  We study non-standard number systems with negative base $-\beta$. Instead of
the Ito-Sadahiro definition, based on the transformation $T_{-\beta}$ of the
interval $\big[-\frac{\beta}{\beta+1},\frac{1}{\beta+1}\big)$ into itself, we
suggest a generalization using an interval $[l,l+1)$ with $l\in(-1,0]$. Such
generalization may eliminate certain disadvantages of the Ito-Sadahiro system.
We focus on the description of admissible digit strings and their periodicity.
