  Given an integer $m\geq1$. Let $\Sigma^{(m)}=\{1,2, \cdots, m\}^{\mathbb{N}}$
be a symbolic space, and let $\{(b_{k},D_{k})\}_{k=1}^{m}:=\{(b_{k},
\{0,1,\cdots, p_{k}-1\}t_{k}) \}_{k=1}^{m}$ be a finite sequence pairs, where
integers $| b_{k}| $, $p_{k}\geq2$, $|t_{k}|\geq 1$ and $ p_{k},t_{1},t_{2},
\cdots, t_{m}$ are pairwise coprime integers for all $1\leq k\leq m$. In this
paper, we show that for any infinite word
$\sigma=\left(\sigma_{n}\right)_{n=1}^{\infty}\in\Sigma^{(m)}$, the infinite
convolution $$ \mu_{\sigma}=\delta_{b_{\sigma_{1}}^{-1} D_{\sigma_{1}}} *
\delta_{\left(b_{\sigma_{1}} b_{\sigma_{2}}\right)^{-1} D_{\sigma_{2}}} *
\delta_{\left(b_{\sigma_{1}} b_{\sigma_{2}}
b_{\sigma_{3}}\right)^{-1}D_{\sigma_{3}}} * \cdots $$ is a spectral measure if
and only if $p_{\sigma_n}\mid b_{\sigma_n}$ for all $n\geq2$ and $\sigma\notin
\bigcup_{l=1}^\infty\prod_{l}$, where $\prod_{l}=\{i_{1}i_{2}\cdots
i_{l}j^{\infty}\in\Sigma^{(m)}: i_{l}\neq j, |b_{j}|=p_{j}, |t_{j}|\neq1\}$.
