When there are inter-harmonics in the grid signal, the fast detection using discrete spectrum analysis causes the accuracy to decrease due to the negative frequency spectrum leakage interference after the discrete Fourier transform(DFT). In order to effectively eliminate the negative frequency spectrum leakage interference, the analytical expression of the spectrum containing the negative frequency component is established, and the improved three-line interpolation DFT based on the maximum side-lobe attenuation window is derived to realize the fast and high-precision detection of inter-harmonics. The systematic error and noise sensitivity of the new algorithm are analyzed. The simulation results show that the new algorithm can not only effectively eliminate the negative frequency spectrum leakage interference, but also has strong robustness to additive white noise. When the inter-harmonics are lower than one frequency resolution, the new algorithm can still achieve high detection accuracy.