Relying on nonredundant diagonal precoding and independent and identically distributed (i.i.d.) source assumption, this paper proposes 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 received 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 for improving solution accuracy when imperfect data estimation occurs. We consider two models of data mismatch, from both deterministic and statistical points of view, and propose the associated design criteria. The optimization problems are formulated to take advantage of the BCCB system matrix property and are solved analytically. The proposed optimal precoder aims to optimize solution robustness against deterministic error perturbation and also minimize the mean-square error when the data mismatch is modeled as a white noise. Pairwise error probability analysis is conducted for investigating the equalization performance. Numerical examples are used to illustrate the performance of the proposed method.