A time-frequency analysis method for high-order instantaneous frequency non-stationary signals

Abstract

The invention discloses a time-frequency analysis method for a high-order instantaneous frequency non-stationary signal, including: for an N-order instantaneous frequency modulation signal model, calculating 2N-1 improved short-time Fourier transforms based on Gaussian window functions of different transformation forms; constructing a generalized time-frequency matrix A; solving the generalized instantaneous frequency explicit expression of the N-order instantaneous frequency modulation signal; using the generalized instantaneous frequency explicit expression and the synchronous extraction operator, extracting the time-frequency energy of the short-time Fourier transform time-frequency distribution at the generalized instantaneous frequency estimation, and obtaining the time-frequency distribution of the high-order instantaneous frequency modulation signal; constructing a reconstruction operator matrix based on the high-order differential of the frequency domain Gaussian window function, and reconstructing the high-order instantaneous frequency non-stationary signal. The invention can accurately calculate the two-dimensional instantaneous frequency distribution of an instantaneous frequency modulation signal of any order, thereby obtaining an accurate time-frequency distribution result of a high-order instantaneous frequency non-stationary signal.

Description

Technical Field

[0001] The invention belongs to the technical field of high-order instantaneous frequency non-stationary signals, and in particular relates to a time-frequency analysis method for high-order instantaneous frequency non-stationary signals. Background Art

[0002] For non-stationary signals with constant instantaneous frequency, the synchronous extraction transform (SET) can estimate its two-dimensional instantaneous frequency distribution, and extract the energy at the ridge of the two-dimensional time-frequency distribution of the signal through the synchronous extraction operator, thereby obtaining a more concentrated signal time-frequency distribution. The following takes a single-component fixed-frequency signal as an example to introduce the working principle of the synchronous extraction transform. Assume that the complex sinusoidal signal model of a single-component fixed-frequency signal is as follows:

[0003]

[0004] In the formula, A is the amplitude of the signal model, and ω0 is the frequency of the signal model. The signal model is subjected to a Short-time Fourier Transform (STFT) to obtain its frequency domain signal form, and the results are as follows:

[0005]

[0006] Where δ(·) is the Dirac function. Then the improved short-time Fourier transform () of the frequency domain signal is calculated as follows:

[0007]

[0008] In the formula, is the frequency domain Gaussian window function. The synchronous extraction transform uses the relationship between the signal improved short-time Fourier transform and its partial derivative to construct the synchronous extraction operator. The partial derivative of both sides of formula (3) with respect to the time shift parameter t is calculated:

[0009]

[0010] In the formula, To find the partial derivative of the function with respect to t. Thus, the two-dimensional instantaneous frequency distribution ω0 of the signal x(t) is obtained, which can be expressed as follows:

[0011]

[0012] To ensure that the result of the above formula is a real number, the time-frequency energy of the signal is concentrated on ω0. The two-dimensional instantaneous frequency required for the time-frequency synchronization extraction transformation is It is an operation to take the real part of a complex number. The specific form is as follows:

[0013]

[0014] In the formula, It represents the operation of taking the imaginary part of a complex number. T(t,ω)≠0 is to ensure that the denominator will not be zero in the numerical calculation process.

[0015] The synchronous extraction transformation performs energy extraction transformation on the improved short-time Fourier transform time-frequency distribution according to the two-dimensional instantaneous frequency distribution. The transformation process is as follows:

[0016]

[0017] In the formula, is a synchronization extraction operator constructed using the Dirac function. According to the inverse transformation method of the short-time Fourier transform, by inversely transforming the time-frequency distribution energy of the synchronization extraction transform, its signal reconstruction algorithm can be obtained as follows:

[0018]

