A high-precision time-varying spectrum analysis method based on fractional S transform

Abstract

The application provides a high-precision time-varying spectrum analysis method based on a fractional S transform, wherein the method is characterized in that: the angle of the fractional Fourier transform domain in which the signal energy is best concentrated is selected, the fractional S transform of the signal is calculated, the signal representation determined by the fractional S transform based on the angle in which the energy is best concentrated is obtained, the signal energy is concentrated on a small number of fractional S transform coefficients, the sparse representation of the signal in the time-fractional frequency plane is realized, and the operation efficiency of the algorithm is improved. Then, the relationship between the fractional frequency and the frequency is used, the signal representation of the joint time and frequency is constructed based on the fractional S transform, and the evolution characteristics of the time-varying spectrum of the signal along with time are shown. Compared with the classical S transform, the time-frequency analysis method based on the fractional S transform can further improve the precision of the time-varying spectrum analysis.

Description

Technical Field

[0001] This invention belongs to the field of signal and information processing technology, and in particular relates to a high-precision time-varying spectrum analysis method based on fractional S-transform. Background Technology

[0002] In practical applications, many signals are non-stationary, such as radar echo signals, electrocardiogram signals, speech signals, gravitational wave signals, seismic signals, and geological exploration signals. With the deepening of applications, non-stationary signal processing has become a bottleneck for further improving the performance of electronic information systems. Time-varying spectrum is a typical characteristic of non-stationary signals. Time-frequency analysis can reflect the characteristics of signal spectrum changes over time and is an effective means of analyzing the time-varying spectrum of non-stationary signals. Among them, the S-transform integrates the advantages of the two fundamental transforms in time-frequency analysis (i.e., short-time Fourier transform and wavelet transform) and has become a highly regarded time-frequency analysis method in recent years, widely used in radar, optics, acoustics, medicine, and seismic signal processing. However, the S-transform is essentially equivalent to using a frequency-domain multiplicative filter to filter the signal, and is only suitable for processing signals with optimal frequency domain energy concentration. For signals with non-optimal frequency domain energy concentration, the processing result is not optimal. For example, linear frequency modulated signals, which are widely present in artificial environments and in nature, are typical signals with non-optimal frequency domain energy concentration. Therefore, a series of new time-frequency analysis methods have emerged based on the S-transform. The fractional S-transform, as a generalized form of the S-transform, combines the advantages of two novel signal transforms: the short-time fractional Fourier transform and the fractional wavelet transform, and has attracted increasing attention. However, the fractional S-transform is essentially a signal representation that combines time and fractional frequency, and cannot directly characterize the time-varying spectrum of a signal. Therefore, this invention utilizes the fractional S-transform to construct a signal representation that combines time and frequency, and proposes a high-precision time-varying spectrum analysis method based on the fractional S-transform. Summary of the Invention

[0003] To address the problems in the prior art, this invention proposes a high-precision time-varying spectrum analysis method based on fractional S-transform.

[0004] This invention is achieved through the following technical solution: This invention proposes a high-precision time-varying spectrum analysis method based on fractional S-transform, the method comprising the following steps:

[0005] Step 1: Given the signal to be analyzed, i.e., an arbitrary finite-energy signal f(t), calculate its fractional Fourier transform F. α (u), where the range of the fractional Fourier transform angle is α∈(0,2π];

[0006] Step 2: Determine the optimal angle α corresponding to the optimal energy concentration in the fractional Fourier transform domain of any finite-energy signal f(t). opt ,Right now

[0007]

[0008] Step 3: Calculate the optimal energy concentration fractional Fourier transform angle α of any finite-energy signal f(t). opt The fractional S-transform below, i.e.

[0009]

[0010] Step 4: Based on the relationship between frequency ω and fractional frequency u, i.e., ω = u cscα - tcotα, perform the fractional S-transform at the optimal angle in Step 3. Calculate the fractional S-transform in terms of joint time t and frequency ω. Right now

[0011]

[0012] Step 5: Use the fractional S-transform represented by the joint time t and frequency ω obtained in Step 4. Calculate the time-varying spectrum of an arbitrary finite-energy signal f(t) Right now

[0013]

[0014] Furthermore, the method also includes the process of recovering the original signal from the time-frequency analysis results determined by the fractional S-transform, specifically as follows:

[0015] Step 6: Utilize the fractional S-transform expressed using the joint time t and fractional frequency ω processed in Step 4. Based on the relationship between frequency ω and fractional frequency u, i.e., ω = ucscα - tcotα, the fractional S-transform represented by the joint time t and fractional frequency u can be calculated. Right now

[0016]

[0017] Step 7: Using the joint time t and fractional frequency u obtained in Step 6, perform the fractional S-transform. By combining the inverse formula of the fractional S-transform, the original signal after time-frequency analysis can be recovered.

