  High-order harmonic generation (HHG) results from strong-field laser matter
interaction and it is one of the main processes that are used to extract
electron structural and dynamical information about the atomic or molecular
targets with sub-femtosecond temporal resolution. Moreover, it is the workhorse
for the generation of attosecond pulses. Here we develop an analytical
description of HHG, which extends the well established theoretical strong-field
approximation (SFA). Our approach involves two innovative aspects: i) First,
using a model non-local, but separable potential, we calculate the bound-free
dipole and the rescattering transition matrix elements analytically for both
atomic and molecular multicenter systems. In comparison with the standard
approaches to the HHG process, these analytic derivations of the different
matrix elements allows us to study directly how the HHG spectra depend on the
atomic target and laser pulse features. We can turn on and of contributions
having distinct physical origins or corresponding to different physical
mechanisms. Our SFA results are compared, when possible, with the direct
numerical integration of the time-dependent Schr\"odinger equation (TDSE) in
reduced and full dimensionality. Excellent agreement is found for single and
multielectronic atomic systems, modeled under the single active electron
approximation, and for simple diatomic molecular systems. Our model captures
also the interference features, ubiquitously present in every strong-field
phenomenon involving a multicenter target.
