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 PDFInfo
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Abstract
The invention provides a multiple time-frequency synchronous extrusion method based on short-time fractional Fourier transform, which is characterized in that the short-time fractional Fourier transform of a signal is calculated under the angle corresponding to the fractional Fourier transform domain of optimal aggregation of signal energy, then complex sine function modulation is carried out on a calculation result, the instantaneous frequency estimation of the signal is obtained, and further the instantaneous frequency estimation of multiple synchronous extrusion is calculated. And then, extruding the energy of the signal to a time-frequency point determined by the multiple instantaneous frequency estimation on a time-frequency surface, so as to obtain a multiple time-frequency synchronous extrusion result based on short-time fractional Fourier transform. Compared with the traditional short-time Fourier transform-based multiple time-frequency synchronous extrusion method, the short-time fractional Fourier transform-based multiple time-frequency synchronous extrusion method can further improve the resolution of time-frequency analysis through selection of the free parameter alpha, and can obtain a time-frequency analysis result of high resolution of signals.
Description
Technical Field
The invention belongs to the technical field of signal and information processing, and particularly relates to a multiple time-frequency synchronous extrusion method based on short-time fractional Fourier transform.
Background
The signal is the carrier of information and the acquisition of useful information from the signal is the basic task of signal processing. The stationary signal processing theory and technology lays the foundation of various electronic information systems such as communication, radar, navigation, detection and the like, and promotes the breakthrough of core technology in the field of electronic information. However, with the continuous popularization of application range, non-stationary signal processing gradually becomes a bottleneck that restricts further improvement of performance of the electronic information system.
Frequency variation is a typical feature of non-stationary signals. A simple and efficient way to analyze non-stationary signals is a short-time fourier transform, which provides a way to describe the signal in the time-frequency plane, which can show a change in frequency of the signal over time. However, the time-frequency resolution of the short-time fourier transform depends on the size of the time-frequency window it determines in the time-frequency plane. Due to the limitation of uncertainty principle, the time-frequency resolution of short-time Fourier transform has a contradiction with each other. Therefore, a time-frequency synchronous extrusion method based on short-time Fourier transformation is generated. The method requires that the components of the signal are separated from each other and do not overlap each other on the time-frequency plane determined by the short-time fourier transform. This precondition is often difficult to meet in practical applications due to the low time-frequency resolution of short-time fourier transforms. In view of this, the invention will propose a multiple time-frequency synchronous extrusion method based on short-time fractional Fourier transform.
Disclosure of Invention
The invention aims to solve the problems in the prior art and provides a multiple time-frequency synchronous extrusion method based on short-time fractional Fourier transform.
The invention is realized by the following technical scheme, and provides a multiple time-frequency synchronous extrusion method based on short-time fractional Fourier transform, which comprises the following steps:
step one, giving a signal to be analyzed, namely a linear frequency modulation signal F (t), and calculating fractional Fourier transform F α (u) wherein the angle has a value in the range of alpha E (0, 2 pi)]；
Step two, determining a linear adjustmentOptimal angle alpha of frequency signal f (t) energy optimal aggregation fractional Fourier transform domain opt I.e.
Step three, selecting a window function g (t) to meet the following requirementsWherein->Calculating the corresponding angle alpha of the optimal aggregation fractional Fourier transform domain of the energy of the linear frequency modulation signal f (t) opt The short-time fractional Fourier transform below, i.e
Step four, the optimal energy aggregation angle alpha of the linear frequency modulation signal f (t) opt The short-time fractional Fourier transform and complex sine function e jtucscα Multiplying, i.e.
Step five, according to the product in step fourCalculating the optimal energy aggregation angle alpha of the linear frequency modulation signal f (t) opt Estimation of the lower instantaneous frequency->I.e.
Step six, according toIn step fiveCalculating the optimal energy aggregation angle alpha of the linear frequency modulation signal f (t) opt Representation of estimated joint time t and frequency ω for lower temporal frequency +.>I.e.