[0018]

[0019] The beneficial effects of this invention are:

[0020] The method described in this invention calculates the fractional-order S-transform of the signal by selecting the angle in the fractional-order Fourier transform domain where the signal energy is optimally concentrated. This yields a signal representation with joint time and fractional-order frequency determined by the fractional-order S-transform at the optimal energy concentration angle. Consequently, the signal energy is concentrated on a small number of fractional-order S-transform coefficients, achieving a sparse representation of the signal in the time-fractional-order frequency plane, which improves the algorithm's computational efficiency. Compared to the classical S-transform, the time-frequency analysis method based on the fractional-order S-transform can further enhance the accuracy of time-varying spectral analysis. Attached Figure Description

[0021] Figure 1 This is a block diagram illustrating the principle of time-varying spectrum analysis based on fractional S-transform.

[0022] Figure 2 This is a block diagram illustrating the principle of recovering the original signal based on the time-frequency analysis results of the fractional S-transform.

[0023] Figure 3 This is a schematic diagram of the time-varying spectrum analysis results based on the classical S-transform.

[0024] Figure 4 This is a schematic diagram of the time-varying spectrum analysis results based on the fractional S-transform. Detailed Implementation

[0025] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0026] For ease of analysis, we first introduce the definition of the fractional Fourier transform. For any finite-energy signal f(t)∈L... 2 (R), the fractional Fourier transform is defined as

[0027]

[0028] In the formula, the kernel function The expression is

[0029]

[0030] In the formula, k∈Z, α is the fractional Fourier transform angle, and u is usually called the fractional frequency, whose coordinate axis corresponds to the fractional Fourier transform domain. Correspondingly, the inverse transform of the fractional Fourier transform is...

[0031]

[0032] In the formula, the superscript symbol* This indicates taking the complex conjugate. Specifically, when α = π / 2, the fractional Fourier transform degenerates into the classical Fourier transform. Further, we introduce the definition of the fractional S-transform. For any finite-energy signal f(t) ∈ L... 2 (R), the fractional S-transform is defined as

[0033]

[0034] In the formula, g(t) represents a normalized Gaussian window function with variance σ = 1 / |ucscα|, i.e.

[0035]

[0036] Correspondingly, the inverse transform of the fractional S-transform is:

[0037]

[0038] Furthermore, the fractional S-transform can also be expressed in the form of the fractional Fourier transform domain, i.e.

[0039]

[0040] It can be seen that the fractional S-transform is equivalent to filtering the signal using a multiplicative filter in the fractional Fourier transform domain. Therefore, for signals whose energy is optimally concentrated in the fractional Fourier transform domain (a special case where the frequency domain is at angle α = π / 2), the fractional S-transform can provide the optimal processing result. For example, a linear frequency modulated (LFM) signal is a typical example of a signal with optimally concentrated energy in the fractional Fourier transform domain; however, its energy is diffuse in the frequency domain, and this type of signal has wide applications in electronic information systems such as radar, communication, and wireless detection.

[0041] It should be noted that, as can be seen from the definition, the fractional-order S-transform is essentially a signal representation that combines time t and fractional-order frequency u. In practical signal analysis, it is usually necessary to understand the evolution characteristics of the signal spectrum over time. Therefore, the desired signal representation combines time t and frequency ω. This requires the ability to construct a signal representation combining time t and frequency ω based on the fractional-order S-transform. To this end, it is necessary to derive the intrinsic relationship between fractional-order frequencies. Therefore, we first introduce the definition of the fractional-order Wegener-Weil distribution. For any finite-energy signal f(t)∈L... 2 (R), the fractional-order Wegener-Vell distribution is defined as

[0042]

[0043] It can be seen that the fractional-order Wegener-Weil distribution provides a signal representation with joint time t and fractional-order frequency u. Specifically, when α = π / 2, it degenerates into the classical Wegener-Weil distribution, i.e.

[0044]

[0045] This indicates that the classical Wegener-Weil distribution provides a representation of the joint time t and frequency ω. By comparing the definitions, we can find the following relationship between the fractional-order Wegener-Weil distribution and the classical Wegener-Weil distribution:

[0046]

[0047] Therefore, the intrinsic relationship between the fractional frequency u and the frequency ω can be obtained, namely...

