- Correlated double sampling (CDS) circuits are essential to processing the X-ray charge-coupled devices (CCDs) that have been used in the modern X-ray astronomical field. For spectroscopy observations, the energy resolution is of decisive importance. In this paper, a comprehensive study of the CDS algorithm based on the incremental Delta Sigma technique, named Delta Sigma-CDS algorithm hereafter, is studied. Theoretical and practical aspects for modulations and demodulations for applications with different pixel rates (PRs) are discussed. The concepts of differential cascade-of-integrator filter and truncated cascaded-integrator-comb filter are extensively investigated from both the time-domain and frequency-domain analyses. The operating principles, prototype design, and experimental results are addressed. As a prototype for 100-kHz PR, a readout integrated circuit (ROIC) is designed. When tested with the target X-ray CCD-CCD236 using a 1 mu Ci Fe-55 radioisotope, the ROIC achieves an energy resolution characterized by the full-width-at-half-maximum as 132.4 +/- 2.4 eV at 5.9 keV, which is very close to the 121-eV limit of CCD236.