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

Abstract

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

A multiple time-frequency synchronous extrusion method based on short-time fractional Fourier transform

Technical field

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.

Background technique

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.

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.

Contents of the invention

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.

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:

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];

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

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

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

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

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

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

In the formula,

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

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,

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

The beneficial effects of the present invention are:

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.

Description of drawings

Figure 1 is a functional block diagram of multiple simultaneous extrusion based on short-time fractional Fourier transform.

Figure 2 is a theoretical graph of the simulated signal frequency changing with time.

Figure 3 is a schematic diagram of the results of multiple simultaneous extrusion based on the existing short-time Fourier transform.

Figure 4 is a schematic diagram of the results of synchronous extrusion based on short-time fractional Fourier transform.

Figure 5 is a schematic diagram of the results of multiple simultaneous extrusion based on short-time fractional Fourier transform.

Detailed ways

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.

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

In the formula, the kernel function The expression of

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

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

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

Correspondingly, the inverse transform of the short-time fractional Fourier transform is

In addition, the short-time fractional Fourier transform can also be expressed in the form of fractional Fourier transform domain, that is

In the formula, G(ucscα) represents the Fourier transform of the window function g(t) (the transformation element is scaled cscα).

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

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

Based on this, the short-time fractional Fourier transform of the chirp signal f(t) can be further obtained as

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

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

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

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

In particular, when α=π/2, the synchronous squeezing short-time fractional Fourier transform degenerates into the classic synchronous squeezing short-time Fourier transform.

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.

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

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

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

It can be obtained that the instantaneous frequency estimate based on short-time fractional Fourier transform is

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

In particular, when α=π/2, the fractional Wegener-Weil distribution degenerates into the classic Wegener-Weil distribution, that is

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

Therefore, the relationship between the fractional frequency u and the frequency ω can be obtained as

ω=ucscα-tcotα (22)

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

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

In the formula, N≥2. Therefore, when N=2, there is

In the formula, Represents the instantaneous frequency estimate of the short-time fractional Fourier transform of the second synchronous squeeze. The specific expression is:

Further, it can be verified

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

In the formula, represents the instantaneous frequency estimate of the Nth synchronous squeeze short-time fractional Fourier transform, and Its specific calculation process is

In the formula, N≥1. Therefore, it can be verified

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.

Correspondingly, the inverse transform of the N-fold synchronous extrusion short-time fractional Fourier transform can be derived through calculation as

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.

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:

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];

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

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

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

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

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

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

In the formula,

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

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,

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

The effect of the present invention can be further explained through the following simulation:

The expression of the simulated signal is

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. A multiple time-frequency synchronous extrusion method based on short-time fractional Fourier transform, characterized in that the method includes the following steps: 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]; 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 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 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 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 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 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 In the formula, 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 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, 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