Vital sign detection using linear frequency-modulated continuous-wave (LFMCW) radar may be subject to the proximity stationary clutters. This paper presents a novel technique to synthesize the slow-time I/Q signals, which are equivalent to those in a single tone quadrature CW radar, from a single-channel LFMCW radar. It correlates the two types of radars in such a way that the proximity stationary clutters are translated to direct current (DC) offsets in the synthesized I/Q signals across slow-time. The circle-fitting based DC offsets calibration (DCcal) technique, which was developed for CW radar, can now be applied to eliminate the impact of the proximity stationary clutters in LFMCW radars for accurate vital sign detection. Moreover, the modified differentiate and cross-multiply (MDACM) algorithm can also be leveraged to eliminate the phase ambiguity issue. Thorough theoretical analysis and working principles are presented. Simulations are performed to validate the proposed technique. Moreover, exhaustive experiments are carried out with a millimeter-wave 79 GHz FMCW radar in the office environment. Mechanical vibration and vital signs are extracted with micrometer-level accuracy in the existence of proximity stationary clutters.