  Many real-world networks are characterized by directionality; however, the
absence of an appropriate Fourier basis hinders the effective implementation of
graph signal processing techniques.
  Inspired by discrete signal processing, where embedding a line digraph into a
cycle digraph facilitates the powerful Discrete Fourier Transform for signal
analysis, addressing the structural complexities of general digraphs can help
overcome the limitations of the Graph Fourier Transform (GFT) and unlock its
potential.
  The Discrete Fourier Transform (DFT) serves as a Graph Fourier Transform for
both cycle graphs and Cayley digraphs on the finite cyclic groups
$\mathbb{Z}_N$. We propose a systematic method to identify a class of such
Cayley digraphs that can encompass a given directed graph. By embedding the
directed graph into these Cayley digraphs and opting for envelope extensions
that naturally support the Graph Fourier Transform, the GFT functionalities of
these extensions can be harnessed for signal analysis.
  Among the potential envelopes, optimal performance is achieved by selecting
one that meets key properties. This envelope's structure closely aligns with
the characteristics of the original digraph. The Graph Fourier Transform of
this envelope is reliable in terms of numerical stability, and its columns
approximately form an eigenbasis for the adjacency matrix associated with the
original digraph.
  It is shown that the envelope extensions possess a convolution product, with
their GFT fulfilling the convolution theorem. Additionally, shift-invariant
graph filters (systems) are described as the convolution operator, analogous to
the classical case. This allows the utilization of systems for signal analysis.