[0019] The synchronous extraction transform only retains the energy of the time-frequency distribution at the ridge position through the time-frequency extraction operator, and can obtain a more concentrated time-frequency distribution of non-stationary signals. However, the inference basis of the synchronous extraction transform is a non-stationary signal with a constant instantaneous frequency, and it is impossible to obtain the accurate instantaneous frequency of a higher-order instantaneous frequency modulated signal with a more complex instantaneous frequency, which leads to the distortion of the two-dimensional time-frequency distribution. Secondly, the synchronous extraction transform uses the inverse short-time Fourier transform to integrate the time-frequency distribution along the frequency direction and reconstruct the signal. Its reconstruction method uses the inverse transformation principle of the short-time Fourier transform, but for the time-frequency distribution of high-order instantaneous frequency modulated signals, its time-frequency energy is greatly lost during the synchronous extraction process, resulting in distortion of the reconstructed signal. Therefore, the synchronous extraction transform signal reconstruction method is no longer applicable to high-order instantaneous frequency modulated signals. Summary of the invention

[0020] Technical problem to be solved: Although the synchronous extraction transform can obtain a more concentrated time-frequency distribution compared to other time-frequency analysis methods when processing non-stationary signals. However, when processing non-stationary signals with more complex instantaneous frequencies, the synchronous extraction transform cannot accurately estimate the high-order instantaneous frequencies of the signal, resulting in distortion of the time-frequency distribution of high-order instantaneous frequency modulated signals. To this end, the present invention proposes a time-frequency analysis method for high-order instantaneous frequency non-stationary signals. By deriving a generalized instantaneous frequency explicit expression for an N-order instantaneous frequency modulated signal, the two-dimensional instantaneous frequency distribution of an instantaneous frequency modulated signal of any order can be accurately calculated, thereby obtaining an accurate time-frequency distribution result of a high-order instantaneous frequency non-stationary signal.

[0021] Technical solution:

[0022] A high-order instantaneous frequency non-stationary signal time-frequency analysis method, the non-stationary signal time-frequency analysis method comprising the following steps:

[0023] S1, for the N-order instantaneous frequency modulation signal model, calculate 2N-1 improved short-time Fourier transforms based on Gaussian window functions of different transformation forms;

[0024] S2, constructing a generalized time-frequency matrix A based on the improved short-time Fourier transform result of step S1;

[0025] S3, solving the generalized instantaneous frequency explicit expression of the N-order instantaneous frequency frequency modulated signal according to the constructed generalized time-frequency matrix A;

[0026] S4, using the generalized instantaneous frequency explicit expression and the synchronous extraction operator, extract the time-frequency energy of the short-time Fourier transform time-frequency distribution at the generalized instantaneous frequency estimate, and obtain the time-frequency distribution of the high-order instantaneous frequency modulation signal;

[0027] S5, constructs a reconstruction operator matrix based on the high-order differentials of the frequency-domain Gaussian window function to reconstruct the high-order instantaneous frequency non-stationary signal.

[0028] Furthermore, in step S1, the formula of the improved short-time Fourier transform based on the Gaussian window function of different transformation forms is:

[0029]

[0030] In the formula, t is the time variable, ω is the frequency variable, is the time domain Gaussian window function, σ is the Gaussian window function parameter, n∈N + It is the improved short-time Fourier transform parameters based on Gaussian window function with different transformation forms.

[0031] Furthermore, in step S2, the generalized time-frequency matrix A is:

[0032]

[0033] Wherein, the generalized time-frequency matrix A is a symmetric matrix, and each element thereof is the result of an improved short-time Fourier transform based on a Gaussian window function in different forms.

[0034] Furthermore, in step S3, the generalized instantaneous frequency explicit expression of the N-order instantaneous frequency modulation signal is:

[0035]

[0036] In the formula, is the Gaussian window function parameter, A 1,n is the algebraic cofactor of the generalized time-frequency matrix A, Indicates taking the imaginary part.

[0037] Further, in step S4, the time-frequency energy of the short-time Fourier transform time-frequency distribution at the generalized instantaneous frequency estimate is extracted using the following formula:

[0038]

[0039] Where δ(·) is the Dirac function.

