Blind channel identification based on the geometry of the received signal constellation

We study the geometric properties of the constellation of the points z(k)=[x(k),x(k-1)]^T formed by an ISI corrupted M-ary PAM source transmitted through a narrowband channel. We show that in the absence of noise, the convex hull of the constellation contains all the information pertaining to the channel taps. In particular, the edges of the convex hull can be used to recover the channel uniquely (up to the sign). These results provide a set of geometric identifiability conditions leading to a novel clustering-based channel identification algorithm