Graph representation of random signal and its application for sparse signal detection

In this paper, a novel graph-based adequate and concise signal representation paradigm is explored. Amplitudes of a signal can be quantized first. Then the quantization levels and transitions among these levels can be specified as vertices and edges so that the signal waveform (spectrum) can be converted to a graph thereby. This new signal representation framework can provide a promising alternative for manifesting the essential structure of random signals in the graph domain. The randomness of signal samples can be characterized by the topological features of the converted graph. In this work, we explore the connectivity of a graph to measure such randomness. A typical application, namely sparse communication signal detection, can be undertaken by the graph-connectivity metric. New pertinent theoretical analyses are also conducted here. First, the analysis on the minimum sample size for constructing a fully-connected graph from random samples is established. Second, the probability analysis of constructing a fully-connected graph with respect to a particular sample size is also derived. Third, if a graph is not fully connected, the probability distribution function of the number of edges is derived. According to Monte Carlo simulations, our proposed graph-based signal-detection method leads to outstanding performance, compared to a popular existing signal-detection technique especially when the signal-to-noise ratio is rather small.