Let the infinite convolutions \(\begin{aligned} \mu _{\{R_{k}\},\{D_{k}\}}=\delta _{R_{1}^{-1}D_{1}}*\delta _{R_{1}^{-1}R_{2}^{-1}D_{2}}*\delta _{R_{1}^{-1}R_{2}^{-1}R_{3}^{-1}D_{3}}*\cdots \end{aligned}\) be generated by the sequence of pairs \(\{ (R_k,D_k) \}_{k=1}^{\infty } \) , where \( R_k\in M_n(\mathbb {Z})\) is an expanding integer matrix, \(D_k\) is a finite integer vector sets that satisfies the following two conditions: (i) \( \# D_k = m \) and \( m>2 \) is a prime; (ii) \( \{x: \sum _{d\in D_{k}}e^{-2\pi i\langle d,x \rangle }=0\} =\cup _{i=1}^{\phi (k)}\cup _{j=1}^{m-1}(\frac{j}{m}\nu _{k,i}+\mathbb {Z}^{n}) \) for some \( \nu _{k,i} \in \{ (l_1, \cdots , l_n)^t : l_i \in [1, m-1] \cap \mathbb {Z}, 1\le i\le n \} \) . In this paper, we study the spectrality of \(\mu _{\{R_{k}\},\{D_{k}\}}\) and present some necessary and sufficient conditions for the existence of an infinite orthogonal exponential basis in \( L^{2}(\mu _{\{R_{k}\},\{D_{k}\}}) \) .
