  In this paper, we develop two kinds of novel closed-form decompositions on
phase shift keying (PSK) constellations by exploiting linear congruence
equation theory: the one for factorizing a $pq$-PSK constellation into a
product of a $p$-PSK constellation and a $q$-PSK constellation, and the other
for decomposing a specific complex number into a difference of a $p$-PSK
constellation and a $q$-PSK constellation. With this, we propose a simple
signal design technique to blindly and uniquely identify frequency selective
channels with zero-padded block transmission under noise-free environments by
only using the first two block received signal vectors. Furthermore, a
closed-form solution to determine the transmitted signals and the channel
coefficients is obtained. In the Gaussian noise and Rayleigh fading
environment, we prove that the newly proposed signaling scheme enables
non-coherent full diversity for the Generalized Likelihood Ratio Test (GLRT)
receiver.