[0040] Further, in step S5, a reconstruction operator matrix is ​​constructed based on the high-order differential of the frequency domain Gaussian window function to reconstruct the high-order instantaneous frequency non-stationary signal, and the reconstruction result is:

[0041]

[0042] In the formula, C is the constructed reconstruction operator matrix, and its construction form is as follows:

[0043]

[0044] Where, the matrix element is the frequency domain Gaussian window function The high-order differential matrix C is a symmetric matrix, C = C T , whose N-order form consists of 2N-1 elements, C 1,n is the algebraic cofactor of the generalized time-frequency matrix C; It is an improved short-time Fourier transform based on Gaussian window function with different transformation forms.

[0045] Beneficial effects:

[0046] The high-order instantaneous frequency non-stationary signal time-frequency analysis method of the present invention proposes a signal reconstruction method of generalized synchronous extraction transform from a new perspective. Compared with the reconstruction method of synchronous extraction transform, the generalized synchronous extraction transform has higher accuracy and versatility. It has been verified that the signal component reconstructed by the generalized synchronous extraction transform of the present invention matches the original signal better than the synchronous extraction transform, especially for non-stationary signal components with strong time-varying instantaneous frequency. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Figure 1 A schematic diagram comparing the high-order instantaneous frequency non-stationary signal time-frequency analysis method of the present invention with the prior art;

[0048] Figure 2 Schematic diagram of a multi-component non-stationary signal with different time-varying instantaneous frequencies; (a) represents the time domain waveform, and (b) represents the theoretical instantaneous frequency;

[0049] Figure 3 The time-frequency distribution results of different time-frequency analysis methods; (a) represents short-time Fourier transform, (b) represents synchronous compression transform, (c) represents synchronous extraction transform, and (d) represents generalized synchronous extraction transform;

[0050] Figure 4 It is a local enlarged diagram of the results of different time-frequency distributions; among them, (a) represents the synchronous compression transformation result of the constant frequency component, (b) represents the synchronous compression transformation result of the linear instantaneous frequency component, (c) represents the synchronous compression transformation result of the strongly time-varying instantaneous frequency component, (d) represents the synchronous extraction transformation result of the constant frequency component, (e) represents the synchronous extraction transformation result of the linear instantaneous frequency component, (f) represents the synchronous extraction transformation result of the strongly time-varying instantaneous frequency component, (g) represents the generalized synchronous extraction transformation result of the constant frequency component, (h) represents the generalized synchronous extraction transformation result of the linear instantaneous frequency component, and (i) represents the generalized synchronous extraction transformation result of the strongly time-varying instantaneous frequency component;

[0051] Figure 5 The time-frequency distribution slices of different signal components at t = 0.177s, where (a) represents the time-frequency slice of the constant frequency component, (b) represents the time-frequency slice of the linear instantaneous frequency component, and (c) represents the time-frequency slice of the strongly time-varying instantaneous frequency;

[0052] Figure 6 Reconstruction effects of different time-frequency analysis methods on different signal components; among them, (a) represents the synchronous extraction transform reconstructed signal of the constant frequency component, (b) represents the generalized synchronous extraction transform reconstructed signal of the constant frequency component, (c) represents the synchronous extraction transform reconstructed signal of the linear instantaneous frequency component, (d) represents the generalized synchronous extraction transform reconstructed signal of the linear instantaneous frequency component, (e) represents the synchronous extraction transform reconstructed signal of the strongly time-varying instantaneous frequency component, and (f) represents the generalized synchronous extraction transform reconstructed signal of the strongly time-varying instantaneous frequency component. DETAILED DESCRIPTION

[0053] The following examples will enable those skilled in the art to more fully understand the present invention, but are not intended to limit the present invention in any way.