[0048] ω=u cscα-tcotα (11)

[0049] Based on this, using the relationship between fractional frequency u and frequency ω, the definition of the fractional S-transform can be rewritten as a representation combining time t and frequency ω, i.e.

[0050]

[0051] Therefore, the time-varying spectrum of the signal can be expressed as:

[0052]

[0053] Based on the above analysis, the following describes a high-precision time-varying spectrum analysis method based on fractional S-transform proposed in this invention.

[0054] This invention proposes a high-precision time-varying spectrum analysis method based on fractional S-transform, the method comprising the following steps:

[0055] Step 1: Given the signal to be analyzed, i.e., an arbitrary finite-energy signal f(t), calculate its fractional Fourier transform F. α (u), where the range of the fractional Fourier transform angle is α∈(0,2π];

[0056] Step 2: Determine the optimal angle α corresponding to the optimal energy concentration in the fractional Fourier transform domain of any finite-energy signal f(t). opt ,Right now

[0057]

[0058] Step 3: Calculate the optimal energy concentration fractional Fourier transform angle α of any finite-energy signal f(t). opt The fractional S-transform below, i.e.

[0059]

[0060] Step 4: Based on the relationship between frequency ω and fractional frequency u, i.e., ω = u cscα - tcotα, perform the fractional S-transform at the optimal angle in Step 3. Calculate the fractional S-transform in terms of joint time t and frequency ω. Right now

[0061]

[0062] Step 5: Use the fractional S-transform represented by the joint time t and frequency ω obtained in Step 4. Calculate the time-varying spectrum of an arbitrary finite-energy signal f(t) Right now

[0063]

[0064] The method also includes a process of recovering the original signal from the time-frequency analysis results determined by the fractional-order S-transform, specifically:

[0065] Step 6: Utilize the fractional S-transform expressed using the joint time t and fractional frequency ω processed in Step 4. Based on the relationship between frequency ω and fractional frequency u, i.e., ω = ucscα - tcotα, the fractional S-transform represented by the joint time t and fractional frequency u can be calculated. Right now

[0066]

[0067] Step 7: Using the joint time t and fractional frequency u obtained in Step 6, perform the fractional S-transform. By combining the inverse formula of the fractional S-transform, the original signal after time-frequency analysis can be recovered.

[0068]

[0069] The effects of this invention can be further illustrated by the following simulations:

[0070] Simulated signals It can be seen that the simulated signal f(t) contains four signal components, namely and Figure 3 and Figure 4 Time-varying spectral analysis results for signal f(t) based on classical S-transform and fractional S-transform are presented respectively. It can be seen that, compared with classical S-transform, time-varying spectral analysis based on fractional S-transform can clearly show the four signal components contained in signal f(t).

Claims

1. A high-precision time-varying spectral analysis method based on fractional S-transform, characterized in that, The method includes the following steps: Step 1: Given the signal to be analyzed, i.e., an arbitrary finite-energy signal f(t), calculate its fractional Fourier transform F. α (u), where the range of the fractional Fourier transform angle is α∈(0,2π]; Step 2: Determine the optimal angle α corresponding to the optimal energy concentration in the fractional Fourier transform domain of any finite-energy signal f(t). opt ,Right now Step 3: Calculate the optimal energy concentration fractional Fourier transform angle α of any finite-energy signal f(t). opt The fractional S-transform below, i.e. Step 4: Based on the relationship between frequency ω and fractional frequency u, i.e., ω = u cscα - t cotα, perform the fractional S-transform at the optimal angle in Step 3. Calculate the fractional S-transform in terms of joint time t and frequency ω. Right now Step 5: Use the fractional S-transform represented by the joint time t and frequency ω obtained in Step 4. Calculate the time-varying spectrum of an arbitrary finite-energy signal f(t) Right now 2. The method according to claim 1, characterized in that, The method also includes a process of recovering the original signal from the time-frequency analysis results determined by the fractional-order S-transform, specifically: Step 6: Utilize the fractional S-transform expressed using the joint time t and fractional frequency ω processed in Step 4. Based on the relationship between frequency ω and fractional frequency u, i.e., ω = u cscα - t cotα, the fractional S-transform represented by the joint time t and fractional frequency u can be calculated. Right now Step 7: Using the joint time t and fractional frequency u obtained in Step 6, perform the fractional S-transform. By combining the inverse formula of the fractional S-transform, the original signal after time-frequency analysis can be recovered.

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