Bimodal Wilson systems in $L^2(\mathbb R)$

Given a window $\phi \in L^2(\mathbb R),$ and lattice parameters $\alpha, \beta>0,$ we introduce a bimodal Wilson system $\mathcal{W}(\phi, \alpha, \beta)$ consisting of linear combinations of at most two elements from an associated Gabor $\mathcal{G}(\phi, \alpha, \beta)$. For a class of window functions $\phi,$ we show that the Gabor system $\mathcal{G}(\phi, \alpha, \beta)$ is a tight frame of redundancy $\beta^{-1}$ if and only if the Wilson system $\mathcal{W}(\phi, \alpha, \beta)$ is Parseval system for $L^2(\mathbb R).$ Examples of smooth rapidly decaying generators $\phi$ are constructed. In addition, when $3\leq \beta^{-1}\in \mathbb N$, we prove that it is impossible to renormalize the elements of the constructed Parseval Wilson frame so as to get a well-localized orthonormal basis for $L^2(\mathbb R)$.