Time-frequency analysis is commonly used for fault detection in induction motors. A variety of signal decomposition techniques have been proposed in the literature, such as Wavelet transform, Empirical Mode Decomposition (EMD), Multiple Signal Classification (MUSIC), among others. They have been successfully used in many works related with the topic. Nevertheless, the studied signals present amplitude changes and chirp-type frequency components that are difficult to track and isolate with the aforementioned techniques. The contribution of this work is the introduction of a novel technique for time-frequency signal decomposition that is based on an adaptive band-pass filter and the Short Time Fourier Transform (STFT), namely Fourier-Based Adaptive Signal Decomposition (FBASD) method. This method is capable of tracking and extracting nonstationary time-frequency components within a user-selected frequency band. With these components, a methodology for detecting and classifying broken rotor bars in induction motors using the startup transient current is also proposed.