Through analyzing the relationship of function in wavelet transforms and the wavelet data, and meeting the conditions of the admissibility of Morlet wavelet, Marr wavelet, DOG (difference of Gaussian) wavelet, Haar wavelet and Daubechies wavelet, it is challenging to simultaneously achieve the smoothness and compactness. Moreover, it is difficult to find a suitable wavelet basis function for acoustic emission (AE) signals of rocks by wavelet analysis. According to the characteristics of AE signals of rocks, the impulse function of the 2nd-order underdamp vibration unit pulse is used as the basic function of the wavelet analysis. In order to fit the AE signals, three tectonic conditions are proposed, and the optimization is proved by each other. Finally, the wavelet based functions with rock characteristic parameters are constructed. It is proved that the new wavelet basic function has substantial advantages in comparison with normal wavelet based functions by dealing with AE signals of rocks. This method provides a theoretical foundation for the application of wavelet analysis in the field of rock acoustic emission.