Frames provide redundant, stable representations of data which have important applications in signal processing. We introduce a connection between symplectic geometry and frame theory and show that many important classes of frames have natural symplectic descriptions. Symplectic tools seem well-adapted to addressing a number of important questions about frames; in this paper, we focus on the frame homotopy conjecture posed in 2002 and recently proved by Cahill, Mixon, and Strawn, which says that the space of finite unit norm tight frames is connected. We give a simple symplectic proof of a generalization of the frame homotopy conjecture, showing that spaces of complex frames with arbitrary prescribed norms and frame operators are connected. To spark further investigation, we also suggest a number of fundamental questions in frame theory which seem amenable to a symplectic approach.

帧提供了冗余，稳定的数据表示形式，这些数据在信号处理中具有重要的应用。我们介绍了辛几何和框架理论之间的联系，并表明许多重要的框架类别都有自然的辛描述。辛工具似乎很适合解决框架的许多重要问题。在本文中，我们关注于2002年提出的框架同构猜想，最近由Cahill，Mixon和Strawn证明，它们表示有限单位范数紧框架的空间是连通的。我们给出了框架同伦猜想推广的一个简单辛证明，表明具有任意规定范数和框架算子的复杂框架的空间是相连的。为了进一步调查，