CN117743817A - Multiple time-frequency synchronous extrusion method based on short-time fractional Fourier transform - Google Patents

Multiple time-frequency synchronous extrusion method based on short-time fractional Fourier transform Download PDF

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CN117743817A
CN117743817A CN202311616849.7A CN202311616849A CN117743817A CN 117743817 A CN117743817 A CN 117743817A CN 202311616849 A CN202311616849 A CN 202311616849A CN 117743817 A CN117743817 A CN 117743817A
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fourier transform
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fractional fourier
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史军
门子俊
石硕
郭庆
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Harbin Institute of Technology Shenzhen
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Abstract

本发明提出一种基于短时分数阶傅里叶变换的多重时频同步挤压方法,所述方法在信号能量最佳聚集的分数阶傅里叶变换域对应的角度下,计算信号的短时分数阶傅里叶变换,然后对计算结果进行复正弦函数调制,并以此获得信号的瞬时频率估计,进而计算出多重同步挤压的瞬时频率估计。然后,将信号的能量挤压到多重瞬时频率估计在时频面上所确定的时频点上,得到基于短时分数阶傅里叶变换的多重时频同步挤压结果。与基于传统短时傅里叶变换的多重时频同步挤压方法相比,基于短时分数阶傅里叶变换的多重时频同步挤压方法通过自由参数α的选择,可以进一步提升时频分析的分辨率,能够获得信号高分辨率的时频分析结果。

The present invention proposes a multiple time-frequency synchronous squeezing method based on short-time fractional Fourier transform. The method calculates the short-time signal at an angle corresponding to the fractional Fourier transform domain where the signal energy is optimally gathered. Fractional Fourier transform is performed, and then the calculation result is modulated by a complex sinusoidal function to obtain the instantaneous frequency estimate of the signal, and then the instantaneous frequency estimate of the multiple simultaneous squeezes is calculated. Then, the energy of the signal is squeezed to the time-frequency points determined by the multiple instantaneous frequency estimates on the time-frequency plane, and the multiple time-frequency synchronous squeeze results based on the short-time fractional Fourier transform are obtained. Compared with the multiple time-frequency synchronous squeezing method based on the traditional short-time Fourier transform, the multiple time-frequency synchronous squeezing method based on the short-time fractional Fourier transform can further improve the time-frequency analysis through the selection of the free parameter α. The resolution can obtain high-resolution time-frequency analysis results of the signal.

Description

一种基于短时分数阶傅里叶变换的多重时频同步挤压方法A multiple time-frequency synchronous extrusion method based on short-time fractional Fourier transform

技术领域Technical field

本发明属于信号与信息处理技术领域,尤其涉及一种基于短时分数阶傅里叶变换的多重时频同步挤压方法。The invention belongs to the technical field of signal and information processing, and in particular relates to a multiple time-frequency synchronous extrusion method based on short-time fractional Fourier transform.

背景技术Background technique

信号是信息的载体,从信号中获取有用信息是信号处理的基本任务。平稳信号处理理论与技术奠定了通信、雷达、导航、探测等各种电子信息系统的基础,推动了电子信息领域核心技术的突破。然而,随着应用范围的不断推广,非平稳信号处理逐渐成为制约电子信息系统进一步提升性能的瓶颈。Signals are the carrier of information, and obtaining useful information from signals is the basic task of signal processing. The theory and technology of stationary signal processing lay the foundation for various electronic information systems such as communications, radar, navigation, and detection, and promoted breakthroughs in core technologies in the field of electronic information. However, with the continuous promotion of application scope, non-stationary signal processing has gradually become a bottleneck restricting the further improvement of the performance of electronic information systems.

