Based on chirp transform and pulse compression technology, chirp transform spectrometers (CTSs) can be used to perform high-resolution and real-time spectrum measurements. Nowadays, they are widely applied for weather and astronomical observations. The surface acoustic wave (SAW) filter is a key device for pulse compression. The system performance is significantly affected by the dispersion characteristics match and the large insertion loss of the SAW filters. In this paper, a linear phase sampling and accumulating (LPSA) algorithm was developed to replace the matched filter for fast pulse compression. By selecting and accumulating the sampling points satisfying a specific periodic phase distribution, the intermediate frequency (IF) chirp signal carrying the information of the input signal could be detected and compressed. Spectrum measurements across the entire operational bandwidth could be performed by shifting the fixed sampling points in the time domain. A two-stage frequency resolution subdivision method was also developed for the fast pulse compression of the sparse spectrum, which was shown to significantly improve the calculation speed. The simulation and experiment results demonstrate that the LPSA method can realize fast pulse compression with adequate high amplitude accuracy and frequency resolution. Compared to existing digital pulse compression technology, this method can significantly reduce the number of required calculations, especially for measurements of sparse signals.