Acoustic signals detection attracts considerable attention in many areas, such as structural healthy monitoring, biomedicine, communication and earthquake prediction. However, the useful signals are always very weak and overwhelmed by background noise. In this paper, we proposed a method based on the gradient acoustic-grating metamaterial (GAGM) to detect harmonic and periodic impulse signals for the first time. The analytic analysis indicates that the incident acoustic waves with various frequencies gradually slow down and eventually stop at different positions in the GAGM. In the numerical study, we show that acoustic rainbow trapping, realized by the GAGM, can spatially separate different frequency components accompanied by the local enhancement of their amplitudes. Moreover, in the experimental investigation, we demonstrate that the GAGM is capable of amplifying pressure amplitudes about 30 times even at different measured positions. With this acoustic system, the harmonic and periodic impulse signals submerged by background noise in the GAGM can be detected much more easily. All the results indicate that the GAGM can be regarded as a functional material or an enhanced acoustic sensing device to overcome the detection limit of conventional acoustic systems. This work opens up new vistas for weak signals detection of many areas.