Step seven, calculating the optimal energy aggregation angle alpha of the linear frequency modulation signal f (t) opt Under the condition, the first resynchronous extrusion short-time fractional Fourier transform, namely
In the method, in the process of the invention,
step eight, calculating the optimal energy aggregation angle alpha of the linear frequency modulation signal f (t) opt Under the condition of second synchronous extrusion, short-time fractional Fourier transform, namely
Step nine, repeating the step eight, and calculating the optimal energy aggregation angle alpha of the linear frequency modulation signal f (t) opt Under the condition, the Nth synchronous extrusion short-time fractional Fourier transform, namely
Step ten, from step nine, angle α opt Lower Nth resynchronous extrusion short-time fractional Fourier transformRecovering the original signal to be analyzed, i.e. the chirp signal f (t), i.e
The invention has the beneficial effects that:
the invention provides a multiple time-frequency synchronous extrusion method based on short-time fractional Fourier transform, which is used for calculating short-time fractional Fourier transform of a signal under an angle corresponding to a fractional Fourier transform domain with optimal aggregation of signal energy, then carrying out complex sine function modulation on a calculation result, obtaining instantaneous frequency estimation of the signal according to the complex sine function modulation, and further calculating the instantaneous frequency estimation of multiple synchronous extrusion. And then, extruding the energy of the signal to a time-frequency point determined by the multiple instantaneous frequency estimation on a time-frequency surface, so as to obtain a multiple time-frequency synchronous extrusion result based on short-time fractional Fourier transform. Compared with the traditional short-time Fourier transform-based multiple time-frequency synchronous extrusion method, the short-time fractional Fourier transform-based multiple time-frequency synchronous extrusion method can further improve the resolution of time-frequency analysis through selection of the free parameter alpha, and can obtain a time-frequency analysis result of high resolution of signals.
Drawings
FIG. 1 is a schematic block diagram of multiple simultaneous extrusion based on short-time fractional Fourier transform.
Fig. 2 is a theoretical graph of simulated signal frequency versus time.
Fig. 3 is a schematic diagram of the result of multiple simultaneous extrusion based on the prior short-time fourier transform.
Fig. 4 is a schematic diagram of the results of synchronous extrusion based on short-time fractional fourier transform.
Fig. 5 is a schematic diagram of the result of multiple simultaneous extrusion based on short-time fractional fourier transform.
Detailed Description
The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present invention, but not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
To facilitate analysis, the concept of fractional fourier transforms is first introduced. For any energy limited signal f (t) ∈L 2 (R), definition of fractional Fourier transform is
In the formula, a kernel functionThe expression of (2) is
Where k e Z, α represents the angle of the fractional fourier transform, the variable u is usually called the fractional frequency, and the coordinate axis on which it is located is usually called the fractional fourier transform domain. Accordingly, the inverse transform of the fractional Fourier transform is formulated as
In the formula, superscript symbol indicates conjugate operation. In particular, when α=pi/2, the fractional fourier transform is degraded to a conventional fourier transform. Further, the concept of short-time fractional fourier transform is introduced. For any energy limited signal f (t) ∈L 2 (R) short-term fractional Fourier transform is defined as
In the formula, the kernel function g α,t,u (τ) is defined by a window function g (t) (and) Generating the expression of
Accordingly, the inverse transform of the short-time fractional Fourier transform is
In addition, the short-time fractional Fourier transform can be expressed in the form of fractional Fourier transform domain, namely
In the formula, G (ucsc α) represents fourier transform of the window function G (t) (the transform element scales cscα).
In order to clarify the basic principle of synchronous extrusion based on short-time fractional fourier transform, analysis is developed below taking a chirp signal as an example. The expression of the chirp signal is
Wherein, eta, omega 0 C represent the amplitude, start frequency and tone frequency of the chirp signal f (t), respectively. The energy of the chirp signal in the fractional fourier transform domain at the angle α= -arcot (c) is optimally concentrated according to the definition of the fractional fourier transform, and the corresponding fractional fourier transform is
Based on this, it is further possible to obtain a short-time fractional Fourier transform of the chirp signal f (t) into
It can be seen that the magnitude modulus value can be obtained by short-time fractional Fourier transformThe frequency-time-varying behavior of the signal f (t) is described, but its resolution depends on the fourier transform G (ucsc alpha) of the window function G (t) (the transform element scales csc alpha). In view of this, the short-time fractional fourier transform of the chirp signal f (t) is rewritten as
This shows that, through complex sinusoid e jtucscα Modulated short-time fractional Fourier transform, i.eThe phase of which contains all the information of the original chirp signal f (t). Thus, according to the definition of the instantaneous frequency in the time-frequency analysis, one can go from +.>To obtain an estimate of the instantaneous frequency of the original chirp signal, i.e
In the method, in the process of the invention,representing the real part. Through calculation, the originalThe estimated value of the instantaneous frequency of the initial linear frequency modulation signal f (t) isConsistent with the theoretical values. 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. Thereby obtaining the synchronous extrusion short-time fractional Fourier transform, namely
In the method, in the process of the invention,accordingly, the inverse transform that can be calculated to derive the synchro-extrusion short-time fractional Fourier transform is defined as
In particular, when α=pi/2, the synchrocompression short-time fractional fourier transform is degraded to a classical synchrocompression short-time fourier transform.
