In this letter, a necessary and sufficient condition is developed to check whether a set of polynomial equations is solvable. Furthermore, under the sufficient condition, a closed-form solution is attained using linear Diophantine equation theory. With this, a novel signal design and a row-circular transmission scheme of phase shift keying (PSK) constellations for MIMO flat fading channels are proposed so that the channel coefficients can be uniquely identified. When certain high-order moments of the received signals are available, a closed-form solution to uniquely determine the channel coefficients is given. When only finite received data are available, a numerical algorithm is provided to efficiently and effectively estimate the channel.