频率时变是非平稳信号的典型特征。一种简单有效分析非平稳信号的方法便是短时傅里叶变换,其提供一种在时频平面上描述信号的方式,能够展示出随时间变化信号频率的变化情况。然而,短时傅里叶变换的时频分辨率取决于其在时频平面上所确定的时频窗的大小。由于受到不确定性原理的限制,短时傅里叶变换的时频分辨率存在相互制约的矛盾。为此,在短时傅里叶变换的基础上时频同步挤压方法便应运而生。该方法要求在短时傅里叶变换所确定的时频面上信号各个分量成分相互分离的,彼此不存在交叠。由于短时傅里叶变换的时频分辨率较低,在实际应用中通常难于满足这一前提要求。鉴于此,本发明将提出一种基于短时分数阶傅里叶变换的多重时频同步挤压方法。Time-varying frequency is a typical characteristic of non-stationary signals. A simple and effective method for analyzing non-stationary signals is the short-time Fourier transform, which provides a way to describe signals on the time-frequency plane and can show changes in signal frequency over time. However, the time-frequency resolution of the short-time Fourier transform depends on the size of the time-frequency window determined on the time-frequency plane. Due to the limitation of the uncertainty principle, the time-frequency resolution of short-time Fourier transform has mutual constraints. For this reason, the time-frequency synchronous extrusion method came into being based on the short-time Fourier transform. This method requires that the various components of the signal on the time-frequency plane determined by the short-time Fourier transform are separated from each other and do not overlap with each other. Due to the low time-frequency resolution of short-time Fourier transform, it is usually difficult to meet this prerequisite in practical applications. In view of this, the present invention will propose a multiple time-frequency synchronous extrusion method based on short-time fractional Fourier transform.

发明内容Contents of the invention

本发明目的是为了解决现有技术中的问题,提出了一种基于短时分数阶傅里叶变换的多重时频同步挤压方法。The purpose of the present invention is to solve the problems in the prior art by proposing a multiple time-frequency synchronous extrusion method based on short-time fractional Fourier transform.

本发明是通过以下技术方案实现的,本发明提出一种基于短时分数阶傅里叶变换的多重时频同步挤压方法,所述方法包括以下步骤:The present invention is realized through the following technical solutions. The present invention proposes a multiple time-frequency synchronous extrusion method based on short-time fractional Fourier transform. The method includes the following steps:

步骤一、给定待分析信号即线性调频信号f(t),计算其分数阶傅里叶变换Fα(u),其中角度取值范围为α∈(0,2π];Step 1. Given the signal to be analyzed, that is, the linear frequency modulation signal f(t), calculate its fractional Fourier transform F α (u), where the angle range is α∈(0,2π];

步骤二、确定线性调频信号f(t)能量最佳聚集分数阶傅里叶变换域的最佳角度αopt,即Step 2: Determine the optimal angle α opt in the fractional Fourier transform domain where the energy of the chirp signal f(t) optimally gathers, that is

步骤三、选择窗函数g(t),满足其中/>计算线性调频信号f(t)能量最佳聚集分数阶傅里叶变换域对应角度αopt下的短时分数阶傅里叶变换,即Step 3. Select the window function g(t) to satisfy Among them/> Calculate the short-time fractional Fourier transform of the linear frequency modulation signal f(t) energy at the optimal accumulation fractional Fourier transform domain corresponding to the angle α opt , that is

步骤四、将线性调频信号f(t)能量最佳聚集角度αopt下的短时分数阶傅里叶变换与复正弦函数ejtucscα进行相乘,即Step 4: Multiply the short-time fractional Fourier transform at the optimal energy concentration angle α opt of the linear frequency modulation signal f(t) with the complex sine function e jtucscα , that is

步骤五、根据步骤四中的乘积计算线性调频信号f(t)能量最佳聚集角度αopt下瞬时频率的估计/>Step 5. According to the product in step 4 Calculate the estimation of the instantaneous frequency at the optimal energy concentration angle α opt of the chirp signal f(t)/> Right now

步骤六、根据步骤五中的计算线性调频信号f(t)能量最佳聚集角度αopt下瞬时频率的估计联合时间t和频率ω的表示/>Step 6. According to the steps in Step 5 Calculate the estimation of the instantaneous frequency at the optimal energy concentration angle α opt of the chirp signal f(t) and express the combined time t and frequency ω/> Right now

步骤七、计算线性调频信号f(t)能量最佳聚集角度αopt下,第一重同步挤压短时分数阶傅里叶变换,即Step 7: Calculate the first synchronization squeeze short-time fractional Fourier transform under the optimal energy gathering angle α opt of the linear frequency modulation signal f(t), that is

式中, In the formula,

步骤八、计算线性调频信号f(t)能量最佳聚集角度αopt下,第二重同步挤压短时分数阶傅里叶变换,即Step 8: Calculate the second synchronous squeeze short-time fractional Fourier transform under the optimal energy gathering angle α opt of the linear frequency modulation signal f(t), that is

步骤九、重复步骤八,计算线性调频信号f(t)能量最佳聚集角度αopt下,第N重同步挤压短时分数阶傅里叶变换,即Step 9. Repeat step 8 to calculate the Nth synchronous squeeze short-time fractional Fourier transform under the optimal energy gathering angle α opt of the linear frequency modulation signal f(t), that is,

