Graph Signal Representations

Signal representations form the foundation of signal processing. Expressing the time signal <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$s$</tex> in different bases allows for different signal analyses, giving rise to numerous transforms in signal processing, each with their own uses and applications. For example, in Discrete Signal Processing (DSP), we have the Discrete Fourier transform and the z-transform. These transforms express <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$s$</tex> using different signal representations, providing different ways to look at the same signal, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$s$</tex>. In Graph Signal Processing (GSP), we generally only consider two signal representations: the vertex signal representation using the identity matrix and the spectral signal representation using the Graph Fourier Transform (GFT). In this paper, we explore the companion signal representation, introduced in [1], and its applications. Through examples, we show that by choosing the correct signal representation (the companion signal representation) for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$s$</tex>, we gain additional insights in GSP that are not possible using just the GSP vertex and spectral signal representations.