This study introduces a processing method based on multi-resolution analysis (MRA) to characterize the multi-scale structures of gravity waves (GWs) with vertical wavelengths less than 13 km in lidar vertical profiles of temperature and wind in the middle atmosphere. The MRA approach is evaluated against conventional techniques, including polynomial fitting, spectral filtering, and nighttime temporal averaging, and applied to a case study of GWs observed on November 20, 2023. Among these methods, MRA demonstrates superior performance by enhancing the signal-to-noise ratio through signal decomposition and selective filtering. This targeted filtering improves the detection and extraction of GW-induced perturbations, particularly for dominant vertical wavelengths around 5 km. In terms of GW potential energy (GWPE), the MRA-based method yields values comparable to those derived from the variance method, except at the stratopause, where it estimates nearly twice the GWPE. However, the variance-based estimate remains within the MRA-derived confidence interval, indicating good agreement. In contrast, the Butterworth low-pass filter produces energy densities an order of magnitude higher than the variance method, suggesting possible overestimation of perturbation amplitudes. Polynomial fitting and nighttime mean methods appear insensitive to small-scale GW structures near the stratopause, where wave dissipation may occur. Beyond energy estimation, the MRA method offers a distinct advantage for analyzing GW propagation and scale interactions due to its multi-scale decomposition capability. It reveals GW features and structures that remain obscured by common techniques, establishing it as a valuable tool for advancing the study of GW dynamics in the middle atmosphere.