步骤十、由步骤九中角度αopt下第N重同步挤压短时分数阶傅里叶变换恢复出原始待分析信号即线性调频信号f(t),即Step 10. Use the Nth synchronization extrusion short-time fractional Fourier transform at the angle α opt in step 9. Recover the original signal to be analyzed, that is, the chirp signal f(t), that is

本发明具有的有益效果是:The beneficial effects of the present invention are:

本发明提出了一种基于短时分数阶傅里叶变换的多重时频同步挤压方法,所述方法在信号能量最佳聚集的分数阶傅里叶变换域对应的角度下,计算信号的短时分数阶傅里叶变换,然后对计算结果进行复正弦函数调制,并以此获得信号的瞬时频率估计,进而计算出多重同步挤压的瞬时频率估计。然后,将信号的能量挤压到多重瞬时频率估计在时频面上所确定的时频点上,得到基于短时分数阶傅里叶变换的多重时频同步挤压结果。与基于传统短时傅里叶变换的多重时频同步挤压方法相比,基于短时分数阶傅里叶变换的多重时频同步挤压方法通过自由参数α的选择,可以进一步提升时频分析的分辨率,能够获得信号高分辨率的时频分析结果。The present invention proposes a multiple time-frequency synchronous squeezing method based on short-time fractional Fourier transform. The method calculates the short-term frequency of the signal at the angle corresponding to the fractional Fourier transform domain where the signal energy is optimally gathered. The time-fractional Fourier transform is performed, and then the calculation result is modulated by a complex sinusoidal function to obtain the instantaneous frequency estimate of the signal, and then the instantaneous frequency estimate of the multiple simultaneous squeezes is calculated. Then, the energy of the signal is squeezed to the time-frequency points determined by the multiple instantaneous frequency estimates on the time-frequency plane, and the multiple time-frequency synchronous squeeze results based on the short-time fractional Fourier transform are obtained. Compared with the multiple time-frequency synchronous squeezing method based on the traditional short-time Fourier transform, the multiple time-frequency synchronous squeezing method based on the short-time fractional Fourier transform can further improve the time-frequency analysis through the selection of the free parameter α. The resolution can obtain high-resolution time-frequency analysis results of the signal.

附图说明Description of drawings

图1是基于短时分数阶傅里叶变换的多重同步挤压原理框图。Figure 1 is a functional block diagram of multiple simultaneous extrusion based on short-time fractional Fourier transform.

图2是仿真信号频率随时间变化的理论曲线图。Figure 2 is a theoretical graph of the simulated signal frequency changing with time.

图3是基于现有短时傅里叶变换多重同步挤压的结果示意图。Figure 3 is a schematic diagram of the results of multiple simultaneous extrusion based on the existing short-time Fourier transform.

图4是基于短时分数阶傅里叶变换同步挤压的结果示意图。Figure 4 is a schematic diagram of the results of synchronous extrusion based on short-time fractional Fourier transform.

图5是基于短时分数阶傅里叶变换多重同步挤压的结果示意图。Figure 5 is a schematic diagram of the results of multiple simultaneous extrusion based on short-time fractional Fourier transform.

具体实施方式Detailed ways

下面将结合本发明实施例中的附图对本发明实施例中的技术方案进行清楚、完整地描述,显然,所描述的实施例仅仅是本发明一部分实施例,而不是全部的实施例。基于本发明中的实施例,本领域普通技术人员在没有做出创造性劳动前提下所获得的所有其他实施例,都属于本发明保护的范围。The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only some of the embodiments of the present invention, rather than all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts fall within the scope of protection of the present invention.

为便于分析,首先引入分数阶傅里叶变换的概念。对于任意能量有限信号f(t)∈L2(R),分数阶傅里叶变换的定义为To facilitate analysis, the concept of fractional Fourier transform is first introduced. For any energy finite signal f(t)∈L 2 (R), the fractional Fourier transform is defined as

式中,核函数的表达式为In the formula, the kernel function The expression of

式中,k∈Z,α表示分数阶傅里叶变换的角度,变量u通常称为分数阶频率,其所在的坐标轴通常称为分数阶傅里叶变换域。相应地,分数阶傅里叶变换的逆变换的公式为In the formula, k∈Z, α represents the angle of the fractional Fourier transform, the variable u is usually called the fractional frequency, and the coordinate axis where it is located is usually called the fractional Fourier transform domain. Correspondingly, the formula for the inverse transform of the fractional Fourier transform is

