Let \(0<\rho <1, N\in {\mathbb {N}}^+, D_{4N}=\{0,1,2,\cdots ,4N-1\}\) , and let \(\{f_j\}_{j=0}^{4N-1}\) be an iterated function system defined by \(f_j(x)=(-1)^{\sigma (j)}\rho (x+j), x\in {\mathbb {R}}\) , where \(\sigma (j)=0\) if \(j\in 4{\mathbb {N}}\cup (4{\mathbb {N}}+1)\) and \(\sigma (j)=1\) if \(j\in (4{\mathbb {N}}+2)\cup (4{\mathbb {N}}+3)\) . \(\mu _{\{(-1)^{\sigma (j)}\rho \},D_{4N}}\) is the self-similar measure defined by \( \mu _{\{(-1)^{\sigma (j)}\rho \},D_{4N}}(\cdot )=\frac{1}{4N}\sum _{j=0}^{4N-1}\mu _{\{(-1)^{\sigma (j)}\rho \},D_{4N}}(f_j^{-1}(\cdot )). \) In this paper, we prove that \(L^2(\mu _{\{(-1)^{\sigma (j)}\rho \},D_{4N}})\) admits an exponential orthonormal basis if and only if \(\rho ^{-1}=q\) with \(q\in {\mathbb {Z}}^+\) and \(4N\mid q\) .
