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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 
$\unicode[STIX]{x1D707}_{\{M_{n}\},\{{\mathcal{D}}_{n}\}}=\unicode[STIX]{x1D6FF}_{M_{1}^{-1}{\mathcal{D}}_{1}}\ast \unicode[STIX]{x1D6FF}_{(M_{1}M_{2})^{-1}{\mathcal{D}}_{2}}\ast \cdots \,$ is a Borel probability measure (Cantor–Dust–Moran measure). In this paper, we investigate whenever there exists a discrete set 
$\unicode[STIX]{x1D6EC}$ such that 
$\{e^{2\unicode[STIX]{x1D70B}i\langle \unicode[STIX]{x1D706},x\rangle }:\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}\}$ is an orthonormal basis for 
$L^{2}(\unicode[STIX]{x1D707}_{\{M_{n}\},\{{\mathcal{D}}_{n}\}})$.
