  We present a novel form of Fourier analysis, and associated signal processing
concepts, for signals (or data) indexed by edge-weighted directed acyclic
graphs (DAGs). This means that our Fourier basis yields an eigendecomposition
of a suitable notion of shift and convolution operators that we define. DAGs
are the common model to capture causal relationships between data values and in
this case our proposed Fourier analysis relates data with its causes under a
linearity assumption that we define. The definition of the Fourier transform
requires the transitive closure of the weighted DAG for which several forms are
possible depending on the interpretation of the edge weights. Examples include
level of influence, distance, or pollution distribution. Our framework is
different from prior GSP: it is specific to DAGs and leverages, and extends,
the classical theory of Moebius inversion from combinatorics. For a
prototypical application we consider DAGs modeling dynamic networks in which
edges change over time. Specifically, we model the spread of an infection on
such a DAG obtained from real-world contact tracing data and learn the
infection signal from samples assuming sparsity in the Fourier domain.
