  The first aim of this article is to give information about the algebraic
properties of alternate bases $\boldsymbol{\beta}=(\beta_0,\dots,\beta_{p-1})$
determining sofic systems. We show that a necessary condition is that the
product $\delta=\prod_{i=0}^{p-1}\beta_i$ is an algebraic integer and all of
the bases $\beta_0,\ldots,\beta_{p-1}$ belong to the algebraic field ${\mathbb
Q}(\delta)$. On the other hand, we also give a sufficient condition: if
$\delta$ is a Pisot number and $\beta_0,\ldots,\beta_{p-1}\in {\mathbb
Q}(\delta)$, then the system associated with the alternate base
$\boldsymbol{\beta}=(\beta_0,\dots,\beta_{p-1})$ is sofic. The second aim of
this paper is to provide an analogy of Frougny's result concerning
normalization of real bases representations. We show that given an alternate
base $\boldsymbol{\beta}=(\beta_0,\dots,\beta_{p-1})$ such that $\delta$ is a
Pisot number and $\beta_0,\ldots,\beta_{p-1}\in {\mathbb Q}(\delta)$, the
normalization function is computable by a finite B\"uchi automaton, and
furthermore, we effectively construct such an automaton. An important tool in
our study is the spectrum of numeration systems associated with alternate
bases. The spectrum of a real number $\delta>1$ and an alphabet $A\subset
{\mathbb Z}$ was introduced by Erd\H{o}s et al. For our purposes, we use a
generalized concept with $\delta\in{\mathbb C}$ and $A\subset{\mathbb C}$ and
study its topological properties.
