Recently, convolutional neural networks (CNNs) for classification by time domain data of multi-signals have been developed. Although some signals are important for correct classification, others are not. The calculation, memory, and data collection costs increase when data that include unimportant signals for classification are taken as the CNN input layer. Therefore, identifying and eliminating non-important signals from the input layer are important. In this study, we proposed a features gradient-based signals selection algorithm (FG-SSA), which can be used for finding and removing non-important signals for classification by utilizing features gradient obtained by the process of gradient-weighted class activation mapping (grad-CAM). When we defined $n_{\mathrm{s}}$ as the number of signals, the computational complexity of FG-SSA is the linear time $\mathcal{O}(n_{\mathrm{s}})$ (i.e., it has a low calculation cost). We verified the effectiveness of the algorithm using the OPPORTUNITY dataset, which is an open dataset comprising of acceleration signals of human activities. In addition, we checked the average of 6.55 signals from a total of 15 signals (five triaxial sensors) that were removed by FG-SSA while maintaining high generalization scores of classification. Therefore, FG-SSA can find and remove signals that are not important for CNN-based classification. In the process of FG-SSA, the degree of influence of each signal on each class estimation is quantified. Therefore, it is possible to visually determine which signal is effective and which is not for class estimation. FG-SSA is a white-box signal selection algorithm because it can understand why the signal was selected. The existing method, Bayesian optimization, was also able to find superior signal sets, but the computational cost was approximately three times greater than that of FG-SSA. We consider FG-SSA to be a low-computational-cost algorithm.