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In this paper, we develop two kinds of 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 p- and q-PSK constellations and the other for decomposing a specific complex number into a difference of a point in p-PSK constellation and a point in q-PSK constellation. With this, we present a novel and simple signal design technique to blindly and uniquely identify frequency selective channels with zero-padded block transmission by only processing the first two block received signals. In a noise-free case, a closed-form solution to determine the transmitted signals and the channel coefficients is obtained. In a Gaussian noise and Rayleigh fading environment, we prove that our scheme enables full diversity for the generalized likelihood ratio test (GLRT) receiver. When only finite received data are given, the linearity of our signal design allows us to use iterative sphere decoders to approximate GLRT detection so that the joint estimation of the channel and symbols can be efficiently implemented.