Revisit Sampling Theory of Bandlimited Graph Signals: One Bridge Between GSP and DSP

Sampling of bandlimited (BL) graph signals is one fundamental problem in graph signal processing (GSP), whose underlying kernel is an irregular graph rather than regular 1-D time-series kernel in classical discrete signal processing (DSP). Though there were amounts of sampling objectives and algorithms proposed for BL graph signals, the essential relationship between those sampling objectives in GSP and Nyquist sampling theorem in DSP is still undiscovered. In this paper, we bridge this gap by revisiting sampling theory in GSP thoroughly. In specific, we first figure out that the minimal sample size used in GSP for unique recovery can derive exact Nyquist sampling frequency in DSP when eigenvector matrix is discrete Fourier transform (DFT) matrix. Then, we propose a graph sampling objective for directed cyclic graph as one bridge, which leads to uniform sampling pattern in DSP and has the same optimal solution as other popular graph sampling objectives in GSP, thus connecting GSP and DSP closely via sampling. Finally, we present simulations to demonstrate potential applications inspired by our study, such as fast reconstruction of BL graph signals.