Measurements acquired from batteries, such as voltage-time and capacity-time signals, offer valuable insights into their cyclability performance. While analyzing these signals in the time domain provides an understanding of global characteristics, scrutinizing local behavior is crucial for detecting issues like battery fading. In this context, we delve into the application of short-time Fourier transform for time-frequency deconstruction of galvanostatic charge/discharge signals, using lithium-sulfur batteries as an illustrative example. Spectrograms resulting from this analysis unveil the evolving frequency content of signals throughout the battery's lifespan, facilitating the identification of critical changes in the response and early detection of issues leading to fading and potential default. Simultaneously, the stability analysis of the dynamic response of red phosphorus and sulfide polyacrylonitrile (RP-SPAN) composite, a promising anode material for lithium batteries, can also be studied using Fourier transform analysis. Our study employs transfer function stability analysis, Kramers-Kronig integral relations, and differential capacity analysis to assess the cell's behavior in both frequency and time domains. Parameters such as stationarity, stability, linearity, dissipation, and degradation are scrutinized over extended charge/discharge cycles. Results reveal a highly nonlinear and time-variant system at low frequencies, aligning with the observed 0.21% average capacity loss per cycle. Our results indicate that short-time Fourier transform could provide insightful information for analyzing battery health.