We apply the efficient inverse Cholesky factorization to Alamouti matrices in Groupwise Space-time Block Code (G-STBC) and Alamouti-like matrices in Orthogonal Matching Pursuit (OMP) for the sub-Nyquist sampling system. By utilizing some good properties of Alamouti or Alamouti-like matrices, we save about half the complexity. Then we propose the whole square-root algorithms for G-STBC and OMP, respectively. The proposed square-root G-STBC algorithm has an average speedup of 2.96~3.6 with respect to the sub-optimal G-STBC algorithm. On the other hand, when comparing the complexities of all the steps except the projection step, the complexity for the proposed square-root OMP algorithm is about 30% of that for the fast OMP algorithm by matrix inverse update.