  Aims. A modulation equation relates the observed data to the object where the
observation is approximated by a linear system. Reconstructing the object from
the observed data is therefore is equivalent to solving the modulation
equation. In this work we present the synthetic direct demodulation (synDD)
method to reduce the dimensionality of a general modulation equation and solve
the equation in its sparse representation.
  Methods. A principal component analysis is used to reduce the dimensionality
of the kernel matrix and k-means clustering is applied to its sparse
representation in order to decompose the kernel matrix into a weighted sum of a
series of circulant matrices. The matrix- vector and matrix-matrix
multiplication complexities are therefore reduced from polynomial time to
linear-logarithmic time. A general statistical solution of the modulation
equation in sparse representation is derived. Several data-analysis pipelines
are designed for the Hard X-ray modulation Telescope (Insight-HXMT) based on
the synDD method.
  Results. In this approach, a large set of data originating from the same
object but sampled irregularly and/or observed with different instruments in
multiple epochs can be reduced simultaneously in a synthetic observation model.
We suggest using the proposed synDD method in Insight-HXMT data analysis
especially for the detection of X-ray transients and monitoring time-varying
objects with scanning observations.