式中,上标符号*表示共轭运算。特别地,当α=π/2时,分数阶傅里叶变换便退化为传统傅里叶变换。进一步地,引入短时分数阶傅里叶变换的概念。对于任意能量有限信号f(t)∈L2(R),短时分数阶傅里叶变换定义为In the formula, the superscript * represents the conjugate operation. In particular, when α=π/2, the fractional Fourier transform degenerates into the traditional Fourier transform. Furthermore, the concept of short-time fractional Fourier transform is introduced. For any finite energy signal f(t)∈L 2 (R), the short-time fractional Fourier transform is defined as

式中,核函数gα,t,u(τ)是由窗函数g(t)(且)生成,其表达式为In the formula, the kernel function g α,t,u (τ) is determined by the window function g(t) (and ) is generated, and its expression is

相应地,短时分数阶傅里叶变换的逆变换为Correspondingly, the inverse transform of the short-time fractional Fourier transform is

此外,还可以将短时分数阶傅里叶变换表示成分数阶傅里叶变换域的形式,即In addition, the short-time fractional Fourier transform can also be expressed in the form of fractional Fourier transform domain, that is

式中,G(ucscα)表示窗函数g(t)的傅里叶变换(变换元做了尺度cscα伸缩)。In the formula, G(ucscα) represents the Fourier transform of the window function g(t) (the transformation element is scaled cscα).

为了阐明基于短时分数阶傅里叶变换同步挤压的基本原理,下面以线性调频信号为例展开分析。线性调频信号的表达式为In order to illustrate the basic principle of synchronous extrusion based on short-time fractional Fourier transform, the linear frequency modulation signal is taken as an example for analysis. The expression of chirp signal is

式中,η、ω0、c分别表示线性调频信号f(t)的幅度、起始频率和调频率。根据分数阶傅里叶变换的定义,该线性调频信号在角度α=-arccot(c)的分数阶傅里叶变换域能量最佳聚集,且对应的分数阶傅里叶变换为In the formula, η, ω 0 and c respectively represent the amplitude, starting frequency and modulation frequency of the linear frequency modulation signal f(t). According to the definition of fractional Fourier transform, the energy of the chirp signal is optimally concentrated in the fractional Fourier transform domain of the angle α = - arccot (c), and the corresponding fractional Fourier transform is

基于此,进一步可以得到线性调频信号f(t)的短时分数阶傅里叶变换为Based on this, the short-time fractional Fourier transform of the chirp signal f(t) can be further obtained as

可以看出,可以通过短时分数阶傅里叶变换的幅度模值来描述信号f(t)的频率时变行为,但其分辨率取决于窗函数g(t)的傅里叶变换G(ucscα)(变换元做了尺度cscα伸缩)。鉴于此,将线性调频信号f(t)的短时分数阶傅里叶变换改写为It can be seen that the amplitude modulus of the short-time fractional Fourier transform can be To describe the frequency time-varying behavior of the signal f(t), but its resolution depends on the Fourier transform G(ucscα) of the window function g(t) (the transformation element is scaled cscα). In view of this, the short-time fractional Fourier transform of the chirp signal f(t) is rewritten as

这表明,经过复正弦ejtucscα调制后的短时分数阶傅里叶变换,即其相位中包含原始线性调频信号f(t)所有的信息。于是,根据时频分析中瞬时频率的定义,可以从/>中得到原始线性调频信号瞬时频率的估计,即This shows that the short-time fractional Fourier transform after complex sinusoid e jtucscα modulation is Its phase contains all the information of the original chirp signal f(t). Therefore, according to the definition of instantaneous frequency in time-frequency analysis, it can be obtained from/> The estimate of the instantaneous frequency of the original chirp signal is obtained, that is

式中,表示取实部。经过计算,原始线性调频信号f(t)瞬时频率的估计值为与理论值一致。这表明,短时分数阶傅里叶变换保留了信号的相位信息,从中可以提取出信号的瞬时频率。从而得到同步挤压短时分数阶傅里叶变换,即In the formula, Indicates taking the real part. After calculation, the estimated value of the instantaneous frequency of the original chirp signal f(t) is consistent with the theoretical value. This shows that the short-time fractional Fourier transform retains the phase information of the signal, from which the instantaneous frequency of the signal can be extracted. Thus, the synchronous squeeze short-time fractional Fourier transform is obtained, that is

式中,相应地,通过计算可以导出同步挤压短时分数阶傅里叶变换的逆变换定义为In the formula, Correspondingly, the inverse transform of the short-time fractional Fourier transform of synchronous extrusion can be derived through calculation and is defined as

特别地,当α=π/2时,同步挤压短时分数阶傅里叶变换便退化为经典同步挤压短时傅里叶变换。In particular, when α=π/2, the synchronous squeezing short-time fractional Fourier transform degenerates into the classic synchronous squeezing short-time Fourier transform.

