the study of orthonormal bases (ONB) of L2[0, 1] introduced in Dutkay et al. (J. Math. Anal. Appl. 409(2):1128-1139, 2014) by means of Cuntz algebra ON representations on L2[0, 1]. For N = 2, one obtains the classic Walsh system. We show that the ONB property holds precisely because the ON representations are irreducible. We prove an uncertainty principle related to these bases. As an application to discrete signal processing we find a fast generalized transform and compare this generalized transform with the classic one with respect to compression and sparse signal recovery.