Let \(\mu \) be a Borel probability measure on \({\mathbb {R}}^n\). We call \(\mu \) a spectral measure if there exists a countable set \(\Lambda \subset {\mathbb {R}}^n\) such that \(E_\Lambda :=\{e^{2\pi i<\lambda ,x>}:\lambda \in \Lambda \}\) forms an orthogonal basis for the Hilbert space \(L^2(\mu )\). Let the measure \(\mu _{\{M,{\mathcal {D}}_n\}}\) be defined by the following expression \(\mu _{\{M,{\mathcal {D}}_n\}}=\delta _{M^{-1}{\mathcal {D}}_1}*\delta _{M^{-2}{\mathcal {D}}_2}*\cdots \), where \(M=\text {diag}(\rho ^{-1},\rho ^{-1})\) with \(|\rho |<1\), and \({\mathcal {D}}_n=\left\{ (0,0)^t,(a_n,b_n)^t,(c_n,d_n)^t)\right\} \) with \(|a_n d_n-b_n c_n|=1\) for all \(n\ge 1\). This paper focuses on the spectrality of a class of Moran measures \(\mu _{\{M,{\mathcal {D}}_n\}}\) in \({\mathbb {R}}^2\). We give some sufficient conditions for \(\mu _{\{M,{\mathcal {D}}_n\}}\) to be a spectral measure. Also some necessary conditions are obtained. Moreover, the spectrality of the above Moran measures are also used to determine the spectrality of certain self-affine measures.

令\（\ mu \）为\（{\ mathbb {R}} ^ n \）的Borel概率度量。我们称\（\亩\）的光谱测量，如果存在一个可数集\（\ LAMBDA \子集{\ mathbb {R}} ^ N \） ，使得\（E_ \ LAMBDA：= \ {E 1 {2 \ pi i <\ lambda，x>}：\ lambda \ in \ Lambda \} \）构成希尔伯特空间\（L ^ 2（\ mu）\）的正交基础。让度量\（\ mu _ {\ {M，{\ mathcal {D}} _ n \}} \）由以下表达式\（\ mu _ {\ {M，{\ mathcal {D}} _ n \}} = \ delta _ {M ^ {-1} {\ mathcal {D}} _ 1} * \ delta _ {M ^ {-2} {\ mathcal {D}} _ 2} * \ cdots \），其中\（M = \ text {diag}（\ rho ^ {-1}，\ rho ^ {-1}）\）与\（| \ rho | <1 \），和\（{\ mathcal {D}} _ n = \ left \ {（0,0）^ t，（a_n，b_n）^ t，（c_n，d_n）^ t）\ right \} \）\）和\（| a_n对于所有\（n \ ge 1 \）d_n-b_n c_n | = 1 \）。本文重点研究的一类Moran测度的幽灵\（\亩_ {\ {M，{\ mathcal {d}} _Ñ\}} \）在\（{\ mathbb {R}} ^ 2 \）。我们给出一些足够的条件以使\（\ mu _ {\ {M，{\ mathcal {D}} _ n \}} \）成为频谱度量。还获得了一些必要的条件。此外，上述Moran度量的频谱也用于确定某些自仿射度量的频谱。