综上分析,同步挤压短时分数阶傅里叶变换只能得到α角度分数阶傅里叶变换域能量最佳聚集的线性调频信号瞬时频率的准确估计。因此,对于具有相同调频率的多分量线性调频信号而言,只需要一次同步挤压短时分数阶傅里叶变换就能获得高分辨率时频表示。然而,对于具有不同调频率的多分量线性调频信号,单次同步挤压短时分数阶傅里叶变换并不能得到最优结果,即对于那些在α角度分数阶傅里叶变换域能量非最佳聚集的信号分量,无法获得高分辨率的时频表示。为此,下面将提出一种基于短时分数阶傅里叶变换的多重同步挤压方法。In summary, the synchronous squeeze short-time fractional Fourier transform can only obtain an accurate estimate of the instantaneous frequency of the chirp signal with optimal energy accumulation in the α-angle fractional Fourier transform domain. Therefore, for multi-component chirp signals with the same modulation frequency, only one synchronized squeeze short-time fractional Fourier transform is needed to obtain a high-resolution time-frequency representation. However, for multi-component linear FM signals with different modulation frequencies, a single synchronization squeeze short-time fractional Fourier transform cannot obtain optimal results, that is, for those in the alpha angle fractional Fourier transform domain whose energy is not optimal, It is impossible to obtain high-resolution time-frequency representation due to optimally gathered signal components. To this end, a multiple simultaneous extrusion method based on short-time fractional Fourier transform will be proposed below.

为了阐明基于短时分数阶傅里叶变换多重同步挤压的思想,不妨假设在α角度分数阶傅里叶变换域,信号分量的能量不是最佳聚集的,并将信号建模为调幅-调频的形式,即In order to illustrate the idea of multiple simultaneous squeezing based on short-time fractional Fourier transform, we might as well assume that the energy of the signal components is not optimally gathered in the alpha angle fractional Fourier transform domain, and model the signal as AM-FM form, that is

式中,A(t)和分别表示信号幅值和相位,并且满足:存在足够小的常数ε≥0,对于任意时间t,有|A′(t)|≤ε和/>于是,可以将短时分数阶傅里叶变换定义中信号f(τ)在时间t处进行泰勒展开,即In the formula, A(t) and represent the signal amplitude and phase respectively, and satisfy: there is a small enough constant ε≥0, and for any time t, there are |A′(t)|≤ε and/> Therefore, the signal f(τ) in the definition of short-time fractional Fourier transform can be Taylor expanded at time t, that is

为了便于计算,并考虑到高斯函数具有最佳的时频分辨率,这里选择高斯函数作为窗函数,其中高斯函数标准方差/>基于此,经过复正弦ejtucscα调制后的短时分数阶傅里叶变换可以表示为For ease of calculation and considering that the Gaussian function has the best time-frequency resolution, the Gaussian function is chosen here. As a window function, where Gaussian function standard deviation/> Based on this, the short-time fractional Fourier transform after complex sinusoid e jtucscα modulation can be expressed as

由此可得,基于短时分数阶傅里叶变换的瞬时频率估计为It can be obtained that the instantaneous frequency estimate based on short-time fractional Fourier transform is

需要指出的是,以上瞬时频率估计中得到的是瞬时频率估计与时间t和分数阶频率u的关系,为了得到瞬时频率估计与频率ω之间的关系,需要首先导出分数阶频率与频率的内在联系。为此,需要引入分数阶魏格纳-维尔分布概念。对于任意能量有限信号f(t),分数阶魏格纳-维尔分布定义为It should be pointed out that what is obtained in the above instantaneous frequency estimation is the relationship between the instantaneous frequency estimate and time t and fractional frequency u. In order to obtain the relationship between the instantaneous frequency estimate and frequency ω, it is necessary to first derive the intrinsic relationship between fractional frequency and frequency. connect. To this end, the concept of fractional Wegener-Weil distribution needs to be introduced. For any finite energy signal f(t), the fractional Wegener-Weil distribution is defined as

