  This paper continues the study of orthonormal bases (ONB) of $L^2[0,1]$
introduced in \cite{DPS14} by means of Cuntz algebra $\mathcal{O}_N$
representations on $L^2[0,1]$. For $N=2$, one obtains the classic Walsh system.
We show that the ONB property holds precisely because the $\mathcal{O}_N$
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.
