Self-attention (SA), which encodes vector sequences according to their pairwise similarity, is widely used in speech recognition due to its strong context modeling ability. However, when applied to long sequence data, its accuracy is reduced. This is caused by the fact that its weighted average operator may lead to the dispersion of the attention distribution, which results in the relationship between adjacent signals ignored. To address this issue, in this paper, we introduce relative-position-awareness self-attention (RPSA). It not only maintains the global-range dependency modeling ability of self-attention, but also improves the localness modeling ability. Because the local window length of the original RPSA is fixed and sensitive to different test data, here we propose Gaussian-based self-attention (GSA) whose window length is learnable and adaptive to the test data automatically. We further generalize GSA to a new residual Gaussian self-attention (resGSA) for the performance improvement. We apply RPSA, GSA, and resGSA to Transformer-based speech recognition respectively. Experimental results on the AISHELL-1 Mandarin speech recognition corpus demonstrate the effectiveness of the proposed methods. For example, the resGSA-Transformer achieves a character error rate (CER) of 5.86% on the test set, which is relative 7.8% lower than that of the SA-Transformer. Although the performance of the proposed resGSA-Transformer is only slightly better than that of the RPSA-Transformer, it does not have to tune the window length manually.