  In intelligent reflecting surface (IRS) assisted communication systems, the
acquisition of channel state information (CSI) is a crucial impediment for
achieving the beamforming gain of IRS because of the considerable overhead
required for channel estimation. Specifically, under the current beamforming
design for IRS-assisted communications, $KMN+KM$ channel coefficients should be
estimated, where $K$, $N$ and $M$ denote the numbers of users, IRS reflecting
elements, and antennas at the base station (BS), respectively. To accurately
estimate such a large number of channel coefficients within a short time
interval, we propose a novel three-phase pilot-based channel estimation
framework in this paper for IRS-assisted uplink multiuser communications. Under
this framework, we analytically prove that a time duration consisting of
$K+N+\max(K-1,\lceil (K-1)N/M \rceil)$ pilot symbols is sufficient for the BS
to perfectly recover all the $KMN+KM$ channel coefficients for the case without
receiver noise at the BS. In contrast to the channel estimation for
conventional uplink communications without IRS where the minimum channel
estimation time is independent of the number of receive antennas at the BS, our
result reveals the crucial role of massive MIMO (multiple-input
multiple-output) in reducing the channel estimation time for IRS-assisted
communications. Further, for the case with receiver noise, the user pilot
sequences, IRS reflecting coefficients, and BS linear minimum mean-squared
error (LMMSE) channel estimators are characterized in closed-form, and the
corresponding estimation mean-squared error (MSE) is quantified.
