The concise fractional Fourier transform and its application in detection and parameter estimation of the linear frequency-modulated signal

The passive detection of the linear frequency-modulated (LFM) signal in low signal to noise ratio (SNR) condition is a significant problem. The concise fractional Fourier transform (CFRFT) is presented to solve this issue. The CFRFT only rotates the frequency axis in time-frequency plane to get the spectrum of the signal in different angles by chirp multiplication and Fourier transform (FT). For the LFM signal which distributes' as a straight line in time- frequency plane, the CFRFT in specific angle can gather its energy as a peak value to improve the SNR, thus the LFM signal can be detected. Meanwhile, the location of the peak value relates to the parameters of the LFM signal. Numerical simulation and experimental results show that, the proposed method can detect the LFM signal masked by the noise efficiently and estimate the signal's parameters accurately. Compared with the conventional fractional Fourier transform (FRFT), the CFRFT reduces the transform complexity to provide the faster computation speed. Hence, the proposed method has a better performance in real-time detection of the LFM signal.