  To achieve the more promising passive beamforming gains in the
double-intelligent reflecting surface (IRS) assisted system over the
conventional single-IRS system, channel estimation is practically indispensable
but also a more challenging problem to tackle, due to the presence of not only
the single- but also double-reflection links that are intricately coupled. In
this paper, we propose a new and efficient channel estimation scheme for the
double-IRS assisted uplink multiple-input multiple-output (MIMO) communication
system to resolve the cascaded channel state information (CSI) of both its
single- and double-reflection links. First, for the single-user case, the
higher-dimensional double-reflection channel is efficiently estimated at the
multi-antenna base station (BS) with low training overhead by exploiting the
fact that its cascaded channel coefficients are scaled versions of those of a
lower-dimensional single-reflection channel. Then, the proposed channel
estimation scheme is extended to the multi-user case, where given an arbitrary
user's cascaded channel estimated as in the single-user case, the other users'
cascaded channels are scaled versions of it and thus can be estimated with
reduced training overhead. Simulation results verify the effectiveness of the
proposed channel estimation scheme as compared to the benchmark scheme.
