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 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.

Measuring quasar distance through joint analysis of spectroastrometry and reverberation mapping observations is a new method for driving the development of cosmology. In this paper, we carry out detailed simulation and analysis to study the effect of four basic observational parameters (baseline length, exposure time, equivalent diameter and spectral resolution) on the data quality of differential phase curves (DPCs), and furthermore on the accuracy of distance measurement. In our simulation, we adopt an axisymmetrical disk model of a broad line region (BLR) to generate differential phase signals. We find that the differential phases and their Poisson errors could be amplified by extending the baseline, while the influence of optical path difference errors can be reduced during fitting the BLR model. Longer exposure time or larger equivalent diameter helps reduce the absolute Poisson error. Therefore, the relative error of DPCs could be reduced by increasing any of the above three parameters, then the accuracy of distance measurement could be improved. In contrast, the uncertainty of absolute angular distances ( D _A ) could be improved with higher spectral resolution, although the relative error of DPCs would be amplified. We show how the uncertainty of distance measurement varies with the relative error of DPCs. For our specific set of model parameters, without considering more complicated structures and kinematics of BLRs in our simulation, it is found that the relative error of DPCs < 20% is a limit for accurate distance measurement. The relative error of DPCs has a lower limit (roughly 5%) and the uncertainty in distance measurement can be better than 2%.