[0054] The present invention discloses a high-order instantaneous frequency non-stationary signal time-frequency analysis method, the steps of which are as follows:

[0055] Step 1: First, calculate the improved short-time Fourier transform based on different transformation forms of Gaussian window functions. For the N-order instantaneous frequency modulation signal model, it is necessary to calculate 2N-1 improved short-time Fourier transforms based on different forms of Gaussian window functions. The calculation process is as follows:

[0056]

[0057] In the formula, is the time domain Gaussian window function, n∈N + .

[0058] Step 2: Construct a generalized time-frequency matrix A to solve the generalized instantaneous frequency explicit expression of the N-order instantaneous frequency frequency modulated signal. The generalized time-frequency matrix is ​​constructed as follows:

[0059]

[0060] Where A is the constructed generalized time-frequency matrix, which is a symmetric matrix, and each element is the improved short-time Fourier transform result based on different forms of Gaussian window functions in equation (9).

[0061] Step 3: Solve the generalized instantaneous frequency explicit expression of the N-order instantaneous frequency frequency modulated signal based on the constructed generalized time-frequency matrix. The solution formula is as follows:

[0062]

[0063] In the formula, is the Gaussian window function parameter, A 1,n is the algebraic co-factor of the generalized time-frequency matrix A above.

[0064] Step 4: Use the generalized instantaneous frequency explicit expression and the synchronization extraction operator to extract the time-frequency energy of the time-frequency distribution of the high-order instantaneous frequency frequency modulated signal, further improve the energy concentration of the time-frequency distribution, and obtain the accurate time-frequency distribution of the high-order instantaneous frequency frequency modulated signal. The generalized synchronization extraction transform (Generalized Synchroextracting Transform, GSET) uses the synchronization extraction operator to extract the time-frequency energy of the short-time Fourier transform time-frequency distribution at the generalized instantaneous frequency estimate, thereby obtaining a more accurate time-frequency distribution. The generalized synchronization extraction transform algorithm expression is as follows:

[0065]

[0066] Step 5: The time-frequency distribution energy obtained by the generalized synchronous extraction transform algorithm is more sparse, so there will be energy loss, and the signal reconstruction cannot be achieved through the time-frequency integration form. Therefore, the present invention adopts more time-frequency distributions and combines synchronous extraction operators to construct a more accurate reconstruction algorithm. The formula is as follows:

[0067]

[0068] In the formula, C is the constructed reconstruction operator matrix. Its construction form is as follows:

[0069]

[0070] Where, the matrix element is the frequency domain Gaussian window function The matrix C is a symmetric matrix, that is, C = C T , whose N-order form consists of 2N-1 elements.

[0071] The time-frequency analysis method of the present invention is further described in detail below through an example.

[0072] First, a multi-component non-stationary signal model with different orders of instantaneous frequencies is constructed. The formula is as follows:

[0073] x(t)=x1(t)+x2(t)+x3(t) (15);

[0074] in,

[0075]

[0076] The sampling frequency of the signal is 1000 Hz and the sampling time is 1 s.

[0077] The time waveform of the signal is as follows Figure 2 As shown in (a) in the figure, the theoretical instantaneous frequency is Figure 2 As shown in (b) of . It can be found that the instantaneous frequency of component x1 remains constant, the instantaneous frequency of component x2 is a linear function of time, and the instantaneous frequency of component x3 has a strong time-varying characteristic.

