In order to solve the problem of low stability of quantum cascade laser (QCL) in low-concentration laughing gas (N₂O) detection, and further reduce the detection limit of concentration, this paper proposes an analytical method based on multi-cycle low-rank reconstruction. method. This method performs matrix reconstruction on the modulated absorption spectrum signal under high-speed sampling, extracts the low-rank features of the signal and removes redundant information, thereby improving the signal-to-noise ratio of the signal. Compared with the traditional orthogonal phase-locked amplification method, the standard deviation of the proposed method at 100 ppb concentration can be improved to 2.82, and within 10 seconds sampling time, R² and RMSE are improved to 0.931 and 0.024 respectively, which is a significant improvement. Accuracy and repeatability of low-concentration N₂O detection. This innovative method provides a new solution for the development of trace gas detection technology and is expected to be widely used in fields such as environmental monitoring and industrial safety.In order to solve the problems of poor repeatability and accuracy of orthogonal phase-locked amplification technology in low concentration detection, this paper developed a new QCL modulation signal analysis method. In view of the characteristics of stable period and obvious difference of the QCL wavelength modulation spectrum, this paper proposed The multi-period low-rank reconstruction analytical method can significantly reduce noise interference by mining the low-rank characteristics of the modulated signal. The sampling is based on the Alternating Least Squares (ALS) algorithm to estimate the maximum singular value of the matrix, and at the singular value Iterative optimization is implemented in the estimation to obtain the optimal singular value distribution curve, thereby achieving signal noise filtering.The normalized second harmonic peak of the signal under each gas concentration was calculated using the orthogonal phase-locked amplification method and the multi-cycle low-rank reconstruction analytical method respectively, and the repeatability of the two methods was verified through the standard deviation. The data selected here of the collection time is 2 minutes (Fig.7). The results show that the standard deviation based on the orthogonal lock-in amplification method is 21.13, while the standard deviation of the multi-cycle low-rank reconstruction analytical method is 2.82, showing a significant difference in stability between the two.This study proposes and verifies a N₂O trace concentration detection technology based on a multi-cycle low-rank reconstruction analysis method. The QCL detection system is used to achieve accurate and stable detection of N₂O in the mid-infrared band, and the detection sensitivity can reach 100 ppb. Compared with traditional orthogonal phase-locked amplification methods, our method exhibits significant advantages in stability and accuracy. Experimental results show that the standard deviation of the multi-period low-rank reconstruction analytical method is 2.82, which is much lower than the 21.13 of the orthogonal phase-locked amplification method, proving the method's ability to effectively suppress noise and improve the stability of detection results during the signal reconstruction process. In practical applications, the proposed multi-cycle low-rank reconstruction analytical method significantly improves detection accuracy while greatly reducing the required sampling time. In a sampling time of only 10 seconds, this method can achieve a coefficient of determination (R²) of 0.931 and a root mean square error (RMSE) of 0.024, demonstrating its fast and efficient detection capabilities.