Studies from the literature show that the prevalence of sleep disorder in children is far higher than that in adults. Although much research effort has been made on sleep stage classification for adults, children have significantly different characteristics of sleep stages. Therefore, there is an urgent need for sleep stage classification targeting children in particular. Our method focuses on two issues: The first is timestamp-based segmentation (TSS) to deal with the fine-grained annotation of sleep stage labels for each timestamp. Compared to this, popular sliding window approaches unnecessarily aggregate such labels into coarse-grained ones. We utilize DeConvolutional Neural Network (DCNN) that inversely maps features of a hidden layer back to the input space to predict the sleep stage label at each timestamp. Thus, our DCNN can yield better classification performances by considering labels at numerous timestamps. The second issue is the necessity of multiple channels. Different clinical signs, symptoms or other auxiliary examinations could be represented by different Polysomnography (PSG) recordings, so all of them should be analyzed comprehensively. We therefor exploit multivariate time-series of PSG recordings, including 6 electroencephalograms (EEGs) channels, 2 electrooculograms (EOGs) channels (left and right), 1 electromyogram (chin EMG) channel and two leg electromyogram channels. Our DCNN-based method is tested on our SDCP dataset collected from child patients aged from 5 to 10 years old. The results show that our method yields the overall classification accuracy of 84.27% and macro F1-score of 72.51% which are higher than those of existing sliding window-based methods. One of the biggest advantages of our DCNN-based method is that it processes raw PSG recordings and internally extracts features useful for accurate sleep stage classification. We examine whether this is applicable for sleep data of adult patients by testing our method on a well-known public dataset Sleep-EDFX. Our method achieves the average overall accuracy of 90.89% which is comparable to those of state-of-the-art methods without using any hand-crafted features. This result indicates the great potential of our method because it can be generally used for timestamp-level classification on multivariate time-series in various medical fields. Additionally, we provide source codes so that researchers can reproduce the results in this paper and extend our method.