## 1 Introduction  
In nature, a large number of nonlinear chirp (NC) signals are generated every day, such as human communication sounds, and the ultrasonic waves from bats, dolphins, and whales, etc. [1; 2; 3; 4]. In the radar system [5; 6], if the sinusoidal signal is transmitted and the target is in uniform-acceleration motion, the echo signal can be approximately regarded as the NC signal. In communication systems [7; 8; 9], some clutter and interference can be regarded as NC signals. As NC signals and their parameters have practical physical meaning in many fields, the research on NC signals detection and parameter estimation is of great significance. Most decomposition methods extract each signal mode directly in the time domain, of which empirical mode decomposition (EMD) [10; 11; 12; 13] is the most famous one. EMD is a completely data-driven method that can adaptively decompose complex NCs into simple modes. Although EMD is sensitive to noise and lacks mathematical models, it has been widely used in various fields. On the basis of EMD, the methods of EMD and ensemble empirical mode decomposition (EEMD) have been proposed;however, these approaches only address part of the problem, and some challenging issues remain to be addressed. For instance, these improved methods based on EMD have a poor resolution for split-off mode. To further understand the characteristics of NC signals, it is necessary to extract each nonlinear chirp model from the signal data. In the past few decades, many methods have been developed to decompose NCs. These methods implement signal decomposition in the time, frequency, and time-frequency (TF) domains. From the TF plane, we can observe time-varying features and perform signal decomposition. However, due to the limitation of the Heisenberg uncertainty principle, the TF representation generated by conventional methods is usually fuzzy, and it is impossible to provide accurate TF representation for time-varying signals. The disadvantages of traditional TF methods severely limit their application in actual signal processing. In the past decades, TF has gradually developed many advanced methods, such as the reassignment method (RM) [14; 15], synchrosqueezing transform (SST) [16; 17], demodulated SST (DSST) [18; 19], and high-order SST [20; 21]. RM technology stimulates post-processing of traditional TF methods and is an effective way to obtain clearer TF results. However, the RM framework is based on a spectrogram, which means that RM results lose the ability to reconstruct the signal. The SST method not only enhances the TF resolution but also allows the signal to be reconstructed. On the contrary, when processing signals that change over time (such as a linear frequency-modulated signal or a non-linear frequency-modulated (FM) signal), SST cannot generate a concentrated TF result. The time-frequency transform maps the chirp signal to the time-frequency domain. It is easy to observe the rule of frequency changes with time, which is conducive to signal detection and parameter estimation. The time-frequency transform can be used to describe the energy density of the signal at different times and frequency by constructing the joint function of time and frequency. At present, time-frequency transforms commonly used are the short-time Fourier transform (STFT) [22; 23; 24; 25], Wigner-Ville distribution (WVD) [26; 27], fractional Fourier transform [28; 29], cubic phase function [30], and Zak transform (ZT) [31; 32]. In addition, as the chirp signal usually presents an impact function of frequency linearly varying with time in the time and frequency domain, the linear detection and extraction method in image processing can be combined with the commonly used time-frequency transformation to estimate the initial frequency and frequency modulation slope. [PERSON] [33] applied the VMD method to nonlinear chirp signals and achieved good results. However, as it processes the signals based on VMD, it is still limited to the determination of the number of signal decomposition layers.