  Given a domain $\Omega\subset\Bbb R^d$ with positive and finite Lebesgue
measure and a discrete set $\Lambda\subset \Bbb R^d$, we say that $(\Omega,
\Lambda)$ is a {\it frame spectral pair} if the set of exponential functions
$\mathcal E(\Lambda):=\{e^{2\pi i \lambda \cdot x}: \lambda\in \Lambda\}$ is a
frame for $L^2(\Omega)$. Special cases of frames include Riesz bases and
orthogonal bases. In the finite setting $\Bbb Z_N^d$, $d, N\geq 1$, a frame
spectral pair can be similarly defined. %(Here, $\Bbb Z_N$ is the cyclic
abelian group of order.) We show how to construct and obtain new classes of
frame spectral pairs in $\Bbb R^d$ by "adding" frame spectral pairs in $\Bbb
R^{d}$ and $\Bbb Z_N^d$. Our construction unifies the well-known examples of
exponential frames for the union of cubes with equal volumes. We also remark on
the link between the spectral property of a domain and sampling theory.
