Given a frame (or a Riesz basis) $F=\{f_j\}_{j\in I}$ in a space of square integrable functions $H$, we consider the set $\varphi F$ obtained by multiplying the $f_j$ by piecewise constant functions $\varphi_j$. We find conditions that ensure that $\varphi F$ is still a frame (or a Riesz basis) in some special and significant case. When $F$ is finite, we present necessary and sufficient conditions that ensure that $\varphi F$ is a Parseval frame in $H$. The results presented in this talk are part of joint papers with J. Edward, P. Casazza and T. Tran