We propose a blind channel estimation method for space-time block-coded single-carrier with frequency-domain equalization (SC-FDE) systems based on a simple, nonredundant precoding. This method of precoding allows us to better structure the relation between the covariance matrices of the received data and the products of channel coefficients. Using this structure, we can easily obtain the products of channel coefficients by using only subtraction and division. Then each of the two channel impulse response vectors is obtained by computing the eigenvector that is associated with the maximal eigenvalue of a Hermitian matrix formed from the corresponding products of channel coefficients. The proposed framework gaurantees the unique recovery of all the products of channel coefficients for colored or white noise cases. The impact of the precoding on channel estimation and equalization is also discussed. Simulations are used to demonstrate the performance of the proposed method.