特别地,当α=π/2时,分数阶魏格纳-维尔分布便退化为经典魏格纳-维尔分布,即In particular, when α=π/2, the fractional Wegener-Weil distribution degenerates into the classic Wegener-Weil distribution, that is

可以看出,分数阶魏格纳-维尔分布确定的是联合时间t和分数阶频率u的信号表示,而经典魏格纳-维尔分布提供的是联合时间t和频率ω的信号表示。比较两者定义,可以进一步得到分数阶魏格纳-维尔分布和经典魏格纳-维尔分布之间关系,即It can be seen that the fractional Wegener-Wegener distribution determines the signal representation of the joint time t and the fractional frequency u, while the classical Wegener-Weil distribution provides the signal representation of the joint time t and frequency ω. Comparing the two definitions, we can further obtain the relationship between the fractional Wegener-Weil distribution and the classic Wegener-Weil distribution, namely

于是,可以得到分数阶频率u和频率ω之间的关系为Therefore, the relationship between the fractional frequency u and the frequency ω can be obtained as

ω=ucscα-tcotα (22)ω=ucscα-tcotα (22)

基于此,得到同步挤压短时分数阶傅里叶变换的瞬时频率估计联合时间t和频率ω的表示,即Based on this, the instantaneous frequency estimate of the synchronous squeeze short-time fractional Fourier transform is obtained by combining the time t and the frequency ω, namely

基于此,下面给出多重同步挤压短时分数阶傅里叶变换的推导过程。记第n次同步挤压短时分数阶傅里叶变换为且/>那么经过N次同步挤压之后,可以得到Based on this, the derivation process of the short-time fractional Fourier transform of multiple simultaneous squeezes is given below. Denote the nth synchronous extrusion short-time fractional Fourier transform as and/> Then after N times of synchronous extrusion, we can get

式中,N≥2。于是,当N=2时,则有In the formula, N≥2. Therefore, when N=2, there is

式中,表示第二次同步挤压短时分数阶傅里叶变换的瞬时频率估计,具体的表达式为In the formula, Represents the instantaneous frequency estimate of the short-time fractional Fourier transform of the second synchronous squeeze. The specific expression is:

进一步的,可以验证Further, it can be verified

这表明,二次同步挤压短时分数阶傅里叶变换的瞬时频率估计比一次同步挤压短时分数阶傅里叶变换的瞬时频率估计/>更加接近信号的瞬时频率因此,二次同步挤压短时分数阶傅里叶变换得到的时频表示与信号的时频结构更加接近。以此类推,可以得到多重同步挤压短时分数阶傅里叶变换为This shows that the instantaneous frequency estimate of the short-time fractional Fourier transform of the quadratic synchronous squeeze Instantaneous frequency estimation by fractional Fourier transform shorter than a single synchronous squeeze/> Closer to the instantaneous frequency of the signal Therefore, the time-frequency representation obtained by the short-time fractional Fourier transform of the secondary synchronization squeeze is closer to the time-frequency structure of the signal. By analogy, the short-time fractional Fourier transform of multiple simultaneous squeezes can be obtained as

式中,表示第N次同步挤压短时分数阶傅里叶变换的瞬时频率估计,且其具体的计算过程为In the formula, represents the instantaneous frequency estimate of the Nth synchronous squeeze short-time fractional Fourier transform, and Its specific calculation process is

式中,N≥1。于是,可以验证In the formula, N≥1. Therefore, it can be verified

这表明,多重同步挤压短时分数阶傅里叶变换能够获得更加准确的瞬时频率估计,由此得到的时频表示也更加逼近信号的时频结构。This shows that multiple synchronization squeezed short-time fractional Fourier transform can obtain a more accurate instantaneous frequency estimate, and the resulting time-frequency representation is also closer to the time-frequency structure of the signal.

相应地,通过计算可以导出N重同步挤压短时分数阶傅里叶变换的逆变换为Correspondingly, the inverse transform of the N-fold synchronous extrusion short-time fractional Fourier transform can be derived through calculation as

特别地,当α=π/2时,多重同步挤压短时分数阶傅里叶变换便退化为传统的多重同步挤压短时傅里叶变换。In particular, when α=π/2, the multiple simultaneous squeezing short-time fractional Fourier transform degenerates into the traditional multiple simultaneous squeezing short-time Fourier transform.

