In this paper, the filter- and frequency-to-time mapping (FTTM)-based photonics-assisted time and frequency acquisition methods are comprehensively analyzed and the accuracy and resolution limitation in the fast sweep scenario is broken by broadening the filter bandwidth. It is found that when the sweep speed is very fast, the width of the generated pulse via FTTM is mainly determined by the impulse response of the filter. In this case, appropriately increasing the filter bandwidth can significantly reduce the pulse width, so as to improve the measurement accuracy and resolution. FTTM-based short-time Fourier transform (STFT) and microwave frequency measurement using the stimulated Brillouin scattering (SBS) effect is demonstrated by comparing the results with and without SBS gain spectrum broadening and the improvement of measurement accuracy and frequency resolution is well confirmed. The frequency measurement accuracy of the system is improved by around 25 times compared with the former work using a similar sweep speed, while the frequency resolution of the STFT is also much improved compared with our former results.