We study how the frequency spectrum of the background isotropic monopole emission is modified and transferred to higher multipoles by boosting effects due to the observer peculiar motion.

The method, based on a linear system, is suitable for various background radiation models and here applied from the radio to the far-IR to several types of cosmic microwave background (CMB) distorted photon distribution functions and extragalactic background signals superimposed to the CMB Planckian spectrum.

We derive explicit solutions for the spherical harmonic coefficients up to any desired multipole, lmax, in terms of linear combinations of the signals at just N = lmax + 1 colatitudes. For appropriate choices of these colatitudes, the symmetry property of the associated Legendre polynomials with respect to π/2 allows the separation of the system into two subsystems, one for l = 0 and even multipoles and the other for odd multipoles, and improves the solution accuracy.

The simplicity and efficiency of this method can significantly reduce the computational cost needed for accurate predictions on the whole sky and for the scientific analysis of data from future projects. Moreover, in the presence of CMB spectral distortions, this formalism, combined with the representation of CMB intrinsic anisotropies, provides a new test to constrain the intrinsic dipole embedded in the kinematic dipole.

**Theme**– Multi-Messenger Astronomy