[0078] Figure 3 The time-frequency distribution diagram obtained by different time-frequency analysis methods is shown in Figure 2. Figure 4 A local zoomed view of the time-frequency distribution of each signal component (the red line is the theoretical instantaneous frequency). Figure 3 It can be seen that the time-frequency aggregation of the short-time Fourier transform result is poor, and other time-frequency post-processing methods have relatively good time-frequency aggregation. Figure 4 As can be observed from (a)-(c) in , Synchrosqueezing Transform (SST), as the most classic time-frequency post-processing method, can generate a uniform time-frequency distribution for component x1. However, when processing signal components with nonlinear instantaneous frequencies (such as x2 and x3), its time-frequency aggregation is poor. Similarly, Figure 4 As shown in (d)-(f) in , although the synchronous extraction transform can achieve a sparser time-frequency distribution compared to the synchronous compression transform, it still cannot generate accurate time-frequency distributions for signal components x2 and x3 with nonlinear instantaneous frequencies. Since both the synchronous compression transform and the synchronous extraction transform are calculated based on the first-order instantaneous frequency estimation, they cannot accurately characterize the time-frequency distribution of signal components with nonlinear instantaneous frequencies. The time-frequency analysis results of the generalized synchronous extraction transform are shown in Figure 4As shown in (g)-(i) in the figure, the generalized synchronous extraction transform is based on the high-order instantaneous frequency estimation, which can provide unbiased estimation for the high-order instantaneous frequency signal components and accurately characterize the components x2 and x3. By comparing the different time-frequency distribution results of component x3, it can be noted that compared with the existing classical time-frequency analysis methods, the time-frequency distribution calculated by the generalized synchronous extraction transform is not only more accurate, but also more sparse.

[0079] In order to further verify the above conclusions, Figure 5 The normalized time-frequency slice of the signal at time t = 0.177s can clearly compare the time-frequency aggregation and accuracy of different time-frequency analysis methods. Figure 5 As shown in (a) in , for the constant frequency signal component x1, the time-frequency aggregation and time-frequency accuracy of each time-frequency post-processing method are almost the same. Figure 5 As shown in (b) in Figure 1, for the signal component x2 with linear instantaneous frequency, the time-frequency aggregation and accuracy of the synchronous compression transform are reduced, and the synchronous extraction maintains accurate time-frequency aggregation, but the time-frequency accuracy is significantly reduced. Figure 5 As shown in (c) in the figure, for the signal component x3 with complex instantaneous frequency, the time-frequency distribution result of the synchronous compression transform has relatively poor time-frequency aggregation and time-frequency accuracy. Although the synchronous extraction transform maintains the time-frequency aggregation, the time-frequency accuracy is significantly reduced. On the contrary, the generalized synchronous extraction transform proposed in the present invention can not only generate a clustered time-frequency distribution for the complex instantaneous frequency signal component x3, but also retain the accuracy of the time-frequency distribution.

[0080] The Rayleigh Entropy (RE) has been used as a quantitative indicator to measure the clustering of time-frequency distribution. The smaller its value, the more concentrated the time-frequency distribution. Table 1 shows the Rayleigh Entropy values ​​of the time-frequency distribution obtained by calculating each signal component using different time-frequency analysis methods. By comparison, it is obvious that the time-frequency distribution of the synchronous extraction transform is sparser than that of the synchronous compression transform because they only extract time-frequency energy on the two-bit instantaneous frequency estimate. Therefore, the generalized synchronous extraction transform has obvious advantages in processing frequency-modulated signals, especially those with high-order instantaneous frequencies. Figure 1 It is a schematic diagram comparing the time-frequency analysis method of the high-order instantaneous frequency non-stationary signal of the present invention with the prior art.

[0081] Table 1 Rayleigh entropy of time-frequency distribution obtained by different time-frequency analysis methods for each component

[0082]

[0083] The global Earth Mover's Distance (EMD) is a quantitative indicator for measuring time-frequency accuracy and is often used to measure the accuracy of time-frequency analysis methods. The smaller the value, the higher the time-frequency analysis method has. Table 2 shows the global deviation distance index of the time-frequency distribution obtained by different time-frequency analysis methods for each signal component. It can be found that the generalized synchronous extraction transform has higher time-frequency accuracy than other time-frequency analysis methods due to its accurate generalized instantaneous frequency estimation. For the signal component x3 with a strong time-varying instantaneous frequency, the effect is more obvious. This analysis is consistent with Figure 3 and Figure 5 The results in (c) are consistent. Therefore, the generalized synchronous extraction transform performs better than other time-frequency analysis methods when analyzing strong time-varying signals.

