  Asymmetric propagation of acoustic waves is theoretically reported in a
chirped phononic crystal made of the combination of two different nonlinear
solids. The dispersion of the system is spatially dependent and allows the
rainbow trapping inside the structure. Nonlinearity is used to activate the
self-demodulation effect, which is enhanced due to the particular dispersion
characteristics of the system. The main feature of the device is that
integrates the nonlinear and the filtering effects in a single propagating
medium. The performed numerical study reveals an efficient generation of the
demodulated wave, up to 15\% in terms of the pressure amplitude, as well as
strong attenuation for undesired frequency components above the cut-off
frequency. The obtained energy rectification ratio is in the order of $10^4$
for the whole range of amplitudes employed in this work, indicating the
robustness of the asymmetry and non-reciprocity of the proposed device for a
wide operational range.
