A simple proof of a known blind channel identifiability result

An alternative simple proof for a well-known result encountered in blind identification and equalization of communication channels is derived. It is shown that the coprimeness condition of a number of FIR channels is equivalent to the full column rank condition of a certain block Sylvester matrix, whose entries are the channel coefficients. As a corollary, it is also shown that in general, the null space of a block Sylvester matrix is spanned by the set of (generalized) Vandermonde vectors associated with the common zeros present in all the channels.