  Graph Fourier transform (GFT) is a fundamental concept in graph signal
processing. In this paper, based on singular value decomposition of Laplacian,
we introduce a novel definition of GFT on directed graphs, and use singular
values of Laplacian to carry the notion of graph frequencies. % of the proposed
GFT. The proposed GFT is consistent with the conventional GFT in the undirected
graph setting, and on directed circulant graphs, the proposed GFT is the
classical discrete Fourier transform, up to some rotation, permutation and
phase adjustment. We show that frequencies and frequency components of the
proposed GFT can be evaluated by solving some constrained minimization problems
with low computational cost. Numerical demonstrations indicate that the
proposed GFT could represent graph signals with different modes of variation
efficiently.
