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One may be interested in Gabor frames and its related theory if one looks further into the frame framework. First, denote a lattice by $\Lambda=A \mathbb{Z}^{2d}$ , where $A$ is an invertible matrix, and let $\pi(\xi,\phi)f=e^{2 \pi i \xi x}f(x-\phi)$
Definition 1 Let $g \in L^2(\mathbb{R}^d)$ be a nonzero window, and let $\lambda \in \Lambda$ , then \begin{equation*} G(g,\lambda)=\left\{ \pi(\lambda)g : \lambda \in \Lambda \right\} \end{equation*}is a Gabor system. If $G(g,\lambda)$ is a frame, it's called a Gabor frame for $L^2(\mathbb{R}^d)$
Supose now that one wants to look at a more general framework, and work with functions in $L^2(\mathbb{R}^d,\mathbb{C}^n)$ . Then the definition above generalises to
Definition 2 Let $\boldsymbol{g} \in L^2(\mathbb{R}^d,\mathbb{C}^n)$ be a nonzero window and let $\lambda \in \Lambda$ , then
\begin{equation*} \boldsymbol{G}(\boldsymbol{g},\lambda)=\left\{ \pi(\lambda) \boldsymbol{g} : \lambda \in \Lambda \right\} \end{equation*} is a Gabor super-frame if the frame inequalities hold, where$$ \pi( \xi, \phi ) \boldsymbol{g}=e^{2 \pi \i x \cdot \xi}\left( g_1(x-\phi),g_2(x-\phi),...,g_n(x-\phi) \right)$$
and for $ \boldsymbol{f},\boldsymbol{h} \in L^2(\mathbb{R}^d,\mathbb{C}^n) $ $$ \left\langle \boldsymbol{f},\boldsymbol{h} \right\rangle_{L^2(\mathbb{R}^d,\mathbb{C}^n)} = \sum_{i=1}^n \left\langle f_i , h_i \right\rangle_{L^2(\mathbb{R}^d)}$$
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- Karlheinz Gröchenig, "Foundations of Time-Frequency Analysis," Birkhhäuser (2000)
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