Summary: Let \(\{M_n\}_{n=1}^{\infty}\) be a sequence of expanding matrices with \(M_n=\operatorname{diag}(p_n,q_n)\), and let \(\{\mathcal{D}_n\}_{n=1}^{\infty}\) be a sequence of digit sets with \(\mathcal{D}_n=\{(0,0)^t,(a_n,0)^t,(0,b_n)^t,\pm (a_n,b_n)^t\} \), where \(p_n\), \(q_n\), \(a_n\) and \(b_n\) are positive integers for all \(n\geqslant 1\). If \(\sup_{n\geqslant 1}\{\frac{a_n}{p_n},\frac{b_n}{q_n}\}<\infty\), then the infinite convolution \(\mu_{\{M_n\},\{\mathcal{D}_n\}}=\delta_{M_1^{-1}\mathcal{D}_1}\ast\delta_{(M_1M_2)^{-1}\mathcal{D}_2}\ast \cdots\) is a Borel probability measure (Cantor-Dust-Moran measure). In this paper, we investigate whenever there exists a discrete set \(\Lambda\) such that \(\{e^{2\pi i\langle\lambda,x\rangle}:\lambda\in\Lambda\}\) is an orthonormal basis for \(L^2(\mu_{\{M_n\},\{\mathcal{D}_n\}})\).