  In this paper, we investigate a double-active-reconfigurable intelligent
surface (RIS)-aided downlink wireless communication system, where a
multi-antenna base station (BS) serves multiple single-antenna users with both
double reflection and single reflection links. Due to the signal amplification
capability of active RISs, the mutual influence between active RISs, which is
termed as the "inter-excitation" effect, cannot be ignored. Then, we develop a
feedback-type model to characterize the signal containing the inter-excitation
effect. Based on the signal model, we formulate a weighted sum rate (WSR)
maximization problem by jointly optimizing the beamforming matrix at the BS and
the reflecting coefficient matrices at the two active RISs, subject to power
constraints at the BS and active RISs, as well as the maximum amplification
gain constraints of the active RISs. To solve this non-convex problem, we first
transform the problem into a more tractable form using the fractional
programming (FP) method. Then, by introducing auxiliary variables, the problem
can be converted into an equivalent form that can be solved by using a
low-complexity penalty dual decomposition (PDD) algorithm. Finally, simulation
results indicate that it is crucial to consider the inter-excitation effect
between active RISs in beamforming design for double-active-RIS-aided
communication systems. Additionally, it prevails over other benchmark schemes
with single active RIS and double passive RISs in terms of achievable rate.
