In nature and the technology world, acquired signals and time series are usually affected by multiple complicated factors and appear as multi-component non-stationary modes. In many situations it is necessary to separate these signals or time series to a finite number of mono-components to represent the intrinsic modes and underlying dynamics implicated in the source signals. Recently the synchrosqueezed transform (SST) was developed as an empirical mode decomposition (EMD)-like tool to enhance the time-frequency resolution and energy concentration of a multi-component non-stationary signal and provides more accurate component recovery. To recover individual components, the SST method consists of two steps. First the instantaneous frequency (IF) of a component is estimated from the SST plane. Secondly, after IF is recovered, the associated component is computed by a definite integral along the estimated IF curve on the SST plane. The reconstruction accuracy for a component depends heavily on the accuracy of the IFs estimation carried out in the first step. More recently, a direct method of the time-frequency approach, called signal separation operation (SSO), was introduced for multi-component signal separation. While both SST and SSO are mathematically rigorous on IF estimation, SSO avoids the second step of the two-step SST method in component recovery (mode retrieval). The SSO method is based on some variant of the short-time Fourier transform. In the present paper, we propose a direct method of signal separation based on the adaptive continuous wavelet-like transform (CWLT) by introducing two models of the adaptive CWLT-based approach for signal separation: the sinusoidal signal-based model and the linear chirp-based model, which are derived respectively from sinusoidal signal approximation and the linear chirp approximation at any time instant. A more accurate component recovery formula is derived from linear chirp local approximation. We present the theoretical analysis of our approach. For each model, we establish the error bounds for IF estimation and component recovery.

在自然界和技术界，采集的信号和时间序列通常受多种复杂因素的影响，并以多分量非平稳模式出现。在许多情况下，有必要将这些信号或时间序列分离为有限数量的单分量，以表示源信号中蕴含的固有模式和基础动力学。最近，同步压缩变换（SST）被开发为类似于经验模式分解（EMD）的工具，以增强多分量非平稳信号的时频分辨率和能量集中，并提供更准确的分量恢复。要恢复单个组件，SST方法包括两个步骤。首先，从SST平面估算组件的瞬时频率（IF）。其次，中频恢复后 相关分量由沿SST平面上估计的IF曲线的定积分计算。组件的重构精度在很大程度上取决于第一步中执行的IF估计的精度。最近，引入了时频方法的直接方法，称为信号分离操作（SSO），用于多分量信号分离。尽管SST和SSO都在数学上严格地进行IF估计，但SSO避免了组件恢复（模式检索）中两步SST方法的第二步。SSO方法基于短时傅立叶变换的某些变体。在本文中，我们通过介绍两种基于自适应CWLT的信号分离方法模型，提出了一种基于自适应连续小波变换（CWLT）的直接信号分离方法：基于正弦信号的模型和基于线性chi的模型，它们分别从任意时刻的正弦信号近似和线性chi近似推导而来。从线性线性调频局部逼近可以得出更准确的组分回收公式。我们介绍了我们的方法的理论分析。对于每个模型，我们为IF估计和组件恢复建立误差范围。