To sum up, the synchronous extrusion short-time fractional Fourier transform can only obtain accurate estimation of the instantaneous frequency of the linear frequency modulation signal with the optimal aggregation of the energy of the alpha angle fractional Fourier transform domain. Thus, for multi-component chirp signals with the same modulation frequency, only one simultaneous squeeze of the short-time fractional fourier transform is required to obtain a high resolution time-frequency representation. However, for multi-component chirp signals with different modulation frequencies, a single simultaneous extrusion of the short-time fractional fourier transform does not give optimal results, i.e. for those signal components whose energy is not optimally concentrated in the alpha angle fractional fourier transform domain, a high resolution time-frequency representation is not obtained. For this purpose, a multiple simultaneous extrusion method based on short-time fractional fourier transform will be proposed below.
To elucidate the concept of multiple synchronous extrusion based on short-time fractional fourier transform, it is not just assumed that the energy of the signal components is not optimally concentrated in the alpha-angle fractional fourier transform domain and the signal is modeled as amplitude-frequency modulated, i.e
Wherein A (t) andrespectively representing the signal amplitude and phase, and satisfying: there is a sufficiently small constant ε.gtoreq.0, and for any time t there are |A' (t) |.ltoreq.ε and +.>The signal f (τ) in the short-time fractional Fourier transform definition can then be Taylor-expanded at time t, i.e
For ease of calculation and considering that the gaussian function has the best time-frequency resolution, the gaussian function is chosen hereAs a window function, wherein the standard deviation of the Gaussian function +.>Based on this, complex sine e is passed jtucscα The modulated short-time fractional fourier transform can be expressed as
It follows that the instantaneous frequency estimation based on short-time fractional Fourier transform is
It should be noted that, in the above instantaneous frequency estimation, the relation between the instantaneous frequency estimation and the time t and the fractional order frequency u is obtained, and in order to obtain the relation between the instantaneous frequency estimation and the frequency ω, the inherent relation between the fractional order frequency and the frequency needs to be derived first. For this purpose, a fractional-order wigner-wilt distribution concept needs to be introduced. For any energy limited signal f (t), the fractional order wigner-wiener distribution is defined as
In particular, when α=pi/2, the fractional-order wigner-wilt distribution is degraded to a classical wigner-wilt distribution, i.e
It can be seen that the fractional-order wigner-wirl distribution determines the signal representation of the joint time t and the fractional frequency u, whereas the classical wigner-wirl distribution provides the signal representation of the joint time t and the frequency ω. By comparing the definition of the two, the relation between fractional order Wigner-Weibull distribution and classical Wigner-Weibull distribution can be further obtained, namely
Thus, the relationship between fractional order frequency u and frequency ω can be obtained as
ω＝ucscα-tcotα (22)
Based on this, a representation of the instantaneous frequency estimate joint time t and frequency ω of the synchro-extrusion short-time fractional Fourier transform is obtained, i.e.
Based on this, the derivation of the multiple simultaneous extrusion short-time fractional fourier transform is given below. Recording the nth synchronous extrusion short-time fractional Fourier transform asAnd->Then after N times of synchronous extrusion, can be obtained
Wherein N is more than or equal to 2. Thus, when n=2, there is
In the method, in the process of the invention,instantaneous frequency estimation representing a second synchro-extrusion short-time fractional Fourier transform, expressed in particular as
Further, it is possible to verify that
This suggests that instantaneous frequency estimation of the secondary synchro-squeeze short-time fractional fourier transformInstantaneous short-time fractional Fourier transform than one-time synchronous extrusionFrequency estimation->More closely to the instantaneous frequency of the signalTherefore, the time-frequency representation obtained by the secondary synchronous extrusion short-time fractional Fourier transform is more similar to the time-frequency structure of the signal. Similarly, the multiple synchronous extrusion short-time fractional Fourier transform can be obtained
In the method, in the process of the invention,instantaneous frequency estimation representing Nth synchro-squeeze short-time fractional Fourier transform, andthe specific calculation process is that
Wherein N is more than or equal to 1. Thus, it can be verified that
This shows that multiple simultaneous extrusion short-time fractional Fourier transforms can obtain more accurate instantaneous frequency estimates, and the resulting time-frequency representation also more approximates the time-frequency structure of the signal.
Accordingly, the inverse transform of the N-ary-synchronous-squeeze short-time fractional Fourier transform can be derived by calculation
In particular, when α=pi/2, the multiple simultaneous extrusion short-time fractional fourier transform is degraded to the conventional multiple simultaneous extrusion short-time fourier transform.
Specifically, the invention provides a multiple time-frequency synchronous extrusion method based on short-time fractional Fourier transform, which comprises the following steps:
step one, giving a signal to be analyzed, namely a linear frequency modulation signal F (t), and calculating fractional Fourier transform F α (u) wherein the angle has a value in the range of alpha E (0, 2 pi)]；
Step two, determining the optimal angle alpha of the optimal aggregation fractional Fourier transform domain of the energy of the linear frequency modulation signal f (t) opt I.e.
