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The observer peculiar motion produces boosting effects in the backgroundanisotropies with frequency spectral behaviours related to its spectrum. Westudy how the frequency spectrum of the background isotropic monopole emissionis modified and transferred to the frequency spectra at higher multipoles, l.We perform the analysis in terms of spherical harmonic expansion up to acertain lmax, for various models from radio to far-IR. We derive a system oflinear equations to obtain spherical harmonic coefficients and provide explicitsolutions up to lmax=6 as linear combinations of the signals at N=lmax+1colatitudes. The associated Legendre polynomials symmetry with respect to{\pi}/2 is used to separate the system into two subsystems, one for l=0 andeven l, the other for odd l. This improves the solutions accuracy with respectto an arbitrary colatitudes choice. We apply the method to analytical orsemi-analytical representions of monopole spectra, i.e. to four types of CMBdistortions, four types of extragalactic backgrounds superimposed to the CMBPlanckian spectrum and some combinations of them. We present our results interms of spherical harmonic coefficients, relationships between the observedand intrinsic monopoles, maps, angular power spectra. We compare the methodresults with the ones obtained using more computationally demanding numericalintegrations or map generation/inversion. The method is generalized to includethe effect of the observer motion relative to the Sun. Its simplicity andefficiency can significantly alleviate the computational effort needed foraccurate predictions and for the analysis of future data. We discuss thesuperposition of the CMB intrinsic anisotropies and of the effects induced bythe observer motion, exploring for the possibility of constraining theintrinsic dipole embedded in the kinematic dipole, in the presence of CMBspectral distortions.

Effects of observer peculiar motion on the isotropic background frequency spectrum: from monopole to higher multipoles