Sparse time-frequency (T-F) representations have been an important research topic for more than several decades. Among them, optimization-based methods (in particular, extensions of basis pursuit) allow us to design the representations through objective functions. Since acoustic signal processing utilizes models of spectrogram, the flexibility of optimization-based T-F representations is helpful for adjusting the representation for each application. However, acoustic applications often require models of \textit{magnitude} of T-F representations obtained by discrete Gabor transform (DGT). Adjusting a T-F representation to such a magnitude model (e.g., smoothness of magnitude of DGT coefficients) results in a non-convex optimization problem that is difficult to solve. In this paper, instead of tackling difficult non-convex problems, we propose a convex optimization-based framework that realizes a T-F representation whose magnitude has characteristics specified by the user. We analyzed the properties of the proposed method and provide numerical examples of sparse T-F representations having, e.g., low-rank or smooth magnitude, which have not been realized before.

翻译：稀疏时频表示已成为数十年来的重要研究课题。其中，基于优化的方法（特别是基追踪的扩展）允许我们通过目标函数设计表示。由于声学信号处理利用谱图模型，基于优化的时频表示的灵活性有助于针对每个应用调整表示。然而，声学应用通常需要对离散Gabor变换获得的时频表示的**幅度**进行建模。将时频表示调整到这样的幅度模型（例如，DGT系数的幅度平滑性）会导致难以求解的非凸优化问题。本文不处理困难的非凸问题，而是提出一种基于凸优化的框架，该框架能够实现幅度具有用户指定特性的时频表示。我们分析了所提出方法的性质，并提供了具有例如低秩或平滑幅度（此前未实现过）的稀疏时频表示的数值示例。