Millimeter wave radar has become a research hotspot in vital sign monitoring due to its advantages of non-contact operation, high penetration, high precision, and real-time capability. It has already been applied in medical cardiovascular health assessment, tumor detection and localization, and health monitoring for newborns and children. However, during indoor monitoring, the millimeter wave radar echo signals are susceptible to subtle human body movements and multipath effects, resulting in various noise components that complicate the extraction of heartbeat information during sleep. This paper proposes a vital sign detection method for millimeter wave radar based on Empirical Wavelet Transform and Variational Mode Decomposition (EWT-VMD) for detecting respiratory and heartbeat signals from the echo signals. By establishing a vital sign echo signal model for millimeter wave radar detection, the frequency composition of the signal is analyzed. Experiments are conducted using a 77 GHz millimeter-wave radar to acquire echo signals, and frequency-domain coherent accumulation is applied to two types of chirp signals within the same signal frame, which simultaneously suppresses noise interference and enhances the amplitude of useful signals. The empirical wavelet transform (EWT) is employed to decompose the vital sign-containing signal, followed by signal reconstruction after clutter elimination. The wavelet-reconstructed signal is further processed by variational mode decomposition (VMD) to extract the human respiratory rate (RR) and heart rate (HR) respectively. The extracted results are compared with the signals collected by an electrocardiogram (ECG) to assess the extraction accuracy. Experimental results across various scenarios demonstrate that the proposed frequency-domain coherent accumulation method effectively enhances echo signal SNR. The combined CWT-VMD algorithm successfully detects RR and HR with a detection accuracy reaching 94%.