We study a specific correlation in spherical harmonic multipole domain for cosmic microwave background ( CMB ) analysis .
This group of correlation between \Delta { \ell } = 4 n , n = 1 , 2 \ldots 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 .