具体地,本发明提出一种基于短时分数阶傅里叶变换的多重时频同步挤压方法,所述方法包括以下步骤:Specifically, the present invention proposes a multiple time-frequency synchronous extrusion method based on short-time fractional Fourier transform. The method includes the following steps:

步骤一、给定待分析信号即线性调频信号f(t),计算其分数阶傅里叶变换Fα(u),其中角度取值范围为α∈(0,2π];Step 1. Given the signal to be analyzed, that is, the linear frequency modulation signal f(t), calculate its fractional Fourier transform F α (u), where the angle range is α∈(0,2π];

步骤二、确定线性调频信号f(t)能量最佳聚集分数阶傅里叶变换域的最佳角度αopt,即Step 2: Determine the optimal angle α opt in the fractional Fourier transform domain where the energy of the chirp signal f(t) optimally gathers, that is

步骤三、选择窗函数g(t),满足其中/>计算线性调频信号f(t)能量最佳聚集分数阶傅里叶变换域对应角度αopt下的短时分数阶傅里叶变换,即Step 3. Select the window function g(t) to satisfy Among them/> Calculate the short-time fractional Fourier transform of the linear frequency modulation signal f(t) energy at the optimal accumulation fractional Fourier transform domain corresponding to the angle α opt , that is

步骤四、将线性调频信号f(t)能量最佳聚集角度αopt下的短时分数阶傅里叶变换与复正弦函数ejtucscα进行相乘,即Step 4: Multiply the short-time fractional Fourier transform at the optimal energy concentration angle α opt of the linear frequency modulation signal f(t) with the complex sine function e jtucscα , that is

步骤五、根据步骤四中的乘积计算线性调频信号f(t)能量最佳聚集角度αopt下瞬时频率的估计/>Step 5. According to the product in step 4 Calculate the estimation of the instantaneous frequency at the optimal energy concentration angle α opt of the chirp signal f(t)/> Right now

步骤六、根据步骤五中的计算线性调频信号f(t)能量最佳聚集角度αopt下瞬时频率的估计联合时间t和频率ω的表示/>Step 6. According to the steps in Step 5 Calculate the estimation of the instantaneous frequency at the optimal energy concentration angle α opt of the chirp signal f(t) and express the combined time t and frequency ω/> Right now

步骤七、计算线性调频信号f(t)能量最佳聚集角度αopt下,第一重同步挤压短时分数阶傅里叶变换,即Step 7: Calculate the first synchronization squeeze short-time fractional Fourier transform under the optimal energy gathering angle α opt of the linear frequency modulation signal f(t), that is

式中, In the formula,

步骤八、计算线性调频信号f(t)能量最佳聚集角度αopt下,第二重同步挤压短时分数阶傅里叶变换,即Step 8: Calculate the second synchronous squeeze short-time fractional Fourier transform under the optimal energy gathering angle α opt of the linear frequency modulation signal f(t), that is

步骤九、重复步骤八,计算线性调频信号f(t)能量最佳聚集角度αopt下,第N重同步挤压短时分数阶傅里叶变换,即Step 9. Repeat step 8 to calculate the Nth synchronous squeeze short-time fractional Fourier transform under the optimal energy gathering angle α opt of the linear frequency modulation signal f(t), that is,

步骤十、由步骤九中角度αopt下第N重同步挤压短时分数阶傅里叶变换恢复出原始待分析信号即线性调频信号f(t),即Step 10. Use the Nth synchronization extrusion short-time fractional Fourier transform at the angle α opt in step 9. Recover the original signal to be analyzed, that is, the chirp signal f(t), that is

本发明的效果可通过以下仿真进一步说明:The effect of the present invention can be further explained through the following simulation:

