We study a specific correlation in spherical harmonic multipole domain for cosmic microwave background (CMB) analysis. This group of correlation between Delta l=4n, n=1,2... is caused by symmetric signal in the Galactic coordinate system. An estimator targeting such correlation therefore helps remove the localized bright point-like sources in the Galactic plane and the strong diffused component down to the CMB level. We use 3 toy models to illustrate the significance of these correlations and apply this estimator on some derived CMB maps with foreground residuals. In addition, we show that our proposed estimator significantly damp the phase correlations caused by Galactic foregrounds. This investigation provides the understanding of mode correlations caused by Galactic foregrounds, which is useful for paving the way for foreground cleaning methods for the CMB.
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