  In this paper, the minimum mean square error (MMSE) channel estimation for
intelligent reflecting surface (IRS) assisted wireless communication systems is
investigated. In the considered setting, each row vector of the equivalent
channel matrix from the base station (BS) to the users is shown to be Bessel
$K$ distributed, and all these row vectors are independent of each other. By
introducing a Gaussian scale mixture model, we obtain a closed-form expression
for the MMSE estimate of the equivalent channel, and determine analytical upper
and lower bounds on the mean square error. Using the central limit theorem, we
conduct an asymptotic analysis of the MMSE estimate, and show that the upper
bound on the mean square error of the MMSE estimate is equal to the asymptotic
mean square error of the MMSE estimation when the number of reflecting elements
at the IRS tends to infinity. Numerical simulations show that the gap between
the upper and lower bounds are very small, and they almost overlap with each
other at medium signal-to-noise ratio (SNR) levels and moderate number of
elements at the IRS.
