Generalized frames and their redundancy

Let $h$ be a generalized frame in a separable Hilbert space $H$ indexed by a measure space $(M,\mathcal{ S},\mu)$, and assume its analysing operator is surjective. It is shown that $h$ is essentially discrete; that is, the corresponding index measure space $(M,\mathcal{ S},\mu)$can be decomposed into atoms $E_1,E_2,\cdots$ such that $L^2(\mu)$ is isometrically isomorphic to the weighted space $\ell^2_w$ of all sequences $\{c_i\}$ of complex numbers with $\vert\vert\{c_i\}\vert\vert^2=\sum \vert c_i\vert^2 w_i<\infty$, where $w_i=\mu(E_i), i=1,2,\cdots.$ This provides a new proof for the redundancy of the windowed Fourier transform as well as any wavelet family in $L^2(\mathbb{R})$.