  Prior studies on intelligent reflecting surface (IRS) have mostly considered
wireless communication systems aided by a single passive IRS, which, however,
has limited control over wireless propagation environment and suffers severe
product-distance path-loss. To address these issues, we propose in this paper a
new multi-active multi-passive (MAMP)-IRS aided wireless communication system,
where a number of active and passive IRSs are deployed to assist the downlink
communication in complex environment, by establishing a multi-hop reflection
path across active and passive IRSs. An optimization problem is formulated to
maximize the achievable rate of a typical user by designing the
active-and-passive IRS routing path as well as the joint beamforming of the BS
and selected active/passive IRSs. To draw useful insights into the optimal
design, we first consider a special case of the single-active multi-passive
(SAMP)-IRS aided system. For this case, we propose an efficient algorithm to
obtain its optimal solution by first optimizing the joint beamforming given any
SAMP-IRS routing path, and then optimizing the routing path by using a new path
decomposition method and graph theory. Next, for the general MAMP-IRS aided
system, we show that its challenging beam routing optimization problem can be
efficiently solved by a new two-phase approach. Its key idea is to first
optimize the inner passive-IRS beam routing between each two active IRSs for
effective channel power gain maximization, followed by an outer active-IRS beam
routing optimization for rate maximization. Last, numerical results are
provided to demonstrate the effectiveness of the proposed MAMP-IRS beam routing
scheme.
