  One of the key challenges in the area of signal processing on graphs is to
design dictionaries and transform methods to identify and exploit structure in
signals on weighted graphs. To do so, we need to account for the intrinsic
geometric structure of the underlying graph data domain. In this paper, we
generalize one of the most important signal processing tools - windowed Fourier
analysis - to the graph setting. Our approach is to first define generalized
convolution, translation, and modulation operators for signals on graphs, and
explore related properties such as the localization of translated and modulated
graph kernels. We then use these operators to define a windowed graph Fourier
transform, enabling vertex-frequency analysis. When we apply this transform to
a signal with frequency components that vary along a path graph, the resulting
spectrogram matches our intuition from classical discrete-time signal
processing. Yet, our construction is fully generalized and can be applied to
analyze signals on any undirected, connected, weighted graph.
