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
$L^{2}(\unicode[STIX]{x1D707}_{\{M_{n}\},\{{\mathcal{D}}_{n}\}})$
.