  Double-reconfigurable intelligent surface (RIS) is a promising technique,
achieving a substantial gain improvement compared to single-RIS techniques.
However, in double-RIS-aided systems, accurate channel estimation is more
challenging than in single-RIS-aided systems. This work solves the problem of
double-RIS-based channel estimation based on active RIS architectures with only
one radio frequency (RF) chain. Since the slow time-varying channels, i.e., the
BS-RIS 1, BS-RIS 2, and RIS 1-RIS 2 channels, can be obtained with active RIS
architectures, a novel multi-user two-timescale channel estimation protocol is
proposed to minimize the pilot overhead. First, we propose an uplink training
scheme for slow time-varying channel estimation, which can effectively address
the double-reflection channel estimation problem. With channels' sparisty, a
low-complexity Singular Value Decomposition Multiple Measurement Vector-Based
Compressive Sensing (SVD-MMV-CS) framework with the line-of-sight (LoS)-aided
off-grid MMV expectation maximization-based generalized approximate message
passing (M-EM-GAMP) algorithm is proposed for channel parameter recovery. For
fast time-varying channel estimation, based on the estimated large-timescale
channels, a measurements-augmentation-estimate (MAE) framework is developed to
decrease the pilot overhead.Additionally, a comprehensive analysis of pilot
overhead and computing complexity is conducted. Finally, the simulation results
demonstrate the effectiveness of our proposed multi-user two-timescale
estimation strategy and the low-complexity Bayesian CS framework.
