Sleep apnea (SA) is a significant respiratory condition that poses a major global health challenge. Deep Learning (DL) has emerged as an efficient tool for the classification problem in electrocardiogram (ECG)-based SA diagnoses. Despite these advancements, most common conventional feature extractions derived from ECG signals in DL, such as R-peaks and RR intervals, may fail to capture crucial information encompassed within the complete ECG segments. In this study, we propose an innovative approach to address this diagnostic gap by delving deeper into the comprehensive segments of the ECG signal. The proposed methodology draws inspiration from Matrix Profile algorithms, which generate an Euclidean distance profile from fixed-length signal subsequences. From this, we derived the Min Distance Profile (MinDP), Max Distance Profile (MaxDP), and Mean Distance Profile (MeanDP) based on the minimum, maximum, and mean of the profile distances, respectively. To validate the effectiveness of our approach, we use the modified LeNet-5 architecture as the primary CNN model, along with two existing lightweight models, BAFNet and SE-MSCNN. Our experiment results on the PhysioNet Apnea-ECG dataset (70 overnight recordings), and the UCDDB dataset (25 overnight recordings) revealed that our new feature extraction method achieved per-segment accuracies of up to 92.11% and 81.25%, respectively. Moreover, using the PhysioNet data, we achieved a per-recording accuracy of 100% and yielded the highest correlation of 0.989 compared to state-of-the-art methods. By introducing a new feature extraction method based on distance relationships, we enhanced the performance of certain lightweight models in DL, showing potential for home sleep apnea test (HSAT) and SA detection in IoT devices. The source code for this work is made publicly available in GitHub: https://github.com/vinuni-vishc/MPCNN-Sleep-Apnea.