Step three, selecting a window function g (t) to meet the following requirementsWherein->Calculating the corresponding angle alpha of the optimal aggregation fractional Fourier transform domain of the energy of the linear frequency modulation signal f (t) opt The short-time fractional Fourier transform below, i.e
Step four, the optimal energy aggregation angle alpha of the linear frequency modulation signal f (t) opt The short-time fractional Fourier transform and complex sine function e jtucscα Multiplying, i.e.
Step five, according to the product in step fourCalculating the optimal energy aggregation angle alpha of the linear frequency modulation signal f (t) opt Estimation of the lower instantaneous frequency->I.e.
Step six, according to the step fiveCalculating the optimal energy aggregation angle alpha of the linear frequency modulation signal f (t) opt Representation of estimated joint time t and frequency ω for lower temporal frequency +.>I.e.
Step seven, calculating the optimal energy aggregation angle alpha of the linear frequency modulation signal f (t) opt Under the condition, the first resynchronous extrusion short-time fractional Fourier transform, namely
In the method, in the process of the invention,
step eight, calculating the optimal energy aggregation angle alpha of the linear frequency modulation signal f (t) opt Under the condition of second synchronous extrusion, short-time fractional Fourier transform, namely
Step nine, repeating the step eight, and calculating the optimal energy aggregation angle alpha of the linear frequency modulation signal f (t) opt Under the condition, the Nth synchronous extrusion short-time fractional Fourier transform, namely
Step ten, from step nine, angle α opt Lower Nth resynchronous extrusion short-time fractional Fourier transformRecovering the original signal to be analyzed, i.e. the chirp signal f (t), i.e
The effect of the invention can be further illustrated by the following simulations:
the expression of the simulation signal is
It can be seen that the emulated signal contains four signal components, i.e And->Fig. 2 shows a theoretical plot of its frequency over time from a signal analytical expression. FIGS. 3, 4 and 5 show the prior art short-time Fourier transform-based multiplexing, respectivelySynchronous extrusion (weight 10), short time fractional fourier transform synchronous extrusion, and short time fractional fourier transform multiple synchronous extrusion (weight 10). Compared with short-time Fourier transform, the multiple synchronous extrusion result obtained by short-time fractional order transform can clearly show the component contained in the signal f (t), and the resolution of time-frequency analysis can be further improved. Multiple synchronous extrusion results obtained by short-time fractional Fourier transform are more energy-concentrated than short-time fractional Fourier transform synchronous extrusion.
Claims (1)
1. The multiple time-frequency synchronous extrusion method based on short-time fractional Fourier transform is characterized by comprising the following steps of:
step one, giving a signal to be analyzed, namely a linear frequency modulation signal F (t), and calculating fractional Fourier transform F α (u) wherein the angle has a value in the range of alpha E (0, 2 pi)]；
Step two, determining the optimal angle alpha of the optimal aggregation fractional Fourier transform domain of the energy of the linear frequency modulation signal f (t) opt I.e.
Step three, selecting a window function g (t) to meet the following requirementsWherein->Calculating the corresponding angle alpha of the optimal aggregation fractional Fourier transform domain of the energy of the linear frequency modulation signal f (t) opt The short-time fractional Fourier transform below, i.e
Step four, linearly adjustingOptimum collection angle alpha of frequency signal f (t) energy opt The short-time fractional Fourier transform and complex sine function e jtucscα Multiplying, i.e.
Step five, according to the product in step fourCalculating the optimal energy aggregation angle alpha of the linear frequency modulation signal f (t) opt Estimation of the lower instantaneous frequency->I.e.
Step six, according to the step fiveCalculating the optimal energy aggregation angle alpha of the linear frequency modulation signal f (t) opt Representation of estimated joint time t and frequency ω for lower temporal frequency +.>I.e.
Step seven, calculating the optimal energy aggregation angle alpha of the linear frequency modulation signal f (t) opt Under the condition, the first resynchronous extrusion short-time fractional Fourier transform, namely
In the method, in the process of the invention,
step eight, calculating the optimal energy aggregation angle alpha of the linear frequency modulation signal f (t) opt Under the condition of second synchronous extrusion, short-time fractional Fourier transform, namely
Step nine, repeating the step eight, and calculating the optimal energy aggregation angle alpha of the linear frequency modulation signal f (t) opt Under the condition, the Nth synchronous extrusion short-time fractional Fourier transform, namely
Step ten, from step nine, angle α opt Lower Nth resynchronous extrusion short-time fractional Fourier transformRecovering the original signal to be analyzed, i.e. the chirp signal f (t), i.e
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