  This paper investigates a channel estimator based on Gaussian mixture models
(GMMs) in the context of linear inverse problems with additive Gaussian noise.
We fit a GMM to given channel samples to obtain an analytic probability density
function (PDF) which approximates the true channel PDF. Then, a conditional
mean estimator (CME) corresponding to this approximating PDF is computed in
closed form and used as an approximation of the optimal CME based on the true
channel PDF. This optimal CME cannot be calculated analytically because the
true channel PDF is generally unknown. We present mild conditions which allow
us to prove the convergence of the GMM-based CME to the optimal CME as the
number of GMM components is increased. Additionally, we investigate the
estimator's computational complexity and present simplifications based on
common model-based insights. Further, we study the estimator's behavior in
numerical experiments including multiple-input multiple-output (MIMO) and
wideband systems.
