Abstract: The performance loss of conventional Grassmannian codebook is analyzed. According to the analysis of unitary perturbations effects on channel capacity as well as the BER, a novel hierarchical codebook consisting of a Grassmannian main-codebook and a perturbation subcodebook proposed for multi-antenna systems with linear receivers is established, and corresponds with the two-step codeword selection scheme that uses the product of two codewords respectively selected from the main-codebook and the sub one as the ultimate precoder. Specifically to the design of subcodebook, two schemes is brought forward: rotation-based and householder transformation-based, and the superiority of the proposed codebook in terms of both the channel capacity and BER is further confirmed by computer simulations.