Relying on non-redundant diagonal precoding and i.i.d. source assumption, we propose a blind channel estimation scheme for single-carrier frequency-domain equalization based space-time block coded systems. The proposed method exploits the precoding-induced linear signal structure in the conjugate cross-correlation between the two temporal block receive signals as well as the circulant channel matrix property, and can yield exact solutions whenever the channel noise is circularly Gaussian and the receive data statistic is perfectly obtained. The channel estimation formulation builds on rearranging the set of linear equations relating the entries of conjugate cross-correlation matrix and products of channel impulse responses into one with a distinctive block-circulant with circulant-block (BCCB) structure. This allows a simple identifiability condition depending on precoder parameters alone, and also provides a natural yet effective optimal precoder design framework against imperfect data estimation. We consider the data mismatch as a deterministic perturbation and propose to minimize the condition number of the BCCB system matrix, leading to a simple closed-form optimal solution. Numerical examples are used to illustrate the performance of the proposed method.