[0084] Table 2 Global deviation distance index of time-frequency distribution obtained by different time-frequency analysis methods for each component

[0085]

[0086] The synchronous extraction transform only uses the energy at the ridge of the time-frequency distribution to reconstruct the signal, which can effectively avoid noise mixing in the signal reconstruction process, which will also lead to inevitable energy loss in the reconstruction process. In order to solve this problem, this patent proposes a signal reconstruction method of the generalized synchronous extraction transform from a new perspective. Compared with the reconstruction method of the synchronous extraction transform, the generalized synchronous extraction transform has higher accuracy and versatility. The time domain signal components reconstructed by the synchronous extraction transform and the generalized synchronous extraction transform are as follows: Figure 6 As shown, the signal components reconstructed by the generalized synchronous extraction transform match the original signal better than the synchronous extraction transform, especially for the non-stationary signal component x3 with a strongly time-varying instantaneous frequency.

[0087] The above are only preferred embodiments of the present invention, and the protection scope of the present invention is not limited to the above embodiments. All technical solutions under the concept of the present invention belong to the protection scope of the present invention. It should be pointed out that for ordinary technicians in this technical field, some improvements and modifications without departing from the principle of the present invention should be regarded as the protection scope of the present invention.

Claims

1. A time-frequency analysis method for high-order instantaneous frequency non-stationary signals, characterized in that: The non-stationary signal time-frequency analysis method comprises the following steps: S1, for the N-order instantaneous frequency modulation signal model, calculate 2N-1 improved short-time Fourier transforms based on Gaussian window functions of different transformation forms ;in, is the time variable, is a frequency variable; S2, constructing a generalized time-frequency matrix A based on the improved short-time Fourier transform result of step S1; S3, solving the generalized instantaneous frequency explicit expression of the N-order instantaneous frequency frequency modulated signal according to the constructed generalized time-frequency matrix A; S4, using the generalized instantaneous frequency explicit expression and the synchronous extraction operator, extract the time-frequency energy of the short-time Fourier transform time-frequency distribution at the generalized instantaneous frequency estimate, and obtain the time-frequency distribution of the high-order instantaneous frequency modulation signal; S5, constructs a reconstruction operator matrix based on the high-order differential of the frequency-domain Gaussian window function to reconstruct the high-order instantaneous frequency non-stationary signal; In step S2, the generalized time-frequency matrix A is: In the formula, the generalized time-frequency matrix A is a symmetric matrix, and each element is the result of the improved short-time Fourier transform based on different forms of Gaussian window functions; In step S3, the generalized instantaneous frequency explicit expression of the Nth order instantaneous frequency modulation signal is: In the formula, , is the Gaussian window function parameter, is the algebraic cofactor of the generalized time-frequency matrix A, It means taking the imaginary part; In step S4, the time-frequency energy of the short-time Fourier transform time-frequency distribution at the generalized instantaneous frequency estimate is extracted using the following formula: In the formula, is the Dirac function; In step S5, a reconstruction operator matrix is ​​constructed based on the high-order differential of the frequency domain Gaussian window function to reconstruct the high-order instantaneous frequency non-stationary signal. The reconstruction result is: In the formula, C is the constructed reconstruction operator matrix, and its construction form is as follows: Where, the matrix element is the frequency domain Gaussian window function The high-order differentials of , the matrix C is a symmetric matrix, , whose N-order form is given by Elements is the algebraic cofactor of the generalized time-frequency matrix C; It is an improved short-time Fourier transform based on Gaussian window function with different transformation forms.

2. The high-order instantaneous frequency non-stationary signal time-frequency analysis method according to claim 1, characterized in that: In step S1, the formula of the improved short-time Fourier transform based on the Gaussian window function of different transformation forms is: In the formula, is the time domain Gaussian window function, It is the improved short-time Fourier transform parameters based on Gaussian window function with different transformation forms.

Images