Modeling and Recovery of Graph Signals and DifferenceBased Signals In this paper, we consider the problem of representing and recovering graph signals with a nonlinear measurement model. We propose a twostage graph signal processing (GSP) framework. First, a GSP representation is obtained by finding the graph filter that best approximates the known measurement function. The new GSP representation enables performing tractable operations over graphs, as well as gaining insights into the signal graphfrequency contents. Then, we formulate the signal recovery problem under the smoothness constraint and derive a regularized leastsquares (LS) estimator, which is obtained by applying the inverse of the approximated graph filter on the nonlinear measurements. In the second part of this paper, we investigate the proposed recovery and representation approach for the special case of graph signals that are influenced by the differences between vertex values only. Differencebased graph signals arise, for example, when modeling power signals as a function of the voltages in electrical networks. We show that any differencebased graph signal corresponds to a filter that lacks the zeroorder filter coefficient, and thus, these signals can be recovered up to a constant by the regularized LS estimator. In our simulations, we show that for the special case of state estimation in power systems the proposed GSP approach outperforms the stateoftheart estimator in terms of total variation.

On Folded Graph Signals Graph sampling allows a multidimensional signal generated on a graph to be represented by the signal at a smaller set of sampled nodes. On the other hand, selfreset analogtodigital converters (ADCs) are used to sample high dynamic range signals resulting in modulooperation based folded signals at the sampled nodes. In this paper, we study the problem of continuoustime graph signal recovery from the folded signals at discrete samples. We present a theoretical graph sampling rate that is sufficient for successful reconstruction of the graph signals from the folded signals. We deduce an optimal sample rate to recover a bandlimited continuoustime graph signal, such that integer programming can be applied for small graphs. To resolve the scalability issue of integer programming, we propose a sparse optimization based recovery method for graph signals satisfying certain conditions. Such an approach requires a novel graph sampling scheme that selects nodes with small signal variation. The proposed algorithm emphasizes that in our spatiotemporal sampling scenario, the inherent relationship among the graph nodes should be exploited in addition to the temporal correlation in the graph signal at different nodes to recover the signal.