Sleep staging is essential for the assessment of sleep quality and the diagnosis of sleep-related disorders. Conventional polysomnography (PSG), while considered the gold standard, is intrusive, labor-intensive, and unsuitable for long-term monitoring. This study evaluates the performance of the Sleepal AI Lamp, a contactless, radar-based consumer-grade sleep tracker, in comparison with gold-standard polysomnography (PSG), using a large-scale dataset comprising 1022 overnight recordings. We extract multi-scale respiratory and motion-related features from radar signals to train a frequency-augmented deep learning model. For the binary sleep-wake classification task, experimental results demonstrated that the model achieved an accuracy of 92.8% alongside a macro-averaged F1 score of 0.895. For four-stage classification (wake, light NREM (N1 + N2), deep NREM (N3), REM), the model achieved an accuracy of 78.5% with a Cohen's kappa coefficient of 0.695 in healthy individuals and maintained a stable accuracy of 77.2% with a kappa of 0.677 in a heterogeneous population including patients with varying severities of obstructive sleep apnea (OSA). These experimental results demonstrate that the sleep staging performance of the contactless Sleepal AI Lamp is in high agreement with expert-labeled PSG sleep stages. Our findings suggest that non-contact radar sensing, combined with advanced temporal modeling, can provide reliable sleep staging performance without requiring physical contact or wearable devices. Owing to its unobtrusive nature, ease of deployment, and robustness to long-term use, the contactless Sleepal AI Lamp shows strong potential for clinical screening, home-based sleep assessment, and continuous longitudinal sleep monitoring in real-world medical and healthcare applications.