仿真信号的表达式为The expression of the simulated signal is

可以看出,仿真信号含有四个信号分量,即 和/>图2根据信号解析表达式给出了其频率随时间变化的理论曲线。图3、图4和图5分别给出了基于现有短时傅里叶变换多重同步挤压(重数为10)、短时分数阶傅里叶变换同步挤压和短时分数阶傅里叶变换多重同步挤压(重数为10)的结果。可以看出,与短时傅里叶变换相比,由短时分数阶变换得到的多重同步挤压结果能够清晰展现出信号f(t)含有的分量成分,可以进一步提高时频分析的分辨率。与短时分数阶傅里叶变换同步挤压相比,由短时分数阶变换得到的多重同步挤压结果能量更加聚集。It can be seen that the simulation signal contains four signal components, namely and/> Figure 2 gives the theoretical curve of its frequency changing with time according to the signal analytical expression. Figure 3, Figure 4 and Figure 5 respectively show the multi-synchronous squeezing based on the existing short-time Fourier transform (multiplicity is 10), short-time fractional Fourier transform synchronous squeezing and short-time fractional Fourier transform. The result of leaf transform multiple simultaneous extrusions (multiplicity 10). It can be seen that compared with the short-time Fourier transform, the multiple synchronous squeeze results obtained by the short-time fractional transform can clearly show the components contained in the signal f(t), which can further improve the resolution of time-frequency analysis. . Compared with the short-time fractional Fourier transform synchronous squeezing, the energy of the multiple synchronous squeezing obtained by the short-time fractional Fourier transform is more concentrated.

Claims (1)

1.一种基于短时分数阶傅里叶变换的多重时频同步挤压方法,其特征在于,所述方法包括以下步骤:1. A multiple time-frequency synchronous extrusion method based on short-time fractional Fourier transform, characterized in that the method includes the following steps: 步骤一、给定待分析信号即线性调频信号f(t),计算其分数阶傅里叶变换Fα(u),其中角度取值范围为α∈(0,2π];Step 1. Given the signal to be analyzed, that is, the linear frequency modulation signal f(t), calculate its fractional Fourier transform F α (u), where the angle range is α∈(0,2π]; 步骤二、确定线性调频信号f(t)能量最佳聚集分数阶傅里叶变换域的最佳角度αopt,即Step 2: Determine the optimal angle α opt in the fractional Fourier transform domain where the energy of the chirp signal f(t) optimally gathers, that is 步骤三、选择窗函数g(t),满足其中/>计算线性调频信号f(t)能量最佳聚集分数阶傅里叶变换域对应角度αopt下的短时分数阶傅里叶变换,即Step 3. Select the window function g(t) to satisfy Among them/> Calculate the short-time fractional Fourier transform of the linear frequency modulation signal f(t) energy at the optimal accumulation fractional Fourier transform domain corresponding to the angle α opt , that is 步骤四、将线性调频信号f(t)能量最佳聚集角度αopt下的短时分数阶傅里叶变换与复正弦函数ejtucscα进行相乘,即Step 4: Multiply the short-time fractional Fourier transform at the optimal energy concentration angle α opt of the linear frequency modulation signal f(t) with the complex sine function e jtucscα , that is 步骤五、根据步骤四中的乘积计算线性调频信号f(t)能量最佳聚集角度αopt下瞬时频率的估计/>Step 5. According to the product in step 4 Calculate the estimation of the instantaneous frequency at the optimal energy concentration angle α opt of the chirp signal f(t)/> Right now 步骤六、根据步骤五中的计算线性调频信号f(t)能量最佳聚集角度αopt下瞬时频率的估计联合时间t和频率ω的表示/>Step 6. According to the steps in Step 5 Calculate the estimation of the instantaneous frequency at the optimal energy concentration angle α opt of the chirp signal f(t) and express the combined time t and frequency ω/> Right now 步骤七、计算线性调频信号f(t)能量最佳聚集角度αopt下,第一重同步挤压短时分数阶傅里叶变换,即Step 7: Calculate the first synchronization squeeze short-time fractional Fourier transform under the optimal energy gathering angle α opt of the linear frequency modulation signal f(t), that is 式中, In the formula, 步骤八、计算线性调频信号f(t)能量最佳聚集角度αopt下,第二重同步挤压短时分数阶傅里叶变换,即Step 8: Calculate the second synchronous squeeze short-time fractional Fourier transform under the optimal energy gathering angle α opt of the linear frequency modulation signal f(t), that is 步骤九、重复步骤八,计算线性调频信号f(t)能量最佳聚集角度αopt下,第N重同步挤压短时分数阶傅里叶变换,即Step 9. Repeat step 8 to calculate the Nth synchronous squeeze short-time fractional Fourier transform under the optimal energy gathering angle α opt of the linear frequency modulation signal f(t), that is, 步骤十、由步骤九中角度αopt下第N重同步挤压短时分数阶傅里叶变换恢复出原始待分析信号即线性调频信号f(t),即Step 10. Use the Nth synchronization extrusion short-time fractional Fourier transform at the angle α opt in step 9. Recover the original signal to be analyzed, that is, the chirp signal f(t), that is
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