From DSP to GSP: Sampling in Both Domains

In Discrete Signal Processing (DSP), sampling in the time domain chooses time samples from which the original bandlimited signal can be perfectly reconstructed. Uniform sampling in the time domain can be interpreted in the frequency domain as replication using a LSI filtering. Sampling in Graph Signal Processing (GSP) has been explored in either the vertex and spectral domains, but not both. Current GSP literature [1] recognizes that sampling in the vertex domain is not the same as frequency replication in the spectral domain, leading to two different sampling methods (one in the vertex domain and the other in the spectral domain). Recently [2], we showed that one can indeed develop an unified GSP sampling theory with equivalent interpretations in both the vertex and spectral domains like in DSP. We explore the steps of DSP sampling in GSP: subsampling, decimation, upsampling, and interpolation. In DSP, each of these steps has interpretations in both the time and frequency domain. We interpret each step in both the vertex and spectral domain for GSP. The unified GSP sampling theory provides many different, but valid choices for the sampling set. This paper considers the impact of these choices on GSP sampling. We show that DSP frequency replication is a result of specific choices and conditions made in the GSP sampling theory, providing new insight and intuition on DSP sampling. As a result, we show that replication in frequency and, by extension, proposed spectral sampling methods, e.g., [3], in the GSP literature, are not generalizable to arbitrary graphs and GSP, in the sense that they do not sample the original signal in the vertex domain, but rather, distort it.