The conventional frequency modulated continuous wave (FMCW) radar accuracy range detection algorithm is based on the frequency estimation and additional phase evaluation which contains Fourier transform and frequency refining analysis in each chirp, so it has the disadvantages of being computationally expensive, and not being suitable for real-time motion measurement. In addition, if there are other objects near the target, the spectra of the clutter and the target will be adjacent and affect each other, making it more challenging to estimate the frequency of the target. In this paper, the analytical expression of the Fourier transform of the beat signal is presented and it can be seen that spectrum leakage makes the phase of Fourier transform no longer consistent with the real phase of signal. The change regularities of real and imaginary parts of Fourier transform are studied, and the corrected phase of ellipse approximation is given in the industrial, scientific, and medical (ISM) band. Accurate displacement can be obtained by accurate phase. The algorithm can filter the direct current (DC) offset which is mainly caused by stationary objects. The performance of the algorithm is evaluated by a radar system whose center frequency is 24.075 GHz and the bandwidth is 0.15 GHz; the measurement accuracy of displacement is 0.087 mm and the accuracy of distance is 0.043 m.

<sec>Real time detection of object motion is widely used in industrial activities and daily life. The contactless measurement is a flexible way, which has no effect on the state of movement of the object. Compared with the optical, ultrasonic and laser sensors, microwave radar has the advantages of high measurement accuracy and being unaffected by the environment such as smoke, dust, fog, and rain. </sec><sec>The frequency modulated continuous wave (FMCW) radar is a widely used radar system, the echo of which contains abundant information, and there is no blind zone in the range because the transmitter and receiver work at the same time. The algorithm of movement detection of FMCW radar is commonly based on the peak estimation of signal spectrum, in order to achieve high accuracy, it is necessary to increase the frequency and bandwidth, resulting in high hardware complexity, a large amount of calculation, poor real-time response and poor anti-jamming ability. The proposed algorithm is based on the discrete Fourier transform with specific frequency of the beat signal. The real part and imaginary part of discrete Fourier transform are superposed in two perpendicular directions, and the resultant trajectory is approximately elliptical. The relative displacement of the object is proportional to the cumulative phase change of the corresponding points on the ellipse, the phase of each trajectory point can be calculated to restore the motion state of the object. The proposed algorithm does not need Fourier transform for the beat signal of each chirp, so the time complexity is low. The beat signal of the static object is processed into a fixed direct-current signal, which has no influence on the measurement of the moving object, therefore the algorithm has the ability to resist the interference of the static object. The measurement is limited to relative motion, because the phase obtained is relative. It has great potential applications in the fields of measuring relative displacement, such as mechanical vibration frequency, vital signal detection, mechanical arm control, etc.. </sec><sec>An experimental setup with a center frequency of 24 GHz, bandwidth of 0.15 GHz and frequency modulation period of 4 ms is used to test the hypothesis. The experimental results are in good agreement with the theoretical results. The displacement measurement accuracy is 0.27 mm, and the linearity is 0.05% with 500 mm as the displacement measurement range. The measurement accuracy of velocity is 1.11 mm/s.</sec>